Modelling and Perception for Autonomous Ground Vehicles in Non-Uniform Terrain

Modelling and Perception for Autonomous Ground Vehicles in Non-uniform Terrain

Michael Richard Woods

A Thesis presented for the degree of Doctor of Philosophy

School of Mechanical and Manufacturing Engineering University of New South Wales Australia October 2015 PLEASE TYPE THE UNIVERSITY OF NEW SOUTH WALES Thesis/Dissertation Sheet

Surname or Family name: Woods

First name: Michael Other name/s: Richard

Abbreviation for degree as given in the University calendar: PhD

School: Mechanical and Manufacturing Engineering Faculty: Engineering

Title: Modelling and Perception for Autonomous Ground Vehicles in Non-uniform Terrain

Abstract 350 words maximum: (PLEASE TYPE)

This thesis researched the sources of uncertainty that an AGV is subject to whilst moving through an unstructured and non-uniform terrain. A number of key sources of uncertainty were identified, with changes in terrain type and thus the coefficient of friction being one key component, as well as the dynamic behaviour of the tyre interacting with the terrain, especially during cornering maneuvers.

Furthermore, it develops a novel terrain perception method which utilises purely non-semantic spatial information. The Extended Range Texture Analysis (ERTA) method is capable of providing an accurate terrain model that accounts for the identified terrain frictional and geometric characteristics, which reduces the uncertainties in terrain frictional characteristics that exists in the system model. The ERTA method can provide a solution for classifying the terrain types in an environment even in the absence of visual or semantic spatial features in a scene.

Two novel tyre models were also developed in this thesis: a simplified Friction Dependant tyre model, and a 3D Analytical Dynamic tyre model. The purpose of developing these models is to enable the incorporation of the sources of uncertainties, induding both frictional characteristics provided by the terrain model and tyre dynamic effects. The simulation results of both the developed tyre models were compared against the widely accepted experimentally derived Magic Formula tyre model.

Following on from this research, control system designers can incorporate the developed tyre models and perception method to reduce the uncertainties in the system parameters and disturbances to design significantly improved controllers.

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Date ...... ?..( .. ./.. 7. .. /.? ..c?..r ....?...... Modelling and Perception for Autonomous Ground Vehicles in Non-uniform Terrain

Michael Richard Woods Submitted for the degree of Doctor of Philosophy March 2016

Abstract

Autonomous Ground Vehicles (AGVs) are increasingly being used in more complex, harsh and remote environments because of their ability to replace a human driver, removing them from danger. However, the environments that are being targeted for the application of AGVs frequently occur in areas that have unstructured, uneven and non-uniform terrain. Safe and accurate navigation through such environments requires reliable terrain perception methods capable of identifying friction characteristics and an accurate system model, both are required in order to reduce the system uncertainties. Currently, methods for reducing these uncertainties are not well established. This thesis researched the sources of uncertainty that an AGV is subject to whilst moving through an unstructured and non-uniform terrain. A number of key sources of uncertainty were identified, with changes in terrain type and thus the coefficient of friction being one key component, as well as the dynamic behaviour of the tyre interacting with the terrain, especially during cornering maneuvers. Furthermore, it develops a novel terrain perception method which utilises purely non-semantic spatial information. The Extended Range Texture Analysis (ERTA) method is capable of providing an accurate terrain model that accounts for the identified terrain frictional and geometric characteristics, which reduces the uncertainties in terrain frictional characteristics that exists in the system model. The ERTA method can provide a solution for classifying the terrain types in an environment even in the absence of visual or semantic spatial features in a scene. Two novel tyre models were also developed in this thesis: a simplified Friction Dependant tyre model, and a 3D Analytical Dynamic tyre model. The purpose of

iii developing these models is to enable the incorporation of the sources of uncertainties, including both frictional characteristics provided by the terrain model and tyre dynamic eﬀects. The simulation results of both the developed tyre models were compared against the widely accepted experimentally derived Magic Formula tyre model. Following on from this research, control system designers can incorporate the developed tyre models and perception method to reduce the uncertainties in the system parameters and disturbances to design signiﬁcantly improved controllers.

iv Acknowledgements

To both of my supervisors at UNSW, Associate Professor Jayantha Katupitiya and Dr Jos´eGuivant who even when I doubted myself were always there to encourage and push me, and who were always there to talk with if I needed them, to talk about progress, personal issues, or more often than not world politics. Alfred Hu, who sadly, passed away during the writing of my thesis, was exceptionally helpful in assisting when I needed to arrange and experiment or order any component for my research or for one of the classes. He was always enthusiastic in providing help where ever he could. Thanks goes out to Steve Cossell who was always available to listen to any idea or problem that I was working on and able to lend a fresh perspective, as well as helping to proof-read parts of my thesis, helping me break out of my procrastination. Thanks goes out to all of my thesis proof-readers, Steve, Hiranya Jayakody, Karan Narula and Anton Lohr, who have together been able to shape my thesis into something to be proud of. Another thanks goes out to Jared Wong, Mark Whitty, Wayne Yuan and Hiranya who were also willing to listen to my current problem and help point me in the right direction. Throughout the majority of my turmoils at UNSW, I have had the help of Scarlett Liu and Hiranya to reassure me that the world in-fact was not ending, and that everything would be ﬁne, and also to help lighten the mood. As well as the rest of the UNSW Mechatronics group who had to put up with my strangeness. Edward, Hin and Mike from the Computational Mechanics and Robotics (CMR) group at UNSW, who along with Mark helped shape my appreciation for research. For everyone who has been involved with MAVSTAR over the years and including all

v of the undergraduate students that I have had the pleasure of co-supervising; Yimin Xu, Wesley Dowling, Chen Qian, Deng Deng, Matt Corke, Andrew Hammond, Nathaniel Lai, Prasheedh Sivapiragasam, Chris Willshire, Chris Williams and Karan Narula. Some of who I was able to take with me into PhD life. To Josh Yen, Will Crowe, Jeﬀ Peng as well as Hiranya and Karan for the opportunity to take part in two successful High Altitude Balloon launches, to prove that I can still succeed if I put my mind to it, as well as being friends who can relate to my PhD experiences. Finally, I would like to thank my parents and family, who have always stayed positive about my thesis and have supported the decisions that I have made leading up to its completion, even if it didn’t make any sense to them.

vi Contents

Abstract iii

Acknowledgements v

Table of Contents x

Nomenclature xi

1 Introduction 1 1.1 Motivation ...... 1 1.2 Objectives ...... 4 1.3 Contributions ...... 4 1.4 Published Papers ...... 6

2 Current State of Research 7 2.1 Vehicle Modelling ...... 7 2.1.1 Kinematic Modelling ...... 8 2.1.2 Dynamic Modelling ...... 9 2.2 Tyre Modelling ...... 15 2.2.1 Empirical Formulae Models ...... 17 2.2.2 Analytical Models ...... 20 2.3 Terrain Perception ...... 27 2.3.1 Geometry Estimation ...... 28 2.3.2 Friction Estimation ...... 32 2.4 Summary ...... 38

vii 3 Vehicle Modelling with Friction Dependence 41 3.1 3D Dynamic Model ...... 42 3.1.1 Vehicle Body Dynamics ...... 42 3.1.2 Suspension Dynamics ...... 46 3.1.3 Tyre Dynamics ...... 49 3.2 Friction Dependant Tyre Force ...... 52 3.3 Simulation ...... 58 3.3.1 Single Terrain Type ...... 59 3.3.2 Diﬀerent Terrain Types ...... 67 3.4 Discussion ...... 71 3.5 Conclusion ...... 72

4 A 3D Analytical Dynamic Tyre Model 75 4.1 Friction Dependant Analytical Tyre Model ...... 77 4.1.1 Distributed Tyre Model ...... 77 4.1.2 Steady State Conditions ...... 82 4.1.3 Parameter ﬁtting ...... 85 4.1.4 Average lumped model ...... 91 4.1.5 Simulation ...... 98 4.2 Tyre Force Distribution ...... 103 4.2.1 Variable force distribution ...... 104 4.2.2 Steady-State Fitting ...... 110 4.2.3 Simulation ...... 114 4.3 Conclusion ...... 117

5 Terrain Perception using Non-semantic Range Data 121 5.1 Introduction ...... 122 5.1.1 Spatial Features ...... 124 5.2 Range Texture Analysis ...... 125 5.2.1 Segmentation ...... 126 5.2.2 Terrain Features ...... 129 5.2.3 Classiﬁcation ...... 132 5.2.4 Results ...... 132 5.2.5 Discussion ...... 135 5.2.6 Summary ...... 137 5.3 Extended Range Texture Analysis ...... 137 5.3.1 Feature Image Generation ...... 141 5.3.2 Segmentation ...... 150 5.3.3 Stationary Platform Results ...... 156 5.3.4 Moving Platform Results ...... 169 5.3.5 mu-Patch Model ...... 171 5.3.6 Discussion ...... 177 5.3.7 Summary ...... 179 5.4 Conclusion ...... 180

6 Application of the 3D Analytical Dynamic Tyre Model 183 6.1 Model Integration ...... 185 6.1.1 3D Vehicle Dynamic Model ...... 185 6.1.2 Analytical Dynamic Tyre Model ...... 186 6.1.3 Terrain Model ...... 187 6.1.4 Full System Model ...... 187 6.2 Simulation ...... 188 6.2.1 Single Terrain Type ...... 189 6.2.2 Diﬀerent Terrain Types ...... 198 6.3 Discussion ...... 202 6.4 Conclusion ...... 204

7 Conclusions 207 7.1 Technical Contributions ...... 211 7.2 Future Work ...... 212

References 213

A Planar Vehicle Model Derivations 235 B Tyre Model Derivations 241 B.1 Force Distributions ...... 241 B.1.1 Constant Force Distribution ...... 242 B.1.2 Parabolic distribution ...... 242 B.1.3 Trapezoidal distribution ...... 243 B.1.4 Continuous Variable distribution ...... 245 B.2 Frictional Coeﬃcient Extraction ...... 247 B.3 MF-SWIFT 6.1 model parameters ...... 248

C Terrain Model Further Explanation 251 C.1 Spatial Features ...... 251 C.1.1 Gray-Level Co-occurrence Matrix Features ...... 251 C.1.2 Gabor Filter Features ...... 254 C.1.3 Power Spectral Density Features ...... 255 C.2 Weighted Majority Voting with Dominance ...... 256 C.3 Dimensionality Reduction ...... 258 C.3.1 Principal Component Analysis ...... 258 C.3.2 Multiple Discriminant Analysis ...... 260 Nomenclature

Symbol Description Units

αj slip angle of the jth tyre rad

ǫ¯i(t) weighted mean tyre element deﬂection in i = x, y m t¯d,s(k) Mean Haralick feature over all θ angles −

−1 v¯rx, v¯ry weighted mean relative velocity over contact patch ms

βs Stribeck exponent -

δj steering angle of the jth tyre rad

−1 ǫ¯˙i(t) weighted mean tyre element deﬂection velocity in i = ms x, y

−1 δ˙j steering angular velocity of the jth tyre rads

−1 ǫˆ˙i(t) cumulative weighted mean tyre element deﬂection ms velocity in i = x, y

ǫx,y(t, ζ, η) contact patch element deﬂection m

ǫˆi(t) cumulative weighted mean tyre element deﬂection in m i = x, y

Iˆ(ξ) 2DFouriertransformofaninputmatrix I −

−1 vˆrx, vˆry cumulative weighted mean relative velocity over the ms contact patch

−1 −1 κi mean internal deﬂection state correction for i = x, y m rad

µi(t) Eﬀective coeﬃcient of friction for pneumatic tyre -

µkx,µky kinetic coeﬃcients of friction -

µsx,µsy static coeﬃcients of friction -

−1 −1 νi weighted mean internal deﬂection state correction for m rad i = x, y

−1 ωj Wheel angular speed of the jth tyre rads

xi Nomenclature

−1 σ0i lumped rubber stiﬀness in i = x, y, z dimensions m

−1 σ1i lumped rubber damping in i = x, y, z dimensions sm

−1 σ2i lumped rubber viscous damping in i = x, y, z sm dimensions

τf,j Motor friction torque for wheel j Nm ζ, η distance along the contact patch in the x and y axes m respectively

2 Ai Area for the ith axis m b Vehicle body wheel base m

−1 bs spring damper coeﬃcient of the suspension Nsm

−1 bt spring damper coeﬃcient of the tyre Nsm

CD,i Drag coeﬃcient for the ith axis −

−1 cs,j damping coeﬃcient of the suspension on wheel j Nsm F Feature space image − fθ,d,s(k) Haralick feature for θ angle, d step distance, s sensor − type, k =1 ... 7

Fx,j Longitudinal force acting on wheel j N

Fy,j Lateral force acting on wheel j N

Fz,j Vertical force acting on wheel j N

−2 fz,j Distributed vertical force on the contact patch of Nm wheel j gs,µ(a, k) Mean of Gabor Filter features over current window − gs,σ(a, k) Standard deviation of Gabor Filter features over − current window gs,span(a, k) Span of Gabor Filter features over current window − hc Vehicle body height m hd,s(k) Maximum Haralick feature over all θ angles − hs Vertical distance from vehicle center of mass to m suspension connection point

2 Ib,i body moment of inertia in the principle axes i = x,y,z kgm

2 Iw,i wheel moment of inertia in the principle axes i = kgm x, y, z

xii Nomenclature

−1 ks,j stiﬀness coeﬃcient of the suspension on wheel j Nm

−1 ks spring constant of the suspension Nm

−1 kt spring constant of the tyre Nm L Contact patch length m l Vehicle body length m ls uncompressed length of the suspension m mb mass of vehicle body kg mt mass of the tyre kg

Mx,j Overturning moment acting on wheel j Nm

My,j Rolling resistance moment acting on wheel j Nm

Mz,j Self-aligning moment actin on wheel j Nm

Pµ Terrain model patch for mean coeﬃcient of friction −

Pσ Terrain model patch for standard deviation of the − coeﬃcient of friction

Ppatch Terrain model for geometric patch −

Ps(k) Radially averaged Power Spectral Density for the kth − bucket r uncompressed height of the tyre m

R(i, j) Residual image matrix − re Eﬀective radius of the wheel m sd,s(k) Span for Haralick feature over all θ angles −

Tclass Terrain classiﬁcation image −

Tconfidence Terrain conﬁdence image − v linear velocity of the wheel ms−1 vd,s(k) Variance for Haralick feature over all θ angles −

−1 vrx, vry relative velocities in the x and y directions ms

−1 vs Stribeck speed ms W Contact patch width m

xiii Nomenclature

xiv Chapter 1

Introduction

1.1 Motivation

Autonomous Ground Vehicles (AGVs) are being used in increasingly more complex, harsh and remote environments because of their ability to replace a human driver and remove them from danger as well as being able to maintain a level of persistent autonomy, with minimal human intervention. The environments that are being targeted for the application of AGVs however, frequently occur in areas that have unstructured, uneven and non-uniform terrain. Safe and accurate navigation through such environments requires an accurate model of the system so that stable controllers can be designed, which is commonly achieved through the use of robust control techniques. This requires assumptions to be made about the type of terrain that an AGV will encounter and interact with. Currently, the assumptions made grossly oversimplify the environment and as such contribute to a reduced overall reliability of the system to respond to changes in the environment. For persistent autonomy, maintaining a high system reliability is crucial, especially during prolonged operations, this thesis addresses the assumptions that are made in the system model uncertainty to help improve the system reliability. With Autonomous Farming becoming an important aspect of the mechanisation of farming [1–3], agricultural AGVs are being required to operate within these harsh and remote environments where the environment is more unstructured and uncertain, whilst confronting ever increasing demands on the level of both navigational accuracy

1 Chapter 1. Introduction and system autonomy. A key challenge in autonomous farming is the requirements, that the AGV is expected to maintain the same level of precision throughout all variations in the environment as well as being able to exhibit a level of persistent autonomy in order to successfully mechanise the farm.

For an AGV to operate over a variety of unstructured and widely varying terrain conditions through the use of robust control techniques requires a greater level of uncertainty to be placed on the model of the vehicle and the model of the terrain. For a system with a high level of uncertainty, the controller may be unable to reach the desired accuracy and maintain trajectory under certain conditions [4–7], in addition to this, high uncertainties in a system increase the robust controllers oﬀset errors that are experienced once the system has converged to the desired set point [8]. Therefore, it is important to provide a model that is as accurate as possible so as to reduce the sources of uncertainty and improve the controller response accordingly. A controller that is designed with uncertainties and assumptions that do not reﬂect the reality of the situation risks the robust controller behaving in a manner not intended, and thus can result in poor system performance.

The ﬁeld of robust control of agricultural vehicles is important, as the high level of uncertainties involved the system structure and parameters as well as the uncertainties in the system disturbances can be taken into account, with these techniques implemented to improve the control of AGVs in these areas [9–12]. The robust control techniques work best when the uncertainties in the system are better known, however the methods for reducing these uncertainties are not as well established. To improve vehicle autonomy and the precision of AGV operations in these conditions, methods for reducing the uncertainties in the system model are proposed in this thesis.

To maintain path tracking precision and to demonstrate persistent autonomy it is important to minimise the amount of uncertainty in the system model. The dominant mechanism by which the AGV interacts with the environment is through the interface between the tyres of the AGV and the terrain, it is through this interface that the controllable forces that inﬂuence the motion of the AGV are generated. Knowledge about this tyre-terrain mechanism allow the forces involved to be determined accurately which makes it possible to improve the accuracy of the path tracking and the level of

2 1.1. Motivation persistent autonomy.

For an AGV that is required to operate on a farm persistently, and that uses robust controllers containing high levels of uncertainty, the reliability of the AGV to react accordingly to changes in the environment is reduced. Depending on the day, the terrain on a number of required AGV passageways might be soaked and water logged, or dry and dusty, or perhaps eroded away. The ability of the AGV to be able to navigate through these changed conditions requires that the control system be responsive to the changes. For a system with high uncertainties these conditions might be too much for the vehicle, and the controller might get the vehicle bogged down in the mud, or it might bottom out the AGV or roll over when attempting to navigate the eroded path. For a normal farm this would be ﬁxed by the farmer travelling out to the AGV and correcting it, but for a very large farm, or for an remotely operated autonomous farm this might take days for the situation to be corrected, potentially crippling the farm if the incident occurred on a main passageway on the farm.

The terrain characteristics that exist on a farm can vary widely and so providing a method by which these characteristics can be modelled and observed is important, and thereby reduce the amount of uncertainty in the vehicle model. To be able to develop a model of the terrain it is important to choose characteristics of the terrain that are observable and dominant. The main observable properties of the terrain that aﬀect the way a vehicle behaves as it traverses across its surface are the frictional properties of that surface and the dominant geometry of the surface. Therefore, it is desirable to develop a model of the terrain that encompasses these characteristics.

A model of the tyre-terrain interaction that depends on the observable terrain frictional characteristics is also desirable to be able to reduce the uncertainties in the system model. For autonomous farming it has been shown that the steering dynamics of the tyre influence the forces and moments that are generated in the contact patch [13], where the width of the contact patch between the tyre and the terrain is included. Additionally, the effect of the weather in agricultural AGV operations plays an important role as it significantly alters the terrain conditions and so a tyre-terrain model that incorporates such variation is desirable.

3 Chapter 1. Introduction

1.2 Objectives

The primary objectives of this thesis are listed as follows:

• To develop a terrain classiﬁcation technique that can perform terrain identiﬁcation under varying conditions so that the terrain frictional and geometric properties can be gathered.

• To propose a new model for tyre-terrain interactions that depends on the frictional properties of the terrain and that incorporates the width of the tyre contact patch as well as the eﬀects of steering on the forces generated.

• To develop a dynamic model representation of the states of the vehicle including forces experienced.

• To validate the developed models through the use of experimentally equivalent tyre models.

1.3 Contributions

The main contributions of this thesis are listed as follows:

• A static tyre force model has been developed that is able to estimate the forces that each tyre is subject to during the motion of the vehicle through diﬀerent terrain types with a dependency on the coeﬃcient of frictions of that terrain.

• A three-dimensional dynamic analytical tyre model has been developed for modelling the tyre-terrain interactions that incorporates the eﬀects of the frictional properties of the terrain. The developed model utilises both the length and the width of the tyre contact patch to be able to account for the eﬀect of the steering rate on the forces and moments generated.

• A velocity and camber angle dependant tyre force distribution is also presented which was used by the developed dynamic tyre model in order to more accurately reﬂect the empirically observed force distribution.

4 1.3. Contributions

• A novel method of terrain type classiﬁcation using purely non-semantic range data has been developed which was used to estimate the frictional properties of the terrain as well as extract the terrain geometry.

• A novel method of developing a confidence in classification metric to be able to quantify the confidence in the current classification result.

• A three-dimensional dynamic model of a 4-Wheel Drive 4-Wheel Steered (4WD4WS) vehicle has been developed that enables an analysis of different tyre models including the developed 3D Analytical Dynamic tyre model and the developed Friction Dependant tyre model, subjected to a variety of different terrain configurations and geometry.

This thesis is organised into chapters as follows: Chapter 2 is a thorough examination of the previous literature that is available in the realm of vehicle dynamics, tyre dynamics as well as terrain segmentation and classification. Chapter 3 develops a three dimensional model for a 4WD4WS vehicle platform, as well as a novel friction dependant tyre model. Simulation results are then presented and examined for a number of different navigational and environmental situations to identify potential sources of uncertainties in the system model. Chapter 4 introduces a new analytical dynamic tyre model that has been developed for modelling the dynamics of tyre terrain interactions so that all the forces and moments that are involved can be calculated. A new force distribution that is velocity and camber angle dependant has also been developed for use with the developed dynamic tyre model. The chapter also includes a series of simulation configurations, with analysis and results that are compared against the widely accepted experimentally derived Magic Formula tyre model, as well as existing experimental results. Chapter 5 discusses and explains two newly developed methods for terrain perception which utilise only non-semantic spatial information about a scene. These methods can identify the terrain that a vehicle may travel over so that terrain characteristics can be extracted and be used with the developed tyre model. Chapter 6 combines the previously developed models for tyre forces as well as the

5 Chapter 1. Introduction terrain model together with the vehicle dynamic model of a 4-Wheel Drive 4-Wheel Steered (4WD4WS) vehicle platform. Simulation results are then shown of the implementation of the tyre and terrain models for a number of diﬀerent navigational and environmental situations. Finally, Chapter 7 summarises the work performed in the production of this dissertation and provides suggestions for the direction of future work that could be explored.

1.4 Published Papers

The following papers were published during the production of this thesis

1. M. Woods, and J. Katupitiya, ”Modelling of a 4WS4WD vehicle and its control for path tracking”. In Computational Intelligence in Control and Automation (CICA), 2013 IEEE Symposium

2. M. Woods, J. Guivant, and J. Katupitiya, ”Terrain Classiﬁcation using Depth Texture Features”. In Proceedings of Australasian Conference on Robotics and Automation, 2013. (ACRA2013)

6 Chapter 2

Current State of Research

This chapter reviews and groups past contributions to modelling and perception for Autonomous Ground Vehicles (AGVs) operating in non-uniform and uneven terrain. Section 2.1 surveys the current methods that exist for the dynamic modelling of an AGV with emphasis given to terrain dependent modelling. This is then followed in Section 2.2 by a review of the current tyre models which are used to model the forces generated in the tyre. A survey of the current terrain perception techniques that form a basis for the terrain perception methods proposed in this dissertation is presented in Section 2.3. Finally, in Section 2.4, a conclusion is presented which speciﬁes that no solution has yet been produced which fully achieves the objectives of this thesis.

2.1 Vehicle Modelling

There are a vast amount of vehicle configurations that have been developed and modelled in the literature, with each providing different advantages and disadvantages for the control of the vehicle trajectory. The models that have been proposed include; a bicycle model [14], a two-wheel inverted pendulum model [15, 16], the more conventional car type vehicles, with simple Ackermann steering and two wheel drive [17], or four wheel drive [18], tracked vehicles [19, 20] and differentially steered vehicle models [21–23]. Additionally, more complicated vehicle models have been proposed, such as four wheel drive, four wheel steered configurations [24, 25], as well as less standard configurations [26]. Some configurations benefit from additional control inputs that can

7 Chapter 2. Current State of Research be used to provide extra stability in the system as a whole. However these configurations pose a greater challenge due to their increased complexity and thus may require a larger amount of uncertainty to be included for the system, with greater uncertainty resulting in less accurate control for completing specific desired tasks. The vehicle configurations that will be examined in more detail will be limited to those with four wheels, as these are the dominant vehicle configuration types that are in use by the general public, including both passenger and agricultural vehicles. In order to achieve the robust control that is necessary for persistent autonomy, a vehicle configuration must be chosen such that it has complete state controllability. The robustness of the controller that is used should be able to deal with the unknown conditions of the terrain. There exists a large number of different methods for controlling AGVs, at low and high speeds, these can rely on the kinematics and/or dynamics of the AGV. At low speed the kinematics are generally used resulting in a simpler model with most slip conditions deemed neglible [27]. At high speeds, the vehicle dynamics are used together with the kinematics in order to account for the slip experienced.

2.1.1 Kinematic Modelling

In a large number of cases, the use of pure kinematics to gain knowledge of how a vehicle moves through a terrain is adequate. The main assumption that is used to justify the accuracy of the kinematic model for predicting the motion over the terrain is that the terrain conditions vary slowly with the wheels maintaining a constant grip on the terrain so to reduce slip between the vehicle and the terrain. Additionally, vehicles that are moving at low speeds also can be assumed to move according to kinematics, under certain terrain conditions, this is due to the forces generated in the tyre being heavily reliant on the vehicle moving at a high speed. The kinematic models that have been developed for vehicles depend on the configuration of that vehicle, not all configurations provide a valid kinematic model. The kinematic model is invalid when a steering configuration that results in non-zero slip angles is used as the vehicle moves through a path, making it necessary to include slip in the tyres. For the majority of cases, slip of the vehicle through the environment cannot be ignored, and so corrections

8 2.1. Vehicle Modelling in the true motion of the vehicle are necessary, with vehicles experiencing more slip requiring more frequent corrections. The model that has been proposed by Madhavan [28] utilises a simple kinematic model with modifications so that the position, speed and heading of the vehicle can be estimated. The modifications that have been introduced to the kinematic model are the addition of wheel slip and skid parameters, which are estimated as a function of the wheel radius and the skid speed. The model is used as a process model for an Extended Kalman Filter (EKF), with update steps providing corrections to the process model at 15Hz. The experimental data used to validate the method was from a 4WD vehicle moving through an unstructured and undulating terrain. Although the modified kinematic model includes the effects of slip and skid on the system after the vehicle has experienced them, it does not allow for the motion of the vehicle through a terrain to be predicted as the slip is estimated based on state observations. A vehicle kinematic model was developed by Feng et al. [29], the model consisted of a bicycle with a trailer, this model was then modified for a tractor with two independently driven rear wheels by Eaton et al. [3]. This model consisted of three wheels, with the single front wheel steering the vehicle and the two back wheels providing the driving forces. The kinematic model assumes that the vehicle is travelling at a constant longitudinal speed. The additional effect of wheel slip in the vehicle model is predominantly developed by the side slip of the vehicle and hence the steering and trajectory of the vehicle is to be controlled. This model is then used to develop a number of different sliding mode control methods by Taghia [12], with the aim of reducing the effect of slip and external disturbances on the trajectory control. The results that are presented are for idealised simulation environments where the slip is assumed to only depend on the front steered wheel. The presented simulation also does not take into account variations that occur in the terrain conditions that can significantly effect the amount of slip that the vehicle is subject to.

2.1.2 Dynamic Modelling

For the majority of cases for four wheeled AGVs, the presence of slip in the terrain is a dominant factor in how the vehicle interacts with the terrain [30–32]. The inclusion

9 Chapter 2. Current State of Research of slip in the vehicle model means that the kinematic model only provides an ideal representation for how the vehicle moves through the terrain when subjected to slippage [33]. For a realistic vehicle model to be achieved, it is necessary to use a dynamic model for the vehicle motion. Dynamic models exist for vehicles that range in complexity depending on the different environments that the vehicle is to operate in. These different terrain conditions include indoor environments where the terrain is flat [22, 23], a typical road surface where the terrain is relatively flat [34, 35], and much more rugged terrain such as an off-road track or agricultural land [36]. There are a number of different approaches for tackling these different terrain types, including; using a planar vehicle model, or implementing a multi-body model. The simpler planar model is useful for developing path planning, path tracking and control applications for AGVs over relatively flat terrain, where as the multi-body model is useful for path tracking and control over terrain where the normal forces experienced by the tyres change significantly as the vehicle performs a maneuver over this terrain.

Planar Models

The planar models that are proposed in the literature are generally developed so that the main forces and moments that inﬂuence the motion of the vehicle are included [37–39].

The longitudinal force Fx, the lateral force Fy, as well as the self-aligning moment

Mz, are the dominant forces and moment that ground vehicles are subjected to, in combination with other external forces such as gravity and wind resistance. Modelling the effects of these forces and moments leads to less complicated controller designs and implementation as there is a lower number of system parameters involved in the vehicles motion. Planar models can provide an adequate approximation for both flat and relatively flat terrain at relatively low speeds with the controllers designed for these environments providing reasonable responses to setpoint changes and disturbance rejection. The planar vehicle dynamic model that was presented by Pota et al. [40] formulates the dynamics of the system as a series of vector and matrix calculations, with the forces applied as a function of the system geometry. The geometry of the system is based on

10 2.1. Vehicle Modelling the kinematic model of Feng et al. [29]. The vehicle is modelled as a bicycle towing a single wheeled implement, which pivots about a hitch point located directly behind the rear wheel of the driving vehicle. The forces that are involved in the calculation are the control inputs for the system, but the eﬀect of longitudinal slip on the forces is not present.

The developed structure for evaluating the equations of motion of the vehicle are extremely useful as they can be easily adapted to a vehicle with many independently steered wheels. However, the simulation results presented show an issue with the model — the lateral forces developed in the wheels do not react to the lateral disturbance as would be expected, with the tractor accelerating laterally over the course of the simulation. The dynamic model presented does not include the model of the tractor tyres, which would react to the lateral disturbance and limit the lateral slip signiﬁcantly. A realistic dynamic model of a tractor would incorporate the dynamics of the tractor tyres so that given a similar disturbance, the model would act consistently with a real system.

The model proposed by Macek et al. [41], assumes that the vehicle conﬁguration is that of a normal Ackermann-like steered vehicle with the two driven wheels situated at the back of the vehicle. This work builds on previous work and focuses on being able to provide a dynamic model for a vehicle that can be used with driving manoeuvres on a road. The model is experimentally veriﬁed through the use a purpose built vehicle platform.

A planar model was developed by Wang and Longoria [37] that incorporates the dynamic effects that are experienced in a 4WD4WS vehicle. The vehicle is driven through the use of four separate driving motors as well as four separate steering motors. The forces that are experienced in the vehicle are modelled through the use of the Dugoff [42] friction model that assumes a linear relationship between the wheel slip and the longitudinal force as well as between the side slip angle and the lateral wheel force. The self-aligning torque of the steered wheels is neglected as it is assumed that the steering controllers act to counter these effects. This model is then extended by Chen [39] to also include the actuator response of the driving wheels for the vehicle.

The simulation results presented by Wang show the eﬀect of the forces that the

11 Chapter 2. Current State of Research vehicle experiences as it traverses through a region where the terrain friction coefficient changes rapidly from a high coefficient to a low coefficient of friction on only one wheel. This simulation was developed to show the effects of how a vehicle goes through a maneuver with one of its wheels subject to an icy road surface. The trajectory of the vehicle using open loop control showed a significant divergence from the desired trajectory as the vehicle moved through the icy road region. The implemented controller estimates the terrain friction coefficient as it traverses over it, in a reactionary way. The dynamic model developed by Yu et al. [38] focuses on a skid steered vehicle with four wheels. It is primarily a planar model, which is then extended to work in a 3D environment by altering the gravity vector for the vehicle as it traverses sloped terrain. The tyre forces developed in the model utilise the coefficient of friction of the terrain as well as the terrain shear modulus and the relative velocity of the tyre. The developed model can be used to predict the motion of a vehicle through different terrains, however the use of the terrain shear modulus is a difficult terrain parameter to estimate. For a general AGV dynamic model to be useful for predicting the motion of a vehicle over non-uniform and uneven terrain, an important influence on the vehicle motion needs to be taken into account, that is, the vehicle body angle. The vehicle body angle influences the distribution of the weight of the vehicle over its tyres, with changes in body angle, and subsequently weight distributed, experienced for even purely longitudinal actions like accelerating and braking. For the vehicle system uncertainties to be reduced accordingly, planar dynamic models are not enough, even though they allow for simpler controllers to be developed.

Multi-body Models

The multi-body model is an extension of the planar model that takes into account the diﬀerence in height of both the center of mass of the vehicle body as well as the position and velocity of each of the wheels as it goes through a maneuver. Generally, the wheels of a vehicle are connected to the main vehicle model through the use of spring damper suspension systems, with a similar conﬁguration shown in Fig. 2.1. The motions that the vehicle body undergoes also include changes in the body roll and pitch angles as well as the height and vertical velocity as the body pivots and interacts with the suspension

12 2.1. Vehicle Modelling systems that connect the wheels and the body. The emergent phenomena from this system is that the wheel normal forces change as the vehicle executes maneuvers. As the forces and moments that are developed in the tyre are proportional to the normal force, this signiﬁcantly eﬀects the way the vehicle moves.

Fig. 2.1: Simple multi-body vehicle conﬁguration with four wheels supported as unsprung-masses connected via suspension elements, image taken from [43]

The different types of multi-body models that exist vary in how the suspension is modelled as well as how the body pivots through different maneuvers. Some pivot about a point inline with the axles from the wheels directly below the center of mass. With the more realistic models rotating the vehicle body so that it effects the suspension lengths. Models with reduced complexity for multi-body vehicles have been developed [44] which employ active control on the suspension systems of the vehicles so that the normal loads felt through each tyre on the vehicle are the same. This allows the system to be modelled as a simpler planar model, allowing for ease of design for controllers, however, the constraint that is used to make this simplification is a steady-state constraint. The planar reduction is invalid when the vehicle is going through a bumpy section of road and so the model and load balancing constraint is not applicable for a system that should maintain operation over unstructured and uneven terrain. A vehicle dynamic model simulation has been developed called Automated Dynamic Analysis of Mechanical Systems (ADAMS) which utilises the MF model along with additional analytical constructs to allow AGV systems to be evaluated as they traverse through different terrain conditions [45]. This simulation platform is developed to allow for system designers to evaluate designed parts to see how they interact and behave when subject to different road conditions. It is also used to evaluate how well

13 Chapter 2. Current State of Research a designed controller behaves when subject to dynamic road conditions, when utilising the ADAMS /Tire model [46]. The base model that is used to assess these interactions is predominantly the MF model [47] although other models can be used. The MF gives the best experimentally accurate results, under normal road conditions.

The ADAMS software allows for the changes in normal forces experienced in the tyre contact patches to be generated as the suspension systems are modelled within the ADAMS simulation environment. This means that during cornering maneuvers the calculated tyre longitudinal and lateral forces as well as the self-aligning moment are still valid. The disadvantage of using this software is that it requires the use of the MF tyre model parameters. The results generated from using the ADAMS software together with the Adams/Tire model can only be used under speciﬁc known conditions.

The multi-body vehicle dynamic model proposed by Anderson [48] is a model that separates the axles and the body of the vehicle so that relative motion between the two masses can occur. This allows for the vehicle body to undergo rotations about its center of mass. The side slip angle that the vehicle experiences is then altered as the slip angle rotation is relative about the center of mass of the body. The effects of the tyre normal force changing as the vehicle body goes through rotation have been neglected, with the tyres not been treated as a body with mass. The model is also further restricted to only roll angle rotation as the vehicle is modelled to move at a fixed speed. The simplified model was used to implement a controller so to avoid hazards and obstacles steering the vehicle. The controller was then run through the ADAMS vehicle dynamic model simulator.

A more complicated model that is present in [49] and further in [50] is that of a realistic model of a High Mobility Multipurpose Wheeled Vehicle (HMMWV) or Humvee. The vehicle kinematics and dynamics were calculated from basic principles and using the Simulink Matlab toolset a simulation was created that was able to model the behaviour of the vehicle over diﬀerent terrain types. The model was generated analytically, and because of this there is no need for deriving and solving the slower numerical solutions. This enables the simulation to run a single time step in 60 (ms), instead of several seconds which would be the case for numerical methods. The vehicle then is controlled through the use of Model Predictive Control techniques.

14 2.2. Tyre Modelling

From these models a controller can be designed that can generate the required response of the steering, the wheel angular velocity or both, for any vehicle configuration. There are a number of control approaches for an unmanned ground vehicle, which include, position, trajectory and to an extent path planning tracking techniques. The low level control focuses mainly on achieving particular set points within a certain settling time, with no concept of what these set points correspond to. The medium level control provides set points to the lower level controllers, based on optimising a particular cost function. Examples include, minimising the time of transit, maximising fuel efficiency or minimising path deviation, etc. such as [51]. High level control, on the other hand is focused on path planning and obstacle avoidance. Each different control approach provides the AGV with a different behaviour depending on how the different levels are implemented and integrated. There exists many methods of achieving the low and medium control when the environmental conditions are known, however there is insufficient work in situations in which the environment is unknown. The usual method for achieving path and trajectory tracking in unknown terrain is to estimate the mechanical parameters of the terrain and then feed this information back into a model of the system for the state to be determined. Another method is to estimate the amount of slip that the vehicle undergoes when passing through a particular terrain type, by sensing additional information and determining the true state of the system and then correcting the odometry estimation based on this.

2.2 Tyre Modelling

Knowledge of the tyre-terrain contact forces for a vehicle is an important aspect when developing a dynamic model of the system, as the main control mechanism of most AGVs is through the tyres. If the forces that the tyres are experiencing can be determined then the motion of the vehicle can be estimated as well as controlled. However, it is diﬃcult to predict the slip conditions of terrain in advance, as the behaviour is quite complex and shown by [52] to be a non-linear system of interactions between the tyre and terrain.

15 Chapter 2. Current State of Research

A common misconception in longitudinal force development in tyres is that the forces that are applied by the wheel on the ground are simply the product of the input torque and the radius of the tyre of the driven wheel. This can be true for solid, rigid wheels that have very large normal forces on very small contact patches as is the case with rail wheels, where the force required to overcome static cohesion for the wheel is quite high.

However, this is not the case for pneumatic tyres, the main reason for this eﬀect is that pneumatic tyres ﬂex and deform considerably more than that of a rigid wheel. Therefore, the assumption of a rigid connection between the wheel hub and the wheel tread is not valid.

The most widely held understanding of how the force is developed between a tyre and the terrain is that it is a function of the wheel longitudinal slip κ, with longitudinal force depending on the slip ratio and the lateral force depending on the slip angle α that the tyre is subject to.

A variety of different approaches used to model the tyre-terrain interaction exist, including: empirical formulaes from [47, 53, 54] which are accurate for specific tyre terrain combinations in steady-state conditions, semi-analytical models such as [55–59] which use separate models for braking/accelerating as well as critical/non-critical slip conditions, pure analytical methods based from first principles such as [60–65] and lastly, finite element models similar to [66–68] for the interaction which requires significant computational resources.

The dynamics of a wheeled vehicle including the slip magnitude and direction as well as the effective developed force of the vehicle can be determined by observing the interaction between the tyre and terrain. The interaction of the vehicle with the terrain is referred to as terramechanics [52, 69] and contributes significantly to the controllability of any vehicle configuration. In an environment in which the terrain is unknown, this method of vehicle dynamics estimation can only be useful if information on the current terrain is available.

The methods that have been proposed in the current literature on this topic include the identiﬁcation of the necessary terrain parameters, the prediction of terrain parameters, as well as the identiﬁcation of the wheel dynamics. All of these topics can

16 2.2. Tyre Modelling be utilised together to form a more comprehensive estimate of the vehicle dynamics in relation to the wheel and terrain interaction. The terrain parameter identification methods and techniques that have been proposed and used in various systems can be further broken down into the offline and online terrain parameter identification techniques. The methods that the offline and online techniques utilise to obtain the terrain parameter estimations generally differ in complexity with the offline methods determining a significantly greater amount of terrain parameter values [70, 71] , whereas the online techniques generally simplify the terrain parameters into easily observable parameters [72]. These methods work well for terrain that is consistent in large regions, where the estimation methods can be utilised.

2.2.1 Empirical Formulae Models

Magic Formula

Of the many models that have been developed to capture and describe the forces and moments that a tyre is subject to, the most commonly known model is that of the Magic Formula (MF) as developed by Pacejka [73]. This tyre model was the ﬁrst to describe the longitudinal, lateral and self-aligning moment using a single parameterised equation, and was an extension of the polynomial ﬁt that had been used previously but with multiple regions [74]. The original MF model that was developed was valid only for pure longitudinal slip and pure cornering slip, with the equation that formed the backbone of the MF in (2.1). The terms Y and X can represent either the combination of the longitudinal force Fx and longitudinal slip κ, or the lateral force Fy and slip angle

α or the self-aligning moment Mz and the slip angle.

Y = D sin C tan−1 (Bφ) and (2.1a)

E φ = (1 − E) X + arctan (BX) (2.1b) B

where

(X,Y )=(κ, Fx) OR (α, Fy) OR (α, Mz) (2.1c)

17 Chapter 2. Current State of Research

The main benefit of this model is that the parameters B,C,D and E are determined so that the MF curve gives the best fit with the experimentally measured data. The parameters represent the stiffness factor B, shape factor C, peak factor D and curvature factor E. The combinations of these parameters allow for a very close fit between experimental data and the MF curve as shown in Fig. 2.2. A different combination of the parameters B,C,D and E are found separately for Fx, Fy and Mz. This allows for variations in the maximum longitudinal and lateral forces that the tyre is able to achieve.

Fig. 2.2: Magic Formula fit to experimental data with different tyre normal forces [75] showing the quality of fit with the data for longitudinal tyre force, image taken from [73]

The inclusion of dominant phenomena experienced in the tyre such as tyre stiffness at zero slip and peak forces and torques values, allows for the MF to be used in developing Anti-lock Braking Systems (ABS) for improving vehicular safety as the model is simple and easy to implement. This model however is limited to pure slip conditions and so the model was subsequently expanded to include combined slip conditions [47], which also included the effects of the tyre camber angle on the forces and moments developed. The model was then further expanded to include the transient effects of the tyre as it undergoes a dynamic behaviour [76]. The key aspect that this paper introduced to account for these transient effects was that only translations of the contact patch

18 2.2. Tyre Modelling deformed the patch, relative to a rigid wheel rim. The transient forces and moments that are developed are due to a point mass of the tyre tread that is connected to the centre of the tyre contact patch.

The model was then subsequently expanded to take into account the eﬀect of tyre spin on the forces and moments [77]. The spin of the tyre is a combination of both the camber angle change as well as the tyre turning through a steady state turn. Motion of the wheel over bumps in the terrain was then incorporated in the model [78], which is now referred to as Magic Formula - Short Wavelength Intermediate Frequency Tyre (MF-SWIFT) model. With the most recent addition to the model, adding a dependency of the forces and moments on the inﬂation pressure of the tyre.

The MF model provides a great level of accuracy in predicting the forces and moments that a tyre will be subjected to, with the disadvantage that it is only valid for a specific combination of tyre and terrain. It is favourable for applications where a normal sized car is travelling along a normal sealed road. The drawback with this model arises when the vehicle begins travelling on a different terrain, causing the estimated forces and moments to no longer be correct. For an adjustment to the model to be made to correct for the new terrain conditions, a raft of off-line experimentation is required in order to generate the new set of parameters for the new tyre-terrain combination. In the MF-SWIFT model, this adjustment is achieved primarily by changing the λx and λy parameters, however, the values that are found for these are not the friction coefficients of the terrain, but rather a relative change from the original frictional coefficients of the terrain that the data was fitted with.

The parameters that are used in the model also have no direct relationship to terrain parameters, as the parameters are a simplification of the systems state parameters for a specific tyre-terrain combination. Before the tyre model can be used on a different terrain type that tyre-terrain combination must be analysed and new magic formula parameters chosen so as to fit the curve. This means that although accurate for a specific tyre-terrain combination, the model is not readily transferable between different terrain conditions, and thus inappropriate for use in a non-uniform environment.

In addition to the MF tyre model, there exists a number of other similar models that attempt to reproduce similar results, but with a signiﬁcantly less complicated model.

19 Chapter 2. Current State of Research

These models are used predominantly in state estimation, and attempt to remove a number of different singularities that exist within the MF model in order to provide a continuously differentiable function for use in state estimation techniques. The tyre model that is proposed by Ward [56] was for the purpose of a dynamic state estimation for a mobile robot subject to slip to check for immobilisation of the vehicle. The importance of Ward’s proposed model is that the traction and rolling resistance models proposed are explicitly differentiable, which allows for singularity free estimation for all movement forms. The model that is used is different to the normal MF model in that the relative velocity between the wheel speed and the ground speed is used in place of the wheel slip. A major strength in the method is that slip and immobilisation detection can be carried out without relying on GPS measurements, which is needed in a vehicle that is required to have persistent autonomy. The tyre model parameters also do not contain terrain dependant terms and so this provides difficulties when the vehicle transitions between different terrain types.

2.2.2 Analytical Models

Analytical models attempt to explain the interaction between the tyre and the terrain using fundamental principles, these models provide a single set of equations for all vehicle motions, however they do require additional information about the tyre-terrain system to be known. This single set of equations allows for only a small number of assumptions to be made for the model to work accurately. These attributes allow for safer and more accurate traction and braking controllers to be designed. Analytical tyre models help explain and describe the behaviour of the tyre over terrain, with the work from Julien [79] representing the initial eﬀorts at presenting an analytical tyre model. The original tyre model was based on the elastic deformation of an elastic band. Since this original analytical model a number of diﬀerent models have been proposed that attempt to accurately describe the behaviour of a pneumatic tyre as it travels over a terrain surface. There are many types of analytical models that exist, including: the point, string, beam and brush models, for all these models the fundamentals that are assumed is that of spring deformation. The original models used simple linear spring deformation,

20 2.2. Tyre Modelling with the main differences between the tyre model types being the explanatory power for explaining the lateral force developed [80]. The effect of spring damping on the tyre and the forces developed was then explored, as well as the effect of tyre contact patch size on the tyre forces [81]. The progression of these models have been towards more comprehensive models that take into account dynamic effects such as the hysteresis of the tyre element as the tyre element undergoes elastic and plastic deformation [60, 82, 83], the relationship between the coefficient of friction and the slip speed, as well as the dependency of the friction and spring coefficients on the loading force of the spring. The models for beam, string and point models do not include enough of the required degrees of freedom to be able to adequately model the forces generated when the vehicle is subjected to combined slip conditions with both longitudinal and lateral wheel slip.

Brush Model

The brush model is an analytical tyre model that was first described as a row of elastic bristles that are in contact with the terrain and deflect along the direction of the tyre relative velocity. The deformation of the bristle represents the amount of equivalent elastic motion through which the carcass, belt and tread of the actual tyre are deformed. As the tyre rolls over a surface, the model assumes that the leading edge of the tyre contains bristles that are not deformed, which make perpendicular contact with the terrain and are then gradually deformed as the bristle moves from the leading to the trailing edge of the contact patch. A three-dimensional representation of the brush model is presented in Fig. 2.3, this model shows both the longitudinal tyre bristle deformation as it grows over the contact patch as well as the lateral deformation that is developed as a result of a non-zero slip angle. The maximum lateral force that is developed in the contact patch is related to how the tyre is moving with respect to the ground. The limit on the brush model deflection under load is a function of the forward speed of the tyre so that a stationary tyre is able to get the maximum lateral deflection with the deflection limited by the longitudinal slip of the tyre.

21 Chapter 2. Current State of Research

Fig. 2.3: Brush Model visualisation, image taken from [75], showing the lateral force deformation along the contact patch when subject to a side slip

The main draw back of the brush model is that it doesn’t accurately replicate the lateral and longitudinal forces that are developed at larger slip values, with the model asymptotically approaching a maximum value. This eﬀect cannot be observed in experimental data, where the maximum force developed in the tyre contact patch occurs when the longitudinal and lateral slips are around 10-20%. This means that the model is suitable for developing control laws as long as the slip ratio is no greater than 10%, with any greater than that resulting in an unpredictable behaviour as well as loss of control.

The self-aligning torque that is developed from this model does not coincide with the response as seen in the experimental data. This means that during steering the brush tyre model is unable to correctly predict the self-aligning moment that is in aﬀect, which can lead to the loss of control through a maneuver for the vehicle or at the very least a loss of control accuracy.

22 2.2. Tyre Modelling

To be able to have an analytical model that is capable of replicating the relationship between the force and the slip that is present in the experimental data, another tyre model is required. The model needs to take into account the maximum value of the longitudinal and lateral forces that are felt as well as the eﬀect of the changing sign of the self-aligning torque during a steering maneuver. The model that attempt to accomplish this is the LuGre [61] dynamic tyre model.

LuGre Tyre Model

The LuGre dynamic tyre friction model was first developed in [61] with the emphasis on utilising the new friction model in control applications that are influenced by frictional errors. The developed tyre friction model includes the effect of the coefficient of friction changing as the tyre contact patch slides at different relative velocities with respect to the true ground speed. The LuGre tyre model also allows for the use of a distributed tyre load to be included to calculate the experienced forces and moments allowing for the generation of the self-aligning torques that change sign at larger slip angles.

The LuGre model builds upon the dynamic friction model that was developed by Dahl [60], which describes the force F developed in a similar way to the force developed by a spring under deformation with ǫ being the internal bristle deflection and k being the bristle spring stiffness coefficient, and has the form

dǫ k|vr| = vr − ǫ and (2.2a) dt FC F = kǫ, (2.2b)

where vr is the relative velocity between the bristles and the ground and FC is the Coulomb friction which is assumed to be independent of load or relative velocity. The

dǫ velocity of the bristle deﬂection dt describes the relationship of the input relative velocity to the relaxation of the bristle deﬂection over time. The model (2.2) with the condition

FC vr =6 0 reduces at steady state to the Coulomb friction, ǫss = ( k ) sgn(vr) and Fss =

FC sgn(vr). The LuGre model builds on the structure of (2.2) by replacing FC with a new function

23 Chapter 2. Current State of Research

1/2 −|vr/vs| g(vr)= µk +(µs − µk)e (2.3) which incorporates the Stribeck effect, or the slip velocity on the transition between static and kinetic frictional forces. The parameters µs and µk are the static and kinetic friction coefficients, and vs is the Stribeck velocity. The Stribeck velocity is the rate at which the friction coefficient transitions from the static frictional coefficient value to the kinetic coefficient of friction [82]. For low values of the stribeck velocity, the transition from static to kinetic friction occurs at a lower relative velocity. The final form of the LuGre friction model [63] takes into account the dependency of the normal force on the friction as well as the elastic, damping and viscous effects, and is shown in (2.4).

dǫ σ0|vr| = vr − ǫ and (2.4a) dt g(vr)

dǫ F = F σ ǫ + σ + σ v (2.4b) n 0 1 dt 2 r The importance of the LuGre model is its ability to take into account a number of dynamic and static frictional phenomena [84]: stiction, the Stribeck eﬀect, pre-sliding displacement and hysteresis. The model continues to provide the same steady state behaviour as did the Dahl model. With the associated steady state force being,

Fss = Fn (sgn(vr)g(vr)+ σ2vr) (2.5) which shows that the steady state force developed in the LuGre model is dominated by the term g(vr)Fn, which is in accordance with the force predicted by the Dahl model. The two dimensional extension of the LuGre model that was developed in [65] follows a similar structure except that it has expanded to include motion in the lateral direction as well as the longitudinal direction. In order to accommodate for this addition, (2.3) is replaced by a function that relies on both the longitudinal and lateral components of relative velocity and the anisotropic nature of the contact patch of the tyre. The tyre properties that give rise to the anisotropic behaviour of the tyre are due

24 2.2. Tyre Modelling to the construction of the tyre. The tyre is generally made up of a few different key components; rubber, fabric and steel cords [85]. These key components all vary between different tyres and will give rise to different behaviours in the longitudinal or lateral directions [86, 87]. The coefficients of friction depend on how well the surfaces can mesh together, with many tyre designs being anisotropic in nature result in the longitudinal and lateral frictional coefficients being different, with the longitudinal component generally being larger to improve the longitudinal force generated by the tyre.

The static and kinetic coeﬃcients of friction that are used in the derivation are µs and µk, containing both the x and y components for each of the friction coeﬃcients.

µsx 0 µs =   > 0 (2.6) 0 µsy    

µkx 0 µk =   > 0 (2.7) 0 µky     The coefficient of friction for a surface is coupled together with the lateral and longitudinal relative velocities as shown in [65], where the combined coefficient of friction lies on the boundary of an ellipse shown in Fig. 2.4 illustrating the combination of separate coefficients of friction and the relative velocities in the separate directions. This constraint arises from the requirement of a maximal dissipation rate for the combination of both directions.

Let the relative velocity vector vr = [vrx, vry], with vs as the stribeck speed and βs to be the stribeck exponent, this allows for the normalised coefficient of friction g(vr) to be formed in 2D from Eq (2.3). The terms are used to normalise the coefficients of friction depending on the relative velocities in the longitudinal and lateral directions. This results in the frictional coefficients that give the maximal dissipation rate.

2 2 2 βs kµ vrk kµ vrk kµ vrk kvr k k s k − v g(vr)= + − e s (2.8) kµ vrk kµ vrk kµ vrk k s k This relative velocity and the normalised coeﬃcient of friction is then used to determine the form of the equation for the deformation of the tyre elements. The tyre

25 Chapter 2. Current State of Research

Fig. 2.4: Frictional ellipse showing the maximum permissible friction coeﬃcient, image taken from [65]

elements are deformed through elastic deformation and are limited by the maximal dissipation rate of the tyre bristles, which act to limit the magnitude of deformation that the bristle can undergo before the bristle begins to slip across the contact patch. The two dimensional model is described in (2.9) and shows that the two dimensional case collapses back to the same solution as the one dimensional case with pure longitudinal motion.

ǫ˙i = vri − C0i(vr)ǫi (2.9a)

µi = −σ0iǫi − σ1iǫ˙i − σ2ivr (2.9b) where

2 kµkvrkσ0i C0i(vr)= 2 (2.10) µkig(vr) An additional extension that was proposed by Velenis [65], is that of the steering rate dependency on the self-aligning torque. The effect of including this extra state is that the tyre deforms to resist a steering torque when stationary. This effect is apparent in reality and so including this effect allows for a much more realistic tyre model to be

26 2.3. Terrain Perception realised. The model developed by Velenis, however, was only valid for the condition of the tyre being stationary. The majority of cases where steering is performed however is when the vehicle is in motion, and considering the significant amount of torque that is generated at rest it is necessary for this effect to be modelled at speed. A tyre model that is able to include the effects of steering at speed has been developed in this thesis.

2.3 Terrain Perception

Control of AGVs in a persistent manner requires the use of terrain perception methods to predict and estimate the forces that the vehicle experiences as it traverses the terrain. Estimating the terrain conditions that the vehicle encounters is therefore important for ensuring that the AGV can be adequately controlled and modelled. The terrain conditions that significantly influence the motion of a vehicle are the frictional coefficients of the terrain as well as the dominant surface geometry. The three key steps in all surveyed approaches are; the segmentation of the terrain data to reduce its complexity, the generation of feature descriptors from the segmented terrain data using various different description techniques, and the classification of the encountered terrain segments into terrain types utilising the information from the feature descriptors. Terrain segmentation forms the basis for the majority of terrain perception and prediction algorithms. Depending on the type of data that is being used for the terrain classification the segmentation is usually achieved through a number of different approaches. The type of data that has been used to sense the terrain in previous methods includes, using visual information from video data, point cloud information from a laser range finder or a fusion of both of these methods, with either stereo cameras or a laser and camera combination. The segmentation of the sensed terrain data is usually performed so that, of the total terrain dataset, a smaller subset is handled at any one time, and when successfully implemented, results in better description and classification results. The usual approach for performing segmentation of the terrain data attempts to group as many similar elements in the dataset as possible. This can be achieved in a variety of different ways

27 Chapter 2. Current State of Research which all depend on the sensors that are being used. Another reason for terrain segmentation apart from terrain type classification is to allow for a terrain model to be constructed from the sensed terrain data. With the generation of an accurate terrain model this allows a vehicle that is traversing through the terrain to perform realistic path planning and obstacle avoidance in order to prevent instances where the vehicle might become immobilised. The traversability of the terrain is also very important with regards to not only knowing if the vehicle will be able to navigate the terrain but also how it will behave whilst on the terrain. While building a terrain model of the AGV’s surrounding environment, it is important to accurately perceive boundaries between different terrain types. The perception of terrain boundaries are important as they serve as the basis behind many path planning algorithms, with the boundaries being used as a way of guiding the vehicle through a path or for weighting certain paths as more or less expensive relative to a metric. There are many ways that these boundaries in the terrain can be identified, and the predominant way of sensing these boundaries is through the use of camera colour data, and also geometric changes in the terrain.

2.3.1 Geometry Estimation

Traversability

The segmentation algorithms that are used for the majority of applications utilise a ﬁxed grid size projection into the global environment, by utilising a either 2D grid overlay on the terrain for two-dimensional systems [88, 89] or a voxel, volume based representation for the 3D cases [90, 91]. In contrast to this, graph based algorithms have also been used to represent terrain [92–95]. The majority of the systems that implement this type of grid based terrain identiﬁcation use it primarily for traversability analysis of the surrounding terrain of a vehicle, so that this information can be used by the vehicle’s path planner to chose the easiest path to traverse. Terrain traversability is the dominant form of terrain modelling that currently exists since the most important control aspect of an AGV is to chose a safe and appropriate surface to drive over, so that damage to the vehicle can be avoided. However, in a large number of cases an AGV may be asked to traverse through a terrain for which there is

28 2.3. Terrain Perception no obvious path, or that all of the terrain in the vicinity are not easily traversable. For these cases it is important to not just identify traversability, but also what the terrain surfaces are, so that a control algorithm can be employed to navigate the vehicle through the terrain. Terrain traversability classification that exists in the current literature is generally constrained to three different cases: traversable, obstacle and unknown[96–98]. The algorithms that employ this method assume that there is a path that can be followed through the environment and that the path is not dependent on the type of terrain that the path consists of. Alternative methods exists that try and classify the traversability of a terrain into incremental stages of traversability so that a vehicle wont get stuck as it traverses an environment [99–102]. Both of these methods aim to try and avoid steep slopes, dips, holes, steps and ledges as well as other hazardous terrain such as water. The vehicle’s resulting behaviour tends to follow roads or smooth dirt surfaces rather than jagged terrain. However, in some cases it may be a requirement for the AGV to traverse a region of the terrain that is not one that is easily traversable. In these cases the greater the amount of information known about the terrain allows for a better estimation of the effects that the terrain will have on the movement of the vehicle.

Dominant Surface Estimation

Traversability of the terrain is important, however path planning is not limited to simple ﬂat surfaces in many cases [103, 104]. For a vehicle to traverse over non-planar terrain, the angle of the surfaces in that environment must be estimated in order to maintain safe navigation through that environment. The main structure contained in the terrain data is the dominant surface of that region, this is in comparison to smaller deviations that occur in the surface due to the type of terrain that the surface covers such as small rocks, clumps of dirt or tufts of grass. For each surface, there are a few scales that need to be considered: the dominant surface or mega structure which contains features that of the scale of meters, the macro structure of the terrain which contains features at the centimeter scale, and the micro structures which contains features of submillimeter scale, an image illustrating the diﬀerences of these scales can be seen in Fig. 2.5. The

29 Chapter 2. Current State of Research features that exist on these scales all contribute diﬀerently to how a vehicle acts whilst traversing through it.

Fig. 2.5: Terrain structures and their scales, image from [105]

There exist a number of different methods for extracting the dominant surface geometry of a terrain. These methods can be separated into two main groups: geometry estimation that depends purely on 3D point cloud data, and methods that use a combination of point cloud data and some other terrain observation such as vision. One of the main methods of estimating surface geometry from the purely point cloud information include the use of Random Sample Consensus (RANSAC) [106], which has proven effective at detecting planar features in both 2D and 3D data in the presence of large amounts of sensor noise and spurious objects. This approach is used to estimate the plane of best fit for any grouping of 3D point clouds [107, 108] as well as other more complicated shapes [109], with the dominant surface composed of inlier data giving this surface. This method utilises a probabilistic approach by approximating the fit of the data whilst neglecting outliers. There are other terrain geometry segmentation algorithms that extract the ground from point cloud data, the implementation that Douillard [90] has developed implements an altered form of RANSAC as well as another clustering type terrain surface method. The first developed method is that of the Gaussian Process Incremental Sample Consensus (GP-INSAC) method, which works in an iterative approach with a starting seed selected as an inlier for the process. From these seeds the surface is approximated incrementally until the data inliers for a region have been incorporated into the terrain

30 2.3. Terrain Perception surface. This requires a number of assumptions about the terrain surface to be made in order to determine an appropriate seed surfaces, which in the implementation are taken to be the surface that is directly in front of the AGV below a certain height.

Douillard also presents a second terrain segmentation method that clusters the point cloud into a mesh based on the gradient over the diﬀerent areas of the point cloud being below gradient thresholds. The two methods generate terrain surfaces from the data, however, both methods link each grid to their neighbours and so incorporate macrostructures into the terrain representation.

Stereo vision based methods of estimating dominant surface geometry have been implemented by Tang [110], where stereo vision is utilised for generating the 3D point cloud information, after which the points are grouped based on the surface colour and then the dominant surface is estimated for this collection of points using RANSAC. The method however relies on there being some form or structure in the image to be able to extract out the segments to begin with from the image data, in the situation of an unstructured environment this assumption is easily broken and thus would have a reduced ability to extract the dominant surfaces in that environment.

A method that estimates the surface slope has been developed by Robledo et al. [111], where a surface range data is taken using a RGB-D camera. The depth data is then processed through iterations by fitting a Piece-Wise Multi-Linear (PWML) surface approximation to the data. The PWML surface patch is then compared against the maximum deviation of the depth data to the surface. If the deviation is greater than a threshold then the region is divided into four smaller regions, where the process repeats. When the deviation of the depth data from the PWML surface patch is below the threshold distance then the environment surface has been found. The PWML patch contains the slope information for the region of depth data that was used to form it. The limitations in this approach is that when a patch is on the boundary between two dominant surfaces, and the deviation in the patch is still within the defined threshold, then the patch is not split and the dominant surfaces present in the scene are not properly extracted. This effect can be reduced by reducing the maximum error allowable in each PWML patch, in order to create a more accurate map of the environment. The disadvantage of reducing the maximum allowable error, is that the size of the patches

31 Chapter 2. Current State of Research also reduces, which causes not only the dominant surfaces to be extracted, but also the macro-structure of the terrain. Research has been conducted into dominant surface estimation as a fusion between sparse laser scans and vision as presented by Howarth [112, 113]. The method employed determines the dominant planar features in a scene by extracting planar features from the laser scan data once at least two separate 2D line segments have been recorded. A plane is fitted to these points using a plane model with Principal Components Analysis, once this plane is found the corners and boundaries of the plane are determined from the camera image. The method works well at extracting planar surfaces from sparse point cloud data, however limitations are discussed about significant influences from biases in the laser scanner caused by surface properties and angle of incidence. The use of image data to determine the boundaries of the planar surfaces also means that for a poorly lit environment or one with many shadows this approach would perceive the boundaries incorrectly. In addition to these methods there has been a number of segmentation methods that have been developed for depth image data, with the initial work carried out by Jiang and Bunke [114]. This work segments the range data by performing edge detection over the depth image, this is generated through the use of gradient based edge detection methods. This form of depth image segmentation is favourable at identifying edges and planes of a surface. However, the method not only captures the dominant surfaces in the depth image, but also the erroneous surfaces created by the texture of the terrain in the image.

2.3.2 Friction Estimation

Terrain friction coefficients are known to depend on terrain characteristics such as terrain cohesion and internal friction angle [71, 72, 115, 116]. These terrain characteristics are known to vary depending on the type of terrain. Terrain friction coefficient estimation can be either direct or indirect in how the friction parameters are predicted. The direct methods of friction estimation rely on information learned through the experience of traversing over a terrain [117], whereas indirect estimation is achieved through identification and classification of the type of terrain for a given

32 2.3. Terrain Perception surface [118, 119] and then assigning that terrain surface with the appropriate friction coeﬃcients [120].

Direct Measurement

A few methods exist for directly measuring the coefficients of friction of a particular terrain surface, these methods predominantly rely on either acquiring accurate measurement of the terrain surface [121, 122] or from measuring the forces felt through the tyres as the terrain is traversed [117, 123–125]. Through these methods the terrain friction coefficients can be estimated, so that the uncertainties in the motion of the vehicle can be reduced. An analysis of the relationship between the surface roughness of the terrain and the static and kinetic coefficients of friction has been carried out by Ergun [121, 122]. With the results showing that there is a significant relationship between the Mean Profile

Depth (MPDmac) of the macrotexture as well as the average wavelength Lamic of the microtexture profile with the value of the static coefficient of friction. The coefficient of static friction is dependant on MPDmac and Lamic, with the speed regression slope depending on MPDmac and Lamic. The required values for each of the terms are in the magnitude of 0.1mm, so to be able to use this approach to estimate the static and kinetic coefficient of friction the sensor that is used would need to have a resolution at least in that order of magnitude. The majority of Laser Measurement Systems (LMS) however are designed for measuring distances in the medium and long range with resolutions in centimeters and so would not be suitable for gathering adequate data. A microscope is used [121, 126] to determine the surface properties of the terrain sample in laboratory conditions, and so such an approach would not be suited for mounting on an AGV. The frictional coefficient estimation techniques that rely on observing how the vehicle traverses through a terrain provide a good estimation of that coefficient. One of the first attempts to estimate the friction conditions of the ground was developed by Eichhorn [127]. This procedure involved using a number of specialised sensors for the tyre to measure the stress and strain experienced in the tyre as it undergoes motions. Another method developed by Gustafsson [123], estimates the frictional coefficients of a

33 Chapter 2. Current State of Research terrain region by using sensors in normal Anti-lock Brake System (ABS) wheels. Both methods provide estimations for the frictional coeﬃcients of the terrain, however, these estimations are only carried out after the vehicle has begun to traverse the region. To further increase the accuracy of control of the vehicle through such a terrain it would be beneﬁcial to be able to predict the surface friction before the vehicle enters the terrain.

The method proposed by Müller et al. [117] employs an estimation technique that utilises only the first section of the slip versus µ curve. Müller et al. take advantage of a link between the maximum friction coefficient and the steepness of the initial linear component of the slip curve. Using a simple linear friction equation taken from Gustafsson [123] and performing a least squares fit of the experimental data, the friction coefficient was able to be estimated. For the friction estimate to be able to identify differences between terrain types, the acceleration of the vehicle needs to be above 4 ms−2. This means that the vehicle is required to be driven relatively aggressively for the maximum friction coefficient to be identified. Driving with such acceleration is possible but it is risky, with the possibility of damage to the vehicle and loss of control.

Friction coeﬃcients of a surface are an abstract concept that attempts to combined a number of the internal sources of resistance to motion. The method developed by Hutangkabodee [116, 128, 129] is used to estimate a number of these internal friction sources; the soil cohesion, internal friction angle, shear modulus and pressure sinkage coeﬃcients. From these soil parameters, it is possible to implement an analytical model based on a non-deformable tyre and deformable terrain. The model allows for the longitudinal driving force of the wheel to be determined.

The later methods of the parameter estimation operate online [130], and also include the estimation of the friction coeﬃcient for the surface. Although the model gives accurate predictions of the longitudinal force and the subsequent friction coeﬃcient of the terrain, the model does provide a method of predicting the lateral force and moment that acts about the rotation of the tyre. The experimental data that was used to validate the model was also for a very slowly moving vehicle, which is valid for the case of a interplanetary robot, but not for an AGV operating at reasonable speeds in a terrestrial environment.

The majority of the methods reviewed above operate as reactive methods that

34 2.3. Terrain Perception estimate the frictional state of a terrain after the vehicle has traversed the terrain. These methods do not provide information about what the vehicle should be expecting to experience and so do not provide any reduction in the uncertainties of the system so that a control or path planning algorithm might beneﬁt from.

Terrain Classiﬁcation

There has been a number of different approaches to terrain classification in recent years, with the three main approaches being, camera only [131–133], LiDAR only [120, 134, 135], a fusion of both [136] as well as the use of stereo vision [103]. These approaches generally utilise additional sensory data such as accelerometers, GPS and wheel encoders for state estimation. Terrain classification using camera only data has been shown to classify terrain well in most environments using colour and image texture data [131, 137, 138] with reasonable classification rates. However, these methods have to contend with variances in the illumination of the terrain, which in unstructured environments is non-controllable. This may lead to errors in classification when the apparent colours of the terrain do not match up with the expected colours. Vision based terrain classification generally utilises a combination of features including colour and texture data which can be in many different forms. The descriptors are calculated from regions of the image which can be group in a number of different ways in order to contain only similar elements so that the classification step can be improved. The susceptibility of the vision only methods to varying light conditions on the accuracy of terrain classification has attempted to be addressed and these approaches are discussed and evaluated. The vision only segmentation is performed utilising the image data and can be extracted by merging similar pixels together to form a larger segment, this is usually referred to as a super-pixel [139]. The information that is used to simplify the computation and classification of feature descriptors as the segment that is developed has a consistent value. A terrain classification method that was developed by Kim et al. [118] utilises the idea of a super-pixel for use in classification of image data. The image data is

35 Chapter 2. Current State of Research compared to a small number of learning images which consist of basis local descriptor vectors that represent diﬀerent terrain types, soil, small gravel, gravel and asphalt. The local descriptors that are used consist of colour information as well as the material texture information that are taken from the Leung and Malik (LM) set [140].

Once the terrain has been classified, it is assigned a friction coefficient for use in path planning. The effect of lighting conditions on the terrain classification method is attempted to be taken into account by including the LM filter responses in the description vector. The results that are discussed however do not include any mention of lighting conditions.

Kim et al. [96] implements a similar terrain classification method that compares the difference between representing the terrain as either a rectangular patch over the image data or groups the pixels into super-pixels. The descriptors that are used in this implementation also come from the LM set including colour values in both the RGB and HSV colour spaces as well as statistical information about the hue and saturation of the segment. The classifications that are given for the terrain types are simply traversable and non-traversable with the results for the super-pixel based segmentation giving the better results. This shows that the variation of terrain type that exist within artificially generated patches significantly affects the accuracy of terrain classification as multiple terrain types can exist within the same patch.

The majority of terrain modelling approaches that exist in the literature utilise a Laser Measurement System (LMS) in order to collect data from the environment. The majority of LMS that are used, perform 2D scanning of the environment, which limits the amount of data that is visible at any one point in time. For the laser scans that are collected to be usable in 3D, the laser has to sweep through the environment. This can be achieved either by the vehicle’s movement through the environment or by using a servomotor to rotate the laser about an axis. There are also full 3D scanning laser systems that can capture the environment in one system from the perspective of the vehicle without needing any additional systems and without needing the vehicle to be in motion.

Wang [120] implemented terrain segmentation and classiﬁcation utilising a downwards facing LMS, which used the laser range and remission data to classify

36 2.3. Terrain Perception the terrain types. The terrain data that was collected included laser range data and remission data from asphalt, concrete, grass and gravel roads, as well as additional inertial and GPS data for the estimation of speed of the vehicle, so that the data can be fused together to produce a model of the terrain surface. The feature descriptors that are generated, however are only calculated based on one single scan, and only on a simple planar surface.

The method yields good classification accuracies by using a combination of range and laser remission data, however the feature descriptors that are used are a high-dimensional set of spatial frequencies of the terrain surface. The features consist of the power spectral density of different spatial frequencies for each lateral scan as well as the mean remission values for each scan. The effects of varying the speed of the vehicle were explored with the conclusion that the LMS data related to terrain identification in their method was speed independent.

The subclass of approaches that utilises both range data in the form of either 3D laser data or stereo vision data as well as the visual data from a camera allows the system to take information from both the limited range of the laser scanner as well as the longer range ability of a camera. This allows for a system that can predict terrain characteristics far ahead of time, allowing for a faster system as the required response time for a control action is increased as more of the approaching environment is visible.

One application of the fusion of both laser and camera data is that presented by Dahlkamp [135], where laser data is first used to estimate the angle of the road so that the data from the camera can be correctly projected onto the driving surface. The laser data is also used to evaluate the environment and classify it into either traversable or non-traversable surface which is detailed in [141]. This approach operates on the height differences within the point cloud being above or below a critical threshold height difference, which is then used to classify the terrain into either obstacle or traversable terrain.

The terrain descriptors that have been used include information of the image texture features such as surface roughness [142] and various spatial frequency features based on the colour image, using texture features generated from the gray-level co-occurrence matrix (GLCM) [143, 144] and normal colour data.

37 Chapter 2. Current State of Research

Feature classification has been carried out using the image texture features developed from the GLCM in combination with colour data as used by Zhao [145] to be able to recognise red and green apples hanging on an apple tree. The benefit of this approach is to utilise the texture properties of the apple, which is significantly different to the texture properties of the rest of the tree. This is due to the apples having a relatively smooth surface in comparison to the branches and leaves of the tree. The ability of the GLCM to be used to identify a surface regardless of the colour of that surface is important when looking for a surface classification approach to use for a persistent solution as the colour of the surface will not be constant as the time of day changes. The terrain classification methods that have been discussed above allow for a terrain type to be classified and given a learned frictional coefficients for a surface of that type, however these methods currently do not perform reliably enough in conditions that contain shadows or where the lighting conditions significantly affect the data recorded. The methods that utilise laser range finder data do not have the same operational limitations as the visual based classification, however they are unable to distinguish between terrain types that are not significantly different and focus mainly on identifying a viable path for a vehicle.

2.4 Summary

This chapter has surveyed the efforts of terrain perception researchers with an emphasis on the estimation of physical properties of the terrain. The terrain perception work that has been presented does not focus on reducing the uncertainties in the system model. Much of the work that has been done, determines the terrain conditions after the vehicle has encountered a section of terrain and so would not be useful for reducing the uncertainties for a control system, which requires prior knowledge. Techniques serveyed that attempt to map terrain conditions do so primarily with vision based sensors. These vision based methods provide solutions when there are favourable lighting conditions. However when there are shadows in the field of view or when different terrains have similar colours, it becomes difficult to identify these terrains and where the boundaries between different terrain are. The proposed terrain perception method in Chapter 5

38 2.4. Summary addresses these issues by classifying the terrain through the use of purely range data. A review of the literature also shows the importance of implementing a realistic vehicle model in order to adequately model the behaviour of a vehicle. A multi-body dynamic vehicle model is developed for this purpose in Chapter 3. The literature review of the existing tyre models also highlighted the need to include the effect of the frictional conditions of the terrain, as the main interaction between the terrain and the tyre is the primary source of vehicle control. It is therefore, necessary to have a tyre model that is able to adapt to the changing frictional conditions of the terrain to reduce the uncertainties required in the system model. This is also addressed in Chapter 3. Furthermore, the research into different tyre models shows the importance of having a descriptive tyre model in order to reduce the uncertainties in the system model. Currently, the descriptive models are not able to adequately model the forces and moments generated by a tyre when subject to normal driving conditions. The most accurate tyre models that exist are semi-empirical models, that provide little insight into the reasons for the tyre behaviour. A source of uncertainty in the system is the lack of descriptive power of the tyre models. To address this, a tyre model that is developed from first principles is presented in Chapter 4.

39 Chapter 2. Current State of Research

40 Chapter 3

Vehicle Modelling with Friction Dependence

Improving the accuracy of path tracking for an AGV when traversing through an unstructured or uneven environment is one of the prominent goals in field robotics. An important aspect of achieving this goal is in reducing the uncertainties that are present in the system model. The system model thus needs to be descriptive of the physical system so that the model can be used accurately in prediction of the kinematic and the dynamic effects that influence the physical system. To achieve this an accurate generalised system model of a vehicle should be developed so that the different aspects of the system uncertainties can be investigated and identified.

This chapter presents, in Section 3.1, a 3-dimensional multi-body vehicle dynamic model for a generalised four wheel vehicle configuration, with each wheel individually driven and steered, a Four Wheel Drive Four Wheel Steer (4WD4WS) configuration. The configuration is generalised in that the model can be adapted to a normal car with Ackermann steering, by fixing the steering angles and the driving torques of the wheels. The 4WD4WS configuration is the most interesting, however, as it represents the potential to implement new control schemes for path tracking and trajectory control.

In Section 3.2, a novel tyre force model is presented that is dependant on the terrain frictional coeﬃcients. A suite of simulation experiments, in Section 3.3, are performed with the 3D vehicle dynamic model, with a comparison between the proposed friction dependant force model and the Magic Formula SWIFT model, which was

41 Chapter 3. Vehicle Modelling with Friction Dependence experimentally derived by Besselink [78]. The dynamic model results are also compared with the kinematic motion of the vehicle model. A discussion is presented in Section 3.4 of the developed vehicle and tyre models, which examines the sources of uncertainties that exists within the developed model as well as its limitations.

3.1 3D Dynamic Model

The dynamic model that best represents the motion of the 4WD4WS vehicle is the one that includes motion in 6 Degrees of Freedom (DoF). A realistic 3D vehicle model consists of the vehicle body, the four tyres as well as the suspension system. The forces and moments that are generated in the tyres are transmitted through to the vehicle body via the suspension system. The suspension system that is used by the 4WD4WS is that of a simple suspension system for the 3D vehicle model. The vehicle 3D dynamic model is expressed as a combination of the body, suspension and tyre dynamics, with the combination of the individual states seen below.

Xv = [Xb,Xs,1,Xs,2,Xs,3,Xs,4,Xt,1,Xt,2,Xt,3,Xt,4] (3.1) where Xb corresponds to the body states, which is derived in Section 3.1.1, Xs,i corresponds to the suspension states for the ith wheel, which is derived in Section

th 3.1.2 and Xt,i represents the i tyre model states, which is derived in Section 3.1.3.

3.1.1 Vehicle Body Dynamics

The motion of the vehicle body for 6DoF consists of translational movement of the center of mass in x, y and z, as well as rotation φ,θ,ψ, these values are with respect to the global reference frame. The forces and moments that are generated by the vehicle are expressed in the body local frame. The local frame linear [u,v,w]T and angular [p,q,r]T velocities are represented as a vector of the individual velocities for the local body frame along the x, y and z axes.

T Xb = [x,y,z,φ,θ,ψ,u,v,w,p,q,r] (3.2)

42 3.1. 3D Dynamic Model

The free body diagram for the vehicle body is shown in Fig. 3.1, which provides the locations of the forces that act on the vehicle with reference to the center of mass of the vehicle body, with the connection points to the suspension system for each wheel located a distance hs vertically from the center of gravity of the vehicle. The numbering for the wheels are as follows: left front (1), right front (2), left back (3) and then ﬁnally right back (4).

Fz,4 b

Fz,3 l Fy,4 Fx,4 Fy,3

hs Fx,3 Fz,2 hc

Fz,1 y Fy,2 q Fx,2 Fy,1 p Fx,1 l x 2 b 2

Fig. 3.1: Vehicle Body Free Body Diagram showing the geometry and location of the forces acting from the suspension system

The effect of the self-aligning moment, the steering torque and the driving torque are not included in the free body diagram. The reason being that the connection between the steering mechanism and the wheel hub acts at a small distance, and with the expected torque from the steering to not be significant in amplitude, the force that is transmitted between the wheel hub and the vehicle body is also not significant. The vehicle dynamic models that have been presented before in the past [25, 36, 39, 41, 48, 146], also do not include the effect of the steering and driving torques or the self-aligning moment on the body model, with the majority making the assumption that the wheel hub and the body are a pseudo-decoupled system.

43 Chapter 3. Vehicle Modelling with Friction Dependence

The motion of the vehicle body can then be calculated through the useof (3.3) and (3.4), with the knowledge of the total force vector F and total moment vector M.

u˙ p u   1     v˙ = F − q × v (3.3)   m                  w˙   r   w              p˙ p p

  −1      q˙ = I M − q × I q (3.4)                       r˙    r   r                where m is the mass of the vehicle body and I is the body inertial tensor dependant on the physical design of the 4WD4WS platform. The forces F and moments M that the vehicle body experiences originates from the tyre-terrain interaction as well as other external forces that the vehicle is subject to. These external forces include the vehicle weight as well as aerodynamic drag and wind gusts. The force that acts on the vehicle body from the tyre and suspension systems

T can be expressed as Fi = [Fx,i, Fy,i, Fz,i] for i =1, 2, 3, 4

F = F1 + F2 + F3 + F4 + W + D (3.5)

where the vectors W and D represent the force due to gravity and the aerodynamic drag, as expressed in (3.6) and (3.7). With the weight acting in the negative direction, as the coordinate system for the vehicle body has the z-axis aligned upwards.

− sin θ   W = −mg cos θ sin φ (3.6)        cos θ cos φ      1 2 2 cD,xAxu   1 2 D = cD,yAyv (3.7)  2     1 c A w2   2 D,z z      44 3.1. 3D Dynamic Model

where cD,i and Ai correspond to the drag coeﬃcient and area for the ith axis. Similarly the moments that the vehicle body is subject to can be evaluated from the relationship between the geometry and forces.

M = Fi × lb,i, (3.8) i=1 X where lb,i is the lever arm distance from the center of mass to the mounting position of the suspension to the vehicle body. The values of lb,i for the vehicle geometry shown in Fig. 3.1, are expressed as below,

L L L L 2 2 − 2 − 2         B B B B lb,1 = lb,2 = − lb,3 = lb,4 = − (3.9)  2   2   2   2                   hs   hs   hs   hs                  The remaining states that are required to be determined are both the global positions and the global attitude, the global linear and angular velocities can be determined from the local linear and angular velocities by transforming the velocities into the global coordinate frame. The rotation matrix for the linear velocities is expressed as A in (3.10).

cθcψ sφsθcψ − cφsψ cφsθcψ + sφsψ   A = cθsψ sφsθsψ + cφcψ+ cφsθsψ − sφcψ (3.10)        −sθ sφcθ cφcθ      with this rotation matrix, the global linear velocities for the vehicle body can be determined through the use of (3.11).

x˙ u˙     y˙ = A v˙ (3.11)              z˙   w˙          The transformation matrix for transforming the local angular rates to the global

45 Chapter 3. Vehicle Modelling with Friction Dependence angular rates requires a diﬀerent transformation matrix because of the use of Euler angles, the required transformation matrix R, is shown in (3.12)

1 sin φ tan θ cos φ tan θ   R = 0 cos φ − sin φ (3.12)        0 sin φ sec θ cos φ sec θ      The global angular velocities for the vehicle body can now be determined through the use of the following equation (3.13).

φ˙ p˙     θ˙ = R q˙ (3.13)          ˙     ψ   r˙          write up a linking paragraph for between sections

3.1.2 Suspension Dynamics

The purpose of the suspension system for ground vehicles is to filter out large motions of the tyre so that the impact that they have on the vehicle body is reduced. The majority of suspension systems consist of a passive spring-damper arrangement, although others can also consist of active components. The suspension arrangement that is in place for the 4WD4WS platform is a simplified configuration as shown in Fig. 3.2, showing the separation of the masses of the wheels and the vehicle body. The motion of both systems are semi-decoupled, the x and y axes remained coupled between the body and the suspension system through the compliant constraints from the suspension members. The representation is an approximation, as the suspension system is not

In the suspension system the forces Fx and Fy generated by the tyre, shown in

Fig. 3.3, are transmitted directly into the vehicle motion, with the vertical force Fz from the tyre transferred through the suspension to the car body. The moment Mz acts about the centre of the wheel and aﬀects the steering through the pivot point that the steering arms connect to. The rolling resistance moment My is conﬁned within

46 3.1. 3D Dynamic Model

ks bs ls mt

kt bt r

Fig. 3.2: Quarter Vehicle Suspension Free Body Diagram, showing the separation of the suspension system and the tyre hub as well as the relationship between the suspension system and the road height zr

the rotation of the wheel and eﬀects the force Fx that is generated. The overturning moment Mx is decoupled and constrained through the suspension members. The states that are needed to be able to model the behaviour of the suspension system Xs,i are shown in (3.14), with independent states given for each of the tyres. As the independent motion from the tyres are only in the z axis this results in a system state of only the position zs,i and velocityz ˙s,i of the height of the bottom of the suspension linkage.

T Xs,i = [zs,i, z˙s,i] (3.14)

The values of zs,i andz ˙s,i are determined through the relationship between the attitude of the vehicle body and the suspension states. The vehicle body is responsible for the values of zb,i andz ˙b,i, which correspond to the heights and velocities of the top of the spring-damper where it connects to the body of the vehicle.

If zb is the height of the connection point of the spring-damper to the body of the

47 Chapter 3. Vehicle Modelling with Friction Dependence

Fig. 3.3: 6 DoF tyre showing the forces and moments acting on the tyre together with the co-ordinate system for the tyre, from Wong [55]

car and zs is the height of the connection point of the spring-damper to the tyre hub, then the force exerted by the suspension is equal to

Fs,i = −ks,i(zb,i − zs,i − ls,i) − bs,i(z ˙b,i − z˙s,i) (3.15)

where ks,i, bs,i and ls,i are the spring constant, damping constant and no load suspension length for the suspension of the ith tyre. The vertical force that is felt by the suspension system through the body also means that the same force is felt by the body, thus the force Fs,i = Fz,i, from (3.5).

The vertical position zb,i and vertical velocityz ˙b,i of the vehicle body where the suspension connects, need to be calculated. These values can be found through the use of the vehicle body geometry, attitude and motion, which can be found in the current vehicle body state Xb.

The vertical positions of the vehicle body connection points zb,i are evaluated by applying a rotation and translation about the center of mass of the vehicle body, the

48 3.1. 3D Dynamic Model rotation goes through the roll and pitch angles and the translation is by the height of the center of mass.

zb,i = [− sin θ, cos θ sin φ, cos θ cos φ, z] · [lb,i; 1], i =1, 2, 3, 4 (3.16)

Similarly, the vertical velocity of the suspension connection points on the body are eﬀected by the velocities of the vehicle body. The velocity of the body at the suspension connection point is found by adding the linear velocity of the car body plus the cross product between the angular velocity of the body with each lever arm distance lb,i for each wheel. The vertical component of that calculation is expressed in 3.17.

z˙b,i = w + lb,i · [−q, p, 0] i =1, 2, 3, 4 (3.17)

Now that the position and velocity of the suspension systems components is known the resultant force on the tyre hub can be determined. The relationship between the tyre hub vertical height and velocity and the vertical height and velocity of the suspension base is expressed in the following equations.

zt,i = zs,i + ht (3.18)

similarly,

z˙t,i =z ˙s,i (3.19)

3.1.3 Tyre Dynamics

Each tyre consists of a number of diﬀerent states, as the tyre consists of both its vertical position and velocity as well as the wheel rotational speed, steering angle and steering rate. Therefore, the state Xt,i for each tyre can be formulated as below,

T Xt,i = [ωi, δi, δ˙i] (3.20)

th ˙ where ωi is the wheel speed of the i tyre, δi is the steering angle and δi is the steering rate, of the ith styre.

49 Chapter 3. Vehicle Modelling with Friction Dependence

The tyre rotational speed is dependant on the tyre moment of inertia in the tyre y yy axis, It,i , the driving torque τd,i, as well as the longitudinal tyre force and the rolling resistance moment that the tyre experiences. The forces and moments that act on the tyre can be seen in Fig. 3.4, with the diﬀerential equation for the tyre angular velocity

ωi shown in (3.21).

τd,i ωi

My,i

Fx,i

Fig. 3.4: Forces and moments acting about the y-axis of the vehicle tyre

1 ω˙ i = yy (τd,i − Fx,ire,i − My,i) (3.21) It,i

where, re,i is the eﬀective tyre radius which is determined based on

zt,i − zr,i ; zt,i − zr,i ≤ r re,i =  (3.22)  0 ; zt,i − zr,i >r

Similarly, the steering mechanism for the tyre consists of the steering torque acting about the z-axis of the tyre, with the steering rate δ˙i dependant on the moment of zz inertia of the tyre about the z-axis, It,i , and the self-aligning moment Mz,i and the steering torque τs,i. The moments acting on the tyre are visualised in Fig. 3.5, with the diﬀerential equations for the motion of the steering mechanism formulated in (3.23) and the state δi is found directly from δ˙i.

50 3.1. 3D Dynamic Model

τs,i

M x z,i

Fig. 3.5: Forces and moments acting on the z-axis of the vehicle tyre

¨ 1 δi = zz (τs,i − Mz,i) (3.23) It,i The tyre tread itself acts as a spring damper system as well, and so the forces that the tyre hub is subject to do not pass directly through to the tread. To determine the normal force that is acting on the tyre-terrain interface this relationship needs to be modelled. The tyre-road interaction depends on the height of the tyre hub, zt,i as well as the road height zr,i, with the tyre no-load radius being r, with the road providing an upwards force on the tyre, this interaction is modelled using (3.24).

Ft,i = −kt,i(zt,i − zr,i − r) − bt,i(z ˙t,i − z˙r,i) (3.24)

th where kt,i and bt,i are the spring and damping constants of the i tyre. This vertical tyre force model, however is only appropriate when the tyre is in contact with the ground. When contact is lost with the ground the only force acting on the tyre is that of gravity acting on its mass and the force exerted by the car body.

Thus a revision is to be made to Ft,i such that

−k (z − z − r) − b (z ˙ − z˙ ); z − z ≥ 0 ′ t,i t,i r,i t,i t,i r,i r,i tr,i Ft,i =  (3.25)  0 ; zr,i − ztr,i < 0

 th where ztr,i the height of the tread of the i tyre, with zr,i − ztr,i ≥ 0 meaning that the

51 Chapter 3. Vehicle Modelling with Friction Dependence tyre and the road are in contact. The net upwards force that is acting on the car tyre is equal to the force exerted by the tyre on the ground minus the force of the car’s suspension minus the force due to gravity. The equation (3.26) shows the tyre vertical acceleration.

1 ′ z¨t,i = (Ft − Fs) − g (3.26) mt

With the force from the suspension Fs,i acting on the tyre, giving the opposite signed force.

3.2 Friction Dependant Tyre Force

The developed 3D multibody dynamic model of the AGV depends on the forces and moments that are felt in the tyre as it interacts with the terrain, subject to a vertical tyre force. The vertical force that is felt by the body and subsequently the suspension and the tyre systems can be known and determined from the attitude and behaviour of the vehicle body as it goes through an environment. The longitudinal and lateral tyre forces Fx,i and Fy,i as well as the rolling resistance My,i and the self-aligning moment

Mz,i can be determined through a number of different tyre models. The empirical tyre models such as the Magic Formula - Short Wavelength Intermediate Frequency Tyre (MF-SWIFT) model [75] use the kinematic motion of the vehicle to calculate the forces and moments, however they require that both the tyre and terrain conditions are known ahead of time. There are also simpler tyre force models that require only a few parameters to reproduce a similar force model such as the one proposed by Ward [56], with the benefit of being able to alter the parameters in order to account for the effects of different terrains on the forces that are developed. The tyre models that has been used extensively in the past, are predominantly based on the Pacejka model, the ’Magic Formula’ [47]. This model decouples the longitudinal and lateral forces, with the tractive effort and side-slip force dependant on the slip ratio and the relationships are expressed in an empirical formulae. They work well for a number of combinations of tyre and terrain types. In their work, the slip ratio is defined in two separate regimes; accelerating (3.27) and braking (3.28), which has a

52 3.2. Friction Dependant Tyre Force singularity at slip ratio = 0.

v i =1 − , (3.27) rω rω i =1 − . (3.28) s v

An analytical model has been described in [55], where not only is there a distinction between accelerating and braking but also the model has two additional transitional points described at the critical slip and skid values, which are shown in (4.20).

Cii, ; i ≤ ic Fx =  (3.29) µpN  µpN 1 − ; i > ic  4C i i  The transition points in both the empirical and analytical model outlined above make it necessary to know additional information about the system including the characteristic contact length of the tyre terrain interaction, which is dependant on speciﬁc interaction characteristics and so is diﬀerent for each tyre-terrain combination.

The approach taken is an adaptation of a method proposed by Ward [56], where the tractive force is taken as a function of the relative velocity between the ground and the rotating wheel. The relative velocity is calculated as shown in (3.30), where ωi is the angular velocity of wheel i and vw,i is the actual linear velocity of the wheel along the longitudinal axis of the wheel,

vr,i = ωire − vw,i. (3.30)

The relative velocity is then used to determine the maximum tractive effort on any given terrain. Ward [56] suggests using the relative velocity to simplify the equation and remove the singularity that exists at zero slip. However an issue that arises in the model used by Ward shown in (3.31) is that the coefficients of the traction function at large relative velocities can result in a tractive effort in a direction opposite to what is intended. If the coefficient C2 is set to zero to avoid this effect, then the shape of the response generated by the model does not fit with the empirical model for large slip values,

53 Chapter 3. Vehicle Modelling with Friction Dependence

−At|vr| Ft,i = Fz,i sign(vr) C1(1 − e )+ C2vr . (3.31) The proposed Friction Dependant (FD) model used to generate the tractive force depends on the static µs,i and kinetic µk,i coefficients of friction between the terrain and wheel i as well as At, which is the tangential stiffness of the tyre tread. The equation used to generate this force is shown in (3.32). The resulting longitudinal force is shown in Fig. 3.6, where the Magic Formula longitudinal force is compared to the developed FD model. The figure shows different combinations of longitudinal static and kinetic coefficients of friction, showing the relationship between the frictional coefficients and the longitudinal tyre force that is developed. The conversion between the relative velocity and the slip κ is achieved from (3.33), with ω = 1. At high relative velocities, the FD model asymptotically approaches the kinetic coefficient of friction,

−At|vr,i| −As|vr,i| Fx,i = Fz,i sign(vr,i) µsx,i(1 − e ) − (µsx,i − µkx,i)(1 − e ) . (3.32) with,

v κ = r , (3.33) ωre

The rolling resistance that the wheel experiences is also calculated using the model proposed by Ward [56] and is shown in Equation (3.34), where the force is a function of the structure of the tyre, its pressure and the longitudinal velocity of the wheel in the direction that the wheel is traveling in. The equation clumps the structure and pressure of the tyre as well as the drive system resistive forces into three variable R1, R2 and Ar. Where R1 is a positive constant representing the maximum rolling resistance coeﬃcient, R2 is the slope of the formula in the high velocity range, and Ar is a positive constant for the slope of the rolling resistance curve in the low velocity range,

−Ar|vw,i| Ri = Fz,i sign(vw,i) R1(1 − e )+ R2|vw,i| . (3.34) The motion of the tyre, not only along the longitudinal axis but also laterally impacts

54 3.2. Friction Dependant Tyre Force

4500

4000 µ = 1.46 µ = 0.78 s k 3500

3000 µ = 1.0 µ = 0.6 s k 2500 (N) x µ = 0.8 µ = 0.5 s k 2000 Force F

µ = 0.6 µ = 0.3 1500 s k

1000

500 MF-SWIFT Proposed 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 longitudinal slip κ

Fig. 3.6: Proposed friction dependant model for the tyre longitudinal force, showing the MF-SWIFT lateral force compared to the developed friction dependant model subject to four diﬀerent longitudinal static and kinetic coeﬃcients of friction

the forces the wheel experiences during interaction with the ground. The motion of the wheel in the direction that it is steered, is of utmost importance, Fig. 3.7 shows this relationship.

δi Vf x − y : wheel ﬁxed coordinate xw

αi

αi : slip angle

δi : steering angle yw

Vf : velocity of wheel center

Fig. 3.7: wheel coordinates showing slip angle and velocity of the wheel center together with the steering angle

The slip angle αi of the wheel depends on the diﬀerence between the steered angle

δi of the wheel and the inertial motion of the wheel coordinates, which is a combination of the vehicle body velocity as well as the angular velocity of the vehicle. The slip angle

55 Chapter 3. Vehicle Modelling with Friction Dependence

αi for each wheel is expressed in (3.35).

u α = δ − tan−1 w,i (3.35) i i v w,i The lateral force the ground exerts on the wheel is generated by the pneumatic tyre being deformed at the αi angle with the amplitude of the deformation being a function of the angle as well as the velocity of the wheel perpendicular to its heading uwi, Equation (3.36) shows the relationship between these variables.

2F µ F = z,i ky,i tan−1(C α ) (3.36) y,i π 1 i

The relationship between the proposed lateral force model and the experimentally derived MF model can be seen in Fig. 3.8. The figure shows the behaviour of the lateral force and its accuracy in comparison to the MF experimental results as well as the lateral force that is felt for terrains with different frictional coefficients. The data for the Magic Formula model was taken on a surface with a coefficient of kinetic friction being µk,y =0.78, which for the FD model of the same coefficient of friction matches the low slip angle values, and then diverges beyond that. The developed friction dependant lateral model is only accurate for slip angles less than 6degrees. This shortcoming is still acceptable under normal driving conditions as the slip angle usually doesn’t exceed this angle. The self-aligning torque that is developed in the tyre is dependant on the pneumatic trail tp,i which corresponds to a displacement of the normal force in the tyre contact patch. Coupled together with the lateral force Fy,i this pneumatic trail is responsible for the generation of the self-aligning torque Mz. The proposed formula for the pneumatic trail is shown in (3.37). The derivation of the self-aligning moment follows a similar form to that developed in [75, pp. 93-95], with an adaption made to take into account the frictional characteristics of the terrain.

3 1−3|ζ|+3ζ2−|ζ| L ( ) Ca tan αi 6 1 2 ; 3µ F > 1 (1−|ζ|+ 3 ζ ) ky,i z tp,i =  (3.37)  0 ; Ca tan αi ≤ 1  3µky,iFz

where L is the length of the tyre contact patch. The value of ζ is dependant on the

56 3.2. Friction Dependant Tyre Force

4500

4000

µ = 0.78 k 3500

µ = 0.6 3000 k

2500 (N)

y µ = 0.4 k

2000 Force F

1500 µ = 0.2 k

1000

500 MF-SWIFT Proposed 0 0 5 10 15 20 25 30 35 40 45 slip angle α (deg)

Fig. 3.8: Proposed friction dependant model for the tyre lateral force, showing the MF-SWIFT lateral force compared to the developed friction dependant model subject to four diﬀerent lateral kinetic coeﬃcients of friction

constant term Ca, the tangent of the slip angle αi, as well as the kinetic coeﬃcient of friction and the normal force acting on the tyre.

C tan α ζ = a i (3.38) 3µky,iFz

Together with the lateral tyre force Fy,i, this gives the proposed self-aligning torque, shown in (3.39). Similar to the lateral model, the self-aligning moment model is accurate only for small slip angles, which are experienced during normal road conditions.

Mz,i = Fy,i tp,i (3.39)

The behaviour of the self-aligning torque in comparison to the Magic Formula can be seen in Fig. 3.9, with the self-aligning torque for a number of diﬀerent kinetic coeﬃcients of friction shown. The self-aligning torque of the developed model provides a good qualitative comparison to the Magic Formula self-aligning torque, however, at larger slip angles the self-aligning torque of the developed model goes to zero and does not exhibit the change of sign that is seen with the experimental results.

57 Chapter 3. Vehicle Modelling with Friction Dependence

40 µ = 0.78 k

µ = 0.6 30 k

µ = 0.4 20 k µ = 0.2 k 10 (Nm) z 0

Torque M -10

-20

-30

-40 MF-SWIFT Proposed -50 -50 -40 -30 -20 -10 0 10 20 30 40 50 slip angle α (deg)

Fig. 3.9: Proposed friction dependant model for self-aligning moment, with the same corresponding colours and kinetic friction coeﬃcients as the lateral force

3.3 Simulation

To examine the effects of different terrain conditions on the behaviour of an AGV in order to be able to identify the sources of uncertainty in the system model a series of simulation configurations have been developed. These simulations include different factors that effect the behaviour of an AGV, these factors include; steering angle, steering changes, speed changes, changes in terrain type, surface inclines, as well as smoothly varying hills.

A suite of four different simulation environments have been developed that incorporate these factors so that the effect on the vehicle behaviour can be examined. The environments include a loop track that looks at the effects of the steering angle, steering rate as well as speed changes, with the same loop track moving over an inclined plane, as well as smoothly varying hilly surfaces. The effects of terrain type changes are also examined through the use of the loop track, with terrain changes during a cornering maneuver. The effects of the inclined plane and hilly surface is also examined with regards to terrain type changes. The zig-zag track maneuver examines the vehicle speed dependance on cornering behaviour, for different terrain geometry, including inclines and smoothly varying surfaces. The effect of the different terrain types on the cornering of the vehicle are also examined.

58 3.3. Simulation

The path that the vehicle takes through each simulation is determined by the dynamic model, with the corresponding longitudinal and lateral tyre forces as well as the self-aligning moment that the vehicle is subjected to during the maneuver. The purpose of these controls is to allow the comparison between different tyre models to be easily made, for the same forward velocity and steering controls, the different force models react differently causing a different behaviour for the same dynamic model. The two different tyre models that are examined through the simulation include the Magic Formula (MF) tyre model and the proposed Friction Dependant (FD) tyre model. The full vehicle dynamic simulation results for the FD tyre model are compared against those of the MF tyre model results and analysed.

3.3.1 Single Terrain Type

Flat Plane Geometry with a Single Terrain

The first set of environments that the vehicle travels through includes a smooth flat plane of only one terrain type. Two different steering maneuvers are used for the simulation, these include, travelling in a rounded rectangular loop track with the parallel sides running at different vehicle velocities, as well as a zig-zag maneuver. The first configuration starts off at the origin and follows a rounded rectangular path, with the first pair of parallel sides running at 2ms−1, then the vehicle steers so that it turns 90 degrees and continues on the second pair of parallel sides with a speed of 10ms−1, the vehicle then continues the loop until it passes back through the original path. The path that the vehicle takes is presented in Fig. 3.10, where the path is also compared to the path that the open loop vehicle dynamic response takes. The paths from both the MF and the FD tyre models are very similar, showing only a small amount of deviation between the responses. The force responses of the dynamic tyre models as the vehicle negotiates the kinematic path are presented in Fig. 3.11, with the subfigures comparing the different responses throughout the maneuver for the longitudinal and lateral forces as well as the self-aligning moment. The longitudinal and lateral forces that are generated during the maneuver for both models show that the FD model provides a good method of simplifying the MF for these simple motions. The self-aligning moment, however, is

59 Chapter 3. Vehicle Modelling with Friction Dependence

0 Y distance (m) -5

-10 Kinematic -15 Dynamic - MF Dynamic - FD -20

0 10 20 30 40 50 60 X distance (m)

Fig. 3.10: Comparison of kinematic and dynamic responses on the loop track on a ﬂat plane, with the vehicle under-steering through each corner, for both the Magic Formula and the developed Friction Dependant model, almost the same amount

different between the MF and the FD, as shown in Fig. 3.13(c), with the MF model providing a moment that reduces in magnitude during the peak of the turn, whereas the FD model maintains a positive value. The differences between the models show that the MF generates a higher moment at the beginning and end of each of the corners. The response of both of the tyre models is similar for this simple case, with the effect of cornering evident in the increase in both forces and moments for the outside wheels during the corner.

The open loop response of the zig-zag maneuver is then presented in Fig. 3.12, where dynamic responses of the different tyre models are compared to the kinematic response with no slip. The setup for the zig-zag maneuver includes the vehicle beginning with a forward velocity of 2ms−1, followed by a corner, after the vehicle has come out of the corner the vehicle forward speed is increased to 10ms−1 and the vehicle performs a turn by the same magnitude and length as the first corner except in the opposite direction. The effect of slip during the cornering is evidence in the difference between the path of the non-slip kinematic and the path of the dynamic responses.

The eﬀect of taking the corners at diﬀerent speeds can also be seen in Fig. 3.12, with the under-steering of the dynamic model increased for the higher speeds. The

60 3.3. Simulation

1000 4000

3000 500 2000 (N) (N) x y 0 1000

0 Force F -500 Force F Fx - MF Fx - FD -1000 Fy - MF Fy - FD -1000 -2000 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 time (s) time (s) (a) Longitudinal Force (b) Lateral Force

0.1 60

(Nm) Roll z 40 Pitch 0.05 20

0 0 Angle (deg) -20 Mz - MF Mz - FD -40 -0.05 Self-Aligning Moment M 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 time (s) time (s) (c) Self-aligning Moment (d) Body angles

Fig. 3.11: Comparison between the Magic Formula and the proposed friction dependant model on a ﬂat plane, with the black lines corresponding to the MF tyre model, and the red corresponding to the FD tyre model

Y distance (m) 10

0 Kinematic -10 Dynamic - MF Dynamic - FD -20 0 10 20 30 40 50 60 70 80 90 100 X distance (m) Fig. 3.12: Comparison of kinematic and dynamic responses for the zig-zag maneuver on a ﬂat plane, with both MF and FD tyre models providing similar results

forces and moments that are experienced during the zig-zag maneuver are shown in Fig. 3.13. The effect of the different self-aligning moments between the models can be seen in Fig. 3.13(c), where the effect of the MF model reaching the peak of the moment at smaller slip angles can be seen.

61 Chapter 3. Vehicle Modelling with Friction Dependence

1000 4000

500 2000 (N) (N) x y 0 0

Force F -500 Force F -2000 Fx - MF Fx - FD Fy - MF Fy - FD -1000 -4000 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 time (s) time (s) (a) Longitudinal Force (b) Lateral Force

0.2 60

(Nm) Roll z 40 0 Pitch 20 -0.2 0 -0.4

-20 Angle (deg) -0.6 -40 Mz - MF Mz - FD -60 -0.8 Self-Aligning Moment M 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 time (s) time (s) (c) Self-aligning Moment (d) Body Angles Fig. 3.13: Comparison of kinematic and dynamic responses for the zig-zag maneuver on a ﬂat plane, showing the similarity in the longitudinal and lateral forces between the tyre models

Inclined Plane Geometry with a Single Terrain

The second set of environments for the vehicle to travel through are a smooth flat plane on an incline with only one terrain type, shown in the background fo Fig. 3.14. The vehicle performs the same rectangular loop as well as the zig-zag maneuver. The response of the vehicle during the motion is analysed, with the focus being on how the vehicle moves when perpendicular to the incline, the angles that the body makes in the global coordinate system and hence the force distribution under the vehicle tyres. The kinematic model response is compared to the dynamic model responses, which can be seen in Fig. 3.14. The responses of the dynamic tyre models as the vehicle travels through the turns on the inclined plane are presented in Fig. 3.15, the sub-figures show the forces and moments that the vehicle is subject to. Additionally, the vehicle body roll and pitch angles are presented so that the effects of the turning and the incline plane are observed. The effects of the incline plane on the cornering response of the vehicle show that the under-steering of the vehicle is reduced when the body roll angle is reduced. The open loop response of the zig-zag maneuver, with the same control inputs as the first simulation configuration is presented in Fig. 3.16, where dynamic responses of the different terrain models are compared to the kinematic response with no slip. The plane that the vehicle travels on is on an incline, which is seen as an overlay on the vehicle trajectories. The cornering of the vehicle at different forward velocities on the

62 3.3. Simulation

Fig. 3.14: Comparison of kinematic and dynamic responses on the loop track on an incline plane, with the gradient representing the height of the terrain that the vehicle is moving on

1000 4000

3000 500 2000 (N) (N) x y 0 1000

0 Force F -500 Force F Fx - MF Fx - FD -1000 Fy - MF Fy - FD -1000 -2000 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 time (s) time (s) (a) Longitudinal Force (b) Lateral Force

10 60

(Nm) Roll z 40 5 Pitch

20 0 0 Angle (deg) -5 -20 Mz - MF Mz - FD -40 -10 Self-Aligning Moment M 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 time (s) time (s) (c) Self-aligning Moment (d) Body angles Fig. 3.15: Comparison between the forces and moments of the Magic Formula and the proposed terrain dependant model on an inclined plane loop track, showing the match between the MF and FD for changing tyre normal force as the vehicle undergoes body rotations

inclined plane is shown to drastically alter the trajectory of the vehicle when travelling over the inclined plane, or more practically when driving along a surface like a hill.

The forces and moments that the vehicle experiences during the zig-zag maneuver can be seen in Fig. 3.17. The forces that are felt by the vehicle include the eﬀects of the inclined plane and subsequently the weight vector of the vehicle. This results in a change in the body angles, and subsequently, a change in the normal forces acting on the tyre contact patch.

63 Chapter 3. Vehicle Modelling with Friction Dependence

Fig. 3.16: Comparison of kinematic and dynamic responses for the zig-zag maneuver on an incline plane, the vehicle is travelling up an inclined slope as shown by the gradient

1000 6000

4000 500 (N) (N)

x y 2000 0 0

Force F -500 Force F -2000 Fx - MF Fx - FD Fy - MF Fy - FD -1000 -4000 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 time (s) time (s) (a) Longitudinal Force (b) Lateral Force

2 60

(Nm) Roll z 40 0 Pitch 20 -2 0 -4

-20 Angle (deg) -6 -40 Mz - MF Mz - FD -60 -8 Self-Aligning Moment M 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 time (s) time (s) (c) Self-aligning Moment (d) Body Angles Fig. 3.17: Comparison of kinematic and dynamic responses for the zig-zag maneuver on an inclined plane, the tyre forces for both models maintain similarity during the maneuvers at diﬀerent steering rates and vehicle speeds

Smoothly Varying Geometry with a Single Terrain

The third set of environments for the vehicle to travel through is a smoothly varying surface similar to hills, with the track sets identical to the first set, a rectangular loop and a zig-zag track. The path that the dynamic model travels through on the rectangular loop track when subject to a surface with smoothly varying height is compared to the kinematic response, which can be seen in Fig. 3.18. The figure shows the height of the terrain as an overlay over the figure with the travelled paths, with the

64 3.3. Simulation height seen to vary smoothly over the course of the vehicle maneuver.

Fig. 3.18: Comparison of kinematic and dynamic responses on the loop track on a smoothly varying surface, which is shown in the terrain height as a gradient that is overlaid on the vehicle path

The motion of the vehicle through the smoothly varying terrain alters the forces and moments that the vehicle is subject to during the trajectory, these can be seen in Fig. 3.19. The longitudinal and lateral forces as well as the self-aligning moment vary as the vehicle moves along the surface as a function of both the geometry as well as the steering behaviour of the vehicle. The response of the vehicle is different from the inclined plane case through the corners as the tyre normal forces are not aligned along any particular direction and so the normal forces act about the vehicle differently. A similar behaviour can be seen in the vehicle when negotiating the zig-zag track whilst on the smoothly varying surface, which can be seen in Fig. 3.20. With the forces generated during the turning maneuver at the different cornering speeds giving a large difference in the forces that are experienced in the tyres. The path that the vehicle takes along the smoothly varying zig-zag track is affected by the forces and moments shown in Fig. 3.21. The effect of the hills on the forces are only apparent in the changes in normal forces that the vehicle experiences as it goes over the surface. These first three simulation sets demonstrate the behaviour of the vehicle for changes in speed, cornering as well as changes in terrain geometry, whilst maintaining a constant value for the frictional characteristics of the terrain. The next three simulation sets

65 Chapter 3. Vehicle Modelling with Friction Dependence

1000 4000

3000 500 2000 (N) (N) x y 0 1000

0 Force F -500 Force F Fx - MF Fx - FD -1000 Fy - MF Fy - FD -1000 -2000 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 time (s) time (s) (a) Longitudinal Force (b) Lateral Force

10 60

(Nm) Roll z 40 5 Pitch

20 0 0 Angle (deg) -5 -20 Mz - MF Mz - FD -40 -10 Self-Aligning Moment M 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 time (s) time (s) (c) Self-aligning Moment (d) Body angles Fig. 3.19: Comparison between the forces and moments of the Magic Formula and the proposed terrain dependant model on a smoothly varying loop track, with close match between the two tyre models present for both forces

Fig. 3.20: Comparison of kinematic and dynamic responses for the zig-zag maneuver on a smoothly varying surface, the height of the terrain is overlaid on the vehicle path

analyse the same behaviours as before, whilst introducing changes in the frictional characteristics of the terrain. In all sets the response of the developed friction dependant tyre model closely followed the behaviour of the dynamic response of the Magic Formula.

66 3.3. Simulation

1000 6000

4000 500 (N) (N)

x y 2000 0 0

Force F -500 Force F -2000 Fx - MF Fx - FD Fy - MF Fy - FD -1000 -4000 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 time (s) time (s) (a) Longitudinal Force (b) Lateral Force

10 50

(Nm) Roll z 5 Pitch 0 0

-50 Angle (deg) -5 Mz - MF Mz - FD -100 -10 Self-Aligning Moment M 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 time (s) time (s) (c) Self-aligning Moment (d) Body Angles Fig. 3.21: Comparison of kinematic and dynamic responses for the zig-zag maneuver on a smoothly varying surface, showing the FD tyre model accurately following the lateral force response of the MF tyre model

3.3.2 Diﬀerent Terrain Types

Flat Plane Geometry with Multiple Terrain Types

Following on from the previous three simulation environments, which were evaluated on a single terrain type, the next three simulation environments will include a combination of different terrain types at varying places along the vehicle path. The fourth set of environments that the vehicle will travel along is a smooth flat plane with a number of different terrain types. The set of tracks in these environments are identical to that of the first set, and include the loop and the zig-zag tracks. The different terrain types are road, dirt and sand, the differences in these terrains are felt in the model as changes in the static and kinetic coefficients of friction as the vehicle negotiates the terrain. The vehicle trajectory and terrain type locations can be seen in Fig. 3.22, with the three different terrain types located along different parts of the trajectory. The vehicle begins the path on road at the origin, which then changes through the first corner into a dirt terrain as indicated by the dark brown colour. The vehicle then continues and moves through the second corner, which coincides with another terrain change between dirt and sand, with the sand indicated by the light brown colour. The next corner that the vehicle goes through occurs during a terrain change from sand to road, with the vehicle transitioning between dirt and road terrain on the final corner. The terrain changes have been designed to take place during each of the corners for the vehicle so

67 Chapter 3. Vehicle Modelling with Friction Dependence that the maximum eﬀect of the friction change can be observed.

Fig. 3.22: Comparison of kinematic and dynamic responses on the loop track on a ﬂat plane over diﬀerent terrain types, with black representing road, dark brown representing dirt and light brown representing sand

During cornering on the different terrain surfaces it can be seen that the amount of under-steer that the vehicle experiences whilst cornering coincides in the reduction of the coefficients of friction for each of the different terrain types. The third and fourth corners that the vehicle moves through however coincide with increases in the coefficients of friction during that corner, with the effect being the reduction of the amount of under-steering through the corner. The forces and moments that the vehicle experiences on the different terrain can be seen in Fig. 3.23, with the different amounts of under-steering being as a result of the changes in lateral forces that the vehicle experiences in accordance with the change in terrain type. The terrain type transition that the vehicle experiences as it moves through all of the corners has a large impact on both the lateral force and the self-aligning moment for the vehicle. With a sudden loss in lateral force through the corner, for corners one and two, and then increase in lateral forces for corners three and four, the motion of the vehicle becomes more complicated as each wheel transitions into the terrain separately. Similar changes in longitudinal and lateral forces were observed in an experiment done by Erdogan [147] as well as [148, 149], where a truck was moving from a road surface to a surface with low friction. The lateral forces that were experienced in the experiment reduced in amplitude as the tyre crossed the transition between the two

68 3.3. Simulation terrain types. The vehicle was moving straight ahead in the experiment, and so a larger deviation would be expected for a friction change when the slip angle is at a larger value, which corresponds to a larger lateral force.

1000 4000

3000 500 2000 (N) (N) x y 0 1000

0 Force F -500 Force F Fx - MF Fx - FD -1000 Fy - MF Fy - FD -1000 -2000 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 time (s) time (s) (a) Longitudinal Force (b) Lateral Force

0.15 60 (Nm) z 40 0.1

20 0.05 0 Angle (deg) 0 Roll -20 Mz - MF Mz - FD Pitch -40 -0.05 Self-Aligning Moment M 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 time (s) time (s) (c) Self-aligning Moment (d) Body angles Fig. 3.23: Comparison between the forces and moments of the Magic Formula and the proposed terrain dependant model on a flat loop track over different terrain, showing the effect of the terrain changes in the lateral forces, with the first two corners showing a larger difference and the last two showing only a small difference

The motion of the vehicle as it performs the zig-zag maneuver can be seen in Fig. 3.24, where the trajectories that the vehicle undergoes when subject to different terrain conditions are shown. In all cases the vehicle experiences greater amount of under-steer on the faster corner, with a gradual increasing in under-steering, which is larger for surfaces with lower frictional characteristics, as the vehicle transitions through the corner. The effects of the different coefficients of friction for the trajectory of the vehicle can be seen in the forces and moments that the vehicle experiences as it goes through a corner, these forces and moments are shown in Fig. 3.25. The figure shows that for the lower friction surfaces that the vehicle has a lower magnitude but more sustained lateral force, which has the effect of initially under-steering but due to the length of time that the force acts, causes under-steering in the vehicle in comparison to the trajectory on surfaces with higher frictional characteristics. From the simulation results it can be seen that the terrain frictional characteristics has a large impact on the motion of the vehicle, with cornering maneuvers resulting in the largest deviations from the kinematic trajectory. The speed that a vehicle takes

69 Chapter 3. Vehicle Modelling with Friction Dependence

Y distance (m) 10

0 Kinematic Road -10 Dirt Sand -20 0 10 20 30 40 50 60 70 80 90 100 X distance (m) Fig. 3.24: Comparison of kinematic and dynamic responses for the zig-zag maneuver on the flat track over different terrain, the FD tyre model is used on three different terrain types: road, dirt and sand.

1000 4000 Fx - road Fy - road 500 Fx - dirt 2000 Fy - dirt

(N) Fx - sand (N) Fy - sand x y 0 0

Force F -500 Force F -2000

-1000 -4000 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 time (s) time (s) (a) Longitudinal Force (b) Lateral Force

0.2 40 (Nm) z Mz - road 0 20 Mz - dirt Mz - sand -0.2 0 -0.4

Angle (deg) Roll -20 -0.6 Pitch

-40 -0.8 Self-Aligning Moment M 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 time (s) time (s) (c) Self-aligning Moment (d) Body Angles Fig. 3.25: Comparison of dynamic responses for the zig-zag maneuver on a ﬂat plane for diﬀerent terrain types, with the forces and moments of the FD tyre model shown for the three terrain types: road, dirt and sand

through a corner is also related to the amount of under or over-steering that a vehicle experiences, but the amount that this eﬀect is seen depends on the terrain frictional characteristics.

70 3.4. Discussion

3.4 Discussion

The simulation results show that a significant uncertainty that the vehicle can be subjected to is a change in terrain type and thus a change in frictional coefficients during a cornering maneuver. This results in a large change in the available lateral force that can be applied by the tyre. This change, coupled with the change in the normal force for each tyre through a turn results in under-steering through the corner. If the slip angle of the tyre is not reduced, then this results in a gradual under-steering through the corner, as seen in the zig-zag maneuver sets. The developed friction dependant tyre model was able to show changes in the longitudinal and lateral forces, as well as the self-aligning moment as the terrain type changed. The friction model closely followed the lateral friction of the Magic Formula during the cornering maneuvers whilst on the road surface, however, the friction model did not provide a good fit for the self-aligning moment. The effect of this difference is if a controller was utilised could result in a poorly controlled steering angle, as well as possible under or over steering of the vehicle through a corner. The developed friction dependant self-aligning moment is also unable to replicate the effect of the change in sign of the self-aligning moment for large slip angles. The steering rate of the tyre has also been shown to effect the self-aligning torque especially at low speeds [65], and so the inclusion of this effect into the model is desirable, which is addressed in the next chapter. These results were gathered with the help of the dynamic model, however the model only utilises static tyre models. The result of this is that if the vehicle model comes to a stop, then the slip angle felt in the tyre becomes unstable, as the slip angle is a function of both the longitudinal velocity and the lateral velocity of the tyre. At low speeds the magnitude of the longitudinal and lateral velocities become similar resulting in an unstable slip angle calculation, this results in instability of the lateral forces and self-aligning moments for the tyres for this model. This effect was avoided within these simulation configurations by preventing the longitudinal velocity from becoming too low, but for a realistic situation the effect would be unavoidable with the current system model. To address this limitation it is often assumed that at low speeds a vehicle moves kinematically, and so a transition between the two models is made at a predetermined

71 Chapter 3. Vehicle Modelling with Friction Dependence determined ’low speed’. Doing so avoids the instability but also introduces an ad-hoc model, with an arbitrary switch over low speed set point. Another way to address this limitation is to utilise a dynamic tyre model, so that an arbitrary transition is not required as the dynamic model is able to adjust to the current state of the system over a period of time. Preventing the instability that is observed in the slip angle at low forward speeds and only requiring one single system model so that a single controller can be designed. The dynamic model also allows for dynamic effects such as hysteresis in the tyre to be observed, which also affects the amount of force that the tyre is capable of producing. Due to the impact that cornering maneuvers have on the dynamics of the system it is also necessary to be able to successfully model these affects. The simulation was configured so that the effects of steering control were removed. This was done by directly controlling the steering angles that each tyre is at, through a geometric configuration so that each tyre was steered about the same instantaneous center of rotation. The effects of the changing self-aligning torque have not been included in the calculation of the steering angle and so it would be expected that the steering angle and steering rate would be different.

3.5 Conclusion

A 3D dynamic vehicle model has been developed and evaluated with respect to a kinematic trajectory. The behaviour of the vehicle as developed through the use of the Magic Formula tyre model and the developed Friction Dependant tyre model, when compared against each other for the same terrain conditions shows that the developed tyre model is able to replicate the results of the experimentally derived Magic Formula model. The simulation results of the vehicle traversing from one terrain to another showed that the lateral force and self-aligning moment changes significantly, resulting in the vehicle taking a different trajectory. This shows that a source of uncertainty in the system model is where there is a transition between different terrain types, especially during a cornering maneuver. Although, the friction dependant model wasn’t able

72 3.5. Conclusion to replicate the Magic Formula results exactly, it was still able to provide a good representation of what forces are experienced through the same terrain type. The dominance of the cornering maneuver over the development of forces in the simulation shows the importance of including the eﬀects that contribute to changes in both the lateral force and the self-aligning torque in the tyres. Both the steering angle and the steering rate are acting on the tyres during the steering maneuver and so examining the eﬀects of both of these system inputs is important in reducing the uncertainty in the system model, which is addressed in the next chapter.

73 Chapter 3. Vehicle Modelling with Friction Dependence

74 Chapter 4

A 3D Analytical Dynamic Tyre Model

In order to reduce the amount of uncertainties that a vehicle is subject to during a maneuver, an accurate model of the forces and moments that the vehicle experiences as it traverses through unstructured and unknown terrain. The interaction between the vehicle tyres and the terrain is the dominant mechanism through which the vehicle interacts with the terrain, it is therefore important to have an accurate model of this interface. The interface can consist of many diﬀerent types of terrain and it is also necessary for the model to be able to adjust for these changes in the environment. The proposed tyre model needs to be able to provide an accurate force estimation as well as the expected uncertainties of the forces and moments that are generated as the vehicle traverses through the changing frictional conditions.

The existing tyre models typically fall into three categories; empirical, analytical and a hybrid of both. The majority of tyre modelling can be achieved using the empirical tyre models, as the operating conditions for these models can be assumed to be mostly constant, ie, sealed roadways and racetrack conditions. Anti-lock braking systems (ABS) and traction control systems operate using the empirical models to control the maximum braking or tractive force based on desired slip rate, or in many cases are implemented using accelerometers to measure the slip/force based on constant normal force and forward speed conditions.

The empirical models have been in existence for a number of decades, but more

75 Chapter 4. A 3D Analytical Dynamic Tyre Model recently there has been a push for an adequate analytical model to be developed in order to model the system in a more generalised setting. A number of analytical tyre models have been developed and expanded over the past decade, with each new model becoming more and more comprehensive. The models developed so far have been focused on normal force dependency as well as extending the model into the lateral direction, with some tyre steering rotation being considered under certain conditions.

So far in the literature the rolling resistance moment that is generated by the wheel has been neglected as an important dynamic effect, with the majority of focus being on the longitudinal and lateral forces as well as the self-aligning moment. The rolling resistance generally is thought to be a reactionary moment with a constant value that depends on the type of tyre tread. Literature exists that shows that there is a velocity dependency on the rolling resistance of the tyre with the moment increasing at higher velocities [150]. This effect has been shown to have a significant impact on the maximum speed of a vehicle, apart from chassis aerodynamic drag effects [151–153]. At high speeds the rolling resistance is therefore an important moment that must be taken into account if a vehicle must perform persistent autonomous actions.

The previous analytical tyre models that exist are based on the LuGre friction model, which has been used to model the realistic motion that a pneumatic tyre goes through and the accompanied forces that are developed. The proposed model attempts to extend these analytical models to be able to predict the rolling resistance and the overturning moment felt on the tyre as well as be able to describe the self aligning moment in the general case, taking into account the wheel steer.

The overall purpose of the proposed model is to give a generalised dynamic model that can be used for both driven and non-driven wheels under all tyre motion and terrain conditions and which can be used to predict all six involved forces and moments. No distinction needs to be made between the separate driving conditions, allowing the model to be used to provide a single model for use in control tasks.

This chapter presents the developed analytical dynamic tyre model, the derivation of the model is detailed in Section 4.1, the forces and moments that the tyre is subject to are derived and compared against the forces and moments that the empirical Magic Formula model generates through a series of simulations. This process is then replicated

76 4.1. Friction Dependant Analytical Tyre Model for a proposed force distribution in Section 4.2 which takes into account the wheel speed and camber angle to improve the accuracy of the force prediction. Simulation results for the various tyre-terrain interaction phenomenon is presented in Section 4.1.5 and Section 4.2.3, including the relationship between the terrain type and the rolling resistance as well as the longitudinal force developed.

4.1 Friction Dependant Analytical Tyre Model

The analytical dynamic tyre model that is developed in this section incorporates the rolling resistance as well as the camber angle effects of the wheel on the tyre terrain dynamics, a more comprehensive and accurate vehicle model can then be achieved through the inclusion of these effects. The proposed model formally consists of introducing the full area of the contact patch in order to adequately model the effects of wheel steer, wheel speed and camber angle on the longitudinal and lateral forces as well as the self aligning moment developed in the contact patch.

4.1.1 Distributed Tyre Model

In this section, a basic LuGre friction modelling approach is taken in order to derive a six degrees-of-freedom (6dof) analytical tyre model for the tyre-terrain contact forces and moments, the approach taken is similar to that used in [63, 65]. This method allows for the longitudinal and lateral forces developed in the contact patch to be calculated, as well as the self-aligning moment, the overturning moment and the rolling resistance. The vertical force is an input to this model and is a function of the characteristics of the tyre as well as the height above the road surface. The tyre contact patch for this model is assumed to be roughly rectangular in shape and is divided into inﬁnitesimal contact patch elements. The 2D point contact LuGre model, shown in (4.1), is applied for each element of the contact patch which is then integrated over the entire contact patch to calculate the frictional forces and moments generated. Where the terms ǫi andǫ ˙i are the internal friction state position and velocity for i = x, y with vri being the relative velocity of the ith dimension.

77 Chapter 4. A 3D Analytical Dynamic Tyre Model

ǫ˙i = vri − C0i(vr)ǫi (4.1a)

µi = −σ0iǫi − σ1iǫ˙i − σ2ivr (4.1b) where

2 kµkvrkσ0i C0i(vr)= 2 (4.2) µkig(vr)

The terms, µk and µki are the matrix and scalar terms for the kinetic coeﬃcient of friction as shown in (2.7).

The terms σ0i, σ1i and σ2i are the spring coefficient, damping coefficient and viscous relative damping coefficient for the tyre element and µi is the resulting friction coefficient that the normal force acts on, the remaining terms in the equation are described in Section 2.2.2. The local frame has a fixed axis about the leading edge of the contact patch, and is orientated in the direction tyre rotation. The contact patch coordinate system is in ζ and η, which represent the longitudinal and lateral directions and can be seen in Fig. 4.1. The frictional forces and moments that are developed in the contact patch of the tyre are due to the deformation of the tyre elements from the relative velocities between the tyre and the ground. Let v be the vehicle velocity and let ω be the angular velocity of the wheel of radius r. Let α be the slip angle of the wheel and δ˙ is the angular rate of the local frame, signifying the steering rate of the wheel.

vrx(η)= ωre − v cos α − ηδ˙ (4.3a) L v (ζ)= −v sin α − ( − ζ)δ˙ (4.3b) ry 2

The friction acting at each of the contact patch elements can be calculated using the point contact LuGre model from before. The internal friction states ǫi for i = x, y are functions of time t, the longitudinal position of the element ζ as well as the lateral

78 4.1. Friction Dependant Analytical Tyre Model

W ǫ(t + dt,ζ + dζ,η + dη) L y y α v α ǫ(t, ζ, η) δ˙ dη η + dη η ζ ζ + dζ x x dη dζ ωr ωr dζ

time: t time: t + dt

Fig. 4.1: Contact Patch Area Diﬀerential

position of the element η on the contact patch.

During the time interval from t to t+dt the tyre element has moved to (ζ +dζ,η+dη)

and using (4.1) this gives the total tyre element deﬂection dǫi shown in (4.4).

dǫi = ǫ(t + dt,ζ + dζ,η + dη) − ǫ(t, ζ, η)= vri − C0i(vr)ǫi(t, ζ, η)dt (4.4)

∂ǫi ∂ǫi ∂ǫi and since dǫi = ∂ζ dζ + ∂η dη + ∂t dt this gives

dǫ ∂ǫ ∂ǫ ∂ǫ i = i ζ˙ + i η˙ + i = v − C (v )ǫ (t, ζ, η) (4.5) dt ∂ζ ∂η ∂t ri 0i r i

The term ζ˙ is the speed at which the frame of reference of ζ are being introduced due to the rolling of the wheel, giving ζ˙ = |ωr|.η ˙ represents the speed of the lateral axis and since the wheel only rolls about its longitudinal axis this term is η˙ = 0. The time derivatives of the contact patch coordinate system ζ˙ andη ˙ are then substituted back into (4.5) to give

∂ǫ (t, ζ, η) ∂ǫ i = v − C (v )ǫ (t, ζ, η) − i |ωr| (4.6a) ∂t ri 0i r i ∂ζ

µi(t)= −σ0iǫi(t) − σ1iǫ˙i(t) − σ2ivri (4.6b)

79 Chapter 4. A 3D Analytical Dynamic Tyre Model

where i = x, y. The total forces for the longitudinal and lateral directions that act on the tyre contact patch are calculated as function of the force distribution across the contact patch as well as the LuGre coeﬃcient of friction µi(t, ζ, η) across the contact patch, this behaviour is modelled using (4.7),

W L 2 Fi(t)= µi(t, ζ, η)fz(t, ζ, η)dηdζ (4.7) W Z0 Z− 2

where L is the length of the contact patch, W is the width of the contact patch and fz(t, ζ, η) is the force distribution across the contact patch as force per unit area. The force distribution across the width of the contact patch results in a moment about the centre of the patch which is the overturning moment of the tyre, this is given by (4.8),

W L 2 Mx(t)= − ηfz(t, ζ, η)dηdζ (4.8) W Z0 Z− 2

similarly the force distribution along the length of the contact patch resulting in the generation of the rolling resistance moment about the centre of the contact patch, this is given by (4.9),

W L 2 L My(t)= − − ζ fz(t, ζ, η)dηdζ (4.9) W 2 Z0 Z− 2

The combination of the frictional coefficients µx and µy across the entire contact patch together with the force distribution about the center of the patch results in the self-aligning moment of the tyre. The self-aligning moment is taken about the center of the contact patch. Only the tangential components of the frictional coefficients contribute to the self-aligning moment, so it is therefore necessary to extract the tangential components as well as the distance between the element and the center of rotation. The distance between the center of rotation and the element is shown in (4.10), and the tangential component of the frictional coefficients is shown in (4.11).

2 L 2 dt = η +(ζ − ) (4.10) r 2 80 4.1. Friction Dependant Analytical Tyre Model

µt = µx sin α + µy cos α L η (ζ − 2 ) = µx + µy (4.11) 2 L 2 2 L 2 η +(ζ − 2 ) η +(ζ − 2 ) q q

These terms are then incorporated together to be able to evaluate the self-aligning moment, which is given by (4.12).

W L 2 Mz(t)= − dtµt(t, ζ, η)fz(t, ζ, η)dηdζ W Z0 Z− 2 W L 2 = − (−σ0xǫx − σ1xǫ˙x − σ2xvrx)η W Z0 Z− 2 L +(−σ ǫ − σ ǫ˙ − σ v )( − ζ) f (t, ζ, η)dηdζ (4.12) 0y y 1y y 2y ry 2 z

for completion the force Fz acting vertically on the tyre patch can be found using

W L 2 Fz(t)= fz(t, ζ, η)dηdζ (4.13) W Z0 Z− 2

To evaluate the distributed model the steady state responses of the model will be compared against the responses of Pacejka’s Magic Formula (MF) model [73] for the forces Fx and Fy as well as the self-aligning moment Mz while there is no steering angle change. The MF model is based on experimental data and accurately captures tyre forces under steady-state conditions. The overturning moment Mx and the rolling resistance My will be compared against the MF based MF-SWIFT 6.1 model from [54] using the tyre parameters from [55], which are in Table B.1.

The self-aligning moment Mz is then also validated against the 2D LuGre model from [65] with the inclusion of the steering speeds. The steady state behaviour of the developed distributed model will be derived and then compared against the MF model for validation.

81 Chapter 4. A 3D Analytical Dynamic Tyre Model

4.1.2 Steady State Conditions

In order to validate (4.6) its steady-state characteristics are required. These are

∂ǫi(t,ζ,η) ˙ obtained by setting ∂t = 0, and by keeping v, α, ω and δ thus vri constant. For this case equation (4.6) becomes,

∂ǫ 1 i = (v − C (v )ǫ (t, ζ, η)) , i = x,y, ω =6 0 (4.14) ∂ζ |ωr| ri 0i r i

For ω = 0, equation (4.6) reduces to the point contact LuGre model (4.1) for ω = 0. Which is just the case of pure translational movement and thus the point contact LuGre model is used for dry contact friction modelling. Enforcing the boundary condition of zero deﬂection at the start of the tyre contact patch ǫi(t, 0, η) = 0 along with vrx and vry being kept constant for the steady-state conditions, and by integrating (4.14) we get

− ζ ss C2 ǫi (ζ, η)= C1i 1 − e i (4.15) where, 2 0 kµkvrkσ i C0i = 2 g(vr )µki

vri C1i = (4.16) C0i C = |ωr| 2i C0i

The expression for the steady-state tyre element deformation is shown in Figures 4.2, 4.3 and 4.4, The values of L and W are equal to 0.08 and 0.12 respectively in the following examples. The results in Fig. 4.2 shows the deformation of the tyre elements when subjected to slip as well as slip angle so that the relative velocities in both x and y are non zero. The deformation in both x and y are generated as a build up of relative motion along the contact patch, with the leading edge of the contact patch where ζ =0 showing zero deformation, the tyre rotational velocity ω = 0.1 with the tyre radius r =0.5. The slip conditions were κ =0.1 and α =2◦.

The tyre element deformation for x and y, shown in Fig. 4.3, when the tyre is subjected to a steering rate change about the center of the tyre contact patch. In this example ω = v = 0 with δ˙ =0.2rads−1 which corresponds to a stationary tyre turning

82 4.1. Friction Dependant Analytical Tyre Model

(a) ǫx (b) ǫy Fig. 4.2: Tyre element deformation caused by wheel slip

(a) ǫx (b) ǫy

Fig. 4.3: Tyre element deformation caused by steering rate δ˙

about its center. With the greatest deformation occurring closer to the center of the contact patch.

The tyre element deformation when subject to a combination of both non-zero wheel speed ω =6 0 and non zero steering rate δ˙ =6 0 is shown in Fig. 4.4. The conditions that are used are a combination of the conditions from both the normal slip conditions as well as the steering rate conditions from above.

The steady-state expressions for the forces and the torques developed in the contact patch using (4.7), (4.8), (4.9), (4.12) and (4.13) give the following equations. With the steady state forces developed in the longitudinal and lateral directions in the contact patch to be

W L 2 ss ss Fi = − (σ0iǫi + σ2ivri)fz(t, ζ, η)dηdζ (4.17) W Z0 Z− 2 83 Chapter 4. A 3D Analytical Dynamic Tyre Model

(a) ǫx (b) ǫx Fig. 4.4: Tyre element deformation caused by a combination of both wheel slip and steering rate δ˙

and the steady-state expression for the self-aligning torque shown below.

W L 2 L ss η (ζ − 2 ) Mz = dt µx + µy dηdζ (4.18) W L L 0 − 2  2 2 2 2  Z Z η +(ζ − 2 ) η +(ζ − 2 )  q q 

The steady-state expressions for forces and moments Fz, Mx and My depending on the steady state condition of only the force distribution function that has been developed to take these behaviours into account, it is assumed that the force distribution that is used is a steady-state function and does not exhibit dynamic responses. Before the analysis of the steady-state forces and moments can proceed the force distribution fz(t, ζ, η) in the contact patch must be developed, a simple constant force distribution is utilised in the following sections so that the eﬀects of the steering rate can be observed without complicating the results with a more realistic force distribution. This means that the forces and moments developed are not as realistic as they can be, which will be discussed, the constant force distribution is shown in (4.19).

F −W W f (ζ, η)= z , 0 ≤ ζ ≤ L, ≤ η ≤ (4.19) z LW 2 2 Using the deﬁnition of the longitudinal slip κ, for braking and accelerating from [55]

v 1 − ωr , ωr>v κ =  (4.20) ωr  1 − v ωr ≤ v

 84 4.1. Friction Dependant Analytical Tyre Model

4000 Fss x 3500 Fss y ° 3000 α = 2

2500

2000 Force (N) 1500

1000

500

0 0 0.2 0.4 0.6 0.8 1 longitudinal slip κ

ss ss Fig. 4.5: Steady-state forces Fx and Fy with Fz = 4000N

ss ss ss with the constant force distribution with steady-state behaviours Fx , Fy and Mz . The forces and moments that are shown in Fig. 4.5 and Fig. 4.6 are in qualitative agreement with similar curves found in the literature [54, 63, 64, 154].

4.1.3 Parameter ﬁtting

For the model that has been developed (4.6) to be used there are several parameters that need to be identified. The unknown parameters are identified by comparing the steady-state behaviours of the model to the steady-state data found in the literature. The identification of the unknown parameters for the model was done by fitting the generated steady-state plots to the steady-state data of the Magic Formula - SWIFT 6.1 model. The MF-SWIFT model parameters are identified through the use of curve fitting to fit closely the experimental data and have a base form of the equations that is similar to that of the Magic Formula model. The parameters that are used for parameter identification can be found in Table B.1. The MF-SWIFT model is also used for the validation of the effect of the steering rate on the self-aligning moment at both rest and at varying forward speeds. The

85 Chapter 4. A 3D Analytical Dynamic Tyre Model

20 κ = 0.03

0 Moment (Nm) −10

−20

−30 −60 −40 −20 0 20 40 60 slip angle α (deg)

ss Fig. 4.6: Steady-state moment Mz with Fz = 4000N

4500

α ° 4000 = 2 α ° = 5 ss 3500 ° F α = 8 x ° 3000 α = 12

2500

2000 Force (N) α ° 1500 = 12 ° α = 8 ss ° F α y 1000 = 5 ° α = 2 500

0 0 0.2 0.4 0.6 0.8 1 longitudinal slip κ

ss ss Fig. 4.7: Steady-state forces Fx and Fy with Fz = 4000N at varying slip angles

86 4.1. Friction Dependant Analytical Tyre Model

MF-SWIFT 6.1 model takes into account the effect of turn-slip on the forces and moments acting on the tyre which is validated against experimental data in [75] for the case of a parking maneuver involving steering angle rate at rest as well as at speed. The values of the static and kinetic coefficients of friction for use in the proposed model are calculated from the slip force curve generated from the MF-SWIFT model. From the force developed in the longitudinal and lateral directions it is obvious that the coefficients of friction are anisotropic with the values of the static and kinetic coefficients of friction differing, as the peak forces that are developed do not provide similar longitudinal and lateral forces. The coefficients of friction that are used in fitting the analytical friction model onto the MF steady-state forces are obtained through the use of the method that is described in Section B.2. The method is used to remove the problem of trying to fit the analytical tyre model without fixing the coefficients of friction first, as the frictional coefficients should coincide with the same coefficients of friction that are present in the MF results. The identification of the parameters for the 3D dynamic tyre was done by comparing

ss ss ss the plots of the steady-state expressions for Fx , Fy and Mz for the constant force distribution. The identiﬁed parameters are shown in Table 4.1.

σ0x(1/m) σ2x(s/m) µkx µsx 755 -0.0005 0.74 1.46

σ0y(1/m) σ2y(s/m) µky µsy 485 0.0011 0.78 1.18

L(m) W (m) vs(m/s) β 0.0874 0.1208 4.91 1 Table 4.1: Identiﬁed Parameters

ss ss The constant force distribution inevitably results in zero values for Mx and My because of the shape of the force distribution, as well as the self-aligning moment not reversing sign at higher slip angles which is expected from the MF-SWIFT model, as

ss ss shown in Fig. 4.10. The forces Fx and Fy match very well against the MF-SWIFT model as shown in Fig. 4.8 and Fig. 4.9. The impact of the steering rate δ˙ on the forces and moments is insigniﬁcant as the graphs are creating using a fast forward velocity.

87 Chapter 4. A 3D Analytical Dynamic Tyre Model

4500

4000

3500

3000

2500 (N) x

2000 Force F

1500

1000

500 MF−SWIFT Proposed 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 longitudinal slip κ

Fig. 4.8: Slip vs Fx comparison between MF-SWIFT and the proposed model

The forces and moments that are felt due to the steering rate at high speeds are, therefore, not signiﬁcant, and only come into prominence during low speed maneuvers. As well as these plots, the additional dependance of the steering rate on the self-aligning moment is also modelled. The MF-SWIFT is also compared to the predicted forces of the 3D dynamic tyre model in steady-state. This steering rate dependant torque was explored in [65] with the results for ω = v = 0 case being determined and discussed, however the eﬀect was not integrated into the model leaving the steering rate torque only valid for ω = 0. With the dynamic model comparison with [65] requiring that ω = 0 and v = 0, with these conditions the equation (4.15) becomes

ss ǫi (ζ, η) = sgn vriC1i, which is consistent with the model in [65]. The proposed model places no limitation on the value of ω or v and when compared to the MF-SWIFT model at varying forward speeds, it can be seen that the new model can be used to approximately model this phenomenon. The result that is shown in Fig. 4.11 is compared to that of the Pacejka Magic Formula response. The Magic Formula model that has been used is the one that incorporates the turn spin into the calculations for the self-aligning torque. Through

88 4.1. Friction Dependant Analytical Tyre Model

4000

3000

2000

1000 (N) y 0 Force F −1000

−2000

−3000 MF−SWIFT Proposed −4000 −50 −40 −30 −20 −10 0 10 20 30 40 50 slip angle α (deg)

Fig. 4.9: α vs Fy comparison between MF-SWIFT and the proposed model

10 (Nm) z 0

−10 Torque M

−20

−30

−40 MF−SWIFT Proposed −50 −50 −40 −30 −20 −10 0 10 20 30 40 50 slip angle α (deg)

Fig. 4.10: α vs Mz comparison between MF-SWIFT and the proposed model

89 Chapter 4. A 3D Analytical Dynamic Tyre Model

1000

800

600

400

200

Moment (Nm) −200

−400

−600

−800 MF−SWIFT Proposed −1000 −100 −80 −60 −40 −20 0 20 40 60 80 100 steering rate dδ/dt (rad/s)

Fig. 4.11: δ˙ vs Mz for ω = 0

the use of this method the steering rate can be introduced as a wheel tyre spin with radius R = inf. The result can also be compared to the result that is presented in [65] for the steering rate dependant torsional moment generated by steering the tyre about the center of the contact patch. It can be seen that at higher steering rates the moment reduces, this effect is also present in the model presented in [65]. The parameters that are used in the model to generate Fig. 4.11 are the same parameters that were identified in Figures 4.8,4.9 and 4.10. This allows for the proposed model to predict the forces and moments due to additional effects whilst not requiring the use of any additional parameters. The difference between the proposed model and the MF model for Fig. 4.10, is due to the unrealistic force distribution that was used to generate the self-aligning moment. This force distribution was a constant force distribution, this problem is addressed in Section 4.2. The self-aligning torque that is developed due to the steering rate, at different forward speeds, with slip κ = 0 is shown in Fig. 4.12. The self-aligning torque that the proposed model generates at the different forward velocities qualitatively matches the

90 4.1. Friction Dependant Analytical Tyre Model

1000 v = 0 ms−1 800 v = 0.01 ms−1 −1 v = 0.1 ms −1 600 v = 1 ms

400

200 v = 10 ms−1

−200 Moment (Nm)

−400

−600

−800 MF−SWIFT Proposed −1000 −100 −50 0 50 100 steering rate dδ/dt (rad/s)

Fig. 4.12: δ˙ vs Mz for diﬀerent ω

MF-SWIFT model at the diﬀerent steering rates. The general case where the steering rates are low can be seen in Fig. 4.13, and as the forward velocity increases the self-aligning torque that is generated from the steering rate decreases. The inconsistency in self-aligning torque for both the side slip and the steering rate moments that are generated as compared to the torque predicted from the Magic Formula can best be explained by the force distribution that is used to generate the plots. The force distribution that was used in generating the moments was that of a constant force distribution with the force being equal throughout the entire contact patch, which is a simple force distribution but is not a very realistic model.

4.1.4 Average lumped model

To be a useful model for analysis and control the model must be able to be implemented to work with a ﬁnite number of states, the distributed model in (4.6) therefore must be converted to a form that can accomplish this. Following a similar method as in [65] and [63] a lumped model is developed to best approximate the distributed model for use in analysis and control.

91 Chapter 4. A 3D Analytical Dynamic Tyre Model

1000 v = 0 ms−1 800 −1 600 v = 0.01 ms

400 v = 0.1 ms−1 200 v = 1 ms−1

0 v = 10 ms−1 −200 Moment (Nm)

−400

−600

−800 MF−SWIFT Proposed −1000 −1 −0.5 0 0.5 1 steering rate dδ/dt (rad/s)

Fig. 4.13: δ˙ vs Mz for diﬀerent ω

The lumped model attempts to capture the mean behaviour of the the internal friction states, in a series of single ordinary diﬀerential equations. These mean states are then used to predict the developed longitudinal and lateral forces as well as the self-aligning moment of the tyre, in at least the steady-state conditions.

The longitudinal and lateral forces Fx(t) and Fy(t) that are developed at the contact patch from the distributed model (4.7) is shown below

W L 2 ∂ǫi(t, ζ, η) Fi(t)= − σ0iǫi(t, ζ, η)+ σ1i + σ2ivri fz(ζ, η)dηdζ (4.21) W ∂t Z0 Z− 2

Let the weighted mean internal friction statesǫ ¯i along the x and y directions be deﬁned as follows.

W 1 L 2 ¯ǫi(t)= ǫi(t, ζ, η)fz(ζ, η)dηdζ ,i = x, y (4.22) F W z Z0 Z− 2 With the time derivative of the weighted mean internal friction states being,

92 4.1. Friction Dependant Analytical Tyre Model

W L 2 dǫ¯i(t) 1 ∂ǫi(t, ζ, η) = fz(ζ, η)dηdζ ,i = x, y (4.23) dt F W ∂t z Z0 Z− 2 Similarly the weight mean relative velocities for both x and y are shown below.

W L 2 1 ˙ v¯rx = ωre − v cos α − ηδ fz(ζ, η)dηdζ Fz 0 − W Z Z 2 = ωre − v cos α − δ˙λ¯x (4.24)

W 1 L 2 L v¯ry = −v sin α − ( − ζ)δ˙ fz(ζ, η)dηdζ Fz W 2 Z0 Z− 2 ˙¯ = −v sin α − δλy (4.25)

With the terms λ¯x and λ¯y determined based on the force distribution on the tyre contact patch. For the case of a symmetric force distribution the corresponding λ¯i will go to zero leaving the terms that are constant across the contact patch, for the case L where the tyre is steered about η = 0 and ζ = 2 . The total longitudinal and lateral forces that are felt at the contact patch can now be represented as a function of the mean statesǫ ¯i, ¯ǫ˙i andv ¯ri as follows,

Fi(t)= −Fz (σ0iǫ¯i(t)+ σ1i¯ǫ˙i(t)+ σ2iv¯ri) (4.26)

To complete the model we need to determine the dynamics of the mean deﬂection states, to achieve this we substitute (4.5) into (4.23)

W 1 2 L kµ2v kσ ∂ǫ ˙ k r 0i i ǫ¯i(t)= vri − 2 ǫi(t, ζ, η) − |ωr| fz(ζ, η)dηdζ Fz W µ g(vr) ∂ζ Z− 2 Z0 ki

ǫ¯˙i =v ¯ri − C¯0i(vr)¯ǫi − κi|ωr|ǫ¯i, i = x, y (4.27)

where,

93 Chapter 4. A 3D Analytical Dynamic Tyre Model

W 2 L ′ W ǫifzdζdη − 2 0 κi = − W , i = x, y (4.28) R 2 R L W ǫifzdζdη − 2 0 R R For the cases where fz is deﬁned by a function that has a value of zero at the boundaries, the third term goes to zero. The average lumped model friction force can then be written as follows

¯ǫ˙i =v ¯ri − C¯0i(vr)+ κi|ωr| ǫ¯i (4.29a)

F¯i(t)= −Fz (σ0iǫ¯i(t)+ σ1i¯ǫ˙i(t)+ σ2iv¯ri) , i = x, y (4.29b)

the function κi(t) is chosen so that the friction force generated through the lumped model F¯i approximate the forces that are felt in the distributed model Fi. The solution to κi(t) however requires the solution to the distributed internal deﬂection ǫi(t, ζ, η).

To proceed we approximate κi such that the steady-state solution of the lumped model ¯ss ss Fi is equal to the steady-state solution for the distributed model Fi , as it was done in [63], [62] and[65]. For constant ω, v and δ˙, the steady state solution for the lumped model can be found by setting ¯ǫ˙i(t) = 0 in (4.29a), giving,

v¯ ǫ¯ss = ri , i = x, y (4.30) i ¯ ss C0i(vr)+ κi |ωr|

ss where κi is deﬁned by

W 2 L ss ′ W 0 ǫi fzdζdη ss − 2 κi = − W (4.31) R 2 R L ss W ǫi fzdζdη − 2 0 R R ss ss and where ǫi from (4.15). Calculating κi directly from (4.31) is not a straight ss forward matter and so instead κi calculated so that the steady-state forces produced by both distributed and lumped model are equal, yielding,

W L 2 ss 1 ss ǫ¯i = ǫi (ζ, η)fz(ζ, η)dηdζ, i = x, y (4.32) F W z Z0 Z− 2 94 4.1. Friction Dependant Analytical Tyre Model

ss The expression forǫ ¯i varies depending on the force distribution fz(ζ, η) that ss is acting on the contact patch. Onceǫ ¯i is calculated from (4.32), κi(t) can be ss approximated with κi , giving

1 v¯ κss = ri − C¯ (¯v ) (4.33) i |ωr| ǫ¯ss 0i r i The lumped model for the self-aligning moment is derived from the distributed self-aligning moment model (4.12) and the distributed internal friction state (4.6).

Substituting the mean internal friction stateǫ ¯i from (4.22) into Mz(t) where appropriate gives,

W W L 2 L 2 ∂ǫx(t, ζ, η) Mz(t)= σ0x ǫx(t, ζ, η)fz(ζ, η)ηdηdζ + σ1x fz(ζ, η)ηdηdζ W W ∂t Z0 Z− 2 Z0 Z− 2 W L 2 LFz +σ2x vrx(η)fz(ζ, η)ηdηdζ + (σ0yǫ¯y(t)+ σ1yǫ¯˙y(t)+ σ2yv¯ry) W 2 Z0 Z− 2 W W L 2 L 2 ∂ǫy(t, ζ, η) −σ0y ǫy(t, ζ, η)fz(ζ, η)ζdηdζ − σ1y fz(ζ, η)ζdηdζ W W ∂t Z0 Z− 2 Z0 Z− 2 W L 2 −σ2y vry(ζ)fz(ζ, η)ζdηdζ W Z0 Z− 2 (4.34)

Define now the cumulative weighted mean internal deflectionǫ î for the aligning torque as follows

W 1 L 2 ˆǫx(t)= ǫx(t, ζ, η)fz(ζ, η)ηdηdζ (4.35) FzW W Z0 Z− 2 and

W 1 L 2 ˆǫy(t)= ǫy(t, ζ, η)fz(ζ, η)ζdηdζ (4.36) FzL W Z0 Z− 2 With the corresponding cumulative weighted mean internal deﬂection state derivatives being,

95 Chapter 4. A 3D Analytical Dynamic Tyre Model

W L 2 dǫˆx(t) 1 ∂ǫx(t, ζ, η) ˆǫ˙x(t)= = fz(ζ, η)ηdηdζ (4.37) dt F W W ∂t z Z0 Z− 2 and

W L 2 dǫˆy(t) 1 ∂ǫy(t, ζ, η) ˆǫ˙y(t)= = fz(ζ, η)ζdηdζ (4.38) dt F L W ∂t z Z0 Z− 2 In much the same manner the cumulative weight mean relative velocities in the contact patch can be expressed as follows,

W 1 L 2 vˆrx = ωre − v cos α − ηδ˙ fz(ζ, η)ηdηdζ (4.39) FzW 0 − W Z Z 2

W L 2 1 L ˙ vˆry = −v sin α − ( − ζ)δ fz(ζ, η)ζdηdζ (4.40) FzL W 2 Z0 Z− 2

The creation of these new terms allows for the total lumped model aligning torque

Mz(t) to take the following form,

Mz(t) 1 = σ0xǫˆx(t)+ σ1xǫˆ˙x(t)+ σ2xvˆrx FzLW L 1 1 1 1 + σ ¯ǫ (t) − ǫˆ (t) + σ ¯ǫ˙ (t) − ǫˆ˙ (t) + σ v¯ − vˆ W 0y 2 y y 1y 2 y y 2y 2 ry ry (4.41)

Finally to take advantage of the aligning torque model in (4.41) the dynamics of the cumulative weighted mean stateǫ ˆi need to be derived. From equations (4.35) and (4.36) we have,

W L 2 µ2 ˙ 1 k kvrkσ0 ∂ǫi ǫˆx(t)= vri − 2 ǫi(t, ζ, η) − |ωr| fz(ζ, η)ηdηdζ (4.42) F W W µ g(v ) ∂ζ z Z0 Z− 2 ki r

simplifying

96 4.1. Friction Dependant Analytical Tyre Model

W µ2 L 2 ˙ k kvˆrkσ0x |ωr| ∂fz ǫˆx(t)=ˆvrx − 2 ǫˆx(t)+ ǫx ηdηdζ µ g(ˆvr) FzW W ∂ζ kx Z0 Z− 2

=v ˆrx − Cˆ0xˆǫx(t) − νx|ωr|ǫˆx (4.43)

where

W 2 L ′ W ǫxfzηdζdη − 2 0 νx = − W (4.44) R 2 R L W ǫxfzηdζdη − 2 0 ˙ R R similarly the term ǫˆy(t) can be calculated, giving the following,

|ωr| ˆǫ˙ (t)=ˆv − Cˆ ǫˆ (t) − ν |ωr|ǫˆ + ǫ¯ (4.45) y ry 0y y y y L y where

W 2 L ′ W ǫyfzζdζdη − 2 0 νy = − W (4.46) R 2 R L W ǫyfzζdζdη − 2 0 R R These equations now allow for the self-aligning moment Mz to be determined from the combination ofǫ î andǫ ¯y. However, calculatingǫ î from equations (4.43) and (4.45) is difficult as it requires knowledge of νi from (4.44) and (4.46). The value of νi can be ss approximated as a constant value νi in a similar way to how the lumped model for forces was approximated. This is done by assuming a steady-steady condition and by setting the predicted lumped model self-aligning torque Mz to be equal to the steady-state self-aligning moment for the distributed model given in (4.18). ˙ ˙ The steady-state value of the lumped model is determined by setting ˆǫx = 0, ǫˆy = 0, ss ¯ǫ˙y =0 as well asǫ ¯y =¯ǫy in equation (4.41). This gives the following equation,

M ss W ǫ¯ss σ 1 ǫˆss = − z + (ˆǫssσ +ˆv σ )+ y + 2y v¯ − vˆ (4.47) y F Lσ σ L x 0x rx 2x 2 σ 2 ry ry z 0y 0y 0y

ss ss The expression forǫ ˆy contains the termǫ ˆx which also needs to be identiﬁed. ss However, this term is dependant on νx , which is diﬃcult to derive. The values of

97 Chapter 4. A 3D Analytical Dynamic Tyre Model

ss ss νx and νy need to be identiﬁed, both of which can be found be found by setting ˙ ˙ ss ss ǫˆx = 0, ǫˆy = 0, ¯ǫ˙y = 0 as well asǫ ¯x = ǫx andǫ ¯y = ǫy from the equations (4.43) and (4.45) to obtain,

1 vˆ νss = rx − Cˆ (4.48) x |ωr| ǫˆss 0x x and, 1 1 |ωr|ǫ¯ss νss = vˆ + y − Cˆ (4.49) y |ωr| ǫˆss ry L 0y y ss in order to solve for Mz an additional constraint needs to be put in place, there ss are four unknowns and three equations. The last constraint is chosen to beǫ ˆx , this is chosen as it has the least impact on the solution in terms of accuracy and is a straight forward equation.

W L 2 ss 1 ss ǫˆx = ǫx fz(ζ, η)ηdηdζ (4.50) F W W z Z0 Z− 2 ss ss The explicit expression forǫ ˆx and Mz is dependant on the force distribution.

4.1.5 Simulation

To evaluate the dynamic responses of the proposed model and its comparison to the existing models, three different open loop simulations are designed. The first simulation series look at the dynamic responses for ramp and step braking control on both the longitudinal and lateral forces felt at the contact patch. The second simulation series examines the relationship between the self-aligning moment and the steering rates in a stationary case. The last series of simulations validates the proposed model in a parking maneuver, which shows the successful incorporation of the steering rate on the self-aligning moment as the model transitions between stationary and forward motion. The first simulation series contain both the dynamic responses of the tyre as a braking torque is applied, both ramped and step responses. The configuration for the simulation is such that the forward velocity is kept constant, with the wheel speed reducing as the braking force increases, which subsequently increases the wheel slip. The simulation setup is identical to the one that was performed experimentally in [155] as well as replicated in [65]. As in the previous experimental and simulation

98 4.1. Friction Dependant Analytical Tyre Model

0 0

−200

−400

−500 −600

−800 Torque (Nm) Torque (Nm) −1000 −1000

−1200

−1400

−1600 −1500 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 time (s) time (s) (a) (b)

0.8 1.2

0.7 1

0.6

0.8 0.5

0.6 0.4

0.3 0.4 Longitudinal Slip Longitudinal Slip

0.2 0.2

0.1

0 0

−0.1 −0.2 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 time (s) time (s) (c) (d)

4500 4500

4000 4000

3500 3500

3000 3000

2500 2500

(N) 2000 (N) 2000 x x F F

1500 1500

1000 1000

500 500

0 0

−500 −500 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 Longitudinal Slip Longitudinal Slip (e) (f)

3500 3500

3000 3000

2500 2500

2000 2000 (N) (N) y y F F 1500 1500

1000 1000

500 500

0 0 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 Longitudinal Slip Longitudinal Slip (g) (h) Fig. 4.14: Open loop simulation results for both ramp and step increases in braking torque

99 Chapter 4. A 3D Analytical Dynamic Tyre Model methods, the forward speed is kept constant, v = 8ms−1 and the wheel slip angle is also kept constant α = 4◦. With two different braking cases being applied on these configurations, the first being a linearly increasing braking torque and the second being a braking force that increases in a series of discrete increases in braking torque. The proposed model was configured to run through these setups and the results can be seen in Fig. 4.14, with the results of the proposed model being in agreement with the experimental results in [155]. The first column of Fig. 4.14 shows the dynamic effects of the ramp braking torque, and the second column shows the dynamic effects of the stepped braking torque, which show the direct effects of the system dynamics on the developed torque. Both Fig. 4.4(a) and Fig. 4.4(b) are excellent examples of the dynamics that are present in the system as they show the presence of a hysteretic loop, which would not be present through the use of steady-state models.

1.5 Mz δ

Mz 1 1000 (Nm) 0.5

δ −0.5 (rad) −1

−1.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time (s)

Fig. 4.15: Evolution of the steering rate dependant self-aligning torque over time

The second simulation series consists of a stationary tyre and the relationship between the self-aligning moment and the steering rate of the tyre. The conﬁguration for the simulation sets the forward velocity and the wheel velocity both to zero, only the steering angle is controlled. The steering angle begins to move in a cyclic motion,

100 4.1. Friction Dependant Analytical Tyre Model between a steering angle of 20◦ and −20◦ with a period of 2 seconds. The cycle is repeated three full times, with the steering rate and the self-aligning moment during the motion recorded. The time dependant results of the self-aligning moment, steering rate and the steering angle is shown in Fig. 4.15, which shows a similar response as to what is observed through the use the dynamic MF-SWIFT model as described in [75, pp. 455-458]. The Fig. 4.16 shows the eﬀect of hysteresis on the self-aligning moment as the steering angle progresses through the turn, with the maximum self-aligning moment observed to occur later into the turn as is very similar to the experimental response to the same maneuver, shown in [156].

1500

1000

500

0 (Nm)

−500

−1000

−1500 −25 −20 −15 −10 −5 0 5 10 15 20 25 δ(◦)

Fig. 4.16: δ vs Mz for a non-rolling tyre turning on the spot

The last series of simulations for the dynamic tyre model is a demonstration of the eﬀects of both the steering rate and the wheel speed on the forces and moments. The simulation consists of starting in a stationary situation with the tyre steering through two full steering cycles between 20◦ and −20◦ with a period of 2 seconds, as before. The steering cycle is continued and the vehicle begins to move forward at a linearly increasing speed, the speed increases up till v =2ms−1. The vehicle speed is controlled

101 Chapter 4. A 3D Analytical Dynamic Tyre Model in this application and so the tyre rolls with the motion, however the eﬀect of the tyre steer produces a periodically changing slip angle, which impact on the self-aligning torque and lateral forces that are developed in the tyre. The self-aligning moment and the lateral force that is generated through the parking maneuver is shown in Fig. 4.17 with respect to time. Additionally, the steering angle and forward speed are also displayed in the same ﬁgure allowing for the relationship between the steering rate and the speed to be observed as the tyre transitions between stationary and high speed dynamics. With the magnitude of the self-aligning moment decreasing as the forward velocity of the tyre increases.

1.5

Mz δ 1 Mz ss 1000 Mz (Nm) 0.5

-0.5 δ (rad) -1

-1.5 0 1 2 3 4 5 6 7 8 time (s)

Fig. 4.17: Self-aligning moment transient through the parking maneuver, showing the steady state self-aligning moment and the steering angle changing over time

The relationship between the developed self-aligning moment and the steering angle is also directly displayed in Fig. 4.18, where the eﬀect of the forward speed can be directly seen to alter the maximum developed self-aligning torque throughout the maneuver. The response of the dynamic model is similar to that of the response which is described in [75, pp. 442-444], which is based on a dynamic version of the MF-SWIFT model, with the response of the MF-SWIFT model considered to be a realistic response

102 4.2. Tyre Force Distribution from an experimental setup.

1000

800

600

400 M z 200

0 (Nm) -200

-400

-600

-800 Mz -1000 -25 -20 -15 -10 -5 0 5 10 15 20 25 δ(◦)

Fig. 4.18: Self-aligning moment transient through the parking maneuver, showing transition between the stationary moment and the evolving moment with forward speed.

4.2 Tyre Force Distribution

The effect of having a realistic force distribution model for the entire contact patch of the tyre enables the prediction of the longitudinal and lateral forces and the self-aligning moment that more closely resemble the forces and moments that a tyre is subject to when moving through the same maneuver. The proposed three dimensional tyre model requires a wheel velocity and camber angle dependant force distribution in order for the tyre model to be able to replicate the forces and moments that are physically acting on the tyre. There have been a number of different attempts at providing a realistic force distribution on the tyre contact patch. The majority of analytical models that exist in the literature so far, have utilised a trapezoidal force distribution under the tyre contact patch [64, 65], however this shape is fixed and is not dependant on the tyre speed, which is what is observed. Additionally,

103 Chapter 4. A 3D Analytical Dynamic Tyre Model the force distribution under the contact patch has thus far been limited to the x axis of the contact patch and so does not adequately account for effects such as the self aligning moment experienced while turning the wheel and does not account for camber angle changes in the wheel, which are effects that require both a two dimensional contact patch. There are others who propose a tyre model that is extremely realistic, which also generates a realistic tyre force distribution [157]. The system dynamics that are generated rely on utilising the natural frequencies and modes of oscillation for a tyre under a disturbance. The number of modes required to attain a steady state solution for the disturbance and thus a realistic representation is in the order of 30 modes. This places a large computational cost on calculating both the natural frequencies for each wheel as well as the final force distribution, with the provided solution only along the x axis of the wheels in the direction of rotation. The tyre force distributions that have been considered for the proposed tyre model are the constant, parabolic, trapezoidal and variable force distributions, which are detailed in Appendix B.1. The constant force distribution was used in the previous section and this section will present the velocity and camber angle dependant variable force distribution.

The force distribution function fz(t, ζ, η) so that the following function is satisﬁed

W L 2 Fz = fz(ζ, η)dηdζ (4.51) W Z0 Z− 2

′ fz(ζ)= ∂fz/∂ζ (4.52)

The steady-state expressions for the force and moment equations (4.7), (4.8), (4.9) and (4.12) can now be evaluated using the now developed force distribution function fz(t, ζ, η)

4.2.1 Variable force distribution

To improve the ﬁt between the expected steady-state forces and moments and the MF experimental tyre model a more descriptive force distribution needs to be developed,

104 4.2. Tyre Force Distribution for the force distribution to be appropriately descriptive it needs to vary depending on the conditions of the tyre [66, 68]. The force distribution over the contact patch needs to depend on the tyre forward speed as well as the camber angle that the tyre makes with the ground, in addition to depending on the normal force acting on the tyre.

A few diﬀerent methods have been examined for use as a descriptive variable force distribution, these include; an altered form of the continuous model by Guo [158, 159] as well as an altered piece-wise trapezoid function similar to that proposed by Deur et al [62]. The proposed 3D-AD tyre model require these models to be extended into two dimensions in order to account for the eﬀects of a steering rate and the change in camber angle of the tyre.

The continuous model allows for a much more accurate ﬁt between the force distribution and the equation through the use of shaping parameters, however the equation that is used to provide the shape for the function is not simple, and there are only a few valid values of the shaping parameters, with many others creating invalid force distributions. The piece-wise trapezoidal model by contrast is very simple, with a couple of direct shaping parameters. The closeness between the modelled force distribution and an experimental force distribution are not as strong but there is still a good ﬁt between the forces and moments generated using a trapezoidal model and the MF tyre model results.

The diﬃculty of the continuous force distribution model to be tuned so that the generated dynamics reﬂect the experimental dynamics manifests itself as an unrealistic force distributions that involves having negative (pulling) force. This outcome is due to the underlying equations of the continuous force distribution model, that utilises a number of non-linear functions that do not guarantee the overall shape of the force distribution. Although the force distribution is able to adapt to accurately recreate the experimental forces and moments, this is achieved through the use of a physically impossible force distribution.

A simple and robust descriptive force distribution for the contact patch is developed using a piece-wise trapezoidal distribution, the distribution is asymmetric about the centre of the tyre to account for the change of sign of the self-aligning torque at higher slip angles. To extend this model into the two dimensional contact patch, an additional

105 Chapter 4. A 3D Analytical Dynamic Tyre Model trapezoid structure is developed for the second dimension, η axis, which provides both axes with asymmetry to account for self-aligning moment sign change.

The structure of the variable force distribution fz(ζ, η) that is developed incorporates two separate one dimensional trapezoidal distributions, fz,ζ(ζ) for the ζ axis, and fz,η(η) for the η axis. The terms, ζR and ζL represent the distances in the ζ axis of the right and left edges of the top side of the trapezium, with the terms ηR and ηL, similarly representing the dimensions for the right and left edges of the trapezium in the η axis. The equation for the developed variable force distribution can be seen below,

α1ζ for 0 ≤ ζ ≤ ζL,  fz,ζ(ζ)=  1 for ζL ≤ ζ ≤ ζR, (4.53a)   α2ζ + β2 for ζR ≤ ζ ≤ L,    −W α3η + β3 for 2 ≤ η ≤ ηL,  f (η)=  (4.53b) z,η  1 for ηL ≤ η ≤ ηR,   W α4η + β4 for ηR ≤ η ≤ 2 ,    fz(ζ, η)= fmaxfz,ζfz,η (4.53c) with

1 1 L α1 = ,α2 = − , β2 = (4.54a) ζL L − ζR L − ζR

2 W 2 W α3 = , β3 = ,α4 = − , β4 = (4.54b) W +2ηL W +2ηL W − 2ηR W − 2ηR with the value of fmax generated as to solve across the distribution for a given Fz as follows,

W L 2 1 Fz = fz(ζ, η)dηdζ = fmax(W + ηR − ηL)(L − ζL + ζR) (4.55) W 4 Z0 Z− 2

The trapezoidal model was altered to have the variables ζL and ηL changed to depend

106 4.2. Tyre Force Distribution

Fig. 4.19: Asymmetrical trapezoid force distribution

on dr(vfwd) and dc(γ) for the distance between the center of the wheel and the centroid of the force distribution for both the velocity dependant case (rolling resistance) and the camber angle dependant case. The structure that has been chosen to represent the velocities dependance has been taken from [59], which allows for the ζ axis centroid of the trapezium dr to depend on the forward velocity of the tyre. The centroid dr represents the mean of the force distribution, with the oﬀset of the normal force acting on the axle of the wheel, resulting in the generation of the rolling resistance, the equation for the centroid placement is below,

−µkx −|vfwd| dr(vfwd)= − sgn vfwd Cr1e (1 − e )+ Cr2vfwd (4.56) where, Cr1 and Cr2 are the rolling resistance coeﬃcients which are used to tune the behaviour of the equation to the experimental results.

Similarly, for the lateral case, the distance dc is used to evaluate the overturning

107 Chapter 4. A 3D Analytical Dynamic Tyre Model

moment, with Cc1 and Cc2 being the overturning moment coeﬃcients. The developed model is dependant on the camber angle γ, and is developed in a similar way to the rolling resistance term dr, except with µky instead of µkx. The form of dc(γ) is given as follows,

−µky −|γ| dc(γ)= − sgn γ Cc1e (1 − e )+ Cc2γ (4.57) The equation for the centroid cx of a trapezium is calculated using (4.58), with a the length of the top side, b the length of the bottom side and c the distance from the origin to the beginning of the top edge, as shown in Fig. 4.20.

Fig. 4.20: Centroid of a trapezium showing the geometry of the trapezium

2ac + a2 + cb + ab + b2 c = (4.58) x 3(a + b)

To determine the shape of the trapezium in the ζ axis, the values for ζL and ζR need to be determined. The terms dr and ζT control the shape of the trapezium, with

ζR = ζL + ζT . The centroid of the trapezium in the ζ axis can be found using (4.59), with the bottom of the trapezium being L, the leading edge distance being ζL and the top side of the trapezium being ζT . The value of dr from before is used to represent the oﬀset from the centre of the bottom side, representing the distance along the contact patch in the longitudinal direction. The value for ζL can be determined by rearranging

108 4.2. Tyre Force Distribution

(4.59).

2 2 L 2ζT ζL + ζT + ζLL + ζT L + L dr + = (4.59) 2 3(ζT + L)

The force distribution along the ζ axis for a given forward speed is shown in Fig. 4.21. This shows that as the forward speed changes the force distribution also changes in response.

1 v = -20 0.8 fwd v = -5 fwd 0.6 v = 0 fwd 0.4 v = 5 fwd 0.2 v = 2 fwd Normalised force (N/m) 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Longitudinal contact patch position (m)

Fig. 4.21: Normalised force distribution for contact patch in the longitudinal direction with dependance on the forward speed vfwd

The term dc controls the position of the centroid along the η axis, the equation for the centroid of the trapezium along the η axis is shown in (4.60). The equation that is used for calculating the centroid relies on origin being at the edge of the trapezium, where as for the lateral direction the origin exists in the middle of the contact patch.

The bottom side of the trapezium is W , the top side is ηT and the leading edge distance is ηL. The term ηR is then calculated using ηR = W − ηL − ηT with the value of ηL found from rearranging (4.60).

2 2 W 2ηT ηL + ηT + ηLW + ηT W + W W dc + = + (4.60) 2 3(ηT + W ) 2

The force distribution for the η axis using the developed approach for a given set of camber angles are shown in Fig. 4.22. The relationship is similar to that of the ζ axis results, with the force distribution dependant on the camber angle of the tyre.

Incorporating both the ζ and η axis components together to form a two-dimensional force distribution that is dependant on the wheel speed ω as well as the camber angle of the wheel γ.

109 Chapter 4. A 3D Analytical Dynamic Tyre Model

0.8 γ = -20deg 0.6 γ = -5deg 0.4 γ = 0deg γ = 5deg 0.2 γ = 20deg Normalised force (N/m) 0 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 Lateral contact patch position (m)

Fig. 4.22: Normalised force distribution for contact patch in the lateral direction for diﬀerent camber angles γ

4.2.2 Steady-State Fitting

The analytical dynamic tyre model when using the variable force distribution not only allows for the evaluation of the steady-state behaviour of the tyre for the longitudinal and lateral forces and the self-aligning moment, but also for the overturning moment and for the rolling resistance. The evaluation of these extra moments are straight forward, in that the variable force distribution is already a steady-state representation of the force distribution felt in the tyre. This allows for the steady-state overturning moment

ss ss and rolling resistance to be readily calculated, with Mx = Mx and My = My. The steady-state representation of the longitudinal and lateral forces and the self-aligning moment is the same as in (4.17) and (4.18). To simplify the evaluation of the steady-state values for the force and moments the integral presented in (4.17) and (4.18) can be further broken down through the different areas that correspond to the different areas of the trapezoidal force distribution. The identification of the parameters for the 3D dynamic tyre was done by comparing

ss ss ss ss ss the plots of the steady-state expressions for Fx , Fy , Mx , My and Mz for the developed variable force distribution. The identified parameters are shown in Table 4.2. The parameter fitting of the rolling resistance is of great importance as this allows for the shape of the force distribution to be determined through the fitting process. The MF-SWIFT 6.1 model provides a method of calculating the rolling resistance moment My, which is then compared to the rolling resistance that the proposed model generates, in addition to these models, the RSAE J2452 standard [151] is also used. This comparison, can be seen in Fig. 4.23, where the proposed model rolling resistance value begins at vfwd = 0 with a zero rolling resistance, while the MF-SWIFT model

110 4.2. Tyre Force Distribution

σ0x(N/m) σ2x(Ns/m) µkx µsx 866 0.0021 0.74 1.46

σ0y(N/m) σ2y(Ns/m) µky µsy 818 -0.0011 0.78 1.18

L(m) W (m) vs(m/s) β 0.0874 0.1208 3.7 1

Cr1 Cr2 ζT /L 0.0001 0.0001 0.7

Cc1 Cc2 ηT /W 0.1480 -0.0001 0.5 Table 4.2: Identiﬁed Tyre Parameters

has a fixed offset rolling resistance torque in place. The J2452 standard specifies the equation, and thus the structure that the rolling resistance parameters are fitted to from experimental results. The RSAE standard, which has been fitted to the MF tyre data, shows a similar shape as the proposed tyre model rolling resistance value.

ss Similarly the reliance on the camber angle γ of the overturning moment Mx provides a similar ﬁgure, shown in Fig. 4.24. The MF-SWIFT results for the overturning moment are almost linearly related to the camber angle and the magnitude of the moment is considerably larger than that of the rolling resistance. The proposed model is able to closely represent the overturning moment as a function of camber angle.

ss ss The forces Fx and Fy match very well against the MF-SWIFT model, these are shown in Fig. 4.25 and Fig. 4.26. Through the use of the variable force distribution a better representation of the self-aligning moment that the tyre experiences is achieved, this improvements are that at high slip angles the self-aligning moment is able to reverse sign as in experiments, this is shown in Fig. 4.27. The self-aligning torque for the proposed model in Fig. 4.27, with the red torques developed through the use of the parameters in Table 4.2. As expected there is a much better fit between the MF model and the proposed model for the fitted data, with the self-aligning torque changing sign for the self-aligning moment for large slip angles. This affect causes a larger number of elements on the edges and near the corners of the contact patch to contribute more torque than they actually do to the self-aligning moment, reducing the value of the maximum self-aligning torque as well as preventing

111 Chapter 4. A 3D Analytical Dynamic Tyre Model

0 MF-SWIFT Proposed -2 RSAE

-4

-6 (N/m) y

-8 Moment M

-10

-12

-14 0 2 4 6 8 10 12 14 16 18 20 Wheel Velocity v (m/s) fwd

Fig. 4.23: ω vs My comparison between MF-SWIFT and the proposed model

800 MF−SWIFT Proposed 600

400

200 (N/m) x 0

Moment M −200

−400

−600

−800 −10 −8 −6 −4 −2 0 2 4 6 8 10 camber angle γ (deg)

Fig. 4.24: γ vs Mx comparison between MF-SWIFT and the proposed model

112 4.2. Tyre Force Distribution

4500

4000

3500

3000

2500 (N) x

2000 Force F

1500

1000

500 MF−SWIFT Proposed 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 longitudinal slip κ

Fig. 4.25: Slip vs Fx comparison between MF-SWIFT and the proposed model

4000

3000

2000

1000 (N) y 0 Force F −1000

−2000

−3000 MF−SWIFT Proposed −4000 −50 −40 −30 −20 −10 0 10 20 30 40 50 slip angle α (deg)

Fig. 4.26: α vs Fy comparison between MF-SWIFT and the proposed model

113 Chapter 4. A 3D Analytical Dynamic Tyre Model

50 MF−SWIFT Proposed 40

10 (Nm) z 0

−10 Torque M

−20

−30

−40

−50 −50 −40 −30 −20 −10 0 10 20 30 40 50 slip angle α (deg)

Fig. 4.27: α vs Mz comparison between MF-SWIFT and the proposed model the change of sign of the self-aligning moment at large slip angles. A new approach for this derivation is therefore needed.

4.2.3 Simulation

The simulation scenarios that are used to evaluate the dynamic model with the new variable force distribution are precisely the same as the scenarios in Section 4.1.5. The parameters that are used for the variable force distribution have been evaluated in the steady-state parameter fitting. The dynamic results of the variable force distribution are compared to the constant force distribution forces and moments. The first simulation scenario, with the results from the variable force distribution seen in Fig. 4.28, shows a comparison between the constant force distribution and the variable force distribution, is only a slight difference between the two force distribution results. The results of the proposed model for both force distributions are in agreement with the experimental results in [155, 160]. The second simulation scenario, which examines the effects of the steering rate on

114 4.2. Tyre Force Distribution

1.2 0

0.8

−500

0.6

0.4 Torque (Nm) Longitudinal Slip

−1000 0.2

−1500 −0.2 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 time (s) time (s) (a) (b)

5000 4000

3500 4000

3000

3000 2500

(N) 2000 (N) 2000 x y F F

1500 1000

1000

0 500

−1000 0 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 Longitudinal Slip Longitudinal Slip (c) (d) Fig. 4.28: Open loop simulation comparison between variable and constant force distributions, for step increases in braking torque

development of the self-aligning moment as a tyre steers back and forth between two angles. A comparison between the self-aligning moment that was developed with the variable force distribution and the constant force distribution over time are shown in Fig. 4.29, with the steady-state self-aligning moments for both cases normalised. The evolution of the self-aligning moment for the variable force distribution shows that it reaches the steady-state at a faster time interval than the constant force distribution.

The transient behaviour of the self-aligning moments are shown in Fig. 4.30, with the comparison between the variable and the constant force distributions showing that the variable force distribution is able to provide a more realistic account of the progression of the self-aligning moment as seen in the experimental results [77, 156]. The biggest diﬀerences between the simulation results for the simple tyre force model and the variable tyre force model is that the self-aligning moment is able to exhibit the more realistic behaviour of the decrease of the self-aligning moment for higher slip angles.

115 Chapter 4. A 3D Analytical Dynamic Tyre Model

1.5 Mz δ

Mz 1 1000 (Nm) 0.5

δ −0.5 (rad) −1

−1.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time (s)

Fig. 4.29: Evolution of the steering rate dependant self-aligning torque over time

This eﬀect is visible in Fig. 4.30, where both the steady-state and dynamic self-aligning moments both reduce in magnitude at the height of the steering angle.

The third simulation scenario, the parking maneuver, is performed using the variable force distribution and shows a comparison between the results of the parking maneuver with the constant force distribution. The self-aligning torque and the lateral force that is generated through the parking maneuver is shown in Fig. 4.31 with respect to time. The relationship between the developed self-aligning moment and the steering angle is also directly displayed in Fig. 4.32, where the eﬀect of the vehicle speed can be directly seen to alter the maximum developed self-aligning torque throughout the maneuver. The ﬁgure shows a comparison between the self-aligning moment evolution for both the variable and constant force distributions, with both responses behaving in a realistic manner.

116 4.3. Conclusion

1000

800

600

400

Mz 200

0 (Nm) −200

−400

−600

−800

−1000 −25 −20 −15 −10 −5 0 5 10 15 20 25 δ(◦)

Fig. 4.30: δ vs Mz for a non-rolling tyre turning on the spot

4.3 Conclusion

This chapter has presented a new analytical dynamic tyre model that includes the eﬀects of steering rate on the forces and moments developed in the contact patch. A novel force distribution is also presented that is dependant on the tyre camber angle and rotation speed. This combination allows for the proposed model to provide an accurate estimation for the longitudinal and lateral tyre forces as well as the self-aligning and overturning moments and the rolling resistance of the tyre.

The analytical dynamic tyre model is validated against the empirical Magic Formula, showing that the proposed model is able to correctly predict the forces that are experienced in the tyre. The ability of the proposed model to quickly change when the frictional characteristics of the terrain change allow it to be able to reduce the uncertainty by providing a more comprehensive system model, in order to better predict the available force and moments.

The proposed analytical dynamic tyre model is also applicable as a general case as there is no requirements on whether the wheel is driven or not, this allows the model to

117 Chapter 4. A 3D Analytical Dynamic Tyre Model

1.5

Mz δ 1 Mz ss 1000 Mz (Nm) 0.5

-0.5 δ (rad) -1

-1.5 0 1 2 3 4 5 6 7 8 time (s)

Fig. 4.31: Self-aligning moment transient through the parking maneuver, showing the steady state self-aligning moment and the steering angle changing over time be used for all wheels easily without needing to designate driving wheels. The inclusion of a generalised center of rotation for the tyre steering also allows for the developed model to be used with most steering conﬁgurations including customs setups where the center of rotation for the steering is not about the center of the contact patch.

118 4.3. Conclusion

1000

800

600

400 M z 200

0 (Nm) -200

-400

-600

-800 Mz con Mz -1000 -25 -20 -15 -10 -5 0 5 10 15 20 25 δ(◦)

Fig. 4.32: Self-aligning moment transient through the parking maneuver, showing transition between the stationary moment and the evolving moment with forward speed, with a comparison to the constant force distribution self-aligning moment.

119 Chapter 4. A 3D Analytical Dynamic Tyre Model

120 Chapter 5

Terrain Perception using Non-semantic Range Data

The terrain that Autonomous Ground Vehicles often encounter in farming and non-suburban environments is generally non-uniform and uneven, leading to a greater amount of uncertainty in system behaviour. It is necessary, therefore, to be able to identify the characteristics of the terrain in order to reduce the modelled uncertainties so that navigation of AGVs can be carried out safely.

This chapter presents the methods developed for classifying terrain types using non-semantic range data for the purpose of assigning frictional coefficients and terrain slope properties in order to improve the terrain model in the following aspects: (i) to reduce the uncertainties present in the system model; (ii) to assist in improving the controller behaviour; (iii) to improve path planning and; (iv) to improve path tracking for an AGV. The two methods that have been developed to provide solutions to the problem of terrain classification estimate both the dominant surface geometry of the environment as well as the terrain type. The first method, discussed in Section 5.2, evaluates them separately, whilst the second method, discussed in Section 5.3, evaluates the dominant surface geometry and the terrain type together in order to segment the terrain according to both geometry and terrain type.

The terrain classification results for both methods are collected and analysed, with the results compared against ground truth. The terrain classification methods are also qualitatively compared against each other. The classified terrain is then combined

121 Chapter 5. Terrain Perception using Non-semantic Range Data together with the geometry of the terrain to form a model of the terrain for use in uncertainty reduction.

5.1 Introduction

Terrain perception is carried out for many different reasons within the field of AGV research, with the most prominent reasons being mapping, path traversability and disturbance rejection for improved controllability. Most terrain mapping methods that have been developed thus far focus primarily on the traversability of the terrain that is sensed, to allow for path planning methods to react if there is a perceived obstacle in the path of the vehicle. The majority of methods that classify terrain types, predominantly use the information to help evaluate whether a vehicle is on a desired path or not, and also to allow for basic information about how the vehicle should be controlled whilst on that terrain. The boundaries between different terrain types are of great importance as the greatest amount of uncertainties are present in the terrain model at these boundaries [161, 162]. Currently, terrain mapping methods estimate the boundary between terrain types through the use of camera colour data or semantic geometry changes that are present. These methods provide good estimation of these boundaries under most conditions, however, effects such as shadows present in the field of view significantly alter the perceived colours of those surfaces, making it difficult to determine whether a terrain change has taken place or whether it is just the effects of the shadow. Similarly, methods that rely on distinct geometry changes, such as bushes or trees, to perceive terrain boundaries are unable to perceive changes in the terrain type without such distinct geometry changes are not present. Another method for terrain classification and segmentation is through the use of non-semantic range data, with current implementations [163] providing only moderately good classification results. A method that is able to provide reliable terrain classification using non-semantic range data can be utilised during situations where visual or semantic spatial information is degraded due to environmental conditions. A colour camera is a straight forward method of gathering the visual information

122 5.1. Introduction about an environment, however the spatial information of an environment is not as simple to record. There are a number of different technologies and methods for gathering the spatial information of a terrain, including 2D and 3D LiDAR systems, structured light and time-of-flight (ToF) cameras, and stereo vision systems. A comparison between the different sensing technologies and their abilities on the current application can be seen in Table 5.1.

Technology 2D LiDAR 3D LiDAR structure ToF stereo light camera vision camera operating long range long range shortrange shortrange long range range distance cm cm mm - µm cm cm - mm precision low light good good good good weak performance bright light good good weak medium good performance Limitations extreme extreme bright bright needs directed directed sunlight sunlight salient sunlight sunlight visual features

Table 5.1: Generalised Spatial sensing technology comparison

Therefore, in order for the proposed terrain model to be useful in reducing the uncertainties for the system, it is important to use a sensor that is capable of providing spatial information detailed enough to facilitate terrain classification. A few papers have analysed these requirements and have determined that measurements of the order of 0.5mm is adequate to be able to determine the coefficient of friction of a road surface adequately [121, 122]. This requirement for the sensor accuracy is used as an initial baseline in order to start the algorithm with enough information to be able to perform the terrain classification satisfactorily. Although stereo vision is able to achieve the required accuracy and can work outdoors, it cannot achieve object matching on surfaces that have very few salient features, such as a road or trail. This means that the stereo vision disparity maps

123 Chapter 5. Terrain Perception using Non-semantic Range Data generated are not complete and the depth image that is created contains large areas where the depth is unknown. The only technology available that can measure the terrain at such ﬁne accuracy and on surfaces with low or no salient features is the 3D structured light camera. Even though the 3D camera does not work in sunlight, it is able to capture the spatial information to the required accuracy, in order to guarantee appropriate classiﬁcation results.

5.1.1 Spatial Features

The spatial features that can be gathered from the environment includes a variety of different sources, including the dominant geometry of the terrain, or mega-texture, the macro and micro-textures of the terrain as well as semantic spatial features such as identified objects or shapes. The definitions for the size of the different textures can be seen in Fig. 5.1, which is found in [122], and adapted from [105]. Many methods detailing the use of the dominant geometry as well as the semantic spatial features exist [94], but there are few that generate the macro-textures of the terrain. The macro/micro-texture of the terrain is one of the most important spatial properties of the terrain and represents the feeling we experience when touching a surface. In addition to the colour of the terrain these properties are the most influential in terrain classification.

Fig. 5.1: Texture Wavelength of terrain, expressed in meters (m), adapted from [122]

The current macro-texture terrain feature methods that exist in literature include primarily the use of fourier transformed surface geometry which was transformed into a series of Power Spectral Density (PSD) features [120]. The terrain classiﬁcation results using only the spatial PSD features had only limited success in providing a reliable terrain prediction. The method required the use of remission data to provide a reliable terrain prediction, which is highly dependant on the colour of the terrain.

124 5.2. Range Texture Analysis

There are a number of different surface texture features that exist as well as power spectral density features, including the texture features derived from the Gray-Level Co-Occurrence Matrix method as described by Haralick [164], the application of the Gabor filter bank [165], as well as the texture features that were developed by Tamura [166]. The texture features that are developed using these methods all give differing levels of image retrieval based on the individual features as shown in [167], however, this study compared only single features. These methods are explained and the generated features that are used within this chapter are shown in section C.1.

5.2 Range Texture Analysis

The Range Texture Analysis (RTA) method is the first method developed for assessing whether terrain types can be classified through the use of 2D non-semantic spatial information of a terrain surface. The method covers the creation of residual images from the dominant surfaces of the segmented terrain in order to generate spatial texture features for the terrain. The segmented surfaces are then classified using a combination of different classification techniques. Visual features of the terrain are also assessed on how well they are able to classify the terrain types and then a combination between spatial and visual features are used in providing an improved terrain type classification. The advantages and limitations of this method are then discussed. The range information for the experiments in this section are provided using a downwards looking PrimeSense Carmine 1.09, similar to that shown in Fig. 5.8, which was also used to collect the visual data over a variety of terrain types, i.e. grass, road, dirt, gravel, rocks and sand as shown in Fig. 5.7. This sensor uses structured light to capture depth data from the environment. This method of depth capture is affected heavily by noise from the ambient and direct sunlight, so all data was taken in shadowy areas. The Carmine 1.09 is setup so that it is facing downwards at a shallow angle about 800mm off the ground with a field of view of 57.5x45 at 30Hz sampling rate. The sensor has a depth resolution of 0.1mm and an accuracy of 0.3mm at 600mm distance. The depth data is returned as 640x480 array of depth data as well as colour data. The

125 Chapter 5. Terrain Perception using Non-semantic Range Data

(a) grass (b) road (c) rock

(d) gravel (e) rock (f) sand Fig. 5.2: Collected visual data of the terrain types sensor is thus capable of observing the necessary precision of the macro-texture of the surface. Segmentation of the range data allows for the separation of the dominant surface geometry from the surface characteristics or material properties of the surface. This reduces the dimensionality of the depth data so that the physical characteristics and descriptors of a patch of terrain can be easily compared, without the need to include the angles of the terrain. The segmentation of the data allows for more resolution in terrain classiﬁcation when identifying boundaries between terrain types. A tree based segmentation method is preferred so that boundaries in 3D space can be easily identiﬁed and greater detail can be applied when describing the boundary areas.

5.2.1 Segmentation

The data segmentation method utilised for finding the dominant geometry is a quad-tree based method and implements a Piece-Wise Multi-Linear (PWML) [111] approximation from the 3D data points to geometric surface planes is shown in figure 5.3. The PWML method allows for the boundaries of these planes to be identified automatically. The range data gathered by the sensor is registered together with the colour data,

126 5.2. Range Texture Analysis

(a) 3D point cloud (b) PWML segmentation Fig. 5.3: Automatic PWML segmentation of point cloud using τ = 5cm

and a subset of the range data that contains the useable data from the frame is chosen, the usable data consists of regions for which only small patches of missing data exists. Once the usable range data has been identiﬁed, any missing data in this subset is interpolated using a linear interpolation method. Linear interpolation is adequate for this approach, as the holes that potentially exist within the usable data are small by deﬁnition. Any holes that exist in the region that are larger, are instead removed from the environment model through the PWML approximation of the data.

The PWML approximation for 2D surfaces, evaluated on the 2D depth image, is deﬁned as follows,

fˆ(u, v) − f (u, v) < τ

∀ (u, v) ∈ Ω

fˆ(u, v)= ak · u · v + bk · u + ck · v + dk (5.1)

∀ (u, v) ∈ Ωk

N N {Ωk}k=1 / ∪ Ωk = Ω k=1

Given a function f (u, v), which is evaluated in the Ω ∈ R2 domain, this can be approximated by a Piece-Wise Multi-Linear function fˆ(u, v). The Ω domain can be N N ˆ partitioned into square regions {Ωk}k=1 / ∪ Ωk = Ω. The f (u, v) function is k=1 implemented in each region by a multi-linear expression fˆ(u, v) = ak · u · v + bk ·

127 Chapter 5. Terrain Perception using Non-semantic Range Data

u + ck · v + dk, with a set of parameters [ak, bk,ck,dk] describing the surface. The function fˆ(u, v) is calculated for each patch by minimising the error between the patch and the 3D data points. A PWML patch is a valid patch if fˆ(u, v) − f (u, v) is less than the threshold value τ, else the data set is sub-divided into smaller subsets where the same process repeats until the size of the patch is 8x8 or that a PWML patch can be fitted adequately. The PWML process allows for useful information about the terrain to be gathered, including the orientation and position of the patch, allowing for the information in each patch to be normalised. After a PWML patch has been fit to the depth data for a particular region, a matrix containing the residuals between the range data and the PWML patch is then generated. The residuals between the 3D point cloud and the PWML patch have been visualised from the top and side in figure 5.4, the colour of the dots corresponds to how far away a data point is from the plane. The data used to generate the figure is the depth data of a rock surface similar to Fig. 5.2(e).

(a) top view (b) side view Fig. 5.4: PWML patch with residuals formed from a rock surface data

The typical residual images are visualised for each terrain type in figure 5.5, showing that there are considerable differences between grass, rock and sand, but there is little difference between the road, dirt and gravel terrain types. These residual images are then used to generate the depth texture features. It is assumed that the spatial displacement between adjacent depth data points on

128 5.2. Range Texture Analysis

0.05

0.04

0.03

0.02

0.01 (a) grass (b) road (c) dirt 0

−0.01 residuals (m)

−0.02

−0.03

−0.04

(d) gravel (e) rock (f) sand −0.05 Fig. 5.5: Typical Depth residuals for each of the encountered terrain types

the top and bottom rows of the residual image is negligible because the perspective angle of the sensor is small. If, however, the perspective angle was shallower there would be a diﬀerence in the spatial displacement between the range data on the top and bottom rows which might lead to the terrain features that are calculated to be inconsistent over the same terrain types. This eﬀect is currently avoided in the proposed method by controlling the perspective angle so that spatial displacement throughout the image is similar.

5.2.2 Terrain Features

Previously, the terrain features that were developed from an environment were taken primarily from the colour and textural information of that scene using a camera, as well as some sparse information about the geometry of the environment using semantic spatial data, with the grayscale colour image used to analyse the texture on the surface [164]. The texture features developed by Haralick utilise the Gray-Level Co-occurrence Matrix (GLCM) method and is described in Section C.1.1. This approach can work for mobile robotics, however classiﬁcation using this method becomes erroneous when the lighting conditions on the terrain type vary signiﬁcantly from the conditions in the training dataset.

129 Chapter 5. Terrain Perception using Non-semantic Range Data

Range Features

The main problem with vision based terrain interpretation is that it is heavily reliant on either a constant or known lighting condition, and so does not work well under conditions where it is dark or when the image becomes overexposed. The colour of the terrain is an easily identifiable feature of a terrain type but becomes difficult to include as a feature, as the value for a terrain type varies widely due to variance in the lighting conditions as well as natural variation in the terrain colour, such as dry grass or different coloured tiles or dirt. The approach that is adopted to address this problem is to use depth measurements to identify texture in the non-semantic range data instead of using visual texture information. The non-semantic range texture data is generated by using the residual images from the previous section, so that each dominant surface in a scene has an accompanying series of texture features based on the depth data. In order to generate these range texture features, a GLCM is then generated from the residual images. The range GLCM is generated by discretising the residual images into 256 separate levels with 5cm variance about the surface, which coincides with the depth resolution of the Carmine sensor and is within the required resolution of the terrain macro-texture. By fixing the number of levels and the separation between levels, this allows for easier direct comparison between surfaces that have larger standard deviation on the surface to ones that have a surface that is very flat. The GLCM provides a set of features from the input data which can be used as a statistical feature for the terrain. The GLCM for each dominant surface segment is calculated using (C.1) with R(i, j) as the residual image for each segment, as calculated from the PWML approximation using (5.2). The residual image is then used to calculate a number of different GLCM features taken from Section C.1.1.

R(i, j)= fˆk (i, j) − f (i, j) (5.2) ∀(i, j) ∈ Ωk

The texture features that are used in the descriptor for each data point is the maximum value for the feature type over all the four primary orientation combinations as shown in equation (C.16), with the step distance d = 1. Additionally the standard

130 5.2. Range Texture Analysis deviation for each subregion of the surface patch is also used as a feature in the descriptor from (5.3). The standard deviation for the subregion is calculated from

(xmin,ymin) in a square subregion of size N × N.

x +N y +N 1 min min σ¯ = (R(i, j))2 (5.3) vN 2 u i=xmin j=ymin u X X t These spatial features have been chosen to maximise the classiﬁcation rate of the classiﬁers, with the features used being, energy, contrast, correlation, homogeneity, information measure of correlation 1 and 2, as well as the maximum probability. There are other statistical texture features that could be evaluated in order to have additional features of the terrain, however, the list of features shown above have the greatest level of distinction between terrain type groups, with other features providing redundant feature information.

Xrange = [¯σ, h1,range(k)] k =1 ... 7 (5.4)

Visual Features

The lighting conditions of the environment affects the perceived colour of the terrain, however, there are still some features that can be extracted from the data that are partially independent of the ambient conditions. These features can be used to improve the terrain classification results under reasonable lighting conditions. The features that have been used in this section are the GLCM texture properties as described in (C.2), (C.3), (C.4) and (C.5) with the data source being a grayscale conversion of the colour information of the region, the mean redr ¯, mean greeng ¯ and mean blue ¯b colour channels that were recorded are also used in the classification approach.

Xcolour = r,¯ g,¯ ¯b, h1,colour(k) k =1 ... 4 (5.5) Range and Visual Features

The augmentation of the colour and the non-semantic range features together further improves classiﬁcation results, with the descriptor for each point being Xhybrid. The

131 Chapter 5. Terrain Perception using Non-semantic Range Data results from the combination of both the depth and the colour information can provide a greater amount of separation between the diﬀerent terrain types for the cases where the visual data is valid.

Xhybrid = [Xrange,Xcolour] (5.6)

5.2.3 Classiﬁcation

The terrain classification approach detailed in this section employs a multi-variate classification fusion system that fuses the outputs of three different classifier methods together, so as to improve the classification accuracy of the method. The output of the fusion algorithm has been shown to generate results that are better than the best individual results when fused together [168]. The algorithm for the Weighted Majority Voting with Dominance (WMVD) method is presented in appendix C.2, the algorithm uses a combination of k-Nearest Neighbours, the Mahalanobis distance as well as a Decision Tree classifier, which are all detailed in Appendix C.

5.2.4 Results

Range Texture Features

The results of the terrain classification based exclusively on the non-semantic range data can be seen in Table 5.2, the classification method accuracy is 94.2%. These results are for the WMVD classification method and show that there is correlation between the different terrain types, namely road, dirt and gravel, as well as rocks and sand. The misclassification of sand and rocks can be attributed to the sand dunes seen in Fig. 5.7f, which shows that the dominant surface geometry of the sand dune was still present in the data. This is the case as the τ used to segment the range image into patches to represent the dominant geometry was too large. The road, dirt and gravel surfaces are very similar, as shown in Fig. 5.5. The individual classifier results can be seen in Table 5.3. The classifiers are successfully able to provide reasonable classification results for all 6 terrain types. The WMVD classifier shows that it is able to combine the results of the individual classifiers

132 5.2. Range Texture Analysis

Grass Road Dirt Rock Gravel Sand Grass 99.9 0 0 0 0 0 Road 0 87.6 8.2 0 4.1 0 Dirt 0 4.2 94.8 0 1.0 0 Rock 0 0 0 95.8 0 4.2 Gravel 0 2.0 2.0 0 96.1 0 Sand 0 0 0 9.3 0 90.7 Average Accuracy: 94.2% Table 5.2: Non-semantic range only WMVD Classification results together effectively so that a better overall terrain type classification can be achieved. The statistical distance metric, the Mahalanobis distance, performed the worst out of all of the classifiers. This result can be seen to originate from the heavily overlapping feature values for the different terrain types, as seen in Fig. 5.6.

0.95

0.9

0.85

0.8 grass rock dirt 0.75 road gravel sand 0.7 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

Fig. 5.6: Dominant spatial features showing separation between groups

The data that was used in performing the classiﬁcation only consisted of regions that had a PWML patch size of 32x32. This was due to the considerable non-linear relationship that exists between the size of the patch used and the texture features that the patch generated. The issue of the size of the region used to generate the texture

133 Chapter 5. Terrain Perception using Non-semantic Range Data

kNN Mahalanobis DT WMVD Grass 99.9 98.7 99.9 99.9 Road 89.9 76.9 82.3 87.6 Dirt 93.4 73.7 85.3 94.8 Gravel 92.1 86.8 78.5 95.8 Rock 90.7 73.7 87.3 96.1 Sand 92.3 62.5 90.1 90.7 Total 93.1 78.7 87.2 94.2 Table 5.3: True positive classification results properties comes from the number of elements that are required in order for the texture properties to be statistically significant so that the effects of any outlier areas in the data set can be reduced, as well as certain features specifically using the size of the region in the feature calculations.

Visual Terrain Features

Utilising the descriptor Xcolour which uses the colour and visual texture features from section 5.2.2, the total accuracy for the classification is 92.3% shown in Table 5.6. This results reveals that using non-semantic range only features can be as effective or better at terrain type classification than visual texture features.

Grass Road Dirt Rock Gravel Sand Grass 93.0 0 3.5 0 3.5 0 Road 2.8 90.3 4.1 0 2.8 0 Dirt 2.8 0 93.4 0 3.8 0 Rock 0 0 2.8 90.6 6.6 0 Gravel 5.0 6.1 0 0 88.9 0 Sand 0 2.2 0 0 0 97.8 Average Accuracy: 92.3% Table 5.4: Colour only WMVD Classiﬁcation results

The Xcolour descriptors that were used in the classiﬁcation as with the range only classiﬁcation results utilised only the descriptors from dominant surfaces that had a region of interest size of 32. This is once again to remove the non-linear relationship that exists between the region size and the texture features.

134 5.2. Range Texture Analysis

Range and Colour Terrain Features

The classiﬁcation results when using the descriptor Xcombo improves to 98.9% classiﬁcation accuracy, as shown in Table 5.5. These features are only partially susceptible to the ambient lighting conditions and so can be used when the lighting conditions are not in their extremes. The results of using both the non-semantic range and the visual feature descriptors together can be seen in Table 5.6

Grass Road Dirt Rock Gravel Sand Grass 99.9 0 0 0 0 0 Road 0 98.0 0 0 2.0 0 Dirt 0 0 99.0 0 1.0 0 Rock 0 0 0 99.0 1.0 0 Gravel 0 1.9 0 0 98.1 0 Sand 0 0 0 0 0 99.9 Average Accuracy: 98.9% Table 5.5: Non-semantic range and colour data WMVD Classiﬁcation results

kNN Mahalanobis DT WMVD Grass 99.9 99.9 99.0 99.9 Road 95.7 94.1 97.5 97.7 Dirt 99.9 87.5 91.9 96.0 Gravel 99.9 97.9 82.4 99.9 Rock 95.8 94.4 95.2 99.9 Sand 99.9 99.9 97.9 99.9 Total 98.6 95.6 94.0 99.0 Table 5.6: True positive classiﬁcation results for the combination descriptor of both non-semantic range and colour features

5.2.5 Discussion

There are a number of beneﬁts that are gained by being able to classify terrain types using only non-semantic range data, so that for environments where visual data is poor or invalid, terrain prediction can be made through the use of a diﬀerent sensing technology. The augmentation of both range and visual based methods allows for

135 Chapter 5. Terrain Perception using Non-semantic Range Data more robust terrain classification that can maintain reliability throughout a number of different operating conditions. However, the current implementation and the way that the data is first segmented based on the dominant surface geometry means that the method is limited to be able to classify different terrain only when they are on distinctly different terrain geometries.

The data to be classified was already separated into one of the six distinct terrain types and for each image a number of data points were taking from within the frame of the image. The data that was used were only the PWML patches that had a size of 32x32, this requirement was used as it was observed that there was a non-linear relationship between the patch size and the texture features. An additional feature was proposed that included the size of the patch to attempt to take into account the relationship between the patch size and the feature value. However, the outcome was similar to not having the additional patch size as the feature, as the features by themselves were already very difficult to get an accurate classification from. The addition of the extra feature did not extract any additional usable information from the small patches.

Improvements in both the colour and combination based classification results can be achieved through the use of better colour features. The features that were used in this work included the means of the raw red, green and blue colour channels, however this colourspace has been shown to not be very robust to the variations in the illumination of the scene from the ambient light source. Alternative colour spaces that could be used would be the Hue-Saturation-Intensity (HSV) colour space, which can give good classification results under different lighting condition [169]. The dataset that is used consists of only one lighting condition, and so the HSV colour space would provide a similar result to the RGB colour space.

The data sets that were used in this section consisted of only good lighting conditions, and so the colour based classification results performed well. The depth camera that is being used restricts the ability to gather more data sets that are of different lighting conditions, especially in brightly lit environments, where there is a big contrast between light and shadow areas. Data sets that contain significant differences in lighting across the image would be of greater importance in showing the performance

136 5.3. Extended Range Texture Analysis of the non-semantic range based classification method. To reduce the uncertainty in the system model by better defining the uncertainty in the terrain model requires a method of assigning a classification confidence onto the classification results. Through the use of a classification confidence metric, it is possible to provide an estimate of the frictional characteristics of all regions regardless of the confidence in the classification. The current method does not provide a classification confidence that can be used in this way.

5.2.6 Summary

This chapter has presented a method for classifying terrain types based on non-semantic range data. The terrain segmentation was carried out using a Piece-Wise Multi-Linear approximation to a subset of 3D point cloud data points which allows for the separation of the dominant surface geometry from the surface macro/micro-textures. The PWML patches that are found from this method are used to create residual images, which were then used in a new texture feature application using the range data. Three separate classifiers are then implemented to classify the six terrain types with a Weighted Majority Voting with Dominance combinational classifier combining the classification results of the three individual classifiers to yield better classification results. Through the use of non-semantic range texture features an overall terrain classification accuracy of 94.2% is achieved on the six different terrain types. With each terrain type a friction coefficient can be assigned to each patch via a lookup table, allowing for improved controllability over a surface. The terrain classification accuracy can be further improved by including four additional colour texture features which results in a classification accuracy of 98.9%, and colour only classification has an accuracy of 92.3%.

5.3 Extended Range Texture Analysis

The purpose of developing a terrain model of the environment is to be able to reduce the uncertainties in the behaviour that a vehicle exhibits whilst traversing through

137 Chapter 5. Terrain Perception using Non-semantic Range Data the environment, so that an AGV can perform more accurate path tracking as well as improved path planning. This can be achieved by incorporating the observed dominant surface geometry as well as the characteristic frictional coeﬃcients of the environment into the terrain model.

The RTA method of terrain classification from the previous section is able to provide a good terrain model that consists of the dominant surface geometry and the classified coefficients of friction. This is achieved by associating the classified terrain types to existing known frictional coefficients for that terrain type. However, the RTA method only segments the terrain based on the dominant surface geometry, and not on terrain type changes. The texture features that are generated in the RTA method are inconsistent between the different segment sizes as well as not providing any level of confidence in the classified terrain types.

To address the limitation in the RTA method, a new method has been developed that evaluates the non-semantic range information differently so that it can be used to segment the terrain. A method of processing the range data to provide an adequate feature map for use in terrain type classification and boundary identification is to perform a pixel by pixel implementation of the RTA method called Extended Range Texture Analysis (ERTA). Where each range data point is assumed to be the center of a fixed size region of interest, instead of segmenting a region based on its dominant geometry and then evaluating the texture features for that region as a whole. Using this region of interest to generate the range texture features for each range data point provides consistency in the texture features generated as they all use the same region size. The scene is then classified based on the range texture features and segmentation is then performed with both the dominant geometry of the surface as well as the identified terrain types.

For validating the approach, a platform retrofitted with 3D perception capabilities was employed. The method can be utilised with many different 3D sensors, with the sensor used for this experiment being a structured light 3D camera. The range images of the different terrain types; grass, artificial turf, gravel, tile and concrete, shown in Fig. 5.7, were captured with the 3D camera facing downwards at a shallow angle, at a distance of 500mm from the ground, with the 3D camera, a PrimeSense Carmine 1.09,

138 5.3. Extended Range Texture Analysis

(a) concrete (b) artiﬁcial turf

(e) tile Fig. 5.7: Typical depth image of the diﬀerent terrain types

mounted on the robotic platform, as shown in the schematic in Fig. 5.8. The range images were taken while the robot platform was stationary, and in areas that were not in direct sunlight as this is a limitation of the sensor used with the lighting conditions being held constant for all data sets.

The robotic platform, shown in Fig. 5.9, was provided by Jayantha Katupitiya from

139 Chapter 5. Terrain Perception using Non-semantic Range Data the School of Mechanical and Manufacturing Engineering, from the University of New South Wales (UNSW) Australia. Additions to the platform have been made in order to mount various sensors The software that was used to record the datasets is Possum, which was used in [170], this software has been provided by Jose Guivant of the School of Mechanical and Manufacturing Engineering, UNSW Australia.

Robotic Platform Carmine

500mm

Fig. 5.8: Schematic of the experimental setup showing the structured light camera mounted on a moving platform

3D-LiDAR GPS

Carmine

IMU LiDAR

Fig. 5.9: Actual experimental platform with the structured light camera mounted at the front of the vehicle and additional sensors, the sensors used are highlighted

140 5.3. Extended Range Texture Analysis

5.3.1 Feature Image Generation

In order to generate a feature image of the scene, an important step is to generate a residual image from the range data. There are a number of different approaches for generating the residual image of the scene, these include the use of a PWML geometric estimation, high-pass spatial filtering, and locally approximated residual patches. The aim of all the residual image methods is to provide an image of the scene that reduces the amount of spatial dependence on the derived range texture features as well as to provide a method that can be applied on different terrain types using a generalised approach. The different residual image generation techniques contain different advantages and disadvantages, however, it is not clear which residual method is the best method for generation of the range texture features. To be able to determine which method is best, the three approaches will be evaluated with respect to the effects they have on the terrain classification results.

Residual Image

Piece-Wise Multi-Linear geometry A number of steps are required to generate the residual image of a scene from the non-semantic range data using the PWML dominant surface geometry estimation, these steps are shown in Fig. 5.10. These steps are as follows: perform PWML geometric estimation on the range data to get the dominant surfaces for the environment, followed by taking the difference between the PWML plane of best fit and the corresponding range data for each PWML patch, which are then used to generate a residual matrix R, as defined in (5.7). The dominant surface corrected residual image is then ready to be used to develop the range texture features.

ˆ R(i, j)= fk (i, j) − f (i, j) (5.7) ∀(i, j) ∈ Ωk

This method of residual image generation is useful as it does not require a priori information about the spatial frequencies of the terrain. The dominant surface geometry segmentation that is performed on the range data is already utilised to estimate the

141 Chapter 5. Terrain Perception using Non-semantic Range Data

(a) colour image (b) depth image

geometry of the terrain. Requiring only an additional step of calculating the diﬀerence between the PWML estimation and the range data for a scene.

The parameter that is used for generating the residual image from the depth image through the use of the PWML patch estimation is the threshold value τ. This parameter determines the maximum error between the PWML patch and the depth data in that region, and can be chosen to try to remove the effects of discontinuities that may exist in the terrain geometry. The smaller the value of τ, the smaller the allowable error is inside a residual patch, which allows for smaller discontinuities to be identified and removed. However, when τ is made too small, it begins to segment the terrain much more aggressively, which leads to inaccurate residual image generation. The effect of the value τ has on the removal of errors in the residual can be seen in Fig. 5.11.

Locally deﬁned patch of best ﬁt

This method generates the patch of best ﬁt centered about a speciﬁc pixel and includes a Region of Interest (RoI) around the pixel that is used to calculate the non-semantic features. The local region that is used to generate the GLCM for that

142 5.3. Extended Range Texture Analysis

(a) τ = 100mm (b) τ = 50mm

pixel is fitted with a PWML patch of best fit, regardless of the maximum error τ that the patch contains. This patch is used to calculate the residuals for the current region of interest. This local patch data is determined purely on the current region of interest, giving rise to the plane estimation algorithm providing poor estimations as it approaches changes in the surface geometry. This results in very poor generation of the residual image with large regions of texture properties being invalid around where there is a change in the surface geometry. The effects of this is that near areas where there is more than one dominant surface geometry, the classification results from these areas are unreliable. The benefit of using the PWML segmentation to determine the dominant surface geometry for the entire image over just taking the region of interest and determining the plane of best fit is that this approach allows for differences in depth caused by dominant geometry changes to be removed prior to developing the residual image. High-pass Filtered Image

143 Chapter 5. Terrain Perception using Non-semantic Range Data

Another method for generating a residual image of a scene is to apply a high-pass filter to the image. The high-pass filter is tuned so that the high frequency spatial information in the scene is conserved whilst removing the low spatial frequency information such as slow undulations and offsets in the environment. The residual image that is generated is similar to the residual image generated by the PWML method, except that it is capable of providing a better residual when there is a step in the depth data. There is still errors that are found in the filtered residual image when a step is encountered, with the effect being an increased value of the residual near to the boundary.

The high-pass spatial filter response needs to be tuned for the expected spatial frequencies that are encountered. This is achieved by fitting the spatial filter response to the output of a Piece-Wise Multi-Linear (PWML) surface estimation [111]. The residual image that is generated from this method can be seen in Fig. 5.12(b), where the frequency that is chosen to filter the original image is found by reducing the difference between the frequency filtered residual image and the residual image generated from the PWML method on a similar scene. This method of choosing the cut-off frequency that are used is dependent on the PWML method and the frequencies that are in the scene.

If the PWML method is unable to provide a good representation of the dominant surface geometry of the terrain, then the residual image that it generates will be a poor representation of the terrain and, thus, the frequency filtered residual image is a bad representation of the environment. It is therefore necessary to choose a scene to use that has very little changes in the dominant surface geometry so that the PWML residual image gives a good response. This means that the frequencies that are chosen for the high-pass filter may not include enough of the frequencies to adequately remove discrete steps in the spatial data. The effects of using both simple and complicated scenes to generate PWML based residual images to tune the high-pass filter are shown with a typical scene for both simple and complicated cases in Fig. 5.12. The large range discontinuities in the scene shown in Fig. 5.12(d) show that, for a filter that is tuned for a planar surface, rounding in the residual steps within the scene is observed.

144 5.3. Extended Range Texture Analysis

(a) simple surface colour (b) simple surface residual

Feature Space Images

The feature space images of the terrain are then calculated from the residual image R that have been previously generated, with each pixel of the feature image calculated by using the different feature types of GLCM, Gabor filters and the radially averaged PSD. The features that can be extracted from the local residual image and the raw depth data include the standard deviation (5.8) of the error, which is calculated for each pixel of the depth image based on RoI associated to that pixel. The RoI surrounding a pixel is defined in this implementation as being a square area centered about the pixel, with the length of the sides of the square being W . The value of W can be changed so that a greater region about the pixel can be included in the calculation. The resulting feature images from each of the three different techniques can be seen in Fig. 5.13, where a typical residual image is shown as well as the typical output residual images from the GLCM textures, the Gabor filter responses and the Radial Power Spectral Density features. The feature images that are shown are only indicative of only one of the many feature dimensions.

145 Chapter 5. Terrain Perception using Non-semantic Range Data

(a) residual image (b) GLCM energy image

(c) Gabor gµ(1, 1) image (d) RPSD P (2) image Fig. 5.13: Images showing a typical residual and the corresponding non-semantic feature images

146 5.3. Extended Range Texture Analysis

The additional effect of the RoI size that is used, is that it acts to smooth out the transition between terrain types as well as possible spurious terrain values that are present in the environment. The effect of this is that classification of a single pixel value can be guaranteed to be related to that of its neighbours, which means that at low scales terrain type classification will result in consistent terrain types.

As found in the previous section, there is a significant relationship between the size of the RoI and the quality of the texture features. To be able to account for the effects of choosing a specific RoI size to generate the depth features, it is necessary to fix the RoI size so that the features can be kept consistent. It is not obvious what RoI size is optimal, and so it is necessary to establish this based on two objectives: the effects that the RoI size has on the classification results, and the effect the RoI size has on misclassification near to the boundary between two or more terrain types. Once the relationship between the RoI and these objectives are found, the optimal RoI for both objectives can be chosen.

Spatial Features

The spatial features that are used in classifying the scene into the different terrain types are defined for each of the three different spatial feature types: GLCM textures, Gabor filter outputs and radially averaged PSD. In addition to the different spatial features that are gathered the standard deviation of the region of interest is also found, shown in (5.8).

1 W W W W σ¯ = (R(i − , j − ))2 (5.8) vW 2 2 2 u i=1 j=1 u X X t

The GLCM texture features that are generated using the range residual image for a ﬁxed region of interest size that are quantised into 32 diﬀerent levels. These levels are chosen so that the maximum and minimum levels correspond to 5mm above the dominant surface and 5mm below the dominant surface.

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XT,range = [¯σ,

t¯1,range(k), v1,range(k),s1,range(k),

t¯2,range(k), v2,range(k),s2,range(k)] k =1 ... 7 (5.9)

The Gabor ﬁlter based spatial features that were generated through the use of each residual matrix method can be seen in (5.10).

XG,range = [¯σ,grange,µ(a, k),grange,σ(a, k),grange,span(a, k)] k =1 ... 3, a =1 ... 8 (5.10)

The output from the Radially averaged Power Spectral Density (RPSD) as deﬁned in (C.18). The output from the power spectral density is taken and placed into 16 diﬀerent bins that the spatial frequency information is placed into, shown in (5.11).

XP,range = [¯σ,Prange(1),Prange(2), ··· ,Prange(16)] (5.11)

Colour Features

The colour features that are used in scene classification are derived from the colour image of the scene in the RGB colour-space. This is the default colour-space of the camera that is used in the data acquisition. The features that are used in classification are based on the same feature extract methods as the spatial features, namely GLCM textures, gabor filter features and radially averaged power spectral density. The GLCM textures are generated by converting the colour image to gray-scale and then quantising the data into 256 different levels and then following the same methodology that was used by the spatial information to get the feature information. The colour texture feature data that is used is shown in (5.12).

148 5.3. Extended Range Texture Analysis

ˆ XT,colour = r,ˆ g,ˆ b, h t¯1,colour(k), v1,colour(k),s1,colour(k),

t¯2,colour(k), v2,colour(k),s2,colour(k)] k =1 ... 7 (5.12)

The gabor filter based colour features are found using the same gabor filter bank as the spatial features, however they use the grayscale converted colour data. The descriptor vector of features that are used for the Gabor filter based features can be seen in (5.13)

XG,colour = r,ˆ g,ˆ ˆb, grange,µ(a, k),grange,σ(a, k),grange,span(a, k) k =1 ... 3, a =1 ... 8 h i (5.13)

Similarly the RPSD features for the colour data are found in a similar way to the depth based ones, with the frequency responses being divided into 17 frequency bins in the spatial frequency domain and with the inclusion of the mean red, green and blue values this forms the descriptor as shown in (5.14).

XP,colour = r,ˆ g,ˆ ˆb,Pc(1),Pc(2), ··· ,Pc(16) (5.14) h i Spatial and Colour Feature Combination

The features that are combined together for terrain type classification purposes are the depth and colour features for each of the three different feature types: GLCM texture features, Gabor filter features and the RPSD spatial frequency features. The three different combinations can be seen in equations (5.15),(5.16) and (5.17).

XT,hybrid = [XT,range,XT,colour] (5.15)

XG,hybrid = [XG,range,XG,colour] (5.16)

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XP,hybrid = [XP,range,XP,colour] (5.17)

Dimensionality Reduction

The feature space vectors that are generated using the range and colour data: Haralick textures, Gabor Filter and Radial Power Spectral Density descriptors are of a very high dimension. This high dimensionality in the feature space means that generating the mean and covariance for each region in order to classify it becomes costly. In order to both increase the reliability of the classiﬁcation results and to reduce the computational time, dimensionality reduction should be carried out. Methods of dimensionality reduction exist with the aim of reducing the dimensions of the feature space down so that they can provide a better basis vector for classiﬁcation methods. These methods include Principal Component Analysis (PCA) as well as Multiple Discriminant Analysis (MDA) and are described in Section C.3.

5.3.2 Segmentation

Classiﬁcation

This section discusses the variety of different statistical methods as well as the non-parametric methods that have been explored with regards to classification of the terrain types into their desired groups. For scene classification and segmentation there are two way of approaching this problem: top-down or bottom-up. The top-down approach classifies the entire region and then splits, and continues to split until all components are classified the same. The bottom-up approach classifies all of the base level components and then attempts to merge components together that are classified into the same terrain type. There are advantages and disadvantages for both approaches, with the main difference between the approaches being the types of classifiers that can be used to perform the classification. In the top-down approach, it is not a single descriptor that is being classified but a collection of descriptors, thus a statistical classification method is required. Alternatively for the bottom-up approach, the base pixel values can only

150 5.3. Extended Range Texture Analysis be classiﬁed through the use of non-statistical methods, such as the previously used classiﬁer in the RTA method, as described in Section 5.2.

For the bottom-up approach it is therefore necessary to be able to classify in the order of hundreds of thousands of pixel values per image, which is a large amount of points to classify for any method, even though if the resolution is small. The types of classification that can be used in this case are supervised classifiers, such as the non-parametric methods of k-Nearest Neighbours, Support Vector Machines (SVM) and Decision Trees along with parametric methods of Artificial Neural Networks and Naive Bayes. These methods are all used extensively in the current literature and are able to perform well, however, their main disadvantage is that for such a large amount of data points it becomes computationally expensive with conventional implementation methods.

Conversely, in the top-down approach the mean and covariance are required to be recalculated for each new region. This is also costly, but these recalculations are only required when a region needs to be split. The number of elements that are generally required to represent a scene using this manner are usually only of the order of a few hundred instead of the millions required in the previous approach.

The method of classiﬁcation that is used in the top-down approach is to classify based on the lowest statistical distance of the current region to the class distributions. A statistical distance is used as there is a large amount of data points available in each region and by calculating the distribution of the region it is less likely to be inﬂuenced by outliers.

Statistical Distances

There are a number of different statistical distances that can be used, many metrics measure the distance directly, such as the Bhattacharyya distance, whereas others measure the divergence between two distributions, both the asymmetric and symmetric Kullback-Leibler divergences and the Jensen-Shannon divergence. The Bhattacharyya distance is a measure of the difference between a distribution P and a target distribution Q. The distance was developed for use in determining the relationship between different multivariate distributions of data, with the definition of the distance shown in (5.18), which is taken from [171]. The equation depends on the means of the distribution P

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and a target distribution Q, µP and µQ, as well as the covariances of the distributions

ΣP and ΣQ,

1 T −1 1 |Σ| db = µQ − µP Σ µQ − µP + ln (5.18) 8 2 |ΣP |.|ΣQ|! 1 p where Σ = 2 (ΣP + ΣQ), which represents the combination of the covariances of the distributions P and Q. The Bhattacharyya distance as calculated from (5.18) relies on an accurate calculation of the determinant of the covariance matrices Σ, ΣP and ΣQ. For large matrix dimensions the determinant becomes difficult to calculate and can suffer from numerical precision limitations as well as being computationally expensive. There are two potential optimisations that can be used for improving the Bhattacharyya distance to avoid this effect, these are provided from [172, pp. 455-458]. The first is to remove the influence of the determinants as in the first term of (5.19), which makes the assumption of dominance in the difference of the means, and the second optimisation is when there is dominance in the difference of covariances, shown in the second term of (5.19)[173].

N 2 1 T −1 1 ul 1+ λl 1 do = µQ − µP Σ µQ − µP + + ln( ) − lnλl (5.19) 8 2 2(1 + λl) 2 2 l=1 X −1 Where ul and λl are the lth eigenvector and eigenvalue of the matrix ΣQ ΣP , where the eigenvalues are arranged in descending order along with their corresponding eigenvectors and N is the dimension of the mean vector µP . Another metric for statistical distributions is the divergence between distributions, where these metrics quantify the difference between two different distributions. There are a few different divergence metrics, with the principal divergence being the KullbackLeibler (KL) divergence, which is expressed in the generalised form shown in (5.20) and generates both an asymmetric and symmetric divergence as well as the Jensen-Shannon (JS) divergence. The asymmetric KL divergence can be seen in (5.21) and the symmetric KL divergence is seen in (5.22)

inf p(x) d (P, Q)= p(x)ln dx, (5.20) kl q(x) Z− inf 152 5.3. Extended Range Texture Analysis

where p and q are used to represent the probability densities of P and Q respectively.

1 |Σ | T d (P, Q)= log( Q − d + tr Σ−1Σ + µ − µ Σ−1 µ − µ (5.21) kl 2 |Σ | Q P Q P Q Q P P

d (P, Q)+ d (Q, P ) d = kl kl (5.22) kls 2

with dkl representing the asymmetric KL divergence and dkls being the symmetric KL divergence, which is used in the distance evaluation. The Jensen-Shannon 1 divergence is shown in (5.23), where C = 2 (P + Q) is the average of both P and Q distributions.

d (P,C)+ d (Q, C) d = kl kl (5.23) js 2 The main issue in the use of these statistical distance measures is that they require enough data points to be able to generate a realistic distribution for the region. It is therefore necessary to limit the minimum size of each region to guarantee that each region contains enough information. Another disadvantage of using the statistical distances, is that the distributions of the base classes are not equally spaced apart. Classes that have similar features are closer in the feature space than the other classes and may overlap in regions. This limitation does not effect the classification results as the full distribution is used during the classification, but it set up biases between similar classes.

Conﬁdence Metric

In order to provide an accurate representation of a terrain type in a scene an additional source of information can be generated which provides a measure of confidence in a classification result. The uncertainties that exist with the terrain model are not just limited to correctly classifying the type of terrain in order assign a coefficient of friction, but also how confident the classifier is that a particular terrain type classification is correct. This additional metric allows for the uncertainty in a classified surface to be taken into account in the terrain model.

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There are a number of different confidence metrics that have been investigated, including: relative inverse distance, weighted confidence metric as well as a comparative distance [174]. The inverse distance is meant to give a confidence metric that relates to the classified class and how close that classified terrain is to the other terrain types, this metric can be calculated as shown in (5.24).

N d (i) χ = i m (5.24) min(d ) P m

Where dm is a vector of statistical distances m to all current terrain classes N. With the m statistical distances as detailed in (5.18 - 5.23).

Algorithm

The terrain classification method utilises the transformed feature space images that were generated in the previous section, using either the PCA or the MDA transformation methods. The feature space image is then used to calculate the distances between a subregion of the image and the available terrain classes. The subregions are classified through the use of the Bayesian distance metrics as described in the previous section, with the classifier selecting the class with the lowest distance for that region. The minimum size of the segments used for classification is 8 × 8, as this amount provides enough data so that accuracy in the mean and covariance can still be maintained [175]. If the size of the segments are too small, then there is only a small amount of data to draw from to create an accurate mean vector µk and covariance matrix Σk. This can be seen in the algorithm as limiting the parent node to a size of no smaller than 16×16, which guarantees that the children of that node will be at least 8 × 8. The proposed Algorithm 1, shown below, is the method by which the feature space image is segmented and classified. The resulting terrain model T contains the classification results from the implementation of this algorithm. The method is applied over the generated feature space image, where the frame of the feature space is used to constrict the splitting of the resulting quad-tree. For each classified parent and children subregions the classification confidence is calculated. This confidence metric is then used to distinguish between confidently

154 5.3. Extended Range Texture Analysis classified terrain areas and not confident classified areas. Areas where the classification confidence is less than a specified threshold χth are left unclassified. This method is taken so that for areas where the terrain classification is not confident that although it can be classified, this classification has a large probability of being incorrect. The two main objectives from this method is to be able to reduce the uncertainties of the model by providing an accurate coefficient of friction and to also provide the boundaries of this coefficient of friction. For the cases where it is not classified, it is far better for the information given to the controller to have a comparatively larger uncertainty, than it is for the controller to be given incorrect information about the environment.

Algorithm 1 Segment and Classify the Feature Space Image 1: Calculate the feature space image F (∗) 2: set stack queue Q to empty 3: Push F onto Q 4: while Q is not empty do 5: Pop parent subregion p from Q 6: Calculate children subregion vector c from p 7: if p is partially in frame then 8: if p is not completely in frame and pwidth > 16 then 9: Push children subregion c on Q 10: else 11: Calculate µ and Σ for p and c 12: for all Class types do 13: Evaluate the distance dm for n and c 14: end for 15: Classify p and c using dm to each class 16: if pclass =6 cclass or |cclass| > 1 then 17: Push children subregion c vector onto Q 18: else 19: calculate conﬁdence χ 20: Set Tclass∀(i, j) ∈ p to pclass 21: Set Tconfidence∀(i, j) ∈ p to χ 22: end if 23: end if 24: end if 25: end while ∗ computationally expensive calculation

where m, is a statistical distance, b, o, kl, kls or js and pclass is the classiﬁed class of the parent node, and similarly cclass is the classiﬁed class of the child nodes

155 Chapter 5. Terrain Perception using Non-semantic Range Data

5.3.3 Stationary Platform Results

The results that were gathered are generated from a series of terrain scenes that were collected using the setup shown in figure 5.8 as described previously. With the same set of data being used for all of the different cases. The data is labelled depending on the terrain type that is present so that a ground truth can be established for assessing classification accuracy. The terrain scenes consist of single and multiple terrain types, and contain both range and colour information.

Non-semantic Range only

To evaluate the classification results it is necessary to observe the relationships between the classification accuracy and the different variables that can be altered for the terrain classification. The variables that are being examined are the type of texture descriptors, region of interest sizes, dimension reduction method, residual image extraction method as well as the type of statistical distance that is employed. The effects on the classification accuracy as well as computation time are examined for the different variable combinations. The results that are shown are for a confidence threshold of zero, indicating that all classifications are accepted, regardless of the confidence. The dimension reduction methods can be compared against each other directly and can be seen in Figure 5.14, the blue bars represent the MDA classification results, with the red bars representing the PCA results. The classification accuracies are also displayed as a function of the statistical distances, shown along the bottom axis. The classification accuracies are found by averaging the accuracies found for all regions of interest sizes, residual image extraction methods and all descriptors; including the colour and combination descriptors. It is clear in the comparison that the MDA method outperforms the PCA method, with the majority of statistical distances showing a classification accuracy reduction of around 10%. Further evidence for this can be seen in Figure 5.15, as the comparison between the MDA and PCA accuracies together with the classification accuracies from the different descriptors; GLCM, Gabor and RPSD. The figure shows only one combination of the extraction and classification techniques, being the windowed residual image and the use of the Bhattacharyya distance to classify. This combination outperforms the

156 5.3. Extended Range Texture Analysis

Classification Accuracy for MDA vs PCA over the different statistical distances 1

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0 db do dkl djs dkls Fig. 5.14: Difference between MDA and PCA dimensionality reduction on the average classification accuracy of all RoI sizes, extraction methods and all descriptors other combinations, with the behaviour of the classification accuracy over the region of interest sizes being similar to that of Figure 5.15. The relationship between the classification accuracy and the region of interest size seems to indicate a slight increase as the region of interest size increases, however this effect varies considerably.

The relationship between the classification accuracy, the statistical distances as well as the residual image extraction techniques can be seen in Figure 5.16. The results are gathered for an average over the different region of interest sizes for only the XT,range descriptor with a confidence threshold of zero. The blue bars represent the high-pass extraction technique, the green bars represent the PWML extraction technique and the red bars are the windowed residual results. It is clear that for this combination that the accuracy of the PWML results are not as good as the frequency or windowed results, and that the windowed results give slightly better classification accuracies.

The diﬀerent statistical distances that are used show that the Bhattacharyya distance, asymmetric KL, symmetric KL and JS divergence based classiﬁers gives the best results for the average of the region of interest sizes for both the high-pass and windowed residual technique. Whereas the optimised Bhattacharyya distance performs

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Classification Accuracy using Windowed Residual and d b 1

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Fig. 5.15: Classification results for the three range descriptors using a Windowed Residual and the Bhattacharrya distance metric poorly for all of the extraction techniques. The Bhattacharyya distance, however, does perform slightly better than the other techniques over the majority of the different combinations of classifier parameters. So far the classification accuracy for the range only classification has been examined without utilising confidence thresholding. The best classification results for the range only descriptors coming from the windowed residual, using the Bhattacharyya classifier and MDA, for a region of interest size of W = 96 is 87.9 % . This result can be further enhanced by using different confidence threshold values. The relationship between the confidence threshold and the classification accuracy is seen in Fig. 5.17, the figure also shows the relationship between the confidence threshold and the percentage of the scene that is classified. As the confidence threshold becomes more restrictive the classification accuracy approaches 100% accuracy, however more of the scene is left unclassified. A compromise exists between the classification accuracy and the percentage scene classification, with the aim of this work to reduce uncertainty in the terrain. It is necessary sometimes to remove potential classification candidates if there is not enough confidence in their classification accuracy. A good compromise for the classification

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Classification Accuracy for different Statistical Distance 1

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χth = 60% is used, this compromise results in a classification accuracy of 91.4%. The misclassification of the different terrain types can be seen to consist of terrain types that have similar features, the tile and concrete, and the artificial turf and gravel surfaces all consist of similar sized features and so these provide greater sources of error. The grass terrain type was the most varied type and so there were a large number of false positives, and few false negatives.

concrete artiﬁcial turf grass gravel tile precision concrete 0.888 0.000 0.000 0.000 0.112 0.888 artiﬁcial turf 0.000 1.000 0.000 0.000 0.000 1.000 grass 0.060 0.049 0.891 0.000 0.000 0.891 gravel 0.000 0.021 0.000 0.962 0.016 0.962 tile 0.165 0.000 0.000 0.004 0.830 0.830 recall 0.823 0.943 1.000 0.996 0.858 0.914

Table 5.7: Confusion Matrix for XT,range,W = 96 with MDA and windowed residual and χth = 60%

The typical scene classiﬁcation using this method can be seen in Fig. 5.18, which

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Classification Accuracy and Confidence Threshold 1 1 0.95 0.9 0.85 0.8 0.8

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Fig. 5.17: Classification accuracy and percentage scene classification as the confidence threshold changes has the same configuration as the data shown in the confusion matrix in Table 5.7.

The configuration includes; descriptor vector XT,range, using the windowed residual extraction technique, with MDA dimensionality reduction and using the Bhattacharyya distance as the classifier, with a confidence threshold of 40%. These figures show that the algorithm is able to adequately classify the terrain as well as segment the terrain classification such that an unknown boundary that exists between different terrain types can be extracted. The environment that is present in these figures consists of gravel and tile terrain types with a boundary between the two that splits the image into two distinct sections. The colour image of the environment is shown as well as the boundary and segmentation results from the visual boundary identification method.

Colour data only

The classification accuracy for the colour based descriptors is different from that of the range only data. The difference can be seen in Fig. 5.19, where previously the descriptor that provided the best classification results was that of the GLCM textures, whereas now the best descriptor is that of the RPSD, as shown with the red solid line,

160 5.3. Extended Range Texture Analysis

(a) Ground Truth (b) Terrain Classiﬁcation overlay

none concrete art. turf grass gravel tile

Fig. 5.18: Example terrain classification with two different terrain types, on the right is concrete and on the left is artificial turf areas that are grey are not confident enough in their initial classification closely followed by the Gabor filter. The difference between the classification accuracy of MDA and PCA continues to show that MDA provides the best classification results.

Classification Accuracy using High−pass Filter and d b 1

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Fig. 5.19: Classiﬁcation results for the three colour descriptors

161 Chapter 5. Terrain Perception using Non-semantic Range Data

The effects of the statistical distances metrics on the colour only classification accuracy for the different statistical distances follows a similar pattern to the results of the range only classification. The results in Fig. 5.20, shows that the Bhattacharyya distance continues to provide better classification results than the other statistical distances, with the windowed residual method also providing the better classification accuracy.

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The relationship between the colour descriptor with the best classiﬁcation accuracy

TF,colour and the confidence threshold is similar to the range based descriptors. Figure 5.21 shows this relationship as well as the effect of the confidence threshold on the percentage scene classification. The configuration for the results shown in the figure are; windowed residual, Bhattacharyya distance classifier, a region of interest size of W = 96 and MDA dimension reduction. Without the effect of the confidence threshold the classification accuracy of this configuration is 79.0%, however by employing the use of a confidence threshold of χth = 50% the classification accuracy increases to 91.4%.

The confusion matrix for the same conﬁguration at a conﬁdence threshold of χth =

162 5.3. Extended Range Texture Analysis

Classification Accuracy and Confidence Threshold 1 1 0.95 0.9 0.85 0.8 0.8 0.75

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Fig. 5.21: Classification accuracy and percentage scene classification as the confidence threshold changes

50% is shown in Table 5.8, where there is still strong correlation between the artificial turf and grass, as well as between gravel and tile. This strong correlation was expected as both terrain pairs were selected to be able to test the difference between the range and the colour based classification, with the pairs being both similar colours as well as similar visual textures.

concrete artiﬁcial turf grass gravel tile precision concrete 0.953 0.000 0.036 0.000 0.011 0.953 artiﬁcial turf 0.000 1.000 0.000 0.000 0.000 1.000 grass 0.001 0.147 0.852 0.000 0.000 0.852 gravel 0.000 0.023 0.000 0.849 0.129 0.849 tile 0.000 0.000 0.000 0.000 1.000 1.000 recall 0.999 0.793 0.960 1.000 0.757 0.914

Table 5.8: Confusion Matrix for XF,colour,W = 96 with MDA and windowed residual and χth = 50%

In comparison to the range only classiﬁcation results the colour only results are slightly less accurate without utilising the conﬁdence threshold. With the best descriptor set for the colour data being the Radially Averaged Power Spectral Density

163 Chapter 5. Terrain Perception using Non-semantic Range Data information.

Range and Colour

The classification accuracy of the range and colour descriptor combinations performs well, with the relationship between classification accuracy and region of interest size for all configurations similar to that shown in Fig. 5.22. The best performing descriptor is not easy to identify, this result is due to combination of descriptors that was used being that of only GLCM textures, only RPSD and only Gabor descriptors.

Classification Accuracy using High−pass Filter and d b 1

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Fig. 5.22: Classiﬁcation results for the three range and colour combination descriptors

The best performing descriptors in the range only case was that of the GLCM textures with the RPSD and Gabor descriptors not performing well, whereas the RPSD and Gabor descriptors performed well in the colour only case with the GLCM textures not performing well. As the combination of descriptors didn’t combine the best cases for both the results are slightly worse than the range only and the colour only descriptor classiﬁcation accuracies. The relationship between the classiﬁcation accuracy, the statistical distance metric

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High−pass Filter 0.1 Piece−Wise Multi−Linear Windowed Residual 0 db do dkl djs dkls Fig. 5.23: Accuracy of terrain type classification using different statistical distance metrics as an average of all region of interest sizes with the XT,hybrid descriptor and the extraction technique, is shown in Fig. 5.23. The statistical distance metric results behaved similar to the results collected by both the range only and the colour only classification configurations, however, the Bhattacharyya distance did not perform as well as the previous cases. Instead the divergence based statistical distances performed better.

The best combination descriptor conﬁguration provided a classiﬁcation accuracy of

84.0% without utilising the confidence threshold with the XT,hybrid descriptor, a region of interest size of W = 72, windowed residual image extraction, Bhattacharyya distance classifier and MDA dimension reduction. The Gabor filter response in Fig. 5.23 at W = 72 is however slightly greater than that of the GLCM texture. However when considering the effect of the confidence threshold this difference in classification accuracy is reversed, with the GLCM texture providing the better classification accuracy at

χth = 50% of 88.9%. The relationship between the classiﬁcation accuracy and the scene classiﬁcation is shown in Fig. 5.24.

The confusion matrix shown in Table 5.9 shows considerable correlation between the terrain types of gravel and artiﬁcial turf, with only a small amount of correlation

165 Chapter 5. Terrain Perception using Non-semantic Range Data

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Fig. 5.24: Classification accuracy and percentage scene classification as the confidence threshold changes

concrete artiﬁcial turf grass gravel tile precision concrete 0.922 0.000 0.004 0.000 0.074 0.922 artiﬁcial turf 0.005 0.995 0.000 0.000 0.000 0.995 grass 0.035 0.095 0.870 0.000 0.000 0.870 gravel 0.033 0.168 0.004 0.768 0.027 0.768 tile 0.000 0.000 0.000 0.000 1.000 1.000 recall 0.916 0.659 0.990 1.000 0.867 0.889

Table 5.9: Confusion Matrix for XT,combo,W = 72 with MDA and windowed residual and χth = 50% between the grass and artiﬁcial turf. This result is most likely due to the similarity between the terrain types appearances giving rise to strongly correlated colour GLCM textures.

Terrain Boundary Misclassiﬁcation

The misclassification of terrain near to a boundary between different terrain types was investigated to observe whether a significant relationship exists between the size of the region of interest and the classification accuracy near to a boundary. To extract the

166 5.3. Extended Range Texture Analysis relationship from the classiﬁcation results it was necessary to be able to calculate the discretised distance from a boundary in each of the scenes, which was counted in pixels. The total number of elements for each discretised distance was determined as well as the total amount of correctly classiﬁed elements for each distances was calculated.

The terrain classification around the boundary between different terrain types can be seen in Fig. 5.25, it can be seen that there is no clear relationship between the distance from a boundary and the classification accuracy. The data that was used to produce the figure originates from the GLCM texture range only descriptor XT,range, using the Bhattachrayya distance metric. All region of interest sizes have similar improvements in their classification accuracies as the distance from the boundary increases. The variation of classification accuracy over all distances behaves in a similar way to the classification accuracy observed previously in Fig. 5.15.

Region of Interest dependance on Classification near Boundary 1

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167 Chapter 5. Terrain Perception using Non-semantic Range Data

Computation Time

The main contribution of the ERTA method is the quality of the terrain classification results, however, it is also of use to analyse the computation time for the method which currently runs off-line, so that future improvements can be identified. The computation time results that are recorded are for the three different texture features for both the range and colour source images. The results for each of the methods are shown as a function of the region of interest window size, and are shown below, in Table 5.10. The implementation for the GLCM and the RPSD methods is performed entirely within C implementation, however, the Gabor filters are only partially implemented in C. The computer that was used to perform the computation was an Intel i5-2500 CPU with a clock speed of 3.3GHz running Windows 7 64bit. The timing is found by recording the time it takes for 10000 calculations and then finding the average computation time, then applying it to a frame of size 320 × 240.

32 48 64 72 96 128 GLCM range 41.47 52.22 68.35 76.8 122.9 176.64 GLCM colour 92.16 122.9 138.2 184.3 230.4 307.2 Gabor range 271.9 463.9 618.2 758.0 1305 2872 Gabor colour 277.2 480.8 604.4 798.7 1298 2826 RPSD range 75.87 89.86 110.6 123.6 156.7 243.5 RPSD colour 70.35 91.39 110.6 132.1 156.8 246.5 Table 5.10: Frame computation time for a frame size 320 × 240, in seconds, as a function of the region of interest size

The results show that as the region of interest size increases the computation time of all methods increases. This effect is because as the region of interest size increases more data is included in the feature extraction and so it increases the amount of time required to evaluate that pixel. The computation time increase as the region of interest sizes increases is not to the same magnitude for all of the methods. The Gabor filter implementation is an adapted implementation provided from [176], which has a significantly greater computation time than the other two methods, as it is necessary to apply the 24 separate filters to the image. The difference in computation time between the range and colour

168 5.3. Extended Range Texture Analysis implementations of the GLCM method are due to the number of used levels that the GLCM uses, in the depth case there are 32 levels whereas the colour case there are 256 levels. For a region of interest size of 64 the computation time for one pixel using the GLCM depth data is less than 1ms and by implementing the algorithm on a diﬀerent system architecture such as a Graphics Processing Unit it is possible that this implementation may be able to work in real-time.

5.3.4 Moving Platform Results

The stationary dataset used in Section 5.3.3, was captured primarily to limit the sources of error that might effect the terrain classification results. However, requiring the robot to periodically stop in order to survey the terrain is not a desired requirement. For this purpose an additional dataset was also captured which aims to prove the developed ERTA method for a more realistic situation. The moving dataset is captured while the vehicle is traversing over different terrain types at reasonable speed. The dataset also contains uneven surfaces, which result in the incident angle of the camera varying over the environment. The experimental platform that was used to collect the scene data was arranged in the same configuration as was used for the stationary dataset, shown in Fig. 5.9. The terrain types that were captured for the moving dataset include: artificial turf, concrete, grass, tile, gravel and bark. The range images were taken while the robot platform was travelling forward at varying speeds, the data was taken just after dusk, with no direct sunlight, with the visual data collected being very dark caused by the lack of light. The data that was captured for this dataset includes roughly 120 frames of depth and visual images each, as well as odometry, inertial and horizontal LiDAR data. Of the 120 frames, 12 key frames were chosen to extract the belief about the terrain types, with only 20% of the data in the key frames used to provide the training dataset. The ground truth for the scenes were labelled manually. The total scene classification for non-semantic range data with a zero confidence threshold can be seen in Table 5.11, where the configuration that was used consisted of

169 Chapter 5. Terrain Perception using Non-semantic Range Data

a region of interest of size W = 72, the Bhattacharyya statistical distance db, a high-pass filter residual and MDA for dimensionality reduction. The classification accuracy that was achieved for a moving platform was 73%, with the majority of the error in the dataset originating from the misclassification between the surfaces tile and concrete. This misclassification can be explained by the similarity in surface conditions between the two terrain types, with the tiles that were used being very similar to concrete.

artiﬁcial turf concrete grass tile gravel bark precision artiﬁcial turf 0.87 0.16 0.00 0.07 0.06 0.00 0.73 concrete 0.04 0.31 0.00 0.20 0.02 0.00 0.54 grass 0.00 0.03 0.89 0.00 0.00 0.00 0.95 tile 0.02 0.44 0.00 0.70 0.00 0.00 0.62 gravel 0.02 0.02 0.00 0.02 0.86 0.03 0.92 bark 0.04 0.05 0.11 0.02 0.06 0.97 0.64 recall 0.87 0.31 0.89 0.70 0.86 0.97 0.73

Table 5.11: Confusion Matrix for XT,range using MDA, db, W = 72 using the high-pass filter residual, with a zero confidence threshold, showing the strong misclassification between concrete and tile

The classification accuracy for the non-semantic range only classification can be further enhanced by implementing confidence thresholding on the scene classification. The relationship between the confidence threshold and the classification accuracy is seen in Fig. 5.26. A good compromise between the classification accuracy and the percentage scene classification can be seen to exist at χth = 40%, this results in a classification accuracy of 82% for the dataset. To maintain local consistency between frames, the platform is localised through a combination of the range sensor as well as other sensors that are available on the platform, these sensors include LiDAR, an Inertia Measurement Unit (IMU) as well as odometry data for the platform. Once the platform is localised a map of the region can be generated, a typical region that the vehicle has traversed through, is shown in Fig. 5.27, showing the colour and depth data as well as the labeled classification results. The colours shown in the classification is the same as was shown in Fig. 5.18, with the addition of the bark terrain type represented by brown. The motion of the vehicle can also be examined by looking at both the linear velocity

170 5.3. Extended Range Texture Analysis

Classification Accuracy and Confidence Threshold 1 1

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0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Confidence threshold (%) Fig. 5.26: Classification accuracy and percentage scene classification as the confidence threshold changes and angular velocities that the vehicle is subject to during the scene traversal. The linear velocity is found through the use of encoders that are mounted on the platforms rear wheels, with the angular velocities found through the use of an IMU, these velocities are shown in Fig. 5.28. The speed that the vehicle is travelling at can be seen to peak at about 0.7 ms−1, with the angular velocities indicating that the roll and pitch angle of the platform was varying considerably during the maneuver. Despite these complicating factors the classification accuracies for the terrain types was found to be 73%, indicating that the developed ERTA method can accommodate for both the blur due to the camera motion as well as camera incident angle.

5.3.5 mu-Patch Model

The ERTA terrain classification results can then be utilised for a large range of applications. The classification method can be used to provide a terrain model that is usable for the 3D-AD tyre model that was developed in Chapter 4. The classification results are used to generate a terrain model that includes both the frictional characteristics as well as the dominant surface geometry of the scene. This

171 Chapter 5. Terrain Perception using Non-semantic Range Data

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bark concrete art. turf grass gravel tile

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172 5.3. Extended Range Texture Analysis

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-20 -1 0 1 2 3 4 5 6 7 8 Time (s) Fig. 5.28: Linear and angular velocities that the platform is subject to during the moving platform experiment allows for the tyre model to be able to identify the expected forces and moments as well as the minimum and maximum values that can be experienced. This interpretation is only one example of how the results of the terrain classification can be utilised, it could be used to limit an AGV to traversing only over terrain that it is confident in. The output of the terrain classification method consists of splitting the observed environment into classified regions of different known terrain types, with borders between different terrain types being identified. Each different terrain type is associated with its own coefficient of friction µk and the standard deviation of that coefficient of friction σk for the kth terrain type. There are a number of different methods that exist for estimating the frictional characteristics of the terrain, through which the mean and standard deviation of the friction coefficients can be identified [177–179]. The mean and standard deviation of the coefficients of friction that is being utilised in this implementation can be seen in Table 5.12. This information is provided to the proposed method and fused together with the terrain classification method and dominant geometry output in order to achieve the µ-Patch model. Additionally, it is important to be able to identify, the assigned friction characteristics for the case where the confidence in the original classification result

173 Chapter 5. Terrain Perception using Non-semantic Range Data was less than the desired conﬁdence threshold. In order to reduce the uncertainties in the system model it is better to assume a realistic worst case friction characteristics for this region. The worst case friction characteristics for a given operating area can be assumed to be an unknown combination of all the terrain that are expected to be encountered in this terrain. This worst case combination, is chosen so that the standard deviation for the surface is the maximal value. To be able to calculate the worst case scenario mean and standard deviation the following mathematical optimisation is used, as shown in (5.25).

N 2 2 maximise σ = pi (σi +(µi − µ0) ) σ0 v u i=1 uX N t (5.25) subject to pi =1 i=1 X 0 ≤ pi ≤ 1 i =1,...,N

where the worst case mean friction µ0 is calculated in series with the objective function using pi

µ0 = piµi (5.26) i=1 X The worst case mean and standard deviation with the base terrain type means and standard deviations, as shown in Table 5.12, provides a worst case for the frictional characteristics which is also shown in the table. The frictional characteristics of the terrain model Tf,µ and Tf,σ are set as the mean and standard deviation of the coefficient of friction µk and σk, for terrain type k, where k is the terrain type number which is classified for a particular region, with µ0 representing the worst case friction characteristics. The worst case friction coefficient distribution is no longer a gaussian distribution, however, the important aspect of this calculation is the identification of the minimum and maximum limits on the coefficient of friction so that the uncertainty in the terrain model can be adequately accounted for. Once a scene has been classified and the appropriate frictional characteristics have been assigned then both the friction and geometric characteristics of the terrain can be fused together to obtain a model of the terrain. The method by which this fusion takes

174 5.3. Extended Range Texture Analysis

µ σ Concrete 0.9 0.02 Artiﬁcial Turf 0.7 0.02 Grass 0.75 0.1 Gravel 0.9 0.05 Tile 0.65 0.04 Worst 0.7767 0.133 Table 5.12: Terrain Type Friction Characteristics place is detailed in Algorithm 2, where the geometric representation for the terrain is taken as the PWML geometric model for the scene and the friction model is the

Tf,µ and Tf,σ as determined earlier, which is a quad-tree based representation of the environment.

Algorithm 2 mu-patch model generation

1: Retrieve all Ωk of Ω 2: Set Q to empty set 3: Push Ωk onto Q 4: while Q is not empty do 5: Pop Parent node p from Q 6: if |Tclass|> 1 ∀(i, j) ∈ p then 7: Calculate children vector c from p 8: Calculate PWML geometric patch ∀c 9: Push c vector onto Q 10: else 11: Set Pµ to Tf,µ∀(i, j) ∈ p 12: Set Pσ to Tf,σ∀(i, j) ∈ p ˆ 13: Set Ppatch to f (i, j) ∀(i, j) ∈ p 14: end if 15: end while

The output of the algorithm can be seen for different terrain scenes in Fig. 5.29. The mu-patch model is represented as a 3D geometry that is overlaid with the mean coefficient of friction of the particular terrain that is classified. The standard deviation of the coefficient of friction is represented as an intensity in the second part of the figure, showing the areas that are classified as worst case having a much greater value of standard deviation of the friction coefficients.

175 Chapter 5. Terrain Perception using Non-semantic Range Data

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176 5.3. Extended Range Texture Analysis

5.3.6 Discussion

The application of confidence thresholds for the scene classification allows for a greater use in terrain classification. The costly part of this method is the data extraction and classification. Along with the terrain classification results for each pixel, an accompanying confidence measure is provided as well. This extra information about the terrain classification allows for other applications to take place, for instance in allowing this information to inform a path planner so that only areas that have a high confidence measure are traversed over in order to reduce the uncertainty in the control, or more directly, in order to avoid areas in the scene for which the classifier has low confidence. The ERTA method allows for easy addition to the number of classes that are available for classification with the required steps providing a set of training data which can be included into the training data from the other terrain types. The MDA method is then re-run with the additional class data added resulting in a new transformation matrix to be applied to the descriptor data with an additional dimension to account for the new terrain class. The mean and covariances for all of the classes are then recalculated and used in the same statistical distance metrics. The use of the confidence metric allows for the expansion of the method into identifying when a new terrain type is encountered, however if a bottom-up approach was used the current classification methods are not able to identify an objective confidence in the observed areas, instead providing a relative confidence between the currently observable classes. The classification results show that using the statistical distance metrics result in reasonable classification accuracies, however they are dependant on the underlying terrain type feature space distributions being at least partially separated so that the interclass distances are large enough to make the classification results reasonable. Although efforts were made to reduce the overlap, large amounts of overlap still exists between the different classes in the feature space. The confidence metric that was used exhibits correlation with the size of the region used to calculate the distances that are used in the classification, with smaller regions providing lower confidence than the larger regions. This effect seems to be due to the reduction in the number of data points that are available to form the region mean and covariance, which results in a coarser mean and covariance and hence a larger distance

177 Chapter 5. Terrain Perception using Non-semantic Range Data than what it should be.

The feature extraction method that was used requires that the spatial frequency in the depth image be relatively consistent within the image as well as consistent with the spatial frequency of the trained terrain classes, and therefore, surfaces that are at larger angles or at a diﬀerent height no longer satisfy this assumption. This limitation aﬀects the way that the sensor can be used to extract information of the ground as the spatial frequency throughout the image needs to be maintained.

The method for feature extraction that is used, pixel by pixel, in the current implementation is extremely slow. Each pixel takes 0.5ms to 3ms to be evaluated depending on the size of the region of interest and the type, GLCM or RPSD. Considering that for a 640 × 480 resolution image there are roughly 3 million pixels this results in a considerable time taken to only extract the features from the original image. This limitation eﬀectively means that the method in its current implementation is not capable of operating in any online applications.

The classiﬁcation accuracies of the GLCM texture descriptors are diﬀerent between the range and the colour results, the cause of this is the way in which the GLCM is calculated. For the range data the GLCM has a clear reference by which to measure the deviations from, this reference being that of the dominant surface. Whereas the colour data did not have this clear reference, which resulted in an attempt to provide the reference by means of expanding the possible levels of colour to 256.

The classification results point towards the windowed residual extraction technique as being the best method to use in order to provide the highest classification accuracy, however in the comparisons the windowed method only provided slightly better classification results than the high-pass residual extraction technique. With consideration for an online application however, the windowed method takes a considerable amount of time to calculate as it needs to be done individually for each of the pixels. Whereas the high-pass residual image can be extracted through the use of a simple filter and only needs to be applied once for the entire scene. The best residual image extraction technique to use for a real application would be that of the high-pass residual image.

The mu-patch terrain model that is generated from this method is able to provide a

178 5.3. Extended Range Texture Analysis reasonable terrain model so that a path planning system or path tracking system may be able to use it successfully to attain its goals. However, there are a few limitations in the way that the model is constructed that can cause issues with the model. The main problem is the use of a quad-tree in the overall structure of the model, the use of this structure means that the multi-linear planes that are fitted to the data for each of the quad-tree regions are not guaranteed to have a continuous surface when crossing into different regions. The effect of this is that the boundaries between regions often contain a large difference between both surfaces and thus there is often a step between these regions. In addition to this limitation the way that the confidence metric is used to classify a terrain component into an unknown area is also problematic, as often the reason for low confidence in one classification is that there is another similar terrain type, so in this case the worst case friction characteristics are not the best that can be achieved for this region but instead a combination of the two similar terrain types would be the better option.

5.3.7 Summary

The terrain classification method of range only data using a windowed residual image, with MDA dimension reduction and a distance metric of Bhattacharyya distance for a region of interest size of 72 pixels, results in a zero threshold classification result of 87.9% and with a classification accuracy of 91.4% classification accuracy with a 60% confidence threshold. The boundary between different terrains was able to be identified with this configuration, with a clear boundary between different terrain types visible. In addition, a classification accuracy of 73% was obtained with a zero threshold for the moving dataset, with a classification accuracy of 82% for a confidence threshold of 40%. The terrain classification using the colour only descriptors provided a best zero threshold classification accuracy of 79.0% and a 91.4% classification accuracy with a 50% confidence threshold. The descriptors that contained combinations of both the spatial and the colour descriptors performed reasonably, with a classification accuracy of 84.0%, however the confidence threshold provided only a slight increase in classification accuracy with a confidence threshold of 50% producing only 88.9%

179 Chapter 5. Terrain Perception using Non-semantic Range Data classification accuracy. The three different texture descriptors that were compared within the results showed that the GLCM textures performed best for the range only results while the RPSD descriptors performed best for the colour data. A combination of these two different descriptors was not performed so as to evaluate the best combination, as the importance of the work was directed at showing the viability of the range only texture descriptors in terrain classification. The results show that a similar classification accuracy between the range and the colour descriptors using the current classification method. The effects of the region of interest size, was that as the region size increases the classification results slightly improved. with no significant difference in misclassification near the boundary between terrain types. This result seems to indicate a weak relationship between the region of interest and the overall classification accuracy. However there was a strong relationship between the region of interest size and the amount of time that is required to calculate the individual pixel descriptor values. For an online application of this method it is therefore necessary to compromise between the larger region of interest sizes for classification accuracy as well as calculation time.

5.4 Conclusion

Through the use of experimental results, two different terrain classification methods have been proposed which are able provide terrain classification results for a scene from non-semantic spatial information of the scene alone. The second method is the most advanced version of the two, which along with classifying the scene into the available terrain types it is also able to identify areas of a scene where there exists multiple terrain types and divide these regions so that a more accurate representation of the scene is produced. The RTA method introduced the concept of using texture features of the range data in order to classify a scene into different terrain types from a set of homogenous terrain scenes. This method was then expanded into the ERTA method, which along with providing the classification results for a scene that is capable of dealing with heterogenous terrain scenes, was also capable of generating a classification confidence

180 5.4. Conclusion over that scene. This allows for areas in the scene which are not well defined to be left unclassified, preventing misclassification of a scene and reducing the uncertainty in the terrain conditions. An environmental model has also been developed to incorporate the information gathered from the scene so that it can be used to inform control or localisation algorithms. The proposed µ-patch model incorporates the frictional characteristics as well as geometric information of the terrain, and provides an interpretation of unclassified scene elements, by associating the unclassified regions with a worst case frictional condition. The information that is provided within the µ-patch model can enable a reduction in the amount of uncertainties that a system is subject to by better identifying the terrain conditions through both the classification and confidence.

181 Chapter 5. Terrain Perception using Non-semantic Range Data

182 Chapter 6

Application of the 3D Analytical Dynamic Tyre Model

The overarching goal of this thesis has been to reduce the uncertainties in the system model so that the performance of robust controllers can be enhanced, so that the goal of AGVs operating with persistent autonomy through non-uniform terrain may be realised. This chapter presents the application of the developed 3D Analytical Dynamic (3D-AD) tyre model with the dynamic vehicle model, with the simulation results compared with the previously calculated results of the Magic Formula (MF) and the developed Friction Dependant (FD) tyre models from Chapter 3. The 3D-AD tyre model includes the eﬀects of both the terrain friction as well as the steering behaviour of the vehicle.

Previously, in Chapter 3, a vehicle dynamic model was developed for a 4WD4WS platform, so that the behaviour of the vehicle can be adequately modelled. A friction dependant tyre model was developed to examine the eﬀects of the changing terrain conditions on the forces and movement of the vehicle through that terrain. A number of sources of uncertainty were identiﬁed, with a more descriptive tyre model needed that was able to replicate better the experimentally derived force model, as well as a terrain perception method that enables the vehicle to better perceive the environment around it.

For this purpose, more descriptive dynamic analytical tyre model has been developed in Chapter 4, that is able to adapt to the changing frictional characteristics of the terrain. The developed tyre model is also responsive to dynamic eﬀects on the tyre,

183 Chapter 6. Application of the 3D Analytical Dynamic Tyre Model and the model also takes into account the eﬀects that the steering rate has on the deformation of the tyre. A velocity and camber angle dependant tyre force distribution was also developed that allows for the rolling resistance and the overturning moment that is felt in the tyre to be modelled.

Additionally, a terrain perception method was also developed in Chapter 5, that allows for the system to be able to perceive the state of the terrain around the current location of the AGV, so that the coefficients of friction for the local terrain can be known to the system. The developed method predominantly utilises range only data to perform the terrain classification, and as such is able to assist in classification where the visual information in the scene might be lacking. The classification method also provides a classification confidence metric so that scene classification can have a minimum confidence in the classification results in order to reduce the uncertainty in the terrain model.

Through the integration of all of these components, the motion that an AGV experiences through a terrain can be better modelled. This chapter integrates the developed models and methods in order for the motion of the simulated vehicle through an environment to more accurately reﬂect the true behaviour of the vehicle. This can allow for a robust controller to be designed so that the AGV can be more reliably controlled as it traverses through a non-uniform terrain area.

In Section 6.1, the developed dynamic vehicle and tyre models are integrated together, with a new system model developed based on these separate models, as well as a discussion about the inﬂuence that the terrain perception method has on the behaviour of the tyre model. A set of simulation conﬁgurations, similar to the simulation sets shown in Chapter 3, are applied to the developed vehicle dynamic model in Section 6.2, with the developed model compared to both the Magic Formula experimental model and the kinematic behaviour of the vehicle. A discussion of the integrated vehicle model as well as the simulation results are presented in Section 6.3, with emphasis on the improvements of the model, as well as the limitations.

184 6.1. Model Integration

6.1 Model Integration

To realise a system model that is able to more accurately represent the reality of a 4WD4WS vehicle operating through a non-uniform terrain environment, the individually developed dynamic vehicle and dynamic tyre models can be integrated together, with the terrain perception method providing a terrain model through which knowledge about the environment is obtained. The incorporation of these models allows for the motion of the vehicle to be better estimated through the environment through the use of additional system states and a simplified belief about the nature of the environment. The base that is used for the model integration is that of the 3D vehicle dynamic model, with additional system states added for the developed 3D dynamic tyre model so that the forces and moments that the vehicle experiences through the tyre-terrain interaction can be better modelled. The behaviour of the vehicle alters as it traverses over different terrain types and for this purpose the terrain perception method is able to provide a terrain model through which the coefficients of friction and the height of the terrain can be known to the system model.

6.1.1 3D Vehicle Dynamic Model

The 3D dynamic vehicle model that is utilised in the model integration is comprised of the models in equations (3.2), (3.14) and (3.20), these variables were discussed in Chapter 3 and listed in the Nomenclature. These system states are incorporated within the model for the system Xv, as previously, and are expressed as shown below.

Xv = [Xb,Xs,1,Xs,2,Xs,3,Xs,4,Xt,1,Xt,2,Xt,3,Xt,4] (6.1)

The main inclusion in the motion of the new system is that the forces and moments that the vehicle experiences are modelled through the use of the newly developed 3D analytical dynamic tyre model. With the vehicle system model being the same model that was previously used in Chapter 3. The forces and moments that are developed in the tyre for this new model are not statically dependant on the current state of the vehicle model as was the case for the Magic Formula based vehicle model, or the

185 Chapter 6. Application of the 3D Analytical Dynamic Tyre Model

Friction Dependant model. The developed 3D analytical dynamic tyre model requires that additional system states be used to keep track of the deformation that the tyre is subject to.

6.1.2 Analytical Dynamic Tyre Model

The analytical dynamic tyre model that was developed in Chapter 4, is used to model to forces and moments that the vehicle undergoes as it moves through a terrain. The developed model is dependant on both the terrain frictional characteristics as well as the steering rate and speed of the tyre. The dynamic model described in (4.29), (4.35) and (4.36) is used to model the tyre deformation for each wheel of the vehicle, with the additional tyre deformation states shown below, where i =1, 2, 3, 4

T Xǫ,i = [¯ǫx,i, ǫˆx,i, ¯ǫy,i, ǫˆy,i] (6.2)

One of the eﬀects of having a 3D dynamic vehicle model is that the force that the vehicle exerts on the ground changes, depending on the geometry of the terrain, how the vehicle moves as well as the suspension that the vehicle has. The eﬀect of this on the tyre model is that the size of the contact patch changes as the physical distance between the wheel hub and the terrain changes. Currently the Magic Formula model is able to take into account the force exerted on the ground by the wheel at the contact patch and calculate the contact patch dimensions, the tyre pressure is also taken into account whilst performing this calculation.

So that the developed dynamic tyre model is able to account for the changes in size of the contact patch as the vehicle moves over the terrain, the same approach is taken. The MF-SWIFT model [75] proposes the equations (6.3)and (6.4) to model the changes in the contact patch size for the length and width, respectively. The calculated contact patch dimensions are used by the developed analytical tyre model to better reﬂect the physical tyre system.

F F L =2r q z + q z (6.3) 0 ra2 c r ra1 c r z0 0 r z0 0 ! 186 6.1. Model Integration

1 F F 3 W =2w q z + q z (6.4) rb2 c r rb1 c r z0 0 z0 0 ! where the parameters that are used in the above equation can be found in Appendix B in Table B.1

6.1.3 Terrain Model

The terrain perception method is responsible for informing the dynamic tyre model of the state of the environment that the tyre is traveling over. The terrain perception method classifies the environment around the vehicle with the frictional characteristics of that terrain type then being applied for the newly classified terrain regions. The coefficients of friction for each terrain type are assumed to be known for each of the terrain types, or provided to the system by another method. The ERTA method that was developed in Chapter 5 is also able to utilise the classification confidence of the terrain in order to better account for the uncertainty in the classification. Together with the geometry of the terrain that is extracted, the coefficients of friction for each terrain type is incorporated into the terrain model for the vehicle. This model is utilised by the vehicle to provide the height of the ground as well as the coefficients of friction that each wheel is subject to whilst in motion. The terrain height estimation is used to provide the tyre normal force that each tyre experiences, which is related to the suspension of the system. This normal force in turn dictates the behaviour of the tyre deformation, as the deformation is directly related to the amount of normal force that the tyre is subject to. The coefficients of friction for the tyre are then utilised to determine the amount of longitudinal and lateral deformation that the tyre is subject to as the tyre tread slides across a surface.

6.1.4 Full System Model

The full system model for the 4WD4WS vehicle platform operating through a non-uniform terrain environment, which is a combination of the dynamic vehicle and the dynamic tyre model. The state is expressed as X, and is a combination of the body state Xv and the tyre states Xǫ,i for each tyre, the state is expressed in the expanded

187 Chapter 6. Application of the 3D Analytical Dynamic Tyre Model form below,

X = [x,y,z,φ,θ,ψ,u,v,w,p,q,r,

zs,1, zs,2, zs,3, zs,4, z˙s,1, z˙s,2, z˙s,3, z˙s,4,

ω1,ω2,ω3,ω4, δ1, δ2, δ3, δ4, δ˙1, δ˙2, δ˙3, δ˙4, (6.5)

ǫ¯x,1, ¯ǫx,2, ǫ¯x,3, ǫ¯x,4, ǫ¯y,1, ¯ǫy,2, ¯ǫy,3, ǫ¯y,4,

T ǫˆx,1, ˆǫx,2, ǫˆx,3, ǫˆx,4, ǫˆy,1, ˆǫy,2, ˆǫy,3, ǫˆy,4] .

The system state X of the dynamic vehicle model that utilises the developed 3D analytical dynamic tyre model, this system state includes the motion of the vehicle body, the suspension and tyre dynamics, as well as the internal deformation states of the tyres, which are used to calculate the forces and moments that are generated in the tyre. The system response is evaluated in simulation and compared to the responses of the Magic Formula tyre model as well as the developed Friction Dependant tyre model.

6.2 Simulation

Once the three dimensional model for the vehicle has been developed the vehicle can then be evaluated in a suite of four simulation environments. The simulation environments are identical to the environments and conﬁgurations of those in Chapter 3, with the addition of the application of the developed 3D-AD tyre model.

The simulations present the developed 3D-AD tyre model results as able to replicate the behaviour of the MF tyre model over the single terrain type and that it is able to demonstrate similar friction dependance as the developed FD tyre model. Additionally, the simulations demonstrate the behaviour of the 3D-AD tyre model being able to take into account additional system parameters, such as the steering rate of the tyre as well as the dynamic behaviour of the tyre in responding to changing conditions.

188 6.2. Simulation

6.2.1 Single Terrain Type

Flat Plane Terrain Geometry with a Single Terrain

The first set of environments that the vehicle will travel through includes a smooth flat plane on only one terrain type. The set includes various open loop steering maneuvers, such as travelling in a circular loop track, as well as travelling in a oval loop track, and then zig-zag track. In all cases the simulation results are compared to that of the non-slip kinematic model, the magic formula and the friction dependant model, which was developed in Chapter 3. The first configuration that is examined is the simple loop track on the single terrain type, with the same configuration settings as the first configuration in Chapter 3. This first configuration examines the effects of the 3D-AD tyre model on the behaviour of the vehicle through simple maneuvers through simple terrain conditions. The path that the vehicle takes around the loop track is shown in Fig. 6.1, where the track is compared to the Magic Formula and the Friction Dependant models, with the differences for the dynamic analytical tyre model is the way in which the cornering is achieved.

Fig. 6.1: Comparison of kinematic and dynamic responses on the loop track on a ﬂat plane incorporating the developed analytical dynamic tyre model, with the trajectory compared against both the MF and FD vehicle responses

The model originally under-steers through the corner, which is caused by the diﬀerences between the lateral forces that are generated in the dynamic tyre model

189 Chapter 6. Application of the 3D Analytical Dynamic Tyre Model and those that are generated with the Magic Formula or the Friction Dependant models. The dynamic responses of the tyre models as the vehicle negotiates the kinematic path are presented in Fig. 6.2, with the sub-figures comparing the different responses throughout the maneuver for the longitudinal and lateral forces as well as the self-aligning moment. The meanǫ ¯ and weighted meanǫ ˆ tyre deformations in both the x and y directions in tyre contact patch is also shown. These show the effect of the steering rate on the tyre deformation as the vehicle negotiates a turn, with the deformation dependant on the motion of the tyre, including the steering rate.

The longitudinal force that the 3D-AD tyre model exhibits is similar to the longitudinal force exerted on the tyres from the Magic Formula. The lateral force that the tyre experiences through the corner is similar to that of both the Magic Formula and the Friction Dependant force models, except that during the turn into the corner there is a spike in the applied lateral force, as well as the tyre deformation, after which the force applied returns to the same value as the other models. This phenomenon arises due to the eﬀects of the steering rate on the way that the tyre is deformed laterally. Similarly, at the end of the cornering the lateral force does not become negative, as it does in the other models, which is once again an eﬀect of the steering rate on the behaviour of the tyre deformation.

The self-aligning moment that is generated from the dynamic tyre model is slightly different to the other two force models. The first difference is that of a delay in the generation of the self-aligning moment caused by the steering rate of the tyre as the vehicle goes into the turn. The second difference is the value that the self-aligning moment has during the middle of the turn, which is larger in amplitude than both the Friction Dependant and the Magic Formula models. This is primarily due to the steady-state value of the self-aligning moment having a comparably larger moment for a greater amount of slip angle values as shown in Section 4.2.2. The third difference is once again caused by the steering rate of the tyre turning back, with the effect being that a change in the self-aligning moment is once again delayed.

The beneﬁt of the 3D-AD model over the existing models is that the steering rate of the wheel is incorporated into the tyre model. This allows for the eﬀect of the tyre steering through cornering maneuvers to be observed with more than just current

190 6.2. Simulation steering angle eﬀecting the amount of lateral force as well as self-aligning moment that the tyre is subject to.

0.15 25

0.1 epsbx 20 epsby epshx epshy 0.05 15

0 10

-0.05 5

-0.1 0 tyre deformation (mm) tyre deformation (mm) -0.15 -5 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 time (s) time (s) (a) Tyre longitudinal deformation (b) Tyre lateral deformation

0.1 0.04 Roll - MF Pitch - MF 0.08 Roll - FD 0.02 Pitch - FD 0.06 Roll - 3D-AD Pitch - 3D-AD 0 0.04 Angle (deg) Angle (deg) -0.02 0.02

0 -0.04 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 time (s) time (s) (c) Roll angles (d) Pitch angles

1000 4000

3000 500 2000 (N) (N) x y 0 1000

0 Force F -500 Force F Fx - MF Fx - FD Fx - 3D-AD -1000 Fy - MF Fy - FD Fy - 3D-AD -1000 -2000 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 time (s) time (s) (e) Longitudinal Force (f) Lateral Force

80 (Nm) z 60

-20 Mz - MF Mz - FD Mz - 3D-AD -40 Self-Aligning Moment M 0 5 10 15 20 25 30 35 40 45 50 time (s) (g) Self-aligning Moment

Fig. 6.2: Dynamic responses of the MF, FD and the analytical dynamic tyre model on a ﬂat plane, showing the tyre internal deformation states, with the MF, FD and the analytical dynamic tyre models corresponding to the black, red and blue lines, respectively

The second configuration for the vehicle to travel through is that of the zig-zag track on a single terrain type over a flat terrain geometry. The motion that the vehicle goes through during the cornering on this track can be seen in Fig. 6.3. The most obvious difference between the path of the dynamic model with the dynamic tyre responses and that of the MF and FD force models is the under-steering that the vehicle takes through the first corner, resulting in a different trajectory as the dynamic tyre response provides less lateral force through the turn. The forces and moments that the vehicle experiences through the zig-zag maneuver

191 Chapter 6. Application of the 3D Analytical Dynamic Tyre Model

Fig. 6.3: Comparison of kinematic and dynamic responses for the zig-zag maneuver on a flat plane with dynamic tyres, with the 3D-AD tyre model showing a larger amount of under-steering is shown in Fig. 6.4. The cause of the under-steering through the first corner can be seen to be a result of the lateral tyre forces through the first corner. The differences between the generated lateral force and self-aligning moment for each of the three tyre models is much more exaggerated through the first corner where the vehicle steering rate is higher and the vehicle moves at a lower speed, compared to the second corner where the steering velocity is lower and the vehicle speed is much higher. This shows that the developed 3D-AD tyre model is able to take into account the vehicle velocity when moving through a corner, allowing for the effect of the steering rate to be correctly modelled as an effect that reduces with increasing vehicle speed. The self-aligning moment that the tyre generates through the corner from the 3D-AD tyre model can be seen to behave in a similar way to that of the MF model, where the magnitude of the self-aligning moment decreases. Although, the self-aligning moment for the 3D-AD tyre model still behaves in a similar way as in the loop track results for the same reason as before, in that at the self-aligning moment at large slip angles in the model is more positive than that of the MF model.

Inclined Plane Terrain Geometry with a Single Terrain

The second set of environments for the vehicle to travel along is that of a smooth inclined plane surface consisting of a single terrain type. The vehicle performs both

192 6.2. Simulation

0.15 30 epsbx epsby 0.1 20 epshx epshy 0.05 10 0 0 -0.05

-0.1 -10 tyre deformation (mm) tyre deformation (mm) -0.15 -20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 time (s) time (s) (a) Tyre longitudinal deformation (b) Tyre lateral deformation

0.2 0.04 Roll - MF Pitch - MF 0 0.02 Roll - FD Pitch - FD 0 -0.2 Roll - 3D-AD Pitch - 3D-AD -0.02 -0.4 -0.04 Angle (deg) Angle (deg)

-0.6 -0.06

-0.8 -0.08 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 time (s) time (s) (c) Roll angles (d) Pitch angles

1000 6000

4000 500 (N) (N)

x y 2000 0 0

Force F -500 Force F -2000 Fx - MF Fx - FD Fx - 3D-AD Fy - MF Fy - FD Fy - 3D-AD -1000 -4000 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 time (s) time (s) (e) Longitudinal Force (f) Lateral Force

100 (Nm) z 50

-50 Mz - MF Mz - FD Mz - 3D-AD -100 Self-Aligning Moment M 0 2 4 6 8 10 12 14 16 18 20 time (s) (g) Self-aligning Moment Fig. 6.4: Comparison of kinematic and dynamic responses for the zig-zag maneuver on a ﬂat plane, showing the tyre internal deformation states, with MF, FD and the analytical tyre model represented by black, red and blue lines, respectively the rectangular loop and zig-zag tracks on this surface. The dynamic results for the developed trajectory are then analysed, with an emphasis on the forces and motion that the vehicle undergoes whilst traveling along the inclined surface. The 3D-AD model trajectory is then compared against both of the MF and FD tyre models, as well as the kinematic path of the vehicle.

For the rectangular loop track maneuver, the motion of the vehicle through the configuration is shown in Fig. 6.5, with the inclined plane seen as a height overlaid on the vehicle trajectories. The main difference is that the vehicle performs less of a turn on the first corner when compared against the same model on the flat plane environment.

The forces and moments that are developed in the tyre due to the motion around

193 Chapter 6. Application of the 3D Analytical Dynamic Tyre Model

Fig. 6.5: Comparison of kinematic and dynamic responses on the loop track on an inclined plane incorporating the developed 3D-AD tyre model, with comparison of the trajectory to that of the MF and FD tyre model vehicle responses the track are shown in Fig. 6.6, with the longitudinal and lateral tyre deformation also shown. The change in the amount of steering compared to the flat plane environment is mainly due to the reduction in the lateral force as the vehicle rounds the corner, and then the lateral force increases again as the vehicle makes the second turn. In between the first and second turn the lateral force is non-zero which is due to the shifted weight of the vehicle causing the force to be vectored in the lateral direction. The force that the tyre generates is in response to the vehicle sliding down the incline. The residual effect of the lateral weighted tyre deformation on the self-aligning moment can be seen, this occurs due to the motion of the vehicle as it changes heights during the vehicle motion.

The values of the forces and moments for the 3D-AD tyre model as compared the MF tyre model show that the proposed 3D Analytical Dynamic tyre model is able to replicate the results for situations where the vehicle is moving over different terrain when subject to changes in body angles and changes in tyre normal forces. The differences between the models through this terrain condition is that the effect of the steering rate on the generated lateral force and self-aligning moment is included in the dynamic case and not just as a steady-state condition.

The path that the vehicle goes through during the zig-zag maneuver whilst moving over the inclined plane surface is shown in Fig. 6.7, where once again the under-steer

194 6.2. Simulation

1000 4000

3000 500 2000 (N) (N) x y 0 1000

0 Force F -500 Force F Fx - MF Fx - FD Fx - 3D-AD -1000 Fy - MF Fy - FD Fy - 3D-AD -1000 -2000 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 time (s) time (s) (a) Longitudinal Force (b) Lateral Force

6 80

(Nm) Roll - MF z 60 4 Roll - FD 40 2 Roll - 3D-AD

20 0

-2

0 Angle (deg)

-20 Mz - MF Mz - FD Mz - 3D-AD -4 -40 -6 Self-Aligning Moment M 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 time (s) time (s) (c) Self-aligning Moment (d) Roll angles

10 Pitch - MF 5 Pitch - FD Pitch - 3D-AD 0 Angle (deg) -5

-10 0 5 10 15 20 25 30 35 40 45 50 time (s) (e) Pitch angles

0.15 25

0.1 epsbx 20 epsby epshx epshy 0.05 15

0 10

-0.05 5

-0.1 0 tyre deformation (mm) tyre deformation (mm) -0.15 -5 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 time (s) time (s) (f) Longitudinal tyre deformation (g) Lateral tyre deformation

Fig. 6.6: Comparison between the Magic Formula and the proposed friction dependant model on an inclined plane showing the developed internal friction states of the tyre, as well as the generated forces and moments, with the MF, FD and the 3D-AD tyre models represented by black, red and blue lines, respectively. through the ﬁrst corner can be seen. The forces and moments that are experienced through the maneuver is very similar to that of the ﬂat plane surface.

The effect of the incline plane on the forces and tyre deformation can be seen in Fig. 6.8, where the lateral tyre force that is generated begins to drop as it moves through the first corner. Once again the self-aligning moment developed in with the 3D-AD tyre model replicates the behaviour of the MF tyre model self-aligning moment in both corners, with the effects of the steering rate through the first corner more pronounced.

195 Chapter 6. Application of the 3D Analytical Dynamic Tyre Model

Fig. 6.7: Comparison of kinematic and dynamic responses for the zig-zag maneuver on an inclined plane incorporating the 3D-AD tyre model, showing a larger amount of under-steering as compared to that of the MF and FD tyre model vehicle responses

Smoothly Varying Terrain Geometry with a Single Terrain

The third set of environments for the vehicle to travel over consists of a smoothly varying surface with a single terrain type, with the shape of the smoothly varying surface the same as the one that was previously used in Chapter 3. The rectangular loop and zig-zag path maneuvers are carried out on this surface with the results described below. The motion of the vehicle as it moves over the surface for the rectangular loop track is shown in Fig. 6.9, with the behaviour similar to that of the ﬂat and inclined tracks.

The forces and moments that the vehicle experiences through the maneuver are shown in Fig. 6.10, with the longitudinal and lateral forces for the dynamic vehicle with the 3D-AD tyre model showing similar results to that of the other two tyre models. The initial spikes in the lateral force entering a corner are larger than before, caused by the vehicle encountering the hilly terrain geometry. The self-aligning moment that the 3D-AD tyre model generates is similar to that of the MF, with the second corner showing that the MF self-aligning moment is experiencing a lower value, which is caused by the reduction in torque at larger slip angles. The ability of the 3D-AD tyre model to account for the same parameters that the MF tyre model accounts for, as evident in the similarity between the two tyre model behaviours allows the 3D-AD tyre model to be used as a descriptive model for how the system works. Together with the same parameters as the MF tyre model, the 3D-AD tyre model also takes into account the

196 6.2. Simulation

0.15 25

0.1 epsbx 20 epsby epshx epshy 0.05 15

0 10

-0.05 5

6 10

4 Roll - MF Pitch - MF Roll - FD 5 Pitch - FD 2 Roll - 3D-AD Pitch - 3D-AD 0 0

-2 Angle (deg) Angle (deg) -5 -4

-6 -10 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 time (s) time (s) (c) Roll angles (d) Pitch angles

1000 4000

3000 500 2000 (N) (N) x y 0 1000

80 (Nm) z 60

-20 Mz - MF Mz - FD Mz - 3D-AD -40 Self-Aligning Moment M 0 5 10 15 20 25 30 35 40 45 50 time (s) (g) Self-aligning Moment Fig. 6.8: Comparison of kinematic and dynamic responses for the zig-zag maneuver on an inclined plane incorporating the 3D Analytical Dynamic tyre model, showing the internal tyre deformation, with a comparison to the MF and FD tyre model vehicle responses eﬀects of the steering rates on the developed tyre forces and moments.

The motion of the vehicle takes through the zig-zag maneuver over the smoothly varying surface is shown in Fig. 6.11. The vehicle has a very similar trajectory to that of the previous two surfaces. The motion of the vehicle in comparison with the MF and FD force models shows that the vehicle under-steers through both corners, as before.

The forces acting on the vehicle through the maneuver are shown in Fig. 6.12, with both the longitudinal and lateral forces developed in the dynamic tyre model providing a similar response to that of both the MF and FD tyre models. The self-aligning moment is also comparable to both the MF and FD moments, however the self-aligning moment generated by the dynamic tyre model begins to deviate as the vehicle increases in speed from 2 to 10ms−1, and during the turn a jump in the value of the self-aligning

197 Chapter 6. Application of the 3D Analytical Dynamic Tyre Model

Fig. 6.9: Comparison of kinematic and dynamic responses on the smoothly varying loop track that incorporates the developed 3D Analytical Dynamic tyre model, with a comparison to the trajectory responses of the MF and FD tyre model vehicles moment is seen at the point where the wheel starts steering in the opposite direction. The sudden change in the steering rate direction causes a discontinuity in the dynamic model, resulting in a spike of the weighted mean lateral deformationǫ ˆy.

6.2.2 Diﬀerent Terrain Types

Flat Plane Terrain Geometry with Multiple Terrain Types

The fourth environmental set for the simulation suite consists of a flat geometric plane for the vehicle to travel along with three different terrain types that exist in different regions of the environment. The path and forces that the vehicle experiences when subject to this environment are examined. The path that the vehicle dynamic model takes when the forces and moments for the tyres are generated based on the developed dynamic tyre model is shown in Fig. 6.13, with the transitions between the different terrain types of road, dirt and sand visible. The terrain transitions between road and dirt through the first corner, and from dirt to sand in the second corner, then from sand to road in the third corner and finally from dirt to road in the final corner. The path that the vehicle takes through the first corner appears to have a smaller amount of under-steer, compared to the other models. This can be seen to originate from the lateral force that the vehicle experiences through the corner, which is shown in

198 6.2. Simulation

0.2 25

epsbx 20 epsby 0.1 epshx epshy 15

0 10

5 -0.1 0 tyre deformation (mm) tyre deformation (mm) -0.2 -5 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 time (s) time (s) (a) Tyre longitudinal deformation (b) Tyre lateral deformation

10 10 Roll - MF Pitch - MF 5 Roll - FD 5 Pitch - FD Roll - 3D-AD Pitch - 3D-AD 0 0 Angle (deg) Angle (deg) -5 -5

-10 -10 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 time (s) time (s) (c) Roll angles (d) Pitch angles

1000 4000

3000 500 2000 (N) (N) x y 0 1000

80 (Nm) z 60

-20 Mz - MF Mz - FD Mz - 3D-AD -40 Self-Aligning Moment M 0 5 10 15 20 25 30 35 40 45 50 time (s) (g) Self-aligning Moment

Fig. 6.10: Comparison of the forces and moments on a smoothly varying surface between the proposed 3D Analytical Dynamic tyre model with the MF and FD tyre models, including the internal deformation states of the 3D-AD tyre model, with the black, red and blue lines, representing the MF, FD and 3D-AD responses respectively.

Fig. 6.14. As the vehicle transitions through the ﬁrst corner, both the 3D-AD and FD tyre models responses change, predominantly in the lateral force and the self-aligning moment, which both reduce in amplitude as the tyre transitions from the road surface onto the dirt surface. Through the second corner the vehicle transitions through the dirt-sand border and as a result the lateral force and self-aligning moment once again reduces in amplitude, with the lateral force exhibiting a spike as the tyre transitions between the terrain types. The third corner transitions between the sand and the road terrain types, with the amplitude for the 3D-AD tyre model response increasing in magnitude, where the diﬀerence in path for the vehicle with the FD tyre model transitions between the terrain types prior to taking the corner. The dynamic vehicle

199 Chapter 6. Application of the 3D Analytical Dynamic Tyre Model

Fig. 6.11: Comparison of dynamic trajectory of the 3D Analytical Dynamic tyre model based vehicle model for the zig-zag maneuver on a smoothly varying surface, with the MF and FD tyre model vehicle trajectories model transitions from the dirt to the road surface at the end.

The path that the vehicle takes through the zig-zag maneuver is shown in Fig. 6.15, where the vehicle traverses over the different terrain types independently. The motion of the dynamic vehicle using the 3D-AD tyre model is compared against the path of the dynamic vehicle using the FD tyre model for each different terrain type. The effect of the different terrain types on the motion of the vehicle due to the differences in frictional characteristics is seen in the increase in under-steering through both corners.

The 3D-AD tyre model trajectory results are compared against the results of the friction dependant tyre model, with similar trajectories in the dynamic vehicle model through the corner. The friction dependant tyre model results also exhibits a small spin-out through the ﬁnal corner whilst travelling over the sandy surface.

The forces and moments that the vehicle experiences through the maneuver are shown in Fig. 6.16. The lateral forces that the vehicle experiences through both corners show a similar force reduction as in the FD tyre model, with the effects of the steering rate through the first corner showing that the higher steering rate effects all of the terrain types similarly. The self-aligning moment that is experienced through the corners shows that the 3D-AD tyre model can also change amplitude dependant on the terrain frictional characteristics. The spikes in the self-aligning moment though each of the turns is caused by the effect of a small tyre deformation being acted on by

200 6.2. Simulation

0.15 30 epsbx epsby 0.1 20 epshx epshy 0.05 10 0 0 -0.05

-0.1 -10 tyre deformation (mm) tyre deformation (mm) -0.15 -20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 time (s) time (s) (a) Tyre longitudinal deformation (b) Tyre lateral deformation

4 10 Roll - MF Pitch - MF 2 Roll - FD 5 Pitch - FD 0 Roll - 3D-AD Pitch - 3D-AD 0 -2 Angle (deg) Angle (deg) -5 -4

-6 -10 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 time (s) time (s) (c) Roll angles (d) Pitch angles

1000 6000

4000 500 (N) (N)

x y 2000 0 0

100 (Nm) z 50

-50 Mz - MF Mz - FD Mz - 3D-AD -100 Self-Aligning Moment M 0 2 4 6 8 10 12 14 16 18 20 time (s) (g) Self-aligning Moment Fig. 6.12: Comparison of forces and moments generated by the 3D-AD tyre model through the zig-zag maneuver on a smoothly varying surface, showing the internal tyre deformation states, with the dynamic response of the MF and the FD tyre models. the steering speed of the wheel, with the spike originating from a change in sign of the steady-state self-aligning moment.

The difference of the lateral force and the self-aligning moment for the different terrain types between the first and the second corners show that there is a significant relationship between the steering rate and the driving speed. With the first corner having a higher steering rate, and subsequently higher difference between the terrain types than the second corner, having a lower steering rate and with a higher vehicle speed through the corner.

The simulation configurations that have been examined have shown that the effects of the steering rate on the development of both the lateral force as well as the self-aligning moment has been significant. The terrain type that the vehicle is traveling

201 Chapter 6. Application of the 3D Analytical Dynamic Tyre Model

Fig. 6.13: Comparison of kinematic and dynamic responses of the developed 3D-AD tyre model on the loop track on a flat plane over different terrain types, with the terrain represented by black for road, dark brown for dirt and light brown for sand over also provides a great source of difference in the developed lateral force and self-aligning moment as well.

6.3 Discussion

The main difference between the dynamic response of the developed dynamic tyre model and that of the Magic Formula force model is the way in which the steering rate of the tyres influenced the deformation in the tyres, especially the lateral deformation of the tyres. This had the effect of significantly changing the trajectory of the vehicle as it traversed along the path. The other difference between the developed dynamic tyre model based vehicle results was the way in which the self-aligning moment value was consistently more positive through the corners than the Magic Formula force model. This difference arises from the steady-state response of the developed dynamic tyre model, which follows the behaviour of the experimentally derived Magic Formula model, but begins reducing the value of the self-aligning moment at a larger value of the slip angle. The influence of the steering rate of the tyre on both the developed lateral force and self-aligning moment is seen in the zig-zag maneuver with the effects of the steering rate on the deformation under different frictional characteristics shown to be the dominant

202 6.3. Discussion

0.15 25

0.1 epsbx 20 epsby epshx epshy 0.05 15

0 10

-0.05 5

0.15 0.04 Roll - MF Pitch - MF 0.1 Roll - FD 0.02 Pitch - FD Roll - 3D-AD Pitch - 3D-AD 0.05 0 Angle (deg) Angle (deg) 0 -0.02

-0.05 -0.04 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 time (s) time (s) (c) Roll angles (d) Pitch angles

1000 4000

3000 500 2000 (N) (N) x y 0 1000

80 (Nm) z 60

-20 Mz - MF Mz - FD Mz - 3D-AD -40 Self-Aligning Moment M 0 5 10 15 20 25 30 35 40 45 50 time (s) (g) Self-aligning Moment

Fig. 6.14: Comparison of the forces and moments generated by the 3D-AD tyre model as it negotiates the through the loop maneuver and the dynamic responses of the MF and the FD tyre models, the internal tyre deformation states are shown for the maneuver, showing the eﬀect of the change in terrain friction has on the tyre deformation factor for surfaces with high frictional coeﬃcients.

Although the simulation results are able to show the eﬀects of the steering rate and wheel speeds on the developed forces and moments, a full experimental setup is required to show that these forces and moments are actually experienced. An experimental platform has been in construction for this purpose, the experimental platform is a 4WD4WS vehicle, with each wheel independently steered and driven. A 6DoF force sensor is mounted on each wheel hub so that the forces and moments that the tyre experiences can be sampled while the vehicle undergoes maneuvers. The experimental platform, however, is still not ready to be used in experiments, this is the reason for the lack of experimental comparison.

203 Chapter 6. Application of the 3D Analytical Dynamic Tyre Model

Y distance (m) 10

0 Kinematic Road -10 Dirt Sand -20 0 10 20 30 40 50 60 70 80 90 100 X distance (m) Fig. 6.15: Comparison of kinematic and dynamic responses for the zig-zag maneuver on a flat plane over different terrain types, incorporating the 3D-AD tyre model, showing the trajectory difference between the MF and FD tyre models and the developed 3D-AD tyre model

Once the experimental platform is available for use in experiments, the plan is to validate the results of the simulation against the experimental results. The tyres that are currently mounted on the experimentally platform do not match the same characteristics of the tyre that has been used in the simulation, and so the steady-state behaviour of the tyre will need to be recorded and tuned for.

6.4 Conclusion

A 3D dynamic vehicle model that incorporated the developed analytical dynamic tyre model was evaluated with respect to both the kinematic behaviour of the vehicle, as well as the dynamic behaviour of the vehicle when the tyre forces are generated through the use of the Magic Formula model as well as the developed Friction Dependant tyre model. The dynamic behaviour of the vehicle with the developed analytical dynamic tyre model showed that the effect of including the steering rate on the developed tyre forces was significant enough to alter the behaviour of the simulated vehicle as the vehicle negotiated corners. The effect of the terrain conditions on the generated forces was also observed, with significant changes in the developed forces and moments during transitions between terrain types. Through the inclusion of the terrain frictional characteristics and the tyre steering

204 6.4. Conclusion

0.15 30 epsbx epsby 0.1 20 epshx epshy 0.05 10 0 0 -0.05

-0.1 -10 tyre deformation (mm) tyre deformation (mm) -0.15 -20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 time (s) time (s) (a) Tyre longitudinal deformation (b) Tyre lateral deformation

0.2 0.04

0 0.02 0 -0.2 -0.02 -0.4 -0.04 Angle (deg) Angle (deg) -0.6 Roll - road Roll - dirt Roll - sand -0.06 Pitch - road Pitch - dirt Pitch - sand -0.8 -0.08 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 time (s) time (s) (c) Roll angles (d) Pitch angles

1000 6000

4000 500 (N) (N)

x y 2000 0 0

Force F -500 Force F -2000 Fx - road Fx - dirt Fx - sand Fy - road Fy - dirt Fy - sand -1000 -4000 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 time (s) time (s) (e) Longitudinal Force (f) Lateral Force

100 (Nm) z 50

-50 Mz - road Mz - dirt Mz - sand -100 Self-Aligning Moment M 0 2 4 6 8 10 12 14 16 18 20 time (s) (g) Self-aligning Moment Fig. 6.16: Comparison of the forces and moments for the zig-zag maneuver on a flat plane between the 3D-AD and the FD tyre model for different terrain types, showing the internal tyre deformation states for each of the terrain types, the red, blue and cyan lines represent the road, dirt and sand terrain types respectively rate on the behaviour of forces generated in the tyres, the uncertainty in the disturbance is reduced. The modelling of these phenomena allows for the effects to be taken into account when a vehicle is being controlled so that during a cornering or terrain transition that the bounds on the system uncertainty don’t need to be unnecessarily inflated.

205 Chapter 6. Application of the 3D Analytical Dynamic Tyre Model

206 Chapter 7

Conclusions

The robust control of Autonomous Ground Vehicles (AGVs) traversing through unstructured and uneven terrain can be improved by reducing the uncertainties of how the system interacts with diﬀerent terrain conditions. This requires reliable perception of the terrain and an accurate system model in order to reduce the uncertainties. Such a reduction in the of uncertainties in the system parameters and disturbances, will assist control system designers to design signiﬁcantly improved controllers.

This thesis has presented research into the sources of uncertainty that an AGV is subject to whilst moving through an unstructured and non-uniform terrain. A number of key sources of uncertainty were identified, with changes in terrain type and thus the coefficient of friction being one key component, as well as the dynamic behaviour of the tyre interacting with the terrain. A simplified Friction Dependant tyre model was developed that takes into account changes in the frictional coefficient of the terrain as the vehicle traverses between different terrain types.

An 3D Analytical Dynamic (3D-AD) tyre model was then developed to reduce the uncertainty in the system structure by providing an analytical basis for the model. The presented terrain type classiﬁcation methodology using purely non-semantic range information was shown to be able to accurately classify the terrain types. This enables the creation of a terrain model to inform the analytical tyre model of the coeﬃcients of friction that the tyre is subject to. The combination of the perception and modelling for an AGV allows for the system structure and the parameter uncertainties to be reduced.

The application of the developed (3D-AD) tyre model allows for a descriptive system

207 Chapter 7. Conclusions model that can be used by control system designers to design improved controllers. A controller designed using this model can utilise the forces and moments that the wheel generates so that a vehicle can be more accurately controlled through an environment by vectoring the forces generated in the tyres. Controlling the motion of a vehicle in this manner allows, not only more accurately control, but also greater rejection of disturbances the vehicle is subjected to during a maneuver.

The application of developed 3D-AD tyre model would allow for a vehicle travelling over a terrain that is subjected to a sudden loss in friction to be adequately controlled by vectoring the available forces so that the disturbance has minimal impact on the motion of the vehicle. This type of mechanism is important for vehicles that are moving through unstructured and non-uniform terrain as increasing disturbance rejection allows for the vehicle to be able to maintain reliable control over a wide variety of terrain frictional characteristics.

The development of the Extended Range Texture Analysis (ERTA) methodology provides another source of information for generating a map of an environment so that an AGV can predict the type of terrain that the AGV might encounter. The ERTA method is able to provide a solution for classifying the terrain types in an environment even in the presence of minimal or no visual or semantic spatial features in a scene. An accurate terrain model can be developed through the use of the ERTA method, which reduces the uncertainties that exist in the system model.

In this thesis it was necessary to be able to identify the sources of uncertainty for an AGV travelling through a non-uniform environment. This was achieved in Chapter 3 through the development of a three-dimensional 4-Wheel Drive 4-Wheel Steer (4WD4WS) vehicle model. In combination with the dynamic vehicle model a simple static tyre model was developed that included the effects of changing terrain conditions on the forces and subsequent trajectory that the vehicle experiences. Although the simple tyre model did not portray the real case, it was still able to be used to identify the sources of uncertainty in the model with the tyres that transitioned into a different terrain providing different longitudinal and lateral forces. These different forces resulted in undesirable motion of the vehicles in most cases.

The transition between terrain types was identiﬁed as a main source of uncertainty

208 for the vehicle model and so a three-dimensional analytical dynamic tyre model was developed in Chapter 4 that was more capable at replicating the experienced forces and moments, with the developed analytical dynamic tyre model able to successfully replicate experimental data presented elsewhere in the form of the Magic Formula tyre model. The developed model is an extension of the LuGre dynamic tyre model that had previously been developed for two dimensions, with the introduction of the tyre contact patch width for the developed model. This added the ability of the tyre model to take into account the eﬀects of steering rate on the development of forces and moments within the tyre contact patch.

In addition to the development of a three-dimensional dynamic tyre model, a novel force distribution has been developed which is velocity and camber angle dependant. The developed force distribution includes the width of the contact patch, which allows for the overturning moment caused by a tyre subject to the inﬂuence of a non-zero camber angle to be generated. The developed overturning moment and rolling resistance were able to replicate the Magic Formula tyre model experimental results.

The requirements of the developed tyre model were that the coefficients of friction for a particular terrain be known. Two novel methodologies were developed to be able to achieve this as shown in Chapter 5, firstly the Range Texture Analysis method and then the Extended Range Texture Analysis method. Through the use of experimental results, two different terrain classification methods had been proposed which are able provide terrain classification results for a scene purely from spatial information alone. The second method is the most advanced version of the two which along with classifying the scene into available terrain types it is also able to identify areas of a scene where there exists multiple terrain types and divide these regions so that a more accurate representation of the scene is produced.

Through the use of experimental results, the Extended Range Texture Analysis method was shown to be able to achieve a base classification accuracy of 87.9% and 91.4% classification accuracy with a 40% confidence threshold. It was also shown that in a comparison between the use of the GLCM texture features, Radial Power Spectral Density features and Gabor features, that GLCM texture features provided the best classification performance for all three features.

209 Chapter 7. Conclusions

An environmental model was also developed to incorporate the information gathered from the scene so that it can be used to inform control or localisation algorithms. The µ-patch terrain model incorporates the frictional characteristics as well as geometric information of the terrain, and provides an interpretation of unclassified scene elements. This allows for uncertainty in the system to be quantified, with the regions of terrain with low classification confidence to be treated separately from regions with high classification confidence so that control over these areas can be tuned appropriately. Finally, in Chapter 6, the incorporation of the developed analytical dynamic tyre model and the developed terrain model into the 3D vehicle dynamic model, allowed for a direct comparison between the developed dynamic model trajectory and the kinematic and Magic Formula based trajectories. The comparison showed that in regions where changes in frictional conditions exist that the developed dynamic tyre model is able to replicate similar behaviour as seen in a number of experimental conditions.

210 7.1. Technical Contributions

7.1 Technical Contributions

The major technical contributions for this thesis are as follows:

• A novel method of developing a confidence in classification metric to be able to quantify the confidence in the current classification result.

• A three-dimensional dynamic model of a 4WD4WS vehicle has been developed that enables an analysis of different tyre models including the developed 3D Analytical Dynamic tyre model and the developed Friction Dependant tyre model, subjected to a variety of different terrain configurations and geometry.

211 Chapter 7. Conclusions

7.2 Future Work

The work presented in this thesis contains a number of different avenues for future work. The developed 3D analytical tyre model can be extended to include the effects of changing normal forces on the spring and damper coefficients in the tyre. These additional parameters would then allow for the tyre model to be able to maintain accuracy for different tyre normal forces. The 3D analytical tyre model can also be extended to allow for the tyre to be steered about any center of rotation, not just the geometric center of the contact patch. This would allow a vehicle with any steering configuration to be able to utilise the developed 3D Analytical Dynamic tyre model to account for the effect of the steering rate on the developed tyre forces and moments. The research for the Extended Range Texture Analysis methodology can also be expanded to be able to provide a more robust confidence metric that is able to take into account the similarities that exist between the available terrain class statistical information. By taking the effects of the similarities between the available classes in providing the confidence metric, it is possible to also predict when the encountered terrain region may contain an entirely new terrain type that can be included in set of terrain types. The current implementation of the texture generation does not work in real-time — this process needs to be optimised in order to reduce the processing time so that the ERTA methodology is capable of running in real-time. This can be achieved by re-using repeated calculations as well as running the process concurrently.

212 References

[1] Jayantha Katupitiya, Ray Eaton, Anthony Cole, Craig Meyer, and Guy Rodnay. Automation of an agricultural tractor for fruit picking. In Robotics and Automation, 2005. ICRA 2005. Proceedings of the 2005 IEEE International Conference on, pages 3201–3206. IEEE, 2005.

[2] R. Eaton, J. Katupitiya, K. W. Siew, and K. S. Dang. Precision guidance of agricultural tractors for autonomous farming. In Systems Conference, 2008 2nd Annual IEEE, pages 1–8, April 2008.

[3] Ray Eaton, Jay Katupitiya, Kheng Wah Siew, and Blair Howarth. Autonomous farming: Modelling and control of agricultural machinery in a uniﬁed framework. International journal of intelligent systems technologies and applications, 8(1):444–457, 2010.

[4] Dong Sang Yoo and Myung-Jin Chung. A variable structure control with simple adaptation laws for upper bounds on the norm of the uncertainties. Automatic Control, IEEE Transactions on, 37(6):860–865, Jun 1992.

[5] J. S. Park, Y. A Jiang, T. Hesketh, and D.J. Clements. Trajectory control of manipulators using adaptive sliding mode control. In Southeastcon ’94. Creative Technology Transfer - A Global Aﬀair., Proceedings of the 1994 IEEE, pages 142–146, Apr 1994.

[6] Ying-Jeh Huang, Tzu-Chun Kuo, and Shin-Hung Chang. Adaptive sliding-mode control for nonlinearsystems with uncertain parameters. Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on, 38(2):534–539, 2008.

213 REFERENCES

[7] B.L. Cong, Z. Chen, and X.D. Liu. On adaptive sliding mode control without switching gainoverestimation. International Journal of Robust and Nonlinear Control, 24(3):515–531, 2014.

[8] ST Venkataraman and S Gulati. Control of nonlinear systems using terminal sliding modes. Journal of dynamic systems, measurement, and control, 115(3):554–560, 1993.

[9] Gabriel Elkaim, Michael O’Connor, Thomas Bell, and BW Parkinson. System identiﬁcation and robust control of a farm vehicle using cdgps. In Proceedings of ION GPS, volume 10, pages 1415–1426. Citeseer, 1997.

[10] Hao Fang, Roland Lenain, Benoit Thuilot, and Philippe Martinet. Robust adaptive control of automatic guidance of farm vehicles in the presence of sliding. In Robotics and Automation, 2005. ICRA 2005. Proceedings of the 2005 IEEE International Conference on, pages 3102–3107. IEEE, 2005.

[11] R. Eaton, H. Pota, and J. Katupitiya. Path tracking control of agricultural tractors with compensation for steering dynamics. In Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on, pages 7357–7362, Dec 2009.

[12] Javad Taghia and Jayantha Katupitiya. A sliding mode controller with disturbance observer for a farm vehicle operating in the presence of wheel slip. In Advanced Intelligent Mechatronics (AIM), 2013 IEEE/ASME International Conference on, pages 1534–1539. IEEE, 2013.

[13] Ray Eaton, Jayantha Katupitiya, Himanshu Pota, and Kheng Wah Siew. Robust sliding mode control of an agricultural tractor under the inﬂuence of slip. In Advanced Intelligent Mechatronics, 2009. AIM 2009. IEEE/ASME International Conference on, pages 1873–1878. IEEE, 2009.

[14] Qichang He, Xiumin Fan, and Dengzhe Ma. Full bicycle dynamic model for interactive bicycle simulator. Journal of Computing and Information Science in Engineering, 5(4):373–380, 2005.

214 REFERENCES

[15] Michael Baloh and Michael Parent. Modeling and model veriﬁcation of an intelligent self-balancing two-wheeled vehicle for an autonomous urban transportation system. In The conference on computational intelligence, robotics, and autonomous systems, pages 1–7, 2003.

[16] Arnoldo Castro. Modeling and dynamic analysis of a two-wheeled inverted-pendulum. Master’s thesis, Georgia Institute of Technology, 2012.

[17] Mohsen Bayani Khaknejad, Amir Ali Janbakhsh, Arash Keshavarz, and Reza Kazemi. Developing an analytical tyre model that estimates tyre forces using a least squares method. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 226(9):1193–1207, 2012.

[18] Shin-ichiro Sakai, Hideo Sado, and Yoichi Hori. Motion control in an electric vehicle with four independently driven in-wheel motors. Mechatronics, IEEE/ASME Transactions on, 4(1):9–16, 1999.

[19] YH Zweiri, LD Seneviratne, and K Althoefer. Modelling and control of an unmanned excavator vehicle. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 217(4):259–274, 2003.

[20] S. Al-Milli, K. Althoefer, and L.D. Seneviratne. Dynamic analysis and traversability prediction of tracked vehicles on soft terrain. In Networking, Sensing and Control, 2007 IEEE International Conference on, pages 279–284, April 2007.

[21] Elie Maalouf, Maarouf Saad, and Hamadou Saliah. A higher level path tracking controller for a four-wheel diﬀerentially steered mobile robot. Robotics and Autonomous Systems, 54(1):23 – 33, 2006.

[22] Laura R Ray, Devin C Brande, and James H Lever. Estimation of net traction for diﬀerential-steered wheeled robots. Journal of Terramechanics, 46(3):75–87, 2009.

[23] Laura E Ray. Estimation of terrain forces and parameters for rigid-wheeled vehicles. Robotics, IEEE Transactions on, 25(3):717–726, 2009.

215 REFERENCES

[24] Yew Keong Tham, Han Wang, and Eam Khwang Teoh. Multi-sensor fusion for steerable four-wheeled industrial vehicles. Control engineering practice, 7(10):1233–1248, 1999.

[25] Shou-Tao Peng, Chau-Chin Chang, and Jer-Jia Sheu. On robust bounded control of the combined wheel slip with integral compensation for an autonomous 4ws4wd vehicle. Vehicle System Dynamics, 45(5):477–503, 2007.

[26] Imad Khan and Matthew Spenko. Dynamics and control of an omnidirectional unmanned ground vehicle. In Intelligent Robots and Systems, 2009. IROS 2009. IEEE/RSJ International Conference on, pages 4110–4115. IEEE, 2009.

[27] Kim Son Dang. Design and control of autonomous crop tracking robotic weeder: Greenweeder. Master’s thesis, University of New South Wales, Australia, 2009.

[28] R. Madhavan and C. Schlenoﬀ. The eﬀect of process models on short-term prediction of moving objects for unmanned ground vehicles. In Intelligent Transportation Systems, 2004. Proceedings. The 7th International IEEE Conference on, pages 471–476, Oct 2004.

[29] Lei Feng, Yong He, Yidan Bao, and Hui Fang. Development of trajectory model for a tractor-implement system for automated navigation applications. In Instrumentation and Measurement Technology Conference, 2005. IMTC 2005. Proceedings of the IEEE, volume 2, pages 1330–1334, May 2005.

[30] Lauro Ojeda, Daniel Cruz, Giulio Reina, and Johann Borenstein. Current-based slippage detection and odometry correction for mobile robots and planetary rovers. Robotics, IEEE Transactions on, 22(2):366–378, 2006.

[31] Xiaojing Song, Zibin Song, Lakmal D Seneviratne, and Kaspar Althoefer. Optical ﬂow-based slip and velocity estimation technique for unmanned skid-steered vehicles. In Intelligent Robots and Systems, 2008. IROS 2008. IEEE/RSJ International Conference on, pages 101–106. IEEE, 2008.

[32] Xiaojing Song, Lakmal D Seneviratne, Kaspar Althoefer, and Zibin Song. A robust slip estimation method for skid-steered mobile robots. In Control,

216 REFERENCES

Automation, Robotics and Vision, 2008. ICARCV 2008. 10th International Conference on, pages 279–284. IEEE, 2008.

[33] Matthew Spenko, Yoji Kuroda, Steven Dubowsky, and Karl Iagnemma. Hazard avoidance for high-speed mobile robots in rough terrain. Journal of Field Robotics, 23(5):311–331, 2006.

[34] Z. Shiller and Y.-R. Gwo. Dynamic motion planning of autonomous vehicles. Robotics and Automation, IEEE Transactions on, 7(2):241–249, Apr 1991.

[35] Jan Van Der Burg and Pierre Blazevic. Anti-lock braking and traction control concept for all-terrain robotic vehicles. In Robotics and Automation, 1997. Proceedings., 1997 IEEE International Conference on, volume 2, pages 1400–1405. IEEE, 1997.

[36] SangHyun Joo, JeongHan Lee, YongWoon Park, WanSuk Yoo, and Jihong Lee. Real time traversability analysis to enhance rough terrain navigation for an 6 6 autonomous vehicle. Journal of Mechanical Science and Technology, 27(4):1125–1134, 2013.

[37] Junmin Wang and R.G. Longoria. Coordinated vehicle dynamics control with control distribution. In American Control Conference, 2006, pages 6 pp.–, June 2006.

[38] Wei Yu, O.Y. Chuy, Jr. Collins, E.G., and P. Hollis. Analysis and experimental veriﬁcation for dynamic modeling of a skid-steered wheeled vehicle. Robotics, IEEE Transactions on, 26(2):340–353, April 2010.

[39] Yan Chen and Junmin Wang. Energy-eﬃcient control allocation with applications on planar motion control of electric ground vehicles. In American Control Conference (ACC), 2011, pages 2719–2724, June 2011.

[40] Hemanshu Pota, Jayantha Katupitiya, and Ray Eaton. Simulation of a tractor-implement model under the inﬂuence of lateral disturbances. In Decision and Control, 2007 46th IEEE Conference on, pages 596–601. IEEE, 2007.

217 REFERENCES

[41] K. Macek, K. Thoma, R. Glatzel, and R. Siegwart. Dynamics modeling and parameter identiﬁcation for autonomous vehicle navigation. In Intelligent Robots and Systems, 2007. IROS 2007. IEEE/RSJ International Conference on, pages 3321–3326, Oct 2007.

[42] Howard Dugoﬀ, PS Fancher, and Leonard Segel. An analysis of tire traction properties and their inﬂuence on vehicle dynamic performance. Technical report, SAE Technical Paper, 1970.

[43] David R. Ingham. Car diagram from wikipedia entry of unsprung mass. https://commons.wikimedia.org/wiki/File:Car_diagram.jpg, 2005. [Online; accessed April 27, 2013].

[44] Gary Witus. Mobility and dynamics modeling for unmanned ground vehicle motion planning. volume 3693, pages 32–43, 1999.

[45] MSC Software. Using Adams/Tire, 7 2014. Rev. 3.

[46] MSC Software. Adams/Tire help - Adams 2013, 7 2014. Rev. 3.

[47] Hans B. Pacejka and Egbert Bakker. The magic formula tyre model. Vehicle System Dynamics, 21(sup001):1–18, 1992.

[48] SJ Anderson, SC Peters, Karl Iagnemma, and J Overholt. Semi-autonomous stability control and hazard avoidance for manned and unmanned ground vehicles. Technical report, DTIC Document, 2010.

[49] Randy Sleight and Sunil K Agrawal. Dynamic model of a four-wheel-drive hmmwv. In ASME 2004 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pages 1183–1191. American Society of Mechanical Engineers, 2004.

[50] Randy Sleight. Modeling and Control of an Autonomous HMMWV. PhD thesis, University of Delaware, 2004.

[51] A.S. Matveev, M. Hoy, and A.V. Savkin. Mixed nonlinear-sliding mode control of an unmanned farm tractor in the presence of sliding. In Control Automation

218 REFERENCES

Robotics Vision (ICARCV), 2010 11th International Conference on, pages 927–932, Dec 2010.

[52] M.G. Bekker. Theory of land locomotion: the mechanics of vehicle mobility. University of Michigan Press, 1956.

[53] RS Sharp and M Bettella. Shear force and moment descriptions by normalisation of parameters and the magic formula. Vehicle system dynamics, 39(1):27–56, 2003.

[54] IJM Besselink, AJC Schmeitz, and HB Pacejka. An improved magic formula/swift tyre model that can handle inﬂation pressure changes. Vehicle System Dynamics, 48(S1):337–352, 2010.

[55] J.Y. Wong. Theory of ground vehicles, 4th ed. John Wiley & Sons Inc., NJ, USA, 2008.

[56] C.C. Ward and K. Iagnemma. A dynamic-model-based wheel slip detector for mobile robots on outdoor terrain. Robotics, IEEE Transactions on, 24(4):821 –831, August 2008.

[57] Claeys Xavier, Yi Jingang, L. Alvarez, R. Horowitz, and C. C. de Wit. A dynamic tire/road friction model for 3d vehicle control and simulation. In Intelligent Transportation Systems, 2001. Proceedings. 2001 IEEE, pages 483–488, 2001.

[58] J. Yi, L. Alvarez, R. Horowitz, and C. C. de Wit. Adaptive emergency braking control using a dynamic tire/road friction model. In Decision and Control, 2000. Proceedings of the 39th IEEE Conference on, volume 1, pages 456–461 vol.1, 2000.

[59] M.R. Woods and J. Katupitiya. Modelling of a 4ws4wd vehicle and its control for path tracking. In Computational Intelligence in Control and Automation (CICA), 2013 IEEE Symposium on, pages 155–162, 2013.

[60] PR Dahl. A solid friction model. Technical report, DTIC Document, 1968.

[61] C. Canudas-de Wit, H. Olsson, K.J. Astrom, and P. Lischinsky. A new model for control of systems with friction. Automatic Control, IEEE Transactions on, 40(3):419–425, Mar 1995.

219 REFERENCES

[62] Joˇsko Deur, Jahan Asgari, and Davor Hrovat. A dynamic tire friction model for combined longitudinal and lateral motion. In Proceedings of the ASME-IMECE World Conference, 2001.

[63] Carlos Canudas-de Wit, Panagiotis Tsiotras, Efstathios Velenis, Michel Basset, and Gerard Gissinger. Dynamic friction models for road/tire longitudinal interaction. Vehicle System Dynamics, 39(3):189–226, 2003.

[64] Joko Deur, Jahan Asgari, and Davor Hrovat. A 3d brush-type dynamic tire friction model. Vehicle System Dynamics, 42(3):133–173, 2004.

[65] E. Velenis, P. Tsiotras, C. Canudas-de Wit, and M. Sorine. Dynamic tyre friction models for combined longitudinal and lateral vehicle motion. Vehicle System Dynamics, 43(1):3–29, 2005.

[66] Sally A Shoop. Finite element modeling of tire-terrain interaction. Technical report, DTIC Document, 2001.

[67] H. Holscher, M. Tewes, N. Botkin, M. Lohndorf, KH Hoﬀmann, and E. Quandt. Modeling of pneumatic tires by a ﬁnite element model for the development a tire friction remote sensor. Center of Advanced European Studies and Research (CAESAR), Ludwig-Erhard-Allee, 2:53175, 2004.

[68] Hao Li and Christian Schindler. Three-dimensional ﬁnite element and analytical modelling of tyre–soil interaction. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, page 1464419312464183, 2012.

[69] Mieczyslaw Gregory Bekker. Introduction to terrain-vehicle systems. University of Michigan Press, 1969.

[70] JR Matijevic, J Crisp, DB Bickler, RS Banes, BK Cooper, HJ Eisen, J Gensler, A Haldemann, F Hartman, KA Jewett, et al. Characterization of the martian surface deposits by the mars pathﬁnder rover, sojourner. Science, 278(5344):1765–1768, 1997.

220 REFERENCES

[71] Y Nohse, K Hashiguchi, M Ueno, T Shikanai, H Izumi, and F Koyama. A measurement of basic mechanical quantities of oﬀ-the-road traveling performance. Journal of Terramechanics, 28(4):359–370, 1991.

[72] K. Iagnemma, Shinwoo Kang, H. Shibly, and S. Dubowsky. Online terrain parameter estimation for wheeled mobile robots with application to planetary rovers. Robotics, IEEE Transactions on, 20(5):921–927, 2004.

[73] Hans B Pacejka, Egbert Bakker, and Lars Nyborg. Tyre modelling for use in vehicle dynamics studies. SAE paper, 870421, 1987.

[74] Amnon Sitchin. Acquisition of transient tire force and moment data for dynamic vehicle handling simulations. Technical report, SAE Technical Paper, 1983.

[75] HB Pacejka. Tire and Vehicle Dynamics. Butterworth-Heinemann, 2012.

[76] H. B. Pacejka and I. J. M. Besselink. Magic formula tyre model with transient properties. Vehicle System Dynamics, 27(sup001):234–249, 1997.

[77] Hans B Pacejka. Spin: camber and turning. Vehicle System Dynamics, 43(sup1):3–17, 2005.

[78] IJM Besselink, HB Pacejka, AJC Schmeitz, and STH Jansen. The mf-swift tyre model: extending the magic formula with rigid ring dynamics and an enveloping model. JSAE review, 26(2), 2005.

[79] R Hadekel. The mechanical characteristics of pneumatic tires. Clearinghouse Fed Sci & Tech Info, 1952.

[80] John Ronaine Ellis. Vehicle handling dynamics. Mechanical Engineering Publication Limited, 1994.

[81] KM Captain, AB Boghani, and DN Wormley. Analytical tire models for dynamic vehicle simulation. Vehicle System Dynamics, 8(1):1–32, 1979.

[82] Brian Armstrong-Helouvry. Control of machines with friction, volume 128. Springer, 1991.

221 REFERENCES

[83] Brian Armstrong-H´elouvry, Pierre Dupont, and Carlos Canudas De Wit. A survey of models, analysis tools and compensation methods for the control of machines with friction. Automatica, 30(7):1083–1138, 1994.

[84] H. Olsson, K.J. strm, C. Canudas de Wit, M. Gfvert, and P. Lischinsky. Friction models and friction compensation. European Journal of Control, 4(3):176 – 195, 1998.

[85] Tom French. Tyre technology. 1989.

[86] A. Y. Coran, K. Boustany, and P. Hamed. Unidirectional ﬁberpolymer composites: Swelling and modulus anisotropy. Journal of Applied Polymer Science, 15(10):2471–2485, 1971.

[87] Xianglan Zhang, S Rakheja, and R Ganesan. Modal analysis of a truck tyre using fe tyre model. International Journal of Heavy Vehicle Systems, 11(2):133–154, 2004.

[88] Lucian Drgu, Thomas Schauppenlehner, Andreas Muhar, Josef Strobl, and Thomas Blaschke. Optimization of scale and parametrization for terrain segmentation: An application to soil-landscape modeling. Computers & Geosciences, 35(9):1875 – 1883, 2009.

[89] M. Himmelsbach, T. Luettel, and H. Wuensche. Real-time object classiﬁcation in 3d point clouds using point feature histograms. In Intelligent Robots and Systems, 2009. IROS 2009. IEEE/RSJ International Conference on, pages 994–1000, Oct 2009.

[90] B. Douillard, J. Underwood, N. Kuntz, V. Vlaskine, A. Quadros, P. Morton, and A. Frenkel. On the segmentation of 3d lidar point clouds. In Robotics and Automation (ICRA), 2011 IEEE International Conference on, pages 2798–2805, May 2011.

[91] B. Douillard, A. Quadros, P. Morton, J.P. Underwood, M. De Deuge, S. Hugosson, M. Hallstrom, and T. Bailey. Scan segments matching for pairwise 3d alignment.

222 REFERENCES

In Robotics and Automation (ICRA), 2012 IEEE International Conference on, pages 3033–3040, May 2012.

[92] F. Moosmann, O. Pink, and C. Stiller. Segmentation of 3d lidar data in non-ﬂat urban environments using a local convexity criterion. In Intelligent Vehicles Symposium, 2009 IEEE, pages 215–220, June 2009.

[93] Frederick Pauling, Michael Bosse, and Robert Zlot. Automatic segmentation of 3d laser point clouds by ellipsoidal region growing. In Australasian Conference on Robotics and Automation, page 10, 2009.

[94] Xiaolong Zhu, Huijing Zhao, Yiming Liu, Yipu Zhao, and Hongbin Zha. Segmentation and classiﬁcation of range image from an intelligent vehicle in urban environment. In Intelligent Robots and Systems (IROS), 2010 IEEE/RSJ International Conference on, pages 1457–1462, Oct 2010.

[95] J. Strom, Andrew Richardson, and E. Olson. Graph-based segmentation for colored 3d laser point clouds. In Intelligent Robots and Systems (IROS), 2010 IEEE/RSJ International Conference on, pages 2131–2136, Oct 2010.

[96] Dongshin Kim, Sang Min Oh, and J.M. Rehg. Traversability classiﬁcation for ugv navigation: a comparison of patch and superpixel representations. In Intelligent Robots and Systems, 2007. IROS 2007. IEEE/RSJ International Conference on, pages 3166–3173, 2007.

[97] Andres Huertas, Larry Matthies, and Arturo Rankin. Stereo-based tree traversability analysis for autonomous oﬀ-road navigation. In Application of Computer Vision, 2005. WACV/MOTIONS’05 Volume 1. Seventh IEEE Workshops on, volume 1, pages 210–217. IEEE, 2005.

[98] Quan Qiu and JianDa Han. 2.5-dimensional angle potential ﬁeld algorithm for the real-time autonomous navigation of outdoor mobile robots. Science China Information Sciences, 54(10):2100–2112, 2011.

[99] Ayanna Howard and Homayoun Seraji. Vision-based terrain characterization and traversability assessment. Journal of Robotic Systems, 18(10):577–587, 2001.

223 REFERENCES

[100] A Shirkhodaie, R Amrani, N Chawla, and T Vicks. Traversable terrain modeling and performance measurement of mobile robots. Technical report, DTIC Document, 2004.

[101] Amir Shirkhodaie, Rachida Amrani, and Edward Tunstel. Soft computing for visual terrain perception and traversability assessment by planetary robotic systems. In Systems, Man and Cybernetics, 2005 IEEE International Conference on, volume 2, pages 1848–1855. IEEE, 2005.

[102] Pierre Sermanet, Raia Hadsell, Marco Scoﬃer, Matt Grimes, Jan Ben, Ayse Erkan, Chris Crudele, Urs Miller, and Yann LeCun. A multirange architecture for collision-free oﬀ-road robot navigation. Journal of Field Robotics, 26(1):52–87, 2009.

[103] Jiajun Gu, Qixin Cao, and Yi Huang. Rapid traversability assessment in 2.5 d grid-based map on rough terrain. International Journal of Advanced Robotic Systems, 5(4), 2008.

[104] Gregory Broten, David Mackay, and Jack Collier. Probabilistic obstacle detection using 2 1/2 d terrain maps. In Computer and Robot Vision (CRV), 2012 Ninth Conference on, pages 17–23. IEEE, 2012.

[105] Jim W Hall, Kelly L Smith, Leslie Titus-Glover, James C Wambold, Thomas J Yager, and Zoltan Rado. Guide for pavement friction. National Cooperative Highway Research Program, Transportation Research Board of the National Academies, 2009.

[106] Martin A Fischler and Robert C Bolles. Random sample consensus: a paradigm for model ﬁtting with applications to image analysis and automated cartography. Communications of the ACM, 24(6):381–395, 1981.

[107] Michael Ying Yang and Wolfgang F¨orstner. Plane detection in point cloud data. In Proceedings of the 2nd int conf on machine control guidance, Bonn, volume 1, pages 95–104, 2010.

224 REFERENCES

[108] Faisal Mufti, Robert Mahony, and Jochen Heinzmann. Robust estimation of planar surfaces using spatio-temporal RANSAC for applications in autonomous vehicle navigation. Robotics and Autonomous Systems, 60(1):16 – 28, 2012.

[109] R. Schnabel, R. Wahl, and R. Klein. Eﬃcient ransac for point-cloud shape detection. Computer Graphics Forum, 26(2):214–226, 2007.

[110] Hao Tang, Zhigang Zhu, and Jizhong Xiao. Stereovision-based 3d planar surface estimation for wall-climbing robots. In Intelligent Robots and Systems, 2009. IROS 2009. IEEE/RSJ International Conference on, pages 97–102, Oct 2009.

[111] Alicia Robledo, Stephen Cossell, and Jose Guivant. Outdoor ride: Data fusion of a 3d kinect camera installed in a bicycle. In Proceedings of the 2011 Australasian Conference on Robotics & Automation, Melbourne, Australia, 2011.

[112] Blair Howarth, Jayantha Katupitiya, Jose Guivant, and Andrew Szwec. Novel robotic 3d surface mapping using range and vision fusion. In Intelligent Robots and Systems (IROS), 2010 IEEE/RSJ International Conference on, pages 1539–1544. IEEE, 2010.

[113] Blair Howarth, Mark Whitty, Jayantha Katupitiya, and Jose Guivant. Extraction and grouping of surface features for 3d mapping. In Proc. Australasian Conference on Robotics and Automation. Melbourne, pages 1–6, 2011.

[114] Xiaoyi Jiang and Horst Bunke. Edge detection in range images based on scan line approximation. Computer Vision and Image Understanding, 73(2):183–199, 1999.

[115] I Shmulevich, D Ronai, and D Wolf. A new ﬁeld single wheel tester. Journal of Terramechanics, 33(3):133–141, 1996.

[116] Suksun Hutangkabodee, YahyaHashem Zweiri, LakmalDasarath Seneviratne, and Kaspar Althoefer. Soil parameter identiﬁcation for wheel-terrain interaction dynamics and traversability prediction. International Journal of Automation and Computing, 3(3):244–251, 2006.

225 REFERENCES

[117] Steﬀen Muller, Michael Uchanski, and Karl Hedrick. Estimation of the maximum tire-road friction coeﬃcient. Journal of dynamic systems, measurement, and control, 125(4):607–617, 2003.

[118] Jayoung Kim, D. Kim, Jonghwa Lee, Jihong Lee, H. Joo, and In-So Kweon. Non-contact terrain classiﬁcation for autonomous mobile robot. In Robotics and Biomimetics (ROBIO), 2009 IEEE International Conference on, pages 824–829, 2009.

[119] Anelia Angelova, Larry Matthies, Daniel Helmick, and Pietro Perona. Slip prediction using visual information. In Proceedings of Robotics Science and Systems, 12:15, 2006.

[120] Shifeng Wang, Sarath Kodagoda, and Ravindra Ranasinghe. Road terrain type classiﬁcation based on laser measurement system data. Proceedings of the 2012 Australasian Conference on Robotics & Automation, 2012.

[121] M. Ergun, S. Iyinam, and A. Iyinam. Prediction of road surface friction coeﬃcient using only macro- and microtexture measurements. Journal of Transportation Engineering, 131(4):311–319, 2005.

[122] Eyad Masad, Arash Rezaei, Arif Chowdhury, and Pat Harris. Predicting asphalt mixture skid resistance based on aggregate characteristics. Technical report, Texas Transportation Institute, Texas A&M University System, 2009.

[123] Fredrik Gustafsson. Slip-based tire-road friction estimation. Automatica, 33(6):1087–1099, 1997.

[124] Luis Alvarez, Jingang Yi, Roberto Horowitz, and Luis Olmos. Dynamic friction model-based tire-road friction estimation and emergency braking control. Journal of dynamic systems, measurement, and control, 127(1):22–32, 2005.

[125] Chang Sun Ahn. Robust estimation of road friction coeﬃcient for vehicle active safety systems. PhD thesis, The University of Michigan, 2011.

226 REFERENCES

[126] A. Gendy and A. Shalaby. Mean proﬁle depth of pavement surface macrotexture using photometric stereo techniques. Journal of Transportation Engineering, 133(7):433–440, 2007.

[127] U Eichhorn and J Roth. Prediction and monitoring of tyre/road friction. In XXIV FISITA Congress, 7-11 June 1992, London. Held at the Automotive Technology Servicing Society. Technical Papers. Safety, The Vehicle and The Road. Volume 2 (IMECHE No C389/321 and FISITA No 925226), 1992.

[128] Suksun Hutangkabodee, Yahya H Zweiri, Lakmal D Seneviratne, and Kaspar Althoefer. Validation of soil parameter identiﬁcation for track-terrain interaction dynamics. In Intelligent Robots and Systems, 2007. IROS 2007. IEEE/RSJ International Conference on, pages 3174–3179. IEEE, 2007.

[129] Suksun Hutangkabodee, Yahya Zweiri, Lakmal Seneviratne, and Kaspar Althoefer. Soil parameter identiﬁcation and driving force prediction for wheel-terrain interaction. International Journal of Advanced Robotic Systems, 5(4):425A432,´ 2008.

[130] Lakmal Seneviratne, Yahya Zweiri, Suksun Hutangkabodee, Zibin Song, Xiaojing Song, Savan Chhaniyara, Said Al-Milli, and Kaspar Althoefer. The modelling and estimation of driving forces for unmanned ground vehicles in outdoor terrain. International Journal of Modelling, Identiﬁcation and Control, 6(1):40–50, 2009.

[131] P. Jansen, W. van der Mark, J.C. van den Heuvel, and F.C.A. Groen. Colour based oﬀ-road environment and terrain type classiﬁcation. In Intelligent Transportation Systems, 2005. Proceedings. 2005 IEEE, pages 216–221, 2005.

[132] C.A. Brooks and K.D. Iagnemma. Self-supervised classiﬁcation for planetary rover terrain sensing. In Aerospace Conference, 2007 IEEE, pages 1–9, 2007.

[133] Mingjun Wang, Jun Zhou, Jun Tu, and CL Liu. Learning long-range terrain perception for autonomous mobile robots. International Journal of Advanced Robotic Systems, 7(1):55–66, 2010.

227 REFERENCES

[134] David Stavens and Sebastian Thrun. A self-supervised terrain roughness estimator for oﬀ-road autonomous driving. In Conference on Uncertainty in AI (UAI), July 2006.

[135] Hendrik Dahlkamp, Adrian Kaehler, David Stavens, Sebastian Thrun, and Gary R Bradski. Self-supervised monocular road detection in desert terrain. In Robotics: science and systems. Philadelphia, 2006.

[136] Marcel H¨aselich, Marc Arends, Nicolai Wojke, Frank Neuhaus, and Dietrich Paulus. Probabilistic terrain classiﬁcation in unstructured environments. Robotics and Autonomous Systems, 2012.

[137] Rebecca Castano, Roberto Manduchi, and Justin Fox. Classiﬁcation experiments on real-world texture. In Third Workshop on Empirical Evaluation Methods in Computer Vision, Kauai, Hawaii, December 10, 2001. Pasadena, CA: Jet Propulsion Laboratory, National Aeronautics and Space Administration, 2001., 2001.

[138] A Talukder, R. Manduchi, R. Castano, K. Owens, L. Matthies, A Castano, and R. Hogg. Autonomous terrain characterisation and modelling for dynamic control of unmanned vehicles. In Intelligent Robots and Systems, 2002. IEEE/RSJ International Conference on, volume 1, pages 708–713 vol.1, 2002.

[139] Radhakrishna Achanta, Appu Shaji, Kevin Smith, Aurelien Lucchi, Pascal Fua, and Sabine Susstrunk. Slic superpixels compared to state-of-the-art superpixel methods. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 34(11):2274–2282, 2012.

[140] Thomas Leung and Jitendra Malik. Representing and recognizing the visual appearance of materials using three-dimensional textons. International Journal of Computer Vision, 43(1):29–44, 2001.

[141] Sebastian Thrun, Michael Montemerlo, and Andrei Aron. Probabilistic terrain analysis for high-speed desert driving. In Robotics: Science and Systems, pages 16–19, 2006.

228 REFERENCES

[142] E.S. Gadelmawla. A vision system for surface roughness characterization using the gray level co-occurrence matrix. NDT & E International, 37(7):577 – 588, 2004.

[143] Muguras Mocofan and Vasiu Radu. A 3d approach to co-occurrence matrix features used in dynamic textures indexing. In Advanced Engineering Forum, volume 8, pages 516–526. Trans Tech Publ, 2013.

[144] Dong-Min Woo, Dong-Chul Park, Seung-Soo Han, and Quoc-Dat Nguyen. Application of 3d co-occurrence features to terrain classiﬁcation. In Long-Wen Chang and Wen-Nung Lie, editors, Advances in Image and Video Technology, volume 4319 of Lecture Notes in Computer Science, pages 305–313. Springer Berlin Heidelberg, 2006.

[145] Jun Zhao, Joel Tow, and Jayantha Katupitiya. On-tree fruit recognition using texture properties and color data. In Intelligent Robots and Systems, 2005.(IROS 2005). 2005 IEEE/RSJ International Conference on, pages 263–268. IEEE, 2005.

[146] J Boot. ATV control:regulating a 4WD/4WS autonomous guided vehicle. Master’s thesis, Technische Universiteit Eindhoven, 2005.

[147] Gurkan Erdogan, Lee Alexander, and Rajesh Rajamani. Friction coeﬃcient measurement for autonomous winter road maintenance. Vehicle System Dynamics, 47(4):497–512, 2009.

[148] Laura R. Ray. Nonlinear tire force estimation and road friction identiﬁcation: Simulation and experiments1,2. Automatica, 33(10):1819 – 1833, 1997.

[149] Kyongsu Yi, Karl Hedrick, and Seong-Chul Lee. Estimation of tire-road friction using observer based identiﬁers. Vehicle System Dynamics, 31(4):233–261, 1999.

[150] J Harned, L Johnston, and G Scharpf. Measurement of tire brake force characteristics as related to wheel slip (antilock) control system design. SAE Transactions, 78(690214):909–925, 1969.

[151] David E. Hall and Cal M. J. Fundamentals of rolling resistance. Rubber Chemistry and Technology, 74(3):525, Jul 2001.

229 REFERENCES

[152] Gai-ling Ma, Hong Xu, and Wen-yong Cui. Computation of rolling resistance caused by rubber hysteresis of truck radial tire. Journal of Zhejiang University SCIENCE A, 8(5):778–785, 2007.

[153] JR Cho, HW Lee, WB Jeong, KM Jeong, and KW Kim. Numerical estimation of rolling resistance and temperature distribution of 3-d periodic patterned tire. International Journal of Solids and Structures, 50(1):86–96, 2013.

[154] Wei Liang, Jure Medanic, and Roland Ruhl. Analytical dynamic tire model. Vehicle System Dynamics, 46(3):197–227, 2008.

[155] P. Fancher, J. Bernard, C. Clover, and C. Winkler. Representing truck tire characteristics in simulations of braking and braking-in-a-turn maneuvers. Vehicle System Dynamics, 27(sup001):207–220, 1997.

[156] Fan Bai, Konghui Guo, and Dang Lu. Tire model for turn slip properties. SAE International Journal of Commercial Vehicles, 6(2):353–361, 2013.

[157] S. Gong. Study of in-plane dynamics of tires. PhD thesis, Technische Univ., Delft (Netherlands)., December 1993.

[158] K. Guo and D. Lu. Unitire: uniﬁed tire model for vehicle dynamic simulation. Vehicle System Dynamics, 45(sup1):79–99, 2007.

[159] Nan Xu, Konghui Guo, Xinjie Zhang, and Hamid Reza Karimi. An analytical tire model with ﬂexible carcass for combined slips. Mathematical Problems in Engineering, 2014, 2014.

[160] P.W.A. Zegelaar and H.B. Pacejka. Dynamic tyre responses to brake torque variations. Vehicle System Dynamics, 27(sup001):65–79, 1997.

[161] P.P. Lin, Maosheng Ye, and Kuo-Ming Lee. Intelligent observer-based road surface condition detection and identiﬁcation. In Systems, Man and Cybernetics, 2008. SMC 2008. IEEE International Conference on, pages 2465–2470, Oct 2008.

[162] Wanki Cho, Jangyeol Yoon, Seongjin Yim, Bongyeong Koo, and Kyongsu Yi. Estimation of tire forces for application to vehicle stability control. Vehicular Technology, IEEE Transactions on, 59(2):638–649, Feb 2010.

230 REFERENCES

[163] Krzysztof Walas. Terrain classiﬁcation using vision, depth and tactile perception. Proceedings of the 2013 RGB-D: Advanced Reasoning with Depth Cameras in Conjunction with RSS, Berlin, Germany, 27, 2013.

[164] R.M. Haralick, K. Shanmugam, and Its’Hak Dinstein. Textural features for image classiﬁcation. Systems, Man and Cybernetics, IEEE Transactions on, SMC-3(6):610–621, 1973.

[165] B.S. Manjunath and W.Y. Ma. Texture features for browsing and retrieval of image data. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 18(8):837–842, Aug 1996.

[166] Hideyuki Tamura, Shunji Mori, and Takashi Yamawaki. Textural features corresponding to visual perception. Systems, Man and Cybernetics, IEEE Transactions on, 8(6):460–473, June 1978.

[167] Peter Howarth and Stefan R¨uger. Evaluation of texture features for content-based image retrieval. In Image and Video Retrieval, pages 326–334. Springer, 2004.

[168] Prasanna Tamilselvan, Pingfeng Wang, and Michael Pecht. A multi-attribute classiﬁcation fusion system for insulated gate bipolar transistor diagnostics. Microelectronics Reliability, 53(8):1117 – 1129, 2013.

[169] L. Sigal, S. Sclaroﬀ, and V. Athitsos. Skin color-based video segmentation under time-varying illumination. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 26(7):862–877, July 2004.

[170] Jose Guivant, Stephen Cossell, Mark Whitty, and Jayantha Katupitiya. Internet-based operation of autonomous robots: The role of data replication, compression, bandwidth allocation and visualization. Journal of Field Robotics, 29(5):793–818, 2012.

[171] T. Kailath. The divergence and bhattacharyya distance measures in signal selection. Communication Technology, IEEE Transactions on, 15(1):52–60, February 1967.

231 REFERENCES

[172] Keinosuke Fukunaga. Introduction to statistical pattern recognition. Academic press, 1990.

[173] Euisun Choi and Chulhee Lee. Feature extraction based on the bhattacharyya distance. Pattern Recognition, 36(8):1703–1709, 2003.

[174] S.J. Smith, M.O. Bourgoin, K. Sims, and H.L. Voorhees. Handwritten character classiﬁcation using nearest neighbor in large databases. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 16(9):915–919, Sep 1994.

[175] Juliane Sch¨afer and Korbinian Strimmer. A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics. Statistical applications in genetics and molecular biology, 4(1), 2005.

[176] Mohammad Haghighat, Saman Zonouz, and Mohamed Abdel-Mottaleb. Cloudid: Trustworthy cloud-based and cross-enterprise biometric identiﬁcation. Expert Systems with Applications, 42(21):7905 – 7916, 2015.

[177] Gurkan Erdogan. New Sensors and Estimation Systems for the Measurement of Tire-Road Friction Coeﬃcient and Tire Slip Variables. PhD thesis, University of Minnesota, 2009.

[178] Gurkan Erdogan, Lee Alexander, and Rajesh Rajamani. Estimation of tire-road friction coeﬃcient using a novel wireless piezoelectric tire sensor. Sensors Journal, IEEE, 11(2):267–279, 2011.

[179] Sanghyun Hong and J.K. Hedrick. Tire-road friction coeﬃcient estimation with vehicle steering. In Intelligent Vehicles Symposium (IV), 2013 IEEE, pages 1227–1232, June 2013.

[180] R.P. Kruger, William B. Thompson, and A Franklin Turner. Computer diagnosis of pneumoconiosis. Systems, Man and Cybernetics, IEEE Transactions on, SMC-4(1):40–49, Jan 1974.

[181] L.-K. Soh and C. Tsatsoulis. Texture analysis of sar sea ice imagery using gray level co-occurrence matrices. Geoscience and Remote Sensing, IEEE Transactions on, 37(2):780–795, Mar 1999.

232 REFERENCES

[182] Hum Y Chai, Lai K Wee, Tan T Swee, Sh-Hussain Salleh, AK Ariﬀ, et al. Gray-level co-occurrence matrix bone fracture detection. American Journal of Applied Sciences, 8(1):26, 2011.

[183] Fionn Murtagh and A Heck. Multivariate data analysis with fortran, c and java code. Northern Ireland: Queens University Belfast, Astronomical Observatory Strasbourg, page 272, 2000.

233 REFERENCES

234 Appendix A

Planar Vehicle Model Derivations

The planar model that is developed here lays the ground work for how the structure of the dynamic model will evolve showing the relationships between the input forces

Fx,i and Fy,i. The planar model is constrained to move only along a plane, and as such does not include the roll and pitch angles of the body, nor does it include the eﬀects of suspension, and thus the change in normal force acting on the tyre contact patch.

Longitudinal and lateral forces as well as the self-aligning moment and the rolling resistance are generated at the tyre-terrain contact patch. Longitudinal forces act in the x axis of the rotating wheel, the lateral forces act along the y axis. The self-aligning moment acts about the z axis of the wheel, and the rolling resistance acts about the y axis of the rotating wheel. The forces that the tyre is subjected to is a function of both the relative velocity between the tyre and the terrain as well as the terrain conditions. The dynamic model used for each of the wheels is shown in (A.1), where the angular velocity of the wheel is determined by the contribution of the longitudinal force Fx,j, the rolling resistance My,j, the bearing frictional torque τf,j and the drive torque τd,j. Note that the drive torques are control inputs.

1 ω˙ j = (τd,j − τf,j − Fx,jre − My,j) (A.1) Iw,y

1 δ¨j = (τs,j − Mz,j) (A.2) Iw,z

These additional states are used to predict the frictional forces for each of the wheels.

235 Chapter A. Planar Vehicle Model Derivations

The dynamic model presented here consists of five subsystems, the main vehicle body and four independently steered and driven wheels. In this model the wheels are assumed to be rigidly attached to the vehicle body and the motion of the vehicle is constrained to the horizontal plane. The coordinate system for the vehicle shown in Fig. A.1 is defined with respect to the vehicle’s coordinate frame which is fixed about the center of gravity (CoG) of the vehicle. Each wheel has its own coordinate system and it is assumed that the wheels are always in contact with the horizontal plane and their coordinates are a fixed offset from the CoG as denoted with the xw,i and yw,i for all i =1, 2, 3, 4 wheels. Y

xw1 yw1

xw3 yw3 y' x'

xw2 yw2

xw4 yw4

Fig. A.1: Vehicle coordinate system

The wheels of the system interact with the ground through point contacts through which the forces are assumed to act upon. The wheel steering torque τs,j and driving torque τd,j for all four wheels are the physical inputs to the vehicle as shown in (A.3).

236 T U = [τs,1, τs,2, τs,3, τs,4, τd,1, τd,2, τd,3, τd,4] (A.3)

The vehicle states that are used in the model are given in (A.4). The x, y, ψ, ψ˙ are globally referenced variables that give the vehicles position and heading in the global coordinates, with respect to some externally deﬁned datum. The remaining state variables are deﬁned with respect to the local coordinate system, where v and u are the vehicle’s local longitudinal and lateral velocities, respectively. The wheel states

δj, δ˙j and ωj represent the current steered angle and angular velocity of wheel j.

T X = x,y,ψ,u,v,r,ω1,ω2,ω3,ω4, δ1, δ2, δ3, δ4, δ˙1, δ˙2, δ˙3, δ˙4 (A.4) h i

The motion of the vehicle body is influenced by the forces acting on the wheels as well as the external forces acting on the system such as wind, aerodynamic drag and components of gravitational forces. The geometry of the forces that the vehicle is subjected to can be seen in Fig. A.1 with the longitudinal and lateral forces at the point contact of the wheels. These forces act in the directions of xw,j and yw,j axis. These forces acting on the vehicle will determine the acceleration and hence the velocities, specifically the lateral and longitudinal velocities and the angular rates, specifically the yaw rate about the center of gravity of the vehicle.

The vehicle body’s local longitudinal and lateral velocity as well as the angular velocity is then determined by the forces that the vehicle is subject to about the CoG of the vehicle. The equations that govern the motion of the vehicle are shown in equations (A.5), (A.6) and (A.7) and the complete state equation is given in (A.8).

Mc(u ˙ + v.r)= (Fx,i cos δi − Fy,i sin δi) (A.5) i=1 X

Mc(v ˙ − u.r)= (Fx,i sin δi + Fy,i cos δi) (A.6) i=1 X 237 Chapter A. Planar Vehicle Model Derivations

4 b I .r˙ = (−1)i (F cos δ − F sin δ ) + c 2 x,i i y,i i i=1 X 4 i l (−1)⌈ 2 +1⌉ (F sin δ + F cos δ ) (A.7) 2 x,i i y,i i i=1 X

u˙

  −1 v˙ = −D [G0 + G1Fx + G2Fy + G3Mz + G4FD] (A.8)        r˙      where D is the inertia matrix. Fx, Fy consists of a vector of the tractive eﬀort and the lateral forces that are generated in each of the wheels. The Gj, for j = 0, 1, 2, 3 matrices transform forces onto the local frame. The matrix G4 represents the external forces, that act at an angle of γ to the centroid of the vehicle body. It is assumed that the forces experienced on each of the wheels is observable as the desired experimental platform is ﬁtted with 6DoF force sensors on each of the wheels.

The global coordinates of the vehicle are given in (A.9), where the local velocities are used to determine the global velocities of the vehicle and hence the global positions, through integration.

x˙ = v. cos ψ − u. sin ψ

y˙ = v. sin ψ + u. cos ψ (A.9)

ψ˙ = r

DX˙ + G0 + G1Fx + G2Fy + G3M + G4FD = 0 (A.10)

Solve for X˙ using the following form

−1 X˙ = −D (G0 + G1Fx + G2Fy + G3M + G4FD) (A.11) where

238 T X˙ = u,˙ v,˙ ψ,¨ ω˙1, ω˙2, ω˙3, ω˙4 (A.12) h i

Fx,1 Fy,1 −Mrr,1       Fx,2 Fy,2 −Mrr,2 F =   , F =   , M =   (A.13) x   y      F   F   τ − M   x,3   y,3   d,3 rr,3               Fx,4   Fy,4   τd,4 − Mrr,4             

Mb 000000 −Mbvψ˙     ˙ 0 Mb 00 0 0 0 Mbuψ          0 0 J 0 0 0 0   0   b            D =  0 0 0 Jw 0 0 0  , G0 =  0  (A.14)              0 0 00 Jw 0 0   0           0 0000 J 0   0   w             0 00000 Jw   0         

0 0 0 0 − cos γ     0 0 0 0 − sin γ          0 0 0 0   0              G3 =  −10 0 0  , G4 =  0  (A.15)              0 −1 0 0   0           0 0 −1 0   0               0 0 0 −1   0          239 Chapter A. Planar Vehicle Model Derivations

− cos δ1 − cos δ2 − cos δ3 − cos δ4   − sin δ1 − sin δ2 − sin δ3 − sin δ4      ( b sδ − l cδ ) −( b sδ + l cδ ) ( b sδ + l cδ ) −( b sδ − l cδ )   2 1 2 1 2 2 2 2 2 3 2 3 2 4 2 4      G1 =  1 0 0 0  (A.16)        0 1 0 0       0 0 1 0         0 0 0 1     

sin δ1 sin δ2 sin δ3 sin δ4   − cos δ1 − cos δ2 − cos δ3 − cos δ4      ( b sδ + l cδ ) −( b sδ + l cδ ) ( b sδ − l cδ ) −( b sδ − l cδ )   2 1 2 1 2 2 2 2 2 3 2 3 2 4 2 4      G2 =  0 0 0 0  (A.17)        0 0 0 0       0 0 0 0         0 0 0 0     

240 Appendix B

Tyre Model Derivations

This appendix contains the steady-state calculations for the diﬀerent force distribution functions that are used for generating the simulation data for Chapter 4, of which only the constant force and a variation of the trapezoidal force distribution are used. Additionally, the method for extracting the kinetic and the static coeﬃcients of friction from the Magic Formula force plots, that is utilised in the tyre model chapter is presented and described. Lastly, the parameters that were used in the MF-SWIFT model for validating the proposed model is presented.

B.1 Force Distributions

The force distribution function fz(t, ζ, η) so that the following function is satisﬁed

W L 2 Fz = fz(ζ, η)dηdζ (B.1) W Z0 Z− 2

′ fz(ζ)= ∂fz/∂ζ (B.2)

The steady-state expressions for the force and moment equations (4.7), (4.8), (4.9) and (4.12) can now be evaluated using the now developed force distribution function fz(t, ζ, η)

241 Chapter B. Tyre Model Derivations

B.1.1 Constant Force Distribution

The simple case force distribution is that of the constant force spread out over the area of the contact patch is

F −W W f (ζ, η)= z , 0 ≤ ζ ≤ L, ≤ η ≤ (B.3) z LW 2 2 for the steady state forces this gives,

L ss C2i − F = F (sgn v σ C 1 − e C2i + σ v ) (B.4) i z ri 0i 1i L 2i ri ss ss the overturning and rolling resistance are Mx = My = 0 and the steady-state self aligning moment is seen to be,

L L ss C1yC2y − − M = sgn v F σ e C2y 2C + L + e C2y (−2C + L) (B.5) z ry z 0y L 2y 2y B.1.2 Parabolic distribution a parabolic force distribution that follows the following model is used

2 9N ζ − L W 2 f (ζ, η)= 1 − 2 − η2 (B.6) z LW 3  L  2 2 ! " #   This force distribution gives the longitudinal and lateral steady state forces as

L ss Fz 3 2 3 2 − 3 F = sgn v σ C 12C − 6C L + L − 6C e C2i (2C + L) + σ v L i L3 ri 0i 1i 2i 2i 2i 2i 2i ri (B.7)

− L 2 C2 3C1yC e y M ss = sgn v F σ 2y z ry z 0y L3 L 2 2 C2y 2 2 12C2y +6C2yL + L − e 12C2y − 6C2yL + L (B.8) with the overturning and rolling resistance moments equal to zero this continues to be the case because both moments require a non-symmetric force distribution.

242 B.1. Force Distributions

Fig. B.1: Parabolic force distribution

The case where the speed is zero is

ss Fi = 0 (B.9)

1 M ss = δF˙ σ L2 − σ W 2 (B.10) z 20 z 2y 2x

B.1.3 Trapezoidal distribution

A more realistic force distribution for the contact patch is a trapezoid distribution as proposed by Deur et al [62], the distribution is asymmetric about the centre of the tire to account for the change of sign of the self-aligning torque at higher speeds. The asymmetry in the 2D trapezoid force distribution proposed is only in the longitudinal

243 Chapter B. Tyre Model Derivations axis, with the force distribution being symmetric in the lateral axis.

α1ζ for 0 ≤ ζ ≤ ζL,  fz,ζ(ζ)=  1 for ζL ≤ ζ ≤ ζR, (B.11a)   α2ζ + β2 for ζR ≤ ζ ≤ L,    −W α3η + β3 for 2 ≤ η ≤ −ηS,  f (η)=  (B.11b) z,η  1 for −ηS ≤ η ≤ −ηS ,   W α4η + β4 for −ηS ≤ η ≤ 2 ,   

fz(ζ, η)= fmaxfz,ζfz,η (B.11c)

Fig. B.2: Asymmetrical trapezoid force distribution

244 B.1. Force Distributions

with

1 1 L α1 = ,α2 = − , β2 = (B.12a) ζL L − ζR L − ζR

2 W 2 W α3 = , β3 = ,α4 = − , β4 = (B.12b) W − 2ηS W − 2ηS W − 2ηS W − 2ηS

with the value of fmax generated as to solve across the distribution for a given Fz as follows,

W L 2 1 Fz = fz(ζ, η)dηdζ = fmax(2ηS + W )(L − ζL + ζR) (B.13) W 4 Z0 Z− 2 The steady-state forces and moments from this force distribution become quite long in comparison to the previous two distributions, as such they are not included, but can be derived through such tools such as Mathematica.

B.1.4 Continuous Variable distribution

Another force distribution that was examined for use with the developed tyre model was that of a continuous distribution that varied depending on the camber angle and the forward velocity.

F 2ζ 2ζ 2ζ f (ζ, η)= z L (1 − ( − 1)2L3 )(1 + L ( − 1)2L3 )(1 − 2L ( − 1)d ) z LW 1 L 4 L 2 L r (B.14) 2η 2η 2η W (1 − ( )2W3 )(1 + W ( )2W3 )(1 − 2W ( )d ) 1 W 4 W 2 W c (B.15)

where L1 and L2 are,

245 Chapter B. Tyre Model Derivations

(2L3 + 1)(4L3 + 1) L1 = , (B.16) 2L3(4L3 +1+ L4)

3(2L3 + 3)(4L3 + 3)(4L3 +1+ L4) L2 = − (B.17) (2L3 + 1)(4L3 + 1)(4L3 +3+3L4)

where L4 is used to represent the eﬀect of the edges on the force distribution, L3 is an integer that represents how spread out the force distribution is, and is a function of the stiﬀness of the tyre. The variables W1, W2, W3 and W4 all follow the same structure as the length based variables, with the representations being valid along the width of the tyre instead of along the length of the tyre contact patch.

0.8

0.6

0.4

normalised force 0.2

0 0.1 0.05 0.05 0

ζ 0 −0.05 (m) η (m)

Fig. B.3: velocity and camber angle dependant asymmetric force distribution

This model was ultimately not included as the variable force distribution, as parameters that are used for ﬁtting need to be integers and they require strict control over their values. Often the force distributions that are found in order to accommodate the rolling resistance moment behave in a way that has a negative force, with a physical

246 B.2. Frictional Coeﬃcient Extraction meaning that the tyre is pulling on the terrain, this is obviously false. For this reason the force distribution was not included in the chapter.

B.2 Frictional Coeﬃcient Extraction

The method by which the kinetic and the static coefficients of friction are determined for use in model fitting is described below. The way in which the coefficients of friction are extracted from the MF parameters, can be achieved by recognising that the of the current analytical dynamic friction models that exists there are two main phenomenon; peak friction µp and peak slip κp or slip angle αp.

The peak coefficient of friction for x µpi can be determined from the term D from the MF formula. This term is then used to determine the static coefficient of friction given the kinetic coefficient of friction µki determined as the friction coefficient that the slip force plot asymptotically approaches at higher slip values, and can be found from the MF-SWIFT formulas by calculating the longitudinal force at κ = 1, and similarly, π the lateral force at α = 2 .

With the knowledge of the speed that the vehicle is travelling at v0 and the peak slip value κp the value of µsx can be determined, and similarly through the knowledge of peak slip angle αp the value of µsy can be determined. The value is determine by ﬁtting an exponential function through both the peak and the pure sliding points, with the form of the function being calculated with as per (B.18), for i = x, y.

c f(c)= ai.bi (B.18)

where ai and bi have been determined by fitting the exponential function shown in (B.18), with c is equal to κ for the longitudinal friction coefficients and equal to α for the lateral friction coefficients. The static coefficient of friction for the longitudinal and separately for the lateral case can be determined by setting κ = 0 and α = 0, as shown in (B.19)

0 µsi = ai.bi (B.19)

247 Chapter B. Tyre Model Derivations

B.3 MF-SWIFT 6.1 model parameters

Table B.1: TNO MF-SWIFT 6.1 Tyre Model Parameters

Tyre Designation 205/60R156 91V

[model]

Vo = 16.7m/s, Vx,low =1m/s

[dimension]

ro =0.3135m, w =0.205m, rrim =0.1905m

[operating conditions]

pio = 220, 000P a

[inertia]

2 2 mtire =9.3kg,Ixx,tire =0.391kgm ,Iyy,tire =0.736kgm

2 2 mbelt =7.247kg,Ixx,belt =0.3519kgm ,Iyy,belt =0.5698kgm

[vertical]

Fzo = 4000N

czo = 209651N/m,kz = 50Ns/m, qv2 =0.04667, qFz2 = 15.4, qFcx =0, qFcy =0,

pFz1 =0.7098

Breff =8.386,Dreff =0.25826, Freff =0.07394, qreo =0.9974, qV 1 =7.742e-4

[structural]

cxo = 358066N/m,pcfx1 =0.17504,pcfx2 =0,pcfx3 =0,

cyo = 102673N/m,pcfy1 =0.16365,pcfy2 =0,pcfy3 =0.24993,

cψ = 4795Nm/rad,pcmz1 =0,

flong = 77.17Hz,flat = 42.41Hz,fyaw = 53.49Hz,fwindup = 58.95Hz,

ζlong =0.056,ζlat =0.037,ζyaw =0.0070,ζwindup =0.050,

qbvx =0.364, qbvθ =0.065

Continued on next page

248 B.3. MF-SWIFT 6.1 model parameters

Table B.1 TNO MF-SWIFT 6.1 Tyre Model Parameters - Cont’d

Tyre Designation 205/60R156 91V

[contact patch] qra1 =0.671, qra2 =0.733, qrb1 =1.059, qrb2 = −1.1878, pls =0.8335,pae =1.471,pbe =0.9622,ce =1.5174

[longitudinal coeﬃcients] pCx1 =1.579,pDx1 =1.0422,pDx2 = −0.08285,pDx3 =0, pEx1 =0.11113,pEx2 =0.3143,pEx3 =0,pEx4 =0.001719, pKx1 = 21.687,pKx2 = 13.728,pKx3 = −0.4098, pHx1 =2.1615e-4,pHx2 =0.0011598,pVx1 =2.0283e-5,pVx2 =1.0568e-4, ppx1 = −0.3485,ppx2 =0.37824,ppx3 = −0.09603,ppx4 =0.06518, rBx1 = 13.046,rBx2 =9.718,rBx3 =0,rCx1 =0.9995, rEx1 = −0.4403,rEx2 = −0.4663,rHx1 = −9.968e-5

[overturning coeﬃcients] qsx1 = −0.007764, qsx2 =1.1915, qsx3 =0.013948, qsx4 =4.912, qsx5 =1.02, qsx6 = 22.83, qsx7 =0.7104, qsx8 = −0.023393, qsx9 =0.6581, qsx10 =0.2824, qsx11 =5.349, qsx12 =0, qsx13 =0, qsx14 =0,ppmx1 =0

[lateral coeﬃcients] pCy1 =1.338,pDy1 =0.8785,pDy2 = −0.06452,pDy3 =0, pEy1 = −0.8057,pEy2 = −0.6046,pEy3 =0.09854,pEy4 = −6.697,pEy5 =0, pKy1 = −15.324,pKy2 =1.715,pKy3 =0.3695,pKy4 = −6.697,pKy5 =0, pHy1 = −0.001806,pHy2 =0.00352,pKy6 = −0.8987,pKy7 = −0.23303, pVy1 = −0.00661,pVy2 =0.03592,pVy3 = −0.162,pVy4 = −0.4864, ppy1 = −0.6255,ppy2 = −0.06523,ppy3 = −0.16666,ppy4 =0.2811,ppy5 =0, rBy1 = 10.622,rBy2 =7.82,rBy3 =0.002037,rBy4 =0,rCy1 =1.0587,

Continued on next page

249 Chapter B. Tyre Model Derivations

Table B.1 TNO MF-SWIFT 6.1 Tyre Model Parameters - Cont’d

Tyre Designation 205/60R156 91V

rEy1 =0.3148,rEy2 =0.004867,rHy1 =0.009472,rHy2 =0.009754,

rVy1 =0.05187,rVy2 =4.853e-4,rVy3 =0,rVy4 = 94.63,rVy5 =1.8914,rVy6 = 23.8

[rolling coeﬃcients]

qsy1 =0.00702, qsy2 =0, qsy3 =0.001515, qsy4 =8.514e-5, qsy5 =0,

qsy6 =0, qsy7 =0.9008, qsy8 = −0.4089

[aligning coeﬃcients]

qBz1 = 12.035, qBz2 = −1.33, qBz3 =0, qBz4 =0.176,

qBz5 = −0.14853, qBz9 = 34.5, qBz10 =0,

qCz1 =1.2923, qDz1 =0.09068, qDz2 = −0.00565,

qDz3 =0.3778, qDz4 =0, qDz6 =0.0017015,

qDz7 = −0.002091, qDz8 = −0.1428, qDz9 =0.00915, qDz10 =0, qDz11 =0,

qEz1 = −1.7924, qEz2 =0.8975, qEz3 =0, qEz4 =0.2895, qEz5 = −0.6786,

qHz1 =0.0014333, qHz2 =0.0024087, qHz3 =0.24973, qHz4 = −0.21205,

ppz1 = −0.4408,ppz2 =0,

ssz1 =0.00918,ssz2 =0.03869,ssz3 =0,ssz4 =0

[turnslip coeﬃcients]

pDxφ1 =0.4,pDxφ2 =0,pDxφ3 =0,

pKyφ1 =1,pDyφ1 =0.4,pDyφ2 =0,pDyφ3 =0,pDyφ4 =0,

pHyφ1 =1,pHyφ2 =0.15,pHyφ3 =0,pHyφ4 = −4,

pEγφ1 =0.5,pEγφ2 =0,

qDtφ1 = 10, qCrφ1 =0.2, qCrφ2 =0.1, qBrφ1 =0.1, qDrφ1 =1, qDrφ2 = −1.5

250 Appendix C

Terrain Model Further Explanation

C.1 Spatial Features

C.1.1 Gray-Level Co-occurrence Matrix Features

The GLCM method that was developed by Haralick operates by recording the jump between different pixel values in the direction and scale that is specified. The outcome from this method is the generation of a square matrix that increments a counter in each cell of the matrix corresponding to the appropriate pixel value transitions. From this matrix, a number of different texture features can be calculated that take into account the statistical nature of the method. The gray-level co-occurrence matrix as defined in (C.1), which is used for all subsequent features and is the definition used in [164], where I is the input 2D matrix.

1 if I(k,l)= i and n n  p (i, j)=  (C.1) ∆x,∆y  I(k + ∆x, l + ∆y)= j k=1 l=1  X X  0 otherwise   where ∆x and ∆y are used to determine the direction and step size that the speciﬁc gray-level co-occurrence matrix is based on, and can also be expressed in a combination of angles θ and step distances d as pθ,d(i, j). The four primary orientations combinations of ∆x and ∆y with a step distance d of 1 grid gives the angles θ as 0◦, 45◦, 90◦ and 135◦, with symmetric features being recorded so that other orientations, ie 180◦, are

251 Chapter C. Terrain Model Further Explanation also included. The diﬀerent step directions and sizes are visualised in Fig. C.1.

135◦[−d, −d] 90◦[−d, 0] 45◦[−d,d]

Pixel of Interest 0◦[0,d]

Fig. C.1: GLCM cell evaluation showing orientation and step size

There are a number of diﬀerent texture features that can be extracted from the

GLCM matrix pθ,d, with a select list of features taken from [164] presented below,

1. Energy 2 fθ,d,s(1) = pθ,d(i, j) (C.2) i,j X 2. Contrast 2 fθ,d,s(2) = |i − j| pθ,d(i, j) (C.3) i,j X 3. Correlation (i − µ i)(j − µ j)p (i, j) f (3) = x y θ,d (C.4) θ,d,s σ σ i,j x y X 4. Homogeneity p (i, j) f (4) = θ,d (C.5) θ,d,s 1+ |i − j| i,j X 5. Information Measure of Correlation 1

Hxy − Hxy1 fθ,d,s(5) = (C.6) max{Hx,Hy}

252 C.1. Spatial Features

6. Information Measure of Correlation 2

1 2 fθ,d,s(6) = (1 − exp [−2(Hxy2 − Hxy)]) (C.7)

where Hx and Hy are entropies of px and py respectively as shown in [164], which represent the marginal probabilities for the rows and columns respectively of

pθ,d(i, j) and

Hxy = − pθ,d(i, j)log(pθ,d(i, j)) (C.8) i,j X Hxy1 = − pθ,d(i, j)log(px(i)py(j)) (C.9) i,j X Hxy2 = − px(i)py(j)log(px(i)py(j)) (C.10) i,j X

7. Maximum Probability

fθ,d,s(7) = max{pθ,d(i, j)} (C.11)

The maximum probability feature is a measure of the most common gray-level co-occurrence throughout the region.

To group the related features, the matrix td,s(k, a) is deﬁned as given in (C.12).

Where td,s(k) is a matrix as shown in (C.12), with s representing the source of the texture value, either the range or colour information as the source.

td,s(k, a)= fθ(a),d,s(k) θ(a)=0◦, 45◦, 90◦, 135◦ a =1, 2, 3, 4 (C.12)

The GLCM features are generated for each of the four directions θ that are used in the GLCM generation. There exists a number of different methods [180–182] for reducing the dimensionality of data in order to make classification less expensive. The methods focus on metrics that compare the different θ angles for each of the different texture features, the metrics that are utilised are the mean, variance, span as well as

253 Chapter C. Terrain Model Further Explanation

the maximum value as described in equations (C.13)-(C.16), where td,s(k, a) represents the vector of features k, of the step distance d in the ath direction as before.

1 4 t¯ (k)= t (k, a) (C.13) d,s 4 d,s a=1 X

1 4 v (k)= (t (k, a) − t¯ (k))2 (C.14) d,s 4 d,s d,s a=1 X

sd,s(k) = max{td,s(k, a)} − min{td,s(k, a)} (C.15)

hd,s(k) = max{td,s(k, a)} (C.16)

C.1.2 Gabor Filter Features

The Gabor filter bank operate by looking at the different orientations and scales for the image and taking the filtered response for image data. Gabor filter based textures are developed by convolving the scene with a variety of different Gabor filters and then taking the filtered responses as features in order to distinguish different textures from a scene. A number of different Gabor filters can be seen in Fig. C.2, where the different filters correspond to different orientations and wavelengths of the filter.

Fig. C.2: Gabor Filter Bank with 8 diﬀerent orientations and 3 diﬀerent scales

The ﬁlter can be applied over the image as a whole or on a segment of the image

254 C.1. Spatial Features as a window filter. The gabor filter response on a windowed segment contains a large amount of information and so to reduce the amount of data the response of a region can be reduced to the mean, standard deviation and span between the maximum and minimum of the filter response, the gabor filtered features can be seen in (C.17).

1 W W g (a, k)= g (i, j) (C.17a) s,µ W 2 s,a,k i=1 j=1 X X

1 W W g (a, k)= (g (i, j) − g (a, k))2 (C.17b) s,σ W 2 s,a,k s,µ i=1 j=1 X X

gs,span(a, k) = max{gs,a,k(i, j)} − min{gs,a,k(i, j)} (C.17c)

with gs,a,k being the response of the gabor ﬁlter using s sensor source, a orientation and k scale, i and j being local coordinates and W being the width of the square region that the ﬁlter is applied to.

C.1.3 Power Spectral Density Features

The fourier transform features for a surface are straightforward to evaluate; however, the responses are in 2D as the transformations are performed in 2D. This means that there are a large number of features if all of the orientation dependent frequency responses were to be used to generate the feature space. In the 1D fourier transform case, the Power Spectral Density (PSD) is most often used to form the features of the signal, and these features were used in the terrain classiﬁcation method proposed by Wang [120]. For comparison to previous work, it is therefore necessary to be able to generate PSD based features but from the 2D fourier transform. This can be achieved through the use of a radially averaged PSD, where the 2D image is converted into polar coordinates with bins at diﬀerent radii being used to reduce the dimension of the fourier transform output.

1 2 Ps(k)= Iˆ(ξ = i) (C.18) Nk rk

where Iˆ(ξ) is the fourier transform of the input 2D matrix, Nk is the number of

255 Chapter C. Terrain Model Further Explanation

elements i that are within the boundaries rk and rk + δr for each set k. The above features can be implemented to gather a number of spatial features from the terrain so that these features may be used in providing a good means of classification of the terrain types. There are a number of different ways that these feature gathering methods can be implemented. The first method that is developed uses the dominant surface geometry that is present in a scene and classifies each region. This method is described in Section 5.2. The second method utilises a different approach for generating the spatial features that does not rely on the dominant surface geometry of the scene and the method is described in Section 5.3.

C.2 Weighted Majority Voting with Dominance

The algorithm is called Weighted Majority Voting with Dominance (WMVD), which was taken adapted from [168] and the implementation relies on the evaluation of classification accuracy. The validation method used is a k-Fold cross validation, with the original data set being split into k mutually exclusive subsets. One of the subsets is used for validation and the other k-1 are used for training. The index i represents the subset, the index j represents the descriptor number, the index m represents the classifier function, and the index l represents the data points in a subset and ranges from 1 to r. The matrix CCij represents the classifier decision of the jth terrain type for the ith subset, where dl,m is the classifier decision of the mth classifier for the lth training point. The vector ACij represents the target classification for the jth terrain type with for the ith subset and al is the target classification for the lth training point.

d1,1 ··· d1,c  . . .  CCij = . .. . (C.19)        dr,1 ··· dr,c      T . ACij = a1, ., ar (C.20) The accuracy of prediction for the individual classiﬁcation functions are then

256 C.2. Weighted Majority Voting with Dominance measured against the true results. (C.21) shows that if the classiﬁer results are true then its corresponding R term is 1, otherwise the term is 0.

1 if CCij(l, m)= ACij(l, 1) R(CCij(l, m), ACij(l, 1)) =  (C.21)  0 otherwise

The R value is then used to determine the classiﬁcation index wmj, for each classiﬁer and each descriptor. The values of wmj are then used to determine the weight of the contribution of each member algorithm in the WMVD approach. The value of wmj is calculated as shown in (C.22).

1 k r w = R(CC (l, m), AC (l, 1)) (C.22) mj kr ij ij i=1 " # X Xl=1 from the value of wmj the weight of the mth member algorithm is determined.

The normalised weight wtmj is determined based on (C.23), which is dependent on the current training data set. If wtmj needs to be updated then additional members need to be added to the training data set to be able to generate a new normalised weight.

wmj wtmj = n c (C.23) j=1 m=1 wmj The classiﬁcation output of the WMVDP forP the next data point is then calculated based on the individual predictions for each of the classiﬁers used. This is then collated into the form of incmj as shown in (C.24)

1 if classifier m classified the incoming  incmj =  point as the jth terrain type (C.24)   0 otherwise   The incoming prediction results incmj are then combined with the normalised weighting wtmj of the individual classifiers as shown in (C.25).

Cj = wtmjincmj (C.25) m=1 X The class j that corresponds to the maximum value of the weighted sum Cj is

257 Chapter C. Terrain Model Further Explanation determined to be the class of the terrain surface that is being evaluated. The three classifiers that are combined together in the WMVD classifier are the k-Nearest Neighbour classifier with k = 5, a probability based Mahalanobis distance classifier and a decision tree classifier. k-Nearest Neighbours The k-Nearest Neighbours classifier operates by being given a known sample set and then comparing any new samples to this set and performing a weighted average of the k nearest neighbours to the new sample to estimate the terrain type, which in our implementation utilises a Mahalanobis distance metric for determining the nearest neighbours to the new point. Mahalanobis The Mahalanobis distance (C.26) is a unit-less measure of how close a sample is to a known sample set. It is the single point variant of the Bhattachrayya distance, where an assumption is made that the point has a similar covariance to the distribution being compared against. The distance measure accounts for correlations in the data set and can be used in a multivariate feature space where the features have correlation and whose units are not equal in their distribution.

−1 dM = (X − µk)Σk (X − µk) (C.26) q where µk and Σk are the mean and covariance of the kth group Decision Tree The decision tree classifier generates a number of discrete rules for classification using the test data. From there, classification is done by comparing the features in the descriptor to that in the branches of the tree, and this is done until a leaf of the decision tree has been found.

C.3 Dimensionality Reduction

C.3.1 Principal Component Analysis

The basis behind PCA is that the method reorders the feature space in order of the most signiﬁcant variation across the data set. This is found calculating the eigenvalues

258 C.3. Dimensionality Reduction and eigenvectors of the set such that

uT XT Xu u = max{ } (C.27) (1) uT u

where X is a matrix of data points for the set, and XT X is recognised as being proportional to the covariance matrix of the dataset. The Lagrangian maximisation problem in (C.27) can thus be rewritten as (C.28) with the Lagrangian multiplier λ.

XT Xu = λu (C.28)

Solving for the eigenvalues and eigenvectors of (C.28) gives the new set of basis vectors by which to transform the data into. The eigenvalues λ can be sorted into descended order with the more dominant eigenvectors ﬁrst, and a minimum value for the eigenvalue can be chosen so that eigenvalues less than this value are not used, and, as such, the accompanying eigenvector can be excluded when transforming the data into the new basis vector. This step eﬀectively reduces the dimensionality of the dataset such that it only includes basis vectors that contribute a minimum amount of useful data in discriminating the dataset.

This method requires that the dataset have a mean of zero, after which the new basis vectors that are used for the dataset are that of the eigenvectors in descending order of eigenvalue. The outcome of this method is that the new basis vector is dominated by data that has large variation. In a large number of cases for classification methods, a large variation in the data is useful as it can be used to effectively separate the underlying classes along dimensions that have higher variability. This is, however, only the case when the classes that are being classified have significant enough differences such that the dimension with high variation contains the different classes in equal proportions. Additionally, if the data points exist on a non-linear manifold, this manifold will not be represented correctly, and there may be errors in identifying the dominant dimension.

This is a problem for classiﬁcation with classes that contain data that do not have similar magnitudes of variation, as the dominant dimension as found using PCA will be signiﬁcantly dominated by the class that has the most variation with the other

259 Chapter C. Terrain Model Further Explanation

1 concrete fakeGrass 0.8 grass gravel tile 0.6

0.4

0.2

−0.2

−0.4 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4

Fig. C.3: PCA dimension reduction using High-pass filter residual and a RoI of 72 classes being clustered together. This effect can be seen in Fig. C.3 in the extended depth feature dataset of the different terrain types, where the data from the grass class dominates the most significant dimension as calculated by PCA.

C.3.2 Multiple Discriminant Analysis

The other method of dimension reduction is that of multiple discriminant analysis from [183]. This method works by taking into account both the variance within the separate classes as well as the variance between classes. The method minimises the covariances within the classes and maximises the variance between the classes. The MDA method works as you have a set of objects i ∈ I with a set of parameters j ∈ J. The parameter mean over the entire data set I is (C.29). Within the set I there exists a number of diﬀerent groups of mutually exclusive objects, let P be the set of these groups and whose union gives I.

1 g = x (C.29) j n ij Xi∈I

1 ypj = i ∈ P xij (C.30) np X 260 C.3. Dimensionality Reduction

where n is the cardinality of set I and np is the cardinality of the set in P of the pth class.

1 t = (x − g )(x − g ) (C.31) jk n ij j ik k i∈I X

1 w = (x − y )(x − y ) (C.32) jk n ij pj ik mk p∈P i∈p X X

n b = p (y − g )(y − g ) (C.33) jk n pj j pk k p∈P X

where the matrices T , W and B that consist of the elements tjk, wjk and bjk respectively. The main improvement of this method is that there is no necessity to remove the mean of the diﬀerent classes so that the variances for each of the features can be assessed. As PCA requires that there is zero mean for the data. For MDA the total covariance matrix T ; the within classes covariance W ; and the between classes covariances B. All three covariance matrices are of dimension m × m where m is the number of dimensions in the feature space. With the total covariance T being a combinations of the within and between class covariances W and B as in (C.34).

T = W + B (C.34)

The sum of squared projections on the points in Rm on any axis vector u is given by uT Tu, and similarly also, uT Bu and uT Wu. With the relationship from (C.34) giving the following.

uT Tu = uT Wu + uT Bu (C.35)

We then choose the vector u which maximise the spread of the class means, whiles restraining the compactness of the classes. These constraints can be seen to have the form in (C.36).

uT Bu max{ } (C.36) uT Tu 261 Chapter C. Terrain Model Further Explanation

The Lagrangian maximisation problem, along with its accompanying Lagrange multiplier λ and diﬀerentiating with respect to u can thus be written as (C.37)

T −1Bu = λu (C.37)

However for eigenvalues to be found using this method it is necessary for T −1B to be symmetric, which is not necessarily the case. To be able to solve this problem, it is noted that B is necessarily positive deﬁnite and symmetric and so can be rewritten as CCT with C being the lower or upper triangular matrix that is given by Cholesky decomposition of B, giving CCT u = λTu. Deﬁne a new vector a such that u = T −1Ca, which when substituted back into the previous formula gives

CT T −1C a = λa (C.38) so now we can solve for the eigenvectors a and substitute back into u, with the new basis vector being of rank P − 1, which effectively reduces the dimensionality of the feature space, as only the first P − 1 eigenvectors u being the discriminating basis vectors. Similar to PCA, MDA will also perform poorly if the data points for each of the classes exists on a non-linear manifold as MDA is a linear transformation. These effects can be taken into account through the use of non-linear transformation methods, however these require a lot of computational power. To reduce the probability that such a situation occurs, the number of available feature space dimensions are increased so that a separation between classes can be achieved through a higher dimension. The results of implementing MDA as described above and applied to the dataset of the terrain features can be seen in Fig. C.4. The figure plots only the first two dominant axes and the different classes are represented in the different colours, with each group consisting of points that are similarly distributed and with the between class distances maximised through MDA.

262 C.3. Dimensionality Reduction

5.4 concrete fakeGrass 5.3 grass gravel 5.2 tile

5.1

4.9

4.8

4.7

4.6 −149 −148 −147 −146 −145 −144 −143

Fig. C.4: MDA dimension reduction using High-pass ﬁlter residual and a RoI of 72

263