GLOCAL CONTROL FOR MECANUM-WHEELED VEHICLE WITH SLIP COMPENSATION
BY
JIRAYU UDOMSAKSENEE
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ENGINEERING (INFORMATION AND COMMUNICATION TECHNOLOGY FOR EMBEDDED SYSTEMS) SIRINDHORN INTERNATIONAL INSTITUTE OF TECHNOLOGY THAMMASAT UNIVERSITY ACADEMIC YEAR 2018
Ref. code: 25615922040554LTX GLOCAL CONTROL FOR MECANUM-WHEELED VEHICLE WITH SLIP COMPENSATION
BY
JIRAYU UDOMSAKSENEE
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ENGINEERING (INFORMATION AND COMMUNICATION TECHNOLOGY FOR EMBEDDED SYSTEMS) SIRINDHORN INTERNATIONAL INSTITUTE OF TECHNOLOGY THAMMASAT UNIVERSITY ACADEMIC YEAR 2018
Ref. code: 25615922040554LTX
Acknowledgements
This research is financially supported by Thailand Advanced Institute of Science and Technology (TAIST), National Science and Technology Development Agency (NSTDA), Tokyo Institute of Technology and Sirindhorn International Institute of Technology (SIIT) under the Excellent Thai Students (ETS) program, and Thammasat University (TU).
I am gratefully indebted to my thesis advisor, Asst. Prof. Dr. Itthisek Nilkhamhang of SIIT at TU, for his helpful advice and consistency support along this thesis. We worked hard together during day and night on my thesis. He is very kind to let Mr. Hendi Wicaksono brief me on the fundamental concept of the glocal control system at the beginning of my thesis. In addition to that, he also introduced me to Prof. Shinji Hara of Tokyo University, the expert of glocal control concept that I have an opportunity to learn more and obtain his advice on my thesis.
I also would like to thank Assoc. Prof. Masaki Yamakita and Yamakita lab members, especially Mr. Rin Takano, for their lectures, helpful advice, and kind hospitality during my ten-week stay at Tokyo Institute of Technology.
I would like to gratefully thank Assoc. Prof. Dr. Waree Kongprawechnon. With her kind advice and support, I have a chance to be a part of TAIST program. In addition, I have also got her continuing support and valuable advice on this thesis. I also would like to acknowledge valuable comments from Dr. Pished Bunnun, the Chairperson of Examination Committee.
Last but not least, I would also like to thank my friends namely Mr. Apisit Pinitnanthakorn, Mr. Peammawat Chantevee, and Ms. Panatda Nalinnopphakhun for providing additional information.
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Abstract
GLOCAL CONTROL FOR MECANUM-WHEELED VEHICLE WITH SLIP COMPENSATION
by
JIRAYU UDOMSAKSENEE
Bachelor of Engineering, Sirindhorn International Institute of Tecnology, Thailand, 2015
Master of Engineering, Sirindhorn International Institute of Tecnology, Thailand, 2018
This thesis proposes a hierarchical decentralized controller for a mecanum- wheeled vehicle represented as a homogenous multi-agent system. The equations of motion for each individual mecanum wheel and the entire vehicle with slip are analyzed to construct a linear time-varying interconnected model. The global objective is the position trajectory tracking of the vehicle as a result of the effective forces produced by each wheel. The local objective is the driving force and slip controls of each wheel, with consideration of the interconnection between all agents. The proposed hierarchical linear quadratic regulator (LQR) control ensures satisfaction of both global and local objectives, according to the concepts of glocal control. Simulation results of a mecanum-wheel vehicle are shown that verify the performance and validity of the method.
Keywords: Mecanum wheel, glocal control, decentralized hierarchical control, slip control
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Table of Contents
Chapter Title Page
Signature Page i Acknowledgements ii Abstract iii Table of Contents iv List of Figures vi List of Tables viii
1 Introduction 1
1.1 Motivation 1 1.1.1 Practical Applications 2 1.1.2 Problems of Mecanum-Wheeled Vehicle 5 1.1.3 Anti-Slip Control System for Conventional-Wheeled Vehicles 5 1.1.4 Anti-Slip Control System for Mecanum-Wheeled Vehicles 6 1.2 Objective 7 1.3 Thesis Scope 7 1.4 Thesis Structure 8
2 Literature Review 9
2.1 Mecanum-wheel Vehicle 9 2.1.1 Fundamental Vehicle Dynamics. 12 2.2 Anti-Slip Control Techniques 14 2.2.1 Slip Ratio 11 2.2.2 Anti-Slip Control via Kinematic Control 15 2.2.3 Anti-SlipControl via Driving Force Control 15 2.3 Hierarchically Decentralized Optimal Control 14
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2.3.1 Homogeneous Multi-Agent Dynamical Systems 17
2.3.2 Multi-Agent Dynamical Systems with physical interconnection 21
3 Dynamical Modelling of Mecanum-Wheel Vehicle 24
3.1 Fundamental Vehicle Dynamics 24 3.2 Slip Ratio 26 3.3 Driving Force Dynamics 26 3.4 Group and Layering 28 3.4.1 Upper Layer: Position Control 29 3.4.2 Lower Layer: Driving Force Control 29 3.5 Hierarchical Decentralized Structure 29
4 Hierarchically Decentralized Optimal Control 31
4.1 Trajectory Tracking Controller 32 4.2 Hierarchically Decentralized Optimal Control 32 4.3 Performance Index 34 4.4 Hierarchical State Feedback LQR design 36 4.5 Numerical Simulation 37 4.5.1 Linear Translation along x-y Axis 40 4.5.2 Trajectory with Diagonal Cornering 53
5 Conclusions and Recommendations 58
References 60
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List of Figures
Figures Page 1.1 Omnidirectional robots: (a) mecanum drive (b) omni drive (c) swerve drive. 2 1.2 Mecanum wheel. 3 1.3 NASA OmniBot 2. 4 1.4 Industrial robots: (a) Airtrax ATX-3000 industrial forklifts (b) Mecanum- wheeled vehicle with container and trolley 2. 4 1.5 Medical mecanum-wheeled vehicle: OMNI 6, CIIPS wheelchair 7, iRW 8 (from left to right). 4 1.6 The interactive shopping trolley 9. 5 2.1 Mecanum wheel. 10 2.2 Mecanum-wheeled vehicle. 10 2.3 Single wheel model. 11 2.4 Mecanum-wheeled vehicle model. 12 2.5 Driving and slip forces developed on a roller of the i wheel. 12 2.6 Friction coefficient versus slip ratio 12. 14 2.7 Relaxation length versus slip ratio 22. 16 2.8 Block diagram of hierarchical networked control system. 19 2.9 Block diagram of state feedback controller. 19 3.1 Single wheel model. 25 3.2 Mecanum-wheeled vehicle model. 25 3.3 Driving and slip forces developed on a roller of the i wheel 25 3.4 Friction coefficient versus slip ratio 12. 27 3.5 Relaxation length versus slip ratio 22. 27 3.6 Hierarchical network system of the mecanum-wheeled vehicle. 29 4.1 Overall control system. 31 4.2 P-Controller. 32 4.3 Block diagram of hierarchical networked control system with physical interconnection. 33 4.4 Block diagram of state feedback controller with physical interconnection. 33 4.5 Friction coefficient versus slip ratio of rubber rollers on dry tarmac surface. 38 vi
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4.6 Relaxation length versus slip ratio 22. 38 4.7 Overall control system with state feedback controller. 39 4.8 Manually-tuned state feedback controller. 39 4.9 X-Y trajectory of manually-tuned state feedback controller. 40 4.10 Velocity and position responses of manually-tuned state feedback controller. 41 4.11 Driving force responses of manually-tuned state feedback controller. 41 4.12 Slip responses of manually-tuned state feedback controller. 42 4.13 Torques generated by manually-tuned state feedback controller. 42 4.14 Position error of manually-tuned state feedback controller. 43 4.15 Driving Force error of manually-tuned state feedback controller. 43 4.16 X-Y trajectory of HLQR controller with global gains. 44 4.17 Velocity and position responses of HLQR controller with global gains. 45 4.18 Driving force responses of HLQR controller with global gains. 45 4.19 Slip responses of HLQR controller with global gains. 46 4.20 Torques generated by HLQR controller with global gains. 46 4.21 Position error of HLQR controller with global gains. 47 4.22 Driving Force error of HLQR controller with global gains. 47 4.23 X-Y trajectory of HLQR controller without global gains. 48 4.24 Velocity and position responses of HLQR controller without global gains. 49 4.25 Driving force responses of HLQR controller without global gains. 49 4.26 Slip responses of HLQR controller without global gains. 50 4.27 Torques generated by HLQR controller without global gains. 50 4.28 Position error of HLQR controller without global gains. 51 4.29 Driving Force error of HLQR controller without global gains. 51 4.30 X-Y trajectory of diagonal cornering. 54 4.31 Velocity and position responses of diagonal cornering. 55 4.32 Driving force responses of diagonal cornering. 55 4.33 Slip responses of diagonal cornering. 56 4.34 Torques generated during diagonal cornering. 56 4.35 Position error during diagonal cornering. 57 4.36 Driving Force error during diagonal cornering. 57
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List of Tables
Tables Page 2.1 Combination of wheel motion and resulting vehicle direction 19. 11 4.1 System parameters and description. 31 4.2 Vehicle ppecification. 37 4.3 Controller gains. 39 4.4 Weighting matrices. 44 4.5 Integral square error of the first path. 53 4.6 Control effort comparison. 53 4.7 Integral square error of diagonal cornering. 53
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Chapter 1 Introduction
1.1 Motivation
Nowadays, autonomous mobile robots are widely used in many industrial and household applications. These robots are typically non-holonomic systems that have limited maneuverability in tight, confined workspaces due to the minimum steering angle. This may necessitate multiple readjustments of the orientation to navigate through narrow pathways and around corners. In these situations, holonomic or omni- directional wheeled robots would provide better performance and maneuverability 1. This includes mobile robots that employ mecanum wheels shown in Fig. 1.1(a) with the ability to move in any direction without changing the orientation of the vehicle. Other mechanisms that allow for omni-directional movements include omni wheels and swerve drives shown in Fig. 1.1(b) and Fig. 1.1(c), respectively. A comparison of these omnidirectional wheel types is as follows 2. Both omni drives and mecanum wheels have compact designs that simplify control system development when vehicle dynamic and kinematic are available. However, mecanum drives provide more traction force and higher load capacity compared to omni drives. In comparison, the swerve drive has a much more complex design. Moreover, mecanum-wheeled vehicles do not suffer from high friction and scrubbing caused by wheel steering that is common in swerve drives. As our aim is to develop a control system of a robot designed primarily for industrial applications with high load capacity, this research focuses on mecanum-wheeled vehicles. However, mecanum drives suffer from slippage and require an adequate anti- slip control system due to high sensitivity to floor conditions.
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(a) (b) (c) Figure 1.1 Omnidirectional robots: (a) mecanum drive (b) omni drive (c) swerve drive.
The design of mecanum wheels looks like a conventional wheel with free- moving rollers attached to the circumferences at angle of πΌ to the main rotational axis, as shown in Fig. 1.2. The number of rollers can vary depending on the design and size of the wheel. There are different types of rollers made from materials such as rubber and polyurethane which are found in heavy duty applications 3. The angle between the free-moving rollers and the rotational axis of the wheel is typically 45β and allows for omnidirectional mobility of the vehicle.
1.1.1 Practical Applications
Omni-directional mobility allows mecanum-wheeled vehicles to operate well in congested environments. Moreover, they are very suitable for applications that require a high degree of maneuverability. For these reasons, mecanum-wheeled vehicles and robots find practical applications in various fields, such as exploration, industrial, and service 2.
Mecanum-wheeled vehicles are often used to support search-and-rescue missions and planetary explorations. When navigating through unknown or rough terrains, the omni-directional capabilities of mecanum wheels allow it to travel efficiently pass obstacles and narrow spaces with increased maneuverability. An example where these robots are used in place of human beings to explore hazardous environments is the NASA OmniBot, shown in Fig. 1.3.
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In industrial applications, mecanum robots have been developed for transporting and handling materials, as well as for automated inspection. Material handling robots are used to transport cargo or workpieces inside of factories and warehouses. These busy environments often have narrow or congested pathways that require omni-directional movement. One example is the Airtrax ATX-3000 industrial forklift shown in Fig. 1.4 (a) that is capable of carrying heavy loads in environment with limited space. Alternatively, the mecanum wheels can be installed on a container or trolley, as shown in shown in Fig. 1.4 (b), to move small goods 4. In 5, a mecanum- wheeled vehicle installed with an ultrasonic scanner is developed for pipe inspection on uneven surfaces while avoiding gravitational slip and also maintaining its velocity and alignment.
Lastly, mecanum-wheeled vehicles are also used in robotics for health-care applications and customer service. Powered wheelchairs utilizing mecanum wheels have been developed to help the elderly, handicapped people or those who have difficulties in walking. These wheelchairs allow the user to move around and assist their daily-living activities. Three examples are shown in Fig. 1.5, and includes the Office Wheelchair for High Maneuverability and Navigational Intelligence for People with Severe Handicap (OMNI) 6, Center for Intelligent Information Processing Systems (CIIPS) omni-directional wheelchair developed at the University of Western Australia 7, and intelligent Robotic Wheelchair (iRW) 8. Another example of service applications is the Interactive Behavior Operated Trolley (InBOT) shopping cart shown in Fig. 1.6 can help customers shop and find desired products 9. Those customers can even control the movement of the shopping cart without pushing it.
πΌ
Figure 1.2 Mecanum wheel.
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Figure 1.3 NASA OmniBot 2.
(a) (b)
Figure 1.4 Industrial robots: (a) Airtrax ATX-3000 industrial forklifts (b) Mecanum-wheeled vehicle with container and trolley 2.
Figure 1.5 Medical mecanum-wheeled vehicle: OMNI 6, CIIPS wheelchair 7, iRW 8 (from left to right).
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Figure 1.6 The interactive shopping trolley 9.
1.1.2 Problems of Mecanum-Wheeled Vehicle
However, mecanum-wheeled robots become difficult to control effectively when moving over surfaces with very low or very high coefficient of friction, such as an oily floor or rough concrete. In these conditions, undesired phenomenon such as slipping or skidding can be observed due to the reduced traction of mecanum wheels when compared with conventional wheels 10. This is caused by a reduction in the effective driving forces due to the orientation of the rollers. To alleviate this problem, several hardware solutions have been proposed. An alternative mecanum wheel was developed by Bengt Ilon to solve the problem when the wheel is operating on an uneven surface, where each roller is split into two parts and centrally mounted to ensure that they will always touch the floor. Likewise, an improved design was proposed in 11 that add extra twist mechanism capable of transforming itself by adjusting the orientation of rollers so the robot can travel in a specific direction with higher driving force using the same torque when compared to the traditional design.
1.1.3 Anti-Slip Control System for Conventional-Wheeled Vehicles
The problem of slip also occurs in vehicles with conventional wheel drives. To compensate for this, improved control systems have been developed. In 12, a driving force distribution controller for in-wheel motor electric vehicles (IWM-EVs) was proposed to ensure straight motion when subjected to a strong wind disturbance. The 5
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main concept involved the implementation of driving force control in order to suppress the slip ratio of electric vehicles. A similar work examines the effect of acceleration on split-friction surfaces and utilizes glocal control with a decentralized hierarchical structure 13. Lastly, 14 introduced a limiter with driving stiffness estimation in order to maintain the driving force within controllable conditions.
1.1.4 Anti-Slip Control System for Mecanum-Wheeled Vehicles
Several dynamic and kinematic models have been developed for mecanum- wheeled vehicles. A popular method for velocity control of omni-directional robots is based on kinematic equations 15. Kinematic controllers use pure geometrical motion and neglect forces and torques, thus simplifying analysis and design. However, omitting these dynamic effects also decreases the performance of the system. Alternatively, dynamic models can be obtained using Lagrangian equations 16 17 or combining Newtonian mechanics with Lagrange method 18 to give a more accurate representation of the vehicle. These previous researches mention the effect of wheel traction but do not incorporate slip into either the kinematic or the dynamic models. An example of a model of a mecanum-wheeled vehicle with slip is derived based on Newtonβs law in 19. As mentioned previously, the problem of slip greatly effects conventional-wheeled vehicles and is especially important for mecanum wheels that have reduced traction.
Currently, the number of research related to slip and slip control of omni- directional vehicle is limited, especially for mecanum drives. A summary of anti-slip techniques is provided in 20, where a velocity adjustment method is implemented in order to control the wheel slip but resulted in instability of the closed loop control system. Due to the limited numbers of researches related to mecanum-wheeled vehicles, some implementations related to conventional non-holonomic robots can be adapted for mecanum-wheeled robot. In 21, a well-known conventional wheel tire model, such as George Rillβs tire model 22 and Pacejkaβs tire model 23, is adapted to describe a mecanum wheel with slip ratios. Velocity feedback control, which is independent from platform kinematics, is developed to make braking of the mecanum-wheeled vehicle safer. Another research explored slip control for mecanum-wheeled vehicles by using position rectification control with kinematic-based symptomatic and preventive
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rectifications of position and orientation 24. Other methods include a fuzzy inference control system based on kinematic modelling 25 and position corrective control 26, but experimental results demonstrate considerable difficulty in achieving trajectory tracking. These methods treat each mecanum wheel as separate agents and typically neglect the physical interconnection between them. Therefore, this research proposes a decentralized hierarchical controller that considers the physical interconnection of the wheels through the vehicle chassis to achieve higher accuracy and performance by encouraging collaborative control 12272829.
1.2 Objective
This paper investigates the dynamic model of each individual mecanum wheel and equations of motion of the vehicle with slip in order to establish a decentralized hierarchical structure. The upper layer consists of the global objective for position control of the vehicle. The lower layer is used for driving force control of each wheel. The interconnection between layers is determined and a glocal control strategy 30 based on hierarchical LQR (HLQR) is proposed. Since the direction of motion of the vehicle depends on the coordination between all mecanum wheels, each driven by independent motors and subjected to different slip ratios, the concept of glocal control will be applied to achieve consensus between all agents and improve performance. The validity of the proposed controller is shown by simulation of a four-wheel mecanum robot.
1.3 Thesis Scope
1. Assume that the model is made for the linear translation only. 2. Assume that the viscous friction coefficient between shaft and bearing is omitted. 3. Assume that all the states of the agents can be measured. 4. Only force adjustment controllers will be used in order to suppress slip. 5. Assume that the platform is travelling on a horizontal plane.
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1.4 Thesis Structure
The organization of the thesis is as follows. Chapter 2 presents the literature review of this research. Chapter 3 the dynamic modelling of the mecanum-wheeled vehicle. Chapter 4 presents the hierarchically decentralized optimal control. Chapter 5 presents the conclusion of this research.
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Chapter 2 Literature Review
This chapter provides an overview of dynamical modelling for mecanum- wheeled vehicles, anti-slip control techniques, and hierarchically decentralized optimal control for time-varying systems. A general description of the mechanics for a mecanum-wheeled robot is given, followed by derivation of dynamical equations using Newtonβs Second Law of Motion. A review of anti-slip control techniques is conducted, including a detailed analysis of slip and how to suppress it. The last section summarizes existing control systems based on hierarchically decentralized optimal control that are relevant to this thesis.
2.1 Mecanum-Wheeled Vehicle
Omni-directional mobility is required to provide better performance and maneuverability for autonomous mobile vehicles (AMVs) in certain applications such as material handling and transport. AMVs equipped with omni-directional drives, such as mecanum wheels, can perform tasks that are not suitable for other non-holonomic mobile vehicles, especially in narrow environments that require free translational and rotational movements. Depending on the objective, high load capacity is also necessary.
Mecanum-wheels are invented by Bengt Ilon in 1973. The design of mecanum wheels looks like a conventional wheel with free-moving rollers attached to the circumferences at angle of πΌ, which is typically 45β in practice, to the main rotational axis, as shown in Fig. 2.1. The number of rollers can vary depending on the design and size of the wheel. There are different types of rollers made from materials such as rubber and polyurethane which are found in heavy duty applications 3. By installing such wheels to a platform, omnidirectional mobility of the vehicle can be realized.
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Mecanum wheel.
Mecanum-wheeled vehicle.
Omni-directional mobility can be achieved when these wheels are mounted to each corner of the platform as shown in Fig. 2.2. The number of the wheels are identified differently in different researches. Fig. 2.4 shows the model of the vehicle with a specific set of wheel numbers. The wheels are numbered starting from the front right wheel to the back right wheel in a counter clockwise direction. However, 18 identified the number of the wheel in another way. Nevertheless, most of the mecanum- wheeld vehicles studied are written in the form similar to Fig. 2.4 as the characteristic of the wheels can be distinguished using even and odd numbers. Because of the orientation that the rollers, which are appeared on the mecanum wheels, make with the motor shafts, the direction of driving forces of each wheel making with the surface are shifted as shown in Fig. 2.5. Hence, different combinations of motor torques applied to the wheels result in different movement of the vehicle as each wheel has its own motor and could be driven independently. The relationship of the vehicle movement and the combination of the motor torque with the same wheel number identification as in Fig. 2.4 can be found in Table 2.1. Note that the table shows the direction of angular velocity of each wheel where the arrows in the figure are in positive direction. + refers to positive direction of motion. - refers to negative direction of motion. 0 means that the wheel is not rotate at that moment.
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Table 2.1: Combination of wheel motion and resulting vehicle direction 19.
Wheel No. 1 2 3 4
Direction of the vehicle North + + + + + + 0 0 0 0 + + South ------0 0 0 0 - - East - + - + - 0 0 + 0 + - 0 West + - + - + 0 0 - 0 - + 0 North-east 0 + 0 + South-west 0 - 0 - North-west + 0 + 0 South-east - 0 - 0
π π πΉ
π πΉ
Single wheel model.
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π¦ π¦ π£ πΉ π¦β² πΉ 2 π¦β πΉ π½ πΌ 1 πΉ π 3
π π₯β² 4 π₯β
π₯ π₯
Mecanum-wheeled Driving and slip forces developed vehicle model. on a roller of the π wheel. 2.1.1 Fundamental Vehicle Dynamics
Several dynamic and kinematic models have been developed for mecanum- wheeled vehicles. A popular method for velocity control of omni-directional robots is based on kinematic equations 15. Kinematic controllers use pure geometrical motion and neglect forces and torques, thus simplifying analysis and design. However, omitting these dynamic effects also decreases the performance of the system. Alternatively, dynamic models can be obtained using Lagrangian equations 16 17 or combining Newtonian mechanics with Lagrange method 18 to give a more accurate representation of the vehicle. These previous researches mention the effect of wheel traction but do not incorporate slip into either the kinematic or the dynamic models. An example of a model of a mecanum-wheeled vehicle with slip is derived based on Newtonβs law in 19. As mentioned previously, the problem of slip greatly effects conventional-wheeled vehicles and is especially important for mecanum wheels that have reduced traction.
In 19, Newtonβs law has been applied in order to find the dynamic equation. Fig. 2.4 shows a mecanum-wheeled vehicle of total mass π with π in-wheel motors (IWMs), where π is assumed to be 4. The πth wheel is depicted in Fig. 2.3 and has a
radius of π , which is assumed to be the same for all wheels and will henceforth be referred to simply as π. Around the circumference of the wheel are rollers, positioned such that the rotational axis of each roller makes an angle of πΌ 45Β° with the rotational axis of the wheel. Fig. 2.4 shows the body of the vehicle with reference to the world frame, where π₯, π¦, π§ refers to the stationary coordinate axis and π₯β², π¦β², π§β² refers to the body-attached coordinate axis at the geometrical center of the vehicle. Here, π is the wheel number, π is the orientation of the vehicle and π½ is the direction of the vehicle
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velocity with respect to the vehicle orientation. πΉππ is the force developed on the roller
th due to the motor torque ππ at the circumference of the π mecanum wheel.. The developed torque can be written as:
π π πΉ (2.1)