Robust Post Impact Vehicle Motion Control Using Torque Vectoring

DEGREE PROJECT IN VEHICLE ENGINEERING, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2019

Robust post impact vehicle motion control using torque vectoring

NIKHIL JAIN

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES Robust post impact vehicle motion control using torque vectoring Nikhil Jain

Supervised by: Jenny Jerrelind, Department of Aeronautical and Vehicle Engineering Lars Drugge, Department of Aeronautical and Vehicle Engineering Mustafa Ali Arat, Motion control and software, NEVS Eduardo Simoes˜ da Silva, Motion control and software, NEVS

Master thesis 2017 Department of Aeronautical and Vehicle Engineering KTH Royal Institute of Technology SE-100 44, Stockholm, Sweden Telephone: + 46 8 790 60 00 i

Abstract

Several statistical studies have suggested that the risk of injury is signiﬁcantly higher in multiple event accidents (MEAs) than in single event crashes. Improper driving in such scenarios leads to hazardous vehicle heading angles and excessive lateral deviations from the vehicle path, resulting in severe secondary crashes. In these situations, the vehicle becomes highly prone to side impacts and such impacts are more harmful to the occupants since the sides of the vehicle have less crash energy absorbing structures than the front and rear ends.

Signiﬁcant advancements have been made in the area of automotive safety to ensure passenger safety. Active safety systems, in particular, are becoming more advanced by the day with vehicles becoming over actuated with electric propulsion and x-by-wire systems. Keep- ing that in mind, in this master thesis a post impact vehicle motion control strategy for an electric vehicle is suggested based on a hierarchical control structure which regulates the lateral deviation of the affected vehicle while maintaining safe heading angles after an impact.

Sliding Mode Control (SMC) has been utilized in the higher controller which generates a virtual output used as an input for a lower controller performing torque allocation. The allocation methods were based on optimization, aimed to minimize tire utilization, and a normal force based approach. The performance of the controller was first evaluated with single track and two track model of the vehicle because of their simplicity making them easy to debug and also since they allowed for quick simulations. This was followed with evaluation with a high fidelity vehicle model in IPG CarMaker for fine tuning the controller. It was observed that the use of SMC strategy to generate virtual yaw moment to be used in torque vectoring for controlling vehicle trajectory post impact proved to be a robust strategy managing to control the vehicle even in cases of actuator failure. So it can be concluded that the hierarchical control structure with the higher Sliding mode controller, generating a virtual yaw moment, and a lower controller doing torque allocation using a normal force based strategy and an optimization approach worked as intended.

Sammanfattning

Flera statistiska studier har foreslagit¨ att risken for¨ skador ar¨ signiﬁkant hogre¨ vid ﬂerfoljd-¨ skollisioner an¨ vid enskilda kollisioner. Felaktig korning¨ i sadana˚ scenarier leder till farliga fordonspositioner och stora avvikelser fran˚ fordonets ursprungliga fardriktning,¨ vilket resul- terar i svara˚ sekundara¨ kollisioner. I dessa situationer blir fordonet ofta utsatt for¨ sidokol- lisioner vilka ar¨ mer skadliga for¨ passagerarna, eftersom sidorna pa˚ fordonet har mindre energiabsorberande strukturer vid deformation an¨ framre¨ och bakre delarna pa˚ fordonet.

Betydande framsteg har gjorts inom omradet˚ bilsakerhet¨ for¨ att sakerst¨ alla¨ passagerarsaker-¨ heten. Aktiva sakerhetssystem,¨ i synnerhet, blir alltmer avancerade i och med att fordon blir overaktuerade¨ med bland annat elektrisk framdrivning och x-by-wire system. Med tanke pa˚ detta foresl¨ as˚ i detta examensarbete en strategisk styrning av fordonets rorelse¨ for¨ ett elfor- don baserat pa˚ en hierarkisk reglerstruktur som reglerar laterala avdriften hos det drabbade fordonet samtidigt som det uppratth¨ aller˚ saker¨ fordonsriktning efter en kollision.

Sliding Mode Control (SMC) har anvants¨ i den hogre¨ kontrollenheten som genererar ett virtuell utvarde¨ som anvands¨ som invarde¨ for¨ en lagre¨ styrenhet som utfor¨ momentalloker- ing. Allokeringsmetoderna baserades pa˚ optimering, som syftade till att minimera dackut-¨ nyttjandet och ett normalkraftsbaserat tillvagag¨ angss˚ att.¨ Styrenhetens prestanda utvarder-¨ ades forst¨ med en ”singel track model” och en ”two track model” av fordonet pa˚ grund av deras enkelhet som gor¨ dem enkla att felsoka¨ och mojligg¨ or¨ snabba simuleringar. Darefter¨ gjordes en utvardering¨ med en mer detaljerad och komplex fordonsmodell i IPG CarMaker for¨ ﬁnjustering av regulatorn. Det observerades att anvandningen¨ av SMC-strategin for¨ att generera ett virtuellt girmoment for¨ att anvandas¨ i momentvektoriseringen for¨ reglering av fordonsriktningen efter kollision visade sig vara en robust strategi som klarar av att styra fordonet aven¨ i handelse¨ av aktuatorfel. Slutsatsen ar¨ att den hierarkiska reglerstrukturen med en hogre¨ ”sliding mode controller”, som genererar ett virtuellt girmoment och en lagre¨ styrenhet som utfor¨ momentallokeringen med en normalkraftbaserad strategi samt optimering fungerade som det var tankt.¨

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Thesis contributors

This master thesis work was performed at the company National Electric Vehicle Sweden (NEVS) situated inTrollhattan¨ in collaboration with Mr. David Nigicser, a master thesis stu- dent from the System, Control and Robotics Masters program at KTH Royal Institute of Technology. The designing and analysis of Model Predictive Control (MPC) [36] as a higher controller for vehicle motion control post impact has been performed by Mr. Nigicser and the literature study for this work has been a joint effort. Since the problem statement was the same, few portions of this work like vehicle models used for simulation, which were provided by NEVS, are similar in the work presented by Mr. Nigicser.

This thesis work focuses on using Sliding mode control strategy in the higher controller with torque allocation methods based on optimization and normal forces for the lower controller.

Acknowledgements

I would like to start by thanking my thesis supervisors at NEVS, Mustafa Ali Arat and Ed- uardo Simoes˜ Da Silva for their immense support during the course of this thesis. It were the discussions with them which helped shape the project and built it in a way such that it could be useful for the industry. I would also like to extend my thanks to the Vehicle Motion Control team at NEVS for being so supportive and taking time to contribute to this work.

I am very grateful to my supervisors at KTH, Lars Drugge and Jenny Jerrelind for their support, your suggestions in the theoretical and the practical aspect of this project were of immense help. I would also like to thank my friends David, Anand and Hjortur¨ for their suggestions and sharing their knowledge. Interactions and discussions with you all were really informative and inspiring.

Lastly, my master degree studies in Sweden would have been impossible without the support of my family. My deepest appreciation to my parents and my sister for being there all along the way and keeping me motivated.

List of Figures

1 Accident statistics [11]...... 5 2 Free body diagram of a single track model...... 9 3 Global and local coordinate system...... 10 4 Free body diagram of a two track model...... 11 5 Slip angle...... 12 6 Non-linear tire force behavior [23]...... 13 7 Data on impulse magnitude and velocities [33]...... 17 8 Single track model in Simulink...... 18 9 Two track model in Simulink...... 20 10 Yaw rate as a function of time for an impact of 2 KNs...... 21 11 Controller setup in IPG CarMaker...... 21 12 Vehicle states after impact...... 23 13 Torque outputs on the four wheels and corresponding tire slip ratios...... 24 14 Vehicle trajectory with and without controller at 8 kNs...... 24 15 Vehicle trajectory with and without controller at 4 kNs...... 24 16 Force outputs with rule based approach. Impact = 8 kNs...... 25 17 Force outputs with optimization. Impact = 8 kNs...... 25 18 Force outputs with optimization. Impact = 8 kNs, actuator 2 failed...... 26 19 Vehicle trajectories with and without actuator failure, with and without controller action. Impact = 8 kNs, actuator 2 failed...... 26

Contents

1 Introduction 1 1.1 Scope and assumptions ...... 2 1.2 Limitations ...... 2 1.3 Research questions ...... 3 1.4 State of the art ...... 3 1.4.1 Safe heading angles ...... 5 1.4.2 Fault tolerant systems ...... 6 1.5 Structure of the report ...... 7

2 Theory 8 2.1 Vehicle modelling ...... 8 2.1.1 Single track model ...... 8 2.1.2 Two track model ...... 10 2.1.3 Tire modelling ...... 12 2.2 Sliding Mode Control ...... 13 2.2.1 Concept ...... 14

3 Method 16 3.1 Scenario selection ...... 17 3.2 Integration with single track plant model ...... 18 3.3 Integration with two track plant model ...... 19 3.4 Integration with IPG CarMaker plant model ...... 20 3.5 Actuator failure - two track plant model ...... 21

4 Results 22 4.1 Integration with IPG CarMaker plant model ...... 22 4.2 Actuator failure - two track plant model ...... 24

5 Conclusions 26 5.1 Feasibility of the system ...... 27 5.2 Future work ...... 27

1 Introduction

Automotive safety is of prime importance to drivers, manufacturers and the society in general. Signiﬁcant technological advancements have been made in this area till date to ensure that the passenger safety is always guaranteed. Safety systems are categorized into two main classes: active and passive safety systems. Active safety systems involve systems for accident prevention and avoidance whereas passive safety systems deals with minimizing the occupant injury during an accident. In the European Union (EU) alone there have been approximately 26000 fatalities due to road accidents in 2015 [1] and this number has been steadily decreasing over the years due to new innovations introduced to the automotive industry as well as increased safety standards from the governments [3][2].

The global passenger car fleet is estimated to be around 1.7 billion by 2035 [4] in which the highest growth rate is projected to be after 2020. With these high number of vehicles on the road, there is an increasing concern of environmental pollution along with the availability of fossil fuels required to run these vehicles. Owing to these concerns, there has been an advent of electric vehicles on roads because of lesser harmful impact on the environment along with the possibility of depending on renewable sources for the generation of electricity, thereby decreasing pollution levels [5]. In the last two decades, electrification of automotive powertrain systems have enabled extended functionalities and more refined active safety systems. Recent studies have shown a reduction of single vehicle accidents of about 50% for vehicles equipped with Electronic Stability Control systems (ESC) [2], and thereby highlighting the potential of these systems. This electrification of chassis and drivetrain systems have led to a high degree of over actuation in vehicles, since there are more actuators than required to control a certain number of degrees of freedom, leading to more flexibility of the vehicle behavior.

Several statistical studies [6], [7], [8] based on vehicle crash data suggest that the risk of severe injuries is much higher in multiple event crashes than in single impact crashes. Mul- tiple event crashes become much more dangerous from side and rear impacts since the driver only focuses on the front view after the initial impact. Improper driving in such scenarios leads to hazardous vehicle heading angles and excessive lateral deviations from the vehicle path, resulting in severe secondary crashes. A crash analysis report by the National High- way Trafﬁc Safety Administration (NHTSA) [9] shows the vehicle heading angle distribution in the most harmful secondary event crashes and indicates that secondary crashes at around 90 degree (broadsiding, T-Bone) are the most severe. In these situations, the vehicle becomes highly prone to side impacts and as shown in [10], such impacts are more harmful to the occupants since the sides of the vehicle have less crash energy absorbing structures than the front and rear ends. An analysis investigating the vehicle dynamic motion after an impact shows that excessive lateral lane deviations play a key role in the risk of secondary collisions [11].

Since Electronic Stability Control [3] is not designed for secondary collision mitigation and even though previous works have proposed solutions to minimize yaw motion and lateral deviation from the original path [11] the threat of vulnerable heading angle to subsequent collisions with another moving vehicle or stationary object still exists. Thus, new active safety systems have to be designed. 2

1.1 Scope and assumptions Based on the above mentioned statistics and control methods, the goal of the project is to develop a post-impact vehicle stability control system that regulates both heading angle and lateral deviation from the original driving path, so that the severity of possible subsequent (secondary) crashes can be reduced. This area of study is rather broad and therefore the work presented here is limited to certain key points listed as follows:

• Proposition of a robust active safety control system for performing post impact vehicle motion control.

• Suggestion of an activation logic for the proposed system.

• Testing of the strategy for certain actuator failure modes to analyze robustness.

For the proposed system, certain assumptions were made, listed as follows:

• The considered plant is an electric vehicle with four individually controllable wheels available for both propulsion and braking (Brake-by-wire).

• Electric Power Assisted Steering (EPAS) is assumed.

• For the manoeuvre, a straight road is assumed with the vehicle velocity as 100 kmph.

1.2 Limitations The subjects which are closely related to the proposed secondary collision mitigation active safety function development but not covered in the present thesis are:

• The design and evaluation of the proposed systems are to be implemented in MAT- LAB/Simulink [12] environments. A high ﬁdelity vehicle model is available in IPG CarMaker [13] and a generic vehicle model is used. The complete study is performed in a simulation environment and no on road testing is done.

• Force estimation [14] and friction estimation [15] of the vehicle are key in active and passive systems. In this work, values directly available from IPG CarMaker are considered rather than implementing the estimation algorithms.

• Constraints of real time signal processing as well as hardware compatibility consider- ations.

• In order to ensure that the proposed active safety system is realizable a comprehensive cost and environmental analysis has to be performed which is not considered in this work.

• Human computer interaction and control hand-over to the driver is not thoroughly researched and values available from other studies are directly considered. 3

1.3 Research questions The goal of this master thesis is to carry out a study on vehicle motion control in emergency situations with special focus on secondary collision mitigation. In this study, the following points were investigated:

• Develop a method to achieve post-impact vehicle stability and determine what can be the safe and stable vehicle states which should be attained for the post-impact stable behavior.

• Evaluate the use of torque vectoring for post impact vehicle motion control.

• Perform a review of a hierarchical control structure to solve the motion control problem. Evaluate the performance of Sliding Mode Control (SMC) as a higher controller and a normal force based/optimization based lower controller. Sliding Mode Control (SMC) is a robust control method that permits to neglect the model of the system to be controlled and represents a robust solution that can be adopted without knowing the perfect representation of the plant.

1.4 State of the art Recently, electric vehicles (EV) have attracted a great deal of interest from the automotive manufacturers as they are a powerful solution to the environmental and energy concerns. EVs have two big advantages over vehicles with internal combustion (IC) engines. Firstly, electric motors can be controlled more accurately with lower response times and secondly, in an EV with four in-wheel motors, each drive wheel can be controlled independently offering more opportunities and ﬂexibility in vehicle motion control.

The aim of this project is to propose a post impact control system, for a electric vehicle with four in-wheel motors, which can bring the vehicle back to a stable state or safe stop after an initial impact in order to maximize passenger safety after an initial collision. One of the key aspects of such a strategy is having an effective torque allocation system which ensures that the tires can be utilized in the best possible way, with optimal slip ratios to bring the vehicle under control in the shortest amount of time. A novel dynamic yaw moment control (DYC) system is suggested in [16] with special attention on an anti-skid system during torque allocation which does not require data on wheel or chassis velocities making the system quite robust. A driver assistance system has been proposed in [17] which uses automatic steering for course correction in the event of an accident. The approach adopted here assumes control from the driver to stabilize the vehicle and then gives the control back to driver after a latency period. The paper also gives quantiﬁed data in terms of yaw-rates, yaw accelerations and body side slip angle values as criterion for when to start and stop the assistance system.

A hierarchical motion control architecture with a high level and low level controller has been introduced in [18]. The higher level controller, a Lyapunov based nonlinear control approach, calculates the required virtual total ground forces and the force split between the left and right sides of the vehicle. The lower level controller distributes the higher level control efforts to the four wheels, using a numerical optimization based control allocation algorithm. This adaptive controller does not need accurate values of vehicle parameters like 4 mass and drag coefﬁcient as simulation results prove its effectiveness.

An accident statistical study performed by the German insurance association conﬁrms that even a vehicle involved in a light impact is likely to experience a severe secondary crash, and one-third of all accidents with severe injuries consist of multiple events [19]. [20] intro- duces a vehicle stabilization approach in response to external disturbances which brings the vehicle to a state such that it avoids a secondary collision. It details a sensing and validation scheme to classify if an external disturbance has actually occured or if it is just a sensor error. Full ABS activation, differential braking and differential braking along with active steering is suggested for controlling the unstable behavior while assuming that all sensors and actuators are undamaged.

Yang et al. [11] used the GIDAS database to collect the accident statistics to locate the ar- eas on a vehicle body which were most prone to accidents (Figure 1) and it was deduced that most of the accidents occur on the front side of the vehicle, either a head on collision or an angled impact. The author further describes that there is a safe zone of kinematic variables even after the impact from which the vehicle states can be stabilized, but outside that, very little or nothing can be done. A combination of differential braking and active steering is shown to be the best way to stabilize the vehicle states. Numerical optimization methods gives details on which approach proves better in what conditions. The author has used different control strategies like torque vectoring and full ABS braking to control the vehicle motion but concludes that even within a single accident a combination of these two techniques may have to be used to achieve the optimal solution faster. The author has described cost functions for crash risk and for crash severity and more specifically defined a cost function to minimize the maximum lateral deviation from the original path. Then, in order to limit the path deviation a Quasi Linear Optimal Controller (QLOC) is developed for online closed loop implementation. Noting that the post impact initial kinematics is used as initial conditions. Sensitivity analysis is done as well to find out what range of control inputs can be used to achieve optimal responses but it is later found that it is very much dependent on actuator bandwidth. In the end, QLOC is compared with Post Impact Braking (PIB) controller for various test scenarios and the author has detailed the techniques which were used to emulate the collision scenarios. 5

Figure 1: Accident statistics [11].

1.4.1 Safe heading angles According to a crash analysis report by the National Highway Trafﬁc Safety Administration (NHTSA) [9], vehicle heading angle distribution is among the most harmful reasons for secondary event crashes. It shows that secondary crashes at around 90°(broadsiding, T-Bone) are the most severe and that excessive lateral lane deviations after the initial impact plays a key role in the risk or secondary collisions [11]. Around 85% of vehicles involved in multiple- event accidents (MEAs) have a heading angle of 90°, either clockwise or counterclockwise, when suffering a secondary event crash, resulting in the most severe injuries [9].

Kim et al. [21] propose a post-impact vehicle stability control system that controls both heading angle and path lateral deviation to mitigate the possibility and severity of a secondary crash. Consequently, the proposed system predicts the heading angle after a collision and controls vehicle spin to reach a safe heading angle (multiples of 180°). The proposed method relies on the following assumptions: the event occurs on a straight road, only two vehicles are involved and the sensors and actuators are fully functioning after the crash. Using a crash dynamics model, magnitude and location of impulse can be computed [22]. This calculated impact force is then used to predict the vehicle motion. Vehicle side slip angles up to 360°are studied, considering tire characteristics based on Pacejka’s Magic Formula [23], con- cluding that signiﬁcant yaw rate reduction can be achieved when the heading angle crosses through 180°and 360°. 6

The control synthesis is formulated as an optimization with the gradient descent method, with the objective function to minimize lateral deviation while achieving desirable heading angles. Control inputs – longitudinal slip ratios – are obtained by minimizing the objective function under the slip ratio constraints. The authors argue that reducing yaw rate as quickly as possible might not be the safest secondary collision mitigation approach, due to the fact that the vehicle is very sensitive to side impacts, hence considering safe heading angles is crucial. The safe heading angle in a post-crash scenario might be determined by the vehicles perception of its surrounding.

1.4.2 Fault tolerant systems The increasing electrification of drivetrain in vehicles gives more flexibility to control the vehicle motion with faster response times but the chances of faults coming in these components can also give rise to dire situations. Faults such as sensor failure, electrical motor failure, in- verter shutdown are some examples in which the control strategies mentioned above might not work. Wanner et al. [24] gives a good introduction to all the accident statistics around the world with providing a good terminology for different kinds of faults and malfunctions within a vehicle. The authors have described methods to model a particular fault in [24] and the evaluation criterion for controller performance for dealing with faults is described. The authors have described the most common faults which can occur in the vehicle with a special focus on faults in the electric driveline. Almost 250 faults are considered and the faults resulting in a similar vehicle response are grouped in one for which a single counter strategy is used. Fault classification is performed in a detailed manner with quantitative evaluation for longitudinal, lateral and yaw disturbances. For describing the controllability of the vehicle, three different indices are used namely, the vehicle stability index, lane keeping index and collision avoidance index. The data on these indices was obtained by performing the run in a normal manner first and then a second run in which a fault was introduced so that a healthy and faulty parameter is obtained at a particular time instant. Following this, the authors give a detailed description of reaction times and general numbers regarding those and used that data to create a driver model so that the experiments on driver reaction times and tests on controllers for stabilizing the vehicle during faults could be simulated in a virtual environment with more precision. Fault detection and isolation are not considered but rather it is assumed that the knowledge about the fault is available as soon as it happens and ways to deal with it are suggested through different control strategies.

In [25], three control allocation strategies are described comprising of an optimization approach using least squares method and two analytical approaches namely, Pseudoinverse Control Allocation (PCA) and Approximate Pseudoinverse Control Allocation (APCA). It is pointed out that optimal control using Recursive Least Squares (RLS) is computationally heavy and so its concluded to use PCA or APCA for onboard implementation. The control allocation strategies are compared with each other as well as with an ESC equipped vehicle which is considered to be the benchmark. The experiment is performed by braking one of the wheels in straight driving and then applying a torque allocation strategy to control the destabilizing behavior. In simulations, however, other faults can be introduced and tested but this will require a more robust controller. 7

1.5 Structure of the report The report is divided into the sections of Theory, Methods, Results and Conclusions. The Theory chapter comprises all prerequisite theoretical background pertaining to vehicle dynamics and the controller strategies used. Methods consists of the details about the application and experimentation with the proposed strategy. Following this, in Results, the performance of the strategy is evaluated with the limitations of the controller discussed in detail. The Conclusions chapter summarizes the results obtained and connecting them with the research questions put forward. This is followed with the possible extensions to the strategy to improve its performance and feasibility. 8

2 Theory

2.1 Vehicle modelling For evaluating the performance of the controller strategy, tests were first performed on sim- pler vehicle models with fewer degrees of freedom. Since the impact conditions will force the tires into non-linear region (large slip angles), a non-linear single track (Figure 2) and two track model (Figure 4) were used to perform initial tests with the controller before the evaluation with more complex, high fidelity models in IPG CarMaker. Since only straight line manoeuvres were considered in this study, it was assumed that δ (wheel steer angle), is zero. A simplified tyre model [26] based on the Magic tyre formula was used for simulations with the single track and two track model. The vehicle dynamics equations derived in the following sections for the single track and two track model were used in creating simulink models of the same. The common parameters used in the modelling of the single track and two track model are listed in Table 1. Table 1: Parameters in common between single track and two track model.

Parameter Description m Mass of the vehicle vx Longitudinal velocity of the vehicle vy Lateral velocity of the vehicle ψ˙ Yaw rate of the vehicle Fx Longitudinal force on the vehicle Fy Lateral force on the vehicle Fz Normal force on the vehicle Jz Moment of inertia of the vehicle L Wheelbase of the vehicle g Acceleration due to gravity h Center of gravity height from the ground

2.1.1 Single track model The free body diagram of the single track model can be seen in Figure 2. The equations of motion being used in the single track model are explained in Equations 1 - 3 as follows:

Local x dynamics : m(v˙x − ψ˙vy) = Fx1 + Fx2 (1)

Local y dynamics : m(v˙y + ψ˙vx) = Fy1 + Fy2 (2)

Yaw dynamics : Jzψ¨ = Fy1l1 − Fy2l2 (3)

Since this is a single track model, the vehicle roll is not considered. Further, the rolling resistance of the tire and the wind resistance of the vehicle have been neglected. The static load on each wheel (Fz) is given by,

mg(L − l ) F = i i = 1, 2 (4) z,i L 9

Figure 2: Free body diagram of a single track model.

Where, li corresponds to the distance from the center of gravity, that is, l1 is the front axle distance from the COG. L is the wheelbase and g is the acceleration due to gravity. Here, m is the mass of the vehicle, ψ˙ and ψ¨ is the yaw rate and yaw acceleration respectively, vx and vy are velocities in the lateral direction respectively. v˙x and v˙y are accelerations in the longitudinal and lateral direction. Fx and Fy is the longitudinal and lateral force on each tire with the subscript 1 and 2 corresponding to the front and rear tires. The load transfer as a result of the pitch motion is given as,

mh ∆F = a i = 1, 2 (5) zx,i 2L x

Here, h corresponds to the Center of gravity height from the ground and ax is the longitudinal acceleration. Therefore, the vertical force on each wheel will be the sum of the static loads and longitudinal load transfer, given by:

mg(L − l ) mh F = i ± a i = 1, 2 (6) z,i L 2L x The relation between the global and local coordinate system can be seen in Figure 3. The vehicle velocities in the global frame of reference is given in Equations 7 - 8.

Y˙ = xsin˙ ψ + ycos˙ ψ (7) X˙ = xcos˙ ψ − ysin˙ ψ (8) 10

Figure 3: Global and local coordinate system.

x˙ and y˙ correspond to the vehicle velocities in the local frame of reference. ψ refers to the yaw or heading angle of the vehicle obtained from the integration of yaw rate. Integration of X˙ and Y˙ provides us with the vehicle position in global frame which is used for deﬁning the reference trajectory of the vehicle in the controller.

2.1.2 Two track model The equations of motion of the two track model are derived and described in the following section. It offers three degrees of freedom and a more realistic vehicle model. The single track model is used to conﬁrm if the controller’s concept is correct mathematically. Further testing is done with the two track model in which application of the controller is possible in a more practical and realistic manner. The equations of motion for the two track model are given in Equations 9 - 11 below:

Local x dynamics : m(v˙x − ψ˙vy) = Fx f l + Fx f r + Fxrl + Fxrr (9)

Local y dynamics : m(v˙y + ψ˙ vx) = Fy f l + Fy f r + Fyrl + Fyrr (10) t Yaw dynamics : J ψ¨ = (F + F )l + (F − F )l + (F + F − F − F ) w (11) z y f r y f l 1 yrr yrl 2 x f r xrr x f l xrl 2

Here, tw is the track-width of the vehicle. 11

Figure 4: Free body diagram of a two track model.

Since this is the two track model, roll dynamics are now considered. The static load distribution for this model is deﬁned as,

mg(L − l ) F = i i = f ( f ront), r(rear) and j = r(right), l(le f t) (12) zij 2L

l f and lr correspond to the front and rear axle distance from the center of gravity respectively. When the vehicle accelerates or decelerates, there is a longitudinal load transfer given by,

mh ∆F = a i = f , r (13) zx,i 2L x h corresponds to the height of the center of gravity from the ground and L is the wheelbase. Similar to longitudinal accelerations, load transfer also happens due to lateral motion of the vehicle and is given by,

! 1 (L − Li) mheCi ∆Fzy,i = )mRi + ay i = f , r (14) twi L Cf + Cr − mghR

R corresponds to the roll centre height of an axle, tw is the track-width, hR is the perpendicular distance between the center of gravity and roll axis, Ci is the roll stiffness of the front and rear axles. Combining the static loads, longitudinal and lateral load transfers, the actual load on each wheel at any instant is given by the expression, 12

! mg(L − li) mh 1 (L − Li) mheCi Fzij = ± ax ± )mei + ay (15) 2L 2L twi L Cf + Cr − mghe

i = f , r and j = l, r

Any other external forces are neglected in the model and no aerodynamic or rolling resistance is taken into account. The global velocities and positions are calculated here in the same way as done for the single track model to get a reference trajectory.

2.1.3 Tire modelling

The slip angle denoted by αij is used to determine the lateral force acting on a wheel. Slip angle is the angle between the direction in which a wheel is pointing and the direction in which it is actually traveling (Figure 5). The slip angle on a wheel is deﬁned as follows,

vy αij = δ − arctan( ) i = f , r and j = l, r (16) |vx|

Figure 5: Slip angle.

Here, vy and vx correspond to the lateral and longitudinal velocity at a particular wheel. δ is the wheel steer angle, which in our case is always zero. In reality, the longitudinal and lateral forces are interdependent and are proportional to longitudinal and lateral slip which is known as the combined slip model. However, to simplify things, in this study the lateral and longitudinal forces are assumed to be interdependent but are only proportional to the lateral slip which is a good approximation of the combined slip model. In Figure 6a, it can be clearly seen that lateral forces are not only directly proportional to the lateral slip but also depend on how much longitudinal slip is present. The lesser the longitudinal slip, the more lateral force can be produced by a wheel at a particular value of lateral slip. In the adjacent ﬁgure (Figure 6b), the same behavior can be observed for the longitudinal forces. The tire can generate only particular combinations of lateral and longitudinal force, the magnitude 13 of which is decided according to the circle of forces which in reality resembles an ellipse (Figure 6c). It can also be concluded that beyond a certain value of slip (both lateral and longitudinal), the tire cannot produce a larger force, which is known as tire saturation. The behavior shown in Figure 6 shows the trend of tire forces but this greatly varies from tire to tire which makes tire modelling a complex task.

Figure 6: Non-linear tire force behavior [23].

A simpliﬁed tire model [26] is used in this study based on the Magic Tyre Formula [23]. The model captures the saturation of the tires well but does not show a digressive behavior with varying load and has no distinct peak in lateral force.

Fyij = Dyij.sin(C.arctan(Bα − E(Bα − arctan(Bα)))) i = f , r and j = l, r (17)

Here, D is the peak factor, C is the shape factor, B is the stiffness factor and E is the curvature factor. Dij = µFzij, if pure lateral forces are considered (µ is the coefﬁcient of friction). The peak factor was expressed using combined slip model assuming the longitudinal forces are known as,

q 2 2 Dyij = (µFzij) − Fxij i = f , r and j = l, r (18)

Here, the longitudinal forces are calculated by the controller and are restricted within the range −µFzijcosα ≤ Fxij ≤ µFzij.

2.2 Sliding Mode Control Sliding Mode Control (SMC) is a robust control method that permits to neglect the model of the system to be controlled and represents a robust solution that can be adopted without knowing the perfect representation of the plant. In this way the approximations performed to build the state space model of the vehicle are no more required because SMC can handle non-linear systems, even under a certain degree of uncertainty in the known dynamics. The robustness of the method is a direct consequence of the control law derivation that relies only on the error dynamics, without involving any parameter of the controlled system in the main control law. 14

In [27] the author has suggested the use of sliding mode control for stabilizing the vehicle in a way that it always follows a reference yaw rate and side slip angle. A 3 DOF two track model is used as a plant and a 3 DOF single track model is used for generating the reference values. Here, a hierarchical controller structure is proposed with the higher level controller generating the values of stabilizing yaw moment and driving torques while the lower level controller allocates the driving force to the four wheels keeping the tire force and actuator constraints in mind.

In [28] the author has proposed a preemptive steering control strategy that takes into con- sideration post-impact vehicle stability. The author proposes a strategy which starts reacting before the collision happens which results in stabilizing the vehicle faster. A combination of steering and differential braking is used for post impact stabilization by controlling the yaw rates and lateral velocity. The controller supplies the control inputs of steering correction and brake signals to counteract the motion generated after the impact for which it uses the sum of results from a feedforward and feedback controller. The feedback controller shown in this work uses SMC to generate a value of steer angle which adds to the response of feedforward controller as well as a value of yaw moment which is used to produce differential braking by using a simple rule based approach.

2.2.1 Concept SMC systems are designed to drive the system states onto a particular surface in the state space, named sliding surface. Once the sliding surface is reached, sliding mode control keeps the states on the close neighbourhood of the sliding surface. Hence the sliding mode control is a two part controller design. The first part involves the design of a sliding surface so that the sliding motion satisfies design specifications. The second is concerned with the selection of a control law that will make the switching surface attractive to the system state [29]. The general form of sliding surface S is defined as:

d S = ( + p)ke (19) dt Where, e is the error between the actual and reference value of a state and p is a scalar constant.

k = d − 1, d is the relative degree of the system that is, the order of the derivative of e required to have an expression that includes the control input explicitly.

There are several algorithms available within Sliding mode control to deal with systems having different relative degrees.

2.2.1.1 Classical SMC Classical SMC approach is mainly suitable for relative degree one systems, however the formulation can be modiﬁed to utilize it in systems with relative degree two. The control output u with classical SMC for this system takes the form, 15

u = Usgn(S) + f (x¨, x˙, t) (20)

Here, U is a constant. This formulation leads to a chattering effect (discontinuous high frequency switching control) in the control output which is completely unsuitable for automotive applications.

2.2.1.2 Super twisting algorithm Super twisting algorithm [30] is mainly meant to control the systems with relative degree two. The sliding surface used for this approach is same as the one used in classical SMC. Based on the equivalent control method, the control output u is written as:

u = ueq + ust (21) where ueq and usw are called the equivalent control and the switching control, respectively. The equivalent control ueq is determined based on the assumption that the system trajectory is staying on the sliding surface. Thus, it is simply obtained by setting S˙ = 0. The switching control ust is designed to guarantee that the system trajectory moves towards the sliding surface and stays on it. It takes the following form, q Z ust = −η |S|sgn(S) − Wsgn(S) (22)

Here, η and W are constants.

Thus, the control output from the controller is the sum of ueq and ust and that is supplied to the plant. One problem with the super twisting algorithm is of chattering like classical SMC which makes it impractical for automotive use. This chattering is because of the ’sgn’ term used in the equation of control input. However, this algorithm is better suited for dealing with relative degree two systems than classical SMC.

2.2.1.3 Sub-optimal algorithm The idea of Sub optimal approach is derived from time-optimal control of a double integrator system [30], here represented using the states variables y1 and y2:

y˙1 = S˙ = y2 (23)

y˙2 = S¨ = λ(t) + τ(t)u(t) (24) 1 τ(t)u(t) = −r sgn(S − S(t )) + r sgn(S(t )) (25) 1 2 c 2 c

S(tc) is the most recent value of S when S˙ = 0. r1 and r2 are scalar constants which are tunable. τ(t) is a function attached to the control input. The function u(t) can be the control input itself, in the case of system of relative degree d = 2, or its derivative. The formulation of equation was made such that u(t) is the derivative of the control input so that in the end, the ﬁnal output will be an integration of u(t) which will integrate the ’sgn’ functions which 16 results in no or atleast highly reduced chattering. This approach produced the best result for our system and its implementation is described as follows:

e = (ψ − ψre f ) + ζ(Y − Yre f ) (26) d S = ( + p)ke (27) dt

Here, ψ and ψre f are the actual and reference yaw angle of the vehicle respectively. Y and Yre f are the actual and reference lateral deviation of the vehicle respectively. ζ is a constant behaving as a weight on the error of lateral deviation.

Now, from the expression of e and S, S˙ and S¨ can be calculated.

S = (ψ˙ − ψ˙re f ) + ζ(Y˙ − Y˙re f ) + p((ψ − ψre f ) + ζ(Y − Yyre f )) (28) ...... ¨ ¨ ¨ S = (ψ − ψre f ) + ζ(Y − Yre f ) + p((ψ¨ − ψ¨re f ) + ζ(Y − Yyre f )) (29) From the yaw dynamics equation for the single track model (Equations 1 - 3), it is obtained that:

Jψ¨ = Fy1l1 − Fy2l2 + M (30) ... Jψ = F˙y1l1 − F˙y2l2 + M˙ (31) ... Substituting the value of ψ from the above equation, the expression of S¨ becomes,

F˙y1l1 − F˙y2l2 M˙ ...... S¨ = ( + − ψ ) + ζ(Y − Y ) + p((ψ¨ − ψ¨ ) + ζ(Y¨ − Y¨ )) (32) J J re f re f re f yre f Therefore, M˙ τ(t)u(t) = (33) J 1 M˙ = −r Jsgn(S − S(t )) + r Jsgn(S(t )) (34) 1 2 c 2 c It is comprehensible that the value of the expression τ(t)u(t) given by Equations 33 - 34 will be the same, be it any single track or two track model. Sliding mode control is computationally light and easy to tune. As can be seen from the above equations, SMC does not require the prefect representation of the plant which makes it very robust to model changes and in particular for this application can be very well applied to a fault tolerant system design without any changes. Therefore, it can perform vehicle motion control even in cases of actuator failure for example, as long as the torque allocation strategy is able to accommodate the modiﬁed model which can be done by using static control allocation [31], for example.

3 Method

For testing the Sliding Mode Control strategy, different vehicle models with increasing com- plexity were used to follow a systematic approach in the testing and tuning of the controller. The tuning of the controller began with its application to a single track model, followed by a two track model and ﬁnally to IPG CarMaker model which represented the actual vehicle (Volkswagen Beetle). 17

3.1 Scenario selection The controller is tuned for light impacts meaning the sensors and actuators are in proper working condition after the impact and the location of the impact is assumed to fall on the vehicle periphery instead of an arbitrary position inside the vehicle. Based on the crash statistics collected and organized in Yang et. al. [11], four scenarios were deﬁned taking the largest impact forces, perpendicular to the direction of travel. The fours scenarios are side impacts at the front and rear axles, from both left and right direction. The angle of the impact is assumed to be perpendicular to the vehicle’s longitudinal axis, since this will induce a moment of maximum amplitude on the car. Thus the impacts are introduced as a moment-force pair, where the moment (Mimpact) acts on the CoG and the force (Fimpact) acts in the lateral direction.

The scenarios deﬁned above is based on data from the German In-Depth Accident Study (GIDAS) [32], analysed by Yang [11] as well as J. Beltran and Y. Song [33]. The energy of the impulse for the main use case is considered to be 8 kNs based on the analysis done in [33] which shows that the impulse energy ranging from 0 - 8 kNs represent 61 % of all the accidents considered. The velocity for the test is considered to be 100 km/h which represents 78 % of the cases (0 - 100 km/h) (Figure 7). The impact duration is considered to be 0.2 s based on the data presented in [33].

Figure 7: Data on impulse magnitude and velocities [33]. 18

3.2 Integration with single track plant model A simple three DOF single track model was created to analyze the performance of a sliding mode controller. The controller took the difference of actual and reference yaw angle along with the lateral deviation from the line of travel as inputs and generated a yaw moment as the output. This yaw moment was used as an input to the yaw dynamic equations of single track model as follows:

m(v˙y + ψ˙vx) = Fy1 + Fy2 + Fimpact (35)

Jzψ¨ = Fy1l1 − Fy2l2 + Yaw moment + Mimpact (36)

In the above equations, Fimpact and Mimpact are the lateral force and moment generated by the collision. For maximum impact, it was assumed that the point of impact is at either end of the vehicle. The methodology to control the vehicle shown here is only theoretically possible since it is not possible to directly generate a yaw moment without application of longitudinal forces in the vehicle. However, since only a single track model is considered in a straight line scenario, a direct application of yaw moment was the only way possible to control the vehicle. Steering was not considered (δ = 0) for simplicity and was out of scope for this project. The collision forces considered throughout the thesis are taken from GIDAS database [11]. The architecture of the single track model with the controller is shown in Figure 8.

Figure 8: Single track model in Simulink.

As it can be seen in Figure 8, the model calculates the slip angle on the front and rear tires from the wheel steer angle and vehicle states information. The slip angles, longitudinal force requests (considered zero for single track model) and normal forces on each wheel 19 is fed to the tire model to generate the longitudinal and lateral force value on each wheel. These forces and yaw moment output from the controller are fed to the vehicle model which outputs the vehicle states. The model starts with an initial velocity while all other states have an initial value of zero. The vehicle state information is fed back to the load transfer block, for generating the value of normal force on each tire, and to the controller for generating the value of counter yaw moment.

3.3 Integration with two track plant model The analysis of the controller in the single track model was followed with an analysis in the two track model whose architecture is presented in Figure 9. A simulink model with equations for the two track model as mentioned in Section 2.1.2 was made which provides a good representation of the whole vehicle. Unlike the single track model, the counter yaw moment generated from the controller is indirectly applied to the vehicle by distributing it to the four wheels as longitudinal forces. An allocation approach based on normal forces was designed for this purpose whose equations are mentioned as follows:

• The virtual control input, that is the counter yaw moment generated by the higher controller is divided between the moment produced from the front and rear axle. The division is based on the ratio of normal forces on the front and rear axle.

Mz = Mz f + Mzr (37) Fz f Mz f = (Mz) (38) Fztot Fzr Mzr = (Mz) (39) Fztot

• When the moments from the front and the rear axle are derived, the next step is to determine the combination of propulsive and brake force on each axle which will generate the respective moment. The longitudinal forces to be produced on each wheel of an axle is then divided according to the ratio of normal forces as follows: t M = (F − F ) w (40) z f x2 x1 2

Fz1 tw (Mz f ) = Fx1 (41) Fz1 + Fz2 2 Fz2 tw (Mz f ) = Fx2 (42) Fz1 + Fz2 2

Mz f Fz1 Fx1 = 2( )( ) (43) tw Fz1 + Fz2 Mz f Fz2 Fx2 = 2( )( ) (44) tw Fz1 + Fz2

The parameter tw introduced above is the track-width. Please note here that Fx1 and Fx2 will be opposite in sign. This can be proved by putting the expressions for Fx1 and 20

Fx2 in Equation 40 above while assuming the magnitude of Fx1 to be negative. This assumption makes the left hand side and right hand side of Equation 40 to be equal, hence proving the assumption to be correct. Similar equations were derived for the longitudinal forces of the rear axle as well.

Figure 9: Two track model in Simulink.

The two track model works in a similar way as the single track model. As it can be seen in Figure 9, the model calculates the slip angle on the front and rear tires from the wheel steer angle and vehicle states information. The slip angles, longitudinal force requests and normal forces on each wheel is fed to the tire model to generate the longitudinal and lateral force value on each wheel. These forces are fed to the vehicle model which outputs the vehicle states. The model starts with an initial velocity while all other states have an initial value of zero. The vehicle state information is fed back to the load transfer block, for generating the value of normal force on each tire, and to the controller for generating the value of counter yaw moment from which the longitudinal force requests are calculated for the tire model.

3.4 Integration with IPG CarMaker plant model After completing the simulations with the single track model and two track model, the tuning parameters for the controller were ﬁnalized. Further simulations were carried out in IPG CarMaker using a high ﬁdelity fourteen DOF vehicle model. A generic Volkswagen Bee- tle was considered for the simulations whose powertrain was replaced with four in-wheel motors and the vehicle was equipped with a steer-by-wire and brake-by-wire system. An algorithm based on the yaw rates was designed to activate the system. Since a straight line scenario was considered, it was assumed that when the yaw rate becomes more than 25 deg/s, the controller is activated and from that moment, throttle, brake and steering signals are made to be zero and only the torque values generated by the controller are sent to the motors. The controller is deactivated when the yaw rate error is less than 2 deg/s for one second after which, control is handed back to the driver. The threshold value of yaw rate as 21

25 deg/s was obtained after observing the yaw rates at the lowest impact magnitude considered amongst the use cases, that is, 2 kNs. This impact induced the highest yaw rate of 35 deg/s to the vehicle while operating without controller action (Figure 10). Therefore, a suitable yaw rate which is below 35 deg/s had to be chosen as a threshold since the controller needed to be active in this lowest level impact. The yaw rate level of 25 deg/s was then considered to be appropriate based on simulation results. The higher controller generating the moment requests had to be ﬁne tuned to give the best performance with the new plant and since the lower controller gave force requests, they had to be converted to torque requests by using the relationship Torque = Force ∗ Radius. The setup can be seen in Figure 11.

Yawrate behavior at impact maginitude - 2 kNs 40

Yawrate (deg/s) 0

-10

-20 0 2 4 6 8 10 12 14 16 Time (seconds)

Figure 10: Yaw rate as a function of time for an impact of 2 KNs.

Figure 11: Controller setup in IPG CarMaker.

3.5 Actuator failure - two track plant model A special case was considered in which one of the actuators fail because of the impact, in which case, the request from the higher controller needs to be distributed among the remaining three wheels. Optimization for torque allocation was tried in Simulink with a two track model and has produced good results. The main motivation behind using optimization was to do a proper distribution of torques in case an actuator failure happens and rather than 22 making a different set of rules for torque allocation in case of actuator failure, the optimization algorithm is used for the task by creating an appropriate cost function. The function fmincon was used to minimize the cost function and the aim here was to prove the concept that the higher controller can handle the situations even during actuator failures. The cost function for the algorithm is inspired from the one presented in [34] and is as follows:

2 2 u = (Bu − y) + (W(u − ud)) (45)

Where,

B = [−tw/2 tw/2 − tw/2 tw/2] 0 u = [Fx1 Fx2 Fx3 Fx4] 1 W = diag( ) i = f l, f r, rl, rr µFzi 0 ud = [0 0 0 0] y = Virtual moment from higher controller

The ﬁrst part of the cost function ensures that the moment generated by the four longitudinal forces matches the virtual control moment from the higher controller and the second part ensures that there is minimum utilization of tire in the longitudinal direction while doing so. In Simulink, this was achieved through an interpreted Matlab function which required the virtual control output from the higher controller and the normal forces on each wheel as inputs. The elements of vector ’B’ could also be made as inputs and modiﬁed at every time instant since the force request for a particular actuator can be made zero by making the corresponding element in the B matrix to be zero. However, this was manually performed for this project. The number of iterations with fmincon were limited to 100 and initialization values for all four outputs were set to zero.

4 Results

4.1 Integration with IPG CarMaker plant model The simulations with the proposed solution are made in IPG CarMaker and produced posi- tive results. For the simulations, multiple impact magnitudes were considered as mentioned in Section 3.1 ranging from 2 kNs to 8kNs and the impact duration was considered to be 0.2s. It was observed that the controller was able to manage the vehicle trajectory as intended after impacts while trying to achieve minimum lateral deviations and heading angle in the multiples of 180°. The results with the impact magnitude of 8 kNs can be seen in the following section.

As it can be seen in Figure 12a and Figure 12b which shows the vehicle states which are inputs for the controller, the vehicle is brought under control in approximately 2 seconds. The impact is injected at t = 1s and lasts for 0.2 seconds. The vehicle without controller goes out of the road at approximately t = 5s after which it is deemed uncontrollable by the driver and the simulation is terminated at that point. However, one thing to note is the torque values required by the system (Figure 13a) for bringing the vehicle in control is very 23 high (1000-1500 Nm) and also very oscillatory. It seems that to bring the vehicle in control in such short amount of time, such outputs will be necessary. The torque outputs, although high and oscillatory, should be achievable through electric motors (at peak power outputs) and should not be a matter of concern in such emergency scenarios [35]. Looking at the slip ratios in Figure 13b, it can be seen that there is high slipping for the ﬁrst two seconds after the impact following which the slip ratio values are considerably lower. This result can be used to conclude that the torque allocation strategy based on normal forces performs well in such scenarios and helps to utilize the tire better. The vehicle trajectory with and without controller action can be seen in Figure 14 and Figure 15 for impact level of 8 kNs and 4 kNs respectively. The arrows in the ﬁgure correspond to the heading angle of the vehicle after impact. For the impact level of 8 kNs, the controller behaves as intended and keeps the vehicle in a suitable orientation with a maximum lateral deviation of 5.4 meters after the impact. In the case of impact magnitude as 4 kNs, the lateral deviation of the vehicle after impact is less when the controller is inactive. The reason for this behavior is that in this particular scenario, the driver model in IPG CarMaker was able to control the vehicle better. How- ever, this cannot be generalized since the driver model in IPG CarMaker is not the perfect representation of a human driver.

Vehicle state behavior with and without controller, Impact = 8 kNs Vehicle state behavior with and without controller, Impact = 8 kNs 250 20 Without controller Without controller With Controller 15 With Controller 200 10

150 5

0 100 -5 Yawrate(deg/s) 50 -10 Lateral velocity (m/s)

-15 0 -20

-50 -25 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 Time(s) Time(s) (a) Yaw rate after impact (b) Lateral velocity after impact

Figure 12: Vehicle states after impact. 24

Torque outputs during controller action, Impact = 8 kNs Slip ratios during controller action, Impact = 8 kNs 1500 1 FL FL FR 0.8 FR 1000 RL RL RR 0.6 RR

0.4 500 0.2

0 0

Slip ratios -0.2

Torque output (Nm) -500 -0.4

-0.6 -1000 -0.8

-1500 -1 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 Time(s) Time(s) (a) Torque output from the controller (b) Slip ratios after impact

Figure 13: Torque outputs on the four wheels and corresponding tire slip ratios.

Figure 14: Vehicle trajectory with and without controller at 8 kNs.

Figure 15: Vehicle trajectory with and without controller at 4 kNs.

4.2 Actuator failure - two track plant model The scenario with one of the actuators failing due to impact was simulated with the two track model in Simulink. The main reason for running this scenario with a simpliﬁed plant model and not the IPG CarMaker model was to make the simulations faster. As explained in Section 3.5, the purpose of this test was to test if Sliding Mode Control is able to control the vehicle trajectory in such cases. Since in these tests, optimization was used to allocate the 25 forces to the actuators, a comparison was made between the force allocation using the rule based and optimization approach which gave some insight into the efﬁciency of the normal force based torque allocation, the results of which are presented as follows:

• The force outputs with the rule based and optimization approach can be seen in Figure 16 and Figure 17 respectively. As can be seen, the force outputs show very similar trend with the rule based and the optimization approach signifying that the longitudinal force allocation based on normal forces is very efﬁcient and utilizes the tires in an effective manner.

Figure 16: Force outputs with rule based approach. Impact = 8 kNs.

Figure 17: Force outputs with optimization. Impact = 8 kNs.

• To see if SMC is able to handle the situation in case of actuator failure, a simulation was performed in which motor supplying torque to the front right wheel was assumed to be failed. The results are shown in the following ﬁgures. The force outputs from the controller can be seen in Figure 18. It can be noted that since the information regarding the failure of front right actuator is fed to the controller, the force output for that actuator is zero. The counter moment is thus generated by allocating the forces to the remaining three actuators. It can be seen that the controller is able to handle the situations pretty well (Figure 19) even when there are actuator failures. Since the output of SMC is only dependent on state trajectory errors, if the torque allocation is proper then SMC can manage the situation well. 26

Figure 18: Force outputs with optimization. Impact = 8 kNs, actuator 2 failed.

Vehicle trajectory with and without actuator failure

6 Without actuator failure, With controller With actuator failure, With controller With actuator failure, Without controller 4

-2 Lateral deviation (m)

-4

-6

0 50 100 150 200 250 Distance travelled (m)

Figure 19: Vehicle trajectories with and without actuator failure, with and without controller action. Impact = 8 kNs, actuator 2 failed.

5 Conclusions

The methodology of minimizing lateral deviation while maintaining the yaw angles of the vehicle at multiple of 180° resulted in being an efﬁcient strategy to perform post impact vehicle motion control. The use of SMC strategy to generate virtual yaw moment to be used in torque vectoring for controlling vehicle trajectory post impact proved to be a robust strategy managing to control the vehicle even in cases of actuator failure. A hierarchical control structure with the higher SMC, generating a virtual yaw moment, and a lower controller doing torque allocation using a normal force based strategy and an optimization approach works as intended. It was also noted that the results of torque allocation using the normal force based strategy and optimization approach were similar indicating towards the efﬁciency of the former. 27

5.1 Feasibility of the system The concept shown in the thesis is theoretically possible but is practical only for a premium car. Four in-wheel motors along with steer-by-wire and brake-by-wire systems are a neces- sity for the concept to work efﬁciently and since these features are not common for vehicles yet, their manufacturing is expensive. The torque outputs on each wheel requested by the controller are high and oscillatory and may be possible only during peak power outputs meaning that the behavior can be achieved only by a few selected electric motors.

5.2 Future work The concept developed only highlights the controller and the strategy however for a complete system, attention needs to be given to other systems as well which were beyond the scope of this work, some of which are explained in the following paragraphs.

In the present work, vehicle motion control is through torque vectoring without any steering inputs, the inclusion of which can alter the results drastically. As mentioned in the above section, the torque outputs from the controller are unrealistic, however they can be signiﬁcantly improved with the help of steering actuation and can even help in stabilizing the vehicle in a shorter amount of time by applying the longitudinal forces in the most optimum direction.

The proposed solution is only designed to work in straight line scenarios and for curved roads more sensors and algorithms need to be in place which can perform path planning in case of impact on such roads. Path planning will also form an integral part of the system in case it needs to be decided if the vehicle should maintain a particular lateral deviation from its original path, for example to avoid crashes with the oncoming vehicles or if the impact magnitude is too high.

The present system works with a fixed set of tuning parameters irrespective of the collision forces which makes the controller to compromise in some situations (it may be too strong in some cases and too weak in others) affecting the time to control the vehicle. Knowing the impact forces can help to dynamically adjust the tuning parameters in different scenarios to make the system more efficient [28]. Simulations can be performed to find the heading angles which would be most suitable for the vehicle during different levels of impact, thereby adjusting the tuning parameters in case of different impacts to achieve smooth convergence to a particular heading angle. 28

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