Predictive vehicle motion control for post-crash scenarios

Transcription

1 DEGREE PROJECT IN ELECTRICAL ENGINEERING, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2017 Predictive vehicle motion control for post-crash scenarios DÁVID NIGICSER KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL ENGINEERING

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3 Predictive vehicle motion control for post-crash scenarios. Dávid Nigicser August 31, 2017 Master Thesis Automatic Control KTH School of Electrical Engineering Industrial supervisors: Mustafa Ali Arat Eduardo Simões da Silva Academic examiner and supervisor: Jonas Mårtensson Valerio Turri

4 Abstract The aim of the project is to design an active safety system for passenger vehicles for mitigating secondary collisions after an initial impact. The control objective is to minimize the lateral deviation from the known original path while achieving a safe heading angle after the initial collision. A hierarchical controller structure is proposed: the higher layer is formulated as a linear time varying model predictive controller that defines the virtual control moment input; the lower layer deploys a rule-based controller that realizes the requested moment. The designed control system is then tested and validated in Simulink as well as in IPG CarMaker, a high fidelity vehicle dynamics simulator. Keywords: Vehicle Motion Control, Multiple Event Accidents, Secondary Collision Mitigation, Linear Time Varying Model Predictive Control, Torque Vectoring, Sliding Mode Control, NEVS I

5 Sammanfattning Syftet med projektet är att för personbilar designa ett aktivt säkerhetssystem för att undvika följdkollisioner efter en första kollision. Målet är att minimera den laterala avvikelsen från den ursprungliga färdvägen och att samtidigt uppnå en säker kurs efter den första kollisionen. En hierarkisk regulatorstruktur föreslås. Det övre skiktet i regulatorn är formulerat som en linjär tidsvarierande modell prediktiv kontroller som definierar den virtuella momentinmatningen. Det nedre skiktet använder en regelbaserad regulator som realiserar det begärda momentet. Det konstruerade styrsystemet testades och validerades sedan i Simulink samt i IPG CarMaker, en simulator med hög precision för fordonsdynamik. Nyckelord: Fordonsstyrning, Rörelsestyrning, Följdolyckor, Sekundär kollisionsbekämpning, Linjär Tidsvarierande Modellprediktiv Reglering, Momentvektorering, Sliding mode-reglering, NEVS II

6 Acknowledgements Firstly, I owe a debt of gratitude to my academic supervisor, Valerio Turri for giving me thoughtful guidance throughout the whole thesis project. Thank you for your engagement and patience during our long conversations. Your constant feedback and advice has enabled me to reach very high standards with the project. Also, I would like to thank my academic examiner, Jonas Mårtensson for facilitating and supporting the thesis project from the side of KTH. Secondly, I am grateful for the guidance of my industrial supervisors, Mustafa Ali Arat and Eduardo Simões da Silva. Your supervision assured the project s relevance to industry and provided me the perfect ground to grow professionally. It has been a pleasure to work alongside people of the Software and Controls department at NEVS, lunch time conversations helped me to relax and stay focused. I am also very grateful for my friends and colleagues in Göteborg. Nikhil, Hjörtur and Anand, it has been a real pleasure to work and share an episode of my life with you guys. Thank you for all those conversations on the bus and train rides, and for your motivation and camaraderie through good days and bad days. Likewise, I am grateful for your friendship and constant support Paolo. I am glad we made that move to the city which has great things in store for us! Lastly and most importantly, I want to express my deepest appreciation to my father, mother, sister and brother. Without your unconditional love and selfless support I could not have finished this thesis and my master degree. Thank you for always inspiring and challenging me to fulfil my potential! Soli Deo gloria III

7 List of Figures 2.1 Top view of a vehicle indicating the physical distribution of impacts analysed in [9] The road-fixed, inertial coordinate system, i.e. global frame, denoted by capitals XY Z. And the vehicle-fixed coordinate system, i.e. local frame, denoted by lower-case xyz [ISO 6487] Two track model of a vehicle in the XY plane with a reference lane. Original figure taken from B. J. Kim et al. [14] Simplified semi-empirical Pacejka tire model of equation (3.13), and its modified version in equation (3.14) capturing the nonlinear lateral force - tire slip angle relation The proposed hierarchical control architecture Open loop vehicle heading angle trajectory - subject to an impulse of 8 kns Illustration of a collision in a generic four-way intersection between two passenger vehicles [ITE, 2010] Frequency distribution of impact magnitude by original velocity, presented in [41] Illustration of the crash validation scheme, including the controller activation timing The plant s global lateral velocity (V Y ) and the linearized (around ψ = 0) prediction model s lateral velocity - 8 kns impact Performance of the linear MPC, compared to an SMC scheme - 4 kns impact Performance of the linear MPC, compared to an SMC scheme - 8 kns impact Longitudinal and lateral tire forces as functions of the longitudinal slip ration and tire slip angle. Original plot can be found in Kim et al. [14] Comparison of the linear MPC and linear time varying MPC controllers - 8 kns impact. The LTV-MPC is plotted for both zero and π heading angle references Vehicle states after an 8 kns impact simulated in IPG Car- Maker with the LTV-MPC deployed Longitudinal slip ratios and torque requests on the wheels (motors) after an 8 kns impact simulated in IPG CarMaker with the LTV-MPC deployed IV

8 5.11 Vehicle trajectories after a 6 and 8 kns impact simulated in IPG CarMaker with the LTV-MPC deployed. The open loop trajectories are included as a reference Lateral acceleration profiles at 6 kns and 8 kns impacts List of Tables 4.1 Final vehicle heading angles in case of open loop simulation of crashes at various initial impulses. Maximum heading angles when the control loop is closed, for the same crashes Solver calculation times in seconds using different QP solvers and tracking different heading angle references. The simulation time is t = 6 seconds Performance comparison of the linear MPC and LTV-MPC controllers. With the LTV-MPC, two different reference heading angle is tracked IPG CarMaker simulations of different magnitude impacts. The settling time is defined as settling around the reference in a band of ± 10 degrees V

9 Nomenclature CoG Center of Gravity DoF Degrees of Freedom LTV Linear Time Varying LTV-MPC Linear Time Varying Model Predictive Control MEA Multiple Event Accident MPC Model Predictive Control MUC Main Use Case NEVS National Electric Vehicle Sweden QP Quadratic Program QLOC Quasi-linear Optimal Controller SMC Sliding Mode Control VI

10 Contents 1 Introduction Motivation Problem formulation Thesis contribution Outline Background Post-crash vehicle motion control Vehicle motion control with MPC Vehicle dynamics modelling Vehicle coordinate systems Two track plant model High fidelity vehicle dynamics model - as plant model Prediction models Linearized bicycle model Extended bicycle model Vehicle motion control Architecture of the proposed control system Linear MPC Cost function Model constraints MPC formulation Linear Time Varying MPC Cost function Model constraints MPC formulation Reference generation Implementation strategy Experiments Definition of the main use case Linear MPC results LTV-MPC results Performance comparison Simulations with IPG CarMaker Human factors VII

11 6 Conclusion and further work Extensions of the controller architecture Extensions of the MPC formulation VIII

12 1 Introduction This chapter provides an overview of the problem intended to solve and describes its relevance to current industrial trends. 1.1 Motivation Automotive safety is of vital importance to drivers, passengers, manufacturers and society in general. Aiming to guarantee passenger safety at all times, the automotive sector has seen significant technological advancements of the field in recent years. These safety systems can be categorised as either active or passive safety systems. Passive safety systems help to guarantee the safety of the driver and passengers without intervening in vehicle dynamics through actuators, e.g. airbags, seat-belt tensioners and vehicles physical structure. Active safety systems, however, actively monitor vehicle states to prevent and mitigate the effects of a crash through actuation, e.g. traction control system, anti-lock braking system and various advanced driver assistance systems. In 2016, on the roads of the European Union alone there have been fatal road accidents recorded [1] and this number has displayed a distinct decreasing tendency over the past decade thanks to the introduction of increasingly advanced safety systems. Recent studies have shown a reduction of single vehicle accidents of about 50% for vehicles equipped with electronic stability control systems [20], highlighting the potential of such systems. The increasing concern over sustainability and environmental pollution along with the availability of fossil fuels led to a shift in mobility from the present traditional internal combustion engines to propulsion with electric motors. In the last two decades, electrification of automotive powertrain systems has enabled extended functionalities and more refined active safety systems. This electrification of chassis and drivetrain systems have led to a high degree of over actuation, thus allowing more flexibility in the design of active safety systems. An analysis by Yang et al. shows that approximately 30% of road accidents are so called multiple event accidents (MEAs) [2], where the vehicle is subject to more than one harmful event. Statistical studies [24, 25, 26] based on vehicle crash data, indicate that the risk of severe injuries is much higher in MEAs than in single event accidents: more than 50% of the MEAs leave occupants with serious injuries. Moreover, a crash analysis report by the National Highway Traffic Safety Administration [19] investigates the vehicle heading angle distribution in MEAs and concludes that the majority of 1

13 MEAs (85%) resulting in serious injuries happen at a clockwise or counterclockwise 90 degree heading angle. In these situations, the vehicle becomes highly prone to side impacts, also called broadsiding or T-boning, as shown in [27]. The main reason why such accidents bare a higher injury risk is that sides of vehicles have less energy absorbing structures than the front and rear ends. An analysis investigating the vehicle dynamic motion after an impact shows that excessive lateral deviation from the original lane play a key role in the risk of secondary collisions [9]. Furthermore, since electronic stability control is not designed for secondary collision mitigation [20, 21] and even though previous works have proposed solutions to minimize yaw motion and lateral deviation from the original path the threat of vulnerable heading angle to subsequent collisions with another moving vehicle or stationary object still exists [9]. Consequently, new active safety systems have to be designed. 1.2 Problem formulation The goal of this master thesis is to conduct a study on vehicle dynamics and motion control for post-crash scenarios with special focus on secondary collision mitigation. In particular a post-crash control system is designed that minimizes lateral deviation from the original path while settling the vehicle at a safe heading angle. The vehicle s motion is controlled through controlling the individual wheel torques. This approach leaves a certain level of manoeuvrability to the driver since interaction through the steering wheel is left available. Nevertheless, since the topic is rather broad, the following assumptions are made: the vehicle is equipped with four individually controllable electric in-wheel motors facilitating sophisticated torque allocation; vehicle states such as heading angle is available at all times; the driver does not apply any steering action when the proposed post-crash controller is activated; an impact detection scheme is in place; MATLAB and Simulink is used for the design and evaluation interfacing a dynamic vehicle plant model in IPG CarMaker. Furthermore, subjects which are closely related to the proposed active safety system but not covered in the present thesis are: tire force and friction estimation; constraints of real time signal processing as well as hardware implementability constraints; cost and environmental analysis. Human computer interaction and control hand-over and handback to and from the driver are not covered in this thesis either. Note that the primary focus is controlling the vehicle s lateral dynamics and thus longitudinal dynamics control is only indirectly considered. 2

14 The present thesis investigates weather model predictive control (MPC) is a suitable higher layer controller in a hierarchical controller architecture to provide control inputs in the described post-crash vehicle motion control problem. 1.3 Thesis contribution The present thesis work contains the following contributions: ˆ Vehicle dynamics modelling. Constructing both plant and prediction models tailored to the introduced problem. ˆ Formulation and implementation of linear and linear time varying model predictive controllers. ˆ Controller performance analysis and validation. Extensive simulations of the proposed active safety system, validating control objectives. This master thesis has been carried out at National Electric Vehicle Sweden (NEVS) in Trollhättan, Sweden. Given the rather broad topic, some specific parts of the proposed active safety system have been investigated by myself, others by Nikhil Jain, a peer student of mine. Nikhil s main contribution is the elaboration of an alternative controller for the post impact vehicle motion control problem, namely a sliding mode controller (SMC) [39]. This means that the project s framework has been developed as a joint effort and is therefore common in both reports. Specifically, the following concepts have been developed in collaboration: the control objective, definition of the main use case, vehicle dynamic plant models, impact injection and controller activation schemes. Nevertheless, these common parts have different significance in the two reports and have been documented individually, and thereby, non of the chapters or sections are an exact copy of each other in the two master thesis reports. 1.4 Outline In chapter 2 an overview of the theoretical background of the formulated problem is described. Chapter 3 contains a set of vehicle dynamics models. After the introduction of a three degrees of freedom (DoF) two track plant model and the fourteen DoF IPG CarMaker plant model is briefly described. The same chapter contains the description of two prediction models: a simple, augmented two DoF linear bicycle model and a more elaborate, three 3

15 DoF extended bicycle model. Chapter 4 introduces the proposed controller architecture and elaborates the design of the linear MPC and the linear time varying MPC (LTV-MPC) controllers. In chapter 5, the main use case is defined and the sequence of simulations is described. This chapter also describes the main design decisions and the necessity of the more sophisticated LTV-MPC formulation. Last, chapter 6 contains the conclusions and details possible further extensions of the project. 4

16 2 Background This chapter is an overview of various vehicle dynamics and motion control problems, focusing on the proposed control architectures and approaches as well as torque allocation. Primarily the focus is on problems with similar assumptions to this thesis. First, an overview of various post-crash stability controllers is given. Second, relevant vehicle dynamics control problems are reviewed, focusing on solutions utilizing MPC in particular. 2.1 Post-crash vehicle motion control Sakai et al. [3] propose a novel robust dynamics yaw-moment control system, describing a vehicle attitude control method generating yaw from torque differences between the left and right wheels, assuming four independently driven in-wheel motors. The described controller is a robust-model matching controller based on linear-quadratic regulator theory using state feedback. Through simulations, the authors identified stability problems on low friction roads, due to saturated traction tire force and therefore propose a new, robust skid detection method which does not require wheel and chassis longitudinal velocities. Instead, observing the ratio of tractive force and motor torque (the slope of the traction force - motor torque curve) wheel skidding is classified. Simulations then show that skids during cornering are successfully suppressed. An adaptive stability control system is proposed by Wang et al. [6] to address the reference tracking problem of a four-wheel independently actuated electric vehicle. A hierarchical controller structure is designed where both the longitudinal speed and yaw rate are controlled simultaneously. The higher layer, an analytical nonlinear control approach using a Lyapunov function, calculates a virtual total ground force request and the force split between the left and right sides of the vehicle. The lower layer distributes the higher layer s control efforts to the four wheels, using a numerical optimization based control allocation algorithm. Yoshimoto et al. investigate the deactivation conditions of a vehicle stability assistance system in a study [4], based on a two wheel nonlinear vehicle model and a linear-quadratic regulator realizing active steering. An instantaneous yaw rate is injected in simulation to the vehicle travelling at a constant speed testing seven different deactivation timings, both on a straight and curved course. The analysis concludes that a latency time is necessary before the assistance system returns control to the driver. This latency time however, depends on the drivers skills. 5

17 A hierarchical stability controller, utilizing direct yaw moment control technique is investigated on a four-wheel independently actuated electric vehicle by Song et al. [28]. The authors propose a novel sliding mode control (SMC) technique for the motion control problem, tracking a desired, reference vehicle motion. The SMC is deployed as the higher layer, tracking both the yaw rate and sideslip where the sliding surface is defined as the weighted combination of the yaw rate and sideslip angle errors. The calculated total driving force and yaw moment requests are the inputs of the lower layer, distributing individual wheel torques considering adhesion limits and actuator constraints, formulated as an optimization problem. The designed control system is tested in a lane-change manoeuvre in simulation. Results show improved reference tracing and steerability while meeting the design requirements. For post-crash vehicle motion control however, yaw rate and driving torque reference generation is a crucial question. In [29], Kim et al. propose a preemptive steering control strategy, calculating a counter-steering action by a feedforward and feedback controller, attenuating vehicle motion after an external impact based on the predicted collision strength. Here, the sliding mode control concept is applied, driving both lateral velocity and yaw rate to the reference. For actuation, differential braking is considered, stabilizing the vehicle post impact, using a simple rule based approach. In her PhD. thesis, Yang [9] investigates post impact vehicle motion control, proposing a post impact controller aiming to avoid or mitigate multiple event accidents. The study is based on an analysis of the German In-Depth Accident Study identifying areas of the vehicle s body which are most prone to accidents. Figure 2.1 shows the physical distribution of the analysed impacts, indicating both impact angles and crash energy. The author further analyses phase portraits of the post-crash vehicle states suggesting that a nonlinear vehicle model is prone to approach an unstable region especially in case of large steering actions. However the stable region may be extended by a feedback algorithm. Furthermore the causes of secondary events were identified according to which, reduction of kinetic energy and lateral deviation from the original path were found to be the most beneficial measures for secondary collision mitigation. It is also found that a combination of active steering and differential braking is the best way to stabilize the vehicle states and to minimize lateral deviation. Though, the concept of four-wheel independent actuation was not investigated. A Quasi Linear Optimal Controller (QLOC) is developed using nonlinear optimal control theory defining a quadratic objective function to minimize the maximum lateral deviation from the original path in a road fixed coordinate system. The optimal control 6

18 Figure 2.1: Top view of a vehicle indicating the physical distribution of impacts analysed in [9]. problem is formulated as a two-point boundary value problem (2pt-BVP) and it is fully determined by the active force and moment constraints of the system. The designed controller is verified in various test scenarios comparing to a generic post impact braking controller [10] as well as to a simple anti-lock brake system. It is found that the maximum lateral deviation can be considerably reduced when the QLOC is deployed as compared to the post impact braking function, significantly outperforming cases when only the anti-lock brake system intervenes. An accident statistical study performed by the German Insurance Association confirms that even vehicles involved in a light impact are highly likely to experience a severe secondary crash and a third of all accidents with severe injuries consist of multiple events [18]. In a paper by Zhou et al. [5], vehicle stabilization in response to exogenous impulsive disturbances is studied. The authors propose a post-impact stabilizing controller with the objectives to attenuate sideslip angle, yaw rate and roll rate immediately after a crash disturbance, recovering vehicle stability. The controller is assumed to have access to all necessary vehicle states and it is based on a planar two track three DoF vehicle model. The controller is an extension of sliding mode control theory, using multiple sliding surface theory. A detailed crash detection and validation scheme is introduced, monitoring the rate of change of certain vehicle states. When above 7

19 an empirical threshold for three consecutive sampling intervals, the control system is activated triggering a validation scheme to avoid misdetection. To evaluate controller performance, the proposed system is tested against three different control approaches (differential braking, differential braking with active steering, full braking without steering action) in simulation subject to the same impulsive disturbances. Results show that the proposed controller can effectively recover post-event vehicle stability outperforming all the other approaches. In a follow-up study, Kim et al. [13] propose a post-impact vehicle stability control system that controls both heading angle and path lateral deviation to mitigate the severity of secondary crashes. Consequently, the proposed system predicts the heading angle after a collision and controls vehicle spin to reach a safe heading angle, defined as multiples of 180 degrees. 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. Relying on a collision model for vehicle motion prediction [12], magnitude and location of impulses can be estimated. This calculated impact force is then used to predict the vehicle s motion. Vehicle side slip angles up to 360 degrees are studied, considering tire characteristics based on Pacejka s Magic Formula [17], concluding that significant yaw rate reduction can be achieved when the heading angle crosses 180 and 360 degrees. The control synthesis is formulated as an optimization problem using gradient descent with an objective function minimizing lateral deviation from the path while achieving the desirable heading angle. 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. Safe heading angles in a post-crash scenario might be determined by the vehicles perception of its surrounding and extensive simulations. 2.2 Vehicle motion control with MPC Lately, model predictive control has gained popularity in the control community thanks to its attractive features and the availability of faster and faster embedded platforms. Numerous research fields apply the concept with increasing success, including the field of vehicle dynamics and motion control. 8

20 Model predictive control is a model based, online, constrained finite horizon optimization method, capable of handling nonlinear time-varying models and constraints systematically [33], optimizing future plant behaviour. Despite its considerably heavy computational burden [34], nonlinear MPC is a very attractive control technique, especially in applications where the process is required to work in wide operating regions and close to the boundary of admissible states and inputs [35]. To ease the computational burden, MPC formulation based on successive online linearization of the nonlinear plant model is commonly used, leading to a linear time varying model predictive control formulation. By linearizing the prediction model before each MPC optimization step the nonlinear MPC problem is transformed into a LTV-MPC problem, yielding simpler optimization problems in real-time [36]. Controlling an active front steering system of an autonomous vehicle is presented in [34] by Falcone et al. A known trajectory is followed on a slippery road at the highest possible entry speed. Two different methods are presented, first using a three DoF nonlinear vehicle plant model, formulating a nonlinear MPC problem. The second method is based on successive online linearization of the plant model, formulating an LTV-MPC problem. The performance of the two controllers is tested in a double lane-change manoeuvre. Results show that with both formulations the manoeuvre is successfully executed, stabilising the vehicle up to an entry speed of 21 m/s on snow covered roads. From a computational point of view, the superiority of the LTV-MPC formulation is apparent. A follow-up study of the above publication is carried out by Falcone et al. [35] using the LTV-MPC approach on the integrated vehicle dynamics problem in autonomous systems. Here, in addition to active steering, active braking and active differentials are controlled, calculating the optimal combination of front steering angle, brake and tractive force inputs to best follow a desired trajectory on slippery roads at certain entry speeds. The LTV dynamics is derived based on a three DoF two track vehicle model, assuming constant normal tire load and using a nonlinear tire model. The controller is again tested in a double lane-change manoeuvre showing an overall good tracking of the desired path. Moreover, the current formulation displays better tracking performance without slowing down the vehicle excessively. In a further, extended publication [36], Falcone et al. investigate the stability of LTV-MPC formulation applied to active steering systems. Even under no model mismatch, the stability of this control scheme is difficult to prove. The authors introduce a sufficient condition for the uniform asymp- 9

21 totic stability of the of the origin of the closed-loop system. The proposed condition is validated in the active front steering problem. Results show consistency with what can be achieved by an ad hoc MPC scheme developed by experts of the process. Turri et al. address lane keeping and obstacle avoidance on low curvature roads (e.g. highways) based on linear MPC control architecture [15]. First the lateral vehicle dynamics is derived as a function of the longitudinal braking or acceleration profile, based on an extended bicycle model under the assumption of the road s large curvature radius. Expressing the lateral tire forces, the nonlinear semi empirical Pacejka formula [17] is used. The nonlinear expression, for a given braking ratio, is bounded by two linear functions accounting for a conservative and a overreacting lateral dynamics. Based on this, a conservative and an overreacting lateral dynamics model is derived. The proposed control architecture consist of: longitudinal profiles generation; parallel MPC problems and post-computation. The MPC formulation predicts the vehicle s motion using both dynamics models. For the state prediction the conservative lateral dynamics model plays the main role both in cost function and constraints. The overreacting lateral dynamics model represents an auxiliary role, with a shorter prediction horizon. The post-computation block evaluates both cost functions and returns the optimal braking ratio and the corresponding steering rate with the minimal cost. Hardware-in-the-loop simulations avoiding one and two consecutive obstacles show the effectiveness of solving the low complexity sub-problems, allowing real-time implementation with long prediction horizons. In a follow-up publication by Kim et al. [14] the same vehicle motion control problem is approached as in [13]. Here an LTV-MPC is formulated as higher layer in a hierarchical control architecture. Based on a three DoF vehicle plant model, a six-state nonlinear first-order system is formulated, considering differential braking as actuation. Taking tire saturation constraints of Pacejka s Magic Formula tire model [17] into consideration, a linear time varying model is derived, formulating an LTV-MPC controller for the control task. The nonlinear vehicle model is successively linearized using Taylor series expansion around non equilibrium-operating points, formulating quadratic programs (QPs) to set up the optimization. With this method, the nonlinear design problem is decomposed into several linear subproblems. MPC then finds a cost-minimizing control sequence over the prediction horizon, while it incorporates feasible control bounds. The proposed, overall hierarchical control structure consists of four functional blocks. The first block compares the desired states with the current states. Then in the second block, representing the higher layer, the LTV-MPC determines 10

22 a desired virtual controls input. The third block maps the virtual control demand onto individual wheel break forces, acting as the lower layer, by an optimal control allocation process using least-squares. In the fourth block, the actuators manipulate actual physical variables to achieve the desired tire forces. The control objective is to minimize both the lateral deviation from the original course and to achieve a safe heading angle while minimizing control efforts. Hence the cost function is defined as the sum of weighted state deviation and the sum of the weighted control input sequence. The desired states are predetermined by off-line optimal computation, determining safer heading angle which minimizes the lateral deviation for the given initial impact. This paper relies on differential braking as actuation, so the outputs of the allocation module (lower layer) are longitudinal wheel brake forces of each tire. The module aims to find the optimal control input to achieve the virtual control sequence. Based on Pacejka s Magic tire model [17], the longitudinal tire forces are mapped onto lateral forces, expressed by a linear equation. To validate the proposed control system, simulations in CarSim software were carried out. It is considering two cars with an initial longitudinal velocity of 30 m/s in a collision with a given initial condition: lateral velocity of 5 m/s, heading angle of 9.2 degrees and yaw rate of deg/s. During the crash no sensor or actuator failure is assumed. The sampling time of the discrete time system is 0.01 seconds, the sampling time for every linearization of the MPC is set to 0.2 seconds, and the horizon is 20 steps, predicting 4 seconds ahead in time. First, the vehicle trajectories in case of four different control strategies are compared. The results suggest that the vehicle with the proposed control strategy settles into a safe final heading angle 180 degrees in this use case and returns to its original lane, as opposed to the vehicles with other control strategies which show larger lateral deviations from the original lane and are subject to broadsiding by vehicles in other lanes. The robustness of the control system is analysed by running the simulations with different initial yaw rate and initial heading angle, corresponding to different magnitudes of the initial impact. On the whole, the proposed control system minimizes maximum lane deviation and brings the vehicle back to the original lane with a favoured heading angle of 180 or 360 degrees. 11

23 3 Vehicle dynamics modelling This chapter presents the mathematical description of the vehicle dynamics which is later used as a plant for controller design. The vehicle coordinate systems are also introduced in brief. 3.1 Vehicle coordinate systems As one of the control objectives is to minimize post-crash lateral deviation from the original path, a road-fixed, inertial coordinate system referred to as global frame and a vehicle-fixed coordinate system referred to as local frame are defined, according to Figure 3.1. Throughout this thesis the local frame is fixed to the vehicle s center of gravity (CoG) where the the x coordinate vector denotes the longitudinal and the y coordinate vector denotes the lateral directions. The global frame is fixed to the road where X and Y denote the global longitudinal and lateral directions. Consequently, the yaw angle ψ denotes the vehicle s heading angle, coinciding in the two frames. Figure 3.1: The road-fixed, inertial coordinate system, i.e. global frame, denoted by capitals XY Z. And the vehicle-fixed coordinate system, i.e. local frame, denoted by lower-case xyz [ISO 6487]. 3.2 Two track plant model The equations of motion of a planar, two track vehicle model is derived using Newton s law of motion, based on [14], in the global frame, where the steering wheel angle is assumed to be zero, i.e. δ = 0. A schematic drawing 12

24 of the model is presented on Figure 3.2 and the equations yield mẍ = F x f,l+r cos ψ + F xr,l+r cos ψ F yf,l+r sin ψ F yr,l+r sin ψ mÿ = F x f,l+r sin ψ + F xr,l+r sin ψ (3.1a) + F yf,l+r cos ψ + F yr,l+r cos ψ (3.1b) I z ψ = l1 F yf,l+r l 2 F yr,l+r + w ( ) (F xfr F xfl ) + (F xrr F 2 xrl ) (3.1c) where Ẋ and Ẏ are lateral and longitudinal velocities in the global frame, ψ is the yaw rate. Constants m and I z denote the vehicle s mass and moment of inertia around the z axis. Distances l 1 and l 2 denote the front and rear axles distance to the center of gravity, w is the track width. Moreover, the longitudinal and lateral tire forces are denoted by F x and F y, where {f, r} denotes the front and rear axles and {l, r, l + r} marks the left and right side of the vehicle and their sum. The tire forces for each tire can be expressed as, F y = f y (α, λ, µ, F z ), F x = f x (α, λ, µ, F z ) (3.2) For this plant, Pacejka s Magic Formula tire model [17] is used, describing the combined longitudinal and lateral forces in (3.2), denoted by f y and f x, as a function of the slip angles α, slip ratio λ, coefficient of friction µ and vertical loads F z, all of which are used as defined in [40]. Figure 3.2: Two track model of a vehicle in the XY plane with a reference lane. Original figure taken from B. J. Kim et al. [14]. 13

25 3.3 High fidelity vehicle dynamics model - as plant model A 14 degrees of freedom vehicle model, defined in IPG CarMaker [42] is used for simulations. Components such as the chassis, tires, powertrain, suspension, vehicle aerodynamics are modelled in detail. Hence, this model accurately represents the dynamics of a vehicle and therefore provides results close to reality. 3.4 Prediction models To exploit the predictive power of dynamic models two different vehicle models are used as prediction models of the MPC. This section first describes a very simple bicycle model, linearized around a specific working point and using a linear approximation of the tire dynamics, used in the linear MPC formulation. Then a more complex, extended bicycle model is derived with a nonlinear tire model, which is used in the LTV-MPC formulation Linearized bicycle model A rather simple 2 DoF bicycle model is derived, based on [8] where the equations of motion in the local frame are written as, mÿ = mẋ ψ + F yf + F yr I z ψ = l1 F yf l 2 F yr + M z (3.3a) (3.3b) where ẏ is the lateral velocity and ψ denotes the yaw rate. Constants m and I z are the vehicle s mass and moment of inertia around the z axis. M z is the virtual control moment input around the z axis. The lateral tire forces are denoted by F y and are calculated by a static, linear approximation of the tire dynamics, F y = C α α (3.4) where C α denotes the corresponding cornering stiffness parameters [8] and {f, r} denotes the front and rear axles, just as before and α is the corresponding slip angle defined by equation (3.12). Note that ẋ = v x is considered to be constant, thus the lateral dynamics is not directly considered. The system dynamics is augmented with the global frame s lateral dynamics, which is then later included in the control problem s minimization, Ẏ = ẋ sin ψ + ẏ cos ψ (3.5) 14

26 Defining the state vector and control variables, ξ(t) = [Y ẏ ψ ψ] T, u(t) = M z (3.6) and linearizing the system dynamics (3.3) around ψ 0, ẋ = v x a linear state space model is derived on the form, where, ξ(t) = Aξ(t) + Bu(t) (3.7a) Ẏ 0 1 v x 0 Y 0 ÿ ψ = 0 a 22 0 a 24 ẏ ψ + 0 M 0 z (3.7b) ψ 0 a 42 0 a 44 ψ 1/I z }{{}}{{} A B a 22 = 2C αf + 2C αr, a 24 = 2l 2C αr 2l 1 C αf v x m v x m a 42 = 2l 2C αr 2l 1 C αf v x I z, a 44 = 2l2 1 C αf + 2l 2 2 C αr v x I z v x (3.8a) (3.8b) The output map including Y, ψ and ψ as in [34] is then defined as, y(t) = C ξ(t) = ξ(t) (3.9) Note that in the beginning of the control synthesis, a plant model based on the same linear bicycle model is used for decision making. It is however not listed with the other plant models Extended bicycle model An extended bicycle model is derived to model the dynamics of the vehicle, as a more elaborate prediction model. Considering a two track model is necessary to be able to express the wheel longitudinal forces as a function of the normal forces and the virtual control output M z R a counter moment acting around the z axis. Figure 3.2 shows the diagram of the vehicle model. Notice that the vehicle dynamics may be described in the reference frame and in the vehicle-fixed coordinates system too. 15

27 The equations of motion, expressed in the local frame, assuming zero steering wheel angle (δ = 0) can be written as [34], mẍ = mẏ ψ + F xfl + F xfr + F xrl + F xrr mÿ = mẋ ψ + F yfl + F yfr + F yrl + F yrr I z ψ = l1 (F yfl + F yfr ) l 2 (F yrl + F yrr ) + M z (3.10a) (3.10b) (3.10c) where ẋ and ẏ are the lateral and longitudinal velocities in the local frame, ψ is the yaw rate. Constants m and I z denote the vehicle s mass and moment of inertia around the z axis. The front and rear axles distance to the center of gravity is denoted by l 1 and l 2 respectively. M z represents the controller action, a virtual moment. Furthermore, F x and F y denote the longitudinal (or tractive ) and lateral (or cornering ) tire forces, where {f, r} denotes the front and rear axles and {l, r} marks the left and right side of the vehicle. The vehicle s equations of motion in the global frame [34] are, as in equation (3.5), Ẋ = ẋ cos ψ ẏ sin ψ, Ẏ = ẋ sin ψ + ẏ cos ψ (3.11) The slip angles at zero steering wheel angle of the front and rear wheels can be expressed as, α f = ẏ + l 1 ψ, α r = ẏ l 2 ψ (3.12) ẋ ẋ Due to the fact that the aim is to control the vehicle s motion in emergency situations, the nonlinear behaviour of the tires is essential to be modelled. Therefore, the commonly used linear approximation of the tire dynamics as in equation (3.4) is not suitable in this study. Hence a more sophisticated model, the simplified semi-empirical Pacejka formula [15], [17] is used, F y = (µf z ) 2 Fx 2 sin(c arctan(bα )) (3.13) where B and C are tire parameters, namely the stiffness factor and the shape factor respectively. The coefficient of friction is assumed to be one i.e. µ = 1. This tire model captures the saturation of the lateral forces, however, for extremely large slip angles around 90 degrees, which appear when the 16

28 3000 arctan(b*sin( )) arctan(b* ) Lateral force, F y [N] Slip angle, [deg] Figure 3.3: Simplified semi-empirical Pacejka tire model of equation (3.13), and its modified version in equation (3.14) capturing the nonlinear lateral force - tire slip angle relation. vehicle is spinning it fails to handle the true behaviour of the slip angle. Thus a slightly modified version of the above Pacejka model is used, on the form of, F y = (µf z ) 2 Fx 2 sin(c arctan(b sin(α ))) (3.14) The difference between the two models is demonstrated on Figure 3.3. Note that the stiffness factor and shape factor influence the slope and saturation values of the function respectively, and can be identified from the slip curve of the plant. The vertical load on each tire is assumed to be static, hence not accounting for the effects of roll and pitch. Rather, the vertical loads are calculated as a split between the front and rear axle, proportional to the axles distances to the center of gravity [35], F zf = l 2 mg l 1 + l 2 2, F z r = l 1 mg l 1 + l 2 2 (3.15) where g denotes the gravitational constant. The longitudinal tire forces are expressed based on the rule-based low level torque allocation strategy [38]. The virtual control input is first split between the front and rear axle, M z = M zf + M zr (3.16) 17

29 and M zf = 2F z f F ztotal M z, M zr = 2F z r F ztotal M z (3.17) On each axle then, the requested torque is divided onto the left and right wheels, thus the longitudinal forces can be calculated using F zfl = F zfr = 1 2 F z f as, F xfl = 2 M z f w F zfl F zfl + F zfr = M z f w (3.18) where w denotes the track width of the vehicle. The remaining longitudinal forces can similarly be expressed, yielding, F xfr = M z f w, F x rl = M z r w, F x rr = M z r w (3.19) With the above expressions, the lateral forces F yfl, F yfr, F yrl and F yrr can be expressed from equation (3.14). Using the equations (3.10) (3.19), the nonlinear vehicle dynamics can be described with the following compact equation [35], ξ(t) = f ( ξ(t), u(t) ) (3.20a) η(t) = h ( ξ(t) ) (3.20b) where the state vector and input are ξ(t) = [ẋ, ẏ, ψ, ψ, Y, X] T and u(t) = M z respectively. f : R 6 R R 6 is the update function and the origin of the state space is an equilibrium point. The output map including ψ, ψ and Y [34] is given by, h(ξ(t)) = h A ξ(t) = ξ(t) (3.21) In the present chapter three separate mathematical descriptions of a road vehicle are presented. In this thesis, these models are either used as a plant model or as a prediction model. 18

30 4 Vehicle motion control This chapter describes the design process of the active safety system for passenger vehicles, mitigating secondary collisions after an initial impact. As stated in chapter 1, the control objective is to minimize path lateral deviation while achieving safe heading angles. 4.1 Architecture of the proposed control system The overall control structure is shown on Figure 4.1. When activated, the higher layer first computes a desired, stabilizing counter moment around the z axis of the vehicle referred to as virtual control moment input. The lower layer then divides the requested moment to the four wheels, calculating the individual tire forces utilizing the available momentary traction. Due to the modular design of the control system, both the higher and lower layers may be implemented using different control strategies. In this thesis model predictive control is used to calculate the moment request in the higher layer, optimizing future vehicle motion in accordance with the control objective. Figure 4.1: The proposed hierarchical control architecture. An SMC based higher layer controller is designed with the same objective along with a simple rule based lower layer torque allocation, by Nikhil Jain [39]. Results of the SMC based higher layer are included for comparison in chapter 5 while the same lower layer is used in all cases. 4.2 Linear MPC Providing proof of concept and keeping the design simple, first a linear MPC is designed based on the linearized bicycle model as prediction model, described in section 3.4. The state vector and input of the prediction model are ξ(t) = [ẋ, ẏ, ψ, ψ, Y, X] T and u(t) = M z. 19

31 Discretizing system (3.7) using zero order hold with fixed sampling time t yields, ξ(k + 1) = A d ξ(k) + B d u(k) (4.1) representing the discrete time linear approximation for a time horizon k = t,..., t + N. Where matrices A d and B d are calculated [32] as, A d = e A t, B d = ˆ t 0 e As B ds (4.2) where matrices A and B are defined in equation (3.7) and the superscript d refers to representation in discrete time Cost function Consider the quadratic cost function defined as, V N ( ξ(t), U(t), ψref ) = N i=1 N 1 i=0 ( ) q 1 Yk 2 + q 2(ψ k ψ ref ) 2 + q 3 ψ k 2 + N 1 Ru 2 t+i,t + i=0 S u 2 t+i,t (4.3) where U(t) = [u t,t,..., u t+n 1,t ] and ψ ref are the optimization vector and constant reference state, respectively. N denotes both the prediction horizon and the control horizon. R and S are weighting matrices of appropriate dimensions and q 1, q 2 and q 3 are weighting scalars. The first summand in equation (4.3) reflects the desired performance on reference tracking, the second and third summands are a measure of virtual counter moment effort penalizing both the control input and control input rate Model constraints Considering the average deceleration of a passenger vehicle while emergency braking denoted by a dec and applying Newton s second law, the total braking force yields, F braking = m a dec (4.4) where m denotes the vehicle s mass. Meaning an average braking force of F braking /4 at each wheel. Then, considering full braking torque on one side of the vehicle (e.g. both front and rear right wheels) and full propulsive torque on the other side 20

32 (assuming that approximately similar propulsive longitudinal forces can be generated), the generated moment around the CoG gives the constraint, u k = 4 F braking /4 w/2 u max (4.5) where w denotes the track width. The constraint on the rate of control input change, namely the slew rate constraint has been chosen to reflect actuator constraints suggested by NEVS professionals, expressed as, u t+1 u t slew (4.6) The optimization is also subject to the state dynamics constraint, ensuring appropriate state evolution according to equation (4.1). Note that no state constraints have been used in the optimization MPC formulation The following problem is then considered, minimize V N (ξ t, U t, ψ ref ) subject to ξ k+1 = A d ξ k + B d u k u k u max u t+1 u t slew (4.7) The optimization yields the optimal input sequence Ut = [u t,t,..., u t+n 1,t ] over the prediction horizon, predicted at time t. Then the first element of Ut, u ( t, ξ(t) ) = u t,t (4.8) is applied to the system (3.1). The same calculation is repeated in all consecutive steps, iteratively. Note that since V N is a convex quadratic function and both the model and the constraints are linear, this MPC problem can be formulated as a quadratic program [33]. 4.3 Linear Time Varying MPC Formulation of the LTV-MPC is presented in this section, describing the linearization and discretization process of the prediction model. In order to derive the system s LTV model, the nonlinear vehicle dynamic equations first have to be linearized around a state trajectory ξ 0 21

33 (non-equilibrium points), where this state trajectory is obtained by applying the input sequence u(k) = u 0 for k 0 (k N is an arbitrary discrete point in time), to system (3.20) with ξ(0) = ξ 0, i.e [35], ξ 0 (t + 1) = f ( ξ 0 (t), u(t) ) u(t) = u 0 (4.9) ξ 0 (0) = ξ 0 The first order Taylor series expansion gives, ξ(t) ξ 0 (t + 1) + A c ( t,0 ξ(t) ξ0 (t) ) + Bt,0( c ) u(t) u0 = A c t,0ξ(t) + Bt,0u(t) c + d c t,0(t) (4.10) defining A c t,0 R6 6 and B c t,0 R6 1 as, A c t,0 = f( ξ(t), u(t) ), ξ ξ0 (t),u 0 and d c t,0 (t) R6 1 as, Bt,0 c = f ( ) ξ(t), u(t) (4.11) u ξ0 (t),u 0 d c t,0(t) = ξ 0 (t + 1) A c t,0ξ 0 (t) B c t,0u 0 (4.12) Note that the superscript c refers to representation in continuous time and the second subscript 0 refers to the first iteration. As the second step, using forward Euler method, the differential equations are discretized as, ξ(k + 1) ξ(k) t ξ(k) (4.13) Expressing ξ(k + 1) and using equation (4.10) yields the LTV model of the vehicle dynamics on which a linear MPC is built [35], ξ(k + 1) = ξ(k) + t ξ(k) = ξ(k) + tξ 0 (k + 1) + ta c tk,0( ξ(k) ξ0 (k) ) + ( ) u(k) u0 tb c tk,0 (4.14) yielding, where, ξ(k + 1) = A k,0 ξ(k) + B k,0 u(k) + d k,0 (k) (4.15) A k,0 = 1 + ta c tk,0, B k,0 = tb c tk,0 (4.16a) 22

34 ( ) d k,0 (k) = t ξ 0 (k + 1) A c tk,0 ξ 0(k) B tk,0 c u 0 (4.16b) using the notation tk = t. Replacing the fixed index 0 by t in equations (4.15) and (4.16), consider the system at each time t, ξ(k + 1) = A k,t ξ(k) + B k,t u(k) + d k,t (k) (4.17) representing the general LTV model approximation of the system (3.20) for a time horizon k = t,..., t + N. System (4.17) is subject to state and input constraints, expressed as, where X R 6 and U R are polytopes [33] Cost function ξ(t) X, u(t) U, (4.18) Consider the quadratic cost function [35] defined as, V N ( ξ(t), U(t), Ξref (t) ) = N ( ) 2 η t+i,t η reft+i,t i=1 N 1 i=0 Q + N 1 ( ) u t+i,t 2 i=0 R + ( ) u t+i,t 2 (4.19) S where U(t) = [u t,t,..., u t+n 1,t ] and Ξ ref (t) = [ξ reft+1,..., ξ reft+n ] are the optimization vector and reference state trajectory at time t, respectively, η t+i,t denotes the output vector prediction at time t + i obtained by starting from the state ξ t,t = ξ(t) and applying the input sequence u t,t,..., u t,t+i to system (4.17). N denotes both the prediction horizon and the control horizon. Q, R and S are weighting matrices of appropriate dimensions. The first summand in equation (4.19) reflects the desired performance on reference tracking, the second and third summands are a measure of virtual counter moment effort penalizing both the control input and control input rate Model constraints The LTV-MPC formulation is subject to the same model constraints as described in section 4.2, with the appropriate dimensions. 23

35 4.3.3 MPC formulation Then at the first time step, t = 1 the following problem is considered, minimize V N (ξ t, U t, Ξ reft ) subject to ξ k+1 = A k,t ξ k,t + B k,t u k,t + d k,t ξ k,t X, k = t + 1,..., t + N u k,t U, k = t + 1,..., t + N 1 (4.20) where the initial state trajectory is obtained by off-line open-loop simulation by zero control input of the vehicle motion subject to the corresponding impulse magnitude. The computed solution yields the optimal input sequence Ut = [u t,t,..., u t+n 1,t ] and the sequence of states ε t = [ξt,t,..., ξt+n,t ] over the prediction horizon predicted at time t. Then the first element of Ut, u ( t, ξ(t) ) = u t,t (4.21) is applied to the system (3.20). From the second step t = 2, system (3.20) is linearized around the previous optimal input sequence U t and state sequence ε t obtained by the previous optimization, solving the MPC problem in equation (4.20) iteratively. Since V N is a convex quadratic function and both the models and constraints are linear, every MPC problem can be formulated as a quadratic program [33]. For the stability of the above formulation, readers are referred to Paolo Falcone et al. [37]. 4.4 Reference generation In order to gain some intuitive understanding of the vehicle s motion in crash scenarios, open loop simulations without control action have been carried out for different impulses. The goal was to see the final heading angles of the vehicle and compare it to the maximum heading angles when the linear MPC is deployed. Based on an analysis [41] of the German In-Depth Accident Study [47] database, 82% of all crashes happen at a maximum impulse of 12 kns, see Figure 5.2. Thus, open loop simulations for impulses between 2-12 kns with increments of 2 kns have been run, simulating a collision from the rear left side of the vehicle (chosen arbitrarily). The results of the simulations are demonstrated in Table 4.1. Figure 4.2 shows the trajectory of the settling vehicle heading angle. 24

36 Table 4.1: Final vehicle heading angles in case of open loop simulation of crashes at various initial impulses. Maximum heading angles when the control loop is closed, for the same crashes. Impact [kns] magnitude ψ final,ol [deg] ψ max,cl [deg] Heading angle [deg] Open loop heading angle trejectory - 8 kns 600 Heading angle Time [s] Figure 4.2: Open loop vehicle heading angle trajectory - subject to an impulse of 8 kns. Determining the reference heading angle requires the interpretation on the closed loop simulation results. It is intuitive to say that the settling time and the magnitude of overshoot of the heading angle reference tracking are the most crucial measures when determining the reference settling angle of the vehicle in a post-crash scenario. It is also clear that the optimal heading angle to track is a function of the impact magnitude. To determine this, it is logical to use the most detailed plant available for the simulation. Therefore, the heading angle references have been determined empirically through extensive simulation using the vehicle plant model in IPG CarMaker. 4.5 Implementation strategy All described algorithms have been implemented in MATLAB using Simulink for simulations, interfacing IPG CarMaker in the specified cases. The optimization problem was set up with the help of the YALMIP toolbox [46]. 25

37 Table 4.2: Solver calculation times in seconds using different QP solvers and tracking different heading angle references. The simulation time is t = 6 seconds. Impact [kns] magnitude ψ reference [rad] MOSEK Gurobi CPLEX π π π π For solving the formulated quadratic programs, three different solvers are used, namely the MOSEK 8 [43], Gurobi [44] and the IBM CPLEX [45] QP solvers. As expected, the LTV formulation of the MPC controller is computationally more expensive than the linear formulation of the controller. This is mostly caused by the constant linearisation of the model. Simulation times in Simulink have increased approximately by a factor of six. There is however, no significant difference in computation speeds between the MOSEK, Gurobi and CPLEX solvers. Table 4.2 demonstrates the solver calculation times of simulating an impact of different impulses in Simulink for 6 seconds. Throughout the extensive simulations with all the different vehicle plant models, as described in chapter 3, the MOSEK and Gurobi solvers exhibited a number of errors in certain cases whereas the CPLEX solver proved to be the most reliable in all situations. Hence, for obtaining simulation results, IBM s solver has been used. 26

38 5 Experiments Following the implementation of the MPC control algorithms, extensive simulation has been carried out, both in Simulink only and through IPG Car- Maker. For the simulations, as described in chapter 3, two different plants have been used, gradually adjusting the controller parameters: A. Four wheel model with a nonlinear Pacejka tire model [17], in Simulink B. Fourteen DoF model in IPG CarMaker In all simulation set-ups, a fixed step size of t step = 0.01 seconds is used with an ode3 (Bogacki-Shampine) solver. The vehicle is assumed to travel on a straight road with 100 km/h initial longitudinal velocity at the time instance of the impact injection. 5.1 Definition of the main use case The controller is designed to act in light impacts which do not cause severe damage to the vehicle and its components i.e. no damage occurs to sensors, actuators as well as the vehicle periphery [5]. Based on the crash statistics analysed by Yang et. al. [9], the main use case (MUC) is defined as a forcemoment pair imposed on the target vehicle by the bullet vehicle in a crash which lasts for a time duration of t. Where this duration of the crash is defined as t = 0.2 seconds [41]. Figure 5.1 illustrates a crash-scenario where A is the target vehicle and B denotes the bullet vehicle. Figure 5.1: Illustration of a collision in a generic four-way intersection between two passenger vehicles [ITE, 2010]. 27

39 Figure 5.2: Frequency distribution of impact magnitude by original velocity, presented in [41]. Furthermore, because the longitudinal dynamics of the vehicle is not controlled directly by the higher layer controller, the impact angle is assumed to be perpendicular to the vehicle s longitudinal axes i.e. 90 degrees. This will induce a moment of near maximum amplitude on the car and a force in the lateral direction only. The location of the side impact is assumed to be at the front or rear axles, from either left or right direction, thus yielding four sub-cases. Hence the impacts are introduced as a force-moment pair, where the moment (M impact ) acts on the CoG and the force (F y,impact ) acts in the lateral direction. The impact magnitude of the MUC is based on data from the German In-Depth Accident Study database [47], analysed by J. Beltran and Y. Song [41]. According to which the impulse magnitude range of 0-8 kns represent 61 % of all accidents considered, shown on Figure 5.2. Therefore the impulse magnitude of the MUC is chosen to be 8 kns, representing a rather large impact. The initial longitudinal velocity of the target vehicle is chosen to be 100 km/h, representing 78 % of the cases (0-100 km/h) studied in the database. 28