Motion planning for autonomous vehicles with the inclusion of post-impact motions for minimising collision risk

The introduction of more automation into our vehicles is increasing our ability to avoid or mitigate the effects of collisions. Early systems could brake when a likely collision was detected, while more advanced systems will be able to steer to avoid or reconfigure a collision during the same circumstances. This paper addresses how the post-impact motion of an impacted vehicle could be included in the decision-making process of severity minimisation motion planning. A framework is proposed that builds on previous work by the authors, combining models from motion planning, vehicle dynamics, and accident reconstruction. This framework can be configured for different contexts by adjusting its cost function according to relevant risks. Simulations of the unified system are carried out and analysed from the perspective of vehicle model complexity and collision parameters sensitivity. Additionally, effects are highlighted concerning different modelling decisions, with respect to vehicle dynamics models and collision models, that are important to consider in further research.


Introduction
Autonomous vehicles (AVs) see rapid development, with advanced functionality added to them for introduction in complicated environments. These environments may give rise to more ambiguous and unexpected scenarios that could make it impossible to avoid crashes. This is even more challenging when considering vehicles experiencing faults and failures. Therefore, AVs should have strategies to mitigate severity when an unavoidable crash is eminent, i.e. strategies for collision mitigation.
The introduction of more automation into road vehicles is increasing our ability to avoid or mitigate the effects of collisions. Early systems could brake when a likely collision was detected, while more advanced systems can steer to avoid or reconfigure a collision during the same circumstances. Such advanced functionality relies on characterising what constitutes an acceptable outcome. This can either focus on the collision itself [1][2][3], with the system for instance striving to choose the impact location to minimise injury [4]. Or, the system can also include other factors, such as the post-impact behaviour of the involved vehicles. The latter option can consider injuries that are caused after an initial collision, perhaps even involving other road users. However, designing such a system is also more complex. This paper is mainly concerned with the latter case.
Currently, a large part of this complexity stems from the need to bring together and overcome knowledge gaps in two disparate research fields.
The first field concerns itself with topics such as trajectory planning and motion control, i.e. focusing on predicting and avoiding crashes. From this research field the area of collision mitigation can learn about advanced control of road vehicles, but little about collisions. These are of less interest, as the goal is to avoid them. When collisions are considered, the primary focus is on the ego vehicle, with post-impact trajectories predicted based on for instance sensing immediate post-impact states of that vehicle [5].
The second field concerns accident reconstruction, i.e. how close a simulated scenario can come to a real accident by investigating different collision models or collision parameters such as contact friction and angle of impact [6]. In this case, the post-impact information such as trajectories of vehicles or the area of deformation is known to analysts and no prediction of the future motion or collision mitigation/reconfiguration is intended. From this research field the area of collision mitigation can learn how to model collisions well, but little about how to predict their characteristics. Collision parameters are mostly assumed to be known, with the analysis taking place after-the-fact.
Furthermore, although vehicle models and collision models that are used in trajectory generation/prediction or accident reconstruction are well-established in the literature, they are designed with different purposes. Models in accident reconstruction software (e.g. PC-Crash) are based on principles of impact, while models in planning and control are focused on predicting the motions of vehicles as accurately as possible or as needed.
Therefore, this paper brings together these two research fields in a collision mitigation system to take post-impact behaviour into account. This unified system builds upon our previous work presented in [7], which proposes a data-driven collision mitigation system that prioritises one trajectory over the other to reduce injury severity for occupants of two vehicles involved in an unavoidable collision. This paper redesigns this approach by adopting collision modelling originally designed for after-the-fact analysis. Simulations of the unified system are then carried out and analysed from the perspective of vehicle model complexity and collision modelling sensitivity. We highlight how to configure the handling of unavoidable collisions for different driving contexts, and how the method sensitivity is bounded by the characteristics of unavoidable collisions and will in the future be decreased through improved sensing.

Theoretical background
This section starts by providing an introduction to motion planning and control. This is followed by investigating motion planning and control fields focusing on post-impact controllers, and accident reconstruction focusing on collision modelling and parameters.

Motion planning and control
A great deal of research in motion planning and control for AVs for collision mitigation has primarily focused on collision avoidance applications [8][9][10]. A summary of several papers addressing this concept is presented in this subsection.
Numerical optimisation methods with Model Predictive Control (MPC) have been used to address collision avoidance in critical situations. For instance, in [10], collision avoidance in autonomous driving is formulated as an optimal control problem. The collision avoidance constraints are on vehicle position, and steering angle and longitudinal tyre forces are control inputs. The optimisation is solved by using quadratic programming for MPC control. With the proposed method, the control inputs that satisfy the collision avoidance constraints are calculated in one step online. Autonomous steering and differential braking are used in [11] for collision avoidance in a two-step control strategy. In the first step, the vehicle is guided to the adjacent lane to avoid collisions, and in the second step, the vehicle tracks the centreline of the adjacent lane. The first step applies a feed-forward controller and the second step uses an MPC controller to calculate the desired steering manoeuvre. Differential braking is used in the second step for stabilisation. In [12], front steering and differential braking are combined for avoiding collisions, vehicle stability, and tracking the desired path in emergency collision avoidance scenarios for AVs. An MPC controller addresses these objectives and calculates the control inputs (front lateral tyre force and yaw moment). This controller was compared with the case of only having front steering in scenarios where the limits of vehicles are pushed. Their results showed that the latter controller (only steering) failed to provide enough yaw moment for collision avoidance. Therefore, the integrated design improved the yaw response for collision avoidance. In [13], motion planning and control for AVs in critical situations are addressed under the condition of locally varying traction. The optimisation problem is a constrained finite-time optimal control problem (CFTOC). The variations in traction are modelled by varying the tyre forces constraints. The results showed that with this design, the ability of the vehicle to avoid collisions is improved by adapting to low and high traction scenarios.
Despite the proposed methods and the complexity of these manoeuvres, collisions cannot always be avoided due to for instance the uncertainty in the environment of AVs. Motion planning and control for AVs in unavoidable collisions have gained attention recently. As an example, [1] proposed an MPC-based motion planning problem for crash mitigation in an unavoidable collision in real-time. A crash severity index is developed that is included in the cost function of the optimisation. Factors included in the cost function are the relative speed between the vehicles, the relative heading angle, and the mass ratio. The authors of [14] extended [1] by developing a switching function that is based on time measurements addressing collision avoidance and collision mitigation in an unavoidable collision. For collision mitigation, the formulation of the severity index is also extended.

Post-impact control
The importance of post-impact motion control to avoid secondary impacts is addressed in [5,[15][16][17][18][19][20][21][22]. In [17], an algorithm to stabilise the vehicle after an impact is developed to decrease the side slip angle, yaw rate and roll rate by differential braking and active steering based on sliding surface control. Kim et al. [19] solved an optimisation problem that minimises the lateral deviation and aligns the vehicle's heading with the original lane. Desired headings are defined as multipliers of 180 • . A linear time-varying Model Predictive Control (MPC) is also proposed in [23], which showed better results than [19] in terms of safely positioning the vehicle. The best results were achieved by preemptive control actions, defined as steering control, that generate a yaw motion before or at the early stages of the crash [5]. This stabilised the vehicle and reduced lateral deviations. Furthermore, a combination of MPC and active front steering is applied in [24], with the same objectives as others, which imitates race car drivers e.g. steering corrections in dangerous situations. A hierarchical controller is introduced in [18]. The sub-optimal second-order sliding mode controller is used for the upper level that generates a yaw moment. The lower level controller is optimisation-based and changes the moment to longitudinal forces and maximises the utilisation of them. Furthermore, by active front steering and differential torque vectoring a feed-forward-feedback controller is proposed in [25] to mitigate the extreme values of yaw rate and side slip angle.
In [26], the effect of different vehicle dynamic and control systems (VDCS) is investigated on collision mitigation for both the ego and another vehicle. Only the ego vehicle is equipped with VDCS. Results showed positive effects of such systems including improving occupant protection by reducing deformation, pitch angle, acceleration, and yaw angle. None of the investigated VDCS had a significant effect on the deformation of the other vehicle. Furthermore, equipping an ego vehicle with VDCS did not have any effect on the pitch behaviour of the other vehicle (struck) but affected its yaw motion. The maximum yaw rate, yaw acceleration, and yaw angle increased for the other vehicle in many cases in comparison to no control. The authors further extended the model to include the occupant kinematics [27]. Improvements in the crash situation and occupant behaviour were observed for the ego vehicle. However, pitch angle, acceleration, and deceleration were slightly increased for the other vehicle compared to free rolling.

Collision modelling and parameters
Different ways to model a collision between two vehicles are developed in the literature. We categorise them into methods that apply the conservation of momentum, mainly used in accident reconstruction [28][29][30], and the ones that integrate vehicles' equations of motion for short collision duration [26,31,32]. Both methods estimate the resulting impulses and immediate post-crash vehicle states by solving a set of equations. However, the former does not consider tyre forces during a collision, and according to [31] allows for significant errors in the post-impact vehicle states.
One of the important yet difficult factors to determine is contact friction or impulse ratio (μ = P t /P n ) [6]. During a collision a force is generated by friction, with an associated friction coefficient, and by 'interlocking of the deformed parts' [28]. This coefficient relates the impulse along the tangential (P t ) and normal direction (P n ). In sliding impact, inaccurate estimation of it can lead to unrealistic results [28]. However, μ is not always considered to be friction-related [33] or even warned against [6]. It can take positive or negative values and is a way to control the impulse along the tangential direction (t) and thus the sliding [6].
Inclusion of μ for collision modelling has been part of both methods [28,31]. Kudlich-Slibar is an impulse-momentum-based method that is used in PC-Crash, where μ is only included in sliding impacts.
Furthermore, a critical value is defined for μ, related to the condition where the relative tangential velocity at the point of impact becomes zero [28]. This critical value can differ. In [28], it is a parameter of the vehicle's inertia, impact location, coefficient of restitution ( ), contact plane angle and velocities at the point of impact. It does not depend on in PC-Crash [6]. For friction values above the critical value, a full-impact occurs while for below it a sliding impact occurs [6,29,34].
For collisions between two vehicles, μ = 0.4 is suggested by [35]. To distinguish between sliding and non-sliding limits, μ = 0.5 ± 0.1 is proposed by [36]. They also recommended μ 0.6 for sliding impacts and 0.4 < μ < 0.6 for full-impacts. However, others have recommended μ > 0.6 for non-sliding impacts [37]. Therefore, it is difficult to decide on a limit for contact friction where sliding and full-impacts are distinguished. Furthermore, experiments showed that this coefficient can vary depending on collision configuration, e.g. side collision or head-on [33].
Furthermore, the relationship between pre-and post-impact velocities depends on set values for μ, , mass ratio, vehicle geometry and vehicle orientation at time of impact [35]. Moreover, different combinations of these parameters can lead to the same post-impact motion [6]. Accident reconstruction analysts have more freedom in setting these values as long as the resulting post-impact motions match the real accident. This is not the case for post-impact motion prediction.

Modelling
This section presents the theory and modifications required to extend collision mitigation through reconfiguration of collisions by taking post-impact behaviour into account. Firstly, the vehicle models for pre-and post-impact motions and collision models used in this study and simulated in MATLAB are presented. Secondly, a collision mitigation framework is proposed.

Pre-impact vehicle motion
In this study, pre-impact trajectories of the ego vehicle are calculated offline by solving an optimal control problem using a dynamic single-track model (see Appendix 1). These trajectories are saved in a trajectory library and used during run-time. This trajectory generation is based on a previous study [4]. The pre-impact trajectory of the target vehicle is calculated by a kinematic single-track model [38]. We assume that the trajectory of the target vehicle is known.

Post-impact vehicle motion
To predict the motion of a vehicle after an impact, a four-degrees-of-freedom (4DOF) vehicle model considering longitudinal, lateral, yaw and roll motion is employed [Equations (1)-(5)], and is based on [5,31,39]. Even though we are only interested in the planar motion of vehicles after collision, the roll motion is included to improve accuracy [31]. This 4DOF model is used for both the ego and target vehicle in Subsection 4.1.1.
is the local coordinate of the point of impact; F x and F y are longitudinal and lateral impact forces. The longitudinal and lateral tyre forces are: F xi and F yi with i = {fl, fr, rl, rr}, i.e. lf is front left tyre, rr is rear right tyre, etc. A simplified magic formula tyre model [40] is applied as Equation (6).
Tyre slip angles are defined as Equation (7), where δ is the wheels steering angle. Normal tyre forces are according to Equation (8), where a x denotes longitudinal acceleration, a y lateral acceleration, and K j = K s /2 with j = {f , r} is roll stiffness.
All the values and descriptions of the parameters included in the vehicle models presented above are listed in Appendix 1 and Table A1. These parameter values have been extracted from the vehicle dynamics software CarMaker 8 and from the standard vehicle model 'Demo_MB_MClass.'

Collision models
Below follows a presentation of different collision models, where several past research works are combined to create models useful in the framework of Subsection 3.3.

4DOF model
To estimate impact forces and vehicle states, a 4DOF collision model is used [31], which is derived by integrating Equations (1)-(4) over the collision duration and applying the trapezoidal rule. The impact force profile is approximated as a triangle according to [41,42]. Given the relationship between impulse and force (P x = t 2 t 1 F x dt and P y = t 2 t 1 F y dt, where t 1 is the start and t 2 is the end of collision), Equations (10)-(13) are derived. These equations are similar for both ego and target vehicle, where subscripts 1 and 2 refer to the target and ego vehicle, respectively. The same set of equations is derived for the ego vehicle. The pre-impact states (small letters) are known, while the post-impact states (capital letters) and the impulses are unknown. The longitudinal tyre forces, steering angle and rolling resistance are disregarded during impact, and lateral tyre forces are assumed to be linear as Equation (9).
The remaining equations are derived based on collision characteristics. Coefficient of restitution is defined as Equation (14), where pre-and post-impact velocities are derived in n−t coordinate system and located at the point of impact (POI). Furthermore, Equation (15) is derived based on the definition of impulse ratio, where is contact plane angle (see Subsection 3.2.4). The relationship between impulses in ego vehicle's coordinate system (P x , P y ) and the target vehicle's coordinate systems (P x , P y ) is according to Equation (16).
The resulting 12 equations are solved by fsolve solver in MATLAB. The outputs are postimpact states and impulses in each vehicle's coordinate system. This model is used to solve the collision in Subsection 4.1.1. When this collision model is used, a 4DOF motion model is applied for trajectory generation [Equations (1)- (5)].

Three-degrees-of-freedom (3DOF) model
Since unavoidable collision situations require fast decision-making, the equations of motion of the 4DOF four track model from Subsection 3.1.2 are simplified to a 3DOF model. Consequently, to arrive at the collision model, the terms containing roll velocity are considered zero in Equations (11)- (13). When this collision model is used, a 3DOF motion model is applied for trajectory generation (see Subsection 4.1.1, 4.2, 4.3 and Appendix 1). A comparison between these two models in terms of post-impact motions is given in Subsection 4.1.1.

Impulse-momentum model
Kudlich-Slibar is a collision model that is used in PC-Crash for accident reconstruction and is derived based on [29,43]. This model applies the impulse-momentum principle and unlike the model given by [28], distinguishes between a sliding and a full-impact. Conservation of linear and angular momentum along the n−t axis is derived for the target (1) and ego vehicle (2) according to Equation (17). Small and capital letters represent pre-and post-impact vehicle states, respectively. Furthermore, velocity is denoted with V and v, yaw rate is denoted with and ω, and t i is the distance between CoG of the vehicle and n axis while n i is the distance between CoG of the vehicle and t axis. T and N represent impulses along t and n axis, respectively.
Velocity at POI (v A and v A ) along t and n axis is derived as Equation (19). The POI for the target vehicle and ego vehicle is A and A , respectively. v 1 and v 2 are the pre-impact velocity of the CoG of the target and ego vehicle, respectively.
By replacing ω 1 and ω 2 from Equation (19) into Equation (18), the post-impact velocities at POI (V A and V A ) are as Equation (20).
For full-impacts, the relative velocities at the end of the compression phase are zero along the tangential (V A|A t ) and normal directions (V A|A n ) [29]. Therefore, the impulses for compression phase (T c and N c ) and the total exchanged momentum (T and N) are calculated as Equation (22).
However, for sliding impacts only V A|A n is zero at the end of the compression phase [29]. While T c ≤ μN c , the impulses are according to Equation (22). While sliding occurs for T c > μN c the impulses are calculated according to Equation (23). Furthermore, the coefficient of restitution is according to Equation (24) [43].

Contact plane and point of impact
To calculate the point of impact, i.e. the point where the momentum is exchanged between the two vehicles, a geometrical method is applied, and the POI is defined as the centroid of the colliding polygons of the vehicles [29,34]. Here, vehicles are considered as rectangles.
The overlapping area between the colliding rectangles is a polygon, with vertices calculated by the Sutherland and Hodgman algorithm [44]. The vertices are then used to calculate the centroid (POI) [34], and are denoted as (x A , y A ). Here, POI is calculated in this way regardless of the applied collision model.
The contact plane (CP) can be calculated geometrically as well [29]. The calculated vertices of the intersecting polygon that do not belong to either of the rectangles of vehicles are identified. The line that passes through them is calculated, and translated in parallel in a way that passes the POI. This line is the contact plane and is parallel to t axis. Furthermore, the angle between the CP and Y global axis is the CP angle and denoted as . However, in [31], is defined as the heading angle of the striking vehicle (here the ego vehicle) in combination with a 4DOF vehicle model.
To calculate in this study, the former method is used mainly in combination with the Kudlich-Slibar model (e.g. Subsection 4.1.2), and the latter is used in combination with a 4DOF or 3DOF collision model (Subsection 4.1.1, 4.2 and 4.3.1). To investigate the sensitivity of post-impact vehicle states to the definition of CP, the geometrical method is used with the 3DOF model (Subsection 4.3.2, 4.3.3 and Appendix 1).

Collision mitigation framework
The proposed collision mitigation framework bridges a gap between what is available, in terms of information, to the accident analyst and vehicle control engineers. The aim of this framework is to take advantage of this information to make predictions on how a collision scenario will develop after an impact and to use the gained perspective to make safer preimpact decisions. This framework, detailed in this section, combines what is presented in previous sections.

Combination of models
Whether an impact is sliding or non-sliding affects the severity of collision [45,46]. Including tyre forces in collision modelling contributes to the accuracy of post-impact motion prediction [31]. The Kudlich-Slibar model (Subsection 3.2.3) distinguishes between these impacts, and the 4DOF and 3DOF vehicle collision models (Subsections 3.2.1 and 3.2.2) consider tyre forces. To account for both, the vehicle collision model is used in combination with the Kudlich-Slibar model. First, the impact type is predicted by Kudlich-Slibar model. Then, if the impact is estimated as sliding, the collision is solved with the Kudlich-Slibar model, meaning that impulse values and immediate post-impact states are calculated with this model. However, if the impact is identified as full-impact, a vehicle collision model is used. The 3DOF model in this framework is preferred since only planar motion of the vehicles is considered in the calculation of collision cost [Equation (25)]. It is also shown in Subsection 4.1.1 that the post-impact states of vehicles with the 3DOF model have a good enough accuracy when compared with the 4DOF model.
Then, to calculate the cost of collision the post-impact trajectories and states of both vehicles are simulated by solving the vehicles differential equations numerically. Consequently, these motions are used as inputs for Equation (25).
Collision parameters such as contact friction and angle of contact plane are kept constant ; however, they can vary between sliding and full-impact due to different impact type characteristics. The specific values are given in Section 4 for each case.

Cost of collision
When an unavoidable collision is determined, a collision cost is calculated for each trajectory of the ego vehicle from the trajectory library and also for the target vehicle. The trajectory that has the lowest collision cost is then followed by the ego vehicle. This collision cost is operationalised by designing a cost function that is based on post-impact motions of the vehicles. A weighted combination of lateral deviation from pre-impact position, heading angle, yaw stability and controllability through yaw rate and side slip angle (β) deviations constitute this cost function. This is presented in Equation (25) When the heading angle for the post-impact trajectory is a multiplier of 180 • , the cost of secondary crash is the lowest (frontal or rear-end) while the cost is the highest when this angle is perpendicular to the longitudinal direction of the vehicle (side impact) [42]. This observation is included in ψ ref − ψ k and is implemented by a modulo operation of ψ k and π , while penalised by how close ψ k is to π 2 for the scenarios applied in this paper. Controllability of the vehicle after impact is considered by cost terms on yaw rate and side slip angle. Higher values of yaw rate and side slip angle require more control efforts to bring the vehicle to its original position or to stabilise it [42]. Therefore, the third term in Equation (25) implicates that a higher post-impact yaw rate has a higher cost than a lower post-impact one. Furthermore, analysis of (ω z − β) phase plane [42] showed that yaw rate values quickly reduce around multipliers of 180 • or vehicle states naturally converge to other equilibrium with a side slip angle of 360 • . Another aspect to consider for allocating cost on β is when this angle is 90 • . This happens when the longitudinal velocity of the vehicle is almost zero or very small (tanβ = v y v x ). Therefore, the highest cost of postimpact side slip angle is assigned to values close to 90 • . These are implemented by the term Ideally, we want to choose a trajectory that leads to the lowest collision cost for both vehicles involved in the collision. However, due to the following reasons, the cost is minimised for the target vehicle.
a. It is assumed that only the ego vehicle is equipped with the collision mitigation system. Therefore, this vehicle can make more informed decisions and perform post-impact control strategies in unavoidable collisions. Furthermore, in real scenarios, more information is available for one's own vehicle and the behaviour of other vehicles is more uncertain. b. The target vehicle is struck by the ego vehicle. Being struck leads to higher injury severity for vehicle occupants [47].

Simulation results
In this section, we demonstrate two aspects of this work.
(a) We present the post-impact motions of both vehicles resulting from impact while comparing these results for a 4DOF against a 3DOF vehicle model, and these comparison results are used in subsequent subsections. Then, we demonstrate the output of the collision mitigation framework that decides the least severe collision in terms of trajectories of the ego vehicle. (b) We perform a set of sensitivity tests to address the challenges that are associated with assumptions on collision parameters or models, meaning that their values could affect the post-impact motion predictions and consequently the choice of the least severe trajectory.
The post-impact motions for the above-mentioned aspects are simulated in MATLAB for t = 3s, which is a suitable time frame to show the evolution of the vehicle states when comparing the models and performing the sensitivity analysis. Furthermore, in all preimpact scenarios of this section, the target vehicle is following a straight trajectory initially and the ego vehicle is striking it with different trajectories.

General scenario: post-impact motions of vehicles
In  Table A1.

4DOF and 3DOF vehicle models
Here, the considered scenario is presented in Figure 1   −160.65 kN and F y = −64.26 kN for ego vehicle. Therefore, the difference in impact force values between these two models is quite small.
Post-impact states of both vehicles are presented in Figure 2 for the 4DOF model and Figure 3 for the 3DOF model. As shown, the post-impact trajectories and vehicle states in these two cases are quite similar for planer motions. Incorporating the roll behaviour in modelling had some effects on lateral velocity, lateral acceleration and yaw rate. However, due to the already saturated tyre forces, seen in Figures 2 and 3, the impact on vehicle trajectory is not affected much. In a less severe crash, the choice of vehicle model could have a larger impact on vehicle trajectory.
Given the results of this section and considering the short-time scenario of an unavoidable collision, in the following subsections a 3DOF vehicle is used for generating the post-impact states and trajectories of each vehicle, or whenever a vehicle model is used for solving the collision, e.g in full-impact collisions.

Kudlich-Slibar model
Here, the considered scenario is presented in Figure 1

Choice of least severe trajectory
Here, we present an example scenario where the application of the proposed framework is demonstrated. During a detected unavoidable collision, the ego vehicle predicts the post-impact motions for both vehicles and calculates the cost of collision based on Equation (25). For this subset of trajectories, four different impact scenarios are presented in Figure 5. The ones presented in Figure 5   full-impact and thus a 3DOF vehicle model is used to solve the collision. Contact friction is μ = 0.4 in all scenarios in this section. The angle of contact plane in sliding impact is based on geometrical rule and for full-impact is the ego vehicle's yaw angle.
Collision cost, based on Equation (25), for ego and target vehicle for each scenario is presented in Table 1. These costs are calculated for equal values of weighting factor, meaning  w i = 1. Some of the states of vehicles that directly or indirectly affect the cost are presented in Figure 6. Considering the cost for the ego vehicle, the collision between trajectory 4 of the ego vehicle and the target vehicle has the lowest cost for the ego vehicle. Following this trajectory leads to the least lateral deviation; least exposure to secondary side impacts and has the lowest cost of post-impact yaw rate. On the other hand, trajectory 2 has the highest cost and the largest lateral deviation. Trajectory 1 has a slightly lower cost than trajectory 2. Furthermore, the highest yaw rate cost and the lowest side slip cost belong to trajectory 3.
Considering the cost for target vehicle, the lowest total cost for it is predicted if the ego vehicle follows trajectory 1, which also has the lowest cost for all terms. In contrast following trajectory 3 by ego vehicle leads to the highest total cost and largest lateral deviation for target vehicle. The highest cost for yaw rate, heading angle and side slip is predicted for target vehicle if ego vehicle follows trajectory 4. According to Subsection 3.3.2, trajectory 1 is thus followed by the ego vehicle.

Value of contact friction
To investigate the sensitivity of post-impact motions of vehicles to the value of μ, some tests are performed. The pre-impact scenario used for this purpose is shown in Figure 1(a). Furthermore, we assume that the contact plane angle is the heading angle of the ego vehicle and = 0.1. Values of contact friction are varied as μ = 0.1, 0.3, 0.5, 0.6. The post-impact trajectories are presented in Figure 7, which can be compared with the full-impact in Figure 5(d). Note that the post-impact trajectories are sensitive to the value of contact friction. Furthermore, the resulting impact forces are given in Table 2.

Contact plane angle
Results in Subsections 4.1.1, 4.3.1 and Figure 5(d) are derived by defining the angle of contact plane as the ego vehicle's yaw angle. Here, we investigate the sensitivity of post-impact For example, in Figure 8(a), the ego vehicle is less deviated from its pre-crash trajectory compared to Figure 5(d), while the target vehicle deviates more from its original path in Figure 8(a). Differences between the states of vehicles in these two cases can also be seen by comparing Figure 8 = 0.233 rad. Therefore, a combination of different collision parameters can lead to similar post-impact motions, which was also concluded in [6].

Collision models
Post-impact motions considering the Kudlich-Slibar and the 3DOF models are compared to solve the same collision configuration. The pre-impact scenario is Figure 1(a). In both cases, = 0.4569 rad based on geometrical rule and = 0.2661 according to Equation (24).
By applying the Kudlich-Slibar method, T c = −119.40 kgm/s 2 and N c = −9.87 × 10 03 kgm/s 2 . As mentioned in Subsection 2.3, for values of contact friction/impulse ration more than the critical value (μ c = T c N c ), a full-impact occurs. Here, this value is calculated   as μ c = 0.0121. We assume that the value of contact friction given by analyst is more than μ c , and the impact is predicted as full-impact. Note that impulse values are independent from the value of contact friction for full-impacts. Thus the total impulses along n−t axis are calculated according to Equation (22). Therefore, the impact forces in each vehicle's coordinate system are calculated as

Discussion
We presented a collision mitigation framework that through a cost function reduces collision severity based on post-impact motion prediction. Furthermore, the sensitivity of these motions to the values of collision parameters was investigated. The cost function's design and the choice of these parameters are discussed here considering the example scenario of Subsection 4.2.

Design of cost function
Results in Subsection 4.2 are derived assuming the same weighting factor for all cost terms of Equation (25). The choice of trajectory to follow by the ego vehicle with the lowest cost may be affected if the cost terms affect the total cost differently. Considering the example scenario, this choice (trajectory 1) is not affected by changing the weighting factors since this trajectory has considerably lower values for all cost terms.
However, this is not always the case. For example, by defining w 1 = 0.01 and w 2 = 10, the effect of lateral deviation on the total cost is decreased while the effect of heading angle is increased. Considering the target vehicle and this change, the total cost of trajectory 3 is reduced to J = 24.1876, while it is increased to J = 33.60 for trajectory 4. This is because trajectory 3 has the highest cost of Y c while trajectory 4 has the highest cost of ψ c .
This means that trajectories in which the vehicle has a larger lateral deviation with no exposure to secondary impacts may be chosen over trajectories with lower lateral deviation but an unsafe yaw angle. Therefore, there exists a trade-off between for instance lateral deviation and safely positioning the vehicle. Consequently, the weighting factors should be dynamically adjusted accordingly based on the context. In other words, any information that can be used to evaluate risk could be directly used by the proposed framework. If the vehicle is aware of whether it is driving on a highway, in a residential neighbourhood, or in a road maintenance area, it can adjust its risk mitigation manoeuvres to the most likely post-impact risks. With the emerging use of vehicle-to-vehicle (V2V) and vehicleto-infrastructure (V2I), this can be extended to consider the traffic conditions, traffic rules, and current behaviour of other vehicles [48]. As an example, if there are a lot of surrounding vehicles travelling too close to each other on a highway, giving a higher weight factor to lateral deviation might avoid secondary impacts with vehicles in adjacent lanes. Furthermore, even if it will come with its own challenges [49], V2V and V2I will be used to share the position of road users and sensor data in the future. This could be used to micromanage the cost function. If the only nearby road user is a pedestrian further down the road from a potential collision, this could be an argument for decreasing the weight factor of lateral deviation (avoiding the pedestrian but steering the vehicle towards otherwise empty parts of the road).

Setting the values of collision parameters
As shown in For trajectory 4 and considering the cost for target vehicle, by setting μ = 0.5 all cost terms, except Y c , are increased in comparison to μ = 0.4. Consequently, by setting μ = 0.5, the total cost is increased to J = 27.41. For the set of investigated trajectories, this change, however, did not affect the choice of least severe trajectory.
Furthermore, as mentioned in Subsection 2.3, contact friction or its critical value among other factors depends on impact type, collision configuration, and the mass property of vehicles. Therefore, choosing an accurate value for it will depend on the proper estimation of the target vehicle's geometry, position and type.
Similarly, the angle of contact plane also affects post-impact motions (see Subsection 4.3.2). We compare Figure 5(d) with Figure 8(a), where is the yaw angle of ego vehicle in the former while in the latter is based on geometrical rule. Considering trajectory 4, when the geometrical rule is applied, the cost for all terms increased for target vehicle and consequently, total cost is increased to J = 48.88. However, for this specific scenario, this change did not affect the choice of least severe trajectory.
Calculating this angle based on geometrical rules requires information on position and orientation of both vehicles. Unfortunately, accurate estimation of these values for the target vehicle is essential but more uncertain. This uncertainty leads to uncertainty in the angle estimation. Consequently, it may affect the cost of post-impact motions. This uncertainty is somewhat reduced due to characteristics of unavoidable collisions, i.e. that target vehicles often do not have enough time for extreme manoeuvres.

Further research in modelling
The choice of collision model can also affect post-impact motions as shown in Subsection 4.3.3. We compare Figure 5(d) with Figure 10(a). The collision in the former is solved with a 3DOF vehicle model while in the latter with the Kudlich-Slibar. In the latter and for target vehicle, cost terms of Y c and ψ c increased while ω zc and β c decreased. Consequently, the total cost for target vehicle increased to J = 104.69 for trajectory 4. For the investigated scenario, this change in modelling did not affect the choice of least severe trajectory; however, that is not always the case.
In the worst case, modelling a sliding impact with 3DOF could associate a trajectory with a higher cost than it actually is, while modelling a full-impact with Kudlich-Slibar may produce less accurate results due to , for instance, exclusion of tyre forces. Therefore, further research is required to bridge this gap in collision modelling at least for collision mitigation applications.

Limitation
The proposed collision mitigation framework and the consequent analysis are performed under certain assumptions, which can have some limitations for the study. For example, it is assumed that the pre-impact position and states of the target vehicle is known, as these are used for instance for calculating impact forces and POI. However, the behaviour of others are uncertain and thus the choice of least severe trajectory can be affected by this uncertainty.
Beside μ and , other collision parameters for which the sensitivity may need to be investigated are coefficient of restitution ( ), collision duration and shape of the collision pulse. For example, depends on impulse direction, orientation of the contact surface (in rear impacts) [50], and closing velocity [51]. Some crash types have a longer duration than others or have different pulse shapes [52]. Therefore, given the relationship between impulse and force, a different impulse shape may affect the value of impact force. Furthermore, a different collision duration than the one assumed here may affect the value of impulses since this parameter is part of solving the collision. Therefore, the choice of these parameters can affect the post-impact motions and consequently the chosen least severe trajectory. More experiments with real crash data is also required to test the effects of different parameters.

Conclusion
We have proposed a collision mitigation framework for an autonomous vehicle's motion planner that incorporates predictions of the post-impact motions of an impacted vehicle. This framework can be configured for different contexts by adjusting its cost function according to relevant risks. To set the values for the collision parameters in its collision model, accurate prediction of position and properties of target vehicle and also collision configuration before an impact is essential. However, given that this framework is proposed for an unavoidable collision, uncertainty in the behaviour of the target vehicle is reduced due to the limited time window in these extreme manoeuvres. Additionally, through sensitivity analysis, important effects are highlighted concerning different modelling decisions, with respect to vehicle dynamics models and collision models, that are important to consider in further research.

Appendix 1. Dynamic single-track model for trajectory library
Pre-impact trajectories of the ego vehicle are calculated offline by solving an optimization problem using the toolbox CasADi and are saved in a trajectory library. This is based on a previous study [7]. Equations of motion are presented here for clarification purposes. These equations are constraints of the optimization and are according to Equation (A1). In this equation, (X, Y) are the global position at the centre of gravity (CoG) of the ego vehicle and ψ is the heading angle. Other states are: longitudinal velocity (v x ), lateral velocity (v y ), yaw rate (ψ), steering angle (δ), longitudinal (F xi ) and lateral tyre force (F yi ). Notations f and r represent front and rear axles, respectively. A Pacejka tyre model [40] is implemented as Equation (A2), where F zi is the normal force, C i is the tyre stiffness factor and α i is the slip angle.

Appendix 2. Sensitivity of contact friction under geometrical rule
We previously showed that post-impact motions of vehicles are sensitive to collision parameters such as contact friction and the angle of contact plane. Here, we present the sensitivity of post-impact trajectories to the value of μ with the assumption that is calculated based on geometrical rule as opposed to Subsection 4.3.1. Values of μ = 0.1, 0.5 are considered and the post-impact trajectories are presented in Figure A1(a ,b), and can be compared with Figure 7 Figure A1(c ,d). In Figure A1(c), is equal to the yaw angle of ego vehicle while in Figure A1(d), it is calculated by geometrical rule. The total impact forces for each vehicle are also shown in Figure A1(c ,d) with the same scaling factor. As shown, impact forces have higher values along X axis in Figure A1(c) while they have higher values along Y axis in Figure A1(d).