Evaluation of a momentum based impact model in frontal car collisions for the prospective assessment of ADAS
Abstract
Motivation
The advent of active safety systems calls for the development of appropriate testing methods that are able to assess their capabilities to avoid accidents or lower impact speeds and thus, to mitigate the injury severity. Up to now the assessment is mostly based on the decrease of the collision speed due to CMS (collision mitigation systems). In order to assess the effects on injury severity developing methods, that are able to predict collision parameters correlating with the risk of getting injured, such as deltav, for different impact situations is a mandatory task.
Objective
In this study a momentum based impact model is assessed in terms of reliability to solve the collision mechanics and therefore to predict deltav for frontal car collisions.
Method
Real accidents were resimulated using predefined input parameters for the impact model (virtual forward simulation – VFS). Subsequently the impact model was analyzed for its sensitivity to specific input parameters.
Conclusion
It was shown that VFS works for full impacts while improvements and optimizations are required for impacts that include a sliding movement in the contact zone of the vehicles.
Keywords
Driver assistance systems Collision mitigation systems Effectiveness assessment Impact mechanics Point of impact Momentum based impact modelAbbreviations
 ADAS
Advanced driver assistance system
 AEB
Autonomous emergency braking
 CEDATU
Central Database for InDepth Accident Study
 CMS
Collision mitigation system
 COG
Center of gravity
 CP
Contact plane
 CS
Coordinate system
 EES
Energy equivalent speed
 NCAP
New Car Assessment Programme
 POI
Point of impact
 VFS
Virtual forward simulation
 VT
Virtual testing
 XRATE
Extended Effectiveness Rating of Advanced Driver Assistance Systems
1 Introduction
In 2010 the European Commission [1] released a series of precautionary measures (focusing on vehicle safety, the safety of infrastructure and road users’ behavior) to halve the number of road deaths until 2020. Salmon et al. [2] point out that between 75% and 95% of all traffic accidents are – at least partially – caused by human error. In this context, the integration of collision mitigation systems (CMS) in new cars is a reasonable step towards enhanced occupant safety. The purpose of CMS is to monitor the surrounding traffic, detect critical events, to brief the driver on the current traffic situation or to take actions if the driver is not responding. Hence, the driver can be partially unburdened from mental stress and his failure probability can be reduced. CMS, such as autonomous emergency braking systems (AEB), aim to mitigate injuries in traffic accidents, usually by reducing the impact velocity and therefore the kinetic energy of the vehicle. However, automatically triggered braking manoeuvers also affect the collision configuration and the impact pattern. E.g. if the overlap in frontal collisions is smaller, a lower deltav can cause more severe injuries, due to high intrusion to the passenger compartment [3].
A number of different approaches is available to determine the efficacy of CMS – retrospective analysis of real accident data and prospective analysis based either on testing (driving simulator studies, field operational tests, naturalistic driving studies) or simulation (virtual testing). Especially for newer ADAS retrospective analyses are difficult, because the number of accidents involving cars equipped with CMS is still too low to provide results of high statistical significance. Thus, prospective investigations are needed. One possibility is to do physical testing, using cars equipped with appropriate systems, similar to the Euro NCAP (New Car Assessment Programme) tests. In these tests, restrictions are applied to reduce the testing effort and to increase the repeatability. E.g. in the Euro NCAP AEB (autonomous emergency braking) testing procedure [4] only the reduction of the collision velocity is considered when calculating the final rating. Instead of physical testing, virtual testing (VT) can be done. VT offers the possibility to simulate a high number of different accident scenarios in a short period of time. Thus, more accident scenarios can be addressed. In VT, it is necessary to introduce surrogate parameters for the injury severity. One of these surrogate parameters is deltav, the change in velocity between the pre and postcrash trajectories of a vehicle [5]. It is obtained by resolving the collision mechanics.
In contrast to VT, retrospective analyses use statistical approaches for determining the influence of specific safety systems on the accident and injury occurrence. The basic data for these analyses is usually gathered via accident reconstruction. The main task of accident reconstruction is to reproduce the accident sequence in time as well as in space. Detailed descriptions can be found in Johannsen [6] or Burg and Moser [7]. Accident reconstruction utilizes data such as the damage patterns, the final positions of the vehicles, etc. Usually it is necessary to tune the preimpact collision parameters (e.g. point of impact (POI), coefficient of restitution, etc.) after setting up the simulation to obtain the final positions of the vehicles. The final vehicle positions and/or an estimation of EES (energy equivalent speed [8]) can thus be used to check the plausibility of the results [9]. The reconstruction provides data like impact velocities, post crash movements or deltav.
VT can be done using the so called virtual forward simulation (VFS). VFS can be used to simulate traffic accidents, starting within the precrash phase (with a sufficient time history before the collision, so that the CMS can take actions [10]). Within VFS, the vehicles can be equipped with generic ADAS to simulate their behavior. Studies carried out by Zauner et al. [11], Tomasch et al. [10] or Kolk et al. [12, 13] used the method of VFS to rate the effects of CMS. Hence the collision configuration, like the relative vehicle positions or the impact velocities, are manipulated by the CMS it is necessary to predict the preimpact collision parameters for the impact model. The collision mechanics are then solved using these preimpact parameters. In VFS, the data of a real accident i.e. final vehicle positions, damage patterns, damage depth, etc. are not known. Thus, they cannot be used to validate the results of the simulation.
2 Objective
The first objective of this study is to assess the performance of a momentum based impact model in terms of accuracy of the postcrash parameters e.g. deltav, postcrash velocity or final positions of the vehicles. The second objective is to analyze the model on its sensitivity to its input parameters, like initial speeds, position of the POI, angle of the contact plane (CP), coefficient of restitution or coefficient of friction.
3 Method
3.1 Momentum based impact model

The crash force between the interacting vehicles is exchanged in one discrete point – the POI

Tire forces and the gravitational force are neglected during the crash

The duration of the crash phase is infinitely small, therefore no acceleration pulses or vehicle deformations are computed

The vehicles do not move during the crash phase

The vehicle bodies have linear stiffness behavior
3.1.1 Conservation of momentum – Equations
The conservation of momentum can be derived by combining Newton’s 2^{nd} and 3^{rd} law and states that the total momentum in a closed system of n interacting bodies remains constant, if no external forces are applied. The total momentum equals the sum of momentums \( \overset{\rightharpoonup }{p_i} \) of the individual bodies (1):
The same equation must also be satisfied by the angular momentum \( \overset{\rightharpoonup }{L} \) (2):
Analyzing equations (1) and (2) shows that the total momentum after the collision is still the same, but the energy is transformed and transferred. During the first phase of the collision, the kinetic energy is partially transformed into vehicle deformation (phase of compression). As in most physical systems, some of the deformation is retransferred into kinetic energy. This is called restitution. Thus, the collision can be split in two phases – the phase of compression and the phase of restitution. The compression phase extends from the initial contact of the vehicles until the velocities of the contact points in normal direction are equal. The latter extends from the end of compression until the bodies separate again. Following this definition, the momentum for each phase can be obtained by:
Equation (3) shows the calculation of the total momentum during the compression phase, where \( \overset{\rightharpoonup }{F}(t) \) represents the acting force and t_{0} and t_{1} represent the beginning and the end of the compression phase. The momentum for the restitution phase S_{R} can be evaluated using t_{1} and t_{2}. At time t_{2} the vehicles separate again. To incorporate the elastic deformation the impulse ratio ε (coefficient of restitution) is introduced as the ratio between S_{R} and S_{C} (4). Using this definition, the entire exchanged momentum can be calculated as shown in (5).
Taking the conservation of energy into account, there are two limits for the impulse ratio. The first one is ε = 0 representing a totally plastic impact, the second one is ε = 1 referring to a totally elastic impact. In car accidents usually a default value of ε = 0.1 is used, because it is applicable for a wide range of closing speeds. This value depends on the geometry of the bodies, the closing speed and the material properties [6, 15, 20]. Additionally the POI, the angle of the CP and the coefficient of friction in the contact zone must be defined.
In accident reconstruction the preimpact parameters of the impact model (collision velocities, POI, angle of the CP, coefficient of restitution, coefficient of friction) need to be refined in a manner, that the vehicles reach their final positions. Additionally estimated parameters such as the EES value can be used to verify the results.
3.1.2 Full impact
A full impact occurs when the condition T ≤ μN is satisfied, meaning that the collision partners do not slide along each other in the contact zone. It also means that the contact points reach the same velocity at the end of the compression phase in normal and tangential direction. In this case, the angle of the CP does not influence the final results.
3.1.3 Sliding impact
The second option is that – in theory – T > μN holds. Thus this is not possible, T = μN holds, meaning that a sliding movement in the contact zone of the vehicles occurs. As a result, the contact points do not reach a common velocity in tangential direction at the end of the compression phase. In this case, the definition of the CP and the coefficient of friction must be estimated very carefully, because the results (final positions, trajectories) show a high dependency on these parameters [21]. Sliding impacts are identified as follows. The opening angle of the friction cone is calculated according to equation (6).
If the angle between crash force and normal direction n in Fig. 2(a) is equal to the opening angle of the friction cone ρ, a sliding impact occurs.
Many studies have been carried out to evaluate the suitability of this impact model for accident reconstruction. Bailey et al. [22] staged five different collisions and recorded data such as deltav, postcrash trajectories, impact speeds etc. After the tests, they validated the impact models using two different methods. First, they calculated the input parameters for the model from the recorded test data and compared the final positions and postcrash trajectories. Second, they used the final positions and postcrash trajectories to determine the initial conditions. Finally, they compared the obtained results with the measured data and found that the error of the model stays within reasonable limits. Other investigations on this topic were carried out by Cliff and Montgomery [23] and Cliff and Moser [24].
3.2 Virtual forward simulation
The VFS method has become more popular for the assessment of ADAS in recent years. One of the advantages of VFS is that a large number of scenarios can be evaluated in a short period of time. When implementing generic ADAS into a vehicle in VFS, the simulation starts within the precrash phase to be able to simulate the behavior of the ADAS during the precrash phase. In case the ADAS takes action, the collision configuration (velocities, impact pattern, etc.) might change. The location of the POI and the angle of the CP might thus also be changed. Additionally it is not possible to use final positions or estimation of EES to verify the results (as these parameters could be used in accident reconstruction). Therefore, the user has to find a way of determining reasonable values for the input parameters (POI, CP, coefficient of restitution, etc.) of the impact model.
In this study, the VFS was done using the simulation tool XRATE (Extended Effectiveness Rating of Advanced Driver Assistance Systems) developed by Kolk [25]. XRATE is a control platform that is capable of setting up driving dynamics simulations and run them automatically. It is based on MATLAB® and utilizes PCCrash™ as solver for the vehicle dynamics and collision mechanics. The crash related parameters are then gathered and saved.
3.2.1 Definition of the POI and the CP using geometrical rules
For reasons of simplicity, the vehicle outline is represented by a rectangle with the same length and width as the real vehicle. As required by the impact model, the vehicles penetrate each other before they reach the collision configuration and the momentum is exchanged. The overlap region, shown in Fig. 2 (b), is represented by a polygon that can be calculated using the algorithm of Sutherland and Hodgman [26]. The corners of this polygon either represent the corners of the vehicle or the intersection points between the vehicle outlines. The centroid of the overlap region can be obtained as described by Nürnberg [27]. The area of the overlap polygon and the coordinates of the nodes of the polygon are used to obtain the coordinates of the centroid. In addition to the POI the CP is also defined using geometrical rules only. The direction of the CP is defined by the intersection points of the simplified vehicle contours. It is then translated in parallel so that it is coincident with the POI, see Fig. 2 (b).
4 Material
The indepth accident database CEDATU (Central Database for InDepth Accident Study) developed by Tomasch [28, 29] was the source for the basic accident data. The CEDATU data fields are based on the STAIRS (Standardization of Accident and Injury Registration Systems) protocol [30]. The fields were enhanced and extended in accordance with the results obtained in different research projects, such as PENDANT (PanEuropean Coordinated Accident and Injury Databases, [31]), RISER (Roadside Infrastructure for Safer European Roads, [32]) and ROLLOVER (Improvement of Rollover Safety for Passenger Vehicles, [33]). Additionally data fields of the national statistics in Austria are incorporated to make a direct comparison possible [34].
5 Sampling criteria
The following sampling criteria were applied for this study:

Only fatalities involving two passenger cars proceeding in opposite directions

No subsequent collisions of the vehicles when moving to the final positions

The penetration depth is set constant

Frontal damage of the vehicles (according to the STAIRS protocol [30])

PDoF (Principal Direction of Force): 1, 2, 10, 11, 12 (STAIRS protocol [30])

Random selection of accidents – no frequency correlation with national statistics (bias of results)
All of the accidents were reconstructed and resimulated by VFS. Thirtysix accidents involving 72 vehicles were analyzed.
6 Results and discussion
In the context of the final positions in full impacts, similar results are obtained for reconstruction and VFS. For sliding impacts there is quite a big difference. Using VFS, the postcrash movement is shorter than in reconstruction.
Results for the pairedttests applied for the values of Fig. 6
Number of samples n  Mean value reconstruction  Mean value VFS  pvalue  

Deltav, full impacts [km/h]  26  51.42  50.54  0.482 
EES, full impacts [km/h]  26  50.19  48.85  0.398 
Change of velocity angle, full impacts [°]  26  92.42  93.71  0.723 
Deltav, sliding impacts [km/h]  12  24.25  31.92  0.255 
EES, sliding impacts [km/h]  12  44.75  55.33  0.180 
Change of velocity angle, sliding impacts [°]  12  12.83  17.33  0.301 
When looking at sliding impacts, the hypothesis that the mean values are different (for any of the three parameters) cannot be neglected (p > 0.05), although the differences appear to be large when examining the plots of Fig. 6b. However, as the number of samples is relatively low, the validity of the test is also low. When comparing the results, however, it can be stated that the differences are much larger than in full impacts. Thus, a determination of the POI and the CP in sliding impacts by geometrical rules only, could be nonvalid.
7 Conclusion
The results of this study show that the use of VFS is reasonable for full impacts. The comparison between accident reconstruction and VFS showed that the differences concerning the final positions of the vehicles, deltav, EES and the change of the velocity angle are small. Due to the good results for the deltav value, an integration of risk functions based on deltav is possible. This study also shows a good correlation with the results of Kolk et al. [14], where the same impact model was investigated for junction accidents.
For sliding impacts, the use of VFS requires optimization and improvements. Within the comparison of accident reconstruction and VFS the final vehicle positions and deltav showed a high sensitivity to the definition of the POI and the CP and thus were quite different. This is due to the fact, that the angle of the contact plane has a great influence on the normal and tangential forces acting in the POI as well as on the levers for the angular momentum.
8 Outlook
Future research should also include the investigations on the penetration depth that was set to a fixed value of 30 ms in this study. The penetration influences the levers of the exchanged angular momentum in the impact model. In reconstruction, the depth of penetration can be obtained from the damage patterns. In VFS, it would be necessary to find a correlation between the closing speed, the overlap and the penetration depth to be used.
Further, the algorithm for the determination of the POI and the CP in VFS needs further investigations specifically because the use of geometrical relationships often results in sliding impacts being identified instead of full impacts.
The prediction of the POI and CP for sliding impacts must be analyzed in detail, because these impacts show a very sensitive behavior. In addition, the number of accidents investigated should be increased to provide results that are more reliable. Additionally, the analyses carried out in this context should also be applied for other types of vehicles and accident scenarios.
An ANOVA could be used to find the most sensitive precollision impact parameters in order to incorporate multiple parameters i.e. overlap, POI, angle of the CP, penetration depth, masses of the vehicles and others in the sensitivity analysis.
Notes
Acknowledgements
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Funding
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Availability of data and materials
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Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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