Mathematical analysis of braking in rear-end crash scenarios

Recently, a test-based assessment method for forward-looking crash avoidance and mitigation systems (pre-crash systems) was developed within the EU project ASSESS. In this method, injury outcome is correlated with the change of velocity (Delta-V) during the crash and the effect of the pre-crash system on this quantity is investigated. Delta-V may be related to relative impact speed (i.e., the speed difference between the vehicles at the moment of collision) by certain assumptions. It is therefore a central question in the ASSESS assessment method to determine the relative impact speed in all rear-end test scenarios in the test protocol. This is most challenging in the scenario when the vehicle under test (striking vehicle) is approaching a braking lead vehicle. In this paper, the corresponding question is addressed in a much more general setting. In particular, instead of the fixed values used in the test protocol for the ASSESS method, all possible values for the initial speeds and decelerations of the vehicles and the initial headway of the lead vehicle are considered in the important case when the two vehicles have equal initial speeds and decelerations. The question addressed here is the following: given arbitrary values of all these parameters, is it possible to determine the speed difference between the two vehicles at the time of collision from the impact speed of the vehicle under test using mathematically rigorous methods? The results show that under simplifying assumptions, the relative impact speed may be expressed as a function of the impact speed of the vehicle under test. The set of valid test results is characterized for all possible parameter values together with the requirements on how early braking needs to be performed in order to avoid the crash. Possible generalizations of the method (including the feasibility of removing some of the theoretical assumptions) are also discussed. The results may be of use in the design of future test protocols for radar-based braking systems as well as in the development of such systems