On computing time-to-collision for automation scenarios

Time to collision (TTC) has been a key vehicle safety metric for decades. With the increasing prevalence of advanced driver assistance systems and vehicle automation, TTC and many related metrics are being applied to the analysis of more complicated scenarios, as well as being integrated into automation algorithms. While the TTC metric was originally conceived to be inclusive of generic two-dimensional situations, its applications has been mostly limited to one-dimensional scenarios. This paper derives general equations and algorithms using two-dimensional information. Additionally, methods from computational geometry, a field that didn't exist when TTC was first used, are employed for the general case of computing TTC between bounding boxes. Parametric equations for lines play a prominent role and offer an elegant way to express the geometry of the scenarios described in this paper. Throughout, the approach is not to derive specific algebraic conditions as in previous efforts. Rather, the focus in on developing general algorithms for computation. The techniques presented are not necessary for traditional car following scenarios; but offer options for more complex situations that trade off analytic solutions for computational flexibility.