ABSTRACT
Before the collision phase of a crash can be analyzed, the run-outs of all the involved vehicles (the postcollision phase) must be reconstructed. In other words, their linear and angular velocities must be determined at the start of their run-outs (when they separate from each other at the end of the collision phase). One way to proceed is to perform successive time-forward simulations until all run-out trajectories are predicted, at least to some reasonable extent. But why try to predict the trajectories when they are already known from evidence?
A more direct way is to move the vehicles over their trajectories going backwards in time, from rest to start of run-out, as is often done by reconstructionists. However, the usual straight lines are often not suitable for representing trajectories. The challenge is to place the vehicles on known trajectory positions (in terms of spatial center of mass coordinates and heading angles), and to find a suitable independent variable for expressing how these inputs vary throughout the run-outs. Time cannot be used for this purpose, since it is not known. However, progress along a path can be expressed in terms of arc length, since it is known to a reasonable degree of accuracy. Instead of traveling backwards in time, the vehicles move backwards in terms of path length distance.
Constructing a run-out trajectory may be thought of as a process of connecting the dots that represent the set of key positions and heading angles. For actual trajectories, though, graphs of X, Y, Z, and ψ (versus arc length) form smooth curves, not piecewise linear ones. A smoothing function (that can represent arbitrarily shaped curves) is needed. Cubic splines can be fitted exactly through all the key points, are smooth in nature, and can be subdivided into arbitrarily small piecewise-linear segments within which relatively simple calculations can be made.
Not that the segments need be all that small (in contrast to simulations). This chapter shows how to set up a publicly available cubic spline function in a spreadsheet, and how to use a spline to subdivide a trajectory into a reasonable number of piecewise-linear segments that represent, to within an arbitrary degree of accuracy, the trajectory established from evidence.
