Collision prediction for polyhedra under screw motions

The prediction of collisions amongst N rigid objects may be reduced to a series of computations of the time to first contact for all pairs of objects. Simple enclosing bounds and hierarchical partitions of the space-time domain are often used to avoid testing object-pairs that clearly will not collide. When the remaining pairs involve only polyhedra under straight-line translation, the exact computation of the collision time and of the contacts requires only solving for intersections between linear geometries. When a pair is subject to a more general relative motion, such a direct collision prediction calculation may be intractable. The popular brute force collision detection strategy of executing the motion for a series of small time steps and of checking for static interferences after each step is often computationally prohibitive. We propose instead a less expensive collision prediction strategy, where we approximate the relative motion between pairs of objects by a sequence of screw motion segments, each defined by the relative position and orientation of the two objects at the beginning and at the end of the segment. We reduce the computation of the exact collision time and of the corresponding face/vertex and edge/edge collision points to the numeric extraction of the roots of simple univariate analytic functions. Furthermore, we propose a series of simple rejection tests, which exploit the particularity of the screw motion to immediately decide that some objects do not collide or to speed-up the prediction of collisions by about 30%, avoiding on average 3/4 of the root-finding queries even when the object actually collide.