We present an efficient method for certifying non-collision for piecewise-polynomial motion plans in algebraic reparametrizations of configuration space. Such motion plans include those generated by popular randomized methods including RRTs and PRMs, as well as those generated by many methods in trajectory optimization. Based on Sums-of-Squares optimization, our method provides exact, rigorous certificates of non-collision; it can never falsely claim that a motion plan containing collisions is collision-free. We demonstrate that our formulation is practical for real world deployment, certifying the safety of a twelve degree of freedom motion plan in just over a second. Moreover, the method is capable of discriminating the safety or lack thereof of two motion plans which differ by only millimeters.

Many computations in robotics can be dramatically accelerated if the robot configuration space is described as a collection of simple sets. For example, recently developed motion planners rely on a convex decomposition of the free space to design collision-free trajectories using fast convex optimization. In this work, we present an efficient method for approximately covering complex configuration spaces with a small number of polytopes. The approach constructs a visibility graph using sampling and generates a clique cover of this graph to find clusters of samples that have mutual line of sight. These clusters are then inflated into large, full-dimensional, polytopes. We evaluate our method on a variety of robotic systems and show that it consistently covers larger portions of free configuration space, with fewer polytopes, and in a fraction of the time compared to previous methods.