# Finding and Optimizing Certified, Collision-Free Regions in Configuration Space for Robot Manipulators

Published in *Workshop on the Algorithmic Foundations of Robotics 2022, to appear*, 2022

Configuration space (C-space) has played a central role in collision-free motion planning, particularly for robot manipulators. While it is possible to check for collisions at a point using standard algorithms, to date no practical method exists for computing collision-free C-space regions with rigorous certificates due to the complexities of mapping task-space obstacles through the kinematics. In this work, we present the first to our knowledge method for generating such regions and certificates through convex optimization. Our method, called C-Iris (C-space Iter- ative Regional Inflation by Semidefinite programming), generates large, convex polytopes in a rational parametrization of the configuration space which are guaranteed to be collision-free. Such regions have been shown to be useful for both optimization-based and randomized motion plan- ning. Our regions are generated by alternating between two convex op- timization problems: (1) a simultaneous search for a maximal-volume ellipse inscribed in a given polytope and a certificate that the polytope is collision-free and (2) a maximal expansion of the polytope away from the ellipse which does not violate the certificate. The volume of the el- lipse and size of the polytope are allowed to grow over several iterations while being collision-free by construction. Our method works in arbitrary dimensions, only makes assumptions about the convexity of the obsta- cles in the task space, and scales to realistic problems in manipulation. We demonstrate our algorithmâ€™s ability to fill a non-trivial amount of collision-free C-space in a 3-DOF example where the C-space can be vi- sualized, as well as the scalability of our algorithm on a 7-DOF KUKA iiwa and a 12-DOF bimanual manipulator.