Computing Large Convex Regions of Obstacle-Free Space Through Semidefinite Programming

This paper presents iris (Iterative Regional Inflation by Semidefinite programming), a new method for quickly computing large polytopic and ellipsoidal regions of obstacle-free space through a series of convex optimizations. These regions can be used, for example, to efficiently optimize an objective over collision-free positions in space for a robot manipulator. The algorithm alternates between two convex optimizations: (1) a quadratic program that generates a set of hyperplanes to separate a convex region of space from the set of obstacles and (2) a semidefinite program that finds a maximum-volume ellipsoid inside the polytope intersection of the obstacle-free half-spaces defined by those hyperplanes. Both the hyperplanes and the ellipsoid are refined over several iterations to monotonically increase the volume of the inscribed ellipsoid, resulting in a large polytope and ellipsoid of obstacle-free space. Practical applications of the algorithm are presented in 2D and 3D, and extensions to N -dimensional configuration spaces are discussed. Experiments demonstrate that the algorithm has a computation time which is linear in the number of obstacles, and our matlab [18] implementation converges in seconds for environments with millions of obstacles.