  Motion planning methods for autonomous systems based on nonlinear programming
offer great flexibility in incorporating various dynamics, objectives, and
constraints. One limitation of such tools is the difficulty of efficiently
representing obstacle avoidance conditions for non-trivial shapes. For example,
it is possible to define collision avoidance constraints suitable for nonlinear
programming solvers in the canonical setting of a circular robot navigating
around M convex polytopes over N time steps. However, it requires introducing
(2+L)MN additional constraints and LMN additional variables, with L being the
number of halfplanes per polytope, leading to larger nonlinear programs with
slower and less reliable solving time. In this paper, we overcome this issue by
building closed-form representations of the collision avoidance conditions by
outer-approximating the Minkowski sum conditions for collision. Our solution
requires only MN constraints (and no additional variables), leading to a
smaller nonlinear program. On motion planning problems for an autonomous car
and quadcopter in cluttered environments, we achieve speedups of 4.8x and 8.7x
respectively with significantly less variance in solve times and negligible
impact on performance arising from the use of outer approximations.
