A Biconvex Method for Minimum-Time Motion Planning Through Sequences of Convex Sets

Abstract

We consider the problem of designing a smooth trajectory that traverses a sequence of convex sets in minimum time, while satisfying given velocity and acceleration constraints. This problem is naturally formulated as a nonconvex program. To solve it, we propose a biconvex method that quickly produces an initial trajectory and iteratively refines it by solving two convex subproblems in alternation. This method converges quickly to low-cost trajectories, returns a feasible solution even if stopped early, and does not require the selection of any line-search or trust-region parameter. Exhaustive experiments show that our method can find high-quality trajectories in a fraction of the time of state-of-the-art solvers for nonconvex optimization. Additionally, tested on the problem of transferring packages between bins using two robot arms, our method achieves a fifty percent increase in throughput compared to waypoint-based motion planners that are common in industry.