Semidefinite Relaxations for Collision-Free Motion Planning

2026-06-11

AI Summary

This work addresses the non-convex problem of planning high-order smooth (e.g., minimum-snap), collision-free trajectories for point robots amidst spherical obstacles. The authors formulate the task as a non-convex optimization over polynomial trajectories and present, for the first time, a theoretical analysis of its semidefinite programming (SDP) relaxation. Key contributions include establishing necessary and sufficient conditions for relaxation tightness, revealing the equivalence between relaxed solutions and globally optimal trajectories in an augmented space, and leveraging symmetry to reduce the SDP dimensionality so that it scales linearly with the polynomial degree—irrespective of the ambient environment dimension. Integrated into an RRT framework as a convex steering function, the method achieves 10–100× speedups over SNOPT/IPOPT in quadrotor C⁴-continuous minimum-snap planning, significantly reduces solution time variance, and reliably yields high-quality locally optimal trajectories.

Abstract

We study semidefinite relaxations for collision-free motion planning. We focus on a point robot moving from start to goal through spherical obstacles in $\mathbb{R}^n$, subject to path continuity constraints and squared derivative costs; a setting that is conceptually simple yet captures the hardness of collision-free motion planning. We formulate this problem exactly as a nonconvex problem over polynomial curves, and present a natural semidefinite relaxation. We contribute two key theoretical insights; to our knowledge this is the first theoretical analysis of semidefinite relaxations for collision-free motion planning. First, we show that solving the convex relaxation is equivalent to solving, to global optimality, a related motion planning problem in a potentially higher-dimensional space. This geometric interpretation yields necessary and sufficient conditions for tightness, and a clear intuition for when the relaxation is loose. Second, we show that the relaxation admits a symmetry reduction that makes it significantly smaller than one might expect, with positive semidefinite cone sizes that scale linearly with the polynomial degree and are independent of the ambient dimension. The resulting relaxation is 10 to 100 times faster than direct nonlinear programming transcriptions solved with SNOPT and IPOPT, exhibits significantly lower variance in solve times, and reliably finds a locally optimal path for the original problem. We demonstrate its effectiveness as a convex steering function in an RRT planner for minimum-snap quadrotor planning with $C^4$ continuous trajectories.

Problem

• collision-free motion planning
• semidefinite relaxation
• polynomial curves
• path continuity
• squared derivative costs

Innovation

• semidefinite relaxation
• collision-free motion planning
• symmetry reduction
• polynomial trajectories
• convex steering function