Biconvex Optimization for Smooth Minimum-Time Trajectories around Convex Obstacles

2026-08-03

AI Summary

This work addresses the problem of generating minimum-time smooth trajectories subject to high-order derivative constraints in environments with convex obstacles. The authors propose a biconvex optimization framework that jointly convexifies the time-optimal objective and dynamic constraints through a change of variables, while modeling collision avoidance via time-varying separating hyperplanes. This formulation yields an alternatingly optimizable biconvex structure that supports arbitrary-order derivative constraints, permits interruption at any iteration, and guarantees convergence while effectively escaping local minima. Notably, the method requires only a simple collision-free piecewise-linear path for initialization yet reliably converges to high-quality solutions. Evaluated on drone navigation and dual-arm box unloading tasks, the approach achieves trajectory quality and computational efficiency comparable to state-of-the-art decoupled planners, with broader applicability and strong robustness to poor initial guesses.

Abstract

We present a biconvex approach for minimum-time motion planning around convex obstacles that is guaranteed to converge, is anytime, and supports derivative constraints to arbitrary order. We jointly convexify the minimum-time objective and all derivative constraints through a change of variables, and handle collision avoidance via time-varying separating planes, reducing the problem to a biconvex program. This program is solved by alternating between computing maximum-margin separating planes and optimizing the trajectory. By only adding planes for obstacles that the current iterate collides with, the trajectory can jump around obstacles and escape local minima. The method is guaranteed to converge starting from a simple collision-free polygonal curve. In our experiments on drone navigation and dual-arm bin unloading, we find that the proposed method reliably produces high-quality trajectories with computation times comparable to state-of-the-art decomposition-based motion planners, while handling a larger class of problems and being substantially more robust to bad initialization. Project page:https://wernerpe.github.io/bmtp-website/

Problem

• minimum-time trajectory
• convex obstacles
• motion planning
• biconvex optimization
• collision avoidance

Innovation

• biconvex optimization
• minimum-time trajectory
• convex obstacles
• separating planes
• derivative constraints