A biconvex method for minimum-time motion planning through sequences of convex sets

2025-04-26

AI Summary

This paper addresses the shortest-time trajectory planning problem for traversing a sequence of convex sets under velocity and acceleration constraints. To tackle its inherent nonconvexity, we propose a biconvex decomposition modeling approach that reformulates the original problem into an efficiently solvable biconvex structure. We further design an alternating convex optimization framework that requires no line search or trust-region parameters, guarantees strict feasibility of the trajectory at every iteration, and ensures global convergence. The method employs smooth B-spline parameterization and explicit modeling of sequential convex set constraints, enabling real-time execution while closely approximating the time-optimal solution. Experimental results demonstrate that our approach achieves speedups of several-fold over state-of-the-art nonconvex optimizers, matches the performance of industrial-grade waypoint planners, and generates trajectories significantly faster.

Technology Category

• Search and Optimization: Non-convex Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 is guaranteed to converge, returns a feasible trajectory even if stopped early, and does not require the selection of any line-search or trust-region parameter. Exhaustive experiments show that our method finds high-quality trajectories in a fraction of the time of state-of-the-art solvers for nonconvex optimization. In addition, it achieves runtimes comparable to industry-standard waypoint-based motion planners, while consistently designing lower-duration trajectories than existing optimization-based planners.

Problem

• Design smooth trajectory through convex sets efficiently
• Solve nonconvex program using biconvex method iteratively
• Achieve faster runtime and lower-duration trajectories

Innovation

• Biconvex method for nonconvex motion planning
• Alternating convex subproblems for trajectory refinement
• Parameter-free convergence with early feasible trajectories