Online Trajectory Optimization for Arbitrary-Shaped Mobile Robots via Polynomial Separating Hypersurfaces

2026-01-14

AI Summary

This work addresses the limitations of traditional trajectory optimization methods that rely on linear separating hyperplanes and convex approximations, which often yield overly conservative solutions in narrow or non-convex environments. The authors propose a novel approach that introduces polynomially parameterized nonlinear separating hypersurfaces, jointly optimizing both the robot trajectory and the hypersurface coefficients to enable collision-free planning around arbitrarily shaped non-convex obstacles. They theoretically prove that any two disjoint bounded closed sets can be separated by a polynomial hypersurface, thereby extending the classical separating hyperplane theorem to the nonlinear setting for the first time. Implemented within a standard nonlinear programming (NLP) framework, the method efficiently computes smooth, agile, and safe trajectories, demonstrating significant performance improvements over existing convex-approximation baselines in both simulation and real-world robotic experiments.

Technology Category

• Search and Optimization: Non-convex Optimization
• Intelligent Robots: Motion and Path Planning
• Planning, Routing, and Scheduling: Replanning and Plan Repair

Application Category

• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

An emerging class of trajectory optimization methods enforces collision avoidance by jointly optimizing the robot's configuration and a separating hyperplane. However, as linear separators only apply to convex sets, these methods require convex approximations of both the robot and obstacles, which becomes an overly conservative assumption in cluttered and narrow environments. In this work, we unequivocally remove this limitation by introducing nonlinear separating hypersurfaces parameterized by polynomial functions. We first generalize the classical separating hyperplane theorem and prove that any two disjoint bounded closed sets in Euclidean space can be separated by a polynomial hypersurface, serving as the theoretical foundation for nonlinear separation of arbitrary geometries. Building on this result, we formulate a nonlinear programming (NLP) problem that jointly optimizes the robot's trajectory and the coefficients of the separating polynomials, enabling geometry-aware collision avoidance without conservative convex simplifications. The optimization remains efficiently solvable using standard NLP solvers. Simulation and real-world experiments with nonconvex robots demonstrate that our method achieves smooth, collision-free, and agile maneuvers in environments where convex-approximation baselines fail.

Problem

• trajectory optimization
• collision avoidance
• nonconvex robots
• separating hypersurfaces
• convex approximation

Innovation

• polynomial separating hypersurfaces
• nonconvex collision avoidance
• trajectory optimization
• nonlinear programming
• geometry-aware planning