Graph-Theoretic B\'ezier Curve Optimization over Safe Corridors for Safe and Smooth Motion Planning

TL;DR

This paper introduces a graph-theoretic framework for optimizing Bézier curves in robotic motion planning, comparing physical, geometric, and statistical objectives over safe corridors to improve smoothness and safety.

Contribution

It presents a unifying graph-theoretic perspective on Bézier curve optimization objectives, including new geometric and statistical consensus distances, and analyzes their effectiveness in safe motion planning.

Findings

Finite difference-based objectives yield simpler interaction graphs.

Variance-based objectives are more intuitive than derivative norms.

All objectives produce similar robot motion profiles.

Abstract

As a parametric motion representation, B\'ezier curves have significant applications in polynomial trajectory optimization for safe and smooth motion planning of various robotic systems, including flying drones, autonomous vehicles, and robotic manipulators. An essential component of B\'ezier curve optimization is the optimization objective, as it significantly influences the resulting robot motion. Standard physical optimization objectives, such as minimizing total velocity, acceleration, jerk, and snap, are known to yield quadratic optimization of B\'ezier curve control points. In this paper, we present a unifying graph-theoretic perspective for defining and understanding B\'ezier curve optimization objectives using a consensus distance of B\'ezier control points derived based on their interaction graph Laplacian. In addition to demonstrating how standard physical optimization objectives define a consensus distance between B\'ezier control points, we also introduce geometric and statistical optimization objectives as alternative consensus distances, constructed using finite differencing and differential variance. To compare these optimization objectives, we apply B\'ezier curve optimization over convex polygonal safe corridors that are automatically constructed around a maximal-clearance minimal-length reference path. We provide an explicit analytical formulation for quadratic optimization of B\'ezier curves using B\'ezier matrix operations. We conclude that the norm and variance of the finite differences of B\'ezier control points lead to simpler and more intuitive interaction graphs and optimization objectives compared to B\'ezier derivative norms, despite having similar robot motion profiles.