Adaptive B\'ezier Degree Reduction and Splitting for Computationally Efficient Motion Planning

TL;DR

This paper introduces an efficient adaptive method for approximating high-order Bézier curves with multiple low-order segments, improving motion planning in robotics by reducing computational costs and increasing accuracy.

Contribution

It presents a novel parameterwise matching reduction method and a new Bézier metric for better curve approximation and analysis in motion planning.

Findings

The method achieves more accurate Bézier curve approximations than standard techniques.

An $n$-th order Bézier curve can be approximated by 3(n-1) quadratic segments.

The approach significantly reduces computational costs in robotic motion planning.

Abstract

As a parametric polynomial curve family, B\'ezier curves are widely used in safe and smooth motion design of intelligent robotic systems from flying drones to autonomous vehicles to robotic manipulators. In such motion planning settings, the critical features of high-order B\'ezier curves such as curve length, distance-to-collision, maximum curvature/velocity/acceleration are either numerically computed at a high computational cost or inexactly approximated by discrete samples. To address these issues, in this paper we present a novel computationally efficient approach for adaptive approximation of high-order B\'ezier curves by multiple low-order B\'ezier segments at any desired level of accuracy that is specified in terms of a B\'ezier metric. Accordingly, we introduce a new B\'ezier degree reduction method, called parameterwise matching reduction, that approximates B\'ezier curves more accurately compared to the standard least squares and Taylor reduction methods. We also propose a new B\'ezier metric, called the maximum control-point distance, that can be computed analytically, has a strong equivalence relation with other existing B\'ezier metrics, and defines a geometric relative bound between B\'ezier curves. We provide extensive numerical evidence to demonstrate the effectiveness of our proposed B\'ezier approximation approach. As a rule of thumb, based on the degree-one matching reduction error, we conclude that an $n^\text{th}$-order B\'ezier curve can be accurately approximated by $3(n-1)$ quadratic and $6(n-1)$ linear B\'ezier segments, which is fundamental for B\'ezier discretization.