Adaptive Bézier Degree Reduction and Splitting for Computationally Efficient Motion Planning

Abstract

As a parametric polynomial curve family, Bézier 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 Bézier 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 article we present a novel computationally efficient approach for adaptive approximation of high-order Bézier curves by multiple low-order Bézier segments at any desired level of accuracy that is specified in terms of a Bézier metric. Accordingly, we introduce a new Bézier degree reduction method, called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">parameterwise matching reduction</i> , which approximates Bézier curves more accurately compared to the standard least squares and Taylor reduction methods. We also propose a new Bézier metric, called the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">maximum control-point distance</i> , that can be computed analytically, has a strong equivalence relation with other existing Bézier metrics, and defines a geometric relative bound between Bézier curves. We provide extensive numerical evidence to demonstrate the effectiveness of our proposed Bézier approximation approach. As a rule of thumb, based on the degree-one matching reduction error, we conclude that an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ n^{\rm{th}} $</tex-math></inline-formula> -order Bézier curve can be accurately approximated by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$3(n-1)$</tex-math></inline-formula> quadratic and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$6(n-1)$</tex-math></inline-formula> linear Bézier segments, which is fundamental for Bézier discretization.