Adaptive Bézier Degree Reduction and Splitting for Computationally Efficient Motion Planning

As a parametric polynomial curve family, B\ézier 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\ézier 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\ézier curves by multiple low-order B\ézier segments at any desired level of accuracy that is specified in terms of a B\ézier metric. Accordingly, we introduce a new B\ézier degree reduction method, called parameterwise matching reduction, that approximates B\ézier curves more accurately compared to the standard least squares and Taylor reduction methods. We also propose a new B\ézier metric, called the maximum control-point distance, that can be computed analytically, has a strong equivalence relation with other existing B\ézier metrics, and defines a geometric relative bound between B\ézier curves. We provide extensive numerical evidence to demonstrate the effectiveness of our proposed B\ézier approximation approach. As a rule of thumb, based on the degree-one matching reduction error, we conclude that an $n^\text{th}$-order B\ézier curve can be accurately approximated by $3(n-1)$ quadratic and $6(n-1)$ linear B\ézier segments, which is fundamental for B\ézier discretization.