Adaptive Bézier Degree Reduction and Splitting for Computationally Efficient Motion Planning

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 manipulators. In such planning settings, the critical features high-order as length, distance-to-collision, maximum curvature/velocity/acceleration either numerically computed at high computational cost or inexactly approximated by discrete samples. To address these issues, this article we present novel computationally efficient approach for adaptive approximation multiple low-order segments any desired level accuracy that is specified terms metric. Accordingly, introduce new degree reduction method, called parameterwise matching reduction , which approximates more accurately compared standard least squares Taylor methods. We also propose metric, xmlns:xlink="http://www.w3.org/1999/xlink">maximum control-point distance can be analytically, has strong equivalence relation with other existing metrics, defines geometric relative bound between curves. provide extensive numerical evidence demonstrate effectiveness our proposed approach. rule thumb, based on degree-one error, conclude an $ n^{\rm{th}} $ -order notation="LaTeX">$3(n-1)$ quadratic notation="LaTeX">$6(n-1)$ linear segments, fundamental discretization.