2022/01/19 by Ömür Arslan, Arslan, Ömür, Aron Tiemessen +1
Computer Science · Engineering · Medicine · #3D Shape Modeling and Analysis #65D17 #68T40 #Advanced Numerical Analysis Techniques #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Electrical engineering #I.2.9 #I.3 #J.6 #Robotics (cs.RO) #Spine and Intervertebral Disc Pathology #Systems and Control (eess.SY) #acm:65D17 #acm:68T40 #cs.CG #cs.RO #cs.SY #eess.SY #electronic engineering #information engineering #msc:65D17 #msc:68T40
paper · pdf · doi:10.48550/arxiv.2201.07834
23 pages, 13 figures, submitted to a journal publication
arxiv created 2022/01/19 · openalex publication_date 2022/01/19 · arxiv updated 2022/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 paper 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 parameterwise matching reduction, that approximates Bézier curves more accurately compared to the standard least squares and Taylor reduction methods. We also propose a new Bézier metric, called the maximum control-point distance, 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 nth-order Bézier curve can be accurately approximated by 3(n-1) quadratic and 6(n-1) linear Bézier segments, which is fundamental for Bézier discretization.