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.

翻译：B\'ezier曲线作为参数多面曲线,广泛用于智能机器人系统的安全和平稳运动设计,从飞行无人机到自主飞行器,到机器人操纵器。在这种运动规划环境中,高阶B\\'ezier曲线的关键特征,如曲线长度、距离到球度、最大曲线/速度/加速度等,要么以高计算成本进行数字计算,要么以离散样本为精确度近似值。为了解决这些问题,我们在本文件中提出了一个新的计算高效方法,通过多个低序B\\\'ezier曲线的精度,以任何理想的精度水平调适近B\'ezer曲线。因此,我们采用了一个新的B\'ezier度降幅方法,称为参数匹配降幅,比标准最低平面和降低方法更精确。我们还提出了一个新的B\'ez更精确度指标,称为最大控制距离,可以计算比值最高值的精度,比值的精确度为比值递减幅度,比比比值低值的比值的直值,比值的比值是比值。我们现有的基值递比值递的比值直比值的比值,比值的比值比值的比值为基值。