论文标题
自适应Bézier学位降低和计算有效的运动计划
Adaptive Bézier Degree Reduction and Splitting for Computationally Efficient Motion Planning
论文作者
论文摘要
作为一个参数多项式曲线家族，Bézier曲线被广泛用于智能机器人系统的安全，平稳的运动设计，从飞行无人机到自动驾驶汽车再到机器人操纵器。在这样的运动计划设置中，高阶Bézier曲线的关键特征，例如曲线长度，距离碰撞，最大曲率/速度/加速度要么以高计算成本进行数值计算，要么通过离散样品不可到而是不可到的。为了解决这些问题，在本文中，我们提出了一种新颖的计算有效方法，可通过多个低阶Bézier节段适应高阶Bézier曲线，该片段以任何所需的准确性水平，根据bézierMier指定。因此，我们引入了一种新的Bézier学位降低方法，称为参数匹配降低，该方法与标准最小二乘和泰勒还原方法相比，该方法更准确地近似Bézier曲线。我们还提出了一个新的Bézier度量，称为最大控制点距离，可以通过分析进行计算，与其他现有的bézier指标具有很强的对等关系，并定义了Bézier曲线之间的几何相对界面。我们提供了广泛的数值证据，以证明我们提出的Bézier近似方法的有效性。根据经验，根据学位的匹配降低错误，我们得出结论，可以将$ n^\ text {th} $ - 订购bézier曲线准确地近似于$ 3（n-1）$ quadratic $ 6（n-1）$ 6（n-1）$linearearearearearearearbézierSgments，这对于BézierIdiveratival是基础。
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 $n^\text{th}$-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.
