提出高效计算机器人轨迹距离的新方法，加速嵌入式系统路径追踪
Efficient Computation of Distance Functions for Navigation Vector Fields in Lie Groups
- 利用G-多项式曲线结构，将距离计算转化为少量多项式求根
- 相比传统优化方法，计算速度显著提升且保持高精度
- 适用于机械臂等实际机器人系统，已开源实现
基于向量场的控制方法广泛用于机器人路径跟踪。某些方法需反复计算机器人位姿与路径曲线间的距离及最近点。近期该方法被推广至李群空间，但此类计算在嵌入式平台高频控制下代价高昂。本文提出一种针对G-多项式曲线（矩阵李群上的多项式曲线推广）的高效距离计算方法，利用曲线结构将问题简化为少量多项式求根。仿真表明，该方法显著降低计算时间并保持精度。文中还给出了SE(3)情形下的实用公式，并在机械臂上实验验证。方法已实现于在线计算包中。
原文摘要 · Abstract (English)
Vector-field-based methods are widely used for robot control and are often applied to the path-tracking problem. Some vector field approaches require repeatedly computing the distance between the robot configuration and the curve, as well as the corresponding closest point. Recently, vector fields have been extended to Lie Groups. In this case, this computation can be expensive, especially when performed at high control frequencies on embedded platforms. This paper proposes a method for efficiently computing the distance between a point and a curve represented as what is called a G-polynomial curve, which is a curve representation that generalizes polynomial curves to matrix Lie groups. The proposed approach exploits the structure of these curves to reduce the problem to a small number of polynomial root-finding computations. Simulation results show that the method significantly reduces computation time while maintaining accuracy compared to existing optimization-based approaches. Practical formulas are also provided for the case of the group SE(3), and the method is validated experimentally on a robotic manipulator. The methodology is implemented in a computational package, available online.
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