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.

翻译：基于向量场的方法被广泛应用于机器人控制，并常被用于路径跟踪问题。一些向量场方法需要反复计算机器人构型与曲线之间的距离以及相应的最近点。近年来，向量场已被推广至李群。在这种情况下，这种计算可能代价高昂，尤其是在嵌入式平台上以高控制频率执行时。本文提出了一种高效计算点与所谓的G-多项式曲线（一种将多项式曲线推广到矩阵李群的曲线表示）之间距离的方法。所提出的方法利用这些曲线的结构，将问题简化为少量多项式求根计算。仿真结果表明，与现有的基于优化的方法相比，该方法在保持精度的同时显著减少了计算时间。此外，还给出了SE(3)群情况下的实用公式，并在机器人机械臂上进行了实验验证。该方法的实现已封装为计算软件包，可在线获取。