Efficient Computation of Distance Functions for Navigation Vector Fields in Lie Groups

2026-06-03

AI Summary

This work addresses the computational burden of distance evaluation and nearest-point projection in vector-field-based path-following control on Lie groups such as SE(3), where conventional iterative optimization methods are too costly for high-frequency embedded control. Focusing on G-polynomial curves, the paper exploits their algebraic structure to reformulate the distance computation as a small set of low-degree polynomial root-finding problems, thereby eliminating iterative procedures and substantially reducing computational complexity. Analytical formulas are derived specifically for SE(3), accompanied by an open-source, efficient implementation. Simulations and robotic arm experiments demonstrate that the proposed method achieves significant speedups while preserving accuracy, making it well-suited for real-time motion control applications.

Abstract

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.

Problem

• distance computation
• vector fields
• Lie groups
• robot control
• G-polynomial curves

Innovation

• Lie Groups
• G-polynomial curves
• distance computation
• vector fields
• robot control