2026-06-05原文2 min readUpdated: 2026-06-05

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

This paper proposes an efficient method for computing distances between points and curves on Lie groups, using G-polynomial curves to reduce the problem to polynomial root finding. It significantly cuts computation time while maintaining accuracy, with practical formulas for SE(3) and experimental validation on a robotic manipulator. The code is publicly available.

SourcearXiv RoboticsAuthor: Vinicius M. Gon\c{c}alves, Jo\~ao Bai\~ao, Felipe Bartelt, Douglas G. Macharet, Gustavo M. Freitas, H\'ector Azp\'urua, Luciano C. A. Pimenta

[2606.05372] Efficient Computation of Distance Functions for Navigation Vector Fields in Lie Groups

[Submitted on 3 Jun 2026]

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

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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.

Subjects:

Robotics (cs.RO); Computational Geometry (cs.CG)

Cite as: arXiv:2606.05372 [cs.RO]

(or arXiv:2606.05372v1 [cs.RO] for this version)

https://doi.org/10.48550/arXiv.2606.05372

arXiv-issued DOI via DataCite (pending registration)

Submission history

From: Felipe Bartelt [view email] [v1] Wed, 3 Jun 2026 19:18:59 UTC (5,056 KB)

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