Constructive Vector Fields for Path Following in Fully-Actuated Systems on Matrix Lie Groups

2026-02-24

AI Summary

This work addresses the problem of path-following and convergence control for fully actuated systems evolving on matrix Lie groups, such as SE(3). A novel vector field approach is proposed that extends Euclidean path-following strategies to general connected matrix Lie groups by leveraging their intrinsic geometric structure. The method constructs a control vector field comprising a component orthogonal to the desired path—ensuring convergence—and a tangential component aligned with the path direction, while employing non-redundant control inputs that exactly match the system’s degrees of freedom. In particular, a computationally efficient vector field algorithm tailored for SE(3) is developed. Experimental validation on a robotic manipulator platform demonstrates high-precision path tracking, confirming the method’s effectiveness for systems with coupled translational and rotational dynamics, such as omnidirectional drones.

Technology Category

• Intelligent Robots: Motion and Path Planning
• Machine Learning: Matrix & Tensor Methods
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

This paper presents a novel vector field strategy for controlling fully-actuated systems on connected matrix Lie groups, ensuring convergence to and traversal along a curve defined on the group. Our approach generalizes our previous work (Rezende et al., 2022) and reduces to it when considering the Lie group of translations in Euclidean space. Since the proofs in Rezende et al. (2022) rely on key properties such as the orthogonality between the convergent and traversal components, we extend these results by leveraging Lie group properties. These properties also allow the control input to be non-redundant, meaning it matches the dimension of the Lie group, rather than the potentially larger dimension of the space in which the group is embedded. This can lead to more practical control inputs in certain scenarios. A particularly notable application of our strategy is in controlling systems on SE(3) -- in this case, the non-redundant input corresponds to the object's mechanical twist -- making it well-suited for controlling objects that can move and rotate freely, such as omnidirectional drones. In this case, we provide an efficient algorithm to compute the vector field. We experimentally validate the proposed method using a robotic manipulator to demonstrate its effectiveness.

Problem

• vector fields
• path following
• matrix Lie groups
• fully-actuated systems
• SE(3)

Innovation

• vector fields
• matrix Lie groups
• non-redundant control
• path following
• SE(3)