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

Felipe Bartelt, Vinicius M. Gonçalves, Luciano C. A. Pimenta

TL;DR

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, and providing an efficient algorithm to compute the vector field.

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.

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

TL;DR

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, and providing an efficient algorithm to compute the vector field.

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.
Paper Structure (21 sections, 13 theorems, 43 equations, 6 figures, 1 table)

This paper contains 21 sections, 13 theorems, 43 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mathematical notation
  4. Vector field review in Euclidean space
  5. Lie groups and Lie algebras
  6. Considered control system
  7. Vector Field in Matrix Lie Groups
  8. Normal component
  9. Tangent component
  10. Orthogonality of components
  11. Local minima and gradients in distance function
  12. Convergence results
  13. An explicit construction for the EE-distance
  14. Particular case of Euclidean space
  15. Experimental Validation
  16. ...and 6 more sections

Key Result

Lemma 2.1

Let $\mathbf{G}:\mathbb{R}\to G$ be a differentiable function. Then, there exists a function $\mathbf{A}:\mathbb{R} \to \mathfrak{g}$ such that

Figures (6)

  • Figure 1: One application of our strategy is the control of omnidirectional drones, like (left) a Voliro drone (image taken from https://voliro.com) and (right) the design from HamandiOmni. Since they accept arbitrary $6$ DoF twists, our methodology can be used to control them to track arbitrary differentiable paths in $\text{SE}(3)$.
  • Figure 2: Example showing the vector field and the components for a point $\mathbf{h}\in \mathbb{R}^m$ and curve $\mathcal{C}$.
  • Figure 3: Depiction of the proof in \ref{['propos:D-NO-local-minima']}.
  • Figure 4: Depiction of Statement (ii) in the proof of \ref{['thm:convergence-vector-field']}.
  • Figure 5: The solid cyan line depicts the target curve, with target orientation frames shown by RGB axes. Intermediary configurations are shown as translucent representations, while the final configuration for each time period is opaque. The end effector orientation frame is shown by RGB axes.
  • ...and 1 more figures

Theorems & Definitions (41)

  • Lemma 2.1
  • proof
  • Definition 2.2: $\!\;\mathcal{S} \!\left(\right)$ map
  • Lemma 2.3
  • proof
  • Definition 2.4: Inverse $\!\;\mathcal{S} \!\left(\right)$ map
  • Definition 2.5: $\Xi$ operator
  • Definition 2.6: $\mathop{\mathrm{L}}\nolimits$ operator
  • Definition 2.7: Differentiable function acting on a Lie Group
  • Lemma 2.8
  • ...and 31 more