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On the Stability of Consensus Control under Rotational Ambiguities

Zhonggang Li, Changheng Li, Raj Thilak Rajan

TL;DR

This letter conducts a thorough analysis of the stability of consensus control in the presence of localization-induced rotational ambiguities, in several scenarios including, e.g., proper and improper rotation, and the homogeneity of rotations.

Abstract

Consensus control of multiagent systems arises in various robotic applications such as rendezvous and formation control. For example, to compute the control inputs of individual agents, the difference in the positions in aligned coordinate frames i.e., the pairwise displacements are typically measured. However, the local coordinate frames might be subject to rotational ambiguities, such as a rotation or a reflection, particularly if the positions of the agent are not directly observed but reconstructed from e.g. pairwise Euclidean distances. This rotational ambiguity causes stability issues in practice, as agents have rotated perceptions of the environment. In this work, we conduct a thorough analysis of the stability in the presence of rotational ambiguities in several scenarios including e.g., proper and improper rotation, and the homogeneity of rotations. We give stability criteria and stability margin on the rotations, which are numerically verified with two traditional examples of consensus control.

On the Stability of Consensus Control under Rotational Ambiguities

TL;DR

This letter conducts a thorough analysis of the stability of consensus control in the presence of localization-induced rotational ambiguities, in several scenarios including, e.g., proper and improper rotation, and the homogeneity of rotations.

Abstract

Consensus control of multiagent systems arises in various robotic applications such as rendezvous and formation control. For example, to compute the control inputs of individual agents, the difference in the positions in aligned coordinate frames i.e., the pairwise displacements are typically measured. However, the local coordinate frames might be subject to rotational ambiguities, such as a rotation or a reflection, particularly if the positions of the agent are not directly observed but reconstructed from e.g. pairwise Euclidean distances. This rotational ambiguity causes stability issues in practice, as agents have rotated perceptions of the environment. In this work, we conduct a thorough analysis of the stability in the presence of rotational ambiguities in several scenarios including e.g., proper and improper rotation, and the homogeneity of rotations. We give stability criteria and stability margin on the rotations, which are numerically verified with two traditional examples of consensus control.
Paper Structure (14 sections, 10 theorems, 13 equations, 5 figures, 1 table)

This paper contains 14 sections, 10 theorems, 13 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Fundamentals and Problem Formulation
  3. Consensus Systems
  4. Problem Formulation
  5. Stability under ambiguities
  6. General conditions for Stability
  7. Stability under Homogeneous Ambiguities
  8. Stability under Heterogeneous Ambiguities
  9. Examples and Simulations
  10. Case 1: Rendezvous Control
  11. Case 2: Distributed Formation Control
  12. Discussion
  13. Conclusion
  14. Alternative Proof for Stability under Homogeneous Rotations

Key Result

Lemma 1

(Eigenvalue distributions of a product of matrices) Given a product of two square matrices $\bm{A} = \bm{G}\bm{Q}$ with $\bm{Q}$ being symmetric positive-definite, $\gamma(\bm{A})\prec 0$ if and only if $\gamma(\bm{G})\prec 0$.

Figures (5)

  • Figure 1: A numerical example of the eigenvalues under homogeneous and proper rotations.
  • Figure 2: A numerical example of the eigenvalues under homogeneous and improper rotations.
  • Figure 3: A numerical example of the eigenvalues under heterogeneous rotations.
  • Figure 4: The graphs for (a) rendezvous control and (b) distributed formation control, where the orange nodes are leaders.
  • Figure 5: The convergence in error $\delta(t)$ across time $t$ for the rendezvous control (top) and the affine formation control algorithm (bottom) under homogeneous ambiguities (left) and heterogeneous (right).

Theorems & Definitions (18)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • ...and 8 more