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Extending the Leader-First Follower Structure for Bearing-only Formation Control on Directed Graphs

Jiacheng Shi, Daniel Zelazo

TL;DR

This work addresses bearing-only formation control on directed graphs by extending the leader-first follower (LFF) paradigm through ordered LFF (OLFF) graphs. It first analyzes a 1-to-many scenario (one follower, many leaders) to derive a unique, globally exponentially stable follower equilibrium $p_n^*=(\sum_{nj} P_{\bm{g}_{nj}})^{-1}\sum_{nj} P_{\bm{g}_{nj}} p_j$ and shows that adding leaders speeds convergence. It then introduces OLFF graphs that permit additional forward edges while preserving a cascade-like dynamic, proving almost global exponential convergence to the realizable target bearing and showing faster convergence with more forward edges. Simulations corroborate the theoretical results, demonstrating improved performance and illustrating the potential to design more flexible directed BOFC networks, with future work addressing robustness and dynamic topologies.

Abstract

This work proposes an extension to the leader-first follower (LFF) class of graphs used to solve the bearing-only formation control problem over directed graphs. The first contribution provides an equilibrium, stability, and convergence analysis for a one-follower, multi-leader system (which is not an LFF graph). We then propose an extension to the LFF structure, termed \emph{ordered} LFF graphs, that allows for additional forward directed edges to be included. Using the results of the one-follower multi-leader system we show that the ordered LFF graphs can be used to solve the directed bearing-only formation control problem. We also show that these structures offer improved convergence speed as compared to the LFF graphs. Numerical simulations are provided to validate the results.

Extending the Leader-First Follower Structure for Bearing-only Formation Control on Directed Graphs

TL;DR

This work addresses bearing-only formation control on directed graphs by extending the leader-first follower (LFF) paradigm through ordered LFF (OLFF) graphs. It first analyzes a 1-to-many scenario (one follower, many leaders) to derive a unique, globally exponentially stable follower equilibrium and shows that adding leaders speeds convergence. It then introduces OLFF graphs that permit additional forward edges while preserving a cascade-like dynamic, proving almost global exponential convergence to the realizable target bearing and showing faster convergence with more forward edges. Simulations corroborate the theoretical results, demonstrating improved performance and illustrating the potential to design more flexible directed BOFC networks, with future work addressing robustness and dynamic topologies.

Abstract

This work proposes an extension to the leader-first follower (LFF) class of graphs used to solve the bearing-only formation control problem over directed graphs. The first contribution provides an equilibrium, stability, and convergence analysis for a one-follower, multi-leader system (which is not an LFF graph). We then propose an extension to the LFF structure, termed \emph{ordered} LFF graphs, that allows for additional forward directed edges to be included. Using the results of the one-follower multi-leader system we show that the ordered LFF graphs can be used to solve the directed bearing-only formation control problem. We also show that these structures offer improved convergence speed as compared to the LFF graphs. Numerical simulations are provided to validate the results.
Paper Structure (12 sections, 13 theorems, 62 equations, 11 figures)

This paper contains 12 sections, 13 theorems, 62 equations, 11 figures.

Table of Contents

  1. INTRODUCTION
  2. Bearing-Only Formation Control
  3. BOFC for Undirected Graphs
  4. BOFC for Directed Graphs
  5. Extending the LFF Structure
  6. BOFC with 1 Follower and Many Leaders (1-to-many)
  7. BOFC with Ordered LFF Graphs
  8. Simulation Results
  9. One-to-Many BOFC Example
  10. Performance Improvement of Ordered LFF Graphs
  11. Further Extensions to the LFF Graphs
  12. Conclusion

Key Result

Lemma 1

Assume that $g,\bm{\mathrm g} \in \mathcal{C}_{1}\bigcap \mathcal{C}_{F_B}(\mathcal{G})$. Then

Figures (11)

  • Figure 1: Example demonstrating the challenge of formation control with directed sensing.
  • Figure 2: An example of an LFF graph.
  • Figure 3: The directed graph for the 1-to-many setup.
  • Figure 4: An example of an ordered LFF graph.
  • Figure 5: Examples of two symmetric configurations with respect to the point $\mathrm p_1$.
  • ...and 6 more figures

Theorems & Definitions (30)

  • Definition 1
  • Definition 2
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 20 more