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Formation Maneuver Control Based on the Augmented Laplacian Method

Xinzhe Zhou, Xuyang Wang, Xiaoming Duan, Yuzhu Bai, Jianping He

TL;DR

The paper addresses the problem of formation maneuver control in 2-D and 3-D spaces, enabling simultaneous translation, scaling, and rotation with arbitrary orientation while preserving the formation's intrinsic configuration. It introduces an augmented Laplacian with matrix-valued edge weights that commute with rotation, along with rotation-axis adjustment and dynamic agent reconfiguration to achieve flexible, scalable maneuvers. Key contributions include a constructive weight design $w_{ij} = a_{ij} I_d + b_{ij} \zeta\zeta^T + c_{ij} \zeta^{\times}$, proofs that target formations remain in $\ker(W)$ during maneuvers, and leader/follower control laws with Lyapunov guarantees; plus a demonstrated link to the 2-D complex Laplacian as a special case. Practical impact lies in reduced neighbor requirements, applicability to arbitrary orientation rotations, and seamless joining of new agents, with validations via 2-D and 3-D simulations.

Abstract

This paper proposes a novel formation maneuver control method for both 2-D and 3-D space, which enables the formation to translate, scale, and rotate with arbitrary orientation. The core innovation is the novel design of weights in the proposed augmented Laplacian matrix. Instead of using scalars, we represent weights as matrices, which are designed based on a specified rotation axis and allow the formation to perform rotation in 3-D space. To further improve the flexibility and scalability of the formation, the rotational axis adjustment approach and dynamic agent reconfiguration method are developed, allowing formations to rotate around arbitrary axes in 3-D space and new agents to join the formation. Theoretical analysis is provided to show that the proposed approach preserves the original configuration of the formation. The proposed method maintains the advantages of the complex Laplacian-based method, including reduced neighbor requirements and no reliance on generic or convex nominal configurations, while achieving arbitrary orientation rotations via a more simplified implementation. Simulations in both 2-D and 3-D space validate the effectiveness of the proposed method.

Formation Maneuver Control Based on the Augmented Laplacian Method

TL;DR

The paper addresses the problem of formation maneuver control in 2-D and 3-D spaces, enabling simultaneous translation, scaling, and rotation with arbitrary orientation while preserving the formation's intrinsic configuration. It introduces an augmented Laplacian with matrix-valued edge weights that commute with rotation, along with rotation-axis adjustment and dynamic agent reconfiguration to achieve flexible, scalable maneuvers. Key contributions include a constructive weight design , proofs that target formations remain in during maneuvers, and leader/follower control laws with Lyapunov guarantees; plus a demonstrated link to the 2-D complex Laplacian as a special case. Practical impact lies in reduced neighbor requirements, applicability to arbitrary orientation rotations, and seamless joining of new agents, with validations via 2-D and 3-D simulations.

Abstract

This paper proposes a novel formation maneuver control method for both 2-D and 3-D space, which enables the formation to translate, scale, and rotate with arbitrary orientation. The core innovation is the novel design of weights in the proposed augmented Laplacian matrix. Instead of using scalars, we represent weights as matrices, which are designed based on a specified rotation axis and allow the formation to perform rotation in 3-D space. To further improve the flexibility and scalability of the formation, the rotational axis adjustment approach and dynamic agent reconfiguration method are developed, allowing formations to rotate around arbitrary axes in 3-D space and new agents to join the formation. Theoretical analysis is provided to show that the proposed approach preserves the original configuration of the formation. The proposed method maintains the advantages of the complex Laplacian-based method, including reduced neighbor requirements and no reliance on generic or convex nominal configurations, while achieving arbitrary orientation rotations via a more simplified implementation. Simulations in both 2-D and 3-D space validate the effectiveness of the proposed method.
Paper Structure (15 sections, 6 theorems, 45 equations, 4 figures)

This paper contains 15 sections, 6 theorems, 45 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Notations
  4. Invariant Constraints
  5. Target Formation and Objectives
  6. Methodology
  7. Augmented Laplacian Matrix
  8. Control Protocols
  9. Rotation Axis Adjustment
  10. Dynamic Agent Reconfiguration
  11. Relationships with 2-D Complex Laplacian Method
  12. Simulation
  13. Analysis of 2-D Simulation Results
  14. Analysis of 3-D Simulation Results
  15. Conclusions

Key Result

Theorem 1

Given rotation axis $\zeta$, the weight $w$ defined in form of wij commutes with the rotation matrix $R_\zeta$, i.e.,

Figures (4)

  • Figure 1: Trajectories in 2-D space (The dark areas stand for obstacles.)
  • Figure 2: Tracking errors in 2-D space.
  • Figure 3: Trajectories in 3-D space (The dark areas stand for obstacles.)
  • Figure 4: Tracking errors in 3-D space.

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 1: Lemma 3 b17
  • Lemma 2
  • proof
  • ...and 4 more