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Decentralized Swarm Control via SO(3) Embeddings for 3D Trajectories

Dimitria Silveria, Kleber Cabral, Peter Jardine, Sidney Givigi

TL;DR

The paper tackles generating stable, closed 3D swarm trajectories for UAVs using only local position data. It introduces a Lie-group embedding on $SO(3)$ to deform a circular reference into diverse 3D shapes and employs a two-tier control scheme: a phase controller to enforce uniform angular spacing and a PD-based position controller to track the embedding-derived targets. A Lyapunov-based analysis proves stability of the embedding, and a one-parameter homeomorphism argument shows that stability translates to the actual 3D trajectories. The method scales to large swarms and is validated through simulations with up to 50 agents and physical experiments with five Crazyflie drones, demonstrating robustness to disturbances and model mismatch while remaining independent of the underlying low-level controller. This approach reduces the need for velocity inputs, enabling broader applicability to resource-constrained platforms and facilitating decentralized collision avoidance and formation control in 3D space.

Abstract

This paper presents a novel decentralized approach for achieving emergent behavior in multi-agent systems with minimal information sharing. Based on prior work in simple orbits, our method produces a broad class of stable, periodic trajectories by stabilizing the system around a Lie group-based geometric embedding. Employing the Lie group SO(3), we generate a wider range of periodic curves than existing quaternion-based methods. Furthermore, we exploit SO(3) properties to eliminate the need for velocity inputs, allowing agents to receive only position inputs. We also propose a novel phase controller that ensures uniform agent separation, along with a formal stability proof. Validation through simulations and experiments showcases the method's adaptability to complex low-level dynamics and disturbances.

Decentralized Swarm Control via SO(3) Embeddings for 3D Trajectories

TL;DR

The paper tackles generating stable, closed 3D swarm trajectories for UAVs using only local position data. It introduces a Lie-group embedding on to deform a circular reference into diverse 3D shapes and employs a two-tier control scheme: a phase controller to enforce uniform angular spacing and a PD-based position controller to track the embedding-derived targets. A Lyapunov-based analysis proves stability of the embedding, and a one-parameter homeomorphism argument shows that stability translates to the actual 3D trajectories. The method scales to large swarms and is validated through simulations with up to 50 agents and physical experiments with five Crazyflie drones, demonstrating robustness to disturbances and model mismatch while remaining independent of the underlying low-level controller. This approach reduces the need for velocity inputs, enabling broader applicability to resource-constrained platforms and facilitating decentralized collision avoidance and formation control in 3D space.

Abstract

This paper presents a novel decentralized approach for achieving emergent behavior in multi-agent systems with minimal information sharing. Based on prior work in simple orbits, our method produces a broad class of stable, periodic trajectories by stabilizing the system around a Lie group-based geometric embedding. Employing the Lie group SO(3), we generate a wider range of periodic curves than existing quaternion-based methods. Furthermore, we exploit SO(3) properties to eliminate the need for velocity inputs, allowing agents to receive only position inputs. We also propose a novel phase controller that ensures uniform agent separation, along with a formal stability proof. Validation through simulations and experiments showcases the method's adaptability to complex low-level dynamics and disturbances.

Paper Structure

This paper contains 16 sections, 2 theorems, 24 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Controller Design
  4. Lie Group Embedding
  5. Controller Overview
  6. Phase Control
  7. Position Control
  8. Stability Analysis
  9. Stability of the Embedding
  10. Stability of the Trajectory
  11. Experimental Methodology
  12. Results
  13. Simulation with 50 Agents
  14. Simulation Inserting One Agent
  15. Physical Experiment
  16. ...and 1 more sections

Key Result

Theorem 1

Let $n\geq3$ be the number of agents in a swarm that rotate in a common plane around a fixed point. If each agent's angular velocity $\omega_{z,d,i}$ is governed by the control law in eq:phase-control, then the system is Lyapunov stable, and all agents converge to a steady-state angular velocity $\o

Figures (13)

  • Figure 1: Examples of 3D geometries generated using the method described in Section \ref{['sec:embedding']} along with $XY$ and $YZ$ projections.
  • Figure 2: Platform used for physical experiments.
  • Figure 3: Experimental setup containing the vehicle (orange box), and the control strategy implemented on each UAV (green boxes).
  • Figure 4: 3D reference and simulated trajectories of Agent 1 following \ref{['eq:angular-speeds']}, with $XY$ and $YZ$ projections (red dotted lines).
  • Figure 5: Desired vs. simulated positions of Agent 1, in $X$,$Y$ and $Z$ axes, as a function of time.
  • ...and 8 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof