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Meanshift Shape Formation Control Using Discrete Mass Distribution

Yichen Cai, Yuan Gao, Pengpeng Li, Wei Wang, Guibin Sun, Jinhu Lü

TL;DR

This paper tackles shape formation for large robot swarms under varying swarm sizes by replacing continuous density representations with a discrete mass-distribution over sample points, enabling decentralized control and scalability.A decentralized meanshift controller, fed by mass estimates, drives robots to align the swarm's mass distribution with the sample-point distribution, while a collision-avoidance term maintains safe inter-robot distances.A decentralized mass estimator ensures every robot converges to the true sample-point masses, and the overall system is analyzed with LaSalle's invariance principle, showing convergence to the desired convex shape under reasonable assumptions.Extensive simulations and real-world TurtleBot experiments validate the method's ability to form complex, non-convex shapes and adapt to additions or removals of robots, indicating strong practical relevance for scalable swarm-formation tasks.

Abstract

The density-distribution method has recently become a promising paradigm owing to its adaptability to variations in swarm size. However, existing studies face practical challenges in achieving complex shape representation and decentralized implementation. This motivates us to develop a fully decentralized, distribution-based control strategy with the dual capability of forming complex shapes and adapting to swarm-size variations. Specifically, we first propose a discrete mass-distribution function defined over a set of sample points to model swarm formation. In contrast to the continuous density-distribution method, our model eliminates the requirement for defining continuous density functions-a task that is difficult for complex shapes. Second, we design a decentralized meanshift control law to coordinate the swarm's global distribution to fit the sample-point distribution by feeding back mass estimates. The mass estimates for all sample points are achieved by the robots in a decentralized manner via the designed mass estimator. It is shown that the mass estimates of the sample points can asymptotically converge to the true global values. To validate the proposed strategy, we conduct comprehensive simulations and real-world experiments to evaluate the efficiency of complex shape formation and adaptability to swarm-size variations.

Meanshift Shape Formation Control Using Discrete Mass Distribution

TL;DR

This paper tackles shape formation for large robot swarms under varying swarm sizes by replacing continuous density representations with a discrete mass-distribution over sample points, enabling decentralized control and scalability.A decentralized meanshift controller, fed by mass estimates, drives robots to align the swarm's mass distribution with the sample-point distribution, while a collision-avoidance term maintains safe inter-robot distances.A decentralized mass estimator ensures every robot converges to the true sample-point masses, and the overall system is analyzed with LaSalle's invariance principle, showing convergence to the desired convex shape under reasonable assumptions.Extensive simulations and real-world TurtleBot experiments validate the method's ability to form complex, non-convex shapes and adapt to additions or removals of robots, indicating strong practical relevance for scalable swarm-formation tasks.

Abstract

The density-distribution method has recently become a promising paradigm owing to its adaptability to variations in swarm size. However, existing studies face practical challenges in achieving complex shape representation and decentralized implementation. This motivates us to develop a fully decentralized, distribution-based control strategy with the dual capability of forming complex shapes and adapting to swarm-size variations. Specifically, we first propose a discrete mass-distribution function defined over a set of sample points to model swarm formation. In contrast to the continuous density-distribution method, our model eliminates the requirement for defining continuous density functions-a task that is difficult for complex shapes. Second, we design a decentralized meanshift control law to coordinate the swarm's global distribution to fit the sample-point distribution by feeding back mass estimates. The mass estimates for all sample points are achieved by the robots in a decentralized manner via the designed mass estimator. It is shown that the mass estimates of the sample points can asymptotically converge to the true global values. To validate the proposed strategy, we conduct comprehensive simulations and real-world experiments to evaluate the efficiency of complex shape formation and adaptability to swarm-size variations.
Paper Structure (23 sections, 14 theorems, 39 equations, 8 figures, 1 table, 1 algorithm)

This paper contains 23 sections, 14 theorems, 39 equations, 8 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation and Overview
  3. Robot Model
  4. System Overview
  5. Distribution Description of Shape Formation
  6. Mass-Distribution Based Formation Definition
  7. Distribution-Similarity Error Metric of Shape Formation
  8. Position and Orientation of Formation Shape
  9. Meanshift Strategy for Shape Formation
  10. Decentralized Mass Estimation
  11. Meanshift Controller via Mass Feedback
  12. Meanshift Velocity Command
  13. Collision-Avoidance Velocity Command
  14. Convergence Analysis
  15. Performance Evaluation
  16. ...and 8 more sections

Key Result

Lemma 1

The metric $F_{\mathrm{uni}}( \mathbf{p} ) \geq 0$ for all $\mathbf{p} \in \mathbb{R}^{dn}$, and $F_{\mathrm{uni}}(\mathbf{p}) = 0$ if and only if $P_k(\mathbf{p}) = P_l(\mathbf{p})$ for all $k,l \in \{1,\ldots,m\}$.

Figures (8)

  • Figure 1: Overview of the system. (a) Illustration of the proposed model and control objective. (b) Intuitive explanation of the shape interpretation negotiation process. (c) Intuitive explanation of the decentralized mass estimation. (d) Intuitive explanation of the meanshift control strategy.
  • Figure 2: Intuitive explanation of the control objectives. (a) Desired shape characterized by sample points. (b) The intuitive explanation of the mass maximization objective. (c) The intuitive explanation of the mass uniformity objective.
  • Figure 3: Intuitive explanation of the metric $F$. (a) 4 cases of different swarm configurations with respect to the arrow shape. (b) The values of $F, F_{\mathrm{max}}$ and $F_{\mathrm{uni}}$ corresponding to cases shown in (a).
  • Figure 4: Simulations of our proposed strategy for different shapes and swarm sizes. (a) Case 1: Trajectories and metrics of triangle, pentagon, and hexagon shapes formed by 20 robots. (b) Case 2: Trajectories and metrics of airplane, arrow, and jigsaw shapes formed by 50 robots. (c) Case 3: Trajectories and metrics of letter-A, pictograph, and fish-bone shapes formed by 100 robots.
  • Figure 5: Comparative results of our strategy and the state-of-the-art Liu2024Self, each achieving a bunny-head shape with 50 robots. (a) Snapshots of our proposed strategy. (b) Snapshots of strategy Liu2024Self with 8th-order and 12th-order moment. (c) Statistical results of the performance metrics, averaged over 20 trials of random initialization.
  • ...and 3 more figures

Theorems & Definitions (36)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • ...and 26 more