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A Centralized Planning and Distributed Execution Method for Shape Filling with Homogeneous Mobile Robots

Shuqing Liu, Rong Su, Karl H. Johansson

TL;DR

The paper tackles shape filling with homogeneous robots that have limited localization by introducing a centralized planning and distributed execution framework. It develops the add-subtract algorithm on a hexagonal lattice, using ribbonization to decompose a target shape $S$ with holes $D$ into a ribbon tree rooted at $R_0$, and prescribes additive and subtractive movement sequences that ensure unobstructed paths and localizability without requiring global coordinates. Key contributions include formal problem formulation on $S \setminus D$, a centralized planner for feasibility and sequence generation, a rigorous ribbon-based correctness proof (Part I and Part II), and simulation validation showing scalability and robustness for hole-containing shapes. The approach advances shape formation under limited localization and communication constraints and provides a foundation for future distributed implementations and extensions to 3D shapes.

Abstract

The pattern formation task is commonly seen in a multi-robot system. In this paper, we study the problem of forming complex shapes with functionally limited mobile robots, which have to rely on other robots to precisely locate themselves. The goal is to decide whether a given shape can be filled by a given set of robots; in case the answer is yes, to complete a shape formation process as fast as possible with a minimum amount of communication. Traditional approaches either require global coordinates for each robot or are prone to failure when attempting to form complex shapes beyond the capability of given approaches - the latter calls for a decision procedure that can tell whether a target shape can be formed before the actual shape-forming process starts. In this paper, we develop a method that does not require global coordinate information during the execution process and can effectively decide whether it is feasible to form the desired shape. The latter is achieved via a planning procedure that is capable of handling a variety of complex shapes, in particular, those with holes, and assigning a simple piece of scheduling information to each robot, facilitating subsequent distributed execution, which does not rely on the coordinates of all robots but only those of neighboring ones. The effectiveness of our shape-forming approach is vividly illustrated in several simulation case studies.

A Centralized Planning and Distributed Execution Method for Shape Filling with Homogeneous Mobile Robots

TL;DR

The paper tackles shape filling with homogeneous robots that have limited localization by introducing a centralized planning and distributed execution framework. It develops the add-subtract algorithm on a hexagonal lattice, using ribbonization to decompose a target shape with holes into a ribbon tree rooted at , and prescribes additive and subtractive movement sequences that ensure unobstructed paths and localizability without requiring global coordinates. Key contributions include formal problem formulation on , a centralized planner for feasibility and sequence generation, a rigorous ribbon-based correctness proof (Part I and Part II), and simulation validation showing scalability and robustness for hole-containing shapes. The approach advances shape formation under limited localization and communication constraints and provides a foundation for future distributed implementations and extensions to 3D shapes.

Abstract

The pattern formation task is commonly seen in a multi-robot system. In this paper, we study the problem of forming complex shapes with functionally limited mobile robots, which have to rely on other robots to precisely locate themselves. The goal is to decide whether a given shape can be filled by a given set of robots; in case the answer is yes, to complete a shape formation process as fast as possible with a minimum amount of communication. Traditional approaches either require global coordinates for each robot or are prone to failure when attempting to form complex shapes beyond the capability of given approaches - the latter calls for a decision procedure that can tell whether a target shape can be formed before the actual shape-forming process starts. In this paper, we develop a method that does not require global coordinate information during the execution process and can effectively decide whether it is feasible to form the desired shape. The latter is achieved via a planning procedure that is capable of handling a variety of complex shapes, in particular, those with holes, and assigning a simple piece of scheduling information to each robot, facilitating subsequent distributed execution, which does not rely on the coordinates of all robots but only those of neighboring ones. The effectiveness of our shape-forming approach is vividly illustrated in several simulation case studies.

Paper Structure

This paper contains 40 sections, 43 theorems, 3 equations, 30 figures, 9 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Problem Description
  4. Setup of the System
  5. The Concept of Epoch
  6. The Add-subtract Algorithm
  7. Ribbon Structure
  8. Planning of Movement Sequences
  9. Execute of the Algorithm
  10. Proof of Correctness
  11. Part I
  12. Part II
  13. Additive Stage
  14. Subtractive Stage
  15. Simulation Case Studies
  16. ...and 25 more sections

Key Result

Proposition 1

Given a lattice point $x \in S$ with a positive hop count $k$, the neighborhood of $x$ can only be one of the two types:

Figures (30)

  • Figure 1: Left: example of a user-defined shape $S$ and the collection of holes $D$; Right: a proper lattice point representation of $S \setminus D$
  • Figure 2: Illustration of the workplace with two half-plane and underlying hexagonal lattices
  • Figure 3: Illustration of epoch, sub-epoch
  • Figure 4: Flow diagram of the add-subtract algorithm
  • Figure 5: The ribbons defined in $S$ and the idle ribbons defined in $H_2$
  • ...and 25 more figures

Theorems & Definitions (130)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Theorem 2
  • Definition 7
  • Definition 8
  • ...and 120 more