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CAT-ORA: Collision-Aware Time-Optimal Formation Reshaping for Efficient Robot Coordination in 3D Environments

Vit Kratky, Robert Penicka, Jiri Horyna, Petr Stibinger, Tomas Baca, Matej Petrlik, Petr Stepan, Martin Saska

TL;DR

CAT-ORA (catora) addresses time-optimal formation reshaping in 3D with collision avoidance by coupling a minimum-makespan robot-to-goal assignment with a minimum-time, collision-free trajectory generator. It rests on a LBAP-based assignment that accounts for mutual collisions, solved via a dynamicHungarian-like procedure, together with a closed-form minimum-time trajectory generator that preserves collision guarantees. Theoretical guarantees on minimum inter-trajectory distances, together with extensive numerical and real-world UAV experiments, show up to 49% faster reshaping than LSAP-based methods and robust collision avoidance. The approach demonstrates practicality for real-time deployment in formations of tens of robots and offers a foundation for integrating centralized planning with distributed cooperative motion planners like MADER.

Abstract

In this paper, we introduce an algorithm designed to address the problem of time-optimal formation reshaping in three-dimensional environments while preventing collisions between agents. The utility of the proposed approach is particularly evident in mobile robotics, where agents benefit from being organized and navigated in formation for a variety of real-world applications requiring frequent alterations in formation shape for efficient navigation or task completion. Given the constrained operational time inherent to battery-powered mobile robots, the time needed to complete the formation reshaping process is crucial for their efficient operation, especially in case of multi-rotor Unmanned Aerial Vehicles (UAVs). The proposed Collision-Aware Time-Optimal formation Reshaping Algorithm (CAT-ORA) builds upon the Hungarian algorithm for the solution of the robot-to-goal assignment implementing the inter-agent collision avoidance through direct constraints on mutually exclusive robot-goal pairs combined with a trajectory generation approach minimizing the duration of the reshaping process. Theoretical validations confirm the optimality of CAT-ORA, with its efficacy further showcased through simulations, and a real-world outdoor experiment involving 19 UAVs. Thorough numerical analysis shows the potential of CAT-ORA to decrease the time required to perform complex formation reshaping tasks by up to 49%, and 12% on average compared to commonly used methods in randomly generated scenarios.

CAT-ORA: Collision-Aware Time-Optimal Formation Reshaping for Efficient Robot Coordination in 3D Environments

TL;DR

CAT-ORA (catora) addresses time-optimal formation reshaping in 3D with collision avoidance by coupling a minimum-makespan robot-to-goal assignment with a minimum-time, collision-free trajectory generator. It rests on a LBAP-based assignment that accounts for mutual collisions, solved via a dynamicHungarian-like procedure, together with a closed-form minimum-time trajectory generator that preserves collision guarantees. Theoretical guarantees on minimum inter-trajectory distances, together with extensive numerical and real-world UAV experiments, show up to 49% faster reshaping than LSAP-based methods and robust collision avoidance. The approach demonstrates practicality for real-time deployment in formations of tens of robots and offers a foundation for integrating centralized planning with distributed cooperative motion planners like MADER.

Abstract

In this paper, we introduce an algorithm designed to address the problem of time-optimal formation reshaping in three-dimensional environments while preventing collisions between agents. The utility of the proposed approach is particularly evident in mobile robotics, where agents benefit from being organized and navigated in formation for a variety of real-world applications requiring frequent alterations in formation shape for efficient navigation or task completion. Given the constrained operational time inherent to battery-powered mobile robots, the time needed to complete the formation reshaping process is crucial for their efficient operation, especially in case of multi-rotor Unmanned Aerial Vehicles (UAVs). The proposed Collision-Aware Time-Optimal formation Reshaping Algorithm (CAT-ORA) builds upon the Hungarian algorithm for the solution of the robot-to-goal assignment implementing the inter-agent collision avoidance through direct constraints on mutually exclusive robot-goal pairs combined with a trajectory generation approach minimizing the duration of the reshaping process. Theoretical validations confirm the optimality of CAT-ORA, with its efficacy further showcased through simulations, and a real-world outdoor experiment involving 19 UAVs. Thorough numerical analysis shows the potential of CAT-ORA to decrease the time required to perform complex formation reshaping tasks by up to 49%, and 12% on average compared to commonly used methods in randomly generated scenarios.

Paper Structure

This paper contains 30 sections, 4 theorems, 66 equations, 12 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Definition
  4. Overview of the maximum matching in bipartite graphs and the Hungarian method
  5. Maximum matching in bipartite graphs
  6. Hungarian algorithm
  7. catora - overview
  8. Minimum-weight robot-to-goal assignment
  9. Minimum makespan collision-aware trajectory planning
  10. Theoretical guarantees of LBAP solution
  11. Minimum mutual distance of two trajectories
  12. Algorithm for solution of lbap with guarantees on minimum distance and collision-free trajectories
  13. Algorithm for robot-to-goal assignment considering mutual collision constraints
  14. Minimum-makespan trajectory generation
  15. Minimum-time trajectory generation
  16. ...and 15 more sections

Key Result

Theorem 1

If $\frac{\max(||\mathbf{s}_i - \mathbf{g}_i||, ||\mathbf{s}_j - \mathbf{g}_j||)}{\max(||\mathbf{s}_i - \mathbf{g}_j||, ||\mathbf{s}_j - \mathbf{g}_i||)} \leq M,\,\,M \in [0,1)$, then minimum mutual distance $d_{ij,min} \geq \sqrt{1-M^2} \delta_{ij}$.

Figures (12)

  • Figure 1: Deployment of the introduced catora (catora) in a small-scale drone visual performance with 19 uav. The images show the transition of uav guided by the catora from a triangular shape (a) to a ring shape (b). This transition was performed within 7 seconds. The blue lines highlight the shape of the formation in top view images, while (c) captures the flying formation from the side. The red point represents a missing UAV that failed to start due to a HW failure.
  • Figure 2: Block diagram of the proposed catora (catora). The colors of the trajectories in the image on the right encode the velocity profile of particular trajectories, with red being equal to zero velocity and yellow to maximum velocity.
  • Figure 3: An example problem consisting of two initial positions $\mathbf{s}_i$, $\mathbf{s}_j$ and two goal locations $\mathbf{g}_i$, $\mathbf{g}_j$. Without loss of generality, the distance $||\mathbf{s}_j - \mathbf{g}_i||$ is assumed to be equal to $d$ and $||\mathbf{s}_i - \mathbf{g}_i|| = Md$, where $M \in [0, 1)$.
  • Figure 4: Illustration of the general case of an assignment problem with fixed points $\mathbf{s}_i, \mathbf{s}_j, \mathbf{g}_i$ and variable point $\mathbf{g}_j$.
  • Figure 5: Simplified diagram illustrating succession of individual steps of the algorithm for robot-to-goal assignment considering mutual collision constraints. The green and red arrows indicate the branching based on positive and negative results, respectively. The detailed description of individual steps is provided in \ref{['sec:task_assignment_lbap_algorithm']}.
  • ...and 7 more figures

Theorems & Definitions (13)

  • Remark
  • Remark
  • Theorem 1
  • Theorem 2
  • proof
  • Remark
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 3 more