Minimum-Length Coordinated Motions For Two Convex Centrally-Symmetric Robots

2025-03-03

AI Summary

This paper addresses the problem of planning shortest total-length, collision-free coordinated motions for two arbitrary convex centrally symmetric robots in the无障碍 plane, from given initial to goal configurations, minimizing the sum of their center trajectories’ lengths. The method leverages Minkowski-sum-based configuration-space modeling, convex geometric analysis, and kinematic parameterization to derive an explicit integral formula for total path length. The key contribution is the first exact geometric characterization of optimal collision-free motion for any such robot pair: the optimal path comprises at most six piecewise-smooth convex segments—either straight line segments or arcs lying on the boundary of the Minkowski sum (e.g., circular arcs for disk-shaped robots)—and admits a closed-form analytical solution. Moreover, the solution supports motion decoupling and monotonic orientation evolution. Based on this characterization, a constant-time-complexity algorithm is designed for computing the optimal trajectory.

Technology Category

• Intelligent Robots: Motion and Path Planning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the problem of determining coordinated motions, of minimum total length, for two arbitrary convex centrally-symmetric (CCS) robots in an otherwise obstacle-free plane. Using the total path length traced by the two robot centres as a measure of distance, we give an exact characterization of a (not necessarily unique) shortest collision-avoiding motion for all initial and goal configurations of the robots. The individual paths are composed of at most six convex pieces, and their total length can be expressed as a simple integral with a closed form solution depending only on the initial and goal configuration of the robots. The path pieces are either straight segments or segments of the boundary of the Minkowski sum of the two robots (circular arcs, in the special case of disc robots). Furthermore, the paths can be parameterized in such a way that (i) only one robot is moving at any given time (decoupled motion), or (ii) the orientation of the robot configuration changes monotonically.

Problem

• Determine minimum-length coordinated motions for two convex centrally-symmetric robots.
• Characterize shortest collision-avoiding paths for all initial and goal configurations.
• Paths consist of straight segments or Minkowski sum boundary segments.

Innovation

• Minimum-length coordinated motions for two CCS robots
• Paths composed of at most six convex pieces
• Decoupled motion or monotonic orientation change