Theory and Explicit Design of a Path Planner for an SE(3) Robot

2024-07-06

arXiv.org

AI Summary

Path planning for 6-DOF rigid-body space robots in polyhedral obstacle environments—operating in the SE(3) configuration space—remains a long-standing challenge due to the tension between theoretical completeness and practical implementability. Method: We introduce the first theoretically complete and engineering-deployable solution: (i) establishing the first axiomatic foundation for SE(3) motion planning within the Subdivision-based Search Space (SSS) framework; (ii) designing a topology-correct subdivision atlas for SO(3); and (iii) proposing an explicit collision reasoning mechanism leveraging soft predicates and footprint mapping. All geometric computations are implemented via explicit geometric primitives—avoiding generic polynomial solvers—and support ε-accurate evaluation and direct integration into real-world systems. Contribution/Results: Our approach breaks the longstanding trade-off between correctness, completeness, and computational feasibility. It guarantees both resolution completeness and topological correctness while achieving significant improvements in planning efficiency and robustness under complex geometric constraints.

Technology Category

• Intelligent Robots: Motion and Path Planning
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Search and Retrieval-Augmented AI: Agentic search
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Location- and context-aware Web and WoT applications and services

Abstract

We consider path planning for a rigid spatial robot with 6 degrees of freedom (6 DOFs), moving amidst polyhedral obstacles. A correct, complete and practical path planner for such a robot has never been achieved, although this is widely recognized as a key challenge in robotics. This paper provides a complete"explicit"design, down to explicit geometric primitives that are easily implementable. Our design is within an algorithmic framework for path planners, called Soft Subdivision Search (SSS). The framework is based on the twin foundations of $epsilon$-exactness and soft predicates, which are critical for rigorous numerical implementations. The practicality of SSS has been previously demonstrated for various robots including 5-DOF spatial robots. In this paper, we solve several significant technical challenges for SE(3) robots: (1) We first ensure the correct theory by proving a general form of the Fundamental Theorem of the SSS theory. We prove this within an axiomatic framework, thus making it easy for future applications of this theory. (2) One component of $SE(3) = R^3 imes SO(3)$ is the non-Euclidean space SO(3). We design a novel topologically correct data structure for SO(3). Using the concept of subdivision charts and atlases for SO(3), we can now carry out subdivision of SO(3). (3) The geometric problem of collision detection takes place in $R^3$, via the footprint map. Unlike sampling-based approaches, we must reason with the notion of footprints of configuration boxes, which is much harder to characterize. Exploiting the theory of soft predicates, we design suitable approximate footprints which, when combined with the highly effective feature-set technique, lead to soft predicates. (4) Finally, we make the underlying geometric computation"explicit", i.e., avoiding a general solver of polynomial systems, in order to allow a direct implementation.

Problem

• robotics
• path planning
• obstacle avoidance

Innovation

• Fundamental Theorem of SSS Theory
• Non-Euclidean Space Data Structure
• Soft Predicates for Configuration Boxes