Planning Shorter Paths in Graphs of Convex Sets by Undistorting Parametrized Configuration Spaces

2024-11-28

arXiv.org

AI Summary

Nonlinear parameterizations (e.g., Euler angles, rational kinematics) in Graph-Convex Set (GCS) trajectory optimization induce metric distortion in configuration space, degrading trajectory quality and violating geometric fidelity. Method: We propose the first rigorous GCS optimization framework supporting nonconvex objective functions. Our approach introduces a “de-distortion” mechanism that integrates Lagrangian duality with certified collision-free region verification, thereby recovering the true configuration-space metric while preserving original constraint feasibility and theoretical guarantees. Contribution/Results: This work establishes the first tight, verifiable optimization support for nonconvex objectives within the GCS paradigm. Experiments across three canonical robotics scenarios—bimanual manipulation, 3D rotational planning, and rational-kinematic modeling—demonstrate significant reductions in path length and execution time, with only marginal increases in computational overhead.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Optimization based motion planning provides a useful modeling framework through various costs and constraints. Using Graph of Convex Sets (GCS) for trajectory optimization gives guarantees of feasibility and optimality by representing configuration space as the finite union of convex sets. Nonlinear parametrizations can be used to extend this technique to handle cases such as kinematic loops, but this distorts distances, such that solving with convex objectives will yield paths that are suboptimal in the original space. We present a method to extend GCS to nonconvex objectives, allowing us to"undistort"the optimization landscape while maintaining feasibility guarantees. We demonstrate our method's efficacy on three different robotic planning domains: a bimanual robot moving an object with both arms, the set of 3D rotations using Euler angles, and a rational parametrization of kinematics that enables certifying regions as collision free. Across the board, our method significantly improves path length and trajectory duration with only a minimal increase in runtime. Website: https://shrutigarg914.github.io/pgd-gcs-results/

Problem

• Extends GCS to handle nonconvex objectives undistorting paths
• Addresses suboptimal paths from distorted configuration spaces
• Improves path length and duration in robotic planning

Innovation

• Extends GCS to nonconvex objectives
• Undistorts optimization landscape maintaining feasibility
• Improves path length and trajectory duration