Customize Harmonic Potential Fields Via Hybrid Optimization Over Homotopic Paths
Shuaikang Wang, Tiecheng Guo, Meng Guo
AI summary
Problem
Existing harmonic potential field methods ensure safe, globally convergent navigation but cannot be customized to produce paths with desired topological (homotopic) properties, limiting user control in cluttered environments.
Approach
The method maps complex obstacle forests to a simplified point world via diffeomorphic transformation, then uses a hybrid optimizer to search homotopy classes, select obstacle tree structures, and tune continuous weights to synthesize valid potentials for target paths.
Key results
- Formal proof of homology invariance under diffeomorphic transformation
- Multi-directional D-signature for efficient homotopy classification
- Hybrid optimization framework for automatic potential customization
- Validation via simulations and hardware experiments in complex scenarios
Why it matters
Empowers robot developers and users to specify desired path topologies for navigation tasks without sacrificing the mathematical guarantees of safety and convergence.
Abstract
Safe navigation within a workspace is a fundamen- tal skill for autonomous robots to accomplish more complex tasks. Harmonic potentials are artificial potential fields that are analytical, globally convergent and provably free of local minima. Thus, it has been widely used for generating safe and reliable robot navigation control policies. However, most existing methods do not allow customization of the harmonic potential fields nor the resulting paths, particularly regarding their topo- logical properties. In this paper, we propose a novel method that automatically finds homotopy classes of paths that can be generated by valid harmonic potential fields. The considered complex workspaces can be as general as forest worlds consisting of numerous overlapping star-obstacles. The method is based on a hybrid optimization algorithm that searches over homotopy classes, selects the structure of each tree-of-stars within the forest, and optimizes over the continuous weight parameters for each purged tree via the projected gradient descent. The key insight is to transform the forest world to the unbounded point world via proper diffeomorphic transformations. It not only facilitates a simpler design of the multi-directional D-signature between non-homotopic paths, but also retain the safety and convergence properties. Extensive simulations and hardware experiments are conducted for non-trivial scenarios, where the navigation poten- tials are customized for desired homotopic properties. Project page: https://shuaikang-wang.github.io/CustFields.