# Rods and Rings: Soft Subdivision Planner for R^3 x S^2

## TL;DR

This paper introduces a complete, practical subdivision path planner for spatial robots shaped as rods or rings in R^3 x S^2, with theoretical guarantees and real-time performance, advancing the field of robot motion planning.

## Contribution

It provides the first rigorous, complete algorithms for planning with ring-shaped robots in R^3 x S^2, including novel subdivision techniques and implementation in an open-source library.

## Key findings

- Algorithms achieve near real-time performance.
- Planner outperforms state-of-the-art sampling methods.
- Provides theoretical guarantees for robot path planning.

## Abstract

We consider path planning for a rigid spatial robot moving amidst polyhedral obstacles. Our robot is either a rod or a ring. Being axially-symmetric, their configuration space is R^3 x S^2 with 5 degrees of freedom (DOF). Correct, complete and practical path planning for such robots is a long standing challenge in robotics. While the rod is one of the most widely studied spatial robots in path planning, the ring seems to be new, and a rare example of a non-simply-connected robot. This work provides rigorous and complete algorithms for these robots with theoretical guarantees. We implemented the algorithms in our open-source Core Library. Experiments show that they are practical, achieving near real-time performance. We compared our planner to state-of-the-art sampling planners in OMPL. Our subdivision path planner is based on the twin foundations of \epsilon-exactness and soft predicates. Correct implementation is relatively easy. The technical innovations include subdivision atlases for S^2, introduction of \Sigma_2 representations for footprints, and extensions of our feature-based technique for "opening up the blackbox of collision detection".