# Certified Polyhedral Decompositions of Collision-Free Configuration Space

## TL;DR

This paper introduces C-IRIS, a novel method that rigorously decomposes the collision-free configuration space into certified convex polyhedral regions using convex optimization, enabling scalable motion planning in complex robotic systems.

## Contribution

It presents the first practical, certified polyhedral decomposition method for C-free spaces using rational parametrization and semidefinite programming, applicable in arbitrary dimensions.

## Key findings

- Successfully decomposes C-free space in 2-DOF examples
- Scales to high-dimensional manipulators like 7-DOF KUKA iiwa
- Provides open-source implementation and interactive examples

## Abstract

Understanding the geometry of collision-free configuration space (C-free) in the presence of task-space obstacles is an essential ingredient for collision-free motion planning. While it is possible to check for collisions at a point using standard algorithms, to date no practical method exists for computing C-free regions with rigorous certificates due to the complexity of mapping task-space obstacles through the kinematics. In this work, we present the first to our knowledge rigorous method for approximately decomposing a rational parametrization of C-free into certified polyhedral regions. Our method, called C-IRIS (C-space Iterative Regional Inflation by Semidefinite programming), generates large, convex polytopes in a rational parameterization of the configuration space which are rigorously certified to be collision-free. Such regions have been shown to be useful for both optimization-based and randomized motion planning. Based on convex optimization, our method works in arbitrary dimensions, only makes assumptions about the convexity of the obstacles in the task space, and is fast enough to scale to realistic problems in manipulation. We demonstrate our algorithm's ability to fill a non-trivial amount of collision-free C-space in several 2-DOF examples where the C-space can be visualized, as well as the scalability of our algorithm on a 7-DOF KUKA iiwa, a 6-DOF UR3e and 12-DOF bimanual manipulators. An implementation of our algorithm is open-sourced in Drake. We furthermore provide examples of our algorithm in interactive Python notebooks.