# Probabilistic Completeness of Randomized Possibility Graphs Applied to Bipedal Walking in Semi-unstructured Environments

## TL;DR

This paper proves that the Randomized Possibility Graph method is probabilistically complete for bipedal walking in semi-unstructured environments, offering faster convergence in finding feasible routes through a hierarchical planning approach.

## Contribution

It provides the first theoretical proof of probabilistic completeness for this motion planning method applied to bipedal robots in complex environments.

## Key findings

- Probabilistic completeness is established for the method in semi-unstructured environments.
- Hierarchical decomposition improves convergence rate of the planning algorithm.
- Simulated scenarios demonstrate the effectiveness of the approach.

## Abstract

We present a theoretical analysis of a recent whole body motion planning method, the Randomized Possibility Graph, which uses a high-level decomposition of the feasibility constraint manifold in order to rapidly find routes that may lead to a solution. These routes are then examined by lower-level planners to determine feasibility. In this paper, we show that this approach is probabilistically complete for bipedal robots performing quasi-static walking in "semi-unstructured" environments. Furthermore, we show that the decomposition into higher and lower level planners allows for a considerably higher rate of convergence in the probability of finding a solution when one exists. We illustrate this improved convergence with a series of simulated scenarios.