Motion Planning of Legged Robots

TL;DR

This paper introduces efficient algorithms for computing the free space of a simple legged robot, considering accessibility and stability constraints, with near-optimal complexity for discrete footholds and polygonal regions.

Contribution

It presents novel algorithms for calculating the robot's free space with proven near-optimal complexity bounds for different foothold configurations.

Findings

Algorithms run in near-quadratic time for discrete footholds.

The algorithms handle polygonal foothold regions efficiently.

Results are close to theoretical lower bounds for free space computation.

Abstract

We study the problem of computing the free space F of a simple legged robot called the spider robot. The body of this robot is a single point and the legs are attached to the body. The robot is subject to two constraints: each leg has a maximal extension R (accessibility constraint) and the body of the robot must lie above the convex hull of its feet (stability constraint). Moreover, the robot can only put its feet on some regions, called the foothold regions. The free space F is the set of positions of the body of the robot such that there exists a set of accessible footholds for which the robot is stable. We present an efficient algorithm that computes F in O(n2 log n) time using O(n2 alpha(n)) space for n discrete point footholds where alpha(n) is an extremely slowly growing function (alpha(n) <= 3 for any practical value of n). We also present an algorithm for computing F when the foothold regions are pairwise disjoint polygons with n edges in total. This algorithm computes F in O(n2 alpha8(n) log n) time using O(n2 alpha8(n)) space (alpha8(n) is also an extremely slowly growing function). These results are close to optimal since Omega(n2) is a lower bound for the size of F.