A General Stance Stability Test Based on Stratified Morse Theory With Application to Quasi-Static Locomotion Planning

Abstract

This paper considers the stability of an object supported by several frictionless contacts in a potential field such as gravity. The bodies supporting the object induce a partition of the object's configuration space into strata corresponding to different contact arrangements. Stance stability becomes a geometric problem of determining whether the object's configuration is a local minimum of its potential energy function on the stratified configuration space. We use stratified Morse theory to develop a generic stance stability test that has the following characteristics. For a small number of contacts - less than three in 2D and less than six in 3D - stance stability depends both on surface normals and surface curvature at the contacts. Moreover, lower curvature at the contacts leads to better stability. For a larger number of contacts, stance stability depends only on surface normals at the contacts. The stance stability test is applied to quasi-static locomotion planning in two dimensions. The region of stable center-of-mass positions associated with a <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -contact stance is characterized. Then, a quasi-static locomotion scheme for a three-legged robot over a piecewise linear terrain is described. Finally, friction is shown to provide robustness and enhanced stability for the frictionless locomotion plan. A full maneuver simulation illustrates the locomotion scheme.