Internal Report

Passive Walking: Transition from 2D to 3D

Abstract

The inherent dynamics of bipedal, passive mechanisms are studied to investigate the relation between motions constrained to two-dimensional planes and those free to move in a three-dimensional environment. In particular, we develop numerical and analytical techniques using dynamical-systems methodology to address the persistence and stability changes of periodic, gait-like motions due to the relaxation of configuration constraints and the breaking of problem symmetries. Symmetry properties of such mechanism are discussed and a few special symmetries are defined. Using these symmetries, we classify a number of different gaits. The results indicate the limitations of a two-dimensional analysis to predict the dynamics in the three-dimensional environment. For example, it is shown how the loss of constraints may introduce characteristically non-2D instability mechanisms, and how small symmetry-breaking terms may result in the termination of solution branches. A center-manifold technique is implemented to arrive at a normal-form for the natural dynamics of a passive, bipedal rigid-body mechanism in the vicinity of infinite foot width and near-symmetric body geometry. In particular, numerical schemes are developed for finding approximate forms of the relevant invariant manifolds and the near-singular dynamics on these manifolds. The normal-form approximations are found to be highly accurate for relative large foot widths with a range of validity extending to widths on the order of the mechanisms` height.