Rebound, Compliance, and Configuration in the Modeling and Analysis of Discrete Impacts in Legged Locomotion

An energetic approach for analysis of the impact problem in two and three dimensional multi-legged robot locomotion is hereby presented. The equations of motion for impact are developed for the two and three dimensional cases. The two dimensional case yields algebraic expressions. The expressions obtained in the three dimensional have to be numerically integrated. However, a special mathematical artifice is needed to proceed with the integration when slip in the impacting end halts. The equations for impact are developed in terms commonly known to the robotics community.

The first part of the approach resembles traditional impact analysis. Here, it can be determined if slip in the end effector halts and remains halted before impact termination, a condition known as {em sticking}. Achieving sticking depends upon the coefficient of friction between the ground and the end effector, as well as the impact configuration

The most important contribution of this thesis is in going beyond the traditional way of analyzing impact by looking at the problem in an energetic framework where rigid and deformable body approaches naturally come together. Formerly, impact analysis methods always predict rebound for a {em coefficient of restitution} greater than zero. The proposed hybrid approach can predict both rebound and non-rebound for a coefficient of restitution greater than zero. Currently, the robotics community makes the {em ad-hoc} assumption that the coefficient of restitution equals zero to simulate impact in walking robots. This assumption may incorrectly predict non-rebound when the impact energy goes beyond what the system can safely store or dissipate.

This thesis presents a novel method for analyzing impact for legged robots as well as any other rigid body. Here the energetic definition cite{stronge} of the coefficient of restitution is primarily used but other may be used as well. This analysis will help determine if the level of compliance in the system at a given configuration is adequate to insure a no-slip, no-rebound impact.