Beyond Basins of Attraction: Quantifying Robustness of Natural Dynamics

TL;DR

This paper introduces a viability theory-based method to quantify the robustness of natural dynamics in systems like spring-mass models, enabling better control policy design and robustness optimization.

Contribution

It extends the analysis of natural dynamics beyond basins of attraction by quantifying robust sets in state-action space applicable to all control policies.

Findings

Viability-based robust sets can be computed for simple running models.

Optimizing natural dynamics increases system robustness.

Effective stiffness correlates with improved robustness.

Abstract

Properly designing a system to exhibit favorable natural dynamics can greatly simplify designing or learning the control policy. However, it is still unclear what constitutes favorable natural dynamics and how to quantify its effect. Most studies of simple walking and running models have focused on the basins of attraction of passive limit-cycles and the notion of self-stability. We instead emphasize the importance of stepping beyond basins of attraction. We show an approach based on viability theory to quantify robust sets in state-action space. These sets are valid for the family of all robust control policies, which allows us to quantify the robustness inherent to the natural dynamics before designing the control policy or specifying a control objective. We illustrate our formulation using spring-mass models, simple low dimensional models of running systems. We then show an example application by optimizing robustness of a simulated planar monoped, using a gradient-free optimization scheme. Both case studies result in a nonlinear effective stiffness providing more robustness.