Formal Connections between Template and Anchor Models via Approximate Simulation

TL;DR

This paper establishes a formal connection between simplified template models like LIP and full humanoid robot models using approximate simulation, enabling reliable control and planning.

Contribution

It introduces a formal framework based on approximate simulation to connect template and full-body models, with practical constraints for contact maintenance.

Findings

LIP approximately simulates the balancer with guarantees.

Quadratic programming enables rapid planning with contact constraints.

Successful push recovery simulation demonstrates the approach.

Abstract

Reduced-order template models like the Linear Inverted Pendulum (LIP) and Spring-Loaded Inverted Pendulum (SLIP) are widely used tools for controlling high-dimensional humanoid robots. However, connections between templates and whole-body models have lacked formal underpinnings, preventing formal guarantees when it comes to integrated controller design. We take a small step towards addressing this gap by considering the notion of approximate simulation. Derived from simulation relations for discrete transition systems in formal methods, approximate similarity means that the outputs of two systems can remain $\epsilon$-close. In this paper, we consider the case of controlling a balancer via planning with the LIP model. We show that the balancer approximately simulates the LIP and derive linear constraints that are sufficient conditions for maintaining ground contact. This allows for rapid planning and replanning with the template model by solving a quadratic program that enforces contact constraints in the full model. We demonstrate the efficacy of this planning and control paradigm in a simulated push recovery scenario for a planar 4-link balancer.