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Reduced-order template models like the Linear Inverted Pendulum (LIP) andSpring-Loaded Inverted Pendulum (SLIP) are widely used tools for controllinghigh-dimensional humanoid robots. However, connections between templates andwhole-body models have lacked formal underpinnings, preventing formalguarantees when it comes to integrated controller design. We take a small steptowards addressing this gap by considering the notion of approximatesimulation. Derived from simulation relations for discrete transition systemsin formal methods, approximate similarity means that the outputs of two systemscan remain $\epsilon$-close. In this paper, we consider the case of controllinga balancer via planning with the LIP model. We show that the balancerapproximately simulates the LIP and derive linear constraints that aresufficient conditions for maintaining ground contact. This allows for rapidplanning and replanning with the template model by solving a quadratic programthat enforces contact constraints in the full model. We demonstrate theefficacy of this planning and control paradigm in a simulated push recoveryscenario for a planar 4-link balancer.

Formal Connections between Template and Anchor Models via Approximate Simulation