  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.
