False Feasibility in Variable Impedance MPC for Legged Locomotion

Abstract: Variable impedance model predictive control (MPC) formulations that treat joint stiffness as an instantaneous decision variable operate on a feasible set strictly larger than the physically realizable set under first-order actuator dynamics. We identify this as a formulation error rather than a modeling approximation, formalize the distinction between the parameter-based feasible set Fparam and the realizable set Freal, and characterize the regime of mismatch via the dimensionless parameter alpha = omega_sT (actuator bandwidth times task timescale). For the 1D hopping monoped, we prove that below an analytical threshold alpha_crit derived in closed form from task physics, no admissible stiffness command realizes the parameter-based prediction. Numerical validation in 1D shows monotonic deviation growth as alpha decreases, with the predicted scaling holding across ten parameter combinations (log-log R2 = 0.99). Mechanism transfer to planar spring-loaded inverted pendulum dynamics confirms center-of-mass and stance-timing deviation as the primary consequence, with regime-dependent friction effects as a tertiary observable. A second threshold alpha_infeas < alpha_crit establishes a floor below which restricting the admissible stiffness range cannot repair realizability, closing the conservative-tuning objection on structural grounds. Augmenting the prediction state with stiffness closes the mismatch by construction.