False Feasibility in Variable Impedance MPC for Legged Locomotion

Abstract

Variable impedance model predictive control (MPC) formulations often treat joint stiffness as an instantaneous decision variable. The resulting feasible set strictly contains 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 F_param and the realizable set F_real, 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 = 0.986). 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. Augmenting the prediction state with stiffness closes the mismatch by construction.