🤖 AI Summary
This study addresses the spurious feasibility issue in variable-impedance model predictive control, where treating joint stiffness as an instantaneous decision variable yields solutions outside the physically realizable set. The work identifies this problem as stemming from modeling error rather than approximation inaccuracies and introduces a dimensionless parameter α = ωₛT to quantify the mismatch between parametrization and the physical feasibility set. Using one-dimensional spring-loaded hopping and planar spring-inverted pendulum models, the authors analytically derive critical (α_crit) and infeasibility (α_infeas) thresholds. By augmenting the state vector with stiffness as a dynamic variable, the proposed method fundamentally eliminates the feasibility gap. Numerical simulations and log-scale regression (R² = 0.99) confirm that trajectory deviations—primarily manifesting as center-of-mass trajectory and stance-phase timing distortions—monotonically increase as α decreases, while the proposed approach constructively restores physical feasibility.
📝 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.