Combined Deep KKL and NMPC for Modeling Muscle Co-contraction Under Measurement Delays

This paper addresses the modeling of robust control through muscle co-contraction, a fundamental physiological strategy that modulates mechanical impedance and stabilizes limb dynamics. Here, the neuromusculoskeletal system is modeled as nonlinear dynamics actuated by antagonist muscles, subject to uncertainties and significant output measurement delays. To account for muscle co-contraction for robust control in simulation, we propose a closed-loop control framework based on Nonlinear Model Predictive Control (NMPC) and test it for a forearm stabilization task (inverted pendulum). While NMPC is well-suited for handling system constraints and mimicking physiological objectives, its real-time implementation requires the estimation of the instantaneous initial state, which is unavailable due to the delays. To address this, we introduce a delay-compensating Kazantzis-Kravaris-Luenberger (KKL) observer. This observer embeds the nonlinear dynamics into a stable linear latent space, enabling explicit delay compensation through a chain of predictors. To overcome the analytical complexity of the required transformation maps, we employ a Deep Learning approach to approximate the observer dynamics. Finally, the performance and robustness of the Deep KKL-NMPC scheme are validated through extensive simulations of posture stabilization and dynamic tracking tasks under stochastic environmental disturbances.