Robust Control under Servo Constraint Following via Nash Equilibrium Theory for Bimanual Humanoid Manipulation

Trajectory tracking in bimanual humanoid robots, whose closed-chain kinematic structures inherently amplify the effects of modeling errors, external disturbances, and time-varying parameters, is a challenging task. To address this, we reformulate the dual-arm tracking task as a servo constraint-following problem and derive the system dynamics under approximate constraints using the Udwadia-Kalaba method. The humanoid system is modeled as a constrained mechanical structure subjected to fast-varying, bounded uncertainties with unknown limits. On this basis, we propose a robust control framework that guarantees both uniform boundedness (UB) and uniform ultimate boundedness (UUB) of the tracking error, ensuring stability and performance even under severe parametric and dynamic uncertainties. To reconcile the trade-off between transient dynamics and steady-state accuracy—essential for service-oriented tasks such as door opening or coffee pouring—we integrate a Nash equilibrium-based optimization mechanism into the controller design. By formulating a two-player non-cooperative game over the controller's key tuning parameters, we analytically derive the existence, uniqueness, and closed-form solutions of the game, achieving an optimal balance between competing objectives. Comprehensive simulations on a reduced-order bimanual humanoid model validate the proposed approach, demonstrating superior tracking accuracy, disturbance rejection, and energy efficiency compared to benchmark methods. The proposed strategy offers a theoretically grounded and practically implementable solution for robust, constraint-compliant humanoid manipulation.