Lyapunov Equation and Integral Sliding Mode-Based Robust Stabilization of a Periodic Trajectory for a Three-Link Planar Biped Robot

This paper addresses the problem of orbital stabilization of a periodic walking gait for a model or a digital twin of a three-link planar biped robot with a single actuator. A Lyapunov equation-based approach is proposed for the synthesis of a stabilizing controller for the corresponding impulsive mechanical system. The method ensures exponential vanishing of transverse coordinates, defining deviations from the nominal periodic trajectory, by solving Lyapunov matrix inequalities, which provide sufficient conditions for orbital stability of the closed-loop dynamics in the nominal case of no disturbances. The proposed approach allows systematic feedback controller design for impulsive systems, taking into account the discontinuities associated with a simplified model of the impact phase of walking.
To ensure robustness against matched disturbances, an additional integral sliding mode (ISM) control law is introduced. The ISM component guarantees exact disturbance compensation (for a solution understood in the Filippov’s sense) from the initial moment of motion, ensuring that the perturbed system behaves identically to the nominal model from the very start. Theoretical results are validated through numerical simulations on a model of a three-link biped robot. The obtained results demonstrate that the proposed control law ensures stable periodic walking and significant reduction of deviations from the nominal gait, even under external perturbations.