Leonid Freidovich

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.

Motivated by problems in robotic interaction control, we present a model-based method for robust orbital stabilization. Our objective is to design a time-invariant feedback law for a model of a nonlinear system, or for its digital twin, that makes the distance between its solutions and a planned periodic trajectory decay exponentially. The method uses transverse coordinates, which are functions that vanish on the orbit and remain independent in the firstorder approximation. We regulate the linearized dynamics of transverse coordinates to zero. The novelty of the method is that it replaces the projection-based modification of a stabilizing time-periodic controller with a combination of a time-invariant control law for a subsystem and a discontinuous sliding-mode term. The sliding-mode part forces the state to a switching manifold in finite time and provides robustness to matched uncertainties. We develop a stepby- step procedure and demonstrate its use by an academic example that consists of two masses coupled by a spring and actuated by an external control force. Although the procedure usually requires numerical approximations, this example allows all steps to be carried out analytically. We also discuss the corresponding design for the velocity-controlled case.