Leonid Fridman

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.

Sliding mode control (SMC) provides strong robustness against matched disturbances. Among SMC schemes, the super-twisting algorithm (STA) provides continuous control action and finite-time convergence for systems of relative degree one. However, in real applications, actuator imperfections and unmodeled dynamics prevent true finite-time convergence and cause high-frequency oscillations called chattering. The chattering effect can be mitigated by tuning control parameters, for instance, through frequency-domain analysis. Yet, most existing methods rely on simplified system models, limiting their applicability to complex systems. This work proposes a generalized frequency-domain framework for STA gain tuning based on a first-order plus dead time (FOPDT) model. The method identifies FOPDT parameters from the reaction curve and employs describing function and harmonic balance analysis to predict analytically the chattering amplitude, average energy, and frequency. The resulting relations provide explicit guidelines for tuning STA gains to minimize chattering while maintaining robust performance. Validation through both simulations and experiments on a DC motor position control system confirms that the proposed approach achieves improved robustness and accuracy compared with standard STA and PI tuning methods.