An Example of Robust Orbital Stabilization of a Periodic Behavior: Subspace and Sliding-Mode-Based Stabilization of Transverse Linearization

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.