Robust Stabilization of Periodic Orbits for a Spherical Robot via Servoconstraints

This paper treats the problem of finding feasible periodic motions and that of their robust orbital stabilization for a spherical robot. The first problem is tackled using the servoconstraint approach based on the mathematical model of a pendulum-type spherical robot moving on a plane under the no-slip and no-loss-of-contact assumptions. The second problem is addressed through a generalization of the well-known Andronov – Vitt theorem, specifically by introducing the transverse coordinates of co-dimension one that are zero on a desired orbit. Linearizing the dynamics in the transverse coordinates results in a linear system with periodic coefficients (LSPC). A rigorous procedure is suggested for stabilizing the LSPC in the presence of structured uncertainties and matched disturbances using vector relay control. This control strategy forces the LSPC solutions onto a predefined time-periodic family of subspaces, where the exponential stability is ensured. The effectiveness of the overall framework is demonstrated through simulations.