Generating stable periodic motion in underactuated systems in the presence of parameter uncertainty: Theory and experiments

Virtual holonomic constraints (VHCs) are extensively used in robotic applications such as bipedal walking. Although it is well-known that VHCs result in periodic motion of underactuated systems, achieving the same is challenging in physical systems due to parameter uncertainty. VHCs are typically imposed using feedback linearization and simulations show and experiments confirm that the internal dynamics can become unstable in the presence of parameter uncertainty. To address the challenging problem of generating stable periodic motion, we propose an extended high-gain observer (EHGO) based controller to enforce the VHCs. The proposed solution successfully recovers marginal stability of the internal dynamics. To stabilize a desired periodic orbit, we use the impulse controlled Poincaré map (ICPM) approach, where impulsive inputs are intermittently applied on a Poincaré section. Experimental results demonstrate that the model-based VHC controller and discrete ICPM controller together fail to stabilize the desired periodic motion but the EHGO-based VHC controller and ICPM controller successfully achieve stable periodic motion.