Two-level control scheme for stabilisation of periodic orbits for planar monopedal running

This study presents an online motion planning algorithm for generating reference trajectories during flight phases of a planar monopedal robot to transfer the configuration of the mechanical system from a specified initial pose to a specified final one. The algorithm developed in this research is based on the reachability and optimal control formulations of a time-varying linear system with input and state constraints. A two-level control scheme is developed for asymptotic stabilisation of a desired period-one orbit during running of the robot. Within-stride controllers, including stance and flight phase controllers, are employed at the first level. The flight phase controller is a feedback law to track the reference trajectories generated by the proposed algorithm. To reduce the dimension of the full-order model of running, the stance phase controller is chosen to be a parameterised time-invariant feedback law that produces a family of two-dimensional finite-time attractive and invariant submanifolds. At the second level, the parameters of the stance phase controller are updated by an event-based update law to achieve hybrid invariance and stabilisation. To illustrate the analytical results developed for the behaviour of the closed-loop system, a detailed numerical example is presented.