Feedback motion planning of legged robots by composing orbital Lyapunov functions using rapidly-exploring random trees

We present a sampling-based framework for feedback motion planning of legged robots. Our framework is based on switching between limit cycles at a fixed instance of motion, the Poincaré section (e.g., apex or touchdown), by finding overlaps between the regions of attraction (ROA) of two limit cycles. First, we assume a candidate orbital Lyapunov function (OLF) and define a ROA at the Poincaré section. Next, we solve multiple trajectory optimization problems, one for each sampled initial condition on the ROA to minimize an energy metric and subject to the exponential convergence of the OLF between two steps. The result is a table of control actions and the corresponding initial conditions at the Poincaré section. Then we develop a control policy for each control action as a function of the initial condition using deep learning neural networks. The control policy is validated by testing on initial conditions sampled on ROA of randomly chosen limit cycles. Finally, the rapidly-exploring random tree algorithm is adopted to plan transitions between the limit cycles using the ROAs. The approach is demonstrated on a hopper model to achieve velocity and height transitions between steps.