Quasi-periodic gaits of passive one-legged hopper

In this paper, we present a novel controller for a passive one-legged hopping robot. First, based on the dynamics of this nonlinear hybrid system, we derive a simple control law to ensure the total energy preservation and continuation. Simulation results show that the robot can hop from the wide set of initial conditions. The generated hopping gaits are found to be quasi-periodic orbits, which can be seen in some Hamiltonian systems. Next, we propose a simple parameter adaptation law to asymptotically stabilize the quasi-periodic gaits to the periodic gaits of arbitrary period, and spring stiffness adaptation law to minimize control inputs. Simulation results show that the robot eventually hops without any control inputs, especially for 1-periodic gait. We believe that the controllers we invented have much potential for energy-efficient control of legged running robot.