Abstract: The Spring-Loaded Inverted Pendulum (SLIP) has been regarded as a canonical model for hopping and running dynamics of legged robots. This paper presents a novel control of the actuated-SLIP hopping on unknown terrains. We propose that in the neighborhood of the desired stable hybrid limit cycle, the local dynamical behavior of a hybrid system can be expressed by a set of phase coordinates and transverse coordinates. Under some acceptable assumptions, the hybrid averaging theorem is applied on the SLIP non-integrable dynamics to simplify the controller design. Using the inherent symmetry of SLIP dynamics, a control Lyapunov function-based hybrid averaging controller is developed to ensure the exponential stability of the desired gait orbit. This results in a set of linear constraints on the control signal, which can be readily solved by a quadratic programming optimization. Furthermore, a novel method is introduced to improve the robustness against unknown disturbances through the online constraint adjustment. The proposed controller is evaluated in various simulations, demonstrating the SLIP hopping on diverse terrains, including flat, sin-wave, and unregular terrains. The performance of the approach is also validated on a quadruped robot SCIT Dog for generating dynamic gaits such as pronking.

Key words: Legged robots , · Spring-Loaded Inverted Pendulum , · Control Lyapunov function , · Dynamic hopping