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The ability to realize nonlinear controllers with formal guarantees ondynamic robotic systems has the potential to enable more complex roboticbehaviors -- yet, realizing these controllers is often practically challenging.To address this challenge, this paper presents the end-to-end realization ofdynamic bipedal locomotion on an underactuated bipedal robot via hybrid zerodynamics and control Lyapunov functions. A compliant model of Cassie isrepresented as a hybrid system to set the stage for a trajectory optimizationframework. With the goal of achieving a variety of walking speeds in alldirections, a library of compliant walking motions is compiled and thenparameterized for efficient use within real-time controllers. Control Lyapunovfunctions, which have strong theoretic guarantees, are synthesized to leveragethe gait library and coupled with inverse dynamics to obtain optimization-basedcontrollers framed as quadratic programs. It is proven that this controllerprovably achieves stable locomotion; this is coupled with a theoretic analysisdemonstrating useful properties of the controller for tuning andimplementation. The proposed theoretic framework is practically demonstrated onthe Cassie robot, wherein 3D walking is achieved through the use ofoptimization-based torque control. The experiments highlight robotic walking atdifferent speeds and terrains, illustrating the end-to-end realization oftheoretically justified nonlinear controllers on dynamic underactuated roboticsystems.

Control Lyapunov Functions for Compliant Hybrid Zero Dynamic Walking