  The ability to realize nonlinear controllers with formal guarantees on
dynamic robotic systems has the potential to enable more complex robotic
behaviors -- yet, realizing these controllers is often practically challenging.
To address this challenge, this paper presents the end-to-end realization of
dynamic bipedal locomotion on an underactuated bipedal robot via hybrid zero
dynamics and control Lyapunov functions. A compliant model of Cassie is
represented as a hybrid system to set the stage for a trajectory optimization
framework. With the goal of achieving a variety of walking speeds in all
directions, a library of compliant walking motions is compiled and then
parameterized for efficient use within real-time controllers. Control Lyapunov
functions, which have strong theoretic guarantees, are synthesized to leverage
the gait library and coupled with inverse dynamics to obtain optimization-based
controllers framed as quadratic programs. It is proven that this controller
provably achieves stable locomotion; this is coupled with a theoretic analysis
demonstrating useful properties of the controller for tuning and
implementation. The proposed theoretic framework is practically demonstrated on
the Cassie robot, wherein 3D walking is achieved through the use of
optimization-based torque control. The experiments highlight robotic walking at
different speeds and terrains, illustrating the end-to-end realization of
theoretically justified nonlinear controllers on dynamic underactuated robotic
systems.
