  This paper presents the design and validation of controlling hopping on the
3D bipedal robot Cassie. A spring-mass model is identified from the kinematics
and compliance of the robot. The spring stiffness and damping are encapsulated
by the leg length, thus actuating the leg length can create and control hopping
behaviors. Trajectory optimization via direct collocation is performed on the
spring-mass model to plan jumping and landing motions. The leg length
trajectories are utilized as desired outputs to synthesize a control Lyapunov
function based quadratic program (CLF-QP). Centroidal angular momentum, taking
as an addition output in the CLF-QP, is also stabilized in the jumping phase to
prevent whole body rotation in the underactuated flight phase. The solution to
the CLF-QP is a nonlinear feedback control law that achieves dynamic jumping
behaviors on bipedal robots with compliance. The framework presented in this
paper is verified experimentally on the bipedal robot Cassie.
