This paper presents the design and validation of controlling hopping on the 3D bipedal robot Cassie. A springmass 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.

A Hybrid passive Linear Inverted Pendulum (H-LIP) model is proposed for characterizing, stabilizing and composing periodic orbits for 3D underactuated bipedal walking. Specifically, Period-1 (P1) and Period-2 (P2) orbits are geometrically characterized in the state space of the H-LIP. Stepping controllers are designed for global stabilization of the orbits. Valid ranges of the gains and their optimality are derived. The optimal stepping controller is used to create and stabilize the walking of bipedal robots. An actuated Spring-loaded Inverted Pendulum (aSLIP) model and the underactuated robot Cassie are used for illustration. Both the aSLIP walking with P1 or P2 orbits and the Cassie walking with all 3D compositions of the P1 and P2 orbits can be smoothly generated and stabilized from a stepping-in-place motion. This approach provides a perspective and a methodology towards continuous gait generation and stabilization for 3D underactuated walking robots.

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In this paper, we present a Hybrid-Linear Inverted Pendulum (H-LIP) based approach for synthesizing and stabilizing 3D underactuated bipedal walking. The H-LIP model is proposed to capture the essential components of the underactuated part and actuated part of the robotic walking. The walking gait of the robot is then synthesized based on the H-LIP. We comprehensively characterize the periodic orbits of the H-LIP and provably derive their stepping stabilization. The step-to-step (S2S) dynamics of the H-LIP is then utilized to approximate the S2S dynamics of the horizontal state of the center of mass (COM) of the robotic walking, which results in a H-LIP based stepping controller to provide desired step sizes to stabilize the robotic walking. By realizing the desired step sizes, the robot achieves dynamic and stable walking. The approach is evaluated in both simulation and experiment on the 3D underactuated bipedal robot Cassie, which demonstrate dynamic walking behaviors with both versatility and robustness.

In this paper, we propose an efficient approach to generate dynamic and versatile humanoid walking with nonconstant center of mass (COM) height. We exploit the benefits of using reduced order models (ROMs) and stepping control to generate dynamic and versatile walking motion. Specifically, we apply the stepping controller based on the Hybrid Linear Inverted Pendulum Model (H-LIP) to perturb a periodic walking motion of a 3D actuated Spring Loaded Inverted Pendulum (3D-aSLIP), which yields versatile walking behaviors of the 3D-aSLIP, including various 3D periodic walking, fixed location tracking, and global trajectory tracking. The 3D-aSLIP walking is then embedded on the fully-actuated humanoid via the task space control on the COM dynamics and ground reaction forces. The proposed approach is realized on the robot model of Atlas in simulation, wherein versatile dynamic motions are generated.