Bipedal Hopping: Reduced-order Model Embedding via Optimization-based Control Xiaobin Xiong and Aaron D. Ames Abstract— This paper presents the design and validation of ݖ controlling hopping on the 3D bipedal robot Cassie. A spring- ,ŝƉLJĂǁ ݕ mass model is identified from the kinematics and compliance of ݔ the robot. The spring stiffness and damping are encapsulated ,ŝƉƌŽůů by the leg length, thus actuating the leg length can create and ,ŝƉƉŝƚĐŚ control hopping behaviors. Trajectory optimization via direct collocation is performed on the spring-mass model to plan <ŶĞĞƉŝƚĐŚ jumping and landing motions. The leg length trajectories are utilized as desired outputs to synthesize a control Lyapunov ^ŚŝŶƉŝƚĐŚ function based quadratic program (CLF-QP). Centroidal an- gular momentum, taking as an addition output in the CLF- dĂƌƐƵƐƉŝƚĐŚ 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 dŽĞƉŝƚĐŚ compliance. The framework presented in this paper is verified experimentally on the bipedal robot Cassie. I. INTRODUCTION Fig. 1. Hopping on Cassie [1] (left) and its coordinate system (right). Reduced-order models such as the canonical Spring Loaded Inverted Pendulum (SLIP) have been widely applied for controlling walking [2] [3] [4], running [5] and hopping that achieve this behavior on the full-order model. Roughly [6] of legged robots. One important benefit of using low speaking, the springs on the physical robot are expected to order dynamical systems for control is that it renders the gait behave similarly to the spring in the spring-mass model when and motion generation problems for legged robots computa- the robot tracks the leg length trajectory. This motivates tionally tractable. However, reduced-order models are often defining the leg length trajectory as a desired output on directly implemented on the full-order model of the robot, each leg and thus formulate a control Lyapunov function e.g., through inverse kinematics [7] or inverse dynamics [8], based quadratic program (CLF-QP) [11] [12] for output without a faithful connection to the structure and morphology stabilization. The end result is a nonlinear optimization-based of the robot. controller that represents the reduced-order dynamics in the In this paper, we present an approach to identifying the full-order model of the robot. spring-mass model for bipedal robots with mechanical com- The QP formulation for hopping is inspired by the ap- pliance, and synthesizing nonlinear controllers by embedding proach for walking in [4], wherein the SLIP dynamics is the spring-mass model into the full-order dynamics. Specifi- embedded onto the center of mass (COM) dynamics of the cally, the spring in the spring-mass model comes from view- full robot via an equality constraint in the QP. The difference ing each leg as a deformable prismatic spring as motivated of our approach is that the hopping dynamics is embedded by the mechanical design of robots with compliance [9]. We by taking the leg length trajectory as a desired output, which borrow the idea of end-effector stiffness from manipulation becomes an inequality CLF constraint in the QP, rendering community [10], and formally derive the stiffness/damping a more feasible QP formulation. Additionally, the hopping of the leg spring from the compliant components in the leg as motion naturally requires a consideration on momentum functions of robot configurations. This facilitates the spring- regulation due to the conservation law on centroidal angular mass model being virtually actuated by changing robot momentum [13] in flight phase. This is done by including the configurations, i.e., by changing the leg length on the spring- angular momentum as an output to stabilize in the CLF-QP. mass model. Trajectory optimization can thus be utilized to The proposed approach is successfully implemented on create hopping behaviors on the spring-mass model. the 3D underactuated bipedal robot Cassie (see Fig. 1) in The planned leg length trajectory from the spring-mass both simulation and experiment [1], achieving the hopping model encodes the underactuation of the leg compliance of on Cassie with ground clearance of ∼ 7 inches and air-time the robot, and can therefore be used to synthesize controllers of ∼ 0:423s. The ground reaction force and toe-off timing of hopping motions on the robot match closely with these *This work is supported by NSF grant NRI-1526519. of the spring-mass model. This further indicates a faithful The authors are with the Department of Mechanical and Civil Engineering, California Institute of Technology, Pasadena, CA 91125 construction and embedding of the reduced-order model onto [email protected], [email protected] the full order model of the robot. (a) II. ROBOT MODEL (b) hip pitch achilles rod hip link The Cassie-series robot from Agility Robotics [14] is a full 3D bipedal robot that is designed to be agile and robust. knee link Like its predecessor ATRIAS [9], Cassie is designed with heel spring concentrated mass at its pelvis and lightweight legs with leaf shin spring shin link springs and closed kinematic chains. The mechanical design, toe motor thus, embodies the SLIP model [2]. From the perspective main chain tarsus link of model-based control, the compliant closed chain on each leg can, however, create additional complexities. Therefore, toe link plantar rod rigid model [4] or overly simplified model [9] is oftentimes motor joints spring joints pure passive joints applied. Here, we present the full body dynamics model with justifiable simplifications. Fig. 2. (a,b) Cassie’s leg and its model. As shown in Fig. 1, Cassie has five motor joints (with the axis of rotation shown in red) on each leg, three of which locate at the hip and the other two are the knee and toe pitch. spring-mass model. It is expected that the stiffness of the leg Fig. 2(a) and (b) provide a close look at the leg kinematics spring changes with different robot configurations, thus we and the abstract model, respectively. We model the shin and explicitly derive the leg stiffness Kleg as a function of joint heel springs as torsion springs at the corresponding deflection angles, the analogy of which is the end-effector stiffness for robotic(a) manipulators [10].(b) With an eye towards the motion axes. Therefore the spring torques are: leg spring planning for the spring-mass model, Kleg is approximated by τshin/heel = kshin/heelqshin/heel + dshin/heelq_shin/heel; (1) a polynomial function of leg length L. Lastly, we present the trajectory optimization via direct collocation for the spring- where kshin/heel; dshin/heel are the stiffness and damping, pro- vided by the manufacturer [14]. Since the achilles rod is mass model. stance width very lightweight, we ignore the achilles rod and replace it by A. Leg Stiffness and Leg Length setting a holonomic constraint hrod on the distance between The leg stiffness Kleg is the resistance of the leg to the connectors (one locates on the inner side of hip joint, external forces. The complementary concept is called leg the other locates at the end of the heel spring). The plantar −1 compliance Cleg = Kleg . When the leg is under external rod is also removed and the actuation is applied to the toe load at the foot, the leg deforms due to compliant elements pitch directly thanks to the parallel linkage design. These in the leg. Assuming that we only consider the transitional two simplifications removed unnecessary passive joints and deformations, the external force can be calculated by, associated configuration variables. As a consequence, the configuration of the leg can be described only by five motor Fext = Klegδ; (4) joints, two spring joints and a passive tarsus joint. The total 3 3×3 3 where Fext 2 R ;Kleg 2 R and δ 2 R . Under the number of degrees of freedom of the floating base model is assumption that the deformation is small and only happens then n = 8 × 2 + 6 = 22. The dynamics can be derived from at the joints, the leg deformation δ can be mapped from joint the Euler-Lagrange equation with holonomic constraints as: deformations ∆q by the foot Jacobian as, T T M(q)¨q + H(q; q_) = Bu + Js τs + Jh;vFh;v; (2) δ = J∆q; (5) Jh;v(q)¨q + J_h;v(q)_q = 0; (3) where J 2 R3×n and ∆q 2 Rn. Let τ denotes the moments n T where q 2 R , M(q) is the mass matrix, H(q; q_) is the at the joints caused by the external load, thus τ = J Fext. 10 Coriolis, centrifugal and gravitational term, B and u 2 R If the stiffness at each joint is ki with i = 1; :::; n, then the are the actuation matrix and the motor torque vector, τs and joint stiffness matrix is defined as KJ = diag(k1; :::; kn), and Js are the spring joint torque vector and the corresponding τ = KJ ∆q. The joint stiffness matrix KJ and leg stiffness n Jacobian, and Fh;v 2 R h;v and Jh;v are the holonomic Kleg are hence related by the joint moments, force vector and the corresponding Jacobian respectively. The K ∆q = τ = J T F = J T K δ = J T K J∆q: (6) subscript v is used to indicate different domains which have J ext leg leg different numbers of holonomic constraints. For instance, Then the leg stiffness can be calculated from the joint when the robot has no contact with the ground, nh;v = 2 as stiffness matrix by, there are two holonomic constraints on h .