This paper develops feedback controllers for walking in 3D, on level ground, with energy efficiency as the performance objective. Assume The Robot Is A Sphere (ATRIAS) 2.1 is a new robot that has been designed for the study of 3D bipedal locomotion, with the aim of combining energy efficiency, speed, and robustness with respect to natural terrain variations in a single platform. The robot is highly underactuated, having 6 actuators and, in single support, 13 degrees of freedom. Its sagittal plane dynamics are designed to embody the spring loaded inverted pendulum (SLIP), which has been shown to provide a dynamic model of the body center of mass during steady running gaits of a wide diversity of terrestrial animals. A detailed dynamic model is used to optimize walking gaits with respect to the cost of mechanical transport (CMT), a dimensionless measure of energetic efficiency, for walking speeds ranging from 0.5 (m/s) to 1.4 (m/s). A feedback controller is designed that stabilizes the 3D walking gaits, despite the high degree of underactuation of the robot. The 3D results are illustrated in simulation. In experiments on a planarized (2D) version of the robot, the controller yielded stable walking.

This paper presents a systematic approach for the design of continuous-time controllers to robustly and exponentially stabilize periodic orbits of hybrid dynamical systems arising from bipedal walking. A parameterized family of continuous-time controllers is assumed so that (1) a periodic orbit is induced for the hybrid system, and (2) the orbit is invariant under the choice of controller parameters. Properties of the Poincaré map and its first- and second-order derivatives are used to translate the problem of exponential stabilization of the periodic orbit into a set of bilinear matrix inequalities (BMIs). A BMI optimization problem is then set up to tune the parameters of the continuous-time controller so that the Jacobian of the Poincaré map has its eigenvalues in the unit circle. It is also shown how robustness against uncertainty in the switching condition of the hybrid system can be incorporated into the design problem. The power of this approach is illustrated by finding robust and stabilizing continuous-time feedback laws for walking gaits of two underactuated 3D bipedal robots.

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Abstract-Models of robotic bipedal walking are hybrid, with a differential equation describing the stance phase and a discrete map describing the impact event, that is, the non-stance leg contacting the walking surface. The feedback controllers for these systems can be hybrid as well, including both continuous and discrete (event-based) actions. This paper concentrates on the eventbased portion of the feedback design problem for 3D bipedal walking. The results are developed in the context of robustly stabilizing periodic orbits for a simulation model of ATRIAS 2.1, a highly underactuated 3D bipedal robot with series-compliant actuators and point feet, against external disturbances as well as parametric and nonparametric uncertainty. It is shown that leftright symmetry of the model can be used to both simplify and improve the design of event-based controllers. Here, the eventbased control is developed on the basis of the Poincaré map, linear matrix inequalities (LMIs), and robust optimal control (ROC). The results are illustrated by designing a controller that enhances the lateral stability of ATRIAS 2.1.

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