Asymptotically stable walking control of a 3D biped robot via potential energy shaping approach

Control of biped robots based on the concept of asymptotical stable periodic motions have become of interest of researchers nowadays. Potential energy shaping, one of the most significant approaches in this regard, has been presented and evaluated well on planar 2D models, so far. In this paper, this concept is developed and investigated for general three-dimensional case, in the presence of non-holonomic constraints. At First, the considered biped model is a 3D compass gait model with finite hip width and arc shaped feet whose stable passive walking has been shown in previous researches. In this approach, the passive periodic gaits which may be adopted for a particular ground slope can be reproduced on any arbitrary ground slope such as flat surface. In fact, thanks to the invariance property of kinetic energy as well as equivariance property of collision map with respect to slope changing action, this important goal is reached only by compensating the potential energy similar to that of passive walker. In another word, inducing a controlled symmetry to the system Lagrangian, we impose a virtual gravity in a new direction resembling the gravity direction of passive walker with respect to the ground. At the end, regarding practical challenges about the implementation of arc feet model, a compass gait model with flat feet and springs at the ankle joint has been proposed instead and the aforementioned control approach is applied again. Simulation results show the effectiveness of the presented approach for both models well.