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Abstract 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.

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Once the concept of passive walking was appeared, control of biped robots based on dynamically stable periodic gaits around a stable limit cycle became of interest of many researchers and it has been further accelerated nowadays. The authors have previously shown that in addition to passive walking, a passive, biped walker could interestingly show asymptotically stable turning motion over a novel 3D surface called "helical slope". In this paper, based on passive turning concept, a control method would be offered which is effective for 3D biped robots. The approach is based on potential energy shaping that is usually applied for walking control. In the proposed method, asymptotically stable passive turning motions that are performed on a certain helical slope are projected to 3D motions over flat ground and along a circular path (which is the image of the helical slope on the ground). The biped model used in this study, is a 3D model of compass gait robot with flat feet and flexible ankles that could generated stable passive turning motions. The simulation results show the effectiveness of the proposed method as well.