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Maintaining the balance when a robot is pushed forward is fundamental for legged robots. An essential topic in the literature is capture point; it is the place where the robot can step to in order to recover from the push. In this work, we study the Variable-Height Inverted Pendulum (VHIP) as the model for a pushed robot. We found all of the points that are allowed to be a capture point for given initial velocities of the center of mass and actuator limitations. We also develop a controller to reach a capture point using the reaction force to the ground as input variable. We pay attention to the unilateral contact and the maximum-value of a function of the reaction force. First, we obtain the necessary conditions that must be satisfied to be able to achieve balance by providing a decomposition of the VHIP into a new Divergent Component of Motion and a Convergent Component of Motion. Then we present two control laws to stabilize the system and we show that the region of attraction is equivalent to the region of necessary condition for balance. Finally, we briefly discuss the physical places where balance can be achieved.
G. Garcia-Chavez (Tue,) studied this question.