A global stability analysis of legged balance controllers under ZMP constraints

Two different basic ideas to control the balance of biped walking robots are: (i) constraining the ground reaction force by the body motion; and (ii) body motion control by the ground reaction force. In (i), the control problem is formulated as body motion planning considering the Zero Moment Point (ZMP). Essentially, this method is not applicable for real-time situations because motion planning and control execution are separated from each other. In contrast, the control formulation (ii) which controls the body motion by manipulating the ground reaction force is applicable for real-time control. However, the global stability under ZMP constraints has not yet been fully studied. This paper analyzes the stability of biped walking robots by formulating the control problem using the simple models based on COM and ZMP. First, we explain our motivation by showing the difficulty of the balance control of biped robots based on (i). We demonstrate the deviation of the ZMP from desired path during transient response numerically. Then, we propose a new control algorithm which is based on formulation (ii). In our method we emphasize the importance of ground vertical reaction force for the global stability. Unlike conventional methods which limit the horizontal forces (i.e., limit the horizontal COM acceleration), we constrain the vertical force. The proposed control method is globally stable, while allowing large horizontal acceleration. Therefore, it is applicable for fast dynamic robot motion. The stabilization mechanism is explained by using the motion trajectories of COM in the phase plane.