Towards Fully Reactive Multi-step Generation for Humanoids against Instantaneous Push: A Case of Walking in Place in Sagittal Plane

In this paper, we address the problem of generating a trajectory of the zero-moment point (ZMP) and the rate of angular momentum for a bipedal robot in the sagittal plane, with which the balance of the robot is recovered from external push. Unlike most previous works that adjusted a pre-designed ZMP or solved (possibly too heavy) nonlinear optimization problems, our main purpose is to develop a fully reactive step generator in the sense that (a) no pre-calculation of nominal trajectory is required, and (b) the algorithm is simple enough to operate in real time, only by utilizing the current state of the robot. For the design, it is seen by reinterpreting the centroidal dynamics in the hybrid model framework that the balance recovery problem can be recast as the problem of stabilizing a hybrid-type (linear) inverted pendulum model. On the basis of the concept of the divergent component of motion, a simple hybrid control law is then constructed to stabilize the hybrid system, which serves as a step generator that automatically determines where and when to step. This paper briefly sketches a mathematical proof on the performance of the proposed generator from a control-theoretic perspective, which is also supported by simulation results.