Predicting foot placement for balance through a simple model with swing leg dynamics

Stepping is one important strategy to restore balance against external perturbations. Although current literature have proposed models to predict the recovery foot placement, swing leg actuation is rarely taken into account. In this paper, we combine the capturability-based analysis with swing leg dynamics and seek to contribute to the following problem: for a biped system recovering balance from external perturbations, how to choose a step position and duration in minimizing swing actuation cost? We expand the linear inverted pendulum model with an actuated linear pendulum mounted on the pelvis, the addition of which is proposed to describe the swing leg dynamics. The closed-form expression of swing actuation with constraints is derived from the explicit formulations of the pelvis and swing foot motion. We calculate the optimal step position and duration to minimize swing cost under various perturbations. Results show that the optimal step duration keeps constant, while the optimal step position is linearly proportional to the magnitude of perturbations. Such findings match well with experimental data from ten subjects delivered with waist-perturbations. These current results demonstrate that our proposed model with swing dynamics suggests an effective alternative to predict recovery foot placement of biped systems following unexpected perturbations.