A Theoretical Framework for Stability Regions for Standing Balance of Humanoids Based on Their LIPM Treatment

The aim of this paper is to construct a theoretical framework for stability analysis relevant to standing balance of humanoids on top of the linear inverted pendulum model, in which their dynamics between the center of mass (CoM) and the zero moment point (ZMP) is dealt with. Based on the well-known sufficient condition that the contact between the ground and the support leg is stable if the corresponding ZMP is always inside the supporting region, this paper aims at characterizing three types of the associated stability regions. More precisely, assuming no external force disturbances affecting the motion of the humanoids, the stability region of the initial CoM position and velocity values can be explicitly computed by solving a finite number of linear inequalities. The stability regions of time-invariant force disturbances such as impulsive force and constant force disturbances are also dealt with in this paper, where the former is exactly obtained through a finite number of linear inequalities while the latter is approximately derived by using an idea of truncation. Furthermore, time-varying force disturbances of finite energy and finite amplitude are concerned with, and their maximum admissible ${l} _{2}$ and ${l} _{\infty }$ norms are computed in this paper, where the former can be exactly obtained by solving the discrete-time Lyapunov equation while the latter is approximately derived through an idea of truncation. It is further shown for both the truncation ideas that the approximately obtained stability regions converge to the exact stability regions with an exponential order of ${N}$ , where ${N}$ is the truncation parameter. Finally, the effectiveness of the computation methods proposed in this paper is demonstrated through some simulation results.