A New Stability Framework for Trajectory Tracking Control of Biped Walking Robots

This article proposes a new stability criterion for biped walking systems on the linear inverted pendulum model, in which the dynamic relationship between the center of mass (CoM) and the zero moment point (ZMP) is dealt with. More precisely, based on the fact that a biped walking robot is stable if its ZMP is always located in the supporting region, we consider whether the ZMP error between its reference and real values stays inside a certain area to guarantee the stability condition. To this end, a norm-based new stability criterion is introduced, in which the temporal supremum of the ZMP error is concerned. Regarding the applicability of the stability criterion, we propose two control approaches to biped walking systems with the consideration of the norm-based stability criterion. In other words, full-state and observer-based feedback control approaches are analyzed in this article, and the initial CoM conditions with respect to the stability criterion are obtained for the two control approaches. We call the sets derived by such conditions the stability regions. Toward a more practical significance, we also deal with the effect of unknown disturbances and sensor noises on the stability of biped walking systems. More importantly, even though computing stability regions intrinsically involves an infinite number of linear inequalities, all the stability regions are shown to be explicitly obtained through only finite numbers of computations in this article. Finally, some simulation results are provided to demonstrate the validity as well as the practical applicability of the developed computation methods.

I. Introduction

As The applications of biped robots for practical tasks are studied widely in [1]–[7], the stability problem of biped walking has been considered deeply in the literature. More importantly, a biped robot is shown in [8]–[10] to be stable if its zero moment point (ZMP) trajectory is always located inside the supporting regions. In this regard, the maximal output admissible (MOA) set [11], [12] is used in [13] to characterize the initial conditions for the center of mass (CoM), with which the aforementioned ZMP condition holds. This pioneering study proposes a method for designing feedback controllers based on the linear inverted pendulum model (LIPM) of biped robots [14], in which the dynamics between the CoM and the ZMP of the biped robots is described by a linear time-invariant system. Subsequently, more sophisticated arguments on the capture point [15] and foot position control have been recently developed in [16] and [17], respectively, with the MOA set-based treatment of biped walking systems.