Section-Map Stability Criterion for Biped Robots Part I: Theory

Stability criterion is the key problem in the theoretical framework of biped robots, and it is the precondition of walking patterns planning and real-time control. ZMP (zero moment point) criterion and Poincare return map criterion are the two representative stability criteria for biped walking; however, they are not applicable to dynamic walking and non-periodic walking respectively. To establish a coherent stability criterion, a rigorous mathematical definition of biped walking stability is presented by combining the character of biped locomotion with the notion of classical stability from the view of hybrid trajectory. It is pointed out that, under some assumption, stability of the hybrid trajectory is equivalent to that of the section sequence at switch section in the task space of biped walking. Based on this result, section-map stability criterion is presented. This criterion is applicable not only to dynamic walking which ZMP criterion can not solve, but also to non-periodic walking which Poincare return map criterion can not solve. The applications and experiments of the proposed criterion are presented in the companion paper.