Stability analysis of passive compass gait using linearized model

The inherent self-stabilization mechanism of passive gait is one of the most fundamental question in the area of studies on limit cycle walking. It is well known that the limit cycle stability can be tested by calculating the eigenvalues of the approximated Poincare return map, and they have been calculated numerically in previous studies. Hirata and Kokame, however, firstly succeeded to derive the Jacobian matrix of the Poincare return map using linearization of the dynamic equation. In this paper, we reconsider their method and propose an easier approach to derive the discrete-time linear error system. Our approach enables to derive the error system without integral terms. Through the theoretical investigations, we show that a passive compass-gait is formed by the combination of unstable stance phases and marginally stable collision phases. The validity is investigated by numerical simulations.