Quadrupedal bounding with a segmented flexible torso: passive stability and feedback control

In this paper, the implications of torso flexibility on the dynamics of quadrupedal running are examined in a template setting. In the same vein with the spring loaded inverted pendulum, a reductive sagittal-plane model with a segmented flexible torso and compliant legs is introduced to capture the dynamics of bounding in the presence of torso flexibility via a minimum number of variables and parameters. Numerical return map studies of the system in dimensionless setting reveal that a large variety of cyclic bounding motions can be realized passively, through the natural interaction of the model with its environment. Despite the simplicity of the model, the resulting motions correspond to torso bending movements that resemble those in galloping mammals without explicit reliance on the fine structural and morphological details. Furthermore, for certain combinations of the system parameters-in particular the torso and leg relative stiffness-self-stable bounding motions emerge. The implications of the existence of such self-stable bounding orbits to control design are also discussed and a hybrid control law is derived that is capable of stabilizing the system as it encounters significantly large disturbances using only a single actuator located at the torso joint.