Touch-down condition control for the bipedal spring-mass model in walking

Behaviors of animal bipedal locomotion can be described, in a simplified form, by the bipedal spring-mass model. The model provides predictive power, and helps us understand this complex dynamical behavior. In this paper, we analyzed a range of gaits generated by the bipedal spring-mass model during walking, and proposed a stabilizing touch-down condition for the swing leg. This policy is stabilizing against disturbances inside and outside the same energy level and requires only internal state information. In order to generalize the results to be independent of size and dimension of the system, we nondimensionalized the equations of motion for the bipedal spring-mass model. We presented the equilibrium gaits (a.k.a fixed point gaits) as a continuum on the walking state space showing how the different types of these gaits evolve and where they are located in the state space. Then, we showed the stability analysis of the proposed touch-down control policy for different energy levels and leg stiffness values. The results showed that the proposed touch-down control policy can stabilize towards all types of the symmetric equilibrium gaits. Moreover, we presented how the peak leg force changes within an energy level and as it varies due to the type of the gait since peak force is important as a measurement of injury or damage risk on a robot or animal. Finally, we presented simulations of the bipedal spring-mass model walking on level ground and rough terrain transitioning between different equilibrium gaits as the energy level of the system changes with respect to the ground height. The analysis in this paper is theoretical, and thus applicable to further our understanding of animal bipedal locomotion and the design and control of robotic systems like ATRIAS, Cassie, and Digit.