Stable periodic motion of a controlled segmented leg model of pedal locomotion with inelastic ground-foot collision

Human and robotic legged locomotion can be described with complex multi-degree-of-freedom dynamic models, whose bifurcation or parameter analysis may explain some features of typical patterns during motion. In this paper, we focus on the effect of kinematic parameters and foot placement techniques on the ground-foot impact intensity. The work is based on a multibody dynamic model of a segmented leg, which possesses some fundamental characteristics of locomotion systems: (a) distinct topology in the flight and ground phases; (b) kinetic energy absorption due to partially/fully inelastic ground-foot collision; (c) an active control strategy for maintaining a prescribed mechanical energy level; and (d) different control strategies for the flight and ground phases. We obtain a quantitative measure for the foot collision intensity by analytic calculations. The pre-impact velocity conditions are obtained from a hopping three-segmented planar leg model that imitates pedal locomotion. The single-legged model contains the foot, the shank, the thigh and a reaction wheel attached to the hip, which models the effect of the upper body. The existence of stable periodic motion associated with hopping is shown for a wide range of parameters by means of finding suitable control torques in the ankle, the knee and the hip joint. The parameters of the linear feedback controller are tuned to optimize different cost functions, such as running speed, energy efficiency and impact intensity. We also investigate how the stability of the periodic motion depends on the control gains.