A new method for finding the proper initial conditions in passive locomotion of bipedal robotic systems

In this paper, the walking motion of a biped robot consisting of upper and lower body members (arms, legs, upper trunk and neck) on an inclined surface is analyzed kinematically and dynamically. Some of the challenges faced by this research include the dual-phase dynamics that govern the periodic gait of this robotic system and the consideration of all the body parts that are involved in the walking motion; which make it extremely difficult to derive the kinetic equations of such robotic systems. To deal with these challenges, the Gibbs-Appell formulation and the Newton's impact laws are employed to derive the most general form of the system's dynamic equations in the swing and transient phases of motion. Subsequently, by using the obtained motion equations and implementing a systematic procedure, we achieve an eigenvalue problem. By solving this problem, the suitable initial conditions that are necessary for the passive gait of this biped robot on a sloping surface are determined. By considering the obtained initial conditions as well as the dynamic response of the system to the Earth's gravity, we describe a general method for designing a biped robot's walking trajectory on an inclined surface. The most important feature of the designed trajectory is its close agreement with the trajectory followed by a biped robot walking passively down a sloping ramp. This close match between the said trajectories enables us to guide the robot in tracking a desired path by applying very small control torques to the robot joints at the start of each step. This claim is corroborated by the simulations carried out for a biped walking robot composed of 6 rigid links, in which the control torques that are applied to the actuated robot joints are obtained by employing a feedback linearization method and via a designed LQR controller.