3D Passive Walkers: Finding Periodic Gaits in the Presence of Discontinuities

This paper studies repetitive gaits found in a 3D passivewalking mechanism descending an inclined plane. By using directnumerical integration and implementing a semi-analytical scheme forstability analysis and root finding, we follow the corresponding limitcycles under parameter variations. The 3D walking model, which is fullydescribed in the paper, contains both force discontinuities andimpact-like instantaneous changes of state variables. As a result, thestandard use of the variational equations is suitably modified. Theproblem of finding initial conditions for the 3D walker is solved bystarting in an almost planar configuration, making it possible to useparameters and initial conditions found for planar walkers. The walkeris gradually transformed into a 3D walker having dynamics in all spatialdirections. We present such a parameter variation showing the stabilityand the amplitude of the hip sway motion. We also show the dependence ofgait cycle measurements, such as stride time, stride length, averagevelocity, and power consumption, on the plane inclination. The paperconcludes with a discussion of some ideas on how to extend the present3D walker using the tools derived in this paper.