3D passive walkers: Finding periodic gaits in the presence of discontinuities

This paper studies repetitive gaits found in a 3D passive walking mechanism descending an inclined plane. By using direct numerical integration and implementing a semi-analytical scheme for stability analysis and root finding, we follow the corresponding limit cycles under parameter variations. The 3D walking model, which is fully described in the paper, contains both force discontinuities and impact-like instantaneous changes of state variables. As a result, the standard use of the variational equations is suitably modified. The problem of finding initial conditions for the 3D walker is solved by starting in an almost planar configuration, making it possible to use parameters and initial conditions found for planar walkers. The walker is gradually transformed into a 3D walker having dynamics in all spatial directions. We present such a parameter variation showing the stability and the amplitude of the hip sway motion. We also show the dependence of gait cycle measurements, such as stride time, stride length, average velocity, and power consumption, on the plane inclination. The paper concludes with a discussion of some ideas on how to extend the present 3D walker using the tools derived in this paper.