Study on bipedal running on compliant ground using hybrid zero dynamics controller

Abstract We present a hybrid zero dynamics control design to develop stable, energy-efficient running gaits for a bipedal robot on compliant ground, extending previous work focused on rigid surfaces. The robot, modeled with five rigid segments and four actuated joints with point feet, has a running gait represented as a periodic hybrid dynamical system with two continuous phases and two discrete transitions. Non-slipping contact during single support is modeled using nonlinear viscoelastic contacts. Using virtual constraints, we represent the gait through reduced-order hybrid invariant zero dynamics. Periodic solutions are found via a multiple shooting method, and energy-efficient gaits are optimized by adjusting the reference trajectory within a sequential quadratic programming framework, incorporating stability and feasibility constraints. Orbital stability is assessed using Floquet theory. Ground compliance increases the dimension of the invariant zero dynamics manifold and introduces sensitivity in the contact phase equations. We overcome these challenges through a multiple shooting formulation and orthogonal projection of the monodromy matrix, isolating critical Floquet multipliers. This enables stable, efficient running gaits on compliant ground. Ground compliance does not inherently reduce energy efficiency and can enhance it despite additional contact dissipation. Moreover, compliant ground reduces impulsive landing forces, mitigating mechanical stress on the robot.