Contact Stability Analysis of a One Degree-of-Freedom Robot

AbstractThe aim of this note is to examine the conditions of stability of a simple robotic task: we consider a one degree-of-freedom (dof) robot that collides with a spring-like environment with stiffness k, the goal being to stabilize the system in contact with the environment. We study conditions on the feedback gains that guarantee quadratic Lyapunov stability of the task with a well-conditioned solution to the Lyapunov equation. It is shown that when the environment's stiffness k grows unbounded, those conditions yield unbounded values of the gains. Motivated by the stability analysis of the impact Poincaré map in the perfectly rigid case ( $$k = + \infty$$ ), we propose an analysis that is independent of k. It enables us to conclude on global asymptotic convergence of the system's state towards the equilibrium point. This work can also be seen as the study of stability of a contact (force control) phase, taking into account the unilateral feature of the constraint.

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