On the Damping-Induced Self-Recovery Phenomenon in Mechanical Systems with Several Unactuated Cyclic Variables

TL;DR

This paper extends the understanding of damping-induced self-recovery in underactuated mechanical systems to multiple cyclic variables, identifying new conserved quantities and demonstrating stability and trajectory bounds through analytical examples.

Contribution

It generalizes previous results to systems with several unactuated cyclic variables and introduces damping-induced momenta that lead to stable dynamics.

Findings

Damping-induced momenta create conserved quantities in multi-variable systems.

Unactuated cyclic variables exhibit asymptotic stability under viscous damping.

Damping imposes bounds on the trajectories of cyclic variables.

Abstract

The damping-induced self-recovery phenomenon refers to the fundamental property of underactuated mechanical systems: if an unactuated cyclic variable is under a viscous damping-like force and the system starts from rest, then the cyclic variable will always move back to its initial condition as the actuated variables come to stop. The regular momentum conservation phenomenon can be viewed as the limit of the damping-induced self-recovery phenomenon in the sense that the self-recovery phenomenon disappears as the damping goes to zero. This paper generalizes the past result on damping-induced self-recovery for the case of a single unactuated cyclic variable to the case of multiple unactuated cyclic variables. We characterize a class of external forces that induce new conserved quantities, which we call the damping-induced momenta. The damping-induced momenta yield first-order asymptotically stable dynamics for the unactuated cyclic variables under some conditions, thereby inducing the self-recovery phenomenon. It is also shown that the viscous damping-like forces impose bounds on the range of trajectories of the unactuated cyclic variables. Two examples are presented to demonstrate the analytical discoveries: the planar pendulum with gimbal actuators and the three-link planar manipulator on a horizontal plane.