Self recovery phenomenon of mechanical systems with an unactuated cyclic variable

Conservation laws in nature correspond to symmetries of related physical systems (Noether´s theorem). Conservation of momentum, for instance, can be interpreted by the symmetric property of a certain motion variable known as the cyclic variable. If a symmetry-breaking force such as a dissipative force or the gravitational force, is applied to the cyclic variable, then the momentum is not conserved any longer in general. The main objective of this paper is to show that there exists a particular type of viscous damping-like force that breaks the symmetry but induces a new conserved quantity in place of the original momentum map. This new conserved quantity can be constructed by combining the time integral of a force linear in velocity and the original momentum map associated with the symmetry. In terms of stability theory of dynamical systems, it can be shown that the existence of the new conserved quantity implies that the corresponding motion variable possesses, as we define in this paper, the self recovery phenomenon. More specifically, the corresponding motion variable will be globally attractive to the initial condition of the variable.We discover that what is fundamental in this self recovery phenomenon is not the positivity of the coefficient of the force linear in the velocity, but certain properties of the time integral of the coefficient function, which can encompass a wide range of viscous damping forces. The self recovery effect and theoretical discoveries are demonstrated by simulation results using two examples: Elroy´s beanie, and the torque-controlled inverted pendulum on a passive cart. The results in this paper will be useful in designing and controlling mechanical systems with underactuation.