Nonlinear Position-Dependent Circuits: A Language for Describing Motions of Nonlinear Mechanical Systems

Abstract

The analogy between the dynamics of a linear lumped-parameter electric circuit consisting of resistor-inductor-capacitor (RLC) in series or in parallel and that of a linear one degree-of-freedom mechanical system of dashpotmass-spring is well known and has been pointed out widely in the literature. This analogy was extensively made use of in robotics by Mason (1981) and Hogan (1985), who introduced new concepts of “compliance control” and “impedance control” respectively in control of physical interactions of a manipulator end-effector with objects or robot task environments. However, this impedance or compliance control concept in robotics has not yet been generalized to cope with nonlinearities in dynamics of the objective system. On the other side, generalization of electric circuits towards nonlinear circuits was attempted by Brayton and Moser (1964), who introduced a new concept called the “mixed potential function” and demonstrated a procedure to construct Lyapunov-type functions from the potential function to prove stability under certain conditions. However, this approach and successive ones such as the EL(Euler-Lagrange) formalism of an electric circuit by Meisel (1966) are limited to treatments of only “electric circuits” and do not take into consideration a large class of nonlinear dynamics related to “mechanical” motion. In other words, to express motions of mechanical systems, not only velocity vectors but also position vectors are key variables on which the potential or the kinetic energy itself depends. That is, the position q is always and explicitly paired with the velocity q̇, and hence some nonlinear elementary blocks must be considered to be position-dependent.