On the dynamics and Lyapunov stability of constrained and embedded rigid bodies

Abstract

Lyapunov stability of constrained and embedded rigid bodies is considered. The constraints are of the equality type where the desired motion is to take place on an a priori defined submanifold of movement. Special and augmented state spaces for the representation of systems of rigid bodies are presented. A systematic method of stabilizing these augmented systems and a procedure for constructing Lyapunov functions are presented. The representation is applicable to augmented as well as reduced state spaces of the system defined by the constraints. The augmented state space results from the embedding of the free rigid body system in the larger state space of free rigid body and position control states, and in which the Lyapunov function is constructed. The reduced state space results when the system is restricted and is reduced to lie on the submanifold of movement. It is shown that, for the class of rigid bodies and the physical constraints considered, the projected feedback structures, and the reduced Lyapunov function constitute appropriate stabilizing structures for the constrained system. It is shown that the method applies equally to holonomically constrained and visco-elastically coupled rigid bodies. Digital computer simulations of one single rigid body system are presented to demonstrate the feasibility and effectiveness of the method. Applications to natural systems and the role of cartilage, ligaments and muscles in maintaining the integrity and stability of the joints are noted.