Non-Minimal Order Model of Mechanical Systems With Redundant Constraints for Simulations and Controls

Abstract

This technical note presents a non-minimal order dynamics model for many analysis, simulation, and control problems of constrained mechanical systems passing through singular configurations during their motion by making use of linear projection operator. The distinct features of this model describing dynamics of the dependent coordinates are: i) The mass matrix $\bar{M}(q)$ is always positive definite even at singular configurations; ii) matrix $\dot{\bar{M}}-2\bar{C}$ is skew symmetric, where all nonlinear terms are lumped into vector $\bar{C}(q,\dot{q})\dot{q}$ after elimination of constraint forces. Eigenvalue analysis shows that the condition number of the constraint mass matrix can be minimized upon adequate selection of a scalar parameter called “virtual mass” thereby reducing the sensitivity to round-off errors in numerical computation. It follows by derivation of two oblique projection matrices for computation of constraint forces and actuation forces. It is shown that projection-based model allows feedback control of dependent coordinates which, unlike reduced-order dependent coordinates, uniquely define spatial configuration of constrained systems.