Non-Minimal Systems with Switching Topology: Dynamics and Controls

TL;DR

This paper introduces a non-minimal order dynamics model for constrained mechanical systems with switching topology, featuring positive definite mass matrices and skew symmetric properties, enabling improved numerical stability and feedback control.

Contribution

The paper develops a novel non-minimal dynamics model using linear projection operators that ensures positive definiteness and skew symmetry, facilitating better analysis and control of switching topology systems.

Findings

Mass matrix remains positive definite at singularities

Eigenvalue analysis reduces numerical sensitivity

Projection-based control enables spatial configuration feedback

Abstract

This paper presents a non-minimal order dynamics model for many analysis, simulation, and control problems of constrained mechanical systems with switching topology 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.