Dynamics and control of constrained mechanical systems in terms of reduced quasi-velocities

A unified formulation for deriving the equations of motion of constrained or unconstrained multi-body systems (MBS) in terms of (reduced) quasi-velocities is presented. In this formulation, the square-root of the mass matrix is used to transform disparate units into homogeneous units for all the quasi-velocities resulting the gauge invariance. We show that the square-root factorization of mass matrix and hence the quasi-velocities are not unique, rather they are related by unitary transformations. Subsequently, we show that a particular transformation leads to significant simplification of the dynamic modeling. The number of differentiations required to derive the equations of motion is reduced. This fact combined with the fact that the expression of the inverse of the mass matrix factorization can be given in a closed-form make the formulation suitable for symbolic manipulation or numerical computation. Moreover, in this formulation the equations of motion are decoupled from those of constrained force and each system has its own independent input (that is not attainable by other formulations). This allows the possibility to develop a simpler force control action that is totally independent from the motion control action. The structure of the formulation is also suitable for control purposes. Tracking control and regulation control of constrained multi-body systems based on a combination of feedbacks on the vectors of the quasi-velocity and the configuration variables (which may contain redundant variables) are presented.