On First-Order Decoupling of Equations of Motion for Constrained Dynamical Systems

Abstract

In this paper we present a method for obtaining first-order decoupled equations of motion for multirigid body systems. The inherent flexibility in choosing generalized velocity components as a function of generalized coordinates is used to influence the structure of the resulting dynamical equations. Initially, we describe how a congruency transformation can be formed that represents the transformation between generalized velocity components and generalized coordinate derivatives. It is shown that the proper choice for the congruency transformation will insure generation of first-order decoupled equations of motion for holonomic systems. In the case of nonholonomic systems, or holonomic systems with unreduced configuration coordinates, we incorporate an orthogonal complement in conjunction with the congruency transformation. A pair of examples illustrate the results. Finally, we discuss numerical implementation of congruency transformations to achieve first-order decoupled equations for simulation purposes.

Non-Adaptive Sliding Mode Controllers in Terms of Inertial Quasi-Velocities

“…From the above derivation arises that Herman (2005b;2006) is an extension of the method Loduha & Ravani (1995) and it is based on the NGVC with M(θ)=Φ T Φ. Hence the two first-order equations (the diagonalized equation of motion and the velocity transformation equation) for rigid manipulator can be rewritten in the form:…”

“…The first of here considered decomposition methods is based on the generalized velocity components (GVC) Loduha & Ravani (1995). In this method M(θ)=Υ −T NΥ −1 .The equations were proposed by Loduha & Ravani (1995).…”

“…The IQV mean that the quasi-velocities contain the kinematic and dynamic parameters of a rigid manipulator as well as its geometrical dimensions. In spite that there exist several IQV, only some of them are considered here, namely: the GVC described in Loduha & Ravani (1995), the NQV given in Jain & Rodriguez (1995), and the NGVC presented in Herman (2005b;2006). It is because these kind of IQV very well explain the idea of non-adaptive sliding mode control in terms of the quasi-velocities.…”

“…There exist several methods which enable such decomposition (e.g. Hurtado (2004) ;Jain & Rodriguez (1995); Junkins & Schaub (1997); Loduha & Ravani (1995); Sovinsky et al (2005)). The method presented in Hurtado (2004) is associated with the Cholesky decomposition Sovinsky et al (2005).…”

Velocity Controller for a Class of Vehicles

“…There are various decomposition methods of the matrix M . In this paper, we refer to the method described in [14] which is based on generalized velocity components introduced in [19] and applied for robotics manipulators in [15]. The decomposition strategy is easy comprehensible and convenient for numerical implementation.…”