Coordinate representations for rigid parts in multibody dynamics

Abstract

The paper is concerned with coordinate representations for rigid parts in multibody dynamics. The discussion is based on the general theory of the dynamics of multibody systems under constraints. General configuration coordinates are introduced and the requirement of their regularity is discussed. The use of Euler angles, quaternions and linear coordinates, where quaternions and linear coordinates require constraint conditions, is analysed in detail. These coordinate systems are all shown to be regular. Equations of motion are formulated, using Lagrange’s as well as Euler’s equations, and they are supplemented by the appropriate constraint conditions in the cases of quaternions and linear coordinates. Mass matrices are derived, and in terms of the Euler angles the mass matrix components are products of trigonometric functions whereas in terms of quaternions the matrix components are quadratic polynomials. Using linear coordinates gives rise to a constant mass matrix. Thus, there is a decreasing degree of complexity, regarding mass matrix components, when going from Euler angles to linear coordinates. This is obtained at the expense of an increasing gross number of degrees of freedom and the necessary introduction of constraint conditions. The different equations of motion obtained are compared with respect to their structural complexity. In all representations the components of the angular velocity are explicitly calculated. This is not always the case in previous investigations of this subject. The paper also gives a new proof of the well-known relation between angular velocity and unit quaternions and their time derivative.