Dual Vectors of the Kinematic and the Dynamic Kind in Matrices

Abstract

Dual vectors are a very efficient mathematical tool for the treating of line vectors and screws. Their great advantage is the consequent analogy to the algebra with the usual vectors. The real parts of dual vectors are in total accordance with the usual vectors. As there does not yet exist commonly a software using the dual unit we apply either subroutines or matrices. Matrices are into the bargain very useful in mechanics. Therefore we decide in this paper on the application of matrices and transfer the dual arithmetic and the dual vector algebra to matrix operations. The transition from kinematics to dynamics and vice versa is performed by conversion matrices for instance of inertia or of elastic compliance which convert real parts into dual parts and dual parts into real parts with changed physical dimensions (correlation of screws in ray coordinates with screws in axis coordinates). The dual vectors for the kinematic state of a rigid body with respect to another are represented in a coordinate system fixed in the considered body instantaneously coinciding in the base system. The absolute kinematic dual quantities in the body fixed system have to be multiplied after transformations with the inertia matrix of the body and also regarded the absolute turn of the body. The results are matrix formulas which yield kinetic wrenches as dual vectors verifiable by other methods [11], [13], [14], [17].