Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren
On the other hand, according to Eq. (1.25),
(1.48)
Hence, Eq. (1.47) becomes
(1.49)
Equation (1.49) implies that
(1.50)
By using the definition of εijk given by Eq. (1.26), Eq. (1.49) can be worked out to what follows:
(1.51)
Upon comparing the coefficients of the basis vectors of
(1.52)
Equation (1.52) can be written again as follows by factorizing the column matrix expression on its right side:
(1.53)
Furthermore, Eq. (1.53) can be written compactly as
(1.54)
In Eq. (1.54),
Considering an arbitrary column matrix
(1.55)
The inverse of the ssm operator is the colm (column matrix) operator, which is defined so that
(1.56)
Coming back to the cross product operation, Eqs. (1.47) and (1.54) imply the following mutual correspondence, which shows how the cross product of two vectors can be equivalently expressed by using the matrix representations of the vectors in a reference frame such as
(1.57)
1.7 Mathematical Properties of the Skew Symmetric Matrices
The skew symmetric matrices have several mathematical properties that turn out to be quite useful especially in the symbolic matrix manipulations. These properties are shown and explained below by concealing the frame indicating superscripts for the sake of brevity.
(1.58)
(1.59)
(1.60)
(1.61)
In Eq. (1.61),
(1.62)
(1.63)