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 ε_ijk 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 on each side of Eq. (1.51), the following column matrix equation can be written.

(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), is defined as the cross product matrix (cpm) corresponding to the column matrix . When Eqs. (1.53) and (1.54) are compared, it is seen that happens to be a skew symmetric matrix generated from the column matrix .

      Considering an arbitrary column matrix , the corresponding skew symmetric matrix is generated by means of the ssm (skew symmetric matrix) operator as described below.

      (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.

       Since is a skew symmetric matrix,

      (1.58)

       Since ,

(1.59)

       Since ,

      (1.60)

       The product of two skew symmetric matrices can be expanded as follows:

(1.61)

In Eq. (1.61), is the inner product and is the outer product of and . As for , it is the identity matrix. That is,

      (1.62)

       Let be a unit column matrix so that . Then,

(1.63)

       Equations (1.63) and (1.59)