Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

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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) imply that is exponentiated as follows:

      (1.64)

      (1.65)

       The ssm operation is applied as shown below on the products and .

      (1.66)

(1.67)

      Here, it is to be noted that Eq. (1.67) is valid if is a rotation matrix, i.e. an orthonormal matrix with .

       It happens that is a singular matrix and its rank is two. That is,

(1.68)

(1.69)

1.8 Examples Involving Skew Symmetric Matrices

      1.8.1 Example 1.1

      This example is about the expansion of the following triple vector product.

(1.70)

      In a selected reference frame , the matrix version of Eq. (1.70) can be written as

(1.71)

      In Eq. (1.71), all the matrices are expressed in the same frame . Therefore, the frame indicating superscript (a) is concealed for the sake of brevity. By expanding the product according to Eq. (1.61)Eq. 1.71 can be written again as follows:

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