Читать книгу Kinematics of General Spatial Mechanical Systems - M. Kemal Ozgoren - Страница 24

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)