Читать книгу Kinematics of General Spatial Mechanical Systems - M. Kemal Ozgoren - Страница 32

A rotation may be carried out about one of the coordinate axes of a reference frame . Such a rotation is defined as a basic rotation with respect to . More specifically, the kth basic rotation with respect to takes place about the kth coordinate axis of . Therefore, the unit vector of the rotation axis of this basic rotation is the kth basis vector of , i.e. . The operator of this basic rotation is denoted as

(2.27)

The kth basic rotation operator associated with is represented in by the matrix , which is designated as the kth basic rotation matrix. It is expressed as follows:

(2.28)

Referring to Section for the discussion about the basic column matrix , it is to be noted that, just like , the basic rotation matrix is also an entity that is not associated with any reference frame. This is because represents the rotation operator in its own frame , whatever is. In other words,

(2.29)

By using Eqs., can be expressed in three equivalent ways as shown in the following equations.

(2.30)

(2.31)

(2.32)

Upon inserting the expressions of the basic column matrices into Eq. (2.30), the basic rotation matrices can be expressed element by element as shown below.

(2.33)

(2.34)

(2.35)