M. Kemal Ozgoren - Kinematics of General Spatial Mechanical Systems

(2.24)

Based on the above analogy, the function can be expressed exponentially as

(2.25)

Hence, the rotation matrix can also be expressed exponentially as

(2.26)

Note that the exponential expression of is not only very compact but it also indicates the angle and axis of the rotation explicitly. Therefore, the exponentially expressed rotation matrices can be used quite conveniently in the symbolic manipulations required in the analytical treatments within the scope of rotational kinematics. For the sake of verbal brevity, an exponentially expressed rotation matrix is simply called here an exponential rotation matrix .

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 k th basic rotation with respect to takes place about the k th coordinate axis of . Therefore, the unit vector of the rotation axis of this basic rotation is the k th basis vector of , i.e. . The operator of this basic rotation is denoted as

(2.27)

The k th basic rotation operator associated with is represented in by the matrix , which is designated as the k th 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)

Suppose a vector is first rotated into a vector and then is rotated into another vector . These two successive rotations can be described as indicated below.