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2.3 Exponentially Expressed Rotation Matrix

The expression of the rotation matrix function , which is given by Eq. (2.17), can be manipulated further as explained below.

      Recall that sin θ and cos θ can be expressed as follows by using their Taylor series expansions:

(2.19)

(2.20)

When Eqs. (2.19) and (2.20) are substituted into Eq. (2.17), the Taylor series expansion of is obtained as

(2.21)

      On the other hand, as shown in Section, has the following exponentiation properties.

      (2.22)

Hence, Eq. (2.21) can be written again as

(2.23)

Note that the series expression on the right‐hand side of Eq. (2.23) is analogous to the Taylor series expansion of the exponential function of a scalar x, which is written below.

      (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.

2.4 Basic Rotation Matrices

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)

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