Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

of and θ as expressed below.

(2.14)

In Eq. (2.14), is defined as the rotation matrix function of the arguments and θ. In other words, generates a rotation matrix out of the arguments and θ as shown below, where may be any column matrix such that .

(2.15)

      An alternative expression can be derived for the function by recalling the following equation from Section.

(2.16)

Upon substituting Eq. (2.16) into Eq. (2.15), the alternative expression is obtained as

(2.17)

      Hence, with , can also be expressed as

      (2.18)

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