Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

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The rotation of into is actually achieved by rotating the basis vector of into the basis vector of for all k ∈ {1, 2, 3}. That is,

      (3.55)

      As indicated above, the considered rotation is achieved by means of the rotation operator rot(a, b), which is shown below with its angle‐axis detail.

      (3.56)

      Consequently, for k ∈ {1, 2, 3}, the basis vectors of and are related to each other as follows according to the Rodrigues formula:

(3.57)

Suppose that the rotation of into is observed in a third reference frame . Then, the following matrix equation can be written in in correspondence to Eq. (3.57).

(3.58)

In Eq. (3.58), is the rotation matrix that represents the rotation operator rot(a, b) in . In other words,

      (3.59)

      By introducing the transformation matrices and , Eq. (3.58) can also be written as shown below.

(3.60)

Recalling that and , Eq. (3.60) can be manipulated into

(3.61)

Equation (3.61) implies that or

(3.62)

      On the other hand,

      (3.63)

Therefore, in either of the two special cases with and , Eq. (3.62) leads to the result that

(3.64)

      However, unless or , Eq. (3.62) implies the inequality that

      (3.65)

      3.5.2 Distinction Between the Rotation and Transformation Matrices

      Here, it has been shown that the transformation matrix between two reference frames and can be expressed as a rotation matrix if that rotation matrix represents the rotation operator rot(a, b) in either or but not in a third different reference frame .

3.6 Relationship Between the Matrix Representations of a Rotation Operator in Different Reference Frames

Consider the following rotation.

      (3.66)

      This rotation can be described by the following matrix equations expressed in two different reference frames and .

(3.67)

(3.68)

      In