Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

the above equations, and are the rotation matrices that represent the same rotation operator in and , respectively. That is,

      (3.69)

      (3.70)

      As for the vectors and , their matrix representations are related to each other as follows by the transformation matrix :

(3.71)

(3.72)

When Eqs. (3.71) and (3.72) are substituted, Eq. (3.68) becomes

(3.73)

When Eqs. (3.67) and (3.73) are compared, it is seen that the matrices and are related to each other as follows by using both and :

(3.74)

3.7 Expression of a Transformation Matrix in a Case of Several Successive Rotations

      Suppose a reference frame is rotated into another reference frame through several successive rotations as described below.

(3.75)

      In such a rotational sequence, the overall transformation matrix can be obtained as follows:

(3.76)

The matrix can be related to the rotation operators that appear in Description (3.75) through two different commonly used formulations depending on the reference frames in which the rotation operators are expressed. These two formulations are explained below.

      3.7.1 Rotated Frame Based (RFB) Formulation

In this case, rot(p, q) is expressed as a rotation matrix in one of the two relevant reference frames, i.e. either in the pre‐rotation frame or in the post‐rotation frame . Thus, the transformation matrix is related to rot(p, q) as follows according to Eq. (3.64):

      (3.77)

Then, Eq. (3.76) gives with one of the following equivalent expressions.

      (3.78)

      (3.79)

      On the other hand, is also equal to the overall rotation matrix expressed in either or . That is,

      (3.80)

      As noted above, in the rotated frame based (RFB) formulation, the rotation matrices are multiplied in the same order as the order of the rotation sequence indicated in Description (3.75).

      3.7.2 Initial Frame Based (IFB) Formulation

      In this case, all the rotation operators are expressed as the following rotation matrices in the initial reference frame .

      (3.81)

Of course, in such a formulation, except , none of the above rotation matrices is a transformation matrix. Therefore, the transformation matrices required in Eq. (3.76) can be obtained as shown below by means of Eq. (3.74).

      (3.82) Скачать книгу