Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

rel="nofollow" href="#ulink_ede7b3df-52f5-59aa-8935-f65affc1b523">(3.95)

      3.8.3 RFB (Rotated Frame Based) Euler Angle Sequences

      In an RFB sequence, e.g. the RFB i‐j‐k sequence, each of the unit vectors of the rotation axes is specified as one of the basis vectors of the reference frames , , and , respectively. That is,

      (3.96)

      The specified unit vectors must be such that j ≠ i and j ≠ k. However, since the rotation axes between the pre‐rotation and post‐rotation frames are common, the following equations can also be written for the unit vectors of the rotation axes.

      (3.97)

      Such a rotation sequence can be described as shown below.

      (3.98)

      In a sequence that has the axis unit vectors specified as shown above, i.e. as the basis vectors of the pre‐rotation frames, the matrix representations of the rotation operators are also expressed naturally in the pre‐rotation frames. In other words,

      (3.99)

      (3.100)

      (3.101)

      Hence, according to the RFB formulation explained in Section 3.7, is obtained as follows:

(3.102)

      3.8.4 Remark 3.4

In an Euler angle sequence, irrespective of whether it is an IFB i‐j‐k or an RFB i‐j‐k sequence, the indices must be such that j ≠ i and j ≠ k in order to keep the angles φ₁, φ₂, and φ₃ independent. Otherwise, these angles can no longer be independent.

For example, if j = i, the three‐factor expression in Eq. (3.102) degenerates into the following two‐factor expression.

(3.103)

      Similarly, if j = k, the three‐factor expression in Eq. (3.102) degenerates this time into the following two‐factor expression.

(3.104)

Equation (3.103) shows that has a missing parameter and it is expressed in terms of only two independent parameters, which are φ₄ and φ₃. This is because φ₁ and φ₂ happen to be indistinguishable and indefinite rotation angles about the same axis with a combined effect that can actually be achieved by a single rotation angle φ₄. In other words, φ₁ and φ₂ happen to be dependent on each other because they complement each other to the effective rotation angle φ₄, that is, φ₁ + φ₂ = φ₄.

Equation (3.104) shows a similar situation with a different effective rotation angle φ₅. In this case, φ₂ and φ₃ happen to be indistinguishable and indefinite rotation angles, which are dependent because they complement each other to the effective rotation angle φ₅, that is, φ₂ + φ₃ = φ₅.

      On the other hand, it is possible to have k = i ≠ j. Based on this possibility, an Euler angle sequence is called symmetric if k = i and asymmetric if k ≠ i. For example, the RFB 1‐2‐3 sequence is asymmetric, whereas the RFB 3‐1‐3 sequence is symmetric.

      3.8.5 Remark 3.5

The comparison of Eqs. (3.95) and (3.102) shows that any transformation matrix obtained by an IFB sequence can also be obtained by an RFB sequence applied in the reversed order.

      For example, the IFB 1‐2‐3 sequence (with the Euler angles φ₁, φ₂, and φ₃) and the RFB 3‐2‐1 sequence (with the Euler angles , , and ) give the same transformation matrix with the following relationships between the Euler angles.

      The IFB and RFB sequences mentioned above can be described as shown below.

      Both of the above sequences lead to the same transformation matrix, which is

      (3.105)

      Note that, although and are the same in the two sequences described above, the corresponding intermediate frames are obviously different. That is, and .

      3.8.6