Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

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alt="images"/> in the original definition of the Euler angles.

      Although , , and may be specified arbitrarily in general, in almost all the practical cases, each of them is specified as a selected basis vector of a selected reference frame. Thus, different Euler angle sequences arise depending on the selected reference frames and their selected basis vectors. All such Euler angle sequences are grouped into two main categories, which are designated as IFB and RFB sequences. These sequences are described and explained below.

      3.8.2 IFB (Initial Frame Based) Euler Angle Sequences

      In an IFB sequence, e.g. the IFB i‐j‐k sequence, each of the unit vectors of the rotation axes is specified as one of the basis vectors of the initial reference frame . That is,

      (3.90)

      The specified unit vectors must be such that j ≠ i and j ≠ k. Such a rotation sequence can be described as shown below.

      (3.91)

In such a sequence, the matrix representations of all the rotation operators are expressed naturally in . In other words,

      (3.92)

      (3.93)

      (3.94)

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

(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

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