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Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren
Читать онлайн.Название Kinematics of General Spatial Mechanical Systems
Год выпуска 0
isbn 9781119195764
Автор произведения M. Kemal Ozgoren
Жанр Математика
Издательство John Wiley & Sons Limited
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
(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,
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.
Similarly, if j = k, the three‐factor expression in Eq. (3.102) degenerates this time into the following two‐factor expression.
Equation (3.103) shows that
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
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
3.8.6