Читать книгу Kinematics of General Spatial Mechanical Systems - M. Kemal Ozgoren - Страница 68

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.