M. Kemal Ozgoren - Kinematics of General Spatial Mechanical Systems

(3.101)

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

(3.102)

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 φ 1, φ 2, and φ 3independent. 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 φ 4and φ 3. This is because φ 1and φ 2happen 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 φ 4. In other words, φ 1and φ 2happen to be dependent on each other because they complement each other to the effective rotation angle φ 4, that is, φ 1+ φ 2= φ 4.

Equation (3.104)shows a similar situation with a different effective rotation angle φ 5. In this case, φ 2and φ 3happen to be indistinguishable and indefinite rotation angles, which are dependent because they complement each other to the effective rotation angle φ 5, that is, φ 2+ φ 3= φ 5.

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.

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 φ 1, φ 2, and φ 3) 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 .

Relying on Remark 3.5, the IFB sequences are almost never used in practice. One reason for this may be the difficulty of visualizing the rotational steps of an IFB sequence while is rotated into . The visualization of the rotational steps of an RFB sequence happens to be much easier. Another reason why the IFB sequences are not preferred may be the reversal in the order of the rotational steps and the order of multiplying the corresponding rotation matrices.

On the other hand, since the RFB sequences are used almost always in practice, the qualifier RFB is often omitted. In other words, an RFB i ‐ j ‐ k sequence is often referred to simply as an i ‐ j ‐ k sequence.

1 (a) RFB 1‐2‐3 Sequence

This sequence is generally known as a roll‐pitch‐yaw sequence . The angles of this sequence are generally named and denoted as roll angle ( φ 1= φ ), pitch angle ( φ 2= θ ), and yaw angle ( φ 3= ψ ). As such, the transformation matrix is formed as follows:

(3.106)