= 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
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 Remark 3.6: Preference Between IFB and RFB Sequences
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
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.
3.8.7 Commonly Used Euler Angle Sequences
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 (φ1 = φ), pitch angle (φ2 = θ), and yaw angle (φ3 = ψ). As such, the transformation matrix is formed as follows:
(3.106)
This sequence is not used very often with the general designations indicated above.
On the other hand, it is used quite often in the area of robotics especially for the purpose of describing the orientation of the end‐effector of a manipulator with respect to the base frame. However, when it is used for this purpose, it is designated differently as a yaw‐pitch‐roll sequence. The angles are also named and denoted differently as yaw or swing angle (φ1 = ψ), pitch or bent angle (φ2 = θ), and roll or twist angle (φ3 = φ). With these designations, the transformation matrix is formed differently as follows:
(3.107)
1 (b) RFB 3‐2‐1 Sequence
This sequence is generally known as a yaw‐pitch‐roll sequence. The angles of this sequence are conventionally named and denoted as yaw angle (φ1 = ψ), pitch angle (φ2 = θ), and roll angle (φ3 = φ). For this sequence, the transformation matrix is formed as follows:
(3.108)
This sequence is used very commonly in the area of vehicle dynamics in order to describe the orientations of all sorts of land, sea, and air vehicles with respect to selected reference frames.
1 (c) RFB 3‐1‐3 Sequence
This sequence is generally known as a precession‐nutation‐spin sequence. The angles of this sequence are conventionally named and denoted as precession angle (φ1 = φ), nutation angle (φ2 = θ), and spin angle (φ3 = ψ). For this sequence, the transformation matrix is formed as follows:
(3.109)
This sequence is used very commonly in the kinematic and dynamic studies that involve spinning bodies such as tops, rotors of gyroscopes, celestial bodies, etc. Actually, this is the sequence that was originally introduced by Leonhard Euler.
1 (d) RFB 3‐2‐3 Sequence
This sequence is sometimes used as an alternative to the 3‐1‐3 sequence in the studies involving spinning bodies. When it is used so, it is also designated as a precession‐nutation‐spin sequence. The angles of this sequence are then similarly named and denoted as precession angle (φ1 = φ), nutation angle (φ2 = θ), and spin angle (φ3
