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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.

      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 = ψ). In such a usage, the transformation matrix is formed as

      (3.110)

      This sequence is also used in the area of robotics as an alternative to the RFB 1‐2‐3 sequence in order to describe the orientation of an end‐effector with respect to the base frame. When it is used so, it is generally designated as a yaw‐declination‐roll sequence. The angles are then named and denoted as yaw or swing angle (φ₁ = ψ), declination angle (φ₂ = θ), and roll or twist angle (φ₃ = φ). In such a usage, the transformation matrix is formed as follows:

      (3.111)

      3.8.8 Extraction of Euler Angles from a Given Transformation Matrix

      Suppose a transformation matrix is somehow given as

      (3.112)

      Then, the Euler angles of a selected sequence can be extracted from by using the procedure explained here. The procedure is explained here for two typical sequences. One of them is the RFB 3‐2‐3 sequence, which is symmetric, and the other one is the 1‐2‐3 sequence, which is asymmetric. However, the same procedure can be used similarly for any other sequence, too.

      1 (a) Extraction of the 3‐2‐3 Euler Angles

      If the RFB 3‐2‐3 sequence is used, is expressed as

(3.113)

By using the formulas presented in Chapter 2 about the mathematical properties of the rotation matrices, the following set of five scalar equations can be derived from Eq. (3.113) by picking up the appropriate elements of .

(3.114)

(3.115)

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