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
1 (a) Extraction of the 3‐2‐3 Euler Angles
If the RFB 3‐2‐3 sequence is used,
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
From Eq. (3.114), sinφ2 and φ2 can be found as follows with an arbitrary sign variable σ:
(3.119)
(3.120)
(3.121)
If sin φ2 ≠ 0, i.e. if d33 > 0, φ1 and φ3 can be found as follows, respectively, from Eq. Pairs (3.115)–3.116 and (3.117)–3.118 consistently with σ, without introducing any additional sign variable.
(3.122)
(3.123)
(3.124)
(3.125)
(3.126)
(3.127)
1 Selection of the Sign Variable
Based on the solution obtained above for d33 > 0, the following analysis can be made concerning the sign variable σ.
If σ = + 1 leads to
(3.128)
Here,
