Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren
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Equation (3.133) implies that
(3.134)
Hence,
(3.135)
In order to visualize the singularity of the 3‐2‐3 sequence, the unit vectors of the first and third rotation axes can be expressed as follows in the initial reference frame
(3.136)
(3.137)
When the singularity occurs with φ2 = 0,
In this singularity, according to Eq. (3.138), the rotations by the angles φ1 and φ3 take place about two axes that have become codirectional. Therefore, only the resultant rotation by the angle φ13 = φ1 + φ3 can be recognized but the angles φ1 and φ3 become obscure and they cannot be distinguished from each other.
When the singularity occurs with
In this singularity, according to Eq. (3.139), the rotations by the angles φ1 and φ3 take place about two axes that have become oppositely directed. Therefore, only the resultant rotation by the angle
1 (b) Extraction of the 1‐2‐3 Euler Angles
If the RFB 1‐2‐3 sequence is used,
Similarly as done above for the 3‐2‐3 Euler angles, the following five scalar equations can be derived from Eq. (3.140) by picking up the appropriate elements of
From Eq. (3.141), cosφ2 and φ2 can be found as follows with an arbitrary sign variable σ:
(3.146)
(3.147)
In Eq. (3.148), σ2 is different from σ. It is defined as follows if c13 ≠ 0.
(3.149)