According to Eq. (2.87) of Chapter 2 about the three successive half rotations,
Hence, Eq. (3.157) reduces to
Equation (3.158) suggests that σ can again be selected as σ = + 1 without a significant loss of generality.
1 Singularity Analysis
If cosφ2 = 0, i.e. if d13 = 0, the 1‐2‐3 sequence becomes singular and the angles φ1 and φ3 cannot be found from Eq. Pairs, which all reduce to 0 = 0. Such a singularity occurs if
In the singularity with
Equation (3.159) implies that
(3.160)
Hence, φ13 is found as
(3.161)
In order to visualize the singularity of the 1‐2‐3 sequence, the unit vectors of the first and third rotation axes can be expressed as follows in the initial reference frame
(3.162)
(3.163)
When the singularity occurs with
In this singularity, according to Eq. (3.164), the rotations by the angles φ1 and φ3 take place about two axes that have become parallel, either codirectionally if
3.9 Position of a Point Expressed in Different Reference Frames and Homogeneous Transformation Matrices
3.9.1 Position of a Point Expressed in Different Reference Frames
Figure 3.3 shows a point P, which is observed in two different reference frames
