Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

alt="equation"/>

(3.133)

Equation (3.133) implies that

      (3.134)

      Hence, is found as

      (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, becomes

(3.138)

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 , becomes

(3.139)

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 can be recognized but the angles φ₁ and φ₃ become obscure and they cannot be distinguished from each other.

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

      If the RFB 1‐2‐3 sequence is used, is expressed as

(3.140)

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 .

(3.141)

(3.142)

(3.143)

(3.144)

(3.145)

From Eq. (3.141), cosφ₂ and φ₂ can be found as follows with an arbitrary sign variable σ:

      (3.146)

      (3.147)

(3.148)

In Eq. (3.148), σ₂ is different from σ. It is defined as follows if c₁₃ ≠ 0.

      (3.149)