Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

(3.129)

Equation (3.129) suggests that σ can be selected as σ = + 1 without a significant loss of generality.

If sinφ₂ = 0, i.e. if d₃₃ = 0, then the 3‐2‐3 sequence becomes singular and the angles φ₁ and φ₃ cannot be found from Eq. Pairs 3.115,3.116,3.117,3.118 and (3.117)–(3.118), which all reduce to 0 = 0. Such a singularity occurs either if φ₂ = 0 or if with . In either case, φ₁ and φ₃ become indefinite and indistinguishable. So, they cannot be found separately. Nevertheless, their combinations denoted as φ₁₃ = φ₁ + φ₃ and can still be found depending on whether φ₂ = 0 or . The way of finding them is explained below.

      If φ₂ = 0, Eq. (3.113) can be manipulated as follows:

(3.130)

Equation (3.130) implies that

      (3.131)

      Hence, φ₁₃ is found as

      (3.132)

      If , Eq. (3.113) can be manipulated as follows:

(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

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