Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

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(3.116)

(3.117)

(3.118)

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 , then σ = − 1 leads to , where

      (3.128)

      Here, and are two independent sign variables, that is, and but they are not necessarily equal. Although and look different, they are actually completely equivalent because they both provide the same transformation matrix as shown below by using the rotation matrix formulas given in Chapter 2.

(3.129)

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

      1  Singularity Analysis

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:

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