Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

(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)

If cosφ₂ ≠ 0, i.e. if d₁₃ > 0, φ₁ and φ₃ can be found as follows, respectively, from Eq. Pairs 3.142,3.143,3.144,3.145 and (3.144)–(3.145) consistently with σ, without introducing any additional sign variable.

      (3.150)

      (3.151)

      (3.152)

      (3.153)

      (3.154)

      (3.155)

Based on the solution obtained above for d₁₃ > 0, the following analysis can be made concerning the sign variable σ.

      If σ = + 1 leads to , then σ = − 1 leads to , where

      (3.156)

      Here, σ₂ = sgn(φ₂) as introduced before. As for and , they 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 similarly as done before for the 3‐2‐3 sequence.

(3.157)

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