Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

imply the following sign variable equations.

      (2.147)

      (2.148)

      (2.149)

When two of the preceding three equations are multiplied side by side, the following additional equations are obtained by noting that .

(2.150)

      (2.151)

(2.152)

Hence, it is seen that Eqs. (2.150)–(2.152) provide the three sign variables as follows in terms of a single arbitrary sign variable σ′ = ± 1.

(2.153)

(2.154)

(2.155)

      Case 2: One of the components of is zero.

      Let ijk ∈ {123, 231, 312} and suppose that n_k = 0, while n_i ≠ 0 and n_j ≠ 0. In such a case, Eqs. (2.138)–(2.143) imply that r_ik = r_ki = r_kj = r_jk = 0, r_kk = − 1, and

(2.156)

Equation (2.156) can be satisfied if σ_i and σ_j are expressed as follows in terms of a single arbitrary sign variable σ′ = ± 1:

(2.157)

(2.158)

      Case 3: Two of the components of are zero.

Let ijk ∈ {123, 231, 312} again and this time suppose that n_i = n_j = 0, while n_k ≠ 0. In such a case, Eqs. (2.138)–(2.143) imply that r_kk = + 1, r_ij = r_ji = r_jk = r_kj = r_ki = r_ik = 0, and r_ii = r_jj = − 1. Then, as the only nonzero component,

(2.159)

      2.9.4 Discussion About the Optional Sign Variables

Considering the sign variables σ and σ′ that occur in Sections 2.9.2 and 2.9.3, it is possible to select them as σ = + 1 and σ′ = + 1 without much loss of generality. A discussion about this statement is presented below.

      1 (a) General Case with sin θ ≠ 0

In such a case, according to Eqs. (2.131)–(2.136), if σ = + 1 leads to θ and , then σ = − 1 leads to θ′ = − θ and . However, the pair is equivalent to the pair as confirmed by the following equation.

(2.160)

Due to Eq. (2.160), σ has no effect on the rotation matrix . Therefore, in a case such that θ and are required to be determined only once in a while or in a somewhat special case such that θ and are required to be determined frequently but the successive values of θ never become zero, the sign ambiguity may be eliminated by selecting the option with σ = + 1 so that θ > 0. However, in a case such that θ and are required to be determined frequently and the successive values of θ turn out to be fluctuating in the vicinity of zero, it may be more appropriate to have θ change its sign (i.e. to have σ switching between +1 and −1) rather than having change its orientation from one direction to the opposite one abruptly and frequently. In other words, it may be more preferable to have rather than .

      1 (b) Special Case with a Half Rotation

In such a