Название | Kinematics of General Spatial Mechanical Systems |
---|---|
Автор произведения | M. Kemal Ozgoren |
Жанр | Математика |
Серия | |
Издательство | Математика |
Год выпуска | 0 |
isbn | 9781119195764 |
(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 .
(2.151)
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.
Case 2: One of the components of is zero.
Let ijk ∈ {123, 231, 312} and suppose that nk = 0, while ni ≠ 0 and nj ≠ 0. In such a case, Eqs. (2.138)–(2.143) imply that rik = rki = rkj = rjk = 0, rkk = − 1, and
Equation (2.156) can be satisfied if σi and σj are expressed as follows in terms of a single arbitrary sign variable σ′ = ± 1:
Case 3: Two of the components of are zero.
Let ijk ∈ {123, 231, 312} again and this time suppose that ni = nj = 0, while nk ≠ 0. In such a case, Eqs. (2.138)–(2.143) imply that rkk = + 1, rij = rji = rjk = rkj = rki = rik = 0, and rii = rjj = − 1. Then, as the only nonzero component,
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.
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