M. Kemal Ozgoren - Kinematics of General Spatial Mechanical Systems

(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)

Considering the sign variables σ and σ ′ that occur in Sections 2.9.2and 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 case, θ = σπ with σ = ± 1. On the other hand, as observed in Eqs. ( 2.153, 2.154, 2.155, 2.156, 2.157, 2.158, 2.159), σ ′ (i.e. the sense of ) is not related to σ = sgn( θ ). As a matter of fact, the angle‐axis pair leads to the same rotation matrix, whatever σ and σ ′ are. This statement is confirmed as shown below.

(2.161)

Owing to Eq. (2.161), whatever σ is, the sign ambiguity caused by σ ′ can again be eliminated by selecting the option with σ ′ = + 1, if θ and are required to be determined only once in a while. However, if θ and are required to be determined frequently in a case such that the values of θ happen to be σπ at certain successive instants, then it may be more appropriate to prevent the possibility that changes its direction abruptly as soon as θ becomes σπ . In other words, instead of insisting on the choice σ ′ = + 1, it may be preferable to choose σ ′ so that with θ ( t ) = σπ and with are almost codirectional, i.e. .

The double argument arctangent function, which is denoted as atan 2( η , ξ ), is defined so that it satisfies the following identity for any ρ > 0.

(2.162)

The most characteristic feature of the function atan 2( η , ξ ) is that it gives θ without any quadrant ambiguity in the following interval, whenever η ≠ 0 and ξ ≠ 0.