M. Kemal Ozgoren - Kinematics of General Spatial Mechanical Systems

(2.137)

1 Special Case with Half Rotation

In this case, θ = ± π . However, unlike the first two cases, does not become indefinite in this case, even though it cannot be found from Eqs. (2.125)– (2.127). Indeed, after a half rotation of a vector, the bisector of the angle between the initial and final orientations of the vector coincides with the rotation axis. This fact is illustrated in Figure 2.2for two different half rotations of the same vector. Owing to this fact, happens to be definite and therefore it can be found somehow, e.g. as explained below, if not from Eqs. (2.125)– (2.127).

In the case of a half rotation, with sin θ = 0 and cos θ = − 1, Eqs. (2.116)– (2.124)reduce to the following forms.

(2.138)

(2.139)

(2.140)

(2.141)

(2.142)

(2.143)

Equations (2.138)– (2.140)provide the components of as follows with three arbitrary sign variables, which are σ 1= ± 1, σ 2= ± 1, and σ 3= ± 1:

(2.144)

(2.145)

(2.146)

However, the three sign variables are not independent from each other because the components of obtained above must also satisfy Eqs. (2.141)– (2.143). Here, another set of three cases arises depending on whether all the components of are nonzero or not.

Figure 2.2Half rotations of the same vector about different axes.

Case 1: All the components of are nonzero.

In this case, Eqs. (2.141)– (2.146)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 iand σ jare expressed as follows in terms of a single arbitrary sign variable σ ′ = ± 1:

(2.157)