M. Kemal Ozgoren - Kinematics of General Spatial Mechanical Systems

(2.114)

Most typically, is specified as an overall rotation matrix in a case of m successive rotations so that . Here, it is required to find the angle and axis, i.e. θ and , of the overall rotation operator. In order to find them, Eq. (2.114)can be written again in the following detailed form.

(2.115)

Equation (2.115)provides the following scalar equations.

(2.116)

(2.117)

(2.118)

(2.119)

(2.120)

(2.121)

(2.122)

(2.123)

(2.124)

Furthermore, the following three equations can be obtained from Eqs. (2.119)– (2.124).

(2.125)

(2.126)

(2.127)

Note that , i.e.

(2.128)

Therefore, the side‐by‐side addition of Eqs. (2.116)– (2.118)leads to the following equation.

(2.129)

Hence, cos θ and sin θ are found as follows:

(2.130)

(2.131)

In Eq. (2.131), σ is an arbitrary sign variable, i.e.

(2.132)

With the availability of sin θ and cos θ , θ can be found by means of the double argument arctangent function . That is,

(2.133)

The definition and the properties of the atan 2function can be seen in Section 2.10.

1 (a) General Case with sin θ ≠ 0

In a general case such that sin θ ≠ 0, Eqs. (2.125)– (2.127)give the components of as follows:

(2.134)

(2.135)

(2.136)

1 (b) Special Cases with sin θ = 0

However, if sin θ = 0, then Eqs. (2.125)– (2.127)cannot give the components of . There are three distinct cases, in which sin θ = 0. These cases are discussed below by considering θ to be in the minimal rotation range , without any loss of generality. This range is defined so that 0 ≤ | θ | ≤ 2 π .

1 Special Case with No Rotation

In this case, θ = 0 and becomes indefinite. Indeed, if there is not any rotation at the moment, then represents any arbitrary unit vector that forms the axis of a prospective rotation, which is yet unknown.

1 Special Case with Full Rotation

In this case, θ = ± 2 π and becomes indefinite again. Indeed, after a full rotation, a rotated vector comes back to its initial orientation no matter what the rotation axis is. In other words, according to the mathematical property expressed by Eq. (2.68)in Section 2.7, the following equation is satisfied for any arbitrary .