Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren
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According to Eq. (2.68),
Hence, it is seen that Eq. (2.110) is verified owing to Eq. (2.113).
2.9 Determination of the Angle and Axis of a Specified Rotation Matrix
2.9.1 Scalar Equations of Rotation
Consider the following equation, in which
Most typically,
Equation (2.115) provides the following scalar equations.
(2.117)
(2.120)
(2.121)
(2.122)
(2.123)
Furthermore, the following three equations can be obtained from Eqs. (2.119)–(2.124).
(2.126)
2.9.2 Determination of the Angle of Rotation
Note that
(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)
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