Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

It can be manipulated as follows:

(2.111)

According to the shifting formulas expressed by Eqs. (2.83)–(2.86), Eq. (2.111) can be manipulated further as shown below.

(2.112)

According to Eq. (2.68), . Therefore, Eq. (2.112) reduces to

(2.113)

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 is a specified rotation matrix expressed in an observation frame .

(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)

      2.9.2 Determination of the Angle of Rotation

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)