Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

The preceding equation can be arranged as follows:

(2.9)

Equation (2.9) is known as the Rodrigues formula, which is named after the French mathematician Benjamin Olinde Rodrigues (1795–1851).

2.2 Matrix Equation of Rotation and the Rotation Matrix

      The matrix form of the Rodrigues formula, i.e. Eq. (2.9), can be written as follows in a selected reference frame :

(2.10)

By noting that , Eq. (2.10) can be factorized so that

(2.11)

Equation (2.11) can be written compactly as

(2.12)

In Eq. (2.12), is defined as the rotation matrix expressed in . It is the matrix representation of the rotation operator in . In other words,

      (2.13)

      Equations (2.11) and (2.12) show that is a function of and θ as expressed below.

(2.14)

In Eq. (2.14), is defined as the rotation matrix function of the arguments and θ. In other words, generates a rotation matrix out of the arguments and θ as shown below, where may be any column matrix such that .

(2.15)

      An alternative expression can be derived for the function by recalling the following equation from Section.

(2.16)

Upon substituting Eq. (2.16) into Eq. (2.15), the alternative expression is obtained as

(2.17)

      Hence, with , can also be expressed as

      (2.18)

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