Читать книгу Kinematics of General Spatial Mechanical Systems - M. Kemal Ozgoren - Страница 30

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)