Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

and θ as explained below.

      Referring to Figure 2.1, the vectors and can be expressed as follows:

(2.3)

(2.4)

In Eqs. (2.3) and (2.4), is the common projection of and on the axis of rotation. Note that is not affected by the rotation operator because lies on the rotation axis. The vector is related to and as expressed below.

      (2.5)

      Referring again to Figure 2.1, the vector , which is perpendicular to and , is obtained by the following cross product.

(2.6)

Upon substituting Eq. (2.3) and noting that , Eq. (2.6) becomes

      (2.7)

      On the other hand, the projectional view on the right‐hand side of Figure 2.1 implies that the vector , which is coplanar with the vectors and , can be expressed as the following linear combination of and .

(2.8)

After the previously obtained equations concerning , , and are substituted, Eq. (2.8) becomes

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