M. Kemal Ozgoren - Kinematics of General Spatial Mechanical Systems

(3.89)

In Description (3.89), and are the intermediate frames on the way from the initial frame to the final or terminal frame . As for the unit vectors , , and , they represent the specified rotation axes. They may be specified arbitrarily as desired in the generalized definition of the Euler angles, whereas they are specified as , , and in the original definition of the Euler angles.

Although , , and may be specified arbitrarily in general, in almost all the practical cases, each of them is specified as a selected basis vector of a selected reference frame. Thus, different Euler angle sequences arise depending on the selected reference frames and their selected basis vectors. All such Euler angle sequences are grouped into two main categories, which are designated as IFB and RFB sequences. These sequences are described and explained below.

In an IFB sequence, e.g. the IFB i ‐ j ‐ k sequence, each of the unit vectors of the rotation axes is specified as one of the basis vectors of the initial reference frame . That is,

(3.90)

The specified unit vectors must be such that j ≠ i and j ≠ k . Such a rotation sequence can be described as shown below.

(3.91)

In such a sequence, the matrix representations of all the rotation operators are expressed naturally in . In other words,

(3.92)

(3.93)

(3.94)

Hence, according to the IFB formulation explained in Section 3.7, is obtained as follows:

(3.95)

In an RFB sequence, e.g. the RFB i ‐ j ‐ k sequence, each of the unit vectors of the rotation axes is specified as one of the basis vectors of the reference frames , , and , respectively. That is,

(3.96)

The specified unit vectors must be such that j ≠ i and j ≠ k . However, since the rotation axes between the pre‐rotation and post‐rotation frames are common, the following equations can also be written for the unit vectors of the rotation axes.

(3.97)

Such a rotation sequence can be described as shown below.

(3.98)

In a sequence that has the axis unit vectors specified as shown above, i.e. as the basis vectors of the pre‐rotation frames, the matrix representations of the rotation operators are also expressed naturally in the pre‐rotation frames. In other words,

(3.99)

(3.100)