Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

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      (3.83)

(3.84)

      (3.85)

(3.86)

The pattern observed in Eqs. (3.84) and (3.86) implies that

(3.87)

      As noted above, in the initial frame based (IFB) formulation, the rotation matrices are multiplied in an order opposite to the order of the rotation sequence indicated in Description (3.75).

On the other hand, the rotation matrix (that describes the rotation of into ) is mathematically equivalent to the orientation matrix (that describes the orientation of with respect to ). Based on this equivalence, Eq. (3.87) can also be written as follows:

      (3.88)

3.8 Expression of a Transformation Matrix in Terms of Euler Angles

      3.8.1 General Definition of Euler Angles

      The Euler angles are named after the Swiss mathematician Leonhard Euler (1707–1783). With a modification of what Euler originally introduced, the definition of the Euler angles was later generalized so that they consist of three rotation angles (φ₁, φ₂, φ₃) about three specified rotation axes. The three axes must be specified so that they are neither coplanar nor successively parallel or coincident. Thus, the Euler angles constitute a set of three independent parameters for the transformation matrix . When a set of Euler angles is used, the reference frame is obtained by rotating the reference frame through the following sequence of three rotations.

(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.

      3.8.2 IFB (Initial Frame Based) Euler Angle Sequences

      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: