M. Kemal Ozgoren - Kinematics of General Spatial Mechanical Systems

In this case, rot( p , q ) is expressed as a rotation matrix in one of the two relevant reference frames, i.e. either in the pre‐rotation frame or in the post‐rotation frame . Thus, the transformation matrix is related to rot( p , q ) as follows according to Eq. (3.64):

(3.77)

Then, Eq. (3.76)gives with one of the following equivalent expressions.

(3.78)

(3.79)

On the other hand, is also equal to the overall rotation matrix expressed in either or . That is,

(3.80)

As noted above, in the rotated frame based (RFB) formulation, the rotation matrices are multiplied in the same order as the order of the rotation sequence indicated in Description (3.75).

In this case, all the rotation operators are expressed as the following rotation matrices in the initial reference frame .

(3.81)

Of course, in such a formulation, except , none of the above rotation matrices is a transformation matrix. Therefore, the transformation matrices required in Eq. (3.76)can be obtained as shown below by means of Eq. (3.74).

(3.82)

(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)

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 ( φ 1, φ 2, φ 3) 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.