Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren
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Then, Eq. (3.76) gives
(3.78)
(3.79)
On the other hand,
(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).
3.7.2 Initial Frame Based (IFB) Formulation
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
(3.82)
(3.83)
(3.85)
The pattern observed in Eqs. (3.84) and (3.86) implies that
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
(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
In Description (3.89),