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Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren
Читать онлайн.Название Kinematics of General Spatial Mechanical Systems
Год выпуска 0
isbn 9781119195764
Автор произведения M. Kemal Ozgoren
Жанр Математика
Издательство John Wiley & Sons Limited
(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),
Although
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
(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
(3.92)
(3.93)
(3.94)
Hence, according to the IFB formulation explained in Section 3.7,