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      (3.83)equation

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      (3.85)equation

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      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).

      (3.88)equation

      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 images. When a set of Euler angles is used, the reference frame images is obtained by rotating the reference frame images through the following sequence of three rotations.

      Although images, images, and images 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 ijk sequence, each of the unit vectors of the rotation axes is specified as one of the basis vectors of the initial reference frame images. That is,

      (3.90)equation

      The specified unit vectors must be such that ji and jk. Such a rotation sequence can be described as shown below.

      (3.91)equation

      (3.92)equation

      (3.93)equation

      (3.94)equation

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

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