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(4.182)
(4.183)
(4.184)
(4.185)
where  j  =  vcf. We note that one rootis at the origin and the other two are on the
imaginary axis. Hence, the system represented by Eq. (4.179) is neutrally stable.
Therefore, in numerical computations invoMng updating of direction cosines,
extreme care should be taken because rounding off errors can build up rapidly and
blow out the solution.
4.2.8 Quatemions
       The method of quaternions presents a practical alternative to the Euler angle rate
method and has the advantage that it avoids the singularity that is present in the
Euler angle rate equations when the pitch angle 0 approaches 90 deg. This method
 is also known as the Euler four parameter method.ln the following, we will present
 a briefdiscussion of this method so that the read            in aposition to use this method.
Detailed mathematics is avoided to keep  ahtglj~:e&L:2ition simple. For additional
information, the reader may refer to othe:sour       ',4
    Fundamental to this method is Euler's theorem that ariy given frame of axes
 Ox2Y222 can be made to coincide with another frame OxiYizi by a single rotation
 D about a fixed axis in space malang angles A, B, C with Oxiyizi. Then, we can
define four parameters, eo, ei, e2, e3 as follows:
     D
eo = cos 2
e, = ,os A si. ~
(4.186)
(4.187)
(4.188)
(4.189)
     :. J: . :
-~-,
ECIUATIONS OF MOTION AND ES11MATION OF STABiUTY DERfVATIVES 345
                                              ~
                                                                                                                                                  ~
It can be readily verified that these four parameters are constrained by the following
condition:
                                                      e~ + ef + e~z + e~ = 1                                     (4.190)
The following transformation matrix is presented without proof.lnterested readers
may refer elsewhere.4
               e~ + e? - e?/ - e~     2(ele2 + eOe3)        2(ele3 - eOe2)
   T12 =     2(ele2 - eOe3)     e~ - e~-+elj - e~     2(e2e3+eOel)       (4.191)
                 2(eOe2 + ele3)        2(e2e3 - eoei)     e~ ~ e~ - e~ + e~
With l/r, O, and 4 as Euler angles, the transformation matrix T12 is also given by
Eq. (4.2h22t,Equating the corresponding elements of the two matrices, it can be
shown tha
Let
so that
     11   0  4   V'. 0 . 4
eo = cos ~/ coS 2 cOs ~ + sin r/2 sm 2 sm 2                     (4.192)
    }s/pcos0 . ~  . /r . 0  4
ei - cos -      s - sm -T .                                          (4.193)
　
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