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of skew-matrix as given by the Eq. (4.167).
Equation (4.175) states that if we have a skew-matrix of angular rates of the b
frame relative to the / frame with components in the b frame and we need to find
the components of the sc rectorin the / frame, then we must premultiply it by
C; and postmult:iply it byamCb.:ec
From Eq. (4.166), we have
Ct, = Cj,S?/~ (4.176)
(Cl,)' = (C. bg2b:b)'
~2::b)'(C;))'
Cb, = _C2bb Cb
(4.177)
(4.17 8)
(4.179)
gjii]
-r _,P]
344 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
where we have used the property (S?lb)' = -S-2lb. Equation (4.179) is of the form
X - AX
The characteristic of Eq. (4.180) is given by
A(A/ - A) :0
(4.180)
(4.181)
where A (.) denotes the determinant of the argument matrix. Using this definition,
the characteristic of Eq. (4.179) is given by
so that
or
A(;k,2 + p2 +q2 +r2) = 0
A, - 0
A =+j, -~-
+ j ltDlb I
(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)
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