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The transformation matrix Lo is not an orthogonal matrix because the vectors
lv, O, and 4 are not mutually perpendicular. Hence, the inverse is not equal to the
transpose. Therefore, we have to compute L~l in the usual way as follows:
~:l = (A(1,))adj( ~) (4.60)
The determinant A(L) of matrix Lcn is given by
A(L) = COS2 ~ cos O + S1112 ~ COS 0 (4.61)
N cos O (4.62)
N
The adjoint of matrix L~ is given by
'cos0 sin0 sin~ cos@ sin
adj(Lo,)= O cos~cos0 -s,.,'S,] (4.63)
0 sin~ cos4
so that
or
~:l = [:
tan 0 sin ~
cos 4
sec 0 sin ~
Substituting in Eq, (4.59), we get
[,/i] = [:
tan 0 sin 4
cos 4}
sec O sin 4
cos ~ tan c
-s,.11:] (4.64)
sec O cos ~
4 = p + tan O(q sin 4 + r cos 4)
0 =qcos4-rsin4
Vr = sec O(q sin 4 +r cos 4)
(4.65)
(4.66)
(4.67)
(4.68)
Singularity in Euler angle rates. Equations (4.66-4.68) for Euler angle rates
become singular at O -. 7r]2. The existence of the singularity can be demonstrated
as follows.
EQUATIONS OF MO1-ION AND ESTIMATION OF STABILI-fY DERIVATIVES 333
4=P+
q sin 0 sin ~ + r sin O cos4
cos 0
O -q cos4-r sin~
Vr = q sin~ + r cos4
cOs 0
For O = 7t/2, from Eqs. (4.55-4.57),
. p-4-'rk .
q -o coS4
r - -0 sin~
Substituting in Eq. (4.71), we have
. (l/t) 2 = 0 sin 4 cos 4 - 0 sin4 cos c)b
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PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL2(101)
