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2 2 -sm 2 sm 2 coS 2
v, . 0 4 4r 0 . 4
e2 = ,os 2 sm 2'cos ~ +s,. Y/ ,os 2 sin 2 (4.194)
e3 = _cos/p si. jj] si.~ +sin yr cos ; ,os ~
222
Cll C12 C13
C~= C21 C22 C23
C31 C32 C33
Cll C21
T12=C7f= C12 C": g3;:]
C13 C23
(4.195)
(4.196)
(4.197)
Then equating the relations (4.22) and (4.197) and using Eqs, (4.192-4.195), the
following relations between the elements of the direction cosine matrix, Euler
angles, and the four quaternion parameters can be obtained. For detailed mathe-
matical derivations, the interested reader may refer elsewhere.4
- cos O cos lr = e~ +ef - e~ - e~
C21 - COS t) sin V : 2(e1e2 + eOe3)
C31 - -sin 0 -. 2(ele3 - eOe2)
C12 - siri0 siriq COS l/f - sin Vr cos gb = 2(ele2 - eOe3)
C22 = siri0 siri~ siriltJ +cos p cos ~ : e~ - e/- + at~ - e~
(4.198)
(4.199)
(4.200)
(4.201)
(4.202)
a
.
j
7
~
.
d.
I
..
~.
."
'. .
(.
e2 = ,os B sm ~-)
e3 =.os C sm /j~
:
PERFORMANCE, STABILJTY, DYNAMICS, AND CONTROL
C32 - cos 0 suJ- 4 = 2(e2e3 + eoei)
C13 = sin 0 cos ~ cos p + sin Vr sin ~ : 2(eOe2 + ele3)
C23 = siri0 cos 4 sin ~ - cos yr sin 0 - 2(e2e3 - eoei)
C33 = co\s O cos @ = elji - ef - eg- + e~
(4.203)
(4.204)
(4.205)
(4.206)
The rate equations for the four quaternion parameters are given by the following
relations. For detailed mathematical derivation, the interested reader may refer
elsewhere.4
eo = -2! (ei P + e2q + e3r)
ei = ~ (eoP + e2r - e3q)
e2 = lz (eoq + e3 p - eir)
e3 = ;: (eor + eiq - e2p)
(4.207)
(4.208)
(4.209)
(4.210)
where
eo = -:}: (ei p + e2q + e3r) + A6eo
ei = ;: (eoP + e2r - e3q) + A6et
e2 = ;: (eoq + e3 p - eir) + 1ee2
e3 = ~ (eor + eiq - e2P) + Aee3
e = 1- (eg + el + eg + e~)
(4.211)
(4.212)
(4.213)
(4.214)
(4.215)
Here, A is a free parameter. Usually, A is set equal to a small multiple of the
integration time step.3
A forward integration of Eqs. (4.211-4.214) generates the time history of four
parameters eo, ei, e2, and e3. Then, the Euler angles can be obtained as follows.
From Eq. (4.200), we have
sin O -. -2(ele3 - eOe2) (4.216)
o = sin-l[-2(ele3 - eOe2)J
(4.217)
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 347
Because 0 is supposed to be in the range -7tl2 < 0 < lr]2, the angle 0 is
uniquely determined by Eq. (4.217). Furthermore, we have from Eqs. (4.203) and
Eq. (4.206),
so that
C33 = COS 0 COS 4 = eg - e? - e~ + e32
C32 = c.os 0 siri~ = 2(e2e3 + eoei)
4 =,os-" (,C, o)
(4.218)
(4.219)
(4.220)
=COS-I 9-ei-e+e),lsgn[2(e2e3+eOel)] (4.221)
.jC --eOe2):
! - eg- - e
yr=COS-I ~F4-C,: :--),lsgn[2(e,e2+eOe3)] c4.222)
'i e3 - eoe
As said before, we will not encounter the singularity at 0 -. 7rl2 if we obtain
the Euler angles using the method of quaternions. We can summarize all three
approaches to calculate the Euler angles as follows:
1) Euler angle rates: r/o, 00, 4o body rates p, q, r + Euler rates I4(, 0,
4 p(t), O(t), ~(t).
2) Direction cosine matrices (DCM): yro, 0o, qbo, DCM at t = 0 ~ body rates
p, q, r -* DCM updates equations + DCM at time t -* yr (t), O(t), ~(t).
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