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0 = sir1-1(-C31)
, = cos-, (
7 =,os-, (
C33
)sgn(C32,
)sgn(C2,
3) Quaternions: the four Euler parameters are given by Eqs. (4.192-4.195),
which are reproduced in the following:
Vr 0 4 rit . O . 4
eo = cos ;~/ cos 2 coS ~+ sin ~ sm 2 sm 2
q o . ~ , Vr . 0 *
ei = cos ; cos 2 sm 2 -. sm 2 sm 2 cos 2
/ r
0.4
e2 =,os gsmg,os~+sm g,os;srn 2
v/ . 0 . 4 . v 0 4
e3 = -cos 2 sm 2 sm g +sin 2 .os 2 ,Os 2
eo = -;:(el p + e2q + e3r) + Aee0
ei - }:(eoP + e2r - e3q) + A,€ei
e2 - /z(eoq + e3p - eir) + A,ee2
e3 - ;:(eor + eiq - e2P) + Aee3
where € is given by Eq. (4.215) and is reproduced in the following:
6 - 1 - (e% + ef + eg- + e~z)
We assume A - 0.0001 and integrate the above rate equations for four quaternion
parameters using MATLAB5 coE9e ODE45. Then, the Euler angles are obtained
using Eqs. (4.217), (4.221), and (4.222).
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 357
o
0
'o
f=
o
(,,
a)
-o
.:
rn
a
DCM
N . N - - quLtcrnims
Fig. 4.12 Euler angles (S z = 30 degts, a = 30 deg).
The results are presented in Figs. 4.12-4.14. We observe that all three methods
give identical values of Euler angles except around discontinuous switching. The
spikes in the roll angle 4 for a = 45 deg (Fig. 4.13) are apparently caused by the
fact that Euler angle rates assume large values as 0 approaches 90 deg.
Example 4.6
At a certain time during a continuous motion of an airplane, the following
direction cosine matrix is recorded.Itis suspected that the elements marked xx are
in error and hence are discarded. Determine the missing elements.
0.8999 -0.4323 0.0578 -1
C- xx 0.8665 -0.2496
1 : 0.9666 j
xx xx
Solution. We observe that the missing elements are C21, C31, and C32. We
have rune elements of the direction cosine matrix and six redundancy relations.
o
o
r,
(,j
6
z
~
358 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
o
m
o
ai
6
c
~
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┃ { ┃ . . ┃ . ┃
┃ ┃ ┃~:--: -- ┃
┃ ---: I ┃--: ---'.----- I . ┃ I ┃
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