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https://er.chdtu.edu.ua/handle/ChSTU/8356| Title: | Дослідження інерціальної навігаційної системи |
| Authors: | Трембовецька, Руслана Володимирівна Гузьман, Іван Іванович |
| Keywords: | безплатформна інерціальна навігаційна система;оптико-електронні системи орієнтації та навігації;мінімізація навігаційних помилок;комплексування навігаційних систем;фільтр Калмана;корекція точнісних характеристик |
| Issue Date: | 15-Dec-2025 |
| Abstract: | У роботі досліджено методи мінімізації навігаційних помилок безплатформної інерціальної навігаційної системи шляхом її комплексування з оптико-електронними системами орієнтації та навігації та застосування алгоритмів корекції точнісних характеристик. The work investigates methods for minimizing navigation errors of a strapdown inertial navigation system by integrating it with optical-electronic orientation and navigation systems and applying algorithms for accuracy correction. |
| URI: | https://er.chdtu.edu.ua/handle/ChSTU/8356 |
| Appears in Collections: | 174 Автоматизація, комп'ютерно-інтегровані технології та робототехніка (Робототехнічні системи та автоматизація) |
Files in This Item:
| File | Description | Size | Format | |
|---|---|---|---|---|
| Диплом-магистр_Гузьман І.pdf Restricted Access | КРМ Гузьман І. | 7.63 MB | Adobe PDF | View/Open Request a copy |
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23
g ’ .
:
gx1 = g cosϑ0 sin γ 0
g y1 = −g sinϑ0 (2.1)
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(2.1),
:
g
ϑ = arcsin − y1
0 g
g (2.2)
γ 0 = arctg − x1
gz1
,
O :
ωξ = 0
ωη = Ω cosϕ (2.3)
ωζ = Ω sinϕ
Ω = 7.29 ⋅10−5 / — , —
. , ( )
:
(2.4)
ψ 0 ,
( )
ϑ0 :
24
ω −ω sinϑ
Ψ0 = arccos y1 ζ 0 (2.5)
ωη cosϑ0
(2.5) .
’
ϑ0 γ 0 :
c gx1
31 = cosϑ0 sin γ 0 = −
g
g
c y1
32 = sinϑ0 = − (2.6)
g
c33 = cosϑ0 cosγ gz1
0 = −
g
±0.5° c31,c32 ,c33 , (2.4)
:
(2.7)
, :
c12 = −cosϑ0 sin Ψ0 = c23 ⋅c31 − c21 ⋅c33 (2.8)
(2.6)–(2.8)
:
25
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0 (2.9)
c22 ωy1g −ωζ g y1
(2.9) ’
. ψ 0 ,ϑ0 ,γ 0
, .
, .
.
(2.2) (2.9). Δnx1,Δny1,Δnz1 — , Δωx1,Δωy1,Δωz1 —
. ,
:
∂Fϑ g
ϑ 0
0 = g = − y1
∂g y1
g 2
y1
g 1+ y1
g
∂Fγ 0 ∂F
γ = g + γ 0 gx1 gx gz1
0 g = − +
∂g x1 ∂g z1 2 2 2
x1 z1 g g
x1 z1 + gx1
g 1+ (2.10)
z1 gz1
Ψ = ∂FΨ0 c + ∂FΨ0 c = − c12 − c12 c22
0 ∂c 12 ∂c 22 2 2
12 22
c 1+ c12 2
z1 c 1+ c12
c 22
22 c22
(2.10)
Δϑ0 Δγ 0 .
,
26
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10−2 g 0.
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,
. 2.3 , X1 ,
Δγ 0
27
Δψ 0 ,
,
. ,
.
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(cos Ψ ω + sin Ψ ω )
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Ωcosϕ
Ω — , ϕ = 57.27° — , Δω —
, ψ 0 — .
. 2.4 Δψ 0
:
Ψ 0 = − tgϕ ⋅(cos Ψ0 + sin Ψ0 ) ⋅ ga + ga cos Ψ 0 sin Ψ 0 (2.12)
28
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. 2.5 Δψ 0
( Δga = const , Δω = const ):
(cos Ψ0 + sin Ψ0 ) ω
+ tgϕ ⋅(cos Ψ + sin Ψ )⋅ g
Ψ = − Ωcosϕ 0 0 a (2.13)
+ ga cos Ψ0 sin Ψ0
29
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ζ .
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ω ’ :
⋅ ⋅
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⋅ ⋅
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⋅ ⋅
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’ ψ ,ϑ,γ .
:
⋅ ⋅
γ = ωy − Ψ sinθ (2.16)
33
. cosγ , sinγ ,
:
⋅ ⋅ ⋅ ⋅ ⋅
ωx cosγ +ωz sinγ = −Ψcosθ sinγ cosγ +θ cos2 γ +Ψcosθ cosγ sinγ +θ sin2 γ =θ (2.17)
, −sinγ , cosγ
, :
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:
⋅ ⋅
γ = ωy − Ψ sinθ
⋅
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⋅
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:
⋅
Ψ = (ωz cosγ −ωx sin γ ) 1 (2.20)
cosθ
, :
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- :
34
(2.22)
(2.22)
MATLAB - .
, :
(2.23)
ω — , A— , ε — .
. 2.10
(2.22)
35
.
( ) ϑ = ±90° . «
», . ,
.
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( ).
:
(2.24)
da — , ω p — .
dt
:
(C SP )t dAs = dAs + (ω PS
P ×) ⋅ Ap (2.25)
dt dt
CSP — S P .
S ξηζ , P xyz ,
Ωx :
0 −ω ps
3 p ω ps
2 p
(ω ps x) = ω ps ps
p 3 p 0 −ω1p (2.26)
−ω ps ps
2 p ω1p 0
(2.25), :
(2.27)
36
:
dC SP
= C SP ⋅ (ω PS
p x) (2.28)
dt
, :
dAs dA
= C SP ⋅ p + (ω PS
p ×) ⋅ As (2.29)
dt dt
0 −ω ps ps
3 p ω2s
(ω ps
s x) = ω ps ps
3s 0 −ω1s (2.30)
−ω ps
2s ω ps
1s 0
:
SP
(ω PS
p x) = dC t
⋅ (C SP ) (2.31)
dt
dC SP
= (ω PS
s ×) ⋅C SP (2.32)
dt
(2.23).
37
. 2.11
(2.32)
,
’ W F :
W = F (2.33)
m
F F ( , )
G :
W = F + G (2.34)
m
( ) a = F .
m
G g (R,ϕ)
m
-
R :
2
W = d R
2 (2.35)
dt
38
(2.33),
:
d 2R
2 = a + g (R,ϕ ) (2.36)
dt
,
a .
:
dV = a + g (R,ϕ ) (2.37)
dt
dV = V , (2.38)
dt
V — . ,
Ω ,
:
(2.39)
(2.40)
:
(2.41)
39
u — , λ — .
:
(2.42)
’ , :
Wξ = Ve −VN (u sinϕ +ωζ ) + u cosϕ + V sin Ψ V
R ζ
3
V V
Wη = V 2 N ζ
N −Ve (u sinϕ +ωζ ) + Ru cosϕ sinϕ + (2.43)
R3
(Vcos Ψ)2 + (Vsin Ψ 2
Wζ = V − )
ζ − 2Vu cosϕ sin Ψ − Ru2 cos2 ϕ
R3
, ’ :
Wx = ax + gx
Wy = ay + g y (2.44)
Wz = az + gz
’ ,
.
Wξ = (ax + gx )(cosγ cos Ψ − sin γ sinϑ sin Ψ) − (ay + gy )cosϑ sin Ψ + (az + gz )
(sin γ cos Ψ − cosγ sinϑ sin Ψ)
Wη = (ax + gx ) (cosγ sin Ψ +sinγ sinϑ cosΨ) +(ay + gy )cosϑ cosΨ +(az + gz )(sinγ sin Ψ + cosγ sinϑ cosΨ) (2.45)
Wζ = (ax + gx )sin γ cosϑ + (ay + g y )sinϑ + (az + gz )cosγ cosϑ
40
(2.45) ,
( ϕ,λ ) :
(2.46)
:
• Tk = 600 .
• : - 1- , h = 0.005 .
1: V = 200 / (720 / ),
(ϕ = 50.21° ,λ = 30.3° ), ψ 0 = 0° .
. 2.12 (2.44)
(2.46) ’ , ψ °
0 = 0
41
2: , ψ 0 = 30° .
. 2.13 (2.44)
(2.46) ’ , ψ = 30°
0
2.3
,
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.
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, : ( ), ,
.
.
. ’ ,
42
.
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.
, .
( ).
, (
) , .
,
.
,
Δa , —
:
δ p = badtdt = batdt = 1 bat
2 (2.47)
2
(2.47) ,
.
: ’
200 / 3 (10800 ).
10−4 g . g = 9.8 / 2 R = 6400000 ,
0.01786 , 1.023°.
Δω
, :
43
δθ = bg dt = bgt (2.48)
.
g ⋅Δα .
:
δ p = gbgtdtdt = 1 gb t 2dt = 1
g gb t3
g (2.49)
2 6
, ,
, ,
.
: 0.005 / ( 2.42 ⋅10−8 / )
3 0.007793 ,
0.45°.
, ,
.
N , h ,
( , ).
:
δΨ = hN Ψmϑmγ ω N +1
m sin(ε − N π ) (2.50)
2
, .
44
. 2.14
. 2.15
. 2.16
45
. 2.17
:
X = Fx + Bu (2.51)
Y = Hx (2.52)
x = [Δλ,Δϕ,Δh,ΔvN ,ΔvE ,ΔvD ,Δα ,Δβ ,Δγ ]T —
( , , ), F — , B
— , u — (
).
F ’
.
.
’ , 200 / 4 ,
10−4 g 0.01 / ,
.
46
. 2.18
2
1.
.
.
2. ( ,
)
, .
3.
( ), .
.
47
4. '
, .
5.
( ) (
).
.
6.
,
.
48
3
–
– .
,
, ( ),
.
3.1 -
- :
1. ,
- .
2. ( ) ,
’ .
.
.
,
.
.
,
49
.
( . 3.1):
• 1- : .
• 2- : (2D-
).
• 3- : (3D- ,
« » ).
• 4- : .
. 3.1
,
, .
50
« » .
, ’ ,
.
,
. ,
, .
,
.
( ), ’ .
3.2 -
- :
1. ,
- .
2. ( ) ,
’ .
.
.
,
.
.
,
51
.
( . 3.1):
• 1- : .
• 2- : (2D-
).
• 3- : (3D- ,
« » ).
• 4- : .
. 3.1
,
, .
« » .
52
, ’ ,
.
,
. ,
, .
,
.
( ), ’ .
3.3 -
,
.
,
,
’ .
( . 3.9).
. 3.9 « »
53
A :
A = D (3.1)
2 ⋅h ⋅ tan γ
2
h — ( ), γ —
( ) ( . 3.9 ).
, D
h ( . 3.9 ), (3.1) :
A = 1 (3.2)
2 ⋅h ⋅ tan γ
2
, ,
i , ( . 3.9 ),
.
, ( .
. 3.3), .
, (
α1,α2 ,α3 ) ( β1, β2 , β3 )
.
, ’
:
(Bx cosγ cosϑ + x0 )sin β1 = (Bx sinϑ + y0 ) cosα1 cos β1
(− Bx sinψ cosϑ + z0 )sin β1 = (Bx sinϑ + y0 ) sinα1 cos β1
(By (sin γ sinψ − cosγ cosψ sinϑ) + x0 )sin β2 = (By cosϑ cosγ + y0 ) cosα2 cos β2 (3.3)
(By (sinγ cosψ + cosγ sinψ sinϑ) + z0 ) sin β2 = (By cosϑ cosγ + y0 )sinα2 cos β2
(Bz (cosγ sinψ + sinγ cosψ sinϑ) + x0 )sin β3 = (− Bz cosϑ sin γ + y0 ) cosα3 cos β3
(Bz (cosγ cosψ − sinγ sinψ sinϑ) + z0 )sin β3 = (− Bz cosϑ sin γ + y0 )sinα2 cos β3
54
Bx , By , Bz — ’ ( );
ψ ,ϑ,γ — ; Bx0 , By0 , Bz0 —
’ ; α i — .
, (3.3) :
((c33 + c12 )Bx + x0 + z0 )sin β1 = (cosα1 + sinα1)(Bx c32 + y0 )cos β1
((c23 + c13)By + x0 + z0 )sin β2 = (cosα2 + sinα2 )(By c32 + y0 )cos β2 (3.4)
((c21 + c11)Bz + x0 + z0 )sin β3 = (cosα3 + sinα3)(Bz c32 + y0 )cos β3
Cij — ( ).
(3.4) ,
.
q . ’
r d :
(q ⋅ (dˆ
T T
i i − rˆ)q)x (q ⋅ (dˆ − rˆ)q) F i
i y x F i
h y
cam = ,
(q ⋅ (dˆ − rˆ)q) (q ⋅ (dˆ
=
− rˆ)q) F i ,
F i (3.5)
i z i z z z
i = 0,1,…, N f −1, N f — ,
.
(3.5)
:
−2r ∂Λ + 2r ∂Λ − 2v ∂Λ + 2v ∂Λ − q ∂Λ ∂Λ
y x y x z + qy − q ∂Λ + q ∂Λ
∂r ∂r ∂v ∂v ∂q ∂q x ∂q t ∂q (3.6)
x y x y t x y z
55
’
Λ(rx , ry , rz ,Vx ,Vy ,Vz ,qt ,qx ,qy ,qz ) , ’ ,
, . ’
, :
q q + q q
ψ = arctan 2 t z x y
1− 2(q2 2
y + qz )
ϑ = arctan (2(qtqy − qzqx )) (3.7)
q q + q q
γ = arctan 2 t x y z
1− 2(q2 2
y + qx )
C :
2 ti +1
Ri = d + vi Δt + Δt
i 0 ai − C (τ ) A(τ ) dτ dt C
2 (3.8)
ti
Δt — i - ; ai —
’ ; V0 — .
-
. 3.10.
56
. 3.10
3
1. -
' ,
.
2.
: , - ,
57
,
.
3.
,
,
.
4.
,
,
' .
5. - ,
,
.
58
4
, , -
,
.
( ) '
.
—
,
.
,
( ), - ( ).
-
,
. . 4.1.
. 4.1
, .
59
,
( )
( ) .
,
.
. 4.2.
. 4.2
,
.
' :
( ),
.
(
) .
, ,
60
' .
4.1
, ( )
' - .
, ( ),
, ,
.
.
. -
' ,
, ,
.
,
: ( )
( ).
k
:
• Φk — ( );
• Hk — ( );
• Qk — (
);
• Rk — (
);
61
• Bk — (
).
(k −1) k :
xk = Φk xk−1 + Bkuk + wk (4.1)
xk — , uk — , wk —
,
Qk .
' yk xk
:
yk = Hk xk + vk (4.2)
vk — ( )
R .
x0
wk vk .
,
: ( )
( ).
, (4.1).
'
.
62
:
P (t ) = E{[xk − xˆk ][xk − xˆ T
k ] } (4.3)
E{⋅} — .
:
x (t0 ) = xˆ0 (4.4)
P0 = E{[xˆ(t0 ) − x(t0 )][xˆ(t0 ) − x(t0 )]T } (4.5)
( ):
xˆ− +
k = Φk −1xˆk −1 + Bk −1uk −1 (4.6)
P−
k = Φ + T
k −1Pk −1Φk −1Qk −1 (4.7)
( ) Kk ,
yk :
K = P−H T H − T −1
k k k k Pk Hk + Rk (4.8)
xˆ+
k = xˆ−
k + K −
k yk − H k xˆk (4.9)
P+
k = [I − Kk H k ] P−
k (4.10)
63
. ,
, —
(Extended Kalman Filter — EKF).
,
.
4.2
.
( )
:
Ce e b
b = Cb (ωeb×)
re e
eb = veb (4.11)
ve e b e e e
eb = Cb f − 2ωei ×veb + g
δr,δV —
; ψ — ; f —
; ωie — ; δ g —
; ε —
.
- ( ).
. 4.3.
64
. 4.3
, '
, :
r c
cp = C c
e (r e e
k cp − rec )
k (4.12)
r e
p — - ;
k
r e
c — ; C e
c — ,
.
:
rc
cpk x uk − u0
rc
cp y = λ v − v
k k 0 (4.13)
rc − f
cpk z
65
u,v — ; f —
' .
ηk ,
:
rc
− f cpk x
u η rc + u0
z k uk cp z η
k u
= + = + k
v η rc η (4.14)
k vk − f cpk y vk
rc + v0
cpk z
’
( "lever arm").
(IMU)
. 4.4.
. 4.4 '
. :
66
φ = −ω e
ie ×φ − Ce
bbg
δ re
eb = δ vb
eb (4.15)
δ vb e
eb = (Cb f b )×φ − 2ω e
ie ×δ vb
eb +δ ge + Ce
bba
∇,ε — (
).
:
rc = re + Cerb
ec eb b bc (4.16)
'
Cb
c :
rc
cp = Cc
b Cb
e (re
ep − rb
k k bc ) (4.17)
(4.15, 4.16)
' (4.17) ,
:
r c c
cp = Cb (Cb
e (r e b
ep − r ) − rb )
k k bc bc (4.18)
.
:
67
f ⋅ rc
cpk x
− f 0 c 2
δ z = rc (rcpk z )
c c
k cp z − f ⋅δ r
k f ⋅ rc cp = H1δ r
k cpk
0 rc cp y (4.19)
cpk z k
(rc
cp z )2
k
' ( )
:
b b
δ rc = −C c b c e b δ reb δ reb
cpk b Cn Cb Cb (reb ×) = H
φ 2 φ 4.20)
:
δ δ rb
zk = H eb
1H2 φ (4.21)
' .
:
• : V = 200 / (720 / ).
• : . (ϕ = 50.21° , λ = 30.3° ).
• : 10−4 g .
• : 0.05° / ( ≈ 2.42 ⋅10−7 / ).
.
68
. 4.5 ’
( / ), ,
( / ), .
. 4.6
69
.
, .
. 4.7
.
1.5
1 yaw unaided
yaw SINS vs OESON
0.5
0
-0.5
-1
. 4.8
yaw error in degrees
70
( ) ,
P .
. 4.9 ’
. 4.10 ’
71
. 4.11
’
. 4.12
( - ),
.
72
. 4.13 (
)
. 4.14 ( )
73
. 4.15
( )
4
1.
.
.
2.
.
- ,
.
3.
,
74
,
.
4. -
,
.
5.
:
,
’ .
6. :
5
±0.5° .
180
.
7. ' ,
, :
2 / ,
10 .
75
-
. :
1.
.
, ' . ,
.
2.
,
.
3.
.
( , ).
4. MATLAB/Simulink
,
.
5. -
( ) .
.
6.
.
.
76
7. ’
. ,
. 25
: 4
/ , 2 ,
180 .
'
.
77
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