Инерциальная навигация
.pdf, ; ΘQ % / ! 3 ,@
τ = τ ! ρ = Q ◦ ρ ◦ Q−1'
*, < Vv % C ,@
τ = τ ! ρ = ρ + vτ
% 2 / vτ / vt! t
,'
1 / 2 %
/ /2 ! /
/ X, / '
+, < g Gg %
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τ = τ ! ρ = ρ + gτ 2/2
% ! 2 / /
2 gt2/2,'
6/ / .3
2 2 / C ' - 2 0
2 % ! 2
52 − 42 = 9 2 ,' / 3 ! !
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(1) |
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(5) |
(10) |
(17) |
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(3) |
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(7) |
(12) |
(19) |
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Θ |
(6) |
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(21) |
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(11) |
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(23) |
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(18) |
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(22) |
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(25) |
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; / Q 2 / / ! / / % , /
' ; ! / 2 2 !
/ 2 / C '
Gg ◦ Tt = TtRrVv Gg ! |
r = gt2/2! |
v = gt' |
% (, |
Tt ◦ Gg = Gg Vv RrTt! |
r = gt2/2! |
v = −gt' |
% L, |
Gg Rr = Rr Gg ' |
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% K, |
7
Rr Gg = Gg Rr' |
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Gg |
◦ |
ΘQ = ΘQ |
◦ |
Gg ! |
g = Q−1 |
◦ |
g |
◦ |
Q' |
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ΘQ |
◦ |
Gg = Gg |
◦ |
ΘQ! |
g = Q |
◦ |
g |
◦ |
Q−1' |
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Gg Vv = Vv Gg '
Vv Gg = Gg Vv '
Gg2Gg1 = Gg ! g = g1 + g2'
% ,
% ,
% ,
% ,
% *,
% +,
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Tt Rr Tt Vv! / . ' 8
! ! % (, % L,!
g ! / '
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2 / * X!
/ * !
' 1 X = τ + ρ
% (,
Tt |
Gg |
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τ + ρ −→ (τ + t) + ρ |
−→ (τ + t) + [ρ + g(τ + t)2/2])! |
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X = τ + ρ % (, |
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Gg |
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Vv |
Rr |
τ + ρ −→ τ + (ρ + gτ 2/2) |
−→ τ + (ρ + vτ + gτ 2/2) |
−→ |
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−→ τ + (ρ + r + vτ + gτ 2 |
/2) −→ (τ + t) + (ρ + r + vτ + gτ 2/2) = |
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Rr |
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Tt |
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= (τ + t) + (ρ + gt2/2 + gtτ + gτ 2/2) = (τ + t) + [ρ + g(τ + t)2/2]
F / ! 2
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2 C /
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2 / C
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2 / C !
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0 .3 2 / .
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Tt+dtRr+drVv+dv ◦ ΘQ+dQ ◦ Gg+dg = ΛIE(τ +dτ ) = ΛIE(τ ) ◦ ΛE(τ )E(τ +dτ ) =
= (TtRrVv ◦ ΘQ ◦ Gg ) ◦ (Tdτ Vadτ Θexp(iωdτ /2)Gndτ ) = |
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= TtRrVv ◦ ΘQ ◦ Gg ◦ Tdτ Vadτ ΘSGndτ |
(17) |
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(17) |
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(5),(23) |
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= TtRrVv ◦ ΘQTdτ Vgdτ Gg Vadτ ΘSGndτ |
= |
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(5),(23) |
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(10),(16),(21) |
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TtRrVv ◦ Tdτ ΘQ ◦ Vgdτ Vadτ Gg ◦ ΘSGndτ |
= |
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(10),(16),(21) |
TtRrTdτ Rvdτ Vv ◦ ΘQ ◦ V(a+g)dτ ◦ ΘS ◦ GS¯◦g◦SGndτ |
(2),(15) |
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(2),(15) |
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(1),(4),(9),(16),(25) |
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TtTdτ RrRvdτ VvVQ◦(a+g)◦Q¯ dτ ◦ ΘQ ◦ ΘS ◦ GS¯◦g◦SGndτ |
= |
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(1),(4),(9),(16),(25) |
Tt+dτ Rr+vdτ Vv+Q◦(a+g)◦Q¯ dτ ◦ ΘQ◦S ◦ GS¯◦g◦S+ndτ = |
= |
= T + R + V ¯ ◦ ΘQ+Q ( ) 2 ◦ G + +
t dτ r vdτ v+Q◦(a+g)◦Qdτ ◦ iω dτ / g g×ωdτ ndτ !
Q = eiϑ/2! |
S = eiωdτ /2! |
Q¯ = Q−1 = Q˜ ! S¯ = S−1 = S˜' |
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! dτ / ' E
! /
dτ 2! ' ! /
/ /2 dτ
0 / !
Q ◦ S = Q ◦ eiωdτ /2 ≈ Q ◦ (1 + iωdτ /2) = Q + Q ◦ (iω)dτ /2!
S−1 ◦ g ◦ S = e−iωdτ /2 ◦ g ◦ eiωdτ /2 ≈ (1 − iωdτ /2) ◦ g ◦ (1 + iωdτ /2) =
= (1 −iωdτ /2) ◦(g + ig ◦ωdτ /2) ≈ g + i(g ◦ω −ω ◦g)dτ /2 = g + g ×ωdτ '
6 0 /
/ /
/ / / @
dt/dτ = 1!
dr/dτ = v!
dv/dτ = Q ◦ (a + g) ◦ Q−1! dQ/dτ = iQ ◦ ω/2!
dg/dτ = n + g × ω'
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Tt+dtRr+drVv+dv Gg +dg ◦ ΘQ+dQ =
7
= (TtRrVvGg ◦ ΘQ) ◦ (Tdτ Vadτ Gndτ Θexp(iωdτ /2))'
; / ' 6
2 3
2 / C ! 3 / ' 1 !
g g !
/ .3 2 % ,
g = Q ◦ g ◦ Q−1'
; / Q g /! ]
−1 ˜ − ◦ ˜ − ◦ −1 dQ /dτ = dQ/dτ = iω Q/2 = iω Q /2!
/ g @
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dg |
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d(Q ◦ g ◦ Q−1) |
= |
dQ |
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g |
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Q−1 |
+ Q |
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dg |
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Q−1 + Q |
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dQ−1 |
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dτ |
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Q ◦ ω ◦ g ◦ Q−1 + Q ◦ (n + g × ω) ◦ Q−1 − |
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Q ◦ g ◦ ωQ−1 = Q ◦ n ◦ Q−1' |
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2 |
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! gg !
dv/dt = Q ◦ a ◦ Q−1 + g '
1 g
! / I'
E . . .
/ 0 / ' F !
2 '
1 2
/ C 0 @
Tt+dtRr+dr ◦ ΘQ+dQ ◦ Vv +dv Gg+dg =
= (TtRr ◦ ΘQ ◦ Vv Gg ) ◦ (Tdτ Θexp(iωdτ /2)Vadτ Gndτ )'
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v = Q−1 ◦ v ◦ Q'
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dv /dτ = a + g + v × ω'
! v v
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dr/dτ = Q ◦ v ◦ Q−1!
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v / /
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X = X + t'
, ; Rr@
X = X + r'
, ; Θ
ϑ@
X = eiϑ/2 |
◦ X ◦ e−iϑ/2' |
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1 / / / : ! / / 2
/ C ! * .
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*, F V
ψ@
X = eψ/2 |
◦ X ◦ eψ/2' |
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X = (X−1 − g/2)−1'
! /
0 / . %X = X−1, /
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Q, E 0 ! /
w Ww@
X = (X−1 + w/2)−1'
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X = γX = eαX'
< 2 α
/ γ = eα > 0 %γ > 1 : !
γ < 1 : ,'
< ! .3 2 0
/ / 72 = 49' ' F
0 / /
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; 0 / RR : % ,! :
/ / @
X = X + R!
R= r0 + r = t + r'
! / ;
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6 0 /
BB! ! % ,
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/ % ,!
: C @
◦ ◦ ¯
X = B X B!
B = eψ/2eα/2eiϑ/2 |
= eα/2eψ/2 |
◦ eiϑ/2! |
B¯ = e−iϑ/2 |
◦ eψ/2eα/2' |
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1 0 /
AA / / %
,! : * % / 3
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X = (X−1 + A)˜ −1!
1 |
(w + g)! |
˜ |
1 |
(w − g)' |
A = a0 + a = /2 |
A = a0 |
− a = /2 |
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τ + ρ = |
τ + ρ + (τ 2 − ρ 2)(a0 + a) |
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1 + 2(a0τ − a · ρ ) + (τ 2 − ρ 2)(a02 |
− a 2) |
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! 0 %
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a !
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R |
B |
A |
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R |
(1) |
(2) |
(5) |
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B |
(3) |
(4) |
(7) |
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A |
(6) |
(8) |
(9) |
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; B ! / 2
2 % / / 2 .
/ / 2 ,'
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