Инерциальная навигация
.pdfΛIK = ΛIE ◦ ΛEK !
. . 3
/ 3 % / B ,' ? ! ΛIE ! ΛEK
ΛIK 0 / / B ! / .3
/ .3 !
.3 % / /
,= %◦, . 0 /!
.3 ' ; / /
! /
@
ΛIJK = ΛEKJ ◦ ΛIJE '
? / / 3 /
/ 3
3 '
E 3 / / . .3
/ .
/ 3 0 / ' 8 .3 2
/ B . 0 /
Rr ◦ Θ ϑ!
; Rr exp(εir/2)!
! Θ
exp(iϑ/2)' ; 0 / 3
/ / 3 ! !
ΛIE !
2 2'
ΛIE = eεir/ ◦ eiϑ/
% ! / ,'
! / / @
' 8 0 ' B
/ !
/ / / !
@
2! 2!
ΛIE = ΛII ◦ ΛI E ! ΛII = eεir/ ΛI E = eiϑ/
I : ! . I
' < !
/ / ! .3
. / I' 6 0
2! 2!
ΛIE = ΛII E ◦ ΛII ! ΛII E = eεir/ ΛII = eiϑ/
I : % .3 I ,!
.I '
/
.3 !
2 2 2 2! − 2 2'
ΛIE = eεir/ ◦ eiϑ/ = eiϑ/ ◦ eεir / r = e iϑ/ ◦ r ◦ eiϑ/
1 .3 ! .3
! !
0 '
8 . ! / /
/' 6 0 .3
. !
3 % ! ! , /
/ ' 6 2
! . 3
' ! !
εir/2 |
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iϑ/2! |
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ΛIE = e |
◦ e |
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r ϑ! |
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2 2 2 2 ( + ) 2! eεir / eiϑ/ = eiϑ/ eεir / = ei ϑ εr / r ϑ!
! 3 J %
3 ,! ΛIE
J
'
M 6 ( -
6/ 2 3 .
˜ ◦ ◦
rE = Λ rI Λ
% Λ / /
I E! 0 / ,' 8
/ λ0! λ1! λ2! λ3 % !
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= ! / / |
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Λ = λ0 + λ1i + λ2j + λ3k |
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rI |
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i1! i2! i3 |
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r1I ! r2I ! r3I ! / rE r1E ! r2E ! r3E % 0 /
! 3 / /
/ / ,' 1 /
/ rE .3
/ @
r1E = (λ20 + λ21 − λ22 − λ23)r1I + 2(λ1λ2 + λ0λ3)r2I + 2(λ1λ3 − λ0λ2)r3I ! r2E = 2(λ1λ2 − λ0λ3)r1I + (λ20 − λ21 + λ22 − λ23)r2I + 2(λ2λ3 + λ0λ1)r3I ! r3E = 2(λ1λ3 + λ0λ2)r1I + 2(λ2λ3 − λ0λ1)r2I + (λ20 − λ21 − λ22 + λ23)r3I '
r2E |
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2(λ1λ2 |
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λ0 |
λ3) λ0 |
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λ1 |
+ λ2 |
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λ3 |
2(λ2 |
λ3 |
+ λ0 |
λ1) |
r2I |
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λ2 |
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r3E |
2(λ1λ3 |
+ λ0 |
λ2) |
2(λ2λ3 |
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λ0 |
λ1) λ0 |
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+ λ3 |
r3I |
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! !
RE = CRI !
RE RI : /) / rE rI !
c21 |
c22 |
c23 |
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2(λ1λ2 |
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λ0 |
λ3) λ0 |
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+ λ2 |
λ |
λ3 |
2(λ2 |
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+ λ0 |
λ1) |
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λ2 |
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c31 |
c32 |
c33 |
2(λ1λ3 |
+ λ0 |
λ2) |
2(λ2λ3 |
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λ1) λ0 |
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+ λ3 |
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C :
'
C / . .3 '
6/ / /
. /
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/
/ '
P /
/ / !
2 0 / / . % . ,' ; 0 / .3
% 'P' 7 ,! 2 /
λ0! λ1! λ2! λ3
.
% ! ! !
/ ,' E 0 . . / / !
.3 / .3 @
1 + c11 + c22 + c33 = 4λ20! 1 + c11 − c22 − c33 = 4λ21! 1 − c11 + c22 − c33 = 4λ22! 1 − c11 − c22 + c33 = 4λ23'
6/
% , %
/ λ20 + λ21 + λ22 + λ23 = 1, / .3 /
! / / /
. ! . .3 . / . % . ., @
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1 |
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! λ = |
c23 − c32 |
! λ = |
c31 − c13 |
! λ = |
c12 − c21 |
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λ = |
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1 + c + c + c |
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4λ0 |
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4λ0 |
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4λ0 |
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c23 − c32 |
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! λ = |
c12 + c21 |
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! λ = |
c31 + c13 |
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λ = |
! λ = |
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1 + c |
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4λ1 |
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4λ1 |
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4λ1 |
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c31 − c13 |
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c12 + c21 |
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! λ = |
c23 + c32 |
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λ = |
! λ = |
! λ = |
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c + c |
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4λ2 |
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c12 − c21 |
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c31 + c13 |
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c23 + c32 |
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λ = |
! λ = |
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c |
c + c |
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4λ3 |
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4λ3 |
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4λ3 |
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% / / / /
,'
?
.3 . ' !
/
/ ' 6
!
. ' E
/ ! 0
/ / 3 ! '
> ! / 0 !
3 . /
! ! / !
'
B / / ϑ1! ϑ2! ϑ3! /
.3 λ0! λ1! λ2! λ3
.3 @
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= cos |
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λ0 |
ϑ12 + ϑ22 + ϑ32 |
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λ1 |
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ϑ1 |
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sin |
ϑ12 |
+ ϑ22 + ϑ32 |
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ϑ12 + ϑ22 |
+ ϑ32 |
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λ2 |
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ϑ2 |
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sin |
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ϑ12 |
+ ϑ22 + ϑ32 |
! |
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ϑ12 + ϑ22 |
+ ϑ32 |
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λ3 |
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ϑ3 |
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sin |
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ϑ12 |
+ ϑ22 + ϑ32 |
= |
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ϑ12 + ϑ22 |
+ ϑ32 |
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0 / Λ = exp(iϑ/2)' ; |
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! ! ! ϑ1 = ϑ2 = ϑ3 = 0 |
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ϑ12 + ϑ22 + ϑ32 |
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= 0 /
@ λ0 = 1! λ1 = λ2 = λ3 = 0'
F . ' 7 / / !
/
λ20 + λ21 + λ22 + λ23 = 1!
. . /
' 8 0 / %Λ = 1,
. / !
%Λ = −1,
2π! / ! / !
' 6 / /
! 2 2 2π! /
' ; 0 /
/
! %−1 ≤ λ0 ≤ 1 . ,@
λ0 = 1! ϑi = 0! i = 1, 2, 3=
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−1 < λ0 < 1! |
ϑi = |
2 arccos λ0 |
λi! i = 1, 2, 3= |
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1 − λ02 |
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λ0 = −1! |
kϑ1! ϑ2! ϑ3= |
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= 2πl |
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ϑ12 + ϑ22 + ϑ32 |
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% ϑ1! ϑ2! ϑ3 : . / ! .3 /
.,'
1 ! / .
/ 2 2π % /
, . 0 2 %
2π,'
/
! .3 /
/! . ' /
/ 2 π / /
0 2 '
M$ ( ( &
6 2 ! 3 !
@
dr
dt
= v'
1 / ! /
/ ' ? 3
/ /
. G '
2 ' 7 .
' <
. : 0
/! ! / C /
; '
$
6 r v
. /! / /
' ;
/
I' 1 0 /
! ! /@
drI = vI ' dt
6 J
. G ! ! /
@
drJ = vJ ' dt
/ ! 0 0
' 6/ /
. 3 . / 0
' P / @ .
2 3 / / 2
. ! 3 %
, / / ' 6
/ '
6 3
!
I! G E' 6
@
˜ |
! |
rE = Λ ◦ rI ◦ Λ |
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˜ |
' |
vE = Λ ◦ vI ◦ Λ |
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B / rE ! !
rI !
Λ % , /
vE ! .3 rE
drE = rE × ωE + vE ' dt
+
6/ 2 / . ' 6
0 H A ! drE /dt !
! 0 2 '
/ / . ! 2 . !
! . '
rE 3
2 .
v E = rE × ωE + vE vE = ωE × rE + v E
8 ! / 3 0
/! ! 0
! ! I@
vI = ωI × rI + v I '
6 ! !
v = ω × r + v '
M % ( (
F 3 :
% . ,@
dv |
˜ |
' |
dt |
= Λ ◦ a ◦ Λ + g |
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6 ! / . / ! /
3 0
@
dvI |
˜ |
dt |
= ΛIE ◦ aE ◦ ΛIE + gI ' |
A I!
G
E % / ! .
3 ,' 6 3
/ !
3 !
/ @
7
dvI |
˜ |
˜ |
dt |
= ΛIE ◦ aE ◦ ΛIE + ΛIJ ◦ gJ ◦ ΛIJ ' |
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> ! /
/ 3 ! =
%
, .3
. '
; /
. ! 0
. .
@
dv
dt
= a + g'
C 0
G . ! 0 2
3 3 .'
;
/ . % ,!
.3 !
/ . ! ! / '
E / / ! !
. / 2
F / . .3
% .3 , ' 1
< . @
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dv |
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m |
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= F ' |
dt |
M ; ( (
F @
dΛIE = 1ΛIE ◦ (iωE )' dt 2
A / / / : % !I E,! 3 %
E,'
6 0 / / .
= ! 3
. / !
% / ,' E
0 @
dΛ |
= |
1 |
Λ ◦ ωE = |
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dt |
2 |
/ . /' 8
3 . / / /
dλ0/dt = −(λ1ω1E + λ2ω2E + λ3ω3E )/2!
dλ1/dt = (λ0ω1E + λ2ω3E − λ3ω2E )/2!
dλ2/dt = (λ0ω2E + λ3ω1E − λ1ω3E )/2!
dλ3/dt = (λ0ω3E + λ1ω2E − λ2ω1E )/2!
. 3 /
'
/
/ % 3 . /
, ! E!
! I' 6 / /
@
dΛ |
= |
1 |
ωI ◦ Λ' |
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dt |
2 |
E 0
/' <
/ 2
2 ! / G
. / ωI ! / ωE '
.