Инерциальная навигация

/ 0 / /

' E !

! 3 /

n ! '

6 .

! 3 .

/ C /

@

ΛIE1(t+dt) = ΛIE1(t) ◦ ΛE1(t)E1(t+dt)!

ΛIE1(t+dt) = Tt+dtRr1+dr1 Vv1+dv1!

ΛIE1(t) = TtRr1 Vv1 !

ΛE1(t)E1(t+dt) = Tdt'

1

' 6

n H

H L n J :

(

> n

! .

.' E /!

/ . . C ' E

/! n

/ / % ,!

/ 0 ' <

C !

C )< . ' / ! /

2 C ' 8

' - 0 : C ) < . ! 0 / C )< . '

< 0 3 ' - / C )< . .

/ /! n . '

2 / C

' 8 3

/

Λ = TtRrVv Gg '

< ! ◦

! 0 /

% ! T R!

V G,' E 0 ! ! !

/ C %

! / /,' 7 /

0 ! .3 .

/ C )< . < . ' 8 !

< . 2 /

C / '

1 .3 2 /

< . @

	T	R	V	G
T	(1)	(2)	(5)	(10)
R	(3)	(4)	(7)	(12)
V	(6)	(8)	(9)	(14)
G	(11)	(13)	(15)	(16)

; / 2 2 '

6/ / 2 '

Gg ◦ Tt = TtRrVv Gg !	r = gt2/2!	v = gt'	% ,
Tt ◦ Gg = Gg Vv RrTt!	r = gt2/2!	v = −gt'	% ,
Gg Rr = Rr Gg '			% ,
Rr Gg = Gg Rr'			% ,
Gg Vv = Vv Gg '			% *,
Vv Gg = Gg Vv '			% +,
Gg2Gg1 = Gg ! g = g1 + g2'			% Q,

8 /! / .

/ % , % ,' > .3

< .

! / / n ' ;

0 . 3

% ,' $ 0

/ ai = 0! i : n

' 6 =

t = ti = τi! i = 1 ÷ n' A !

i /

% .3 0 dti/dτi = 1

/ 3 / C )< . ,!

! / n 3

'

6 .3 n @

ΛIEi(t+dt) = ΛIEi(t) ◦ ΛEi(t)Ei (t+dt)!

i = 1 ÷ n!

ΛIEi(t+dt) = Tt+dtRri+dri Vvi+dvi Ggi+dgi !

ΛIEi(t) = TtRri Vvi Ggi !

ΛEi(t)Ei (t+dt) = TdtGnidt'

A ! ! t : ! ri : i

! vi : i '

/

% / I

t, 3 . /

n n @

gi /!

i' ? ! n

gi

n		mj
j
	rj	ri
gi =	|		2
j=i		− |
=1

.3

rj − ri

|rj − ri|!

/ i! mj :

j % ,!

3

% ! 0 / ,'

- .3 2 / < . ! /

/

/ / 2n / /

/ /

dri

dvi

n

rj − ri

= vi,

= gi = mj

,

i = 1

÷

n!

dt

dt

3

j

rj

−

ri

|

j=i

|

=1

0 / 2 n ! 3 n ! .3 / ni

ni =

dgi

=

n

mj

vj − vi

3

3(vj

vi)

(rj

ri)

rj − ri

5

'

dt

j

−

−

·

−

j=i

|

−

|

|

−

|

=1

6 n

0 ! ! ' 8

/

/ ' E 4

: n !

'

1' " / "

M ( " (

. 2

/ / /

rE = Q−IE1 ◦ rI ◦ QIE '

A . /

I E % QIE , % / rI ,! /

I! %

! / , ! /

E % ! ! rE ,' E /

% I E,

. ! /

rI QIE !

/ / rE '

; I E /

QIE I E! QEI

E I' ; QEI = Q−IE1!

0 / @

rE = QEI ◦ rI ◦ Q−EI1'

? / E I

rI = Q−EI1 ◦ rE ◦ QEI = QIE ◦ rE ◦ Q−IE1 '

; / / ! / / /!

.3 / / '

/ / !

3 %

! ,' ; 0

. /

/ / '

.

! 3

3

rE = QEI ◦ rI ◦ QIE ! rI = QIE ◦ rE ◦ QEI '

8 ! ! .3 / !

'

6 / / /2

! ! / 2 /

/ 0 / ! 3

C )< . ! /

. / G ' 1

. /

/ % /

, %

,' 6/ 2

rI !

QIE @

r I = QIE ◦ rI ◦ Q−IE1'

' A 3 / r IrI . / % !

/ ,! /

I' ; I E

! ! 3 @

r E	= QI−E1 ◦ r I ◦ QIE '
; / r I		! ! /
/ @
r E	= rI

% 0 /

,'

M " ( ) &

; 3

G . / ΛJ K % J K, .3 @

ΛJEK = Λ−IE1 ◦ ΛJI K ◦ ΛIE '

$

- 0 /

/ @

ΛIK = ΛIE ◦ ΛEK !

! / @

ΛIJK = ΛEKJ ◦ ΛIJE '

6/ / / : / ! / ' -

/ ! 0

.3 / /

% /

C )< . /

/ 0 /,'

6/ 2

% ,@

QJEK = Q−IE1 ◦ QJI K ◦ QIE '

< !

: ! '

1 /

' < 2

/ % 3 .

/ /! /

! / . ,' > 0

@

QIK = QIE ◦ QEK '

A . / I E

/ E K!

/ I K' E

3 .

QIK QIE QEK ' $

/ 3

A ◦ B = C'

+

7 / . /

! .3

QIE ! QEK QIK @

			!
QIE = α + axi + ay j + az k
			!
QEK = β + bxi + byj + bz k
		'
QIK = γ + cxi + cy j + cz k

1 / .3

/ @

γ = αβ − axbx − ay by − az bz !

cx = αbx + axβ + ay bz − az by!

cy = αby + ay β + az bx − axbz ! cz = αbz + az β + axby − ay bx'

$ / 3

@

QIJ = QIE ◦ QEK ◦ QKJ

2 ' /

/

3 . /

/ / ! 3

%

.3 . /

2 ,'

8 / ! /

! . . /

! / / . 0 / ' ; . Q

2! Q = eiϑ/

ϑ: % ,! .3

/ / % ,

! .3 !

7

! / .' ; %

/ , / 0

'

> ! 3 . ! /

. / ! /

@

QIJK = QEKJ ◦ QIJE '

6 / J ! 3

0 / / ' ;

3 0 % I E! E

K I K, J / J !

. @

QIJK = QEKJ ◦ QIJE '

6 / / ! ! I' 1

@

QIIK = QEKI ◦ QIIE !

! / 2 /

! @

QIK = QEKI ◦ QIE '

1 0 / '

< / / % 3

, ' $

! / !

/ /

!

'

/ !

! / ! 3

2 ! @

QIJJ = QKJ J ◦ QEKJ ◦ QIJE '

. : /

'

7 / / !

2 / ! 3

/ / !

/ !

3 / ' F

QEK

I '

/ E K' 8

IH ; 3 0 '

; E K ' ?

/ !

0 I E K' B !

0 2 ! 0 / @

QEKE = QEK ! QEKK = Q−EK1 ◦ QEKE ◦ QEK = QEK '

! / .

! @ 0 / % /

/ . , / F )C %

/ ,' < /

/ / '

M + " ( * &

; / / / / 3 / /

' B / /

Q Λ! .3 / 3 / /

' ! / /

% , / . / / / ' ?

/ 3 /

% / ,' 6

) !

. ! ! /

!

3 ' / / 2 3 3 .

/

.