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..

.. , .. , .. , .. , .. , ..

-- 2013

.., .., .., .., .., .. () , . . , . . . : , , , . Lavrenov S.M., Michaylin D.A., Tuchin A.G., Tuchin D.A., Fitenko V.V., Yaroshevskiy V.S. DISD mathematical model for the soft Lunar landing projects Doppler measurer of velocity and range (DISD) provides detection and radiolocation capture of the signal reflected form the underlying surface. DISD is used in the soft Lunar landing projects. Mathematical models of range and velocity measurement and of normal angles determination are considered in the paper. The functioning of the DISD device is described. The algorithm converting measurements into range values is presented. Key words: DISD, range measurement, velocity measurement, Lunar landing.

....................................................................................................................... 3 1. .................................................................... 3 2. .................................................................... 5 3. ............................................................................ 8 4. .................................................. 9 5. .............................. 10 6. ........................ 10 ....................................................................................................................... 13 ...................................................................................... 15

() [1, 2]. [3]. [4] : - () ; - ; - OX . , , , . . . . [3] . . , . , .

1.
, . , , , . - - , .

4 13325 ( 1.065 ), : f1 = 61.52 , f 2 = 82.03 , f3 = 102.50 , f 4 = 187.50 , 1 = 2438 , 2 = 1829 , 3 = 1463 , 4 = 800 . , . . 1 .

. 1. f1 , f 2 , f3 , f 4 . . 2 .

. 2. ui , i- , :

5
ui = ( - sin cos i cosi - cos i sini
T ),

i

(1)

i i- OYZ; i i- OYZ, i = 225 - 90 i, i = 1,... , 4 , OY. Vi = (V , ui ) , ui . OX . f1 , f 2 , f3 . . 40.992 . , . 4в40.992=163.968 . 600 f 4 , .

2.
f ,
c , c . 2 f ( ) L . D

L =

D = n L + B ,

(2)

n , B ( D L ). , . (2) :
D1 = n1 L1 + B1 + 1 , D2 = n2 L2 + B2 + 2

(3)

1 2 - . , D1 D2 (3) . , ,

6
max ( L1 , L2 ) nmax = min ( L1 , L2 ) ( nmax + 1) ,

nmax =

min ( L1 , L2 ) max ( L1 , L2 ) - min ( L1 , L2

)

-

. , f1 f 2 nmax = 3 , 7300 , . = 2 - 1 :
n1 L1 + B1 = n2 L2 + B2 + .

(4)

(4) n1 n2 . (4)
n1 = n2 L2 + B2 - B1 + . L1

(5)

, D , . n2 n2max = , L2 [ x ] - , x . n2 n2max , , n1 (5). n1 n1 n1 ,
n1 = n1 +n1 .
D

(6)
max

, , n2 = 0,... , n2 (n1 , n1 ) .
mi n2 n1 = arg min n1 ( n2 mi n2 n2

= arg min n1 ( n2

( (

) )

) )

,

n2 = 0,... , n2

max max

mi mi , n2 = 0,... , (n2 n1 - 1), (n2 n1 + 1),... , n2
min 1 1 min 1 1 min 2 1 min 2 1

.

n , n n , n (5) (6). n1min1 - n1min 2 , n1min1 , n2min1 . . (3) D1min1 , D1min 2

7
mi n1min1 , n2 n1 , min 2 min 2 n1 , n2 ,

Dlast .

D1min1 - Dlast < D1min 2 - Dlast D1min1 - Dl
ast

D1min 2 - Dlast

Dlast ..

. 3. n1 , . . x = k n1 + n2 , k - , : k , , , .. . (4) (5)
n1 L1 + B1 = ( x - k n1 ) L2 + B2 +

(7)

n1 = x L2 + B2 - B1 + L1 + k L2

(8)

8 , (8) k > 0 , (5). . x :
D D xmax = k + L1 L2 .

. . 3 n1 , (8) x 4000 . k =-1 , [3].

3.
V , .. Vi =- (V , ui ) . , . V = ( vx , v y , vz )
=

( (V , u )
i =1 i

4

+ Vi

)

2

. :
- sin 2 i sin i cos i cos i - sin i cos i sin i - Vi sin i = - Vi cos i cos i - V cos sin i i i

sin cos cos
2

i 2

cos i cos i i cos 2 i i sin i cos i

-

sin cos cos
2

i

i 2

cos i sin i vx sin i cos i v y = i sin 2 i vz

(9)

= 1 = 2 = 3 = :

4

i = 225 - 90 i, i = 1,... , 4

(9)

vx vy = v z

V1 + V2 + V3 + V4 4sin V -V -V + V 1234 2 2 cos -V - V + V + V 1234 2 2 cos

.

9 Vi = 70 .

4.
. , 4- , . i, j, k. , , :
nijk = ( D j u j - Di ui ) в ( Dk uk - Di ui ) ,

(10)

Di , D j , Dk ui , u j , uk . , :
n4231 = ( D4u4 - D2u2 ) в ( D3u3 - D1u1 ) .

(11)

OX OXY :
nxy = nijk - ( nijk , ez ) ez ,

(12)

ex , ey , ez . xy nxy OX :
( nxy , ex ) , n , e 0 arccos ( xy x ) nxy xy = . nxy , ex ) ( , ( nxy , ex ) < 0 -arccos nxy

(13)

OX OXZ :
nxz = nijk - ( nijk , ey ) ey .

(14)

xz nxz OX :

10
( nxz , ex ) , n , e 0 ( xz x ) arccos nxz . xz = -arccos ( nxz , ex ) , ( n , e ) < 0 xz x nxz

(15)

nijk =

xy

xz

n142 + n143 + n243 + n214 + n4231 , 5

.

5.
i = 7.50 ± 0.50 . ui . n1i = ui в ex , n2i = n1i в ui , :
q1i q2i q3i q4i q5i = = = = = ui ui ui ui ui + - + - tg tg tg tg

( i ( i ( i ( i

) ) ) )

n1i n1i n2i . n2i

(16)

= 70 .

6.
(8) k = 4 . i = Di , - , 0 0.01. . - f1 f 2 , f 2 f3 f3 f1 .

11 4.5 2 / 50 /. . 4. , .

. 4. 4.5 (2 /) . , ( ). . = 0.05 . . 5 4000 4500 .

12

. 5. 4000 . 2 / . - f 2 f3 , - f3 f1 , - f1 f 2 . . , 450 B2 (+225 ) B1 (-225 ). . 6 . 100 1.5 . . 50 / 4500 500 . 7. .

13

. 6. . 2 / 200 000 4500 2 / 2 809 400 000 . .

1. . . 2. . . 3. . 4. , , .

14

. 7. 4.5 . 50 /

15

1. , / .., .., .., C.., .., .., .., .., .., .., .. // . . -- 2009. -- . 47. -- 4. -- . 332-342. URL: http://elibrary.ru/item.asp?id=12600952 2. / . .. . - .: . 2010. 680 . URL: http://www.vega.su/upload/iblock/551/5510dafe61b43b1e07a116b1afa4e8a7.pdf 3. , / .., .. // VI « », .: « ... », .1, 2012. URL: http://jre.cplire.ru/jre/microwawes/docs/conf6books.rar 4. / .. // « ». .: , 14-18 2011 . URL: http://d902.iki.rssi.ru/theses-cgi/thesis.pl?id=2647