Calculation attitude parameters:
1. Theory.
Axes position of the GSE system relative to the s/c coordinate
system is determinated by 3 angles Alpha, Betha and Gamma.
Alpha and Betha determinate coordinates of a new XGSE axis,
while Gamma determinates YGSE or ZGSE.
Designating constructive axes of s/c
by X, Y and Z with X axis going along nominal
spin axis let us introduce Alpha and Beta angles as
the angles between X axis and Sun direction projection onto XY
and XZ planes respectively.
Alpha and Betha angles can be approximated by the
trigonometric functions of time t :
Alpha = A1 + A2sinW1t + A3cosW1t + A4sinW2t + A5cosW2t,
Betha = B1 + B2sinW2t + B3cosW1t + B4sinW2t + B5cosW2t,
where W1 is a mean spin rate of s/c, W2 is a mean
angular velocity of the angular momentum projection on YZ plane.
As a third attitude parameter Gamma angle has been taken
which is the angle between Y axis of s/c and projection of the North
Pole of ecliptic direction on YZ plane. This
angle is approximated by the linear function of time
Gamma = c1 + c2*t
Let us calculate the coordinates new X,Y and Z axes in GSE
system:
X' = [XXG YXG ZXG]
Z' = [XZG YZG ZZG]
X GSE coordinates are :
XXG = 1/SQRT(1 + tg^2(Alpha) + tg^2(Betha));
YXG = tg(Betha)/SQRT(1 + tg^2(Alpha) + tg^2(Betha));
ZXG = tg(Alpha)/SQRT(1 + tg^2(Alpha) + tg^2(Betha));
Z GSE coordinates are :
XZG = -A/SQRT(A^2 + XXG^2);
YZG = cos(Gamma) * SQRT(1-XXG^2);
ZZG = sin(Gamma) * SQRT(1-XXG^2);
where A = YXG * cos(Gamma) + ZXG * sin(Gamma);
Y GSE coordinates may be calculated
as as vector product of vectors X and Z:
Y' = [XYG YYG XYG] = X' o
Y'
Matrix M can be contructed from X' Y' Z' columns: M
= [X' Y' Z']. Any vector in s/c coordinate system can be transformed
to GSE system :
____
___
Vgse = M
* Vsc
2.Example.
String in the attitude file look like this:
1 1998 3 7 33.377 .890 .205 .232
-1.719 .105 .061 .298 1.716 .253 .063 -.110
52.5669 39.0572 -2.6696 -52.5669
It means that for 1998 March, 7, for time of 33.377
thousand seconds from 0.00 hour, for interval length of 0.890 thousand
seconds, the attitude coefficients are:
A(5) = 0.205 0.232 -1.719 0.105 0.061
B(5) = 0.298 1.716 0.253 0.063 -0.110
W1 = 52.5669 W2 = 39.0572
c1 = -2.6696 c2 = -52.5669
Application of these coefficients to corresponding
experimental data yield following results:
Data table:
|
Input vector
Time
Bx By Bz
980307 09 16 21 604 -665.00 233.00 264.00
980307 09 16 24 604 -666.00 275.00 218.00
980307 09 16 27 604 -668.00 310.00 168.00
980307 09 16 30 604 -669.00 336.00 113.00
980307 09 16 33 604 -669.00 355.00 56.00
980307 09 16 36 604 -670.00 366.00 -2.00
980307 09 16 39 604 -671.00 366.00 -59.00
980307 09 16 42 604 -672.00 359.00 -114.00
980307 09 16 45 604 -672.00 342.00 -166.00
980307 09 16 48 604 -672.00 317.00 -214.00
980307 09 16 51 604 -673.00 285.00 -255.00
980307 09 16 54 604 -673.00 245.00 -290.00
980307 09 16 57 604 -672.00 199.00 -318.00
980307 09 17 00 604 -672.00 149.00 -337.00
980307 09 17 03 604 -672.00 95.00 -348.00
980307 09 17 06 604 -671.00 39.00 -350.00
980307 09 17 09 604 -671.00 -19.00 -343.00
980307 09 17 12 604 -670.00 -75.00 -327.00
980307 09 17 15 604 -670.00 -129.00 -303.00
980307 09 17 18 604 -669.00 -180.00 -271.00
980307 09 17 21 604 -669.00 -226.00 -232.00
980307 09 17 24 604 -669.00 -267.00 -188.00
980307 09 17 27 604 -668.00 -300.00 -138.00
980307 09 17 30 604 -668.00 -327.00 -85.00
980307 09 17 33 604 -669.00 -345.00 -29.00
980307 09 17 36 604 -669.00 -355.00 28.00
980307 09 17 39 604 -669.00 -356.00 84.00
980307 09 17 42 604 -669.00 -348.00 139.00
980307 09 17 45 604 -669.00 -331.00 190.00
980307 09 17 48 604 -669.00 -306.00 237.00 |
Output vector
BxGSE ByGSE BzGSE
-666.1537 217.2564 -274.3152
-667.1761 211.6487 -277.1026
-669.2543 208.6975 -281.2331
-670.4191 206.1493 -285.0732
-670.6675 206.2733 -290.4797
-671.9767 208.9620 -296.0442
-673.2971 212.8188 -298.4244
-674.6415 219.2784 -300.3945
-675.0048 225.2637 -299.5475
-675.3804 231.1883 -297.1232
-676.6790 237.3895 -291.4288
-676.9938 240.6227 -284.3134
-676.2987 241.6170 -276.6772
-676.5246 240.8757 -267.6702
-676.7437 237.2624 -259.6874
-675.9178 231.4441 -252.6475
-676.0935 222.7400 -248.0635
-675.1897 213.5963 -244.8460
-675.2586 204.0340 -244.4298
-674.3103 194.2222 -246.9522
-674.3244 185.4542 -251.7030
-674.3270 179.2297 -259.5260
-673.2791 174.1737 -267.6318
-673.2338 172.5618 -278.1308
-674.1508 173.0241 -288.1151
-674.0475 176.5205 -298.1115
-673.8899 182.9417 -306.1989
-673.7102 190.6397 -312.6614
-673.4671 199.8289 -315.8044
-673.1952 209.3942 -316.7464
|
Angles
Alpha Beta Gamma (rad)
0.0151 -0.9736 -0.2279
0.0196 -0.9972 -0.0722
0.0237 -0.9961 0.0853
0.0274 -0.9702 0.2407
0.0306 -0.9203 0.3901
0.0331 -0.8475 0.5298
0.0349 -0.7536 0.6564
0.0360 -0.6411 0.7666
0.0363 -0.5127 0.8578
0.0359 -0.3715 0.9277
0.0347 -0.2211 0.9746
0.0327 -0.0653 0.9973
0.0301 0.0922 0.9953
0.0269 0.2473 0.9686
0.0231 0.3964 0.9178
0.0188 0.5356 0.8443
0.0143 0.6615 0.7498
0.0095 0.7710 0.6367
0.0046 0.8614 0.5078
-0.0003 0.9305 0.3663
-0.0050 0.9764 0.2158
-0.0095 0.9982 0.0598
-0.0136 0.9951 -0.0977
-0.0173 0.9674 -0.2527
-0.0204 0.9156 -0.4015
-0.0229 0.8412 -0.5403
-0.0246 0.7458 -0.6657
-0.0257 0.6320 -0.7746
-0.0259 0.5024 -0.8642
-0.0254 0.3604 -0.9325 |
Natan Eismont