Approximate method for determining duration of coverage of sunlit earth by anchored IMP optical aspect syste

Chromey, F. R.

English

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AIMP OPTICAL ASPECT COVERAGE - 
THREE DIMENSIONAL APPROXIMATION 
by 
F. R. Chromey 
June, 1966 
Goddard Space Flight Center 
Greenbelt, Maryland 
. -  
AIMP OPTICAL ASPECT COVERAGE - 
THREE DIMENSIONAL APPROXIMATION 
F. R. Chromey 
ABSTRACT 
This study derives an approximate method for determining the 
duration of coverage of the sunlit earth by the AIMP optical aspect 
system. The derivation is designed so that coverage data may be ob- 
tained through minor modifications of an existing computer program 
(QUIMP). Results a r e  applicable to any spin-stabilized vehicle with a 
similar optical aspect system. Plots of the aspect system coverage 
for  several AIMP launches are  included. 
iii 
.. 
CONTENTS 
Page 
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1 
METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1 
Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1 
Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5 
Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  9 
RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  13 
CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  13 
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  13 
. 
V 
AIMP OPTICAL ASPECT COVERAGE - 
THREE DIMENSIONAL APPROXIMATION 
I 
INTRODUCTION 
The Anchored IMP optical aspect system contains three telescopes whose 
optical axes are aligned at 45O, 90°, and 135' to the vehicle spin axis. Determi- 
nation of vehicle orientation involves a consideration of the times during which 
the system can see the sunlit earth through any of these telescopes. 
Reference 1 attempts to find the length of the period immediately after in- 
jection during which data may be obtained from the optical aspect system. It 
does this by calculating the maximum and minimum angles (measured with 
respect to the spin axis and known hereafter as %AX and aMIN ) of telescope 
alignment between which a portion of the sunlit earth will be viewed in the course 
of an instantaneous rotation of the vehicle about its spin axis. The two dimensional 
method used in reference 1, however, has been found to compute values of aMm 
and %IN which yield coverages of significantly shorter duration than would be 
realized in actual fact. The object of this analysis is to provide a more rigorous 
approximation of these angles. The results, of course, may be applied to any 
spin-s tabilized vehicle. 
Since an exact determination of the AIMP telescope viewing times is being 
developed, this analysis limits itself to providing optical aspect coverage data 
through a minimum of computer programming effort. In particular, it is intended 
that all data be obtained through minor modifications of the existing QUIMP 
computer program (QUIMP is itself a version of the Quick-Look Mission Analysis 
Program modified for the AIMP launch philosophy). 
METHOD 
Assumptions : 
The following quantities are  calculated by QUIMP and are considered as 
known at any specified time in the AIMP trajectory: 
SEVA - the sun-earth-vehicle angle, 
-.) 
R - the position vector from the center of the earth to the vehicle, 
1 
s'" - a unit vector along the vehicle spin axis (input o r  computed parallel 
to the velocity vector at  injection), 
RVE - the angle between go and -c. (Spin axis-earth angle). 
z, - the position vector from the center of the sun to the vehicle, 
SDA - the earth semi-disk angle as'seen from the vehicle. These quantities 
are illustrated in Figure 1. 
Figure 1 
Now consider Figure 2, where the plane of the paper is such as to contain 
both s'' and a. The dotted line is the tangent plane and the dashedellipseis the 
terminator. The curve FGF' represents the intersection of a right circular cone 
of half-angle u and axis s'" with the earth. Let us take another view of this 
situation, this time looking antiparallel to if, projecting all lines into the tangent 
2 
t 
Figure 2 
plane. Such a view is presented in Figure 3. In addition to curve G'FG, we have 
drawn in the intersections of the cones of axis go which contain the pointsA , B 
and C. Call the half-angles of these cones a*, ag, and aC, respectively, measured 
from so. Similarly, label the half-angles of the cones passing through points 
D and E a~ aD and a E .  
We now make the following assumption: 
In all cases, both aHAX and uHIN are equal to angles in the following list of 
seven; uA, a , a =, a D ,  a=, Oo, and 180'. B 
Note that there a re  cases where this assumption might be justified only as an 
approximation, as in Figure 4 (i.e. where it is possible to draw a cone-earth 
intersection tangent to the terminator on the dark side at a point other than A , 
B ,  o r C ) .  
" 
Note also that the assumptions made in reference 1 (two-dimensional geometry) 
a re  equivalent to restricting the terminator plane perpendicular to the plane of 
Figure 2, and to eliminating aB and aC as possible values for aMAX and aMIN. 
3 
D 
L 
Figure 3 
Figure 4 
We must now develop both a method for calculating a A ,  ag , aC, aD , and 
a E ,  and a method for  determining which of the seven angles in our list to select 
as a M A X  and u ~ ~ ~ :  
4 
, 
.. 
Mathematics 
Consider Figure 5. We define a coordinate system centered at the vehicle 
as follows: 
-i 
We write S , the representation of io in this new system, as 
where 
- - t . - i  -+ 
We now draw vectors D,, D, and Dc from the vehicle to points A ,  B , and C 
respectively, and define 
VET = 90' - SEVA 
- + 
Now, to find D, and D,, we consider spherical triangle NMB and write, from 
the law of sines: 
sin (180 - p) - sin (VET) 
sin (90 - SDA) 
- 
sin (90) 
so that 
5 
6 
.. 
. If we consider right spherical triangle MBT, we may wri te  
s i n  p = t a n  (VET) ctn p 
p = Sin- t an(VET) ctn ,8] 
Now, we let 
in our coordinate system and write, from Figure 5, 
DBl = [Re cos (p) C O S  (VET)] - 
where R e  is the radius of the earth, 
and 
DB, = R e  c o s  (p) s i n  (VET), 
DB, = IRe sin (p) 1 
-+ 
We find D, through similar procedures: We note 
and, considering right, spherical triangle MTC, write 
sin VET = t a n  E cot  y 
7 
so that 
1 [ S, z; (VET) E = Tan-l 
thus, if  
in our coordinate system, we may write from Figure 5 
D, = [Re c o s  (E) cos (VET)] - 121 
D, = Re cos ( E )  sin (VET) 
D, = R e  sin (E) 
We are  now in a position to compute 
Note also that aD or  aE may be equal to either 
RVE t SDA, 
RVE - SDA, 
8 
or 
Logic 
SDA - RVE, 
360" - RVE - SDA 
Thus, a M I N  and amX may each be calculated in nine different ways (aA, ag, 
aC, O o ,  180°, and the four values of aD and a E ) .  
Of the 81 possible combinations, 51 may be eliminated as having no physical 
significance. The remaining thirty combinations may be conveniently distributed 
in nine cases arising from distinct geometric situations. Once the case is as- 
certained, ayIN and aMAX may be chosen through simple magnitude tests. The 
nine cases follow: 
Case I: 
Case II: 
Case IIb: 
Case IIc: 
Case IIa: 
The sunlit portion of the earth is not visible 
from the vehicle: Set ayIN = aMAX = Oo 
The disk either is completely illuminated 
or  the situation is as shown at the right. 
In either event we may have either 
where -s extended will intersect the disk. 
Set 
a M I N  = RVE - SDA 
aMAx = 180" 
where s will intersect the disk if  extended. 
Set 
aMIN = 0" 
- RVE + SDA a~~~ - 
All other case I1 orientation of s'. Set 
a = R V E  - S D A  MIN 
aMAX = RVE + SDA 
0 
CASE It 
9 
Case IIIa: Situation as shown at right where s will 
not intersect the disk if extended and - g 
will intersect neither the sunlit portion 
nor the portion between V and A if ex- 
tended. Set 
a MIN =RVE-SDA 
aMAX = the l a r g e s t  of (aA aB and a c )  
-i 
Case IIlb: Situation as shown at right where -S ex- 
tended must intersect the dark portion of 
the disk between V and A .  Set 
= the smallest of (aB, aC, and 360 - a MIN 
RVE - SDA) 
am = the largest of (aA, aB,.anda, ) CASE I l l  b 
Case IVa: Situation illustrated at right. -2  extended 
will not intersect the disk. 
2 extend will intersect neither the sunlit 
portion of the disk nor the darkend portion 
betweenV and A. Set 
a MIN = the smallest of (aA, aB, and iC) 
a = R V E + S D A  MAX 
CASE IV a 
Case IVb: Situation illustrated at right. s’ extended 
must intersect the dark portion of the desk 
between V and A .  Set 
aMIN = the smallest of (aAy aB andac ) 
aHAX = the largest of (aB, aC and SDA- RVE) 
10 
CASE IV b 
. 
CASE V: 
Case VI: 
Case VII: 
-I 
Situation illustrated at right. - S extended 
must intersect the sunlit portion of the disk 
between V and E and not between V and A. 
Set 
a M I N  = the smallest of (aA, aB,aC , 360- 
RVE-SDA) 
a =180° M A X  
+ 
Situation illustrated at right, -S must 
intersect the sunlit portion of the disk 
between V and A.  Set 
a = the smallest of @*,a,  ) a C ,  and MIN 
RVE-SDA) 
aMAX = 180° 
Situation illustrated at right. 2 must 
intersect the sunlit portion of the disk 
between V and E.  Set 
a = o o  MIN 
aMAX = The largest of (aA,aB,aC, SDA - 
RVE) 
Case VIII: Situation illustrated at right. 2 must 
intersect the sunlit portion of the disk 
between V and A. Set 
a = o  MIN 
a = the largest of (aA, aB )aC , SDA - MAX 
RVE) 
Case E: Situation illustrated at right. Either 2 
and -5 may o r  may not intersect the 
disk. Set 
a = the smaller of (aA and a,) MIN 
aMAX = the larger of (aA anda,). 
Q 
CASE V 
CASE V I  
Q 
CASE VI1 
Q 
CASE Vlll 
CASE IX 
11 
EXIT FR.  SUBR. O U T  
I 
Y 
COMPUTES r-l aA,  06 ,  ac, e, RVE, % 
Y 
I I 
Figure 6 
12 
The logic used in determining the case of a particular situation in the modi- 
fied version of QUIMP is flowcharted in Figure 6. 
RESULTS 
Plots of aMIN and aMAX versus time for a few sample AIMP launches, simi- 
* lar to those listed in Reference 3,  a r e  included in the appendix. In these cases 
S was taken parallel to the velocity vector at injection. 
CONCLUSION 
Preliminary checks with more exact determination of optical aspect coverage 
indicate the assumptions made in this study to be excellent for the cases considered. 
REFERENCES 
1. Groves, R. T., "AIMP Optical Aspect System Coverage for Early Post-Injection 
Attitude Determination," Systems Analysis Office Technical Brief, June 3,  1965. 
2. "Programmer's Manual for Quick-Look Mission Analysis Program, 'I prepared 
for Goddard Space Flight Center by Philco Corporation, WDL Division. 
3. Groves, F'uchs, and Sandifer, "AIMP Mission Profile and Real Time Computing 
System," GSFC X-500-65-255, June 22, 1965. 
13 
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AIMP optical aspect coverage - Three dimensional approximation

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