A method of optimum orbital averaging was employed to study the long range accuracy potential of polar orbiting applications satellites. This approach involved the determination of the boundary conditions of one set of differential equations of motion by adjusting the initial conditions in a least square sense with the use of data generated by another set of differential equations of motion

Bonavito, N. L.

Foreman, J. C.

English

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Long Range Orbital Error'
 f Estimation For Applications Satealites
(NASA
—TM-78076)	 LONG RANGE ORBITAL ERROR
	 N78-26167
ESTIMATION FOR APPLICATIONS SAXILLITES
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Nat iona l Ae ronautics and	 ^,
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Goddard Space Flight Center	 '9zq S b'^
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LONG RANGE ORBITAL ERROR
F:tiTIMATION FOR AI'I ► I.I('A'I'IONS SA'I'D;LLI'1'ES
N. L. Bonavit<'
J. C. Foreman 
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E..	 i
January 1078
1. Measurements Evaluation Branch, Code 932. Goddard Space Flight
Center, Greenbelt, Maryland 20771
2. Systems and Applied Sciences Cor)oration, Itiverdale. Maryland 20840
GODDARD SPACE FI.1611T CENTER
	
. m	 Greenbelt. Maryland
LONG RANGE ORBITAL ERROR
ESTIMATION FOR APPLICATIONS SATELLPI'ES
N. L. ijONAVITO
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J. C. POREIVIAN
ABSTRACT
In Mis report, we estimate by means of an optimum averaging technique, the
orbital tracking errors for polar orbiting NASA applications satellites. Accu-
rate long range estimations of orbital tracking errors are important to users
in the field as an apriorl condition for orbit updating. The approach we use is
also called the method of precise conversion of elements, and utilizes results
of a two week ephemeris produced by a double precision Cowell numerical in-
tegration calculation, starting with a nominal set of initial orbital elements.
The gravitational field representation in the mmnerical integration is complete
through both order and degree twenty., Included in this model are the effects
of solar radiation pressure, lunisolar perturbations, and atmospheric drag.
Periodic values of inertial Cartesian coordinates of the satellite state are taken
from the first week of Clio eplienieris and used as observational data in an
orbit determination calculation. This data is then used to find the boundary 	
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conditions or epoeb vectors, through a least square processing, of the equations
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of motion representing analytic solutions ,f two separate differential equations
of an orbit; specifically, those of Vintl and Brouwer-Lyddane. These latter con-
tain only a gravitational field representation through the fourth and fifth zonal
harmonies respectively, and are well adapted to applications satellite orbits,
namely low eccentricity, low semi-major axis, and with little or no air drag.
Both the Vinti and Brouwer-Lyddane methods are then used separately to pre-
diet an orbit over a period of eleven to tnirtoen days b eyond opoch, and are com-
pared with the Cowell solution during this time by calculating the in-track errors,
using Cowell as a standard. The Vint! and Brouwer-Lyddane methods give from
eight to thirteen kilometers in-track error respectively for the first week of
free propagation, and approximately twenty--two to twenty-seven kilometers for
the in-track component after thirteen days.
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ORIGINAL PAGE IS
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TABLES
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CONTENTS
page
ABSTRACT......... . ......................................... 	 III
I. INTRODUCTION . ........................................ 	 1
II. STATEMENT C IF THE PROBLEM ......................... 	 4
III. RESULTS AND CONCLUSIONS ............................ 	 7
TJ.	 ACKNO\WLEDGMENTS ...................................	 12
V.	 REFERENCES .......................................... 	 13
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Table 1	 ..................................................... 	 8
Table 2
	
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LONG RANGE ORBITAL ERROR 	
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ESTIMATION FOR APPLICATIONS SATELLITES
I. INTRODUCTION
One of the more important tasks associated with the applications satellite sup-
port systems is pre-flight orbital error estimation. Accurate long term orbit
prediction is useful as an apriori condition at the field sites for updating the
orbit. Crucial to the results of such an analysis are not only the particulars
of the estimation algorithm, but also the physical model used to describe the
problem under investigation. How meaningful these results are will depend to
a great extent upon how fundamental or comprehensive the system is, as well
as the accuracy of the estimated initial conditions, A practical test is to make
a direct comparison with real data from missions of a similar nature, being
careful to consider the essential differences between the systems.
In this report, we shall employ a method of optimum orbital averaging or
precise conversion of orbital elements to study the long range accuracy poten-
tial of polar orbiting applications satellites such as Landsat and TIROS-N.
The essence of this approach involves the determination of the boundary con-
ditions of one set of differential equations of motion, in this case, those
describing the orbital motion of an artificial earth satellite, by adjusting the
initial conditions or constants of integration to a least square sense, with the
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use of 'data' generated by another set of differential equations of motion. in
dwory, so that the model might describe a reasonable approximation to tine
expected behavior of a sy stem such as TIROS-N, one needs at the very least, to
include as data sample commensurate with real cases. For example, one obser-
vational pair of radio direction cosine data at a specific tine is insufficient to
determine in orbit, since an indefinite number of ellipses corresponding to this
piece of information can pass through this point. On the other hand, one Car-
teslan coordinate and velocity set will completely determine the spaw'ecratft
trajectory. As a resuit, it is necessary to process several distinct direction
cosine pairs III 	 to fix the orbital elements of the satellite. However, the
corresponding are of observations will virtualiy never exhibit the quality and
degree of unifor mity possible by interrogating a numerical integration program.
Likewise, the kinds of uncertainties such as noise, bias, and forces of nature, 	 ,
that are Inherent in real observational data, cannot be fully and precisely repro-
ducted bn ,I 	 For the case in which one wishes to transfer orbital ele-
ments from a theory that was fitted to real data to any other, the method em-
I
ployed in this article then becomes a powerful method of calculation. Uniformly
spaced, high density data, of the type provided by complete sets of Cartesian
coordinates would offer one of the best possibilities for exchange of information
from one set of boundary conditions to those of another orbit determination
1
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ORIGIi,'AL PAGE IS
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In this paper, we fit analytic calculations to a unifoimi span of data, spec A
at hourly intervals, covering periods of one week and ton days respectively.
The analytic methods used here are those of Brouwer with modifications for
small eccentricity and inclination by Lyddane (1), and that of Vlnti (2).
The fitted analytic methods are then used to calculate the orbit for a period of
two weeks from its opoeli, and a comparison with the numerical integration
Is made by calculating the tracking error along tht direction of motion up to
thirteen days,
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II. STATEMENT OF THE PROBLEM
The Satellite Aided Search & Rescue Demonstration will be clown on a TIROS-N
spacecraft. In this demonstration, field sites with only minicomputer capabil-
ity may, under certain adverse contingencies, be required to predict orbits
for relatively long periods of time. This in turn will directly affect the
"distress beacon" position location accuracy. Ai the following calculation we
consider a nominal TIROS-N orbit. The same general conclusions however,
apply to other applications spacecraft such as NIMBUS, O) OS, Landsat, cte.
Those nominal initial conditions for the TIROS-Dl satellite are as follows:
a = semi-major ayes = 7200886. 36 meters
e = eccentricity = zero
	
i =Declination	 ORIGINAL PAUL IS98.70 degrees	 OF POOR QUALITY
M = mean anomaly = zero degrees
W = argument of perigee = zero degrees
S2 = longitude of right ascending node = zero degrees
The epoch time is taken to be January 1, 1977. at zero hours, zero minutes, and
zero seconds, and the trajectory time span covers twenty-one days. The effee-
tive cross-sectional diameter and the mass of the spacecraft is taken to be
== i	3.489 meters and 711.687 kilograms respectively. The reflectivity constant
is 1. 2, while the area for radiation pressure is 9. 560 square meters, and the
!	 solar pressure constant 4. 50 x 10 -3 (kilograms/(second) )/kilometer. In
eluded in the force model of the numerical integration method is the effect of
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atmospheric drag. The corresponding atmospheric density model Is that of
Ilarris-Prtester, 1764, with a profile range bataveon one hundred and one
thoustanai ktlameteti°s. Lunisv â car peri°arbations tire also modelled. The gravi-
j	 tational field coefficients are those of the Goddard Earth Modol-I (GEM-1), (3).
The integration phase was performed using a twelfth order Cowell-Adams system
of fixed stepsize, 24 ecconds in duration, since it was anticipated that during
the compare interval with the analytic theories, tlac integration method might
degrade appreciably as opposed to effects of the gravitational field variations.
The base inertial system employed in all calculations is that of the mean
equator and equinox of 1950.
The method of compare used is described as follows: R N , V N , and L N =
R N x VN , are the posit,an, velocity toad total orbital angular momentum
vectors of the spacecraft as functions of time determined by the numerical
Integration, RA is the corresponding satellite position vector at the same
time, and determined by the analytic theory, while the total position error of the
analytic theory at the given time is the magnitude of the vector difference,
411 = R N -
 R A . Then with respect to the numerically integrated orbit plane,
the Instantaneous components of this total position error, along the radial
direction il, perpendicular to the orbit plane c, and along the direction of
j'	 }	 the velocity vector L', are given by,
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respectively.	 I VN
The convergence criterion for the orbit differential correction least squares
solution process is as follows: For a set of to observation residuals, the
current weighted root mean square value of the observation residual column
vector at the ith iteration is EV A Y, )'
	 The corresponding predicted weight-
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ed root 1110:01 square is II A y, - A l A x,+, IV ; where AI Is the coefficient
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matrix of the equations of condition, and A z,+t is the (1 + 1st) correction
to tho state vector. From this, we have that when,
11 AY, 11
	 II A Yi - A, A z, +t II
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	 G e ,	 ORIGINAL PAGE I9.
OF POOR QUA? TZ
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where c = 10 -4 , the differential correction has converged.
For the first orbit fitting calculation, both analytic methods use only a gravita-
tional field representation derived from GEM-1, with Brouwer-Lyddane retain-
Ing those coefficients through the fifth zonal harmonic, and Vinti hrough the
fourth zonal harmonic. In the second case, both analytic methods employ the
Goddard Earth Model-5 (GPM-5), (4), representation during the correction
process, through the same zonal harmonics specified in the first calculation.
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III. RESULTS AND CONCLUSIONS
In Tables 1 and 2 we list the results of post differential correction comparlsons
of in-track calculations of twenty-four hour Intervals, for both the Brouwer-Lyddano
and Vintl analytic orbit t alculation mothods. Table 1 represents the o n-so In
which the analytic theories utilize the GEAl-1 gravitational field representation
and Table 2 gives results involving; the use of the GEM-5 field in the analytic
theories. In both tables. Lime is measured in days, and the in-track error is
measured In kilometers, From Table 1, the comparison over the period of eleven
days shows an in-track erro growth of approximately 2,3 kilometers per dad.
After one week, this orror is approximately 13 kilometers. With use of the
GEM-5 field the results are somewhat Improved (Table 2). The in-track error
g rowth for Bromver-Lyddanc is approximately 2.1 kilometers per day and for
Vinti, 1.7 kilometers per day. Alter one week, Brouwer-Lyddanc shows all 11
kilometer error, and Vinti, 5.5 kilometers. After thirteen days, the in-track
errors are approximately 27 kilometers and 21.5 kilometers for Brouwer-
Lyddane and Vinti respectively. Both analytic methods appear to give consistent
results.
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Time
(Days)
Brouwer-Lyddane
In-Track Error (kilometers)
Vlnti In-Track
Error (kilometers)
1 2.8806 2.8083
2 3.7461 3.4563
3 5.4258 5.2077
+1 7.5062 7.7914
5 0.7210 0.0300
6 11.3436 11.0218
7 13.4276 12.8735
8 16.4265 16.4257
0 10.6-,)7 20.0096
10 22.3330 22.3113
11 24.7967 24.0057
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Table 1
Estimated In Track Error For TIROS-N Using
The OEM-1 field
DRICITNP,L YK ^]
pF Pooh	 eI.N I
Time
(Days)
Brouwer-Lyddane
In-Track Error (kilometers)
Vinti In-Track
Error (kilometers)
1 1.3907 .0096
2 2.5823 1.8987
3 3.5839 2.7664
4 4.5635 3.2121
5 6.3790 4.9396
6 8.7123 6.7720
7 10.9978 8.5331
8 12.7566 10.6874
9 14.9692
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10 18.1158 14.0599
11 21.4810 16.6674
12 24.3299 19.2313
13 26.9314 21.5138
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Table 2
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	 Estimated In Track Error For TIROS-N
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Using The GEM-5 Field
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In addition, since few nonconservative forces are expected to greatly perturb the
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"i	 orbit, an in-track error growth of approximately 1.7 to 2.3 kilometers per day
'	 for an orbit of the TIROS-N class, would appear to be a reasonable a priori esti-
mate. rurthernaore, since this grmvth appears almost linear over eleven to
thirteen days, one might expect a rapid convergence during subsequent orbit
Improvement calculations.
As a final consideration in orbit orror estimation, let us consider two important
error sources. If we compare the in-track behavior of a TIROS-N type satellite,
with no nonconservative perturbations present, using a Brouwer orbit calculation
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with and without the second-order short period terms in the semi-major axis,
we will get an in-track error growth of approximately 320 meters per day. On
the other hand, if we assume an uncertainty of one pHrt in 10 6 for N, the product
of the planetary mass and the gravitational constant, we obtain a value for this
error growth of slightly more than 300 meters per day. Lyddane and Cohen (5),
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j	 have shown that failure to Initially adjust the semi major axis in the second order
short period terms may immediately show as a serious secular like in-track
error manifest through the mean anomaly for which 320 meters per clay would
represent a lower bound. This, together with a built in 'boundary uncertainty'
for the parameter u, might seen to account foe the secular in-track error growth
ORIGNT AL PAGE IS
OF POOR QIIALYry
IV. ACKNOWLEDGMENTS
The authors are grateful to Messrs. IL A. Gordon and G. D. Mistretta of
the Systems Development and Analysis Branch, Goddard Space Plight Center,
for several valuable discussions which contributed to this report.
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V. REFERENCES
1. Lyddane, R. Ii. , 'Small Eccentricities or Inclinations In the Brouwer
Theory of Artificial Satellite Motion,' The Astronomical Journal,
Vol. 68, June 1963. pp. 555-558.
2. Vinti, J. P. , 'Inclusion of the Third Zonal Harmonic in tin Accurate
Reference Orbit of all 	 Satellite.' Journal of Research of
The National Bureau of Standards, Vol. 70B No. 1, January—March
1966.
3. Lerch, Francis J. , et al. , 'Gravitational Field Models for tine
p•
Earth (GENT 1 and 2),' NASA/GSFC X-553-72-146, May 1972. 	 !;
4. Lerch, Francis, J. , ct al. , 'Goddard Earth Models (GENT 5 and 6).
q
NASA/GSFC X-921-74-145. December 1974.
5. Lyddane, R. H., and Cohen, C. J., 'Numerical Comparison between Brouwer's 	 4
Theory and Solution by Cowell's Method for the Orbit of an Artificial Satel- i
hte,' The Astronomical Journal, Vol. 67, No. 5, April 1962, pp. 176-177.
c
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Long range orbital error estimation for applications satellites

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