An empirical orbit determination method is shown to yield highly accurate navigation results when applied to lunar orbit tracking data. Regressions and predictions of free flight Apollo 14 tracking data exhibit minimal residual growth, and the solution orbital elements behave in a very consistent manner. Solutions from data acquired during propulsive maneuvers result in degraded predictions. The residual patterns from free flight processing are shown to be consistent from pass to pass and are correlated with lunar topographic features

Bullock, M. V.

Ferrari, A. J.

English

NASA Technical Reports Server

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date October 1, 1971 955L'EnfantPl_a NoAh,S.W.
Washington,D.C.20024
to Distribution B71 i0001
from A. J. Ferrari, M. V. Bullock
subjectEmpirical Orbit Determination
Using Apollo 14 Data -- Case 310
ABSTRACT
$
>
An empirical orbit determination meLhod has been shown
: to yield highly accurate navigation results when applied to lunar
orbit tracking data. Regressions and F:edictions of free flight
4
[ Apollo 14 tracking data exhibit minimal residual growth, and the
i solution orbital elements behave in a very consistent manner•¢
Solutions from data acquired during propulsive maneuvers result
in degraded predictions. The residual patterns from free flight
• processing are shown to be ccnsistent from pass to pass and are
!__ correlated with lunar topographic features.
¢:
N72- 1483 8 (NASA-CR- 124755) EMPIEICAL ORBIT
DETERMINATION USING APOLLO 14 DATA A.J.
Yerrari, etp al (Bellcomm, Inc.) I Oct. }_0_.,,Uuclas 1971 16 CSCL 22C _. - r)_
:: 11679 G3/30
'_, (CATEGORY) -- _ /_
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1972007188
https://ntrs.nasa.gov/search.jsp?R=19720007188 2020-03-11T21:45:42+00:00Z
JI\
B komm
date October i, 1971 955L'EnfantPlazaNorth,S.W.
Washington,D.C.20024
to Distribution B71 I0001
from A. Jo Ferrari, M. V. Bullock
subjectEmpirical Orbit Determination
Using Apollo 14 Data -- Case 310
MEMORANDUM FOR FILE
INTRODUCTION
Accurate orbit determination and prediction become
especially difficult for the case of a spacecraft in orbit about
a body whose gravitational potential field is not well known.
Most widely used processing methods employ a model of the gravity
• field, and hence the results are constrained by the quality of
the assumed model. An alternative method is an empirical scheme
which models the effects of the gravity field rather than the
causes. The Osculating Lunar Elements Program (OLEP) uses such
an approach for orbit determination in lunar orbit, representing
: the state of the satellite as time-varying orbital elements.
Estimates obtained for the constant and time-dependent parts of
each element are a result of the perturbing effects of the actual
gravity field; no model of the field is assumed.
In this study the orbit determination and prediction
capabilities of OLEP are investigated using Command Service
Module (CSM) Doppler tracking data from the lunar parking orbit
. of Apollo 14. Long and short arc solutions are presented, and
the behavior of the estimated osculating orbital elements is
studied. Correspondence between residuals and topographic fea-
tures is shown.
MATHEMATICAL DESCRIPTION
The OLEP*approach uses time-varying functions for the
low-eccentricity orbital elements, a, e s = e sin _, e = e cos _,; c
I, _, and m = _ + M, to model the motion of a spacecraft in an
*Bullock, M. V. and Ferrari, A. J., "Orbit Determination For
Lunar Parking Orbits Using Time-Varying Orbital Elments,"
i NASA Contractor Report 110008, May 1970.
t
I
4
1972007188-002
',
/
9
_;J _ 2 m
Apollo-type lunar orbit. A typical element is represented as
= + _i t_ _ (t) n°
_ ec, and m in the casewith quadratic terms included for e s,
of regressions of more than two passes of data. The numerical
I singularities associated with nearly equatorial orbits areavoided by defining these low-eccentricity elements in a special
_ selenocentric frame which represents any orbit as a polar orbit.
• _ This transformation is accomplished by rotating the initial
estimate of the selenocentric state at epoch through two of its
i_ associated Euler angles (_,I).
,, The semi-major axis does not appear as an explicit
{ solution parameter. The constant and time-dependent portions of
t the remaining five elements constitute the parameter set for
:_'! which estimates" are obtained during the orbit determination pro-
i cans. The estimate for m I, the linear portion of the modified
:- anomaly, is used to imply a corresponding semi-major axis by
iii_ using the classical Kepler relationship
a = _ ,
m1
where _ is the Newtonian constant times the lunar mass.
,,, DATA ANALYSIS
' Orbit determinations are performed using free flight
[ and minimally corrupted data from Apollo 14. Since propulsive
maneuvers are not modeled in OLEP, the best orbit determination
results are to be expected when free flight data are processed.
Data acquired during coupled attitude maneuvers, such as land-
mark tracking, should be of free flight quality, but in practice
the jets are never perfectly balanced so a slight translation is
imparted to the spacecraft. At least two passes of data are
: processed in each case, since one pass does not contain enough
_: orbital period and time-varying information to enable OLEP to
: predict accurately. The effects of the CSM separation burn
[_; are also shown.
q.
i i ii i
1972007188-003
J4,'
-- 3 --
a. Processing of Free Flight Data
Pesiduals resulting from two-pass regressions of pre-
DOI tracking data are shown in Figures 1 and 2. "Jhis orbit is
characterized by a perilune of 8n.mi. and an apoldne of 60n.mi.
above the lunar surface. The two-pass fit, two-pass predict
residuals for the four sets of data are virtually identical,
with peak-to-peak values of ±.6 feet/second. The excellent
quality of the solutions is apparent in the fact that the pre-
{ diction residuals exhibit almost no secular growth and that they
maintain the characteristic short-periodic shape of the fit
_: residuals. As expected, some growth occurs when a two-pass
solution is predicted for five passes (Figure 2), but the solu-
_ tion still describes the orbit very well for the entire time
span. Peak-to-peak growth in the prediction region is less
than double that of the fit region, and the residuals still
display some short-periodic shape.
High quality results are also obtained when the OLEP
process is extended to multi-pass regressions. Two solutions
were obtained, one from the seven pre-DOI passes discussed above
_ and one from six post-DOI passes; the residuals from both are
shown in Figure 3. The post-DOI orbit was near-circular with
both perilune and apolune about 60n.mi. above the lunar surface.
In peak-to-peak value and residual shape these results are ex-
tremely similar to the two-pass fits.
b. Processing of Corrupted Data
I Figure 4 shows several examples of two-pass processing
of pre-DOI data acquired during various indicated maneuvers. The
solution from passes 3 and 4 was somewhat corrupted by landmark
_ tracking between passes 3 and 4 even though the maneuver was
•_ _ coupled, and the prediction residuals are markedly different from
those which resulted from free flight data processing. The peak-
_ to-peak value is more than double that in the fit region, and
the characteristic shape is lost. Similar results occur when a
solution from free flight data is predicted through data containing
maneuvers. LM RCS jet hot firing tests were performed during pass
ii, and the undocking and separation maneuvers occured in pass
12. These events changed the orbit so that the solutions from
the previous passes no longer describe it adequately.
OSCULATING ELEMENT COMPARISONS
A further indication of the quality of the solutions
obtained by the OLEP method is consistency in the behavior of
the estimated parameters. Local estimates of the orbital elements
i i| i
1972007188-004
fare obtained from two-pass fits using passes 4-10, and a single
fit over the entire data span gives a set of long-term estimates.
The long-arc fit should provide more accurate estimates of the
elements than the short-arc fits because the longer interval
gives the process more information about the time behavior of
the elements.
Figures 5 and 6 show comparisons of eccentricity, incli-
: nation, longitude of the ascending node, and argument of perilune
from short and long-arc fits. The perturbing effects of the earth
and the sun on the elements were analytically removed from the
' long-arc solution. The resulting variations in the elements are
i presented to non-central gravitational fea-
show the effects of
tures on the spacecraft. Extremely accurate estimates of e, _,
: and _ were obtained from the short fits. The local estimates
; follow the slopes of the long-arc estimates very closely. The
inclination estimates are not as consistent, but this is not an
unexpected result since the inclination is the most difficult
parameter to determine.
RESIDUAL ANALYSIS
In Figure 7 the residuals from passes 4-10 are given
as a function of the longitude of the sub-vehicle point, with
"_ the consecutive passes being overlaid. The OLEP process esti-
{ mates only the secular and long-period perturbing effects of
the gravity field, hence the residuals are short-periodic line-
of-sight velocity errors. The consistency of the residual
_ pattern from pass to pass is especially striking. Prominent
• topographic features over which the spacecraft passed are noted
i_ on the figure. Their effects on the orbit can be seen in thecorrespondence between these features and the occurrence of the
!I largest residuals.
I SUMMARY AND CONCLUSIONS
It has been demonstrated that highly accurate orbit
determination and prediction can be performed with no assump-
tions about the lunar gravity field using the OLEP process. The
residuals from the processing of free flight Apollo 14 data
experience minimal secular growth and exhibit a characteristic
short-periodic shape. The orbital elements resulting from two
pass solutions behave in a consistent manner when compared with
elements from a long-arc solution. The quality of the results
is reduced using data acquired during maneuvers; even coupled
maneuvers impart some change to the orbit since the jets are
'j ,
;4
1972007188-005
fi - 5 -
not perfectly balanced during these firings. Consistency in
the residual pattern from consecutive passes of free flight
data is shown, and the residuals are correlated with sub-
vehicle topographic features.
A. J. Ferrari
AJP
2014-MvB-hat M.V. Bullock
i i ii i i i i
1972007188-006
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FIGURE 1 - TWO-PAre FITS OF FREE FLIGHT DATA
i
1972007188-007
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FIGURE 2 - PROCESSING OF PASSES 4 AND §
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1972007188-008

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FIGURE 4 - PflOCIIIING OF COllOID DATA
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1972007188-010
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0 . 10,000 20,000 30,000 40,000
• TIME (SECONDS)
ECCENTRICITY AS A)
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•_ INCLINATION All A
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i_ FIOURE§ - APOLLO14 ORBITAL ELEMENTS,PAIISE84 - 10 ,_
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1972007188-011
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TIME (SECONDS)
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A_ A FUNCTION OF TIME
FIGURE8 - APOLLO14ORBITALELEMENTS,PABSE84 - 10 _i
i i i i i HI I I I r I I nn n r n I I I HI I ] I I H H I I I I I I mHIHr I I
1972007188-012
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PROCELLARUM NECTARIS FECUNDITATIS
LONGITUDE (DEGREES)
FIGURE 7- APOLLO14 RESIDUALSAS A FUNCTIONOF LONGITUDE, PASSES4 - 10
'.
I
__ -_
1972007188-013


Empirical orbit determination using Apollo 14 data

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