RELATIVE ORBIT DETERMINATION OF GEOSYNCHRONOUS SATELLITES USING THE COWPOKE EQUATIONSChris Sabol^{1}, Keric Hill^{2}, Craig McLaughlin^{3}, K. Kim Luu^{2}, Michael Murai^{4}^{1} Air Force Maui Optical and Supercomputing Site (AMOS), Air Force Research Laboratory, 535 Lipoa Pkwy., Ste. 200, Kihei, HI 96753 ^{2} University of Colorado, Boulder, CO 90309 ^{3} University of North Dakota, Grand Forks, ND 58202 ^{4} Air Force Maui Optical and Supercomputing Site (AMOS), Oceanit Laboratories, Inc., 590 Lipoa Pkwy., Kihei, HI 96753
INTRODUCTIONClusters of spacecraft in geosynchronous orbit (GEO) are becoming more common. Orbital
slot allocations in GEO are rapidly being filled, and it is increasingly difficult to acquire slots
for new satellites. Consequently, many organizations opt to collocate their spacecraft in the
same slot. Eutelsat had as many as five satellites collocated in a formation at 13° E by
2001 [1]. In addition, unintentional close approaches have occurred and could become more
frequent as the GEO belt becomes more crowded. In 1997, Telstar 401, a satellite in GEO,
experienced a major failure, and control of the spacecraft was lost [2]. It is drifting in GEO
and has had encounters with other satellites as close as 4 km [3]. Since that time, other GEO
spacecraft have also begun drifting uncontrollably. Close approaches and satellite formations
create a challenge for space surveillance. Identification of the satellites in these clusters can be
difficult, and crosstagging (misidentification) occurs [4]. Resolvable imaging can be used to
identify spacecraft in low Earth orbit but is not currently possible for GEO altitudes. Non
imaging approaches include comparing the brightness and characterizing the color of light
reflected from the spacecraft for identification purposes [5]. However, identifying spacecraft
solely from their dynamics would eliminate the need for special filters or other sensors. A
better understanding of the relative motion of the spacecraft could reduce crosstagging and
improve close approach predictions. Improved determination of minimum approach distances
could eliminate unnecessary collision avoidance maneuvers and minimize propellant usage.
Cluster orbits can be modeled using the Cluster Orbits With Perturbations Of Keplerian
Elements (COWPOKE) equations of motion [6]. The COWPOKE equations predict spacecraft separations in the spherical radial, alongtrack, and crosstrack coordinate frame
based on the Keplerian elements of the reference satellite, the element differences for the
second satellite, and elapsed time. These inputs for COWPOKE can be obtained with relative
metric data from optical sensors or space surveillance products. Chargecoupled device
(CCD) images of clusters of spacecraft should provide very accurate measurements of
spacecraft separation since inframe error sources theoretically cancel out. The United States
Air Force currently operates several optical systems capable of imaging clusters at the Maui
Space Surveillance Complex, such as Raven, Phoenix, and the GroundBased ElectroOptical
Deep Space Surveillance System (GEODSS). Raven uses small, commercially available
telescopes to acquire CCD images of space objects for tracking [7]. Astrometry is used to
match stars in the CCD image to the star catalog. Thus, the pointing accuracy of the images
can approach the accuracy of the star catalog used. Phoenix is a BakerNunn telescope
refurbished to take wideangle CCD images [8]. As many as 21 satellites have been detected
in one Phoenix image. After the Deep Stare upgrade, GEODSS could be used to acquire
relative metrics as well.
This paper is an extension of previous work exploring the application of COWPOKE
towards geosynchronous cluster orbit prediction [9]. First, a perturbation study is conducted
to determine the force modeling required at GEO. Improvements are made to the COWPOKE
equations that allow for better representation of GEO motion. A method is construed to
estimate the Keplerian element differences using optical measurements of relative right
ascension and declination. Finally, results are discussed for which COWPOKE was used to
predict the relative motion of a cluster of satellites in GEO.
USING COWPOKE AT GEOA study was conducted to identify perturbing forces that significantly affect the relative
motion of a cluster of geosynchronous satellites. The effects of central body gravity, third
body gravity, and solar radiation pressure were investigated using the Draper Semianalytic
Satellite Theory (DSST). The Draper research and development version of the Goddard
Trajectory Determination System (DGTDS) was used to propagate the orbits with DSST [10].
The motion of two spacecraft was propagated to generate a “truth” relative motion using an
8x8 JGM2 geopotential field, lunisolar thirdbody pointmass using JPL ephemerides, and
solar radiation pressure (SRP) based on a spherical satellite and cylindrical Earthshadow
model. Next, the relative motion of the two spacecraft was generated while neglecting one of
the sources of perturbations, such as thirdbody gravity. The radial, alongtrack, and cross
track separations obtained from the “truth” relative motion were compared to those generated
with the incomplete force model to find the approximate error that would result from
neglecting that perturbation source in the COWPOKE formulation. These test runs were
conducted two times; the first set of runs included long period effects propagated over 30
days, and the second set investigated short periodic effects propagated over 5 days. Table 1
shows the reference elements and the differential elements used in the propagation.
Table 1. Initial conditions for DSST runs referenced to the mean equator
and equinox of the B1950.0 coordinate system.
Cases that neglected all nonspherical gravity forces were conducted at differing initial
mean anomalies. Those omitting higher order geopotential terms, but including J_{2}, were
conducted with varying longitudes of the node to survey the longitudinal dependencies of the
tesseral harmonics. Tests examining lunisolar effects ran at varying days of the year while
those for SRP effects used different days of the year as well as at various differential areato
mass ratios for the spacecraft. Errors were calculated by dividing the difference in radial,
alongtrack, or crosstrack separation by the maximum separations. Fig. 1 shows the worst
case errors produced by neglecting each of the perturbing forces after 30 days.
The SRP results in Fig. 1 arise from a case in which the areatomass ratio of one
satellite is ten times that of the other. As the separation between the spacecraft decreases to
zero, the effect of neglecting gravitational effects also decreases to zero; however, SRP effects
do not decrease to zero if the areatomass ratios differ. Because of this, SRP effects should be
taken into account if high accuracy is warranted over long periods or if the satellites are
within several kilometers.
The effects of SRP on an orbit were formulated using Keplerian elements and the
element differences, assuming that the latter quantities are small. However, because Keplerian
elements are singular for i = 0 and e = 0, orbits in GEO can have large differences in Ω, ω, and M_{0} and still remain within a few kilometers of each other. Simulations found that this
formulation of SRP did not prove to be useful for GEO and therefore was not used in later
tests. However, for most cases, twobody motion results in acceptably low error.
Fig. 1. Maximum error encountered in the separation of two satellites at GEO due to the neglect
of a perturbing force after a 30day orbit propagation.
Another source of orbit perturbation is stationkeeping and momentum control
maneuvers. Since maneuvers were not modeled in COWPOKE, they present an additional
source of error if occurring during the observation period.
IMPROVED COWPOKE EQUATIONSAlthough COWPOKE has been shown to be an effective predictor of the relative motion of a cluster of satellites, there are some sources of error. For GEO, the right ascension of the
ascending node, argument of perigee, and mean anomaly element differences may not be
small which violates the assumptions of the original COWPOKE derivation. In particular, the approximation of the true anomaly difference was linear in terms of the mean anomaly
difference; this causes error, particularly in the alongtrack direction, when the mean anomaly
difference is significant. The crosstrack term component of COWPOKE also showed error
with large right ascension differences.
Replacing the linear approximation of the true anomaly difference with an exact
difference of the two true anomaly terms substantially reduced the alongtrack error. For the
nearcircular GEO case, the true anomaly of each satellite as a function of time is
approximated by a first order expansion in terms of eccentricity and mean anomaly which was
shown to be effective in [6]. The crosstrack component was improved by an investigation
into the spherical geometry involved. Corrections were made to both amplitude and phasing
of the crosstrack component which are accurate for large values of right ascension of the
ascending node, argument of perigee, and mean anomaly differences. The improved
COWPOKE equations are
(1)
where
(2)
δr is the separation in the radial direction, δtx is the crosstrack separation, and δat is the
alongtrack separation. δν is the difference in true anomaly. a, e, i, Ω, ω, and M are the orbital
elements of the reference satellite, and δa, δe, δi, δΩ, δω, and δM are the differences in the
elements of the two satellites. μ is the gravitational parameter, and t is the time elapsed
since the epoch of M.
The alongtrack term still exhibits error when the δΩ and δi terms are large. To avoid
this as an error source in the relative orbit determination experiments, it was decided to use
the alongtrack component from Vadali’s unit sphere model for relative motion [11]. The
Vadali unit sphere model is very similar in philosophy to COWPOKE; relative motion is
modeled through Keplerian elements and element differences. Vadali’s geometric model,
however, is more rigorous than the simple COWPOKE approach. Here is the alongtrack
component of Vadali’s unit sphere model which was incorporated into COWPOKE for the
remainder of this analysis:
(3)
DETERMINATION OF ELEMENT DIFFERENCESPredicting the relative motion of a cluster of satellites with COWPOKE requires a set of
orbital elements for the reference satellite and the relative elements of the second satellite. A
leastsquares orbit determination method was used to find these element differences [12].
COWPOKE expresses spacecraft separations in the spherical radial, alongtrack and
crosstrack reference frame, but the optical observations used in this effort are in the
topocentric right ascension and declination frame. Therefore, the topocentric observations had
to be converted to geocentric observations, which requires satellite range knowledge as well
as the local sidereal time [13]. For this analysis, a constant range value was used and values of
UT1UTC, precession and nutation angles, and lunar terms were ignored; it is believed that
these approximations do not have significant impact once the observations are differenced.
The COWPOKE crosstrack and alongtrack separations could then be equated with the
geocentric right ascension and declination frame as follows:
(4)
where
(5)
Let Y be the relative observation vector, and X the state vector containing the orbital element differences. Equations (4) and (5) represent a nonlinear mapping between the state
vector and the observations. In order to invert the problem, we must linearize the equations
about a reference trajectory, X*.
(6)
(7)
The δα terms are the observed differences in right ascension, and the δd are the observed differences in declination. Equation (6) can be rearranged and terms can be
redefined as follows:
(8)
where y is the difference between the observed and calculated relative observations, H is the linearized observationstate relationship, and x is the estimated correction to the state matrix. If at least 3 relative observation pairs are included in the y vector, one can estimate the state
deviation as shown below:
(9)
The estimate of the state can be updated in an iterative fashion, as shown below, until
the solution converges.
(10)
Using this method, an estimate of the Keplerian element differences was obtained. This
method requires that the Keplerian elements of the reference satellite are known. The
NORAD twoline element set (TLE) of the reference satellite was used for the reference orbit
in this study. With the element differences, the relative right ascension and declination of the
two satellites can be predicted using COWPOKE.
SIMULATION STUDYIn order to test the feasibility of using COWPOKE to better predict relative motion, a
simulation was performed using 2 collocated geosynchronous satellites. Truth orbits were
propagated using the Cowell Special Perturbations (SP) propagator internal to DGTDS. The
truth orbits spanned from Jan 1 – Feb 5, 2003. Table 2 contains the osculating orbital
elements for the satellites referenced to the mean equator and mean equinox of the B1950.0
coordinate system.
Table 2. The initial orbital elements of the two satellites used in the identification study
with epoch January 1, 2003, 0 hr.
The next step in the process was to develop TLE representations of the truth orbits. This
step was necessary since TLEs are used in the observation correlation process. To do this, an
orbit using the GP4/DP4 propagator internal to DGTDS was fit to the position and velocity
vectors produced by the Cowell truth trajectory. The position and velocity data spanned 1 Jan
to 1 Feb and were spaced at every hour. The resulting fits were accurate to around the 2 km
level RMS. Fig. 2 plots the DP4 trajectories relative to the truth orbit of Sat 1 over February
15, a 5day prediction interval.
One can clearly see in Fig. 2 that the TLE trajectory for Sat 2 comes closer to the Truth
location for Sat 1 than the Truth location for Sat 2 during a significant portion of the orbit.
Similarly, the TLE trajectory for Sat 1 comes close to the Truth location for Sat 2 during a
portion of the orbit. Both of these situations could lead to a crosstag in the observation
correlation process (i.e., observations of Sat 2 could be incorrectly attributed to Sat 1 and vice
versa).
Fig. 2. TLE orbit predictions of the motion of Satellite 2 relative to Satellite 1 truth orbit
compared to the truth relative motion.
The mean orbital elements from the TLE fits (epoch 1 Feb) were differenced and used
to initialize the COWPOKE equations along with the TLE elements for Sat 1. One can see in
Fig. 3 that even using the flawed initial conditions produced by the TLE fits, the relative
motion is very representative of the Truth relative motion. This signifies that the relative
position is a powerful piece of information that can be used to help correlate optical
observations at the sensor.
Fig. 3. COWPOKE relative motion predictions with TLE and truth orbits.
REAL DATA RESULTSObservations from Raven were used to test COWPOKE’s effectiveness. Raven images were
taken of the DirecTV 4S and AMC 4 spacecraft collocated at 101° West longitude during the
nights of July 2324 and July 29August 1, 2003. One of these images is shown in Fig. 4.
Raven images were used to compute the separation of the two satellites. Sat 1, the
reference satellite, was chosen to be AMC4, and DirecTV 4S was designated Sat 2. The
reference orbit for Sat 1 was generated using the TLE from July 20, 2003. Other TLEs might
be available at an epoch closer to the 24^{th}, but for R&D Raven operations, the catalog is only
updated every few days. The observed separation in right ascension and declination on the
night of the 23^{rd} were used to estimate the Keplerian element differences. The observations
from the 23^{rd} didn’t span a long enough period to accurately solve for the difference in
semimajor axis, so an a priori estimate of 0 m was added for δa, with a standard deviation of
1000 km. With the resulting estimate of the element differences, COWPOKE was used to
predict the relative position of the two satellites at the time of each observation taken on the
night of the 24^{th}. The COWPOKE predictions are compared to the positions predicted by the
TLEs at the time of each observation in Fig. 5. The COWPOKE prediction of the location of
Sat 2 was off by an average of 155 microradians, while the TLE predictions differed from the
observations by an average of 576 microradians.
Fig. 4. Raven image of AMC4 and DirecTV 4S.
Fig. 5. COWPOKE’s prediction of the position of Satellite 2 on July 24^{th} compared to the TLE predictions.
The observations from July 23 and 24 were used to estimate the element differences and
predict the relative motion for July 29, when the next telescope images were taken. DP4
predictions and the COWPOKE reference orbit were obtained using TLEs for Sat 1 and Sat 2
from the 26^{th} and 27^{th} respectively. The results for the 29^{th} are shown in Fig. 6. The COWPOKE predictions differed from the truth by an average of 390 microradians, while the
TLE predictions were off by 721 microradians.
Fig. 6. COWPOKE’s prediction of the position of DirecTV 4S on July 29 compared to the TLE
predictions.
The observations from July 23, 24, and 29 were then used to estimate improved element
differences. Those element differences were used to predict the relative position of Sat 2 on
the 30^{th}, and the results are plotted in Fig. 7. COWPOKE had an average error of 300
microradians, and the TLE for Sat 2 averaged 894 microradians. Fig. 8 shows the results of
similar predictions for July 31 using the observations from all previous nights. To be clear,
the TLE prediction span at this point is several days while the COWPOKE prediction is only
one day.
Fig. 7. COWPOKE’s prediction of the position of DirecTV 4S on July 30 compared to the TLE
predictions.
In Fig. 8, one can see that the true position of Sat 1 has shifted away from the TLE
prediction since the night before. Also, the COWPOKE prediction no longer matches the
observation for Sat 2, especially in declination. There is strong evidence that Sat 1 performed
a stationkeeping maneuver between the 30^{th} and the 31^{st}. Even with the possible maneuver,
COWPOKE still provided a better estimate of the relative motion than the TLE. COWPOKE
had an average error of 450 microradians, and TLE for Sat 2 averaged 869 microradians.
Fig. 8. COWPOKE’s prediction of the position of DirecTV 4S on July 31
compared to the TLE predictions.
Whatever the cause of the sudden shift, it was decided to start a new fit span. Only the
observations from the night of the 31^{st} were used to solve for the new element differences,
and these were propagated forward with COWPOKE to the 1^{st} of August. The COWPOKE and TLE predictions for that night are compared to the observations in Fig. 9. COWPOKE
had an average error of 210 microradians, and the TLE for Sat 2 averaged 958 microradians.
One of the advantages of COWPOKE is that the effects of maneuvers can be mitigated by
using a one day fit span while TLEs typically use a longer fit span and will suffer the effects
of maneuvers continuously.
Fig. 9. COWPOKE’s prediction of the position of DirecTV 4S on August 1
compared to the TLE predictions.
While the COWPOKE results are somewhat encouraging when compared to the TLE’s,
the overall performance is not as good as expected. Perturbation analysis indicate that the
equations should be accurate with only a few percent error. The simulation results showed
similar error levels. If the relative orbit estimation algorithms were functioning properly, one
would expect to see results with errors at the few percent level. For the one day fit cases
shown in Fig. 5 and 9, larger errors might be expected due to limited observability over a
short data arc. For the five day prediction case shown in Fig. 6, one might also expect to see
larger errors due to the prediction interval. Then there is the case, shown in Fig. 8, where the
reference satellite appears to maneuver; this would also cause prediction error. However, one
case remains, shown in Figure 7, where several days of data are used in the estimation process
and the prediction interval is only one orbit; the COWPOKE prediction error is as large as all
of the other cases and is around 10% of the separation distance. This is larger than expected
and indicates that there may be an unknown error source in the algorithm or software tools.
Regardless, efforts must be made to better understand the limitations of this approach.
CONCLUSIONS AND FUTURE WORKThis work has shown that the COWPOKE equations can be used to provide meaningful
relative motion of geosynchronous satellite clusters. Perturbation analysis indicated that 2body dynamics are adequate for medium accuracy applications. Improvements were made to
the equations, however, to account for large right ascension of the ascending node, argument
of perigee, and true anomaly element differences.
Estimating the Keplerian element differences and using the COWPOKE equations to
predict the relative motion can supply valuable information in spacecraft identification. Using six nights of Raven images, it is shown that COWPOKE estimated the position of DirecTV
4S relative to AMC4 much better than TLE predictions, even with unmodeled maneuvers.
This indicates that COWPOKE holds the potential to be a valuable space surveillance tool.
Sizable error still remains in the relative orbit prediction results. These errors are larger
than anticipated so care must be taken to detemine the major source of this error and remove
it. If this can be accomplished, the relative orbit estimation approach will be far more
valuable. Beyond those improvements, the effects of differential SRP should be formulated
for circular, equatorial orbits to better predict the motion of GEO clusters.
Acknowledgments. Keric Hill was supported by the AFRL Space Scholars Program. Our sincere
appreciation goes to Dr. Clifford Rhoades of AFOSR for funding Space Scholars at AMOS, and to
Maj. Bill Hilbun, Ph.D., of AFOSR for funding the initial COWPOKE research. Thanks to Chuck
Matson, Paul Kervin, Lt.C. Jeff McCann, Valerie Skarupa, TSgt. Khalid Golden, and Irma Aragon for
supporting COWPOKE research at AMOS. Additional thanks to the Space Vehicles Directorate for
administering and allowing AMOS’s participation in the Space Scholars program.
REFERENCES
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