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<title> Testing of the Java Astrodynamics Toolkit Propagator</title>
<h1 align="center">Testing of the Java Astrodynamics Toolkit Propagator </h1>
<h3 align="center">Dr. David E. Gaylor<a href="#tthFtNtAAB" name="tthFrefAAB"><sup>1</sup></a>
&nbsp;, Richard C. Page III<a href="#tthFtNtAAC" name="tthFrefAAC"><sup>2</sup></a><br />
&nbsp;and Kathryn V. Bradley<a href="#tthFtNtAAD" name="tthFrefAAD"><sup>3</sup></a><br />
<font size="+0"><i>
Emergent Space Technologies, Greenbelt, MD, 20770, USA</i></font><br />
</h3>
<h3 align="center">7 August 2006 </h3>
<h2> Abstract</h2>
The Java Astrodynamics Toolkit (JAT) is an open source software
library of reusable software components for mission design,
trajectory optimization, and simulation of navigation, guidance and
control systems. This paper provides results of comparing various
Earth orbit ephemerides created with JAT with those created using
Analytical Graphics Inc.'s Satellite Tool Kit/High Precision Orbit
Propagator (STK/HPOP) and a.i. solutions, Inc.'s FreeFlyer.
Ephemerides produced using STK and FreeFlyer are also compared to
each other. With careful attention to parameters and models, results
show agreement of millimeter to meter level over several days.
Differences are discussed including potential sources of error.
<div class="p"><!----></div>
<h2><a name="tth_sEc1">
1</a>&nbsp;&nbsp;Introduction</h2>
The Java Astrodynamics Toolkit (JAT) is an open source library of
reusable software components written in the Java programming
language to support rapid development of spacecraft simulations
including 2-D and 3-D visualization capabilities. JAT is distributed
under the GNU General Public License and is available at
http://jat.sourceforge.net. Potential applications of JAT include:
<ul>
<li> Simulations of autonomous spacecraft navigation, guidance and
control
<div class="p"><!----></div>
</li>
<li> Space mission design and analysis including trajectory
optimization
<div class="p"><!----></div>
</li>
<li> Operational orbit determination and orbital events
generation
<div class="p"><!----></div>
</li>
</ul>
<div class="p"><!----></div>
This paper provides results of comparing ephemerides created by
propagating various Earth orbits using JAT with those created using
Satellite Tool Kit/High Precision Orbit Propagator (STK/HPOP) from
Analytical Graphics, Inc. (AGI) and FreeFlyer (FF) from a.i.
solutions, Inc. Ephemerides produced using STK and FreeFlyer are
also compared to each other. FreeFlyer is a spacecraft simulation
package currently in use at the NASA Goddard Space Flight Center.
Version 5.5, build 5.5.0.25 of FreeFlyer and STK version 6.1 were
used for comparison in the propagator validation.
<div class="p"><!----></div>
<h2><a name="tth_sEc2">
2</a>&nbsp;&nbsp;Initial State Vectors</h2> The following orbits were selected for testing: Sun-Synchronous, Geostationary (GEO), Molniya, International Space Station
(ISS) and the Global Positioning Satellite System (GPS) orbit. The
initial epoch was chosen to be June 1, 2004, 12:00:00. The initial
conditions which were used for the test case orbits are presented in
Table&nbsp;<a href="#Tab:Initial">1</a>. The corresponding osculating orbital elements
at the initial epoch are presented in Table&nbsp;<a href="#Tab:OrbitalElements">2</a>.
The orbit naming convention is based on the approximate altitude of
the orbit and does not follow the exact path of any particular
orbiting body. For example, ISS refers to an orbit which
approximates that of the International Space Station, but does not
refer to the actual location of the International Space Station.
<div class="p"><!----></div>
<a name="tth_tAb1">
</a> <center> <center>Table 1: Initial Conditions</center>
<table border="1"><tr><td></td></tr>
<tr><td>Category </td><td align="center">Orbit </td><td align="center">X </td><td align="center">Y </td><td align="center">Z </td><td align="center">Vx </td><td align="center">Vy </td><td align="center">Vz </td></tr>
<tr><td></td><td align="center">Type </td><td align="center">(km)</td><td align="center">(km)</td><td align="center">(km)</td><td align="center">(km/s)</td><td align="center">(km/s) </td><td align="center">(km/s) </td></tr>
<tr><td>LEO </td><td align="center">ISS </td><td align="center">-4453.783586 </td><td align="center">-5038.203756 </td><td align="center">-426.384456 </td><td align="center">3.831888 </td><td align="center">-2.887221 </td><td align="center">-6.018232 </td></tr>
<tr><td>LEO </td><td align="center">Sun-Sync </td><td align="center">-2290.301063 </td><td align="center">-6379.471940 </td><td align="center">0 </td><td align="center">-0.883923 </td><td align="center">0.317338 </td><td align="center">7.610832 </td></tr>
<tr><td>MEO </td><td align="center">GPS </td><td align="center">5525.33668 </td><td align="center">-15871.18494 </td><td align="center">-20998.992446 </td><td align="center">2.750341 </td><td align="center">2.434198 </td><td align="center">-1.068884 </td></tr>
<tr><td>HEO </td><td align="center">Molniya </td><td align="center">-1529.894287 </td><td align="center">-2672.877357 </td><td align="center">-6150.115340 </td><td align="center">8.717518 </td><td align="center">-4.989709 </td><td align="center">0 </td></tr>
<tr><td>GEO </td><td align="center">GEO </td><td align="center">36607.358256 </td><td align="center">-20921.723703 </td><td align="center">0 </td><td align="center">1.525636 </td><td align="center">2.669451 </td><td align="center">0 </td></tr><tr><td></td></tr><a name="Tab:Initial">
</a></table>
</center>
<div class="p"><!----></div>
<div class="p"><!----></div>
<a name="tth_tAb2">
</a> <center> <center>Table 2: Initial Osculating Orbital Elements</center>
<table border="1"><tr><td></td></tr>
<tr><td>Orbit </td><td align="center">a </td><td align="center">e </td><td align="center">i </td><td align="center">Arg of Perigee </td><td align="center">RAAN </td><td align="center">True Anomaly </td></tr>
<tr><td>Type </td><td align="center">km </td><td align="center"></td><td align="center">Deg </td><td align="center">Deg </td><td align="center">Deg </td><td align="center">Deg </td></tr>
<tr><td>ISS </td><td align="center">6747.414767 </td><td align="center">.001646462 </td><td align="center">51.667871 </td><td align="center">151.908712 </td><td align="center">45.649594 </td><td align="center">32.718375 </td></tr>
<tr><td>Sun-Sync </td><td align="center">6778.137291 </td><td align="center">.000000046 </td><td align="center">97.034619 </td><td align="center">337.556006 </td><td align="center">250.251300 </td><td align="center">22.443994 </td></tr>
<tr><td>GPS </td><td align="center">26558.859022 </td><td align="center">.015930402 </td><td align="center">56.286490 </td><td align="center">33.142451 </td><td align="center">52.707850 </td><td align="center">216.676430 </td></tr>
<tr><td>Molniya </td><td align="center">26553.376213 </td><td align="center">.740969399 </td><td align="center">63.4 </td><td align="center">270.000001 </td><td align="center">330.214160 </td><td align="center">359.999998 </td></tr>
<tr><td>GEO </td><td align="center">42164.165718 </td><td align="center">.000000135 </td><td align="center">0 </td><td align="center">103.882708 </td><td align="center">0 </td><td align="center">226.368596 </td></tr><tr><td></td></tr><a name="Tab:OrbitalElements">
</a></table>
</center>
<div class="p"><!----></div>
The integrator step size, propagation length and output frequency
were varied for the different test cases as shown in
Table&nbsp;<a href="#Tab:Frequency">3</a>. The propagation length and output frequency
were chosen based on a previous study performed by The Aerospace
Corporation to validate STK/HPOP[<a href="#Chao:2000" name="CITEChao:2000">1</a>].
<div class="p"><!----></div>
<a name="tth_tAb3">
</a> <center> <center>Table 3: Step Size, Propagation Length and Output
Frequency</center>
<table border="1"><tr><td></td></tr>
<tr><td>Orbit Type </td><td align="center">Integrator </td><td align="center">Propagation </td><td align="center">Output </td></tr>
<tr><td>Type </td><td align="center">Step Size </td><td align="center">Length </td><td align="center">Frequency </td></tr>
<tr><td></td><td align="center">(s) </td><td align="center">(days) </td><td align="center">(mins) </td></tr>
<tr><td>ISS </td><td align="center">5 </td><td align="center">1 </td><td align="center">1 </td></tr>
<tr><td>Sun-Sync </td><td align="center">5 </td><td align="center">1 </td><td align="center">1 </td></tr>
<tr><td>GPS </td><td align="center">60 </td><td align="center">2 </td><td align="center">2 </td></tr>
<tr><td>Molniya </td><td align="center">5 </td><td align="center">3 </td><td align="center">5 </td></tr>
<tr><td>GEO </td><td align="center">60 </td><td align="center">7 </td><td align="center">10 </td></tr><tr><td></td></tr><a name="Tab:Frequency">
</a></table>
</center>
<div class="p"><!----></div>
Twelve test cases were performed for each orbit type. Test cases
were run to verify JAT's ability to perform accurate propagation of
Earth orbits using various force models. The force models considered
in this study were two-body Earth gravity, JGM2 and JGM3 spherical
harmonic gravity models; Solar/Lunar third-body perturbation
effects; atmospheric drag using the Harris-Priester density model,
and the Naval Research Laboratory Mass Spectrometer and Incoherent
Scatter Radar Exosphere (NRLMSISE 2000) atmospheric density model;
and solar radiation pressure (SRP). Each of the force models was run
independently to test their individual accuracy, using two-body
Earth except for spherical harmonic cases. A final test case was run
for each satellite orbit which included an atmospheric drag model,
the SRP model, a spherical harmonic gravity model and third-body
perturbations from both the Sun and the Moon. These last test cases
were performed to validate the capability of JAT to accurately
propagate satellite orbits with multiple models.
<div class="p"><!----></div>
<h2><a name="tth_sEc3">
3</a>&nbsp;&nbsp;Earth Orientation Parameters</h2>
The use of spherical harmonic gravity
models requires precise calculation of the Earth's orientation.
Earth orientation involves parameters for the following
transformations: Nutation, Precession, Pole Wander, and Sidereal
Time, including the exact UT1 time obtained from the UT1-UTC
correction in the International Earth Rotation Service (IERS)
Bulletin A&nbsp;[<a href="#IERS:2004" name="CITEIERS:2004">4</a>]. It is important to understand how the
numerical integrators handle the rotation. FreeFlyer does not appear
to update the perturbations in the Earth's orientation during a
propagation run. Our testing seemed to indicate that STK/HPOP is
updating some information less frequently than every time step.
Currently, JAT updates the Earth orientation every time the
integrator calls for the derivatives. This was done to ensure that
JAT was at its highest accuracy.
<div class="p"><!----></div>
JAT implements the conventions established by the International
Astronomical Union (IAU), in the Explanatory Supplement to the
Astronomical Almanac&nbsp;[<a href="#Seidelmann:1992" name="CITESeidelmann:1992">6</a>] &nbsp;[<a href="#Hohenkerk:1992" name="CITEHohenkerk:1992">7</a>],
and by the IERS 1996 conventions&nbsp;[<a href="#McCarthy:1996" name="CITEMcCarthy:1996">8</a>] . Exhaustive
comparisons were conducted between the Almanac, and IERS 96
conventions for Nutation, Precession, and Greenwhich Hour Angle.
<div class="p"><!----></div>
The Earth orientation computation in FreeFlyer, however, does not
seem to implement pole wander, though it uses the UT1-UTC value from
the IERS. Because of this discrepancy, pole wander was turned off
for the purpose of validation between the three programs.
<div class="p"><!----></div>
<h2><a name="tth_sEc4">
4</a>&nbsp;&nbsp;Integrator</h2> The JAT fixed step-size, 8th order Runge-Kutta numerical integrator was used. The Runge-Kutta 8th order integrator within FreeFlyer was utilized in fixed step size mode for all of the
propagation comparisons. The Runge-Kutta-Fehlberg 7th order
integrator with 8th order error control within STK/HPOP was used as
a fixed step integrator with error control turned off.
<div class="p"><!----></div>
<h2><a name="tth_sEc5">
5</a>&nbsp;&nbsp;Atmospheric Drag</h2> Currently, JAT contains two high
fidelity atmospheric drag models: NRLMSISE2000 and Harris-Priester.
The NRLMSISE2000 model was adapted from code developed by the Naval
Research Laboratory&nbsp;[<a href="#NRLMSISE" name="CITENRLMSISE">13</a>]. The Harris Priester Model is
based on the model in <em>Satellite Orbits</em>&nbsp;[<a href="#Montenbruck" name="CITEMontenbruck">14</a>].
STK/HPOP also has both the NRLMSISE2000 and Harris-Priester models
available in version 6.1 and later. FreeFlyer does not have the
NRLMSISE2000 model, but it does have the Harris-Priester model. The
value used for the F10.7 cm solar flux was 150 W/m<sup>2</sup>/Hz and
a<sub>p</sub> was 14.918648166.
<div class="p"><!----></div>
<h3><a name="tth_sEc5.1">
5.1</a>&nbsp;&nbsp;Ap/Kp Indicies</h3>
When using the NRLMSISE 2000 in STK, the geomagnetic index (k<sub>p</sub>),
was set to 3 for this study. STK allows the user to input k<sub>p</sub> (the
planetary index) and F10.7 values in the force model. However, STK
does not allow the user to enter in an a<sub>p</sub> (planetary amplitude)
value. The NRLMSISE 2000 model in JAT requires the a<sub>p</sub> value in
order to calculate the density in the drag model. Therefore a
correlation between the k<sub>p</sub> value and the a<sub>p</sub> value was needed.
Wertz presents the equation for the correlation as:
[<a href="#Wertz:1978" name="CITEWertz:1978">12</a>]
<a name="Eqn:Vallado apkp">
</a>
<br clear="all" /><table border="0" width="100%"><tr><td>
<table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center">
a<sub>p</sub> = <span class="roman">e</span><sup>[(k<sub>p</sub>+1.6)/1.75]</sup> </td></tr></table>
</td><td width="1%">(1)</td></tr></table>
As previously mentioned, a k<sub>p</sub> value of 3 was used for the test
cases, which according to Equation&nbsp;<a href="#Eqn:Vallado apkp">1</a> above
results in an a<sub>p</sub> value of 13.85396. However, when this value is
entered into JAT the error in position is approximately 500 meters
for the LEO satellite orbits. Therefore, it was determined that the
a<sub>p</sub> was incorrect. In order to validate the value of a<sub>p</sub> which
should be used in the JAT, other sources were investigated.
<div class="p"><!----></div>
According to NOAA, for a k<sub>p</sub> value of 3, the corresponding value
of a<sub>p</sub> is 15&nbsp;[<a href="#NOAA:2005" name="CITENOAA:2005">5</a>]. Since the nature of atmospheric
density is empirical, there are several different analytical
approaches which disagree slightly on the exact value which should
be used in the calculation of density for atmospheric drag. When a
value of 15 is used for the a<sub>p</sub> in JAT, the position error is
approximately 51 meters for the ISS case and 28 meters for Sun-Sync.
<div class="p"><!----></div>
NOAA presents the following equation for the correlation:
&nbsp;[<a href="#NOAA:2005" name="CITENOAA:2005">5</a>]
<a name="Eqn:STK apkp">
</a>
<br clear="all" /><table border="0" width="100%"><tr><td>
<table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center">
28*k<sub>p</sub>+0.03*<span class="roman">e</span><sup>k<sub>p</sub></sup> = a<sub>p</sub> + 100*1<font face="symbol">-</font
><span class="roman">e</span><sup><font face="symbol">-</font
>0.08*a<sub>p</sub></sup> </td></tr></table>
</td><td width="1%">(2)</td></tr></table>
<div class="p"><!----></div>
When a k<sub>p</sub> value of 3 is used in Equation&nbsp;<a href="#Eqn:STK apkp">2</a> the
a<sub>p</sub> value is 14.918648166. A position error of approximately 1
meter over a day is found when this value of a<sub>p</sub> is used with the
ISS orbit.
<div class="p"><!----></div>
<h2><a name="tth_sEc6">
6</a>&nbsp;&nbsp;Third Body Perturbations</h2> Models for Solar and
Lunar perturbations have been created in JAT. These models use the
JPL DE405 ephemerides for the Sun and Moon. FreeFlyer and STK/HPOP
were also set to utilize Sun and Moon positions obtained from the
DE405 ephemeris.
<div class="p"><!----></div>
<h2><a name="tth_sEc7">
7</a>&nbsp;&nbsp;Gravity Models</h2> JAT and STK/HPOP contain both JGM2 and
JGM3 spherical harmonic gravity models. FreeFlyer only contains the
JGM2 model. For all cases using the JGM2 and JGM3 models, the degree
and order were both set to 20 for all three programs. STK/HPOP has
the ability to model solid and ocean tides. JAT currently does not
have this ability and so this was option turned off in STK/HPOP.
STK/HPOP was also set to not include relativistic accelerations. In
STK/HPOP, JGM2 and JGM3 gravity files were applied with a zero
degree and order for the two-body cases.
<div class="p"><!----></div>
<h2><a name="tth_sEc8">
8</a>&nbsp;&nbsp;Solar Radiation Pressure</h2> A solar radiation pressure (SRP)
model is also included in JAT. The SRP model assumes a uniform
spherical spacecraft with a given cross-sectional area. The SRP
model has a discontinuity at the boundary between full illumination
and entering the shadow of the Earth. To handle this, a cylindrical
partial illumination model is implemented to provide some continuity
in the derivatives as the radiation component becomes zero. The
vector to the sun is computed using the JPL DE405 ephemeris. In
FreeFlyer, a basic solar radiation pressure model is available,
although documentation on this model is not very detailed. The
settings for the altitude of the atmosphere for the Earth's shadow
were set to zero in STK/HPOP and JAT.
<h2><a name="tth_sEc9">
9</a>&nbsp;&nbsp;Ephemeris Comparison Results and Discussion</h2>
The ephemerides produced by JAT, STK/HPOP and FreeFlyer were
compared with each other. Table <a href="#Tab:ISS">4</a> presents the maximum
position and velocity differences for the ISS orbit. In all cases,
JAT agrees well with STK/HPOP. The largest differences in this set
of tests was between JAT and FreeFlyer when using the
Harris-Priester model, however, it is interesting to note that
STK/HPOP and FreeFlyer also do not agree well in these cases.
<div class="p"><!----></div>
<div class="p"><!----></div>
<a name="tth_tAb4">
</a> <center> <center>Table 4: Position and Velocity Differences for ISS
Orbit</center>
<table border="1"><tr><td></td></tr>
<tr><td>Description </td><td align="center">JAT-STK </td><td align="center">JAT-STK </td><td align="center">JAT-FF </td><td align="center">JAT-FF </td><td align="center">FF-STK </td><td align="center">FF-STK</td></tr>
<tr><td></td><td align="center">Position </td><td align="center">Velocity </td><td align="center">Position </td><td align="center">Velocity </td><td align="center">Position </td><td align="center">Velocity</td></tr>
<tr><td></td><td align="center">(m) </td><td align="center">(m/s) </td><td align="center">(m) </td><td align="center">(m/s) </td><td align="center">(m) </td><td align="center">(m/s)</td></tr>
<tr><td>2-body </td><td align="center">0.000829 </td><td align="center">0.000808 </td><td align="center">0.000048 </td><td align="center">0.0000008 </td><td align="center">0.000838 </td><td align="center">0.000809 </td></tr>
<tr><td>2-body+Moon </td><td align="center">0.000830 </td><td align="center">0.000815 </td><td align="center">0.000329 </td><td align="center">0.000001 </td><td align="center">0.000996 </td><td align="center">0.000815 </td></tr>
<tr><td>2-body+Sun </td><td align="center">0.000827 </td><td align="center">0.000838 </td><td align="center">0.000037 </td><td align="center">0.0000008 </td><td align="center">0.000826 </td><td align="center">0.000838 </td></tr>
<tr><td>2-body+NRL </td><td align="center">0.269426 </td><td align="center">0.000995 </td><td align="center">- </td><td align="center">- </td><td align="center">- </td><td align="center">- </td></tr>
<tr><td>2-body+HP </td><td align="center">1.031710 </td><td align="center">0.001802 </td><td align="center">2.33001 </td><td align="center">0.002665 </td><td align="center">5.495820 </td><td align="center">0.006752 </td></tr>
<tr><td>2-body+SRP </td><td align="center">0.103515 </td><td align="center">0.000875 </td><td align="center">0.069386 </td><td align="center">0.000071 </td><td align="center">0.136046 </td><td align="center">0.000885 </td></tr>
<tr><td>JGM2 </td><td align="center">0.368138 </td><td align="center">0.001085 </td><td align="center">0.500248 </td><td align="center">0.000552 </td><td align="center">0.836881 </td><td align="center">0.001576 </td></tr>
<tr><td>JGM3 </td><td align="center">0.368124 </td><td align="center">0.001076 </td><td align="center">- </td><td align="center">- </td><td align="center">- </td><td align="center">- </td></tr>
<tr><td>All+JGM2+NRL </td><td align="center">0.857388 </td><td align="center">0.001496 </td><td align="center">- </td><td align="center">- </td><td align="center">- </td><td align="center">- </td></tr>
<tr><td>All+JGM2+HP </td><td align="center">1.053890 </td><td align="center">0.001723 </td><td align="center">4.39748 </td><td align="center">0.005034 </td><td align="center">5.441040 </td><td align="center">0.006492 </td></tr>
<tr><td>All+JGM3+NRL </td><td align="center">0.857146 </td><td align="center">0.001572 </td><td align="center">- </td><td align="center">- </td><td align="center">- </td><td align="center">- </td></tr>
<tr><td>All+JGM3+HP </td><td align="center">1.053360 </td><td align="center">0.001730 </td><td align="center">- </td><td align="center">- </td><td align="center">- </td><td align="center">- </td></tr><tr><td></td></tr><a name="Tab:ISS">
</a></table>
</center>
<div class="p"><!----></div>
Table <a href="#Tab:Sun-Sync">5</a> presents the maximum position and velocity
differences for the Sun Synchronous orbit. JAT and STK/HPOP agree
well for all cases except for the cases that include atmospheric
drag as would be expected based on the previous discussion on the
difficulties in matching atmospheric density models.
<div class="p"><!----></div>
<div class="p"><!----></div>
<a name="tth_tAb5">
</a> <center> <center>Table 5: Position and Velocity Differences for Sun-Sync
Orbit</center>
<table border="1"><tr><td></td></tr>
<tr><td>Description </td><td align="center">JAT-STK </td><td align="center">JAT-STK </td><td align="center">JAT-FF </td><td align="center">JAT-FF </td><td align="center">FF-STK </td><td align="center">FF-STK</td></tr>
<tr><td></td><td align="center">Position </td><td align="center">Velocity </td><td align="center">Position </td><td align="center">Velocity </td><td align="center">Position </td><td align="center">Velocity</td></tr>
<tr><td></td><td align="center">(m) </td><td align="center">(m/s) </td><td align="center">(m) </td><td align="center">(m/s) </td><td align="center">(m) </td><td align="center">(m/s)</td></tr>
<tr><td>2-body </td><td align="center">0.00081 </td><td align="center">0.00080 </td><td align="center">0.00003 </td><td align="center">0.0000008 </td><td align="center">0.00082 </td><td align="center">0.00080 </td></tr>
<tr><td>2-body+Moon </td><td align="center">0.00083 </td><td align="center">0.00083 </td><td align="center">0.00028 </td><td align="center">0.0000009 </td><td align="center">0.00093 </td><td align="center">0.00083 </td></tr>
<tr><td>2-body+Sun </td><td align="center">0.00082 </td><td align="center">0.00082 </td><td align="center">0.00003 </td><td align="center">0.0000008 </td><td align="center">0.00081 </td><td align="center">0.00082 </td></tr>
<tr><td>2-body+NRL </td><td align="center">8.66478 </td><td align="center">0.01035 </td><td align="center">- </td><td align="center">- </td><td align="center">- </td><td align="center">- </td></tr>
<tr><td>2-body+HP </td><td align="center">3.33383 </td><td align="center">0.00439 </td><td align="center">1.61195 </td><td align="center">0.00181 </td><td align="center">1.77066 </td><td align="center">0.00261 </td></tr>
<tr><td>2-body+SRP </td><td align="center">0.09203 </td><td align="center">0.00083 </td><td align="center">0.03162 </td><td align="center">0.00003 </td><td align="center">0.09110 </td><td align="center">0.00083 </td></tr>
<tr><td>JGM2 </td><td align="center">0.17022 </td><td align="center">0.00096 </td><td align="center">0.27362 </td><td align="center">0.000296 </td><td align="center">0.37380 </td><td align="center">0.00111 </td></tr>
<tr><td>JGM3 </td><td align="center">0.16998 </td><td align="center">0.00089 </td><td align="center">- </td><td align="center">- </td><td align="center">- </td><td align="center">- </td></tr>
<tr><td>All+JGM2+NRL </td><td align="center">8.22196 </td><td align="center">0.00970 </td><td align="center">- </td><td align="center">- </td><td align="center">- </td><td align="center">- </td></tr>
<tr><td>All+JGM2+HP </td><td align="center">3.53857 </td><td align="center">0.00461 </td><td align="center">1.8332 </td><td align="center">0.00209 </td><td align="center">1.70622 </td><td align="center">0.00258 </td></tr>
<tr><td>All+JGM3+NRL </td><td align="center">8.22146 </td><td align="center">0.00995 </td><td align="center">- </td><td align="center">- </td><td align="center">- </td><td align="center">- </td></tr>
<tr><td>All+JGM3+HP </td><td align="center">3.53792 </td><td align="center">0.00455 </td><td align="center">- </td><td align="center">- </td><td align="center">- </td><td align="center">- </td></tr><tr><td></td></tr><a name="Tab:Sun-Sync">
</a></table>
</center>
<div class="p"><!----></div>
Table <a href="#Tab:GPS">6</a> presents the maximum position and velocity
differences for the GPS orbit. All three programs have close
agreement in this case due to the nature of the GPS orbit, which was
chosen to minimize the effects of perturbations.
<div class="p"><!----></div>
<div class="p"><!----></div>
<a name="tth_tAb6">
</a> <center> <center>Table 6: Position and Velocity Differences for GPS
Orbit</center>
<table border="1"><tr><td></td></tr>
<tr><td>Description </td><td align="center">JAT-STK </td><td align="center">JAT-STK </td><td align="center">JAT-FF </td><td align="center">JAT-FF </td><td align="center">FF-STK </td><td align="center">FF-STK</td></tr>
<tr><td></td><td align="center">Position </td><td align="center">Velocity </td><td align="center">Position </td><td align="center">Velocity </td><td align="center">Position </td><td align="center">Velocity</td></tr>
<tr><td></td><td align="center">(m) </td><td align="center">(m/s) </td><td align="center">(m) </td><td align="center">(m/s) </td><td align="center">(m) </td><td align="center">(m/s)</td></tr>
<tr><td>2-body </td><td align="center">0.00080 </td><td align="center">0.00083 </td><td align="center">0.000008 </td><td align="center">0.0000008 </td><td align="center">0.00080 </td><td align="center">0.00083 </td></tr>
<tr><td>2-body+Moon </td><td align="center">0.00080 </td><td align="center">0.00085 </td><td align="center">0.03287 </td><td align="center">0.000005 </td><td align="center">0.03298 </td><td align="center">0.00085 </td></tr>
<tr><td>2-body+Sun </td><td align="center">0.00084 </td><td align="center">0.00080 </td><td align="center">0.00064 </td><td align="center">0.0000008 </td><td align="center">0.00122 </td><td align="center">0.00080 </td></tr>
<tr><td>2-body+NRL </td><td align="center">0.00782 </td><td align="center">0.00083 </td><td align="center">- </td><td align="center">- </td><td align="center">- </td><td align="center">- </td></tr>
<tr><td>2-body+HP </td><td align="center">0.00080 </td><td align="center">0.00083 </td><td align="center">0.000008 </td><td align="center">0.0000008 </td><td align="center">0.00080 </td><td align="center">0.00083 </td></tr>
<tr><td>2-body+SRP </td><td align="center">0.08960 </td><td align="center">0.00084 </td><td align="center">0.72128 </td><td align="center">0.00008 </td><td align="center">0.72511 </td><td align="center">0.00088 </td></tr>
<tr><td>JGM2 </td><td align="center">0.01950 </td><td align="center">0.00081 </td><td align="center">0.01660 </td><td align="center">0.000002 </td><td align="center">0.02076 </td><td align="center">0.00081 </td></tr>
<tr><td>JGM3 </td><td align="center">0.01940 </td><td align="center">0.00081 </td><td align="center">- </td><td align="center">- </td><td align="center">- </td><td align="center">- </td></tr>
<tr><td>All+JGM2+NRL </td><td align="center">0.11936 </td><td align="center">0.00083 </td><td align="center">- </td><td align="center">- </td><td align="center">- </td><td align="center">- </td></tr>
<tr><td>All+JGM2+HP </td><td align="center">0.12493 </td><td align="center">0.00083 </td><td align="center">0.61088 </td><td align="center">0.00006 </td><td align="center">0.52284 </td><td align="center">0.00086 </td></tr>
<tr><td>All+JGM3+NRL </td><td align="center">0.11951 </td><td align="center">0.00085 </td><td align="center">- </td><td align="center">- </td><td align="center">- </td><td align="center">- </td></tr>
<tr><td>All+JGM3+HP </td><td align="center">0.12509 </td><td align="center">0.00085 </td><td align="center">- </td><td align="center">- </td><td align="center">- </td><td align="center">- </td></tr><tr><td></td></tr><a name="Tab:GPS">
</a></table>
</center>
<div class="p"><!----></div>
Table <a href="#Tab:Molniya">7</a> presents the maximum position and velocity
differences for the Molniya orbit. The Molniya orbit is the most
challenging for a number of reasons. JAT and STK/HPOP agree well for
the cases that did not include atmospheric drag or a spherical
harmonic gravity model. It is interesting to note that JAT and
FreeFlyer did agree well in the cases with atmospheric drag.
<div class="p"><!----></div>
<div class="p"><!----></div>
<a name="tth_tAb7">
</a> <center> <center>Table 7: Position and Velocity Differences for Molniya
Orbit</center>
<table border="1"><tr><td></td></tr>
<tr><td>Description </td><td align="center">JAT-STK </td><td align="center">JAT-STK </td><td align="center">JAT-FF </td><td align="center">JAT-FF </td><td align="center">FF-STK </td><td align="center">FF-STK</td></tr>
<tr><td></td><td align="center">Position </td><td align="center">Velocity </td><td align="center">Position </td><td align="center">Velocity </td><td align="center">Position </td><td align="center">Velocity</td></tr>
<tr><td></td><td align="center">(m) </td><td align="center">(m/s) </td><td align="center">(m) </td><td align="center">(m/s) </td><td align="center">(m) </td><td align="center">(m/s)</td></tr>
<tr><td>2-body </td><td align="center">0.00090 </td><td align="center">0.00084 </td><td align="center">0.00047 </td><td align="center">0.0000008 </td><td align="center">0.00082 </td><td align="center">0.00084 </td></tr>
<tr><td>2-body+Moon </td><td align="center">0.00082 </td><td align="center">0.00080 </td><td align="center">0.08194 </td><td align="center">0.000068 </td><td align="center">0.08180 </td><td align="center">0.00080 </td></tr>
<tr><td>2-body+Sun </td><td align="center">0.00085 </td><td align="center">0.00082 </td><td align="center">0.03110 </td><td align="center">0.000025 </td><td align="center">0.03129 </td><td align="center">0.00082 </td></tr>
<tr><td>2-body+NRL </td><td align="center">28.39440 </td><td align="center">0.02389 </td><td align="center">- </td><td align="center">- </td><td align="center">- </td><td align="center">- </td></tr>
<tr><td>2-body+HP </td><td align="center">15.16700 </td><td align="center">0.01287 </td><td align="center">0.02214 </td><td align="center">0.000018 </td><td align="center">15.18920 </td><td align="center">0.01289 </td></tr>
<tr><td>2-body+SRP </td><td align="center">0.63727 </td><td align="center">0.00092 </td><td align="center">2.77142 </td><td align="center">0.00229 </td><td align="center">2.13422 </td><td align="center">0.00190 </td></tr>
<tr><td>JGM2 </td><td align="center">5.57807 </td><td align="center">0.00456 </td><td align="center">7.72931 </td><td align="center">.006160 </td><td align="center">2.30217 </td><td align="center">0.00184 </td></tr>
<tr><td>JGM3 </td><td align="center">5.57807 </td><td align="center">0.00481 </td><td align="center">- </td><td align="center">- </td><td align="center">- </td><td align="center">- </td></tr>
<tr><td>All+JGM2+NRL </td><td align="center">27.86270 </td><td align="center">0.02052 </td><td align="center">- </td><td align="center">- </td><td align="center">- </td><td align="center">- </td></tr>
<tr><td>All+JGM2+HP </td><td align="center">7.57025 </td><td align="center">0.00460 </td><td align="center">5.43094 </td><td align="center">0.00417 </td><td align="center">12.97950 </td><td align="center">0.00862 </td></tr>
<tr><td>All+JGM3+NRL </td><td align="center">27.86160 </td><td align="center">0.02082 </td><td align="center">- </td><td align="center">- </td><td align="center">- </td><td align="center">- </td></tr>
<tr><td>All+JGM3+HP </td><td align="center">7.57107 </td><td align="center">0.00493 </td><td align="center">- </td><td align="center">- </td><td align="center">- </td><td align="center">- </td></tr><tr><td></td></tr><a name="Tab:Molniya">
</a></table>
</center>
<div class="p"><!----></div>
Table <a href="#Tab:GEO">8</a> presents the maximum position and velocity
differences for the GEO orbit. In all cases, JAT agrees well with
STK/HPOP. The largest differences in this set of tests was between
JAT and FreeFlyer when including SRP, however, it is interesting to
note that STK/HPOP and FreeFlyer also do not agree well in these
cases.
<div class="p"><!----></div>
<div class="p"><!----></div>
<a name="tth_tAb8">
</a> <center> <center>Table 8: Position and Velocity Differences for GEO
Orbit</center>
<table border="1"><tr><td></td></tr>
<tr><td>Description </td><td align="center">JAT-STK </td><td align="center">JAT-STK </td><td align="center">JAT-FF </td><td align="center">JAT-FF </td><td align="center">FF-STK </td><td align="center">FF-STK</td></tr>
<tr><td></td><td align="center">Position </td><td align="center">Velocity </td><td align="center">Position </td><td align="center">Velocity </td><td align="center">Position </td><td align="center">Velocity</td></tr>
<tr><td></td><td align="center">(m) </td><td align="center">(m/s) </td><td align="center">(m) </td><td align="center">(m/s) </td><td align="center">(m) </td><td align="center">(m/s)</td></tr>
<tr><td>2-body </td><td align="center">0.00070 </td><td align="center">0.00070 </td><td align="center">0.00009 </td><td align="center">0.0000007 </td><td align="center">0.00070 </td><td align="center">0.00070 </td></tr>
<tr><td>2-body+Moon </td><td align="center">0.00090 </td><td align="center">0.00081 </td><td align="center">0.07980 </td><td align="center">0.00005 </td><td align="center">0.07989 </td><td align="center">0.00081 </td></tr>
<tr><td>2-body+Sun </td><td align="center">0.00087 </td><td align="center">0.00081 </td><td align="center">0.05206 </td><td align="center">0.000004 </td><td align="center">0.05230 </td><td align="center">0.00081 </td></tr>
<tr><td>2-body+NRL </td><td align="center">0.00070 </td><td align="center">0.00070 </td><td align="center"></td><td align="center"></td><td align="center"></td><td align="center"></td></tr>
<tr><td>2-body+HP </td><td align="center">0.00070 </td><td align="center">0.00070 </td><td align="center">0.00009 </td><td align="center">0.0000007 </td><td align="center">0.00070 </td><td align="center">0.00070 </td></tr>
<tr><td>2-body+SRP </td><td align="center">0.00080 </td><td align="center">0.00082 </td><td align="center">2.73497 </td><td align="center">0.00012 </td><td align="center">2.89851 </td><td align="center">0.00087 </td></tr>
<tr><td>JGM2 </td><td align="center">0.07287 </td><td align="center">0.00081 </td><td align="center">1.58727 </td><td align="center">0.000116 </td><td align="center">1.65447 </td><td align="center">0.00083 </td></tr>
<tr><td>JGM3 </td><td align="center">0.07276 </td><td align="center">0.00081 </td><td align="center"></td><td align="center"></td><td align="center"></td><td align="center"></td></tr>
<tr><td>All+JGM2+NRL </td><td align="center">0.72280 </td><td align="center">0.00084 </td><td align="center"></td><td align="center"></td><td align="center"></td><td align="center"></td></tr>
<tr><td>All+JGM2+HP </td><td align="center">0.72280 </td><td align="center">0.00084 </td><td align="center">2.99845 </td><td align="center">0.00014 </td><td align="center">3.41299 </td><td align="center">0.00090 </td></tr>
<tr><td>All+JGM3+NRL </td><td align="center">0.72298 </td><td align="center">0.00082 </td><td align="center"></td><td align="center"></td><td align="center"></td><td align="center"></td></tr>
<tr><td>All+JGM3+HP </td><td align="center">0.72298 </td><td align="center">0.00082 </td><td align="center"></td><td align="center"></td><td align="center"></td><td align="center"></td></tr><tr><td></td></tr><a name="Tab:GEO">
</a></table>
</center>
<h2><a name="tth_sEc10">
10</a>&nbsp;&nbsp;Conclusions</h2>
JAT achieves agreement below meter levels for all cases except those
with atmospheric drag. The sensitivity to differences in the density
calculation make an exact match difficult. Atmospheric drag depends
on proper modeling of geomagnetic and molecular behavior influencing
atmospheric density. Care must be taken to model the geomagnetic
index correctly. However, given that the range of modeling error can
be on the order of kilometers or more for low Earth orbits, meters
or tens of meters in the Molniya case is expected.
<div class="p"><!----></div>
The verification of the spherical harmonic gravity models revealed
ambiguities in which IERS convention and update procedure is used in
the rotation from the Earth Centered Inertial frame to the Earth
Centered Earth Fixed frame. Ultimately, due to the complexity of the
IAU and IERS conventions and their subsequent corrections, it is
extremely difficult to match algorithms exactly without knowledge of
all equations and values used or access to the source code.
<div class="p"><!----></div>
This study shows that JAT has the capability to closely match the
ability of commercial packages. However, the uncertainty of the
models for both spherical harmonic and atmospheric drag indicates
that exact agreement does not necessarily correspond to accurate
propagation. It is recommended that propagators seek to ensure the
current, correct algorithms for Earth rotation from the IAU and IERS
as well as the correct atmospheric data from NOAA in addition to
validation against existing codes. Commercial packages such as
STK/HPOP and FreeFlyer often implement different conventions and
data sets in different versions, so care should be taken in
determining the standards used.
<div class="p"><!----></div>
<h2><a name="tth_sEc11">
11</a>&nbsp;&nbsp;Future Work</h2>
The goal of JAT is to provide high-fidelity models for simulation of
guidance, navigation, and control of spacecraft. As an open source
project, the Java Astrodynamics Toolkit will continue to be updated
with new conventions and updated data from the IERS, IAU, and NOAA.
It will grow in both accuracy and functionality as the project
progresses.
<div class="p"><!----></div>
<h2>References</h2>
<dl compact="compact">
<dt><a href="#CITEChao:2000" name="Chao:2000">[1]</a></dt><dd>
C.&nbsp;C.&nbsp;Chao and J.&nbsp;W.&nbsp;Cook et al.
<i>Independent Verification and Validation for Analytical Graphics,
Inc. of Three Astrodynamic Functions of the Satellite Tool Kit:
Version 4.1.0.</i>
Aerospace Corporation. El Segundo, CA. ATR-2000(7605)-1. 15 February 2000.
<div class="p"><!----></div>
</dd>
<dt><a href="#CITECarpenter:2003" name="Carpenter:2003">[2]</a></dt><dd>
J.&nbsp;R.&nbsp;Carpenter, editor.
<i>Global Positioning Service Enhanced Onboard Navigation System,
Mathematical Specifications.</i>
Version 2, Release 2.1 December 2003.
Goddard Space Flight Center, Mission Engineering and Systems
Analysis Division/Code 595.
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEGaylor:2005" name="Gaylor:2005">[3]</a></dt><dd>
D.&nbsp;Gaylor, R. Page, T. Berthold.
Java Astrodynamics Toolkit.
URL: <a href="http://jat.sourceforge.net"><tt>http://jat.sourceforge.net</tt></a>, Accessed July 2006.
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEIERS:2004" name="IERS:2004">[4]</a></dt><dd>
International Earth Rotation and Reference Systems Service.
<i>IERS Bulletin A Reports</i>, 2004.
URL: <a href="ftp://cddisa.gsfc.nasa.gov/pub/reports/iers/bulletin_a/volume_xvii/iers_bulletina.xvii_021"><tt>ftp://cddisa.gsfc.nasa.gov/pub/reports/iers/bulletin_a/volume_xvii/iers_bulletina.xvii_021</tt></a>, Accessed July 2005.
<div class="p"><!----></div>
</dd>
<dt><a href="#CITENOAA:2005" name="NOAA:2005">[5]</a></dt><dd>
NOAA.
Geomagnetic k<sub>p</sub> and a<sub>p</sub> Indices.
URL: <a href="http://www.ngdc.noaa.gov/stp/GEOMAG/kp_ap.html"><tt>http://www.ngdc.noaa.gov/stp/GEOMAG/kp_ap.html</tt></a>
<div class="p"><!----></div>
</dd>
<dt><a href="#CITESeidelmann:1992" name="Seidelmann:1992">[6]</a></dt><dd>
P.K.&nbsp;Seidelmann, B. Guinot and L.E. Doggett.
"Time". <i>Explanatory Supplement to the Astronomical
Almanac.</i>
University Science Books. Mill Valley, CA, 1992:[48-53]
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEHohenkerk:1992" name="Hohenkerk:1992">[7]</a></dt><dd>
C.Y.&nbsp;Hohenkerk, B.D. Yallop, C.A. Smith, and A.T. Sinclair.
"Celestial Reference Systems". <i>Explanatory Supplement to the
Astronomical Almanac.</i>
University Science Books. Mill Valley, CA, 1992:[95-120]
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEMcCarthy:1996" name="McCarthy:1996">[8]</a></dt><dd>
D.D.&nbsp;McCarthy.
IERS Standards. IERS Technical Note 21.
U.S. Naval Observatory. 1996.
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEIERSB" name="IERSB">[9]</a></dt><dd>
International Earth Rotation and Reference Systems Service.
<i>IERS Bulletin B Reports</i>, 2004.
URL: <a href="http://www.iers.org/iers/publications/bulletins/"><tt>http://www.iers.org/iers/publications/bulletins/</tt></a>, Accessed July 2005.
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEVallado:2001" name="Vallado:2001">[10]</a></dt><dd>
D.&nbsp;A.&nbsp;Vallado.
<i>Fundamentals of Astrodynamics and Applications, 2nd Edition.</i>
Microcosm Press. El Segundo, CA. 2001:[528]
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEVallado:2005" name="Vallado:2005">[11]</a></dt><dd>
D.&nbsp;A.&nbsp;Vallado. 2005.
“An Analysis of State Vector Propagation using Differing Flight Dynamics
Programs
Paper AAS 05-199 presented at the AAS/AIAA
Space Flight Mechanics Conference. Copper Mountain, CO.
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEWertz:1978" name="Wertz:1978">[12]</a></dt><dd>
J.&nbsp;R.&nbsp;Wertz.
<i>Spacecraft Attitude Determination and Control.</i>
D.&nbsp;Reidel Publishing Company. Dordrecht, Holland. 1978:[3,8,9]
<div class="p"><!----></div>
</dd>
<dt><a href="#CITENRLMSISE" name="NRLMSISE">[13]</a></dt><dd>
M.&nbsp;Picone, A.E. Hedin, and D. Drob.
<i>NRLMSISE-00 Model 2001.</i>
URL:
<a href="http://modelweb.gsfc.nasa.gov/atmos/nrlmsise00.html"><tt>http://modelweb.gsfc.nasa.gov/atmos/nrlmsise00.html</tt></a>, Accessed
July 2006.
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEMontenbruck" name="Montenbruck">[14]</a></dt><dd>
O.&nbsp;Montenbruck, E. Gill.
<i>Satellite Orbits.</i>
Springer-Verlag Berlin Heidelberg. The Netherlands.
2000:[88]</dd>
</dl>
<div class="p"><!----></div>
<hr /><h3>Footnotes:</h3>
<div class="p"><!----></div>
<a name="tthFtNtAAB"></a><a href="#tthFrefAAB"><sup>1</sup></a>VP Aerospace Systems, dave.gaylor@emergentspace.com
<div class="p"><!----></div>
<a name="tthFtNtAAC"></a><a href="#tthFrefAAC"><sup>2</sup></a>Aerospace Engineer, richard.page@emergentspace.com
<div class="p"><!----></div>
<a name="tthFtNtAAD"></a><a href="#tthFrefAAD"><sup>3</sup></a>Aerospace Engineer, kate.bradley@emergentspace.com
<br /><br /><hr /><small>File translated from
T<sub><font size="-1">E</font></sub>X
by <a href="http://hutchinson.belmont.ma.us/tth/">
T<sub><font size="-1">T</font></sub>H</a>,
version 3.74.<br />On 30 Aug 2006, 11:44.</small>
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