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Earth orbital space suffers from the ever increasing count of space objects, including operational satellites and space debris. Space system operations rely on the management of vast catalogs of objects to avoid any damaging collision. NORAD (North American Aerospace Defense Command) and NASA (National Aeronautics and Space Administration) both maintain a database for a large quantity of orbiting objects. Data are stored as Two Line Elements (TLE) and used along with specific analytical propagation models. Operation centers need Orbit Determination methods to accurately compute conjunctions and collision probabilities. With more and more flying objects, computations must be fast enough to ensure satellite safety. Mixing Orbit Determination and TLE analytical propagation models appears to be an effective way to grant security in space. This paper presents an open-source solution for an Orbit Determination method based on TLE propagation models. The method was implemented and validated inside the Orekit space mechanic library. It was then confronted with a classical numerical Orbit Determination on a GNSS test case.
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Thomas Paulet and Bryan Cazabonne
CS GROUP, Flight Dynamics and Science, 31506 Toulouse, France, Email:;
Earth orbital space suffers from the ever increasing
count of space objects, including operational satellites
and space debris. Space system operations rely on
the management of vast catalogs of objects to avoid
any damaging collision. NORAD (North American
Aerospace Defense Command) and NASA (National
Aeronautics and Space Administration) both maintain a
database for a large quantity of orbiting objects. Data
are stored as Two Line Elements (TLE) and used along
with specific analytical propagation models. Operation
centers need Orbit Determination methods to accurately
compute conjunctions and collision probabilities. With
more and more flying objects, computations must be
fast enough to ensure satellite safety. Mixing Orbit
Determination and TLE analytical propagation models
appears to be an effective way to grant security in space.
This paper presents an open-source solution for an Orbit
Determination method based on TLE propagation models.
The method was implemented and validated inside the
Orekit space mechanic library. It was then confronted
with a classical numerical Orbit Determination on a GNSS
test case.
Keywords: Orbit Determination; TLE; Open Source;
Orekit; Automatic Differenciation; Analytical Orbit
Orbit Determination is a technique used to estimate the
state vector of a space object from a first guess and a set
of observable measurements. The state vector contains
the orbital elements, the propagation dynamic parameters
and the measurement biases. Determined orbit is meant
to be as accurate as possible and accessible within the
shortest computation time. Those two paradigms become
even more challenging with the ever increasing number
of space objects orbiting the Earth. Orbit Determination
is especially necessary for debris tracking and collision
probability computation [
]. Thus, this technique shall be
able to quickly take care of large object collections.
Numerical Orbit Determination is widely used. It needs
a numerical orbit propagator which accurately computes
the orbital perturbations on a space object by equation
of motion numerical integration. It reaches significant
precision level with realistic force models, but requires
high computation time. Analytical orbit propagators rather
sacrifice accuracy to benefit computation speed. They
employ a set of empirical equations for modeling a space
object dynamic instead of a numerical integration of the
equation of motion. This kind of orbit propagators can
be adapted to be employed in an Orbit Determination
process. This kind of application is suitable to address
space surveillance topics, where fast orbit estimation is
TLE data are a widespread way to represent an orbit.
They are generated and freely released by the NASA
and the NORAD for each Earth orbiting object bigger
than a softball ball. A TLE is a set of mean orbital
elements that locates an object in space with a few
kilometer-accuracy [
]. Even though quite imprecise, they
remain deployed in myriad space mechanic applications
] [
]. This kind of data also requires specific propagation
models to be used properly. The realization of an Orbit
Determination application based on TLE orbit propagators
looks promising. It was chosen to implement the process
with both the Simplified General Perturbation 4 (SGP4)
and the Simplified Deep-space Perturbation 4 (SDP4) orbit
propagation algorithms. Those analytical propagation
models are empirical and only take into account the main
orbital perturbations as presented in Figure 1.
A cornerstone of an Orbit Determination algorithm
development is the state transition matrix computation.
This mathematical object, regrouping state vector
partial derivatives, requires an accurate computation and
validation. It becomes important for short determination
arcs and very accurate observations. Automatic
differentiation is able to compute those partial derivatives
by applying chain rule. This method prevents the complex
task to establish and validate all model derivatives. State
transition matrix terms are computed directly from model
evolution equations, instead of creating a new differential
equation to integrate as it is done in Orekit numerical
or semi-analytical orbit determination [
] [
]. Moreover,
automatic differentiation is used to create SGP4/SDP4
compliant TLEs from a state vector.
We tested the analytical orbit determination performances
using real GPS satellite data. A network of six IGS
(International GNSS Service) stations is used. Only
pseudo range measurements were considered. Estimated
orbits were compared with IGS precise products. Station
reference positions were retrieved from SINEX file. We
also compared the analytical orbit determination method
with the Orekit’s numerical method. Final estimated orbit
accuracy and computation time were investigated. It
appeared that computation time is slightly increased with
analytical method while determined position accuracy on
the orbit shares the same the magnitude order.
This paper presents the development and validation
of the TLE orbit determination in the Orekit flight
dynamics library. It also demonstrates Orekit capability
to build improved TLE with an accuracy of a hundred of
meters instead of a kilometer. This promotes developing
additional analytical propagation models in order to
acquire greater diversity within the open-source library.
Orekit is an open-source space flight dynamics library
]. It is written in Java and provides low level elements
for the development of flight dynamics applications. It
was first developed by CS GROUP in 2002 as a private
library for the company collaborators. In 2008, the library
evolved towards an open-source project under Apache
v2.0 License [
]. Orekit is now used worldwide, both by
academics and industries, to realize space applications,
studies and operations.
2.1. The TLE SGP4 and SDP4 propagation
TLE format was created in the 60s, it gathers mean orbital
parameters of a space object. Though, it requires specific
algorithms to be analysed and then propagated [
]. A
collection of analytical orbit propagators were developed
along with this format, seeking fast computations of space
trajectories. These models were named Simplified General
Perturbations (SGP). Among them, SGP4 and SDP4 are
mainly used for TLE manipulations. SGP4 is used for low
Earth orbit propagation while SDP4 handles further space
objects with terrestrial orbit. If orbit period is below 225
minutes, the low orbit model is to be applied, otherwise,
the SDP4 is. These propagation algorithms consider the
main perturbation influences on a satellite: first four zonal
Earth gravity field harmonics, atmospheric drag and solar
radiation pressure. SDP4 adds luni-solar gravity attraction
and deep space secular effects. Perturbation models are
summed up in Figure 1.
For a given object, TLEs are generated periodically. This
refreshment rate along with the use of SGP4 and SDP4
enables to estimate the position of a space object within
a kilometer magnitude precision [
]. The aim of the
study was to improve the accuracy of this well spread
data format, in order to increase its scope.
It is to be noted that atmospheric drag model takes
into account a special coefficient called B*. The B*
represents the ballistic coefficient of the considered space
object. Drag coefficient is usually estimated during
Orbit Determination process. Thus, estimating the B*
coefficient may also improve a TLE.
TLE analytical Orbit Determination will then aim at
estimating the six TLE coefficients that express the orbital
state: mean motion, eccentricity, inclination, longitude
of the ascending node, argument of periapsis and mean
anomaly, in addition of the B*.
2.2. The Batch Least Squares Orbit Determination
The Batch Least Squares algorithm is a classical technique
used for operational orbit determination. For a given
satellite initial state
and for an available observation
arc, the Batch Least Squares algorithm provides an
estimation of the satellite’s state such as
The calculation of Equation 1 is done by an iterative
process solving the non-linear Equation 2 [10]
δy0= (ATW A)1ATW b (2)
A : the partial derivatives matrix
W : the weighting matrix
b : the residual vector
The weighting matrix is initialized, at the beginning of the
estimation, by the user. The residual vector is computed,
for each measurement, by the difference between the
observed and the estimated measurements. Finally, the
partial derivatives matrix can be expressed by the product
of the observation matrix
by the state transition matrix
Ht,t =∂ρt
is an observed measurement at an arbitrary epoch
. In Orekit library, both the observation matrix and the
state transition matrix are calculated using the automatic
differentiation technique, which is detailed in the next
2.3. Computing the state transition matrix with
automatic differentiation
Automatic differentiation is a set of techniques to avoid the
analytical calculation of the derivatives of long equations.
It relies on the fact that every computer program is
decomposed as a sequence of elementary arithmetic
operations (i.e. addition, subtraction, etc.), elementary
functions (i.e; exp, sin, cos, etc.), and control flow
statements [
]. The calculation of the derivatives is
accurate to the precision of the computer system. For
instance, if
denotes an orbital element (e.g. the
eccentricity of the orbit), automatic differentiation gives
derivatives with respect to any parameter (i.e. orbital,
dynamic, or measurement parameters) by using only the
analytical expression of
. The partial derivatives are
stored in an array where the first element is the value
of the parameter and the other elements are its partial
derivatives, as represented in Equation 6.
[Yi∂Yi/∂Y1∂Yi/∂Y2· · · ∂Yi/∂Y6](6)
As a result, automatic differentiation is used to
automatically differentiate all the simplified equations
of SGP4 and SDP4 algorithms in order to build the
state transition matrix needed by the orbit determination
process. Automatic differentiation is also used to
calculate the observation matrix
. Indeed, during
the measurements estimation, the partial derivatives
are calculated simultaneously with the value of
the measurements thanks to automatic differentiation
Once the method is implemented, it remains to be tested
against real data. GNSS (Global Navigation Satellite
System) satellites are interesting because of the availability
of measurements and precise ephemeris. Data are freely
provided by the IGS. A satellite was chosen to simulate
a full orbit determination test case, with wisely selected
measurement stations along orbit ground track. Then, a
robustness test was performed on several GNSS satellites,
with more or less degraded conditions.
3.1. GPS IIR-M 6 test
In order to validate the process on a real operational case,
satellite GNSS IIR-M 6, NORAD ID 32711, is studied.
Considered measurement arc and stations are given on
Figure 2. First guess of the Orbit Determination process
is the following TLE:
1 32711U 08012A 16044.40566026 -.00000039 00000-0 00000+0 0 9991
2 32711 55.4362 301.3402 0091577 207.7302 151.8353 2.00563580 58013
IGS SP3 high precision ephemeris are considered as
the reference for the spacecraft position [
]. The
measurements are also taken from IGS products. They
allowed to gather 8211 pseudo-range measurements on a
5h30m arc [13].
Three configurations are tested. Firstly, only the
six orbital parameters are estimated during the Orbit
Determination process. Then, the B* coefficient is added
to the set of estimated parameters. A more classical
numerical Orbit Determination is finally performed for
comparison. The force models of the latter are fitted on
SDP4. During the computation, drag coefficient is also
estimated. In all three cases, ionospheric and tropospheric
pseudo range biases are also estimated [14].
The analytical method computations return two new
TLEs, that can be immediately propagated by usual
SGP4/SDP4 algorithms. B* is not estimated to produce
the first, however, it is in the second. They are given
1 32711U 08012A 16044.40566026 -.00000039 00000-0 00000-0 0 9992
2 32711 55.4358 301.3401 0091593 207.7154 151.8480 2.00564373 58012
1 32711U 08012A 16044.40566026 -.00000039 00000-0 -13229+5 0 9994
2 32711 55.4358 301.3400 0091600 207.7008 151.8629 2.00564891 58018
The process clearly evolved orbital parameters.
When estimated, the B* coefficient value seems far too
high. It is usually few
at most against
in this
case. Propagation will show either if this estimation is
appropriate or overestimated.
Relevant figures about the outputs of the different
runs are provided in Table 1. This test highlights that
analytical TLE Orbit Determination provides a significant
enhancement for the spacecraft position knowledge
compared to the initial TLE. Indeed, estimated TLEs
are about an order of magnitude more accurate than the
Measurement residuals are presented in Figure 3.
As expected, dispersion is greater with SDP4 usage, yet,
the calculations still manage to oncentrate the residuals
around zero. This reflects the correct behavior of the
Batch Least Square while using analytical propagation.
Analytical method remains less accurate than the
numerical. However, with respect to the batch least
square number of state evaluations, the analytical Orbit
Determination is faster than the numerical.
Stability of the algorithm is assessed studying Root Mean
Square error along the Orbit Determination steps. They
are given for all the three runs in Figure 4. Indeed, RMS
evolution is regular, and does not present any monotony
variation. The orbital state clearly always evolve toward a
better estimation of the measured object.
To complete the study and ensure that the new TLEs
are truly closer to real orbit, propagation comparison is
performed. Once again, the reference orbit is taken from
IGS SP3 ephemeris. Propagation does not exceed the
13th of February 2016 because of discontinuities between
two SP3 files [
]. Results are presented in Figure 5.
Clearly, orbit knowledge is improved not only at the
Orbit Determination epoch, but on 20-hour time span.
Considered that a TLE is usually generated every orbit,
i.e. about 12 hours in a GNSS case, the satellite position
accuracy can always be improved. Minimum position
error is achieved at median measurement epoch, it reaches
few ten meters against about 600 meters for the original
3.2. Expansion and robustness
A final robustness test is performed. It consists in several
TLEs analytical Orbit Determinations on different GPS
satellites, using the same set of measurement stations and
arc than the GPS IIR-M 6 case. Therefore, measurements
may not be suited for all orbits. The aim is to appreciate
how stiff the process is while dealing with data of various
quality. In all cases, it is chosen not to estimate the
B* coefficient but still tropospheric and ionospheric
pseudo range biases. Results are provided in Table 2.
In a nutshell, 9 out of 14 test cases show a position
accuracy enhancement after TLE Orbit Determination is
performed. Most part of the improvements are beyond
50% gain in position precision. Failing cases correspond
to degraded Orbit Determination conditions. Indeed,
either the computation epoch is out of measurement arc
(G09, G19), or the number of measurements is quite
low (G15, G22). However, the algorithm still works out
for some cases with same issues, such as G02 or G06.
It is not obvious to conclude about the robustness, and
precisely characterize how inputs shall be to perform the
analytical TLE Orbit Determination. Still, the method
perfectly acts with regular Orbit Determination conditions,
i.e. significant number of measurements well spread
around computation time.
The presented tests demonstrate that the analytical Orbit
Determination based on the SDP4 model truly improves
the accuracy of public TLE. Witnessed enhancement is
up to an order of magnitude on a GPS satellite position
accuracy at Orbit Determination epoch, and even more
when propagated. Moreover, the method produced a real
TLE that can be used as any NASA and NORAD data.
Orekit is even able to build a corresponding OMM (Orbit
Mean Message) to address other user needs.
Computation time is yet to be improved for the method
to be useful for massive catalog management. As a
matter of fact, the most part of the calculation time,
with numerical or analytical method, is spent in frame
conversion. They are performed for each measurement
at each Batch Least Square step. This explains that with
a lot of measurements, analytical method is hardly faster
than numerical. Orekit community is about to solve this
issue, then the previous test cases should be run again to
experience the computation time gain.
Orbit Determination using an analytical propagator
coupled with automatic differentiation still seems to be a
promising method to manage huge space object catalogs.
Implementing it successfully within the Orekit library
with the TLE models encourages to extend the method to
other analytical propagators. Though, Eckstein-Hechler or
Brouwer-Lyddane models could also be used to perform
fast and adapted Orbit Determinations. This work would
enlarge Orekit capabilities in term of Orbit Determination
and its applications.
Authors would like to acknowledge Mr. Maxime Journot,
Mr. Luc Maisonobe and Mr. Pascal Parraud, all from CS
GROUP, Space division, France. Technical discussions
with them provided a significant and valuable help to
implement this analytical Orbit Determination method in
the Orekit library.
Table 1. TLE analytical and numerical Orbit Determination results on GPS IIR-M 6 case. Errors are given at Orbit
Determination epoch, which is also first guess TLE time.
Only orbital
B* and orbital
orbit determination
First guess error with
respect to reference (m) 914.4 914.4 914.4
Result error with
respect to reference (m) 111.2 130.9 64.2
improvement 87.2% 85.7% 93.0%
mean (m) 3.52 2.20 -1.9e-3
Residual standard
deviation (m) 7.23 3.95 1.10
State evaluation
count 5 9 4
time (s) 36.6 65.3 32.9
Table 2. TLE analytical and numerical Orbit Determination results on multiple GPS satellite cases. Errors are given
at Orbit Determination epoch, which is also first guess TLE time. Green colored lines correspond to cases with posi-
tion accuracy enhancement compared to the initial TLE. In red, Orbit Determination does not lead to a better position
Figure 1. SGP4 and SDP4 analytical propagation force models. SGP4 is to be used when object orbital period is lower
than 225 minutes. SDP4 should be applied either.
Figure 2. GPS II R-M 6 ground track during measurement arc. Satellite goes from West to East. The arc starts on the
13th of February 2016 at 08:30:00 UTC and lasts 5h30min. Ground stations used for the Orbit Determination are also
Figure 3. Measurement residuals after numerical (at the top) and analytical TLE Orbit Determination (both at the middle
and the bottom). A GNSS satellite was considered along with related pseudo range measurements.
Analytical Orbit Determination was perform twice. Firstly estimating only orbital parameters (at the middle). Then, the
B* model parameter was also estimated (at the bottom).
Stations: BADG in blue, BSHM in red, POLV in green, SUTH in yellow, TLSE in light blue, ZECK in purple.
Black vertical line is Orbite Determination first guess epoch.
Figure 4. Estimated state Root Mean Square error along Orbit Determination iteration.
In yellow, numerical Orbit Determination
In blue, TLE Orbit Determination without B* estimation
In red, TLE Orbit Determination with B* estimation
Figure 5. SDP4 analytical propagation of GPS II R-M 6 original and determined TLEs. First guess epoch is 2016/02/13
09:44:09 UTC. Reference orbit is ISG SP3 ephemeris product for the 13th of March 2016.
In yellow, original TLE
In blue, determined TLE without B* estimation
In red, determined TLE with B* estimation
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... In addition, we use the automatic differentiation technique to calculate the gradient of the propagator, i.e. Eq. (8) (Paulet and Cazabonne, 2021). The observation function, Eq. (9) and its gradient are analytically derived. ...
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A simulation has been developed to examine how orbit determination accuracy, size of avoidance maneuvers, encounter geometry and warning time affect the probability of collision. The simulation shows that reasonably sized collision avoidance maneuvers are effective in reducing the probability of collision only if accurate orbit determination data is available for both the active satellite and the threat. Also, a method to determine the maneuvers required to reduce the probability of collision to a desired value is presented.
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In order to explore the precision and accuracy of International GNSS Service (IGS) orbits, we difference geocentric satellite positions midway between successive daily Final orbits for the period starting 5 November 2006, when the IGS switched its method of antenna calibration, through 31 December 2007. This yields a time series of orbit repeatabilities analogous to the classical geodetic test for position determinations. If we compare our average positional discontinuities to the official IGS accuracy codes, root-sum-squared (RSS) for each pair of days, we find the discontinuities are not well correlated with the predicted performance values. If instead the IGS weighted root-mean-square (WRMS) values from the Final combination long-arc analyses are taken as the measure of IGS accuracy, we find the position differences and long-arc values are correlated, but the long-arc values are exaggerated, particularly around eclipses, despite the fact that our day-boundary position differences apply to a single epoch each day and the long-arc analyses consider variations over a week. Our method is not well suited to probe the extent to which systematic effects dominate over random orbit errors, as indicated by satellite laser ranging residuals, but eclipsing satellites often display the most problematic behavior. A better metric than the current IGS orbit accuracy codes would probably be one based on the orbit discontinuities between successive days.
This book presents fundmentals of orbit determination--from weighted least squares approaches (Gauss) to todays high-speed computer algorithms that provide accuracy within a few centimeters. Numerous examples and problems are provided to enhance readers understanding of the material. * Covers such topics as coordinate and time systems, square root filters, process noise techniques, and the use of fictitious parameters for absorbing un-modeled and incorrectly modeled forces acting on a satellite. * Examples and exercises serve to illustrate the principles throughout each chapter. * Detailed solutions to end-of-chapter exercises available to instructors.
Traditional empirical thermospheric density models are widely used in orbit determination and prediction of low-Earth satellites and debris objects Unfortunately these models often exhibit large density errors These translate into orbit errors adversely affecting applications such as re-entry predictions and conjunction event analyses The extensive database of Two-Line Element TLE orbit data contains a wealth of current and historical information on satellite drag This data is available at a sufficiently high spatial and temporal resolution to allow a calibration of existing neutral density models in low degree and order spherical harmonics at at least two altitude levels and with a temporal resolution and latency of around one day We are therefore converting new TLE data for selected objects to satellite drag data automatically on a daily basis taking special care of data editing The resulting drag data is then used in a daily adjustment of calibration parameters which modify the output of an existing empirical density model Using this method the model accuracy can be increased beyond the limits imposed by the traditional use of solar and geomagnetic activity proxies
For precise real time or near real time differential GPS positioning in a wide or global area, precise GPS orbits or, alternatively, precise orbital corrections with respect to a reference orbit, such as GPS broadcast ephemerides, must be used. This work tests orbit interpolation methods, in order to represent the GPS orbits and orbital corrections accurately and efficiently for these and other GPS applications. For precise GPS orbits given in the SP3 format at the 15 min interval, numerical tests were conducted using Lagrange and Chebyshev as well as trigonometric polynomial functions. The results have demonstrated that the 19- or 20-term trigonometric function is apparently the most efficient interpolator for a 12 h GPS orbital arc, achieving 1 cm level 3D interpolation accuracy that can meet the requirements of most precise applications. The test results also demonstrated that the 9-term trigonometric function always yields optimal interpolation for a 2 h GPS orbit arc, in terms of interpolation errors, compared to the results when using a different number of terms for the same function or one of the other tested polynomial functions. This is evident from the minimal performance degradation when using the 9-term trigonometric function to interpolate near or at the end of a data interval. By limiting interpolation to the center 15 min to 1.5 h of a 2 h orbit arc, thereby eliminating the need to interpolate near the ends of that interval, users can opt for more terms (11 and 13) or different interpolators to further improve interpolation accuracy. When interpolating the orbital corrections with respect to the GPS broadcast ephemeris, all the tested interpolation functions of 3- to 9-term yield the same suitably accurate results. Therefore, a 3- to 5-term trigonometric function is arguably sufficiently accurate and more efficient for GPS orbital correction messaging in wide area and real time positioning.