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Towards Accurate Orbit Determination using Semi-analytical Satellite Theory

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Space agencies generally use numerical methods to meet their orbit determination needs. Due to the ever increasing number of space objects, the development of new orbit determination methods becomes essential. DSST is an orbit propagator based on a semi-analytical theory. It combines the accuracy of numerical propagation and the speed of analytical propagation. The paper presents an open-source DSST orbit determination application included in the Orekit library. Accuracy of the DSST orbit determination is demonstrated by comparison with a numerical method. Both the satellite's state vector estimation and the measurement residuals are used as comparison metrics.
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1
TOWARDS ACCURATE ORBIT DETERMINATION USING SEMI-
ANALYTICAL SATELLITE THEORY
Bryan Cazabonne
*
and Paul J. Cefola
Space agencies generally use numerical methods to meet their orbit
determination needs. Due to the ever increasing number of space objects, the
development of new orbit determination methods becomes essential. DSST is an
orbit propagator based on a semi-analytical theory. It combines the accuracy of
numerical propagation and the speed of analytical propagation. The paper
presents an open-source DSST orbit determination application included in the
Orekit library. Accuracy of the DSST orbit determination is demonstrated by
comparison with a numerical method. Both the satellite’s state vector estimation
and the measurement residuals are used as comparison metrics.
INTRODUCTION
Orbit Determination is a technique used to estimate the satellite’s state vector from its
observable measurements. The state vector may be position and velocity or an orbital element set.
It may include dynamical parameters such as the drag coefficient. Orbit determination is an
indispensable tool for many applications such as satellite station-keeping, space traffic
management including collision risk studies, and scientific analysis of data for space geodesy.
Fast and accurate orbit determination is mandatory for these applications. To cope with the ever-
increasing number of space objects, developing new orbit determination methods that address
both requirements is essential. Space agencies generally use the numerical method to meet their
orbit determination needs. The numerical method can be very precise with sufficient force
models, but it requires significant computation time. To get around the computation time issue,
analytical orbit determination methods are possible. Brouwer Theory is the basis of most of the
analytical orbit determination methods.
1
The NAVSPASUR PPT2 algorithm is one operational
implementation of Brouwer.
2
The USAF SGP4 theory, which is used to generate the NORAD
TLE, employs the Brouwer theory together with a power law model for the atmospheric density.
3
,
4
However, these operational analytical orbit determination methods are based on limited
dynamical models and may not meet accuracy requirements. These analytical orbit determination
methods generally aim at measurement intervals of a few days.
*
Bryan Cazabonne is Spaceflight Mechanics Engineer at CS GROUP, 6 Rue Brindejonc des Moulinais, Toulouse,
France, email: bryan.cazabonne@csgroup.eu.
Paul J. Cefola is Research Scientist, Department of Mechanical & Aerospace Engineering, University at Buffalo
(SUNY), Amherst, NY, USA, email: paulcefo@buffalo.edu, paul.cefola@gmail.com. Fellow AAS. Also Consultant in
Aerospace Systems, Spaceflight Mechanics, and Astrodynamics, Vineyard Haven, MA, USA.
(Preprint) AAS 21-309
2
Presented as alternatives to numerical and analytical methods, semi-analytical techniques
combine the accuracy of numerical propagation and the characteristic speed of analytical
propagation. Many semi-analytical techniques exist. One early semi-analytical orbit
determination method is the ROAD algorithm due to Wagner.
5
In ROAD, the dynamical model is
the mean element equations of motion. ROAD assumes short arc mean elements as the
‘observations’. In 1977 the Draper Laboratory proposed the extension of its GTDS semi-
analytical orbit propagator to include detailed short period motion models and improved partial
derivatives models.
6
This paper focuses on the Draper Semi-analytical Satellite Theory (DSST),
which is flexible, complete and applicable to all orbit types.
7
,
8
Therefore, having an orbit
determination solution using DSST is interesting, especially if the solution is open-source. There
are different implementations of DSST orbit determination.
9
,
10
,
11
In 2018, a first open-source
implementation has been included in the Orekit space flight library.
12
However, the results were
not satisfactory in terms of satellite’s state vector accuracy. Yurasov and Nazarenko also
developed a semi-analytical method comparable to the DSST: the Universal Semi-analytical
Method (USM).
13
,
14
,
15
However, we won’t discuss it in the paper.
DSST divides the computation of the osculating orbital elements into two contributions: the
mean orbital elements and the short-periodic terms. Both models are developed in the equinoctial
orbital elements via the Method of Averaging. Both the mean orbital elements and short period
motions are computed using a combination of analytical and numerical techniques. There is also a
semi-analytical formulation for the partial derivatives used in the DSST orbit determination.
16
Accurate computation of partial derivatives is an important step for a precise orbit determination.
It becomes important for short determination arc and with very accurate data. However,
computing derivatives is also a critical step, especially with DSST whose equations are
complicated. Orekit DSST uses automatic differentiation to avoid the calculation of the
derivatives of long equations. It allows calculating derivatives to any order and with any number
of parameters.
17
,
18
Automatic differentiation is equivalent to calculating derivatives by applying
the chain rule without expressing the analytical formulas.
The roadmap of the paper will be to first introduce the Orekit DSST orbit determination
process. This includes a presentation of the DSST development in Orekit, and an introduction to
the batch least squares orbit determination. Furthermore, a particular attention will be paid to the
calculation and validation of the Orekit DSST state transition matrix. Finally, results to
demonstrate the accuracy of the Orekit DSST orbit determination will be presented. Conclusions
and Future Work end the paper.
OVERVIEW OF THE OREKIT DSST ORBIT DETERMINATION PROCESS
Orekit is an open-source space flight dynamics library.
19
It is written in Java and provides low
level elements for the development of flight dynamics applications. Orekit started in 2002 as an
in-house closed project developed by CS GROUP. Since 2008, Orekit is distributed under the
open-source Apache License version 2.0.
20
Since Orekit is distributed under an open-source li-
cense, its popularity continuously increased. Nowadays, Orekit is worldwide used, both by space
industry and academic institutions. Countries with known Orekit users are shown in Figure 1.
Draper Semi-analytical Satellite Theory
DSST is a mean element satellite theory. Its development started in the mid-1970s at the
Computer Sciences Corporation and the Charles Stark Draper Laboratory by a team lead by Paul
J. Cefola. DSST is a semi-analytical theory expressed in non-singular equinoctial
elements. The equinoctial elements are formulated using the classical Keplerian
elements.
21
3
DSST divides the computation of the osculating orbital elements into two contributions: the
mean orbital elements and the short-periodic terms. Both models are developed in the equinoctial
orbital elements via the Method of Averaging. The transformation from mean elements to
osculating element is calculated using Equation (1)
 
  with i = 1, 2, ..., 6
(1)
where
 short-period function, 2 periodic
mean elements
 
Orekit DSST development started in 2011 with the implementation of the equation of motion
for the mean elements. The short periodic contribution was added in 2014. Finally, the first Orekit
DSST orbit determination was released in 2018. During its development, Orekit DSST has been
conscientiously validated against the original FORTRAN version.
22
,
23
,
24
The current version of
Orekit DSST provides a lot of features which are summarized in Table 1.
Table 1. Orekit DSST features
Feature
Description
Force models
Zonal and tesseral harmonics higher than 50 × 50
Third body attraction for the main bodies of the solar system
Solar radiation pressure
Atmospheric drag
Implementation of both the mean elements contribution and the
first order short periodic terms for all force models
Orbit types
Applicable to all orbit types
Numerical integrator
Possibility to use either a fixed step or a variable step integrator
Orbit determination
Possibility to estimate dynamical parameters (e.g. drag coeffi-
cient)
Partial derivatives computed for both mean elements and short-
periodic terms using automatic differentiation
Batch Least Squares Orbit Determination
The Batch Least Squares algorithm is the most widely used orbit determination technique for
space objects catalogue maintenance. For a given satellite initial state and for an available
observations arc, the Batch Least Squares algorithm provides the best estimate of the satellite
state such as

(2)
Calculating the Equation (2) is carried out in an iterative process by solving the non-linear
Equation (3)
25
,
26

(3)
4
where partial derivatives matrix
  weighting matrix
residual vector
The partial derivatives matrix can be expressed by the product of the observation partials
matrix, and the state transition matrix
 
(4)
where
 



(5)
(6)
where  is an observed measurement at an arbitrary epoch . Figure 2 shows the calling
hierarchy of the Orekit DSST Batch Least Squares orbit determination. The figure presents the
different steps of calculation and the integration of the previous equations in the process.
OREKIT DSST STATE TRANSITION MATRIX
One of the most important steps during an orbit determination is the calculation of the state
transition matrix. It can be one of the major resource consuming parts of an orbit determination
application. Therefore, a well-conceived implementation and a strong validation are both
mandatory for an accurate orbit determination.
Mathematical Model
Andrew Green developed a semi-analytical theory for the partial derivatives of the perturbed
motion.
27
Green’s work is compatible with the semi-analytical model of the equation of motions
in DSST method. The state transition matrix in Equation (6) can be expressed by the product of
two other matrices, as in Equation (7)

  
(7)
where is the vector containing the osculating position and velocity of the satellite,  is
the vector containing the osculating equinoctial elements at an arbitrary epoch , and is the
Green’s matrix which is expressed by

 

(8)
In Equation (8), represents the vector containing the mean equinoctial elements at the
epoch time, and the vector containing the estimated dynamical parameters. Therefore, is a
6 × matrix, where denotes the number of estimated parameters (i.e. orbital and dynamic
parameters, measurement parameters are not considered in the matrix). The matrix can be
expanded as

(9)
5
where

  
 
  
 
(10)
The and matrices represent the partial derivatives of the short period motion. They can
be computed by direct application of Equation (10). The and matrices represent the partial
derivatives of the mean elements at arbitrary epoch with respect to the estimated parameters. The
calculation of and matrices is completed using Equations (11) and (12), also called the
variational equations
28
with
(11)
with
(12)
where,
  

 
 
(13)
(14)
In Equations (13) and (14), denote the equinoctial mean element rates. In Orekit library,
Equations (11) and (12) are integrated simultaneously with the set of equations of motion by the
numerical integrator. Furthermore, partial derivatives are not computed by finite differencing
method. Indeed, Orekit DSST uses the automatic differentiation technique to compute all the
necessary partial derivatives. More details about the automatic differentiation technique are given
in the next section.
Automatic Differentiation
Automatic differentiation can be summarized as follows
29
“In mathematics and computer algebra, automatic differentiation is a set
of techniques to evaluate the derivative of a function specified by a
computer program. Automatic differentiation exploits the fact that every
computer program, no matter how complicated, executes a sequence of
elementary arithmetic operations (addition, subtraction, multiplication,
division, etc.), elementary functions (exp, log, sin, cos, etc.) and control
flow statements. Automatic differentiation takes source code of a
function as input and produces source code of the derived function.”
Automatic differentiation has been developed as a mathematical tool to avoid the calculation
of the derivatives of long equations. It is equivalent to calculating derivatives by applying the
chain rule without expressing the analytical formula of the derivatives. For instance, if denotes
an orbital element, automatic differentiation allows the calculation of derivatives with respect
to any parameter (e.g. orbital and dynamic parameters) by using only the analytical expression of
. The results will be stored in a vector of which the first element will be the value of the
parameter, and the following elements its derivatives, as in Equation (15)
6





(15)
The calculation of the derivatives is accurate to the accuracy of the computer system. In other
words, it does not depend on the approximation of the analytical expression of the derivatives.
Therefore, automatic differentiation is used in Orekit DSST orbit determination to calculate
and matrices of Equations (13) and (14) at each integration step. Furthermore, automatic
differentiation is also used to calculate and matrices.
Automatic differentiation tools are available for several programming languages including
C/C++ and Fortran 77, as well as the java-based capability employed in the present work. The
C/C++ and Fortran 77 automatic differentiation tools can be used to improve the State Transition
Matrix capabilities of the C/C++ and Fortran 77 DSST Standalone orbit propagation programs.
Further, we should investigate the application of Automatic Differentiation to: (1) the
construction of the mean element equations of motion from the averaged disturbing potential, and
(2) the construction of the short period motion formulas from the short period generating
functions.
Testing of the DSST State Transition Matrix
The following approaches are used in testing the Orekit DSST partial derivatives calculation
using automatic differentiation:
Testing of the mean element state transition matrix () computed with automatic
differentiation via a comparison with a finite differences calculation (Cases 1 and 2).
Testing of the short period partial derivatives matrix ( computed with automatic
differentiation via a comparison with a finite differences calculation (Cases 3 and 4).
The finite differences calculation is based on the nine-point formula for the derivative at the
central point.
30
The epoch mean orbital elements used for the testing of the mean elements state transition
matrix are given in Table 2. They are identical to those used for GTDS state transition matrix
testing.16 The mean orbital element set is given in EME2000 coordinates.
Table 2. Epoch mean orbital elements (Cases 1 and 2)
Orbit element
Value
semi-major axis
6706966.2 meters
eccentricity
0.0010252154
inclination
87.266393 degrees
argument of perigee
94.431363 degrees
right ascension of the ascending node
64.668178 degrees
mean anomaly
105.69973 degrees
epoch (UTC)
2008-09-15T21:59:46.000
The epoch osculating position and velocity used for the testing of the short period state
transition matrix are given in Table 3. Again, they are identical to those used for GTDS state
transition matrix testing.16 The osculating position and velocity are given in EME2000
7
coordinates. An osculating to mean elements transformation is done before mean elements
integration.
Table 3. Epoch osculating position and velocity (Cases 3 and 4)
Coordinate
Value
X Position
-2595256.643 meters
Y Position
-5741664.984 meters
Z Position
-2321359.682 meters
X Velocity
1450.193597 meters per second
Y Velocity
2258.205121 meters per second
Z Velocity
-7221.683085 meters per second
epoch (UTC)
2008-09-15T21:59:46.000
Details about the test cases used to validate the Orekit DSST state transition matrix are given
in Table 4.
Table 4. Test cases for the validation of the DSST State Transition Matrix
Case
Geo-potential [degree × order]
3rd Body [Moon & Sun]
Drag
SRP
or
Figure
1
2 × 0
NO
NO
NO
Figure 3
2
2 × 0
YES
YES
YES
Figure 4
3
2 × 0
NO
NO
NO
Figure 5
4
2 × 0
YES
YES
YES
Figure 6
For each test case, a 10-day forward propagation is done. A classical Runge-Kutta integration
process is used for mean elements integration. The Harris-Priester model is used to calculate the
atmospheric density. The comparison between automatic differentiation and finite differences
calculations is performed each 15 minutes. The relative difference between both methods is
calculated. Because the addition of non-gravitational forces increases a lot the calculation time of
finite differences method, Cases 2 and 4 calculate the relative difference each 4 hours instead of
each 15 minutes. For each test case, the mean relative difference and the standard deviation are
calculated. Statistics are summarized in Table 5.
Table 5. Statistics on DSST State Transition Matrix Tests
Case
Mean relative difference
Standard deviation
1
1.9322884144335276E-07
7.218497449671177E-07
2
4.1091663072620243E-04
3.8998220658845433E-03
3
1.3422304521968056E-08
5.001927728157451E-07
4
1.5867047496066795E-08
1.883413880920726E-07
Figure 3 to Figure 6 and Table 5 highlight the closure between automatic differentiation and
finite differences methods for the calculation of Orekit DSST state transition matrix. Mean
relative difference is between 10-4 and 10-7 for the calculation of mean elements partial
derivatives and about 10-8 for the calculation of short period partial derivatives.
8
A larger mean relative difference is observed when non-gravitational forces are added in the
propagation (Case 2). Because the satellite orbit is low (i.e. perigee altitude is about 322
kilometers, and apogee altitude is about 335 kilometers), the atmospheric drag effect on the
satellite is significant. Mean elements partial derivatives for atmospheric drag effect are therefore
significant too. Finite differences accuracy is limited by the number of points used to compute the
partial derivatives. Because the dynamic in the Earth’s atmosphere is important, a large number
of points must be used to accurately compute the partial derivatives. However, it would increase
considerably the computation time. In order to have consistency between our test cases and to
keep a reasonable computation time, we used the same number of points for each test case.
Therefore, the accuracy of finite differences calculation is impacted when atmospheric drag is
added. Nevertheless, the mean relative difference between automatic differentiation and finite
differences methods remains low and acceptable for Case 2.
In conclusion, we can fully rely on automatic differentiation technique for the calculation of
the partial derivatives needed by Orekit DSST orbit determination.
OREKIT DSST ORBIT DETERMINATION RESULTS
Orekit DSST capabilities are demonstrated by comparing orbit determination results with the
Orekit numerical method under GPS orbit determination conditions. The estimation process is
based on a Batch Least Squares algorithm. Recursive filters can also be used for DSST orbit
determination.
31
,
32
,
33
,
34
However, there are not considered in this paper.
Testing of the Semianalytical Batch Least Squares
The GPS-07 satellite was chosen for demonstrating Orekit DSST orbit determination
performance on real data. The selection of this satellite was influenced by the availability of GPS
observations.
35
Both GPS pseudo-range and phase measurements can be used for orbit
determination. Because GPS phase modelling is not completed in Orekit, only pseudo-range
measurements are used in the orbit determination process. Observed measurements from a
network of five stations have been used, representing 4009 measurements with a fit interval of 6
hours. The orbit determination was carried out with 12x12 geo-potential terms, lunar-solar point
masses, and solar radiation pressure. GPS satellite altitude allows neglecting atmospheric effects
on the satellite orbit. The six equinoctial orbital elements and the satellite’s reflection coefficient
were estimated during the orbit determination process.
Figure 7 shows the measurement residuals obtained by the DSST orbit determination.
Measurement residuals are very close to those obtained with the numerical method as presented
in Figure 8. First, the shape of the residuals between the two methods is identical. The 1.5-hours
sinusoidal effect observed on the two figures is due to multipath effects affecting GPS pseudo-
range measurements.
36
That effect can be reduced by using smoothing algorithms such as the
Hatch Filter.
37
However, smoothing algorithms are not yet implemented in Orekit. Furthermore,
the two figures highlight that the minimum, the maximum, and the mean values of the residuals
are close and consistent between the two methods. That assumption is confirmed by the residual
statistics presented in Table 6.
Table 6. Statistics on pseudo-range measurement residuals
Method
Minimum value (meters)
Maximum value (meters)
Mean value (meters)
RMS
DSST
-2.905
2.557
~ 0
0.76
Numerical
-2.820
2.529
~ 0
0.71
9
Figure 9 displays the satellite’s position difference between DSST and numerical methods.
The position difference is calculated for each measurement epoch during the orbit determination
process. The minimum difference is equal to 1.8 meters and it is obtained for the orbit
determination epoch. Furthermore, the maximum and the mean differences are equals to 4.9
meters and 2.8 meters, respectively.
The results highlight that the Orekit DSST is able to calculate a satellite position close to the
numerical method during an orbit determination process. However, due to small differences
between the two models, the consistency is not absolute. These small differences are due to
unmodeled effects in Orekit DSST.
One important effect to consider for medium and high altitude satellites is the impact of the
weak-time dependent terms. Weak-time dependent terms are used to consider that a third-body is
not perfectly fixed during the averaging interval. Andrew Green developed a mathematical model
for the weak-time dependent terms where the osculating equations of motion for lunar-solar point
masses perturbation are functions of two periodic phase angles: the satellite mean longitude and
the third body mean longitude. Green highlighted the impact of using weak-time dependent terms
for medium altitude orbits.27 Srinivas Setty discussed the impact of these terms for geostationary
orbits. He also proposed a closed-form implementation adapted to near circular orbits (i.e.
eccentricity lower than 0.2).
38
Therefore, we can make the assumption that weak-time dependent
terms are also needed for GPS orbits. In order to confirm that assumption, additional tests were
performed.
Propagating the Estimated Orbits
The previous results highlighted unmodeled effects on the satellite’s position difference
between DSST and numerical methods. That effect is assumed to be the non-modelling of weak-
time dependent terms in Orekit DSST. In order to confirm this assumption, a 5-days forward
propagation of the estimated orbits (i.e. the estimated orbit of both propagation methods) is
performed. The configuration of the DSST and the numerical orbit propagators is the same as that
employed in the previous study, permitting a consistent interpretation of the results.
Figure 10 shows the equinoctial parameter differences between DSST and numerical methods
for the 5-days forward propagation of the estimated GPS orbits. The equinoctial parameter
differences have interesting order of magnitudes. The semi-major axis difference is between 10
meters and -20 meters, while the difference for the other orbital parameters is about 10-6. The
results show the ability of Orekit DSST to propagate equinoctial parameters close to the
numerical method for the 5-days forward orbit propagation. However, a 12-hours periodic effect
is observed on Figure 10. It is also visible on Figure 11 to Figure 13, representing the radial,
along-track, and cross-track differences between both methods. These differences are consistent
with those observed by Srinivas Setty for the geostationary orbit test case before applying weak-
time dependent terms in third body short-periodic motion.38 Therefore, we can conclude that
weak-time dependent terms are also necessary for GPS orbits. These terms must be considered in
order to perform accurate orbit determination of navigation satellites using DSST method. They
are also needed to improve the consistency between DSST and numerical methods.
CONCLUSION
Results demonstrate that the measurement residuals of the Orekit DSST orbit determination
are consistent with the numerical orbit determination. In addition, the comparison of both
methods shows the ability of Orekit DSST to calculate, at each measurement epoch in the
determination arc, a satellite position close to the numerical solution. Small differences are
nevertheless observed. They are due to unmodeled effects in the dynamic configuration of the
10
Orekit DSST. More details are given in the following section. Finally, the computation of the
state transition matrix using automatic differentiation is consistent with a very accurate finite
differencing method.
FUTURE WORK
There are several areas in which we intend to improve the capabilities of the Orekit DSST
orbit propagator. In particular, we would like to refine the physical models in Orekit DSST by
adding the weak-time-dependent lunar-solar short period motion. These terms can have a
significant impact for GNSS satellite orbits. Therefore, adding this contribution can reduce
significantly the periodic effect observed in the comparison of the Orekit DSST against the
numerical propagation. Furthermore, we would like to add the J2-squared terms for eccentric
orbits via quadrature method or via closed-from method.
39
,
40
Finally, it could be interesting to
formulate a semi-analytical model for empirical forces affecting satellite orbits. These forces are
used to account for the unmodeled forces which act on the satellites. Therefore, they are very
useful and widely used, especially for precise orbit determination applications.
We would like to improve the capabilities of Orekit DSST orbit determination by adding the
Extended Semianalytical Kalman Filter (ESKF). The ESKF reconciles the conflicting goal of the
DSST perturbation theory (i.e. large step size) and the Extended Kalman Filter (EKF) theory (i.e.
re-initialization at each measurement epoch).
Finally, we would like to extend the Orekit DSST orbit determination tests. Thanks to the
availability of laser ranging measurements, it could be possible to perform orbit determination
tests for several orbit types using the Orekit DSST. Furthermore, the Crustal Dynamics Data
Information System (CDDIS) is an interesting source of data for precise satellite orbits and
measurements. We propose to compare the results against a numerical solution to highlight the
performance of Orekit DSST orbit determination.
ACKNOWLEDGMENTS
Bryan Cazabonne would like to acknowledge Mr. Luc Maisonobe, Mr. Pascal Parraud , Mr.
Maxime Journot, and Mr. Thomas Paulet, all of CS GROUP, France. Discussions with them
provided a valuable help to improve the capabilities of Orekit DSST orbit determination.
Paul Cefola would like to acknowledge technical discussions with Prof. Juan Felix San Juan,
University of Rioja, Logrono, Spain, Mr. David Vallado, CSSI/AGI, Colorado Springs, Colorado,
Mr. William Robertson, Draper Laboratory, Cambridge, Massachusetts, Dr. Ronald J. Proulx,
Newton, Massachusetts, Dr. Srinivas Setty, Vyoma Gmbh, Munich, Germany, Mr. Zach Folcik,
MIT Lincoln Laboratory, Lexington, Massachusetts, Prof. Richard Linares, Department of
Aeronautics and Astronautics, MIT, Cambridge, Massachusetts, and Dr. Jim Schatzman,
Augustus Aerospace Company, Lone Tree, Colorado . Paul Cefola would also like to
acknowledge ongoing discussions with Mr. Kye Howell, Mr. Brian Athearn, and Ms. Prudence
Athearn Levy, all of Martha’s Vineyard, Massachusetts.
11
FIGURES
Figure 1. Countries with known Orekit users (representing in blue).
Figure 2. Orekit Batch Least Squares orbit determination principle.
12
Figure 3. Case 1: Relative differences between Automatic Differentiation and Finite Differences
for the calculation of mean elements partial derivatives with respect to the initial mean elements. In
grey are represented the partial derivatives with respect to
, in purple the partial derivatives with
respect to
, in blue the partial derivatives with respect to initial
, in yellow the partial derivatives
with respect to
, and in green the partial derivatives with respect to
.
13
Figure 4. Case 2: Relative differences between Automatic Differentiation and Finite Differences
for the calculation of mean elements partial derivatives with respect to the initial mean elements. In
grey are represented the partial derivatives with respect to
, in purple the partial derivatives with
respect to
, in blue the partial derivatives with respect to initial
, in yellow the partial derivatives
with respect to
, in green the partial derivatives with respect to
, and in red the partial
derivatives with respect to
.
14
Figure 5. Case 3: Relative differences between Automatic Differentiation and Finite Differences
for the calculation of the partial derivatives of the short periodic terms with respect to the epoch
mean elements. In grey are the partial derivatives with respect to
, in purple the partial derivatives
with respect to
, in blue the partial derivatives with respect to initial
, in yellow the partial
derivatives with respect to
, in green the partial derivatives with respect to
, and in red the partial
derivatives with respect to
.
15
Figure 6. Case 4: Relative differences between Automatic Differentiation and Finite Differences
for the calculation of the partial derivatives of the short periodic terms with respect to the epoch
mean elements. In grey are the partial derivatives with respect to
, in purple the partial derivatives
with respect to
, in blue the partial derivatives with respect to initial
, in yellow the partial
derivatives with respect to
, in green the partial derivatives with respect to
, and in red the partial
derivatives with respect to
.
16
Figure 7. Measurement residuals of the Orekit DSST orbit determination.
Figure 8. Measurement residuals of the Orekit numerical orbit determination.
Figure 9. Difference between DSST and numerical methods at each measurement epochs.
17
Figure 10. Equinoctial parameter differences between DSST and numerical methods for a five
days forward propagation of the estimated GPS orbits.
18
Figure 11. Radial difference between DSST and numerical methods for a five days propagation of
the estimated GPS orbits.
Figure 12. Along-track difference between DSST and numerical methods for a five days
propagation of the estimated GPS orbits.
Figure 13. Cross-track difference between DSST and numerical methods for a five days
propagation of the estimated GPS orbits.
19
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... 9,10,11 In 2021, a complete open-source implementation using a batch-least squares algorithm has been included in the Orekit space flight library. 12 During this study, the calculation of the state transition matrix based on automatic differentiation has been presented and strongly validated. The current study focuses on the extension of Orekit DSST orbit determination capabilities by adding a recursive filter theory, the Extended Semi-analytical Kalman Filter (ESKF). ...
... In DSST theory, the equations of motion for the mean equinoctial elements can be written as in Equation (11) and (12). ...
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The paper presents an open-source orbit determination application based on the Draper Semi-analytical Satellite Theory (DSST) and a recursive filter, the Extended Semi-analytical Kalman Filter (ESKF). The ESKF reconciles the conflicting goal of the DSST perturbation theory (i.e., large step size) and the Extended Kalman Filter (EKF) theory (i.e., re initialization at each measurement epoch). Validation of the Orekit ESKF is demonstrated using simulated data. Both the satellite’s state vector estimation and the measurement residuals are used as comparison metrics.
... 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 [5] [6]. Moreover, automatic differentiation is used to create SGP4/SDP4 compliant TLEs from a state vector. ...
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Thesis
This research work focuses on the selection, development and testing of orbit prediction and orbit determination methods: the selection is carried out for the purpose of space object catalogue maintenance. Approximations in the selected propagator method and uncertainties in the selected force models are studied. Methods for uncertainty propagation are investigated. The first objective of this study is to find a propagation method that can be used for several cataloguing tasks, and will offer a favourable trade-off between accuracy and computational effort. The selected propagation method should be a computationally lean alternative to the more common numerical orbit propagation techniques. As a second objective, efforts were concentrated in characterising the model uncertainties and in formulating a method to capture these uncertainties within the propagated covariances.