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OPEN SOURCE ORBIT DETERMINATION WITH SEMI-ANALYTICAL THEORY
Bryan Cazabonne∗, Luc Maisonobe†
CS-SI, 5 rue Brindejonc des Moulinais, 31506 Toulouse CEDEX 5, FRANCE
ABSTRACT
Space objects catalog maintenance demands an accurate and
fast Orbit Determination (OD) process to cope with the ever
increasing number of observed space objects. The develop-
ment of new methods, that answer the two previous problems,
becomes essential.
Presented as an alternative to numerical and analytical
methods, the Draper Semi-analytical Satellite Theory (DSST)
is an orbit propagator based on a semi-analytical theory al-
lowing to preserve the accuracy of a numerical method while
providing the speed of an analytical method. This propaga-
tor allows computing the mean elements and the short-period
effects separately. We reproduced this architecture at the OD
process level in order to be able to return, as desired, the mean
elements or the osculating elements. Two major use cases are
thus possible: fast OD for big space objects catalog mainte-
nance and mean elements OD for station keeping needs.
This paper presents the different steps of development of
the DSST-OD included in the Orekit open-source library [1].
Integrating an orbit propagator into an OD process can be a
difficult process. Computing and validating derivatives is a
critical step, especially with the DSST whose equations are
very complex. To cope with this constraint, we used the au-
tomatic differentiation technique. Automatic differentiation
has been developed as a mathematical tool to avoid the calcu-
lations of the derivatives of long equations. This is equivalent
to calculating the derivatives by applying chain rule without
expressing the analytical formulas. Thus, automatic differen-
tiation allows a simpler computation of the derivatives and a
simpler validation. Automatic differentiation is also used in
Orekit for the propagation of the uncertainties using the Tay-
lor algebra.
Existing OD applications based on semi-analytical theo-
ries calculate only the derivatives of the mean elements. How-
ever, for higher accuracy or if the force models require further
development, adding short-period derivatives improves the re-
sults. Therefore, our study implemented the full contribution
of the short-period derivatives, for all the force models, in
the OD process. Nevertheless, it is still possible to choose
between using the mean elements or the osculating elements
derivatives for the OD.
∗bryan.cazabonne@c-s.fr
†luc.maisonobe@c-s.fr
This paper will present how the Jacobians of the mean
rates and the short-periodic terms are calculated by automatic
differentiation into the DSST-specific force models. It will
also present the computation of the state transition matrices
during propagation. The performance of the DSST-OD is
demonstrated under Lageos2 and GPS Orbit Determination
conditions.
Index Terms—Orbit Determination, Semi-analytical
theory, Open-source, DSST, Automatic differentiation, Orekit.
1. INTRODUCTION
Our needs about the space sector are always evolving. It is
difficult to imagine our everyday lives without the services
provided by satellites like telecommunications or navigation.
However, there are some counterparts to this globalization
of satellite-based systems. Between the first orbital launch
of Sputnik on 4 October 1957 and 1 January 2016 there
have been over 5,160 launches and approximately 200 known
satellite break-ups [2]. Therefore, a particular interest must
be given to the impacts of the increase in the number of space
objects. Especially with the issue of space debris. Wiede-
mann et al [3] estimate that the number of space debris with
a size of 1 cm or more have exceeded 700,000. According to
a NASA study [4], the number of objects in orbit with a size
of 10cm or more is greater than 17,000.
The challenge is then to track and catalog these objects
in order to control the risk of collision and thus preserve the
physical integrity of the satellites and their ability to fulfill
their missions. Orbit Determination (OD) and space objects
catalogs become indispensable tools. OD is used to estimate
the observations on the position and the velocity of an object
in orbit from an estimation algorithm and an orbit propagator.
Setty et al [2] highlighted that the US Joint Space Operations
Center (JSpOC) performed hundreds of thousands of OD per
day to maintain their space objects catalog. This means that
the OD process must be both fast and accurate to cope with
the volume of data to be analyzed.
Three types of orbit propagators exist. First, the analytical
orbit propagator. It has the advantage of being fast but it has
limited accuracy. The second type of orbit propagator is the
numerical. At the opposite of the analytical propagator, it
presents the advantage of being highly accurate. However, its
computing speed does not answer the problem of the volume
of data to be analyzed. The last type of orbit propagator is
the semi-analytical. It is a mix between the numerical orbit
propagator and the analytical. It has the advantages of both,
the accuracy of a numerical propagator and the computing
speed of an analytical one. Therefore, it is natural to pay close
attention to this type of propagator to cope with the important
number of orbit propagations to perform each day to maintain
the space objects catalogs.
The Draper Semi-analytical Satellite Theory (DSST) is an
orbit propagator based on a semi-analytical theory. It is an
interesting tool for a fast and accurate orbit propagation, close
to current needs. Its development started in the mid-1970s at
the Computer Sciences Corporation of Maryland by a team
led by Paul J. Cefola [5].
The DSST is present in the Orekit open-source library [1].
It is implemented for propagating the states based on classical
real numbers (i.e. double precision numbers), limiting its use.
Indeed, the real numbers used for the classical computations
represent only a scalar value, generally one of the components
of a state vector. It is possible, without altering the already
complex structure of the algorithms (i.e. without making it
even more complex and thus limiting the risk of error), to
perform computations on extensions of the real numbers in
order to add additional information to the scalar values of the
state components, typically partial derivatives. This thanks
to the automatic differentiation. The aim our study is to use
the automatic differentiation in order to extend the use of the
Orekit implementation of the DSST. Specifically, to use it in
OD processes.
The structure of this paper is as follows. In section 2,
we present the theoretical elements about the DSST that are
useful for the paper understanding. In section 3, we discuss
the development of the DSST-OD. Section 4 gives the current
numerical results obtained by the DSST-OD. Lageos 2 and
GPS OD are performed. Conclusions and Future Work end
the paper.
2. THE DSST
2.1. The equinoctial elements
One of the peculiarity of the DSST is the orbital parameters
formalism. They are expressed in equinoctial coordinates. If
a,e,i,Ω,ω, and Mdesignate the conventional Keplerian
elements set, the equinoctial elements are given by:
a=a
h=esin(ω+IΩ)
k=ecos(ω+IΩ)
p= [tan( i
2)]Isin Ω
q= [tan( i
2)]Icos Ω
λ=M+ω+IΩ
(1)
The Ielement is called the retrograde factor and has two
possible values:
I=+1 for the direct equinoctial elements
−1for the retrograde equinoctial elements (2)
hand krepresent (in the orbital frame) the components
of the eccentricity vector, pand qare the components of the
inclination vector and λis the phased angle. The equinoctial
elements are a rewrite of the conventional Keplerian orbital
parameters. They have the advantage of avoiding singularities
when the eccentricity or inclination of the orbit tend towards
zero, contrary to the classical Keplerian parameters [6, 7].
2.2. DSST orbital perturbations
In Orekit, five orbital perturbations are implemented for the
DSST: the Atmospheric Drag, the Third Body Attraction, the
Zonal and Tesseral harmonics of the Earth’s potential and the
Solar Radiation Pressure (SRP). These perturbations cause
the variation of the orbital element illustrated on Figure 1.
Fig. 1. Variation of an orbital parameter due to the orbital
perturbations [8].
The linear variation of the element is called the secular
variation. The short periods represent the periodic variations
of the element whose period is lower than the orbital period.
Finally, the long periods variations represent the periodic vari-
ations whose period is greater than the orbital period. All
these variations are used in the mathematical model of the
DSST. They are all implemented in the DSST-specific force
models on Orekit.
2.3. Mathematical model of the DSST
The DSST is based on a very advanced mathematical model.
This complexity makes it a powerful tool. Setty et al exposed
the mathematical model of the DSST [9]. This model can be
summarized by the following equation:
Yi(t) = Yi(t) + PN
j=0 kij ηij (t)i = 1,2,3,4,5,6 (3)
Where,
•Y: [a, h, k, p, q, λ] Equinoctial elements.
•¯
Y: [¯a, ¯
h, ¯
k, ¯p, ¯q, ¯
λ] Mean equinoctial elements.
•PN
j=0 kij ηij (t): Short-periodic terms.
•kij : Multiplying factor.
•N: Degree of precision for short-periodic computation.
•ηij (t): Short-periodic functions.
Within the DSST theory, the variation of an orbital ele-
ment is expressed by the sum of the mean elements and the
short-periodic terms. Mean elements are computed numer-
ically while short-periodic terms are computed analytically
and that for all the DSST-specific force models.
On equation (3) the main contribution comes from the
mean elements. In practice, during the computation of the
partial derivatives, the main contribution also comes from
the derivatives of the mean elements. In that respect, for
many OD applications, computing the partial derivatives of
the mean elements can be sufficient. However, for higher
accuracy or if the force models require further development,
adding short-periodic derivatives improves the results. Our
study implemented both contributions in the OD process,
while leaving the choice to the user to consider only the
contribution of the mean elements derivatives or the total
contribution by adding the short-periodic derivatives.
3. THE SEMI-ANALYTICAL ORBIT
DETERMINATION
3.1. DSST-specific force models configuration
3.1.1. Automatic differentiation
Automatic differentiation has been developed as a mathemat-
ical tool to avoid the computation of the derivatives of long
equations. All the theoretical elements about this theory are
presented in Kalman’s paper [10].
Let Yibe an orbital element. Automatic differentiation
allows the access to all the useful derivatives of Yiwithout
having to find their analytical expressions. This step is very
important when we want to establish an OD process. Indeed,
the computation of the derivatives is a mandatory step. The
vector resulting from using automatic differentiation on equa-
tions of motion is the following one:
[Yi∂Yi/∂Y1∂Yi/∂Y2· · ·
∂Yi/∂Y6∂Yi/∂P1· · · ∂Yi/∂Pn](4)
Where,
•P: The force model parameters (drag coefficient, etc).
•n: The number of force model parameters taken into
account for the OD.
To implement an OD process with the DSST, two types of
derivatives are used. First, the derivatives with respect to the
orbital parameters. Then, the derivatives with respect to the
force model parameters.
3.1.2. Development
In each DSST-specific force model implemented in Orekit
there is a method that allows the computation of the mean
elements rates. Let ˙
Ybe the vector containing these terms.
˙
Y= [ ˙a, ˙
h, ˙
k, ˙p, ˙q, ˙
λ](5)
This method was implemented to provide the vector on
equation (5) for the states based on classical real numbers.
In order to establish a DSST-OD, this function has been also
implemented to provide the Jacobians of the mean elements
rates by automatic differentiation. The resulting element is
then a matrix having the following form:
˙
Y1∂Y1˙
Y1∂Y2˙
Y1· · · ∂Y6˙
Y1∂P1˙
Y1· · · ∂Pn˙
Y1
˙
Y2∂Y1˙
Y2∂Y2˙
Y2· · · ∂Y6˙
Y2∂P1˙
Y2· · · ∂Pn˙
Y2
.
.
..
.
..
.
.....
.
..
.
.....
.
.
˙
Y6∂Y1˙
Y6∂Y2˙
Y6· · · ∂Y6˙
Y6∂P1˙
Y6· · · ∂Pn˙
Y6
(6)
This matrix can be summarized by:
˙
Y∂˙
Y
∂Y
∂˙
Y
∂P (7)
Where,
•∂˙
Y
∂Y : Jacobian of the mean elements rates with respect
to the orbital parameters.
•∂˙
Y
∂P : Jacobian of the mean elements rates with respect
to the force model parameters.
These two Jacobians are used for the computation of the
state transition matrices thanks to the variational equations.
3.2. The variational equations
3.2.1. The state transition matrices
The state transition matrices are used by the OD algorithm
to estimate the new parameters (force models and orbitals).
Consequently, they are of great importance. Computation and
validation of the state transition matrices are significant steps
that should not be neglected.
These matrices are given by equations (8) and (9):
∂Y
∂Y0
=
∂Y1
∂Y01
∂Y1
∂Y02
· · · ∂Y1
∂Y06
∂Y2
∂Y01
∂Y2
∂Y02
· · · ∂Y2
∂Y06
.
.
..
.
.....
.
.
∂Y6
∂Y01
∂Y6
∂Y02
· · · ∂Y6
∂Y06
(8)
∂Y
∂P =
∂Y1
∂P1
∂Y1
∂P2
· · · ∂Y1
∂Pn
∂Y2
∂P1
∂Y2
∂P2
· · · ∂Y2
∂Pn
.
.
..
.
.....
.
.
∂Y6
∂P1
∂Y6
∂P2
· · · ∂Y6
∂Pn
(9)
In order to obtain these matrices, we performed four steps
of development:
1. Extract the Jacobians ∂˙
Y
∂Y and ∂˙
Y
∂P from the matrix on
equation (7).
2. Extract the Jacobians ∂Y
∂Y0and ∂ Y
∂P from the additional
state initialized by the user at the creation of the orbit
propagator.
3. Apply the variational equations. These equations are
a mathematical tool for expressing the variations of a
parameter according to the initial state. They are given
by:
d∂Y
∂Y0
dt =∂˙
Y
∂Y ×∂ Y
∂Y0
d(∂Y
∂P )
dt =∂˙
Y
∂Y ×∂ Y
∂P +∂˙
Y
∂P
(10)
4. Add these differential equations to the set of equations
of motion. They will be integrated simultaneously by
the numerical integrator, benefiting from the additional
equations mechanism of the underlying mathematical
library.
After the numerical integration, a particular attention has
been paid to the validation of the computation of the state
transition matrices.
3.2.2. Validation of the state transition matrices
Validation of the state transition matrices computation, thanks
to the variational equations, was done by computing the same
matrices by finite differences. The goal of the validation was
to compare if the values were identical. Three tests have been
performed.
1. A first test to validate the computation of the matrix ∂ Y
∂P
computing only the derivatives of the central attraction
coefficient.
2. A second test to validate the computation of the matrix
∂Y
∂P computing only the drag coefficient derivatives.
3. A third test to validate the computation of the matrix
∂Y
∂Y0without computing any force model parameter
derivative.
Table 1 gives the force models configuration for each test.
The propagation time was 30 minutes for the three tests.
Table 1. Configuration of the force models for state transition
matrices validation.
Test Tesseral Zonal Drag SRP Moon
1X X
2X X
3X
At first, these tests showed that the Newtonian attraction
derivatives were not taken into account in the computation
of the state transition matrices. They also revealed that the
some dependencies to the central attraction coefficient were
implicit and therefore not differentiated.
Therefore, we added a new force model in order to take
into account the Newtonian attraction during the computa-
tion of the state transition matrices. We also implemented
the DSST-specific force models to have the central attraction
coefficient appear explicitly as a force model parameter.
After these improvements, we obtained consistent state
transition matrices.
3.3. Short-periodic terms derivatives
3.3.1. Development
In addition to the mean elements derivatives, we added the
contribution of the short-periodic terms derivatives into the
OD process. We paid close attention to retain the possibility
to choose between using the mean elements or the osculating
elements derivatives for the OD. This by computing the mean
elements and the short-periodic effects separately.
As for the mean elements rates, there is a method allowing
the computation of the short-periodic terms. This method is
implemented in each DSST-specific force model of Orekit. In
order to add the contribution of the derivatives of these terms
on the DSST-OD, we implemented this method to provide the
Jacobians of the short-periodic terms by automatic differenti-
ation.
The last step consisted in adding the contribution of the
short-periodic terms derivatives to the state transition matrices
(In the case where the user wants to add them). As we said
before, mean elements are computed numerically while short
periodic terms are computed analytically. In that respect, we
added the contribution of the short-periodic terms derivatives
after the numerical integration of the mean elements rates.
3.3.2. Validation
The validation has been performed in the same way as
part 3.2.2. We computed the state transition matrices, with
the contribution of the short-periodic derivatives, following
the three cases of test of Table 1. We again computed the state
transition matrices by finite differences in order to compare
the values obtained. The similarity between both compu-
tations showed that our computation of the state transition
matrices gives correct results.
3.4. Orbit Determination tools
Orekit provides two OD algorithms, which are compatible
with the DSST. The first one is a Batch Least Squares algo-
rithm and the other one is a Kalman Filter. The architecture as
well as the operating principle of these algorithms in Orekit,
are exposed by Maisonobe et al [11].
We paid special attention on the development of the OD
process that uses the Batch Least Squares algorithm to make it
compatible with the DSST. Work is under way to develop also
the DSST-OD with the Kalman Filter. In the results section,
the accuracy and the computation time of both methods are
exposed.
4. RESULTS
4.1. Orbit Determination conditions
4.1.1. Computer characteristics
The tests were performed on a 3.20 GHz Intel Core i5-3470
laptop with a 8 GB RAM.
4.1.2. Lageos2 and GPS force models
The first step was to define the force models used for each test
case. They were adapted from the ones used for the validation
of the OD with the numerical propagator in Orekit. It is not
yet possible, for instance, to take into account the relativity
force model for the GPS OD with the Orekit implementation
of the DSST. However, this force is negligible with respect to
the other force models used.
Table 2 gives the configuration of the force models for
each DSST-OD.
Table 2. Configuration of the force models used for the DSST
Orbit Determination.
OD Tesseral Zonal Drag SRP Moon Sun
Lageos2 X X
GPS X
OD Geo-potential [degree ×order]
Lageos2 8×8
GPS 20×20
4.1.3. Integration step
The second step was to define the integration step used for the
DSST-OD. The DSST has a significant advantage compared
to the numerical propagator in terms of computation time. For
the same level of accuracy, the numerical propagator requires
an integration step around one hundredth of the orbital period
to achieve a correct OD while the DSST needs an integration
step of the order of several orbital periods. This because the
elements computed numerically by the DSST are the mean
elements. Typically, a step equal to one day makes it possible
to obtain satisfactory results [7].
In Orekit, the user has the possibility to not enter directly
the integration step, but instead an interval and a positional
tolerance. The integrator chooses the optimal intergration
step (i.e. the fastest) that fulfills the tolerance requirements
in position imposed by the user. This method is used for both
DSST-OD tests. Table 3 gives the values of the minimum step
and the maximum step used for the DSST-OD tests.
Table 3. Integration steps used for the DSST-OD tests (first
line). These values are used for both Lageos2 and GPS test
cases and for both Batch Least Squares and Kalman Filter
algorithms. The second line gives the values used for the
Numerical-OD tests, with the same configurations.
Minimum step (s) Maximum step (s) Tolerance (m)
6000 86400 10
0.001 300 10
4.1.4. Test cases for the short-periodic terms
The derivatives of the short-periodic terms were not added all
at the same time. We gradually added them to highlight the
main contributions to improving the OD accuracy (Table 4).
For each test, we considered the Lageos2 OD.
Table 4. Lageos2 test cases for highlighting the contribution
of the short-periodic derivatives. Case 1 considers only the
mean elements derivatives for the OD. Case 2 considers the
mean elements derivatives and the short-periodic derivatives
of the Zonal harmonics in the OD process. Case 3 adds the
contribution the Tesseral short-periodic derivatives. Case 4
considers all the derivatives.
Case Zonal Tesseral Third body
1X X X
2X X
3X
4
4.2. Batch Least Squares Orbit Determination
We first performed the DSST-OD with the Batch Least
Squares algorithm present in Orekit. The results obtained
were compared with the Orekit numerical OD in order to
demonstrate the performance of the Orekit DSST-OD.
4.2.1. Lageos2 and GPS OD with mean elements derivatives
Figure 2 and 3 present the results obtained by the Lageos2
and the GPS DSST-OD considering only the mean elements
derivatives. These results were compared with those obtained
by the numerical OD for the same configurations. Our study
analyzed the accuracy of the DSST-OD by computing the
mean relative gap between the computed state vector and the
theoretical one. It is important to note that we considered all
the derivatives of the mean elements for all the force models
used in the simulations.
Fig. 2. Comparison between the numerical and the DSST-OD
in terms of computation time. Batch Least Squares case.
Fig. 3. Comparison between the numerical and the DSST-OD
in terms of accuracy. Batch Least Squares case.
For the computation time issue, Figure 2 demonstrates
that the DSST propagator is about fourteen times faster than
the numerical propagator for the Lageos2 OD. However, it
is about thirty seconds slower for the GPS OD. We explain
this anomaly by the difference in the choice of force models
used by the two Orbit Determinations. Specifically, taking
into account the Solar Radiation Pressure greatly increases
the computation time. We realized this thanks to a software
permitting more in-depth performance testing on GPS OD.
Figure 3 shows that the accuracy of the DSST-OD is
worse than the one with the numerical propagator. However,
the gap between the computed and the theoretical state vector
is low (∼10-4) . This demonstrates that the Orekit DSST-OD
gives satisfactory results. The accuracy difference between
the numerical propagator and the DSST is due to the fact that
the derivatives of the short-periodic terms are not considered
in this tests. Therefore, we added them to the Lageos2 OD.
4.2.2. Lageos2 OD with both types of derivatives
Figure 4 shows the impact of adding the short-periodic terms
derivatives to the Lageos2 OD. Several remarks can be made
thanks to this figure.
First, adding the short-periodic derivatives improves the
accuracy of the DSST-OD. Indeed, the addition of the Zonal
short-periodic derivatives enhances significantly the relative
gap between the theoretical and the computed state vector.
Furthermore, the computation time have not increased sig-
nificantly after adding these terms. Problems start with the
Tesseral contribution for the short-periodic derivatives. The
computation time was multiplied by eight without improving
the accuracy. We again performed more in-depth performance
testing to highlight the critical points (i.e. the ones that cause
the increase of the computation time) into the Tesseral short-
periodic derivatives computation.
Fig. 4. Gradual addition of the short-periodic derivatives to
the Lageos2 OD. Batch Least Squares case.
In conclusion, adding the Zonal short-periodic derivatives
to the mean elements derivatives is sufficient to ensure the
accuracy of the OD and maintain a respectable computation
time. Prospects for improvement in the accuracy and in the
computation time are discussed on section 6.
4.3. Kalman Filter Orbit Determination
In the same way as the Batch Least Squares algorithm, we
carried out the DSST-OD with the Kalman Filter present in
Orekit. Results were once again compared with the Orekit
numerical OD.
Figure 5 and 6 present the results obtained by the Lageos2
DSST-OD considering only the mean elements derivatives.
Fig. 5. Comparison between the numerical and the DSST-OD
in terms of computation time. Kalman Filter case.
Fig. 6. Comparison between the numerical and the DSST-OD
in terms of accuracy. Kalman Filter case.
Figure 7 shows the impact of adding the short-periodic
terms derivatives to the Lageos2 OD.
It is important to note that the integration of the DSST in
the Kalman Filter OD is under development. These results
are only a first approach and improvements are still needed.
However, it is possible to give a first interpretation of these
results.
The conclusions in terms of accuracy and computation
time are identical to section 4.2. At the moment, the results
Fig. 7. Gradual addition of the short-periodic derivatives to
the Lageos2 OD. Kalman Filter case.
obtained by the DSST-OD with the Batch Least Squares algo-
rithm are better than the ones with the Kalman Filter. Indeed,
computation time and accuracy are both better. Future work
on the DSST-OD with the Kalman Filter will maybe reverse
the current trend as show by ˇ
Segan’s paper [12].
5. CONCLUSION
In this paper, we have demonstrated the performance of the
Orekit implementation of the Draper Semi-analytical The-
ory under Lageos2 and GPS Orbit Determination conditions.
The results are encouraging, they highlighted the ability of
the DSST to perform Orbit Determinations with a Batch
Least Squares algorithm or a Kalman Filter. The results also
demonstrated that the user have the choice of using the mean
elements or the osculating elements derivatives for the Orbit
Determination. In addition, the results showed that, in some
cases, the DSST is able to perform the Orbit Determination
faster than the numerical propagator. Accuracy is better for
the numerical propagator but we obtained small relative gaps
with the Orekit implementation of the DSST.
All the work done in this paper will be published with
release 10 of Orekit. They are nevertheless already available
in a dedicated public branch of our source code management
system [13]. In conclusion, open-source and semi-analytical
Orbit Determination are not two opposing worlds !
6. FUTURE WORK
The authors propose to perform further implementation works
in the following directions:
•Improving the performance of the DSST-OD in terms
of computation time. This by optimizing the critical
points highlighted in the Solar Radiation Pressure and
the Tesseral force models.
•Improving the accuracy of the DSST-OD. When the
user wants to initialize the Zonal and the Tesseral force
models, it must enter ten coefficients related to the
short-periodic terms. However, we do not know the
perfect combinations for these coefficients. We just
know that they are sensible to the orbit type. Studies
about the values of these coefficients must be perform
in order to improve the accuracy of the DSST-OD.
For our tests, we put the maximum values allowed for
these coefficients. They correspond to the order and
the degree of the Geo-potential.
•Improving the results of the DSST-OD with the Kalman
Filter.
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