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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.

7. REFERENCES

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Diaz, “Attributes affecting the Accuracy of a Batch

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