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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

REFERENCES

1

Brouwer D. Solution of Problem of Artificial Satellite Theory without Drag, Astronomical J., Vol. 64, No. 1274, pp.

378-397, November 1959.

2

Schumacher, P. W. Jr., and Glover, R. A., Analytical Orbit Model for the U.S. Naval Space Surveillance: An

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August 14-17, 1995.

3

Lane M. H., and Cranford K. H., An Improved Analytical Drag Theory for the Artificial Satellite Problem, AIAA pre-

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Vallado D. A., and Crawford P., SGP4 orbit determination, AIAA Paper 2008-6770, AIAA/AAS Astrodynamics

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Wagner C. A., Earth Zonal Harmonics from Rapid Numerical Analysis of Long Satellite Arc, NASA Coddard Space

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6

Cefola P. J., et al, Demonstration of the Semi-analytical Satellite Theory Approach to Improving Orbit Determination,

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7

Cefola P. J., Long A. C., and Holloway G., The long-term prediction of artificial satellite orbits, AIAA Paper 74-170,

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8

McClain W. D., A recursive formulated first-order semianalytic artificial satellite theory based on the generalized

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9

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11

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16

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21

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Cefola P. J., Bentley B., Maisonobe L., Parraud P., Di-Costanzo R., and Folcik Z., Verification of the Orekit Java

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January 26-30, 2014.

24

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2021 [meeting converted to virtual due to pandemic].