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1

A SEMI-ANALYTICAL APPROACH FOR ORBIT DETERMINATION

BASED ON EXTENDED KALMAN FILTER

Bryan Cazabonne,

*

Julie Bayard,

†

Maxime Journot,

‡

and Paul J. Cefola

§

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 both simulated

data and real data from CDDIS (Crustal Dynamics Data Information System).

The ESKF results are compared with those obtained by the GTDS ESKF.

INTRODUCTION

Orbit determination is used to estimate satellite’s state vector from observed measurements.

The state vector may be position and velocity or an orbital element set. The element set may be

osculating or mean. The state vector may also include dynamical parameters such as the drag

coefficient and the satellite’s reflection coefficient. 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’s Theory is

the basis of most of the analytical orbit determination methods.

1

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.

2

,

3

However, operational analytical orbit determination

methods may not meet accuracy requirements.

Semi-analytical techniques combine the accuracy of numerical propagation and the

characteristic speed of analytical propagation. One early semi-analytical orbit determination

method is the ROAD algorithm due to Wagner.

4

In ROAD algorithm, the dynamical model is the

*

Bryan Cazabonne is Spaceflight Mechanics Engineer at CS GROUP, 6 Rue Brindejonc des Moulinais, Toulouse,

France, email: bryan.cazabonne@csgroup.eu.

†

Julie Bayard is Spaceflight Mechanics Engineer at CS GROUP, 6 Rue Brindejonc des Moulinais, Toulouse, France,

email: julie.bayard@csgroup.eu.

‡

Maxime Journot is Spaceflight Mechanics Engineer at CS GROUP, 6 Rue Brindejonc des Moulinais, Toulouse,

France, email: maxime.journot@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.

AAS 21-614

2

mean element equations of motion. In 1977, the Draper Laboratory proposed the extension of its

GTDS (Goddard Trajectory Determination System) semi-analytical orbit propagator to include

detailed short period motion models and improved partial derivative models.

5

,

6

The current study

focuses on the Draper Semi-analytical Satellite Theory (DSST), which is flexible, complete, and

applicable to all orbit types.

7

,

8

There are different implementations of DSST orbit determination.

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).

13

,

14

,

15

The classical (i.e., based on

numerical propagation) Extended Kalman Filter (EKF) algorithm is already available in the

library. However, the re-initialization of the EKF underlying orbit propagator at each

measurement epoch is a major constraint for semi-analytical satellite theory. The ESKF algorithm

reconciles the conflicting goal of the DSST perturbation theory (i.e., large step size) and the EKF

theory (i.e., re-initialization at each measurement epoch).

The roadmap of the paper will be to first introduce the Orekit’s implementation of the EKF. A

general introduction to the concept of ESFK algorithm is then presented. Particular attention is

drawn to the operations on both the integration and observation grids. Validation of the Orekit

ESKF is demonstrated under orbit determination conditions using both simulated data and real

data from the CDDIS (Crustal Dynamics Data Information System) website.

16

Orekit ESKF orbit

determination results are compared with those obtained by the reference GTDS ESKF.

Conclusion and Future Work end the paper.

MATHEMATICAL PRELIMINARIES

Orekit is an open-source space flight dynamics library.

17

It is written in Java and provides low

level elements for the development of flight dynamics applications. Since 2008, Orekit is

distributed under the Apache License version 2.0.

18

Orekit provides various functionalities related

to coordinate transformations, reading and writing of standardized formats, orbit propagation, and

orbit determination using batch-least squares algorithm and recursive filters.

The Extended Kalman Filter

The Extended Kalman Filter (EKF) is an extremely useful algorithm for problems based on

continuous data streams. It is based on a recursive process where the estimated covariance matrix

and satellite’s state

calculated from the previous observation are used to estimate the

new satellite state for the current observation. The EKF algorithm is composed of two steps, a

prediction step and a correction step. During the prediction step, the predicted covariance matrix

and satellite’s state are calculated following Equation (1) and (2).

(1)

(2)

where

state transition matrix (

process noise matrix

In Equation (2), the notation is used to indicate one choice of orbit propagator.

3

During the correction step, the predicted covariance and satellite’s state are updated using the

satellite’s observation. The calculation of these two elements is done by Equation (3) to (5).

(3)

(4)

(5)

where observation covariance matrix

observation partials matrix

Kalman gain

vector

In Orekit library, the updated covariance matrix

is not calculated using Equation (4). Orekit

uses the Joseph algorithm, as in Equation (6). Joseph algorithm is equivalent to the classical

formula but guarantees the output stays symmetric.

19

(6)

In Equations (1) to (6), the calculation of the observation partials matrix and the state

transition matrix is completed using Equation (7) and (8). Orekit library uses the automatic

differentiation technique to calculate all the necessary partial derivatives.

20

,

21

(7)

(8)

where is an observed measurement at epoch . Figure 1 shows the calling hierarchy of

the Orekit EKF orbit determination. The figure presents the different steps of calculation and the

integration of the previous equations in the process.

The Draper Semi-analytical Satellite Theory

The Draper Semi-analytical Satellite Theory (DSST) is a mean elements satellite theory

expressed in non-singular equinoctial elements.

22

It 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 and

computed using a combination of analytical and numerical techniques.

DSST was developed with an emphasis on accuracy and computational efficiency. It models

the motion due to conservative perturbations using the Lagrangian Variation Of Parameters

formalism in Equation (9). The Gaussian Variation Of Parameters formalism in Equation (10) is

used to model non-conservative perturbations.

(9)

(10)

4

where

satellite's velocity vector

accelerations caused by the non-conservative perturbations

osculating equinoctialelements

disturbing potential for the conservative forces

In DSST theory, the equations of motion for the mean equinoctial elements can be written as

in Equation (11) and (12).

with i = 1, 2, ..., 5

(11)

(12)

where

equinoctialelements

mean mean motion

mean longitude

functions of the slowly varying mean elements

denotes the small magnitude of the element

In Equations (11) and (12), the functions of the slowly varying mean elements for the different

orbital perturbations can be found in McClain.

23

Finally, the transformation from mean equinoctial elements to osculating equinoctial elements

is calculated using Equation (13).

with i = 1, 2, ..., 6

(13)

where

short-period function, 2 periodic

Because the DSST orbit propagator uses large step size to perform the numerical integration

of the equations of motion for the mean equinoctial elements (e.g., half-day for GEO satellites), it

is not suitable for a classical EKF orbit determination. The EKF algorithm needs to re-initialize

the orbital state at each observation epoch. However, the time difference between two

observations is usually much smaller than the DSST step size. In order to take advantage of the

DSST theory within a recursive filter orbit determination, Steve Taylor designed the Extended

Semi-analytical Kalman Filter in 1981.13

THE EXTENDED SEMI-ANALYTICAL KALMAN FILTER IMPLEMENTATION

The Extended Semi-analytical Kalman Filter (ESKF) reconciles the conflicting goals of the

DSST perturbation theory and the EKF theory. Steve Taylor used the concept of the mean

equinoctial elements integration grid. Therefore, the nominal orbital state is updated only at the

integration grid points.

5

The following procedures, mainly based on the Taylor thesis, describe the operations to

perform on both the integration grid and the observation grid. It will be assumed that both the

equinoctial elements and the dynamical parameters are estimated. For simplicity, estimation of

the measurement parameters (e.g., station biases) is ignored.

Initialization of the Extended Semi-analytical Kalman Filter

1. Set the initial covariance matrix

the initial state

and the initial estimated filter

correction

.

2. Set the state transition matrix to the identity matrix, and the partial derivatives of

the mean equinoctial elements with respect to the dynamic parameters to the zero

matrix.

3. Initialize the short periodic functions of all the involved forces.

Operations on the Integration Grid

1. Update the nominal state

2. Integrate to obtain the nominal mean equinoctial elements

, the state transition

matrix , and the partial derivatives of the mean equinoctial elements with respect to

the dynamical parameters .

3. Calculate the Fourier coefficients

and

, and update the short periodic

functions

with respect to the Fourier coefficients.

Operations on the Observation Grid

1. Obtain the new observation at time.

2. Interpolate to obtain the nominal mean equinoctial elements

, the state transition

matrix , and the partial derivatives of the mean equinoctial elements with respect to

the dynamical parameters , all at time.

3. Interpolate for the short periodic coefficients and calculate the short periodic functions

at time.

4. Calculate the transition matrices using Equation (14) and (15). These matrices contain the

partial derivatives of the predicted parameters with respect to the ones at the epoch of the

previous observation .

(14)

(15)

5. Predict the filter corrections

using the corrected corrections

calculated from

the previous observation.

(16)

6

(17)

(18)

6. Calculate the predicted osculating equinoctial elements, as in Equation (19).

(19)

where

(20)

7. Calculate the predicted measurement

, its partial derivatives, and the

observation residual.

8. Calculate the observation partials matrix, as in Equation (21).

(21)

where

(22)

The and matrices in Equation (20) and (22) represent the partial derivatives of the

short period motion. They were introduced by Andrew Green.

24

9. Calculate the predicted covariance matrix

using the estimated covariance matrix

calculated from the previous observation, as in Equation (23).

(23)

In Equation (23), still denotes the user-defined process noise matrix.

10. Perform the correction step of the filter using Equation (24) to (26). The correction step

of the ESKF is very close to the EKF.

(24)

(25)

(26)

As seen in Equation (25), the corrected covariance matrix is also calculated using Joseph

algorithm. In addition, still denotes the observation covariance matrix.

7

11. Calculate the corrected measurement and residual using the corrected osculating elements

given in Equation (27).

(27)

The ESKF continues with step 1 of observation grid until all observations have been processed

or until the next integration step is encountered. If the next integration step is encountered, the

operations on the integration grid are followed. Figure 2 shows the calling hierarchy of the Orekit

ESKF orbit determination.

Figure 3 presents an Unified Modeling Language (UML) diagram of the implementation of

the ESKF in Orekit. The main Java class on this diagram is the SemiAnalyticalKalmanEstimator

class. The Orekit’s users will use this class to execute the ESKF orbit determination. This class is

built from SemiAnalyticalKalmanEstimatorBuilder class. The choice between the operations on

the Integration Grid or the Observation Grid is handled by the ESKFMeasurementHandler class.

This class is added to the user-defined DSSTPropagator in order to highlight integration steps.

The link between the ExtendedKalmanEstimator class of Hipparchus library and Orekit is made

by the SemiAnalyticalKalmanModel class. This class is very important because it performs most

of the steps presented before. It also performs the initialization steps in the class constructor.

The Green’s matrices and are calculated by automatic differentiation using the

DSSTJacobianMapper class. This class is built from the DSSTPartialDerivativesEquations class.

The purpose of the DSSTPartialDerivativesEquations class is also to calculate the and

matrices using variational equations. In Orekit library, the variational equations are

integrated simultaneously with the equations of motion by the DSSTPropagator. Generally, any

additional equation (i.e., additional to the main equations of motion) in Orekit can be integrated

simultaneously with the equations of motion by the DSSTPropagator if it implements the

Orekit’s Java interface AdditionalEquations. Therefore, the two matrices are interpolated when

the nominal mean equinoctial elements are also interpolated.

OREKIT EXTENDED SEMI-ANALYTICAL KALMAN FILTER VALIDATION

Validation against simulated data

The Orekit implementation of the ESKF is first validated using simulated data. The epoch

mean orbital elements used for testing the Orekit ESKF are given in Table 1. The mean orbital

elements set is given in EME2000 coordinates.

Table 1. Epoch mean orbital elements used for ESKF validation against simulated data.

Orbit element

Value

semi-major axis

1.5E7 meters

eccentricity

0.125

inclination

71.619724 degrees

argument of perigee

14.323945 degrees

right ascension of the ascending node

78.781697 degrees

mean anomaly

3.580986 degrees

epoch (UTC)

2000-02-24T11:35:47.000

8

Details about the test cases used to validate the Orekit ESKF are given in Table 2. Three test

cases are used: one two-body case, and two perturbed cases.

Table 2. Test cases for ESKF validation against simulated data.

Case

Force model

Residuals Figure

1

Two-body

Figure 4

2

Two-body +

Figure 5

3

Two-body + + +

Figure 6

For each test case, a one-day forward propagation of the epoch mean orbital elements is done

in order to generate osculated pseudo-range measurements. The three test cases have 445

simulated measurements from two stations. The station coordinates are given in Table 3.

Table 3. Station coordinates used for ESKF validation against simulated data.

Name

Latitude

Longitude

Altitude

Isla Desolación

-53.05388 degrees

-75.01551 degrees

1750.0 meters

Slættaratindur

62.29639 degrees

-7.01250 degrees

880.0 meters

The residuals between the simulated and the estimated values are calculated for each

measurement. For Case 2 and 3, an offset of 1.2 meters is added to the initial value of the semi-

major axis in order to start the estimation process with a small difference compared to the

reference epoch mean orbital elements. The objective is to test the ability of the Orekit ESKF to

estimate a correct orbit. This 1.2 meters value corresponds to the value already used for the

validation of the numerical EKF algorithm against simulated data in Orekit. The residual mean

values and standard deviations are summarized in Table 4.

Table 4. Mean values of the measurement residuals for each test case.

Case

Mean residual value (meters)

Standard deviation (meters)

1

4.9724E-08

9.0858E-08

2

3.8843E-03

4.3080E-02

3

3.9437E-03

4.3336E-02

Figure 4 to Figure 6 highlight the validation of the Orekit ESKF against simulated

measurements. The mean residual value for Case 1 is about 10-8 meters and 10-3 meters for Case 2

and 3. The difference between Case 1 and Case 2 and 3 is expected. The two last test cases are

perturbed. In other words, they introduce the impact of the short periodic terms in the ESKF

execution. The 1.2 meters offset added to the initial value of the semi-major axis has a significant

impact in the accuracy of the residuals.

Table 5 shows the difference between the reference and the estimated positions for the last

measurement epoch. The reference position corresponds to the propagated mean orbital elements

given in Table 1 to the last measurement epoch.

9

Table 5. Position difference between the reference and the estimated orbit.

Case

Initial position difference (meters)

Final position difference (meters)

2

1.0504

0.0454

3

1.0504

0.0407

The initial difference between the epoch mean elements and the first orbit used by the

estimator is about 1.05 meters. It corresponds to the 1.2 meters offset on the semi-major axis. At

the end of the estimation process, the difference between the reference position and the estimated

position is about 4 centimeters. This result highlights the ability of the Orekit ESKF to improve

the knowledge of the orbit during the estimation process.

Validation against real data

The Lageos 2 satellite was chosen for demonstrating Orekit ESKF validation. The selection of

this satellite was influenced by the availability of the satellite’s ephemeris in the CDDIS. Five

days of predicted Lageos 2 positions are used as measurements in the orbit determination process.

The predicted positions are taken from a Consolidated Prediction File (CPF) produced by the

NERC Space Geodesy Facility. Joanna Najder compared the accuracy of Lageos 2 predicted

positions in CPF with the precise orbits contained in the Extended Standard Product - 3 (SP3)

files. She highlighted a mean error of 0.5-1 meter for Lageos 2 prediction files.

25

Therefore, it is

interesting to use Lageos 2 predicted positions for the validation of the Orekit ESKF against real

data. The orbit determination is carried out with 20x20 geo-potential terms, lunar-solar point

masses, and solar radiation pressure. Lageos 2 satellite altitude allows neglecting atmospheric

effects on the satellite orbit. A constant process noise is used. The six equinoctial orbit elements

are estimated during the orbit determination process.

Figure 7 shows the measurement residuals obtained by the Orekit ESKF orbit determination.

They correspond to the differences between the observed and the predicted satellite’s positions

calculated during the Step 7 of the Operations on the Observation Grid. Measurement residuals

are very close to those obtained by GTDS as presented in Figure 9. The amplitude of the residuals

between the two methods is similar (i.e., between ± 5 meters). For the first day of observations,

the amplitude of the GTDS ESKF residuals is greater than Orekit ESKF. An initial error is added

in the GTDS case while no error is added for Orekit. In addition, the fact that the Lageos 2 data

are predicted by a numerical orbit propagator contributes to the small trend in the error growth

over the five days in Orekit ESKF. The statistics on the predicted measurement residuals obtained

by the Orekit ESKF are presented in Table 6.

Table 6. Statistics on Orekit ESKF residuals (observed minus predicted).

Coordinate

Mean residual value (meters)

Standard deviation (meters)

X

1.6199E-02

2.5635

Y

-9.7587E-04

2.3475

Z

2.0427E-02

2.7101

Figure 8 displays the measurement residuals between the observed and the corrected satellite’s

positons calculated during the Step 11 of the Operations on the Observation Grid. This figure

highlights the significant contribution of the correction step of the Orekit ESKF to improve the

10

estimation of the orbit. The statistics on the corrected measurement residuals are presented in

Table 7.

Table 7. Statistics on Orekit ESKF residuals (observed minus corrected).

Coordinate

Mean residual value (meters)

Standard deviation (meters)

X

6.6822E-05

0.0041

Y

6.3195E-05

0.0037

Z

7.0171E-05

0.0042

The results highlight the validation of the Orekit ESKF against real measurements. They show

that the Orekit ESKF is able to estimate accurate satellite positions. The mean residual value of

each coordinate is about 10-5 meters and the standard deviation is about 4 millimeters. The period

of the sinusoidal effect observed on Figure 8 is equal to the orbital period. The statistics on the

corrected measurement residuals are considerably better than the statistics on the predicted

measurement residuals. This demonstrates again the significant impact of the correction step of

the ESKF.

CONCLUSION

Results demonstrate the validation of the Orekit ESKF against both simulated and real data.

First, the measurement residuals for the three simulated test cases show the ability of the Orekit

ESKF to perform an accurate orbit determination based on generated data. These results also

highlight the ability of the Orekit ESKF to improve the knowledge of the orbit during the

estimation process. The validation against real data shows the consistency between Orekit ESKF

and GTDS ESKF implementations. This study offer an improvement compared to the Taylor

thesis. The Equation (27) is a new equation highlighting the contribution of the correction step of

the ESKF. Finally, the validation against the five days of predicted positions for the Lageos 2

satellite demonstrates the meter level agreement between the Orekit DSST and the real world.

FUTURE WORK

There are several areas in which we intend to improve the capabilities of the Orekit ESKF. In

particular, we would like to extend the validation of the Orekit ESKF against real satellite data

from the CDDIS website. Because Lageos-2 geometry is spherical, we would like to validate the

Orekit ESKF using data from satellite with more complex geometry (e.g., box and solar array

spacecraft model). In addition, we would like to test the performance of the Orekit ESKF with

orbits perturbed by atmosphere drag (e.g., CryoSat-2 orbit).

26

Another improvement would be the implementation of the ESKF for multiple satellites.

Indeed, the current implementation is only meant for a single satellite orbit determination.

However, with the development of satellite constellations and multi-satellite missions, an

implementation of multi-satellite orbit determination is interesting.

Finally, we would like to improve the capabilities of Orekit orbit determination by adding new

recursive filters. The Backward Smoothing Extended Kalman Filter (BSEKF) and the Backward

Smoothing Extended Semi-analytical Kalman Filter (BSESKF) are recursive filters that show

more reliable convergence and robustness than the EKF and ESKF, respectively.15,

27

The

implementation of a semi-analytical form of the Unscented Kalman Filter (UKF) is also an

interesting challenge that we would like to address.

28

,

29

11

ACKNOWLEDGMENTS

The authors would like to acknowledge Mr. Luc Maisonobe and Mr. Pascal Parraud, both

from 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, Dr. Ronald J. Proulx, Newton, Massachusetts, Dr. Srinivas

Setty, Munich, Germany, Mr. Zach Folcik, MIT Lincoln Laboratory, Lexington, Massachusetts,

Dr. Jim Schatzman, Augustus Aerospace Company, Lone Tree, Colorado, and Mr. Jacob

Stratford, Brigham Young University, Provo, Utah. 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.

FIGURES

Figure 1. Orekit Extended Kalman Filter orbit determination principle.

12

Figure 2. Orekit Extended Semi-analytical Kalman Filter orbit determination principle.

13

Figure 3. UML diagram of Orekit implementation of the Extended Semi-analytical Kalman Filter.

14

Figure 4. Case 1: Residuals between simulated (i.e., observed) and estimated range measurements

for Orekit Extended Semi-analytical Kalman Filter validation.

Figure 5. Case 2: Residuals between simulated (i.e., observed) and estimated range measurements

for Orekit Extended Semi-analytical Kalman Filter validation.

Figure 6. Case 3: Residuals between simulated (i.e., observed) and estimated range measurements

for Orekit Extended Semi-analytical Kalman Filter validation.

15

Figure 7. Lageos-2 Orekit ESKF ECI measurement residuals between observed and predicted

satellite’s positions. Predicted positions are calculated during Step 7 of the operations on the

observation grid.

16

Figure 8. Lageos-2 Orekit ESKF ECI measurement residuals between observed and corrected

satellite’s positions. Corrected positions are calculated during Step 11 of the operations on the

observation grid.

17

Figure 9. Lageos-2 GTDS DSST ESKF ECEF Measurement Residuals (GGM01S 50x50, Lunar

Solar Point Masses, SRP, SET, J2000 Integration Coordinate System, DSST Short-period model:

SPGRVFRC set to complete model, SRP short period motion, Short-Period J2 partials ) (position

differences are in meters and velocity differences are in cm/sec).

18

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