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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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