CS – Communication & Systèmes / 1CONCEPTEUR, OPÉRATEUR & INTÉGRATEUR DE SYSTÈMES CRITIQUES www.c-s.fr

DSST ORBIT DETERMINATION IN OREKIT

BRYAN CAZABONNE

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AGENDA

�Context and target

�DSST presentation

�Mean Elements derivatives

�State Transition Matrices

�Short-periodic terms derivatives

�Orbit Determination

�Conclusion

AGENDA

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CONTEXT AND TARGET

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OREKIT FEATURES

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Propagation Estimation Frames Times Orbits …

Numerical

Analytical

Semi-Analytical

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TWO USE CASES

�Fast Orbit Determination

• Several hundreds of thousands (Setty et al, 2016)

Number of Orbit Determinations performed by the US Joint SpaceOperation Center per day to maintain their space objects catalog.

Need fast and accurate Orbit Determination

�Mean Elements Orbit Determination

Station keeping needs

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SOLUTION: THE DRAPER SEMI-ANALYTICAL SATELLITE THEORY

�Draper Semi-analytical Satellite Theory (DSST)

• Rapidity of an analytical propagator

• Accuracy of a numerical propagator

�Target

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DSSTOrbit

Determination

AutomaticDifferentiation

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DSST PRESENTATION

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ORBITAL PERTURBATIONS

Third body attraction

Zonal and Tesseral harmonicsof terrestrial potential

Solar radiation pressure Atmospheric drag

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MATHEMATICAL MODEL OF DSST

c� t = c� t + � k�η� t , i = 1, 2, 3, 4, 5, 6�

��

Short-periodic termsMean Elements

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DEVELOPMENT STEPS

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Force models configuation

Computation of the meanelements derivatives and the short-periodic termsderivatives by automaticdifferentiation

State Transition Matrices

Computation of the State Transition Matrices thanksto the variationalequations

Orbit Determination

Perform the DSST-ODwith the Orekit’s BatchLeast Squares algorithmand the Kalman Filter.

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MEAN ELEMENTS DERIVATIVES

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�Each DSST-specific force model on Orekit has a method allowingthe computation of the mean elements rates.

MEAN ELEMENTS DERIVATIVES

� = [������] �� =

������������

�Method implemented for the states based on the real numbers ✔

�Need to be implemented to provide the Jacobians of the meanelements rates by automatic differentiation.

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�GOAL

AUTOMATIC DIFFERENTIATION

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� !� !�"

!� !�#

⋯ !� !�%

!� !&"

⋯ !� !&'

• Yi: Orbital element

• Pk: Force model parameter

• N: The number of force model parameters taken into accountfor the Orbit Determination

�GAIN

• Safer implementation

• Simpler validation

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JACOBIANS

�� =

������������

Automatic Differentiation

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�′� = �� !��!�

!��!&

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STATE TRANSITION MATRICES

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VARIATIONAL EQUATIONS

) !�!�*)+ = !��

!� × !�!�*

) !�!&)+ = !��

!� × !�!& + !��

!&

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�′� = �� !��!�

!��!&

VariationalEquations

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VALIDATION

�Computation of!�!�*

and!�!& matrices by finite differences and

comparison to those previously obtained.

�Newtonian Attraction derivatives were not taken into account in thecomputation of the state transition matrices.

�Some dependencies to the central attraction coefficient wereimplicit and therefore not differentiated.

Problem : Different matrices !

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Problem solved ✔

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SHORT-PERIODIC TERMSDERIVATIVES

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SHORT-PERIODIC TERMS DERIVATIVES

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Automatic Differentiation

Compute the Jacobians of the short-periodic termsinto the DSST-specificforce models

Addition of the contribution

Add the contribution of the short-periodic derivativesafter the numericalintegration of the meanelements rates

Validation

Compute the state transition matrices (withthe short-periodicderivatives) by finitedifferences The user has the

choice to use onlythe mean elementsderivatives or addingthe short-periodicterms derivatives

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ORBIT DETERMINATION

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FORCE MODELS USED: LAGEOS 2

Third body attraction

Zonal and Tesseral harmonicsof terrestrial potential

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Lageos 2

Third body attraction

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FORCE MODELS USED: GNSS

Third body attraction

Zonal and Tesseral harmonicsof terrestrial potential

Solar radiation pressure

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Third body attraction

GNSS

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TEST CASES: INTEGRATION STEP

�The DSST has significant advantage compared to the numericalpropagator for the integration step. This because the elementscomputed numerically by the DSST are the mean elements.

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DSST Numerical

Minimum step (s) 6000

Maximum step (s) 86400

Tolerance (m) 10

Minimum step (s) 0,001

Maximum step (s) 300

Tolerance (m) 10

VS

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TEST CASES: SHORT-PERIODIC TERMS DERIVATIVES

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Case Zonal Tesseral Third Body

1 × × ×

2 ✓ × ×

3 ✓ ✓ ×

4 ✓ ✓ ✓

�Gradual addition of the short-periodic terms derivatives to highlight themain contributions.

�Performed tests for Lageos2 Orbit Determination.

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BATCH LEAST SQUARES / MEAN ELEMENTS

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RAPIDITY

ACCURACY

0

10

20

30

40

50

60

70

80

90

Lageos2 GNSS

Co

mp

uta

tio

n t

ime

(s)

DSST

Numerical

2,9.10-4

3,6.10-5

3,6.10-11

1,4.10-6

Lageos2 GNSS

Rel

ati

ve

Ga

p

DSST

Numerical

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KALMAN FILTER / MEAN ELEMENTS

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RAPIDITY ACCURACY

0

5

10

15

20

25

30

35

40

Lageos2

Co

mp

uta

tio

n t

ime

(s)

DSST

Numerical

0,28

1,4,10-6

Lageos2

Rel

ati

ve

Ga

p

DSST

Numerical

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BATCH LEAST SQUARES / LAGEOS2 SP DERIVATIVES

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0,00E+00

4,00E-05

8,00E-05

1,20E-04

1,60E-04

2,00E-04

2,40E-04

2,80E-04

0

10

20

30

40

50

60

70

80

90

100

Case (1) Case (2) Case (3) Case (4)

Rel

ati

ve

Ga

p

Co

mp

uta

tio

n t

ime

(s)

Computation time (s) Relative gap

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KALMAN FILTER / LAGEOS2 SP DERIVATIVES

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0,00E+00

1,00E-04

2,00E-04

3,00E-04

4,00E-04

5,00E-04

6,00E-04

7,00E-04

8,00E-04

9,00E-04

1,00E-03

0

10

20

30

40

50

60

70

80

90

100

Case (1) Case (2) Case (3) Case (4)

Rel

ati

ve

Ga

p

Co

mp

uta

tio

n t

ime

(s)

Computation time (s) Relative gap

0,28

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CONCLUSION

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OUTLOOKS

DSST

Need acces to « optimal »values for force modelsinitialization.

Need to optimize criticalcomputation steps.

Improve the Kalman Filter Orbit Determination withthe DSST.

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Thank you for your attention

Publication : Open-source Orbit Determination using semi-analytical theory (Cazabonne et al, 2018)