OPEN SOURCE ORBIT DETERMINATION WITH SEMI-ANALYTICAL THEORY

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

OPEN SOURCE ORBIT DETERMINATION WITH SEMI-ANALYTICAL THEORY

BRYAN CAZABONNE & LUC MAISONOBE

CS – Communication & Systèmes / 2 / 2

AGENDA

Context and target

DSST presentation

Mean Elements derivatives

State Transition Matrices computation

Short-periodic terms derivatives

Orbit Determination

Conclusion

AGENDA

/ 2


1

/ 3

CONTEXT AND TARGET

/ 3


OREKIT HISTORY

/ 4

Inception | 2002

Orekit instended as a basis for ground segments bids


OREKIT HISTORY

/ 4

Inception | 2002


Version technically complete | 2006

Exceeds technical expectations, why not propose it by itself ?


OREKIT HISTORY

/ 4

Inception | 2002




Failure of the commercial approach | 2008

OPEN-SOURCE ! Permissive license. Very good reception by space community


OREKIT HISTORY

/ 4

Inception | 2002






Cathedral development model | 2008 - 2011

Project teams decides when to release


OREKIT HISTORY

/ 4

Inception | 2002







Bazaar development model | Since 2011

Collaborative tools External commiters



OREKIT HISTORY

/ 4

Inception | 2002








Bazaar development model | Since 2011

Collaborative tools External commiters

Open governance | 2012

PMC representatives from agencies, academics, private companies


OREKIT FEATURES

/ 5

Propagation Estimation Frames Times Orbits …

Numerical

Analytical

Semi-Analytical


TWO USE CASES

Fast Orbit Determination

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

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

Need fast and accurate Orbit Determination

Mean Elements Orbit Determination

Station keeping needs

/ 6


SOLUTION: THE DRAPER SEMI-ANALYTICAL SATELLITE THEORY

Draper Semi-analytical Satellite Theory (DSST)

• Rapidity of an analytical propagator

• Accuracy of a numerical propagator

Target

/ 7

DSST Orbit

Determination

Automatic Differentiation


2

/ 14

DSST PRESENTATION

/ 8


ORBITAL PERTURBATIONS

Third body attraction

Zonal and Tesseral harmonics of terrestrial potential

Solar radiation pressure Atmospheric drag

/ 9


MATHEMATICAL MODEL OF DSST

ci t = ci t + kijηij t , i = 1, 2, 3, 4, 5, 6

N

j=1

/ 10



ci t = ci t + kijηij t , i = 1, 2, 3, 4, 5, 6

N

j=1

Mean Elements

/ 10



ci t = ci t + kijηij t , i = 1, 2, 3, 4, 5, 6

N

j=1

Short-periodic terms Mean Elements

/ 10


DEVELOPMENT STEPS

• Computation of the mean elements

derivatives and the short-periodic terms

derivatives by automatic

differentiation.

Force models configuration

• Computation of the State Transition

Matrices thanks to the variational equations.

State Transition Matrices

• Perform the DSST-OD with the Orekit’s Batch

Least Squares algorithm and the

Kalman Filter.

Orbit Determination

/ 11


3

/ 20

MEAN ELEMENTS DERIVATIVES

/ 12


GOAL

AUTOMATIC DIFFERENTIATION

/ 13

𝐘𝐢𝛛𝐘𝐢𝛛𝐘𝟏

𝛛𝐘𝐢𝛛𝐘𝟐

⋯𝛛𝐘𝐢𝛛𝐘𝟔

𝛛𝐘𝐢𝛛𝐏𝟏

⋯𝛛𝐘𝐢𝛛𝐏𝐍

• Yi: Orbital element

• Pk: Force model parameter

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

GAIN

• Safer implementation

• Simpler validation


Each DSST-specific force model on Orekit has a method allowing

the computation of the mean elements rates.

MEAN ELEMENTS DERIVATIVES 1/2

𝐘 = [𝐚 𝐡 𝐤 𝐩 𝐪 𝛌] 𝐘 =

𝐚 𝐡

𝐤 𝐩 𝐪

𝛌

Method implemented for the states based on the real numbers ✔

Need to be implemented to provide the Jacobians of the mean

elements rates by automatic differentiation.

/ 14



𝐘 = [𝐚 𝐡 𝐤 𝐩 𝐪 𝛌]

meanElementRate =

/ 15




Mean Elements Rates

𝐘 =

𝐚 𝐡

𝐤 𝐩 𝐪

𝛌

meanElementRate =

/ 15




Mean Elements Rates

𝐘 =

𝐚 𝐡

𝐤 𝐩 𝐪

𝛌

meanElementRate =

𝐘′ =

𝐚 𝛛𝐚𝐚 𝛛𝐡𝐚 … 𝛛𝛌𝐚 𝛛𝐏𝟏𝐚 … 𝛛𝐏𝐍𝐚

𝐡

𝛌

𝛛𝐚𝐡 𝛛𝐡𝐡 … 𝛛𝛌𝐡 𝛛𝐏𝟏𝐡 … 𝛛𝐏𝐍𝐡

𝛛𝐚𝛌 𝛛𝐡𝛌 … 𝛛𝛌𝛌 𝛛𝐏𝟏𝛌 … 𝛛𝐏𝐍𝛌

⋮ ⋮ ⋮ ⋮ ⋮ ⋮


/ 15


4

/ 26

STATE TRANSITION MATRICES COMPUTATION

/ 16


JACOBIANS OF MEAN ELEMENTS RATES

𝐘′ =


𝐡

𝛌



⋮ ⋮ ⋮ ⋮ ⋮ ⋮

/ 17

⋱ ⋱



𝐘′ =


𝐡

𝛌



⋮ ⋮ ⋮ ⋮ ⋮ ⋮

𝐘

/ 17

⋱ ⋱



𝐘′ =


𝐡

𝛌



⋮ ⋮ ⋮ ⋮ ⋮ ⋮

𝐘 𝛛𝐘

𝛛𝐘

/ 17

⋱ ⋱



𝐘′ =


𝐡

𝛌



⋮ ⋮ ⋮ ⋮ ⋮ ⋮

𝐘 𝛛𝐘

𝛛𝐘

𝛛𝐘

𝛛𝐏

/ 17

⋱ ⋱


APPLICATION OF THE VARIATIONAL EQUATIONS

𝐝𝛛𝐘𝛛𝐘𝟎𝐝𝐭

=𝛛𝐘

𝛛𝐘×

𝛛𝐘

𝛛𝐘𝟎

𝐝𝛛𝐘𝛛𝐏

𝐝𝐭=𝛛𝐘

𝛛𝐘×𝛛𝐘

𝛛𝐏+𝛛𝐘

𝛛𝐏

/ 18


VALIDATION

Computation of 𝛛𝐘

𝛛𝐘𝟎 and

𝛛𝐘

𝛛𝐏 matrices by finite differences and

comparison to those previously obtained.

/ 19


VALIDATION


𝛛𝐘𝟎 and

𝛛𝐘



Problem : Different matrices !

/ 19


VALIDATION


𝛛𝐘𝟎 and

𝛛𝐘



Newtonian Attraction derivatives were not taken into account in the

computation of the state transition matrices.

Some dependencies to the central attraction coefficient were

implicit and therefore not differentiated.


/ 19


VALIDATION


𝛛𝐘𝟎 and

𝛛𝐘



Newtonian Attraction derivatives were not taken into account in the

computation of the state transition matrices.

Some dependencies to the central attraction coefficient were

implicit and therefore not differentiated.


/ 19

Problem solved ✔


5

/ 36

SHORT-PERIODIC TERMS DERIVATIVES

/ 20



/ 21


Compute the Jacobians of the short-periodic terms into the DSST-specific force models

Addition of the contribution

Add the contribution of the short-periodic derivatives after the numerical integration of the mean elements rates

Validation

Compute the state transition matrices (with the short-periodic derivatives) by finite differences



/ 21


Compute the Jacobians of the short-periodic terms into the DSST-specific force models

Addition of the contribution

Add the contribution of the short-periodic derivatives after the numerical integration of the mean elements rates

Validation

Compute the state transition matrices (with the short-periodic derivatives) by finite differences The user has the

choice to use only the mean elements derivatives or adding the short-periodic terms derivatives


6

/ 39

ORBIT DETERMINATION

/ 22


FORCE MODELS USED: LAGEOS 2



/ 23

Lageos 2



FORCE MODELS USED: GNSS



Solar radiation pressure

/ 23


GNSS


TEST CASES: INTEGRATION STEP

The DSST has significant advantage compared to the numerical

propagator for the integration step. This because the elements

computed numerically by the DSST are the mean elements.

/ 24

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


TEST CASES: SHORT-PERIODIC TERMS DERIVATIVES

/ 25

Case Zonal Tesseral Third Body

1 × × ×

2 ✓ × ×

3 ✓ ✓ ×

4 ✓ ✓ ✓

Gradual addition of the short-periodic terms derivatives to highlight the

main contributions .

Performed tests for Lageos2 Orbit Determination.


BATCH LEAST SQUARES / MEAN ELEMENTS

/ 26

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 9,3.10-5

3,6.10-11

1,4.10-6

Lageos2 GNSS

Rel

ati

ve

Ga

p

DSST

Numerical


KALMAN FILTER / MEAN ELEMENTS

/ 27

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


BATCH LEAST SQUARES / LAGEOS2 SP DERIVATIVES

/ 28

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


KALMAN FILTER / LAGEOS2 SP DERIVATIVES

/ 29

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


5

/ 48

CONCLUSION

/ 30


OUTLOOKS

DSST

/ 31


OUTLOOKS

DSST

Need acces to « optimal » values for force models initialization

/ 31


OUTLOOKS

DSST

Need acces to « optimal » values for force models initialization

Need to optimize critical computation steps.

/ 31


OUTLOOKS

DSST

Need acces to « optimal » values for force models initialization.

Need to optimize critical computation steps.

Improve the Kalman Filter Orbit Determination with the DSST.

/ 31

CS – Communication & Systèmes / 53 CONCEPTEUR, OPÉRATEUR & INTÉGRATEUR DE SYSTÈMES CRITIQUES www.c-s.fr

CS 22, avenue Galilée - 92350 Le Plessis Robinson Tél. : 01 41 28 40 00 www.c-s.fr

Thank you for your attention

OPEN SOURCE ORBIT DETERMINATION WITH SEMI-ANALYTICAL THEORY

Documents

Transcript of OPEN SOURCE ORBIT DETERMINATION WITH SEMI-ANALYTICAL THEORY