open source orbit determination with semi-analytical theory
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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
CS – Communication & Systèmes / 3 / 3
1
/ 3
CONTEXT AND TARGET
/ 3
CS – Communication & Systèmes / 4 / 4
OREKIT HISTORY
/ 4
Inception | 2002
Orekit instended as a basis for ground segments bids
CS – Communication & Systèmes / 5 / 5
OREKIT HISTORY
/ 4
Inception | 2002
Orekit instended as a basis for ground segments bids
Version technically complete | 2006
Exceeds technical expectations, why not propose it by itself ?
CS – Communication & Systèmes / 6 / 6
OREKIT HISTORY
/ 4
Inception | 2002
Orekit instended as a basis for ground segments bids
Version technically complete | 2006
Exceeds technical expectations, why not propose it by itself ?
Failure of the commercial approach | 2008
OPEN-SOURCE ! Permissive license. Very good reception by space community
CS – Communication & Systèmes / 7 / 7
OREKIT HISTORY
/ 4
Inception | 2002
Orekit instended as a basis for ground segments bids
Version technically complete | 2006
Exceeds technical expectations, why not propose it by itself ?
Failure of the commercial approach | 2008
OPEN-SOURCE ! Permissive license. Very good reception by space community
Cathedral development model | 2008 - 2011
Project teams decides when to release
CS – Communication & Systèmes / 8 / 8
OREKIT HISTORY
/ 4
Inception | 2002
Orekit instended as a basis for ground segments bids
Version technically complete | 2006
Exceeds technical expectations, why not propose it by itself ?
Failure of the commercial approach | 2008
OPEN-SOURCE ! Permissive license. Very good reception by space community
Cathedral development model | 2008 - 2011
Bazaar development model | Since 2011
Collaborative tools External commiters
Project teams decides when to release
CS – Communication & Systèmes / 9 / 9
OREKIT HISTORY
/ 4
Inception | 2002
Orekit instended as a basis for ground segments bids
Version technically complete | 2006
Exceeds technical expectations, why not propose it by itself ?
Failure of the commercial approach | 2008
OPEN-SOURCE ! Permissive license. Very good reception by space community
Cathedral development model | 2008 - 2011
Project teams decides when to release
Bazaar development model | Since 2011
Collaborative tools External commiters
Open governance | 2012
PMC representatives from agencies, academics, private companies
CS – Communication & Systèmes / 10 / 10
OREKIT FEATURES
/ 5
Propagation Estimation Frames Times Orbits …
Numerical
Analytical
Semi-Analytical
CS – Communication & Systèmes / 11 / 11
OREKIT FEATURES
/ 5
Propagation Estimation Frames Times Orbits …
Numerical
Analytical
Semi-Analytical
CS – Communication & Systèmes / 12 / 12
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
CS – Communication & Systèmes / 13 / 13
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
CS – Communication & Systèmes / 14 / 14
2
/ 14
DSST PRESENTATION
/ 8
CS – Communication & Systèmes / 15 / 15
ORBITAL PERTURBATIONS
Third body attraction
Zonal and Tesseral harmonics of terrestrial potential
Solar radiation pressure Atmospheric drag
/ 9
CS – Communication & Systèmes / 16 / 16
MATHEMATICAL MODEL OF DSST
ci t = ci t + kijηij t , i = 1, 2, 3, 4, 5, 6
N
j=1
/ 10
CS – Communication & Systèmes / 17 / 17
MATHEMATICAL MODEL OF DSST
ci t = ci t + kijηij t , i = 1, 2, 3, 4, 5, 6
N
j=1
Mean Elements
/ 10
CS – Communication & Systèmes / 18 / 18
MATHEMATICAL MODEL OF DSST
ci t = ci t + kijηij t , i = 1, 2, 3, 4, 5, 6
N
j=1
Short-periodic terms Mean Elements
/ 10
CS – Communication & Systèmes / 19 / 19
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
CS – Communication & Systèmes / 20 / 20
3
/ 20
MEAN ELEMENTS DERIVATIVES
/ 12
CS – Communication & Systèmes / 21 / 21
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
CS – Communication & Systèmes / 22 / 22
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
CS – Communication & Systèmes / 23 / 23
MEAN ELEMENTS DERIVATIVES 2/2
𝐘 = [𝐚 𝐡 𝐤 𝐩 𝐪 𝛌]
meanElementRate =
/ 15
CS – Communication & Systèmes / 24 / 24
MEAN ELEMENTS DERIVATIVES 2/2
𝐘 = [𝐚 𝐡 𝐤 𝐩 𝐪 𝛌]
Mean Elements Rates
𝐘 =
𝐚 𝐡
𝐤 𝐩 𝐪
𝛌
meanElementRate =
/ 15
CS – Communication & Systèmes / 25 / 25
MEAN ELEMENTS DERIVATIVES 2/2
𝐘 = [𝐚 𝐡 𝐤 𝐩 𝐪 𝛌]
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
⋮ ⋮ ⋮ ⋮ ⋮ ⋮
Automatic Differentiation
/ 15
CS – Communication & Systèmes / 26 / 26
4
/ 26
STATE TRANSITION MATRICES COMPUTATION
/ 16
CS – Communication & Systèmes / 27 / 27
JACOBIANS OF MEAN ELEMENTS RATES
𝐘′ =
𝐚 𝛛𝐚𝐚 𝛛𝐡𝐚 … 𝛛𝛌𝐚 𝛛𝐏𝟏𝐚 … 𝛛𝐏𝐍𝐚
𝐡
𝛌
𝛛𝐚𝐡 𝛛𝐡𝐡 … 𝛛𝛌𝐡 𝛛𝐏𝟏𝐡 … 𝛛𝐏𝐍𝐡
𝛛𝐚𝛌 𝛛𝐡𝛌 … 𝛛𝛌𝛌 𝛛𝐏𝟏𝛌 … 𝛛𝐏𝐍𝛌
⋮ ⋮ ⋮ ⋮ ⋮ ⋮
/ 17
⋱ ⋱
CS – Communication & Systèmes / 28 / 28
JACOBIANS OF MEAN ELEMENTS RATES
𝐘′ =
𝐚 𝛛𝐚𝐚 𝛛𝐡𝐚 … 𝛛𝛌𝐚 𝛛𝐏𝟏𝐚 … 𝛛𝐏𝐍𝐚
𝐡
𝛌
𝛛𝐚𝐡 𝛛𝐡𝐡 … 𝛛𝛌𝐡 𝛛𝐏𝟏𝐡 … 𝛛𝐏𝐍𝐡
𝛛𝐚𝛌 𝛛𝐡𝛌 … 𝛛𝛌𝛌 𝛛𝐏𝟏𝛌 … 𝛛𝐏𝐍𝛌
⋮ ⋮ ⋮ ⋮ ⋮ ⋮
𝐘
/ 17
⋱ ⋱
CS – Communication & Systèmes / 29 / 29
JACOBIANS OF MEAN ELEMENTS RATES
𝐘′ =
𝐚 𝛛𝐚𝐚 𝛛𝐡𝐚 … 𝛛𝛌𝐚 𝛛𝐏𝟏𝐚 … 𝛛𝐏𝐍𝐚
𝐡
𝛌
𝛛𝐚𝐡 𝛛𝐡𝐡 … 𝛛𝛌𝐡 𝛛𝐏𝟏𝐡 … 𝛛𝐏𝐍𝐡
𝛛𝐚𝛌 𝛛𝐡𝛌 … 𝛛𝛌𝛌 𝛛𝐏𝟏𝛌 … 𝛛𝐏𝐍𝛌
⋮ ⋮ ⋮ ⋮ ⋮ ⋮
𝐘 𝛛𝐘
𝛛𝐘
/ 17
⋱ ⋱
CS – Communication & Systèmes / 30 / 30
JACOBIANS OF MEAN ELEMENTS RATES
𝐘′ =
𝐚 𝛛𝐚𝐚 𝛛𝐡𝐚 … 𝛛𝛌𝐚 𝛛𝐏𝟏𝐚 … 𝛛𝐏𝐍𝐚
𝐡
𝛌
𝛛𝐚𝐡 𝛛𝐡𝐡 … 𝛛𝛌𝐡 𝛛𝐏𝟏𝐡 … 𝛛𝐏𝐍𝐡
𝛛𝐚𝛌 𝛛𝐡𝛌 … 𝛛𝛌𝛌 𝛛𝐏𝟏𝛌 … 𝛛𝐏𝐍𝛌
⋮ ⋮ ⋮ ⋮ ⋮ ⋮
𝐘 𝛛𝐘
𝛛𝐘
𝛛𝐘
𝛛𝐏
/ 17
⋱ ⋱
CS – Communication & Systèmes / 31 / 31
APPLICATION OF THE VARIATIONAL EQUATIONS
𝐝𝛛𝐘𝛛𝐘𝟎𝐝𝐭
=𝛛𝐘
𝛛𝐘×
𝛛𝐘
𝛛𝐘𝟎
𝐝𝛛𝐘𝛛𝐏
𝐝𝐭=𝛛𝐘
𝛛𝐘×𝛛𝐘
𝛛𝐏+𝛛𝐘
𝛛𝐏
/ 18
CS – Communication & Systèmes / 32 / 32
VALIDATION
Computation of 𝛛𝐘
𝛛𝐘𝟎 and
𝛛𝐘
𝛛𝐏 matrices by finite differences and
comparison to those previously obtained.
/ 19
CS – Communication & Systèmes / 33 / 33
VALIDATION
Computation of 𝛛𝐘
𝛛𝐘𝟎 and
𝛛𝐘
𝛛𝐏 matrices by finite differences and
comparison to those previously obtained.
Problem : Different matrices !
/ 19
CS – Communication & Systèmes / 34 / 34
VALIDATION
Computation of 𝛛𝐘
𝛛𝐘𝟎 and
𝛛𝐘
𝛛𝐏 matrices by finite differences and
comparison to those previously obtained.
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.
Problem : Different matrices !
/ 19
CS – Communication & Systèmes / 35 / 35
VALIDATION
Computation of 𝛛𝐘
𝛛𝐘𝟎 and
𝛛𝐘
𝛛𝐏 matrices by finite differences and
comparison to those previously obtained.
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.
Problem : Different matrices !
/ 19
Problem solved ✔
CS – Communication & Systèmes / 36 / 36
5
/ 36
SHORT-PERIODIC TERMS DERIVATIVES
/ 20
CS – Communication & Systèmes / 37 / 37
SHORT-PERIODIC TERMS DERIVATIVES
/ 21
Automatic Differentiation
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
CS – Communication & Systèmes / 38 / 38
SHORT-PERIODIC TERMS DERIVATIVES
/ 21
Automatic Differentiation
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
CS – Communication & Systèmes / 39 / 39
6
/ 39
ORBIT DETERMINATION
/ 22
CS – Communication & Systèmes / 40 / 40
FORCE MODELS USED: LAGEOS 2
Third body attraction
Zonal and Tesseral harmonics of terrestrial potential
/ 23
Lageos 2
Third body attraction
CS – Communication & Systèmes / 41 / 41
FORCE MODELS USED: GNSS
Third body attraction
Zonal and Tesseral harmonics of terrestrial potential
Solar radiation pressure
/ 23
Third body attraction
GNSS
CS – Communication & Systèmes / 42 / 42
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
CS – Communication & Systèmes / 43 / 43
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.
CS – Communication & Systèmes / 44 / 44
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
CS – Communication & Systèmes / 45 / 45
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
CS – Communication & Systèmes / 46 / 46
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
CS – Communication & Systèmes / 47 / 47
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
CS – Communication & Systèmes / 48 / 48
5
/ 48
CONCLUSION
/ 30
CS – Communication & Systèmes / 49 / 49
OUTLOOKS
DSST
/ 31
CS – Communication & Systèmes / 50 / 50
OUTLOOKS
DSST
Need acces to « optimal » values for force models initialization
/ 31
CS – Communication & Systèmes / 51 / 51
OUTLOOKS
DSST
Need acces to « optimal » values for force models initialization
Need to optimize critical computation steps.
/ 31
CS – Communication & Systèmes / 52 / 52
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
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