Uncertainty propagation with TDA : The mysterious case of Snoopy

an author's https://oatao.univ-toulouse.fr/27575

Caleb, Thomas and Lizy-Destrez, Stéphanie Uncertainty propagation with TDA : The mysterious case of Snoopy. (2020) In: International Conference on Uncertainty Quantification & Optimisation 2020 (UTOPIAE), 16 November 2020 - 19 November 2020 (Virtual event, Belgium).

Uncertainty propagation with TDA

The mysterious case of Snoopy

Thomas Caleb

Introduction

With every measure comes an uncertainty

Uncertainty propagation is crucial in orbital mechanics

Preventing collisions between spacecraft Avoiding rendezvous failures

Forecasting collisions with asteroids

Growing number of spacecraft =⇒ Need to quantify these uncertainties to avoid major collisions (Iridium 33 And Cosmos 2251)

Objective

Implementing a propagator in Python, able to compute trajectories and quantify their uncertainties, especially for long propagations

Introduction

With every measure comes an uncertainty

Uncertainty propagation is crucial in orbital mechanics

Preventing collisions between spacecraft Avoiding rendezvous failures

Forecasting collisions with asteroids

Growing number of spacecraft =⇒ Need to quantify these uncertainties to avoid major collisions (Iridium 33 And Cosmos 2251)

Objective

Implementing a propagator in Python, able to compute trajectories and quantify their uncertainties, especially for long propagations

Introduction

With every measure comes an uncertainty

Uncertainty propagation is crucial in orbital mechanics

Preventing collisions between spacecraft

Avoiding rendezvous failures Forecasting collisions with asteroids

Growing number of spacecraft =⇒ Need to quantify these uncertainties to avoid major collisions (Iridium 33 And Cosmos 2251)

Objective

Implementing a propagator in Python, able to compute trajectories and quantify their uncertainties, especially for long propagations

Introduction

With every measure comes an uncertainty

Uncertainty propagation is crucial in orbital mechanics

Preventing collisions between spacecraft

Avoiding rendezvous failures

Forecasting collisions with asteroids

Growing number of spacecraft =⇒ Need to quantify these uncertainties to avoid major collisions (Iridium 33 And Cosmos 2251)

Objective

Implementing a propagator in Python, able to compute trajectories and quantify their uncertainties, especially for long propagations

Introduction

With every measure comes an uncertainty

Uncertainty propagation is crucial in orbital mechanics

Preventing collisions between spacecraft Avoiding rendezvous failures

Forecasting collisions with asteroids

Growing number of spacecraft =⇒ Need to quantify these uncertainties to avoid major collisions (Iridium 33 And Cosmos 2251)

Objective

Implementing a propagator in Python, able to compute trajectories and quantify their uncertainties, especially for long propagations

Introduction

With every measure comes an uncertainty

Uncertainty propagation is crucial in orbital mechanics

Preventing collisions between spacecraft Avoiding rendezvous failures

Forecasting collisions with asteroids

Growing number of spacecraft =⇒ Need to quantify these uncertainties to avoid major collisions (Iridium 33 And Cosmos 2251)

Objective

Implementing a propagator in Python, able to compute trajectories and quantify their uncertainties, especially for long propagations

Introduction

With every measure comes an uncertainty

Uncertainty propagation is crucial in orbital mechanics

Preventing collisions between spacecraft Avoiding rendezvous failures

Forecasting collisions with asteroids

Growing number of spacecraft =⇒ Need to quantify these uncertainties to avoid major collisions (Iridium 33 And Cosmos 2251)

Objective

Implementing a propagator in Python, able to compute trajectories and quantify their uncertainties, especially for long propagations

Table of contents I

1 Introduction

2 Taylor Differential Algebra

3 Dynamics

4 Case studies

5 Monte-Carlo estimations

6 Results

7 Conclusion and future work

Introduction to TDA

Taylor Differential Algebra (TDA): Assimilation of functions of n variables with their Taylor expansions of order k [Berz 1999]

Similar as the processing of real numbers into floating points

Figure:Analogy between floating points and TDA (Armellin et al.)

Libraries implementing TDA:

DACE (C++) by Politecnico di Milano [Massari 2017] Audi (C++ and Python) by D. Izzo and F. Biscani [Izzo 2018]

Introduction to TDA

Taylor Differential Algebra (TDA): Assimilation of functions of n variables with their Taylor expansions of order k [Berz 1999]

Similar as the processing of real numbers into floating points

Figure:Analogy between floating points and TDA (Armellin et al.)

Libraries implementing TDA:

DACE (C++) by Politecnico di Milano [Massari 2017] Audi (C++ and Python) by D. Izzo and F. Biscani [Izzo 2018]

Introduction to TDA

Taylor Differential Algebra (TDA): Assimilation of functions of n variables with their Taylor expansions of order k [Berz 1999] Similar as the processing of real numbers into floating points

Figure:Analogy between floating points and TDA (Armellin et al.)

Libraries implementing TDA:

DACE (C++) by Politecnico di Milano [Massari 2017] Audi (C++ and Python) by D. Izzo and F. Biscani [Izzo 2018]

Introduction to TDA

Taylor Differential Algebra (TDA): Assimilation of functions of n variables with their Taylor expansions of order k [Berz 1999] Similar as the processing of real numbers into floating points

Figure:Analogy between floating points and TDA (Armellin et al.)

Libraries implementing TDA:

DACE (C++) by Politecnico di Milano [Massari 2017]

Audi (C++ and Python) by D. Izzo and F. Biscani [Izzo 2018]

Introduction to TDA

Taylor Differential Algebra (TDA): Assimilation of functions of n variables with their Taylor expansions of order k [Berz 1999] Similar as the processing of real numbers into floating points

Figure:Analogy between floating points and TDA (Armellin et al.)

Propagating uncertainties using TDA I

Example

Solving the following Cauchy problem numerically : ( ˙y(t) = f (y(t), t)

y (0) = y0

For real numbers, Euler’s explicit method can be used :

yn+1 =yn+h · f (yn,tn) +O(h2) (1)

If y0is tainted with an error

δ

y0 =⇒ an other propagation is

needed

Propagating uncertainties using TDA I

Example

Solving the following Cauchy problem numerically : ( ˙y(t) = f (y(t), t)

y (0) = y0

For real numbers, Euler’s explicit method can be used :

yn+1 =yn+h · f (yn,tn) +O(h2) (1)

Propagating uncertainties using TDA II

For polynomial maps, : Initial condition [y0]is a polynomial :

[y0](

δ

y0) =y0+

δ

y0. Euler’s method can be adapted:

[yn+1] = [yn] +h · f ([yn],tn) +O(h2) (2)

Then, ([yn])n∈N is a sequence of polynomials

Quantification of the error, since:

f (x +

δ

x ) = f (x ) + f(1)(x )

δ

x + ... +f

(k )

k !

δ

x

k _+o

_δ

_xk +1 ₍₃₎

With x the expansion point and

δ

x the uncertainty on x Then the error is:

(

δ

x ) ≤ C   f (k +1)  (k + 1)!|

δ

x | k +1 ₍₄₎ With C > 0

Propagating uncertainties using TDA II

For polynomial maps, : Initial condition [y0]is a polynomial :

[y0](

δ

y0) =y0+

δ

y0. Euler’s method can be adapted:

[yn+1] = [yn] +h · f ([yn],tn) +O(h2) (2)

Then, ([yn])n∈N is a sequence of polynomials

Quantification of the error, since:

f (x +

δ

x ) = f (x ) + f(1)(x )

δ

x + ... +f

(k )

k !

δ

x

k_+o

_δ

_xk +1 ₍₃₎

With x the expansion point and

δ

x the uncertainty on x

Then the error is:

(

δ

x ) ≤ C   f (k +1)  (k + 1)!|

δ

x | k +1 ₍₄₎ With C > 0

Propagating uncertainties using TDA II

For polynomial maps, : Initial condition [y0]is a polynomial :

[y0](

δ

y0) =y0+

δ

y0. Euler’s method can be adapted:

[yn+1] = [yn] +h · f ([yn],tn) +O(h2) (2)

Then, ([yn])n∈N is a sequence of polynomials

Quantification of the error, since:

f (x +

δ

x ) = f (x ) + f(1)(x )

δ

x + ... +f

(k )

k !

δ

x

k_+o

_δ

_xk +1 ₍₃₎

With x the expansion point and

δ

x the uncertainty on x Then the error is:

(

δ

x ) ≤ C   f (k +1)  (k + 1)!|

δ

x | k +1 ₍₄₎ With C > 0

Propagating uncertainties using TDA III

Main properties

This trajectory can be evaluated on all the uncertainty space. Only one propagation is needed

All Ordinary Differential Equation (ODE) solvers can be adapted to TDA

A single TDA-propagation takes more time than a classic propagation

An approximation error is made when expanding to Taylor series. But the error can be lowered by increasing the order

Propagating uncertainties using TDA III

Main properties

This trajectory can be evaluated on all the uncertainty space. Only one propagation is needed

All Ordinary Differential Equation (ODE) solvers can be adapted to TDA

A single TDA-propagation takes more time than a classic propagation

An approximation error is made when expanding to Taylor series. But the error can be lowered by increasing the order

Propagating uncertainties using TDA III

Main properties

This trajectory can be evaluated on all the uncertainty space. Only one propagation is needed

All Ordinary Differential Equation (ODE) solvers can be adapted to TDA

A single TDA-propagation takes more time than a classic propagation

An approximation error is made when expanding to Taylor series. But the error can be lowered by increasing the order

Propagating uncertainties using TDA III

Main properties

This trajectory can be evaluated on all the uncertainty space. Only one propagation is needed

All Ordinary Differential Equation (ODE) solvers can be adapted to TDA

A single TDA-propagation takes more time than a classic propagation

An approximation error is made when expanding to Taylor series. But the error can be lowered by increasing the order

Compared performances I

Compared performances I

Figure:Uncertainty propagation using real numbers

Compared performances I

Compared performances I

Figure:Uncertainty propagation using real numbers

Compared performances I

Compared performances I

Figure:Uncertainty propagation using real numbers

Compared performances I

Compared performances I

Figure:Uncertainty propagation using real numbers

Compared performances II

Compared performances II

Figure:Uncertainty propagation using TDA

Compared performances II

Compared performances II

Figure:Uncertainty propagation using TDA

Compared performances II

Compared performances II

Figure:Uncertainty propagation using TDA

Compared performances III

Computation time of N trajectories:

Real numbers: N propagations are needed:

∆treal =N∆tprop,real (5)

TDA: 1 propagation and N evaluations are needed:

∆tpoly = ∆tprop,poly+N∆teval (6)

The speed-up rN provided by TDA is:

rN = ∆tprop,poly N∆tprop,real + ∆teval ∆tprop,real → ∆teval ∆tprop,real (7)

Compared performances III

Computation time of N trajectories:

Real numbers: N propagations are needed:

∆treal =N∆tprop,real (5)

TDA: 1 propagation and N evaluations are needed:

∆tpoly = ∆tprop,poly+N∆teval (6)

The speed-up rN provided by TDA is:

rN = ∆tprop,poly N∆tprop,real + ∆teval ∆tprop,real → ∆teval ∆tprop,real (7)

Compared performances III

Computation time of N trajectories:

Real numbers: N propagations are needed:

∆treal =N∆tprop,real (5)

TDA: 1 propagation and N evaluations are needed:

∆tpoly = ∆tprop,poly+N∆teval (6)

The speed-up rN provided by TDA is:

rN = ∆tprop,poly N∆tprop,real + ∆teval ∆tprop,real → ∆teval ∆tprop,real (7)

Compared performances III

Computation time of N trajectories:

Real numbers: N propagations are needed:

∆treal =N∆tprop,real (5)

TDA: 1 propagation and N evaluations are needed:

∆tpoly = ∆tprop,poly+N∆teval (6)

The speed-up rN provided by TDA is:

rN = ∆tprop,poly N∆tprop,real + ∆teval ∆tprop,real → ∆teval ∆tprop,real (7)

Ephemeris Model and SRP

The acceleration−→

γ

is written as: − →

γ

= X body ∈bodies − →

γ

body+−→

γ

SRP (8)

Position of attracting bodies delivered by SPICE [Folkner 2014]: − →

_γ

body =

μ

body − → r body −−→r − → r body −−→r 3 (9)

Solar Radiation Pressure (SRP) expressed as [Georgevic 1971]: − →

γ

SRP= − CRKSRPS m · − → r Sun−−→r − → r Sun− − → r 3 (10)

Ephemeris Model and SRP

The acceleration−→

γ

is written as: − →

γ

= X body ∈bodies − →

γ

body+−→

γ

SRP (8)

Position of attracting bodies delivered by SPICE [Folkner 2014]: − →

_γ

body =

μ

body − → r body −−→r − → r body −−→r 3 (9)

Solar Radiation Pressure (SRP) expressed as [Georgevic 1971]: − →

γ

SRP= − CRKSRPS m · − → r Sun−−→r − → r Sun− − → r 3 (10)

Ephemeris Model and SRP

The acceleration−→

γ

is written as: − →

γ

= X body ∈bodies − →

γ

body+−→

γ

SRP (8)

Position of attracting bodies delivered by SPICE [Folkner 2014]: − →

_γ

body =

μ

body − → r body −−→r − → r body −−→r 3 (9)

Solar Radiation Pressure (SRP) expressed as [Georgevic 1971]: − →

γ

CRKSRPS · − → r Sun−−→r

Modelling SRP uncertainties

The uncertainties of SRP are gathered in a single variable: [−→

γ

SRP] = −[CR0 ] KSRPS m − → r r3 (11) With:

δ

C_R0 C_R0 = s 

δ

CR CR 2 + 

δ

S S 2 + 

δ

m m 2 (12)

The expression of the acceleration can be simplified with: [

μ

Sun] =

μ

Sun− [CR0 ]

K_SRPS

m (13)

Offers a speed-up of at least 2compared to the modeling of the uncertainties on each variable

Modelling SRP uncertainties

The uncertainties of SRP are gathered in a single variable: [−→

γ

SRP] = −[CR0 ] KSRPS m − → r r3 (11) With:

δ

C_R0 C_R0 = s 

δ

CR CR 2 + 

δ

S S 2 + 

δ

m m 2 (12) The expression of the acceleration can be simplified with:

[

μ

Sun] =

μ

Sun− [CR0 ]

K_SRPS

m (13)

Offers a speed-up of at least 2compared to the modeling of the uncertainties on each variable

Modelling SRP uncertainties

The uncertainties of SRP are gathered in a single variable: [−→

γ

SRP] = −[CR0 ] KSRPS m − → r r3 (11) With:

δ

C_R0 C_R0 = s 

δ

CR CR 2 + 

δ

S S 2 + 

δ

m m 2 (12) The expression of the acceleration can be simplified with:

[

μ

Sun] =

μ

Sun− [CR0 ]

K_SRPS

m (13)

Offers a speed-up of at least 2compared to the modeling of the uncertainties on each variable

Snoopy

Apollo 10’s Lunar Module, jettisoned in 1969 in heliocentric orbit [Adamo 2012]

Figure:Apollo 10 LM Snoopy

May have reentered in 2015 under the name WT1190F

Investigated at ISAE-SUPAERO along with CNES [Hautesserres 2020, Villanueva Rourera 2020] Snoopy’s trajectory and WT1190F’s seem close No reentry observed for Snoopy

Has Snoopy reentered the Earth’s Sphere Of Influence (SOI) ?

Snoopy

Apollo 10’s Lunar Module, jettisoned in 1969 in heliocentric orbit [Adamo 2012]

Figure:Apollo 10 LM Snoopy

May have reentered in 2015 under the name WT1190F

Investigated at ISAE-SUPAERO along with CNES [Hautesserres 2020, Villanueva Rourera 2020] Snoopy’s trajectory and WT1190F’s seem close No reentry observed for Snoopy

Snoopy

Apollo 10’s Lunar Module, jettisoned in 1969 in heliocentric orbit [Adamo 2012]

Figure:Apollo 10 LM Snoopy

May have reentered in 2015 under the name WT1190F

Investigated at ISAE-SUPAERO along with CNES [Hautesserres 2020, Villanueva Rourera 2020]

Snoopy’s trajectory and WT1190F’s seem close No reentry observed for Snoopy

Has Snoopy reentered the Earth’s Sphere Of Influence (SOI) ?

Snoopy

Apollo 10’s Lunar Module, jettisoned in 1969 in heliocentric orbit [Adamo 2012]

Figure:Apollo 10 LM Snoopy

May have reentered in 2015 under the name WT1190F

Investigated at ISAE-SUPAERO along with CNES [Hautesserres 2020, Villanueva Rourera 2020]

Snoopy’s trajectory and WT1190F’s seem close

No reentry observed for Snoopy

Snoopy

Apollo 10’s Lunar Module, jettisoned in 1969 in heliocentric orbit [Adamo 2012]

Figure:Apollo 10 LM Snoopy

May have reentered in 2015 under the name WT1190F

Investigated at ISAE-SUPAERO along with CNES [Hautesserres 2020, Villanueva Rourera 2020] Snoopy’s trajectory and WT1190F’s seem close

No reentry observed for Snoopy

Has Snoopy reentered the Earth’s Sphere Of Influence (SOI) ?

Snoopy

Apollo 10’s Lunar Module, jettisoned in 1969 in heliocentric orbit [Adamo 2012]

Figure:Apollo 10 LM Snoopy

May have reentered in 2015 under the name WT1190F

Investigated at ISAE-SUPAERO along with CNES [Hautesserres 2020, Villanueva Rourera 2020] Snoopy’s trajectory and WT1190F’s seem close No reentry observed for Snoopy

Comet C/2013 A1 (Siding Spring)

Propagation of the trajectory of the comet Siding Spring

Similar trajectory as the one of Snoopy [Farnocchia 2016]

Figure:Trajectory of Siding Spring (NASA)

Position available on SPICE for a year

Used as a reference to validate the propagator

Comet C/2013 A1 (Siding Spring)

Propagation of the trajectory of the comet Siding Spring

Similar trajectory as the one of Snoopy [Farnocchia 2016]

Position available on SPICE for a year

Comet C/2013 A1 (Siding Spring)

Propagation of the trajectory of the comet Siding Spring

Similar trajectory as the one of Snoopy [Farnocchia 2016]

Figure:Trajectory of Siding Spring (NASA)

Position available on SPICE for a year

Used as a reference to validate the propagator

Comet C/2013 A1 (Siding Spring)

Propagation of the trajectory of the comet Siding Spring

Application to the case of Snoopy I

The aim is to estimatepR(t)the probability of Snoopy’s presence in

a sphere S(R) centered on the Earth, of radius R at a date t

The integral to estimate is then: pR(t) =

Z

R3

1S(R) kxkEarth
f (x)dx (14)

With f the distribution function of Xi

Monte-Carlo estimator of pR(t) ˆ p_RN(t) = 1 N N X i=1 1S(R) kXikEarth
(15)

Multiples of the radius RSOI =9.29.105km of the sphere of

Application to the case of Snoopy I

The aim is to estimate pR(t) the probability of Snoopy’s presence in

a sphere S(R) centered on the Earth, of radius R at a date t The integral to estimate is then:

pR(t) =

Z

R3

1S(R) kxkEarth
f (x)dx (14)

With f the distribution function of Xi

Monte-Carlo estimator of pR(t) ˆ p_RN(t) = 1 N N X i=1 1S(R) kXikEarth
(15)

Multiples of the radius RSOI =9.29.105km of the sphere of

influence are considered : Rn=nRSOI

Application to the case of Snoopy I

The aim is to estimate pR(t) the probability of Snoopy’s presence in

a sphere S(R) centered on the Earth, of radius R at a date t The integral to estimate is then:

pR(t) =

Z

R3

1S(R) kxkEarth
f (x)dx (14)

With f the distribution function of Xi

Monte-Carlo estimator of pR(t) ˆ pN_R(t) = 1 N N X i=1 1S(R) kXikEarth
(15)

Multiples of the radius RSOI =9.29.105km of the sphere of

Application to the case of Snoopy I

The aim is to estimate pR(t) the probability of Snoopy’s presence in

a sphere S(R) centered on the Earth, of radius R at a date t The integral to estimate is then:

pR(t) =

Z

R3

1S(R) kxkEarth
f (x)dx (14)

With f the distribution function of Xi

Monte-Carlo estimator of pR(t) ˆ pN_R(t) = 1 N N X i=1 1S(R) kXikEarth
(15)

Multiples of the radius RSOI =9.29.105km of the sphere of

influence are considered : Rn=nRSOI

Application to the case of Snoopy II

Since 1S(R) kXikEarth
follows a Bernoulli law of parameter pR, its

variance is known

The error made by the estimator can then be estimated when ˆ p_RN(t) 6= 0:

ε

NR(t) = q Var (ˆpN_R(t)) ˆ pN R(t) = q ₁ ˆ pN R(t) − 1 √ N → 0 (16) When ˆpN_R(t) = 0, the confidence interval at 99.9% is chosen to estimate the error [Hanley 1983]:

ˆ p_RN(t) ∈  0,6.9 N  = h 0, 2.76.10−4 i (17) With N = 2.5.104

Application to the case of Snoopy II

Since 1S(R) kXikEarth
follows a Bernoulli law of parameter pR, its

variance is known

The error made by the estimator can then be estimated when ˆ pN R(t) 6= 0:

ε

NR(t) = q Var (ˆpN_R(t)) ˆ pN R(t) = q ₁ ˆ pN R(t) − 1 √ N → 0 (16)

When ˆpN_R(t) = 0, the confidence interval at 99.9% is chosen to estimate the error [Hanley 1983]:

ˆ p_RN(t) ∈  0,6.9 N  = h 0, 2.76.10−4 i (17) With N = 2.5.104

Application to the case of Snoopy II

Since 1S(R) kXikEarth
follows a Bernoulli law of parameter pR, its

variance is known

The error made by the estimator can then be estimated when ˆ pN R(t) 6= 0:

ε

NR(t) = q Var (ˆpN_R(t)) ˆ pN R(t) = q ₁ ˆ pN R(t) − 1 √ N → 0 (16) When ˆp_RN(t) = 0, the confidence interval at 99.9% is chosen to estimate the error [Hanley 1983]:

ˆ p_RN(t) ∈  0,6.9  = h 0, 2.76.10−4 i (17)

Propagator validation : Siding Spring

Positions are normalized by AU = 1.5.108_{km, and velocities by}

V =qμsun

Snoopy’s trajectory

Figure:Snoopy’s trajectory

Snoopy’s trajectory

Figure:Snoopy’s trajectory

Snoopy’s probability of presence II

Figure:Snoopy’s probability of presence near Earth

Approximation error and speed-up

A set of 200 trajectories of Snoopy were propagated to quantify the error made by TDA-based MC

Order of magnitude of the error : 10−7 _{=⇒ error ≈ 100km}

This error does not affect the results on pR(t)

Approximation error close to the error made compared to SPICE

TDA-based Monte-Carlo is 100 times faster than classic Monte-Carlo on Snoopy

Approximation error and speed-up

A set of 200 trajectories of Snoopy were propagated to quantify the error made by TDA-based MC

Order of magnitude of the error : 10−7 _{=⇒ error ≈ 100km}

This error does not affect the results on pR(t)

Approximation error close to the error made compared to SPICE

TDA-based Monte-Carlo is 100 times faster than classic Monte-Carlo on Snoopy

Approximation error and speed-up

A set of 200 trajectories of Snoopy were propagated to quantify the error made by TDA-based MC

Order of magnitude of the error : 10−7 _{=⇒ error ≈ 100km}

This error does not affect the results on pR(t)

Approximation error close to the error made compared to SPICE

TDA-based Monte-Carlo is 100 times faster than classic Monte-Carlo on Snoopy

Approximation error and speed-up

A set of 200 trajectories of Snoopy were propagated to quantify the error made by TDA-based MC

Order of magnitude of the error : 10−7 _{=⇒ error ≈ 100km}

This error does not affect the results on pR(t)

Approximation error close to the error made compared to SPICE

TDA-based Monte-Carlo is 100 times faster than classic Monte-Carlo on Snoopy

Approximation error and speed-up

A set of 200 trajectories of Snoopy were propagated to quantify the error made by TDA-based MC

Order of magnitude of the error : 10−7 _{=⇒ error ≈ 100km}

This error does not affect the results on pR(t)

Approximation error close to the error made compared to SPICE

TDA-based Monte-Carlo is 100 times faster than classic Monte-Carlo on Snoopy

Conclusion

TDA propagator

Validated thanks to the comet Siding Spring

Faster than classic Monte-Carlo estimations for large samples Approximation error : Does not impact the probability estimations

Did Snoopy reenter the Earth’s sphere of influence ?

Close approach of Snoopy, at least in S(11RSOI)

No statistical evidence that Snoopy entered the Earth’s SOI from 2000 to 2013

Conclusion

TDA propagator

Validated thanks to the comet Siding Spring

Faster than classic Monte-Carlo estimations for large samples Approximation error : Does not impact the probability estimations

Did Snoopy reenter the Earth’s sphere of influence ?

Close approach of Snoopy, at least in S(11RSOI)

No statistical evidence that Snoopy entered the Earth’s SOI from 2000 to 2013

Conclusion

TDA propagator

Validated thanks to the comet Siding Spring

Faster than classic Monte-Carlo estimations for large samples

Approximation error : Does not impact the probability estimations

Did Snoopy reenter the Earth’s sphere of influence ?

Close approach of Snoopy, at least in S(11RSOI)

No statistical evidence that Snoopy entered the Earth’s SOI from 2000 to 2013

Conclusion

TDA propagator

Validated thanks to the comet Siding Spring

Faster than classic Monte-Carlo estimations for large samples

Approximation error : Does not impact the probability estimations

Did Snoopy reenter the Earth’s sphere of influence ?

Close approach of Snoopy, at least in S(11RSOI)

No statistical evidence that Snoopy entered the Earth’s SOI from 2000 to 2013

Conclusion

TDA propagator

Validated thanks to the comet Siding Spring

Faster than classic Monte-Carlo estimations for large samples Approximation error : Does not impact the probability estimations

Did Snoopy reenter the Earth’s sphere of influence ?

Close approach of Snoopy, at least in S(11RSOI)

No statistical evidence that Snoopy entered the Earth’s SOI from 2000 to 2013

Conclusion

TDA propagator

Validated thanks to the comet Siding Spring

Faster than classic Monte-Carlo estimations for large samples Approximation error : Does not impact the probability estimations

Did Snoopy reenter the Earth’s sphere of influence ?

Close approach of Snoopy, at least in S(11RSOI)

No statistical evidence that Snoopy entered the Earth’s SOI from 2000 to 2013

Conclusion

TDA propagator

Validated thanks to the comet Siding Spring

Faster than classic Monte-Carlo estimations for large samples Approximation error : Does not impact the probability estimations

Did Snoopy reenter the Earth’s sphere of influence ?

Close approach of Snoopy, at least in S(11RSOI)

No statistical evidence that Snoopy entered the Earth’s SOI from 2000 to 2013

Future work

TDA propagator improvements

Implementation of an automatic domain splitting method to control the error with more precision [Armellin 2010, Wittig 2015]

Implementation in C++ instead of Python to increase performances

Estimations on Snoopy

If Snoopy did not reenter, what is WT1190F ?

Propagating WT1190F’s trajectory to compare to two objects

Thank You !

Future work

TDA propagator improvements

Implementation of an automatic domain splitting method to control the error with more precision [Armellin 2010, Wittig 2015]

Implementation in C++ instead of Python to increase performances

Estimations on Snoopy

If Snoopy did not reenter, what is WT1190F ?

Propagating WT1190F’s trajectory to compare to two objects

Future work

TDA propagator improvements

Implementation of an automatic domain splitting method to control the error with more precision [Armellin 2010, Wittig 2015]

Implementation in C++ instead of Python to increase performances

Estimations on Snoopy

If Snoopy did not reenter, what is WT1190F ?

Propagating WT1190F’s trajectory to compare to two objects

Thank You !

Future work

TDA propagator improvements

Implementation of an automatic domain splitting method to control the error with more precision [Armellin 2010, Wittig 2015]

Implementation in C++ instead of Python to increase performances

Estimations on Snoopy

If Snoopy did not reenter, what is WT1190F ?

Propagating WT1190F’s trajectory to compare to two objects

Future work

TDA propagator improvements

Implementation of an automatic domain splitting method to control the error with more precision [Armellin 2010, Wittig 2015]

Implementation in C++ instead of Python to increase performances

Estimations on Snoopy

If Snoopy did not reenter, what is WT1190F ?

Propagating WT1190F’s trajectory to compare to two objects

Thank You !

Future work

TDA propagator improvements

Implementation of an automatic domain splitting method to control the error with more precision [Armellin 2010, Wittig 2015]

Implementation in C++ instead of Python to increase performances

Estimations on Snoopy

If Snoopy did not reenter, what is WT1190F ?

Propagating WT1190F’s trajectory to compare to two objects

Future work

TDA propagator improvements

Implementation of an automatic domain splitting method to control the error with more precision [Armellin 2010, Wittig 2015]

Implementation in C++ instead of Python to increase performances

Estimations on Snoopy

If Snoopy did not reenter, what is WT1190F ?

Propagating WT1190F’s trajectory to compare to two objects

Thank You !

References I

Daniel R. Adamo.

Earth Departure Trajectory Reconstruction of Apollo Program Components Undergoing Disposal in Interplanetary Space.

http://www.aiaahouston.org, 2012.

R. Armellin, P. Di Lizia, F. Bernelli-Zazzera et M. Berz.

Asteroid Close Encounters Characterization Using Differential Algebra: The Case of Apophis.

Springer, 2010. Martin Berz.

Modern map methods in particle beam physics.

References II

D. Farnocchia, S. R. Chesley, M. Micheli, A. Delamere, R. S. Heyd, D. J. Tholen, J. D. Giorgini, W. M. Owen et L. K. Tamppari.

High Precision Comet Trajectory Estimates: the Mars Flyby of C/2013 A1 (Siding Spring).

Icarus, 2016.

William M. Folkner, James G. Williams, Dale H. Boggs, Ryan S. Park, et Petr Kuchynka.

The Planetary and Lunar Ephemerides DE430 and DE431.

Rapport technique, Jet Propulsion Laboratory, California Institute of Technology, 2014.

R. M. Georgevic.

Mathematical Model of the Solar Radiation Forceand Torques Acting on the Componentsof a Spacecraft.

Rapport technique, Jet Propulsion Laboratory, 1971.

References III

A. Hanley J A Lippman-Hand.

If Nothing Goes Wrong, Is Everything All Right? Interpreting Zero Numerators.

Journal of the American Medical Association, 1983.

D. Hautesserres, L. Villanueva Rourera et P. Guardabasso.

Research of the History of WT1190F and that of Snoopy.

Rapport technique, Centre National d’Etudes Spatiales (CNES) and Institut Supérieur de l’Aéronautique et de l’Espace

(ISAE-SUPAERO), 2020.

Dario Izzo et Francesco Biscani.

References IV

Mauro Massari, Pierluigi Di Lizia et Mirco Rasotto.

Nonlinear Uncertainty Propagation in Astrodynamics Using Differential Algebra and Graphics Processing Units.

American Institute of Aeronautics and Astronautics, 2017. Christian P. Robert et George Casella.

Monte carlo statistical methods.

Springer, 2004.

Lydia Villanueva Rourera, Stéphanie Lizy-Destrez et Paolo Guardabasso.

Snoopy’s Trajectory - Debris Identification.

Rapport technique, Institut Supérieur de l’Aéronautique et de l’Espace (ISAE-SUPAERO), 2020.

References V

A. Wittig, P. Di Lizia, R. Armellin, K. Makino, F. Bernelli-Zazzera et M. Berz.

Propagation of Large Uncertainty Sets in Orbital Dynamics by Automatic Domain Splitting.

Domain splitting methods I

How to increase the precision on the approximation ?

Increasing the order n of the TDA =⇒ Dramatic growth of the computational time [Wittig 2015, Armellin 2010]

Quantification of the error made by the approximation, since: f (x +

δ

x ) = f (x ) + f(1)(x )

δ

x + ... +f

(n)

n!

δ

x

n_+o

_δ

_xn+1 ₍₁₈₎

With x the expansion point and

δ

x the uncertainty on x Then the error is:

(

δ

x ) ≤ C   f (n+1)  (n + 1)!|

δ

x | n+1 ₍₁₉₎ With C > 0

Domain splitting methods I

How to increase the precision on the approximation ?

Increasing the order n of the TDA =⇒ Dramatic growth of the computational time [Wittig 2015, Armellin 2010]

Quantification of the error made by the approximation, since: f (x +

δ

x ) = f (x ) + f(1)(x )

δ

x + ... +f

(n)

n!

δ

x

n_+o

_δ

_xn+1 ₍₁₈₎

With x the expansion point and

δ

x the uncertainty on x Then the error is:

(

δ

x ) ≤ C   f (n+1)  (n + 1)!|

δ

x | n+1 ₍₁₉₎ With C > 0

Domain splitting methods I

How to increase the precision on the approximation ?

Increasing the order n of the TDA =⇒ Dramatic growth of the computational time [Wittig 2015, Armellin 2010]

Quantification of the error made by the approximation, since: f (x +

δ

x ) = f (x ) + f(1)(x )

δ

x + ... +f

(n)

n!

δ

x

n_+o

_δ

_xn+1 ₍₁₈₎

With x the expansion point and

δ

x the uncertainty on x

Then the error is:

(

δ

x ) ≤ C   f (n+1)  (n + 1)!|

δ

x | n+1 ₍₁₉₎ With C > 0

Domain splitting methods I

How to increase the precision on the approximation ?

Increasing the order n of the TDA =⇒ Dramatic growth of the computational time [Wittig 2015, Armellin 2010]

Quantification of the error made by the approximation, since: f (x +

δ

x ) = f (x ) + f(1)(x )

δ

x + ... +f

(n)

n!

δ

x

n_+o

_δ

_xn+1 ₍₁₈₎

With x the expansion point and

δ

x the uncertainty on x Then the error is:

(

δ

x ) ≤ C   f (n+1)  (n + 1)!|

δ

x | n+1 ₍₁₉₎ With C > 0

Domain splitting methods II

If ∆x > 0 is the size of the uncertainty space, then:

(

δ

x ) ≤ C   f (n+1)  (n + 1)!∆x n+1 _{= } max (20)

Splitting the domain in half reduces the error

(

δ

x ) ≤ max

Domain splitting methods II

If ∆x > 0 is the size of the uncertainty space, then:

(

δ

x ) ≤ C   f (n+1)  (n + 1)!∆x n+1 _{= } max (20)

Splitting the domain in half reduces the error

(

δ

x ) ≤ max

2n+1 (21)

Domain splitting methods III

Figure:Error reduction thanks to Domain Splitting

Domain splitting methods III

Figure:Error reduction thanks to Domain Splitting

Domain splitting methods III

Figure:Error reduction thanks to Domain Splitting

Introduction to Monte-Carlo estimations I

The aim is to estimate the following integral: I =

Z

Rd

φ

(x )f (x )dx (22) With f a density of probability, and d ∈ N the dimension of the problem. [Robert 2004]

The Monte-Carlo (MC) estimator of I is: ˆ IN = 1 N N X i=1

φ

(Xi) (23)

Where N ∈ N and the random variables in the sequence (Xi)i∈N

are independent and identically distributed (IID) random variables following the distribution f .

Introduction to Monte-Carlo estimations I

The aim is to estimate the following integral: I =

Z

Rd

φ

(x )f (x )dx (22) With f a density of probability, and d ∈ N the dimension of the problem. [Robert 2004]

The Monte-Carlo (MC) estimator of I is: ˆ IN = 1 N N X i=1

φ

(Xi) (23)

Introduction to Monte-Carlo estimations II

Theorem (Convegence of the MC estimator)

If ˆIN is the MC estimator of I, then when N → ∞:

ˆ

IN −→ E(

φ

(X1)) =I (24)

With E (·) the expectation.