Can Uncertainty Propagation Solve The Mysterious Case of Snoopy ?

an author's https://oatao.univ-toulouse.fr/27577

Caleb, Thomas and Lizy-Destrez, Stéphanie Can Uncertainty Propagation Solve The Mysterious Case of Snoopy ? (2021) In: COMET ORB 2021, 9 February 2021 - 10 February 2021 (Virtual event, France).

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 C++, 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 C++, 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 C++, 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 C++, 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 C++, 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 C++, 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 C++, able to compute trajectories and quantify their uncertainties, especially for long propagations

Table of contents I

1 Introduction

2 TDA and Dynamics

3 Case studies

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 floats and TDA (from [Armellin 2010])

Libraries implementing TDA:

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

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 floats and TDA (from [Armellin 2010])

Libraries implementing TDA:

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

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 floats and TDA (from [Armellin 2010])

Libraries implementing TDA:

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

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 floats and TDA (from [Armellin 2010])

Libraries implementing TDA:

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 floats and TDA (from [Armellin 2010])

Libraries implementing TDA:

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

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

Propagating uncertainties using TDA

The following equation needs to be solved numerically : (

˙y (t) = f (y (t), t) y (0) = y0

Initial condition [y0]is a polynomial : [y0](

δ

y0) =y0+

δ

y0

The whole uncertainty space is represented by [y0]

Integration methods (Euler’s method here) can be adapted to: [yn+1] = [yn] +h · f ([yn],tn) +O(h2) (1)

The sequence of polynomials ([yn])n∈N is an expansion of the flow

Propagating uncertainties using TDA

The following equation needs to be solved numerically : (

˙y (t) = f (y (t), t) y (0) = y0

Initial condition [y0]is a polynomial : [y0](

δ

y0) =y0+

δ

y0

The whole uncertainty space is represented by [y0]

Integration methods (Euler’s method here) can be adapted to: [yn+1] = [yn] +h · f ([yn],tn) +O(h2) (1)

The sequence of polynomials ([yn])n∈N is an expansion of the flow

around y0

Propagating uncertainties using TDA

The following equation needs to be solved numerically : (

˙y (t) = f (y (t), t) y (0) = y0

Initial condition [y0]is a polynomial : [y0](

δ

y0) =y0+

δ

y0

The whole uncertainty space is represented by [y0]

Integration methods (Euler’s method here) can be adapted to: [yn+1] = [yn] +h · f ([yn],tn) +O(h2) (1)

The sequence of polynomials ([yn])n∈N is an expansion of the flow

Propagating uncertainties using TDA

The following equation needs to be solved numerically : (

˙y (t) = f (y (t), t) y (0) = y0

Initial condition [y0]is a polynomial : [y0](

δ

y0) =y0+

δ

y0

The whole uncertainty space is represented by [y0]

Integration methods (Euler’s method here) can be adapted to: [yn+1] = [yn] +h · f ([yn],tn) +O(h2) (1)

The sequence of polynomials ([yn])n∈N is an expansion of the flow

around y0

Propagating uncertainties using TDA

The following equation needs to be solved numerically : (

˙y (t) = f (y (t), t) y (0) = y0

Initial condition [y0]is a polynomial : [y0](

δ

y0) =y0+

δ

y0

The whole uncertainty space is represented by [y0]

Integration methods (Euler’s method here) can be adapted to: [yn+1] = [yn] +h · f ([yn],tn) +O(h2) (1)

Domain splitting methods I

How to increase the precision on the Taylor approximation ?

Increasing the order k 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

(k )_{(x )}

k !

δ

x

k _+o

_δ

_xk +1 ₍₂₎

With x the expansion point and

δ

x the uncertainty on x Then the error is:

(

δ

x ) ≤ C   f (k +1)_{(x )}  (k + 1)! |

δ

x | k +1 ₍₃₎ With C > 0

Domain splitting methods I

How to increase the precision on the Taylor approximation ?

Increasing the order k 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

(k )_{(x )}

k !

δ

x

k _+o

_δ

_xk +1 ₍₂₎

With x the expansion point and

δ

x the uncertainty on x Then the error is:

(

δ

x ) ≤ C   f (k +1)_{(x )}  (k + 1)! |

δ

x | k +1 ₍₃₎ With C > 0

Domain splitting methods I

How to increase the precision on the Taylor approximation ? Increasing the order k 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

(k )_{(x )}

k !

δ

x

k _+o

_δ

_xk +1 ₍₂₎

With x the expansion point and

δ

x the uncertainty on x

Then the error is:

(

δ

x ) ≤ C   f (k +1)_{(x )}  (k + 1)! |

δ

x | k +1 ₍₃₎ With C > 0

Domain splitting methods I

How to increase the precision on the Taylor approximation ? Increasing the order k 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

(k )_{(x )}

k !

δ

x

k _+o

_δ

_xk +1 ₍₂₎

With x the expansion point and

δ

x the uncertainty on x

Then the error is:

(

δ

x ) ≤ C   f (k +1)_{(x )}  |

δ

x |k +1 (3)

Domain splitting methods II

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

(

δ

x ) ≤ C   f (k +1)_{(x )}  (k + 1)! ∆x k +1_{= } max (4)

Splitting the domain in half reduces the error

(

δ

x ) ≤ max

2k +1 (5)

Domain splitting methods II

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

(

δ

x ) ≤ C   f (k +1)_{(x )}  (k + 1)! ∆x k +1_{= } max (4)

Splitting the domain in half reduces the error

(

δ

x ) ≤ max

Domain splitting methods III

Figure:Domain Splitting principle (inspired by [Wittig 2015])

Domain splitting methods III

Figure:Domain Splitting principle (inspired by [Wittig 2015])

Domain splitting methods III

Figure:Domain Splitting principle (inspired by [Wittig 2015])

Domain splitting methods III

Figure:Domain Splitting principle (inspired by [Wittig 2015])

Why choosing TDA propagation ?

Main properties

Uncertainty propagation

All ODE solvers can be adapted to TDA

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

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

Estimation error

An approximation error is made when expanding to Taylor series The error can be lowered by increasing the order, or by splitting the initial domain as needed

Why choosing TDA propagation ?

Main properties

Uncertainty propagation

All ODE solvers can be adapted to TDA

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

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

Estimation error

An approximation error is made when expanding to Taylor series The error can be lowered by increasing the order, or by splitting the initial domain as needed

Why choosing TDA propagation ?

Main properties

Uncertainty propagation

All ODE solvers can be adapted to TDA

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

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

Estimation error

An approximation error is made when expanding to Taylor series The error can be lowered by increasing the order, or by splitting the initial domain as needed

Why choosing TDA propagation ?

Main properties

Uncertainty propagation

All ODE solvers can be adapted to TDA

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

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

Estimation error

An approximation error is made when expanding to Taylor series The error can be lowered by increasing the order, or by splitting the initial domain as needed

Why choosing TDA propagation ?

Main properties

Uncertainty propagation

All ODE solvers can be adapted to TDA

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

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

Estimation error

An approximation error is made when expanding to Taylor series The error can be lowered by increasing the order, or by splitting the initial domain as needed

Why choosing TDA propagation ?

Main properties

Uncertainty propagation

All ODE solvers can be adapted to TDA

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

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

Estimation error

An approximation error is made when expanding to Taylor series

The error can be lowered by increasing the order, or by splitting the initial domain as needed

Why choosing TDA propagation ?

Main properties

Uncertainty propagation

All ODE solvers can be adapted to TDA

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

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

Estimation error

An approximation error is made when expanding to Taylor series

The error can be lowered by increasing the order, or by splitting the initial domain as needed

Ephemeris Model and SRP

The acceleration−→

γ

is written as: − →

γ

= X body ∈bodies − →

γ

body+−→

γ

SRP (6)

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

_γ

body =

μ

body − → r body −−→r − → r body −−→r 3 (7)

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

γ

SRP= − CRKSRPS m · − → r Sun−−→r − → r Sun− − → r 3 (8)

Ephemeris Model and SRP

The acceleration−→

γ

is written as: − →

γ

= X body ∈bodies − →

γ

body+−→

γ

SRP (6)

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

_γ

body =

μ

body − → r body −−→r − → r body −−→r 3 (7)

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

γ

SRP= − CRKSRPS m · − → r Sun−−→r − → r Sun− − → r 3 (8)

Ephemeris Model and SRP

The acceleration−→

γ

is written as: − →

γ

= X body ∈bodies − →

γ

body+−→

γ

SRP (6)

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

_γ

body =

μ

body − → r body −−→r − → r body −−→r 3 (7)

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

Modelling SRP uncertainties

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

γ

SRP] = −[CR0 ] · KSRPS m · [−→r ] [r ]3 (9) With:

δ

C_R0 C_R0 = s 

δ

CR CR 2 + 

δ

S S 2 + 

δ

m m 2 (10)

The expression of the acceleration can be simplified with: [

μ

Sun] =

μ

Sun− [CR0 ] ·

KSRPS

m (11)

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

Modelling SRP uncertainties

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

γ

SRP] = −[CR0 ] · KSRPS m · [−→r ] [r ]3 (9) With:

δ

C_R0 C_R0 = s 

δ

CR CR 2 + 

δ

S S 2 + 

δ

m m 2 (10) The expression of the acceleration can be simplified with:

[

μ

Sun] =

μ

Sun− [CR0 ] ·

KSRPS

m (11)

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

Modelling SRP uncertainties

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

γ

SRP] = −[CR0 ] · KSRPS m · [−→r ] [r ]3 (9) With:

δ

C_R0 C_R0 = s 

δ

CR CR 2 + 

δ

S S 2 + 

δ

m m 2 (10) The expression of the acceleration can be simplified with:

[

μ

Sun] =

μ

Sun− [CR0 ] ·

KSRPS

m (11)

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

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 Similar trajectory as the one of Snoopy [Farnocchia 2016]

2020 SO

Near Earth Object captured in October 2020 [Talbert 2020]

Identified as the Centaur rocket booster of Surveyor 2 launched on 20th September of 1966 [Nieberding 1968]

Figure:Trajectory of 2020 SO centered on the Earth (JPL)

Similar case study as Snoopy’s

Pre-Apollo era spacecraft Lost in heliocentric orbit

Long term propagation (54 years)

2020 SO

Near Earth Object captured in October 2020 [Talbert 2020]

Identified as the Centaur rocket booster of Surveyor 2 launched on 20th September of 1966 [Nieberding 1968]

Similar case study as Snoopy’s

Pre-Apollo era spacecraft Lost in heliocentric orbit

2020 SO

Near Earth Object captured in October 2020 [Talbert 2020]

Identified as the Centaur rocket booster of Surveyor 2 launched on 20th September of 1966 [Nieberding 1968]

Figure:Trajectory of 2020 SO centered on the Earth (JPL)

Similar case study as Snoopy’s

Pre-Apollo era spacecraft Lost in heliocentric orbit

Long term propagation (54 years)

2020 SO

Near Earth Object captured in October 2020 [Talbert 2020]

Identified as the Centaur rocket booster of Surveyor 2 launched on 20th September of 1966 [Nieberding 1968]

Figure:Trajectory of 2020 SO centered on the Earth (JPL)

Similar case study as Snoopy’s

Lost in heliocentric orbit

2020 SO

Near Earth Object captured in October 2020 [Talbert 2020]

Identified as the Centaur rocket booster of Surveyor 2 launched on 20th September of 1966 [Nieberding 1968]

Figure:Trajectory of 2020 SO centered on the Earth (JPL)

Similar case study as Snoopy’s Pre-Apollo era spacecraft

Lost in heliocentric orbit

Long term propagation (54 years)

2020 SO

Near Earth Object captured in October 2020 [Talbert 2020]

Identified as the Centaur rocket booster of Surveyor 2 launched on 20th September of 1966 [Nieberding 1968]

Figure:Trajectory of 2020 SO centered on the Earth (JPL)

Snoopy

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

Figure:Apollo 10 LM Snoopy (NASA)

May have reentered in 2015 under the name WT1190F

Investigated at ISAE-SUPAERO along with CNES

[Hautesserres 2020, Vagnone 2018, Villanueva Rourera 2020] Snoopy’s trajectory and WT1190F’s seem close

No reentry observed for Snoopy

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

Snoopy

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

Figure:Apollo 10 LM Snoopy (NASA)

May have reentered in 2015 under the name WT1190F

Investigated at ISAE-SUPAERO along with CNES

[Hautesserres 2020, Vagnone 2018, 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 (NASA)

May have reentered in 2015 under the name WT1190F

Investigated at ISAE-SUPAERO along with CNES

[Hautesserres 2020, Vagnone 2018, Villanueva Rourera 2020]

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

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

Snoopy

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

Figure:Apollo 10 LM Snoopy (NASA)

May have reentered in 2015 under the name WT1190F Investigated at ISAE-SUPAERO along with CNES

[Hautesserres 2020, Vagnone 2018, 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 (NASA)

May have reentered in 2015 under the name WT1190F Investigated at ISAE-SUPAERO along with CNES

[Hautesserres 2020, Vagnone 2018, Villanueva Rourera 2020] Snoopy’s trajectory and WT1190F’s seem close

No reentry observed for Snoopy

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

Snoopy

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

Figure:Apollo 10 LM Snoopy (NASA)

May have reentered in 2015 under the name WT1190F Investigated at ISAE-SUPAERO along with CNES

[Hautesserres 2020, Vagnone 2018, Villanueva Rourera 2020] Snoopy’s trajectory and WT1190F’s seem close

Monte-Carlo : The case of Snoopy

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 (12)

With f the distribution function of Xi [Robert 2004]

Monte-Carlo estimator of pR(t) ˆ pR(t) = 1 N N X i=1 1_S(R) kXikEarth
(13)

Multiples of the radius RSOI =10−2AU of the SOI are

Monte-Carlo : The case of Snoopy

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 (12)

With f the distribution function of Xi [Robert 2004]

Monte-Carlo estimator of pR(t) ˆ pR(t) = 1 N N X i=1 1_S(R) kXikEarth
(13)

Multiples of the radius RSOI =10−2AU of the SOI are

considered : Rn=nRSOI

Monte-Carlo : The case of Snoopy

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 (12)

With f the distribution function of Xi [Robert 2004] Monte-Carlo estimator of pR(t) ˆ pR(t) = 1 N N X 1S(R) kXikEarth
(13)

Multiples of the radius RSOI =10−2AU of the SOI are

Monte-Carlo : The case of Snoopy

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 (12)

With f the distribution function of Xi [Robert 2004] Monte-Carlo estimator of pR(t) ˆ pR(t) = 1 N N X i=1 1S(R) kXikEarth
(13)

Multiples of the radius RSOI =10−2AU of the SOI are

considered : Rn=nRSOI

Monte-Carlo : Error estimation

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

Var (1S(R) kXikEarth
) = pR· (1 − pR)

If ∆ˆpR =3.09

√

ˆ

pR_√·(1−ˆpR)

N , the central limit theorem ensures that :

ˆpR− ∆ˆpR, ˆpR+ ∆ˆpR



(14) is an estimation of the confidence interval at 99.9%, if Var (pR) 6=0

Since ˆpR· (1 − ˆpR) ≤ 1₄, the number of trajectories necessary to

estimate the probability can bounded by : ∆ˆpR≤ 3.09 1 2√N < 1.55 √ N (15) For instance N = 2.5.104 =⇒ ∆ˆpR <1%

Monte-Carlo : Error estimation

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

Var (1S(R) kXikEarth
) = pR· (1 − pR)

If ∆ˆpR =3.09

√

ˆ

pR_√·(1−ˆpR)

N , the central limit theorem ensures that :

ˆpR− ∆ˆpR, ˆpR+ ∆ˆpR



(14) is an estimation of the confidence interval at 99.9%, if Var (pR) 6=0

Since ˆpR· (1 − ˆpR) ≤ 1₄, the number of trajectories necessary to

estimate the probability can bounded by : ∆ˆpR≤ 3.09 1 2√N < 1.55 √ N (15) For instance N = 2.5.104 =⇒ ∆ˆpR <1%

Monte-Carlo : Error estimation

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

Var (1S(R) kXikEarth
) = pR· (1 − pR)

If ∆ˆpR =3.09

√

ˆ

pR_√·(1−ˆpR)

N , the central limit theorem ensures that :

ˆpR− ∆ˆpR, ˆpR+ ∆ˆpR



(14) is an estimation of the confidence interval at 99.9%, if Var (pR) 6=0

Since ˆpR· (1 − ˆpR) ≤ 1₄, the number of trajectories necessary to

estimate the probability can bounded by :

1 1.55

Monte-Carlo : Error estimation

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

Var (1S(R) kXikEarth
) = pR· (1 − pR)

If ∆ˆpR =3.09

√

ˆ

pR_√·(1−ˆpR)

N , the central limit theorem ensures that :

ˆpR− ∆ˆpR, ˆpR+ ∆ˆpR



(14) is an estimation of the confidence interval at 99.9%, if Var (pR) 6=0

Since ˆpR· (1 − ˆpR) ≤ 1₄, the number of trajectories necessary to

estimate the probability can bounded by : ∆ˆpR≤ 3.09 1 2√N < 1.55 √ N (15) For instance N = 2.5.104 =⇒ ∆ˆpR <1%

Validation methodology

Python prototype C++ program Dynamics validation Domain splitting validation Probability estimation validation Has Snoopy reentered in the Earth’s SOI ? Siding Spring 2020 SO

Figure:Diagram of the validation methodology

Propagator validation : Siding Spring I

Positions normalized by AU = 1.5.108_{km, velocities by V =}qμsun

AU :

Figure:Normalised error wrt Spice

Propagator validation : Siding Spring II

Propagator validation : Siding Spring III

Impact of domain splitting on the approximation error, at order 3 :

Figure:Evolution of the approximation error

2020 SO’s trajectory

Figure:2020 SO’s trajectory

2020 SO’s trajectory

TDA order : 3

Elapsed time : 115s Step : 1 day

2020 SO’s probability of presence

(a)Time dependence

TDA order : 5 Nb of domains : 8 Size sample : 25K Elapsed time : 58min TDA tolerance : 1e-11 Max TDA error : 1e-10 Step : 1 day

(b)Max probability

Figure:2020 SO’s probability of presence near Earth

Surveyor 2

TDA order : 5 Nb of domains : 16 Size sample : 100 Elapsed time : 100s Step : 1 day

Snoopy’s trajectory

Figure:Snoopy’s trajectory

Snoopy’s trajectory

TDA order : 3 Elapsed time : 99s Step : 1 day

Figure:Snoopy’s trajectory

Snoopy’s probability of presence

TDA order : 5 Nb of domains : 16 Size sample : 25K Elapsed time : 70min TDA tolerance : 1e-11 Max TDA error : 1e-7 Step : 1 day

Approximation error and speed-up

A set of 1000 trajectories of Snoopy were propagated to quantify the error made by TDA-based MC

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

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

Approximation error close to the modelling and integration error

TDA-based Monte-Carlo : 20 times faster than classic Monte-Carlo on Snoopy

Similar precision as classic Monte-Carlo

The code is still fully sequential, domain splitting may be done in parallel

Approximation error and speed-up

A set of 1000 trajectories of Snoopy were propagated to quantify the error made by TDA-based MC

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

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

Approximation error close to the modelling and integration error TDA-based Monte-Carlo : 20 times faster than classic Monte-Carlo on Snoopy

Similar precision as classic Monte-Carlo

The code is still fully sequential, domain splitting may be done in parallel

Approximation error and speed-up

A set of 1000 trajectories of Snoopy were propagated to quantify the error made by TDA-based MC

Order of magnitude of the maximum relative error : 10−7 _{=⇒ Max}

error ≈ 100km

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

Approximation error close to the modelling and integration error TDA-based Monte-Carlo : 20 times faster than classic Monte-Carlo on Snoopy

Similar precision as classic Monte-Carlo

The code is still fully sequential, domain splitting may be done in parallel

Approximation error and speed-up

A set of 1000 trajectories of Snoopy were propagated to quantify the error made by TDA-based MC

Order of magnitude of the maximum relative error : 10−7 _{=⇒ Max}

error ≈ 100km

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

Approximation error close to the modelling and integration error

TDA-based Monte-Carlo : 20 times faster than classic Monte-Carlo on Snoopy

Similar precision as classic Monte-Carlo

The code is still fully sequential, domain splitting may be done in parallel

Approximation error and speed-up

A set of 1000 trajectories of Snoopy were propagated to quantify the error made by TDA-based MC

Order of magnitude of the maximum relative error : 10−7 _{=⇒ Max}

error ≈ 100km

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

Approximation error close to the modelling and integration error

TDA-based Monte-Carlo : 20 times faster than classic Monte-Carlo on Snoopy

Similar precision as classic Monte-Carlo

The code is still fully sequential, domain splitting may be done in parallel

Approximation error and speed-up

A set of 1000 trajectories of Snoopy were propagated to quantify the error made by TDA-based MC

Order of magnitude of the maximum relative error : 10−7 _{=⇒ Max}

error ≈ 100km

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

Approximation error close to the modelling and integration error TDA-based Monte-Carlo : 20 times faster than classic Monte-Carlo on Snoopy

Similar precision as classic Monte-Carlo

The code is still fully sequential, domain splitting may be done in parallel

Approximation error and speed-up

A set of 1000 trajectories of Snoopy were propagated to quantify the error made by TDA-based MC

Order of magnitude of the maximum relative error : 10−7 _{=⇒ Max}

error ≈ 100km

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

Approximation error close to the modelling and integration error TDA-based Monte-Carlo : 20 times faster than classic Monte-Carlo on Snoopy

Similar precision as classic Monte-Carlo

The code is still fully sequential, domain splitting may be done in parallel

Snoopy’s return

Conclusion

TDA propagator

Validated thanks to the comet Siding Spring

Faster than classic Monte-Carlo estimations for large samples Approximation error : Can be tuned so that it does not impact the probability estimations

Methodology validation

2020 SO is computed to be at least in S(R6)a few month after the

launch of Surveyor 2

It seems that Surveyor 2 is likely to be in S(R6)at that time

Did Snoopy reenter the Earth’s sphere of influence ?

Close approach of Snoopy, at least in S(R7)

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 : Can be tuned so that it does not impact the probability estimations

Methodology validation

2020 SO is computed to be at least in S(R6)a few month after the

launch of Surveyor 2

It seems that Surveyor 2 is likely to be in S(R6)at that time

Did Snoopy reenter the Earth’s sphere of influence ?

Close approach of Snoopy, at least in S(R7)

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

Next close encounter with Earth : Between 2025 and 2027

Conclusion

TDA propagator

Validated thanks to the comet Siding Spring

Faster than classic Monte-Carlo estimations for large samples

Approximation error : Can be tuned so that it does not impact the probability estimations

Methodology validation

2020 SO is computed to be at least in S(R6)a few month after the

launch of Surveyor 2

It seems that Surveyor 2 is likely to be in S(R6)at that time

Did Snoopy reenter the Earth’s sphere of influence ?

Close approach of Snoopy, at least in S(R7)

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 : Can be tuned so that it does not impact the probability estimations

Methodology validation

2020 SO is computed to be at least in S(R6)a few month after the

launch of Surveyor 2

It seems that Surveyor 2 is likely to be in S(R6)at that time

Did Snoopy reenter the Earth’s sphere of influence ?

Close approach of Snoopy, at least in S(R7)

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

Next close encounter with Earth : Between 2025 and 2027

Conclusion

TDA propagator

Validated thanks to the comet Siding Spring

Faster than classic Monte-Carlo estimations for large samples Approximation error : Can be tuned so that it does not impact the probability estimations

Methodology validation

2020 SO is computed to be at least in S(R6)a few month after the launch of Surveyor 2

It seems that Surveyor 2 is likely to be in S(R6)at that time Did Snoopy reenter the Earth’s sphere of influence ?

Close approach of Snoopy, at least in S(R7)

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 : Can be tuned so that it does not impact the probability estimations

Methodology validation

2020 SO is computed to be at least in S(R6)a few month after the launch of Surveyor 2

It seems that Surveyor 2 is likely to be in S(R6)at that time

Did Snoopy reenter the Earth’s sphere of influence ?

Close approach of Snoopy, at least in S(R7)

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

Next close encounter with Earth : Between 2025 and 2027

Conclusion

TDA propagator

Validated thanks to the comet Siding Spring

Faster than classic Monte-Carlo estimations for large samples Approximation error : Can be tuned so that it does not impact the probability estimations

Methodology validation

2020 SO is computed to be at least in S(R6)a few month after the

launch of Surveyor 2

It seems that Surveyor 2 is likely to be in S(R6)at that time

Did Snoopy reenter the Earth’s sphere of influence ?

Close approach of Snoopy, at least in S(R7)

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 : Can be tuned so that it does not impact the probability estimations

Methodology validation

2020 SO is computed to be at least in S(R6)a few month after the

launch of Surveyor 2

It seems that Surveyor 2 is likely to be in S(R6)at that time

Did Snoopy reenter the Earth’s sphere of influence ?

Close approach of Snoopy, at least in S(R7)

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

Next close encounter with Earth : Between 2025 and 2027

Conclusion

TDA propagator

Validated thanks to the comet Siding Spring

Faster than classic Monte-Carlo estimations for large samples Approximation error : Can be tuned so that it does not impact the probability estimations

Methodology validation

2020 SO is computed to be at least in S(R6)a few month after the

launch of Surveyor 2

It seems that Surveyor 2 is likely to be in S(R6)at that time

Did Snoopy reenter the Earth’s sphere of influence ?

Close approach of Snoopy, at least in S(R7)

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 : Can be tuned so that it does not impact the probability estimations

Methodology validation

2020 SO is computed to be at least in S(R6)a few month after the

launch of Surveyor 2

It seems that Surveyor 2 is likely to be in S(R6)at that time

Did Snoopy reenter the Earth’s sphere of influence ? Close approach of Snoopy, at least in S(R7)

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

Next close encounter with Earth : Between 2025 and 2027

Conclusion

TDA propagator

Validated thanks to the comet Siding Spring

Faster than classic Monte-Carlo estimations for large samples Approximation error : Can be tuned so that it does not impact the probability estimations

Methodology validation

2020 SO is computed to be at least in S(R6)a few month after the

launch of Surveyor 2

It seems that Surveyor 2 is likely to be in S(R6)at that time

Did Snoopy reenter the Earth’s sphere of influence ? Close approach of Snoopy, at least in S(R7)

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

Future work

Parallelization of the computations

Estimations on 2020 SO

Refining 2020 SO’s trajectory

Thorough propagation of Surveyor 2 launcher to compare it with 2020 SO

Estimations on Snoopy

If Snoopy did not reenter, what is WT1190F ?

Propagating WT1190F’s trajectory to compare to two objects Propagation of Lunar Prospector Trans-lunar injection stage, other candidate for WT1190F

Future work

Parallelization of the computations

Estimations on 2020 SO

Refining 2020 SO’s trajectory

Thorough propagation of Surveyor 2 launcher to compare it with 2020 SO

Estimations on Snoopy

If Snoopy did not reenter, what is WT1190F ?

Propagating WT1190F’s trajectory to compare to two objects Propagation of Lunar Prospector Trans-lunar injection stage, other candidate for WT1190F

Future work

Parallelization of the computations Estimations on 2020 SO

Refining 2020 SO’s trajectory

Thorough propagation of Surveyor 2 launcher to compare it with 2020 SO

Estimations on Snoopy

If Snoopy did not reenter, what is WT1190F ?

Propagating WT1190F’s trajectory to compare to two objects Propagation of Lunar Prospector Trans-lunar injection stage, other candidate for WT1190F

Future work

Parallelization of the computations Estimations on 2020 SO

Refining 2020 SO’s trajectory

Thorough propagation of Surveyor 2 launcher to compare it with 2020 SO

Estimations on Snoopy

If Snoopy did not reenter, what is WT1190F ?

Propagating WT1190F’s trajectory to compare to two objects Propagation of Lunar Prospector Trans-lunar injection stage, other candidate for WT1190F

Future work

Parallelization of the computations Estimations on 2020 SO

Refining 2020 SO’s trajectory

Thorough propagation of Surveyor 2 launcher to compare it with 2020 SO

Estimations on Snoopy

If Snoopy did not reenter, what is WT1190F ?

Propagating WT1190F’s trajectory to compare to two objects Propagation of Lunar Prospector Trans-lunar injection stage, other candidate for WT1190F

Future work

Parallelization of the computations Estimations on 2020 SO

Refining 2020 SO’s trajectory

Thorough propagation of Surveyor 2 launcher to compare it with 2020 SO

Estimations on Snoopy

If Snoopy did not reenter, what is WT1190F ?

Propagating WT1190F’s trajectory to compare to two objects Propagation of Lunar Prospector Trans-lunar injection stage, other candidate for WT1190F

Future work

Parallelization of the computations Estimations on 2020 SO

Refining 2020 SO’s trajectory

Thorough propagation of Surveyor 2 launcher to compare it with 2020 SO

Estimations on Snoopy

If Snoopy did not reenter, what is WT1190F ?

Propagating WT1190F’s trajectory to compare to two objects

Propagation of Lunar Prospector Trans-lunar injection stage, other candidate for WT1190F

Future work

Parallelization of the computations Estimations on 2020 SO

Refining 2020 SO’s trajectory

Thorough propagation of Surveyor 2 launcher to compare it with 2020 SO

Estimations on Snoopy

If Snoopy did not reenter, what is WT1190F ?

Propagating WT1190F’s trajectory to compare to two objects

Propagation of Lunar Prospector Trans-lunar injection stage, other candidate for WT1190F

Future work

Figure:WT1190F’s preliminary trajectory

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. John J. Nieberding.

Atlas-centaur Flight Performance for Surveyor Mission B, Appendix A.

Rapport technique, NASA, 1968. Christian P. Robert et George Casella.

Monte carlo statistical methods.

Springer, 2004.

References V

Tricia Talbert.

New Data Confirm 2020 SO to be the Upper Centaur Rocket Booster from the 1960’s.

https://www.nasa.gov/feature/new-data-confirm-2020-so-to-be-the-upper-centaur-rocket-booster-from-the-1960-s, 2020.

Accessed on the 30/01/2021.

Federica Vagnone.

Snoopy’s Trajectory – Debris Identification.

References VI

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.

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.

Springer, Celest Mech Dyn Astr, 2015.

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) (16)

If y0is tainted with an error

δ

y0 =⇒ an other propagation is

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) (16)

If y0is tainted with an error

δ

y0 =⇒ an other propagation is

needed

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 I

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 II

Compared performances III

Computation time of N trajectories:

Real numbers: N propagations are needed:

∆treal =N∆tprop,real (17)

TDA: 1 propagation and N evaluations are needed:

∆tpoly = ∆tprop,poly+N∆teval (18)

The speed-up rN provided by TDA is:

rN = ∆tprop,poly N∆tprop,real + ∆teval ∆tprop,real → ∆teval ∆tprop,real (19)

Compared performances III

Computation time of N trajectories:

Real numbers: N propagations are needed:

∆treal =N∆tprop,real (17)

TDA: 1 propagation and N evaluations are needed:

∆tpoly = ∆tprop,poly+N∆teval (18)

The speed-up rN provided by TDA is:

rN = ∆tprop,poly N∆tprop,real + ∆teval ∆tprop,real → ∆teval ∆tprop,real (19)

Compared performances III

Computation time of N trajectories:

Real numbers: N propagations are needed:

∆treal =N∆tprop,real (17)

TDA: 1 propagation and N evaluations are needed:

∆tpoly = ∆tprop,poly+N∆teval (18)

The speed-up rN provided by TDA is:

rN = ∆tprop,poly N∆tprop,real + ∆teval ∆tprop,real → ∆teval ∆tprop,real (19)

Compared performances III

Computation time of N trajectories:

Real numbers: N propagations are needed:

∆treal =N∆tprop,real (17)

TDA: 1 propagation and N evaluations are needed:

∆tpoly = ∆tprop,poly+N∆teval (18)

The speed-up rN provided by TDA is:

rN = ∆tprop,poly N∆tprop,real + ∆teval ∆tprop,real → ∆teval ∆tprop,real (19)

Compared performances IV

Figure:Speed-up provided by TDA-based Monte-Carlo

Introduction to Monte-Carlo estimations I

The aim is to estimate the following integral: I =

Z

Rd

φ

(x )f (x )dx (20)

With f a density of probability, and d ∈ N the dimension of the problem. [Robert 2004]

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

φ

(Xi) (21)

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 (20)

With f a density of probability, and d ∈ N the dimension of the problem. [Robert 2004]

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

φ

(Xi) (21)

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 II

Theorem (Convegence of the MC estimator)

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

ˆIN −→ E(

φ

(X1)) =I (22)

Monte-Carlo : Error estimation II

The confidence interval cannot by computed thanks to CLT if Var (pR) =0. Thus, if ˆpR ∈ {0, 1}

If ˆpR=0, the confidence interval at 99.9% is estimated as

[Hanley 1983] : ˆ p_R∈  0,6.9 N  (23) If ˆpR=1, the confidence interval at 99.9% is estimated in the same

way : ˆ pR ∈  1 − 6.9 N ,1  (24)