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69th International Astronautical Congress (IAC), Bremen, Germany, 1-5 October 2018.

Copyright ©2018 by M. Romano, C. Colombo and J. M. Sánchez Pérez,

Published by the IAF, with permission and released to the IAF to publish in all forms

IAC-18-A3.5.10 Page 1 of 9

IAC-18-A3.5.10

Efficient planetary protection analysis for interplanetary missions

Matteo Romanoa*, Camilla Colomboa, Jose Manuel Sánchez Pérez b

a Dept. of Aerospace Science and Technology, Politecnico di Milano, Milano, Italy, matteo1.romano@polimi.it,

camilla.colombo@polimi.it

b European Space Operations Centre (ESOC), ESA, Darmstadt, Germany, Jose.Manuel.Sanchez.Perez@esa.int

* Corresponding Author

Abstract

The process of verifying that interplanetary missions respect the planetary protection requirements must account

for uncertainties in the design parameters of the mission and perform long numerical simulations to estimate the

impact probability of the mission-related objects with celestial bodies that could develop extra-terrestrial life. This

kind of analysis is usually done via Monte Carlo simulation, with high computational cost since the requirements

also include high confidence levels of the probability estimate. In order to reduce the computational load of the

simulation the line sampling method, already analysed in previous works, is used here in order to further characterise

his numerical performance, by providing an approximate formula highlighting the dependency of the method from

the level of probability and the shape of the impact regions in the uncertainty space, and by analysing how its

accuracy changes for different shapes of the initial distribution. The observations made here will allow to identify in

advance in which cases the method will perform better than the standard Monte Carlo according to the expected

impact probability and the shape of the initial distribution.

Keywords: Planetary protection, Orbital propagation, Numerical integration, Monte Carlo sampling

1. Introduction

1.1 Planetary protection

Planetary protection (PP) requirements set stringent

constraints on the design of trajectories in the Solar

System, since they aim to avoid the contamination of

planets and moons where life could develop by limiting

the probability of impact between spacecraft or launcher

stages and these celestial bodies [1]. The process of

verifying that spaceflight missions fulfil the

requirements is typically performed via Monte Carlo

(MC) methods and must account for uncertainties in the

design parameters of the spacecraft, random failures,

errors in the determination of its state, chaotic n-body

dynamics and not modelled effect in the dynamics,

which introduces numerical errors in the propagation of

the trajectory. This is expensive in terms of numerical

resources, since the requirements also include long time

intervals (up to 100 years in most cases) and high

confidence levels of the probability estimates, which

increases the number of propagations to perform.

1.2 Proposed approach

On the side of the statistical analysis, to reduce the

computational load, the impact probability is estimated

through the use of the Line Sampling (LS) method

[2][4], as an alternative to the conventional Monte Carlo

method, which propagates a large number of initial

conditions directly sampled from an uncertainty

distribution; instead, LS samples the initial distribution

in a more efficient way, aimed to provide a probability

estimation with a higher confidence level, or employing

a lower number of samples to reach the desired

accuracy level. In this work, the LS is further

characterised by analysing how the shape of the initial

uncertainty (expressed through a covariance matrix)

affects numerical performance of the method; in

addition, the number of LS runs necessary to reach the

confidence level imposed by the PP requirements is

estimated in advance by using an approximated

analytical formula which was developed starting from

the information already available in the literature.

On the other side, the orbital propagations are

carried out by taking into account the characterisation of

the close approaches with planetary bodies, by obtaining

information about the dynamics using the eigenvalues

of the Jacobian matrix of the equations of motion.

The techniques presented here have been

implemented into SNAPPshot (tool suite for the

verification of the compliance to planetary protection

requirements initially developed at the University of

Southampton in the framework of a study for ESA

[5][6]). The tool follows a Monte Carlo approach, where

the initial uncertainty (over the state or other design

parameters of the spacecraft or launcher) is sampled into

many initial conditions, that are then propagated to

estimate the probability of impact (or orbital resonance)

with other celestial bodies.

1.3 Manuscript content and outline

69th International Astronautical Congress (IAC), Bremen, Germany, 1-5 October 2018.

Copyright ©2018 by M. Romano, C. Colombo and J. M. Sánchez Pérez,

Published by the IAF, with permission and released to the IAF to publish in all forms

IAC-18-A3.5.10 Page 2 of 9

The paper is organised as follows: Section 2 is

dedicated to explaining the advances on the application

of the LS method for MC analysis, and to the approach

proposed to improve the accuracy and efficiency of the

orbital propagation in terms of analysis of the close

approaches; Section 3 considers the planetary protection

analysis for the launcher upper stage of the Solo

spacecraft to show the application of all the techniques,

by defining test cases aimed to showing how different

conditions affect the accuracy of the impact probability

estimate to obtain a criterium to identify in advance

when the LS will be more efficient than the standard

MC; finally, Section 4 will summarise the main results

and conclusions, and anticipate future developments.

2. New developments

2.1 Line Sampling

The LS method was introduced in previous works

[2][3], where a general explanation of the theory behind

it was presented, together with the results of its

application to different test cases, to show how the

choice of this sampling method can improve the

efficiency of the MC simulations for planetary

protection analysis. In short, the main feature of this

method is the analytical estimation of the probability,

obtained by reducing the multi-dimensional integration

problem across the uncertainty domain to many one-

dimensional problems along lines following a reference

direction that are used to sample the initial distribution;

this direction is determined so that it points toward an

impact region of the domain, and, if this is properly

chosen, the method can considerably reduce the number

of required system simulations with respect to a

standard MC.

In this work, the method is further developed by

introducing a way to estimate in advance the number of

runs that are required to reach a desired confidence level

for a given expected impact probability. This is done to

mirror the functionality that is already built in the

SNAPPshot tool for the standard MC analysis.

In the case of the LS, the literature already gives a

qualitative estimation of its efficiency compared with

the standard MC in terms of convergence rate [4]. A

summary of it is reported in Section 2.1.1 to introduce

the notation that will be used in Section 2.1.2 .

2.1.1 Theoretical formulation of the LS method

Following the explanation and the notation

presented in [4], the probability of the event F (which

can be seen as the failure of a system or, in this case, an

impact with a celestial body) can be expressed as the

multidimensional integral in the form

( ) ( ) ( ) ( )

F

P F P F I q d= = X

x x x x

(2.1)

where

( )

1,..., d

d

xx=x

is the vector of the uncertain

variables of the system,

()qXx

is the multidimensional

probability density function (pdf), F is the subdomain of

the variables

x

leading to the event of interest, defined

by a performance function

()gXx

(which is lower than

or equal to zero if

Fx

and greater than zero

otherwise) and

()

F

Ix

is an indicator function such that

( ) 1

F

I=x

if

Fx

and

( ) 0

F

I=x

otherwise.

A coordinate transformation from the physical space

to the standard normal space

:T→

Xθxθ

brings as

advantages the normalisation of the physical variables

through the covariance matrix, and the possibility to

express the multidimensional pdf as a product of d unit

Gaussian standard distributions

()

jj

:

1

( ) ( )

d

jj

j

=

=

θ

(2.2)

With reference to Fig. 1, in the d-dimensional standard

normal space, the domain F is the subspace for which

the samples

( )

1,..., T

d

=θ

satisfy a given property

(e.g. an impact with a planet or a system failure). With

the assumption that

1

points in the direction of the

sampling vector

α

(this can always be assured by a

suitable rotation of the coordinate axes), the subdomain

F can be also expressed as

( )

1 1 1

: ,..., ,...,

d

jd

FF

= θ

(2.3)

with

1

1

d

F−

, in this way the region F corresponds to

the values of

θ

such that the performance function

()gθθ

satisfies the relation

, 1 1 1

( ) ( ) 0gg

−−

= −

θθ

θθ

,

where

( )

1

12

,..., Td

d

−

−=θ

.

69th International Astronautical Congress (IAC), Bremen, Germany, 1-5 October 2018.

Copyright ©2018 by M. Romano, C. Colombo and J. M. Sánchez Pérez,

Published by the IAF, with permission and released to the IAF to publish in all forms

IAC-18-A3.5.10 Page 3 of 9

Fig. 1 - Scheme representing the sampling procedure

along a line (characterised by the ck parameter)

to identify the border (red line) of the region F of

interest (image from [4]).

Considering this change of variables and the

definition in Eq. (2.3), the integral in Eq. (2.1) can be

rewritten as

1

( ) ( ) ( ) ( )

d

F j j F

j

d

P F I d E I

=

==

θ θ θ

(2.4)

(where

[]EX

is defined as the expected value of the

generic random variable

X

) and manipulated as

follows:

( )

( )

( )

1

1

1 1 1 1 1

2

1

1 1 1

2

1

11

( ) ( ) ( )

... ( ) ( ) ( )

... ( ) ( )

()

d

F j j

j

d

d

F j j

j

d

d

jj

j

d

P F I d

I d d

Fd

EF

−

=

−−

=

−

−−

=

−

−

=

=

=

=

θ

θθ

θθ

θθ

θ

(2.5)

where

( ) ( ) ( )

A

A I d

=

x x x

is the definition of the

Gaussian measure of A, where A is the subset of the

random variables

x

which lead to a given result (e.g. an

impact). In case of the standard MC (which could be

considered a Point Sampling method, in relation with

the LS),

( )

11

()F−

θ

is a discrete random variable equal

to

()

F

Iθ

(meaning that

( ) ( )

2

1 1 1 1

( ) ( )FF

−−

= θθ

is

always true

1

1

d−

−

θ

), while for the LS method

( )

11

()F−

θ

is a continuous random variable where

11

()

kk

Fc

−=−θ

(see Fig. 1, where the sampling

procedure is represented highlighting the boundary of

the region corresponding to the event F), meaning that

( )

11

0 ( ) 1F−

θ

and

( ) ( )

2

1 1 1 1

0 ( ) ( )FF

−−

θθ

are always true

1

1

d−

−

θ

.

The consequence of these properties is visible when

considering the definition of variance of an estimator for

the two methods. An estimator

ˆ()PF

of the probability

()PF

as expressed in Eq. (2.5) can be computed as

( )

11

1

1

ˆ( ) ( )

T

Nk

k

T

P F F

N−

=

=

θ

(2.6)

where

, 1,...,

k

T

kN=θ

are independent and identically

distributed samples in the standard normal coordinate

space. Given the generic definition of variance for

()PF

following Eq. (2.5) as

( )

( )

( ) ( )

( )

11

1

2

2

1 1 1 1

22

1 1 1 1

22

11

()

( ) ( ) ( )

( ) ( )

( ) ( )

PF

F P F d

E F E F

E F P F

−−

−

− − −

−−

−

= −

= −

= −

θθ

θ

θ θ θ

θθ

θ

(2.7)

the variance of the estimator

ˆ()PF

is defined as

( )

( ) ( )

2

22

11

ˆ()

( ) ( )

TT

PF

P F N F N

−

= = θ

(2.8)

meaning that the variance of the estimator directly

depends on the variance of the random variable

( )

11

()F−

θ

. Consequently

( )

( ) ( )

( )

( )

( )

11

11

2

11

22

1 1 1 1

2

1 1 1 1

2

()

( ) ( )

( ) ( )

( ) 1 ( ) ( )

F

F

E F E F

E F E F

P F P F I

−−

−−

−

−−

−−

= −

−

= − =

θθ

θθ

θ

θθ

θθ

θ

(2.9)

A coefficient of variation (c.o.v.)

2ˆ

( ( )) ( )P F P F

=

can be defined as a measure of

the efficiency of the sampling method, with lower

values of

meaning a higher efficiency of the method

in converging to the exact value of the probability. Eq.

(2.9) demonstrates that the c.o.v. of estimator in Eq.

(2.6) as given by the LS method is always smaller than

the one given by the standard MC, implying that the

convergence rate of the LS is always faster than, or as

fast as, that of the standard MC.

2.1.2 New developments

69th International Astronautical Congress (IAC), Bremen, Germany, 1-5 October 2018.

Copyright ©2018 by M. Romano, C. Colombo and J. M. Sánchez Pérez,

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IAC-18-A3.5.10 Page 4 of 9

Fig. 2 - Scheme representing the approximations

used to express the variance of the LS method as

a function of the probability estimate.

While in the case of the standard MC Eq. (2.9) is

easy to treat, since

( ) ( )

11

2

1 1 1 1

( ) ( )E F E F

−−

−−

=

θθ

θθ

, in the LS case

( )

1

2

11

()EF

−−

θθ

is a continuous variable, defined

through the integral

( )

( )

1

2

11

2

1 1 1

2

1

()

... ( ) ( )

d

jj

j

d

EF

Fd

−−

−−

=

−

=

θθ

θθ

(2.10)

which cannot be easily manipulated analytically due to

the presence of

( )

2

11

()F−

θ

. For this reason, it is

chosen to express this term with an approximation.

The definition of

( )

11

()F−

θ

given in Eq. (2.5) can

be further expanded as

( )

( ) ( )

1

1 1 1 1 1 1

1 1 1 1 1

()

( ) ( ) ( )

( ) 1 ( ) ( )

F

c

F I d

d c c

−

−−

−

−−

=

= = − = −

θ

θθ

θθ

(2.11)

with

1

()c−

θ

defined as the border of the region F

displayed in Fig. 2 as a red line.

1

()c−

θ

is then

expanded as

11

ˆ

( ) ( )c c c

−−

=+θθ

, with the first term

defined as an “average” value of

1

()c−

θ

(represented as

a dashed blue line in Fig. 2) such that

( )

( ) ( )

11 1 1 1 ˆ

( ) ( ) ( )P F E F F c

−−−

= = = −

θθθ

, and

the second term as a variation with respect to this

average value.

The hypothesis is made that

1

()c

−

θ

represents a

small variation with respect to the average value

ˆ

c

, as

in the case of a quasi rectilinear border of the region F

orthogonal to the sampling direction

α

. Under this

hypothesis, the integral in Eq. (2.11) can be rewritten as

( )

( )

11

1

11

1 1 1 1 1 1

ˆ

( ) ( )

ˆ()

1 1 1 1 1 1

ˆˆ

1

()

( ) ( )

( ) ( )

ˆˆ

( ) ( )

c c c

cc

cc

F

dd

dd

c c c

−−

−

−

+

+

−

==

=−

− −

θθ

θ

θ

θ

(2.12)

resulting in

( )

( )( )

( ) ( )

( ) ( )

1

1

1

1 1 1

11

2

11

2

1

2 2 2

11

2 2 2

11

2 2 2

11

()

ˆˆ

( ) ( )

ˆ ˆ ˆ ˆ

2 ( ) ( ) ( ) ( )

ˆ ˆ ˆ ˆ

2 ( ) ( ) ( ) ( )

ˆˆ

( ) ( ) 2 ( ) ( ) ( ) ( )

EF

E c c c

E c c c c c c

E c E c c c E c c

P F P F c E c c E c

−

−

−

− − −

−−

−

−

−−

−−

−−

− −

= − − − +

= − − − +

= − +

θ

θ

θ

θ θ θ

θθ

θ

θ

θθ

θθ

θθ

(2.13)

Taking expression (2.13) into account, and defining in a

compact way

1

11

( ) ( )c E c

−

−−

=

θ

θθ

, the variance

given by the LS in Eqs. (2.7) and (2.9) becomes

( ) ( )( )

( )

( )

1

1

22

11

22

11

22

11

1

2

( ) ( )

( ) ( )

ˆˆ

( ) 2 ( ) ( ) ( ) ( )

ˆ

( ) 2 ( ) ( )

( ) 1 ( ) ( )

F

P F F

E F P F

P F c c c E c

P F c c

P F P F I

−

−

−

−

−−

−

=

= −

− +

−

− =

θ

θ

θ

θ

θθ

θ

θ

(2.14)

Highlighting the new terms in Eq. (2.15)

( )( )

( )

2

1 1 1

2

ˆ

( ) ( ) 2 ( ) ( )

( ) 1 ( ) ( )

F

F P F c c

P F P F I

−−

−

− =

θθ

θ

(2.15)

this means that a new estimation for the worst

covariance given by the LS method (nominally, from

Eq. (2.9), equal to the one given by the standard MC)

was obtained, which takes into account the probability

level through the term

ˆ

()c

, and the shape of the region

F and the direction of sampling through the term

1

()c−

θ

. When the approximation of small

1

()c

−

θ

is

valid (that is, when the region F has a regular shape and

is distributed across the initial uncertainty, and the

sampling direction is chosen properly so that it points

toward it) and the probability level is low, the term

1

22

1

ˆ

( ) ( )c E c

−−

θθ

is also small, and we can say

that the variance given by the LS is below a value

( )

1

( ), ( )f P F c −

θ

such that

δc(θ)

ˆ

c

69th International Astronautical Congress (IAC), Bremen, Germany, 1-5 October 2018.

Copyright ©2018 by M. Romano, C. Colombo and J. M. Sánchez Pérez,

Published by the IAF, with permission and released to the IAF to publish in all forms

IAC-18-A3.5.10 Page 5 of 9

( ) ( ) ( )

22

1

( ) ( ), ( ) ( )

LS MC

P F f P F c P F

−

θ

, thus

increasing the convergence rate of LS with respect to

standard MC. On the contrary, when the approximation

does not hold (that is in cases with high probability

levels, non-optimal sampling direction, or badly shaped

impact regions),

( )

1

( ), ( )f P F c −

θ

grows toward the

covariance level of the MC.

2.2 Fly-by detection

2.2.1 Explanation

As already pointed out in previous works [2], close

approaches with planetary bodies critically influence the

accuracy of the numerical propagation, due to an

increase of the nonlinearity of the dynamics with respect

to the interplanetary phase of the propagation. This

effect is stronger for very close fly-bys and was

observed to affect in similar ways any integration

method that was examined.

For this reason, a technique that uses information

from the dynamics to identify numerically a fly-by

condition has been developed as a criterion that can be

evaluated automatically during the integration. The

Jacobian of the equations of motion (expressed via its

eigenvalues and their derivatives) is used to detect when

the propagated object is approaching a planet, looking at

both the relative position (already accounted for when

considering distance-based criteria such as Sphere of

Influence SOI radius) and the relative velocity between

the planet and the object. This method was already

introduced in a previous work [2], while here it will be

explained in detail and applied to more test cases.

The equations of motion of the barycentric restricted

n-body problem can be written as

3

0

()

()

nj

j

jj

t

t

=

−

= = − −

rr

ar rr

(2.16)

with the Jacobian matrix defined as

d

d

==

0I

f

JG0

x

(2.17)

where

x

is the state vector, containing position and

velocity vectors

r

and

v

),

I

is the identity matrix, and

G

results from the derivation of the gravitational

accelerations defined in Eq. (2.16):

0

n

j

j=

=

==

GaG

r

(2.18)

where the index j=0 corresponds to the main attractor of

the system. The set of eigenvalues of the complete

Jacobian are given by

det( ) det( )

= − = −GI I G I

(2.19)

with

being the eigenvalue with the maximum

absolute value. In this case, as an approximation, only

the contributions to the Jacobian given by each planet

alone are considered:

det( ), 1,...,

jj jN

= − =GI

(2.20)

with

j

being the set of eigenvalues given by the

contribution of j-th planet, and

j

the maximum

eigenvalue of such set (it is clear that

jj

).

Debatin et al. [7] propose a simplified expression for

that can be estimated as

3

2j

j

j

= −rr

(2.21)

With this criterion, not only the single eigenvalues

are considered, but also their derivative in time:

( ) ( )

5

3

2

T

jj

jj

j

−−

=

−

r r v v

rr

(2.22)

The value of the eigenvalue contributions given by

the single planets are compared with the one given by

the main attractor (the Sun in the case of an

interplanetary trajectory), and the same is done for their

derivatives. A fly-by event is identified when one or

both ratios in Eq. (2.23) reaches a given threshold:

01

02

j

j

(2.23)

For the application of this method, one or both

expressions in (2.23) (ratio of the values and ratio of the

derivatives of the eigenvalues) can be used, separately

or together, as shown in the next section.

2.2.2 Example

The application of this method can be seen in Fig. 3,

which shows the case of multiple close approaches

between the launcher upper stage od Solo and Venus. In

both cases the variations of the eigenvalues and of their

derivatives are compared with the crossing of the SOI

and Hill sphere of Venus, to show the differences

between the two criteria.

69th International Astronautical Congress (IAC), Bremen, Germany, 1-5 October 2018.

Copyright ©2018 by M. Romano, C. Colombo and J. M. Sánchez Pérez,

Published by the IAF, with permission and released to the IAF to publish in all forms

IAC-18-A3.5.10 Page 6 of 9

a)

b)

Fig. 3 - Variation in time of the eigenvalues corresponding to the single contribution of the Sun and Venus

during the propagation of the nominal trajectory of Solo’s launcher upper stage: values (a) and

derivatives (b) of the eigenvalues, compared with the crossing of SOI (red area) and Hill sphere (green

area) of Venus. The vertical dashed lines refer to the epochs where the ratio between the eigenvalues of

Venus and the Sun is equal to 1.0.

Fig. 3 shows the variation in time of the eigenvalues

relative to Venus and the Sun and their associated

derivative (respectively (a) and (b)) during the

propagation of the nominal trajectory of the launcher

upper stage of Solo. In both cases, the comparison is

done on the 100 years propagation (top), with a focus on

the 1st close approach with Venus during the first year

of the mission (centre), and on a second close approach

80 years later (bottom). In both graphs, different

information is reported: the red and green areas

represent, respectively, the crossing of the SOI and the

Hill sphere of Venus; the vertical dashed lines indicate

the epochs where the ratios defined in Eq. (2.23) are

both equal to 1.0 (only one value is used for both ratios

for simplicity).

The plots show that both close approaches can be

successfully identified using the definitions defined

previously. In particular, a threshold value of 1.0 for the

ratios correctly identifies the not only the 1st CA (where

an actual crossing of the SOI occurs), but also the 2nd

CA, which happens at a larger distance from Venus,

with no SOI crossing. This is possible due to the

information about the relative velocity between the

propagated body and the planet contained in the

derivative in Eq. (2.22). Notice also that a tolerance

equal to 1.0 allows to determine initial and final epochs

for the close approach in a broader sense than the ones

defined by the SOI and Hill sphere crossings,

particularly for the derivative ratio, meaning that a

lower value can be also used in the case of

interplanetary trajectories.

3. Planetary protection analysis

3.1 Test case definition

69th International Astronautical Congress (IAC), Bremen, Germany, 1-5 October 2018.

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IAC-18-A3.5.10 Page 7 of 9

The techniques presented in the previous sections

were implemented into the SNAPPshot tool [5][6] and

used to perform a Planetary Protection (PP) analysis for

the Atlas V upper stage of ESA’s SolO (Solar Orbiter)

mission (according to the October 2018 launch option

[8]. The analysis will focus on the fly-by with Venus the

is expected during the first year of the mission with the

given launch option. Although this planet has no

explicit planetary protection requirements, the Venus

fly-by represents an interesting case to test the LS

technique. Initial data are taken from [5], with initial

conditions and covariance matrix expressed in Cartesian

coordinates.

A series of test cases is defined by modifying the

covariance matrix used for the simulation to reproduce

the effect of different shapes of the initial uncertainty

distribution on the performance of the LS method. In

particular, it is chosen to perform a transformation

composed of a rotation into the local-vertical-local-

horizontal frame, followed by a squeezing

transformation in the along-track direction expressed by

the matrix F defined as

1/ 0 0

, 0 0

0 0 1/

f

f

f

==

sq

sq

sq

F0

FF

0F

(3.1)

where the elements of the rotated covariance matrix

referring to the along-track direction (of both the

position and the velocity) are increased by a factor

f

,

while the components in the radial and normal-to-plane

directions are reduced by the same factor, in order to

preserve the total volume of the uncertainty distribution.

This choice was made following the observations

reported in [3], where the method was successfully

applied to different test cases involving the propagation

of NEOs, as in those cases the initial uncertainty

distribution appeared highly elongated in the along-

track direction of the orbit.

3.2 Dynamical model and propagation setup

The propagations are carried out in Cartesian

coordinates with respect to an EME2000 reference

frame centred in the Solar System Barycentre (SSB),

with the inclusion of the gravitational contributions of

the Sun, all the major planets, and the Earth’s moon.

Most of the physical constants (gravitational

parameters, planetary radii, etc.) are obtained from the

JPL Horizons database via the SPICE toolkit

*

.

Propagations are stopped according to 3 conditions:

the maximum time is reached; an impact with one of the

included celestial bodies occurs; an escape from the SOI

of the Sun occurs.

*

https://naif.jpl.nasa.gov/naif/

The propagations are carried out with the use of an

8th order Dormand-Prince RK method, with an

embedded scheme to adapt the time-step (already

available in SNAPPshot) with absolute and relative

tolerances of 10-12.

3.3 Results

In this Section the results of the application of LS to

the selected test case for different shapes of the initial

uncertainty, obtained by setting different values for the

squeezing factor of the covariance matrix.

Fig. 4 shows the uncertainty distributions in two

different cases: the unmodified one in Fig. 4a, and one

elongated in the along-track direction using a squeezing

factor f=16 in Fig. 4b, with the impact region (found via

standard MC) highlighted in red, and its boundary. The

results of the corresponding simulations are reported in

Table 1 and Table 2 respectively, in terms of number of

random samples, number of orbital propagations, impact

probability and the relative standard deviation (which is

used as a measure of the accuracy of the methods, with

smaller values corresponding to a higher accuracy).

Note that in the LS more propagations are performed

than in standard MC due to the numerical iterations

necessary to identify the zeroes of the performance

function that defines the border of the impact region, as

already specified in [2]. In particular, in all cases

presented here 10 iterations were used in order to

identify the border to ensure a correct identification of

the boundary.

a)

69th International Astronautical Congress (IAC), Bremen, Germany, 1-5 October 2018.

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IAC-18-A3.5.10 Page 8 of 9

b)

Fig. 4 – Representations of the initial velocity

dispersions for the launcher upper stage of Solo

mission, shaped differently according to a

squeezing factor f=1 (no squeezing) in (a), and

f=16 in (b). The initial conditions leading to an

impact with Venus (and identified via standard

MC simulation) are shown in red, while the

boundary of the impact region (computed via

LS) is shown in green. The blue arrow represents

the sampling direction.

Nsamples

Nprop

ˆ

P(I)

ˆ

σ

MC

54114

54114

4.34e-2

8.76e-4

LS

~46000

~460000

4.63e-2

4.41e-4

Table 1 – Results of the application of standard MC

and LS to the test case with unmodified initial

distribution (squeezing factor f=1), as in Fig. 4a.

Nsamples

Nprop

ˆ

P(I)

ˆ

σ

MC

54114

54114

2.01e-2

6.15e-4

LS

~46000

~500000

2.01e-2

2.15e-4

Table 2 – Results of the application of standard MC

and LS to the test case with modified initial

distribution (squeezing factor f=16), as in Fig.

4b.

From Fig. 4 one can see that the impact regions with

Venus in the two cases are identified by the same initial

conditions, but while in case (a) the impact region is

lumped and all contained inside the uncertainty

distribution, in case (b) the impact region goes from side

to side of the distribution, thus representing one of the

favourable cases already shown in [3]. This is

confirmed by the results reported in Table 1 and Table

2, showing that the value of standard deviation given by

the LS, already lower than the one of the MC, decreases

when the distribution has an elongated shape.

Similar considerations can be made by considering

Fig. 5, where the variation of the values of impact

probability and the associated standard deviation for

more values of the squeezing factor f. It shows that even

a low elongation of the initial distribution can decrease

the value of the standard deviation, thus improving the

accuracy of the LS.

Fig. 5 – Variation of the impact probability with

Venus (top) and of the associated standard

variation (bottom) with the variation of the

squeezing factor f, comparing standard MC and

LS.

4. Conclusions

The work presented here describes some

developments of the state of research about the LS

method, by providing a better understanding of the

performance of the method with respect to the one of

the standard MC simulations. This is done both

theoretically, by providing an approximated formula

that highlights the dependency of the method both from

the level of impact probability (as already proven by the

existing literature and the previous works related to the

method) and from the shape of the impact region, and

numerically, by providing a test cases devised to show

how the accuracy of the LS depends on the shape of the

initial uncertainty distribution. The information gained

from this work can be used to identify in advance in

which cases the LS (compared with the standard MC)

will be more efficient (in terms of number of random

samples needed to reach a given confidence level)

depending on the expected impact probability and the

shape of the initial distribution.

Future work to further improve tools for PP analysis

will focus on the extension of the LS algorithm to the

case of multiple impact events with different bodies and

69th International Astronautical Congress (IAC), Bremen, Germany, 1-5 October 2018.

Copyright ©2018 by M. Romano, C. Colombo and J. M. Sánchez Pérez,

Published by the IAF, with permission and released to the IAF to publish in all forms

IAC-18-A3.5.10 Page 9 of 9

the improvement of the preliminary analysis to identify

impact conditions. Aside from sampling methods,

different ways and parameterisations to express the

initial uncertainties will be explored to make the

sampling more efficient, together with the direct

propagation of uncertainties.

Acknowledgements

The work performed for this paper has received

funding from the European Research Council (ERC)

under the European Union’s Horizon 2020 research and

innovation programme (grant agreement No 679086 –

COMPASS), and from the European Space Agency

(ESA) though a Networking/Partnering Initiative (NPI).

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