Efficient planetary protection analysis for interplanetary missions

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.

Full text

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

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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
Fx
and greater than zero
otherwise) and
()
F
Ix
is an indicator function such that
( ) 1
F
I=x
if
Fx
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
.

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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

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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

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( ) ( ) ( )
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.

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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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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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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).
References
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Verification of planetary protection requirements
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