WO2016091728A1

WO2016091728A1 - Computer modelling of atmospheric re-entry of objects such as satellites

Info

Publication number: WO2016091728A1
Application number: PCT/EP2015/078595
Authority: WO
Inventors: Cristina DE PERSIS; Simon Paul Wilson; Irene HUERTAS
Original assignee: The Provost, Fellows, Foundation Scholars, And The Other Members Of Board, Of The College Of The Holy And Undivided Trinity Of Queen Elizabeth, Near Dublin
Priority date: 2014-12-10
Filing date: 2015-12-03
Publication date: 2016-06-16

Abstract

A modelling system is implemented by a digital processor. The system models fragmentation of spacecraft during the atmospheric re-entry. It predicts propagation of the surviving fragments, given the physical characteristics of the craft and its re-entry trajectory. Fragmentation is modelled by a stick-breaking type of model, an example of which is the Dirichlet model, and this model is tuned using Bayesian processing in order to better fit the data. Such tuning in particularly effective in the invention because much of the available data is in the form of expert opinion. A probability technique such as Bayesian processing quantifies the expert opinion and other non-hard data into the form of a probability distribution. This approach also accommodates the fact that there is much missing data.

Description

"Computer Modelling Of Atmospheric Re-entry Of Objects Such As Satellites" INTRODUCTION Field of the Invention

The invention relates to modelling of atmospheric re-entry of objects such as satellites.

A satellite, orbiting around the Earth, would continue to orbit forever if gravity were the only force acting on it, but it is not. There exist other forces acting on the satellite to perturb it away from the nominal orbit: the luni-solar attraction, the perturbation due to the non-spherical geo- potential, the solar radiation pressure and the atmospheric drag.

Prior Art Discussion

Atmospheric drag is the major cause of a natural decay under specific circumstances. It is due to the frequent collisions between the satellite and the surrounding air molecules, and for this reason it increases with the density, and the atmospheric density increases as the altitude decreases. Also, the effects of the atmospheric drag are affected by the solar activity. Drag acts in the opposite direction of motion and it is a non-conservative, energy-dissipating perturbation. This energy reduction causes the orbit to get smaller, leading to further increases in drag. Eventually, the altitude of the orbit becomes so small that the satellite re-enters the atmosphere.

This dissipating action is greater at lower altitudes. Objects orbiting in the Low Earth Orbit, or simply "LEO" (hundreds of kilometres in altitude), region may decay naturally by atmospheric drag within weeks, months, or years depending on the object and its altitude. Objects at higher altitudes may remain in orbit for hundreds or thousands of years. Of course, some satellites and launch hardware have propulsive capability that can be used to de-orbit these objects more quickly and eventually to control the re-entry.

Whenever possible, spacecraft or orbital stages that are terminating their operational phases should be de-orbited (controlled re-entry is preferred) or where appropriate manoeuvred into a "graveyard" orbit, so as not to interfere with objects still in operation. This is part of the IADC Space Debris Mitigation Guidelines. Most of the unmanned space missions end up with a destructive re-entry. Of course, this is not the case of crewed missions, where the satellites are protected by specially designed heat shields, or with re-entry vehicles for sample return missions. During the decay the satellite enters into denser regions of the atmosphere at a very high velocity, exceeding speeds twenty times faster than a speeding bullet. This causes the object to be decelerated abruptly, the structure to be exposed to aerodynamic loads that can exceed ten times the acceleration of gravity, and to a friction that heats up the object so that some of its components can reach its melting point or ablate. Eventually the combinations of these events cause the spacecraft to break apart by the failure of critical structural components and, with much less likelihood, by an explosion of fuel or oxidizer remaining in the tanks. The former case is named a low energetic break-up event, while the latter is termed a high energetic break-up event. During the descent, the first components which get detached from the main body and break into fragments are usually the solar panels, the antennas and other protruding elements. This is followed by a major breakup event, lowly energetic or highly energetic, that generally involves all the structure and that occurs at an altitude between 70 and 90 km. The resulting fragments will travel further in the atmosphere and can experience further fragmentation, completely demise, by melting or ablation, as long as sufficient heating and loads exist. The same happens for all of the fragments generated afterwards.

During the descent the surviving debris lose speed and begin to cool until they fall and impact the ground. Even if they hit the Earth's surface at relatively low speed, they can still represent a hazard to people, fauna, or flora. Fortunately no casualties or injuries have been reported from components of re-entering spacecraft until now: based on the available reports, it seems that only one person has been struck-but not injured-by a lightweight fragment of a re-entering satellite, in Oklahoma in 1997. If a spacecraft or orbital stage is to be disposed of by re-entry into the atmosphere, debris that survives to reach the surface of the Earth should not pose an undue risk to people or property. This may be accomplished by limiting the amount of surviving debris and confining the debris to uninhabited regions, such as broad ocean areas.

An analysis of re-entry structures suggests that about ten to forty per-cent of a satellite's mass may survive the re-entry. First of all, the actual percentage depends on the re-entering vehicle features (materials, mass, size, shape). Moreover it depends on the steepness of the re-entry trajectory and the energy released during the break-up. The probability of survival for a specific component is mainly determined by the materials used in its construction, the size and by its position. For example, components made of materials with high melting temperatures, such as stainless steel, titanium, and glass, often survive, while pieces made of aluminium, which has a low melting temperature, do not survive re-entry. Larger debris with moderate melting temperatures have higher chances of survival, because they can radiate a greater amount of heat, taking advantage of their large surface areas. When a component is contained within the body of the vehicle and then protected by surrounding structure, it may survive even if it is made of a low melting temperature material.

In order to minimize the safety risk of the spaceflight projects, it is important to identify the hazardous conditions and accident scenarios as well as the probabilistic assessment of their consequences. To assess the damage caused by these surviving objects, with steadily increasing numbers of space objects and increase of possible candidates for ground impact, it would be advantageous to have an automated tool to predict the mass of impacting fragments, their features, their re-entry trajectory and finally the area where they will hit the Earth.

EP1110862 (Boeing) describes satellite antenna control.

EP0628899 (Space Systems) describes spacecraft orbit altitude control methods.

The invention is directed towards providing improved accuracy of modelling of atmospheric reentry of objects such as satellites to achieve more accurate predictions.

SUMMARY OF THE INVENTION

According to the invention, there is provided a modelling system comprising a digital data processor and data interfaces, in which the processor is configured to:

(a) receive quantitative data about a spacecraft including number of parts, geometry, and materials;

(b) perform fragmentation modelling according to a model to predict manner of spacecraft break-up, (c) receive subjective data representing expert opinion based on historical spacecraft break-up,

(d) tune the fragmentation model and/or modify its outputs by applying statistical probability algorithms based on the subjective data; and

(e) provide outputs predicting manner of break-up.

In one embodiment, the input data for step (a) includes:

data representing physical characteristics of the craft including shape, mass material composition;

motion characteristics data, including re-entry trajectories with angles of incidence; and

data defining state of tanks before starting re-entry; and an

atmospheric model.

In one embodiment, the processor is configured to perform, before step (b), a probability analysis as to whether the break-up will be of high energy or low energy types.

In one embodiment, the processor is configured to apply a Bayesian network model for assessment of probability of a highly or lowly energetic break-up event to occur.

In one embodiment, a proportional hazard model is combined with said probability analysis to assess a conditional probabilities table.

In one embodiment, the processor is configured to perform the fragmentation modelling (b) to provide one or more of:

a predicted number of fragments, and

predicted fragment masses, sizes, material compositions, break-up altitudes, and ejection velocities.

In one embodiment, the processor is configured to perform the fragmentation modelling on the basis that during craft descent the craft splits up into external objects including solar panels and antennas, and a main body.

In one embodiment, the processor is configured to perform the fragmentation modelling on the basis that there is a major break-up event for each external object and one for the main body, and the processor is configured to track separately fragmentation for each of these major break-up events.

In one embodiment, the step (b) includes the processor evaluating a probability distribution of altitude of detachment from the main body, applying a failure mode with inferred parameters, computing a probability distribution of the number of fragments generated by the breaking-up and probability distributions of their masses, and then applying the fragmentation model.

In one embodiment, the processor is configured to, for the fragmentation model, execute different software processes for each craft external object and main body.

In one embodiment, the processor is configured to execute the processes in parallel.

In one embodiment, for the step (b) the processor is configured to assume that the masses of craft fragments are distributed according to a family of continuous multivariate probability distributions.

In one embodiment, the processor is configured to model craft break-up in step (b) as breaking of a stick of unit length at random locations in different steps.

In one embodiment, the processor is configured to model craft break-up in step (b) as a plurality of fragmentation events of nodes splitting into edge nodes in one or more stages until there are leaf nodes representing surviving fragments.

In one embodiment, the processor is configured to use a splitting parameter and shape of a weighted tree to provide a vector of the parameters of a probability distribution.

In one embodiment, the processor is configured to define the distribution of the masses of fragments, and to generate masses of the fragments as random numbers from a Dirichlet distribution providing a parameter vector which defines how the mass of the craft is distributed over fragments including ablated and demised fragments.

In one embodiment, the processor is configured to maintain processing structure flexible and adaptive to different kinds of re-entry or different crafts by providing a one-to-one correspondence between a vector and a rooted weighted tree and consequently between the masses and the tree, in which a number of leaves of the tree is equal to a length of the vector and the length of the vector is equal to the number of generated fragments. In one embodiment, the processor is configured to perform at least some of step (b) by representing masses of the elements into which the craft is fragmented as nodes, and edges from parent to child nodes represent fragmentation steps, in which the children are the masses of the fragments generated by the break-up of the parent, and leaf nodes of the tree represent surviving fragment masses or demised masses.

In one embodiment, the processor is configured to perform Bayesian inference processing in step (d) to determine parameters of a distributions output of the fragmentation model.

In one embodiment, the processor is configured to trigger a module specifically developed for cases of explosions during the atmospheric re-entry.

In one embodiment, the processor is configured to trigger said module according to step (b).

In one embodiment, said module takes account of the possibility to have more than one explosion event.

In another aspect, the invention provides a method of computer modelling of atmospheric reentry of objects such as satellites, the method being performed by a digital data processor and data interfaces, in which the method comprises the steps of:

receiving quantitative data about a spacecraft including number of parts, geometry, and materials;

performing fragmentation modelling according to a model to predict manner of spacecraft break-up,

receiving subjective data representing expert opinion based on historical spacecraft break-up,

tuning the fragmentation model and/or modify its outputs by applying statistical probability algorithms based on the subjective data; and

providing outputs predicting manner of break-up. In another aspect, the invention provides a computer readable medium comprising non-transitory software code to perform the steps of a method as defined in any embodiment when executing on a digital processor.

Additional Statements

(b) perform fragmentation modelling according to a model to predict manner of spacecraft break-up,

(c) receive subjective data representing expert opinion based on historical spacecraft breakup,

(e) provide outputs predicting manner of break-up.

In one embodiment, the input data for step (a) includes data representing physical characteristics of the craft including shape, mass material composition; motion characteristics data, such as trajectories, steep or shallow re-entry, and angles of incidence; description of the state of tanks before starting the re-entry process; and an atmospheric model.

In one embodiment, the step (b) is preceded by a probability analysis as to whether the break-up will be of the high energy or low energy types.

In one embodiment, a Bayesian network model is applied for assessment of the probability for a highly or lowly energetic break-up event to occur and a proportional hazard model is combined with it to assess a conditional probabilities table.

In one embodiment, the fragmentation modelling (b) provides one or more of:

number of fragments, and

fragment masses, sizes, material compositions, break-up altitudes, and ejection velocities. In one embodiment, the processor is configured to perform the fragmentation modelling on the basis that during the craft descent the craft splits up into external objects including solar panels and antennas, and the main body. Preferably, the processor is configured to perform the fragmentation modelling on the basis that there is a major break-up event for each external object and one for the main body, and the processor is configured to track separately the fragmentation story for each of these major break-up events.

In one embodiment, the step (b) includes evaluating a probability distribution of altitude of detachment from the main body, applying a failure mode with inferred parameters, compute the probability distribution of the number of fragments generated by the breaking-up and the probability distributions of their masses, and then applying the fragmentation model.

In one embodiment, the fragmentation model executes different software processes for each spacecraft external object and its main body. Preferably at least some of the processes are executed in parallel.

In one embodiment, for (b) the processor is configured to assume that the masses of craft fragments are distributed according to a family of continuous multivariate probability distributions.

In one embodiment, the processor is configured to model craft break-up in step (b) as a plurality of fragmentation events of nodes splitting into edge nodes in one or more stages until there are leaf nodes representing surviving fragments. In one embodiment, a splitting parameter and shape of a weighted tree provide a vector of the parameters of a probability distribution.

In one embodiment, in step (b) the distribution of the masses of fragments is defined and the masses of the fragments are generated as random numbers from a Dirichlet distribution providing a parameter vector which defines how the mass of the spacecraft is distributed over generated fragments including ablated and demised fragments. Preferably, the processing structure is maintained flexible and adaptive to different kinds of re-entry or different crafts by providing a one-to-one correspondence between a vector and a rooted weighted tree and consequently between the masses and the tree, in which the number of leaves of the tree is equal to the length of the vector and the length of the vector is equal to the number of generated fragments.

In one embodiment, the processor is configured to perform at least some of step (b) by representing masses of the elements into which the craft is fragmented as nodes, and edges represent the fragmentation steps, edge points from a "parent" to a "child" node, where the children are the masses of the fragments generated by the break-up of the parent, and leaf nodes of the tree represent surviving fragment masses or demised masses.

In one embodiment, Bayesian inference processing is performed in step (d) to determine the parameters of the distributions output of the fragmentation model.

In one embodiment, the system includes a module configured specifically for the cases of explosions during the atmospheric re-entry, and the system is arranged to trigger said module according to step (b). In one embodiment, said module takes account the possibility to have more than one explosion event.

In another aspect, the invention provides a method of modelling spacecraft re-entry to earth's atmosphere, the method being implemented by a digital data processor and data interfaces, the method comprising the steps of:

(a) receiving quantitative data about a spacecraft including number of parts, geometry, and materials;

(b) performing fragmentation modelling according to a model to predict manner of spacecraft break-up,

(c) receiving subjective data representing expert opinion based on historical spacecraft break-up,

(d) tuning the fragmentation model and/or modify its outputs by applying statistical probability algorithms based on the subjective data; and

(e) providing outputs predicting manner of break-up.

In a further aspect, the invention provides a computer readable medium comprising non- transitory software code arranged to perform the steps of the method as defined above when executing on a digital processor. Glossary

Fragmentation sequence: the sequence of break-up events occurring during the re-entry Fragmentation event: event which generates fragments or demised masses Fragment: Parts in which the craft breaks-up Demised masses: Parts of the craft which ablate in the atmosphere

Brief Description of the Drawings

The invention will be more clearly understood from the following description of some embodiments thereof, given by way of example only with reference to the accompanying drawings in which :-

Fig. 1 is a diagram illustrating the structure of a model of a craft and dynamics of its reentry, in which fragmentation history includes all of the fragmentation events occurring during re-entry;

Figs. 2 and 3 are diagrams illustrating fragments of the model,

Fig. 4 is a diagram illustrating structure of a Bayesian network for the assessment probability for a highly energetic event to occur,

Figs. 5 and 6 are diagrams showing examples of a fragmentation sequence tree, in which F is a fragment that will generate other fragments in the next level, D is a fragment that will demise completely, N means that nothing will happen, and G means that the fragment will impact the ground (this fragment belonging to the last level),

Fig. 7 is a diagram showing an example of how a fragmentation sequence modelled by Dirichlet fragment distribution, and Figs. 8 to 11 illustrate distributions of generated masses.

Description of the Embodiments

Overview

A modelling system of the invention is implemented by any general purpose computer with a digital processor programmed to implement modelling steps as described below. Depending on desired speed and application, the computer may use some firmware such as FPGAs or ASICs for particular tasks.

The system automatically models fragmentation of spacecraft during the atmospheric re-entry. It predicts propagation of the surviving fragments, given the physical characteristics of the craft and its re-entry trajectory.

Fragmentation is modelled by a stick-breaking type of model, an example of which is the Dirichlet model, and this model is tuned using Bayesian processing in order to better fit the data. Such tuning in particularly effective in the invention because much of the available data is in the form of expert opinion. A probability technique such as Bayesian processing quantifies the expert opinion and other non-hard data into the form of a probability distribution. This approach also accommodates the fact that there is much missing data.

The system decomposes the problem into two main tasks:

1. Assess the probability for an highly or lowly energetic break-up event to occur, considering the various possible causes and the re-entry configuration

2. Simulate the fragmentation sequence and the fragmentation cloud parameters.

The output is the probability of a highly energetic break-up, the number and characterization of fragmentation events during the descent, number of generated fragments, the masses, the material composition, the size, the break-up altitude, the ejection velocity.

The fragmentation model has two different statistical sub-models, an "explosive hazard assessment model" and a "fragmentation model". The explosive hazard assessment model characterizes and quantifies the fragmentation events through the application of failure models, while a variation of the stick breaking process is applied for the characterization of the fragments.

All of the desired outcomes are treated as random variables and the output is a collection of probability distribution functions whose parameters are inferred through the application of the Bayesian statistical methods. The input datasets for the inference and the input datasets of the model are different.

The datasets processed in the Bayesian framework, in order to infer the parameters which fit the model, are observed data coming from previous re-entries, explosive and not, and expert opinions.

The incorporation of the expert opinions in the dataset is possible with Bayesian statistics and it is particularly suitable in this context.

On the other hand the input data for the fragmentation model are:

- data representing physical characteristics of the craft (shape, mass material composition etc.),

- motion characteristics data, such as trajectories (steep or shallow re-entry) and angles of incidence,

- description of the state of tanks before starting the re-entry process, and

- atmospheric model.

The following summarizes what is illustrated by the drawings.

Fig. 1 shows the structure of the fragmentation model.

Figs. 2 and 3 illustrate how break-ups are represented by the system. In more detail, the nodes are a representation of the masses of the elements into which the craft is fragmented, while the edges represent the fragmentation steps. An edge points from a "parent" to a "child" node, where the children are the masses of the fragments generated by the break-up of the parent. The leaves of the tree stand for the surviving fragment masses or the demised masses.

The root node, node 1, represents the dry mass of the object before the starting of the fragmentation process, and the other red nodes (in this case nodes 2, 3, 4 and 22) represent the surviving part of the main object after each fragmentation event, e.g. the root fragmented into three elements here indicated with the nodes 12, 8 and 2. The node 2 represents the biggest element; the nodes 2 and 12 are fragments which break up again, while 8 is a surviving fragment or a demised mass. Fig. 3 is a weighted tree, i.e. a tree to which edge labels are assigned. The label on the edge which connects the parent node v to the child node w indicates which proportion of the parent mass v goes into the child w.

Fig. 4 illustrates a Bayesian network for initial analysis of the probability of a highly energetic event occurring. This precedes the fragmentation modelling, and it improves efficiency and accuracy of the following steps. The events of the Fig. 4 modelling may for example be:

- pressure on a wall being greater than a maximum,

- a component melting gas leak, and

- tank bursting. Figs. 5 and 6 show a sample fragmentation output, and Fig. 7 shows that it can be modeled by a particular example of a stick-breaking model known as a Dirichlet distribution. Figs. 8 to 11 illustrate mass break-up examples as overall system outputs.

More Detail

Referring again to Fig. 1, the structure of the model is illustrated according to the IDEFO Standard. The fragmentation sequence describes all fragmentation events during the re-entry and tracks the generated fragments. Each fragmentation sequence has different fragmentation levels, one for each involved break-up event. In the second level of the model, the "external objects" are for example solar panels and antennae. At the third level there is assessment of the break-up intensity for an object such as the main body. Beneath that there is assessment of the birth altitude for each fragmentation level, and below that prediction of the fragment feature for each fragmentation level. The third level is repeated for each object (both external objects and the main body). These levels take the physical form of datasets. Video of a spacecraft re-entry may be analyzed to determine the fragmentation sequence and give an estimate of fragment masses. Laboratory data yield data on the reliability of the components and their resistance to heat and pressure.

Fragmentation Modelling

Fig. 2 shows the nodes that represent of the masses of the elements into which the craft is fragmented, while the edges represent the fragmentation steps. Fig. 3 is a schematic view of an example fragmentation story tree, which tracks the generated fragments. The red nodes (full circles) represent the main body and the blue nodes (dotted circles) are other fragments that have detached from it (the colour legend is given in the drawings). The processor is configured with the condition that during the descent the spacecraft splits up in two main sets: the external objects (solar panels and antennas) and the main body. Given this assumption it models a major break-up event for each external object and one for the main body. The system tracks separately the fragmentation story for each of these major break-up events (for each external object and for the main body) as they have different starting points.

The fragmentation model consists of different branches that can be run in parallel, the first for each external object (detachment and low energy fragmentation) and the second for the main body (high or low energy fragmentation). The first consists of the following steps:

1. Evaluate the probability distributions of the altitude of detachment from the main body, applying a failure model (such as Cox's Proportional Hazard models) with inferred parameters. Compute the probability distribution of the number of fragments generated by the breaking-up and the probability distributions of their masses, applying the fragmentation model. The parameters of these distributions depend on the object characteristics.

2. Derive the other fragment features.

The second, for the main body, consists of the following steps. 1. Assess the probability for a highly energetic break-up event to occur. This probability can be computed through the implementation of a Bayesian network, which takes in account the combination of the occurrence of all the different events that can lead to a highly energetic break-up event. The probability that each of these basic events occurs can be evaluated applying a failure model (Cox's Proportional Hazard models).

If this probability is large enough the fragmentation is modeled as generated by a highly energetic break-up event, otherwise as generated by a lowly energetic break-up event.

2. Select the right fragmentation model on the basis of the energy level release.

3. Compute the probability distribution of the number of generated fragments and the probability distributions of their masses.

4. Derive the other fragments features.

The general structure of the model is depicted in Fig. 1, and the following is a sample of pseudocode for the model:

Pseudocode for Fragmentation Model

# System set up

elicit expert opinion about a prior of the parameters of the distribution of the masses of the surviving fragments;

elicit expert opinion about the incomplete observations; for each re-entry:

# Predict fragmentation for next re-entry (before it takes

# place)

use Bayesian updating methods to produce prediction for a fragmentation event; observe re-entry data;

# Update state of knowledge in light of observing new

# re-entry

Implement the Bayesian inference for the distribution of the masses of the surviving fragments;

end for

The major fragmentation model inputs are: - Characteristics of the fragmentation event to consider: energy level, parent object etc.

- Estimation of the number of the generated fragments.

The major outputs are:

- Characterization of the generated and surviving fragments, e.g. distribution of the masses of the fragments generated by the fragmentation event, and percentage of demised mass.

The main output of the fragmentation model is the distribution of the masses of the generated fragments. The mass is considered as the main characteristic, from which the others can be derived.

Advantageously, the fragmentation process can be considered as taking a stick of length 1 and breaking it up into ⁿ pieces in different steps, such that the lengths of these pieces follow a given statistical distribution, where is the number of the generated fragments. It is equivalent to sampling from a given distribution a linear partition of the stick of length 1.

One example of a suitable distribution for automated processing is the Dirichlet distribution. The system assumes for the fragmentation model that the masses of the fragments (including the ablated ones) are Dirichlet distributed. It follows the definition of the Dirichlet distribution.

The Dirichlet distribution

The Dirichlet distribution with parameters ^ai- ·>^■> ^ακ has a probability density function given by

for all mi, ..., m r > 0 satisfying ⁷⁷¹¹ + ··^■+ τ^ηκ = 1 _

The normalizing constant is the multinomial Beta function.

It is a distribution supported on a bounded interval. For this reason and for the decimation property, described below, it can be considered suitable as distribution of the masses of the fragments and ablated items.

Decimation property of the Dirichlet distribution

!f ijBf ^,,.,Μ^Οίτξα^ ^,,.,tt_g) _and (¾ T₂)~-Mr(«ii¼. «¾J¾) where fit +Λ (?¾¾, W¾¾B½_# ,„ «1^2, «2 ·· , «*)

Example

The following describes in one embodiment implementation of the invention by a digital processor executing algorithms.

Consider the normalized masses ™i m_N = M_X/M, ..., M /M ₀f the objects (fragments and demised masses) generated by a fragmentation event, where M is the total mass of the parent object and M »i¾ the real masses. «.#»%) is a N-tuple of positive values which sum to 1. Then the Dirichlet distribution can be considered suitable to model the distribution of the masses of the objects generated by a fragmentation event.

{»¾, ,.. ,m_s)^mr( ) This distribution is parameterized by the vector*, which is sensitive to the initial conditions of the fragmentation event, for instance on the energy level (highly energetic or lowly energetic fragmentation event). Intuitively it can be thought of as determining how distributed is the total mass in ^m,v . This parameter vector can be estimated from observed data applying the

Bayesian inference tools.

Once the parameters vector ^a is known, the distribution of the masses of fragments is defined and the masses of the fragments can be generated as random numbers from a Dirichlet distribution. As depicted in Fig.7, applying the decimation property of the Dirichlet distribution, we can assume that:

- at the first fragmentation level there are n = 2 fragments whose masses are ^mi> ^71X2 and they are distributed as follows:

yn_t,m^^Mr(a, a)

- at the second fragmentation level ^mz breaks up into two fragments, whose masses will be called ^m2 and Then there are 2 fractures and = 3 fragments whose masses are mi, m2, rm _ancj they are distributed as follows: at the third fragmentation level m3 breaks up into two fragments, whose masses will be called ¹⁷¹³ and ^m+. Then there are 3 fractures and nfrag ^{= 4} fragments whose masses are ™, »«3 , »* and they are distributed as follows:

at the ^Wt* fragmentation level ^mn breaks up into two fragments, whose masses will be called ^m« and m» * i. Then there are ⁿ fractures and = n + 1 fragments whose masses are ma , ma , . _ mn + i and they are distributed as follows

, a a a a _\ i:m_t, m₂,m_Stm_4f m_m+i)^Dir I ¾ -_» - * .. i

I 4 n + 1 it + V at the ^Λ* fragmentation level ^m»»_? breaks up into ⁿr fragments, whose masses will be called , mm_g ÷« - i,

Then there are ^¾/ra ⁼ «_p + «_r ~ 1 fragments whose masses are distributed as follows

It is more convenient to introduce the Dirichlet distribution starting from the Beta distribution, which is one of its particular cases and which has an easier intuitive interpretation.

From the Beta distribution to the Dirichlet distribution

The Beta distribution is a family of continuous probability distributions defined on the interval [§,1| parameterized by two positive shape parameters, denoted by ^a and fi, that appear as exponents of the random variable ^x and control the shape of the distribution.

The Beta distribution can be intuitively understood considering it as a probability distribution of a probability, where the probability itself is the random variable . It can be applied to assign a probability to all the possible values a probability can take. Indeed the feasible values that a probability can take are included in the set [0,1] , that is equal to the support of the Beta distribution. Similarly it can be applied to model the behaviour of random variables limited to intervals of finite length.

Algebraically, the Beta distribution is given by

where B( > β) is a normalization factor. It is the beta function, which can be expressed in terms of the gamma function:

^{B a) "} r &_xaO

The Dirichlet distribution is the multivariate generalization of the Beta distribution. It is a distribution over multinomials, which are N-tuples {^mi>- ···#»%} which sum to unity and such that m_lf ... ,m_N > 0 On the contrary the Beta distribution is meant for the two dimensional case, where the 2-tuple is just a pair (^mi< ^mz ) such that ^mt + ^m2 ⁼ 1 . Then the Dirichlet distribution is simply the generalization of the Beta distribution to higher dimensions.

Tree-structured fragmentation process

Once the parameters vector is known, the distribution of the masses of fragments is defined and the masses of the fragments can be generated as random numbers from a Dirichlet distribution. This parameter vector defines how the mass of the spacecraft is distributed over the generated fragments (including the ablated or demised ones). It is required that its structure is very flexible and adaptive to the different kinds of re-entry or different crafts. For these reasons a one-to-one correspondence has been created between this vector ^a and a rooted weighted tree and consequently between the masses and the tree. The number of leaves of the tree is equal to the length of the vector ^a and the length of the vector ^a is equal to the number of generated fragments.

An example of a rooted weighted tree is depicted in Fig. 3. The root, the red node number 1, is the dry mass of the object before the break-up event. Every node stands for the mass of a fragment or for a demised mass. The sum of the masses of the children of a vertex ^v is equal to the mass of ^v . The red nodes represent what remains of the main object after the different fragmentation events.

All edges point away from the root. The parent of a vertex (or node) is the vertex connected to it on the path to the root; every vertex except the root has a unique parent. A child of a vertex ^v is a vertex of which ^v is the parent.

The children of each node are the fragments or demised masses generated from it. The generated fragments can break-up again or survive. The surviving fragments are represented by the leaves of the tree, together with the demised masses. The leaves stand for the desired masses (

The processor is configured to operate on the basis that there are two different levels of fragmentation: the primary and the secondary ones. The primary fragmentation event is the event generated by a red node, while all the others are secondary.

Fragmentation height of a node is defined as the length of the downward path between the closest red node and the node. The nodes generated by a primary fragmentation have fragmentation height equal to 1, while the nodes generated by a secondary fragmentation have fragmentation height equal to 2. Depending on the kind of breaking-up event this height can also take larger values.

For example a vector is called a splitting parameter vector Y ⁼ {¾» · · · » ¾} . The elements of F are positive and they sum to 1. The size of this vector is the degree of the tree, being the degree of a tree the maximum degree of any of its nodes. This parameter Y defines how the father node is divided into its children.

Let ^v a parent vertex with mass ^sv and the number of its children.

Every parent vertex is associated to a partition of the splitting parameter {ft* · · · > X_dv- i' 1 - ¾^"* Yi } and * is divided into ^■ masses of the following size:

The processor is configured to treat the edges which connect the parent vertex ^v to its children as being weighted with the correspondent values of the splitting parameter vector. The weight of the edge which connect ^v to the child ^c _v is equal to Yi ^{= s}c_v/^sv.

Each element of the Dirichlet distribution parameter vector corresponds to a leaf of this tree and it is given by the product between the weights of the edges which belong to the shortest path between the leaf and the first node, considering the weighted tree.

Assuming this relationship between the ^a vector, the tree and the splitting parameter vector, it turns out that ^a can be easily deduced by the inference of the other two.

The tree can assume different shapes as shown in the examples in Figs. 8 to 11. The trees represented in these plots have two different kinds of nodes: the red nodes have three children, while the black nodes have only one child. The root is a red node. Each red node has 2 red and 1 black nodes as children, while the child of a black node is ever a black node.

The red leaves of this tree represent the masses of the K generated fragments, while the black leaves are the lost or demised masses.

Considering that the root is at level 0, the level ^ of this tree depends on the number of expected generated fragments as follows:

d = log; K

It follows that the

- the number of leaves N

- the number of generated fragments, or red leaves, is 2^d = K

- the number of demised masses, or black leaves,

2'

Explosive hazard assessment model

Failure Models Failure models are statistical models that assess the uncertainty associated with the occurrence of events and measure the time (or the distance) to a certain event, such as the failure of a mechanical component. The parameters of the failure models can be defined and inferred, taking in account the physics of the failure.

The processor applies this class of statistical methods in order to assess the probability:

- that each break-up event (including the detachment of the external components) occurs before a certain altitude;

- for a highly energetic break-up to occur rather than a lower one.

An example of applicable failure model is the Cox Proportional hazards model, which can be applied to compute the hazard function.

The hazard function ^r( of survival time T measures the instantaneous risk, in that r(t)dt j_s the probability of failu e in the next small interval dt given survival to time t. It is also known as instantaneous failure rate or simply failure rate.

The following is pseudocode for this model:

Pseudocode for Explosive Hazard Assessment Model

# System set up

construct a fault or event tree for the events leading to explosion;

elicit expert opinion about a prior on one or more of the base event probabilities under agreed nominal re-entry conditions;

use simple comparison elicitation (e.g. AHP) to specify priors for other base event probabilities; for each re-entry:

# Predicting risk for next re-entry (before it takes place)

if re-entry conditions are not nominal:

use comparison elicitation to weight likelihood of events according to comparison with nominal re-entry conditions;

end if

if re-entry conditions are nominal: use Bayesian prediction methods to derive posterior probability of

explosion risk given expert opinion and data to date;

else:

use comparison weights to calibrate posterior distribution of event probabilities;

use Bayesian prediction methods to derive posterior probability of explosion risk given expert opinion and data to date;

end if

observe re-entry data

# Update state of knowledge about event probabilities in

# light of observing new re-entry

if re-entry conditions are nominal:

use Bayesian learning methods to update opinion about the probabilities of events;

else:

use comparison weights to calibrate likelihood of data for new re-entry; use Bayesian learning methods to update opinion;

end if

end for

Bayesian Network

The Bayesian Network (or simply "BN") is not a failure model but it can be applied to combine the results of different failure models. A Bayesian network, also known as "belief network", is a probabilistic graphical model aimed to represent a system of random variables that are dependent on each other. For example, a Bayesian network could represent the probabilistic relationship between diseases and symptoms.

It has the structure of a directed acyclic graph (DAG), where nodes denote random variables: they may be observable quantities, unknown parameters or hypotheses. Each variable has a finite set of mutually exclusive states. Arcs, i.e. linking lines, realize a probabilistic and causal dependence between random variables that they connect.

BNs permit an efficient representation and computation of the joint probability distribution (JPD) over a set of random variables. They reflect a simple conditional independence statement: each variable is independent of its non-descendants in the graph given the state of its parents. This property allows to reduce, sometimes significantly, the number of parameters necessary to compute the joint probability distribution of the variables. The conditional probability distribution at each node depends only on its parents, thus the Markovian property is satisfied. It is often represented by a table which contains the local probability that each child node takes on each of the feasible values, for each combination of values of its parents. Therefore the joint distribution of a collection of variables can be determined uniquely by these local conditional probability tables (CPTs).

Bayesian networks help to capture the intuitive understanding of complex systems, since they take advantage of expressing a graphical description of the dependencies between events. Then, in order to apply the Bayesian network model to this problem, the first aim is the identification of the basic events which can trigger an explosion and of the qualitative assumptions about their cause effect relationships.

Explosive Hazard Assessment Model

The scenario of a re-entering spacecraft explosion can be considered as the result of the occurrence and of the simultaneous combination of specific events. The Bayesian network model, because of the network structure, can take in account the logic combinations of the events that can lead to an explosion and their cause-effect relationship.

The probability for a highly energetic break-up event to occur is computed by the processor through the implementation of a Bayesian network, whose nodes or random variables are the different events that can create the environment for a highly energetic break-up event. On the other hand, the probability that each of these basic events occurs is evaluated by the processor applying a failure model (i.e. the Cox's Proportional Hazard model).

Advantageously, the system of the invention in some embodiments applies a proportional hazard model to model the single events involved in the process and the Bayesian network to model the whole system. This means that the PHM model can be applied to assess the conditional probabilities table of the Bayesian network, while the Bayesian network provides an evaluation of the probability of the explosion to occur. For each node of the Bayesian network a different hazard function needs to be formulated, as follows:

r{li) = r_eexp{ _xA^*{¾, h) - fi_wW(h h ) ) } Where:

- ^ro is the baseline hazard function, for K and W null.

β^χρ(β_κΚ{ h) + 0_wW(h *))) is the term dependent on the covariates K and W.

- βκ and $w are the coefficients of the covariates.

- The covariates K and W are the physical entities, which the occurrence of the events that can lead to a highly energetic break-up depend on.

- k _anci h are the altitude and altitude rate

It will be appreciated that the invention provides for the automated simulation of a very complex process in a manner which is possible with computers having a conventional level of processing speed and available RAM resources. It is particularly advantageous that the various stages are broken down into particular tasks as described such as initially determining a probability of break-up. Such modelling advantageously allows prediction of re-entry events, thereby allowing planning of satellite construction and methods of re-entry. It is envisaged that the modelling system of the invention may form part of a craft design system, in which re-entry modelling is performed for each design iteration. Furthermore, it may form part of a craft control system, enabling it to predict the optimum location for re-entry and parameters associated with it in order to achieve the optimum re-entry conditions.

The invention is not limited to the embodiments described but may be varied in construction and detail.

Claims

1. A modelling system comprising a digital data processor and data interfaces, in which the processor is configured to:

(c) receive subjective data representing expert opinion based on historical spacecraft break-up,

(e) provide outputs predicting manner of break-up.

2. A modelling system as claimed in claim 1, wherein the input data for step (a) includes:

data defining state of tanks before starting re-entry; and an

atmospheric model.

3. A modelling system as claimed in claims 1 or 2, wherein the processor is configured to perform, before step (b), a probability analysis as to whether the break-up will be of high energy or low energy types.

4. A modelling system as claimed in claim 3, wherein the processor is configured to apply a Bayesian network model for assessment of probability of a highly or lowly energetic break-up event to occur.

5. A modelling system as claimed in claim 4, wherein a proportional hazard model is combined with said probability analysis to assess a conditional probabilities table.

6. A modelling system as claimed in any preceding claim, wherein the processor is configured to perform the fragmentation modelling (b) to provide one or more of: a predicted number of fragments, and

A modelling system as claimed in any preceding claim, wherein the processor is configured to perform the fragmentation modelling on the basis that during craft descent the craft splits up into external objects including solar panels and antennas, and a main body.

A modelling system as claimed in claim 7, wherein the processor is configured to perform the fragmentation modelling on the basis that there is a major break-up event for each external object and one for the main body, and the processor is configured to track separately fragmentation for each of these major break-up events.

A modelling system as claimed in claims 7 or 8, wherein the step (b) includes the processor evaluating a probability distribution of altitude of detachment from the main body, applying a failure mode with inferred parameters, computing a probability distribution of the number of fragments generated by the breaking-up and probability distributions of their masses, and then applying the fragmentation model.

A modelling system as claimed in any preceding claim wherein the processor is configured to, for the fragmentation model, execute different software processes for each craft external object and main body.

A modelling system as claimed in claim 10, wherein the processor is configured to execute the processes in parallel.

12. A modelling system as claimed in any preceding claim, wherein for (b) the processor is configured to assume that the masses of craft fragments are distributed according to a family of continuous multivariate probability distributions.

13. A modelling system as claimed in any preceding claim, wherein the processor is configured to model craft break-up in step (b) as breaking of a stick of unit length at random locations in different steps.

A modelling system as claimed in any preceding claim, wherein the processor is configured to model craft break-up in step (b) as a plurality of fragmentation events of nodes splitting into edge nodes in one or more stages until there are leaf nodes representing surviving fragments.

A modelling system as claimed in claim 14, wherein the processor is configured to use a splitting parameter and shape of a weighted tree to provide a vector of the parameters of a probability distribution.

A modelling system as claimed in any of claims 13 to 15, wherein the processor is configured to define the distribution of the masses of fragments, and to generate masses of the fragments as random numbers from a Dirichlet distribution providing a parameter vector which defines how the mass of the craft is distributed over fragments including ablated and demised fragments.

A modelling system as claimed in claim 16, wherein the processor is configured to maintain processing structure flexible and adaptive to different kinds of re-entry or different crafts by providing a one-to-one correspondence between a vector and a rooted weighted tree and consequently between the masses and the tree, in which a number of leaves of the tree is equal to a length of the vector and the length of the vector is equal to the number of generated fragments.

A modelling system as claimed in any preceding claim, wherein the processor is configured to perform at least some of step (b) by representing masses of the elements into which the craft is fragmented as nodes, and edges from parent to child nodes represent fragmentation steps, in which the children are the masses of the fragments generated by the break-up of the parent, and leaf nodes of the tree represent surviving fragment masses or demised masses.

A modelling system as claimed in any preceding claim, wherein the processor is configured to perform Bayesian inference processing in step (d) to determine parameters of a distributions output of the fragmentation model.

20. A modelling system as claimed in any preceding claim, wherein the processor is configured to trigger a module specifically developed for cases of explosions during the atmospheric re-entry.

21. A modelling system as claimed in claim 20, wherein the processor is configured to trigger said module according to step (b).

22. A modelling system as claimed in claim 21, wherein said module takes account of the possibility to have more than one explosion event.

23. A method of computer modelling of atmospheric re-entry of objects such as satellites, the method being performed by a digital data processor and data interfaces, in which the method comprises the steps of:

(e) providing outputs predicting manner of break-up.

24. A computer readable medium comprising non-transitory software code to perform the steps of a method of claim 23 when executing on a digital processor.