Track-to-orbit association problem for non-collaborative targets in the presence of manoeuvres

Proyecto Fin de Carrera Ingeniería de Telecomunicación Formato de Publicación de la Escuela Técnica Superior de Ingeniería Autor: F. Javier Payán Somet Tutor: Juan José Murillo Fuentes Dep. Teoría de la Señal y Comunicaciones Escuela Técnica Superior de Ingeniería Universidad de Sevilla Sevilla, 2013 Trabajo Fin de Máster Máster en Ingeniería Aeronáutica Track-to-orbit association problem for non-collaborative targets in the presence of manoeuvres Autor: Antonio Malaver Jambrina Tutores: Rafael Vázquez Valenzuela .Alejandro Pastor Rodríguez Dpto. Ingeniería Aeroespacial y Mecánica de Fluidos Escuela Técnica Superior de Ingeniería Universidad de Sevilla Sevilla, 2024 Trabajo Fin de Máster Máster en Ingeniería Aeronáutica Track-to-orbit association problem for non-collaborative targets in the presence of manoeuvres Autor: Antonio Malaver Jambrina Tutores: Rafael Vázquez Valenzuela Catedrático de Mecánica Orbital en la Universidad de Sevilla Alejandro Pastor Rodríguez Head of SST and STM in Germany at GMV Dpto. Ingeniería Aeroespacial y Mecánica de Fluidos Escuela Técnica Superior de Ingeniería Universidad de Sevilla Sevilla, 2024 Trabajo Fin de Máster: Track-to-orbit association problem for non-collaborative targets in the presence of manoeuvres Autor: Antonio Malaver Jambrina Tutores: Rafael Vázquez Valenzuela Alejandro Pastor Rodríguez El tribunal nombrado para juzgar el trabajo arriba indicado, compuesto por los siguientes profesores: Presidente: Vocal/es: Secretario: acuerdan otorgarle la calificación de: El Secretario del Tribunal Fecha: Agradecimientos L a vida es una odisea, siempre sorprendiéndonos y mostrándonos caminos que no esperábamos recorrer. A lo largo de mi gran viaje, del que Gulliver sentiría celos, he tenido la suerte de compartir mis aventuras con personas maravillosas. Personas que han estado día tras día apoyándome y siendo mis pilares en la Tierra mientras yo trataba de alcanzar los cielos. Al igual que D’Artagnan contaba con Athos, Porthos y Aramis, yo he tenido la increíble suerte de ser ayudado por tres mentores talentosos, amables y siempre dispuestos a ayudarme en todo momento. Gracias a Alejandro, Diego y Rafael por escuchar mis ideas, dar rienda suelta a mi creatividad y nunca cortarme las alas, siendo siempre mi brújula dorada. Los amigos son la familia que uno elige. Esa familia que rema contigo en cada uno de los momentos de tu vida, convirtiéndola cada día en una historia interminable y sacándote una sonrisa cuando más lo necesitas. Aida, Alberto, Alejandro, Álvaro, Amelia, Andrés, Ayelén, Cristina, Gabriel, José María, Juan de Dios, Juan Miguel, Lola, María, Noelia y Santiago. El mundo sería un sitio miserable sin vosotros a mi lado. He aquí mi secreto, que no puede ser más simple: sólo con el corazón se puede ver bien; lo esencial es invisible a los ojos. Y tú, Lucía, eres lo esencial que ha llenado mi vida desde que nos conocimos. El mundo es tan hermoso que era una pena no nacer dos veces para verlo, y al conocerte nací por segunda vez para poder ver toda la belleza que el mundo tenía escondida. Te amo como un torbellino, como un león, como una furia irreprimible. Gracias por enseñarme tanto en tan poco tiempo y por ayudarme cada día a ser la mejor versión de mí. No puedo olvidarme de mis raíces, de quien me vio crecer. Esas personas que convirtieron mi mundo en un mundo feliz durante mi infancia. Gracias a mis dos abuelas, Dolores y Manoli, por estar siempre donde se las necesitaba, a mis tíos por todo el cariño que me han dado estos años, y a mis primos por ser mis mejores amigos durante la infancia. Reír es un acto de rebeldía, una forma de luchar contra la insoportable levedad del ser y decirle al mundo que fuimos felices aun cuando no nos tocaba serlo. Mi hermana Irene es la persona más rebelde que conozco, porque blande la risa como un arma y lleva su felicidad a rincones donde no esperarías encontrarla. Gracias por hacer de muchos días planos una divina comedia. Gracias por ser tú y nadie más. Gracias por quererme tantísimo cada día. Yo he querido llegar a la Luna sólo para ver de cerca cómo tú llegas a las estrellas. El verdadero corazón de esta odisea son mis padres, Antonio y Laura. Sin vuestro amor incondicional, apoyo constante y sacrificios silenciosos, jamás habría logrado todo esto. Lo que soy, o espero ser, se lo debo a la angelical devoción de mi madre y a la fuerza inquebrantable de mi padre. Sois mi hogar en cualquier parte del mundo. Gracias por enseñarme que no importa cuán alto vuele, siempre tendré un lugar donde aterrizar. Vosotros sois mi inspiración y mi mayor logro. Os quiero. Antonio Malaver Jambrina Sevilla, 2024 I Resumen global Título: Problema de asociación traza-órbita para objetos en presencia de maniobras no alertadas Resumen: Este proyecto tiene como objetivo desarrollar un sistema integral para la catalogación de satélites y la estimación de maniobras. Dado el creciente número de satélites, resulta esencial un seguimiento preciso y una predicción de maniobras para evitar colisiones y gestionar el tráfico espacial. El estudio explora diversas metodologías para mejorar la detección de maniobras, centrándose en técnicas estadísticas avanzadas como la entropía y el análisis de componentes principales (PCA) para la evaluación y puntuación de candidatos. El proyecto involucra la implementación (traza-órbita, traza-traza y estimación de maniobras) y desarrollo (promotor de hipótesis conjunto de trazas-satélite) de una cadena completa de catalogación, que integra varios componentes clave, incluyendo el preprocesamiento de datos, la estimación de maniobras y la promoción de hipótesis. Se realiza un análisis detallado del impacto de la proximidad de los satélites, la magnitud de las maniobras y el ruido de radar en la precisión del sistema de seguimiento. Además, el proyecto examina los desafíos que plantea la disponibilidad limitada de datos, como el uso de un número reducido de trazas de observación y la complejidad introducida por satélites con trayectorias que se cruzan. Para respaldar estos análisis, se utilizan observaciones simuladas y planes de maniobras, proporcionando un entorno controlado para probar la efectividad de varias técnicas. La metodología está diseñada con el objetivo de lograr una alta precisión mientras se aseguran actualizaciones oportunas de las posiciones y movimientos de los satélites. Palabras clave: Catalogador espacial, Estimación de maniobras, Detección de maniobras, Correlación de maniobras, Resolución de trazas no correladas, Catálogo de objetos espaciales, Vigilancia y seguimiento espacial, Asociación de trazas Conclusiones: En este proyecto se concluye que, aunque se pueden utilizar múltiples enfoques para la catalogación de satélites y la estimación de maniobras, cada uno presenta ventajas y limitaciones. La entropía, a pesar de su potencial, introduce ruido en el proceso de decisión. El uso de PCA no ha proporcionado resultados óptimos debido a la inestabilidad que introduce la entropía y la dependencia de las variables del escenario simulado. El algoritmo desarrollado utilizando la función SoftMax y los métodos de puntuación muestran unos resultados prometedores. Además, se destaca la importancia de un pre-filtro efectivo para reducir la propagación de errores en el proceso de catalogación. Finalmente se propone la implementación de una metodología mixta que combine la catalogación estacionaria y en tiempo real, que podría ofrecer una solución robusta aprovechando las ventajas de ambas para proporcionar actualizaciones precisas y rápidas sobre la posición y las maniobras de los satélites. III 2Chapter 1. Introduction 1.1 Motivation Although the Age of Discovery ended centuries ago, the exploration of our planet is far from over. Every day, new data on climate, oceans, natural phenomena, etc., bring new insights into the mysteries of Earth, providing crucial information for a better understanding of our home world. Moreover, as more and more research focuses on the exploration and analysis of the cosmos, higher level technologies and instruments are needed. These two separate worlds, Earth study and space exploration, rely heavily on the same objects that allow us to watch television every day and predict the weather: satellites. These objects help us collect billions of bytes of data about our environment and distant worlds, and have become an essential tool for space exploration and Earth monitoring. This need for satellites has led to a sharp increase in the number of them since the launch of Sputnik-1. Nearly 17,000 satellites have been launched, and with 35% of all satellites launched in the last three years and the potential for 100,000 satellites launched in the next decade [ 28 ], this rapid rate of growth demonstrates the need for better understanding, maintenance and tracking of satellites to avoid collisions and keep Earth orbits accessible. In 1978, Donald Kessler delved into the issue of satellite collisions and space debris in an article entitled "Collision frequency of artificial satellites: The creation of a debris belt" [ 8 ]. This publication outlines the implications of a self-sustaining increase in the space debris population. Initially triggered by collisions between intact objects, such as the population of satellites orbiting Earth today, this phenomenon is now recognised as the Kessler Syndrome. As the cascade progresses, collisions between colliding fragments perpetuate the problem, potentially leading to a web of space debris around Earth that could trap us in Earth, preventing us from accessing outer space for decades, even centuries. Space Surveillance and Tracking (SST) addresses the problem of satellite collisions by tracking man-made objects orbiting Earth and providing crucial information on their orbits and potential colliders. For example, the SST segment of the Space Situational Awareness (SSA) programme of the European Space Agency (ESA) focuses on the development of a catalogue of Low Earth Orbits (LEO) with their orbital characteristics and physical properties. In this context, what is often referred to as a catalogue contains information on each object known to the agency or the company that created it. Data on the objects are obtained through networks of sensors (such as radars or telescopes) that constantly scan the sky. Immediately after observing the trajectory of an object in the sky, a question arises: Does this trajectory (usually called track) correspond to an already catalogued object or to a new one? In order to build-up and maintain a catalogue, it is necessary to process the information from the observations provided by the sensors. For this purpose, the flowchart (presented in Figure 1.1) is as follows: 1. A new track is received (set of n observations taken by a single sensor usually over a short period of time, originating from the same resident space object and not sufficient to reliably estimate an orbit). 2. The system tries to correlate this track with any of the existing objects already registered in the catalogue. a) If successful, the observed track belongs to a pre-existing object and the new track information is used to update the catalogued orbit using Orbit Determination. b) Otherwise, this track could belong to a new object or to an object whose trajectory has changed, i.e. a recent launch, fragments due to a recent collision, etc. or a manoeuvred satellite. 1.1 Motivation 3 • A track association process is employed with the objective of establishing a correlation between the newly acquired track and the existing pool of tracks stored in the database. This process is initiated when the previously stored tracks are identified as being uncorrelated, resulting in the formation of a set of tracks that, with a high probability, belong to the same object. • A manoeuvre detection and estimation is performed, trying to associate an object in the catalogue capable of manoeuvring with the set of tracks. If this association works, the tracks correspond to a registered object. If not, it is a new object to be added to the catalogue. Figure 1.1 Flowchart of the catalogue chain. This thesis focuses mainly on the last task mentioned in the flowchart, the manoeuvre detection and estimation process (red box). Its main objective is to try to relate one of the catalogued satellites to a set of tracks whose origin is unknown. As several sets of tracks corresponding to different objects are analysed simultaneously, the estimator must associate each suitable orbit with each track according to certain criteria. To describe it in detail, consider the following problem: sensors provide four sets of uncorrelated tracks, each of which belongs to a different object. The manoeuvre detection and estimation algorithm filters from the catalogue the possible candidates that could match these tracks after performing a (feasible) manoeuvre. The algorithm evaluates each orbit-track association combination between tracks and orbits (which we call hypothesis), assigning each of them a manoeuvre in terms of magnitude and epoch. The algorithm then analyses each combination to decide, based on a certain criterion (e.g., minimum impulse), which is the most likely association. When a hypothesis is promoted, all combinations involving these two elements are removed from the decision tree, and the remaining combinations are analysed in the same way until each set of tracks that passes the filter is related to an orbit. If the remaining candidates do not meet the criterion, the system 4Chapter 1. Introduction considers those tracks as belonging to new objects. This process is depicted in Figure 1.2. (a) (b) (c) (d) Figure 1.2 Correlation process between sets of tracks and catalogued objects. In a) the decision tree is created, relating each orbit to each set of tracks. Each line corresponds to what we call hypothesis. In b) the algorithm selects from all possible candidates the one with the optimal value of the chosen criterion, discarding other combinations involving the selected orbit-track pair. In c) the process starts again with the remaining candidates to try to assign each track to an object. In d) the final state is shown, having correlated three tracks to three existing objects. However, if the remaining pair (Orbit D - Track 3) has not got an optimal value of the selected criteria, this correlation could be discarded. After understanding this process, one question arises naturally: which is the criterion to be used? The effectiveness of the manoeuvre detection and estimation process heavily relies on the chosen criterion, as it determines the accuracy and reliability of the correlation process in the manoeuvre detection and estimation. The selected criterion must balance between computational efficiency and the precision of the correlation to ensure that the resulting catalogue is both comprehensive and accurate. This is the main question this thesis aims to answer. 1.2 State of the art Manoeuvre detection and estimation has been a blooming research topic since the beginning of the space era. Nowadays, due to the importance of this topic and its impact on space sustainability, manoeuvre detection and estimation is a very active area of research. In surveillance scenarios, manoeuvre characterisation faces two major challenges: firstly, different tracks may have long 1.2 State of the art 5 revisit times, which reduces the chances of directly detecting a manoeuvre; secondly, the estimated manoeuvre needs to be integrated into the catalogue chain, which requires a multi-target association framework. Most of the current methodologies are designed for tracking scenarios, while others, discussed below, have been developed for surveillance environments, although the correlation problem is not directly addressed. In this section we deepen into the state of the art to provide an overview of existing methodologies and revise the one that is applied in this thesis. 1.2.1 Optimal Control Optimal control methods are ideal for surveillance scenarios where prior information is lacking. These techniques use performance functions to detect and estimate manoeuvres by calculating control distance distributions. In this way, they allow for efficient monitoring and prediction of dynamic behaviour without the need for prior data. This makes them highly adaptable and effective in real-time applications where conditions and targets can change unpredictably. In addition, these methods can increase the accuracy of surveillance systems, improving their responsiveness to various manoeuvre patterns and ensuring comprehensive monitoring. Holzinger [ 5 ], [ 6 ] assumes that the manoeuvre is performed with the least amount of fuel consumption (which may not always be representative of real-world scenarios) and uses a control distance metric Pto estimate the fuel-optimal manoeuvre ∆Vimp: P=dc(a,b) = Ztf t0 1 2uT(τ)u(τ)dτ.(1.1) This metric provides an upper bound for the minimum possible impulsive manoeuvre ∆Vimp: ∆Vimp ≤∆VC(yb,ya) = q2(tf−t0)P.(1.2) Reachability analysis has been applied to manoeuvre detection, particularly in the context of resolving uncorrelated tracks (UCTs). Singh [ 27 ] introduced an optimal control framework using the total velocity increment ( ∆V ) as a cost measure. This framework accurately estimates ∆V for connecting UCTs through fuel-optimal manoeuvres. Vázquez [ 30 ] and Montilla [ 18 ] present two algorithms that use state reachability analysis for manoeuvre detection. The first algorithm compares sensor range and range rate measurements with those derived from the initial orbit. The initial state is sampled, propagated and converted into radar measurements at specific epochs. These attributable and projected measurements are then compared in terms of mean state and covariance. If a manoeuvre has occurred, the values will differ, which is checked using the Mahalanobis distance or confidence regions. The second algorithm employs optimal control theory, calculating a distribution of the ∆V that connects the orbit and the measurements. Initially, this optimal control problem is solved deterministically using a multi-shot method. Then, a Monte Carlo shooting algorithm generates a cumulative distribution of the cost function for scenarios with and without the manoeuvre. This distribution helps determine the probability of a manoeuvre. 1.2.2 Manoeuvre Detection Filters Manoeuvre detection filters are widely discussed in the literature. These filters are primarily used in tracking scenarios where frequent data updates are available. Woodburn [ 31 ] proposes a sequential filter to process tracking data prior to a manoeuvre, followed by a fixed interval smoother to estimate the post-manoeuvre state. The difference between the pre-manoeuvre and post-manoeuvre states provides the manoeuvre estimate, which is computed as 6Chapter 1. Introduction ˜ x∆V=˜ xk+1−˜ xk,(1.3) ˜ P∆V=˜ Pk+˜ Pk+1,(1.4) where ˜ xk and ˜ xk+1 are the pre-manoeuvre and post-manoeuvre state estimates, and ˜ Pk and ˜ Pk+1 are their respective covariances. Ko [ 10 ] presents a modified sequential Kalman filter that estimates both the acceleration and the orbital state by representing the perturbing acceleration vector UT(t) as a Fourier series in the eccentric anomaly E . This approach effectively models the unknown perturbing accelerations due to manoeuvres using a 14 dimension basis representation. 1.2.3 Heuristics Heuristic methods take advantage of historical orbital data to predict the post-manoeuvre state and support near real-time manoeuvre detection. The main problem with this approach is that a-priori information is often not available, making these methods inapplicable. Lemmens [ 13 ] developed a methodology to detect orbital manoeuvres through two-line element (TLE) history analysis, applying outlier statistics on the time series of TLE data to detect small manoeuvres while avoiding false positives. Siminski’s method [ 26 ] uses kernel density estimation to predict post-manoeuvre states from feature vectors, which include preand post-manoeuvre orbital elements and the difference between them. This probabilistic approach helps to identify the most likely candidate. Lastly, other approaches such as the one proposed by Shabarekh [ 25 ] are based on machine learning algorithms that use the historical data of the objects to identify and estimate the manoeuvres. 1.2.4 Perturbed dynamics The last class of methodologies reviewed in this thesis is the one used in the algorithms of the cataloguing process. Although it is widely reviewed in Section 2.4.4, a brief introduction of these approaches is depicted here. Perturbed dynamics methods approximate the effects of a manoeuvre on satellite dynamics, suitable for survey scenarios with sparse data. Pastor [ 20 ] and Porcelli [ 22 ] propose an algorithm that perturbs the pre-manoeuvre state xA(t) through a state transition matrix. This matrix allows to propagate perturbations under linear dynamics assumption from t0 to t . Therefore, from any given time t to the manoeuvre epoch tM , the final state vector can be obtained using this transition matrix. Residuals are computed as the difference between actual and estimated measurements ρi= z(ti)−h(ti,y(ti)) and the manoeuvre u is estimated via batch least squares method by minimising these residuals through a cost function. 1.2.5 Emerging Trends and Future Opportunities The combination of artificial intelligence (AI) and big data analytics is revolutionising manoeuvre detection. Machine learning and deep learning algorithms automate this process, improving accuracy by analysing large data sets from a variety of sensors. Advances in algorithm design [ 14 ] and data fusion [ 11 ] improve the ability to monitor satellites in various orbital regimes and observation scenarios. AI-based analysis and automated systems [ 16 ] are set to lead future advances in space situational awareness, fostering the safety and sustainability of space activities. 1.3 Aim of the project 7 1.3 Aim of the project This thesis aims to analyse the cataloguing process as well as to explore and identify the most suitable criterion for the correlation process after the manoeuvre estimation described in Section 1.1. By evaluating various potential criteria, such as minimum impulse, minimum residuals, or likelihood maximisation, the goal is to establish a robust method for associating satellite tracks with their corresponding orbits. The chosen criterion will be rigorously tested and validated through simulations to ensure its effectiveness in maintaining and updating satellite catalogues with a Python library developed to carry out the whole cataloguing process. The main objectives of this thesis are: • Analysis and study of the cataloguing process and algorithms: Firstly, the methods used for each of the tasks of the cataloguing process shown in Figure 1.1 are analysed, understanding the algorithms and tools developed in GMV for their implementation. • Development of a Python library capable of maintaining a catalogue: A Python library that integrates these algorithms and implements new ones is presented. • Development of an algorithm to solve the association problem: Subsequently, a novel approach to cover the correlation problem described in Figure 1.2 is presented and different metrics and criteria are proposed to optimise the decision algorithm. • Testing with simulated satellites: Finally, this pipeline is tested to check if it is able to maintain a catalogue and perform the whole cataloguing chain, analysing how the criteria behave and the results obtained. 1.4 Structure The structure of this document is defined as follows: Chapter 1: This chapter serves the reader as an introduction to the cataloguing problem. The sections Motivation, State of the art and Aim of the project are included to define the scope of this project, to present current developments and technology, and to emphasise important key aspects. Chapter 2: An overview of basic aspects of orbital mechanics is presented, along with a brief review of the algorithms of all the tools necessary for the cataloguing chain. Chapter 3: The STARS library is analysed, describing the workflow of its tools and the implementation of the different algorithms. In addition, the correlation method created for this project is deeply explained, revising the criteria used. Chapter 4: Simulated satellites are used to test the library. The simulated cases are presented and described. A validation and verification of the cataloguing library and the correlation method is also performed, concluding with the results obtained. Chapter 5: Conclusions are drawn from the project and future improvements are proposed. 2 Fundamentals In this chapter, the fundamental concepts on which this thesis is based on are defined and discussed. 2.1 Reference frames Locating objects in space is not a trivial task. Up and down do not exist, there are no clear reference directions and the centre of the selected frame can be anywhere. Moreover, depending on the task to be solved, one frame is used instead of another. The reference frames can be classified in two groups: • Non-inertial reference frames: These reference frames could be rotating, accelerating or both at the same time. • Inertial reference frames: These reference frames are the ones that are neither accelerating nor rotating. As specified above, depending on the task to be addressed, one reference frame or another is used. Due to the multi-tasking problem faced in this thesis, three different reference frames are considered, each of them applied in one of the subtasks of the cataloguing process: the Geocentric Celestial Reference Frame (GCRF), the International Terrestrial Reference Frame (ITRF) and the Tangential, Normal and Cross-Track Reference Frame (TNW). 2.1.1 Geocentric Celestial Reference Frame GCRF The Geocentric Celestial Reference Frame (GCFR) is the standard inertial coordinate system of Earth and serves as the geocentric counterpart to the International Celestial Reference Frame (ICRF). The International Celestial Reference Frame (ICRF) is a barycentric inertial reference frame defined by the measured positions of galactic sources, mainly quasars, observed by Very Long Baseline Interferometry (VLBI). Although general relativity states that true inertial frames cannot exist around gravitating bodies, these reference frames are crucial because they do not show any measurable angular rotation due to the extremely far distance to the galactic sources used for their definition. The reference direction used in celestial coordinate systems is the first point of Aries  , which is the direction from Earth to the Sun during the vernal equinox. Thus, the X-axis is defined by the first Aries point, the Z-axis is perpendicular to the equatorial plane pointing towards the North Pole and the Y-axis follows the right hand rule. One of the natural motions of Earth is the so-called precession of Earth’s axis (Figure 2.1). This is the change of orientation of Earth’s rotation axis due to the gravitational pull of the Sun, the 9 10 Chapter 2. Fundamentals inclination of the ecliptic and the aspherical shape of Earth. It has a period of approximately 26000 years. Therefore, depending on the epoch of the first point of Aries there are different reference frames. In particular, the GCRF uses J2000, which is defined by Earth’s mid-equator and the equinox at 12:00 Earth time on 1st January 2000. The GCRF is widely used in the calculation of satellite orbits, where accurate knowledge of Earth’s orientation and motion is crucial. Therefore, in this thesis it is used to define satellite orbits. Figure 2.1 Precession and nutation of Earth’s axis [29]. 2.1.2 International Terrestrial Reference Frame ITRF An International Terrestrial Reference Frame (ITRF) is a reference frame that implements the ITRS. The International Terrestrial Reference System (ITRS) outlines procedures for establishing suitable reference frames for measurements at or near Earth’s surface. The ITRF originates at Earth’s centre of mass, which includes the oceans and the atmosphere. Like celestial reference frames, ITRF solutions are generated periodically, using the latest mathematical models and surveying techniques to achieve the highest possible accuracy. Depending on the reference epoch, these frames are given different names. For example, the ITRF91 has its reference epoch in 1988, while the ITRF2000 has its reference epoch in 1997 (Figure 2.2). This meticulous process ensures that the ITRS remains a reliable standard for global positioning and geospatial measurements, which is its main purpose. In this thesis, this reference frame is used to locate the sensors used to obtain the satellite observations. 2.1.3 Tangential, Normal and Cross-track reference frame TNW A local orbital coordinate Tangential, Normal, Cross-track (TNW) rotating frame, is defined by three orthogonal axes that rotate with the satellite as it orbits Earth. This system is crucial for satellite navigation and control, providing a frame of reference that moves with the satellite. The three axes that form this reference frame are: 1. Tangential (T) axis: The X-axis aligns with the satellite’s velocity vector, pointing in the direction of the satellite’s motion along its orbit. It represents the tangential direction, indicating where the satellite is heading. 2.1 Reference frames 11 Figure 2.2 ITRF 2000 [1]. 2. Normal (N) axis: The Y-axis, also known as the normal axis, completes the right-handed coordinate system. In a circular orbit, this axis points towards the nadir (the point directly beneath the satellite on Earth’s surface). For an eccentric orbit, the normal axis points as close to nadir as possible while maintaining orthogonality to the plane formed by the T and W axes. 3. Cross-track (W) axis: The Z-axis aligns with the satellite’s orbital angular momentum vector, perpendicular to the orbital plane. It points in the direction of the orbit’s pole, defining the cross-track direction. The TNW rotating frame is particularly useful for various satellite manoeuvres, such as position keeping, orbit adjustments and attitude control. It allows engineers and mission planners to understand and predict satellite behaviour in orbit, ensuring accurate positioning and efficient operation. Therefore, the TNW reference frame is used in this thesis to define and estimate the manoeuvres. Figure 2.3 TNW reference frame [24]. 18 Chapter 3. STARS library and methodology • data: This folder contains two sub-folders: the input data and the output data. The input data correspond with the necessary files for running the algorithms (the Space Weather data, the EOPs, the radar-definition file, the tides, the gravitational field, etc.). On the other hand, the output data folder is where all the results of the different cases are stored. • src: The source folder (src) is the main folder of the library. It contains seven Python modules, each of them containing a class or functions that perform an specific task of the whole cataloguing process. As it is shown in Figure 3.1, the source code gets the data to initialise the computations from the input data folder. The configuration folder is used to feed the programmes with the different parameters to be used during the simulations and finally the results are stored in the output data folder. 3.2 STARS features The source folder contains seven modules, each one corresponding with a crucial part of the cataloguing process. This section aims to explain each one in detail while following the simulation process carried out in this thesis. 3.2.1 Satellite As it was mentioned in Sections 1.4 and 2.4.1, in this thesis the cases analysed are simulated satellites. Due to this fact, the STARS library must be capable of creating its own simulated satellites, simulated orbits and simulated tracks. For this task, the file satellite.py is home of the class Satellite , which is able to create the OPM files for the satellites with their manoeuvres and state vector, propagate the orbit and simulate the observations from any sensor of that simulated orbit. The Satellite class main inputs consist on: •kep_elements: The Keplerian elements of the satellite to be simulated. •sv_epoch : The epoch of the Keplerian elements provided, which is to be used to define the state vector of the satellite. •start_time and stop_time : The beginning and end epochs of the simulation of the satellite’s orbit. •sat_mass and sat_area: The mass and area of the satellite. •sat_CD and sat_CR: The drag and solar radiation pressure coefficients of the satellite. •man_array : An array containing the three components of the manoeuvre the satellite performs in TNW reference frame (in case there is one). •man_epoch: The epoch of the manoeuvre. After defining the satellites, the OPM files of each satellite needs to be created to start the simulation. This process is carried out by a method (function defined inside a Python class) called create_opm . It is worth noting that two different OPM files are created for every satellite: one with the manoeuvre and another one without manoeuvre. The first one is what is usually called the ground truth and it contains all the information of the satellite. The second one does not have the information of the manoeuvre and can be used for the estimations. Nevertheless, this file is not used in the pipeline as it is explained later in Section 3.2.3. Once the OPM file of the satellite is defined and stored, the orbit of the satellite must be simulated using an orbit propagator. This algorithm propagates the OPM file of the satellite and it simulates 3.2 STARS features 19 Figure 3.2 STARS satellite simulator workflow. its orbit between the start and stop dates defined in the class. It only needs the configuration and it creates an OEM file with the complete orbit and a .log file containing the information of the propagation process. The cataloguing chain must begin with the analysis of the tracks of the different objects observed by the sensors. The function obssim_processor of satellite.py serves to simulate these observations. This Python function simply simulates the satellite orbit between the start and stop dates defined using the OEM file of the satellites and MAORI . The observations are similar to the ones S3TSR would provide in a real-case scenario (Section 2.3), with its noise modelled as zero-mean Gaussian noise in the range ( σρ=20 m ), range-rate ( σ˙ ρ=0.65 m/s ) and the angles (σδ,α=0.4deg). The simulated tracks are stored in TDM files. Finally, a method called write_csv is used to create a .csv file with the data of all the simulated satellites. All the functions and the workflow of the simulation of satellites and tracks observations are depicted in Figure 3.2 and Algorithm 1. From this point, the STARS library is generic. This is, it can be used for both real and simulated data as the input of the first function of the cataloguing process is just the TDM files of the tracks. Algorithm 1 Satellites simulator Require: Satellites data 1: for Satellite data in Satellites data do 2: Satellite() ←Satellite data 3: OPM file ←Satellite.create_opm() 4: OEM and .log files ←Satellite.propag_orbit() 5: end for 6: TDM files ←Observations simulator(all OEM files) 20 Chapter 3. STARS library and methodology 3.2.2 Track and track-to-track After simulating the satellites (or collecting real-world data) n tracks files are available. The process that must be followed for each new track is the one described in Figure 1.1. However, in this thesis all the tracks are analysed simultaneously to simplify the computations and reduce the computational time a “real-time” cataloguer would require. Figure 3.3 STARS track.py and t2t.py workflow. Having said so, the next task to be fulfilled is to analyse the tracks to try to correlate them without manoeuvre with the existing satellites (the satellites we simulated). If it is not possible, the uncorrelated tracks must be grouped into sets of tracks in which each track, with high probability, belong to the same unknown object. This chain of tasks is done using two different STARS files: track.py and t2t.py . As these two files work almost together, they are explained in this section in tandem for the sake of simplicity. The Python file track.py just contains a class called Track which stores the data of one track. The Track class only needs as input the filename of the TDM of the track. The main part of this class is the method called t2o_processor , which mainly uses MAORI to try to correlate the tracks with registered objects. The output files of the t2o_processor are the .log files from each track-to-orbit process. The key aspect of this process is the variables assigned to each track class after the execution of track.t2o_processor . If this process is successful, the track is assigned a true flag to a parameter called t2o_flag . In addition, if the assigned object corresponds with the one the track belongs to, a true flag is assigned to a parameter called track.t2o_success . To do so, a method called t2o_postprocessor implemented within the class Track is used after 3.2 STARS features 21 executing t2o_processor and it uses the information from the .log files of the track-to-orbit process. After processing the available tracks, many of them could have been correlated with their corresponding satellite. However, after the manoeuvre epoch the track-to-orbit process tends to fail, leaving a set of uncorrelated tracks that we do not know which object they belong to. With this group of uncorrelated tracks we must apply some algorithm to try to correlate one to another to create sets of tracks that, with high probability, belong to the same unknown object. This is done using the t2t.py file of STARS that contains a function called t2t_processor , which performs the track-to-track association (if possible) of the provided uncorrelated tracks after the track-to-orbit process. This is, the tracks that have a false t2o_flag . This function executes the track-to-track algorithm, returning an OPM file for each set of tracks and the .log file of the process. Subsequent to this computation, the chain returns to the Track class to update one internal variable of the class: the track.t2t_code . This unique variable, consisting on two numbers of four digits separated with an underscore symbol (ABCD_WXYZ), serves as a code to relate each track with the ones it has been correlated to during the track-to-track process. In addition, a method within the Track class called write_csv is used to add each track with its information to a .csv file. All the functions and the workflow of track.py and t2t.py are shown in Figure 3.3 and Algorithm 2. Algorithm 2 Tracks analysis Require: TDM files 1: for TDM file in TDM files do 2: Track() ←TDM file 3: .log file ←Track.t2o_processor() 4: Track.t2o_flag,Track.t2o_success ←Track.t2o_postprocessor() 5: end for 6: OPM and .log files ←t2t_processor(for all Tracks if Track.t2o_flag() is False) 7: for Track in Tracks if Track.t2o_flag is False do 8: Track.t2t_code ←Track.t2t_postprocessor() 9: end for 3.2.3 Orbit determination Before heading into the last part of the cataloguing chain, the reader may be wondering what is to be done with the correlated tracks by the track-to-orbit process. With the uncorrelated tracks we created sets of tracks correlated by the track-to-track. But what do we do with the ones that were successfully correlated? In Section 2.4.4 we mentioned that the manoeuvre estimation algorithm needs as input the set of uncorrelated tracks and the OPM of the satellite we are trying to estimate the manoeuvre of. One possible way to do so is to use the OPM generated by Satellite.create_opm , which is the orbit simulated through the dynamical model without any noise. This is not the ideal case, as in a real world scenario the dynamical model should have some error with respect to the real orbit, introducing uncertainty into the computations. Therefore, a trade-off solution was selected for the simulations. Instead of directly using the OPM file generated without error by Satellite.create_opm , we collect all the tracks corresponding to the same satellite that the track-to-orbit process was able to correlate before the manoeuvre epoch. This measures belong to the same object and they have the Gaussian error introduced by the radar during the observations simulation. Then, a new OPM is computed using MAORI. 22 Chapter 3. STARS library and methodology This is done by the function odet_processor implemented in odet.py . It only needs the trackto-orbit correlated tracks that belong to the same satellite an it performs the orbit determination of the pre-manoeuvre tracks. After so, an OPM file is generated with the orbit data with the intrinsic Gaussian noise of the radar, and a .log file with the results of the process. The process implemented in odet.py is depicted in Figure 3.4 and Algorithm 3. Figure 3.4 STARS odet.py workflow. Algorithm 3 Orbit determination Require: Track-to-orbit pre-manoeuvre correlated tracks (t2o_corr) 1: Satellites names ←t2o_corr.get_satellite_names() 2: for Satellite name in Satellites names do 3: Tracks group ←Tracks in t2o_corr with Track.sat_name equal to Satellite name 4: OPM and .log files ←odet_processor() 5: end for 3.2.4 Hypothesis At the end of our cataloguing chain the track-to-orbit and track-to-track algorithms are fully implemented, leaving us with tracks correlated to their corresponding satellites through the track-toorbit process and uncorrelated tracks that have been grouped into sets of tracks using the track-to-track tool. Taking a look at Figure 1.1, the last part of the catalogue is the manoeuvre determination and estimation of the possible manoeuvres and the promotion of the candidates. The current section and the following one (Section 3.2.5) present the main contribution of this thesis: the development of a novel algorithm for finding feasible manoeuvres and promoting the optimal ones in terms of certain criteria. The algorithms improve the accuracy of the models, making it a valuable advancement. The last part of the cataloguing chain is divided in two subtasks as they belong to different problems: • Manoeuvre estimation: It consists on finding and estimating a feasible manoeuvre for each hypothesis (tracks-object pair). • Manoeuvres analysis and promotion: In Section 1.1 (more specifically Figure 1.2) the correlation process for all the possible candidates was explained. This is what corresponds with this subtask of the problem: to decide which one of the possible manoeuvres to promote. 3.2 STARS features 23 The first one, the manoeuvre estimation, is fully tackled by hypothesis.py . This STARS file consists on a class called Hypothesis with two methods within it: •Hypothesis.manest_processor. •Hypothesis.manest_postprocessor. Figure 3.5 STARS hypothesis.py workflow. Hypothesis is a class consisting on one of the set of tracks created by the track-to-track algorithm and one of the objects from the list of available objects. Hence, if there are n sets of uncorrelated tracks and m objects in the catalogue, L=n×m hypotheses are created as it was mentioned in Section 2.4.4. For each hypothesis, the method manest_processor is used to try to estimate a feasible manoeuvre using MAORI . This process results in the creation of a .log file with the information about the convergence and the manoeuvre (if it was estimated), a .csv file with all the possible manoeuvres estimated for that hypothesis and an OPM file (if the process converged) with all the data about the estimated manoeuvre. The results of this estimation are stored within each Hypothesis class using the method manest_postprocessor. Out of all the data collected, the main results are the following ones: •Hypothesis.manoeuvre_dv : The manoeuvre components in TNW reference frame estimated by manest.bin for the hypothesis. This is, {ˆ VT,ˆ VN,ˆ VW}. •Hypothesis.manoeuvre_epoch : The epoch where manest.bin situated the manoeuvre. This is, ˆ tM. •Hypothesis.WRMS : The loss function estimated during the weighted least-squares refinement process of the manoeuvre estimation. •Hypothesis.manoeuvre_cov_matrix : The covariance matrix Σ of the four components of the manoeuvre, which are {ˆ VT,ˆ VN,ˆ VW,ˆ tM} , obtained during the weighted least-squares process. 24 Chapter 3. STARS library and methodology •Hypothesis.entropy : The differential entropy of the least-squares process computed using the expression E=1 2ln (2πe)4|Σ|. This entropy is explained in the next section. Note that these variables are None if there is no manoeuvre estimated. The workflow implemented in hypothesis.py is depicted in Figure 3.5 and Algorithm 4. Algorithm 4 Hypotheses creation and manoeuvres estimation Require: OD OPM files and Sets of uncorrelated tracks 1: for OPM file in odet OPM files do 2: for Set of uncorrelated tracks in Sets of uncorrelated tracks do 3: Hypothesis() ←OPM file, set of uncorrelated tracks 4: OPM, .csv and .log files ←Hypothesis.manest_processor() 5: Hypothesis.variables ←Hypothesis.manest_postprocessor() 6: end for 7: end for 3.2.5 Promoter We have reached the last step of the problem. After cataloguing the tracks and estimating possible manoeuvres, we are left with a set of hypotheses with manoeuvres considered feasible by the manoeuvre estimator. Let us recover the problem posed in Section 1.1: our sensor detected four sets of uncorrelated tracks, each of them belonging to a different object. The manoeuvre estimation algorithm filters from the catalogue the possible candidates that could match these tracks after performing a (reasonable) manoeuvre. The algorithm evaluates each hypothesis assigning a manoeuvre in terms of magnitude and epoch to each one. The same question proposed in Section 1.1 and depicted in Figure 1.2 arises: how can we select from all the possible candidates one and promote it? Which is the criterion to be applied? This section aims to cover this topic, focusing on the methodology first and then in the STARS library implementation. One direct criterion could be the lowest control effort. From all the possible candidates, promote the ones with the lowest ∆ˆ V . Although simple, this method can be effective taking into account that satellites normally perform their manoeuvres using the least amount of fuel possible. However, let us consider the following case to show why this method could fail: there are two satellites in the same orbit with a phase angle of 5deg . The first one (named A1) manoeuvres, approaching the other satellite that is in front of it (named A2). Now assume that the manoeuvre estimation algorithm is able to find a manoeuvre for both satellites that solves the problem and we need to decide which one is the correct. For A1 it estimates a manoeuvre with a magnitude of 10 cm/s , while for A2 the manoeuvre has a magnitude of 9cm/s . With the lowest control effort criterion the promoted hypothesis would be the one corresponding with the case where A2 manoeuvred, although this is not the real case and it might fits worse the observations of the sensors. As it can be seen, selecting the correct candidate utilising only the magnitude of the manoeuvre may lead to errors. Fortunately, as mentioned in Section 3.2.4, the manoeuvre estimation algorithm provides more information about the estimated manoeuvre for each hypothesis, such as the weighted root mean square (WRMS) or the covariance matrix (Σ). The WRMS measures the accuracy of the estimated manoeuvre vector by taking into account the residuals of the satellite’s observations after the manoeuvre. These residuals are the differences between the observed measures of the satellite and the corresponding values predicted by the estimated manoeuvre vector. The WRMS provides a weighted average of these residuals’ squares, indicating the overall accuracy of the post-manoeuvre orbit. A lower WRMS value means the estimated manoeuvre vector more closely matches the actual post-manoeuvre observations. 3.2 STARS features 25 On the other hand, the covariance matrix is a key component in understanding the uncertainty and reliability of the estimated manoeuvre vector. It provides information about the variances of the estimated parameters and the correlations between them. Each diagonal element of the covariance matrix represents the variance of an individual parameter in the manoeuvre vector and the epoch of the manoeuvre, while the off-diagonal elements represent the covariance between pairs of parameters. A smaller variance indicates higher confidence in the estimated value of that parameter. It is important to understand that actual measurement errors do not impact the computation of the covariance matrix, as it is deeply explained in [17]. Figure 3.6 STARS promotion.py workflow. The covariance matrix can be used to compute the differential entropy of the manoeuvre estimation. Differential entropy is a measure of the uncertainty or randomness in a continuous probability distribution. Unlike discrete entropy (namely Shannon’s entropy), which is used for discrete random variables, differential entropy applies to continuous random variables. Differential entropy quantifies the amount of uncertainty or unpredictability. Higher differential entropy indicates greater uncertainty in the variable’s possible values, while lower values indicate lower uncertainty. Entropy is usually referred to as the “surprise”. The lower the surprise (entropy), the higher the certainty. For the case of the least-squares method, the entropy of a multivariate normal distribution is given by [12]: 26 Chapter 3. STARS library and methodology E=1 2ln ((2πe)n|Σ|),(3.1) where nis the dimension of the covariance matrix. Low manoeuvre magnitude indicates low fuel consumption, which is a manoeuvre design objective. Low WRMS indicate that the estimated manoeuvre matches the observations with higher fidelity. And low entropy indicates low uncertainty in the estimated manoeuvre. Therefore, these three parameters, the manoeuvre magnitude ∆ˆ V , the WRMS and the entropy E , are used in the promotion criterion. The first step of our promotion process should be to establish a certain threshold for the variables of the manoeuvres. For example, one possible criterion could be to discard every hypothesis with a higher ∆ˆ Vthan 10 m/sbased on the satellite orbit or characteristics. After doing so, a score value can be assigned to each hypothesis where a manoeuvre is found using these parameters. This score is computed using a weighted sum. As the magnitude of these three variables differ a lot from one to another, a minimum-maximum normalisation is applied to each hypothesis to standardise the values and to get rid of the units as follows: X∗ i=Xmax −Xi Xmax −Xmin ,(3.2) where X denotes ∆ˆ V , WRMS or E and the maximum and minimum value of each variable is obtained from all the hypotheses with manoeuvre. Therefore, the minimum-maximum normalisation gives a zero value to the global maximum ∆ˆ V , WRMS and E and a value of one to the global minimum. This criterion is selected because the lower any of these variables is, the more probable the manoeuvre they are related to is the correct one. Two different scores are defined using the values provided by Equation 3.2 combined in a weighted sum: a first one taking into account the three variables, and a second one without considering the entropy. These scores are computed for each hypothesis as follows: SwE i=WwE ∆ˆ V·∆ˆ V∗ i+WwE WRMS ·WRMS∗ i+WwE E·E∗ i,(3.3) SwoE i=WwoE ∆ˆ V·∆ˆ V∗ i+WwoE WRMS ·WRMS∗ i,(3.4) where the index wE stands for “with entropy”, the index woE stands for “without entropy” and W∆ˆ V , WWRMS,WEare the weights in each pondered sum. These two scores are used separately and each one of them is applied to promote the hypothesis individually. Two different approaches are proposed just to investigate the behaviour of the variables and the methodology. As the same procedure from this point is applied for each of the scores, a generic score Siis considered to explain the remaining methodology. To select the weights assigned to each one of the variables, two methods are proposed. The first one is Principal Component Analysis (PCA), which is a powerful statistical technique used to simplify the complexity in high-dimensional data while retaining trends and patterns. PCA reduces the dimensionality of the data while preserving as much variability as possible. This means that the most significant patterns in the data are captured with fewer variables (principal components), which simplifies the analysis and reduces computational load. The principal components obtained from PCA are orthogonal (uncorrelated). Using these components to weight the original variables ensures that the weights are not redundant and capture unique aspects of the data. This leads to a more efficient and meaningful representation of the data. For a deeper explanation of this method, the reader is referred to [7]. This method provides how much each variable influences in the sum. In the case of the score with entropy, these are the ∆ˆ V , WRMS and E . And in the case without entropy, the directions are 3.2 STARS features 27 ∆ˆ V and WRMS . This is done using the library sklearn that provides a PCA module. Note that the weights must add put to 1. To remove the extreme values (outliers) and use the central distribution of values, the analysis is done considering the 15-85 percentile range of each variable The weights are obtained using the eigenvector of the main principal component in terms of variance. This eigenvector is expressed on the basis of the original variables. Since the eigenvector is unitary, using the value of its squared components provides a weighting of the original variables that meet the condition of adding up to one. Figure 3.7 Principal Components Representation [3]. Another approach is to select the weights manually after some testing and analysing the behaviour of the different variables of the problem. These two weighting approaches (PCA and manual testing) are discussed in Section 4.3 and the first part of Section 4. After defining the weight of each variable in the scores, we end up with scores values that range from zero to one for every hypothesis. The higher the value of the score is, the more likely that estimated manoeuvre is expected to be. Hence, to select which hypothesis is to be promoted, the maximum score criterion is used and the following procedure is used: 1. The maximum score is selected. 2. If the set of tracks present in the hypothesis corresponding with the maximum score is present in other hypotheses, they are considered for the next step. 3. A mark Pi using these hypotheses’ scores is computed using the SoftMax function [ 2 ]. This function transform a vector of N real values and normalise them using an exponential normal- 34 Chapter 4. Results of simulated satellites 10. Orion-cross-1201 •Nominal noise of the radar. • Manoeuvre magnitude of 20 cm/s : For A1 and A3 the manoeuvre is ∆V= [20,0,0]Tcm/s . For A2 and A4 the manoeuvre is ∆V= [−20,0,0]Tcm/s. • Phasing between satellites of 1deg : θ1=45 deg , θ2=46 deg , θ3=47 deg , θ4=48 deg . • In this scenario, the satellites manoeuvre to get closer to each other. A1 gets closer to A2 and A3 gets closer to A4. 11. Orion-3tracks-1201: •Nominal noise of the radar. •Manoeuvre magnitude of 20 cm/s:∆V= [20,0,0]Tcm/s. • Phasing between satellites of 1deg : θ1=45 deg , θ2=46 deg , θ3=47 deg , θ4=48 deg . • In this scenario, the sets of tracks considered have three tracks instead of the four the track-to-track algorithm creates. This is done to test how the number of tracks influences the solution and the manoeuvre estimation process. Finally, as mentioned in Section 3.2.5, a pre-filtering criterion is considered to remove outliers from the final analysis. In these scenarios the satellites have a mass of 260 kg and are in LEO, so ∆ˆ V>1m/s and WRMS >10 are chosen as hypothesis pre-filtering criteria. The former discards hypotheses with an impulse too high for the regime we are analysing (and the mass of the satellites) and the latter focuses on filtering candidates with a high error. Only one is necessary to discard the hypothesis. 4.2 Algorithms metrics Each one of the steps of the catalogue can lead to errors and propagate these error downstream. To analyse how each algorithm is behaving, different metrics for each of them are introduced. 4.2.1 Track-to-orbit metrics The number of analysed (all the available ones) and promoted tracks, as well as the correctly promoted percentage, are studied. The correctly promoted tracks are those who have been correlated with the satellite they belong to. In addition, these parameters are divided in pre-manoeuvre tracks, post-manoeuvre tracks and total tracks to see the influence of the manoeuvres in the track-to-orbit algorithm. Therefore, the following table is used: Table 4.1 Track-to-orbit metrics. Analysed Promoted %Correct Pre-manoeuvre tracks Post-manoeuvre tracks Total tracks 4.2.2 Track-to-track metrics Track-to-track may correlate tracks that do not belong to the same object. To analyse how well this algorithm behaves, the percentage of correctly associated tracks (sets of associated tracks all belonging to the same satellite) is used. This percentage represents the number of uncorrelated 4.2 Algorithms metrics 35 tracks associated with their pairs out of the total number of associated tracks. This percentage is computed as shown in the following equation: %t2t =Correctly associated tracks Total associated tracks ·100.(4.1) 4.2.3 Manoeuvre estimator metrics To evaluate the manoeuvre estimation process, the hypotheses are classified in four categories: • True Negatives (TN): These are hypotheses which set of tracks belong to a different object than the selected one and no manoeuvre is found. • False Positives (FP): These are hypotheses which set of tracks belong to a different object than the selected one and a manoeuvre is found. • False Negatives (FN): These are hypotheses which set of tracks belong to the same object as the selected one and no manoeuvre is found. • True Positives (TP): These are hypotheses which set of tracks belong to the same object as the selected one and a manoeuvre is found. In addition, sensitivity (the percentage of true positives out of all hypotheses with manoeuvre), specificity (the percentage of true negatives out of all hypotheses without manoeuvre) and accuracy (the percentage of correctly predicted cases out of all the cases) are considered. The results are presented in the following tables: Table 4.2 Manoeuvre estimator metrics. No manoeuvre Manoeuvre No predicted TN FN Predicted FP TP Sensitivity: Specificity: Accuracy: 4.2.4 Promotion metrics Although already mentioned in previous sections, it is recalled that the cataloguing process is applied by studying all tracks at the same time, rather than in a timely-ordered manner. Therefore, as the simulation time after the manoeuvre is five days, many sets of tracks may belong to the same object and duplicate hypotheses (in the sense of the objects they belong to) could be created. This is done because no track pool is available and the hypotheses differ from the data available in the real world. In a real scenario, the first set of tracks belonging to a manoeuvred object would be analysed and updated in the catalogue, with the track-to-orbit algorithm being able to correlate from that point new tracks arriving from that object. In our scenarios this is not the case and all observations are analysed at the same time. Thus, many hypotheses not promoted by the promotion algorithm may be correct. To avoid the metrics of the promoter being directly affected by this fact, we only consider two kind of cases: • Corectly promoted hypotheses (True positives): The number of hypotheses whose manoeuvre is correct and whose assigned object is the correct one. • Incorrect promoted hypotheses (False positives): The number of hypotheses where a manoeuvre that did not occur is promoted or whose assigned object is not the correct one. 36 Chapter 4. Results of simulated satellites 4.3 Behaviour of the variables and the scoring weights Now that we have defined all the scenarios and the metrics used to evaluate each part of the cataloguing process, in this section we focus on how each of the proposed variables behave and how well the weights estimated by the PCA algorithm are working. Let us consider the results for the first scenario focusing on the last step of the cataloguer: the promotion algorithm. In Figure 4.1 the distribution of the three variables ( ∆ˆ V , WRMS and entropy) for the hypotheses where a manoeuvre was found is shown. It is worth mentioning that all the hypotheses evaluated in this scenario are true, very likely due to the big spacing of the satellites. In addition, the promoted hypotheses using the score with entropy are surrounded with a black line, while the ones promoted by the score without entropy have a red border. If the two scores promoted the same hypothesis, it has both black and red border. The non-promoted hypotheses have a high transparency. Figure 4.1 Scatter pairplot of the three variables of Orion-1115 hypotheses using PCA to estimate the weights. The first thing we notice is the distribution of the dots. Focusing on the diagonal histograms, for both ∆ˆ V and WRMS the hypotheses tend to have low values and accumulate along the real manoeuvre magnitude ∆V=10 cm/s and WRMS =1 . However, this is not the case for the entropy, whose values are more evenly distributed without a clear tendency to the lowest values. This great discrepancy in the behaviour of the entropy and the other variables play a crucial role in the weighting process. As PCA computes the principal direction in terms of the variance, one would expect that the principal direction is almost parallel to the entropy axis. In fact, the weights obtained in this scenario for the score with entropy are WwE ∆ˆ V=0.026 , WwE WRMS =0.002 , WwE E=0.973 , giving almost all the weight of the score to the entropy, which is neither the ideal nor the expected outcome. One would expect the behaviour of the entropy to be similar to the one of the rest of the variables as for lower entropy the uncertainty of the estimation is also reduced, but this is not the case. 4.3 Behaviour of the variables and the scoring weights 37 Furthermore, the entropy is checked not to be a good discriminator of the optimal hypothesis. The optimal hypotheses are those with the lowest error in both magnitude and epoch. Let us now plot in Figure 4.2 the hypotheses in terms of the estimated manoeuvre epoch error and the estimated manoeuvre ∆ˆ V error, colouring them using their respective scores with entropy (which is almost the entropy as its weight is almost one). It can be seen that the entropy introduces a lot of noise into the estimation process, finally promoting the hypotheses with the highest manoeuvre magnitude error. Figure 4.2 Error plot (colour bar: score with entropy) of the Orion-1115 hypotheses using PCA to estimate the weights. Focusing now on the promoted hypotheses by the score without entropy we see a similar behaviour in terms of the weights assigned to each variable. As it was stated before, the weight given to ∆ˆ V in the entropy score was ten times bigger than the one considered for WRMS. This fact is reflected when these two variables are used alone, assigning the PCA the following weights: WwoE ∆ˆ V=0.998 , WwoE WRMS =0.002 . This result should not be a surprise after having analysed Figure 4.1, as the second most spread out variable is the manoeuvre magnitude while the WRMS values are concentrated around one. Using these weights, the score without entropy behaves almost as the “lowest control effort” criterion mentioned in Section 3.2.5. This criterion, as shown in Figure 4.2, is a much better decision choice than the entropy alone. The practical implications of these findings are noteworthy. The score is shown to behave badly when the weight given to the entropy is too high as this variable has not a clear distribution. Moreover, the PCA analysis highly depends on how spread out each of the variables are, attributing more weight to those with an unclear distribution. These characteristic results are not exclusive of this particular scenario as it can be seen in Figure 4.3, which corresponds with the Orion-1110 scenario. The weights estimated for this case are WwE ∆ˆ V=0.018,WwE WRMS =0.003,WwE E=0.979 and WwoE ∆ˆ V=0.579,WwoE WRMS =0.421. In all the cases considered the most spread out variable is the entropy, followed by the manoeuvre magnitude and the WRMS. Furthermore, depending on the case analysed the weights of ∆ˆ V and WRMS vary tremendously. These results lead us to these two final conclusions: 38 Chapter 4. Results of simulated satellites • The entropy is not behaving as it was initially expected, resulting in a bad estimation of the optimal hypotheses. On account of this, it should be treated with caution and assigned a lower weight in the entropy-based score if we want optimal hypotheses to be promoted. • As a result of the widespread values of the entropy, PCA is attributing almost all the weight to this variable when it is taken into account in the score. Moreover, this method is not proving itself robust to estimate the weights, so from now on the alternative proposed in Section 3.2.5 is applied. Figure 4.3 Scatter pairplot of the three variables of Orion-1110 hypotheses using PCA to estimate the weights. 4.4 Scenarios analysis and results In this section the most interesting outcomes of the scenarios considered are presented. The results set out in Section 4.3 led us to conclude that the entropy must be considered cautiously and that the weights calculated using PCA are very sensitive to the scenario variables. Here is where the other weighting approach proposed in Section 3.2.5 is brought into play. After analysing different cases and testing multiple weights, the following values are used in the analyses: • For the score with entropy, the biggest influence in the sum is given to ∆ˆ V , followed by the WRMS and lastly the entropy, which is assigned a low weighting value due to its noisy behaviour. Therefore, WwE ∆ˆ V=0.60,WwE WRMS =0.30,WwE E=0.10. • In the case of the score without entropy the contribution of the entropy to the weighted sum is split and distributed equally. Hence, WwoE ∆ˆ V=0.65,WwoE WRMS =0.35, From this point, PCA is no longer used and the weights considered for both scores are fixed. 4.4 Scenarios analysis and results 39 4.4.1 Orion-1115 This basic scenario was designed to have as little miscorrelation as possible and to test the behaviour of the promotion using the proposed weights. Miscorrelation refers to the situation where the manoeuvre estimation algorithm finds a feasible manoeuvre of a certain object with a set of associated tracks that do not belong to that hypothetical object. Minimising miscorrelation allows us to analyse how well the new weights behave, as well as having a reference scenario where all the algorithms work almost perfectly. The results for the different algorithms are presented in the following tables, where it is depicted the perfect score obtained by them. Track-to-orbit metrics: Table 4.3 Track-to-orbit metrics of Orion-1115. Analysed Promoted %Correct Pre-manoeuvre tracks 84 84 100% Post-manoeuvre tracks 122 0 0% Total tracks 206 84 100% Track-to-track metric: %t2t =100%. Manoeuvre estimator metrics: Table 4.4 Manoeuvre estimator metrics of Orion-1115. No manoeuvre Manoeuvre No predicted 84 0 Predicted 0 28 Sensitivity: 100% Specificity: 100% Accuracy: 100% Figure 4.4 Error plot (colour bar: score with entropy) of the Orion-1115 hypotheses. As the satellites are so far apart, there is zero confusion and all applications work perfectly. This scenario is of no interest from the point of view of the SoftMax criterion created to select between 40 Chapter 4. Results of simulated satellites hypotheses with the same set of tracks as there are no false hypotheses. However, the correct 28 hypotheses provide a good framework for testing the weights selected for the scores. The first thing that calls our attention about Figure 4.4 is the difference in the epoch of the promoted manoeuvres by the scores with and without entropy (black and red, respectively). It is clear that the entropy, even with a low weight assigned, keeps introducing noise and uncertainty in the selection of the optimal candidates. Figure 4.5 Error plot (colour bar: entropy) of the Orion-1115 hypotheses. To highlight the irregular behaviour of entropy, Figure 4.5 shows the error plot using the values of the entropy in the colour bar. It is clear from this figure that the best-case scenarios do not have the lowest entropy values and that there is no clear pattern when using this variable. Figure 4.6 Error plot (colour bar: score without entropy) of the Orion-1115 hypotheses. 4.4 Scenarios analysis and results 41 Another point to note is the accuracy of the entropy-free score. For all the satellites, the hypotheses selected using this criterion have the lowest error in both magnitude and epoch. The same plot using the no-entropy score as the colour bar is depicted in Figure 4.6. Finally, the hypotheses promoted by each of the scores and their results are shown in the Table 4.5. For A2 and A4 the same hypotheses are promoted using both scores, but got A1 and A3 the promoted ones through the score without entropy are clearly better in terms of manoeuvre magnitude error, epoch error and WRMS. Note that all the promoted hypotheses in both cases are correct, although ones are better than others. Table 4.5 Promoted hypotheses of the Orion-1115 scenario. Promoted hypotheses using the score with entropy Code SwE ∆ˆ V[cm/s]Er(∆ˆ V)%E(ˆ tM)[min] WRMS Correct A1 0046_1218 0.918 [9.8,2.75,−2.31]4.37 −105.13 1.128 ✓ A2 0026_5723 0.9[9.99,−0.56,−0.95]0.52 2.77 0.959 ✓ A3 0030_0905 0.917 [9.46,2.35,−0.03]−2.51 −87.52 1.506 ✓ A4 0042_4671 0.946 [9.99,0.18,−0.65]0.17 1.31 0.973 ✓ Promoted hypotheses using the score without entropy Code SwoE ∆ˆ V[cm/s]Er(∆ˆ V)%E(ˆ tM)[min] WRMS Correct A1 0022_5184 0.945 [10.0,2.66,0.54]3.64 −1.06 1.025 ✓ A2 0026_5723 0.969 [9.99,−0.56,−0.95]0.52 2.77 0.959 ✓ A3 0034_3167 0.952 [9.99,−0.71,−1.49]1.3−1.27 1.115 ✓ A4 0042_4671 0.97 [9.99,0.18,−0.65]0.17 1.31 0.973 ✓ From now onwards, only the metrics of the algorithms, the error plot using the score without entropy as colour bar (as it is the one with a better behaviour) and the promoted hypotheses are analysed in the rest of the chapter. 4.4.2 Influence of the distance between satellites The three ongoing scenarios (Orion-1110, Orion-1105 and Orion-1101) are focused on showing how the distance between satellites affect the algorithms and their performance. 4.4.2.1 Orion-1110 Track-to-orbit metrics: Table 4.6 Track-to-orbit metrics of Orion-1110. Analysed Promoted %Correct Pre-manoeuvre tracks 84 84 100% Post-manoeuvre tracks 121 1 0% Total tracks 205 85 98.82% Track-to-track metric: %t2t =96.3%. The wrongly correlated set of tracks is 0038_3503. Manoeuvre estimator metrics: Table 4.7 Manoeuvre estimator metrics of Orion-1110. No manoeuvre Manoeuvre No predicted 80 1 Predicted 1 26 Sensitivity: 96.30% Specificity: 98.77% Accuracy: 98.15% 42 Chapter 4. Results of simulated satellites With 10 deg of phasing, the distance between satellites is still high enough for the algorithms to be able to distinguish between each of the objects with high accuracy. However, the increased proximity between objects introduced some false positives and false negatives, indicating that as closer the satellites are, the higher the confusion will be. Figure 4.7 Error plot (colour bar: score without entropy) of the Orion-1110 hypotheses. Focusing on Figure 4.7, the same behaviour of the scores seen in the previous cases is maintained: the score with entropy has a lot of entropy-induced noise and the score without entropy selects near-perfect candidates from among all possible candidates. One thing to note about the entropy-free score is that, among all possible candidates, it selects those with the same manoeuvre epoch as the actual one. This behaviour was neither expected nor intended, since selecting a candidate with a manoeuvre occurring ±k orbit periods does not affect the cataloguing process as long as that solution reliably predicts the orbit after the manoeuvre. In fact, if the lowest control effort criterion was used, in the case of the A2 and A4 satellites the chosen solution would be the ones with the negative ∆V error (lower ∆V predicted than actual) that are one orbital period away from the actual manoeuvre epoch. However, the introduction of WRMS in the scoring criterion tends to discard these solutions in favour of those with the correct epoch, as they have the least amount of predicted post-manoeuvre error. The reader may have notice that, although one false positive hypothesis was predicted by the manoeuvre estimation algorithm, the mark criterion of Equation 3.5 was not mentioned in this section. The reason for this is that this false hypothesis was pre-filtered before the promotion process as it had a ∆ˆ V=1159 cm/sand a WRMS =31.36. The results shown in Table 4.8 remark that, although all the promoted candidates are correct, the ones selected by the score without entropy are in fact the optimal solutions among all the possible hypotheses. 4.4 Scenarios analysis and results 43 Table 4.8 Promoted hypotheses of the Orion-1110 scenario. Promoted hypotheses using the score with entropy Code SwE ∆ˆ V[cm/s]Er(∆ˆ V)%E(ˆ tM)[min] WRMS Correct A1 0046_1218 0.932 [9.8,2.75,−2.31]4.37 −105.13 1.128 ✓ A2 0034_3268 0.913 [9.99,4.06,−0.3]7.85 −7.63 1.078 ✓ A3 0038_3778 0.927 [9.7,3.32,−0.38]2.57 −90.95 1.141 ✓ A4 0046_1361 0.925 [9.99,−0.25,−2.43]2.86 −20.17 1.265 ✓ Promoted hypotheses using the score without entropy Code SwoE ∆ˆ V[cm/s]Er(∆ˆ V)%E(ˆ tM)[min] WRMS Correct A1 0022_5184 0.961 [10.0,2.66,0.54]3.64 −1.06 1.025 ✓ A2 0026_4762 0.96 [9.97,−3.26,0.78]5.15 0.06 0.961 ✓ A3 0026_4778 0.967 [9.98,−2.94,0.31]4.07 0.06 0.948 ✓ A4 0034_3167 0.964 [9.99,−0.71,−1.49]1.3−1.27 1.115 ✓ 4.4.2.2 Orion-1105 Track-to-orbit metrics: Table 4.9 Track-to-orbit metrics of Orion-1105. Analysed Promoted %Correct Pre-manoeuvre tracks 84 84 100% Post-manoeuvre tracks 120 1 0% Total tracks 204 85 98.82% Track-to-track metric: %t2t =88% . The wrongly correlated sets of tracks are 0038_0925 (A3 + A2 + A2 + A2), 0042_4684 (A2 + A2 + A1 + A2) and 0046_3435 (A3 + A2 + A3 + A3). Increasing the proximity of the satellites up to 5deg starts to cause errors in the track-to-track, as three sets of tracks are created that do not belong to the same object. As these tracks do not match, the hypotheses created from these sets of tracks have no feasible manoeuvre predicted by the manoeuvre estimator. This situation is repeated for the rest of the cases, as the correlated tracks introduce too much error in the manoeuvre estimation process and the algorithm is not able to find any reasonable manoeuvre. Manoeuvre estimator metrics: Table 4.10 Manoeuvre estimator metrics of Orion-1105. No manoeuvre Manoeuvre No predicted 75 3 Predicted 0 22 Sensitivity: 88.0% Specificity: 100.0% Accuracy: 97.0% In Figure 4.8, the same behaviour of the scores seen in the previous cases is maintained, being the score without entropy the best option. Although there is a promoted candidate by this score that is one orbital period away from the real solution (A4) this is due to the fact that the other candidates for this satellite that are located around the real manoeuvre epoch have a really high error in the estimated manoeuvre. Therefore, the criterion chose the candidate delayed one orbital period as it is the one with the lowest manoeuvre error and WRMS. The promoted hypotheses are again shown in Table 4.11. 50 Chapter 4. Results of simulated satellites variability, especially in scenarios with larger manoeuvres, highlighting the increased difficulty in correlating tracks correctly as the objects occupy the orbit of other satellites. Figure 4.12 Zoom error plot (colour bar: score without entropy) of the Orion-1201 hypotheses. Table 4.23 Promoted hypotheses of the Orion-1201 scenario. Promoted hypotheses using the score with entropy Code SwE ∆ˆ V[cm/s]Er(∆ˆ V)%E(ˆ tM)[min] WRMS Correct A1 0034_5709 0.929 [20.01,8.44,−0.69]8.66 −4.33 1.085 ✓ A2 0026_3293 0.933 [20.02,5.84,−0.2]4.3−2.54 0.975 ✓ A3 0022_1097 0.896 [19.98,−3.23,1.5]1.5 2.18 1.069 ✓ A4 0022_1661 0.895 [20.0,−2.52,0.2]0.78 1.24 1.099 ✓ Promoted hypotheses using the score without entropy Code SwoE ∆ˆ V[cm/s]Er(∆ˆ V)%E(ˆ tM)[min] WRMS Correct A1 0022_5185 0.996 [20.0,2.19,0.49]0.62 −0.36 1.025 ✓ A2 0026_3293 0.992 [20.02,5.84,−0.2]4.3−2.54 0.975 ✓ A3 0022_1097 0.991 [19.98,−3.23,1.5]1.5 2.18 1.069 ✓ A4 0022_1661 0.99 [20.0,−2.52,0.2]0.78 1.24 1.099 ✓ In conclusion, the proximity between satellites along with a higher manoeuvre tends to induce more error in the cataloguing algorithms, increasing the number of wrongly associated tracks in the track-to-track algorithm. Nevertheless, the pipeline remains robust to these errors, correctly finding the manoeuvres. 4.4 Scenarios analysis and results 51 4.4.4 Influence of radar noise The next scenario (Orion-2201) is used to characterise the behaviour of the cataloguer when the intrinsic error of the radar is doubled. In the metrics the values of the results from scenario Orion-1201 of Section 4.4.3.2 are also shown to ease the comparison of results. 4.4.4.1 Orion-2201 Track-to-orbit metrics: Table 4.24 Track-to-orbit metrics of Orion-2201 (in parenthesis Orion-1201). Analysed Promoted %Correct Pre-manoeuvre tracks 84 (84)84 (84)100% (100%) Post-manoeuvre tracks 120 (120)6(1)0% (0%) Total tracks 204 (204)90 (85)93.33% (98.82%) Track-to-track metric: %t2t =61.9% (50%) . The wrongly correlated sets of tracks are 0022_5001, 0026_4923, 0042_4638, 0043_5920, 0045_1299, 0046_1299, 0046_3324 and 0047_4536. Manoeuvre estimator metrics: Table 4.25 Manoeuvre estimator metrics of Orion-2201 (in parenthesis Orion-1201). No manoeuvre Manoeuvre No predicted 9(75)3(7) Predicted 54 (39)18 (15) Sensitivity: 85.71% (68.18%)Specificity: 14.29% (40.91%)Accuracy: 32.14% (47.73%) The complexity of this scenario is heightened by the objects’ maximum proximity, as the objects manoeuvres bring them near the positions of the other objects, compounded by a radar error that is double the standard value. Consequently, the accuracy of the track-to-orbit and manoeuvre estimation is significantly compromised, resulting in 54 incorrect hypotheses being erroneously validated. Again, the pre-filter discards many of these false positives, reducing the number from 54 to 8. The pre-filter again proves its robustness and usefulness, as the noisy hypotheses introduced in the promotion process are reduced to a 13% of the original set. However, as 8 false positives are considered, the SoftMax criterion is again triggered when miscorrelation is found. The highest score comes from the hand of the set of tracks with code 0039_3783. This set of tracks is related in this hypothesis to A2, which is its correct correlation. However, the manoeuvre estimation algorithm found a miscorrelation with the A1 satellite. Table 4.26 Objects candidates for the 0039_3783 set of tracks (correct one: A1). Candidate ∆ˆ V[cm/s]WRMS Entropy SwE Mark PPromoted A1 38.42 0.997 −5.348 0.806 0.461 − A2 21.89 1.04 −5.054 0.939 0.539 No Table 4.26 shows that in terms of ∆V both manoeuvres are indeed close to each other. Moreover, the WRMS of both manoeuvres is almost one, with a smaller value of the error in the false hypothesis. As the entropy in both cases has almost the same value, both manoeuvres could be feasible if we did not know which one is correct. Therefore, as the score of both hypotheses is similar and the mark value is halved, the algorithm decides not to promote the hypothesis. 52 Chapter 4. Results of simulated satellites This is what one would expect in this type of scenario where miscorrelation is found and the manoeuvres are not so far apart in terms of the variables considered. Therefore, the pipeline behaves as expected. Let us now focus on Figure 4.13, where the hypotheses promoted by each score are plotted. Due to the increased noise of the scenario, the error committed by both scores is substantially higher compared to less noisy scenarios. The score without entropy still tends to perform better, promoting more accurate hypotheses, but the overall performance degrades markedly as shown in Table 4.27. Figure 4.13 Zoom error plot (colour bar: score without entropy) of the Orion-2201 hypotheses. Although the WRMS is still around one, this is because this parameter is computed using the residuals of the measurements, which include the intrinsic error of the radar. Consequently, even with increased radar noise, the WRMS remains similar, making it a less reliable indicator of the true accuracy in this scenario. Table 4.27 Promoted hypotheses of the Orion-2201 scenario. Promoted hypotheses using the score with entropy Code SwE ∆ˆ V[cm/s]Er(∆ˆ V)%E(ˆ tM)[min] WRMS Correct A1 0034_5709 0.92 [20.02,14.02,−0.96]22.3−7.94 1.089 ✓ A2 0026_3293 0.924 [20.05,9.6,−0.21]11.13 −4.65 0.976 ✓ A3 0033_4802 0.93 [19.99,4.75,6.36]7.55 22.97 1.076 ✓ A4 0033_3170 0.924 [19.97,−2.13,8.52]9.09 26.02 1.147 ✓ Promoted hypotheses using the score without entropy Code SwoE ∆ˆ V[cm/s]Er(∆ˆ V)%E(ˆ tM)[min] WRMS Correct A1 0038_3478 0.959 [19.39,−7.96,−10.51]17.26 −109.97 1.166 ✓ A2 0030_3170 0.994 [20.02,4.33,0.5]2.42 −0.25 1.067 ✓ A3 0022_3546 0.998 [19.98,−4.76,1.75]3.05 1.5 0.991 ✓ A4 0033_3170 0.976 [19.97,−2.13,8.52]9.09 26.02 1.147 ✓ This scenario illustrates the significant impact of increased radar noise on the cataloguing process. The doubling of radar error leads to a higher number of false hypotheses and complicates the task 4.4 Scenarios analysis and results 53 of distinguishing between correct and incorrect manoeuvre estimates. Despite the robustness of the pre-filter in reducing false positives, the presence of noise required the use of the SoftMax method to further filter out incorrect correlations in counterpart with Section 4.4.3.2, which is the same scenario with nominal radar noise. Overall, the Orion-2201 scenario demonstrates that while current algorithms and methods can manage the increased noise to some extent, their effectiveness is compromised, leading to a reduction in accuracy. 4.4.5 Influence of approaching and crossing objects Until now, the scenarios analysed presented satellites whose manoeuvring direction and magnitude were the same. These satellites were not approaching each other, but their change of position and their proximity induced to errors in the algorithms. This section aims to analyse the influence of approaching satellites (such as those in the Orion-A1A3-1101 and Orion-A1A3-1201 scenarios) and crossing satellites (Orion-cross-1201) on the cataloguing chain. 4.4.5.1 Orion-A1A3-1101 Track-to-orbit metrics: Table 4.28 Track-to-orbit metrics of Orion-A1A3-1101. Analysed Promoted %Correct Pre-manoeuvre tracks 141 141 100% Post-manoeuvre tracks 63 1 0% Total tracks 204 142 99.3% Track-to-track metric: %t2t =83.33% . The wrongly correlated sets of tracks are 0030_0851 and 0038_3816. Manoeuvre estimator metrics: Table 4.29 Manoeuvre estimator metrics of Orion-A1A3-1101. No manoeuvre Manoeuvre No predicted 14 2 Predicted 22 10 Sensitivity: 83.33% Specificity: 38.89% Accuracy: 50.0% As discussed in section 4.1, in this scenario the manoeuvring satellites are A1 and A3, while A2 and A4 remain in their original orbit. Since two satellites do not manoeuvre, track-to-orbit has fewer post-manoeuvre trajectories to correlate, improving its performance and being able to correctly correlate a larger number of trajectories during the simulation. The number of erroneously correlated sets of tracks in the track-to-track algorithm as well as the false hypotheses in the manoeuvre estimator are also significantly reduced compared to the case analysed in Section 4.4.2.3. Again, this is directly due to the reduction of manoeuvribg satellites. Out of all the false positive hypotheses found by the manoeuvre estimation algorithm, the pre-filter was able to discard 18 of them, reducing the total number of false hypotheses to 4. In the promotion process one miscorrelation was analysed, as the hypothesis with the second highest score was cross-correlated with two additional satellites. However, as one of them was previously discarded, it was not taken into account in the SoftMax criterion. Again, the mark obtained by this hypothesis was not sufficiently high to promote it and it was discarded, as shown in Table 4.30. This case is particularly interesting due to the promotion of an incorrect hypothesis. The set of tracks named 0047_4598, which belongs to the A3 satellite, serves as an example. The manoeuvre estimator found feasible manoeuvres with satellites A1, A2, and A3. According to the promotion 54 Chapter 4. Results of simulated satellites Table 4.30 Objects candidates for the 0043_4684 set of tracks (correct one: A3). Candidate ∆ˆ V[cm/s]WRMS Entropy SwE Mark PPromoted A2 69.41 1.911 −11.896 0.504 0.319 − A3 15.43 0.99 −7.436 0.919 0.681 No algorithm, the hypothesis with the highest score is selected for promotion if its mark exceeds 0.9. Once a hypothesis is promoted, all other hypotheses containing the same object are removed from the process. (a) (b) (c) (d) Figure 4.14 Correlation process that resulted in the promotion of an incorrect hypothesis. Initially, the hypothesis with the highest score, which relates 0026_2969 with A3, was promoted, leading to the removal of all other hypotheses containing A3. Subsequently, the hypothesis correlating A1 with 0046_1218 was considered and promoted, causing the removal of the remaining hypotheses involving A1. Consequently, the last remaining hypothesis, which correlates A2 with 0047_4589, was promoted despite being incorrect, as no miscorrelation was identified at this stage of the analysis. In this scenario, the first promoted hypothesis was related to the A3 satellite. As a result, all hypotheses containing A3, including the correct one for 0047_4598, were discarded. The second highest score involved A1, leading to the removal of all hypotheses containing this object, including the false positive that correlated 0047_4598 with A1. After promoting the hypotheses for A3 and A1, the only remaining feasible satellite found by the manoeuvre estimation algorithm was A2, which was incorrectly correlated with 0047_4598. Since no other correlated satellites were left in the process, the mark attributed to the incorrect hypothesis was 1 , resulting in its promotion despite being wrong. Figure 4.14 provides a schematic of this process. It is worth noting that this situation is not expected to arise in a real-time cataloguer, as 0047_4598 would have been processed individually. In such a system, it would not have been incorrectly 4.4 Scenarios analysis and results 55 promoted to A2, since A1 and A3 would still be available for consideration. In this context, the continuous cataloguer offers a distinct advantage by preventing premature elimination of correct hypotheses. By processing tracks individually and continuously, the cataloguer can maintain a more accurate and dynamic picture of satellite positions and manoeuvres. This reduces the risk of false positives and incorrect correlations, ensuring that each track is evaluated with the most complete set of data available. Therefore, although this is an incorrect promotion, it is attributed to the simulation method (processing all tracks simultaneously) rather than a deficiency in the cataloguer itself. The final promoted hypotheses are shown in Table 4.31. As the WRMS value of the incorrectly promoted hypothesis is lower than 2, no caution flag is added in this case. Table 4.31 Promoted hypotheses of the Orion-A1A3-1101 scenario. Promoted hypotheses using the score with entropy Code SwE ∆ˆ V[cm/s]Er(∆ˆ V)%E(ˆ tM)[min] WRMS Correct A1 0046_1218 0.922 [9.8,2.75,−2.31]4.37 −105.13 1.128 ✓ A2 0047_4598 0.737 [21.52,23.12,21.38]281.44 2482.25 1.251 ✘ A3 0026_2969 0.914 [10.04,6.33,0.2]18.71 −5.55 0.964 ✓ Promoted hypotheses using the score without entropy Code SwoE ∆ˆ V[cm/s]Er(∆ˆ V)%E(ˆ tM)[min] WRMS Correct A1 0026_4815 0.995 [10.01,4.05,0.14]8.05 −4.93 0.98 ✓ A2 0047_4598 0.764 [21.52,23.12,21.38]281.44 2482.25 1.251 ✘ A3 0026_2969 0.989 [10.04,6.33,0.2]18.71 −5.55 0.964 ✓ In Figure 4.15 the promoted hypotheses are plotted, showing that the wrongly correlated satellite A2 is far from the correct manoeuvre. Since A2 performed no manoeuvre, the error in ∆ˆ V was computed assuming the same value as the actual manoeuvre carried out by A1 and A3. This significant discrepancy highlights the impact of incorrect promotions on the accuracy of the cataloguing process. Figure 4.15 Error plot (colour bar: score without entropy) of the Orion-A1A3-1101 hypotheses. 56 Chapter 4. Results of simulated satellites Zooming in on the figure (Figure 4.16) and analysing the results for A1 and A3, similar conclusions to those in previous cases can be drawn. The score considering entropy does not perform as well as the entropy-free score, which estimates better the magnitude of the manoeuvre and correctly identifies the epoch of its occurrence. Entropy-free scoring consistently outperforms entropy-influenced scoring, providing more accurate results in both the magnitude and timing of manoeuvres. This analysis highlights the issue of incorrect hypothesis promotion due to the stationary nature of the cataloguing scenario. In reality, tracks would be processed sequentially as they arrive, but in the simulations they are analysed simultaneously. This subjects the cataloguer to a more demanding and challenging situation than what would occur in an actual operational scenario. Figure 4.16 Zoom error plot (colour bar: score without entropy) of the Orion-A1A3-1101 hypotheses. 4.4.5.2 Orion-A1A3-1201 Track-to-orbit metrics: Table 4.32 Track-to-orbit metrics of Orion-A1A3-1201. Analysed Promoted %Correct Pre-manoeuvre tracks 141 141 0% Post-manoeuvre tracks 63 0 100% Total tracks 204 141 100% Track-to-track metric: %t2t =66.67% . The wrongly correlated sets of tracks are 0026_4923, 0030_0819, 0046_1298 and 0046_3521. Manoeuvre estimator metrics: Table 4.33 Manoeuvre estimator metrics of Orion-A1A3-1201. No manoeuvre Manoeuvre No predicted 12 4 Predicted 24 8 Sensitivity: 66.67% Specificity: 33.33% Accuracy: 41.67% 4.4 Scenarios analysis and results 57 The algorithms’ behaviour observed in the previous section is replicated in this scenario, albeit with a increase in the number of incorrectly correlated track sets. The pre-filter is highly effective within this particular framework, decreasing the count of false positives analysed from 24 to just 2. Figure 4.17 Error plot (colour bar: score without entropy) of the Orion-A1A3-1201 hypotheses. During the promotion process, a single miscorrelation was examined, and since the mark achieved by this hypothesis exceeded 0.9 it was promoted, as indicated in Table 4.34. Table 4.34 Objects candidates for the 0034_3218 set of tracks (correct one: A3). Candidate ∆ˆ V[cm/s]WRMS Entropy SwE Mark PPromoted A2 85.44 3.617 −12.108 0.093 0.007 − A3 22.70 0.967 −7.585 0.925 0.993 Yes As occurred in the previous scenario, an incorrect hypothesis is promoted. This hypothesis, attributed to the A2 satellite, actually pertains to A3. However, unlike the previous case, its promotion did not result from the hypothesis elimination process. It occurred because the set of tracks named 0042_5027 had only one hypothesis in the promotion process. This was due to the manoeuvre estimation algorithm’s inability to identify a viable manoeuvre for this set of tracks with A3. Consequently, this erroneous hypothesis was the unique candidate for the A2 satellite. It is important to note this situation, which differs from the previous scenario. In Section 4.4.5.1, the incorrect hypothesis was promoted due to the simulation methodology. In this instance, it was promoted because the cataloguer failed. The same situation as in the previous scenario may be observed in Figures 4.17 and 4.18. Another thing to remark is that the wrong hypothesis promoted is denoted by the algorithm with the caution flag mentioned in Section 3.2.5 as it has a WRMS greater than 2. This may serve to discard this hypothesis or treat it with caution. 58 Chapter 4. Results of simulated satellites Figure 4.18 Zoom error plot (colour bar: score without entropy) of the Orion-A1A3-1201 hypotheses. Table 4.35 Promoted hypotheses of the Orion-A1A3-1201 scenario. Promoted hypotheses using the score with entropy Code SwE ∆ˆ VEr(∆ˆ V)%E(ˆ tM)[min] WRMS Correct A1 0034_5709 0.917 [20.01,8.44,−0.69]8.66 −4.33 1.085 ✓ A2 0042_5027 0.431 [30.74,54.71,23.08]234.34 1617.18 2.199 "✘ A3 0034_3218 0.925 [20.02,10.71,0.26]13.51 −3.93 0.967 ✓ Promoted hypotheses using the score without entropy Code SwoE ∆ˆ VEr(∆ˆ V)%E(ˆ tM)[min] WRMS Correct A1 0022_5185 0.992 [20.0,2.19,0.49]0.62 −0.36 1.025 ✓ A2 0042_5027 0.372 [30.74,54.71,23.08]234.34 1617.18 2.199 "✘ A3 0022_1097 0.985 [19.98,−3.23,1.5]1.5 2.18 1.069 ✓ 4.4.5.3 Orion-cross-1201 All the scenarios considered involved the satellites manoeuvring in the same direction. In this instance, the satellites attempt to approach each other by manoeuvring towards their respective partners (A1 with A2, and A3 with A4). Track-to-orbit metrics: Table 4.36 Track-to-orbit metrics of Orion-cross-1201. Analysed Promoted %Correct Pre-manoeuvre tracks 84 84 100% Post-manoeuvre tracks 120 2 0% Total tracks 204 86 97.67% Track-to-track metric: %t2t =64.0% . The wrongly correlated sets of tracks are 0022_3572, 0026_3287, 0034_3170, 0041_4510, 0043_5923, 0044_1418, 0045_3633, 0046_3523 and 0048_4458. 4.4 Scenarios analysis and results 59 Manoeuvre estimator metrics: Table 4.37 Manoeuvre estimator metrics of Orion-cross-1201. No manoeuvre Manoeuvre No predicted 8 6 Predicted 67 19 Sensitivity: 76.0% Specificity: 10.67% Accuracy: 27.0% The close proximity of the satellites, combined with their approaching and crossing manoeuvres, introduces significant uncertainty into the simulation. This scenario results in the highest count of false positives in manoeuvre estimation, reaching 67. However, the pre-filter effectively reduces this figure to 13 false hypotheses, though this reduced number still poses a substantial challenge for the promotion process. Given the high number of false hypotheses, miscorrelation necessitates the use of the mark for one of the hypotheses, as detailed in Table 4.38. The scoring criterion without entropy is considered, as it is the one that employs the SoftMax criterion. The set of tracks 0035_3665 belongs to the A2 satellite, while the manoeuvre estimation process identified feasible manoeuvres with three satellites: A2, A3, and A4. Any of these three hypotheses could be correct. When the A2 hypothesis is considered, the other two are also analysed using the SoftMax function. The false hypotheses are significantly off in terms of ∆ˆ V , and their WRMS values are greater than the one of the correct hypothesis. Consequently, the SoftMax function assigns a mark of 0.915 to the correct case, which is high enough for the hypothesis to be promoted. Table 4.38 Objects candidates for the 0035_3665 set of tracks (correct one: A2). Candidate ∆ˆ V[cm/s]WRMS SwoE Mark PPromoted A2 20.11 1.055 0.99 0.915 Yes A3 74.70 2.395 0.241 0.04 − A4 75.05 2.067 0.248 0.045 − This scenario demonstrates the robustness of the entire cataloguing chain. Despite the numerous false hypotheses and high uncertainty caused by the satellites’ crossing paths, the promoted manoeuvres predict the epoch and magnitude of the manoeuvres with remarkable accuracy. Additionally, as shown in Table 4.39, the estimated manoeuvres correctly indicate that A1 and A3 are accelerating forwards (positive tangential component), while A2 and A4 are manoeuvring backwards. Table 4.39 Promoted hypotheses of the Orion-cross-1201 scenario. Promoted hypotheses using the score with entropy Code SwE ∆ˆ VEr(∆ˆ V)%E(ˆ tM)[min] WRMS Correct A1 0030_3795 0.907 [20.01,−6.37,0.07]5.0 2.13 0.852 ✓ A2 0030_5172 0.915 [−20.0,−3.06,1.09]1.3−2.13 1.014 ✓ A3 0035_5905 0.906 [20.01,9.41,1.48]10.78 −4.05 1.166 ✓ A4 0026_4923 0.911 [−20.05,−5.82,0.35]4.38 −1.64 0.981 ✓ Promoted hypotheses using the score without entropy Code SwoE ∆ˆ VEr(∆ˆ V)%E(ˆ tM)[min] WRMS Correct A1 0030_3795 0.985 [20.01,−6.37,0.07]5.0 2.13 0.852 ✓ A2 0035_3665 0.99 [−19.99,−2.15,0.44]0.57 −0.73 1.055 ✓ A3 0030_3170 0.969 [20.01,8.3,0.61]8.37 −2.18 1.062 ✓ A4 0026_4923 0.983 [−20.05,−5.82,0.35]4.38 −1.64 0.981 ✓ 66 Chapter 5. Conclusions and future work The challenges with using entropy in the context of satellite cataloguing and manoeuvre come from the fact that it is computed from the covariance matrix obtained through the weighted leastsquares method during each manoeuvre estimation. This covariance matrix, which depends on the a-priori knowledge of the measurement standard deviation and the partial derivatives of the measurement model, may be used as an indicator of the expected uncertainty in the manoeuvre estimation. However, as Montenbruck highlights [17], the covariance matrix is not influenced by the actual measurement errors encountered during data collection. It reflects the theoretical distribution of errors based on the model’s assumptions and the anticipated measurement noise, rather than the quality and accuracy of the tracking data. This distinction is crucial because the covariance matrix, and hence the derived entropy, does not account for systematic errors or inaccuracies present in the actual measurements. As a result, the entropy derived from the covariance matrix fails to provide a reliable a-posteriori measure of the data’s quality. This limitation means that while the entropy metric can theoretically indicate uncertainty, it does not accurately reflect the true conditions under which the data was obtained or the actual error characteristics. The measured entropy thus becomes a theoretical construct that may not align with the practical realities of the tracking data’s quality, leading to potential misinterpretations and unreliable decision-making when it is used in scoring and weighting manoeuvres. The reliance on the covariance matrix for entropy calculation introduces a fundamental disconnect between theoretical uncertainty and actual measurement accuracy. This gap could explain why entropy does not function as an indicator of reliability in the manoeuvre estimation process. For a more accurate assessment, a-posteriori analysis of measurement residuals and systematic error evaluation is necessary, which entropy, as currently computed, does not adequately address. Consequently, the use of entropy as a variable in Principal Component Analysis (PCA) to weight variables has not yielded satisfactory results. PCA, although a powerful statistical tool, does not provide optimal results in this context due to the instability introduced by entropy and the dependency of the values of the variables on the scenario. In contrast, the use of fixed weights for variables, in particular by assigning greater importance to ∆V and weighted root mean square (WRMS), has shown better results. These fixed weights have resulted in more accurate and reliable manoeuvre estimates, underlining the effectiveness of this approach compared to weights derived from PCA. In particular, scoring without entropy has proven to be very effective in identifying the correct manoeuvres, whereas scoring with entropy introduces considerable noise, complicating the decision-making process. The SoftMax criterion, designed to deal with miscorrelation cases, has proven to be effective in not promoting ambiguous candidates. This behaviour is in line with expectations and confirms the robustness of the criterion in filtering out uncertain hypotheses. The proximity of satellites plays a key role in the performance of the cataloguing chain. Closer proximity between satellites significantly increases the complexity and challenge of accurately cataloguing and estimating manoeuvres, as demonstrated by the higher rates of false positives and incorrect correlations in these scenarios. It is important to note that the influence of manoeuvre magnitude on the accuracy of the cataloguing process is complex and nuanced. While larger manoeuvres can indeed introduce more errors, this effect is not straightforward. There exists a trade-off between the accuracy of the manoeuvre estimation method (such as the weighted least-squares method used in our system) and the separation between satellites. Specifically, as manoeuvres become larger, the linear approximation used in the manoeuvre estimation method becomes less valid. This deviation from linearity can lead to inaccuracies in estimating the manoeuvre, as the method assumes small deviations around a linear model. Consequently, larger manoeuvres may be less accurately captured due to this inherent limitation in the modelling approach. Conversely, larger manoeuvres also increase the spatial separation between satellites over time, which can actually simplify the task of track-to-track association. With 5.2 Limitations 67 increased separation, it becomes easier to distinguish between different satellites, reducing the likelihood of incorrect associations. This increased separation can mitigate the effects of the errors introduced by the manoeuvre estimation process, as the distinct tracks make it easier to correctly identify and associate each satellite. In this context, the pre-filter mechanism plays a critical role in balancing these factors. By effectively reducing the number of false positives prior to the promotion process, the pre-filter helps maintain the integrity of the cataloguing chain. It acts as an initial safeguard, preemptively eliminating incorrect assumptions and ensuring that only the most plausible hypotheses proceed to the final stages of analysis. This mechanism is particularly valuable when dealing with scenarios involving significant manoeuvres, where the risk of both estimation errors and association challenges is heightened. Thus, the pre-filter contributes significantly to the robustness and accuracy of the overall cataloguing process, especially in complex and dynamic environments. Scenarios where satellites are approaching and crossing over further highlight the importance of relative motion on the accuracy of satellite cataloguing. These scenarios pose additional challenges due to the increased likelihood of false correlations and incorrect manoeuvre estimates. The number of tracks used in manoeuvre estimation is another critical factor. Reducing the number of tracks, as shown in Section 4.4.6.1, significantly decreases the reliability of the assumptions, leading to a marked increase in false positives. This finding underlines the importance of using a sufficient number of tracks to ensure accurate estimation of manoeuvres. In conclusion, the results highlight the importance of careful selection of the variables (discarding the entropy), the robustness of the pre-filtering mechanism and the critical role of satellite proximity and relative motion in the accuracy of satellite cataloguing. The results also highlight the differences between stationary and online cataloguing methods, each with its own advantages and difficulties. 5.2 Limitations This study has several key limitations that should be addressed in future work to enhance the robustness and applicability of the satellite cataloguing system. Firstly, the study primarily uses a stationary cataloguing approach, which processes all tracks simultaneously, offering a comprehensive evaluation of all potential scenarios. However, this static analysis can lead to the promotion of incorrect hypotheses, as it does not continuously update the catalogue with real-time data. This approach contrasts with an online cataloguing system, which processes tracks as they arrive, providing a more dynamic and current picture of satellite positions and manoeuvres. While online cataloguing can reduce the likelihood of false positives by utilising immediate data, it may also introduce additional noise and errors due to limited information at any given time, over-estimating or under-estimating the manoeuvres’ magnitude and epoch. Secondly, the use of simulated observations throughout the study presents a significant limitation. These simulations, while useful for controlled experiments, do not fully capture the complexities and unpredictabilities of real-world scenarios. The simulated manoeuvre plans, which assume a controlled and predictable set of manoeuvres, may not accurately reflect the variety and unpredictability of manoeuvres in actual satellite operations. This assumption limits the system to show if it is able to adapt to unexpected changes and deviations from the planned manoeuvres. Another critical aspect is the lack of consideration for uncertainties in the dynamics of satellite motion. The same model used for ground truth simulation is also employed in the manoeuvre estimation and other processes, leading to an idealised scenario. In reality, various factors such as atmospheric drag, gravitational perturbations, and satellite-specific characteristics can introduce uncertainties not considered in the dynamical model that are not accounted for in the current study. This oversight could lead to overconfidence in the model’s predictions and underestimation of potential errors. 68 Chapter 5. Conclusions and future work Lastly, the evaluation of the metrics used to assess the accuracy and effectiveness of the cataloguing process relies on solving the problem itself. This approach is inherently limited, as it cannot be applied to real-world scenarios where ground truth data is unavailable. Consequently, the metrics cannot be independently verified against actual satellite states whose manoeuvres are not specifically known, limiting the ability to validate the system’s performance in a realistic operational environment due to the lack of real-world data. These limitations highlight the need for further refinement and testing of the cataloguing system, particularly through the integration of real-world data and the consideration of dynamic uncertainties. Addressing these issues will enhance the reliability and applicability of the system in practical satellite tracking and manoeuvre estimation tasks. 5.3 Future work The present project has provided significant information on the complexities of satellite cataloguing and manoeuvre estimation. However, there remain several avenues for future work and possible improvements to further increase the accuracy and robustness of the cataloguing chain. One of the main areas of work is the refinement of the weights used in the scoring criteria. Given the noisy behaviour of entropy, alternative methods to assess the stability and reliability of this values could be explored. In addition, further research into more sophisticated statistical techniques or machine learning models to dynamically adjust weights based on scenario-specific factors could lead to greater accuracy in the estimation of manoeuvres. Improving the pre-filtering mechanism is another key issue. While the current pre-filter effectively reduces the number of false positives, the development of more advanced filtering algorithms that can adapt to different scenarios could improve its performance. The integration of historic data would help to predict and filter false positives more accurately. The SoftMax criterion has proven effective in dealing with miscorrelation, but there is room for further refinement. Exploring more advanced optimisation and decision-making algorithms that can better distinguish between closely correlated candidates may help reduce the occurrence of ambiguous hypotheses that are discarded while one of them is correct. Another parameter that needs to be reviewed is the value β selected to scale the exponential functions in the SoftMax equation. Although the behaviour of the selected β is good enough in the cases analysed, further testing is needed to validate the feasibility of its value. Needless to say, continuous validation and testing with real-world data and scenarios will be crucial. While the tested satellites provide a good insight into the overall performance of the library, real-world scenarios may give rise to errors that are not present in them due to the uncertain nature of the real data. In addition, the transition from a stationary cataloguer to an online cataloguer that processes the tracks in real time could provide more insight into the overall behaviour of the library. Developing and testing an online cataloguer, along with strategies to mitigate the introduction of noise and address the challenges of real-time data processing, would be an important step forward. Finally, considering the advantages and disadvantages of both stationary and online catalogues, the author proposes developing a hybrid system that runs both methodologies in parallel. The online cataloguer can detect manoeuvres with minimal time delay, providing real-time tracking capabilities. However, it may lack the comprehensive data available to the stationary cataloguer, potentially leading to significant errors in predicted manoeuvres. Conversely, the stationary cataloguer has proven more reliable in accurately estimating the epoch and magnitude of manoeuvres, yet its methodology does not offer continuous updates and may promote false positives. By combining the strengths of both approaches, a hybrid cataloguer could deliver real-time satellite tracking with enhanced accuracy and reliability, minimising the intrinsic delay of the stationary cataloguer and the potential errors of each individual system. List of Figures 1.1 Flowchart of the catalogue chain 3 1.2 Correlation process between sets of tracks and catalogued objects. In a) the decision tree is created, relating each orbit to each set of tracks. Each line corresponds to what we call hypothesis. In b) the algorithm selects from all possible candidates the one with the optimal value of the chosen criterion, discarding other combinations involving the selected orbit-track pair. In c) the process starts again with the remaining candidates to try to assign each track to an object. In d) the final state is shown, having correlated three tracks to three existing objects. However, if the remaining pair (Orbit D - Track 3) has not got an optimal value of the selected criteria, this correlation could be discarded 4 2.1 Precession and nutation of Earth’s axis [29] 10 2.2 ITRF 2000 [1] 11 2.3 TNW reference frame [24] 11 2.4 Keplerian orbital elements 12 2.5 Antenna beams of a monopulse auto-track system [9] 13 2.6 S3TSR 14 2.7 Schematic of the manoeuvre estimation method 16 3.1 STARS folders interaction 17 3.2 STARS satellite simulator workflow 19 3.3 STARS track.py and t2t.py workflow 20 3.4 STARS odet.py workflow 22 3.5 STARS hypothesis.py workflow 23 3.6 STARS promotion.py workflow 25 3.7 Principal Components Representation [3] 27 3.8 STARS catalogue.py workflow 29 4.1 Scatter pairplot of the three variables of Orion-1115 hypotheses using PCA to estimate the weights 36 4.2 Error plot (colour bar: score with entropy) of the Orion-1115 hypotheses using PCA to estimate the weights 37 4.3 Scatter pairplot of the three variables of Orion-1110 hypotheses using PCA to estimate the weights 38 4.4 Error plot (colour bar: score with entropy) of the Orion-1115 hypotheses 39 4.5 Error plot (colour bar: entropy) of the Orion-1115 hypotheses 40 4.6 Error plot (colour bar: score without entropy) of the Orion-1115 hypotheses 40 4.7 Error plot (colour bar: score without entropy) of the Orion-1110 hypotheses 42 69 70 List of Figures 4.8 Error plot (colour bar: score without entropy) of the Orion-1105 hypotheses 44 4.9 Error plot (colour bar: score without entropy) of the Orion-1101 hypotheses 46 4.10 Zoom error plot (colour bar: score without entropy) of the Orion-1101 hypotheses 47 4.11 Zoom error plot (colour bar: score without entropy) of the Orion-1205 hypotheses 48 4.12 Zoom error plot (colour bar: score without entropy) of the Orion-1201 hypotheses 50 4.13 Zoom error plot (colour bar: score without entropy) of the Orion-2201 hypotheses 52 4.14 Correlation process that resulted in the promotion of an incorrect hypothesis. Initially, the hypothesis with the highest score, which relates 0026_2969 with A3, was promoted, leading to the removal of all other hypotheses containing A3. Subsequently, the hypothesis correlating A1 with 0046_1218 was considered and promoted, causing the removal of the remaining hypotheses involving A1. Consequently, the last remaining hypothesis, which correlates A2 with 0047_4589, was promoted despite being incorrect, as no miscorrelation was identified at this stage of the analysis 54 4.15 Error plot (colour bar: score without entropy) of the Orion-A1A3-1101 hypotheses 55 4.16 Zoom error plot (colour bar: score without entropy) of the Orion-A1A3-1101 hypotheses 56 4.17 Error plot (colour bar: score without entropy) of the Orion-A1A3-1201 hypotheses 57 4.18 Zoom error plot (colour bar: score without entropy) of the Orion-A1A3-1201 hypotheses 58 4.19 Error plot (colour bar: score without entropy) of the Orion-cross-1201 hypotheses 60 4.20 Zoom error plot (colour bar: score without entropy) of the Orion-cross-1201 hypotheses 60 4.21 Zoom error plot (colour bar: score without entropy) of the Orion-3tracks-1201 hypotheses 62 List of Tables 4.1 Track-to-orbit metrics 34 4.2 Manoeuvre estimator metrics 35 4.3 Track-to-orbit metrics of Orion-1115 39 4.4 Manoeuvre estimator metrics of Orion-1115 39 4.5 Promoted hypotheses of the Orion-1115 scenario 41 4.6 Track-to-orbit metrics of Orion-1110 41 4.7 Manoeuvre estimator metrics of Orion-1110 41 4.8 Promoted hypotheses of the Orion-1110 scenario 43 4.9 Track-to-orbit metrics of Orion-1105 43 4.10 Manoeuvre estimator metrics of Orion-1105 43 4.11 Promoted hypotheses of the Orion-1105 scenario 44 4.12 Track-to-orbit metrics of Orion-1101 44 4.13 Manoeuvre estimator metrics of Orion-1110 45 4.14 Objects candidates for the 0043_4974 set of tracks (correct one: A3) 45 4.15 Objects candidates for the 0043_4684 set of tracks (correct one: A4) 45 4.16 Objects candidates for the 0038_3125 set of tracks (correct one: A3) 46 4.17 Promoted hypotheses of the Orion-1101 scenario 47 4.18 Track-to-orbit metrics of Orion-1205 47 4.19 Manoeuvre estimator metrics of Orion-1205 48 4.20 Promoted hypotheses of the Orion-1205 scenario 49 4.21 Track-to-orbit metrics of Orion-1201 49 4.22 Manoeuvre estimator metrics of Orion-1201 49 4.23 Promoted hypotheses of the Orion-1201 scenario 50 4.24 Track-to-orbit metrics of Orion-2201 (in parenthesis Orion-1201) 51 4.25 Manoeuvre estimator metrics of Orion-2201 (in parenthesis Orion-1201) 51 4.26 Objects candidates for the 0039_3783 set of tracks (correct one: A1) 51 4.27 Promoted hypotheses of the Orion-2201 scenario 52 4.28 Track-to-orbit metrics of Orion-A1A3-1101 53 4.29 Manoeuvre estimator metrics of Orion-A1A3-1101 53 4.30 Objects candidates for the 0043_4684 set of tracks (correct one: A3) 54 4.31 Promoted hypotheses of the Orion-A1A3-1101 scenario 55 4.32 Track-to-orbit metrics of Orion-A1A3-1201 56 4.33 Manoeuvre estimator metrics of Orion-A1A3-1201 56 4.34 Objects candidates for the 0034_3218 set of tracks (correct one: A3) 57 4.35 Promoted hypotheses of the Orion-A1A3-1201 scenario 58 4.36 Track-to-orbit metrics of Orion-cross-1201 58 71 72 List of Tables 4.37 Manoeuvre estimator metrics of Orion-cross-1201 59 4.38 Objects candidates for the 0035_3665 set of tracks (correct one: A2) 59 4.39 Promoted hypotheses of the Orion-cross-1201 scenario 59 4.40 Track-to-orbit metrics of Orion-3tracks-1201 (in parenthesis Orion-1201) 61 4.41 Manoeuvre estimator metrics of Orion-3tracks-1201 61 4.42 Promoted hypotheses of the Orion-3tracks-1201 scenario 62 4.43 Summary of the results of the scenarios 63 Bibliography [1] Zuheir Altamimi, Patrick Sillard, and Claude Boucher, Itrf2000: A new release of the international terrestrial reference frame for earth science applications, Journal of Geophysical Research: Solid Earth 107 (2002), no. 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