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Ce travail de thèse se veut être une contribution à l'estimation globale des risques de collisions dans l'environnement spatial de la Terre. Deux échelles de temps s'affrontent pour l'estimation de ces risques : le très court terme (quelques heures à quelques jours) où les évolutions notables de probabilité de collisions vont être dues à des événements de caractère catastrophique (collision entre satellites ou explosion), et le long voire le très long terme (où il s'agit d'évaluer l'évolution spatio-temporelle de la population de l'ensemble des débris spatiaux). Dans les deux cas, il s'agit d'évaluer la dangerosité d'une région de l'espace où évoluent des satellites opérationnels. L'assimilation de bases de données des objets en orbite autour de la Terre est fortement présente dans ce travail, tout comme des simulations pour les objets petits, nombreux, et donc non-observables. L'objectif suivi est d'étudier les caractéristiques statistiques générales de la population ainsi que celles d'une famille particulière née d'un événement ponctuel (de type fragmentation), tant par le biais d'outils usuels que novateurs, afin d'aboutir à une estimation des risques engendrés à court et long termes. Une méthode innovante est proposée, basée sur des outils de statistique spatiale utilisés en mathématiques appliquées dans des domaines variés mais jusque là autres que l’astrodynamique. Nous avons déterminé la répartition des débris spatiaux d'un point de vue statistique, en décrivant la dynamique d'agrégation ou de répulsion entre les objets, puis évalué l'influence des particules de petite taille sur la distribution des fragments. L'étude d'une fragmentation particulière et de la formation de son nuage de débris en orbite est également réalisée. Elle passe par la description des caractéristiques géométriques du nuage au cours du temps ainsi que par la déterminationdes temps de fermeture de ses différentes étapes de formation. Cette fragmentation est également utilisée pour déterminer l'influence des paramètres propres à l'explosion et à l'évolution des orbites sur la dispersion du nuage d'un point de vue statistique. Sur la base de l’ensemble de ces travaux, une série de critères nous permet finalement de tracer des voies vers une calibration d'un modèle de fragmentation complet par comparaison avec des données réelles issues d'observations. / This PhD work is a contribution to the global estimation of collision risks in the Earth space environment. To estimate these risks, two time scales complement one another: over very short terms (from a few hours to a few days) the strongest changes of collision probabilities are likely to be due to catastrophic events (collision between satellites or explosions), whereas over long and even very long terms (decades or even centuries) the main goal is to assess the spatio-temporal evolution of a whole population of space debris. In both cases, this is the evaluation of the dangerousness of regions where active satellites evolve which is at stake.Throughout the discussion, we focuse on databases assimilation of on-orbit objects, as well as on simulations for small and (then) numerous objects objects that are unobservable. We follow the goal of characterizing the statistical distribution of the global population, as well as specific families generated after a punctual event (i.e: fragmentation). Some estimations of the incurred risks are provided over both short and long time scales. An innovative method is proposed to characterize the space debris distribution, that is rather commonly used in the fields of applied mathematique but not that frequently in astrodynamics: this method is based on spatial statistics to determine the space debris distribution from a statistical point of view. By defining the notions of aggregation and repulsion dynamics between objects, we have assessed the influence of small particles on the fragments distribution.The study of a real fragmentation and of the corresponding space debris cloud evolution is also conducted. The geometrical characteristics of the cloud over time are supplied as well as the estimation of the closure times corresponding to the different evolution phases of the cloud. This concrete example is also on the basis of a sensitivity analysis: by enlightening the influence of some parameters standing for the explosion and the orbit evolution parameterization, the spreading of the cloud is characterized from a statistical point of view.As a final step of this approach, some criteria are discussed to open a path through the calibration of a complete fragmentation model thanks to comparisons with real data coming from observations.

In the early phase of the design of a space mission it is generally desirable to investigate as many feasible alternative solutions as possible. Traditionally a system margin approach is used in order to estimate the correct value of subsystem budgets. While this is a consolidated and robust approach, it does not give a measure of the reliability of any of the investigated solutions. In addition the mass budget is typically overdimensioned, where a more accurate design could lead to improvements in payload mass. This study will address two principal issues typically associated with the design of a space mission: (i) the effective and efficient generation of preliminary solutions by properly treating their inherent multi-disciplinary elements and (ii) the minimisation of the impact of uncertainties on the overall design, which in turn will lead to an increase in the reliability of the produced results. The representation and treatment of the uncertainties are key aspects of reliable design. An insufficient consideration of uncertainty or an unadapted mathematical representation leads to misunderstanding of the real issues of a design, to delay in the future development of the project or even potentially to its failure. The most common way to deal with uncertainty is the probabilistic approach. However, this theory is not suitable to represent epistemic uncertainties, arising from lack of knowledge. Alternative theories have been recently developed, amongst which we find Evidence Theory which is implemented in this work. Developed by Shafer from Dempster's original work, it is regarded by many as a suitable paradigm to accurately represent uncertainties. Evidence Theory is presented and discussed from an engineering point of view and special attention given to the implementation of this approach. Once mathematically represented, the uncertainties can be taken into account in the design optimisation problem. However, the computational complexity of Evidence Theory can be overwhelming and therefore more efficient ways to solve the reliable design problem are required. Existing methods are considered and improvements developed by the author, to increase the efficiency of the algorithm by making the most of the available data, are proposed and tested. Additionally, a new sample-based approximation technique to tackle large scale problems, is introduced in this thesis. Assuming that the uncertainties are modelled by means of intervals, the cluster approximation method, and especially implemented as a Binary Space Partition, appears to be very well-suited to the task. The performance of the various considered methods to solve the reliable design optimisation problem in the frame of Evidence Theory is tested and analysed. The dependency on the problem characteristics, such as dimensionality, complexity, or multitude of local solutions are carefully scrutinised. The conclusions of these tests enables the author to propose guidelines on how to tackle the problem depending on its specificity. Finally, two examples of preliminary space mission design are used to illustrate how the proposed methodology can be applied. Using realistic and current mission designs, the results show the benefits that could be achieved during the preliminary analyses and feasibility studies of space exploration.

The idea of spacecraft formations, flying in tight configurations with maximum baselines of a few hundred meters in low-Earth orbits, has generated widespread interest over the last several years. Nevertheless, controlling the movement of spacecraft in formation poses difficulties, such as in-orbit high-computing demand and collision avoidance capabilities, which escalate as the number of units in the formation is increased and complicated nonlinear effects are imposed to the dynamics, together with uncertainty which may arise from the lack of knowledge of system parameters. These requirements have led to the need of reliable linear and nonlinear controllers in terms of relative and absolute dynamics. The objective of this thesis is, therefore, to introduce new control methods to allow spacecraft in formation, with circular/elliptical reference orbits, to efficiently execute safe autonomous manoeuvres. These controllers distinguish from the bulk of literature in that they merge guidance laws never applied before to spacecraft formation flying and collision avoidance capacities into a single control strategy. For this purpose, three control schemes are presented: linear optimal regulation, linear optimal estimation and adaptive nonlinear control. In general terms, the proposed control approaches command the dynamical performance of one or several followers with respect to a leader to asymptotically track a time-varying nominal trajectory (TVNT), while the threat of collision between the followers is reduced by repelling accelerations obtained from the collision avoidance scheme during the periods of closest proximity. Linear optimal regulation is achieved through a Riccati-based tracking controller. Within this control strategy, the controller provides guidance and tracking toward a desired TVNT, optimizing fuel consumption by Riccati procedure using a non-infinite cost function defined in terms of the desired TVNT, while repelling accelerations generated from the CAS will ensure evasive actions between the elements of the formation. The relative dynamics model, suitable for circular and eccentric low-Earth reference orbits, is based on the Tschauner and Hempel equations, and includes a control input and a nonlinear term corresponding to the CAS repelling accelerations. Linear optimal estimation is built on the forward-in-time separation principle. This controller encompasses two stages: regulation and estimation. The first stage requires the design of a full state feedback controller using the state vector reconstructed by means of the estimator. The second stage requires the design of an additional dynamical system, the estimator, to obtain the states which cannot be measured in order to approximately reconstruct the full state vector. Then, the separation principle states that an observer built for a known input can also be used to estimate the state of the system and to generate the control input. This allows the design of the observer and the feedback independently, by exploiting the advantages of linear quadratic regulator theory, in order to estimate the states of a dynamical system with model and sensor uncertainty. The relative dynamics is described with the linear system used in the previous controller, with a control input and nonlinearities entering via the repelling accelerations from the CAS during collision avoidance events. Moreover, sensor uncertainty is added to the control process by considering carrier-phase differential GPS (CDGPS) velocity measurement error. An adaptive control law capable of delivering superior closed-loop performance when compared to the certainty-equivalence (CE) adaptive controllers is finally presented. A novel noncertainty-equivalence controller based on the Immersion and Invariance paradigm for close-manoeuvring spacecraft formation flying in both circular and elliptical low-Earth reference orbits is introduced. The proposed control scheme achieves stabilization by immersing the plant dynamics into a target dynamical system (or manifold) that captures the desired dynamical behaviour. They key feature of this methodology is the addition of a new term to the classical certainty-equivalence control approach that, in conjunction with the parameter update law, is designed to achieve adaptive stabilization. This parameter has the ultimate task of shaping the manifold into which the adaptive system is immersed. The performance of the controller is proven stable via a Lyapunov-based analysis and Barbalat’s lemma. In order to evaluate the design of the controllers, test cases based on the physical and orbital features of the Prototype Research Instruments and Space Mission Technology Advancement (PRISMA) are implemented, extending the number of elements in the formation into scenarios with reconfigurations and on-orbit position switching in elliptical low-Earth reference orbits. An extensive analysis and comparison of the performance of the controllers in terms of total Δv and fuel consumption, with and without the effects of the CAS, is presented. These results show that the three proposed controllers allow the followers to asymptotically track the desired nominal trajectory and, additionally, those simulations including CAS show an effective decrease of collision risk during the performance of the manoeuvre.