Sparse implicitization by interpolation: Geometric computations using matrix representations

Based on the computation of a superset of the implicit support, implicitization of a parametrically given hyper-surface is reduced to computing the nullspace of a numeric matrix. Our approach exploits the sparseness of the given parametric equations and of the implicit polynomial. In this work, we study how this interpolation matrix can be used to reduce some key geometric predicates on the hyper-surface to simple numerical operations on the matrix, namely membership and sidedness for given query points. We illustrate our results with examples based on our Maple implementation

Model Predictive Control Applications to Spacecraft Rendezvous and Small Bodies Exploration

The overarching goal of this thesis is the design of model predictive control algorithms for spacecraft proximity operations. These include, but it is not limited to, spacecraft rendezvous, hovering phases or orbiting in the vicinity of small bodies. The main motivation behind this research is the increasing demand of autonomy, understood as the spacecraft capability to compute its own control plan, in current and future space operations. This push for autonomy is fostered by the recent introduction of disruptive technologies changing the traditional concept of space exploration and exploitation. The development of miniaturized satellite platforms and the drastic cost reduction in orbital access have boosted space activity to record levels. In the near future, it is envisioned that numerous artificial objects will simultaneously operate across the Solar System. In that context, human operators will be overwhelmed in the task of tracking and commanding each spacecraft in real time. As a consequence, developing intelligent and robust autonomous systems has been identified by several space agencies as a cornerstone technology. Inspired by the previous facts, this work presents novel controllers to tackle several scenarios related to spacecraft proximity operations. Mastering proximity operations enables a wide variety of space missions such as active debris removal, astronauts transportation, flight-formation applications, space stations resupply and the in-situ exploration of small bodies. Future applications may also include satellite inspection and servicing. This thesis has focused on four scenarios: six-degrees of freedom spacecraft rendezvous; near-rectilinear halo orbits rendezvous; the hovering phase; orbit-attitude station-keeping in the vicinity of a small body. The first problem aims to demonstrate rendezvous capabilities for a lightweight satellite with few thrusters and a reaction wheels array. For near-rectilinear halo orbits rendezvous, the goal is to achieve higher levels of constraints satisfaction than with a stateof- the-art predictive controller. In the hovering phase, the objective is to augment the control accuracy and computational efficiency of a recent global stable controller. The small body exploration aims to demonstrate the positive impact of model-learning in the control accuracy. Although based on model predictive control, the specific approach for each scenario differs. In six-degrees of freedom rendezvous, the attitude flatness property and the transition matrix for Keplerian-based relative are used to obtain a non-linear program. Then, the control loop is closed by linearizing the system around the previous solution. For near-rectilinear halo orbits rendezvous, the constraints are assured to be satisfied in the probabilistic sense by a chance-constrained approach. The disturbances statistical properties are estimated on-line. For the hovering phase problem, an aperiodic event-based predictive controller is designed. It uses a set of trigger rules, defined using reachability concepts, deciding when to execute a single-impulse control. In the small body exploration scenario, a novel learning-based model predictive controller is developed. This works by integrating unscented Kalman filtering and model predictive control. By doing so, the initially unknown small body inhomogeneous gravity field is estimated over time which augments the model predictive control accuracy.El objeto de esta tesis es el dise˜no de algoritmos de control predictivo basado en modelo para operaciones de veh´ıculos espaciales en proximidad. Esto incluye, pero no se limita, a la maniobra de rendezvous, las fases de hovering u orbitar alrededor de cuerpos menores. Esta tesis est´a motivada por la creciente demanda en la autonom´ıa, entendida como la capacidad de un veh´ıculo para calcular su propio plan de control, de las actuales y futuras misiones espaciales. Este inter´es en incrementar la autonom´ıa est´a relacionado con las actuales tecnolog´ıas disruptivas que est´an cambiando el concepto tradicional de exploraci´on y explotaci´on espacial. Estas son el desarrollo de plataformas satelitales miniaturizadas y la dr´astica reducci´on de los costes de puesta en ´orbita. Dichas tecnolog´ıas han impulsado la actividad espacial a niveles de record. En un futuro cercano, se prev´e que un gran n´umero de objetos artificiales operen de manera simult´anea a lo largo del Sistema Solar. Bajo dicho escenario, los operadores terrestres se ver´an desbordados en la tarea de monitorizar y controlar cada sat´elite en tiempo real. Es por ello que el desarrollo de sistemas aut´onomos inteligentes y robustos es considerado una tecnolog´ıa fundamental por diversas agencias espaciales. Debido a lo anterior, este trabajo presenta nuevos resultados en el control de operaciones de veh´ıculos espaciales en proximidad. Dominar dichas operaciones permite llevar a cabo una gran variedad de misiones espaciales como la retirada de basura espacial, transferir astronautas, aplicaciones de vuelo en formaci´on, reabastecer estaciones espaciales y la exploraci ´on de cuerpos menores. Futuras aplicaciones podr´ıan incluir operaciones de inspecci´on y mantenimiento de sat´elites. Esta tesis se centra en cuatro escenarios: rendezvous de sat´elites con seis grados de libertad; rendezvous en ´orbitas halo cuasi-rectil´ıneas; la fase de hovering; el mantenimiento de ´orbita y actitud en las inmendiaciones de un cuerpo menor. El primer caso trata de proveer capacidades de rendezvous para un sat´elite ligero con pocos propulsores y un conjunto de ruedas de reacci´on. Para el rendezvous en ´orbitas halo cuasi-rectil´ıneas, se intenta aumentar el grado de cumplimiento de restricciones con respecto a un controlador predictivo actual. Para la fase de hovering, se mejora la precisi´on y eficiencia computacional de un controlador globalmente estable. En la exploraci´on de un cuerpo menor, se pretende demostrar el mayor grado de precisi´on que se obtiene al aprender el modelo. Siendo la base el control predictivo basado en modelo, el enfoque espec´ıfico difiere para cada escenario. En el rendezvous con seis grados de libertad, se obtiene un programa no-lineal con el uso de la propiedad flatness de la actitud y la matriz de transici´on del movimiento relativo Kepleriano. El bucle de control se cierra linealizando en torno a la soluci´on anterior. Para el rendezvous en ´orbitas halo cuasi-rectil´ıneas, el cumplimiento de restricciones se garantiza probabil´ısticamente mediante la t´ecnica chance-constrained. Las propiedades estad´ısticas de las perturbaciones son estimadas on-line. En la fase de hovering, se usa el control predictivo basado en eventos. Ello consiste en unas reglas de activaci´on, definidas con conceptos de accesibilidad, que deciden la ejecuci´on de un ´unico impulso de control. En la exploraci´on de cuerpos menores, se desarrolla un controlador predictivo basado en el aprendizaje del modelo. Funciona integrando un filtro de Kalman con control predictivo basado en modelo. Con ello, se consigue estimar las inomogeneidades del campo gravitario lo que repercute en una mayor precisi´on del controlador predictivo basado en modelo

Embedded and validated control algorithms for the spacecraft rendezvous

L'autonomie est l'une des préoccupations majeures lors du développement de missions spatiales que l'objectif soit scientifique (exploration interplanétaire, observations, etc) ou commercial (service en orbite). Pour le rendez-vous spatial, cette autonomie dépend de la capacité embarquée de contrôle du mouvement relatif entre deux véhicules spatiaux. Dans le contexte du service aux satellites (dépannage, remplissage additionnel d'ergols, correction d'orbite, désorbitation en fin de vie, etc), la faisabilité de telles missions est aussi fortement liée à la capacité des algorithmes de guidage et contrôle à prendre en compte l'ensemble des contraintes opérationnelles (par exemple, saturation des propulseurs ou restrictions sur le positionnement relatif entre les véhicules) tout en maximisant la durée de vie du véhicule (minimisation de la consommation d'ergols). La littérature montre que ce problème a été étudié intensément depuis le début des années 2000. Les algorithmes proposés ne sont pas tout à fait satisfaisants. Quelques approches, par exemple, dégradent les contraintes afin de pouvoir fonder l'algorithme de contrôle sur un problème d'optimisation efficace. D'autres méthodes, si elles prennent en compte l'ensemble du problème, se montrent trop lourdes pour être embarquées sur de véritables calculateurs existants dans les vaisseaux spatiaux. Le principal objectif de cette thèse est le développement de nouveaux algorithmes efficaces et validés pour le guidage et le contrôle impulsif des engins spatiaux dans le contexte des phases dites de "hovering" du rendez-vous orbital, i.e. les étapes dans lesquelles un vaisseau secondaire doit maintenir sa position à l'intérieur d'une zone délimitée de l'espace relativement à un autre vaisseau principal. La première contribution présentée dans ce manuscrit utilise une nouvelle formulation mathématique des contraintes d'espace pour le mouvement relatif entre vaisseaux spatiaux pour la conception d'algorithmes de contrôle ayant un traitement calculatoire plus efficace comparativement aux approches traditionnelles. La deuxième et principale contribution est une stratégie de contrôle prédictif qui assure la convergence des trajectoires relatives vers la zone de "hovering", même en présence de perturbations ou de saturation des actionneurs. Un travail spécifique de développement informatique a pu démontrerl'embarquabilité de ces algorithmes de contrôle sur une carte contenant un microprocesseur LEON3 synthétisé sur FPGA certifié pour le vol spatial, reproduisant les performances des dispositifs habituellement utilisés en vol. Finalement, des outils d'approximation rigoureuse de fonctions ont été utilisés pour l'obtention des solutions validées des équations décrivant le mouvement relatif linéarisé, permettant ainsi une propagation certifiée simple des trajectoires relatives via des polynômes et la vérification du respect des contraintes du problème.Autonomy is one of the major concerns during the planning of a space mission, whether its objective is scientific (interplanetary exploration, observations, etc.) or commercial (service in orbit). For space rendezvous, this autonomy depends on the on-board capacity of controlling the relative movement between two spacecraft. In the context of satellite servicing (troubleshooting, propellant refueling, orbit correction, end-of-life deorbit, etc.), the feasibility of such missions is also strongly linked to the ability of the guidance and control algorithms to account for all operational constraints (for example, thruster saturation or restrictions on the relative positioning between the vehicles) while maximizing the life of the vehicle (minimizing propellant consumption). The literature shows that this problem has been intensively studied since the early 2000s. However, the proposed algorithms are not entirely satisfactory. Some approaches, for example, degrade the constraints in order to be able to base the control algorithm on an efficient optimization problem. Other methods accounting for the whole set of constraints of the problem are too cumbersome to be embedded on real computers existing in the spaceships. The main object of this thesis is the development of new efficient and validated algorithms for the impulsive guidance and control of spacecraft in the context of the so-called "hovering" phases of the orbital rendezvous, i.e. the stages in which a secondary vessel must maintain its position within a bounded area of space relatively to another main vessel. The first contribution presented in this manuscript uses a new mathematical formulation of the space constraints for the relative motion between spacecraft for the design of control algorithms with more efficient computational processing compared to traditional approaches. The second and main contribution is a predictive control strategy that has been formally demonstrated to ensure the convergence of relative trajectories towards the "hovering" zone, even in the presence of disturbances or saturation of the actuators. Specific computational developments have demonstrated the embeddability of these control algorithms on a board containing a FPGA-synthesized LEON3 microprocessor certified for space flight, reproducing the performance of the devices usually used in flight. Finally, tools for rigorous approximation of functions were used to obtain validated solutions of the equations describing the linearized relative motion, allowing a simple certified propagation of the relative trajectories via polynomials and the verification of the respect of the constraints of the problem

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4

Advances in Robot Kinematics : Proceedings of the 15th international conference on Advances in Robot Kinematics

International audienceThe motion of mechanisms, kinematics, is one of the most fundamental aspect of robot design, analysis and control but is also relevant to other scientific domains such as biome- chanics, molecular biology, . . . . The series of books on Advances in Robot Kinematics (ARK) report the latest achievement in this field. ARK has a long history as the first book was published in 1991 and since then new issues have been published every 2 years. Each book is the follow-up of a single-track symposium in which the participants exchange their results and opinions in a meeting that bring together the best of world’s researchers and scientists together with young students. Since 1992 the ARK symposia have come under the patronage of the International Federation for the Promotion of Machine Science-IFToMM.This book is the 13th in the series and is the result of peer-review process intended to select the newest and most original achievements in this field. For the first time the articles of this symposium will be published in a green open-access archive to favor free dissemination of the results. However the book will also be o↵ered as a on-demand printed book.The papers proposed in this book show that robot kinematics is an exciting domain with an immense number of research challenges that go well beyond the field of robotics.The last symposium related with this book was organized by the French National Re- search Institute in Computer Science and Control Theory (INRIA) in Grasse, France

Implicitization of Curves Parameterized by Generalized Trigonometric Polynomials.

Consider a plane curve given parametrically by a generalized trigonometric polynomial, that is, x + iy = P n k=1 a k e ik` . In this paper, we obtain an implicitization of the curve, that is, an equation in x and y which captures all the points on the curve and, if any, only finitely many more points

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal