Polygon placement under translation and rotation

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External polygon containment problems

AbstractGiven a convex polygonal object P with k vertices and an environment consisting of polygonal obstacles with a total of n corners, we seek a placement for the largest copy of P that does not intersect any of the obstacles, allowing translation, rotation and scaling. We employ the parametric search technique of Megiddo (1983), and the fixed size polygon placement algorithms developed by Leven and Sharir (1987), to obtain an algorithm that runs in time O(k2nλ6(kn)log3(kn)loglog(kn)). We also present several other efficient algorithms for restricted variants of the extremal polygon containment problem, using the same ideas. These variants include: placement of the largest homothetic copies of one or two convex polygons in another convex polygon and placement of the largest similar copy of a triangle in a convex polygon

Study of the Dependence of Solar Radiation Regarding Design Variables in Photovoltaic Solar Installations with Optimal Dual-Axis Tracking

Solar tracking is an efficient strategy to increase the radiative capture of photovoltaic collectors. Within the multiple efforts made in recent decades to improve the production of these facilities, various works have studied solutions to optimize the number of rotation axes (single or dual rotation axes), the degree of collector coverage, the distances between trackers, the geometric arrangement of trackers or the minimization of shading between collectors. However, although in this type of installation it is common to find collectors with geometric shapes other than rectangles, no studies on the influence of the shape of the collectors on the radiative incidence are found in the literature. In this connection, the present work systematically addresses the study of incident solar radiation in photovoltaic installations with dual-axis trackers with collectors of different geometric shapes. By means of the exhaustive study, the conclusion is drawn that, for dual-axis photovoltaic installations with an optimal tracking strategy, the main variables that influence the annual radiative incidence are the spacing between collectors, the coverage ratio (GCR), and the collector surface, while the type of arrangement of collectors and the shape of these do not show predictive values

Optimal clustering of a pair of irregular objects

Cutting and packing problems arise in many fields of applications and theory. When dealing with irregular objects, an important subproblem is the identification of the optimal clustering of two objects. Within this paper we consider a container (rectangle, circle, convex polygon) of variable sizes and two irregular objects bounded by circular arcs and/or line segments, that can be continuously translated and rotated. In addition minimal allowable distances between objects and between each object and the frontier of a container, may be imposed. The objects should be arranged within a container such that a given objective will reach its minimal value. We consider a polynomial function as the objective, which depends on the variable parameters associated with the objects and the container. The paper presents a universal mathematical model and a solution strategy which are based on the concept of phi-functions and provide new benchmark instances of finding the containing region that has either minimal area, perimeter or homothetic coefficient of a given container, as well as finding the convex polygonal hull (or its approximation) of a pair of objects

Non-Decoupled Locomotion and Manipulation Planning for Low-Dimensional Systems

International audienceWe demonstrate the possibility of solving planning problems by inter-leaving locomotion and manipulation in a non-decoupled way. We choose three low-dimensional minimalistic robotic systems and use them to illustrate our paradigm: a basic one-legged locomotor, a two-link manipulator with a manipulated object, and a simultaneous locomotion-and-manipulation system. Using existing motion planning and control methods initially designed for either locomotion or manipulation tasks, we see how they apply to both our locomotion-only and manipulation-only systems through parallel derivations, and extend them to the simultaneous locomotion-and-manipulation system. Motion planning is solved for these three systems using two different methods : (i) a geometric path-planning-based one, and (ii) a kinematic control-theoretic-based one. Motion control is then derived by dynamically realizing the geometric paths or kinematic trajectories under the Couloumb friction model using torques as control inputs. All three methods apply successfully to all three systems, showing that the non-decoupled planning is possible

Planificación automática y supervisión de operaciones de montaje mediante robots

One of the main topics to be solved in order to fully automate a robotized assembly task is the automatic determination of the movements to be done by the robot to properly perform the tasks when the existing uncertainty is not negligible. This problem is of particular theoretical and practical interest when rotational degrees of freedom and friction forces are taken into account. In this thesis, an automatic movement planner that considers these aspects is proposed, including the description of how to execute the plan and supervise the evolution of the task. In order to generate the plan, the task is represented by a finite number of states, which are associated to the nodes of a graph with the links connecting contiguous states. Then, using uncertainty models developed in the thesis, the domains of the possible configurations and reaction forces that can be measured by the corresponding sensors in each task state are determined. From the existing initial conditions of the task and the desired final conditions, an initial and a goal state are determined, and, using the state graph, a sequence of contiguous states joining them is searched. At the same time, state transition operators (movement directions of the robot) that may allow the transition from one state to the next in the sequence are also determined.The execution of the task according to the plan basically consist in the estimation of the current state by matching the sensorial information obtained on-line with the domains of configuration and force of each state, and then, the application of the proper state transition operator to proceed in the state sequence towards the goal state.The main contributions of the thesis are the following: on one side, as a general contribution, the proposed planning procedure that allows the simultaneous consideration of friction forces, rotational degrees of freedom, and the different uncertainty sources that affect a robotized task; on the other side, as more specific contributions, the proposal of task states as the occurrence of a set of basic contacts, and, for movements on a plane, the fusion of the uncertainty models and the determination of the reaction forces possible in any contact situation by using the dual representation of the force lines. The thesis includes the application of the developed concepts to a simple assembly task (the block in the corner problem) considering movements on a plane. Although the implementation is not a general application prototype, it contributes to the validation of the theoretical results of the work.Uno de los principales problemas a resolver en la automatización total de una tarea de montaje robotizada, es la determinación automática de los movimientos que debe realizar el robot para llevar a cabo la tarea cuando la incertidumbre que le afecta es significativa. Este problema es de especial interés teórico y práctico cuando se consideran grados de libertad de rotación y fuerzas de fricción. En la tesis se propone un planificador automático de movimientos que tiene en cuenta estos aspectos. Se describe también cómo llevar a cabo la ejecución del plan y supervisar la evolución de la tarea.Para llevar a cabo la planificación, la tarea se representa mediante un conjunto finito de estados. Considerando la incertidumbre mediante modelos desarrollados en la tesis, se determinan los dominios de observación de configuraciones y de fuerzas de reacción que pueden ser indicadas por los sensores cuando tiene lugar cada estado de la tarea. Los estados de la tarea se representan como nodos de un grafo en el que los arcos unen los estados contiguos.A partir de las condiciones iniciales de la tarea y condiciones finales deseadas se establecen sendos estados inicial y final, y, utilizando el grafo de estados, se determina una secuencia de estados contiguos que los ligue. Paralelamente, se determinan operadores de cambio de estado (direcciones de movimiento del robot) que pueden permitir la transición de un estado a otro de la secuencia.La ejecución de la tarea acorde al plan consiste básicamente en estimar el estado en curso contrastando la información sensorial obtenida en-línea con los dominios de observación de configuración y fuerza, para aplicar entonces el operador de cambio de estado que corresponda, y así sucesivamente hasta alcanzar el estado final.Las principales aportaciones de la tesis son las siguientes. Por un lado, desde un punto de vista general, cabe destacar el procedimiento de planificación propuesto, que permite considerar simultáneamente fuerzas de fricción, grados de libertad de rotación y las incertidumbres que afectan a una tarea de montaje robotizada. Por otra parte, pueden mencionarse como aportaciones particulares, la introducción del concepto de estados de la tarea como ocurrencia de un determinado conjunto de contactos básicos y, para el caso de movimientos en un plano, el modelado y fusión de incertidumbre de una forma más precisa que las descritas en trabajos previos, así como la determinación de las fuerzas de reacción que pueden tener lugar en cualquier contacto mediante el uso de la representación dual de sus rectas de acción. La tesis incluye la aplicación de los conceptos teóricos desarrollados a una tarea de montaje (bloque en la esquina) considerando movimientos de los objetos en un plano. Aunque la implementación no pretende ser un prototipo de aplicación general, contribuye a la validación de los resultados del trabajo

Généralisation du diagramme de Voronoï et placement de formes géométriques complexes dans un nuage de points.

La géométrie algorithmique est une discipline en pleine expansion dont l'objet est la conception d'algorithmes résolvant des problèmes géométriques. De tels algorithmes sont très utiles notamment dans l'ingénierie, l'industrie et le multimédia. Pour être performant, il est fréquent qu'un algorithme géométrique utilise des structures de données spécialisées.Nous nous sommes intéressés à une telle structure : le diagramme de Voronoï et avons proposé une généralisation de celui-ci. Ladite généralisation résulte d'une extension du prédicat du disque vide (prédicat propre à toute région de Voronoï) à une union de disques. Nous avons analysé les régions basées sur le prédicat étendu et avons proposé des méthodes pour les calculer par ordinateur.Par ailleurs, nous nous sommes intéressés aux problèmes de placement de formes , thème récurrent en géométrie algorithmique. Nous avons introduit un formalisme universel pour de tels problèmes et avons, pour la première fois, proposé une méthode de résolution générique, en ce sens qu'elle est apte à résoudre divers problèmes de placement suivant un même algorithme.Nos travaux présentent, d'une part, l'avantage d'élargir le champ d'application de structures de données basées sur Voronoï. D'autre part, ils facilitent de manière générale l'utilisation de la géométrie algorithmique, en unifiant définitions et algorithmes associés aux problèmes de placement de formes.Computational geometry is an active branch of computer science whose goal is the design of efficient algorithms solving geometric problems. Such algorithms are useful in domains like engineering, industry and multimedia. In order to be efficient, algorithms often use special data structures.In this thesis we focused on such a structure: the Voronoi diagram. We proposed a new generalized diagram. We have proceeded by extending the empty disk predicate (satisfied by every Voronoi region) to an arbitrary union of disks. We have analyzed the new plane regions based on the extended predicate, and we designed algorithms for computing them.Then, we have considered another topic, which is related to the first one: shape placement problems. Such problems have been studied repeatedly by researchers in computational geometry. We introduced new notations along with a global framework for such problems. We proposed, for the first time a generic method, which is able to solve various placement problems using a single algorithm.Thus, our work extend the scope of Voronoi based data structures. It also simplifies the practical usage of placement techniques by unifying the associated definitions and algorithms.MULHOUSE-SCD Sciences (682242102) / SudocSudocFranceF

Généralisation du diagramme de Voronoï et placement de formes géométriques complexes dans un nuage de points.

La géométrie algorithmique est une discipline en pleine expansion dont l'objet est la conception d'algorithmes résolvant des problèmes géométriques. De tels algorithmes sont très utiles notamment dans l'ingénierie, l'industrie et le multimédia. Pour être performant, il est fréquent qu'un algorithme géométrique utilise des structures de données spécialisées.Nous nous sommes intéressés à une telle structure : le diagramme de Voronoï et avons proposé une généralisation de celui-ci. Ladite généralisation résulte d'une extension du prédicat du disque vide (prédicat propre à toute région de Voronoï) à une union de disques. Nous avons analysé les régions basées sur le prédicat étendu et avons proposé des méthodes pour les calculer par ordinateur.Par ailleurs, nous nous sommes intéressés aux problèmes de placement de formes , thème récurrent en géométrie algorithmique. Nous avons introduit un formalisme universel pour de tels problèmes et avons, pour la première fois, proposé une méthode de résolution générique, en ce sens qu'elle est apte à résoudre divers problèmes de placement suivant un même algorithme.Nos travaux présentent, d'une part, l'avantage d'élargir le champ d'application de structures de données basées sur Voronoï. D'autre part, ils facilitent de manière générale l'utilisation de la géométrie algorithmique, en unifiant définitions et algorithmes associés aux problèmes de placement de formes.Computational geometry is an active branch of computer science whose goal is the design of efficient algorithms solving geometric problems. Such algorithms are useful in domains like engineering, industry and multimedia. In order to be efficient, algorithms often use special data structures.In this thesis we focused on such a structure: the Voronoi diagram. We proposed a new generalized diagram. We have proceeded by extending the empty disk predicate (satisfied by every Voronoi region) to an arbitrary union of disks. We have analyzed the new plane regions based on the extended predicate, and we designed algorithms for computing them.Then, we have considered another topic, which is related to the first one: shape placement problems. Such problems have been studied repeatedly by researchers in computational geometry. We introduced new notations along with a global framework for such problems. We proposed, for the first time a generic method, which is able to solve various placement problems using a single algorithm.Thus, our work extend the scope of Voronoi based data structures. It also simplifies the practical usage of placement techniques by unifying the associated definitions and algorithms.MULHOUSE-SCD Sciences (682242102) / SudocSudocFranceF

Nuevos desarrollos teóricos en la captación solar en estructuras fijas y móviles

Solar irradiance is a fundamental variable for the characterization of the solar resource as an energy source. The scarcity of data on this variable has encouraged the development of explanatory models (isotropic models, anisotropic models, etc.). However, the models of solar irradiance have not been exploited so far in the generation of knowledge of patterns relating to optimal capture. In particular, this lack of development is evident in the study of solar trackers of photovoltaic facilities and in the radiation incident on buildings in cities. Mathematically, equations governing irradiance models can be derived with respect to position variables and, in this way, generate movement results that optimise capture and, therefore, energy production. Likewise, the exploitation of these models enables to explain known results in the sustainable energy field. A good example is the case of solar trackers based on the maximization of solar capture which have greater energy generation rates than those based on astronomical tracking. This thesis presents the analytical deduction of generic and unified equations of the movement of solar tracking systems. As a novelty, these equations are more generic, thus allowing the optimization of the positioning of photovoltaic (PV) facilities where diffuse and reflected irradiance are usable as opposed to those usually published that just consider the position of the sun (astronomical motion equations). The analysis of the results obtained criticizes the axiomatic idea – widely considered by numerous authors – establishing that the ideal tracking system in PV facilities is that tracker providing the best possible alignment with direct sunbeams. In PV plants based on solar tracking, during low-elevation solar angle hours, shadows appear between the collectors causing a dramatic decrease in production. This thesis presents a novel optimal tracking strategy to prevent the creation of these shadows. The presented method determines whether or not there is shading between collectors. Thus, when the collectors are not shaded, a tracking trajectory for maximum irradiance on the collectors is suggested. When the collectors are shaded, backtracking is proposed. Therefore, energy production in plants with this novel tracking method can be 1.31 % higher than that in PV plants with astronomical tracking. Moreover, this method allows the study of PV facilities for which there have been no published approaches, such as plants with non-rectangular collectors or those located on topographically heterogeneous surfaces. The growing need to improve the environmental and energy sustainability of buildings involves the use of solar radiation incident on their surfaces. However, in cities this task is complicated due to the constructive geometry that leads to shading between buildings. In this context, this work presents a study of solar access to the façades of buildings in cities. The methodology is based on the determination of the incident annual solar radiation in 121 significant points of each façade considering the twelve representative days of the year. To characterize the influence of the different city typologies on solar access, the Urban Solar Coefficient (the ratio of the irradiance received at one point in a building to the total irradiance received in the neighbourhood in which the building is located) is proposed. A study in two neighborhoods in Cordoba (Spain) with different urban settings have been analyzed. Specifically, two typologies of neighborhoods have been compared: one with ”L-shaped” and “U-shaped blocks” and another with “Grouped blocks”. For both of them, the Urban Solar Coefficient has been calculated, obtaining a higher mean value for the neighborhood with ”L-shaped” and “U-shaped blocks” (0.317) than for the one with “Grouped blocks” (0.260). Accordingly, the results show that urban morphology can influence the Urban Solar Coefficient and solar access. Finally, a regression model for each neighborhood has been obtained in order to determine the dependence of the Urban Solar Coefficient on neighborhood geometry factors. This thesis has addressed a characterization of irradiance in mobile and fixed structures, proposing real problems whose solutions are fully usable in the domains of urbanism and photovoltaic energy.La irradiancia solar es una variable fundamental para la caracterización del recurso solar como fuente de energía. La escasez de datos de esta variable ha provocado que se hayan desarrollado modelos explicativos de la misma (modelos isotrópicos, anisotrópicos, etc.). Sin embargo, los modelos de irradiancia solar han sido hasta la fecha poco explotados en la generación de conocimiento de pautas relativas a una captación óptima. De manera particular, esta falta de desarrollo se manifiesta en el estudio de seguidores solares de instalaciones fotovoltaicas y en la radiación incidente sobre edificios de núcleos urbanos. Matemáticamente, las ecuaciones que rigen los modelos de irradiancia pueden ser derivadas respecto de las variables de posición y, de esta manera, generar resultados de movimiento que optimizan la captación y, por tanto, la producción energética. Asimismo, la explotación de estos modelos permite explicar resultados conocidos en el ámbito energético sostenible. Valga como ejemplo el caso de los seguidores solares basados en la maximización de la captación solar que presentan mejores tasas de generación energética que los basados en seguimiento astronómico. Esta tesis presenta la deducción analítica de las ecuaciones genéricas y unificadas de movimiento de seguidores solares. Muestran como novedad ser más genéricas, permitiendo la optimización del posicionamiento en instalaciones fotovoltaicas aprovechando las componentes difusa y reflejada de la irradiancia frente a las habitualmente publicadas que solo tienen en cuenta la posición del sol (ecuaciones de movimiento astronómico). El análisis de los resultados obtenidos refuta la idea axiomática, ampliamente difundida por numerosos autores, que establece como seguidor ideal en instalaciones fotovoltaicas aquel que procura el mejor alineamiento posible con los rayos solares directos. Además, en las instalaciones fotovoltaicas con seguidores solares aparecen durante las horas de altura solar baja, sombreos entre colectores que provocan una drástica caída de producción. Esta tesis presenta una nueva estrategia óptima de seguimiento que evita la creación de estas sombras. El método propuesto determina si hay o no sombra entre los colectores de una instalación. Por lo tanto, cuando los colectores no están sombreados, se propone una trayectoria de seguimiento para obtener la máxima irradiancia en los colectores. Cuando los colectores estuviesen sombreados se propone el retroseguimiento. La producción energética en las plantas con este novedoso método de seguimiento puede ser un 1,31% superior a la de instalaciones fotovoltaicas con seguimiento astronómico y sin intersombreo. Además, este método permite estudiar instalaciones para las que actualmente no existen enfoques publicados, como instalaciones con colectores no rectangulares o aquellas situadas en terrenos con topografía no plana. Por otro lado, la creciente necesidad de mejorar la sostenibilidad ambiental y energética de los edificios implica el aprovechamiento de la radiación solar incidente en sus superficies. Sin embargo, en las ciudades esta tarea se complica debido a la geometría constructiva que provoca el sombreo entre los edificios. En este contexto, esta tesis presenta un estudio del acceso solar a las fachadas de los edificios de las ciudades. La metodología se basa en la determinación de la radiación solar anual incidente en 121 puntos significativos de cada fachada considerando los doce días más representativos del año. Para caracterizar la influencia de las diferentes tipologías de edificaciones respecto al acceso solar, se propone el Coeficiente Solar Urbano (relación entre la irradiancia recibida en un punto de un edifico y la total recibida en el barrio en que se encuentra dicho edificio). Se ha analizado un estudio en dos barrios de Córdoba (España) con diferentes entornos urbanos. En concreto, se han comparado dos tipologías de barrios: uno con "bloques en forma de L" y "bloques en forma de U" y otro con "bloques agrupados". Para ambos se ha calculado el Coeficiente Solar Urbano, obteniendo un valor medio superior para el barrio con "bloques en forma de L" y "bloques en forma de U" (0,317) que para el barrio con "bloques agrupados" (0,260). En consecuencia, los resultados muestran que la morfología urbana puede influir en el Coeficiente Solar Urbano y el acceso solar. Finalmente, se ha obtenido un modelo de regresión para cada barrio con el fin de determinar la dependencia del Coeficiente Solar Urbano respecto a los factores geométricos del barrio. Con esta tesis se ha abordado una caracterización de la irradiancia en aplicaciones móviles y fijas, planteando problemas reales cuyas soluciones son de una aplicabilidad plena en los ámbitos del urbanismo y la energía fotovoltaica