Resolución de sistemas de ecuaciones lineales

Numerosos problemas de la ciencia, la ingenieria o la economia requieren de la busqueda de soluciones de una ecuacion. Desde tiempos remotos se ha tratado de modelizar problemas presentes en la naturaleza con expresiones que, al fin y al cabo, permitan conocer a priori como se va a comportar un sistema. Entre las tecnicas utilizadas para dicha busqueda de soluciones encontramos los metodos iterativos. Iterar a partir de una serie de expresiones nos va a permitir conocer la solucion de una funcion no lineal a partir de esquemas adecuados para ello. Ademas de los conocidos metodos de Newton y Steffensen, se van a implementar metodos con mayor orden de convergencia. Clasificar los metodos iterativos en funcion de sus caracteristicas intrinsecas nos va a permitir valorar la bondad o la conveniencia del uso de un metodo iterativo u otro. Como en todos los problemas de ingenieria y matematicas, tendremos que obtener una solucion de compromiso. Otra de las caracterizaciones existentes, complementaria a la anterior, es el estudio de la dinamica compleja. El operador de punto fijo asociado a cada uno de los metodos iterativos cuando se aplica sobre una funcion no lineal va a permitir que caractericemos cada uno de los esquemas en el plano complejo. Buena parte del trabajo desarrollado se ha centrado en la representacion grafica de la dinamica de los metodos iterativos. El plano dinamico es una herramienta que nos permite visualizar la estabilidad de un metodo, el tamano de sus cuencas de convergencia o la idoneidad de determinados puntos iniciales para comenzar a iterar. Asimismo, para familias de metodos uniparametricas, el plano de parametros va a colaborar en la eleccion del miembro de la familia mas adecuado. Interpretando los planos dinamicos como una aproximacion a los fractales, presentaremos la dimension fractal como un factor de medida de lo intrincado que puede resultar el conjunto de Julia asociado a un metodo iterativo. Los fractales pertenecen a la frontera entre el determinismo y la teoria del caos, de forma que podremos transferir conceptos de ambas disciplinas sobre el estudio fractal. Mostraremos como aplicacion de los metodos iterativos y la dinamica compleja la determinacion de orbitas preliminares de satelites artificiales. A partir de la posicion de un satelite en dos instantes diferentes, es posible determinar los parametros de la elipse que describe. Para ello, utilizaremos un algoritmo en el que se incluye un metodo clasico de resolucion para, a continuacion, mejorar sus prestaciones con nuestras propuestas de metodos iterativos. Basandonos en la busqueda de soluciones y en los metodos iterativos como tecnica de obtencion de soluciones, las aplicaciones abarcan campos mas alla de la mecanica orbital. El diseno de filtros digitales, el procesado digital de imagenes o la caracterizacion de enlaces de radiofrecuencia son algunos de los ejemplos de aplicacion. A partir de los conceptos anteriores, presentamos esta Tesis Doctoral para la obtencion del titulo de Doctor en Matematicas, contextualizando la tematica en los primeros capitulos para, a continuacion, presentar las publicaciones en revistas internacionales como fruto de la investigacion.

Análisis dinámico y aplicaciones de métodos iterativos de resolución ecuaciones no lineales

Synchronous approaching is an important capability for autonomous cooperation of multiple unmanned aerial vehicles (UAVs). In this paper, a homotopy-based trajectory planning method is presented for the multi-UAV synchronous approaching problem. A homotopic trajectory description is employed to construct the trajectory solution space of the UAVs. A novel onion-like homotopy structure is proposed to decouple the performance indexes of the trajectory planning problem. Local trajectory homotopy structures are designed based on the detouring model and the hovering model of UAV. The optimal trajectories for synchronous approaching are searched within the homotopy structures. Simulation results show how synchronous the UAVs are, by using the proposed homotopy-based trajectory planning method. 

Multi-UAV Synchronous Approaching Using Homotopy-Based Trajectory Planning

Achieving the smart motion of any autonomous or semi-autonomous robot requires an efficient algorithm to determine a feasible collision-free path. In this paper, a novel collision-free path homotopy-based path-planning algorithm applied to planar robotic arms is presented. The algorithm utilizes homotopy continuation methods (HCMs) to solve the non-linear algebraic equations system (NAES) that models the robot’s workspace. The method was validated with three case studies with robotic arms in different configurations. For the first case, a robot arm with three links must enter a narrow corridor with two obstacles. For the second case, a six-link robot arm with a gripper is required to take an object inside a narrow corridor with two obstacles. For the third case, a twenty-link arm must take an object inside a maze-like environment. These case studies validated, by simulation, the versatility and capacity of the proposed path-planning algorithm. The results show that the CPU time is dozens of milliseconds with a memory consumption less than 4.5 kB for the first two cases. For the third case, the CPU time is around 2.7 s and the memory consumption around 18 kB. Finally, the method’s performance was further validated using the industrial robot arm CRS CataLyst-5 by Thermo Electron.

https://www.mdpi.com/1424-8220/22/11/4022/pdf?version=1653545766

A Novel Collision-Free Homotopy Path Planning for Planar Robotic Arms

Most existing path-planning algorithms are applied in either trafficable environments or non-trafficable environments. Off-road vehicles (ORVs) are often faced with a mix of trafficable and non-trafficable environments. Therefore, trafficability should be considered in path planning for ORVs. Conventional ant colony algorithms (ACAs) are prone to stagnation and often fail to reach the optimal path. To address these problems, an improved ACA that considers trafficability was proposed in this study, which improved the pheromone distribution rules and adaptively adjusted the pheromone volatility coefficient. Based on this improved ACA, a multilevel adaptive path-planning model was proposed to solve path-planning problems with various scales of area. Experiments and comparative studies revealed that the improved ACA was applicable to path-planning problems in complex environments and achieved better performance and a higher computing efficiency than conventional counterparts.

/pdf/a-multilevel-adaptive-path-planning-model-in-off-road-1nepxu50.pdf

A Multilevel Adaptive Path-Planning Model in Off-Road Environments

This article explains the development of a tool (application) to solve nonlinear algebraic equations systems (NAES). Systems of nonlinear equations are usually complex and can not be solved analytically or in an exact way, therefore numerical methods are used to solve this type of systems. This paper shows the implementation of four numerical algorithms: conjugate gradient method (CGM), Newton-Raphson (N-R), Ostrowski's method (OM) and homotopic continuum methods (HCM). These methods are capable of solving systems of nonlinear algebraic equations of $n\times n$ . In addition, the GiNaC library is used, which is completely written in C++ and allows the user to perform algebraic operations and interact within this language. The application has been programmed considering the following attributes: easy to use, multi-platform, that is, it must be able to operate in different operating systems (S.O.), fast in obtaining the solution, able to configure some fields to try to find the solution.

Gerardo C. Velez-Lopez

Papers

A Tool to Solve Nonlinear Algebraic Equations Systems

A Novel Collision-Free Homotopy Path Planning for Planar Robotic Arms