Efficient and Featureless Approaches to Bathymetric Simultaneous Localisation and Mapping

This thesis investigates efficient forms of Simultaneous Localization and Mapping (SLAM) that do not require explicit identification, tracking or association of map features. The specific application considered here is subsea robotic bathymetric mapping. In this context, SLAM allows a GPS-denied robot operating near the sea floor to create a self-consistent bathymetric map. This is accomplished using a Rao-Blackwellized Particle Filter (RBPF) whereby each particle maintains a hypothesis of the current vehicle state and map that is efficiently maintained using Distributed Particle Mapping. Through particle weighting and resampling, successive observations of the seafloor structure are used to improve the estimated trajectory and resulting map by enforcing map self consistency. The main contributions of this thesis are two novel map representations, either of which can be paired with the RBPF to perform SLAM. The first is a grid-based 2D depth map that is efficiently stored by exploiting redundancies between different maps. The second is a trajectory map representation that, instead of directly storing estimates of seabed depth, records the trajectory of each particle and synchronises it to a common log of bathymetric observations. Upon detecting a loop closure each particle is weighted by matching new observations to the current predictions. For the grid map approach this is done by extracting the predictions stored in the observed cells. For the trajectory map approach predictions are instead generated from a local reconstruction of their map using Gaussian Process Regression. While the former allows for faster map access the latter requires less memory and fully exploits the spatial correlation in the environment, allowing predictions of seabed depth to be generated in areas that were not directly observed previously. In this case particle resampling therefore not only enforces self-consistency in overlapping sections of the map but additionally enforces self-consistency between neighboring map borders. Both approaches are validated using multibeam sonar data collected from several missions of varying scale by a variety of different Unmanned Underwater Vehicles. These trials demonstrate how the corrections provided by both approaches improve the trajectory and map when compared to dead reckoning fused with Ultra Short Baseline or Long Baseline observations. Furthermore, results are compared with a pre-existing state of the art bathymetric SLAM technique, confirming that similar results can be achieved at a fraction of the computation cost. Lastly the added capabilities of the trajectory map are validated using two different bathymetric datasets. These demonstrate how navigation and mapping corrections can still be achieved when only sparse bathymetry is available (e.g. from a four beam Doppler Velocity Log sensor) or in missions where map overlap is minimal or even non-existent

Development of an adaptive navigation system for indoor mobile handling and manipulation platforms

A fundamental technology enabling the autonomous behavior of mobile robotics is navigation. It is a main prerequisite for mobile robotics to fulfill high-level tasks such as handling and manipulation, and is often identified as one of the key challenges in mobile robotics. The mapping and localization as the basis for navigation are intensively researched in the last few decades. However, there are still challenges or problems needed to be solved for online operating in large-scale environments or running on low-cost and energy-saving embedded systems. In this work, new developments and usages of Light Detection And Ranging (LiDAR) based Simultaneous Localization And Mapping (SLAM) algorithms are presented. A key component of LiDAR based SLAM algorithms, the scan matching algorithm, is explored. Different scan matching algorithms are systemically experimented with different LiDARs for indoor home-like environments for the first time. The influence of properties of LiDARs in scan matching algorithms is quantitatively analyzed. Improvements to Bayes filter based and graph optimization based SLAMs are presented. The Bayes filter based SLAMs mainly use the current sensor information to find the best estimation. A new efficient implementation of Rao-Blackwellized Particle Filter based SLAM is presented. It is based on a pre-computed lookup table and the parallelization of the particle updating. The new implementation runs efficiently on recent multi-core embedded systems that fulfill low cost and energy efficiency requirements. In contrast to Bayes filter based methods, graph optimization based SLAMs utilize all the sensor information and minimize the total error in the system. A new real-time graph building model and a robust integrated Graph SLAM solution are presented. The improvements include the definition of unique direction norms for points or lines extracted from scans, an efficient loop closure detection algorithm, and a parallel and adaptive implementation. The developed algorithm outperforms the state-of-the-art algorithms in processing time and robustness especially in large-scale environments using embedded systems instead of high-end computation devices. The results of the work can be used to improve the navigation system of indoor autonomous robots, like domestic environments and intra-logistics.Eine der grundlegenden Funktionen, welche die Autonomie in der mobilen Robotik ermöglicht, ist die Navigation. Sie ist eine wesentliche Voraussetzung dafür, dass mobile Roboter selbständig anspruchsvolle Aufgaben erfüllen können. Die Umsetzung der Navigation wird dabei oft als eine der wichtigsten Herausforderungen identifiziert. Die Kartenerstellung und Lokalisierung als Grundlage für die Navigation wurde in den letzten Jahrzehnten intensiv erforscht. Es existieren jedoch immer noch eine Reihe von Problemen, z.B. die Anwendung auf große Areale oder bei der Umsetzung auf kostengünstigen und energiesparenden Embedded-Systemen. Diese Arbeit stellt neue Ansätze und Lösungen im Bereich der LiDAR-basierten simultanen Positionsbestimmung und Kartenerstellung (SLAM) vor. Eine Schlüsselkomponente der LiDAR-basierten SLAM, die so genannten Scan-Matching-Algorithmen, wird näher untersucht. Verschiedene Scan-Matching-Algorithmen werden zum ersten Mal systematisch mit verschiedenen LiDARs für den Innenbereich getestet. Der Einfluss von LiDARs auf die Eigenschaften der Algorithmen wird quantitativ analysiert. Verbesserungen an Bayes-filterbasierten und graphoptimierten SLAMs werden in dieser Arbeit vorgestellt. Bayes-filterbasierte SLAMs verwenden hauptsächlich die aktuellen Sensorinformationen, um die beste Schätzung zu finden. Eine neue effiziente Implementierung des auf Partikel-Filter basierenden SLAM unter der Verwendung einer Lookup-Tabelle und der Parallelisierung wird vorgestellt. Die neue Implementierung kann effizient auf aktuellen Embedded-Systemen laufen. Im Gegensatz dazu verwenden Graph-SLAMs alle Sensorinformationen und minimieren den Gesamtfehler im System. Ein neues Echtzeitmodel für die Grafenerstellung und eine robuste integrierte SLAM-Lösung werden vorgestellt. Die Verbesserungen umfassen die Definition von eindeutigen Richtungsnormen für Scan, effiziente Algorithmen zur Erkennung von Loop Closures und eine parallele und adaptive Implementierung. Der entwickelte und auf eingebetteten Systemen eingesetzte Algorithmus übertrifft die aktuellen Algorithmen in Geschwindigkeit und Robustheit, insbesondere für große Areale. Die Ergebnisse der Arbeit können für die Verbesserung der Navigation von autonomen Robotern im Innenbereich, häuslichen Umfeld sowie der Intra-Logistik genutzt werden

Contributions to Localization, Mapping and Navigation in Mobile Robotics

This thesis focuses on the problem of enabling mobile robots to autonomously build world models of their environments and to employ them as a reference to self–localization and navigation. For mobile robots to become truly autonomous and useful, they must be able of reliably moving towards the locations required by their tasks. This simple requirement gives raise to countless problems that have populated research in the mobile robotics community for the last two decades. Among these issues, two of the most relevant are: (i) secure autonomous navigation, that is, moving to a target avoiding collisions and (ii) the employment of an adequate world model for robot self-referencing within the environment and also for locating places of interest. The present thesis introduces several contributions to both research fields. Among the contributions of this thesis we find a novel approach to extend SLAM to large-scale scenarios by means of a seamless integration of geometric and topological map building in a probabilistic framework that estimates the hybrid metric-topological (HMT) state space of the robot path. The proposed framework unifies the research areas of topological mapping, reasoning on topological maps and metric SLAM, providing also a natural integration of SLAM and the “robot awakening” problem. Other contributions of this thesis cover a wide variety of topics, such as optimal estimation in particle filters, a new probabilistic observation model for laser scanners based on consensus theory, a novel measure of the uncertainty in grid mapping, an efficient method for range-only SLAM, a grounded method for partitioning large maps into submaps, a multi-hypotheses approach to grid map matching, and a mathematical framework for extending simple obstacle avoidance methods to realistic robots

Adaptive Localization and Mapping for Planetary Rovers

Future rovers will be equipped with substantial onboard autonomy as space agencies and industry proceed with missions studies and technology development in preparation for the next planetary exploration missions. Simultaneous Localization and Mapping (SLAM) is a fundamental part of autonomous capabilities and has close connections to robot perception, planning and control. SLAM positively affects rover operations and mission success. The SLAM community has made great progress in the last decade by enabling real world solutions in terrestrial applications and is nowadays addressing important challenges in robust performance, scalability, high-level understanding, resources awareness and domain adaptation. In this thesis, an adaptive SLAM system is proposed in order to improve rover navigation performance and demand. This research presents a novel localization and mapping solution following a bottom-up approach. It starts with an Attitude and Heading Reference System (AHRS), continues with a 3D odometry dead reckoning solution and builds up to a full graph optimization scheme which uses visual odometry and takes into account rover traction performance, bringing scalability to modern SLAM solutions. A design procedure is presented in order to incorporate inertial sensors into the AHRS. The procedure follows three steps: error characterization, model derivation and filter design. A complete kinematics model of the rover locomotion subsystem is developed in order to improve the wheel odometry solution. Consequently, the parametric model predicts delta poses by solving a system of equations with weighed least squares. In addition, an odometry error model is learned using Gaussian processes (GPs) in order to predict non-systematic errors induced by poor traction of the rover with the terrain. The odometry error model complements the parametric solution by adding an estimation of the error. The gained information serves to adapt the localization and mapping solution to the current navigation demands (domain adaptation). The adaptivity strategy is designed to adjust the visual odometry computational load (active perception) and to influence the optimization back-end by including highly informative keyframes in the graph (adaptive information gain). Following this strategy, the solution is adapted to the navigation demands, providing an adaptive SLAM system driven by the navigation performance and conditions of the interaction with the terrain. The proposed methodology is experimentally verified on a representative planetary rover under realistic field test scenarios. This thesis introduces a modern SLAM system which adapts the estimated pose and map to the predicted error. The system maintains accuracy with fewer nodes, taking the best of both wheel and visual methods in a consistent graph-based smoothing approach

Computation and Time constraints in Localization and Mapping Problems

Research on simultaneous localization and mapping problems has been extensively carried out by robotics community in the last decade and several subproblems –like data association, map representation, dynamic environments or semantic mapping– have been more or less deeply investigated. One of the most important questions is the online execution of localization and mapping methods. Since observations are periodically captured by robot sensors, localization and mapping algorithms are constrained to complete the execution of an update before a new observation is available. In literature, several partial contributions have been presented, most of them focused on the reduction of computational complexity, but no comprehensive discussion of real-time feasibility had been previously proposed. The reasons that make real-time feasibility difficult are different in the case of localization and of mapping problems, but a general criterion can be found. In this thesis we claim that a locality principle is a general design criterion for real-time or incremental execution of localization and mapping algorithms. The probabilistic robotics paradigm provides a unified formulation for the different problems and a conceptual framework for the application of the proposed criterion. Locality may be applied to perform temporal or spatial decomposition of the global estimation. This thesis provides a general perspective of real-time feasibility and the identification of locality principle as a general design criterion for algorithms to meet time constraints. The particular contributions of this thesis correspond to the application of the locality principle to specific problems. The Real-Time Particle Filter is an advanced version of Particle Filter algorithm conceived to achieve a tradeoff between time constraints and filter accuracy depending on the number of samples. This goal is achieved by partitioning the overall samples required to obtain the required accuracy into sets, each of them corresponding to an observation, and by reconstructing the new set at the end of an estimation window. We proposed two main contributions: first, an analysis of the efficiency of the resampling solution of the Real-Time Particle Filter through the concept of effective sample size; second, a method to compute the mixture weights that balances the the effective sample size of partition sets and is less prone to numerical instability. The second specific contribution is an incremental version of a maximum likelihood map estimator. The adopted technique combines stochastic gradient descent and incremental tree parameterization and exploits an efficient optimization technique and organizes the graph into a spanning tree structure suitable for a decomposition. In this thesis, the incremental version of the original algorithm has been adapted using again the locality principle. Local decomposition is achieved selecting the portion of the network perturbed by the addition of a new constraint. Furthermore, the perturbation of gradient descent iteration is limited for the region already converged by adapting the learning rate. Finally, optimization is scheduled with an heuristic rule that controls the error increase in the constraint network. The constraint solver has been integrated with a map builder that extracts the constraint network from laser scans and represents the environment with a metric-topological hybrid map. While real-time feasibility is not granted, the proposed incremental tree network optimizer is suitable for online execution and the algorithm converges faster than the previous version of the same algorithm and in several condition performs better than other state-of-the-art methods. The final contribution is a parallel maximum likelihood algorithm for robot mapping. The proposed algorithm estimates the map iterating a linearization step and the solution of the linear system with Gauss-Seidel relaxation. The network is divided in connected clusters of local nodes and the reorder induced by this decomposition transforms the linearized information matrix in block-border diagonal form. Each diagonal block of the matrix can then be solved independently. The proposed parallel maximum likelihood algorithm can exploit the computation resources provided by commodity multi-core processor. Moreover, this solution can be applied to multi-robot mapping. The contributions presented in this dissertation outline a novel perspective on real-time feasibility of robot localization and mapping methods, thus bringing these algorithmic techniques closer to applications

Sparse Bayesian information filters for localization and mapping

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution February 2008This thesis formulates an estimation framework for Simultaneous Localization and Mapping (SLAM) that addresses the problem of scalability in large environments. We describe an estimation-theoretic algorithm that achieves significant gains in computational efficiency while maintaining consistent estimates for the vehicle pose and the map of the environment. We specifically address the feature-based SLAM problem in which the robot represents the environment as a collection of landmarks. The thesis takes a Bayesian approach whereby we maintain a joint posterior over the vehicle pose and feature states, conditioned upon measurement data. We model the distribution as Gaussian and parametrize the posterior in the canonical form, in terms of the information (inverse covariance) matrix. When sparse, this representation is amenable to computationally efficient Bayesian SLAM filtering. However, while a large majority of the elements within the normalized information matrix are very small in magnitude, it is fully populated nonetheless. Recent feature-based SLAM filters achieve the scalability benefits of a sparse parametrization by explicitly pruning these weak links in an effort to enforce sparsity. We analyze one such algorithm, the Sparse Extended Information Filter (SEIF), which has laid much of the groundwork concerning the computational benefits of the sparse canonical form. The thesis performs a detailed analysis of the process by which the SEIF approximates the sparsity of the information matrix and reveals key insights into the consequences of different sparsification strategies. We demonstrate that the SEIF yields a sparse approximation to the posterior that is inconsistent, suffering from exaggerated confidence estimates. This overconfidence has detrimental effects on important aspects of the SLAM process and affects the higher level goal of producing accurate maps for subsequent localization and path planning. This thesis proposes an alternative scalable filter that maintains sparsity while preserving the consistency of the distribution. We leverage insights into the natural structure of the feature-based canonical parametrization and derive a method that actively maintains an exactly sparse posterior. Our algorithm exploits the structure of the parametrization to achieve gains in efficiency, with a computational cost that scales linearly with the size of the map. Unlike similar techniques that sacrifice consistency for improved scalability, our algorithm performs inference over a posterior that is conservative relative to the nominal Gaussian distribution. Consequently, we preserve the consistency of the pose and map estimates and avoid the effects of an overconfident posterior. We demonstrate our filter alongside the SEIF and the standard EKF both in simulation as well as on two real-world datasets. While we maintain the computational advantages of an exactly sparse representation, the results show convincingly that our method yields conservative estimates for the robot pose and map that are nearly identical to those of the original Gaussian distribution as produced by the EKF, but at much less computational expense. The thesis concludes with an extension of our SLAM filter to a complex underwater environment. We describe a systems-level framework for localization and mapping relative to a ship hull with an Autonomous Underwater Vehicle (AUV) equipped with a forward-looking sonar. The approach utilizes our filter to fuse measurements of vehicle attitude and motion from onboard sensors with data from sonar images of the hull. We employ the system to perform three-dimensional, 6-DOF SLAM on a ship hull

A Drift-Resilient and Degeneracy-Aware Loop Closure Detection Method for Localization and Mapping In Perceptually-Degraded Environments

Enabling fully autonomous robots capable of navigating and exploring unknown and complex environments has been at the core of robotics research for several decades. Mobile robots rely on a model of the environment for functions like manipulation, collision avoidance and path planning. In GPS-denied and unknown environments where a prior map of the environment is not available, robots need to rely on the onboard sensing to obtain locally accurate maps to operate in their local environment. A global map of an unknown environment can be constructed from fusion of local maps of temporally or spatially distributed mobile robots in the environment. Loop closure detection, the ability to assert that a robot has returned to a previously visited location, is crucial for consistent mapping as it reduces the drift caused by error accumulation in the estimated robot trajectory. Moreover, in multi-robot systems, loop closure detection enables finding the correspondences between the local maps obtained by individual robots and merging them into a consistent global map of the environment. In ambiguous and perceptually-degraded environments, robust detection of intra- and inter-robot loop closures is especially challenging. This is due to poor illumination or lack-thereof, self-similarity, and sparsity of distinctive perceptual landmarks and features sufficient for establishing global position. Overcoming these challenges enables a wide range of terrestrial and planetary applications, ranging from search and rescue, and disaster relief in hostile environments, to robotic exploration of lunar and Martian surfaces, caves and lava tubes that are of particular interest as they can provide potential habitats for future manned space missions. In this dissertation, methods and metrics are developed for resolving location ambiguities to significantly improve loop closures in perceptually-degraded environments with sparse or undifferentiated features. The first contribution of this dissertation is development of a degeneracy-aware SLAM front-end capable of determining the level of geometric degeneracy in an unknown environment based on computing the Hessian associated with the computed optimal transformation from lidar scan matching. Using this crucial capability, featureless areas that could lead to data association ambiguity and spurious loop closures are determined and excluded from the search for loop closures. This significantly improves the quality and accuracy of localization and mapping, because the search space for loop closures can be expanded as needed to account for drift while decreasing rather than increasing the probability of false loop closure detections. The second contribution of this dissertation is development of a drift-resilient loop closure detection method that relies on the 2D semantic and 3D geometric features extracted from lidar point cloud data to enable detection of loop closures with increased robustness and accuracy as compared to traditional geometric methods. The proposed method achieves higher performance by exploiting the spatial configuration of the local scenes embedded in 2D occupancy grid maps commonly used in robot navigation, to search for putative loop closures in a pre-matching step before using a geometric verification. The third contribution of this dissertation is an extensive evaluation and analysis of performance and comparison with the state-of-the-art methods in simulation and in real-world, including six challenging underground mines across the United States

Information metrics for localization and mapping

Decades of research have made possible the existence of several autonomous systems that successfully and efficiently navigate within a variety of environments under certain conditions. One core technology that has allowed this is simultaneous localization and mapping (SLAM), the process of building a representation of the environment while localizing the robot in it. State-of-the-art solutions to the SLAM problem still rely, however, on heuristic decisions and options set by the user. In this thesis we search for principled solutions to various aspects of the localization and mapping problem with the help of information metrics. One such aspect is the issue of scalability. In SLAM, the problem size grows indefinitely as the experiment goes by, increasing computational resource demands. To maintain the problem tractable, we develop methods to build an approximation to the original network of constraints of the SLAM problem by reducing its size while maintaining its sparsity. In this thesis we propose three methods to build the topology of such approximated network, and two methods to perform the approximation itself. In addition, SLAM is a passive application. It means, it does not drive the robot. The problem of driving the robot with the aim of both accurately localizing the robot and mapping the environment is called active SLAM. In this problem two normally opposite forces drive the robot, one to new places discovering unknown regions and another to revisit previous configurations to improve localization. As opposed to heuristics, in this thesis we pose the problem as the joint minimization of both map and trajectory estimation uncertainties, and present four different active SLAM approaches based on entropy-reduction formulation. All methods presented in this thesis have been rigorously validated in both synthetic and real datasets.Dècades de recerca han fet possible l’existència de nombrosos sistemes autònoms que naveguen eficaçment i eficient per varietat d’entorns sota certes condicions. Una de les principals tecnologies que ho han fet possible és la localització i mapeig simultanis (SLAM), el procés de crear una representació de l’entorn mentre es localitza el robot en aquesta. De tota manera, els algoritmes d’SLAM de l’estat de l’art encara basen moltes decisions en heurístiques i opcions a escollir per l’usuari final. Aquesta tesi persegueix solucions fonamentades per a varietat d’aspectes del problema de localització i mappeig amb l’ajuda de mesures d’informació. Un d’aquests aspectes és l’escalabilitat. En SLAM, el problema creix indefinidament a mesura que l’experiment avança fent créixer la demanda de recursos computacionals. Per mantenir el problema tractable, desenvolupem mètodes per construir una aproximació de la xarxa de restriccions original del problema d’SLAM, reduint així el seu tamany a l’hora que es manté la seva naturalesa dispersa. En aquesta tesi, proposem tres métodes per confeccionar la topologia de l’approximació i dos mètodes per calcular l’aproximació pròpiament. A més, l’SLAM és una aplicació passiva. És a dir que no dirigeix el robot. El problema de guiar el robot amb els objectius de localitzar el robot i mapejar l’entorn amb precisió es diu SLAM actiu. En aquest problema, dues forces normalment oposades guien el robot, una cap a llocs nous descobrint regions desconegudes i l’altra a revisitar prèvies configuracions per millorar la localització. En contraposició amb mètodes heurístics, en aquesta tesi plantegem el problema com una minimització de l’incertesa tant en el mapa com en l’estimació de la trajectòria feta i presentem quatre mètodes d’SLAM actiu basats en la reducció de l’entropia. Tots els mètodes presentats en aquesta tesi han estat rigurosament validats tant en sèries de dades sintètiques com en reals

Information metrics for localization and mapping

Decades of research have made possible the existence of several autonomous systems that successfully and efficiently navigate within a variety of environments under certain conditions. One core technology that has allowed this is simultaneous localization and mapping (SLAM), the process of building a representation of the environment while localizing the robot in it. State-of-the-art solutions to the SLAM problem still rely, however, on heuristic decisions and options set by the user. In this thesis we search for principled solutions to various aspects of the localization and mapping problem with the help of information metrics. One such aspect is the issue of scalability. In SLAM, the problem size grows indefinitely as the experiment goes by, increasing computational resource demands. To maintain the problem tractable, we develop methods to build an approximation to the original network of constraints of the SLAM problem by reducing its size while maintaining its sparsity. In this thesis we propose three methods to build the topology of such approximated network, and two methods to perform the approximation itself. In addition, SLAM is a passive application. It means, it does not drive the robot. The problem of driving the robot with the aim of both accurately localizing the robot and mapping the environment is called active SLAM. In this problem two normally opposite forces drive the robot, one to new places discovering unknown regions and another to revisit previous configurations to improve localization. As opposed to heuristics, in this thesis we pose the problem as the joint minimization of both map and trajectory estimation uncertainties, and present four different active SLAM approaches based on entropy-reduction formulation. All methods presented in this thesis have been rigorously validated in both synthetic and real datasets.Dècades de recerca han fet possible l’existència de nombrosos sistemes autònoms que naveguen eficaçment i eficient per varietat d’entorns sota certes condicions. Una de les principals tecnologies que ho han fet possible és la localització i mapeig simultanis (SLAM), el procés de crear una representació de l’entorn mentre es localitza el robot en aquesta. De tota manera, els algoritmes d’SLAM de l’estat de l’art encara basen moltes decisions en heurístiques i opcions a escollir per l’usuari final. Aquesta tesi persegueix solucions fonamentades per a varietat d’aspectes del problema de localització i mappeig amb l’ajuda de mesures d’informació. Un d’aquests aspectes és l’escalabilitat. En SLAM, el problema creix indefinidament a mesura que l’experiment avança fent créixer la demanda de recursos computacionals. Per mantenir el problema tractable, desenvolupem mètodes per construir una aproximació de la xarxa de restriccions original del problema d’SLAM, reduint així el seu tamany a l’hora que es manté la seva naturalesa dispersa. En aquesta tesi, proposem tres métodes per confeccionar la topologia de l’approximació i dos mètodes per calcular l’aproximació pròpiament. A més, l’SLAM és una aplicació passiva. És a dir que no dirigeix el robot. El problema de guiar el robot amb els objectius de localitzar el robot i mapejar l’entorn amb precisió es diu SLAM actiu. En aquest problema, dues forces normalment oposades guien el robot, una cap a llocs nous descobrint regions desconegudes i l’altra a revisitar prèvies configuracions per millorar la localització. En contraposició amb mètodes heurístics, en aquesta tesi plantegem el problema com una minimització de l’incertesa tant en el mapa com en l’estimació de la trajectòria feta i presentem quatre mètodes d’SLAM actiu basats en la reducció de l’entropia. Tots els mètodes presentats en aquesta tesi han estat rigurosament validats tant en sèries de dades sintètiques com en reals