The Final Thesis deals with the evaluation of the benefits of the Elbe Cycling Path within the Giant Mountains Tourist Region. The teoretical part is explaining the basic terms of cycling, describing the infrastructure necessary for this activity and also organizations and institutions active in this area. Within one separate chapter there is a description of the Elbe Cycling Path including it's history. The practical part is focusing on particular section of the cycling path between Elbe spring and Dvůr Králové nad Labem. It is based on the results of the qualitative and quantitative research, both undertaken on the Elbe Cycling Path. The benefits are evalueted according to the results of the research and some measures are proposed to use the potential of the path in better way.

In 1948, Richard Feynman published the first paper on his new approach to non-relativistic quantum mechanics. Before Feynman's work there were two mathematical formulations of quantum mechanics. Schrödinger's formulation was based on PDE (the Schrödinger equation) and states representation by wave functions, so it was in the framework of analysis and differential equations. The other formulation was Heisenberg's matrix algebra. Initially, they were thought to be competing. The proponents of one claimed that the other was “ wrong. ” Within a couple of years, John von Neumann had proved that they are equivalent. Although Feynman's theory was not fundamentally new, it nonetheless offered an entirely fresh and different perspective: via a precise formulation of Bohr's correspondence principle, it made quantum mechanics similar to classical mechanics in a precise sense. In addition, Feynman's approach made it possible to explain physical experiments, and, via diagrams, link them directly to computations. What resulted was a very powerful device for computing energies and scattering amplitudes - the famous Feynman's diagrams. In his formulation, Feynman aimed at representing the solution to the non-relativistic Schrödinger equation in the form of an “ average ” over histories or paths of a particle. This solution is commonly known as the Feynman path integral. It plays an important role in the theory but appears as a postulate based on intuition coming from physics rather than a justified mathematical object. This is why Feynman's vision has caught the attention of many mathematicians as well as physicists. The papers of Gelfand, Cameron, and Nelson are among the first, and more substantial, attempts to supply Feynman's theory with a rigorous mathematical foundation. These attempts were followed by many others, but unfortunately all of them were not quite satisfactory. The difficulty comes from a need to define a measure on an infinite-dimensional space of continuous functions that represent all possible paths of a particle. This Feynman's measure has to produce an integral with the properties requested by Feynman. In particular, the expression for the Feynman measure has to involve the non-absolutely integrable Fresnel integrands. The non-absolute integrability of the Fresnel integrands makes the measure fail to be positive and to have the countably additive property. Thus, a well-defined measure in the case of the Feynman path integral does not exist. Extensive research has been done on the methods of relating the Feynman path integral to the integral with respect to the Wiener measure. The method of analytic continuation in mass defines the Feynman path integral as a certain limit of the Wiener integrals. Unfortunately, this method can be used as definition for only almost all values of the mass parameter in the Schrödinger equation. For physicists, this is not a satisfactory result and needs to be improved. In this work we examine those questions which originally led to the Feynman path integral. By now we know that Feynman's “ dream ” cannot be realized as a positive and countably additive measure on the path-space. Here, we offer a new way out by modifying Feynman's question, and thereby achieving a solution to the Schrödinger equation via a different kind of averages in the path-space. We give our version of the question that Feynman “ should have asked ” in order to realize the elusive path integral. In our formulation, we get a Feynman path integral as a limit of linear functionals, as opposed to the more familiar inductive limits of positive measures, traditionally used for constructing the Wiener measure, and related Gaussian families. We adapt here an approach pioneered by Patrick Muldowney. In it, Muldowney suggested a Henstock integration technique in order to deal with the non-absolute integrability of the kind of Fresnel integrals which we need in our solution to Feynman's question. By applying Henstock's theory to Fresnel integrals, we construct a complex-valued “ probability distribution functions ” on the path-space. Then we use this “ probability ” distribution function to define the Feynman path integral as an inductive limit. This establishes a mathematically rigorous Feynman limit, and at the same time, preserves Feynman's intuitive idea in resulting functional. In addition, our definition, and our solution, do not place any restrictions on any of the parameters in the Schrödinger equation, and have a potential to offer useful computational experiments, and other theoretical insights.

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This thesis proposes new algorithms for a group of sensing robots to learn a para- metric model for a dynamic spatio-temporal field, then based on the learned model trajectories are planned for sensing robots to best estimate the field. In this thesis we call these two parts learning and monitoring, respectively. For the learning, we first introduce a parametric model for the spatio-temporal field. We then propose a family of motion strategies that can be used by a group of mobile sensing robots to collect point measurements about the field. Our motion strategies are designed to collect enough information from enough locations at enough different times for the robots to learn the dynamics of the field. In conjunction with these motion strategies, we propose a new learning algorithm based on subspace identification to learn the parameters of the dynamical model. We prove that as the number of data collected by the robots goes to infinity, the parameters learned by our algorithm will converge to the true parameters. For the monitoring, based on the model learned from the learning part, three new informative trajectory planning algorithms are proposed for the robots to collect the most informative measurements for estimating the field. Kalman filter is used to calculate the estimate, and to compute the error covariance of the estimate. The goal is to find trajectories for sensing robots that minimize a cost metric on the error covariance matrix. We propose three algorithms to deal with this problem. First, we propose a new randomized path planning algorithm called Rapidly-exploring Random Cycles (RRC) and its variant RRC* to find periodic trajectories for the sensing robots that try to minimize the largest eigenvalue of the error covariance matrix over an infinite horizon. The algorithm is proven to find the minimum infinite horizon cost cycle in a graph, which grows by successively adding random points. Secondly, we apply kinodynamic RRT* to plan continuous trajectories to estimate the field. We formulate the evolution of the estimation error covariance matrix as a differential constraint and propose extended state space and task space sampling to fit this problem into classical RRT* setup. Thirdly, Pontryagin’s Minimum Principle is used to find a set of necessary conditions that must be satisfied by the optimal trajectory to estimate the field. We then consider a real physical spatio-temporal field, the surface water temper- ature in the Caribbean Sea. We first apply the learning algorithm to learn a linear dynamical model for the temperature. Then based on the learned model, RRC and RRC* are used to plan trajectories to estimate the temperature. The estimation performance of RRC and RRC* trajectories significantly outperform the trajectories planned by random search, greedy and receding horizon algorithms.

This thesis deals with the general problem of robot motion planning and control. It proposes the hypothesis that it should bepossible to create a generic software framework capable of dealing with all robot motion planning and control problems, independent of the robot being used, the task being solved, the workspace obstacles or the algorithms employed. The thesis work then consisted of identifying the requirements and creating a design and implementation of such a framework. This report motivates and documents the entire process. The framework developed was tested on two different robot arms under varying conditions. The testing method and results are also presented.The thesis concludes that the proposed hypothesis is indeed valid.

A Monte Carlo light transport simulation is used in scattering-aware color 3D printing pipeline (Elek et al. [2017], Sumin et al. [2019]) to drive an iterative optimization loop. Its purpose is to find a material arrangement that yields the closest match in terms of surface appearance towards a target. As the light transport prediction takes up about 90% of the time it poses a significant bottleneck towards a practical application of this technology. The dense volumetric textures also require a lot of memory. Explicitly simulating every light interaction is particularly challenging in the setting of 3D printouts due to the heterogeneity, high density and high albedo of the media. In this thesis, we explore existing volumetric rendering techniques (Křivánek et al. [2014], Herholz et al. [2019]) and finally engineer a customized estimator for our setting, improving the performance considerably. Additionally, we investigate various storage solutions for the volumetric data and successfully reduce the memory footprint. All the algorithms are available in the form of Mitsuba renderer plugins.

Degree constraints on the vertices of a path allow for the definitions of uphill and downhill paths. Specifically, we say that a path P = vi, v2,⋯ vk+1 is a downhill path if for every i, 1 ≤ i ≤ k, deg(vi) ≥ deg(v1+1). Conversely, a path π = u1, u2,⋯ uk+1 is an uphill path if for every i, 1 ≤ i ≤ k, deg(ui) ≤ deg(ui+1). The downhill domination number of a graph G is the minimum cardinality of a set S of vertices such that every vertex in V lies on a downhill path from some vertex in S. The uphill domination number is defined as expected. We give a polynomial time algorithm to find a minimum downhill dominating set and a minimum uphill dominating set for any graph.

Placing degree constraints on the vertices of a path yields the definitions of uphill and downhill paths. Specifically, we say that a path π = v1, v2, ⋯ vk+1 is a downhill path if for every i, 1 ≤ i ≤ k, deg(v1) ≥ deg(vi+1). Conversely, a path π = u1, u2, ⋯ uk+1 is an uphill path if for every i, 1 ≤ i ≤ k, deg(u1) ≤ deg(ui+1). The downhill domination number of a graph G is defined to be the minimum cardinality of a set S of vertices such that every vertex in V lies on a downhill path from some vertex in S. The uphill domination number is defined as expected. We explore the properties of these invariants and their relationships with other invariants. We also determine a Vizing-like result for the downhill (respectively, uphill) domination numbers of Cartesian products.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第19121号 / 情博第567号 / 新制||情||100(附属図書館) / 32072 / 京都大学大学院情報学研究科複雑系科学専攻 / (主査)教授 木上 淳, 教授 磯 祐介, 教授 西村 直志 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

Since their introduction in the 1950s, industrial Automated Guided Vehicles (AGV) have gone from automatic machinery limited by hardware to complex robots limited by software, called Autonomous Mobile Robots. Small and medium businesses need to be able to utilize cutting-edge technology. Therefore, this research focuses on deploying mapless AMRs on cheap open source AMRs in dynamic industrial environments. The study implements Dijkstra’s and A-STAR algorithms on a simulated Turtlebot3 model deployed in a Gazebo rendering of an industrial warehouse with moving objects added. The Turtlebot3 model traverses the environment where time and distance results are observed. The results shown in the research indicate that Dijkstra’s algorithm is barely affected by the change of the initial map state, while the A-STAR algorithm performed worse on average. Future work should focus on minimizing the sensors needed and continue testing with more algorithms, but early tests show promising results.