Sampling-based Algorithms for Optimal Motion Planning

During the last decade, sampling-based path planning algorithms, such as Probabilistic RoadMaps (PRM) and Rapidly-exploring Random Trees (RRT), have been shown to work well in practice and possess theoretical guarantees such as probabilistic completeness. However, little effort has been devoted to the formal analysis of the quality of the solution returned by such algorithms, e.g., as a function of the number of samples. The purpose of this paper is to fill this gap, by rigorously analyzing the asymptotic behavior of the cost of the solution returned by stochastic sampling-based algorithms as the number of samples increases. A number of negative results are provided, characterizing existing algorithms, e.g., showing that, under mild technical conditions, the cost of the solution returned by broadly used sampling-based algorithms converges almost surely to a non-optimal value. The main contribution of the paper is the introduction of new algorithms, namely, PRM* and RRT*, which are provably asymptotically optimal, i.e., such that the cost of the returned solution converges almost surely to the optimum. Moreover, it is shown that the computational complexity of the new algorithms is within a constant factor of that of their probabilistically complete (but not asymptotically optimal) counterparts. The analysis in this paper hinges on novel connections between stochastic sampling-based path planning algorithms and the theory of random geometric graphs.Comment: 76 pages, 26 figures, to appear in International Journal of Robotics Researc

Diagnostics of source distribution and particle population in Monte Carlo source iteration methods

This thesis is concerned with the development of mesh-input-free diagnostics for the determination of the iteration at which the source distribution of a Monte Carlo simulation has reached a stationary state, as well as the sufficiency of particle population size for a given tally cell volume, so as to reduce bias and increase accuracy of estimations of physical properties. Such physical properties can include, but are not limited to, neutron effective multiplication, power distribution, neutron flux and various interaction rates. When the physical properties of a Monte Carlo simulation are accurately estimated, they can be used to predict the actual behavior of a nuclear system, only being limited to the assumptions used to create the model. Five methods were used to describe the state of the source distribution. Four of the methods were established indicators of the source distributions state that required the input of a mesh, which divided the geometry into bins. These indicators are the Shannon entropy, Jensen measure, the progressive relative entropy and the posterior relative entropy. The fifth indicator of the source distribution\u27s state was developed as to eliminate the need for the input of a mesh upon the geometry. This method will be identified as the regionwise average position indicator or RAPI and is calculated by taking the sum of the distances of the regionwise average particle positions in the model at each cycle from the corresponding regionwise average particle positions at the first cycle. In conjunction with the Shannon entropy, Jensen measure, progressive relative entropy and the RAPI, an on-the-fly step-refined judgment of the indicators of the source distribution\u27s state will be employed to determine at which cycle or iteration the indicators have reached convergence, signifying that the simulated source distribution has begun to fluctuate around the true source distribution. This step-refined on-the-fly diagnostic of the source distribution was developed from the Wilcoxon rank sum in non-parametric statistics. The posterior relative entropy cycle of convergence is determined to be the cycle at which the posterior relative entropy becomes less than the average value of the posterior relative entropy over the second half of active cycles. The cycle of convergence was determined for three different models by the use of the above described methods. The resulting cycle of convergence obtained by the use of the indicators requiring a mesh input was compared against that obtained from the mesh-input-free indicator, RAPI, for each of the models. It was found that the RAPI was an excellent representation of the source distribution\u27s state and more conservative than the posterior relative entropy diagnosis. The RAPI can be used to determine the cycle at which the source distribution converged to an equilibrium fluctuation range of stationary state, thus eliminating the need for mesh-input for physical property estimation. Applications of graph theory techniques to Monte Carlo methods were also investigated as a means of meshless convergence indication, but drawbacks for such an application led to a particle population diagnostic investigation. This was done because meshless particle population diagnosis for the power distribution has yet to be done in Monte Carlo source iteration methods. In power distribution calculations, tally cells are used to estimate the power distribution in a model. To approach this problem, the concept of Euclidian minimum spanning trees (EMST) was applied to the source distribution to develop a meshless diagnosis of the particle population. One source particle effect is the characteristic volume of one particle and is defined to be the cubic of the average edge length of an EMST. Then using this characteristic volume, weak and strong requirements of the particle population size were defined for minimum tally cell volume. This diagnostic was compared against a verified population diagnostic, which requires a mesh input, termed as PD-MESH in this thesis. These diagnostic methods were used in the analysis of a pressurized water reactor initial full core simulation. The comparison of the EMST-based population diagnosis to PD-MESH showed that it can be used to determine if a population size is of sufficient size for power distribution calculations, eliminating the need for mesh-based diagnosis

Interstellar MHD Turbulence and Star Formation

This chapter reviews the nature of turbulence in the Galactic interstellar medium (ISM) and its connections to the star formation (SF) process. The ISM is turbulent, magnetized, self-gravitating, and is subject to heating and cooling processes that control its thermodynamic behavior. The turbulence in the warm and hot ionized components of the ISM appears to be trans- or subsonic, and thus to behave nearly incompressibly. However, the neutral warm and cold components are highly compressible, as a consequence of both thermal instability in the atomic gas and of moderately-to-strongly supersonic motions in the roughly isothermal cold atomic and molecular components. Within this context, we discuss: i) the production and statistical distribution of turbulent density fluctuations in both isothermal and polytropic media; ii) the nature of the clumps produced by thermal instability, noting that, contrary to classical ideas, they in general accrete mass from their environment; iii) the density-magnetic field correlation (or lack thereof) in turbulent density fluctuations, as a consequence of the superposition of the different wave modes in the turbulent flow; iv) the evolution of the mass-to-magnetic flux ratio (MFR) in density fluctuations as they are built up by dynamic compressions; v) the formation of cold, dense clouds aided by thermal instability; vi) the expectation that star-forming molecular clouds are likely to be undergoing global gravitational contraction, rather than being near equilibrium, and vii) the regulation of the star formation rate (SFR) in such gravitationally contracting clouds by stellar feedback which, rather than keeping the clouds from collapsing, evaporates and diperses them while they collapse.Comment: 43 pages. Invited chapter for the book "Magnetic Fields in Diffuse Media", edited by Elisabete de Gouveia dal Pino and Alex Lazarian. Revised as per referee's recommendation

International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library

Aeronautical Engineering: A continuing bibliography (supplement 158)

This bibliography lists 499 reports, articles and other documents introduced into the NASA scientific and technical information system in January 1983

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008

The roles of random boundary conditions in spin systems

Random boundary conditions are one of the simplest realizations of quenched disorder. They have been used as an illustration of various conceptual issues in the theory of disordered spin systems. Here we review some of these result

Aeronautical engineering: A continuing bibliography with indexes, supplement 100

This bibliography lists 295 reports, articles, and other documents introduced into the NASA Scientific and Technical Information System in August 1978

Sampling-based algorithms for optimal path planning problems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 141-152).Sampling-based motion planning received increasing attention during the last decade. In particular, some of the leading paradigms, such the Probabilistic RoadMap (PRM) and the Rapidly-exploring Random Tree (RRT) algorithms, have been demonstrated on several robotic platforms, and found applications well outside the robotics domain. However, a large portion of this research effort has been limited to the classical feasible path planning problem, which asks for finding a path that starts from an initial configuration and reaches a goal configuration while avoiding collision with obstacles. The main contribution of this dissertation is a novel class of algorithms that extend the application domain of sampling-based methods to two new directions: optimal path planning and path planning with complex task specifications. Regarding the optimal path planning problem, we first show that the existing algorithms either lack asymptotic optimality, i. e., almost-sure convergence to optimal solutions, or they lack computational efficiency: on one hand, neither the RRT nor the k-nearest PRM (for any fixed k) is asymptotically optimal; on the other hand, the simple PRM algorithm, where the connections are sought within fixed radius balls, is not computationally as efficient as the RRT or the efficient PRM variants. Subsequently, we propose two novel algorithms, called PRM* and RRT*, both of which guarantee asymptotic optimality without sacrificing computational efficiency. In fact, the proposed algorithms and the most efficient existing algorithms, such as the RRT, have the same asymptotic computational complexity. Regarding the path planning problem with complex task specifications, we propose an incremental sampling-based algorithm that is provably correct and probabilistically complete, i.e., it generates a correct-by-design path that satisfies a given deterministic pt-calculus specification, when such a path exists, with probability approaching to one as the number of samples approaches infinity. For this purpose, we develop two key ingredients. First, we propose an incremental sampling-based algorithm, called the RRG, that generates a representative set of paths in the form of a graph, with guaranteed almost-sure convergence towards feasible paths. Second, we propose an incremental local model-checking algorithm for the deterministic p-calculus. Moreover, with the help of these tools and the ideas behind the RRT*, we construct algorithms that also guarantee almost sure convergence to optimal solutions.by Sertac Karaman.Ph.D