Many sampling based algorithms have been introduced recently. Among them Rapidly Exploring Random Tree (RRT) is one of the quickest and the most efficient obstacle free path finding algorithm. Although it ensures probabilistic completeness, it cannot guarantee finding the most optimal path. Rapidly Exploring Random Tree Star (RRT*), a recently proposed extension of RRT, claims to achieve convergence towards the optimal solution thus ensuring asymptotic optimality along with probabilistic completeness. However, it has been proven to take an infinite time to do so and with a slow convergence rate. In this paper an extension of RRT*, called as RRT*-Smart, has been proposed to overcome the limitations of RRT*. The goal of the proposed method is to accelerate the rate of convergence, in order to reach an optimum or near optimum solution at a much faster rate, thus reducing the execution time. The novel approach of the proposed algorithm makes use of two new techniques in RRT*-- Path Optimization and Intelligent Sampling. Simulation results presented in various obstacle cluttered environments along with statistical and mathematical analysis confirm the efficiency of the proposed RRT*- Smart algorithm

Lebesgue integration is a fundamental concept in many mathematical theories, such as real analysis, probability and information theories. Reported higher-order-logic formalizations of the Lebesgue integral either do not include, or have a limited support for the Borel algebra, which is the canonical sigma algebra used on any metric space over which the Lebesgue integral is defined. In this report, we overcome this limitation by presenting a formalization of the Borel sigma algebra that can be used on any metric space, such as the complex numbers or the n-dimensional Euclidean space. Building on top of this framework, we have been able to prove some key Lebesgue integral properties, like its linearity and monotone convergence. Furthermore, we present the formalization of the "almost everywhere" relation and prove that the Lebesgue integral does not distinguish between functions which differ on a null set as well as other important results based on this concept. As applications, we present the verification of Markov and Chebyshev inequalities and the Weak Law of Large Numbers theorem.

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Abstract-Reliability analysis has become a tool of fundamental importance to virtually all electrical and computer engineers because of the extensive usage of hardware systems in safety and mission critical domains, such as medicine, military, and transportation. Due to the strong relationship between reliability theory and probabilistic notions, computer simulation techniques have been traditionally used to perform reliability analysis. However, simulation provides less accurate results and cannot handle large-scale systems due to its enormous CPU time requirements. To ensure accurate and complete reliability analysis and thus more reliable hardware designs, we propose to conduct a formal reliability analysis of systems within the sound core of a higher order logic theorem prover (HOL). In this paper, we present the higher order logic formalization of some fundamental reliability theory concepts, which can be built upon to precisely analyze the reliability of various engineering systems. The proposed approach and formalization is then utilized to analyze the repairability conditions for a reconfigurable memory array in the presence of stuck-at and coupling faults.

Rapidly Exploring Random Tree (RRT) is one of the quickest and the most efficient obstacle free path finding algorithm. However, it cannot guarantee finding the most optimal path. A recently proposed extension of RRT, known as Rapidly Exploring Random Tree Star (RRT*), claims to achieve convergence towards the optimal solution but has been proven to take an infinite time to do so and with a slow convergence rate. To overcome these limitations, we propose an extension of RRT*, called RRT*-Smart, which aims to accelerate its rate of convergence and to reach an optimum or near optimum solution at a much faster rate and at a reduced execution time. Our novel algorithm inculcates two new techniques in RRT*: these are path optimization and intelligent sampling. Simulation results presented in various obstacle cluttered environments confirm the efficiency of RRT*-Smart.