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48585 results in Computer Science

Page 1729 of 2430

  • Deconvolution on the Euclidean motion group and planar robotic manipulator design

  • Peter T. Kim, Yan Liu, Zhi-Ming Luo, Yunfeng Wang
  • Journal:
    Robotica / Volume 27 / Issue 6 / October 2009
    Published online by Cambridge University Press:
    15 January 2009, pp. 861-872

  • Several problems of practical interest in robotics can be modelled as the convolution of functions on the Euclidean motion group. These include the evaluation of reachable positions and orientations at the distal end of a robot manipulator arm. A natural inverse problem arises when one wishes to design rather than to model manipulators. Namely, by considering a serial-chain robot arm as a concatenation of segments, we examine how statistics of known segments can be used to select, or design, the remainder of the structure so as to attain the desired statistical properties of the whole structure. This is then a deconvolution density estimation problem for the Euclidean motion group. We prove several results about the convergence of these deconvolution estimators to the true underlying density under certain smoothness assumptions. A practical implementation to the design of planar robot arms is demonstrated.

  • Appendix C - CONCEPTS OF PROBABILITY THEORY

  • from Part D - APPENDICES
  • Philippe Flajolet, Institut National de Recherche en Informatique et en Automatique (INRIA), Rocquencourt, Robert Sedgewick, Princeton University, New Jersey
  • Book:
    Analytic Combinatorics
    Published online:
    11 April 2011
    Print publication:
    15 January 2009, pp 769-778

  • Summary

    This appendix contains entries arranged in logical order regarding the following topics:

    Probability spaces and measure; Random variables; Transforms of distributions; Special distributions; Convergence in law.

    In this book we start from probability spaces that are finite, since they arise from objects of a fixed size in some combinatorial class (see Chapter III and Appendix A.3: Combinatorial probability, p. 727 for elementary aspects), then need basic properties of continuous distributions in order to discuss asymptotic limit laws. The entries in this appendix are related principally to Chapter IX of Part C (Random Structures). They present a unified framework that encompasses discrete and continuous probability distributions alike. For further study, we recommend the superb classics of Feller [205, 206], given the author's concrete approach, and of Billingsley [68], whose coverage of limit distributions is of great value for analytic combinatorics.

    Probability spaces and measure

    An axiomatization of probability theory was discovered in the 1930s by Kolmogorov. A measurable space consists of a set Ω, called the set of elementary events or the sample set and a σ-algebra A of subsets of Ω called events (that is, a collection of sets containing ∅ and closed under complement and denumerable unions). A measure space is a measurable space endowed with a measure µ A ↦ ℝ≥0 that is additive over finite or denumerable unions of disjoint sets; in that case, elements of A are called measurable sets.

  • AN INVITATION TO ANALYTIC COMBINATORICS

  • Philippe Flajolet, Institut National de Recherche en Informatique et en Automatique (INRIA), Rocquencourt, Robert Sedgewick, Princeton University, New Jersey
  • Book:
    Analytic Combinatorics
    Published online:
    11 April 2011
    Print publication:
    15 January 2009, pp 1-12

  • Summary

    διὸ δὴ συμμειγνύμενα αὐτά τε πρὸς αὑτὰ ϰαὶ πρὸς ἂλληλα τὴν

    ποιϰιλίαν ἐστὶν ἂπειρα ἧς δὴ δεῖ ϑεωροὺς γίγνεσϑαι τοὺς

    μέλλοντας περὶ ϕύσεως εἰϰότι λόγῳ

    —Plato, The Timaeus

    Analytic Combinatorics is primarily a book about combinatorics, that is, the study of finite structures built according to a finite set of rules. Analytic in the title means that we concern ourselves with methods from mathematical analysis, in particular complex and asymptotic analysis. The two fields, combinatorial enumeration and complex analysis, are organized into a coherent set of methods for the first time in this book. Our broad objective is to discover how the continuous may help us to understand the discrete and to quantify its properties.

    Combinatorics is, as told by its name, the science of combinations. Given basic rules for assembling simple components, what are the properties of the resulting objects? Here, our goal is to develop methods dedicated to quantitative properties of combinatorial structures. In other words, we want to measure things. Say that we have n different items like cards or balls of different colours. In how many ways can we lay them on a table, all in one row? You certainly recognize this counting problem—finding the number of permutations of n elements.

Page 1729 of 2430