Low-Discrepancy Curves and Efficient Coverage of Space

We introduce the notion of low-discrepancy curves and use it to solve the problem of optimally covering space. In doing so, we extend the notion of low-discrepancy sequences in such a way that sufficiently smooth curves with low discrepancy properties can be defined and generated. Based on a class of curves that cover the unit square in an efficient way, we define induced low discrepancy curves in Riemannian spaces. This allows us to efficiently cover an arbitrarily chosen abstract surface that admits a diffeomorphism to the unit square. We demonstrate the application of these ideas by presenting concrete examples of low-discrepancy curves on some surfaces that are of interest in robotics.

[1]  E. Wright,et al.  An Introduction to the Theory of Numbers , 1939 .

[2]  R. D. Richtmyer.  THE EVALUATION OF DEFINITE INTEGRALS, AND A QUASI-MONTE-CARLO METHOD BASED ON THE PROPERTIES OF ALGEBRAIC NUMBERS , 1951 .

[3]  J. Halton.  On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals , 1960 .

[4]  J. Hammersley,et al.  Monte Carlo Methods , 1965 .

[5]  W. Reiher.  Hammersley, J. M., D. C. Handscomb: Monte Carlo Methods. Methuen & Co., London, and John Wiley & Sons, New York, 1964. VII + 178 S., Preis: 25 s , 1966 .

[6]  I. Sobol.  On the distribution of points in a cube and the approximate evaluation of integrals , 1967 .

[7]  William H. Press,et al.  Numerical recipes in C. The art of scientific computing , 1987 .

[8]  F. A. Seiler,et al.  Numerical Recipes in C: The Art of Scientific Computing , 1989 .

[9]  William H. Press,et al.  Numerical recipes , 1990 .

[10]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[11]  Daniel E. Koditschek,et al.  Exact robot navigation using artificial potential functions , 1992, IEEE Trans. Robotics Autom..

[12]  J. Mccleary,et al.  Geometry from a Differentiable Viewpoint: Recapitulation and coda , 1994 .

[13]  Russel E. Caflisch,et al.  Quasi-Random Sequences and Their Discrepancies , 1994, SIAM J. Sci. Comput..

[14]  J. Mccleary.  Geometry from a Differentiable Viewpoint: Recapitulation and coda , 1995 .

[15]  Godfrey H. Hardy,et al.  An introduction to the theory of numbers (5. ed.) , 1995 .

[16]  F. James,et al.  Multidimensional sampling for simulation and integration: measures, discrepancies, and quasi-random numbers , 1996, hep-ph/9606309.

[17]  W. J. Whiten,et al.  Computational investigations of low-discrepancy sequences , 1997, TOMS.

[18]  Alfred Gray,et al.  Modern differential geometry of curves and surfaces with Mathematica (2. ed.) , 1998 .

[19]  William H. Press,et al.  Numerical recipes in C , 2002 .

[20]  Steven M. LaValle,et al.  Incremental low-discrepancy lattice methods for motion planning , 2003, 2003 IEEE International Conference on Robotics and Automation (Cat. No.03CH37422).

[21]  Steven M. LaValle,et al.  On the Relationship between Classical Grid Search and Probabilistic Roadmaps , 2004, Int. J. Robotics Res..

[22]  Howie Choset,et al.  Coverage for robotics – A survey of recent results , 2001, Annals of Mathematics and Artificial Intelligence.

[23]  Steven M. LaValle,et al.  Incremental Grid Sampling Strategies in Robotics , 2004, WAFR.

[24]  M. Combescure.  Hamiltonian Chaos and Fractional Dynamics , 2005 .

[25]  D. R. Heath-Brown,et al.  An Introduction to the Theory of Numbers, Sixth Edition , 2008 .