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.

Sampling-based algorithms for optimal motion planning

Cellular networks are usually modeled by placing the base stations on a grid, with mobile users either randomly scattered or placed deterministically. These models have been used extensively but suffer from being both highly idealized and not very tractable, so complex system-level simulations are used to evaluate coverage/outage probability and rate. More tractable models have long been desirable. We develop new general models for the multi-cell signal-to-interference-plus-noise ratio (SINR) using stochastic geometry. Under very general assumptions, the resulting expressions for the downlink SINR CCDF (equivalent to the coverage probability) involve quickly computable integrals, and in some practical special cases can be simplified to common integrals (e.g., the Q-function) or even to simple closed-form expressions. We also derive the mean rate, and then the coverage gain (and mean rate loss) from static frequency reuse. We compare our coverage predictions to the grid model and an actual base station deployment, and observe that the proposed model is pessimistic (a lower bound on coverage) whereas the grid model is optimistic, and that both are about equally accurate. In addition to being more tractable, the proposed model may better capture the increasingly opportunistic and dense placement of base stations in future networks.

A Tractable Approach to Coverage and Rate in Cellular Networks

Covering point process theory, random geometric graphs and coverage processes, this rigorous introduction to stochastic geometry will enable you to obtain powerful, general estimates and bounds of wireless network performance and make good design choices for future wireless architectures and protocols that efficiently manage interference effects. Practical engineering applications are integrated with mathematical theory, with an understanding of probability the only prerequisite. At the same time, stochastic geometry is connected to percolation theory and the theory of random geometric graphs and accompanied by a brief introduction to the R statistical computing language. Combining theory and hands-on analytical techniques with practical examples and exercises, this is a comprehensive guide to the spatial stochastic models essential for modelling and analysis of wireless network performance.

/pdf/stochastic-geometry-for-wireless-networks-50d7xggv33.pdf

Stochastic Geometry for Wireless Networks

spatstat is a package for analyzing spatial point pattern data. Its functionality includes exploratory data analysis, model-fitting, and simulation. It is designed to handle realistic datasets, including inhomogeneous point patterns, spatial sampling regions of arbitrary shape, extra covariate data, and "marks" attached to the points of the point pattern. 

A unique feature of spatstat is its generic algorithm for fitting point process models to point pattern data. The interface to this algorithm is a function ppm that is strongly analogous to lm and glm. 

This paper is a general description of spatstat and an introduction for new users.

/pdf/spatstat-an-r-package-for-analyzing-spatial-point-patterns-3xa0wyybp2.pdf

spatstat: An R Package for Analyzing Spatial Point Patterns

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.

Sampling-based Algorithms for Optimal Motion Planning

Mathematical Foundation. Point Processes I--The Poisson Point Process. Random Closed Sets I--The Boolean Model. Point Processes II--General Theory. Point Processes III--Construction of Models. Random Closed Sets II--The General Case. Random Measures. Random Processes of Geometrical Objects. Fibre and Surface Processes. Random Tessellations. Stereology. References. Indexes.

Stochastic Geometry and Its Applications

Stochastic Geometry and Its Applications.

In the paper fibre processes are considered, i.e. discrete sets of curves distributed at random in the plane. For translation invariant fibre processes the PALM distribution of the angle in a typical fibre point is defined, and point processes are studied which are obtained by intersection with lines. As a special case motion invariant, i.e. translation invariant and isotropic, fibre processes are treated. Finally weighted fibre processes are considered. As example segment processes are investigated.

Formulas for stationary planar fibre processes I - general theory

Intersections of stationary fibre processes in the plane with arbitrary curved lines of finite total length are considered. It is shown that the mean number of intersection points can be calculated from the intensity and angle distribution of the fibre process and the angle distribution of the curved line. Conversely, if the mean number of intersection points is known for all positions of the cuived line? then it is possible to determine the intensity and angle distribution of the fibre process.

Joseph Mecke

Papers

Stochastic Geometry and Its Applications

Stochastic Geometry and Its Applications.

Formulas for stationary planar fibre processes I - general theory

Formulas for stationary planar fibre processes III – intersections with fibre systems

On the relationship between the 0-cell and the typical cell of a stationary random tessellation