This paper describes decentralized control laws for the coordination of multiple vehicles performing spatially distributed tasks. The control laws are based on a gradient descent scheme applied to a class of decentralized utility functions that encode optimal coverage and sensing policies. These utility functions are studied in geographical optimization problems and they arise naturally in vector quantization and in sensor allocation tasks. The approach exploits the computational geometry of spatial structures such as Voronoi diagrams.

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Coverage control for mobile sensing networks

This paper presents control and coordination algorithms for groups of vehicles. The focus is on autonomous vehicle networks performing distributed sensing tasks where each vehicle plays the role of a mobile tunable sensor. The paper proposes gradient descent algorithms for a class of utility functions which encode optimal coverage and sensing policies. The resulting closed-loop behavior is adaptive, distributed, asynchronous, and verifiably correct.

Computational geometry

A modern, up-to-date introduction to optimization theory and methods This authoritative book serves as an introductory text to optimization at the senior undergraduate and beginning graduate levels. With consistently accessible and elementary treatment of all topics, An Introduction to Optimization, Second Edition helps students build a solid working knowledge of the field, including unconstrained optimization, linear programming, and constrained optimization. Supplemented with more than one hundred tables and illustrations, an extensive bibliography, and numerous worked examples to illustrate both theory and algorithms, this book also provides: * A review of the required mathematical background material * A mathematical discussion at a level accessible to MBA and business students * A treatment of both linear and nonlinear programming * An introduction to recent developments, including neural networks, genetic algorithms, and interior-point methods * A chapter on the use of descent algorithms for the training of feedforward neural networks * Exercise problems after every chapter, many new to this edition * MATLAB(r) exercises and examples * Accompanying Instructor's Solutions Manual available on request An Introduction to Optimization, Second Edition helps students prepare for the advanced topics and technological developments that lie ahead. It is also a useful book for researchers and professionals in mathematics, electrical engineering, economics, statistics, and business.

An introduction to optimization

In this paper, we introduce the progressive simplicial complex (PSC) representation, a new format for storing and transmitting triangulated geometric models. Like the earlier progressive mesh (PM) representation, it captures a given model as a coarse base model together with a sequence of refinement transformations that progressively recover detail. The PSC representation makes use of a more general refinement transformation, allowing the given model to be an arbitrary triangulation (e.g. any dimension, non-orientable, non-manifold, non-regular), and the base model to always consist of a single vertex. Indeed, the sequence of refinement transformations encodes both the geometry and the topology of the model in a unified multiresolution framework. The PSC representation retains the advantages of PM’s. It defines a continuous sequence of approximating models for runtime level-of-detail control, allows smooth transitions between any pair of models in the sequence, supports progressive transmission, and offers a space-efficient representation. Moreover, by allowing changes to topology, the PSC sequence of approximations achieves better fidelity than the corresponding PM sequence. We develop an optimization algorithm for constructing PSC representations for graphics surface models, and demonstrate the framework on models that are both geometrically and topologically complex. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling surfaces and object representations. Additional

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Progressive simplicial complexes

Dealing with simplicial decompositions which are dimension independent allows for the convergence of disparate viewpoints from computer graphics, solid and geometric modeling. In this framework it is possible to treat in a unified manner several geometric problems, such as solid modeling of articulated objects, simplicial approximation of curved manifolds, motion encoding and interference detection, free configuration space computation, and graphical representation of multidimensional data. In the paper the authora describe the winged scheme, a simple representation based on simplicial decompositions, which can be used for linear polyhedra of any dimension and which allows for “solid” approximation of curved manifolds when combined with curved maps (e.g., NURBS). Various operators and algorithms are discussed, including boundary evaluation, linear and screw extrusion, grid generation, simplicial maps, and set operations. A simple manipulation language is also introduced, and some nontrivial examples are discussed.

Dimension-independent modeling with simplicial complexes

Elements of Differential Geometry

In order either to plan or control the static or dynamic behaviour of models in CAD applications, it is often necessary, to evaluate integral properties of solid models (i.e. volume, centroid, moments of inertia, etc.). This paper deals with the exact evaluation of inertial properties of homogeneous polyhedral objects. In particular, a finite integration method is developed for the computation of various order monomial integrals over polyhedral solids and surfaces in the 3-space. The integration method presented here can also be used for the exact evaluation of domain integrals of trivariate polynomial forms.

Boundary integration over linear polyhedra

This article presents a functional programming approach to geometric design with embedded polyhedral complexes. Its main goals are to show the expressive power of the language as well as its usefulness for geometric design. The language, named PLASM (the Programming LAnguage for Solid Modeling), introduces a very high level approach to “constructive” or “generative” modeling. Geometrical objects are generated by evaluating some suitable language expressions. Because generating expressions can be easily combined, the language also extends the standard variational geometry approach by supporting classes of geometric objects with varying topology and shape. The design language PLASM can be roughly considered as a geometry-oriented extension of a subset of the functional language FL. The language takes a dimension-independent approach to geometry representation and algorithms. In particular it implements an algebraic calculus over embedded polyhedra of any dimension. The generated objects are always geometrically consistent because the validity of geometry is guaranteed at a syntactical level. Such an approach allows one to use a representation scheme which is weaker than those usually adopted in solid modelers, thus encompassing a broader geometric domain, which contains solids, surfaces, and wire-frames, as well as higher-dimensional objects.

Geometric programming: a programming approach to geometric design

From the Publisher:
PLaSM, a general purpose functional language, blends the powerful algebraic approach to programming developed at IBM research with a dimension-independent approach to geometric data structures and algorithms. This book demonstrates how this geometric code is surprisingly compact and easy to write. The authors combine excellent coverage of the theory with well-developed examples so readers can gain a better understanding of the material.

http://www.pascucci.org/pdf-papers/PLASM-book-toc.pdf

Alberto Paoluzzi

Papers

Dimension-independent modeling with simplicial complexes

Elements of Differential Geometry

Boundary integration over linear polyhedra

Geometric programming: a programming approach to geometric design

Geometric Programming for Computer Aided Design