This thesis addresses situated, embodied agents interacting in complex domains. It focuses on two problems: 1) synthesis and analysis of intelligent group behavior, and 2) learning in complex group environments. Basic behaviors, control laws that cluster constraints to achieve particular goals and have the appropriate compositional properties, are proposed as effective primitives for control and learning. The thesis describes the process of selecting such basic behaviors, formally specifying them, algorithmically implementing them, and empirically evaluating them. All of the proposed ideas are validated with a group of up to 20 mobile robots using a basic behavior set consisting of: safe--wandering, following, aggregation, dispersion, and homing. The set of basic behaviors acts as a substrate for achieving more complex high--level goals and tasks. Two behavior combination operators are introduced, and verified by combining subsets of the above basic behavior set to implement collective flocking, foraging, and docking. A methodology is introduced for automatically constructing higher--level behaviors

Abstract—We present a nonparametric algorithm for finding localized energy solutions from limited data. The problem we address is underdetermined, and no prior knowledge of the shape of the region on which the solution is nonzero is assumed. Termed the FOcal Underdetermined System Solver (FOCUSS), the algorithm has two integral parts: a low-resolution initial estimate of the real signal and the iteration process that refines the initial estimate to the final localized energy solution. The iterations are based on weighted norm minimization of the dependent variable with the weights being a function of the preceding iterative solutions. The algorithm is presented as a general estimation tool usable across different applications. A detailed analysis laying the theoretical foundation for the algorithm is given and includes proofs of global and local convergence and a derivation of the rate of convergence. A view of the algorithm as a novel optimization method which combines desirable characteristics of both classical optimization and learning-based algorithms is provided. Mathematical results on conditions for uniqueness of sparse solutions are also given. Applications of the algorithm are illustrated on problems in direction-of-arrival (DOA) estimation and neuromagnetic imaging. I.

The problem of synthesizing and analyzing collective autonomous agents has only recently begun to be practically studied by the robotics community. This paper overviews the most prominent directions of research, defines key terms, and summarizes the main issues. Finally, it briefly describes our approach to controlling group behavior and its relation to the field as a whole.

this article we study the influence of this truncation to the difference between the numerical solution and the exact solution of the perturbed differential equation. Results on the long-time behaviour of numerical solutions are obtained in this way. We present applications to the numerical phase portrait near hyperbolic equilibrium points, to asymptotically stable periodic orbits and Hopf bifurcation, and to energy conservation and approximation of invariant tori in Hamiltonian systems.

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Power electronics circuits are rich in nonlinear dynamics. Their operation is characterized by cyclic switching of circuit topologies, which gives rise to a variety of nonlinear behavior. This paper provides an overview of the chaotic dynamics and bifurcation scenarios observed in power converter circuits, emphasizing the salient features of the circuit operation and the modeling strategies. In particular, this paper surveys the key publications in this field, reviews the main modeling approaches, and discusses the salient bifurcation behaviors of power converters with particular emphasis on the disruption of standard bifurcation patterns by border collisions.

In many applications, the primary objective of numerical simulation of time-evolving systems is the prediction of macroscopic, or coarse-grained, quantities. A representative example is the prediction of biomolecular conformations from molecular dynamics. In recent years a number of new algorithmic approaches have been introduced to extract effective, lower-dimensional, models for the macroscopic dynamics; the starting point is the full, detailed, evolution equations. In many cases the effective low-dimensional dynamics may be stochastic, even when the original starting point is deterministic. This review surveys a number of these new approaches to the problem of extracting effective dynamics, highlighting similarities and differences between them. The importance of model problems for the evaluation of these new approaches is stressed, and a number of model problems are described. When the macroscopic dynamics is stochastic, these model problems are either obtained through a clear separation of time-scales, leading to a stochastic effect of the fast dynamics on the slow dynamics, or by considering high dimensional ordinary differential equations which, when projected onto a low dimensional subspace, exhibit stochastic behaviour through the presence of a broad frequency spectrum. Models whose stochastic microscopic behaviour leads to deterministic macroscopic dynamics are also introduced. The algorithms we overview include SVD-based methods for nonlinear problems, model reduction for linear control systems, optimal prediction techniques, asymptotics-based mode elimination, coarse timestepping methods and transfer-operator based methodologies.

The splitting of separatrices of hyperbolic fixed points for exact symplectic maps of n degrees of freedom is considered. The non-degenerate critical points of a real-valued function (called the Melnikov potential) are associated to transverse homoclinic orbits and an asymptotic expression for the symplectic area between homoclinic orbits is given. Moreover, if the unperturbed invariant manifolds are completely doubled, it is shown that there exist, in general, at least 4 primary homoclinic orbits (4n in antisymmetric maps). Both lower bounds are optimal. Two examples are presented: a 2n-dimensional central standard-like map and the Hamiltonian map associated to a magnetized spherical pendulum. Several topics are studied about these examples: existence of splitting, explicit computations of Melnikov potentials, transverse homoclinic orbits, exponentially small splitting, etc.

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We present a set of advanced techniques for the visualization of 2D Poincar'e maps. Since 2D Poincar'e maps are a mathematical abstraction of periodic or quasiperiodic 3D flows, we propose to embed the 2D visualization with standard 3D techniques to improve the understanding of the Poincar'e maps. Methods to enhance the representation of the relation x $ P (x), e.g., the use of spot noise, are presented as well as techniques to visualize the repeated application of P , e.g., the approximation of P as a warp function. It is shown that animation can be very useful to further improve the visualization. For example, the animation of the construction of Poincar'e map P is inherently a proper visualization. During the paper we present a set of examples which demonstrate the usefulness of our techniques. Keywords: visualization, dynamical systems, Poincar'e maps 1 Introduction Poincar'e sections are an important tool for the investigation of dynamical systems in theory as well a...