Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

This paper provides a theoretical framework for analysis of consensus algorithms for multi-agent networked systems with an emphasis on the role of directed information flow, robustness to changes in network topology due to link/node failures, time-delays, and performance guarantees. An overview of basic concepts of information consensus in networks and methods of convergence and performance analysis for the algorithms are provided. Our analysis framework is based on tools from matrix theory, algebraic graph theory, and control theory. We discuss the connections between consensus problems in networked dynamic systems and diverse applications including synchronization of coupled oscillators, flocking, formation control, fast consensus in small-world networks, Markov processes and gossip-based algorithms, load balancing in networks, rendezvous in space, distributed sensor fusion in sensor networks, and belief propagation. We establish direct connections between spectral and structural properties of complex networks and the speed of information diffusion of consensus algorithms. A brief introduction is provided on networked systems with nonlocal information flow that are considerably faster than distributed systems with lattice-type nearest neighbor interactions. Simulation results are presented that demonstrate the role of small-world effects on the speed of consensus algorithms and cooperative control of multivehicle formations

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Consensus and Cooperation in Networked Multi-Agent Systems

Data Mining Practical Machine Learning Tools and Techniques

This paper reviews some main results and progress in distributed multi-agent coordination, focusing on papers published in major control systems and robotics journals since 2006. Distributed coordination of multiple vehicles, including unmanned aerial vehicles, unmanned ground vehicles, and unmanned underwater vehicles, has been a very active research subject studied extensively by the systems and control community. The recent results in this area are categorized into several directions, such as consensus, formation control, optimization, and estimation. After the review, a short discussion section is included to summarize the existing research and to propose several promising research directions along with some open problems that are deemed important for further investigations.

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An Overview of Recent Progress in the Study of Distributed Multi-Agent Coordination

System Identification I

A PROBABILISTIC FRAMEWORK FOR AIRCRAFT CONFLICT DETECTION Maria Prandini, John Lygeros, Arnab Nilim and Shankar Sastry Department of Electrical Engineering and Computer Sciences University of California at Berkeley Berkeley CA 94720 fprandini, lygeros, nilim, sastryg@robotics.eecs.berkeley.edu ABSTRACT We describe a general con ict detection/resolution scheme, focusing on the con ict detection component for a pair of aircraft ying at the same altitude. The proposed approach is formulated in a probabilistic framework, thus allowing uncertainty in the aircraft positions to be explicitly taken into account when detecting a potential con ict. The computational issues involved in the application of the proposed con ict alerting system are addressed by resorting to randomized algorithms. Finally, the validation of the proposed detection scheme is performed by Monte Carlo simulation on a stochastic ODE model of the aircraft motion. Our simulations show very promising results. The detection scheme will be used as the basis for a con ict resolution scheme in forthcoming work. INTRODUCTION Safety, of which con ict prediction and resolution form an integral part, is the primary concern of all advanced air tra c management systems. Con ict prediction and resolution are considered at three di erent levels in the air tra c management process: Long range: Some form of con ict prediction and resolution is carried out at the level of the entire National Airspace System (NAS), over a horizon of several hours. It involves composing ight plans and airline schedules (on a daily basis, for example) to ensure that airport and sector capacities are not exceeded. This is typically accomplished by large scale integer and linear programming techniques1;2. Research supported by DARPA under grant F33615-98-C-3614, by NASA under grant NAG 2-1039 and by ARO under grant MURI DAAH 04-96-1-0341. The authors would like to thank Dr. George J. Pappas for the stimulating discussions. Copyright c 1999 by the Regents of the University of California. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Sampled estimates of aircraft positions Conflict probability information

A probabilistic framework for aircraft conflict detection.

The probability of conflict between two aircraft is calculated by modeling aircraft motion as a deterministic trajectory plus a (scaled) Brownian motion perturbation. In this formalism, the probability of conflict becomes the probability that a Brownian motion escapes from a time-varying safe region. Approximate expressions for the probability of conflict are obtained in closed form for both the finite and infinite horizon cases. Based on these expressions, an autonomous algorithm is proposed for decentralized conflict resolution. The algorithm generalizes potential field ideas to path planning in a highly dynamic environment.

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Aircraft conflict prediction and resolution using Brownian motion

In this brief, a novel scheme to multi-aircraft conflict detection and resolution is introduced. A key feature of the proposed scheme is that uncertainty affecting the aircraft future positions along some look-ahead prediction horizon is accounted for via a probabilistic reachability analysis approach. In particular, ellipsoidal probabilistic reach sets are determined by formulating a chance-constrained optimization problem and solving it via a simulation-based method called scenario approach . Conflict detection is then performed by verifying if the ellipsoidal reach sets of different aircraft intersect. If a conflict is detected, then the aircraft flight plans are redesigned by solving a second-order cone program resting on the approximation of the ellipsoidal reach sets with spheres with constant radius along the look-ahead horizon. A bisection procedure allows one to determine the minimum radius such that the ellipsoidal reach sets of different aircraft along the corresponding new flight plans do not intersect. Some numerical examples are presented to show the efficacy of the proposed scheme.

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Multi-aircraft Conflict Detection and Resolution Based on Probabilistic Reach Sets

This paper proposes a novel sampling-based motion planner, which integrates in Rapidly exploring Random Tree star (
 $$\hbox {RRT}^{\star }$$
 ) a database of pre-computed motion primitives to alleviate its computational load and allow for motion planning in a dynamic or partially known environment The database is built by considering a set of initial and final state pairs in some grid space, and determining for each pair an optimal trajectory that is compatible with the system dynamics and constraints, while minimizing a cost Nodes are progressively added to the tree of feasible trajectories in the $$\hbox {RRT}^{\star }$$
 algorithm by extracting at random a sample in the gridded state space and selecting the best obstacle-free motion primitive in the database that joins it to an existing node The tree is rewired if some nodes can be reached from the new sampled state through an obstacle-free motion primitive with lower cost The computationally more intensive part of motion planning is thus moved to the preliminary offline phase of the database construction at the price of some performance degradation due to gridding Grid resolution can be tuned so as to compromise between (sub)optimality and size of the database The planner is shown to be asymptotically optimal as the grid resolution goes to zero and the number of sampled states grows to infinity

Sampling-based optimal kinodynamic planning with motion primitives

A model for discrete time stochastic hybrid systems whose evolution can be influenced by some control input is proposed in this paper. With reference to the introduced class of systems, a methodology for probabilistic reachability analysis is developed that is relevant to safety verification. This methodology is based on the interpretation of the safety verification problem as an optimal control problem for a certain controlled Markov process. In particular, this allows to characterize through some optimal cost function the set of initial conditions for the system such that safety is guaranteed with sufficiently high probability. The proposed methodology is applied to the problem of regulating the average temperature in a room by a thermostat controlling a heater.

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Maria Prandini

Papers

A probabilistic framework for aircraft conflict detection.

Aircraft conflict prediction and resolution using Brownian motion

Multi-aircraft Conflict Detection and Resolution Based on Probabilistic Reach Sets

Sampling-based optimal kinodynamic planning with motion primitives

Reachability analysis for controlled discrete time stochastic hybrid systems