This paper addresses the problem of steering a group of underactuated autonomous vehicles along given spatial paths, while holding a desired inter-vehicle formation pattern. For a general class of vehicles moving in either two or three-dimensional space, we show how Lyapunov-based techniques and graph theory can be brought together to yield a decentralized control structure where the dynamics of the cooperating vehicles and the constraints imposed by the topology of the inter-vehicle communications network are explicitly taken into account. Path-following for each vehicle amounts to reducing an appropriately defined geometric error to a small neighborhood of the origin. Vehicle coordination is achieved by adjusting the speed of each vehicle along its path according to information on the positions of a subset of the other vehicles, as determined by the communications topology adopted. The system obtained by putting together the path-following and vehicle coordination strategies adopted takes a cascade form, where the former subsystem is input-to-state stable (ISS) with the error variables of the latter as inputs. Convergence and stability of the overall system are proved formally. The results are also extended to solve the problem of temporary communication failures. Using the concept of "brief instabilities" we show that for a given maximum failure rate, the coordinated path following system is stable and the errors converge to a small neighborhood of the origin. We illustrate our design procedure for underwater vehicles moving in three-dimensional space. Simulations results are presented and discussed

In the stochastic formulation of chemical reactions, the dynamics of the first M-order moments of the species populations generally do not form a closed system of differential equations, in the sense that the time-derivatives of first M-order moments generally depend on moments of order higher than M. However, for analysis purposes, these dynamics are often made to be closed by approximating the needed derivatives of the first M-order moments by nonlinear functions of the same moments. These functions are called the moment closure functions. Recent results have introduced the technique of derivative-matching, where the moment closure functions are obtained by first assuming that they exhibit a certain separable form, and then matching time derivatives of the exact (not closed) moment equations with that of the approximate (closed) equations for some initial time and set of initial conditions. However, for multi-species reactions these results have been restricted to second order truncations, i.e, M = 2. This paper extends these results by providing explicit formulas to construct moment closure functions for any arbitrary order of truncation M. We show that with increasing M the closed moment equations provide more accurate approximations to the exact moment equations. Striking features of these moment closure functions are that they are independent of the reaction parameters (reaction rates and stoichiometry) and moreover the dependence of higher-order moment on lower order ones is consistent with the population being jointly lognormally distributed. To illustrate the applicability of our results we consider a simple bi-molecular reaction. Moment estimates from a third order truncation are compared with estimates obtained from a large number of Monte Carlo simulations

We study the problem of finding the minimum-length curvature constrained closed path through a set of regions in the plane. This problem is referred to as the Dubins Traveling Salesperson Problem with Neighborhoods (DTSPN). An algorithm is presented that uses sampling to cast this infinite dimensional combinatorial optimization problem as a Generalized Traveling Salesperson Problem (GTSP) with intersecting node sets. The GTSP is then converted to an Asymmetric Traveling Salesperson Problem (ATSP) through a series of graph transformations, thus allowing the use of existing approximation algorithms. This algorithm is shown to perform no worse than the best existing DTSPN algorithm and is shown to perform significantly better when the regions overlap. We report on the application of this algorithm to route an Unmanned Aerial Vehicle (UAV) equipped with a radio to collect data from sparsely deployed ground sensors in a field demonstration of autonomous detection, localization, and verification of multiple acoustic events.