Second-Order Agents on Ring Digraphs

The paper addresses the problem of consensus seeking among second-order linear agents interconnected in a specific ring topology. Unlike the existing results in the field dealing with one-directional digraphs arising in various cyclic pursuit algorithms or two-directional graphs, we focus on the case where some arcs in a two-directional ring graph are dropped in a regular fashion. The derived condition for achieving consensus turns out to be independent of the number of agents in a network.Comment: 6 pages, 10 figure

A Cyclic Pursuit Framework for Networked Mobile Agents Based on Vector Field Approach

This paper proposes a pursuit formation control scheme for a network of double-integrator mobile agents based on a vector field approach. In a leaderless architecture, each agent pursues another one via a cyclic topology to achieve a regular polygon formation. On the other hand, the agents are exposed to a rotational vector field such that they rotate around the vector field centroid, while they keep the regular polygon formation. The main problem of existing approaches in the literature for cyclic pursuit of double-integrator multiagent systems is that under those approaches, the swarm angular velocity and centroid are not controllable based on missions and agents capabilities. However, by employing the proposed vector field approach in this paper, while keeping a regular polygon formation, the swarm angular velocity and centroid can be determined arbitrary. The obtained results can be extended to achieve elliptical formations with cyclic pursuit as well. Simulation results for a team of eight mobile agents verify the accuracy of the proposed control scheme

Application of Difference Schemes to Decision the Pursuit Problem

The problem of the pursuit curve construction in the case when the tangent to pursuer’s motion trajectory passes at any time through the point representing the pursued is considered. A new approach to construct the pursuit curves using difference schemes is proposed. The proposed technique eliminates the need to derive the differential equations for the description of the pursuit curves, which is quite difficult task in the general case. In addition, the application of difference methods is justified in a situation where it is complicated to find the analytical solution of an existing differential equation and it is possible to obtain the pursuit curve only numerically. Various modifications of difference schemes respectively equivalent to the Euler, to the Adams – Bashforth and to the Milne methods are constructed. Their software implementation is realized by using the mathematical package Mathcad. We consider the case of a uniform rectilinear motion of the pursued whose differential equation describing the path of the pursuer and its analytical solution are known. We compare the numerical solutions obtained by the different methods with the well-known analytical solution. The error of the obtained numerical solutions is examined. Moreover, an application is considered illustrating the construction of the difference schemes for the case of an arbitrary trajectory of the pursued. Also, we extend the proposed method to the case of cyclic pursuit with several participants in the three-dimensional space. In particular, we construct a difference scheme equivalent to the Euler method for a three-dimensional analogue of the "bugs problem". The results obtained are demonstrated by means of animated examples for either two-dimensional or three-dimensional cases

Cyclic pursuit without coordinates: Convergence to regular polygon formations

Abstract-We study a multi-agent cyclic pursuit model where each of the identical agents moves like a Dubins car and maintains a fixed heading angle with respect to the next agent. We establish that stationary shapes for this system are regular polygons. We derive a sufficient condition for local convergence to such regular polygon formations, which takes the form of an inequality connecting the angles of the regular polygon with the heading angle of the agents. A block-circulant structure of the system's linearization matrix in suitable coordinates facilitates and elucidates our analysis. Our results are complementary to the conditions for rendezvous obtained in earlier work [Yu et al., IEEE Trans. Autom. Contr., Feb. 2012]

Optimal Control of Vehicular Formations With Nearest Neighbor Interactions

We consider the design of optimal localized feedback gains for one-dimensional formations in which vehicles only use information from their immediate neighbors. The control objective is to enhance coherence of the formation by making it behave like a rigid lattice. For the single-integrator model with symmetric gains, we establish convexity, implying that the globally optimal controller can be computed efficiently. We also identify a class of convex problems for double-integrators by restricting the controller to symmetric position and uniform diagonal velocity gains. To obtain the optimal non-symmetric gains for both the single- and the double-integrator models, we solve a parameterized family of optimal control problems ranging from an easily solvable problem to the problem of interest as the underlying parameter increases. When this parameter is kept small, we employ perturbation analysis to decouple the matrix equations that result from the optimality conditions, thereby rendering the unique optimal feedback gain. This solution is used to initialize a homotopy-based Newton’s method to find the optimal localized gain. To investigate the performance of localized controllers, we examine how the coherence of large-scale stochastically forced formations scales with the number of vehicles. We establish several explicit scaling relationships and show that the best performance is achieved by a localized controller that is both non-symmetric and spatially-varying

New decentralized algorithms for spacecraft formation control based on a cyclic approach

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 223-231).When considering the formation control problem for large number of spacecraft, the advantages of implementing control approaches with a centralized coordination mechanism can be outpaced by the risks associated with having a primary vital control unit. Additionally, in formations with a large number of spacecraft, a centralized approach implies an inherent difficulty in gathering and broadcasting information from/to the overall system. Therefore, there is a need to explore efficient decentralized control approaches. In this thesis a new approach to spacecraft formation control is formulated by exploring and enhancing the recent results on the theory of convergence to geometric patterns and exploring the analysis of this approach using the tools of contracting theory. First, an extensive analysis of the cyclic pursuit dynamics leads to developing control laws useful for spacecraft formation flight which, as opposed to the most common approaches in the literature, do not track fixed relative trajectories and therefore, reduce the global coordination requirements. The proposed approach leads to local control laws that verify global emergent behaviors specified as convergence to a particular manifold. A generalized analysis of such control approach by using tools of partial contraction theory is performed, producing important convergence results. By applying and extending results from the theory of partially contracting systems, an approach to deriving sufficient conditions for convergence is formulated. Its use is demonstrated by analyzing several examples and obtaining global convergence results for nonlinear, time varying and more complex interconnected distributed controllers. Experimental results of the implementation of these algorithms were obtained using the SPHERES testbed on board the International Space Station, validating many of the important properties of this decentralized control approach. They are believed to be the first implementation of decentralized formation flight in space. To complement the results we also consider a short analysis of the advantages of decentralized versus centralized approach by comparing the optimal performance and the effects of complexity and robustness for different architectures and address the issues of implementing decentralized algorithms in a inherently coupled system like the Electromagnetic Formation Flight.by Jaime Luís Ramírez Riberos.Ph.D

Problem of uniform deployment on a line segment for second-order agents

Consideration was given to a special problem of controlling a formation of mobile agents, that of uniform deployment of several identical agents on a segment of the straight line. For the case of agents obeying the first-order dynamic model, this problem seems to be first formulated in 1997 by I.A. Wagner and A.M. Bruckstein as "row straightening." In the present paper, the straightening algorithm was generalized to a more interesting case where the agent dynamics obeys second-order differential equations or, stated differently, it is the agent's acceleration (or the force applied to it) that is the control