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Are energy savings the only reason for the emergence of bird echelon formation?
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Abstract
We analyze the conditions under which the emergence of frequently observed echelon formation can be explained solely by the maximization of energy savings. We consider a two-dimensional multi-agent echelon formation, where each agent receives a benefit that depends on its position relative to the others, and adjusts its position to increase this benefit. We analyze the selfish case where each agent maximizes its own benefit, leading to a Nash-equilibrium problem, and the collaborative case in which agents maximize the global benefit of the group. We provide conditions on the benefit function under which the frequently observed echelon formations cannot be Nash equilbriums or group optimums. We then show that these conditions are satisfied by the conventionally used fixed-wing wake benefit model. This implies that energy saving alone is not sufficient to explain the emergence of the migratory formations observed, based on the fixed-wing model. Hence, either non-aerodynamic aspects or a more accurate model of bird dynamics should be considered to construct such formations.
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This note analyzes the stability properties of a group of mobile agents that align their velocity vectors, and stabilize their inter-agent distances, using decentralized, nearest-neighbor interaction rules, exchanging information over networks that change arbitrarily (no dwell time between consecutive switches). These changes introduce discontinuities in the agent control laws. To accommodate for arbitrary switching in the topology of the network of agent interactions we employ nonsmooth analysis. The main result is that regardless of switching, convergence to a common velocity vector and stabilization of inter-agent distances is still guaranteed as long as the network remains connected at all times.
This is the revised second edition of our 1982 book with the same title, which presents a rather comprehensive treatment of static and dynamic noncooperative game theory, with emphasis placed (as in the first and second editions) on the interplay between dynamic information patterns and the structural properties of several different types of equilibria. Whereas the second edition (1995) was a major revision with respect to the original edition, this Classics edition only contains some moderate changes with respect to the second one. There has been a number of reasons for the preparation of this edition:
• The second edition was sold out surprisingly fast.
• After some fifty years from its creation, the field of game theory is still very alive and active, as also reinforced by the selection of three game theorists (John Harsanyi, John Nash and Reinhard Selten) to share the 1994 Nobel prize in economics. Quite a few books on game theory have been published during the last ten years or so (though most of them essentially deal with static games only and are at the undergraduate level).
• The recent interest in such fields as biological games, mathematical finance and robust control gives a new impetus to noncooperative game theory.
• The topic of dynamic games has found its way into the curricula of many universities, sometimes as a natural supplement to a graduate level course on optimal control theory, which is actively taught in many engineering, applied mathematics and economics graduate programs.
• At the level of coverage of this book, dynamic game theory is well established by now and has reached a level of maturity, which makes the book a timely addition to SIAM's prestigious Classics in Applied Mathematics series.
The organized flight of birds is one of the most easily observed, yet challenging to study, phenomena in biology. Birds that fly in organized groups generally do so in one of two fashions: Line formations and Cluster formations. The former groups are typical of large birds such as waterfowl, where birds fly arranged in single lines, often joined together. The scientific questions about these groups usually involve potential adaptive functions, such as why geese fly in a V. Cluster formations are typically made up of large numbers of smaller birds such as pigeons or starlings flying in more irregular arrangements that have a strong three-dimensional character. The groups are defined by synchronized and apparently simultaneous rapid changes in direction. Scientific questions about these groups are usually concerned with mechanism such as how synchrony is achieved. Although field observations about the phenomenon date to the origins of natural history, experimental studies did not begin until the 1970s. Early experimenters and theoreticians were primarily biologists, but more recently aeronautical engineers, mathematicians, computer scientists and, currently, physicists have been attracted to the study of organized flight. Computer modelling has recently generated striking visual representations of organized flight and a number of hypotheses about its functions and mechanisms, but the ability to test these hypotheses lags behind the capacity to generate them. We suggest that a multi disciplinary approach to the phenomenon will be necessary to resolve apparently conflicting current hypotheses.
Fifty-four skeins of pink-footed geese (Anser brachyrhynchus) were photographed from directly underneath to eliminate the effects of perspective distortion, and the wing-tip spacings (the distance between adjacent birds' wing tips perpendicular to the flight path at maximum wingspan) and depths (the distance between adjacent birds' body centres parallel to the flight path) were measured at the same time as local wind speeds. The photographs were used to test for savings in induced power from wing positioning relative to the predicted positions of vortices generated by other wings, using a theoretical model. The mean wing-tip spacing corresponded to a saving in induced power of 14 %, less than one-third of the maximum possible. The saving in total power might be as low as 2.4 %. The high variation in wing-tip spacing suggests that pink-footed geese found difficulty maintaining position and thus adopted a strategy of flying outboard of the optimal position that maximises savings. This may minimise the risk of straying into a zone where savings are negative. There was a significant correlation between depth and wing-tip spacing, supporting an alternative communication hypothesis, whereby the birds position themselves to obtain maximum information on their neighbour's position. In high winds, there was little change in wing-tip spacing variation but a decrease in depth variation, suggesting a shift towards more regularly spaced skeins.
Motivated by the challenging formation stabilization problem for mobile robotic teams wherein no distance or relative position measurements are available but each robot can only measure some of relative angles with respect to its neighbors in its local coordinate frame, we develop the notion of "angle rigidity" for a multi-point framework, named "angularity", consisting of a set of nodes embedded in a Euclidean space and a set of angle constraints among them. Different from bearings or angles defined in a global frame, the angles we use do not rely on the knowledge of a global frame and are signed according to the counter-clockwise direction. Here angle rigidity refers to the property specifying that under proper angle constraints, the angularity can only translate, rotate or scale as a whole when one or more of its nodes are perturbed locally. We first demonstrate that this angle rigidity property, in sharp comparison to bearing rigidity or other reported rigidity related to angles of frameworks in the literature, is not a global property since an angle rigid angularity may allow flex ambiguity. We then construct necessary and sufficient conditi
We present a survey of formation control of multi-agent systems. Focusing on the sensing capability and the interaction topology of agents, we categorize the existing results into position-, displacement-, and distance-based control. We then summarize problem formulations, discuss distinctions, and review recent results of the formation control schemes. Further we review some other results that do not fit into the categorization.
A constrained n-person game is considered in which the constraints for each player, as well as his payoff function, may depend on the strategy of every player. The existence of an equilibrium point for such a game is shown. By requiring appropriate concavity in the payoff functions a concave game is defined. It is proved that there is a unique equilibrium point for every strictly concave game. A dynamic model for nonequilibrium situations is proposed. This model consists of a system of differential equations which specify the rate of change of each player's strategy. It is shown that for a strictly concave game the system is globally asymptotically stable with respect to the unique equilibrium point of the game. Finally, it is shown how a gradient method suitable for a concave mathematical programming problem can be used to find the equilibrium point for a concave game.
This paper studies the distributed control of bearing-constrained multi-agent
formations using bearing-only measurements. In order to analyze
bearing-constrained formations, we first present a bearing rigidity theory that
is applicable to arbitrary dimensions. Based on the proposed bearing rigidity
theory, we analyze two bearing-only formation control problems. In the first,
each agent can measure the relative bearings to their neighbors in a global
reference frame, while in the second problem, each agent can only measure the
bearings and relative orientations of their neighbors in their local frames.
For the two problems, we propose distributed bearing-only control laws and
prove almost global stability for infinitesimally bearing rigid formations.
Formation flight is widely used by migrating birds. Each wing flies in an upwash field generated by all other wings in the formation, and this leads to a reduction in flight power demand for each wing as well as for the whole formation. The benefits of aerodynamic interference can be determined from aerodynamic theory, and comparisons with flight tests on formations of two airplanes show excellent agreement. For maximum power reduction, the rear wing has to be located as close as possible aside the wake of the front wing. For formations in a horizontal plane, the total power reduction of the whole formation depends strongly on the number and the lateral distance of the wings. A longitudinal displacement of the wings in flight direction has no influence on the total flight power reduction but only on its distribution among the involved individuals. The local flight power reduction is highest in the inner parts of the formation and decreases towards the apex and the side edges. Small and light individuals experience larger benefits than larger and heavier birds. The considerable benefit of saving energy by aerodynamic interference is thought to be the most important reason for the occurrence of formation flight of large birds.
Locomotion efficiency for a variety of animals may be improved by group movement in specific formations. A theory of formation flight developed by Lissaman & Shollenberger (1970) predicts birds may obtain energy savings through the use of a vee formation. Measurements of Canada geese (Branta canadensis) (Gould < Heppner, 1974) suggested an aerodynamic advantage was unlikely. However, further analysis of this data to obtain the critical formation parameter (wing tip spacing) suggests the theory may apply. The theory is modified to better account for the characteristics of a vee formation and adapted to the specific case of the Canada goose. The relationships between observed and optimal formations suggest that energy savings could be an important function of vee formations.
This study investigates the influence, on aircraft wake vortex behavior, of atmospheres that are stably stratified (neutral to very strong) and weakly turbulent, by means of large-eddy simulations at very high Reynolds number and on relatively fine grids. The atmospheric fields are first generated using large-eddy simulations of forced and stratified turbulence reaching a statistically stationary state. The obtained fields are shown to be realistic and consistent with the literature. A pair of counter-rotating vortices, with relatively tight cores, is then put in the obtained fields and evolved. The evolution of the wake vortex topology, transport, and decay is analyzed in depth by measuring the wake vortex characteristics in all cross planes. The vortex transport and deformation are related to the stratification and turbulence levels. Stratification combined with weak turbulence is seen to strongly affect the Crow instability development. Different decay mechanisms are identified, related to the interactions with the turbulence, the turbulent baroclinic vorticity, and/or between the vortices themselves. Finally, improved simplified models of vortex altitude evolution and of vortex decay (with two phases) are proposed and calibrated on the present results. They yield good agreement with the large-eddy simulation results and are usefully integrated in our operational models.
We consider in this paper formations of autonomous agents moving in a two-dimensional space. Each agent tries to maintain its distances toward a pre-specified group of other agents constant and the problem is to determine if one can guarantee that the distance between every pair of agents (even those not explicitly maintained) remains constant, resulting in the persistence of the formation shape. We provide here a theoretical framework for studying this problem. We describe the constraints on the distance between agents by a directed graph and define persistent graphs. A graph is persistent if the shapes of almost all corresponding agent formations persist. Although persistence is related to the classical notion of rigidity, these are two distinct notions. We derive various properties of persistent graphs, and give a combinatorial criterion to decide persistence. We also define minimal persistence (persistence with the least possible number of edges), and we apply our results to the interesting special case of cycle-free graphs. Copyright © 2006 John Wiley & Sons, Ltd.
This article sets out the rudiments of a theory for analyzing and creating architectures appropriate to the control of formations of autonomous vehicles. The theory rests on ideas of rigid graph theory, some but not all of which are old. The theory, however, has some gaps in it, and their elimination would help in applications. Some of the gaps in the relevant graph theory are as follows. First, there is as yet no analogue for three-dimensional graphs of Laman's theorem, which provides a combinatorial criterion for rigidity in two-dimensional graphs. Second, for three-dimensional graphs there is no analogue of the two-dimensional Henneberg construction for growing or deconstructing minimally rigid graphs although there are conjectures. Third, global rigidity can easily be characterized for two-dimensional graphs, but not for three-dimensional graphs.
Flocks of birds self-organize into V-formations when they need to travel long distances. It has been shown that this formation allows the birds to save energy, by taking advantage of the upwash generated by the neighboring birds. In this work we use a model for the upwash generated by a flying bird, and show that a flock of birds can self-organize into a V-formation if every bird were to process spatial and network information through an adaptive diffusive process. The diffusion algorithm requires the birds to obtain measurements of the upwash, and also to use information from neighboring birds. The result has interesting implications. First, a simple diffusion algorithm can account for self-organization in birds. The algorithm is fully distributed and runs in real time. Second, according to the model, that birds can self-organize based on the upwash generated by the other birds. Third, that some form of information sharing among birds is necessary to achieve flight formation. We also propose a modification to the algorithm that allows birds to organize into a U-formation, starting from a V-formation. We show that this type of formation leads to an equalization effect, where every bird in the flock observes approximately the same upwash.
In formation flight each wing flies in an upwash field generated by all other wings of the formation. This leads to a reduction in flight power demand for each wing as well as for the whole formation.Methods of theoretical aerodynamics are used to calculate the flight power reduction for arbitrarily shaped flight formations with any number of birds. These methods are applied to homogeneous and inhomogeneous flight formations in which birds of the same kind or birds of different span, aspect ratio and weight may be present.The total flight power reduction of the whole formation strongly depends on the lateral distance of the wings. A longitudinal displacement of the wings in flight direction has no influence on the total flight power reduction but only on their distribution on the involved individuals. The local flight power reduction is highest in the inner parts of the formation and decreases towards the apex and towards the side edges of the formation. Small and light individuals are automatically favoured by larger and heavier birds. It is shown that some minor portion of twist is necessary to fly in a formation without a rolling moment. In addition it turns out that the optimum flight speed of a formation is slightly lower than the optimum flight speed of single individuals.
This paper describes a distributed coordination scheme with local information exchange for multiple vehicle systems. We introduce second-order consensus protocols that take into account motions of the information states and their derivatives, extending first-order protocols from the literature. We also derive necessary and sufficient conditions under which consensus can be reached in the context of unidirectional information exchange topologies. This work takes into account the general case where information flow may be unidirectional due to sensors with limited fields of view or vehicles with directed, power-constrained communication links. Unlike the first-order case, we show that having a (directed) spanning tree is a necessary rather than a sufficient condition for consensus seeking with second-order dynamics. This work focuses on a formal analysis of information exchange topologies that permit second-order consensus. Given its importance to the stability of the coordinated system, an analysis of the consensus term control gains is also presented, specifically the strength of the information states relative to their derivatives. As an illustrative example, consensus protocols are applied to coordinate the movements of multiple mobile robots. Copyright © 2006 John Wiley & Sons, Ltd. Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/56082/1/1147_ftp.pdf
An approximate analysis of atmospheric effects on wake vortex motion and decay is presented. The effects of density stratification, turbulence, and Reynolds number are combined in a single model so that the relative importance of different parameters can be estimated. Predicted wake motion is shown to be in good agreement with limited data from ground facility and flight test measurements taken under low-turbulence conditions. Wake decay was found to depend strongly on both density stratification and turbulence. For typical levels of turbulence, wake decay was found to result from the Crow instability except under strongly stratified conditions.
Many species of large bird fly together in formation, perhaps because flight power demands and energy expenditure can be reduced when the birds fly at an optimal spacing, or because orientation is improved by communication within groups. We have measured heart rates as an estimate of energy expenditure in imprinted great white pelicans (Pelecanus onocrotalus) trained to fly in 'V' formation, and show that these birds save a significant amount of energy by flying in formation. This advantage is probably a principal reason for the evolution of flight formation in large birds that migrate in groups.
This paper considers two cooperative control problems for nonholonomic mobile agents. In the first problem, we discuss the design of cooperative control laws such that a group of nonholonomic mobile agents cooperatively converges to some stationary point under various communication scenarios. Dynamic control laws for each agent are proposed with the aid of sigma-processes and results from graph theory. In the second problem, we discuss the design of cooperative control laws such that a group of mobile agents converges to and tracks a target point which moves along a desired trajectory under various communication scenarios. By introducing suitable variable transformations, cooperative control laws are proposed. Since communication delay is inevitable in cooperative control, in each of the above cooperative control problems, we analyze the effect of delayed communication on the proposed controllers. As applications of the proposed results, formation control of wheeled mobile robots is discussed. It is shown that our results can be successfully used to solve formation control problem. To show effectiveness of the proposed approach, simulation results are included.
In this paper, we present a theoretical framework for design and analysis of distributed flocking algorithms. Two cases of flocking in free-space and presence of multiple obstacles are considered. We present three flocking algorithms: two for free-flocking and one for constrained flocking. A
comprehensive analysis of the first two algorithms is provided. We demonstrate the first algorithm embodies all three rules of Reynolds. This is a formal approach to extraction of interaction rules that lead to the emergence of collective behavior. We show that the first algorithm generically leads to regular fragmentation, whereas the second and third algorithms both lead to flocking. A systematic method is provided for construction of cost functions (or collective potentials) for flocking. These collective potentials penalize deviation from a
class of lattice-shape objects called alpha-lattices.
We use a multi-species framework for construction of collective potentials that consist of flock-members, or alpha-agents, and virtual agents associated with alpha-agents called beta- and gamma-agents. We show that the tracking/migration problem for flocks can be solved using an algorithm with a peer-to-peer architecture. Each node (or macro-agent) of this peer-to-peer network is the aggregation of all three species of agents. The implication of this fact is that ``flocks need no leaders''. We discuss what constitutes flocking and provide a universal definition of ``flocking'' for particle systems that has the same role as ``Lyapunov stability'' for nonlinear dynamical systems. By ``universal'', we mean independent of the method of
trajectory generation for particles. Various simulation results are provided that demonstrate the effectiveness of our novel algorithms and analytical tools. This includes performing 2-D and 3-D flocking, split/rejoin maneuver, and squeezing maneuver for 40 to 150 agents (e.g. particles and UAVs).
Flocks, herds and schools: A distributed behavioral model
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Novel type of phase transition in a system of self-driven particles
- T Vicsek
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T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, "Novel
type of phase transition in a system of self-driven particles," Physical
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Whose energy cost would birds like to save: a revisit of the migratory formation flight
- M Shi
- J M Hendrickx
M. Shi and J. M. Hendrickx, "Whose energy cost would birds like to
save: a revisit of the migratory formation flight," In preparation.