Counting Critical Formations on a Line

نویسندگان

  • Brian D. O. Anderson
  • Uwe Helmke
چکیده

Formation shape control for a collection of point agents is concerned with devising decentralized control laws which ensure that the formation will move so that certain interagent distances approximate prescribed values as closely as possible. Such laws are often derived using steepest descent of a potential function which is invariant under translation and rotation, and then critical formations are those that are fixed under the evolution of the decentralized control dynamics, i.e., those corresponding to equilibrium points of the control dynamics. Using a specific and frequently used potential function for formation control, this paper introduces tools from Morse theory and complex algebraic geometry to estimate the number of critical formations of N agents on a line. We show that there are at least 2N − 1 equilibrium points and at most 3N−1 isolated equilibria. Moreover, bounds on the number of equilibrium points with a k-dimensional stable manifold (the socalled Morse-index) are established. We show that generically there are exactly five critical formations for three agents on a line, and exactly 27 complex critical formations for four agents on a line, where a complex critical formation is defined as an equilibrium point of the gradient flow with complex, not necessarily real, coordinates. Except for a single critical formation, no two or more of the agents in the other 26 critical formations are collocated.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

The Application of fractal dimension and morphometric properties of drainage networks in the analysis of formation sensibility in arid areas (Case Study, Yazd-Ardakan Basin)

Introduction: Many natural phenomena have many variables that make it difficult to find relationships between them using common mathematical methods. This problem, along with the impossibility of measuring all elements of nature, has led to a major evolution in the way of understanding and explaining phenomena. In this way, one can use the fractal geometry with the theory that many natural phen...

متن کامل

Signatures of critical full counting statistics in a quantum-dot chain

We consider current shot noise and the full counting statistics in a chain of quantum dots which exhibits a continuous nonequilibrium phase transition as a function of the tunnel couplings of the chain with the electrodes. Using a combination of analytical and numerical methods, we establish that the full counting statistics is conventional away from the phase transition, but becomes, in a well...

متن کامل

IRWIN AND JOAN JACOBS CENTER FOR COMMUNICATION AND INFORMATION TECHNOLOGIES Animation of Flocks Flying in Line Formations

The coordinated flight of bird flocks is a pleasant and attractive sight. While most previous approaches have focused on animating cluster formations, this paper introduces a technique for animating flocks that fly in certain patterns (socalled line formations). We distinguish between the behavior of such flocks during initiation and their behavior during steady flight. We provide a biologicall...

متن کامل

Negotiated Formations

We present a decentralized, behavior-based approach to assembling and maintaining robot formations. Our approach dynamically grows formations from single robots into line segments and ultimately larger and more complex formations. Formation growth occurs by negotiation when robots are in close proximity. The approach has been validated using simulation as well as experiments with physical robot...

متن کامل

LINE-SEGMENTS CRITICAL SLIP SURFACE IN EARTH SLOPES USING AN OPTIMIZATION METHOD

In this paper the factor of safety (FS) and critical line-segments slip surface obtained by the Alternating Variable Local Gradient (AVLG) optimization method was presented as a new topic in 2D. Results revealed that the percentage of reduction in the FS obtained by switching from a circular shape to line segments was higher with the AVLG method than other methods. The 2D-AVLG optimization meth...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:
  • SIAM J. Control and Optimization

دوره 52  شماره 

صفحات  -

تاریخ انتشار 2014