Coordination of groups of mobile autonomous agents using nearest neighbor rules
Summary (2 min read)
I. INTRODUCTION
- I N [1], Vicsek et al. propose a simple but compelling discrete-time model of autonomous agents (i.e., points or particles) all moving in the plane with the same speed but with different headings.
- Included here is the work of Czirok et al. [3] who propose one-dimensional models which exhibit the same type of behavior as Vicsek's.
- Meanwhile, in modeling biological systems, Grünbaum and Okubo use statistical methods to analyze group behavior in animal aggregations [8] .
- The th follower updates its heading just as in the Vicsek model, using the average of its own heading plus the headings of its neighbors.
- The authors prove that the headings of all agents must converge to the leader's provided all agents are " linked to their leader" together via their neighbors frequently enough as the system evolves.
II. LEADERLESS COORDINATION
- The system studied by Vicsek et al. [1] consists of autonomous agents (e.g., points or particles), labeled 1 through , all moving in the plane with the same speed but with different headings.
- It is natural to say that the agents under consideration are linked together across a time interval if the collection of graph encountered along the interval, is jointly connected.
- Let be fixed and let be a switching signal for which there exists an infinite sequence of contiguous, nonempty, bounded, time-intervals , , starting at , with the property that across each such interval, the agents are linked together, also known as Theorem 2.
- Such and are examples of "primitive matrices" where by a primitive matrix is meant any square, nonnegative matrix for which is a matrix with all positive entries for sufficiently large [28] .
- It is worth noting that existence of a common quadratic Lyapunov function for all discrete time stable matrices in some given finite set , is a much stronger condition than is typically needed to guarantee that all infinite products of the converge to zero.
III. LEADER FOLLOWING
- The authors consider a modified version of Vicsek's discrete-time system consisting of the same group of agents as before, plus one additional agent, labeled 0, which acts as the group's leader.
- Like before, each of these relationships can be conveniently described by a simple undirected graph.
- In other words, the agents are linked to their leader across an interval just when the -member group consisting of the agents and their leader is linked together across .
- In the sequel, the authors outline several ideas upon which the proof of Theorem 4 depends.
- To proceed, the authors need a few more ideas concerned with nonnegative matrices.
IV. LEADER FOLLOWING IN CONTINUOUS TIME
- The authors aim here is to study the convergence properties of the continuous-time version of the leader-follower model discussed in the last section.
- Where now takes values in the real half interval .
- In the sequel the authors will analyze controls of this form subject to two simplifying assumptions.
- Second the authors assume the agents are synchronized in the sense that for all and all .
- Much like before, their goal must also be nonnegative with positive diagonal elements.
V. CONCLUDING REMARKS
- As stated in the abstract, the main objective of this paper has been to provide a theoretical explanation for behavior observed in the simulation studies reported in [1] .
- The theorems in this paper all provide convergence results for rich classes of switching signals and arbitrary initial heading vectors.
- On the other hand, with widely distributed initial agent positions and very small, one would expect to see a bifurcation of the group into distinct subgroups with different steady state headings.
- Of course issues along these lines would not arise at all if the systems we've considered were modeling other physically significant variables such as agent speed or temperature where one could take all of rather than just as the set in which the take values.
- Nonetheless, these models do seem to exhibit some of the rudimentary behaviors of large groups of mobile autonomous agents and for this reason they serve as a natural starting point for the analytical study of more realistic models.
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Cites background or methods from "Coordination of groups of mobile au..."
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Frequently Asked Questions (6)
Q2. What is the meaning of the union of a collection of graphs?
By the union of a collection of simple graphs, , each with vertex set , is meant the simple graph with vertex set and edge set equaling the union of the edge sets of all of the graphs in the collection.
Q3. What is the meaning of the definition of a simple graph?
By the intersection of a collection of simple graphs, , each with vertex set , is meant the simple graph with vertex set and edge set equaling the intersection of the edge sets of all of the graphs in the collection.
Q4. What is the form of the update equations?
The explicit form of the update equations exemplified by (38), depends on the relationships between neighbors which exist at time .
Q5. What is the simplest way to show that a group of agents is linked to its leader?
Then(40)The theorem says that the members of the -agent group all eventually follow their leader provided there is a positive integerwhich is large enough so that the -agent group is linked to its leader across each contiguous, nonempty time-interval of length at most .
Q6. What is the definition of a tight condition for a set of matrices?
This condition is tight in the sense that one can find a finite set of matrices with joint spectral radius , whose infinite products converge to zero despite the fact that there does not exist common quadratic Lyapunov function for the set.