1. Analytic Synchronization
A concept for understanding emergent behaviors in
asynchronous multi-agent systems
A. S. Morse
Yale University
Napa Valley
August 2, 2005

2. Research Objectives SWARMS
To study multi-agent motion control problems with the objective of better
understanding the process of devising provably correct local control
strategies which can coordinate and otherwise induce desired emergent
{group-wide} behaviors.
Formation control: With the objective of understanding how to form, maintain,
and otherwise manage a formation of mobile autonomous agents, we are actively
studying a variety of algorithms based on the classical concept of graph rigidity.
Much of this research has been done with Brian Anderson who will be giving a talk
at the workshop on this subject.
Sensor localization: As a direct spin-off from our original work formation control,
we are also actively studying how global graph rigidity can be used to advantage
to deal with long-standing localization problems which arise in sensor networks.
Distributed control: We are heavily involved in the study of high-level distributed
control strategies of all types with the goal of understanding how to analyze and
synthesize them. Earlier work with Ali Jadbabaie on heading synchronization
as well as more recent work on rendezvousing are representative directions.
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3. We will talk about two problems:
We have been specifically interested in the development of techniques for
analyzing asynchronous multi-agent systems. One such technique, called
analytic synchronization, will be outlined later in the talk.
We will talk about two problems:
The Asynchronous Rendezvous Problem
The Asynchronous Heading Synchronization Problem
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4. The Asynchronous Rendezvous Problem
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5. Unsynchronized Strategies !
The Asynchronous Rendezvous Problem
Consider a set of n mobile autonomous agents which can all move in the plane.
Agents
Each agent is able to continuously sense the relative positions of all other agents in
its “sensing region” where by agent i’s sensing region is meant a closed disk of
radius r centered at agent i’s current position.
agent i r
sensing region
Problem: Devise local control strategies, one for each agent, which without active
communication between agents, cause all members of the group to
eventually rendezvous at a single unspecified point.
Synchronized strategies proposed in
Unsynchronized Strategies !
H. Ando, Y. Oasa, I. Suzuki, M. Yamashita, October 1999 IEEE Trans. Rob. & Auto.
Lin, Morse, Anderson, CDC 2003
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6. Stop and Go Maneuvers
{of Agent i}
A stop and go maneuver takes place within a time interval consisting of two
successive subintervals:
fixed positive numbers
sensing period
maneuvering period
Each agent is stationary during each of its sensing periods.
During a maneuvering period, an agent moves from its current rest
position to its next “way-point” and again comes to rest.
Successive way-points for each agent are constrained to be within r
units of each other.
is an upper bound on the time it takes for agent i to move
from one way-point to the next.
Details of maneuvering to way-points not considered.
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7. successive subintervals:
Stop and Go Maneuvers
{of Agent i}
A stop and go maneuver takes place within a time interval consisting of two
successive subintervals:
sensing period
maneuvering period
tik
kth
Agent i’s event times: ti1, ti2, ….
Not synchronized with other agents’ event times
A design parameter called a sensing time S is chosen to satisfy
Registered neighbors of agent i at event time tik are those agents which
during agent i’s kth sensing period are stationary for at least S seconds at
positions within agent i’s sensing region.
It is assumed that no agent stops for as long as S seconds during
any of its maneuvering periods.
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8. during which agent j is stationary.
sensing period
maneuvering period
tik
kth
Registered neighbor is a symmetric relation: If agent j is a registered
neighbor of agent i at tik and q is the sensing/maneuvering period of agent j
during which registration takes place, then agent i is a registered neighbor
of agent j at tjq.
Registered position of neighbor j at tik is the position of agent j during
the last interval of at length at least S within agent i’s kth sensing period,
during which agent j is stationary.
Registered neighbors of agent i at event time tik are those agents which
during agent i’s kth sensing period are stationary for at least S seconds at
positions within agent i’s sensing region.
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9. j i SWARMS Retaining Neighbors
Agent i is said to satisfy the motion constraint induced by neighbor j at time tik
if the point to which agent i moves at the end of its kth maneuvering period is
within a closed disk of diameter r centered at the mean of its position at tik and the
registered position of neighbor j at tik. i j
Neighbor Retention: Suppose that agents i and j satisfy the motion constraints induced
by their registered neighbors. If agent j is a registered neighbor of agent i at tik,
the agent j is also a registered neighbor of agent i at ti(k+1)
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10. SWARMS Retaining Neighbors
Agents is said to satisfy the motion constraint induced by neighbor j at time tik
if the point to which agent i moves at the end of its kth maneuvering period is
within a closed disk of diameter r centered at the mean of its position at tik and the
registered position of neighbor j at tik.
Cooperation Assumption: Each agent satisfies the motion constrains induced
by each of its registered neighbors.
Neighbor Retention: Suppose that agents i and j satisfy the motion constraints induced
by their registered neighbors. If agent j is a registered neighbor of agent i at tik,
the agent j is also a registered neighbor of agent i at ti(k+1)
Implies that each agent will retain all of its current registered neighbors forever.
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11. SWARMS sensing region constraint disc Agent A neighbors
Agent A’s Motion Constraints Induced by its Neighbors
sensing region
constraint disc
Agent A
neighbors
constraint disc
constraint set
constraint disc
constraint disc
Either contains just agent A or is strictly convex with nonempty interior
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12. SWARMS Way Point Selection Rules
By the kth local convex hull of agent i is meant the convex hull of the set consisting
of agent i’s position at time tik and the registered positions of agent i’s neighbors
at time tik.
Agent i’s kth way point
1. must be the same as its position at time tik if agent i has no neighbors
at time tik.
2. must be within its kth local convex hull.
3. must not be a corner of its kth local convex hull unless the local convex hull
is a single point.
4. must be within the motion constraint sets induced by its neighbors.
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13. SWARMS convex hull target constraint set 0 is in the Interior of

14. SWARMS convex hull target  constraint set 0 is on Boundary of

15. MAIN RESULT
Despite the lack of synchronization it is possible to define a simple undirected
graph at each event time tik of each agent which characterizes which agents
are in which other agents’ sensing regions at time tik.
It can be shown that if there is ever an event time of any one agent at which this
neighbor graph is connected, then the use of the aforementioned type of strategy
by all n agents will cause them to eventually rendezvous at one point.
Proving that this is so is challenging because the n agent system under
consideration is asynchronous.
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16. Rendezvousing Asynchronously
Each agent’s behavior can be modeled as a system Ai evolving on its event time
sequence ti1, ti2,…. where, as we said before, tik is the time at which agent i’s kth
maneuvering period begins.
The n interacting Ai constitute the asynchronous multi-agent system of interest.
Since event time sequences of different agents are not synchronized, standard
dynamical systems analysis tools cannot be directly applied.
ANALYTIC SYNCHRONIZATION
The n agent asynchronous system is analyzed by “embedding” all the Ai
into a single synchronous system S evolving on the time set T which results when
the n individual agent event time sequences are merged into a single ordered set.
This is accomplished by appropriately extending the domain of definition
of each agent subsystem Ai from the event time sequence ti1, ti2,… to all of T.
Convergence is then established using familiar ideas applicable to S.
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17. We will talk about two problems:
The Asynchronous Rendezvous Problem
The Asynchronous Heading Synchronization Problem
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18. The Asynchronous Heading Synchronization Problem
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19. Motivated by simulation results reported in the paper:
The Asynchronous Heading Synchronization Problem
Motivated by simulation results reported in the paper:
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20. A theoretical explanation for this observed behavior can be found in
Vicsek et al. simulated a flock of n agents {particles} all moving in
the plane at the same speed s, but with different headings 1, 2, …. n
i = heading qi
s = speed s
Each agent’s heading is updated using a local rule based on the average
of its own current heading plus the headings of its “neighbors.”
Vicsek’s simulations demonstrated that these nearest neighbor rules can
cause all agents to eventually move in the same direction despite the
absence of a leader and/or centralized coordination and despite the fact
that each agent’s set of neighbors changes with time.
A theoretical explanation for this observed behavior can be found in
Jadbabaie, Lin & Morse, IEEE TAC, June 2003
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21. ri neighbors of agent i agent i SWARMS
each agent is a neighbor of itself
neighbors of
agent i ri
agent i
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22. HEADING UPDATE EQUATIONS
s = speed s
 i = heading
 i
Average at time t of headings of neighbors of agent i.
Ni (t) = set of indices of agents i’s “neighbors” at time t
ni(t) = number of indices in Ni(t)
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23. SWARMS Jadbabaie, Lin & Morse, IEEE TAC, June 2003
Interesting property: Neighbor graphs change with time
Feature of the original Vicsek model considered in the above paper
No leader
All ri = r.
Synchronous operation
No delays in sensing of headings
Main technical tools exploited
Simple {undirected} graphs to describe neighbor relationships
Algebraic graph theory
A theorem of J. Wolfowitz in Proc. AMS, 1963 which gives conditions for an
infinite product of left stochastic matrices to converge to a rank one matrix.
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24. Subsequent research by B. Francis, L. Moreau, V. Blondell, J
Subsequent research by B. Francis, L. Moreau, V. Blondell, J. Tsitsiklis, D. Angeli,
D. Spielman, M. Cao, B. Anderson, G. Tanner, G. Pappas, R. Beard, and others.
No leader
convergence rates
ri  r
All ri = r.
sensing delays
Synchronous operation
No delays in sensing of headings
asynchronous operation
Additional technical tools exploited include
directed graphs
scrambling matrices, Sarymsakov matrices, and random walks
a special partial Lyapunov function tailored for Markov Chains –
Doob {1953}, Senta {1981}, Tsisiklis {1984}, Bertsekas & Tsisiklis {1989}.
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25. The Asynchronous Vicsek Flocking Problem
A modified version of the Vicsek flocking problem in which
each agent updates its heading at times determined by its own clock.
Groups’ clocks are not assumed to be synchronized together.
The times at which any one agent updates its heading are not assumed
to be evenly spaced.
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26. The Asynchronous Vicsek Model
{Here t denotes real continuous time}
There are n agents labeled 1 through n.
Each agent i maintains a constant heading i(t) for ti(k -1) < t · tik, k ¸ 1
where ti0 = 0 and tik is agent i’s kth update event time.
Event time sequences are not assumed to be synchronized
qi = heading qi
s = speed s
Ni (tik) = set of indices of agents i’s neighbors including itself, at time tik
ni(tik) = number of indices in Ni(tik)
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27. Flocking Asynchronously
Each agent’s behavior can be modeled as a system Ai evolving on its event time
sequence ti1, ti2,…. where, as we said before, tik is the time at which agent i
instantaneously changes its heading from one value to the next.
The n interacting Ai constitute the asynchronous multi-agent system of interest.
Since event time sequences of different agents are not synchronized, standard
dynamical systems analysis tools cannot be directly applied.
ANALYTIC SYNCHRONIZATION
The n agent asynchronous system is analyzed by “embedding” all the Ai
into a single synchronous system S evolving on the index set of the time set T
which results when the n individual agent event time sequences are merged into
a single ordered set.
This is accomplished by appropriately extending the domain of definition
of each agent subsystem Ai from the event time sequence ti1, ti2,… to all of T.
Convergence is then established using familiar ideas applicable to S.
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28. SWARMS agent 1 t11 t12 t15 t16 t13 t14 interacting agent 2 t26 t25 t24

29. SWARMS agent 1 T = t3 t5 t10 t13 t7 t8 t11 t9 t6 t1 t12 t4 t2 t11 t12

30. SWARMS Analytic Synchronization
Let T = set of all event times tik of all n agents.
Re-label elements of T as t0, t1, t2, … so that t + 1 > t  = 0, 1, 2,...
Define
Since
it must be true for any value of  for which t is an event time of agent i, that
where  0 is such that t 0 = next event time of agent i after t .
because i(t) is constant for t < t · t 0 .
But
Therefore
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31. SWARMS At this point: if t is an event time of agent i
But i(t) is constant between event times so
if t is not an event time of agent i
So if we define
if t is not an event time of agent i
then
if t is not an event time of agent i
Therefore
all  ¸ 0
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32. SWARMS Summary Let T = set of all event times tik of all n agents.
Re-label elements of T as t0, t1, t2, … so that t + 1 > t  = 0, 1, 2,...
At its event times, agent i has the same neighbors as before embedding
if t is an event time of agent i
Between its event times, agent i has only itself as a neighbor
if t is not an event time of agent i
all  ¸ 0
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33. P = index set of all possible neighbor configurations.
V = agent index set ={1, 2, …, n}
For each p 2 P
Gp = {V, Ap} - a directed graph with vertex set V and arc set Ap
(i, j) 2 A p if agent i is a neighbor of agent j in configuration p
All vertices have self-arcs 7 4 1 3 5 2 6
(1,2)
G = set of all directed graphs
with vertex set V and self-arcs at
all vertices
Since agent i has no neighbors other
than itself between its event times,
at such times vertex i has only one
incoming arc.
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34. Matrix Representation of Gp = {V, Ap }
adjacency matrix Ap =[aij]n£ n
aij(p) = 1 if i is a neighbor of j
aij(p) = 0 otherwise
Dp = diagonal {d1(p), d2(p), …., dn(p)}n£n
di(p) = in-degree of vertex i 7 4 1 3 5 2 6
(1,2)
in-degree = 1
in-degree = 2
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35. adjacency matrix Ap =[aij]n £ n
State Space Equation
adjacency matrix Ap =[aij]n £ n
aij(p) = 1 if i is a neighbor of j
aij(p) = 0 otherwise
Dp = diagonal {d1(p), d2(p), …., dn(p)}n £ n 7 4 1 3 5 2 6
(1,2)
s() = index in P of neighbor
configuration at time .
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36. SWARMS Switching Among Neighbor Graphs
if t is an event time of agent i
if t is not an event time of agent i
Constraints induced by analytic synchronization
If t is not an event time of agent i, then vertex i in G() cannot have any incoming
arcs other than its own.
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37. SWARMS Some Graph Concepts
Call a graph G 2 G rooted if there is at least one vertex v for which, for each vertex i,
there is a directed path from v to i.
Rooted graphs are important because if the graphs in the sequence G(0), G(0) ... encountered along a system trajectory are rooted, then convergence to a common heading occurs.
Although rooted graphs could easily arise along a trajectory in the synchronous
flocking problem, in the unsynchronized version of the problem under consideration
here, this can never occur except possibly in the highly unusual situation when
different agents have some of their event times occurring at exactly the same instant.
By the composition of a graph G1 with a graph G2, written G2 ± G1, is meant that graph
which has an arc from i to j just in case there is a vertex k for which G1 has an arc from
i to k and G2 has an arc from k to j.
Call a finite sequence of graphs G1, G2,….Gm in G jointly rooted, if the composition
Gm± ± G1 is a rooted graph.
Call an infinite sequence of graphs G1,G2,….Gj, … in G repeatedly jointly rooted,
if there is a finite integer m such that each successive subsequence
Gm(k-1) +1, G(m(k-1)+2), … Gmk, k ¸ 1, is jointly rooted.
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38. Reaching a Common Heading Asynchronously
If the sequence of graphs G (0), G(1),... encountered along a trajectory is
repeatedly jointly rooted, then
at an exponentially fast convergence rate.
The hypothesis above is not vacuous: Despite the constraints imposed
on the neighbor sets by analytic synchronization, there are in fact plenty of
trajectories of the asynchronous system of interest which generate sequences of
graphs which are repeatedly jointly rooted.
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39. Observations
New data structures, models, etc. are needed to represent large groups
of mobile autonomous agents and sensors at various degrees of granularity,
for purposes of simulations, management, analysis and control.
Such representations will exploit tools from both graph theory and
from the theory of dynamical systems
At least initially, individual agent/sensor descriptions using simple
kinematic and dynamic models will suffice.
System complexity will stem more from the number of agent/sensor models
being studied than from the detailed properties of the individual
agent models.
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40. SWARMS