IMA Short Course
Distributed Optimization and Control
Flocking Asynchronously
in Continuous Time
A. S. Morse
Yale University
University of Minnesota
June 2, 2014
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
s
s = speed
i
i = heading
at the
the same
same time
time as
as the
the rest
rest using a local rule
Each agent’s heading is updated at
based on the average of its own current heading plus the headings of its
“neighbors.”
Suppose agent’s clocks are not synchronized – what happens?
Update Rule for Agent i0s Heading µi
Agent i0s event times = 0; ti1; ti2; : : : assumed to satisfy
Event times not necessarily evenly spaced or synchronized with other agents
event times.
At tik agent i computes its kth way-point
Indices of neighbors
of agent i at time tik
and updates its heading monotonically on (tik ; ti(k+1)] from µi(tik ) to wi(tik ):
waypoint
i
ti1
ti2
ti3
ti4 ti5
Ni (tik) = set of labels of agent i’s neighbors at time tik
Thus each agent’s neighbors are defined at all of its own event times.
T = {0, t1 , t2, ... } ordered set of event times of all n agents.
To state the convergence result, we stipulate that each agent i has only itself as a
neighbor at each time in T which is not at event time of agent i.
This has no effect on the update rules.
Ni (t) = {i}
for any t 2 T which is not an event time of agent i.
Thus each Ni (t) is well defined for all t 2 T
Extended neighbor graph E(t) is the neighbor graph of index sets N1(t), N2(t), ..., Nn(t)
t 2 T.
3
1
4
2
Extended neighbor graph E(t) at a time t which is an event time of only agents
1 and 3.
Note that agents 2 and 4 have only themselves as neighbors.
CONVERGENCE
Synchronous Case: For any trajectory of the synchronous system
along which the sequence of neighbor graphs N(0), N(1), …. is repeatedly jointly rooted,
there is a constant ss to which each agent’s heading i converges exponentially fast.
Asynchronous Case: For any trajectory on T of the asynchronous system
i 2 {1, 2, … ,n}
along which the sequence of extended neighbor graphs E(0), E(t1), …. is repeatedly
jointly rooted, there is a constant ss to which each agent’s heading i converges
exponentially fast.
How can one prove this?
First develop a more explicit model
i
ti1
i
1
0
ti2
ti3
ti4 ti5
First develop a more explicit model
i
ti1
i
1
0
ti2
ti3
ti4 ti5
First develop a more explicit model
i
ti1
i
1
0
ti2
ti3
ti4 ti5
Can combine agent i’s two update equations to get the familiar update equation
This formula tells how i evolves only on agent i’s event time set.
But to use this formula we need to know values of the j at agent i’s event times
In the synchronous case where event times are the same for all agents,
the tik are independent of i, and the preceding update equations are sufficient.
For the asynchronous case a common time scale is needed …..
A Common Time Scale
T
= set of all event times tik of all n agents
Re-label the elements of T as t0, t1, t2, … where t0 = 0 and t < t +1 for  2 {0, 1, 2, …}
agent
1
t11
t12
t13
t14
t15
t16
interacting
agent
2
t21 t22
t23
t24
t25
t26 t27
agent
1
T =
agent
2
t21
t22
1
2
t11
3
t23
4
t12
5
t24
t13
6
7
t14
8
t25
9
t15
12 t16
11 t27
10 t26
13
Analytic Synchronization
The n mutually unsynchronized processes below, P1, P2, …Pn together constitute
the asynchronous system to be analyzed via “analytic synchronization.”
Merge all event time sequences into a single ordered sequence T.
Define the “synchronized state” of Pi at event times t 2 to be the original
unsynchronized state of Pi at these times plus possibly some additional variables.
At times in T between two successive event times in Ti, define the state
of Pi to be constant at the same value as at the first of these two event times.
Analyze the synchronous system S comprised of the n synchronized Pi
i 2 {1, 2, … ,n}
Synchronizing Pi
For all times tk 2 T = {t0 , t1, .... } between agent i’s qth and (q +1)th event times
tiq and ti(q+1) respectively, including time tiq, define
Can show that these variable evolve on all of T as
where Ti is the set of event times of agent i
Can you do this?
i 2 {1, 2, … ,n}
Defining the Synchronous System S Comprised of the n Synchronized Pi
stochastic matrix
Asynchronous flocking matrix
S
Asynchronous Flocking Matrices
R = set of all lists of n real numbers r = {r1, r2, …., rn} where ri 2 [0, 1]
B = set of all lists of n integers b = {b1, b2, …., bn} where bi 2 {0, 1}
Gsa = set of all self arced directed graphs with n vertices
It is possible to construct a function,
F : Gsa £ R £ B ! set of all 2n £ 2n stochastic matrices
which is continuous on R, such that
where
Note that the set of all asynchronous flocking matrices, namely image of F,
is compact because R is closed.
Example
Suppose tk = T is an event time of agents 2 and 3 in a 4 agent network
Suppose the extended neighbor graph E(T) is
1 & 4:
2 & 3:
µ1
µ2
µ3 µ4 w1 w2 w3 w4
7
3
= ° (F)
1
5
6
8
4
2
Not necessarily rooted
Vertices without self arcs
Summary
For all times tk 2 T = {t0 , t1, .... } between agent i’s qth and (q +1)th event times
tiq and ti(q+1) respectively, including time tiq, we defined
Then we defined
and asserted that
To prove that all i converge to a common heading ss can be shown to be
equivalent to proving that call 2n entries xi in x converge to ss.
Check this!
Thus as before the problem reduces to determining conditions under which
But the graphs of the F(k) do not necessarily have self arcs at all vertices.
So the preceding facts about compositions of self-arced graphs do not apply
Moreover the convergence condition is stated in terms of sequences of extended
neighbor graphs, not sequences of asynchronous flocking matrix graphs.
To prove that all i converge to a common heading ss can be shown to be
equivalent to proving that call 2n entries xi in x converge to ss.