Transcript Distributed Coordination : From Flocking and

Workshop on computational worldview and sciences Caltech 03/15/07
Distributed Motion Coordination in
Networked Dynamic Systems: From Flocking
and Synchronization to Coverage Verification
Ali Jadbabaie
Department of Electrical and Systems Engineering
and GRASP Laboratory
University of Pennsylvania
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Networked dynamical systems
Nonlinear/uncertain
hybrid/stochastic etc.
Complexity
of
dynamics
Single
Agent
Complex
networked
systems
Flocking
Synchronization
Multi-agent
Consensus
systems
Complexity
of interconnection
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Networked dynamical systems
Nonlinear/uncertain
hybrid/stochastic etc.
Complexity
of
dynamics
Single
Agent
Complex
networked
systems
Flocking/synchronization
Consensus/
Multi-agent
Coverage
systems
Complexity
of interconnection
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Nonlinear/uncertain
hybrid/stochastic etc.
Complexity
of
dynamics
Single
Agent
Complex
networked
systems
Flocking/synchronization
consensus
Multi-agent
systems
Jadbabaie et al
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Complexity
of interconnection
Statistical Physics and
emergence of collective behavior
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Multi-agent setting: Vicsek’s kinematic model
• How can a group of moving agents collectively decide on
direction, based on nearest neighbor interaction?
r
neighbors of
agent i
agent i
How does global behavior emerge from local interactions?
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Distributed consensus algorithm for
kinematic agents
= speed
= heading
MAIN QUESTION :
Under what conditions do all headings
converge to the same value and agents reach a consensus on where to go?
For small angles
For angles in (-/2,/2), the nonlinear problem becomes
linear with a coordinate change!
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
The underlying proximity graph
We use graphs to represent
neighboring relations
vertices:
edges:
4
3
5
2
6
switching signal ,
finite set of indices corresponding to all
graphs over n vertices.
1
adjacency matrix
Valence matrix
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Conditions for reaching consensus
Theorem (Jadbabaie et al. 2003): If there is a sequence of
bounded, non-overlapping time intervals Tk, such that over any
interval of length Tk, the network of agents is “jointly connected ”,
then all agents will reach consensus on their velocity vectors.
This happens to be both necessary and sufficient for
exponential coordination, boundedness of intervals not
required for asymptotic coordination. (see Moreau ’04)
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
History of consensus Algorithms
Consensus based on repeated “local” averaging
Estimation-modification process among a group of experts in
a fixed network: average your “opinion” with that of your
neighbors” Degroot’74
Update opinions via a nonhomogeneous Markov chain
Chaterjee and Seneta’74. Based on work of Hajnal’58,
Sarymsakov’61, Paz’71.
Consensus among multi processors in concurrency theory
Lynch, Herlihy, Shostak ,…
Consensus algorithms as examples of asynchronous
distributed gradient methods Tsitsiklis’84, Tsitsiklis et al.’86
In control theory literature, in the context of velocity
alignment among kinematic agents Jadbabaie et al.’03,
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Consensus in random networks
is a random switching signal
Theorem : Assuming graphs are randomly chosen and
independent, reaching consensus is a trivial event, i.e.,
either it happens almost surely or almost never, i.e., it
satisfies the Kolmogorov 0-1 law.
Theorem: necessary and sufficient condition for almost
sure convergence is
,i.e., the average
system needs to reach deterministic agreement.
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
The Laplacian of the graph
1
B is the (n x e)
incidence matrix of
graph G.
2
4
3
The graph Laplacian (n x n) encodes structural properties of the graph
W is diagonal
Some properties of the Laplacian:
It is positive semi-definite
The multiplicity of the zero eigenvalue is the number of connected
components
The kernel (for connected graph) is the span of vector of ones,
First nonzero eigenvalue is called algebraic connectivity.
Its corresponding eigenvector, called the Fiedler vector. Its sign paper
encodes a lot of information about “bottlenecks” and “cutpoints”
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Consensus in Continuous time
As before, (t) is a piecewise constant switching signal
The model is now a hybrid or switching dynamical system
Need to assume a dwell time on each graph to avoid
complications
The result is virtually the same, as exponentials of Laplacians
are stochastic matrices
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Vicsek’s Model with
Periodic Boundary Condition
Autonomous agents with constant
speed and adjustable headings
Local interaction rule:
A consensus algorithm if the graphs
are jointly connected over time
Periodic boundary conditions (Motion
over a flattened unit torus) to avoid
boundary effects.
Location on a torus can be seen as
the fractional part of the location of
an agent moving in an infinite plane.
Reasonable assumption for statistical
physics simulations, for large
populations
Vicsek’s simulations: velocity
alignment without any assumption on
connectivity.
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Kronecker’s Theorem, Weyl’s Theorem
and Number Theory
Kronecker’s Theorem: The numbers
are dense in
the unit interval
if
is an irrational number.
Weyl’s Theorem: If the sequence
grows fast enough,
then for almost every
the numbers
are
equidistributed in the unit interval
.
Equidistributedness implies denseness and more!
Weyl’s theorem can be generalized to higher dimensions.
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Flocking, artificial life, and
computer graphics
Reynolds [Reynolds 87] named the autonomous
systems that behave like members of animal groups
boids (bird + oids)
He developed a descriptive model for flocking
behavior based on the combined action of
alignment and cohesion-separation forces
alignment: steer towards the average heading of flockmates
separation: steer to avoid crowding flockmates
cohesion: steer towards the average position of flockmates
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Distributed coordination with dynamic
models: Flocking with collision avoidance
Double integrator model
Neighbors of i distance
dependent:
Cohesion/Separation
Alignment
For dynamic models, Proximity graph Connectivity implies
emergence of Collective motion (Tanner, Jadbabaie, Pappas)
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Flock artificial potential energy
Each Vij is required to be:
increasing as
unbounded as
unique minimum
The potential energy Vi of
boid i depends on its
neighbors
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Example (Ogren et al ’04)
Topology dictates analysis
Dynamic Topology
Fixed Topology
Fixed (logical) network
Local sensing/communication
Graph is constant
Control is smooth
Classic Lyapunov theory
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Graph changes with time
Unreliable communication, no
dwell time
Control is discontinuous
Non smooth Lyapunov theory
Enforcing connectivity
Treat loss of connectivity as
an obstacle Zavlanos, Jadbabaie, Pappas ’07
Everything else is the same!
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Synchronization
Fireflies Flashing
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Kuramoto Model
di
K N
 i   sin( j  i )
dt
N j 1
N : Number of oscillators
i : Natural frequency of oscillator i , i  1, , N.
All-to-all interaction
i : Phase of oscillator i , i  1, , N.
K : Coupling strength
Model for pacemaker cells in the heart and nervous system,
collective synchronization of pancreatic beta cells, synchronously
flashing fire flies, rhythmic applause, gait generation for bipedal
robots, …
Benchmark problem in physics
Not very well understood over arbitrary networks
Introduced by Kuramoto in 1975 as a toy model of synchronization
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Kuramoto model & graph topology
2
1
3
6
4
5
0
1

1
A
0
0

 1
1 1 0 0 1
0 1 0 0 0 
1 0 1 0 0

0 1 0 1 1
0 0 1 0 1

0 0 1 1 0 
di
K N
 i   Aij sin( j  i )
dt
N j 1
B is the incidence matrix of the graph
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Frequency and phase locking
di K N
  Aij sin( j  i )
dt
N j 1
Frequency entrainment and
phase locking as long as
graph is connected.
Proof same as the continuous-time
agreement model
di
K N
 i   Aij sin( j  i )
dt
N j 1
sin( x )
x
Frequency entrainment possible.
Phase stability, but no phase
locking.
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Kuramoto model, dual decomposition,
and nonlinear utility minimization
min
Minimize the misalignment
1 N N
 Aij 1  cos(i   j ),
2 i 1 j 1
N
s.t.
 Aij sin(i   j ) 
j 1
Ni
K
N
N
N
N i i
1 N N
L   Aij 1  cos(i   j )   Aij i sin(i   j )  
2 i 1 j 1
K
i 1 j 1
i 1
L
 sin(i   j )  ( i  j )cos(i   j )
 i   j 
N
Ni
L
  Aij sin(i   j ) 
 i j 1
K
( i   j )
K L
K N
i  
 i   Aij
N  i
N j 1
1  ( i   j )2
Kuramoto model is the just a gradient algorithm for minimization of a global
utility which measures misalignment between phasors (exactly like TCP!)
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Phase locking in switching networks
with heterogeneous delays
When the natural frequencies are identical, can switch to a rotating
frame, and set i to 0.
Theorem (Papachristodoulou and Jadbabaie’06)
If f is locally passive (e.g., a sine function), and the switching sequence is
such that the union of graphs across uniformly bounded intervals contains
a globally reachable node infinitely often, then the synchronized set is
asymptotically attracting, so long as the delays are finite.
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Beyond Graphs in Networked Systems
Main Idea: understanding global properties with local
information: algebraic topology
For certain problems, e.g. coverage, makes sense to go
beyond graphs and pair-wise interactions
Example: Given a set of sensor nodes in a given domain
(possibly bounded by a fence), is every point of the
domain under surveillance by at least one node?
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Coverage Problems
Problem: Given a set of sensor nodes in a
domain (possibly bounded by a fence), is every
point of the domain under surveillance by at
least one node?
Figure from De Silva and Ghrist ’06
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
From Graphs to Simplicial Complexes
Simplicial Complex: A finite
collection of simplices
Simplex: Given V, an
unordered non-repeating
subset
k-simplex: The number of
points is k+1
Faces: All (k-1)-simplices in
the k-simplex
Orientation
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
From Graphs to Simplicial Complexes
Simplicial complex: made up of simplices of
several dimensions
Properties
Whenever a simplex lies in the collection then so does each of
its faces
Whenever two simplices intersect, they do so in a common face.
Valid Examples
Graphs
Triangulations
Invalid examples
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Rips-Vietoris Simplicial Complex
0-simplices : Nodes
1-simplices : Edges
2-simplices: A triangle in the
connectivity graph ~ 2simplex (Fill in with a face)
K-simplices: a complete
subgraph on k+1 vertices
k-simplex in the Rips
complex ~ (k+1) points
within communication range
of each other
Generalization of r-disk graphs
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Coverage Problems
Intersection of sensing ~ simplicial
complexes
Communication graphs ~ simplicial
complexes
Holes ~ homology of simplicial complexes
A sensor network has coverage hole if there
is a “robust” hole in the simplicial complexes
induced by the communication graphs [Ghrist
et al.]
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Rips and Čech Complexes:
Topological vs. Geometric information
A set of points
•
(Rips complex of radius ): k-simplex, if the pairwise
distance between k points are less than .
- Easy to compute in a dsitributed manner.
- However, does not preserve the topological properties.
•
(Čech complex of radius ): k-simplex, if k coverage
disks of radius overlap
- Hard to compute.
- Preserves the topological properties.
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Topological information from Rips complex
A Čech complex can be bounded by two Rips complexes:
Ghrist and Muhammad’05
Use two communication power levels
,
Check for holes in
by Finding generators for homology
groups.
- A sufficient condition for coverage holes
If a hole exists in
, then check for holes in
- A necessary condition.
There is a gap between the necessary and the sufficient conditions.
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Boundary Maps: Generalization of
Incidence matrices
: the vector space whose basis is the set of oriented ksimplices of X
The boundary map
is the linear transformation
k-cycles:
k-boundaries:
Note:
Homology groups : Hk(X) = Zk (X) / Bk (X)
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Relevance of Homology
dim H0(X) ~ no of connected components of X
dim H1(X) ~ types of loops in X that surround
‘punctures’
dim Hk(X) ~ no of k+1-dimensional ‘voids’ in X
Available software
Plex (Stanford)
CHomP (Georgia Tech)
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Combinatorial k-Laplacians
Since X is finite we can represent the boundary maps in matrix form
incidence matrix
Moreover, we can get the adjoint
[Eckmann 1945] The Combinatorial k-Laplacian
is given by
Note:
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
k-Laplacian at the Simplex Level
Adjacency of a simplex to other
simplices
Upper adjacency if they share a
higher simplex (e.g. 2 nodes
connected by an edge)
Lower adjacency if they share a
common lower simplex (e.g. two
edges share a node)
‘Local’ formula with
orientations
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
k-Laplacian at the Simplex Level
Adjacency of a simplex to other simplices
Upper adjacency if they share a higher simplex (e.g. 2 nodes connected by
an edge)
Lower adjacency if they share a common lower simplex (e.g. two edges
share a node)
‘Local’ formula with orientations
Hodge theory, 1940’s: Kernel of the Laplacian ~ cohomologies
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Laplacian Flows
Laplacian flows : a semi-stable dynamical system
(Recall heat equation for k = 0)
[Muhammad-Egerstedt MTNS’06]
System is asymptotically
stable if and only if
rank(Hk(X)) = 0.
A method to detect
‘no holes’ locally
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Laplacian Flows (contd.)
System converges to
the unique harmonic cycle
if rank(Hk(X)) = 1.
A method to detect
‘proximity to hole’ locally
when single hole
When rank(Hk(X)) > 1 :
System converges to the span of harmonic homology cycles
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Decentralized Computation of
Homology
Compute eigenvector decomposition of the k-Laplacian
All eigenvalues are non-negative
Eigenvectors corresponding to zero eigenvalues ~ harmonic
homology classes
‘Small’ positive eigenvalues ~ near-harmonic
There is a recent decentralized algorithm for eigenvectors
computation. Each entry is computed only based on
information from neighboring nodes.
Use the distributed Algorithm for spectral analysis with a
complexity of
, where mix is the mixing rate of the
random walk on the network. Kempe and McSherry, 2004
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Decentralized computation
Step 1 : Build the simplicial (Rips) complex
Computations take place at node level
Need protocols at the node level for
Simplex membership : What simplices are a node part of ?
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Decentralized computation
Step 1 : Build the simplicial (Rips) complex
Computations take place at node level
Need protocols at the node level for
Simplex membership : What simplices are a node part of ?
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Decentralized computation
Step 1 : Build the simplicial (Rips) complex
Computations take place at node level
Need protocols at the node level for
Simplex membership : What simplices are a node part of ?
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Decentralized computation
Step 1 : Build the simplicial (Rips) complex
Computations take place at node level
Need protocols at the node level for
Simplex membership : What simplices are a node part of ?
Simplex owner-ship : What simplices (and therefore a subcomplex) are
owned by a node?
e.g. the node with smallest
label gets the simplex
Owned sub-complexes have trivial
homology in all dimension
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Example, eigenvectors of 1
Network
2nd homology class
1st homology class
‘Fiedler-like’- eigenvector
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Consensus in Switching Graphs
Mobility, switching graphs and consensus : switched linear
system
Joint connectedness (Jadbabaie’ 2003)
Theorem : Consensus if and only if there is a sequence of
bounded, non-overlapping time intervals, such that over any
interval, the network of agents is jointly connected.
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Coverage in Switching Simplicial
Complexes
Can we repeat similar analysis for switching simplicial complexes?
YES!
Jointly `hole-free’ simplicial complexes
Joint hole-free implies trivial homology
in union complex
Theorem (Muhammad, Jadbabaie ’06): Switched linear system is
globally asymptotically stable if and only if there exists an infinite
sequence of bounded intervals, across each of which the simplicial
complexes encountered are jointly hole-free.
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Computational test for relative
homlogical criterion for coverage
Theorem (De Silva and Ghrist’06): For coverage, H2(R,F) should
contain a nontrivial element which does not vanish on the boundary
Computational test: If the above distributed dynamical system converges
to a nonzero value which does not vanish on boundary, we have coverage
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Union vs. amalgamated complex
Jointly hole free simplicial complex
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Amalgamated complexes
Verification of sweep coverage
Given a set of sensors with a disk footprint, add an edge when 2
sensors overlap. A face when 3 sensors overlap, …
Construct the 1st Laplacian 1
Rips complex is “jointly persistently hole free over time”
intersection
of kernels of Laplacians is zero
Switched dynamical systems converges to 0
Unfortunately This does NOT imply sweep coverage 
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Jointly-hole-free complexes and
sweep coverage
BAD NEWS:
If the node is close to the boundary,
The union of simplicial complexes is
still jointly hole-free and we satisfy the
relative homological criterion, BUT,
If the node is close to the boundary,
we can’t sweep the center 
GOOD NEWS
If, however, we already have a set possible
sensor locations which do verify
Coverage when they are static, we can
verify sweep coverage when they blink on
and off 
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems
Work in Progress ….
Distinguish between multiple homology classes by
decentralized eigenvector decomposition of k-Laplacian
(Kempe’s algorithm)
Combinatorial Laplacian of amalgamated complexes
Quantify ‘proximity to holes’
Quantify fragilities in network : near-harmonic cycles (Fiedler
like characterization such as cutpoints for holes? Properties of
K-Laplacian Eigenvectors and Eigenvalues)
A “spectral theory” for simplicial complexes?
Is there a percolation result similar to geometric graphs?
Statistical physics of simplicial complexes
Consensus and agreement in CAT-0 topological spaces
A. Jadbabaie “Distributed Motion Coordination in Networked Dynamic Systems