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Distributed Motion Coordination: From Swarming to Synchronization

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Hamilton Institute Seminar 06/24/2005

Distributed Motion Coordination:From Swarming to Synchronization

Ali Jadbabaie

Department of Electrical and Systems Engineering

and GRASP Laboratory

University of Pennsylvania

- Flocks, swarms and schools exhibit coordinated group behavior although each animal acts completely autonomously
- How do these behaviors emerge?
- How are they sustained?
- How do individual decisions lead to collective group behavior?

SWARMS

Complexity, Statistical Physics,

emergence of collective behavior

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r

neighbors of

agent i

agent i

Vicsek’s kinematic model

- How can a group of moving agents collectively decide on direction, based on nearest neighbor interaction?

How does global behavior emerge from local interactions?

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= speed

MAIN QUESTION :Under what conditions do all headings converge to the same value and agents reach a consensus on where to go?

= heading

Distributed consensus algorithm

SWARMS

We use graphs to model neighboring relations

- V: A set of vertices indexed by the set of mobile agents.
- E: A set of edges the represent the neighboring relations.
- W: A set of weights over the set of edges.

Agent i’s neighborhood

The neighboring relation is represented by a fixed graph G, or a collection of graphs G1, G2,…Gm

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switching signal ,

adjacency matrix

finite set of indices corresponding to allgraphs overnvertices.

Valence matrix

4

3

5

2

6

1

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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 agentsis “jointly connected ”, then all agents move in a formation.

This happens to be both necessary and sufficient for

exponential coordination, boundedness of intervals not required for asymptotic coordination. (Moreau ’04, Ren & Beard ‘05)

SWARMS

- Asynchronous update (Tsitsiklis et al. ‘84,Cao and Morse ‘04)
- Switching, directed graphs (Moreau ’04, Ren & Beard’04, Zhu, Francis ’04)
- Gossip in networks ( Boyd et a. ’04)
- Balanced, directed graphs, no switching.Olfati & Murray 04
- Consensus +quantization (Savkin ’04).
- Consensus on random graphs (Hatano and Mesbahi ’04)
- No quadratic Lyapunov function exists, but maxji – minjj is a valid Lyapunov function, if connectivity holds. (products of length n-1 of Fi s are pseudo-contractive with respect to a subspace norm. )
- Analysis extends to

Choose x = tan()

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- Double integrator model
- Neighbors of i:

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- The graph Laplacian encodes structural properties of the graph
- Some properties of the Laplacian:
- It is positive semi-definite
- The multiplicity of the zero eigenvalue is the number of connected components
- One corresponding eigenvector is the vector of ones, 1.
- The second smallest eigenvalue quantifies connectivity.

SWARMS

Dynamic Topology

Local sensing/communication

Graph changes with time

Control is discontinuous

Non smooth Lyapunov theory

Fixed Topology

Fixed (logical) network

Graph is constant

Control is smooth

Classic Lyapunov theory

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For both fixed and dynamic topology:If the neighboring graph stays connected, all agent velocity vectors become asymptotically the same, collisions between interconnectedagents are avoided and the system approaches a configuration that minimizes all agent potentials.

We could shape potentials for any desired configuration,

and also update it as the objective changes.

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- Consider a group of N oscillators coupled nonlinearly as
- It is the simplest model of coupled oscillators, simple enough for analysis, but complicated enough to have interesting non-trivial behavior.
- The degree of synchronization, is measured with the magnitude of the average phasor:
- r(t) close to 1 means synchronization, and r(t) close to zero means asynchrony.

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History of the coupled oscillators

- Study of Mutual synchronization of biological oscillators goes back to Weiner in 1950s .
- Examples: pacemaker cells in the heart and nervous system, collective synchronization of pancreatic beta cells, synchronously flashing fire flies.
- Synchronization of oscillators has also been studied in the context of
injection locking in RF circuits

- Good abstraction for studying networks of loads and generators in the power grid.
- All-to-all case with infinite oscillators characterized, finite case and arbitrary topologies open …

See the book by Steven Strogatz

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1

B is the (n x e) incidence matrix of graph G.

2

- Weighted Laplacian
- Some properties of the Laplacian:
- The e dimensional vector space of edges can be decomposed to an n-1 dimensional cut space (span of columns of BT) and m-n+1 dimensional cycle space (Kernel of B).

3

4

W is diagonal

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B is the incidence matrix of the graph representing the

interconnection of oscillators

Simple case: all oscillators are identical

Theorem:Consider the unperturbed Kuramoto Model defined over an arbitrary connected graph with incidence matrix B. For any given 0 and any positive value of the coupling, the vector is a locally asymptotically stable equilibrium solution. Furthermore, the rate of approach to equilibrium is no worse than

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Thus

For |i|</2 for a connected graph, all trajectories will converge to S

Therefore, all velocity vectors will synchronize.

But, this stability result is not global. In the case of the ring topology is not the only equilibrium. This is due to the fact that B and BT have the same null space!

is also stable:

Fixed points:

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- When =0, is an asymptotically stable fixed point.
- The order parameter can be written as
Where e is the number of edges in the graph

- is a Lyapunov function, measuring velocity misalignment.
- Using LaSalle, all trajectories converge to invariant sets. Can extend to the case of changing topology, if the graph is “jointly connected”.
- The speed of synchronization depends on the algebraic connectivity of the graph (2nd smallest eigenvalue of the Laplacian).

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- When the frequencies are non zero, there is no fixed point for small values of coupling.
- Theorem:Bounds on the critical value of the coupling can be determined by maximum deviation of frequencies from the mean, and algebraic connectivity of the graph.
- When is random,
- Can develop a mean field model for general topologies by

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Sum of pair-wise

potentials

Supply at each node

Subject to:

Lagrangian

- Want to globally minimize 1-r2 over the whole network
- Let z= sin(BT)

Kuramoto model is the Subgradient algorithm for solving the dual

Shor 87, Tsitsiklis ’86

Subgradient algorithm

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- Phase information from neighbors arrive with arbitrary time delay ij<1. [Aij] is the adjacency matrix.
- In case of degree regular graphs, linearized model with ij= was studies by Earl and Strogatz
- Using Lyapunov-Krasovskii functionals, can analyze the case of arbitrary connected graphs and non-homogeneous delays.
- We assume K is large enough that the oscillators are synchronized, and linearize the dynamics around the synchronized state, i(t) = , where i = t + i(t)
- with Gij = Aij cos( ij)

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- Theorem: Consider a network of N identical oscillators with linearized dynamics
Gij = Aij cos( ij)and Gij>0 when i and j are neighbors. Synchronized state is stable independent of delay.

Proof sketch: Usethe following

V() as a Lyapunov/Krasovski

functional :

Corollary:If [Gij] is the adjacency matrix of a connected graph, then the continuous time, consensus problem is asymptotically stable with arbitrary time delays

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The input

Minimizes the

misalignment potential

The control law minimizes the potential by following its gradient.

But we can’t measure the headings of neighbors W/O communication

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Yw

qj

j

lij

qi

bij

i

Xw

Knowing relative heading would mean having binocular vision or solving structure from motion. This would require multiple visible features on each agent.

Measured Quantities

- the projection of an agent (bearing) βij
- the speed of the projection (optical flow)
- The time-to-collision or Expansion rate (rate of approaching or receding of an object), measured as the relative rate of change of the projection area

Pigeons and flies are capable of all 3 measurements!.

Wang & Frost, Nature, 1992, Fry and Dickinson , Science 2003

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- Theorem: with the distributed controller
joint connectivity in time flocking

Proof based on construction of Lyapunov function whose derivative is the quadratic form of a state dependent Laplacian, and the following lemma

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Agent i’s neighborhood

Consider a group of N agents with different velocity vectorsand constant, unit speed (extension to dynamic case possible).

θ is the heading and φ is the attitude.

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Zw

TiS

X iq

vj

aij

vi

X if

Xw

Yw

Bullo, Murray and Sarti, “Control on the Sphere and Reduced Attitude Stabilization”, 1995

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Zw

TiS

Yij

Xq

vj

vi

Xf

Yw

For any two agents iand j :

- Geodesic versor Yij shows the geodesic direction from vi to vj
is the component of vj orthogonal to vi.

aij

Xw

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Zw

TiS

Yij

Xq

vj

vi

Xf

Xw

Theorem [ Moshtagh, Jadbabaie and Daniilidis, CDC’05]:

Consider the system of N equations

If the proximity graph of the agents is fixed and connected, then the control laws

result in flocking. Furthermore the consensus state is locally asymptotically stable. A similar result holds in the case of switching graphs, if the union graph is connected in time.

aij

Yw

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Lyapunov function: measure of discrepancy between velocity vectors

velocity vectors will synchronize.

In 2-d this is vT L v

It Can be shown that

Using LaSalle’s invariance principle, all trajectories converge to the largest invariant set within the set:

For 0<|i|<, could also use ||||^2 as a Lyap. Function. For a sublevel set inside |i|</2, all trajectories converge to a set where i =j

Use ||||^2 as Lyapunov function on this set, then all trajectories will converge to

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vj

Qij

Visual Servoing Approach

Equation of Motion

Agent i

We can solve for the input in terms of the measurements.

We can construct distributed control laws for flocking, based

on visual sensing and measurement of bearing, time to collision

and optical flow. No communication or relative distance or heading

information is needed

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Simulations

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- Implementation on ER robots (underway!)
- Measurement of OF and TtoC is noisy!!
(How do flies do it?)

- How to optimize connectivity in a distributed way? Use 2(L(x)) =0 as an obstacle
- 2(L(x)) is matrix concave!!, the corresponding eigenvector gives a subgradient direction
- 2(L+L) ≥2(L)+Trace(G L), G=v2 v2* , Lv2=2 v2
- Can find v2 in a distributed way!!
- How does the graph evolve as a function of the positions?
- Potential-based forces can be used for collision avoidance, but how can we avoid local minima in graphs with cycles?
- Determine which edges have the most impact on 2(L)

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