N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009Slide 1

Spatial Dynamics, Information Flow and Collective Behavior

Naomi Ehrich LeonardMechanical & Aerospace Engineering

Princeton Universitynaomi@princeton.edu

www.princeton.edu/~naomi

N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009Slide 2

Observing Fish Schooling

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with Iain Couzin, Dan Swain, Christos Ioannou, Yael Katz

N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009Slide 3

Robot/Fish Hybrid Experiment

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Replica fish use real-time feedback control

Replica Fish

Real Fish

Camera

Robot with magnets

Bluetooth Link

Tracking/Control Workstation

Replica fish leading real fish (Jens Krause, Univ. Leeds).

Replica predator approaching real fish.

Replica and real fish

with Iain Couzin, Dan Swain, Christos Ioannou, Yael Katz

N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009Slide 4

Observations of Fish with Oscillating Speed

Data:Iain Couzin (Princeton) Albert Kao (Harvard)

2.0 meters

4.5 cm

Banded killifish

Data processing: with Dan Swain

N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009Slide 5

Observed 2-Fish School Trajectories

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N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009Slide 6

Analysis of Killifish DataSubtract mean speed. Take Hilbert transform to get analytic signal.Magnitude of analytic signal.Phase of analytic signal.

Least square fit

Sinusoidal fit

Same frequency for each fish!

Best fit frequency = 3.58 rad/sec = 0.57 HzRelative phase = 3.295 = 1.05*pi

N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009Slide 7

Goal

Use formal models to analytically investigate the coupling of spatial dynamics, information flow and decision-making dynamics in fish schools.

What is the influence on speed, accuracy, robustness of collective decision-making?

Are there classes of information flow patterns (spatial patterns) that yield “better” decision-making?

Is periodically time-varying information flow (spatial dynamics) advantageous?

N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009Slide 8

Spatial Dynamics of Agents with Oscillatory Speed

Steering control

Speed phase controlE.g.,

Extend model of Justh and Krishnaprasad.

N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009Slide 9

Graphs and Information Flow

Edge (j,k) (directed arrow) from node j to node k if agent k can sense agent j.

Undirected edgeDirected edge

Information flow determined by sensing paradigm, e.g., - zone model - direction limited sensing

N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009Slide 10

Controlled Spatial Dynamics with Oscillatory Speed

Two sets of coupled oscillator dynamics

For example:

N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009Slide 11

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Circular Patterns with Oscillating Speed

Prove convergence to these circular patterns with feedback term, depending only on relative measurements, to synchronize instantaneous circle centers

Swain, Leonard, Couzin, Kao, Sepulchre, CDC ‘07Swain and Leonard, ACC ‘09

N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009Slide 12

Effective Sensing Region

Periodically time-varying graph makes analysis tractable:Using effective sensing region compute union of edges over a period to get time-invariant graph. Study collective decision-making dynamics:

Swain, Cao, Leonard, CDC ‘08

Specializes to consensus dynamics if:

N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009Slide 13

Randomly Spaced Agents: Relative motion increases connectivity

Random geometric graph

Periodically varying graphVariation in oscillation phases

Ref: Balister, Bollobas, Walters, Random Structures and Algorithms, 2005.

Variation in oscillation phases

Probability of connectivity

N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009Slide 14

Regularly Spaced Agents:Relative motion increases connectivity

Consensus convergence rate maximal when relative phase of speed oscillations is

σσ

Consider consensus dynamics under this periodically time-varying sensing topology:

N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009Slide 15

Robustness of Collective Decision Dynamics

N identical systems with heterogeneous external inputs and coupling defined by Laplacian L:

with Luca Scardovi

R.U. Verma, 2006

N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009Slide 16

Robustness of Collective Decision Dynamicsm = 1

C.W. Wu, 2005

Alg. connectivity for directed graphs

(Scardovi and Leonard, 2009)

(Extends to state-space models with non-identical init.cond. if zero-reachable.)

N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009Slide 17

Example

Dispersion of setpoints Dispersion of steady state consensus values

Simulation: 4 Oscillating agents Consensus variables vs. Time

Scalar linear dynamical systems with heterogeneous setpoints :

In steady state, we have

N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009Slide 18

Final RemarksOngoing development of to understand and exploit influence of coupling of spatial dynamics, information flow on collective behavior and decision-making dynamics.

Tie in with recent work on dynamics/leadership in swarms with heterogeneous information (micro-groups)(Nabet, Leonard, Couzin, Levin, J. Nonlinear Science, 2009)

New hybrid fish/robot test-bed for further integrated behavioral investigation with controlled experiments (joint with Iain Couzin).

Collective decision dynamics in teams of humans and robots (AFOSR MURI).

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Hybrid Fish/Robot Test-bed: Imaging Processing for Real-time Feedback Control of Replica Fish

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Decision Making and Informed Individuals

Time scale separation, reduced model, bifurcation analysis (Nabet, Leonard, Couzin, Levin, 2006, 2008)

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Properties of Time-Varying Graphs

Neighborhood of agent k at time t:

The time-invariant graph over the interval I corresponds to edge set:

G(t) is uniformly connected if there exists an index k and a time horizon T > 0 such that, for all t, node k is connected to all other nodes for where I = [t, t+T].

A graph is periodic with period T if G(t) = G(t+T) for all t.If a periodic graph is strongly connected over any interval of length T, it is uniformly connected.We write to denote the graph over any interval of length T.