1. N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009 Slide 1 Spatial Dynamics, Information Flow and Collective Behavior Naomi Ehrich Leonard Mechanical & Aerospace Engineering Princeton University naomi@princeton.edu www.princeton.edu/~naomi

2. N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009 Slide 2 Observing Fish Schooling with Iain Couzin, Dan Swain, Christos Ioannou, Yael Katz

3. N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009 Slide 3 Robot/Fish Hybrid Experiment 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

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

5. N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009 Slide 5 Observed 2-Fish School Trajectories

6. N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009 Slide 6 Analysis of Killifish Data Subtract 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 Hz Relative phase = 3.295 = 1.05*pi

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

8. N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009 Slide 8 Spatial Dynamics of Agents with Oscillatory Speed Steering control Speed phase control E.g., Extend model of Justh and Krishnaprasad.

9. N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009 Slide 9 Graphs and Information Flow Edge (j,k) (directed arrow) from node j to node k if agent k can sense agent j. Undirected edge Directed edge Information flow determined by sensing paradigm, e.g., - zone model - direction limited sensing

10. N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009 Slide 10 Controlled Spatial Dynamics with Oscillatory Speed Two sets of coupled oscillator dynamics For example:

11. N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009 Slide 11 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 ‘07 Swain and Leonard, ACC ‘09

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

13. N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009 Slide 13 Randomly Spaced Agents: Relative motion increases connectivity Random geometric graph Periodically varying graph Variation in oscillation phases Ref: Balister, Bollobas, Walters, Random Structures and Algorithms, 2005. Variation in oscillation phases Probability of connectivity

14. N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009 Slide 14 Regularly Spaced Agents: Relative motion increases connectivity Consensus convergence rate maximal when relative phase of speed oscillations is Consider consensus dynamics under this periodically time-varying sensing topology:

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

16. N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009 Slide 16 Robustness of Collective Decision Dynamics m = 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.)

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

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

19. N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009 Slide 19 Hybrid Fish/Robot Test-bed: Imaging Processing for Real-time Feedback Control of Replica Fish

20. N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009 Slide 20 Decision Making and Informed Individuals Time scale separation, reduced model, bifurcation analysis (Nabet, Leonard, Couzin, Levin, 2006, 2008)

21. N.E. Leonard – Block Island Workshop on Swarming – June 3, 2009 Slide 21 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.