1. N.E. Leonard – MBARI – August 1, 2006 Slide 1/46 Cooperative Control and Mobile Sensor Networks in the Ocean Naomi Ehrich Leonard Mechanical & Aerospace Engineering Princeton University naomi@princeton.edu, www.princeton.edu/~naomiwww.princeton.edu/~naomi Derek Paley (grad student), Francois Lekien, Fumin Zhang (post-docs) Dave Fratantoni and John Lund (Woods Hole Oceanographic Inst.) Russ Davis (Scripps Inst. Oceanography) Rodolphe Sepulchre (University of Liege, Belgium) Collaborators:

2. N.E. Leonard – MBARI – August 1, 2006 Slide 2/46 Adaptive Sampling and Prediction (ASAP) Ocean Processes Ocean Model Model Prediction Data Assimilation Adaptive sampling Learn how to deploy, direct and utilize autonomous vehicles most efficiently to sample the ocean, assimilate the data into numerical models in real or near-real time, and predict future conditions with minimal error. Feedback and cooperative control of glider fleet are key tools.

3. N.E. Leonard – MBARI – August 1, 2006 Slide 3/46 ASAP Team Additional Collaboratoring PI’s: Jim Bellingham (MBARI) Yi Chao (JPL) Sharan Majumdar (U. Miami) Mark Moline (Cal Poly) Igor Shulman (NRL, Stennis) MURI Principal Investigators: Russ Davis (SIO) David Fratantoni (WHOI) Pierre Lermusiaux (Harvard) Jerrold Marsden (Caltech) Alan Robinson (Harvard) Henrik Schmidt (MIT) Co-Leaders and MURI Principal Investigators: Naomi Leonard (Princeton) and Steven Ramp (NPS) Funded by a DoD/ONR Multi-Disciplinary University Research Initiative (MURI) with additional funding from ONR and the Packard Foundation.

4. N.E. Leonard – MBARI – August 1, 2006 Slide 4/46 Goals of ASAP Program 1.Demonstrate ability to provide adaptive sampling and evaluate benefits of adaptive sampling. Includes responding to a. changes in ocean dynamics b. model uncertainty/sensitivity c. changes in operations (e.g., a glider comes out of water) d. unanticipated challenges to sampling as desired (e.g., very strong currents) 2.Coordinate multiple assets to optimize sampling at the physical scales of interest. 3.Understand dynamics of 3D upwelling centers a.Focus on transitions, e.g., onset of upwelling, relaxation. b.Close the heat budget for a control volume with an eye on understanding the mixed layer dynamics in the upwelling center. c.Locate the temperature and salinity fronts and predict acoustic propagation.

5. N.E. Leonard – MBARI – August 1, 2006 Slide 5/46 Approach 1.Research that integrates components and develops interfaces. 2.Focus on automation and efficient computational aids to decision making. 3.Five virtual pilot experiments (VPE) run between January and July 2006. Simulation test-bed developed and successfully demonstrated. 4.Major field experiment August 1-31, 2006 (part of MB2006). Preliminary field work in March 2006 in Buzzard’s Bay, MA and May 2006 in Great South Channel.

6. N.E. Leonard – MBARI – August 1, 2006 Slide 6/46 Glider Plan Feedback to large changes and disturbances. Gross re- planning and re-direction. Adjustment of performance metric. Feedback to switch to new GCT. Adaptation of collective motion pattern/behavior (to optimize metric + satisfy constraints). Feedback to maintain GCT. Feeback at individual level that yields coordinated pattern (stable and robust to flow + other disturbances). Many individual gliders. Other observations. Ocean models. GCT = Optimal Coordinated Trajectory = coordinated pattern for the glider fleet. Coordination = Prescription of the relative location of all gliders (as a function of time).

7. N.E. Leonard – MBARI – August 1, 2006 Slide 7/46 Glider Plan OCT for increased sampling in southwest corner of ASAP box. Candidate default OCT with grid for glider tracks. Adaptation SIO glider WHOI glider km

8. N.E. Leonard – MBARI – August 1, 2006 Slide 8/46 Data Flow: Actual and Virtual Experiments Princeton Glider Coordinated Control System (GCCS)

9. N.E. Leonard – MBARI – August 1, 2006 Slide 9/46 Princeton Glider Coordinated Control System

10. N.E. Leonard – MBARI – August 1, 2006 Slide 10/46 Collective Motion Problems Coordinate group of individually controlled systems: Mobile sensor networks Fumin Zhang Derek Paley Reconfigurable formations for feature tracking. Patterns for synoptic area coverage.

11. N.E. Leonard – MBARI – August 1, 2006 Slide 11/46 Patterns for Synoptic Area Coverage Sampling metric Optimization of coordinated tracks Coordinated control of mobile sensors onto tracks Cooperative estimate of flow field which influences motion of mobile sensors. Adaptation of coordinated tracks

12. N.E. Leonard – MBARI – August 1, 2006 Slide 12/46 Sampling Metric: Objective Analysis Error Scalar field viewed as a random variable: Data collected consists of is OA estimate that minimizes is a priori mean. Covariance of fluctuations around mean is

13. N.E. Leonard – MBARI – August 1, 2006 Slide 13/46 AOSN Performance Metric Rudnick et al, 2004

14. N.E. Leonard – MBARI – August 1, 2006 Slide 14/46 Coverage Metric: Objective Analysis Error Rudnick et al, 2004

15. N.E. Leonard – MBARI – August 1, 2006 Slide 15/46 Optimization Optimal elliptical trajectories for two vehicles on square spatial domain. Feedback control used to stabilize vehicles to optimal trajectories. Optimal solution corresponds to synchronized vehicles. Flow shown is 2% of vehicle speed. No flow. Metric = 0.018 Horizontal flow. Metric = 0.020 Vertical flow. Metric = 0.054 No heading coupling. Metric = 0.236

16. N.E. Leonard – MBARI – August 1, 2006 Slide 16/46 Collective Motion for Mobile Sensor Networks Sensor platforms coordinate motion on patterns so data collected minimizes uncertainty in sampled field.

17. N.E. Leonard – MBARI – August 1, 2006 Slide 17/46 Modeling, Analysis and Synthesis of Collective Motion Collective motion patterns distinguished by level of synchrony. Photos: Norbert Wu

18. N.E. Leonard – MBARI – August 1, 2006 Slide 18/46 Collective Motion Stabilization Problem Achieve synchrony of many, individually controlled dynamical systems. How to interconnect for desired synchrony? Use simplified models for individuals. Example: phase models for synchrony of coupled oscillators. Kuramoto (1984), Strogatz (2000), Watanabe and Strogatz (1994) (see also local stability analyses in Jadbabaie, Lin, Morse (2003) and Moreau (2005)) with R. Sepulchre, D. Paley

19. N.E. Leonard – MBARI – August 1, 2006 Slide 19/46 Planar Particle Model: Constant Speed & Steering Control [Justh and Krishnaprasad, 2002]

20. N.E. Leonard – MBARI – August 1, 2006 Slide 20/46 Symmetry and Equilibria symmetry. Reduced space is Fixed points in the reduced (shape) space correspond to 1)Parallel trajectories of the group. 2)Circular motion of the group on the same circle. [Justh and Krishnaprasad, 2002] Let (shape control)

21. N.E. Leonard – MBARI – August 1, 2006 Slide 21/46 Key Ideas Particle model generalizes phase oscillator model by adding spatial dynamics: Parallel motion ⇔ Synchronized orientations Circular motion ⇔ “Anti-synchronized” orientations Assume identical individuals. Unrealistic but earlier studies suggest synchrony robust to individual discrepancies (see Kuramoto model analyses).

22. N.E. Leonard – MBARI – August 1, 2006 Slide 22/46 Key Ideas  is phase coherence, a measure of synchrony, and it is equal to magnitude of average linear momentum of group. [Kuramoto 1975, Strogatz, 2000] Average linear momentum of group: Centroid of phases of group:

23. N.E. Leonard – MBARI – August 1, 2006 Slide 23/46 Synchronized state Balanced state

24. N.E. Leonard – MBARI – August 1, 2006 Slide 24/46 Design Methodology Concept 1.Construct potentials that are extremized at desired collective formations. is maximal for synchronized phases and minimal for balanced phases. 2. Derive corresponding gradient-like steering control laws as stabilizing feedback:

25. N.E. Leonard – MBARI – August 1, 2006 Slide 25/46 Phase Potential: Stabilized Solutions

26. N.E. Leonard – MBARI – August 1, 2006 Slide 26/46 Synchrony of collective measured by relative phasing & spacing of particles: - Phase potential and spacing potential We prove global results on Potentials defined as function of Laplacian L of interconnection graph: decentralized control laws use only available information. For this talk we assume undirected, unweighted, connected graphs. However, our results extend to time-varying, directed, weakly connected interconnections. Low-order parametric family of stabilizable collectives. Use for path planning, optimization, reverse engineering. Design Methodology

27. N.E. Leonard – MBARI – August 1, 2006 Slide 27/46 Interconnection Topology as Graph Example: Ring topology. 1 7 8 6 5 2 4 9 3 Particle = node Edge = communication link See also Jadbabaie, Lin, Morse 2003, Moreau 2005

28. N.E. Leonard – MBARI – August 1, 2006 Slide 28/46 Quadratic Form Induced by Graphs Example: Ring topology. (Olfati & Murray, 2004)

29. N.E. Leonard – MBARI – August 1, 2006 Slide 29/46 Phase Potential Phase Potential: Gradient of Phase Potential: [SPL] (see also Jadbabaie et al, 2004)

30. N.E. Leonard – MBARI – August 1, 2006 Slide 30/46 Spacing Potential Spacing potential: Gradient control:

31. N.E. Leonard – MBARI – August 1, 2006 Slide 31/46 Composite Potential Spacing Phase [SPL]

32. N.E. Leonard – MBARI – August 1, 2006 Slide 32/46 Phase + Spacing Gradient Control: Ring

33. N.E. Leonard – MBARI – August 1, 2006 Slide 33/46 Isolating Symmetric Patterns Consider higher harmonics of the phase differences in the coupling (K. Okuda, Physica D, 1993): 2

34. N.E. Leonard – MBARI – August 1, 2006 Slide 34/46 Phase Potentials with Higher Harmonics

35. N.E. Leonard – MBARI – August 1, 2006 Slide 35/46 Spacing + Phase Potentials: Complete Graph M=1,2,3 M=4,6,12

36. N.E. Leonard – MBARI – August 1, 2006 Slide 36/46 Multi-Scale and Multi-Graph

37. N.E. Leonard – MBARI – August 1, 2006 Slide 37/46 ASAP Virtual Control Room

38. N.E. Leonard – MBARI – August 1, 2006 Slide 38/46

39. N.E. Leonard – MBARI – August 1, 2006 Slide 39/46 Glider Coordinated Trajectories

40. N.E. Leonard – MBARI – August 1, 2006 Slide 40/46 Glider GCT Optimizer

41. N.E. Leonard – MBARI – August 1, 2006 Slide 41/46 Glider Planner Status

42. N.E. Leonard – MBARI – August 1, 2006 Slide 42/46 Glider Positions

43. N.E. Leonard – MBARI – August 1, 2006 Slide 43/46 Glider Prediction

44. N.E. Leonard – MBARI – August 1, 2006 Slide 44/46 Glider OA Error Map

45. N.E. Leonard – MBARI – August 1, 2006 Slide 45/46 Glider OA Flow

46. N.E. Leonard – MBARI – August 1, 2006 Slide 46/46 OA Metrics

47. N.E. Leonard – MBARI – August 1, 2006 Slide 47/46 Final Remarks Derived simply parameterized family of stabilizable collective motions. Optimization of collective behavior (motion, sampling) given constraints of system (energy, communication) and challenges of environment (obstacles, flow field). Glider Coordinated Control System (GCCS) -- software suite for real and virtual experiments. ASAP 2006 field experiment has begun. Five gliders under coordinated control that is fully automated (control running on computer at Princeton).