Cooperative Control and Mobile Sensor Networks in the Ocean

1. Cooperative Control and Mobile Sensor Networks in the Ocean Naomi Ehrich Leonard Mechanical & Aerospace Engineering Princeton University naomi@princeton.edu, www.princeton.edu/~naomi Collaborators: 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)

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

4. Goals of ASAP Program • 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 • Focus on transitions, e.g., onset of upwelling, relaxation. • Close the heat budget for a control volume with an eye on understanding the mixed layer dynamics in the upwelling center. • Locate the temperature and salinity fronts and predict acoustic propagation.

5. Approach Research that integrates components and develops interfaces. Focus on automation and efficient computational aids to decision making. Five virtual pilot experiments (VPE) run between January and July 2006. Simulation test-bed developed and successfully demonstrated. 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. GCT = Optimal Coordinated Trajectory = coordinated pattern for the glider fleet. Coordination = Prescription of the relative location of all gliders (as a function of time). 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. Ocean models. Other observations.

7. km km km km SIO glider WHOI glider Glider Plan OCT for increased sampling in southwest corner of ASAP box. Candidate default OCT with grid for glider tracks. Adaptation

8. Data Flow: Actual and Virtual Experiments Princeton Glider Coordinated Control System (GCCS)

9. Princeton Glider Coordinated Control System

10. Collective Motion Problems Coordinate group of individually controlled systems: Mobile sensor networks Patterns for synoptic area coverage. Reconfigurable formations for feature tracking. Fumin Zhang Derek Paley

11. 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. Scalar field viewed as a random variable: Data collected consists of Sampling Metric: Objective Analysis Error is a priori mean. Covariance of fluctuations around mean is is OA estimate that minimizes

13. AOSN Performance Metric Rudnick et al, 2004

14. Coverage Metric: Objective Analysis Error Rudnick et al, 2004

15. Optimization No flow. Metric = 0.018 Optimal elliptical trajectories for two vehicles on square spatial domain. Feedback control used to stabilize vehicles to optimal trajectories. Horizontal flow. Metric = 0.020 Optimal solution corresponds to synchronized vehicles. Vertical flow. Metric = 0.054 Flow shown is 2% of vehicle speed. No heading coupling. Metric = 0.236

16. Collective Motion for Mobile Sensor Networks Sensor platforms coordinate motion on patterns so data collected minimizes uncertainty in sampled field.

17. Modeling, Analysis and Synthesis of Collective Motion Photos: Norbert Wu Collective motion patterns distinguished by level of synchrony.

18. Collective Motion Stabilization Problem with R. Sepulchre, D. Paley • Achieve synchrony of many, individually controlled dynamical systems. • How tointerconnect for desired synchrony? • Use simplifiedmodels 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))

19. Planar Particle Model: Constant Speed & Steering Control [Justh and Krishnaprasad, 2002]

20. Symmetry and Equilibria Let (shape control) symmetry. Reduced space is Fixed points in the reduced (shape) space correspond to Parallel trajectories of the group. Circular motion of the group on the same circle. [Justh and Krishnaprasad, 2002]

21. 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. Average linear momentum of group: Centroid of phases of group: 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]

23. Synchronized state Balanced state

24. Design Methodology Concept 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. Phase Potential: Stabilized Solutions

26. Design Methodology • 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.

27. 2 9 1 3 8 4 7 5 6 Interconnection Topology as Graph Particle = node Edge = communication link See also Jadbabaie, Lin, Morse 2003, Moreau 2005 Example: Ring topology.

28. Quadratic Form Induced by Graphs (Olfati & Murray, 2004) Example: Ring topology.

29. Phase Potential Phase Potential: [SPL] Gradient of Phase Potential: (see also Jadbabaie et al, 2004)

30. Spacing Potential Spacing potential: Gradient control:

31. Composite Potential Phase Spacing [SPL]

32. Phase + Spacing Gradient Control: Ring

33. 2 Isolating Symmetric Patterns Consider higher harmonics of the phase differences in the coupling (K. Okuda, Physica D, 1993):

34. Phase Potentials with Higher Harmonics

35. Spacing + Phase Potentials: Complete Graph M=1,2,3 M=4,6,12

36. Multi-Scale and Multi-Graph

37. ASAP Virtual Control Room

39. Glider Coordinated Trajectories

40. Glider GCT Optimizer

41. Glider Planner Status

42. Glider Positions

43. Glider Prediction

44. Glider OA Error Map

45. Glider OA Flow

46. OA Metrics

47. 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).