1. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 1 Cooperative Control and Mobile Sensor Networks Application to Mobile Sensor Networks, Part II Naomi Ehrich Leonard Mechanical and Aerospace Engineering Princeton University and Electrical Systems and Automation University of Pisa naomi@princeton.edu, www.princeton.edu/~naomi www.princeton.edu/~naomi

2. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 2 Key References [1] Leonard, Paley, Lekien, Sepulchre, Fratantoni, Davis, “Collective motion, sensor networks and ocean sampling,” Proc. IEEE, 95(1), 2007. [2] F. Lekien and N.E. Leonard, “Non-Uniform Coverage and Cartograms,” preprint, [Online] http://www.princeton.edu/~naomi [3] D. Paley, F. Zhang and N.E. Leonard. Cooperative control for ocean sampling: The Glider Coordinated Control System. IEEE Transactions on Control System Technology, to appear.

3. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 3 5 Scripps Spray Gliders10 WHOI Slocum Gliders AOSN-II Glider Plan

4. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 4 AOSN-II Glider Measurements

5. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 5 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

6. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 6 (with F. Lekien, D. Paley)

7. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 7 Coverage Metric: Objective Analysis Error Rudnick et al, 2004

8. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 8 Coverage Metric: Objective Analysis Error Francois Lekien

9. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 9 Computation of Optimal Trajectories Box: Trajectories: Constraint: Optimality Trajectories:

10. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 10 Maximize Information in Measurements Direct optimization of sampling metric leads to overly complex patterns of data distribution. Direct optimization of sampling metric along “ideal tracks” provides practical basis for automatic steering and inter-vehicle coordination. Family of closed loops as candidate ideal tracks is sufficiently large for multi-scale patterns and information rich sampling plans. Adaptations can be made as changes in ideal tracks. A set of ideal tracks with prescribed relative glider spacing is called a set of Glider Coordinated Trajectories (GCT). adapt

11. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 11 Optimal Solution for Ellipses For ellipses, the optimum is at Corresponds to one glider per region of area

12. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 12 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 Leonard, Paley, Lekien, Sepulchre, Fratantoni, Davis, Proc. IEEE, Jan. 2007.

13. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 13 Glider Plan GCT for increased sampling in southwest corner of ASAP box. Candidate default GCT with grid for glider tracks. Adaptation SIO glider WHOI glider km

14. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 14 Virtual Pilot Experiment: March ‘06; Ocean: August ‘03

15. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 15 Maximize Information in Measurements HOPS ROMS NCOM

16. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 16

17. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 17

18. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 18 Glider Coordinated Trajectories (GCT)

19. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 19 Glider Planner Status

20. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 20 August 2006 Adaptations

21. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 21 Monterey Bay, CA, August 2006

22. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 22 OA error map, 2006

23. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 23 ASAP Sampling Performance, August 2006

24. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 24

25. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 25 Comparing Actual Glider Motion and NCOM Prediction

26. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 26 Comparing Actual Glider Motion and HOPS Prediction

27. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 27 Evaluating Glider Motion Predictions

28. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 28 Non-uniform metric optimization There exist many methods to cover a region uniformly. Example: Cortes and Bullo, 2005. These methods do not extend easily to non-uniform and dynamic metrics. Find a transformation from the physical space where volume elements are given by the non-uniform OA metric (i.e., dA=B(x,y) dx dy) to a virtual space with the Euclidian metric (i.e., dA=dx dy). Examples: the cartograms of Gastner and Newman, 2004. Reproduced from Cortes & Bullo, SIAM J. Control Optim, 44(5), 1543—1574, 2005. Reproduced from Gastner & Newman, Proc. National Academy US, 101(20), 7499—7504, 2004. Francois Lekien

29. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 29 Non-uniform metric optimization Step #1: Extend the function to a larger domain with homogeneous Neumann boundary conditions: Step #2: Solve the diffusion equation:

30. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 30 Non-uniform metric optimization Physical space: non-uniform metric Transformed space: Euclidian metric

31. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 31 OA error map (Cartesian space with nonlinear metric) After transformation (Curved space with Euclidean metric)

32. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 32 Data Flow: Actual and Virtual Experiments Princeton Glider Coordinated Control System (GCCS)

33. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 33 Princeton Glider Coordinated Control System