N.E. Leonard – U. Pisa – 18-20 April 2007Slide 1

Cooperative Control and Mobile Sensor Networks

Application to Mobile Sensor Networks, Part II

Naomi Ehrich Leonard

Mechanical and Aerospace EngineeringPrinceton University

and Electrical Systems and Automation University of Pisa

naomi@princeton.edu, www.princeton.edu/~naomi

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

N.E. Leonard – U. Pisa – 18-20 April 2007Slide 3

5 Scripps Spray Gliders 10 WHOI Slocum Gliders

AOSN-II Glider Plan

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AOSN-II Glider Measurements

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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

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(with F. Lekien, D. Paley)

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Coverage Metric: Objective Analysis ErrorRudnick et al, 2004

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Coverage Metric: Objective Analysis

Error

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Francois Lekien

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Computation of Optimal Trajectories

Box:

Trajectories:

Constraint:

Optimality Trajectories:

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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

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Optimal Solution for Ellipses

For ellipses, the optimum is at

Corresponds to one glider per regionof area

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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 = 0.236Leonard, Paley, Lekien, Sepulchre,

Fratantoni, Davis, Proc. IEEE, Jan. 2007.

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Glider Plan

GCT for increased sampling in southwest corner of ASAP box.

Candidate default GCT with grid for glider tracks. Adaptation

SIO gliderWHOI glider

km

km

km

km

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Virtual Pilot Experiment: March ‘06; Ocean: August ‘03

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Maximize Information in Measurements

HOPS

ROMS

NCOM

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Glider Coordinated Trajectories (GCT)

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Glider Planner Status

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August 2006 Adaptations

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Monterey Bay, CA, August 2006

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OA error map, 2006

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ASAP Sampling Performance, August 2006

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Comparing Actual Glider Motion and NCOM Prediction

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Comparing Actual Glider Motion and HOPS Prediction

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Evaluating Glider Motion Predictions

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Non-uniform metric optimization

There exist many methods to cover a regionuniformly. Example: Cortes and Bullo, 2005.

These methods do not extend easily tonon-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, SIAMJ. Control Optim, 44(5), 1543—1574, 2005.

Reproduced from Gastner & Newman, Proc. National Academy US, 101(20), 7499—7504, 2004.

Francois Lekien

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Non-uniform metric optimization

Step #1: Extend the function to a larger domain with homogeneous Neumann boundary conditions:

Step #2: Solve the diffusion equation:

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Non-uniform metric optimization

Physical space: non-uniform metric Transformed space: Euclidian metric

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OA error map(Cartesian space with nonlinear metric)

After transformation(Curved space with Euclidean metric)

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Data Flow: Actual and Virtual ExperimentsPrinceton Glider Coordinated Control System (GCCS)

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Princeton Glider Coordinated Control System