ASAP Progress Report
Adaptive Sampling and Cooperative Control
Naomi Ehrich Leonard
Francois Lekien, Derek Paley, Fumin Zhang
Mechanical and Aerospace Engineering
Princeton University
naomi@princeton.edu
http://www.princeton.edu/~naomi
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N.E. Leonard – ASAP Progress Meeting – February 17-18, 2005
Overview
At least nine gliders to be used for adaptive sampling on boundaries and in interior of the “box”.
Adaptive sampling should minimize model error by (1) maximizing coverage, (2) finding fronts,
and (3) observing local changes in the heat budget.
Three central tasks:
1.
Design trajectories for glider array. Optimal trajectories will require coordinated design -relative positions of all gliders central to the design.
2.
Design feedback control algorithms to ensure coordination of gliders, in spite of currents,
unexpected events, other perturbations.
3.
Design feedback algorithms for adaptation of trajectories for glider array in response to
changing ocean dynamics and changing operational conditions.
Coordinated
Array Design
Cooperative
Control Law
Gliders
Relative positions, currents
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Sensor
measurements
Model
N.E. Leonard – ASAP Progress Meeting – February 17-18, 2005
1. Glider Array Design: Objectives
Use approximate model to derive optimal trajectories that maximize information to full model.
Requires evaluation that optimal plan from approximate model enhances performance of full model.
Match array design to ocean processes we want to observe including
Migration of warm water offshore during upwelling, onshore migration during relaxation, pattern of
these processes south of Ano Nuevo, 3-D effects, location of horizontal divergence, how deeply
surface and bottom mixing penetrates stratified water column, special patterns around topography.
Design how gliders should be coordinated to realize full potential of array.
Accommodate influence of flow field on glider navigation in array design.
Tradeoffs between optimality and robustness.
Evaluate effectiveness of glider array design.
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N.E. Leonard – ASAP Progress Meeting – February 17-18, 2005
2 . Feedback Control for Glider Coordination: Objectives
Coordinated
Array Design
Cooperative
Control Law
Gliders
Sensor
measurements
Model
Relative positions, currents
Feedback should keep the gliders in
their optimal relative positions despite
currents that push the gliders away from
the prescribed trajectories.
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Constant glider speed relative to flow.
Relative position measurements
computed every two hours and estimates
of other glider positions used.
N.E. Leonard – ASAP Progress Meeting – February 17-18, 2005
3. Adaptation of Array Design: Objectives
Coordinated
Array Design
Cooperative
Control Law
Gliders
Sensor
measurements
Model
Relative positions, currents
To “close the loop”, quantify effect of increased knowledge on design of coordinated
trajectories.
Use sensor data from gliders to re-determine metric and then recompute array design.
Accommodate changes in ocean processes (using updated model estimates) and changes in
operations (e.g., add/subtract glider).
In case of feature tracking, use sensor data directly to influence changes in subarray (path and
shape).
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N.E. Leonard – ASAP Progress Meeting – February 17-18, 2005
1. Glider Array Design: Approach
Derive performance of
sampling array from linear
data assimilation scheme
Choose metric to be
functional of error in the best
linear estimate of final state
(after data has been
assimilated).
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N.E. Leonard – ASAP Progress Meeting – February 17-18, 2005
Glider Array Design: Approach cont.
Search for trajectories corresponding to a
global minimum of the metric.
Design near-optimal trajectories:
Use parametrized family of “simple” shapes.
Parameters include
shape, size, orientation
# simple shapes to cover region.
# of sensors per shape.
Relative positions of all sensors.
Choose near optimal solutions where
value of metric is at plateau for
robustness.
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N.E. Leonard – ASAP Progress Meeting – February 17-18, 2005
Sampling Metric from Objective Analysis
Represent ocean with finite number of random variables:
Measurement matrix represents influence of M measurements on N grid points:
Priori covariance
between two grid points approximated from past observations:
Posteriori covariance of best linear estimate:
Metrics:
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N.E. Leonard – ASAP Progress Meeting – February 17-18, 2005
Computation of Optimal Trajectories
Box:
Trajectories:
Constraint:
Optimality Trajectories:
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N.E. Leonard – ASAP Progress Meeting – February 17-18, 2005
Computation of Optimal Trajectories
A Priori Correlation:
Scaled Trajectories:
Size and shape:
Speed and time:
Constraint:
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Computation of Optimal Trajectories
For
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N.E. Leonard – ASAP Progress Meeting – February 17-18, 2005
Nearly Optimal Trajectories
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N.E. Leonard – ASAP Progress Meeting – February 17-18, 2005
Minimum Error
The value of the metric
() does not depend
on
is a function of
(and ) only!
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N.E. Leonard – ASAP Progress Meeting – February 17-18, 2005
Optimal Solution for Ellipses
For ellipses, the optimum is at
Corresponds to one glider per region
of area
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N.E. Leonard – ASAP Progress Meeting – February 17-18, 2005
Sub-sampling
Sample small scale gradients and keep an acceptable model error?
Dramatic increase below
Sub-sampling limited to
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N.E. Leonard – ASAP Progress Meeting – February 17-18, 2005
Smallest Sub-Sampling Scale
Minimum sub-sampling scale for
:
Minimum side of a triangle of 3 gliders:
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August 16, 2003
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N.E. Leonard – ASAP Progress Meeting – February 17-18, 2005
Level Sets and Front Tracking
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•
Thermal Front Parameter (TFP)
•
Thermal Front
•
Warm / Cold Fronts
N.E. Leonard – ASAP Progress Meeting – February 17-18, 2005
2. Feedback Control for Glider Coordination
• Feedback control laws and convergence proofs for coordinating constant speed
vehicles to move around closed paths with uniform inter-vehicle spacing.
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N.E. Leonard – ASAP Progress Meeting – February 17-18, 2005
Feedback Control for Glider Coordination
vehicle
• Feedback provides robustness to small perturbations. In progress:
• Improving robustness to currents.
• Coordination with reduced “communication”, I.e., each glider to know relative position
with respect to only subset of other gliders.
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N.E. Leonard – ASAP Progress Meeting – February 17-18, 2005
3. Adaptation of Array Design
1)
For metric, evaluate a priori covariance from unstructured data collected by sensors.
2)
Consider inhomogeneous statistics. Reveals importance of currents (e.g., high
correlation along a jet, low correlation across a jet).
3)
Consider capturing inhomogeneity with advection term.
4)
Possibly use potentials to direct array to processes of interest.
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N.E. Leonard – ASAP Progress Meeting – February 17-18, 2005
Example: Double-Gyre Model
Test flow field:
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N.E. Leonard – ASAP Progress Meeting – February 17-18, 2005
Final Remarks and Future Directions
Metric based on objective analysis error map. Starting point is homogeneous, isotropic field.
Considering different ways to augment approach to ensure that arrays are well matched (and can
adapt) to ocean processes that we want to observe.
Near-optimal glider trajectories from parameterization of simple shapes. Parametrization
includes relative positions of gliders.
Numerical studies with elliptical trajectories yield preliminary results on nature of near-optimal
solutions. Work still to be done to find complete near-optimal coordinated array solutions.
Can choose among near-optimal solutions: seek a solution that aids robustness to currents and
contributes to better computing the metric.
Need still to fully address transit problem, I.e., how best to bring glider(s) from deployment to
near-optimal glider array. Approach: combine sampling metric with minimum-time (or energy)
metric (see Jerry’s talk).
Evaluate performance of near-optimal glider arrays first in simulation.
Develop means to evaluate success of glider array and adaptation for the experiment.
Use at least 9 gliders for coverage of “the box”.
Propose testing new level set/front tracking algorithms with gliders (perhaps before or after main
glider array coverage experiment).
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N.E. Leonard – ASAP Progress Meeting – February 17-18, 2005