Transcript Cooperative Control and Mobile Sensor Networks in the Ocean Naomi Ehrich Leonard Mechanical & Aerospace Engineering Princeton University [email protected], www.princeton.edu/~naomi Collaborators: Derek Paley (grad student), Francois Lekien,

Cooperative Control and Mobile Sensor
Networks in the Ocean
Naomi Ehrich Leonard
Mechanical & Aerospace Engineering
Princeton University
[email protected], 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)
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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.
Ocean
Processes
Ocean Model
Model Prediction
Data Assimilation
Adaptive sampling
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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.
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Goals of ASAP Program
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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.
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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.
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Glider
Plan
GCT = Optimal Coordinated Trajectory = coordinated pattern for the glider fleet.
Coordination = Prescription of the relative location of all gliders (as a function of time).
Feedback to large changes and disturbances. Gross replanning 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.
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Ocean models.
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SIO glider
WHOI glider
Glider Plan
km
km
km
km
Candidate default OCT with
grid for glider tracks.
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Adaptation
OCT for increased sampling in
southwest corner of ASAP box.
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Data Flow: Actual and Virtual Experiments
Princeton Glider Coordinated Control System (GCCS)
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Princeton Glider Coordinated Control System
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Collective Motion Problems
Coordinate group of individually controlled systems: Mobile sensor networks
Reconfigurable formations for feature tracking.
Patterns for synoptic area coverage.
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Fumin Zhang
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Derek Paley
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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
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Sampling Metric: Objective Analysis Error
Scalar field viewed as a random variable:
is a priori mean. Covariance of fluctuations around mean is
Data collected consists of
is OA estimate that minimizes
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AOSN Performance Metric
Rudnick et al,
2004
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Coverage Metric: Objective Analysis Error
Rudnick et al,
2004
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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.
No flow.
Metric = 0.018
Horizontal flow.
Metric = 0.020
Vertical flow.
Metric = 0.054
Flow shown is 2% of
vehicle speed.
No heading coupling.
Metric = 0.236
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Collective Motion for Mobile Sensor Networks
Sensor platforms
coordinate motion on
patterns so data
collected minimizes
uncertainty in sampled
field.
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Modeling, Analysis and Synthesis of Collective Motion
Photos:
Norbert Wu
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Collective motion patterns distinguished by level of synchrony.
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Collective Motion Stabilization Problem
with R. Sepulchre, D. Paley
• 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))
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Planar Particle Model: Constant Speed & Steering Control
[Justh and Krishnaprasad, 2002]
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Symmetry and Equilibria
Let
(shape control)
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]
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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).
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Key Ideas
Average linear momentum
of group:
Centroid of phases of group:

[Kuramoto 1975,
Strogatz, 2000]
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is phase coherence, a measure
of synchrony, and it is equal to
magnitude of average linear
momentum of group.
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Synchronized state
Balanced state
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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:
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Phase Potential: Stabilized Solutions
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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.
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Interconnection Topology as Graph
Particle = node
Edge = communication link
See also Jadbabaie, Lin, Morse 2003, Moreau 2005
Example: Ring topology.
1
2
9
3
8
4
7
6
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Quadratic Form Induced by Graphs
(Olfati & Murray, 2004)
Example: Ring topology.
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Phase Potential
Phase Potential:
[SPL]
Gradient of Phase Potential:
(see also Jadbabaie et al, 2004)
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Spacing Potential
Spacing potential:
Gradient control:
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Composite Potential
Phase
Spacing
[SPL]
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Phase + Spacing Gradient Control: Ring
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Isolating Symmetric Patterns
Consider higher harmonics of the phase differences in the coupling
(K. Okuda, Physica D, 1993):
2
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Phase Potentials with Higher Harmonics
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Spacing + Phase Potentials: Complete Graph
M=1,2,3
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Multi-Scale and Multi-Graph
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ASAP Virtual Control Room
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Glider Coordinated Trajectories
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Glider GCT Optimizer
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Glider Planner Status
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Glider Positions
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Glider Prediction
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Glider OA Error Map
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Glider OA Flow
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OA Metrics
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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).
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