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) Slide 1/46 N. E. Leonard – MBARI – August 1, 2006
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 Prediction Data Assimilation Adaptive sampling Slide 2/46 N. E. Leonard – MBARI – August 1, 2006
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 Do. D/ONR Multi-Disciplinary University Research Initiative (MURI) with additional funding from ONR and the Packard Foundation. Slide 3/46 N. E. Leonard – MBARI – August 1, 2006
Goals of ASAP Program 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 3 D 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. 1. Locate the temperature and salinity fronts and predict acoustic propagation. Slide 4/46 N. E. Leonard – MBARI – August 1, 2006
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 MB 2006). Preliminary field work in March 2006 in Buzzard’s Bay, MA and May 2006 in Great South Channel. Slide 5/46 N. E. Leonard – MBARI – August 1, 2006
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. Slide 6/46 Ocean models. N. E. Leonard – MBARI – August 1, 2006
Glider Plan SIO glider WHOI glider km km Candidate default OCT with grid for glider tracks. Slide 7/46 Adaptation OCT for increased sampling in southwest corner of ASAP box. N. E. Leonard – MBARI – August 1, 2006
Data Flow: Actual and Virtual Experiments Princeton Glider Coordinated Control System (GCCS) Slide 8/46 N. E. Leonard – MBARI – August 1, 2006
Princeton Glider Coordinated Control System Slide 9/46 N. E. Leonard – MBARI – August 1, 2006
Collective Motion Problems Coordinate group of individually controlled systems: Mobile sensor networks Reconfigurable formations for feature tracking. Fumin Zhang Slide 10/46 Patterns for synoptic area coverage. Derek Paley N. E. Leonard – MBARI – August 1, 2006
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 Slide 11/46 N. E. Leonard – MBARI – August 1, 2006
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 Slide 12/46 N. E. Leonard – MBARI – August 1, 2006
AOSN Performance Metric Rudnick et al, 2004 Slide 13/46 N. E. Leonard – MBARI – August 1, 2006
Coverage Metric: Objective Analysis Error Rudnick et al, 2004 Slide 14/46 N. E. Leonard – MBARI – August 1, 2006
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 Slide 15/46 N. E. Leonard – MBARI – August 1, 2006
Collective Motion for Mobile Sensor Networks Sensor platforms coordinate motion on patterns so data collected minimizes uncertainty in sampled field. Slide 16/46 N. E. Leonard – MBARI – August 1, 2006
Modeling, Analysis and Synthesis of Collective Motion Photos: Norbert Wu Collective motion patterns distinguished by level of synchrony. Slide 17/46 N. E. Leonard – MBARI – August 1, 2006
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)) Slide 18/46 N. E. Leonard – MBARI – August 1, 2006
![Planar Particle Model: Constant Speed & Steering Control [Justh and Krishnaprasad, 2002] Slide 19/46](https://present5.com/presentation/5a945a69ebe0a9bb85ab54000e7cbf9e/image-19.jpg "Planar Particle Model: Constant Speed & Steering Control [Justh and Krishnaprasad, 2002] Slide 19/46")
Planar Particle Model: Constant Speed & Steering Control [Justh and Krishnaprasad, 2002] Slide 19/46 N. E. Leonard – MBARI – August 1, 2006
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] Slide 20/46 N. E. Leonard – MBARI – August 1, 2006
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). Slide 21/46 N. E. Leonard – MBARI – August 1, 2006
Key Ideas Average linear momentum of group: Centroid of phases of group: [Kuramoto 1975, Strogatz, 2000] Slide 22/46 is phase coherence, a measure of synchrony, and it is equal to magnitude of average linear momentum of group. N. E. Leonard – MBARI – August 1, 2006
Synchronized state Balanced state Slide 23/46 N. E. Leonard – MBARI – August 1, 2006
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: Slide 24/46 N. E. Leonard – MBARI – August 1, 2006
Phase Potential: Stabilized Solutions Slide 25/46 N. E. Leonard – MBARI – August 1, 2006
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. Slide 26/46 N. E. Leonard – MBARI – August 1, 2006
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 Slide 27/46 5 N. E. Leonard – MBARI – August 1, 2006
Quadratic Form Induced by Graphs (Olfati & Murray, 2004) Example: Ring topology. Slide 28/46 N. E. Leonard – MBARI – August 1, 2006
![Phase Potential: [SPL] Gradient of Phase Potential: (see also Jadbabaie et al, 2004) Slide](https://present5.com/presentation/5a945a69ebe0a9bb85ab54000e7cbf9e/image-29.jpg "Phase Potential: [SPL] Gradient of Phase Potential: (see also Jadbabaie et al, 2004) Slide")
Phase Potential: [SPL] Gradient of Phase Potential: (see also Jadbabaie et al, 2004) Slide 29/46 N. E. Leonard – MBARI – August 1, 2006
Spacing Potential Spacing potential: Gradient control: Slide 30/46 N. E. Leonard – MBARI – August 1, 2006
![Composite Potential Phase Spacing [SPL] Slide 31/46 N. E. Leonard – MBARI – August](https://present5.com/presentation/5a945a69ebe0a9bb85ab54000e7cbf9e/image-31.jpg "Composite Potential Phase Spacing [SPL] Slide 31/46 N. E. Leonard – MBARI – August")
Composite Potential Phase Spacing [SPL] Slide 31/46 N. E. Leonard – MBARI – August 1, 2006
Phase + Spacing Gradient Control: Ring Slide 32/46 N. E. Leonard – MBARI – August 1, 2006
Isolating Symmetric Patterns Consider higher harmonics of the phase differences in the coupling (K. Okuda, Physica D, 1993): 2 Slide 33/46 N. E. Leonard – MBARI – August 1, 2006
Phase Potentials with Higher Harmonics Slide 34/46 N. E. Leonard – MBARI – August 1, 2006
Spacing + Phase Potentials: Complete Graph M=1, 2, 3 M=4, 6, 12 Slide 35/46 N. E. Leonard – MBARI – August 1, 2006
Multi-Scale and Multi-Graph Slide 36/46 N. E. Leonard – MBARI – August 1, 2006
ASAP Virtual Control Room Slide 37/46 N. E. Leonard – MBARI – August 1, 2006
Slide 38/46 N. E. Leonard – MBARI – August 1, 2006
Glider Coordinated Trajectories Slide 39/46 N. E. Leonard – MBARI – August 1, 2006
Glider GCT Optimizer Slide 40/46 N. E. Leonard – MBARI – August 1, 2006
Glider Planner Status Slide 41/46 N. E. Leonard – MBARI – August 1, 2006
Glider Positions Slide 42/46 N. E. Leonard – MBARI – August 1, 2006
Glider Prediction Slide 43/46 N. E. Leonard – MBARI – August 1, 2006
Glider OA Error Map Slide 44/46 N. E. Leonard – MBARI – August 1, 2006
Glider OA Flow Slide 45/46 N. E. Leonard – MBARI – August 1, 2006
OA Metrics Slide 46/46 N. E. Leonard – MBARI – August 1, 2006
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). Slide 47/46 N. E. Leonard – MBARI – August 1, 2006
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