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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 1
Cooperative Control and Mobile Sensor Networks
Cooperative Control, Part II
Naomi Ehrich Leonard
Mechanical and Aerospace EngineeringPrinceton University
and Electrical Systems and Automation University of Pisa
[email protected], www.princeton.edu/~naomi
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 2
Collective Motion Stabilization Problem
• 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))
• Interconnected system has high level of symmetry. Consequence: reduction techniques of geometric control.(e.g., Newton, Holmes, Weinstein, Eds., 2002 and cyclic pursuit, Marshall, Broucke, Francis, 2004).
with Rodolphe Sepulchre (University of Liege), Derek Paley (Princeton)
Phase-oscillator models have been widely studied in the neuroscience and physics literature. They represent simplification of more complex oscillator models in which the uncoupled oscillator dynamics each have an attracting limit cycle in a higher-dimensional state space. Under the assumption of weak coupling, higher-dimensional models are reduced to phase models (singular perturbation or averaging methods).
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 3
Overview of Stabilization of Collective Motion
• We consider first particles moving in the plane each with constant speed and steering control.
• The configuration of each particle is its position in the plane and the orientation of its velocity vector.
• Synchrony of collective motion is measured by the relative phasing and relative spacing of particles.
• We observe that the norm of the average linear momentum of the group is a key control parameter: it is maximal for parallel motions and minimal for circular motions around a fixed point.
• We exploit the analogy with phase models of couple oscillators to design steering control laws that stabilize either parallel or circular motion.
• Steering control laws are gradients of phase potentials that control relative orientation and spacing potentials that control relative position.
• Design can be made systematic and versatile. Stabilizing feedbacks depend on a restricted number of parameters that control the shape and the level of synchrony of parallel or circular formations.
• Yields low-order parametric family of stabilizable collective motions: offers a set of primitives that can be used to solve path planning or optimization tasks at the group level.
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 4
Key References
[1] Sepulchre, Paley, Leonard, “Stabilization of planar collective motion: All-to-all communication,” IEEE TAC, June 2007, in press.
[2] Sepulchre, Paley, Leonard, “Stabilization of planar collective motion with limited communication,” IEEE TAC, conditionally accepted.
[3] Moreau, “Stability of multiagent systems with time-dependent communication links,” IEEE TAC, 50(2), 2005.
[5] Scardovi, Sepulchre, “Collective optimization over average quantities,” Proc. IEEE CDC, 2006.
[6] Scardovi, Leonard, Sepulchre, “Stabilization of collective motion in the three dimensions: A consensus approach,” submitted.
[7] Swain, Leonard, Couzin, Kao, Sepulchre, “Alternating spatial patterns for coordinated motion, submitted.
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 5
Planar Unit-Mass Particle Model
Steering control
Speed control
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 6
Planar Particle Model: Constant (Unit) Speed
[Justh and Krishnaprasad, 2002]
Shape variables:
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 7
Relative Equilibria
[Justh and Krishnaprasad, 2002]
Then 3N-3 dimensional reduced space is
If steering control only a function of shape variables:
And only relative equilibria are1. Parallel motion of all particles.2. Circular motion of all particles on the same circle.
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 8
Phase Model
Then reduced model corresponds to phase dynamics:
If steering control only a function of relative phases:
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 9
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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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 10
Key Ideas
is phase coherence, a measure of synchrony, and it is equal to magnitude of average linear momentum of group.[Kuramoto 1975,
Strogatz, 2000]
Average linear momentumof group:
Centroid of phases of group:
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 11
Synchronized state
Balanced state
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 12
Phase Potential
1. Construct potential from synchrony measure, 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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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 13
Phase Potential
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 14
Phase Potential: Stabilized Solutions
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 15
Stabilization of Circular Formations: Spacing Potential
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 16
Stabilization of Circular Formations: Spacing Potential
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 17
Stabilization of Circular Formations: Spacing Potential
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 18
Composition of Phasing and Spacing Potentials
Can also prove local exponential stability of isolated local minima.
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 19
Phase + Spacing Gradient Control
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 20
Stabilization of Higher Momenta
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 21
Stabilization of Higher Momenta
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 22
Symmetric Balanced Patterns
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 23
Symmetric Balanced Patterns
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 24
Symmetric Patterns, N=12
QuickTime™ and aVideo decompressor
are needed to see this picture.
M=1,2,3
M=4,6,12
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 25
Stabilization of Collective Motion with Limited Communication
• Design concept naturally developed for all-to-all communication is recovered in a systematic way under quite general assumptions on the network communication:
Approach 1. Design potentials based on graph Laplacian so that control laws respect communication constraints. (Requires time-invariant and connected communication topology and gradient control laws require bi-directional communication).
Approach 2. Use consensus estimators designed for Euclidean space in the closed-loop system dynamics to obtain globally convergent consensus algorithms in non-Euclidean space. Generalize methodology to communication topology that may be time-varying, unidirectional and not fully connected at any given instant of time. Requires passing of relative estimates of averaged quantities in addition to relative configuration variables.
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 26
Graph Representation of Communication
Particle = node Edge from k to j = comm link from particle k to j
(Jadbabaie, Lin, Morse 2003, Moreau 2005)
1
7
8
6 5
2
4
9
3
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 27
Circulant Graphs
P.J. Davis, Circulant Matrices. John Wiley & Sons, Inc., 1979.
(undirected)
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 28
Time-Varying Graphs
Moreau, 2004
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 29
Phase Synchronization and Balancing: Time Invariant Communication
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 30
Phase Synchronization and Balancing: Time Invariant Communication
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 31
Well-Studied Result in Euclidean Space
See also Moreau 2005, Jadbabaie et al 2004 for local results.
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 32
Achieving Nearly Global Results for Time-Varying, Directed Graphs
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 33
Achieving Nearly Global Results for Time-Varying, Directed Graphs
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 34
Parallel and Circular Formations: Time-Invariant Case
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 35
Parallel and Circular Formations: Time-Varying Case
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 36
Further Results
• Resonant patterns.
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 37
Non-constant Curvature
QuickTime™ and a decompressor
are needed to see this picture.
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 38
Planar Particle Model: Oscillatory Speed Model
Swain, Leonard, Couzin, Kao, Sepulchre, submitted Proc. IEEE CDC, 2007
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 39
Two Sets of Coupled Oscillator Dynamics
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 40
Steady State Circular Patterns
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 41
Steady State Circular Patterns for Individual
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 42
Stabilization of Circular Patterns
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 43
Circular Patterns with Prescribed Relative Phasing
QuickTime™ and a decompressor
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 44
Stabilization of Circular Patterns with Noise
QuickTime™ and aTIFF (LZW) decompressor
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N.E. Leonard – U. Pisa – 18-20 April 2007Slide 45
Convergence with Limited Communication
Definition of blind spot angle
Simulation with blind spot