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

Cooperative Controland

Mobile Sensor Networks

Cooperative Control, Part II

Naomi Ehrich Leonard

Mechanical and Aerospace EngineeringPrinceton University

and Electrical Systems and AutomationUniversity of Pisa

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

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 physicsliterature. They represent simplification of more complex oscillator models in which theuncoupled oscillator dynamics each have an attracting limit cycle in a higher-dimensionalstate space. Under the assumption of weak coupling, higher-dimensional models arereduced 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: itis 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 thatstabilize either parallel or circular motion.

• Steering control laws are gradients of phase potentials that control relative orientation and spacingpotentials that control relative position.

• Design can be made systematic and versatile. Stabilizing feedbacks depend on a restricted numberof 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 canbe used to solve path planning or optimization tasks at the group level.

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: Aconsensus 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

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.

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 suggestsynchrony robust to individual discrepancies (see Kuramoto model analyses).

N.E. Leonard – U. Pisa – 18-20 April 2007Slide 10

Key Ideas

⇔

is phase coherence, a measureof synchrony, and it is equal tomagnitude of average linearmomentum 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

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

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

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

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

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

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

N.E. Leonard – U. Pisa – 18-20 April 2007Slide 24

Symmetric Patterns, N=12

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 wayunder quite general assumptions on the network communication:

Approach 1. Design potentials based on graph Laplacian so that control laws respectcommunication constraints. (Requires time-invariant and connected communication topologyand gradient control laws require bi-directional communication).

Approach 2. Use consensus estimators designed for Euclidean space in the closed-loopsystem dynamics to obtain globally convergent consensus algorithms in non-Euclidean space.Generalize methodology to communication topology that may be time-varying, unidirectionaland not fully connected at any given instant of time. Requires passing of relative estimates ofaveraged quantities in addition to relative configuration variables.

N.E. Leonard – U. Pisa – 18-20 April 2007Slide 26

Graph Representation of Communication

Particle = nodeEdge from k to j = comm link from particle k to j

(Jadbabaie, Lin, Morse2003, 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)

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

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.

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

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

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

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

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

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

N.E. Leonard – U. Pisa – 18-20 April 2007Slide 44

Stabilization of Circular Patterns with Noise

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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