Download NE Leonard – U. Pisa – 18-20 April 2007

Cooperative Control
and
Mobile Sensor Networks
Cooperative Control, Part II
Naomi Ehrich Leonard
Mechanical and Aerospace Engineering
Princeton University
and Electrical Systems and Automation
University of Pisa
[email protected],
www.princeton.edu/~naomi
Slide 1
N.E. Leonard – U. Pisa – 18-20 April 2007
Collective Motion Stabilization Problem
with Rodolphe Sepulchre (University of Liege), Derek Paley (Princeton)
• 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)
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).
(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).
Slide 2
N.E. Leonard – U. Pisa – 18-20 April 2007
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.
Slide 3
N.E. Leonard – U. Pisa – 18-20 April 2007
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.
Slide 4
N.E. Leonard – U. Pisa – 18-20 April 2007
Planar Unit-Mass Particle Model
Steering control
Speed control
Slide 5
N.E. Leonard – U. Pisa – 18-20 April 2007
Planar Particle Model: Constant (Unit) Speed
[Justh and Krishnaprasad, 2002]
Shape variables:
Slide 6
N.E. Leonard – U. Pisa – 18-20 April 2007
Relative Equilibria
If steering control only a function of shape variables:
Then 3N-3 dimensional reduced space is
And only relative equilibria are
1. Parallel motion of all particles.
2. Circular motion of all particles on the same circle.
[Justh and Krishnaprasad, 2002]
Slide 7
N.E. Leonard – U. Pisa – 18-20 April 2007
Phase Model
If steering control only a function of relative phases:
Then reduced model corresponds to phase dynamics:
Slide 8
N.E. Leonard – U. Pisa – 18-20 April 2007
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 9
N.E. Leonard – U. Pisa – 18-20 April 2007
Key Ideas
Average linear momentum
of group:
Centroid of phases of group:

[Kuramoto 1975,
Strogatz, 2000]
Slide 10
is phase coherence, a measure
of synchrony, and it is equal to
magnitude of average linear
momentum of group.
N.E. Leonard – U. Pisa – 18-20 April 2007
Synchronized state
Balanced state
Slide 11
N.E. Leonard – U. Pisa – 18-20 April 2007
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:
Slide 12
N.E. Leonard – U. Pisa – 18-20 April 2007
Phase Potential
Slide 13
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Phase Potential: Stabilized Solutions
Slide 14
N.E. Leonard – U. Pisa – 18-20 April 2007
Stabilization of Circular Formations: Spacing Potential
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N.E. Leonard – U. Pisa – 18-20 April 2007
Stabilization of Circular Formations: Spacing Potential
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Stabilization of Circular Formations: Spacing Potential
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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 2007
Phase + Spacing Gradient Control
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Stabilization of Higher Momenta
Slide 20
N.E. Leonard – U. Pisa – 18-20 April 2007
Stabilization of Higher Momenta
Slide 21
N.E. Leonard – U. Pisa – 18-20 April 2007
Symmetric Balanced Patterns
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Symmetric Balanced Patterns
Slide 23
N.E. Leonard – U. Pisa – 18-20 April 2007
Symmetric Patterns, N=12
M=1,2,3
QuickTime™ and a
Video decompressor
are needed to see this picture.
M=4,6,12
Slide 24
N.E. Leonard – U. Pisa – 18-20 April 2007
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.
Slide 25
N.E. Leonard – U. Pisa – 18-20 April 2007
Graph Representation of Communication
Particle = node
Edge from k to j = comm link from particle k to j
1
2
9
3
8
4
7
6
Slide 26
(Jadbabaie, Lin, Morse
2003, Moreau 2005)
N.E. Leonard – U. Pisa – 18-20 April 2007
5
Circulant Graphs
(undirected)
P.J. Davis, Circulant Matrices. John Wiley & Sons, Inc., 1979.
Slide 27
N.E. Leonard – U. Pisa – 18-20 April 2007
Time-Varying Graphs
Moreau, 2004
Slide 28
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Phase Synchronization and Balancing: Time Invariant Communication
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Phase Synchronization and Balancing: Time Invariant Communication
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Well-Studied Result in Euclidean Space
See also Moreau 2005, Jadbabaie et al 2004 for local results.
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Achieving Nearly Global Results for Time-Varying, Directed Graphs
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Achieving Nearly Global Results for Time-Varying, Directed Graphs
Slide 33
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Parallel and Circular Formations: Time-Invariant Case
Slide 34
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Parallel and Circular Formations: Time-Varying Case
Slide 35
N.E. Leonard – U. Pisa – 18-20 April 2007
Further Results
• Resonant patterns.
Slide 36
N.E. Leonard – U. Pisa – 18-20 April 2007
Non-constant Curvature
QuickTime™ and a
decompressor
are needed to see this picture.
Slide 37
N.E. Leonard – U. Pisa – 18-20 April 2007
Planar Particle Model: Oscillatory Speed Model
Swain, Leonard, Couzin, Kao, Sepulchre,
submitted Proc. IEEE CDC, 2007
Slide 38
N.E. Leonard – U. Pisa – 18-20 April 2007
Two Sets of Coupled Oscillator Dynamics
Slide 39
N.E. Leonard – U. Pisa – 18-20 April 2007
Steady State Circular Patterns
Slide 40
N.E. Leonard – U. Pisa – 18-20 April 2007
Steady State Circular Patterns for Individual
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N.E. Leonard – U. Pisa – 18-20 April 2007
Stabilization of Circular Patterns
Slide 42
N.E. Leonard – U. Pisa – 18-20 April 2007
Circular Patterns with Prescribed Relative Phasing
QuickTime™ and a
decompressor
are needed to see this picture.
Slide 43
N.E. Leonard – U. Pisa – 18-20 April 2007
Stabilization of Circular Patterns with Noise
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Slide 44
N.E. Leonard – U. Pisa – 18-20 April 2007
Convergence with Limited Communication
Definition of blind spot angle
Simulation with blind spot
Slide 45
N.E. Leonard – U. Pisa – 18-20 April 2007