kovan research lab ecal ‘09, hande Çelikkanat, 16.09.2009 a critical review of flocking models...

KOVAN Research LabECAL ‘09, Hande Çelikkanat, 16.09.2009

A Critical Review of Flocking Models

Erol Şahin and Hande Çelikkanat

KOVAN Research LabDepartment of Computer Engineering

Middle East Technical UniversityAnkara, Turkey

ECAL '09, Budapest, HungarySeptember 2009

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Flocking in Nature

• Rapid, directed movement

• No dedicated leader• No collisions• Robust and scalable• Protection against

predators• Energy efficiency• Migration over long

distances

Flocking is one of the miracles of nature in which a group of animals such as birds and fishes move and maneuver as if they were a

single creature

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Reynold's Flocking Algorithm (1987)

Assumption: sense of heading, bearing and range of neighbors

SeparationAlignment Cohesion

Individuals avoid

collisions

with their neighbors

Individuals match their heading to the average heading of their neighbors

Individuals move to the geometric center of

their neighbors

• Realistic-looking simulations of flock of birds

• Depends only on local interactions

synthesis of flocking for the first time

C. Reynolds, “Flocks, herds and schools: A distributed behavioral model,” in SIGGRAPH ’87: Proc. of the 14th annual conference on computer graphics and interactive techniques, pp. 25–34, ACM Press, July 1987

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Statistical Physics: Trying to understand the evident order

• Statistical physics tools to study the emergence of collective behavior

Agents Sensing Noise Neighborhood Environment

Mobile / stationary particles

No inertia

Range, bearing and heading of neighbors

Agent-based sensing

Sensing/actuation Local Periodic boundaries / Open space

Model

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The Mermin-Wagner Theorem: The limits of theory

A theory of equilibrium ferromagnets

Ordered phase (= global alignment of headings)

cannot emerge

in 1- or 2-D systems with no external field (= goal)

having only local interactions

at non-zero temperatures (=non-zero noise)

Short-range interactions cannot produce

long-range order

N. D. Mermin and H. Wagner, “ Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models ,” Physical Review Letters, vol. 17, no. 22, pp. 1133–1136, 1966

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Self-Driven Particles Model (Vicsek et al., 1995)


Mobile, massless particles

Constant speed

No inertia

Headings of neighbors

Actuation noise Local (in range) Periodic boundaries

Collisions allowed

Model

Heading set to the average of neighbors

Instantaneous, synchronous update of headings

Update Rule

increasing noise increasing density

local groupsrandom motion aligned motion

Phase transition from unaligned to aligned motion(disordered to ordered state)

T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen, and O. Shochet, “Novel type of phase transition in a system of self-driven particles,” Physical Review Letters, vol. 75, no. 6, 1995

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A contradiction?

Toner and Tu, 1998• Flocks are non-equilibrium dynamical systems• Hence not constrained by Mermin-Wagner theorem• Flocking is spontaneous symmetry breaking towards an arbitrary direction• d < 4: Fluctuations in local velocity of the flock so large, that motion in one part of the flock relative to the rest beats diffusion, and it becomes the principle means of information transfer

Czirok and Vicsek, 2006The particles are not stationary, but mobile within the flockThe local neighbors of a particle change in time

→ long-range interactions→ long-range order

J. Toner and Y. Tu, “Flocks, herds, and schools: A quantitative theory of flocking,” Physical Review E, vol. 58, no. 4, pp. 4828–4858, 1998A. Czirok and T. Vicsek, “Collective behavior of interacting self-propelled particles,” Physica A, vol. 373, pp. 445–454, 2007

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Vectorial Network Model (Aldana and Huepe, 2003)


Stationary particles

No inertia

Headings of neighbors

Actuation noise Either local or

random (long-range)

Static

Model

Heading set to the average of neighbors


Update Rule

Phase transition from unaligned to aligned motion... if :

1. there exists random connections (even few)

2. noise below critical level

If totally local, Mermin-Wagner Theorem applies

M. Aldana and C. Huepe, “Phase transitions in self-driven many-particle systems and related non-equilibrium models: A network approach,” Journal of Statistical Physics, vol. 112, no. 1-2, pp. 135–153, 2003

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Gregoire et al., 2003


Mobile, massless particles

No inertia

Headings and ranges of neighbors

Actuation noise

Local (Voronoi neighbors) Open space

Model

Heading averaging (α) + attraction / repulsion (β)


Update Rule

Results in coherent motionBehavior defined by α vs. β

β

α

moving droplet

immobile solid

fluid droplet

flying crystal

G. Gregoire, H. Chate, and Y. Tu, “Moving and staying together without a leader,” Physica D, vol. 181, pp. 157–170, 2003.

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Huepe et al., 2008

• Compared original SDP model (Vicsek et al., 1995) extended SDP model with attraction/repulsion

(Gregoire et al., 2003)

Unrealistically high local density values in original SDP

[No repulsive term]

Not suitable for modeling natural or robotic swarms with typically low densities (Ballerini et al., 2008)

C. Huepe and M. Aldana, “New tools for characterizing swarming systems: A comparison of minimal models,” Physica A, vol. 387, pp. 2809–2822, 2008

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Control Theory:Stability through perfect sensing

• Devise flocking algorithms with a set of control laws

• Analytically prove stability

• “Flocking” may refer to:• motion with leader

• movement towards a goal

• formation control with perfect position information

Agents Sensing Noise

Mobile particles

No inertia

Perfect

Agent-based

Heading / range / bearing sensing of neighbors

No noise

Model

Differs from physicists’ view

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A. Jadbabaie, J. Lin, and A. Morse, “Coordination of groups of mobile autonomous agents using nearest neighbor rules,” IEEE Transactions on Automatic Control, vol. 48, no. 6, pp. 988–1001, 2003.

• A theoretical explanation for emergent alignment in SDP model

• Investigated its stability by considering the changing nearest neighbor sets (neighboring graphs)

• Neglected noise

Jadbabaie et al., 2003

Stable when there exists an infinite sequence of contiguous, non-empty,

bounded time-intervals [ti, ti+1), st. across each interval, the n agents are connected to each other

Relaxed conditionthe neighboring graphs are not connected to each other, but

their union is connected

+ =

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Tanner et al., 2003• Stable control law for flocking in free space

• In fixed/dynamic topology cases

• Stability (heading convergence and collision avoidance) proved by Graph Theory and Lyapunov's theorem


Mobile mass-particles

No inertia

Range, bearing and velocity of neighbors

No noise 1. Fixed

2. Varying with time

Open space

Model

Control Law

heading alignmentattraction/repulsion

H. G. Tanner, A. Jadbabaie, and G. J. Pappas, “Stable flocking of mobile agents part i: fixed topology,” in Proceedings of the 42nd IEEEConference on Decision and Control, vol. 2. pp. 2010–2015, 2003

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Olfati-Saber, 2006


Mobile mass-particles

No inertia

ALG 1 Range, bearing and heading of neighbors

No noise Local Open space


Common goal position


Virtual agents on obstacle peripheries

Model

ALG1 equivalent to Reynolds’ algorithm

Leads to fragmentation for large groups (>10)

ALG 2 and ALG 3 generate stable flocking

(at the cost of more unrealistic sensing assumptions)

R. Olfati-Saber, “Flocking for multi-agent dynamic systems: Algorithms and theory,” IEEE Transactions on Automatic Control, vol. 51, no. 3, pp. 401–420, 2006

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What about robots?

Imperfect sensing and actuation

(very high noise)

Asynchronous decision making of agents

Inertial effects

(heading updates not instantaneous)

Not agent-based, but raw sensory readings (Typical of IR and sonar sensors)

Not continuous, but highly discrete sensing

(Typical of IR and sonar sensors)

Systematic and stochastic delays in sensing

Effects of physical volume

(Quasi-static particles which cannot move within the flock easily)


• A good option to unify the theories in a physical environment• We have implemented self-organized flocking on the Kobot robot

platform• and studied

– Leaderless and goal-free flocking– An unacknowledged, informed minority of the group steering the flock– Migration over long distances

H. Celikkanat, A. E. Turgut, and E. Sahin, Guiding a robot flock via informed robots, DARS 2008

H. Celikkanat, Control of a mobile robot swarm via informed robots, M.Sc. Thesis, Middle East Technical University, Ankara, Turkey, 2008

F. Gokce and E. Sahin,To flock or not to flock: The pros and cons of flocking in long-range “migration” of mobile robot swarms, AAMAS 2008

A. E. Turgut, C. Huepe, H. Celikkanat, F. Gokce, and E. Sahin,Modeling phase transition in self-organized mobile robot flocks, ANTS 2008

A. E. Turgut, H. Celikkanat, F. Gokce, and E. Sahin,Self-organized flocking in mobile robot swarms, Swarm Intelligence, vol. 2, no. 2-4, 2008

A. E. Turgut, H. Celikkanat, F. Gokce, and E. Sahin, Self-organized flocking with a mobile robot swarm, AAMAS 2008

Robots in the big picture

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Thank you

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kovan research lab ecal ‘09, hande Çelikkanat, 16.09.2009 a critical review of flocking models...

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