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Hamilton Institute Seminar 06/24/2005. Distributed Motion Coordination: From Swarming to Synchronization. Ali Jadbabaie. Department of Electrical and Systems Engineering and GRASP Laboratory University of Pennsylvania. Distributed Coordination in nature.
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Hamilton Institute Seminar 06/24/2005
Distributed Motion Coordination:From Swarming to Synchronization
Ali Jadbabaie
Department of Electrical and Systems Engineering
and GRASP Laboratory
University of Pennsylvania
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Complexity, Statistical Physics,
emergence of collective behavior
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r
neighbors of
agent i
agent i
Vicsek’s kinematic model
How does global behavior emerge from local interactions?
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= speed
MAIN QUESTION :Under what conditions do all headings converge to the same value and agents reach a consensus on where to go?
= heading
Distributed consensus algorithm
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We use graphs to model neighboring relations
Agent i’s neighborhood
The neighboring relation is represented by a fixed graph G, or a collection of graphs G1, G2,…Gm
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switching signal ,
adjacency matrix
finite set of indices corresponding to allgraphs overnvertices.
Valence matrix
4
3
5
2
6
1
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Theorem (Jadbabaie et al. 2003): If there is a sequence of bounded, non-overlapping time intervals Tk, such that over any interval of length Tk, the network of agentsis “jointly connected ”, then all agents move in a formation.
This happens to be both necessary and sufficient for
exponential coordination, boundedness of intervals not required for asymptotic coordination. (Moreau ’04, Ren & Beard ‘05)
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Choose x = tan()
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Dynamic Topology
Local sensing/communication
Graph changes with time
Control is discontinuous
Non smooth Lyapunov theory
Fixed Topology
Fixed (logical) network
Graph is constant
Control is smooth
Classic Lyapunov theory
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For both fixed and dynamic topology:If the neighboring graph stays connected, all agent velocity vectors become asymptotically the same, collisions between interconnectedagents are avoided and the system approaches a configuration that minimizes all agent potentials.
We could shape potentials for any desired configuration,
and also update it as the objective changes.
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History of the coupled oscillators
injection locking in RF circuits
See the book by Steven Strogatz
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1
B is the (n x e) incidence matrix of graph G.
2
3
4
W is diagonal
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B is the incidence matrix of the graph representing the
interconnection of oscillators
Simple case: all oscillators are identical
Theorem:Consider the unperturbed Kuramoto Model defined over an arbitrary connected graph with incidence matrix B. For any given 0 and any positive value of the coupling, the vector is a locally asymptotically stable equilibrium solution. Furthermore, the rate of approach to equilibrium is no worse than
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Thus
For |i|</2 for a connected graph, all trajectories will converge to S
Therefore, all velocity vectors will synchronize.
But, this stability result is not global. In the case of the ring topology is not the only equilibrium. This is due to the fact that B and BT have the same null space!
is also stable:
Fixed points:
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Where e is the number of edges in the graph
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Sum of pair-wise
potentials
Supply at each node
Subject to:
Lagrangian
Kuramoto model is the Subgradient algorithm for solving the dual
Shor 87, Tsitsiklis ’86
Subgradient algorithm
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Gij = Aij cos( ij)and Gij>0 when i and j are neighbors. Synchronized state is stable independent of delay.
Proof sketch: Usethe following
V() as a Lyapunov/Krasovski
functional :
Corollary:If [Gij] is the adjacency matrix of a connected graph, then the continuous time, consensus problem is asymptotically stable with arbitrary time delays
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The input
Minimizes the
misalignment potential
The control law minimizes the potential by following its gradient.
But we can’t measure the headings of neighbors W/O communication
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Yw
qj
j
lij
qi
bij
i
Xw
Knowing relative heading would mean having binocular vision or solving structure from motion. This would require multiple visible features on each agent.
Measured Quantities
Pigeons and flies are capable of all 3 measurements!.
Wang & Frost, Nature, 1992, Fry and Dickinson , Science 2003
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joint connectivity in time flocking
Proof based on construction of Lyapunov function whose derivative is the quadratic form of a state dependent Laplacian, and the following lemma
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Agent i’s neighborhood
Consider a group of N agents with different velocity vectorsand constant, unit speed (extension to dynamic case possible).
θ is the heading and φ is the attitude.
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Zw
TiS
X iq
vj
aij
vi
X if
Xw
Yw
Bullo, Murray and Sarti, “Control on the Sphere and Reduced Attitude Stabilization”, 1995
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Zw
TiS
Yij
Xq
vj
vi
Xf
Yw
For any two agents iand j :
is the component of vj orthogonal to vi.
aij
Xw
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Zw
TiS
Yij
Xq
vj
vi
Xf
Xw
Theorem [ Moshtagh, Jadbabaie and Daniilidis, CDC’05]:
Consider the system of N equations
If the proximity graph of the agents is fixed and connected, then the control laws
result in flocking. Furthermore the consensus state is locally asymptotically stable. A similar result holds in the case of switching graphs, if the union graph is connected in time.
aij
Yw
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Lyapunov function: measure of discrepancy between velocity vectors
velocity vectors will synchronize.
In 2-d this is vT L v
It Can be shown that
Using LaSalle’s invariance principle, all trajectories converge to the largest invariant set within the set:
For 0<|i|<, could also use ||||^2 as a Lyap. Function. For a sublevel set inside |i|</2, all trajectories converge to a set where i =j
Use ||||^2 as Lyapunov function on this set, then all trajectories will converge to
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vj
Qij
Visual Servoing Approach
Equation of Motion
Agent i
We can solve for the input in terms of the measurements.
We can construct distributed control laws for flocking, based
on visual sensing and measurement of bearing, time to collision
and optical flow. No communication or relative distance or heading
information is needed
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Simulations
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(How do flies do it?)
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