Distributed motion coordination of robotic networks Lecture 4 - PowerPoint PPT Presentation

Distributed motion coordination of robotic networks Lecture 4 – deployment Jorge Cort´ es Applied Mathematics and Statistics Baskin School of Engineering University of California at Santa Cruz http://www.ams.ucsc.edu/˜jcortes Summer School on Geometry Mechanics and Control Centro Internacional de Encuentros Matem´ aticos Castro Urdiales, June 25-29, 2007 ,

Roadmap Lecture 1: Introduction, examples, and preliminary notions Lecture 2: Models for cooperative robotic networks Lecture 3: Rendezvous Lecture 4: Deployment Lecture 5: Agreement ,

Today 1 Deployment – Basic motion coordination capability 2 Non-deterministic continuous-time dynamical systems – nonsmooth stability analysis 3 Robustness – against agents’ arrivals and departures ,

Outline 1 Deployment Expected-value deployment Area deployment Expected-value deployment with limited-range interactions 2 Deployment: basic behaviors Nonsmooth stability analysis Multi-center disk-covering and sphere-packing 3 Conclusions ,

Deployment Objective: optimal task allocation and space partitioning optimal placement and tuning of sensors Constraints: algorithms amenable to implementation in a network adaptive versus static distributed versus centralized formal validation versus heuristics increasingly important w/ complexity of network, task, environment, constraints truly implementable on experimental testbeds asynchronous, delay, limited bandwidth, limited energy, interference ,

Coverage optimization DESIGN of performance metrics 1 how to cover a region with n minimum-radius overlapping disks? 2 how to design a minimum-distorsion (fixed-rate) vector quantizer? (Lloyd ’57) 3 where to place mailboxes in a city / cache servers on the internet? ANALYSIS of cooperative distributed behaviors 4 how do animals share territory? what if every fish in a swarm goes toward center of own dominance region? Barlow, Hexagonal territories, Animal Behavior , 1974 5 what if each vehicle goes to center of mass of own Voronoi cell? 6 what if each vehicle moves away from closest vehicle? ,

Top-down: expected-value deployment Objective: Given sensors/nodes/robots/sites ( p 1 , . . . , p n ) moving in environment Q achieve optimal coverage defined according to ,

Top-down: expected-value deployment Objective: Given sensors/nodes/robots/sites ( p 1 , . . . , p n ) moving in environment Q achieve optimal coverage defined according to Scenario 1 —expected value performance measure given distribution density function φ � � q − p i � 2 � minimize H C ( p 1 , . . . , p n ) = E φ min i ,

Scenario 1: coverage algorithm Name: Coverage behavior Goal: distributed optimal agent deployment (i) own Voronoi cell computation Requires: Definition (ii) centroid computation At each communication round, each agent: 1: acquire neighbors’ positions 2: compute own dominance region Computation 3: follow gradient – move towards centroid Caveat: convergence only to local minimum of H C ,

Simulation initial configuration gradient descent final configuration ,

Scenario 1: technical approach 1 Alternative formulation ( f : R + �→ R + , differentiable, non-decreasing) � � � n � E φ min f ( � q − p i � ) = f ( � q − p i � ) φ ( q ) dq i V i ( P ) i =1 � n � ≤ f ( � q − p i � ) φ ( q ) dq W i i =1 ,

Scenario 1: technical approach 1 Alternative formulation ( f : R + �→ R + , differentiable, non-decreasing) � � � n � E φ min f ( � q − p i � ) = f ( � q − p i � ) φ ( q ) dq i V i ( P ) i =1 � n � ≤ f ( � q − p i � ) φ ( q ) dq W i i =1 2 Compute decentralized gradient � ∂ H C ∂ ( P ) = f ( � q − p i � ) φ ( q ) dq ∂p i ∂p i V i ( P ) ,

Scenario 1: technical approach 1 Alternative formulation ( f : R + �→ R + , differentiable, non-decreasing) � � � n � E φ min f ( � q − p i � ) = f ( � q − p i � ) φ ( q ) dq i V i ( P ) i =1 � n � ≤ f ( � q − p i � ) φ ( q ) dq W i i =1 2 Compute decentralized gradient � ∂ H C ∂ ( P ) = f ( � q − p i � ) φ ( q ) dq ∂p i ∂p i V i ( P ) � f ( � q − p i � ) � n i ( q ) , ∂q + � φ ( q ) dq ∂p i ∂V i ( P ) ,

Scenario 1: technical approach 1 Alternative formulation ( f : R + �→ R + , differentiable, non-decreasing) � � � n � E φ min f ( � q − p i � ) = f ( � q − p i � ) φ ( q ) dq i V i ( P ) i =1 � n � ≤ f ( � q − p i � ) φ ( q ) dq W i i =1 2 Compute decentralized gradient � ∂ H C ∂ ( P ) = f ( � q − p i � ) φ ( q ) dq ∂p i ∂p i V i ( P ) � f ( � q − p i � ) � n i ( q ) , ∂q + � φ ( q ) dq ∂p i ∂V i ( P ) � � f ( � q − p j � ) � n ji ( q ) , ∂q + � φ ( q ) dq ∂p i V j ( P ) ∩ V i ( P ) j neigh i ,

Scenario 1: technical approach 1 Alternative formulation ( f : R + �→ R + , differentiable, non-decreasing) � � � � n E φ min f ( � q − p i � ) = f ( � q − p i � ) φ ( q ) dq i V i ( P ) i =1 � n � ≤ f ( � q − p i � ) φ ( q ) dq W i i =1 2 Compute decentralized gradient � ∂ H C ∂ ( P ) = f ( � q − p i � ) φ ( q ) dq ∂p i ∂p i V i ( P ) � f ( � q − p i � ) � n i ( q ) , ∂q + � φ ( q ) dq ∂p i ∂V i ( P ) � f ( � q − p i � ) � n i ( q ) , ∂q − � φ ( q ) dq ∂p i ∂V i ( P ) ,

Scenario 1: technical approach 1 Alternative formulation ( f : R + �→ R + , differentiable, non-decreasing) � � � � n E φ min f ( � q − p i � ) = f ( � q − p i � ) φ ( q ) dq i V i ( P ) i =1 � n � ≤ f ( � q − p i � ) φ ( q ) dq W i i =1 2 Compute decentralized gradient � ∂ H C ∂ ( P ) = f ( � q − p i � ) φ ( q ) dq ∂p i ∂p i V i ( P ) ,

Scenario 1: technical approach 1 Alternative formulation ( f : R + �→ R + , differentiable, non-decreasing) � � � � n E φ min f ( � q − p i � ) = f ( � q − p i � ) φ ( q ) dq i V i ( P ) i =1 � n � ≤ f ( � q − p i � ) φ ( q ) dq W i i =1 2 Compute decentralized gradient � ∂ H C ∂ ( P ) = f ( � q − p i � ) φ ( q ) dq = 2 M V i ( P ) ( p i − C V i ( P ) ) ∂p i ∂p i � �� � V i ( P ) for f ( x )= x 2 ,

Scenario 1: technical approach 1 Alternative formulation ( f : R + �→ R + , differentiable, non-decreasing) � � � n � min f ( � q − p i � ) = f ( � q − p i � ) φ ( q ) dq E φ i V i ( P ) i =1 � n � ≤ f ( � q − p i � ) φ ( q ) dq W i i =1 2 Compute decentralized gradient � ∂ H C ∂ ( P ) = f ( � q − p i � ) φ ( q ) dq = 2 M V i ( P ) ( p i − C V i ( P ) ) ∂p i ∂p i � �� � V i ( P ) for f ( x )= x 2 critical points for H are centroidal Voronoi configurations ,

Correctness of dispersion laws Distributed: over Delaunay graph Adaptive: changing environment, agent arrivals and departures Verifiably correct: convergence to centroidal Voronoi configurations via LaSalle Invariance Principle Asynchronous implementation: wake up 1 determine local Voronoi diagram (w/ outdated information) 2 determine centroid of own Voronoi region 3 take a step in that direction go to sleep ,

Top-down: area deployment Objective: Given sensors/nodes/robots/sites ( p 1 , . . . , p n ) moving in environment Q achieve optimal coverage defined according to Scenario 2 —area (with limited-range sensor or communication radius r ) given distribution density function φ � � � area φ ( ∪ n maximize 2 ( p i )) = max 1 B r ( p i ) ( q ) φ ( q ) dq i =1 B r i Q 2 ,

Scenario 2: weighted normal Take density function constant, φ = 1 � n B r ( p ) φ arc( r ) 2 ,

Scenario 2: weighted normal Take density function constant, φ = 1 � n B r ( p ) φ arc( r ) 2 If arc( r ) is described by [ θ − , θ + ] ∋ θ �→ p + r 2 (cos θ, sin θ ) ∈ R 2 � θ + � θ + − θ − �� � θ + + θ − � � θ + + θ − �� r (cos θ, sin θ ) dθ = sin cos , sin 2 2 2 2 θ − ,

Scenario 2: weighted normal Take density function constant, φ = 1 � n B r ( p ) φ arc( r ) 2 If arc( r ) is described by [ θ − , θ + ] ∋ θ �→ p + r 2 (cos θ, sin θ ) ∈ R 2 � θ + � θ + − θ − �� � θ + + θ − � � θ + + θ − �� r (cos θ, sin θ ) dθ = sin cos , sin 2 2 2 2 θ − ,

Scenario 2: area coverage algorithm Name: Coverage behavior distributed optimal agent deployment Goal: (i) own cell computation Requires: (ii) weighted normal computation For all i , agent i synchronously performs: 1: determine own cell V i ∩ B r 2 ( p i ) � 2: determine weighted normal arc( r ) n B r ( p ) φ 2 3: move in the direction of weighted normal Caveat: convergence only to local maximum of area φ ( ∪ n i =1 B r 2 ( p i )) ,

Simulation initial configuration gradient descent final configuration ,

Correctness and complexity of dispersion laws Distributed: over r -limited Delaunay graph Adaptive: changing environment, agent arrivals and departures Convergence: Gradient + LaSalle Invariance Principle Complexity: for d = 1 , first-order agents with r -lim Delaunay graph TC( T ( rǫ ) -deplmnt , CC centroid ) ∈ O ( n 3 log( nǫ − 1 )) ,