On Distributed Coordination in Robotic Networksmotion.me.ucsb.edu/joey/website/media/talk-jwd-Feb2010.pdf · On Distributed Coordination in Robotic Networks ... Smooth Nearness Diagram

On Distributed Coordination in Robotic NetworksGossip Coverage and Frontier-based Pursuit

Joseph W. Durham

Center for Control, Dynamical Systems,and Computation

Department of Mechanical EngineeringUniversity of California at Santa Barbaramotion.mee.ucsb.edu/~joey

February 5, 2010

Joseph Durham (UCSB) Distributed Coordination Feb 5 1 / 45

motion.mee.ucsb.edu/~joey

Introduction

Distributed Coordination Algorithms

Team of robotic agents tasked with performing a joint mission in anenvironment

Each individualsenses its immediate surroundingscommunicates with nearby agentsprocesses information gatheredperforms local action in response

Algorithm design goalDesign individual control and communication laws such that the groupreaches a desired goal


Introduction

Distributed Coordination Algorithms

Team of robotic agents tasked with performing a joint mission in anenvironment

Each individualsenses its immediate surroundingscommunicates with nearby agentsprocesses information gatheredperforms local action in response

Algorithm design goalDesign individual control and communication laws such that the groupreaches a desired goal


Introduction

Applications

Ocean monitoring gliders from noc.soton.ac.uk, warehouse robots from KIVA Systems,

hopping planetary explores from NASA


Introduction

Papers in this Talk

J. W. Durham, A. Franchi, and F. Bullo. Distributed pursuit-evasion with limited-visibility

sensors via frontier-based exploration. In IEEE Int. Conf. on Robotics and Automation,

Anchorage, Alaska, May 2010. To appear

J. W. Durham, R. Carli, P. Frasca, and F. Bullo. Discrete partitioning and coverage control

with gossip communication. In ASME Dynamic Systems and Control Conference,

Hollywood, CA, October 2009

J. W. Durham and F. Bullo. Smooth nearness-diagram navigation. In IEEE/RSJ Int. Conf.

on Intelligent Robots & Systems, pages 690–695, Nice, France, September 2008

Collaborators: Ruggero Carli, Antonio Franchi, Paolo Frasca, and myadvisor Francesco Bullo.


Introduction

Outline

1 Introduction

2 Robotic Network Model

3 Gossip CoverageProblem sketchCurrent resultsFuture directions

4 Frontier-based Pursuit-EvasionProblem sketchCurrent resultsFuture directions

5 Conclusion


Robotic Network Model

1 Introduction




5 Conclusion



Hardware



Robot Model

Differential driveTranslational velocity vRotational velocity θ

Physical stateX = (x , y , θ)

In practice measurement of actual ve-locities is imperfect, integrals diverge

Either must accept position errors oruse sensors for localization



Robot Model



In theory state is an integral of veloci-ties




Robot Model







Robot Model







Sensor Model

Sensor footprintS(x , y , θ) is the intersection of visibilitypolygon from (x , y) and the area per-ceivable by the sensor oriented by θ

Sensor footprint can be used for:Obstacle detectionLocalizationIntruder detection


Robotic Network Model Control and Communication

Control and Communication Models

Processor StateW: the state of the robot’s processor – stored data, current behavior

Communication AlphabetL: set of messages a robot can send to other robots


Robotic Network Model Control and Communication

Network Connectivity Models

Communication GraphMany possible models for whichagents can communicate

Combinations of:Network geometryPhysical proximityCurrent robot rolesRandomness


Robotic Network Model Software

Robotic Software Overview

Player interfaces

NavigationSoftware

Control & CommunicationAlgorithm

To/from other agents




Player interfaces

NavigationSoftware



Player/Stage is an open-sourcerobotics software library

FeaturesPlayer provides interfacesfor hardwareEach robot is a server on aTCP/IP networkStage simulates hardware,interfaces to algorithms arethe same




Player interfaces

NavigationSoftware



SND Navigation handles localpath planning and execution

Algorithm design can focus on:Desired positioning ofrobotCommunication forcoordination



Smooth Nearness Diagram Navigation

Evolution of ND+ Nav by J. Mingues,J. Osuna, L. Montano

Input: S, desired pose (x , y , θ)

Output: v and θ

Find gaps in sensor footprint

Pick best gap to drive towardsAdjust commands based onnearby obstacles

Available as the snd driver inPlayer/Stage






Output: v and θ

Find gaps in sensor footprint

Pick best gap to drive towardsAdjust commands based onnearby obstacles







Output: v and θ

Find gaps in sensor footprintPick best gap to drive towards

Adjust commands based onnearby obstacles







Output: v and θ

Find gaps in sensor footprintPick best gap to drive towardsAdjust commands based onnearby obstacles







Output: v and θ

Find gaps in sensor footprintPick best gap to drive towardsAdjust commands based onnearby obstacles




Summary of Robotic Network Model

Algorithm Design Requirements1 Data structures

Correct or account for errors in (x , y , θ)Processor stateW and communication alphabet L

2 Update functionsMessage-generation functionProcessor state transition functionMotion control function to pick desired pose

3 Communication graph model


Gossip Coverage

1 Introduction




5 Conclusion


Gossip Coverage Problem sketch

Motivation

Biological examples of coverage control

Tilapia mossambica, Barlow et al ’74 Sage sparrows, Petersen et al ’87



Related Prior Work I

Lloyd’s Algorithm

take convex environment Q with density function φ : Q → R≥0

place N robots at p = p1, . . . ,pNpartition environment into v = v1, . . . , vNdefine expected quadratic deviation

H(v ,p) =

∫v1

f (‖q − p1‖)φ(q)dq + . . .+

∫vN

f (‖q − pN‖)φ(q)dq

Theorem (Lloyd ’57 “least-square quantization”)1 at fixed partition, optimal positions are centroids2 at fixed positions, optimal partition is Voronoi3 Lloyd algorithm: alternate p-v optimization

−→ convergence to the set of centroidal Voronoi partitions



Related Prior Work I

Lloyd’s Algorithm

take convex environment Q with density function φ : Q → R≥0

place N robots at p = p1, . . . ,pNpartition environment into v = v1, . . . , vNdefine expected quadratic deviation

H(v ,p) =

∫v1

f (‖q − p1‖)φ(q)dq + . . .+

∫vN

f (‖q − pN‖)φ(q)dq

Theorem (Lloyd ’57 “least-square quantization”)1 at fixed partition, optimal positions are centroids2 at fixed positions, optimal partition is Voronoi3 Lloyd algorithm: alternate p-v optimization

−→ convergence to the set of centroidal Voronoi partitions



Related Prior Work II

Distributed Coverage Control

At each comm round:

1: acquire neighbors’ positions2: compute Voronoi region3: move towards centroid of

own Voronoi region

Result: convergence to the setof centroidal Voronoi partitions

J. Cortés, S. Martínez, T. Karatas, and F. Bullo.

Coverage control for mobile sensing networks.

IEEE Transactions on Robotics and Automa-

tion, 20(2):243–255, 2004



Related Prior Work III

Gossip coverage in continuous spacePairwise territory exchange between neighborsRegions may be non-convex during evolutionResult: convergence to the set of centroidal Voronoi partitions

P. Frasca, R. Carli, and F. Bullo. Multiagent coverage algorithms with gossip com-

munication: control systems on the space of partitions, March 2009. Available at

http://arXiv.org/abs/0903.3642


Gossip Coverage Current results

Discretized Environments

Domain is a weighted graph G = (Q,E ,w)

Required propertiesG must be connectedAll edge-weights w must be positive

G can easily represent a non-convex environment with holes



Voronoi Iteration on Graphs

Distances are shortest pathlengths in connected

sub-graphs of G

Vertices join partition of centroidthey are closest to



Cost Function

Centroid pi of sub-graph vi is vertex which minimizes

Hi(h, vi) =∑k∈vi

distvi (h, k)

Total cost

Hmulti-center(p, v) =N∑

i=1

Hi(pi , vi)

Minimize expected distance between random vertex and closest robot



Cost Function

Centroid pi of sub-graph vi is vertex which minimizes


distvi (h, k)

Total cost

Hmulti-center(p, v) =N∑

i=1

Hi(pi , vi)

Minimize expected distance between random vertex and closest robot



Hardware Experiment



Hardware Experiment



Hardware Experiment



Hardware Experiment



Hardware Experiment



Hardware Experiment



Hardware Experiment



Hardware Experiment



Simulation Movie



Gossip Coverage Assumptions

Map assumptions:Team is provided an initial connected N-partition of environment

Initial agent partitions are connectedCover space without overlap

Communication assumptions:Given infinite time, each agent will talk to each of its neighbors aninfinite number of timesTwo options:

There exists a finite upper bound on the time betweenconversations for each pairThere is a non-zero probability for each pairwise communicationoccurring at all times



Gossip Coverage Assumptions

Map assumptions:Team is provided an initial connected N-partition of environment

Initial agent partitions are connectedCover space without overlap

Communication assumptions:Given infinite time, each agent will talk to each of its neighbors aninfinite number of timesTwo options:

There exists a finite upper bound on the time betweenconversations for each pairThere is a non-zero probability for each pairwise communicationoccurring at all times



Algorithm Claims

1 Maintain connectedN-partition during evolution

Each region is connectedNo overlap

2 Total cost decreases wheneveragents exchange territory

3 Provable convergence to asingle centroidal Voronoipartition in finite time



Convergence Theorem

X finite set of connected N-partitions of graph GAlgorithm defines set-valued map T : X → X

Version of the LaSalle Invariance PrincipleRequirements for convergence

1 X is compact, positively invariant under T2 Hmulti-center non-increasing under T , decreasing under T \ id3 Hmulti-center and T are continuous on X4 One of two communication assumptions

There exists a finite upper bound on the time betweenconversations for each pair (i , j)There is a non-zero probability for each pair (i , j) to communicate atall times



Convergence Theorem

X finite set of connected N-partitions of graph GAlgorithm defines set-valued map T : X → X

Version of the LaSalle Invariance PrincipleRequirements for convergence

1 X is compact, positively invariant under T2 Hmulti-center non-increasing under T , decreasing under T \ id3 Hmulti-center and T are continuous on X4 One of two communication assumptions

There exists a finite upper bound on the time betweenconversations for each pair (i , j)There is a non-zero probability for each pair (i , j) to communicate atall times



Computational Complexity


distvi (h, k)

Key computationDistances from h to all k ∈ vi

If edge-weights are uniform, can use BFS in linear timeOtherwise, must use Dijkstra in log-linear time

Computing centroidMost computationally complex piece, three options:

Exhaustive search in O(|vi |2)

Gradient descent in O(|vi | log |vi |)Center of mass approximation in O(|vi |)



Computational Complexity


distvi (h, k)

Key computationDistances from h to all k ∈ vi

If edge-weights are uniform, can use BFS in linear timeOtherwise, must use Dijkstra in log-linear time

Computing centroidMost computationally complex piece, three options:

Exhaustive search in O(|vi |2)

Gradient descent in O(|vi | log |vi |)Center of mass approximation in O(|vi |)



Summary

Chief contributionsConverge to a single centroidal Voronoi partition in finite timeCoverage control which works in non-convex environments withholesComputation can scale well to large areas with many robots


Gossip Coverage Future directions

Ongoing Work in Coverage Control

Current directions

Motion protocolAgents will patrol boundary of territory to meet neighborsCan model need to meet neighbors as tasks on boundary

Local broadcast communicationMore realistic model of wireless communicationRequires overlapping territories during evolution


Frontier-based Pursuit-Evasion

1 Introduction




5 Conclusion


Frontier-based Pursuit-Evasion Problem sketch

Our Clearing Problem

T34 security bot from tmsuk and Alacom in Japan

The Team: Robots with limited-range sensors

The Mission: Guarantee detection of any evaders in an unknownenvironment



Exploration Inspiration

ObservationClearing an environment is a con-strained form of exploration

For stationary evaders,cleared = exploredOtherwise, cleared can berecontaminated




For explorationFrontier: Boundary between ex-plored and unexplored areas

For pursuit-evasionFrontier: Boundary betweencleared and contaminated areas

Our ApproachCompletely cover frontier at alltimesContinuously push backfrontier















Key Issues

Existing methods for computingglobal frontier require:

Global mapGlobal localization (to buildglobal map)

Our new method requires:Complete coverage of frontierat all timesMutual localization betweenneighboring robots



Key Issues

Existing methods for computingglobal frontier require:

Global mapGlobal localization (to buildglobal map)

Our new method requires:Complete coverage of frontierat all timesMutual localization betweenneighboring robots


Frontier-based Pursuit-Evasion Current results

Distributed Algorithm Roles

Expand

Frontier-Guard

Follow

Wander

Leaders

Frontier-Guard: Key role for al-gorithm. Cover local frontier anddispatch agents to expand it.Expand: Agent moving to aviewpoint it was assigned.

Non-Leaders

Follow: Waiting for orders froma guard.Wander: Cleared local area,now searching for a guard to fol-low.



Distributed Global Frontier

Each frontier-guard stores its localoriented frontier arcs

Frontier UpdatingWhen a new guard reaches itsviewpoint, it must:

1 Ask for frontier arcs fromneighboring guards

2 Inform neighbors of frontiersegments inside footprint

3 Classify local frontier based onintersections



Viewpoint Planner

Assumption: Sensor footprintsare circular

Goal: Pick new viewpoints VMinimize |V |Maximize area exposed

Viewpoints required for angularwidth Ω of arc:

Ω ≤ 2π3 : |V | = 1

Ω = 2π: |V | = 3For intermediate, choice of what tooptimize



Example Simulation



Movie



Frontier Coverage

0 5 10 15 20 25 30 350%

20%

40%

60%

80%

100%

Iteration Number

Perc

ent

Are

a C

leare

d

0

30

60

90

120

150

Fro

nti

er

Cell

s P

er

Guard

Frontier cell count per guard does not grow with area clearedDistributed storage requires only constant memory per agent



Empty Space



Summary

Chief contributionsOnline clearing algorithm which works in non-convexenvironments with holesDistributed storage and updating of global frontierRequires only mutual localization


Frontier-based Pursuit-Evasion Future directions

Ongoing Work in Pursuit-Evasion

Current directions

Distributed hardware implementation and experimentsViewpoint planner for circular sector sensor footprintsBounds on number of agents necessary to clear a map


Conclusion

Conclusion

Distributed coordination algorithm framework for hardwareTwo parallel algorithm implementations:

1 Coverage of discretized environments2 Frontier-based pursuit-evasion


Conclusion

The End

Questions?


Documents

On Distributed Coordination in Robotic Networksmotion.me.ucsb.edu/joey/website/media/talk-jwd-Feb2010.pdf · On Distributed Coordination in Robotic Networks ... Smooth Nearness Diagram