Title,%20Introduction

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Title, Introduction
Analysis and Implementation of Distributed
Algorithms for Multi-Robot Systems
James McLurkin jamesm_at_csail.mit.edu MIT Computer
Science and Artificial Intelligence Lab
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(No Transcript)
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Why Multi-Robot Systems?
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Contributions

• Distributed algorithms with provable correctness
  in ideal conditions and robust real-world
  performance
• Complexity metrics for computation on multi-robot
  systems
• Empirical verification of trade-offs between
  algorithm accuracy, communications usage and
  robot speed

With one outstanding conjecture
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example application
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1. Applications, Assumptions, and the
Computational Model
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An Example Application Building Search
1. Disperse throughout a building2. Find an item
of interest3. Lead the user to the item

• GuideBot
• ChargingBot
• InteriorBot
• BoundaryBot

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BlueFrontiers RedInterior GreenCharging
WhiteGuides
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The Algorithms from the Video

• Weve seen two different algorithms
• Weve also seen that
• The robots move around a lot,
• They respond to their neighbors to maintain a
  uniform density
• They sometimes leave the group to go charge.

Navigation
Boundary Detection
Navigation
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canonicalmulti-robot application
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Our Canonical Multi-Robot Application has

• high robot mobility
• short-range local communication multi-hop
  global communication
• a noisy sensor on each robot to measure the
  positions of other robots
• algorithms that are robust to population changes
  and robot failures

we are focused on explain long range-gthigh
bandwidth
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Multi-Robot Applications are Hard Because of

• high robot mobility
• short-range local communication multi-hop
  global communication
• a noisy sensor on each robot to measure the
  positions of other robots
• algorithms that are robust to population changes
  and robot failures

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formal model and apparatus
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Experimental Apparatus SwarmBot

• IR Inter-Robot Communicationsand Localization
• Bump skirt for obstacle avoidance

And theyre all the same with unique ID
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Model Robot State

• We can describe the state, s, of a single robot
  as a tuple of its ID, pose, and private and
  public variables

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Model Robot State

• We can describe the state, s, of a single robot
  as a tuple of its pose and private and public
  variables
• s ?ID, pose, private vars, public vars?

?
posex,y,?
y
x
external global coordinate system
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Model Configuration

• We define a configuration, C, as the states of n
  robots
• All the robots use the same software and hardware

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Model Local Network Geometry

• Each robot can communicate with and localize
  neighboring robots within radius r

r
robot b pose estimateabxab,yab,?ab
robot a local coordinate system
robot c poseacxac,yac,?ac
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Model Configuration Graph

• A configuration C and communication radius r
  produces a
• configuration graph G
• C is valid iff G is connected

r
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Model Periodic Communications

• Each robot broadcasts its public vars every t
  seconds
• We assume local communications are reliable
• This creates a synchronizer, giving us global
  rounds

verify that synchronizer is what I think say
that robots are not synchronized
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Model Algorithm

• An algorithm A runs on every robot and transforms
  any configuration C1 through valid intermediate
  configurations to a new configuration C2, s.t.
  C2 satisfies a given set of properties

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Experimental Apparatus Data Collection

• Radio
• Behavior LEDs
• Beacon for position measurement

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Experimental Apparatus
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message speed and the RSR
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2. Message Speed and the Robot Speed Ratio
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Broadcast Tree Construction

• The main form of communication is multi-hop
  broadcast
• The messages form a broadcast tree on the graph
  as they propagate

source
Intanagonwiwat, Govindan, Estrin. Directed
Diffusion A Scalable and Robust Communication
Paradigm for Sensor Networks, 2000
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Global Communications Structure

• We use five types, color-coded on the plots
• No Communication
• One-Hop Communication
• Broadcast Tree Construction
• Convergecast
• Convergecast Rebroadcast
• Broadcast Tree Construction Physical Routing
• Each type requires a more stable network than the
  previous type

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Broadcast Tree

• Hops are distributed in a spatial way
• Any robot can estimate distance and direction to
  root

1 hop
2 hops
3 hops
source
0 hops (root)
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Network Path Length
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Distance per Communication Hop

• The max crookedness of any path is the
  t-spanner, k
• We compute the average k to estimate the distance
  per hop,

In our networks,
Kleinrock and Silvester, Optimum Transmission
Radii for Packet Radio Networks, 1978
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Message Speed and Robot Speed

• We can compute the propagation speed of the
  messages

0 hops (root)
robot speed
The Robot Speed Ratio
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The Robot Speed Ratio (RSR)

• A dimensionless measure of robot speed in
  multi-robot systems
• A robot moving at a RSR gt 1 away from a message
  source cannot execute an algorithm using
  broadcast communication
• We would expect that robots executing an
  algorithm at a low RSR would be more accurate
  than at a high RSR
• We can compute the RSR from system parameters

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Computing the RSR

• If someone hands you a box of robots, an
  algorithm, and a building

you can express message speed in terms of system
parameters
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RSR Trade-Offs

• For a given constant RSR, you can
• Trade robot speed for communications bandwidth
• Trace CA, algorithm communications complexity,
  for robot speed
• Trade neighbor count for robot speed

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Does the RSR Really Matter?

• This is silly. Arent communications always
  faster than robots?

vs.
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Message Speed
smessage (computed) 6.9 m/s smessage (measured)
4.4 m/s
n 44, t 0.250s, speed 0, RSR 0
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broadcast tree navigation
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3. Multi-Robot Distributed Algorithms
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Algorithm Broadcast Tree Navigation

• Purpose
• To guide an active robot from anywhere in the
  configuration to the root robot
• Requires
• A broadcast routing tree
• An active robot
• Communications Type
• Broadcast Tree
• Running Time
• O(diam(C)) time
• Output Configuration
• Active robot veryclose to root robot
• Accuracy Metric
• Path Efficiency

n 32, t 0.250s, speed 0, RSR 0
Spears, Spears, Hamann, and Heil, Distributed,
Physics-Based Control of Swarms of Vehicles, 2004
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Broadcast Tree Navigation

• A robot uses the broadcast tree to navigate
  towards the source

1 hop
2 hops
3 hops
0 hops (root)
0 hops (root)
Li and Rus, Navigation Protocols in Sensor
Networks, 2005
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But What if the Tree is Moving?

• Motion smears network and reduces the
  correspondence between network topology and
  physical positions

n 33, t 0.250s, speed 0.08, RSR 0.02
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Broadcast Tree Navigation Algorithm
n 25,t 0.250s,speed 0.08m/s,RSR 0.02
n 26,t 4s,speed 0.08m/s,RSR 0.02
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Communications and Navigation Summary
one-hop communications
broadcast tree communications
broadcast convergcast rebroadcast
communications
broadcast tree physical routing
cant say that these lines decay at different
(better) rates
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local boundary classification the c-shape
algorithm
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Boundary Detection

• Three Algorithms
• Local boundary classification
• Boundary subgraph construction
• Global boundary classification
• Useful for
• Perimeter estimation
• Void remediation
• Obstacle characterization
• Target tracking
• Group cohesion

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Boundary Detection Related Work

• Pfisterer, Fischer, Buschmann, Fekete, Kröller,
  Neighborhood-Based Topology Recognition in Sensor
  Networks, 2004
• Edelsbrunner, Kirkpatrick, and Seidel, On the
  Shape of a Set of Points in the Plane, 1983
  (Alpha-Shapes)

give definition of what a boundary is
next defn of a-shape alg for computing
a-shapes alg needs d-triangulation we cant
compute d-triangulation
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Alpha Shapes1

• Any point on a disc of radius 1/a that contains
  no other points is a boundary

1. Edelsbrunner, Kirkpatrick, and Seidel, On the
Shape of a Set of Points in the Plane, 1983
fix oclusion Start with definition then algorithm
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What a to use?
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Cant Compute Distributed a-Shapes

• The robots cannot agree on boundary nodes in the
  concave crossing case

bullet local decisions affect global
algorithmic this is difference between local
c-shape and alpha shape new slide We want to
define a boundary that is similar to the a-shape,
but computable in a dist fashion
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Local Boundary Classification Algorithm C-Shape

• Purpose
• To determine which robots are on the boundary of
  the configuration in a distributed fashion
• Requires
• n/a
• Communications Type
• One-Hop
• Running Time
• O(1) round
• Output Configuration
• boundary robots flagged
• Accuracy Metric

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Local Boundary Classification Algorithm C-Shape

• 1. Sort neighbors in cyclic order by local
  bearing
• 2. Use inferred edges to look for cycle through
  all neighbors in order
• 3. If cycle exists ? not boundary robot
• Handles concave crossing case by admitting all
  robots

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Boundary Detection
fix covered text draw pictures in blackness
mode label local articulation points
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Problems with C-Shapes

• Extra robots in concave crossings
• Incomplete internal boundaries

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Local Boundary Classification
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boundary subgraph construction
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Boundary Subgraph Construction

• Purpose
• To determine which robots are on the same
  boundary
• Requires
• Local Boundary Class
• Communications Type
• Broadcast Tree
• Running Time
• O(diam(G)) rounds
• Output Configuration
• Boundary ID (BID) lowest robot ID on boundary
• Accuracy Metric

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Boundary Subgraph Construction

• Each boundary runs leader election algorithm to
  form subgraph and agree on name
• Accuracy relies on boundary subgraph being stable
  long enough for the message to get all the way
  around

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Boundary Subgraph Construction
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Boundary Helpers

• Boundary subgraph is fragile, with min-cut of 1
• Boundary Helpers patch holes in connectivity

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Boundary Subgraph Construction

• Harsh accuracy metric produces bimodal
  distribution

define metric again explain odd behavior
explain metric did I get word right compared to
did I get whole sentence right
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Conjecture of Contiguous Exterior Boundary
Subgraph
G

• If
• Each exterior boundary robot is adjacent to the
  outer face
• and
• Both of each boundary robots boundary subgraph
  neighbors are also boundaries
• then
• the boundary is contiguous
• The problem? Concave crossings.
• Solution Define G with crossing boundary edges
  removed, and patch boundary between affected
  robots
• Show that all other properties hold

G
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global boundary classification
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Global Boundary Classification

• Purpose
• To determine if boundaries are external or
  internal
• Requires
• Boundary Subgraph
• Communications Type
• Broadcast Tree Convergecast Rebroadcast
• Running Time
• O(diam(G)) rounds
• Output Configuration
• All boundary robots know their boundary type
• Accuracy Metric

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Global Boundary Classification Algorithm

• Computes the exterior angle of the boundary
  polygon
• exterior angle Sturning angles of boundary
  robots
• uses convergecast summation

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Global Boundary Classification
remove data use picture
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Global Boundary Classification
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Boundary Detection Summary

• one-hop communications

broadcast tree communications
broadcast convergcast rebroadcast
communications
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discussion
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4. Conclusions
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All the Algorithms

• Physical accuracy depends on RSR and
  communications type

one-hop communications
broadcast tree communications
broadcast convergcast rebroadcast
communications
broadcast tree physical routing
Rus, Donald, Jennings, Moving Furniture with
Teams of Autonomous Robots, 1995 OKane and
LaValle, On Comparing the Power of Robots
This is about trade-offs rus and spears
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Does This Apply to all Robots?

• Are there conserved quantities in this model?
• Is there a parameterization to facilitate
  comparisons across disparate hardware platforms?

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Future Work

• Geometric Consensus and Algorithm Classes
• Do we have to solve a distributed consensus
  problem to use a global algorithm on a system
  with local network geometry?
• Can we design overlapping consensus regions or
  low-quality, but nimble, geometry?
• Error Rates of Self-Stabilizing Distributed
  Algorithms
• We ultimately want to measure something like
• Network people measure network half-life to
  characterize topology changes

of correct bits computed by Alg/sec of
erroneous bits injected by environment/sec
high-level, abstract
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Contributions

• Algorithms with provable correctness in ideal
  conditions and robust real-world performance
• Complexity metrics for computation on multi-robot
  systems
• Physical Accuracy
• Communication Complexity
• Verification of trade-offs between configuration
  accuracy, communication complexity and the robot
  speed ratio

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• Singapore-MIT Alliance

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Acknowledgements The Fiancé Dara Bourne for
Everything The Advisor Leslie Kaelbling For all
of her patience and advice The Committee Erik
Demaine, Daniela Rus, Seth Teller The man whos
been showing mehow to build stuff right since
1991 Ron Wiken iRobot Jim Frankel and Jennifer
Smith The Graduate Students Aisha Walcott Ed
Olsen Mac Schwager The Brainy Ladies from
G585 Sarah Finney Meg Aycinena Emme
Brunskill Natalia Hernandez Gardiol Kaijen
Hsiao and Luke Zettlemoyer The Many Great
UROPS David Blau Brian Schmidt Alexander
Sanchez Rian Hunter Angelique Moscicki Jeremy
Smith Tony Valderrama
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Extra Slides
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The Standard Answers

• C
• Atmel AT91FR4081 ARM7TDMI core _at_ 40.5mhz
• 1 MB and 3 MB
• 8 Ni-Cd AA
• 4 hours and 1 hour
• ThreadX Real-Time Kernel and Custom OS
• 12 engineer-years
• about 45 minutes

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Why is the problem that I solved hard

• What has been done
• related work is general
• Local coordinates makes it hard

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Talk Outline

• Preliminaries
• Example Application
• Canonical application
• Formal model
• Complexity metrics for multi-robot systems
• Message Speed and the Robot Speed Ratio
• Multi-Robot Distributed Algorithms
• Broadcast tree navigation
• Local boundary detection
• Boundary subgraph construction
• Global boundary detection
• Conclusions

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Model Valid Configuration

• C is valid iff G is connected

r
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Software Design for Multi-Robot Systems
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Multi-Robot Complexity Metrics
Physical Accuracy
Physical Running Time, PTA
pretty pictures go here
pretty pictures go here
Accuracy 0,1
Communications Complexity
Configuration Complexity, GCA
pretty pictures go here
pretty pictures go here
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The Broadcast Tree Root Vector Algorithm

• Computes the approximate location and direction
  to root

Nagpal, Shrobe, Bachrach, Organizing a Global
Coordinate System from Local Information on an Ad
Hoc Sensor Network
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Distributed Boundary Detection C-Shape

• Robots that have missing sectors are on the
  boundary
• Physical accuracy depends on measurement accuracy
  of local network geometry

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Concave Crossing

• Robots that have missing sectors are on the
  boundary
• Physical accuracy depends on measurement accuracy
  of local network geometry

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Algorithm Summary
Running time Communications complexity (bits/robot/cycle) Communication Type
DTA Random-Assignment O(1) - No communication
Local Boundary Classification O(m2) - Network Geometry
DTA Simultaneous-Assignment O(diam(G)) O(n) Broadcast Effectiveness
DTA Sequential-Assignment O(ndiam(G)) O(1) Broadcast Tree Construction Routing
Boundary Subgraph Construction O(diam(G)) O(1) Broadcast Tree Construction Routing
Boundary Classification O(diam(G)) O(1) Broadcast Tree Construction Routing Rebroadcast
DTA Tree-Based-Assignment O(diam(G)) O(m) Broadcast Tree Construction Routing Rebroadcast
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Global Boundary Classification
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Why Multi-Robot Systems?
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Boundary Detection

• We want to be able to tell which robots are on
  the boundary of the network
• We want to be able to tell which robots are on
  the same boundary
• We want to be able to distinguish the external
  boundary from internal internal voids

why is boundary detection useful
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Global Subgraph Classification

• Each boundary subgraph
• 1. Classifies itself based on constituent robots
  convex, concave, convex-concave
• 2. Classifies itself based on sum of angles
  external versus internal
• Runs in O(diam(G)) time, but requires two-way
  network communication, and will only be accurate
  at low speeds

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Related Work

• Multi-Robot Systems
• Maja J Mataric, 1994
• Spears, W., D. Spears, J. Hamann, and R. Heil,
  2004
• Sensor Networks
• C Intanagonwiwat, R Govindan, D Estrin. 2000
• David Culler Alec Woo, Terence Tong. 2003
• Amorphous Computing
• Coore, Daniel, 1994
• Computational Geometry
• Xiang-Yang Li, Gruia Calinescu, and Peng-Jun
  Wang. 2002
• D. Pfisterer, S. Fischer, C. Buschmann, S.P.
  Fekete, A Kröller, 2004

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Convergecast Summation
add algorithm spec sheet
add algorithm description?
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Limitations

• The definition of the accuracy metric is
  application-specific
• e.g. distance accuracy vs. throughput on routing
  trees

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multi-robot complexity metrics
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Related Work Metrics

• Complexity trade-offs between sensors and
  computation
• Rus, Donald, Jennings, Moving Furniture with
  Teams of Autonomous Robots, 1995
• OKane and LaValle, On Comparing the Power of
  Robots
• Donald, On Information Invariants in Robotics,
  1994
• Accuracy Metrics
• Spears, Spears, Hamann, and Heil, Distributed,
  Physics-Based Control of Swarms of Vehicles,
  2004
• Kannan and Parker, Metrics for quantifying
  system performance in intelligent, fault-tolerant
  multi-robot teams, 2007

Rus, Donald, Jennings, Moving Furniture with
Teams of Autonomous Robots, 1995 OKane and
LaValle, On Comparing the Power of Robots
spent too much time on related work add all refs
here, talk about few
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dynamic Task assignment
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Talk Outline
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Dynamic Task Assignment (DTA)

• Group tasks often need to be decomposed into
  subtasks
• Given an input of percentages, assign robots to
  tasks
• For example, the vector (1/6 , 1/3 , 1/2) applied
  to 12 robots would produce the assignment shown
  below

Desired Task Ratio
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DTA Random-Assignment
goal assignment T(1/6 , 1/3 , 1/2)
random-assignment algorithm 1. if task
is unassigned 2. pick task i with probability
ti 3. end
Time O(1) Comms
0 Error Variance O(1/n)
Communications required none
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DTA Sequential-Assignment
goal assignment T(1/6 , 1/3 , 1/2)
sequential-assignment algorithm
1. loop 2. while there are unassigned robots 3.
find lowest ID amongst unassigned robots 4.
wait diam(G) periods to allow message
propagation 5. if lowest select
task else wait until next round 6. endif 7.
endwhile 8. clear task assignments 9. endloop
Time O(ndiam(G))
Comms O(1) Error Variance 0
Communications required Need spatial
relationship to estimate diameter
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DTA Simultaneous-Assignment
goal assignment T(1/6 , 1/3 , 1/2)
simultaneous-assignment algorithm
1. loop 2. receive msg 3. if msg does not
contain my ID 4. add my ID to
msg 5. endif 6. pick task based on ID position in
msg 7. xmit msg 8. endloop
Time O(diam(G)) Comms
O(n) Error Variance 0
Communications required Need effective
communications across network
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DTA Tree-Based-Assignment
goal assignment T(1/6 , 1/3 , 1/2)
tree-based-assignment algorithm
1. loop 2. elect leader 3. if leader 4. gather
global task distribution 5. compute assignment
changes 6. broadcast reassignment msgs 7. elseif
receive reassignment msg 8. change
task 9. endif 10.endloop
Time O(diam(G)) Comms
O(1) Error Variance 0
Communications required Need stable tree for
routing
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Four Dynamic Task Assignment Algorithms
Running time Communications complexity (bits/robot/cycle) Ideal Error Variance Measured Accuracy
Random-Assignment O(1) 0 O(1/n) -
Sequential-Assignment O(ndiam(G)) O(1) 0 1.0
Simultaneous-Assignment O(diam(G)) O(n) 0 1.0
Tree-Based-Assignment O(diam(G)) O(m) 0 0.9
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DTA in Moving Networks
Dynamic Task Assignment Accuracy vs. Network
Speed (data goes here)