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Self organizing robot Self organizing robot gathering Seminar in - - PowerPoint PPT Presentation

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Self organizing robot Self organizing robot gathering Seminar in Distributed Computing Christof Baumann Mentor: T obias Langner Wednesday, 2 March 2011 Self organizing robot gathering Christof Baumann 1/36 What is it all about n

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1/36 Wednesday, 2 March 2011 Self organizing robot gathering – Christof Baumann

Self organizing robot gathering

Christof Baumann Mentor: T

  • bias Langner

Seminar in Distributed Computing

Self organizing robot

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2/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

What is it all about

  • n robots with restricted capabilities
  • 2D plane setting
  • They want to gather in a single point
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3/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Where gathering could be used

  • Mars robots
  • Multiple robot types (to save money)
  • Robots equipped with radio
  • “Dumb” robots
  • Radio robots do jobs for the whole group
  • To exchange data they need to gather
  • Military
  • Mine searching
  • Spy robots
  • Task splitting
  • After gathering the main robot distributes the tasks
  • Distributed Flight Array
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4/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Distributed Flight Array

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5/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Distributed Flight Array

  • Movie
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6/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

The paper

  • Title: A Local O(n²) Gathering Algorithm
  • Bastian Degener
  • Barbara Kempkes
  • Friedhelm Meyer auf der Heide
  • (All at University of Paderborn in Germany)
  • Published
  • at the Symposium on Parallelism in Algorithms and

Architecture (SPAA)

  • in the year 2010
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7/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Overview

  • Motivation
  • Previous work
  • The models
  • The algorithm
  • Conclusions
  • Questions
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8/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Previous Work

  • No runtime bounds with a just local view
  • All runtime bounds known rely on a global view
  • Gathering if malicious robots are involved
  • Robots that are not point sized but have an extent
  • View of robot can be blocked
  • Compass model
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9/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Overview

  • Motivation
  • Previous work
  • The models
  • The algorithm
  • Conclusions
  • Questions
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10/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Robot Model

  • Limited viewing range
  • Do not have a memory
  • No common coordinate

system

  • Assign target positions

to other robots within connection range

  • Measure positions of
  • ther robots within

viewing range

1 2

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11/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Time Model

  • Just one robot active at the time
  • Next robot chosen randomly
  • Round model
  • Each robot is active at least once
  • A round takes steps in expectation
  • Coupon collector

Onlogn

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12/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Active vs. inactive Robots

  • Active Robot
  • See positions of other robots
  • Tell robots target position
  • Move to own target position (max. distance of 2)
  • Inactive Robot
  • Be told a target position
  • Move to the target told (max. distance of 3)
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13/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Overview

  • Motivation
  • Previous work
  • The models
  • The algorithm
  • Conclusions
  • Questions
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14/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

The algorithm

  • The active robot executes one of
  • Termination
  • Just executed once
  • Complete the gathering
  • Fusion
  • Fuse two robots
  • Fused robots are treated as one
  • Reduction
  • Reduce the area of the convex hull of the network
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15/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Termination

  • If all robots are

in connection range

  • If this step is

done we have gathered if the network was connected

Everybody to my position

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16/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Network connectivity after termination step

  • No robots in viewing range
  • Robots just in connection

range

  • Nothing gets

disconnected

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17/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Fusion

  • Fuse two

(or more) robots together

  • If there is a

configuration in which these conditions hold

  • Robots still

contained in the convex hull

  • Still connected

Yellow robot to position

  • f red one
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18/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Network connectivity after fusion step

  • Robots in viewing range stay

connected by definition

  • If it is not possible to

fuse robots the third possibility is executed

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19/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Lower bound for the # of robots in the connection range to have a fusion

  • Define c as the

number of nodes the active robot can see

  • If c > 16 a fusion

is possible

  • Pigeonhole
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20/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Reduction

  • If fusion not

possible

  • Compute

the convex hull

  • Compute

intersections with maximum distance of convex hull and connection range

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21/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Reduction

  • Compute line

segment L between the points

  • Move robots
  • n the same

side as the active robot to their closest point on L

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22/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Network connectivity after reduction step (1/2)

  • Only robots within

the active robots connection range are moved

  • Convex hull of active

robot stays connected

  • By projection the

distance does not increase

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23/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Network connectivity after reduction step (2/2)

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24/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Run time Analysis

  • In each round one of the 3 possibilities is executed
  • Termination
  • Fusion
  • Reduction
  • If there's a bound for the maximum number of

rounds for each of them we have a bound for the algorithm

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25/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Progress Fusion

  • Directly visible progress
  • Easy to bound
  • Maximally n-1 rounds

with fusion

  • Runtime: O(n)
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26/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Progress Reduction

  • Reducing the size of the

global convex hull

  • We will prove that the

area of the global convex hull is decreased in expectation by a constant factor in each round

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27/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Bound the reduction area of a global convex hull vertex robot (1/2)

  • The convex hull is at least

reduced by the area of T

T  sin  2 ⋅cos  2   

  • Because of
  • The global convex hull

contains the viewing range of the active robot at the beginning

  • f a time step

 sin 2 ⋅cos  2 

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28/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

  • Give a bound for the angle

seen by the active robot

1 1 1 1 1  3 P

Bound the reduction area of a global convex hull vertex robot (2/2)

sin 2 ⋅ cos  2 1 2⋅cos  2   3

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29/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Bound the reduction area of a round

  • We want to use the sum of internal angles of the

global convex hull to get the area truncated in one round

  • If a robot that is a vertex
  • f the convex hull is the

first one active in its neighborhood then holds

T  '

∑ i'=⋅

m−2 '

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30/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

  • The expected truncated area by the active vertex

robot is

Bound the expected reduction area of a single step (1/2)

E[a]Pr[robot is the first activated in connection range] ⋅ area truncated =Pr[robot is the first activated in connection range]⋅1 2 cos  2  Pr[robot is the first activated in connection range]⋅1 2 cos ' 2 

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31/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

  • Probability that a vertex robot with c neighbors is

not moved before its activation

Bound the expected reduction area of a single step (2/2)

  • c is the maximum number of robots in viewing

range without a fusion

  • We already know that c<16

E[a]1 c⋅1 2 cos' 2 

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32/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

  • Sum up

Bound the reduction area in a round

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33/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Runtime of the algorithm

  • Fusions
  • maximally n-1
  • Reductions
  • In the beginning the convex hull has maximum area of n²
  • We have a constant reduction in each round
  • We need O(n²) rounds in expectation
  • Expectation comes from the stochastic round model
  • The algorithm itself is deterministic
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34/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Overview

  • Motivation
  • Previous work
  • The models
  • The algorithm
  • Conclusions
  • Questions
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35/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Conclusions / Critics

  • The constraint that the active robots can give
  • rders is very strong
  • The randomized round model is hard to implement

in practice

  • Just one active robot at the time
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36/36 Self organizing robot gathering – Christof Baumann Wednesday, 2 March 2011

Overview

  • Motivation
  • Previous work
  • The models
  • The algorithm
  • Conclusions
  • Questions