Friedhelm Meyer auf der Heide 1
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
Energy-Efficient Local Strategiesfor Robotic Formations
Friedhelm Meyer auf der Heide University of Paderborn
Joint work with
Bastian Degener, Barbara Kempkes,
Peter Kling, Jaroslaw Kutylowski
(Paderborn)
Friedhelm Meyer auf der Heide 2
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
Gathering problem:Gather all robots in one point
Cycle formation problem: Form a cycle
Short chain problem:Minimize the length of a chain of robots between two stations
Robotic formation problems
Friedhelm Meyer auf der Heide 3
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
Challenge: Consider robots with very limited capabilities
Our mobile robots: - can only see neighbors within a constant radius.
Thus, the decision on what to do next is solely based on relative positions of neighbors in the unit disk graph
Simple local rules are used for a global goal.
Friedhelm Meyer auf der Heide 4
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityA simple local rule: Go to the center
- In a step, a robot walks to the center of its neighbors,
i.e. to the center of their smallest enclosing ball.
Friedhelm Meyer auf der Heide 5
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityA simple local rule: Go to the center
- In a step, a robot walks to the center of its neighbors,
i.e. to the center of their smallest enclosing ball.
Friedhelm Meyer auf der Heide 6
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityA simple local rule: Go to the center
- In a step, a robot walks to the center of its neighbors,
i.e. to the center of their smallest enclosing ball.
Friedhelm Meyer auf der Heide 7
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity Discrete time models, efficiency measures
A round finishes as soon as each robot was active at least once. Asynchronous sense-compute-move model
If activation proceeds in random order: Asynchronous random order sense-compute-move model
If in a round, a (suitably chosen) subset of the robots becomes active Synchronous local activation model
Energy consumption: distance travelled, number of rounds
Trade Off: More rounds give more information about the system state, thereby shorter travel distances are possible.
Friedhelm Meyer auf der Heide 8
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityWhat I will talk about
- Local algorithms for Gathering and Short Chains- Discussion of energy-efficiency of discrete time
models- Algorithm for Short Chains in bounded-distance
and continuous time model
Friedhelm Meyer auf der Heide 9
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
Gathering
Friedhelm Meyer auf der Heide 10
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityGathering, simple local strategy
A simple strategy: Go-To-The-Center- In a step, a robot walks to the center of its neighbors,
i.e. to the center of their smallest enclosing ball.
Friedhelm Meyer auf der Heide 11
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityGathering, simple local strategy
A simple strategy: Go-To-The-Center- In a step, a robot walks to the center of its neighbors,
i.e. to the center of their smallest enclosing ball.
Friedhelm Meyer auf der Heide 12
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityGathering, simple local strategy
A simple strategy: Go-To-The-Center- In a step, a robot walks to the center of its neighbors,
i.e. to the center of their smallest enclosing ball.
- If its neighbors are connected,
it fuses with one of them
Friedhelm Meyer auf der Heide 13
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityGathering, simple local strategy
A simple strategy: Go-To-The-Center- In a step, a robot walks to the center of its neighbors,
i.e. to the center of their smallest enclosing ball.
- If its neighbors are connected,
it fuses with one of them
Ando, Suzuki, Yamashita (95), Cohen, Peleg (06),
MadH, Kempkes (08)
Go-To-The-Center performs gathering
in finitely many rounds.
Friedhelm Meyer auf der Heide 14
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityGathering with provable time bounds
Degener, Kempkes, MadH (SPAA2010)
Gathering can be done by a local algorithm in O(n²) rounds, in the asynchronous random order sense-compute-move model and in the synchronous local activation model. Each robot travels distance O(n2).
First algorithm with proven bound for number of rounds in a local model.
Friedhelm Meyer auf der Heide 15
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityThe algorithm
Algorithm for a robot r :
•Sense positions of robots within distance 2.
•If all detected robots are within distance 1, gather them at r’s position.
•Else compute the convex hull of these robots.
•If r is a vertex of the convex hull:
• If the angle of the convex hull at r is smaller then ¼/3, rearrange the robots such that some of them are moved to the same position (are “fused” ) , without destroying the connectivity of the UDG
• Else: see picture
r
2
Start situation:
•n robots have positions in the plane
•Their unit disk graph is connected
•One node is active at a time
Friedhelm Meyer auf der Heide 16
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
Example 2: 1848 nodes, 24 rounds, random order
Friedhelm Meyer auf der Heide 17
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
Forming short chains
Friedhelm Meyer auf der Heide 18
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityThe short chain problem
A winding chain of robots connects a base camp to an explorer. The chain is connected, i.e., neighboring nodes have distance at most 1.
Locality assumption: robots only see predecessor and successor in the chain.
How to transform the chain in a (close to) shortest one by local rules?
base campexplorer
Friedhelm Meyer auf der Heide 19
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
Strategies for the short chain problem
- Go-To-The-Middle- Hopper
These strategies use a discrete round model.
- Move-On-Bisector
This strategy continuously senses and continuously adapts speed and direction.
Friedhelm Meyer auf der Heide 20
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityA chain of length 300
Friedhelm Meyer auf der Heide 21
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityAfter 25 rounds
Friedhelm Meyer auf der Heide 22
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityAfter 150 rounds
Friedhelm Meyer auf der Heide 23
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityAfter 270 rounds
Friedhelm Meyer auf der Heide 24
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityGo-To-The-Middle
Go-To-The-Middle Strategy
In each round:• every robot moves to the middle position
between its neighbors
relay i
relay i+1
relay i+2
Friedhelm Meyer auf der Heide 25
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityGo-To-The-Middle
. . .
explorer base camp
Friedhelm Meyer auf der Heide 26
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityGo-To-The-Middle Analysis
J. Kutylowski, MadH
Go-To-The-Middle needs (n2/²) and O(n2 log(n)/²) rounds for reaching the straight line up to distance ².
Friedhelm Meyer auf der Heide 27
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityThe Hopper strategy
The Hopper strategy is executed in sequential runs, starting at the explorer.
The Hop operation
Friedhelm Meyer auf der Heide 28
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityHopper
remove: dist < 1 shorten: (angle <90)
A run ends with a remove, a shorten, or, if only hops occur, at the base camp.
Friedhelm Meyer auf der Heide 29
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity Hopper
explorer base camp
2 runs
Friedhelm Meyer auf der Heide 30
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityHopper
Note:
Runs of the Hopper strategy can be pipelined,
m runs on a chain of length n need n+3m rounds.
A shortest chain is not reached in general, but a short one.
J. Kutylowski, MadH (TCS 2009)
The Hopper Strategy needs O(n) rounds, each robots travels distance O(n).
It reduce the chain length to at most 21/2 D, and the number of robots to less than 3D.
Friedhelm Meyer auf der Heide 31
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityTime models
We have looked at discrete round models:
In a round, robots can sense their neighborhood, compute, and move a distance of at most 1 (or 2).
But: The closer the final configuration is approached, the smaller the movements become. Rounds do not reflect distance travelled.
Alternative cost measures incorporate the travelled distance.
- Restrict a movement to distance ± per step
! ±-bounded model
- Assume continuous sensing, and continuous adaptation of speed of direction to positions of neighbors (assume speed limit 1)
! continuous model
Friedhelm Meyer auf der Heide 32
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
Continuous and ±-bounded version of Go-To-The-Middle
Degener, Kempkes, Kling, MadH (2010)
±-bounded model ( ±2(0,1)):
(n2+n/±) = #rounds = O(n2log(n) + n/±)
Maximum distance travelled = £(±n2 + n)
± = 1/n : O(n2 log(n)) rounds, £(n) travel distance
Continuous model:
Maximum distance travelled = time = £(n)
Friedhelm Meyer auf der Heide 33
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityShort Chain: The Move-on-Bisector strategy
A robot continuously does the following:- As long as it has not reached the straight line between its neighbors, it
moves with speed 1 in direction of the bisector.
- As soon as it has reached this line, it continuously adapts speed and direction, so that it stays on the line and maintains the ratio between the distances to neighbors.
Friedhelm Meyer auf der Heide 34
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityShort Chain: The Move-on-Bisector strategy
Start ..\..\..\..\Program Files\Continuous Robot Simulator\bin\ContinuousRobotSimulator.exe
Degener, Kempkes, Kling, MadH (Sirocco 2010)
The Move-on-Bisector strategy needs time O(min{n,(Opt+d)log(n)}.
(d = distance between stations,
Opt = optimal time (= max. distance between robots and line.)
Friedhelm Meyer auf der Heide 35
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and ComplexityConclusions
• Designing and analysing local algorithms that modify the network in order to fulfil global tasks is a challenging problem
• Lots of open problems: Which capabilities of robots are necessary for a given
global task, which are suffcient, which are technically feasible?
Swarms: How can certain properties be maintained under dynamics?
Many more ……………..
Friedhelm Meyer auf der Heide 36
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
Thank you for your attention!Thank you for your attention!
Friedhelm Meyer auf der HeideHeinz Nixdorf Institute & Computer Science
DepartmentUniversity of Paderborn
Fürstenallee 1133102 Paderborn, Germany
Tel.: +49 (0) 52 51/60 64 80Fax: +49 (0) 52 51/60 64 82
Mailto: [email protected]://wwwhni.upb.de/en/alg