Maintaining Communication Between an Explorer and a Base Station

1. Maintaining Communication Between an Explorer and a Base Station Miroslaw Dynia Jaroslaw Kutylowski Pawel Lorek Friedhelm Meyer auf der Heide

2. Problem statement • Robots moving through terrain (exploring, working …) • Base station serves as supply robot base station

3. Problem statement • Robots moving through terrain (exploring, working …) • Base station serves as supply • Robots should self-organize to fulfill their tasks • a communication network is a necessary primitive • How to maintain such a communication network?

4. Problem statement • Large distances between robots • Mobile relay stations support communication links robot base station

5. Problem statement • Large distances between robots • Mobile relay stations support communication links • New approach for communication on long distances • Necessary in complicated terrain (mountains…) • Related to backbones in networks (GSM infrastructure) • But: mobile, adaptive and ad-hoc

6. Problem statement • Large distances between robots • Mobile relay stations support communication links • Robots move • Communication network must react to dynamics • Relays are costly • Use as few as possible Need for a strategy for mobile relay stations Self-organizing robots, organic system local strategy, no communication, simple (no memory)

7. Agenda • Model • Go-To-The-Middle strategy • Analysis for static case • proof outline • Analysis for dynamic case • review over experiments • theoretical results • Further results & open questions

8. Model • Plane • One explorer • One base station • Relay stations arranged in a chain explorer base station

9. Model • Plane • One explorer • One base station • Relay stations arranged in a chain • Two neighbored relay stations in distance at most d • Relay stations should arrange on line between explorer and base station • Static setting – Explorer and base station stand still • Dynamic setting – Explorer moves

10. Model robot explorer base station base station • Why one explorer makes sense? • to get an understanding of the problem • for multiple explorers an efficient solution to the one-explorer problem is necessary

11. Go-To-The-Middle • Go-To-The-Middle Strategy • every relay station moves to the middle positionbetween its neighbors • discrete time steps • all stations move in parallel relay i+1 relay i+2 relay i

12. Go-To-The-Middle • Go-To-The-Middle Strategy • every relay station moves to the middle positionbetween its neighbors • discrete time steps • all stations move in parallel • Properties • simple • memoryless • biologically inspired – bird flocks • related to formation control

13. Go-To-The-Middle Analysis (static) • Key question • given a valid configuration of relay stations betweenthe explorer and base station • what is the number of Go-To-The-Middle rounds necessary to get the relays next to the optimal line? explorer base station

14. Go-To-The-Middle Analysis (static) • for each relay consider its distance from the line between explore and base station • describe the distances as a vector • v = (d1,…,dn) di explorer base station

15. Go-To-The-Middle Analysis (static) • vector v after applying one step of Go-To-The-Middle • v’ = v A • n x n matrix A vector v after applying t steps of Go-To-The-Middle v’ = v At

16. Go-To-The-Middle Analysis (static) • At the beginning vi ≤ n • We look for a t such thatvi At ≤ 1 and so At ≤ 1/n • Then the distance of each station to the optimal lineis at most 1 • consider a random walk on a line with reflecting barriers ½ ½ ½ ½ ½ ½ ½ ½ ½ ½

17. Go-To-The-Middle Analysis (static) • each element of line is a state • probability distribution to be in a particular state at beginning • w = (w1,…,wn) • the same probability distribution after t steps of random walk • w’ = w Bt • there are results stating that Bt < 1/n for t=c n2 log n • (elementary Markov Chain theory) ½ ½ ½ ½ ½ ½ ½ ½ ½ ½

18. Go-To-The-Middle Analysis (static) matrix B random walk on a line and GTM have common background in t=c n2 log n we have At<1/n

19. Go-To-The-Middle Analysis (static) • what is the number of Go-To-The-Middle rounds necessary to get the relays next to the optimal line? • quite a lot ≈ n2 log n • maybe such bad configurations do not come up in practice? • analysis of Go-To-The-Middle in the dynamic case

20. Go-To-The-Middle Analysis (dynamic) • Model • base station stands still • explorer moves • explorer starts moving next to base station • whenever needed explorer deploys new relays • one GTM-step for one step of explorer • Analysis goal • monitor the number of relay stations used • compare to the number needed for a perfect line • ratio R

21. Go-To-The-Middle Analysis (dynamic) • Experimental evaluation • explorer moves on a circle around base station • hard case • for every distance, the number of relay stations reaches a stability point • ratio R grows linearly with the distance of explorer to base station

22. Go-To-The-Middle Analysis (dynamic) • Experimental evaluation • explorer performs a (bayesian) random walk on plane • ratio R remains constant

23. Go-To-The-Middle Analysis (dynamic) • Model • explorer deploys new relay stations only when moving away from base station • explorer waits when distance to last relay station is too large • Analysis • what is the speed of the explorer? (how much must he wait?) • Result • speed of explorer ≈1/d with d the distance to base station

24. Further results & open questions • Further (unpublished) results • reducing the locality and simplicity (to some extent) one can obtain much better performances • extension to terrain with obstacles • Open questions • can one improve the performance without sacrificing locality and simplicity? • general lower bound for local strategies? • multiple explorers

25. Thank you for your attention! Heinz Nixdorf Institute & Computer Science Institute University of Paderborn Fürstenallee 11 33102 Paderborn, Germany Tel.: +49 (0) 52 51/60 64 66 Fax: +49 (0) 52 51/62 64 82 E-Mail: jarekk@upb.de http://wwwhni.upb.de/alg