1. Friedhelm Meyer auf der Heide 1 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Energy-Efficient Local Strategies for Robotic Formations Friedhelm Meyer auf der Heide University of Paderborn Joint work with Bastian Degener, Barbara Kempkes, Peter Kling, Jaroslaw Kutylowski (Paderborn)

2. Friedhelm Meyer auf der Heide 2 HEINZ NIXDORF INSTITUTE University 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

3. Friedhelm Meyer auf der Heide 3 HEINZ NIXDORF INSTITUTE University 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.

4. Friedhelm Meyer auf der Heide 4 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity A 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.

5. Friedhelm Meyer auf der Heide 5 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity A 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.

6. Friedhelm Meyer auf der Heide 6 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity A 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.

7. Friedhelm Meyer auf der Heide 7 HEINZ NIXDORF INSTITUTE University 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.

8. Friedhelm Meyer auf der Heide 8 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity What 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

9. Friedhelm Meyer auf der Heide 9 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Gathering

10. Friedhelm Meyer auf der Heide 10 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Gathering, 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.

11. Friedhelm Meyer auf der Heide 11 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Gathering, 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.

12. Friedhelm Meyer auf der Heide 12 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Gathering, 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

13. Friedhelm Meyer auf der Heide 13 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Gathering, 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.

14. Friedhelm Meyer auf der Heide 14 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Gathering 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(n 2 ). First algorithm with proven bound for number of rounds in a local model.

15. Friedhelm Meyer auf der Heide 15 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity The algorithm Algorithm for a robot r : Sense positions of robots within distance 2. If all detected robots are within distance 1, gather them at rs 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

16. Friedhelm Meyer auf der Heide 16 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Example 2: 1848 nodes, 24 rounds, random order

17. Friedhelm Meyer auf der Heide 17 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Forming short chains

18. Friedhelm Meyer auf der Heide 18 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity The 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 camp explorer

19. Friedhelm Meyer auf der Heide 19 HEINZ NIXDORF INSTITUTE University 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.

20. Friedhelm Meyer auf der Heide 20 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity A chain of length 300

21. Friedhelm Meyer auf der Heide 21 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity After 25 rounds

22. Friedhelm Meyer auf der Heide 22 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity After 150 rounds

23. Friedhelm Meyer auf der Heide 23 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity After 270 rounds

24. Friedhelm Meyer auf der Heide 24 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Go-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

25. Friedhelm Meyer auf der Heide 25 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Go-To-The-Middle... explorer base camp

26. Friedhelm Meyer auf der Heide 26 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Go-To-The-Middle Analysis J. Kutylowski, MadH Go-To-The-Middle needs (n 2 /²) and O(n 2 log(n)/²) rounds for reaching the straight line up to distance ².

27. Friedhelm Meyer auf der Heide 27 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity The Hopper strategy The Hopper strategy is executed in sequential runs, starting at the explorer. The Hop operation

28. Friedhelm Meyer auf der Heide 28 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Hopper remove: dist < 1 shorten: (angle <90) A run ends with a remove, a shorten, or, if only hops occur, at the base camp.

29. Friedhelm Meyer auf der Heide 29 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Hopper explorer base camp 2 runs

30. Friedhelm Meyer auf der Heide 30 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Hopper 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 2 1/2 D, and the number of robots to less than 3D.

31. Friedhelm Meyer auf der Heide 31 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Time 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

32. Friedhelm Meyer auf der Heide 32 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Continuous and ±-bounded version of Go-To-The-Middle Degener, Kempkes, Kling, MadH (2010) ±-bounded model ( ±2(0,1)): (n 2 +n/±) = #rounds = O(n 2 log(n) + n/±) Maximum distance travelled = £(±n 2 + n) ± = 1/n : O(n 2 log(n)) rounds, £(n) travel distance Continuous model: Maximum distance travelled = time = £(n )

33. Friedhelm Meyer auf der Heide 33 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Short 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.

34. Friedhelm Meyer auf der Heide 34 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Short Chain: The Move-on-Bisector strategy Start..\..\..\..\Program Files\Continuous Robot Simulator\bin\ContinuousRobotSimulator.exe..\..\..\..\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.)

35. Friedhelm Meyer auf der Heide 35 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Conclusions 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 ……………..

36. Friedhelm Meyer auf der Heide 36 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Thank you for your attention! Friedhelm Meyer auf der Heide Heinz Nixdorf Institute & Computer Science Department University of Paderborn Fürstenallee 11 33102 Paderborn, Germany Tel.: +49 (0) 52 51/60 64 80 Fax: +49 (0) 52 51/60 64 82 Mailto: fmadh@upb.de http://wwwhni.upb.de/en/alg