Distributed Algorithms for Multi-Robot Systems

1. Title, Introduction Analysis and Implementation of Distributed Algorithms for Multi-Robot Systems James McLurkin jamesm@csail.mit.edu MIT Computer Science and Artificial Intelligence Lab

3. |measured performance| |maximum performance| physical accuracy = Contributions • A suite of algorithms with provable performance in ideal conditions and robust real-world • Complexity Metrics for Computation on Multi-Robot Systems: • Physical Accuracy • Physical Running Time • Communication Complexity • Configuration Complexity • Physical accuracy dependence on robot speed • A metric of how “good” a configuration is: 0 ≤ accuracy ≤ 1 • A scalar analog of correctness

4. Assumptions • Local Communications & Multi-Hop Communications • Neighbor Localization • High Mobility • Single Connected Component (rooted to charger) • Same software on all robots • Unique IDs • Implications • Local Communications + High MobilityNeed regular communications to discover local network,Periodic communications can reduce bandwidth and simplify proofs • Neighbor Localization + Local CommunicationsCan use local network geometry in algorithms

5. Multi-Hop Message Broadcast • A robot can use the message broadcast to navigate towards the source • A robot anywhere in the network can make progress to source 1 hop 2 hops source 3 hops Li and Rus, Navigation Protocols in Sensor Networks, 2005

6. spanning tree path Euclidean distance k = Communications and Mobility are Related • Network is used for communication and physical routing • Communications must use multi-hop messages • Relationship of network topology to physical configuration is important

7. The Graph Spanner • The “crookedness” of the path is the spanner • Spanner is a function of density Klienrock and Sylvester, Optimum Transmission Radii for Packet Radio Networks, 1978

8. 2r kt message speed = Message Speed If robots move too fast, the network can’t maintain spatial relationships

9. Broadcast Effectiveness The faster the robots move, the less time they have to receive messages from their neighbors

10. Message Speed and Effectiveness Affects Accuracy • Multi-hop message speed can be comparable to robot speed • Algorithm accuracy depends on message speed, network speed, and communications style

11. Multi-hop Tree Construction: Dynamic Network • Moving robots create dynamic network topologies • Motion “smears” network and reduces the correspondence between network topology and physical positions

12. Multi-hop Navigation: Dynamic Network • Moving robots create dynamic network topologies • Motion “smears” network and reduces the correspondence between network topology and physical positions midpoint routing tree picture goes here

13. Multi-hop Navigation: Dynamic Network

14. Alpha Shapes1 • Any point on a disc of radius a that contains no other points is a-extreme 1. Edelsbrunner, Kirkpatrick, and Seidel, On the Shape of a Set of Points in the Plane, 1983

15. Not Alpha shapes • Show counter examples in sketcher

16. Distributed Boundary Detection: C-Shape • Robots that have missing sectors are on the boundary • Physical accuracy depends on measurement accuracy of local network geometry

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

18. Local Boundary Structure add concave crossing case Talk about inferred edges Accuracy≈0.92 when network speed = 0

19. No Geometric Consensus • The robots cannot compute a consistent triangulation… B B A A Robot A’s Triangulation Robot B’s Triangulation • …so the a-shape they compute is an approximation… B B Robot B’s a-shape Robot A’s a-shape B A • …with additional boundary nodes

20. Boundary Subgraph Construction • Each boundary runs leader election algorithm to form subgraph and agree on name • Runs in O(diam(G)) time • Accuracy depends on network being stable long enough for the message to get all the way around

21. Global Boundary Classification

22. The Main Point • Physical accuracy depends on network speed and communications type • Distributed Algorithm Communication Structure: • No communication • Network Geometry • Broadcast Effectiveness • Broadcast Tree Construction • Broadcast Tree Construction + Routing • Broadcast Tree Construction + Routing + Rebroadcast Beautiful data showing algorithm accuracy vs. communication type (data goes here)

23. Algorithm Summary

24. Does This Apply to all Robots? • Can I describe a model relating communications and mobility? • Are there conserved quantities in this model? • Is there a parameterization to facilitate comparisons across disparate hardware platforms?

25. The Rest of my Thesis: • Physical Accuracy • How well will the algorithm perform and • How robust is it to errors? • Physical Running Time • How long will the algorithm take? • Communication Complexity • How much communications resources are needed? • Configuration Complexity • How many robots are required? • How much information is stored in their physical configuration? • In general, Multi-Robot Distributed Algorithm performance can depend more on physical speed and communications than on processor performance • A framework of cross-platform performance metrics will be useful, but more importantly, will get me out of MIT

26. Future Work • Multi-Robot Multi-Hop Speed Limits • Broadcast + convergeCast + broadcast is this it? • Algorithm Classes Coordinate Systems with errors • (Local  global)  distributed consensus? • Self Stabilizing Distributed Algorithms • Continuum between • dist alg finte # of errors • want theory of rate of errors • control theory