Realistic Issues Using Centroidal Voronoi Tessellations for Multi-Agent Coordination

1. Realistic Issues Using Centroidal Voronoi Tessellations for Multi-Agent Coordination Amanda Belleville acbelleville@gmail.com Lacy Christensen lacychristensen@hotmail.com August 5, 2010 NSF-REU

2. Outline • Introduction • Centroidal Voronoi Tessellations • Realistic Issues • Diff-Mas 2D • Time Delay • Multiple Actuators and Pollution Sources • Evacuation of a mixed Population • Future Work • Acknowledgments • References

3. Introduction • Many experiments have been conducted using CVT’s. • We have summarized work from multiple sources using CVT’s and have determined realistic issues that we wish to address. • Using CVT’s, it is possible to cover an area and use actuators and sensors to analyze what is happening in that area. Whether it be pollution spreading throughout the land that needs to be diffused or a natural disaster wreaking havoc and needing to be assessed to safely evacuate the people involved.

4. Voronoi Tessellations • A Voronoi Tessellation is a plane with n generating points that is divided into n polygons such that each polygon has one generating point. All the points in the polygon are closer to that generating point than to any other generating point. Generating point

5. Centroidal Voronoi Tessellation Generating point is equal to the centroid (center of mass). Centroid and generating point http://www.math.psu.edu/qdu/Res/Pic/cvt-eg.gifGzip.

6. Previous work with Voronoi Diagrams • Consensus between Robots with changing topologies [4] • Distance Formula with time delays and Heterogeneous velocities [11] • Sensor Placement with Voronoi Diagrams [13] • Boundary Expansion (BE) Algorithms [14] • Voronoi Diagrams with heterogeneous sensors [15] • Heterogeneous group of robots [19]

7. Realistic Issues relating to Voronoi Diagrams • Our Focus: • Voronoi Diagrams in a diffusion process. • Implementing time delays in a diffusion process. • Using Multiple Actuators • Using Multiple Pollution Sources • Discussing evacuation of a mixed population.

8. Diff-Mas 2D • Diff-Mas 2D is a Simulation platform in MatLab , which shows how actuators respond to a pollution source using a static mesh sensor network. • Diff-Mas 2D uses two different algorithms for actuator control. One is based on the basic movements of Centroidal Voronoi Tessellations, where each actuator follows it’s centroid based on the pollution density. The other is a consensus CVT algorithm, that has the actuators reach a consensus with each other based on the actuators who are in polluted regions.

9. Color Scheme • To model the pollution density, we created a color scheme. The area where the pollution is the most dense is black ( > 0.15). The next pollution density is marked by green (> 0.08), and the area with the least dense pollution is indicated by yellow (> 0.01). If the actuators over spray an area it is indicated by white (< -0.01). • Blue circles indicate the actuators, red circles are their centroids.

10. This is the first screen during the simulation. It provides the layout of the simulation and a legend with the symbols used. This is the first part of the simulation, the screen changes every 0.1 seconds showing the spreading pollution, the actuators movement, and the changing positions of the centroids.

11. Original Simulation Consensus CVT

12. Consensus CVT vs. CVT Consensus CVT CVT In CVT the actuator will always move towards their centroid, but in consensus CVT they will reach consensus about where to go with other actuators if they do not sense enough pollution.

13. Original Simulation with no Diffusion Consensus CVT CVT

14. Consensus CVT vs. CVT No diffusion Consensus CVT CVT

15. Basic Diffusion With no time delay, the better algorithm to use is the consensus CVT, based on the above graph.

16. With a time delay… • More realistic simulation • Represents delay in communication between sensors and actuators • Uniform time delay represents CVT characteristic of waiting for neighbor to acquire information

17. Implementing a time delay • In the code, we delayed the simulation by providing old information (up to 15 iterations). This prevented the actuators from responding to the pollution for up to .45 seconds. • While implementing a time delay, we also experimented with CVT vs. Consensus CVT. • We were able to determine the delay by outputting the times as it iterated through the program.

18. Delay 15 CVT Delay 15 consensus Delay 10 consensus Delay 10 CVT Delay 15 Consensus Delay 15 CVT Delay 10 Consensus Delay 10 CVT

19. Delay 15 Consensus

20. Delay 15 CVT

21. Delay 10 Consensus

22. Delay 10 CVT

23. Total Pollutant over Time

24. Comparing Spray for .45 Sec Delay CVT-uses spray of 68 Consensus-uses spray of 110

25. Comparing Spray for .3 Sec. Delay CVT-uses spray of 76 Consensus-uses spray of 58

26. Multiple Actuators and Pollution Sources • This part of the experiment uses varying number of actuators (4 and 9) and varying pollution sources (1 and 2).

27. 9 Actuators with Varying Pollution Sources

28. 9 Actuators with Varying Pollution Sources

29. Trajectories One pollution source Two pollution sources Using 9 actuators and 2 pollution sources we find that in a symmetrical case, the actuators that are equidistant from both pollution sources were unable to choose which pollution source to go towards, so they remained stuck in between throughout the entire simulation. Using 9 actuators and 1 Pollution Source, the actuators closest to the pollution source did the majority of the work.

30. Spray: 50

31. 4 Actuators with Varying Pollution Sources

32. 4 Actuators with Varying Pollution Sources

33. Trajectories One Pollution Source Two Pollution Sources Using only 4 actuators with these 2 pollution sources and keeping the spray constant proved to be insufficient to combat the pollution. Three of the actuators followed their centroids to the lower pollution source while the top actuator worked on the top pollution source alone.

34. Spray: 80

35. Evacuation of a Mixed Population • During a natural disaster, you need to evacuate people of all abilities using the least dangerous routes. We believe you can use CVT’s to model this, with obstacles indicating dangerous routes and actuators with differing speeds and capabilities to represent the varying population. • In areas with multiple disasters, you can give the disasters weights depending on their severity, and then use weighted voronoi regions. Then to find the areas least affected by disaster, you follow the Voronoi edges. [7],[8],[9]

36. Future Work • Consensus on spray amount between actuators in cases with limited spray. • Moving pollution sources. • Dynamic Sensor Networks vs. Static Sensor Networks. • Solving problem of symmetrical pollution causing ineffective actuators. • Weighted distance for Voronoi Region. • Time Delay based on actuator’s distance from sensor. • Random time delay instead of uniform. • Evacuation of a mixed population.

37. Acknowledgments • The Authors thank CSOIS for their tremendous support, both financially and intellectually. With special thanks to Dr. Haiyang Chao, Dr. Christophe Tricaud, and Dr. Yangquan Chen. Also, thanks to NSF for funding this research.

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