October 5, Probabilistic Modeling II


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Multi-Robot Systems

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October 5, Probabilistic Modeling II

  1. 1. Multi-Robot Systems<br />CSCI 7000-006<br />Monday, October 4, 2009<br />NikolausCorrell<br />
  2. 2. Last week<br />Probabilistic models<br />Reactive swarm systems: rate equations<br />Deliberative systems: Master equation<br />Enumerate all possible states<br />Probabilistic state transitions reflect state transitions in the system<br />
  3. 3. Today<br />Examples of swarming systems<br />Coverage<br />Aggregation<br />Parameter calibration<br />System identification<br />
  4. 4. Distributed Boundary Coverage<br />Coverage of every point on the boundary of objects in a specified area<br />Applications: Inspection, Maintenance, Painting, …<br />
  5. 5. Baseline: Randomized Coverage without Localization<br />Search<br />Inspect<br />Translate<br />Avoid Obstacle<br />Wall | Robot<br />Obstacle clear<br />Search<br />Inspect<br />Translate<br />along blade<br />pt<br />Blade<br />1-pt<br />Tt expired<br />
  6. 6. Probabilistic Model<br />Avoid Obstacle<br />Inspect<br />Search<br />Translate<br />along blade<br />Robotic System<br />Environment<br />
  7. 7. Encountering Probabilities<br />pw<br />pr<br />pb<br />Probability to encounter an object is proportional to<br />Robot speed<br />Sensor range<br />Size of the object and the arena<br />Assumptions<br />Uniform distribution of robots in the environment<br />Robots encounter only one object at a time<br />1<br />
  8. 8. Macroscopic Equations<br />Every state corresponds to one difference equation<br />Existence of a steady-state distribution can be proven by analyzing the underlying Markov chain<br />
  9. 9. Candidate Model for Swarm Robotic Inspection<br />Robotic System<br />Environment<br />
  10. 10. Model prediction vs. Real Robot Experiments<br />20 robots<br />25 robots<br />30 robots<br />
  11. 11. Aggregation<br />Robots stop probabilistically<br />Probability function of number of neighbors<br />Estimate of neighborhood using infrared sensing<br />Many neighbors, high probability<br />Few neighbors, low probability<br />N. Correll and A. Martinoli. Modeling Self-Organized Aggregation in a Swarm of Miniature Robots. In IEEE 2007 International Conference on Robotics and Automation Workshop on Collective Behaviors inspired by Biological and Biochemical Systems, Rome, Italy, 2007.<br />
  12. 12. Individual Robot Behavior<br />pjoin(Environment)<br />Search<br />Rest<br />pleave(Environment)<br />
  13. 13. Modeling assumptions<br />A robot moves through the environment (random walk) during which it encounters other robots with constant probability<br />The probability to encounter one robot is pc, the probability to encounter a cluster of n robots npc<br /><ul><li>Uniform distribution of objects in the environment and linear super-position of encountering probabilities</li></li></ul><li>Probabilistic Finite State Machine<br />npc pjoin(n)<br />Search<br />n-Cluster<br />npc pleave(n)<br />What other state transitions are possible using this controller? Hint:<br />think about transitions caused by other robots.<br />
  14. 14. “Passive” State Transitions<br />j<br />j+1<br />Example: 4 robots change their state without actually moving<br />
  15. 15. Probabilistic Finite State Machine<br />Robots joining aggregates<br />“Passive” state <br />transitions<br />
  16. 16. Average Number of Agents in a Cluster of j<br />pcNs(k)jNj(k)pjoin(j)<br />Ns(k)pcNj-1(k)pjoin(j-1)*j<br />j-Aggregate<br />Nj+1(k)pleave(j+1)*j<br />Nj(k)pleave(j)*(j-1)<br />Nj(k)pleave(j)<br />
  17. 17. Temporal evolution of the degree distribution<br />What would happen if the communication range changes and what model parameter would be affected?<br />Realistic simulation (left), model prediction (right). 1500 experiments in Webots, communication range 10cm, arena 1m diameter, 12 individuals.<br />
  18. 18. Encountering probability and communication range<br />7cm communication range (left), and 12cm communication range (right). 1500 experiments, 12 individuals.<br />
  19. 19. Limitations of Rate Equation approach<br />Estimation of model parameters using geometric properties potentially inaccurate<br />Rate equations yield only the average performance, not its distribution<br />Probabilistic Finite State machine does not capture all properties of the system<br />
  20. 20. Multi-Level Modeling<br />Ss<br />Sa<br />Ss<br />Sa<br />Ss<br />Sa<br />Ss<br />Sa<br />Rate equations<br />(Macroscopic level)<br />Abstraction<br />Level of Detail<br />Multi-agent models<br />(Microscopic level)<br />Realistic simulation<br />Real System<br />
  21. 21. Coverage: Performance Distribution<br />20 Real Robots<br />Agent-based simulation<br />
  22. 22. Parameter Calibration using realistic simulation<br />
  23. 23. Limitations of parameter calibration<br />Attempt to summarize multi-faceted system dynamics into scalar value<br />Problematic assumptions<br />Uniform distribution<br />Disc-shaped detection ranges<br />Uniform speed<br />…<br />Qualitative better than quantitative prediction<br />
  24. 24. Parameter Estimation<br />Estimating model parameters from real robot experimentation<br />Analytical solutions for linear systems<br />Excite degrees of freedoms separately in experiments<br />Observation of the system<br />Model prediction<br />
  25. 25. Example: Simple linear system<br />The system’s future states can be predicted by a linear combination of the system’s current states.<br />
  26. 26. Simple linear system<br />Model:<br />Prediction error:<br />Parameters minimizing the prediction error:<br />N(k) : system equationsq : system parametersn : length of one experiment <br />
  27. 27. Results<br />Initial guess<br />Experiment<br />Optimal parameterized<br />Model<br />20 experiments per team size<br />
  28. 28. Upcoming<br />System optimization using probabilistic models<br />Discrete Event System (DES) simulation<br />
  29. 29. Scheduling<br />Monday: Lecture<br />Friday: Course project get-together<br />Tuesday, Wednesday, Thursday: individual meetings<br />October 11-15: IROS conference in St. Louis<br />November 30 – December 11: Project presentations (15 min)<br />