October 5, Probabilistic Modeling II
- 1. Multi-Robot SystemsCSCI 7000-006Monday, October 4, 2009NikolausCorrell
- 2. Last weekProbabilistic modelsReactive swarm systems: rate equationsDeliberative systems: Master equationEnumerate all possible statesProbabilistic state transitions reflect state transitions in the system
- 3. TodayExamples of swarming systemsCoverageAggregationParameter calibrationSystem identification
- 4. Distributed Boundary CoverageCoverage of every point on the boundary of objects in a specified areaApplications: Inspection, Maintenance, Painting, …
- 5. Baseline: Randomized Coverage without LocalizationSearchInspectTranslateAvoid ObstacleWall | RobotObstacle clearSearchInspectTranslatealong bladeptBlade1-ptTt expired
- 6. Probabilistic ModelAvoid ObstacleInspectSearchTranslatealong bladeRobotic SystemEnvironment
- 7. Encountering ProbabilitiespwprpbProbability to encounter an object is proportional toRobot speedSensor rangeSize of the object and the arenaAssumptionsUniform distribution of robots in the environmentRobots encounter only one object at a time1
- 8. Macroscopic EquationsEvery state corresponds to one difference equationExistence of a steady-state distribution can be proven by analyzing the underlying Markov chain
- 9. Candidate Model for Swarm Robotic InspectionRobotic SystemEnvironment
- 10. Model prediction vs. Real Robot Experiments20 robots25 robots30 robots
- 11. AggregationRobots stop probabilisticallyProbability function of number of neighborsEstimate of neighborhood using infrared sensingMany neighbors, high probabilityFew neighbors, low probabilityN. 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.
- 13. Modeling assumptionsA robot moves through the environment (random walk) during which it encounters other robots with constant probabilityThe probability to encounter one robot is pc, the probability to encounter a cluster of n robots npcUniform distribution of objects in the environment and linear super-position of encountering probabilitiesProbabilistic Finite State Machinenpc pjoin(n)Searchn-Clusternpc pleave(n)What other state transitions are possible using this controller? Hint:think about transitions caused by other robots.
- 14. “Passive” State Transitionsjj+1Example: 4 robots change their state without actually moving
- 15. Probabilistic Finite State MachineRobots joining aggregates“Passive” state transitions
- 16. Average Number of Agents in a Cluster of jpcNs(k)jNj(k)pjoin(j)Ns(k)pcNj-1(k)pjoin(j-1)*jj-AggregateNj+1(k)pleave(j+1)*jNj(k)pleave(j)*(j-1)Nj(k)pleave(j)
- 17. Temporal evolution of the degree distributionWhat would happen if the communication range changes and what model parameter would be affected?Realistic simulation (left), model prediction (right). 1500 experiments in Webots, communication range 10cm, arena 1m diameter, 12 individuals.
- 18. Encountering probability and communication range7cm communication range (left), and 12cm communication range (right). 1500 experiments, 12 individuals.
- 19. Limitations of Rate Equation approachEstimation of model parameters using geometric properties potentially inaccurateRate equations yield only the average performance, not its distributionProbabilistic Finite State machine does not capture all properties of the system
- 20. Multi-Level ModelingSsSaSsSaSsSaSsSaRate equations(Macroscopic level)AbstractionLevel of DetailMulti-agent models(Microscopic level)Realistic simulationReal System
- 21. Coverage: Performance Distribution20 Real RobotsAgent-based simulation
- 22. Parameter Calibration using realistic simulation
- 23. Limitations of parameter calibrationAttempt to summarize multi-faceted system dynamics into scalar valueProblematic assumptionsUniform distributionDisc-shaped detection rangesUniform speed…Qualitative better than quantitative prediction
- 24. Parameter EstimationEstimating model parameters from real robot experimentationAnalytical solutions for linear systemsExcite degrees of freedoms separately in experimentsObservation of the systemModel prediction
- 25. Example: Simple linear systemThe system’s future states can be predicted by a linear combination of the system’s current states.
- 26. Simple linear systemModel:Prediction error:Parameters minimizing the prediction error:N(k) : system equationsq : system parametersn : length of one experiment
- 27. ResultsInitial guessExperimentOptimal parameterizedModel20 experiments per team size
- 28. UpcomingSystem optimization using probabilistic modelsDiscrete Event System (DES) simulation
- 29. SchedulingMonday: LectureFriday: Course project get-togetherTuesday, Wednesday, Thursday: individual meetingsOctober 11-15: IROS conference in St. LouisNovember 30 – December 11: Project presentations (15 min)