2. Last week Probabilistic models Reactive swarm systems: rate equations Deliberative systems: Master equation Enumerate all possible states Probabilistic state transitions reflect state transitions in the system
3. Today Examples of swarming systems Coverage Aggregation Parameter calibration System identification
4. Distributed Boundary Coverage Coverage of every point on the boundary of objects in a specified area Applications: Inspection, Maintenance, Painting, …
7. Encountering Probabilities pw pr pb Probability to encounter an object is proportional to Robot speed Sensor range Size of the object and the arena Assumptions Uniform distribution of robots in the environment Robots encounter only one object at a time 1
8. Macroscopic Equations Every state corresponds to one difference equation Existence of a steady-state distribution can be proven by analyzing the underlying Markov chain
11. Aggregation Robots stop probabilistically Probability function of number of neighbors Estimate of neighborhood using infrared sensing Many neighbors, high probability Few neighbors, low probability 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.
16. Average Number of Agents in a Cluster of j pcNs(k)jNj(k)pjoin(j) Ns(k)pcNj-1(k)pjoin(j-1)*j j-Aggregate Nj+1(k)pleave(j+1)*j Nj(k)pleave(j)*(j-1) Nj(k)pleave(j)
17. Temporal evolution of the degree distribution What 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 range 7cm communication range (left), and 12cm communication range (right). 1500 experiments, 12 individuals.
19. Limitations of Rate Equation approach Estimation of model parameters using geometric properties potentially inaccurate Rate equations yield only the average performance, not its distribution Probabilistic Finite State machine does not capture all properties of the system
20. Multi-Level Modeling Ss Sa Ss Sa Ss Sa Ss Sa Rate equations (Macroscopic level) Abstraction Level of Detail Multi-agent models (Microscopic level) Realistic simulation Real System
23. Limitations of parameter calibration Attempt to summarize multi-faceted system dynamics into scalar value Problematic assumptions Uniform distribution Disc-shaped detection ranges Uniform speed … Qualitative better than quantitative prediction
24. Parameter Estimation Estimating model parameters from real robot experimentation Analytical solutions for linear systems Excite degrees of freedoms separately in experiments Observation of the system Model prediction
25. Example: Simple linear system The system’s future states can be predicted by a linear combination of the system’s current states.
26. Simple linear system Model: Prediction error: Parameters minimizing the prediction error: N(k) : system equationsq : system parametersn : length of one experiment
27. Results Initial guess Experiment Optimal parameterized Model 20 experiments per team size
29. Scheduling Monday: Lecture Friday: Course project get-together Tuesday, Wednesday, Thursday: individual meetings October 11-15: IROS conference in St. Louis November 30 – December 11: Project presentations (15 min)
