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Multi-Robot Systems CSCI 7000-006 Monday, October 4, 2009 NikolausCorrell
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
Today Examples of swarming systems Coverage Aggregation Parameter calibration System identification
Distributed Boundary Coverage Coverage of every point on the boundary of objects in a specified area Applications: Inspection, Maintenance, Painting, …
Probabilistic Model Avoid Obstacle Inspect Search Translate along blade Robotic System Environment
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
Macroscopic Equations Every state corresponds to one difference equation Existence of a steady-state distribution can be proven by analyzing the underlying Markov chain
Candidate Model for Swarm Robotic Inspection Robotic System Environment
Model prediction vs. Real Robot Experiments 20 robots 25 robots 30 robots
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.
Modeling assumptions A robot moves through the environment (random walk) during which it encounters other robots with constant probability The probability to encounter one robot is pc, the probability to encounter a cluster of n robots npc
Uniform distribution of objects in the environment and linear super-position of encountering probabilities
Probabilistic Finite State Machine npc pjoin(n) Search n-Cluster npc pleave(n) What other state transitions are possible using this controller? Hint: think about transitions caused by other robots.
“Passive” State Transitions j j+1 Example: 4 robots change their state without actually moving
Probabilistic Finite State Machine Robots joining aggregates “Passive” state transitions
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)
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.
Encountering probability and communication range 7cm communication range (left), and 12cm communication range (right). 1500 experiments, 12 individuals.
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
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
Coverage: Performance Distribution 20 Real Robots Agent-based simulation
Parameter Calibration using realistic simulation
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
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
Example: Simple linear system The system’s future states can be predicted by a linear combination of the system’s current states.
Simple linear system Model: Prediction error: Parameters minimizing the prediction error: N(k) : system equationsq : system parametersn : length of one experiment
Results Initial guess Experiment Optimal parameterized Model 20 experiments per team size
Upcoming System optimization using probabilistic models Discrete Event System (DES) simulation
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)