October 5, Probabilistic Modeling II

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Multi-Robot Systems

Multi-Robot Systems

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  • 1. Multi-Robot Systems
    CSCI 7000-006
    Monday, October 4, 2009
    NikolausCorrell
  • 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, …
  • 5. Baseline: Randomized Coverage without Localization
    Search
    Inspect
    Translate
    Avoid Obstacle
    Wall | Robot
    Obstacle clear
    Search
    Inspect
    Translate
    along blade
    pt
    Blade
    1-pt
    Tt expired
  • 6. Probabilistic Model
    Avoid Obstacle
    Inspect
    Search
    Translate
    along blade
    Robotic System
    Environment
  • 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
  • 9. Candidate Model for Swarm Robotic Inspection
    Robotic System
    Environment
  • 10. Model prediction vs. Real Robot Experiments
    20 robots
    25 robots
    30 robots
  • 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.
  • 12. Individual Robot Behavior
    pjoin(Environment)
    Search
    Rest
    pleave(Environment)
  • 13. 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.
  • 14. “Passive” State Transitions
    j
    j+1
    Example: 4 robots change their state without actually moving
  • 15. Probabilistic Finite State Machine
    Robots joining aggregates
    “Passive” state
    transitions
  • 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
  • 21. Coverage: Performance Distribution
    20 Real Robots
    Agent-based simulation
  • 22. Parameter Calibration using realistic simulation
  • 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
  • 28. Upcoming
    System optimization using probabilistic models
    Discrete Event System (DES) simulation
  • 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)