September 30, Probabilistic Modeling


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Multi-Robot Systems

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  • I would like to demonstrate my methodology using a simple example where two robots – robot 1 and robot 2 – have to solve two tasks, A and B.

    For this problem you can think of a series of possible algorithms that we can use, from simple random allocation of tasks to centralized, optimal allocation.

    We also know that our robots key capabilities navigation, localization and communication are subject to uncertainty.

    What we can now do is to write down all possible states of the system – robot 1 doing task A, robot 2 doing task B, both robots doing task A and so forth. We can then write down all possible state transitions. For example for a random algorithm all initial states are equally likely. For a deterministic algorithm, state transitions are given by the uncertainty of the robots subystems, for example the likelihood that a coordination message got lost.

    In this example, it is easy to calculate the expected value for time to completion analytically for a series of coordination mechanisms and assumptions on sensor and actuator reliability.
  • More formally, we associate a probability with each state of the system that varies over time as well probabilities for each possible state transition. You can visualize that using a Markov chain, here showing a two-state system with the time varying probability for the system to change from state omega prime to omega.

    The rate with which the probability to be in a certain state changes per time step is then given by the Master equation from physics – here in discrete time. The probability gets decreased by all probabilistic state transitions leading away from the system and it gets increased by all probabilistic state transitions leading into the system.
  • September 30, Probabilistic Modeling

    1. 1. Multi-Robot Systems CSCI 7000-006 Monday, September 30, 2009 Nikolaus Correll
    2. 2. So far • Reactive and deliberative distributed algorithms • Formal models describing sub-sets of the systems • Deterministic models for deliberative algorithms • Convex cost functions and feedback control for reactive systems
    3. 3. Problems so far • How to model – Sensor uncertainty (localization, vision, range) – Communication uncertainty – Actuation uncertainty (e.g. wheel-slip) • Deterministic algorithms break • Reactive algorithm become unpredictable
    4. 4. Problem Statement • Predict the performance of a system given – Problem – Algorithms – Capabilities/Uncertainty • Find most suitable coordination scheme / set of resources 4 Robot 1 Robot 2 Task A Task B Problem Algorithms Random Deliberative Centralized Decentralized Collaborative Greedy N. Correll. Coordination schemes for distributed boundary coverage with a swarm of miniature robots: synthesis, analysis and experimental validation. EPFL PhD thesis #3919, 2007. Capabilities / Probabilistic Behavior Navigation Localization Communication
    5. 5. Master equations and Markov Chains 5 • The system state ({robot states} X {environment state}) is finite • Non-deterministic elements of the system follow a known statistical distribution : Conditional probability to be in state w when in state w’ a time-step before Transition probability from w’ to w in a Markov Chain : Probability for the system to be in state w at time k w’ w
    6. 6. From Master to Rate Equations 6 : probability to be in state x at time k x can be a robot’s or a system’s state : Total number of robots Average number of robots in state x:
    7. 7. Example 1: Collision Avoidance Two states, search and avoid N0 robots State duration of avoid – Probabilistic – Deterministic Possible implementations: Obstacle “Proximal” Obstacle “180o turn” What are the parameters of this system and what are their distributions? How to get them?
    8. 8. Parameters Encountering probability pR – Probability to encounter another robot per time step Interaction time Ta – Average time a collision lasts – Geometric distribution or Dirac pulse
    9. 9. Interaction time Average time Ta constant regardless whether probabilistic or deterministic Distribution Ta is different depending on – Controller – Model abstraction level Model is only an approximation!
    10. 10. Interaction times • Probabilistic • Deterministic • Non-Parametric Distribution Systematic experiments with 1 or 2 robots.
    11. 11. Deterministic Time-Out “180o turn avoidance” Agent-based simulation Simulation egocentric Simulation allocentric A S S
    12. 12. Example 1a: Collision Avoidance Probabilistic Delay Search Avoidance pR 1/Ta
    13. 13. Example 1b: Collision Avoidance Deterministic Delay Search pR 1 Avoidance Ta
    14. 14. Example 2: Collaboration Two states: search and wait N0 robots M0 collaboration sites State duration of wait – probabilistic: robots wait a random time – deterministic: robots wait a fixed time robot site
    15. 15. Parameters Encountering probability ps – Probability to encounter one site Interaction time Tw – (Average) time a robot waits for collaboration before moving on Robot-Robot collisions are ignored in this example
    16. 16. Example 2a: Collaboration Probabilistic Delay Search Wait psNs(k) 1/Tw ps(M0-Nw(k))
    17. 17. Example 3a: Collaboration Deterministic Delay Search Wait psNs(k) ps(M0-Nw(k)) ps(M0-Nw(k-Tw))G(k;k-Tw)
    18. 18. Summary: Memory-less systems Systems with no or little memory (Time-outs), essentially reactive Master equation for a single robot allows estimation of population dynamics How to deal with deliberative systems that use memory?
    19. 19. Example 3: Task Allocation Scenario: 2 robots, 2 tasks A and B Robots prefer task A over task B Global metric requires solution of both tasks Task evaluation subject to noise, robots choose the wrong task with probability p A B 1-p p A B 1-p 1-p “Greedy” “Coordinated”
    20. 20. Example 3a: Task Allocation Non-Collaborative, Greedy Both robots will go for task A, then B Expected time: 2 time-steps Noise! Effective outcomes might be AA, AB, BB, BA There is a possibility to complete in one time- step (due to noise): AB or BA What is the state transition diagram of this system?
    21. 21. Master-Equations for Deliberative Systems: Greedy algorithm AA AB BBBA Why does this system asymptotically converge to AB or BA?
    22. 22. Example 3b: Task Allocation Collaborative Robots will allocate the tasks among them Robot 1 will go for task A, robot 2 go for task B If only one task is left, both try to accomplish it Effective outcomes AA, AB, BB, BA Expected time to completion 1 time-step for: AB and BA
    23. 23. Expected Time to Completion greedy collaborative
    24. 24. Summary Master equation: change of probability to be in state x Enumerate all possible states of a system Calculate all possible state transition probabilities Solve difference equations (numerically, analytically, Lyapunov, …) Useful for analyzing dominant collaboration dynamics of a system
    25. 25. Upcoming Formal approaches to obtain model parameters How to model systems with large state space?