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Multi-Robot Systems

Multi-Robot Systems

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    October 26, Optimization October 26, Optimization Presentation Transcript

    • Multi-Robot Systems
      CSCI 7000-006
      Monday, October26, 2009
      NikolausCorrell
    • So far
      Probabilistic models for multi-robot systems
      Extract probabilistic behavior of sub-systems
      Small state space: rate equations
      Large state space: DES simulation
    • Today
      System optimization using probabilistic models
      Find optimal control parameters
      Explore new capabilities using models
      Find optimal control and system parameters
    • Comparison of Coordination Schemes
      Too large
      Metrics for comparison
      Speed
      System cost
      System Size
      System Reliability
      Benefits to the User
      Size
      Speed
      Cost
      Too slow
      No Mapping
      Reliability
      Too expensive
      Mapping
      Benefits
      System-design is a constraint optimization problemSolution: Appropriate Models
    • Model-based design
      Size, Cost, …
      Speed, Reliability, …
      Control parameters
      Real System
      Model
      Controller Design
    • Model-based optimization
      Physical simulator
      Simulate controllers and robot designs
      DES simulator
      Simulate controllers and available information
      Optimize using
      Systematic experiments
      Learning/optimization
      Communication
      Navigation accuracy
    • Optimization using analytical models
      Probabilistic state machine is derived from the robot controller
      One difference equation per state
      pc
      Search
      Collision
      Rest
      pr
      Ns
      Tr
      Nc
      1/Tc
    • Optimization using analytical models
      Probabilistic state machine is derived from the robot controller
      One difference equation per state
      System parameters
      pc
      Search
      Collision
      Rest
      pr
      Ns
      Tr
      Nc
      1/Tc
    • Optimization using analytical models
      Probabilistic state machine is derived from the robot controller
      One difference equation per state
      Control parameters
      pc
      Search
      Collision
      Rest
      pr
      Ns
      Tr
      Nc
      1/Tc
    • Optimal Control: Brief Intro
      Find optimal control inputs for a dynamical system to optimize a metric of interest
      Example: Tank reactor, maximize quantity B by tuning inflow and outflow
      Known: system dynamics
      inflow
      A
      outflow
      A->B->C
      A->C
    • Static Optimization
      A
      inflow
      outflow
      Find optimal control inputs (constant)
      Example: inflow 50l/min, outflow 10l/min
      Constraint: Volume of the tank at final time
      flow
      volume
      time
    • Dynamic Optimization
      A
      inflow
      outflow
      Find optimal control input profiles (time-varying)
      Example: max inflow for 10s, outflow off, after 50s and outflow max
      Constraint: Volume of the tank during the whole process
      flow
      volume
      time
    • Optimal Control
      Capture terminal and stage cost as well as constraints using a single cost function
      The optimization problem is then solved by minimizing this cost function
    • Example: Coverage
      Collaboration policy:
      Robots wait at tip for Ts
      Waiting robots inform other robots to abandon coverage
      Trade-Off between additional exploration versus decreased redundancy
      Communication introduces coupling among the robots (non-linear dynamics)
      N. Correll and A. Martinoli. Modeling and Analysis of Beaconless and Beacon-Based Policies for a Swarm-Intelligent Inspection System. In IEEE International Conference on Robotics and Automation (ICRA), pages 2488-2493, Barcelona, Spain, 2005.
    • Optimal Control Problem
      A static beacon policy does not reduce completion time but only energy consumption
      Is there a dynamic policy which improves coverage performance?
      Find the optimal profile for the parameter Tsminimizing time to completion
    • Model
      N. Correll and A. Martinoli. Towards Optimal Control of Self-Organized Robotic Inspection Systems. In 8th International IFAC Symposium on Robot Control (SYROCO), Bologna, Italy, 2006.
    • Optimization Problem
      u=
      Terminal cost: time to completion
      Stage cost: energy consumption
      Constraints: number of virgin blades zero
    • Possible optimization method: ExtremumSeeking Control
      Necessary condition of optimality:
      Optimization as a feedback control problem:
      Gradient Estimate by sinusoidal perturbation:
    • Optimal Marker Policy
      Stationary Marker
      Optimal policy
      “Turn marker onafter around 180s,mark for 5s and go
      On.”
      Method
      fminconusing the macroscopic model and optimal parameters based on base-line experiment.
    • Results/Discussion
      Optimal results when beacon behavior is turned on toward the end of the experiment
      Intuition: Exploration more important in the beginning
      An optimal beacon policy only exists if there are more robots than blades
    • Randomized Coverage with Mobile Marker-based Collaboration
      Translate
      Inspect
      Inspect
      Avoid Obstacle
      Wall | Robot
      Obstacle clear
      Search
      Inspect
      Mobile
      Marker
      pt
      Blade
      1-pt | Marker
      Tt expired
    • g=0 no collaboration
      g=1 full collaboration
    • No Collaboration vs. Mobile Markers
      No Markers
      Mobile Markers
      20 Real Robots
      Agent-based simulation
    • Model-based design: Pitfalls
      m
      e
      l
      Model-based controller design depends on
      accurate parameters
      Ideal model
      Optimization problem(s)
      Find optimal control parameters
      Find optimal model parameters
      Model
      M
      m
      Estimate both model and control parameters simultaneously
      e
      Model
      l
      “Optimal control under uncertainty of measurements”
      M
    • Simultaneous optimization of model and control parameters
      How to select pi when Ti are unknown?
      Optimization algorithm
      Initial guess for model and control parameters
      Run the system and collect data
      Find optimal fit for model parameters
      Find optimal control parameters
      Repeat until error between experiment and model vanishes
      N. Correll. Parameter Estimation and Optimal Control of Swarm-Robotic Systems: A Case Study in Distributed Task Allocation. In IEEE International Conference on Robotics and Automation, pages 3302-3307, Pasadena, USA, 2008.
    • Optimal Control ofSystem and Control Parameters
      Control Parameters
      System Parameters
      All experiments
      Next experiment
    • Case Study: Task Allocation
      Finite number of tasks
      Robots select task iwith probability
      pi =const. (Independent robots)
      pi(k) function of Nj(k)
      Task itakes time Ti in average
      K. Lerman, C. Jones, A. Galstyan, and M. Matarić, “Analysis of
      dynamic task allocation in multi-robot systems,” Int. J. of Robotics
      Research, 2006.
    • 1. Independent Robots
      Model(Number of robots in state i)
      Parameters
      probability to do a task
      System parameters
      Analytical optimization
    • 2. Threshold-based Task Allocation
      Probability to do a task
      Stimulus
      Threshold
      Stimulus: Number of robots doing the task already
      Model (non-linear)
      Optimization: numerical
    • Experiment
      Step 1: Estimate model parameters
      Ti are unknown
      Take random control parameters
      Measure steady state
      Find Ti given known control parameters
      Step 2: Find optimal control parameters
    • System dynamics
      DESSimulation
      Difference Equations
      Independent Robots
      (Linear Model)
      Threshold-based (Non-Linear Model)
      100 robots
      1000 robots
    • Results
      Linear Model
      Non-Linear Model
      25 robots
      25 robots
      100 robots
    • Summary
      System models can be used for finding optimal control policies and parameters
      Models can be physical simulation, DES, or analytical
      More abstract model allow for more efficient search, even analytical
      System parameters can be optimized simultaneously with system in the loop
    • Upcoming
      Multi-Robot Navigation (M. Otte)
      Learning and adaptation in swarm systems
      3 weeks lectures, 1 week fall break
      November 29: reports due
      2 weeks project presentations, random order