October 26, Optimization


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October 26, Optimization

  1. 1. Multi-Robot Systems<br />CSCI 7000-006<br />Monday, October26, 2009<br />NikolausCorrell<br />
  2. 2. So far <br />Probabilistic models for multi-robot systems<br />Extract probabilistic behavior of sub-systems<br />Small state space: rate equations<br />Large state space: DES simulation<br />
  3. 3. Today<br />System optimization using probabilistic models<br />Find optimal control parameters<br />Explore new capabilities using models<br />Find optimal control and system parameters<br />
  4. 4. Comparison of Coordination Schemes<br />Too large<br />Metrics for comparison<br />Speed<br />System cost<br />System Size<br />System Reliability<br />Benefits to the User<br />Size<br />Speed<br />Cost<br />Too slow<br />No Mapping<br />Reliability<br />Too expensive<br />Mapping<br />Benefits<br />System-design is a constraint optimization problemSolution: Appropriate Models<br />
  5. 5. Model-based design<br />Size, Cost, …<br />Speed, Reliability, … <br />Control parameters<br />Real System<br />Model<br />Controller Design<br />
  6. 6. Model-based optimization<br />Physical simulator<br />Simulate controllers and robot designs<br />DES simulator<br />Simulate controllers and available information<br />Optimize using<br />Systematic experiments<br />Learning/optimization<br />Communication<br />Navigation accuracy<br />
  7. 7. Optimization using analytical models<br />Probabilistic state machine is derived from the robot controller<br />One difference equation per state<br />pc<br />Search<br />Collision<br />Rest<br />pr<br />Ns<br />Tr<br />Nc<br />1/Tc<br />
  8. 8. Optimization using analytical models<br />Probabilistic state machine is derived from the robot controller<br />One difference equation per state<br />System parameters<br />pc<br />Search<br />Collision<br />Rest<br />pr<br />Ns<br />Tr<br />Nc<br />1/Tc<br />
  9. 9. Optimization using analytical models<br />Probabilistic state machine is derived from the robot controller<br />One difference equation per state<br />Control parameters<br />pc<br />Search<br />Collision<br />Rest<br />pr<br />Ns<br />Tr<br />Nc<br />1/Tc<br />
  10. 10. Optimal Control: Brief Intro<br />Find optimal control inputs for a dynamical system to optimize a metric of interest<br />Example: Tank reactor, maximize quantity B by tuning inflow and outflow<br />Known: system dynamics<br />inflow<br />A<br />outflow<br />A-&gt;B-&gt;C<br />A-&gt;C<br />
  11. 11. Static Optimization<br />A<br />inflow<br />outflow<br />Find optimal control inputs (constant)<br />Example: inflow 50l/min, outflow 10l/min<br />Constraint: Volume of the tank at final time<br />flow<br />volume<br />time<br />
  12. 12. Dynamic Optimization<br />A<br />inflow<br />outflow<br />Find optimal control input profiles (time-varying)<br />Example: max inflow for 10s, outflow off, after 50s and outflow max<br />Constraint: Volume of the tank during the whole process<br />flow<br />volume<br />time<br />
  13. 13. Optimal Control<br />Capture terminal and stage cost as well as constraints using a single cost function <br />The optimization problem is then solved by minimizing this cost function<br />
  14. 14. Example: Coverage<br />Collaboration policy:<br />Robots wait at tip for Ts<br />Waiting robots inform other robots to abandon coverage <br />Trade-Off between additional exploration versus decreased redundancy<br />Communication introduces coupling among the robots (non-linear dynamics)<br />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.<br />
  15. 15. Optimal Control Problem<br />A static beacon policy does not reduce completion time but only energy consumption<br />Is there a dynamic policy which improves coverage performance?<br />Find the optimal profile for the parameter Tsminimizing time to completion<br />
  16. 16. Model<br />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.<br />
  17. 17. Optimization Problem<br />u=<br />Terminal cost: time to completion<br />Stage cost: energy consumption<br />Constraints: number of virgin blades zero<br />
  18. 18. Possible optimization method: ExtremumSeeking Control<br />Necessary condition of optimality:<br />Optimization as a feedback control problem:<br />Gradient Estimate by sinusoidal perturbation:<br />
  19. 19. Optimal Marker Policy<br />Stationary Marker<br />Optimal policy<br />“Turn marker onafter around 180s,mark for 5s and go<br />On.”<br />Method<br />fminconusing the macroscopic model and optimal parameters based on base-line experiment.<br />
  20. 20. Results/Discussion<br />Optimal results when beacon behavior is turned on toward the end of the experiment<br />Intuition: Exploration more important in the beginning<br />An optimal beacon policy only exists if there are more robots than blades<br />
  21. 21. Randomized Coverage with Mobile Marker-based Collaboration<br />Translate<br />Inspect<br />Inspect<br />Avoid Obstacle<br />Wall | Robot<br />Obstacle clear<br />Search<br />Inspect<br />Mobile<br />Marker<br />pt<br />Blade<br />1-pt | Marker<br />Tt expired<br />
  22. 22. g=0 no collaboration<br />g=1 full collaboration<br />
  23. 23. No Collaboration vs. Mobile Markers<br />No Markers<br />Mobile Markers<br />20 Real Robots<br />Agent-based simulation<br />
  24. 24. Model-based design: Pitfalls<br />m<br />e<br />l<br />Model-based controller design depends on <br />accurate parameters<br />Ideal model<br />Optimization problem(s)<br />Find optimal control parameters<br />Find optimal model parameters<br />Model<br />M<br />m<br />Estimate both model and control parameters simultaneously<br />e<br />Model<br />l<br />“Optimal control under uncertainty of measurements”<br />M<br />
  25. 25. Simultaneous optimization of model and control parameters<br />How to select pi when Ti are unknown?<br />Optimization algorithm<br />Initial guess for model and control parameters<br />Run the system and collect data<br />Find optimal fit for model parameters<br />Find optimal control parameters<br />Repeat until error between experiment and model vanishes<br />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.<br />
  26. 26. Optimal Control ofSystem and Control Parameters<br />Control Parameters<br />System Parameters<br />All experiments<br />Next experiment<br />
  27. 27. Case Study: Task Allocation<br />Finite number of tasks<br />Robots select task iwith probability<br />pi =const. (Independent robots)<br />pi(k) function of Nj(k)<br />Task itakes time Ti in average<br />K. Lerman, C. Jones, A. Galstyan, and M. Matarić, “Analysis of<br />dynamic task allocation in multi-robot systems,” Int. J. of Robotics<br />Research, 2006.<br />
  28. 28. 1. Independent Robots<br />Model(Number of robots in state i)<br />Parameters<br />probability to do a task<br />System parameters<br />Analytical optimization<br />
  29. 29. 2. Threshold-based Task Allocation<br />Probability to do a task<br />Stimulus<br />Threshold<br />Stimulus: Number of robots doing the task already<br />Model (non-linear)<br />Optimization: numerical<br />
  30. 30. Experiment<br />Step 1: Estimate model parameters<br />Ti are unknown<br />Take random control parameters<br />Measure steady state<br />Find Ti given known control parameters<br />Step 2: Find optimal control parameters<br />
  31. 31. System dynamics<br />DESSimulation<br />Difference Equations<br />Independent Robots<br />(Linear Model)<br />Threshold-based (Non-Linear Model)<br />100 robots<br />1000 robots<br />
  32. 32. Results<br />Linear Model<br />Non-Linear Model<br />25 robots<br />25 robots<br />100 robots<br />
  33. 33. Summary<br />System models can be used for finding optimal control policies and parameters<br />Models can be physical simulation, DES, or analytical<br />More abstract model allow for more efficient search, even analytical<br />System parameters can be optimized simultaneously with system in the loop<br />
  34. 34. Upcoming<br />Multi-Robot Navigation (M. Otte)<br />Learning and adaptation in swarm systems<br />3 weeks lectures, 1 week fall break<br />November 29: reports due<br />2 weeks project presentations, random order<br />