September 21, Modeling of Gradient-Based Controllers I

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September 21, Modeling of Gradient-Based Controllers I

  1. 1. Multi-Robot Systems<br />CSCI 7000-006<br />Monday, September 21, 2009<br />NikolausCorrell<br />
  2. 2. So far<br />Reactive Algorithms<br />Local interactions<br />Threshold-based dynamics<br />Gradient-based<br />We know it works, but how to prove it?<br />Deliberative Algorithms<br />Local planning<br />Collaborative planning<br />Tight coordination<br />
  3. 3. Today<br />Part II of the course: Modeling of Multi-Robot Systems<br />Gradient-based models for reactive control<br />Cost-function over position of the robots<br />Basic behaviors, nomode switching <br />M. Schwager, A Gradient Optimization Approach to Adaptive Multi-Robot Control, Ph.D. Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, September, 2009.<br />
  4. 4. Gradient-based approaches<br />“Gradient” estimate<br />Use sensor and position information<br />Compare value and positions with local neighbors<br />Requires local range and bearing for collaboration<br />Gradient can describe<br />Environmental sensing<br />Neighborhood relations<br />
  5. 5. Example: Maximizing Visual Coverage<br />Flying robots are equipped with downward facing cameras<br />Cost function of robot positions encodes information gain<br />Robots move to locally optimize information gain<br />Course question: come up with a reactive controller to do this<br />M. Schwager, B. Julian, and D. Rus, Optimal coverage for multiple hovering robots with downward-facing cameras, In Proc. of the International Conference of Robotics and Automation (ICRA 09), Kobe, Japan, May, 2009.[<br />
  6. 6. M. Schwager, B. Julian, and D. Rus, Optimal coverage for multiple hovering robots with downward-facing cameras, In Proc. of the International Conference of Robotics and Automation (ICRA 09), Kobe, Japan, May, 2009.[<br />Example: Maximizing Visual Coverage<br />
  7. 7. Example 2: Optimally sample an environmental distribution<br />Goal: deploy more robots into regions with high information density<br />Information density unknown at first<br />Learn parameterized model while moving<br />
  8. 8. Example 2: Optimally sample a environmental distribution<br />M. Schwager, J. McLurkin, J.-J. E. Slotine, and D. Rus, From theory to practice: Distributed coverage control experiments with groups of robots, In Proc. of the International Symposium on Experimental Robotics (ISER 08), Athens, Greece, July, 2008.<br />
  9. 9. Multi-Robot System Model<br />State-space of each robot is its position in P<br />Vectoris a single point indescribing the system<br /> Cost function<br />Control input (speed)<br />pi<br />
  10. 10. Closed-Loop Control<br />Speed for robot i calculated such that each robot moves towards a local minima of H<br />Stability depends on properties of H!<br />M. Schwager, A Gradient Optimization Approach to Adaptive Multi-Robot Control, Ph.D. Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, September, 2009.<br />
  11. 11. Stability of Dynamical Systems<br />Entire classes on this topic!<br />Straightforward for linear systems<br />Non-Linear systems:<br />Find function that bounds the system<br />Prove properties of this function<br />Keyword: Lyapunov stability and its variations<br />Convexity and Continuity<br />
  12. 12. Lyapunov Stability<br />System<br />Find function <br />V(x) &gt;= 0 and V(x)=0 only if x=0<br />V’(x(t)) &lt; 0 (negative definite)<br />System is Lyapunov stable if V(x) with the above properties exist<br />V(x) is called the Lyapunov candidate <br />Common analogy, spring-damper, energy of the system the Lyapunov candidate, energy only decays, system stable<br />
  13. 13. Other important concepts: Convexity<br />A set is convex when all points on the line between any two points is also on the set<br />Non-convex sets are concave<br />Local minima of convex functions are global minima<br />Convex Set<br />Concave Set<br />Convex function<br />
  14. 14. Other important concepts: Continuity<br />A continuous function f has no abrupt changes<br />A function is Lipschitz continuous if there is a positive b so that<br />Limits the maximum slope<br />A locally Lipschitz first-order differential equation has a unique solution!<br />Lipschitz:<br />Locally Lipschitz:<br />
  15. 15. Voronoi Cost Function<br />Introduce sensory function fover P<br />Example: Oil spill<br />Goal: more robots where fis high <br />Cost functionwith the cost of measuring a value at q from pi<br />Optimal solution:minimize H(P)<br />Q<br />pi<br />
  16. 16. Voronoi Cost Function<br />Goal:<br />mini has only one solution (closest robot to q)<br />Voronoi tesselation of q with cells Vi<br />Result:<br />V(s) consisting of all points closer tosthan to any other site<br />J. Cortes, S. Martınez, T. Karatas, and F. Bullo. Coverage control for mobile<br />sensing networks. IEEE Transactions on Robotics and Automation, 20(2):243–<br />255, April 2004.<br />
  17. 17. Optimizing the Voronoi Cost Function<br />Solve<br />Calculate 1st derivative<br />Let , e.g. light sensor<br />Define<br />Move pi to the centroid of its Voronoi cell! <br />Mass of Vi<br />First moment of Vi<br />Centroid of Vi <br />S. P. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, 28(2):129–137, 1982.<br />
  18. 18. Course question<br />The gradient is given by<br />What is the control law for robot i<br />ui=<br />
  19. 19. Proof of Convergence (Sketch)<br />Theorem: converges assymptotically to the set of centroidalVoronoi Configurations<br />Tool: LaSalle’s invariance principle<br />Requirements:<br /><ul><li>Q, P are bounded
  20. 20. P invariant under control law -> trajectories are bounded
  21. 21. Control law locally Lipschitz
  22. 22. < 0</li></ul>Q<br />
  23. 23. From theory to practice<br />Voronoi neighbors are not communication neighbors <br />Approximate decomposition (noise on position information)<br />Time discrete execution vs. continuous dynamics<br />Voronoi Neighbors are not necessarily communication neighbors<br />
  24. 24. Summary<br />Convergence can be proven for a subset of reactive control laws<br />Tools: gradient descent on cost function<br />Key: encoding of the problem into an analytically tractable cost function<br />Tricks: Voronoi decomposition<br />Develop applications that can be broken down into known systems<br />
  25. 25. This week<br />Other important classes of reactive multi-robot problems with provable properties<br />Friday: Prairie-Dog, hand in results from last week to <br />Till Sunday: Project proposal, 1 page (max), 12pt<br />
  26. 26. Project Proposal<br />Everybody should think about a project<br />What is the objective?<br />I want to test hypothesis A<br />I want to apply method B to problem C<br />I want to prove conjecture D<br />What is the method?<br />This is about coming up with a scientific research project in the domain of this course<br />You don’t have to do this project! We will put together teams next week to work on feasible projects<br />

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