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September 21, Modeling of Gradient-Based Controllers I
 

September 21, Modeling of Gradient-Based Controllers I

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    September 21, Modeling of Gradient-Based Controllers I September 21, Modeling of Gradient-Based Controllers I Presentation Transcript

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