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September 23, Modeling of Gradient-Based Controllers II

September 23, Modeling of Gradient-Based Controllers II



Multi-Robot Systems

Multi-Robot Systems



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    September 23, Modeling of Gradient-Based Controllers II September 23, Modeling of Gradient-Based Controllers II Presentation Transcript

    • Multi-Robot Systems
      CSCI 7000-006
      Wednesday, September 23, 2009
    • So far (Modeling)
      Deterministic models for deliberative systems
      Gradient-based controllers for reactive systems
      Generating controllers by performing gradient descent on a cost-function
      From global to local optimization problems using Voronoi partitions
    • Today
      More gradient-based control:
      Shape formation
      Introduction to hybrid systems
    • Gradient-based control
      Convergence to minimal sets of a cost function over robot positions
      Minimal sets can also be shapes or isocontours
      Minimal sets can also be temporary and local
    • Gradient-based approach for shape formation
      Goal: distribute all robots along a 2D curve
      Applications: construction, perimeter surveillance
      “Minimum Set” given by an implicit function s(x,y)=0 on a 3D surface
      L. Chaimowicz, Michael, N., and V. Kumar, "Controlling Swarms of Robots Using Interpolated Implicit Functions" Proceedings of the 2005 IEEE International Conference on Robotics and Automation, pp. 2498-2503, Barcelona, Spain, April 2005.
    • Shape formation: Controller
      Letf be a suitable convex function with the desired shape as isocontour with value 0
      Let qi=[xi,yi] be the robot position
      Let vi=qi’ be the robot speed and ui=vi’ its acceleration
      Let Fc and Fr be forces repelling robots from each other
    • Stability
      Lyapunov candidate V(q,q’)>=0
      Course Question: what did we not prove?
    • Problems
      What about the repulsive terms?
      What about too few robots?
      What about too many robots?
      Further reading
      M. A. Hsieh, V. Kumar and L. Chaimowicz. Decentralized Controllers for Shape Generation with Robotic Swarms. Robotica, Vol. 26, Issue 5, September 2008, pp 691-701.
    • Shape generation
      f could be a sum of Radial Basis Functions given a set of constraint points
      RBF i is centered around pi
      Find set of weights wi so that all constraints are satisfied
    • From theory to practice
      Robots get stuck in local minima
      Unreachable shapes (inside of letter P, e.g.) depending on initial position
      Real robots
      No local range and bearing
      Constraints non-holonomic
    • Example: Herding/Flocking
      Agents are attracted to their neighbors
      Agents are repelled by their neighbors
      Agents move voluntarily (random or informed)
    • Model
      Kinematic model:
      Artificial Potential field
      Random noise
      Agent-to-agent force
      M. Schwager, C. Detweiler, I. Vasilescu, D. M. Anderson, D. Rus - Data-Driven Identification of Group Dynamics for Motion Prediction and Control, Journal of Field Robotics 25(6-7):305-324, 2008.
    • What can you do with this model?
      Numerical simulation
      Initialize positions
      Calculate agent-to-agent interaction forces between all agents
      Update positions
      Gradient controller?
      Yes! Only speed is updated
      Can we formulate this as acost function?
    • Generalized Coverage Control
      Cost to service point qin Q:
      New: Team-based cost
      Mixing function: encodes collaboration
      New cost function
      M. Schwager, A Gradient Optimization Approach to Adaptive Multi-Robot Control, Ph.D. Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, September, 2009.
    • Properties of the Mixing function
      Tells how information from different robots should be combined to sense at q
      Course question: What happens for
    • Mixing function
      Results in standard Voronoi cost function (Monday)
    • Mixing function
      Cost function
    • From generalized coverage to flocking
      Cost function
      Let agent-to-agent force be
      Take gradient:
    • Hybrid Systems
      So far: all robots behave according to the same dynamical system
      Hybrid systems: robot dynamics are a function of discrete states
    • Example: Cow Herding
      Continuous part*
      Artificial potential field:
      Far-field attraction
      Near-field repulsion
      Gaussian noise added to force estimates
      Discrete part
      Cows can be in two states: Grazing and Stressed.
      Different potential fields for each state
      *M. Schwager, C. Detweiler, I. Vasilescu, D. Anderson, and D. Rus, “Data-driven identification of group dynamics for motion prediction and control,” Journal of Field Robotics, 2008.
    • Behavioral Hypothesis
      We theoretically study the influence of two potential social effects:
      Animals tend to aggregate more when under stress due to a stimulus
      Stress propagates within the herd [Butler, 2006]
      These hypotheses are implemented in a hybrid dynamical model and tested in simulation.
    • System Description
      Cows and Environment
      Hereford and Hereford x Brangus
      USDA experimental range, 466ha paddock
      900Mhz radio
      Stereo headphones
      Electrical stimulation
    • Formal description
      State-space of agent i
      State transition probabilities
      Control input (stimulus)
      Stress propagation
      Artificial Potential field
      Random noise
      Agent-to-agent force
    • Simulation Environment
      Dynamical simulation
      Initial condition: N cows grazing inside a circular fence of 25m diameter (random distribution)
      Fence moves northwards with constant 20m/h (open loop)
      After 5h simulated time the experiment is stopped
      Investigate different values for a and R
      Speed-up of about x15 between real experiment and dynamical simulation
    • Sample Result: Impact of Increased Gregarious Behavior during Stress
      50 simulations per data point
      R= 0 m
      R= 5 m
      R= 10 m
      For constant stimulus, a(x=S)>a(x=G) necessary condition for aggregation to work
    • Sample Result: Impact of Stress Propagation
      Success: >50% of population within fence
      R= 0 m
      R= 5 m
      R= 10 m
      Moderate stress propagation increases control performance, but potentially leads to instable systems
    • Hybrid Systems
      Analysis of individual dynamics, but unclear what state the other robots are in
      Analysis of discrete dynamics, e.g. Markov chain
      Verification using numerical tools
      OverviewGoebel, Rafal; Sanfelice, Ricardo G.; Teel, Andrew R. (2009), "Hybrid dynamical systems", IEEE Control Systems Magazine29 (2): 28–93
    • Summary
      Gradient descent approaches are a versatile tool for
      Shape formation
      Community is moving unified theory for controller analysis and synthesis
      Analysis of discrete-continuous systems still in its infancy
    • Next Week
      Discussion of course projects
      “develop”, “study”, “explore” are all words that should NOT be in your research objective
      formulate a hypothesis that leads to your method
      Probabilistic Models for reactive and deliberative systems
      Assignment of teams