September 23, Modeling of Gradient-Based Controllers II


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September 23, Modeling of Gradient-Based Controllers II

  1. 1. Multi-Robot Systems<br />CSCI 7000-006<br />Wednesday, September 23, 2009<br />NikolausCorrell<br />
  2. 2. So far (Modeling)<br />Deterministic models for deliberative systems<br />Gradient-based controllers for reactive systems<br />Generating controllers by performing gradient descent on a cost-function<br />From global to local optimization problems using Voronoi partitions<br />
  3. 3. Today<br />More gradient-based control: <br />Shape formation<br />Flocking<br />Introduction to hybrid systems<br />
  4. 4. Gradient-based control<br />Convergence to minimal sets of a cost function over robot positions<br />Minimal sets can also be shapes or isocontours<br />Minimal sets can also be temporary and local<br />
  5. 5. Gradient-based approach for shape formation<br />Goal: distribute all robots along a 2D curve<br />Applications: construction, perimeter surveillance<br />“Minimum Set” given by an implicit function s(x,y)=0 on a 3D surface<br />L. Chaimowicz, Michael, N., and V. Kumar, &quot;Controlling Swarms of Robots Using Interpolated Implicit Functions&quot; Proceedings of the 2005 IEEE International Conference on Robotics and Automation, pp. 2498-2503, Barcelona, Spain, April 2005.<br />
  6. 6. Shape formation: Controller<br />Letf be a suitable convex function with the desired shape as isocontour with value 0<br />Let qi=[xi,yi] be the robot position<br />Let vi=qi’ be the robot speed and ui=vi’ its acceleration<br />Let Fc and Fr be forces repelling robots from each other <br />
  7. 7. Stability<br />Lyapunov candidate V(q,q’)&gt;=0<br />V(q,q’)&lt;0<br />Course Question: what did we not prove?<br />
  8. 8. Problems<br />What about the repulsive terms?<br />What about too few robots?<br />What about too many robots?<br />Further reading<br />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.<br />
  9. 9. Shape generation<br />f could be a sum of Radial Basis Functions given a set of constraint points<br />Constraint<br />RBF i is centered around pi<br />Find set of weights wi so that all constraints are satisfied<br />
  10. 10. From theory to practice<br />Simulation<br />Robots get stuck in local minima<br />Unreachable shapes (inside of letter P, e.g.) depending on initial position<br />Real robots<br />No local range and bearing<br />Constraints non-holonomic<br />
  11. 11. Example: Herding/Flocking<br />Agents are attracted to their neighbors<br />Agents are repelled by their neighbors<br />Agents move voluntarily (random or informed)<br />
  12. 12. Model<br />Kinematic model:<br />Artificial Potential field<br />Random noise<br />Agent-to-agent force<br />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.<br />
  13. 13. What can you do with this model?<br />Numerical simulation<br />Initialize positions<br />Calculate agent-to-agent interaction forces between all agents<br />Update positions<br />Gradient controller?<br />Yes! Only speed is updated<br />Can we formulate this as acost function?<br />
  14. 14. Generalized Coverage Control<br />Cost to service point qin Q:<br />New: Team-based cost<br />Mixing function: encodes collaboration<br />New cost function<br />Q<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 />
  15. 15. Properties of the Mixing function<br />Tells how information from different robots should be combined to sense at q<br />Course question: What happens for <br />
  16. 16. Mixing function<br />For<br />Results in standard Voronoi cost function (Monday) <br />
  17. 17. Mixing function<br />Let<br />Cost function<br />Result:<br />Q<br />
  18. 18. From generalized coverage to flocking<br />Cost function <br />Let agent-to-agent force be<br />Take gradient: <br />
  19. 19. Hybrid Systems<br />So far: all robots behave according to the same dynamical system<br />Hybrid systems: robot dynamics are a function of discrete states<br />Logic<br />X’=f1(X)<br />X’=f2(X)<br />Logic<br />
  20. 20. Example: Cow Herding<br />Continuous part*<br />Artificial potential field:<br />Far-field attraction<br />Near-field repulsion<br />Gaussian noise added to force estimates<br />Discrete part<br />Cows can be in two states: Grazing and Stressed.<br />Different potential fields for each state<br />*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.<br />
  21. 21. Behavioral Hypothesis<br />We theoretically study the influence of two potential social effects:<br />Animals tend to aggregate more when under stress due to a stimulus<br />Stress propagates within the herd [Butler, 2006]<br />R<br />R<br />These hypotheses are implemented in a hybrid dynamical model and tested in simulation.<br />
  22. 22. System Description<br />Cows and Environment<br />Hereford and Hereford x Brangus<br />USDA experimental range, 466ha paddock<br />Sensors<br />GPS<br />Accelerometer<br />Communication<br />900Mhz radio<br />Actuators<br />Stereo headphones<br />Electrical stimulation<br />
  23. 23. Formal description<br />State-space of agent i<br />R4<br />State transition probabilities<br />Control input (stimulus)<br />Stress propagation<br />Artificial Potential field<br />Random noise<br />Agent-to-agent force<br />
  24. 24. Simulation Environment<br />Dynamical simulation<br />Experiment<br />Initial condition: N cows grazing inside a circular fence of 25m diameter (random distribution)<br />Fence moves northwards with constant 20m/h (open loop) <br />After 5h simulated time the experiment is stopped<br />Investigate different values for a and R<br />Speed-up of about x15 between real experiment and dynamical simulation<br />
  25. 25. Sample Result: Impact of Increased Gregarious Behavior during Stress<br />50 simulations per data point<br />R= 0 m<br />R= 5 m<br />R= 10 m<br />For constant stimulus, a(x=S)&gt;a(x=G) necessary condition for aggregation to work<br />
  26. 26. Sample Result: Impact of Stress Propagation<br />Success: &gt;50% of population within fence<br />R= 0 m<br />R= 5 m<br />R= 10 m<br />Moderate stress propagation increases control performance, but potentially leads to instable systems<br />
  27. 27. Hybrid Systems<br />Analysis of individual dynamics, but unclear what state the other robots are in<br />Analysis of discrete dynamics, e.g. Markov chain<br />Verification using numerical tools<br />OverviewGoebel, Rafal; Sanfelice, Ricardo G.; Teel, Andrew R. (2009), &quot;Hybrid dynamical systems&quot;, IEEE Control Systems Magazine29 (2): 28–93<br />
  28. 28. Summary<br />Gradient descent approaches are a versatile tool for<br />Shape formation<br />Flocking<br />Coverage<br />Community is moving unified theory for controller analysis and synthesis<br />Analysis of discrete-continuous systems still in its infancy <br />
  29. 29. Next Week<br />Discussion of course projects<br />“develop”, “study”, “explore” are all words that should NOT be in your research objective<br />formulate a hypothesis that leads to your method<br />Probabilistic Models for reactive and deliberative systems<br />Assignment of teams<br />