Multi-Robot Systems

1. Multi-Robot Systems CSCI 7000-006 Wednesday, September 23, 2009 NikolausCorrell

2. 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

3. Today More gradient-based control: Shape formation Flocking Introduction to hybrid systems

4. 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

5. 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.

6. 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

7. Stability Lyapunov candidate V(q,q’)>=0 V(q,q’)<0 Course Question: what did we not prove?

8. 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.

9. Shape generation f could be a sum of Radial Basis Functions given a set of constraint points Constraint RBF i is centered around pi Find set of weights wi so that all constraints are satisfied

10. From theory to practice Simulation 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

11. Example: Herding/Flocking Agents are attracted to their neighbors Agents are repelled by their neighbors Agents move voluntarily (random or informed)

12. 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.

13. 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?

14. Generalized Coverage Control Cost to service point qin Q: New: Team-based cost Mixing function: encodes collaboration New cost function Q M. Schwager, A Gradient Optimization Approach to Adaptive Multi-Robot Control, Ph.D. Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, September, 2009.

15. Properties of the Mixing function Tells how information from different robots should be combined to sense at q Course question: What happens for

16. Mixing function For Results in standard Voronoi cost function (Monday)

17. Mixing function Let Cost function Result: Q

18. From generalized coverage to flocking Cost function Let agent-to-agent force be Take gradient:

19. Hybrid Systems So far: all robots behave according to the same dynamical system Hybrid systems: robot dynamics are a function of discrete states Logic X’=f1(X) X’=f2(X) Logic

20. 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.

21. 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] R R These hypotheses are implemented in a hybrid dynamical model and tested in simulation.

22. System Description Cows and Environment Hereford and Hereford x Brangus USDA experimental range, 466ha paddock Sensors GPS Accelerometer Communication 900Mhz radio Actuators Stereo headphones Electrical stimulation

23. Formal description State-space of agent i R4 State transition probabilities Control input (stimulus) Stress propagation Artificial Potential field Random noise Agent-to-agent force

24. Simulation Environment Dynamical simulation Experiment 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

25. 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

26. 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

27. 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

28. Summary Gradient descent approaches are a versatile tool for Shape formation Flocking Coverage Community is moving unified theory for controller analysis and synthesis Analysis of discrete-continuous systems still in its infancy

29. 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