1. A Practical Approach to Robotic Swarms IASTED Conference on Control and Applications May 2008 Howard M. Schwartz and Sidney N. Givigi Jr.

2. Slide Name Objectives,Develop a practical approach to robotic swarms.,Must be easy to implement and tractable.,Must appeal to the control engineer’s sense of performance.

3. Slide Name Literature Review,Olfati-Saber, R., “Flocking for Multi-Agent Dynamic Systems: Algorithms and Theory”, IEEE Trans. Auto. Contr. 2006.,Tanner, H.G., Jadbabaie, A., and Pappas G.J., “Stable Flocking of Mobile Agents, Part I: Fixed Topology”, Proc. of CDC, 2003. –These methods require one design an attraction and repulsive function. Designing this function is not clear. Loss of control engineers intuition. Is the system working correctly?

4. Slide Name Our Method,We use an inertial model Define Connected and Unconnected Sets Connected Unconnected

5. Slide Name The Forces on the Robots,The force on unconnected robots is a type of gravity force. The force on the connected robots is a type of spring damper force The total force on a given robot is Where, is the unit vector from i to j And r ij is the distance from i to j

6. Slide Name Simulation Results,20 Robots, 100x100 grid, k p =4, k v =4, d 0 =10, k g =100, and r=12.

7. Slide Name Swarming with obstacle avoidance,Define a potential field. Forces act along negative gradient of field Then the complete force acting on each robot is

8. Slide Name Simulation of robots swarming with obstacle avoidance,k f = 200 all other terms are the same as before.

9. Slide Name Swarm robots with constant motion and obstacle avoidance.,Define specified velocity v xd = 1.0, then the force becomes,

10. Slide Name Stability Analysis,Why does this work. Substituting for k v = 4 and k p = 4, we get the eigenvalues, λ 1 = -1.17, λ 2 = -6.82, λ 3 = 0.

11. Slide Name Stability of 3 Connected Robots,Linearize for small motions about the equilibrium point. The force on robot 1 due to robot 3 due to small motions is, The force in the x direction then becomes,

12. Slide Name Stability of 3 robots,The acceleration of robot i in the x direction is, In the case of 3 connected robots we have 12 states and we can write the linearized equations in the form

13. Slide Name Stability of 20 Robots,Using a computer to evaluate the configuration and recognizing only 3 distinct relationships between robots, we get the following maximum and minimum eigenvalues for the linearized system,,λ max = -19.86, λ min = -0.12±0.47j,Therefore the origin is asymptotically stable.

14. Slide Name Experimental Results,The robots are given positions over bluetooth link.,The robots are controlled by a HC11 Handyboard.,Web cameras installed in the ceiling track the robots.

15. Slide Name Robots Following each other and doing obstacle avoidance

16. Slide Name Conclusion,Practical approach to swarm robots –Connected and unconnected sets, gravity and spring/damper forces Potential fields define obstacles The swarm is locally stable Experimental results validate the method.