Cooperative Control and Mobile Sensor Networks

1. Cooperative Control and Mobile Sensor Networks Cooperative Control, Part I, A-C Naomi Ehrich Leonard Mechanical and Aerospace Engineering Princeton University and Electrical Systems and Automation University of Pisa naomi@princeton.edu, www.princeton.edu/~naomi

2. Natural Groups • Exhibit remarkable behaviors! • Animals may aggregate for • Predator evasion • Foraging • Mating • Saving energy Photo by Norbert Wu

3. Animal Aggregations andVehicle Groups • Animal group behaviors emerge from individual-level behavior. • * Simple control laws for individual vehicles yield versatile fleet. • Coordinated behavior in natural groups is locally controlled: • - Individuals respond to neighbors and local environment only. • - Group leadership and global information not needed. • * Minimal vehicle sensing and communication requirements. • Robustness to changes in group membership. • Herds, flocks, schools: sensor integration systems. • * Adaptive, mobile sensor networks.

4. Animal Group Models and Mechanics • Lagrangian models for fish, birds, mammals: • Locomotion forces • (drag, constant speed) • Aggregation forces • (attraction/repulsion) • Arrayal forces • (velocity/orientation alignment) • Deterministic environmental forces • (gravity, fluid motions) • Random forces • (from behavior or the environment) Artificial potentials, Gyroscopic forces, Symmetry-breaking, Reduction, Energy functions, Stability

5. Artificial Potentials for Cooperative Control • Design potential functions with minimum at desired state. • Control forces computed from gradient of potential. • Potential provides Lyapunov function to prove stability. • Distributed control. • Neighborhood of each vehicle defined by sphere of radius d1 (and h1). • Leaderless, no order of vehicles necessary. Provides robustness to failure. • Vehicles are interchangeable. • Concepts extend from particle models to rigid body body models. (Koditschek, McInnes, Krishnaprasad, …)

6. Outline and Key References Artificial Potentials and Projected Gradients: R. Bachmayer and N.E. Leonard. Vehicle networks for gradient descent in a sampled environment. In Proc. 41st IEEE CDC, 2002. Artificial Potentials and Virtual Beacons: N.E. Leonard and E. Fiorelli. Virtual leaders, artificial potentials and coordinated control of groups. In Proc. 40th IEEE CDC, pages 2968-2973, 2001. Artificial Potentials and Virtual Bodies with Feedback Dynamics: P. Ogren, E. Fiorelli and N.E. Leonard. Cooperative control of mobile sensor networks: Adaptive gradient climbing in a distributed environment. IEEE Transactions on Automatic Control, 49:8, 2004.

7. Outline and Key References D. Virtual Tensegrity Structures: B. Nabet and N.E. Leonard. Shape control of a multi-agent system using tensegrity structures. In Proc. 3rd IFAC Wkshp on Lagrangian and Hamiltonian Methods for Nonlinear Control, 2006. E. Networks of Mechanical Systems and Rigid Bodies: S. Nair, N.E. Leonard and L. Moreau. Coordinated control of networked mechanical systems with unstable dynamics. In Proc. 42nd IEEE CDC, 2003. T.R. Smith, H. Hanssmann and N.E. Leonard. Orientation control of multiple underwater vehicles. In Proc. 40th IEEE CDC, pages 4598-4603, 2001. S. Nair and N.E. Leonard. Stabilization of a coordinated network of rotating rigid bodies. In Proc. 43rd IEEE CDC, pages 4690-4695, 2004. F. Curvature Control and Level Set Tracking: F. Zhang and N.E. Leonard. Generating contour plots using multiple sensor platforms. In Proc. IEEE Swarm Intelligence Symposium, 2005.

8. Coordinating Control with Interacting Potentials Leonard and Fiorelli, CDC 2001

9. Stability of Formation

10. A. Artificial Potential Plus Projected Gradient Bachmayer and Leonard, CDC, 2002 Feedback from artificial potentials and from measurements of environment: Vehicle group in descending Gaussian valley

11. Gradient Descent: Single Vehicle with Local Gradient Information

12. Gradient Descent: Multiple Vehicle with Local Gradient Information

13. Gradient Descent: Multiple Vehicle with Local Gradient Information

14. Gradient Descent: Multiple Vehicle with Local Gradient Information

15. y r i x j Gradient Descent Example: Two Vehicles, T = ½ ||x||2

16. y i rA x j k Gradient Descent Example: Three Vehicles, T = ½ ||x||2

17. y y i rB i rA j x x j k k Gradient Descent Example: Three Vehicles, T = ½ ||x||2

18. Gradient Descent with Projected Gradient Information

19. Gradient Descent with Projected Gradient Information: Single Vehicle Case

20. Gradient Descent with Projected Gradient Information: Single Vehicle Case

21. Gradient Descent with Projected Gradient Information: Single Vehicle Case

22. Gradient Descent with Projected Gradient Information: Multiple Vehicle Case See also Moreau, Bachmayer and Leonard, 2002

23. Gradient Descent with Projected Gradient Information: Multiple Vehicle Case

24. B. Virtual Bodies and Artificial Potentials for Cooperative Control • Virtual beacons (virtual leaders) • Manipulate group geometry: specialized group geometries, symmetry breaking.

25. N vehicles with fully actuated dynamics*: Artificial Potentials and Virtual Beacons M virtual beacons are reference points on a virtual (rigid) body. describes the virtual body c.o.m. Assume all virtual beacons (i.e. virtual body) and reference frame move at constant velocity *Extension to underactuated systems is possible. For example, Lawton, Young and Beard, 2002, consider dynamics of an off-axis point on a nonholonomic robot which by feedback linearization can be made to look like double-integrator dynamics.

26. fI fh x ij Control Law for Vehicle i i k j Vh VI h h1 h0 ik d1 d0 xij h h0 d1 h1 d0 ik

27. 5 Body Simulation

28. Schooling Case: N=2, M=1 h0 h0 h0 d0 d0 v0 v0 v0 Stable (S1 symmetry) Unstable Equilibrium is minimum of total potential.

29. Symmetry Breaking Case: N=M=2 h0 h0 d0 v0 h0 d0 v0 h0

30. v0 v0 d0 d0 d0 d0 Schooling Case: N=3 d0 h0

31. v0 h1 h0 d1 d0 Schooling: Hexagonal Lattice, N > 3

32. v0 d1 d0 Schooling: Special geometries, N>3, M>1

34. Schooling Stability

35. Introduce feedback dynamics for virtual bodies to introduce mission: direct group motion, split/merge subgroups, avoid obstacles, climb gradients. • Configuration space of virtual body is • for orientation, position and dilation factor: • Because of artificial potentials, vehicles in formation will translate, • rotate, expand and contract with virtual body. • To ensure stability and convergence, prescribe virtual body • dynamics so that its speed is driven by a formation error. • Define direction of virtual body dynamics to satisfy mission. • Partial decoupling: Formation guaranteed independent of mission. • Prove convergence of gradient climbing. C. Artificial Potentials and Virtual Body with Feedback Dynamics (Ogren, Fiorelli, Leonard, MTNS 2002 and IEEE TAC, 2004)

36. Stability of Formation

37. Translation, Rotation, Expansion and Contraction

38. Stability of xeq(s) at Any Fixed s

39. Speed of Traversal and Formation Stabilization

42. Simulation of Path in SO(2)

43. Mission Trajectories Let translation, rotation, expansion, and contraction evolve with feedback from sensors on vehicles to carry out mission such as gradient climbing. Augmented state space is (x,s,r,R,k). Express the vector fields for the virtual body motion as: To satisfy mission we choose rules for the directions

44. Adaptive Gradient Climbing

45. 0 Least Squares Estimate of Gradient of Measured Scalar Field

46. Least Squares Estimate of Gradient of Measured Scalar Field Assumed measurement noise

47. Least Squares Estimate of Gradient of Measured Scalar Field

48. Optimal Formation Problem

49. Optimal Formation Problem

50. Optimal Formation Problem: Three-Vehicle Case