1. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 1 Cooperative Control and Mobile Sensor Networks Cooperative Control, Part II Naomi Ehrich Leonard Mechanical and Aerospace Engineering Princeton University and Electrical Systems and Automation University of Pisa naomi@princeton.edu, www.princeton.edu/~naomi www.princeton.edu/~naomi

2. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 2 Collective Motion Stabilization Problem Achieve synchrony of many, individually controlled dynamical systems. How to interconnect for desired synchrony? Use simplified models for individuals. Example: phase models for synchrony of coupled oscillators. Kuramoto (1984), Strogatz (2000), Watanabe and Strogatz (1994) (see also local stability analyses in Jadbabaie, Lin, Morse (2003) and Moreau (2005)) Interconnected system has high level of symmetry. Consequence: reduction techniques of geometric control. (e.g., Newton, Holmes, Weinstein, Eds., 2002 and cyclic pursuit, Marshall, Broucke, Francis, 2004). with Rodolphe Sepulchre (University of Liege), Derek Paley (Princeton) Phase-oscillator models have been widely studied in the neuroscience and physics literature. They represent simplification of more complex oscillator models in which the uncoupled oscillator dynamics each have an attracting limit cycle in a higher-dimensional state space. Under the assumption of weak coupling, higher-dimensional models are reduced to phase models (singular perturbation or averaging methods).

3. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 3 Overview of Stabilization of Collective Motion We consider first particles moving in the plane each with constant speed and steering control. The configuration of each particle is its position in the plane and the orientation of its velocity vector. Synchrony of collective motion is measured by the relative phasing and relative spacing of particles. We observe that the norm of the average linear momentum of the group is a key control parameter: it is maximal for parallel motions and minimal for circular motions around a fixed point. We exploit the analogy with phase models of couple oscillators to design steering control laws that stabilize either parallel or circular motion. Steering control laws are gradients of phase potentials that control relative orientation and spacing potentials that control relative position. Design can be made systematic and versatile. Stabilizing feedbacks depend on a restricted number of parameters that control the shape and the level of synchrony of parallel or circular formations. Yields low-order parametric family of stabilizable collective motions: offers a set of primitives that can be used to solve path planning or optimization tasks at the group level.

4. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 4 Key References [1] Sepulchre, Paley, Leonard, “Stabilization of planar collective motion: All-to-all communication,” IEEE TAC, June 2007, in press. [2] Sepulchre, Paley, Leonard, “Stabilization of planar collective motion with limited communication,” IEEE TAC, conditionally accepted. [3] Moreau, “Stability of multiagent systems with time-dependent communication links,” IEEE TAC, 50(2), 2005. [5] Scardovi, Sepulchre, “Collective optimization over average quantities,” Proc. IEEE CDC, 2006. [6] Scardovi, Leonard, Sepulchre, “Stabilization of collective motion in the three dimensions: A consensus approach,” submitted. [7] Swain, Leonard, Couzin, Kao, Sepulchre, “Alternating spatial patterns for coordinated motion, submitted.

5. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 5 Planar Unit-Mass Particle Model Steering control Speed control

6. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 6 Planar Particle Model: Constant (Unit) Speed [Justh and Krishnaprasad, 2002] Shape variables:

7. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 7 Relative Equilibria [Justh and Krishnaprasad, 2002] Then 3N-3 dimensional reduced space is If steering control only a function of shape variables: And only relative equilibria are 1. Parallel motion of all particles. 2. Circular motion of all particles on the same circle.

8. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 8 Phase Model Then reduced model corresponds to phase dynamics: If steering control only a function of relative phases:

9. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 9 Key Ideas Particle model generalizes phase oscillator model by adding spatial dynamics: Parallel motion ⇔ Synchronized orientations Circular motion ⇔ “Anti-synchronized” orientations Assume identical individuals. Unrealistic but earlier studies suggest synchrony robust to individual discrepancies (see Kuramoto model analyses).

10. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 10 Key Ideas  is phase coherence, a measure of synchrony, and it is equal to magnitude of average linear momentum of group. [Kuramoto 1975, Strogatz, 2000] Average linear momentum of group: Centroid of phases of group:

11. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 11 Synchronized state Balanced state

12. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 12 Phase Potential 1.Construct potential from synchrony measure, extremized at desired collective formations. is maximal for synchronized phases and minimal for balanced phases. 2. Derive corresponding gradient-like steering control laws as stabilizing feedback:

13. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 13 Phase Potential

14. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 14 Phase Potential: Stabilized Solutions

15. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 15 Stabilization of Circular Formations: Spacing Potential

16. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 16 Stabilization of Circular Formations: Spacing Potential

17. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 17 Stabilization of Circular Formations: Spacing Potential

18. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 18 Composition of Phasing and Spacing Potentials Can also prove local exponential stability of isolated local minima.

19. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 19 Phase + Spacing Gradient Control

20. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 20 Stabilization of Higher Momenta

21. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 21 Stabilization of Higher Momenta

22. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 22 Symmetric Balanced Patterns

23. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 23 Symmetric Balanced Patterns

24. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 24 Symmetric Patterns, N=12 M=1,2,3 M=4,6,12

25. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 25 Stabilization of Collective Motion with Limited Communication Design concept naturally developed for all-to-all communication is recovered in a systematic way under quite general assumptions on the network communication: Approach 1. Design potentials based on graph Laplacian so that control laws respect communication constraints. (Requires time-invariant and connected communication topology and gradient control laws require bi-directional communication). Approach 2. Use consensus estimators designed for Euclidean space in the closed-loop system dynamics to obtain globally convergent consensus algorithms in non-Euclidean space. Generalize methodology to communication topology that may be time-varying, unidirectional and not fully connected at any given instant of time. Requires passing of relative estimates of averaged quantities in addition to relative configuration variables.

26. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 26 Graph Representation of Communication Particle = node Edge from k to j = comm link from particle k to j (Jadbabaie, Lin, Morse 2003, Moreau 2005) 1 7 8 6 5 2 4 9 3

27. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 27 Circulant Graphs P.J. Davis, Circulant Matrices. John Wiley & Sons, Inc., 1979. (undirected)

28. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 28 Time-Varying Graphs Moreau, 2004

29. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 29 Phase Synchronization and Balancing: Time Invariant Communication

30. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 30 Phase Synchronization and Balancing: Time Invariant Communication

31. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 31 Well-Studied Result in Euclidean Space See also Moreau 2005, Jadbabaie et al 2004 for local results.

32. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 32 Achieving Nearly Global Results for Time-Varying, Directed Graphs

33. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 33 Achieving Nearly Global Results for Time-Varying, Directed Graphs

34. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 34 Parallel and Circular Formations: Time-Invariant Case

35. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 35 Parallel and Circular Formations: Time-Varying Case

36. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 36 Further Results Resonant patterns.

37. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 37 Non-constant Curvature

38. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 38 Planar Particle Model: Oscillatory Speed Model Swain, Leonard, Couzin, Kao, Sepulchre, submitted Proc. IEEE CDC, 2007

39. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 39 Two Sets of Coupled Oscillator Dynamics

40. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 40 Steady State Circular Patterns

41. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 41 Steady State Circular Patterns for Individual

42. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 42 Stabilization of Circular Patterns

43. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 43 Circular Patterns with Prescribed Relative Phasing

44. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 44 Stabilization of Circular Patterns with Noise

45. N.E. Leonard – U. Pisa – 18-20 April 2007 Slide 45 Convergence with Limited Communication Definition of blind spot angle Simulation with blind spot