Cooperative Control and Mobile Sensor Networks

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

2. Collective Motion Stabilization Problem with Rodolphe Sepulchre (University of Liege), Derek Paley (Princeton) • Achieve synchrony of many, individually controlled dynamical systems. • How tointerconnect for desired synchrony? • Use simplifiedmodels 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 ofsymmetry. • Consequence: reduction techniques of geometric control. • (e.g., Newton, Holmes, Weinstein, Eds., 2002 and cyclic pursuit, Marshall, Broucke, Francis, 2004). 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. 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. 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. Planar Unit-Mass Particle Model Steering control Speed control

6. Planar Particle Model: Constant (Unit) Speed [Justh and Krishnaprasad, 2002] Shape variables:

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

8. Phase Model If steering control only a function of relative phases: Then reduced model corresponds to phase dynamics:

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. Average linear momentum of group: Centroid of phases of group: 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]

11. Synchronized state Balanced state

12. Phase Potential 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. Phase Potential

14. Phase Potential: Stabilized Solutions

15. Stabilization of Circular Formations: Spacing Potential

16. Stabilization of Circular Formations: Spacing Potential

17. Stabilization of Circular Formations: Spacing Potential

18. Composition of Phasing and Spacing Potentials Can also prove local exponential stability of isolated local minima.

19. Phase + Spacing Gradient Control

20. Stabilization of Higher Momenta

21. Stabilization of Higher Momenta

22. Symmetric Balanced Patterns

23. Symmetric Balanced Patterns

24. Symmetric Patterns, N=12 M=1,2,3 M=4,6,12

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. (Jadbabaie, Lin, Morse 2003, Moreau 2005) 2 9 1 3 8 4 7 5 6 Graph Representation of Communication Particle = node Edge from k to j = comm link from particle k to j

27. Circulant Graphs (undirected) P.J. Davis, Circulant Matrices. John Wiley & Sons, Inc., 1979.

28. Time-Varying Graphs Moreau, 2004

29. Phase Synchronization and Balancing: Time Invariant Communication

30. Phase Synchronization and Balancing: Time Invariant Communication

31. Well-Studied Result in Euclidean Space See also Moreau 2005, Jadbabaie et al 2004 for local results.

32. Achieving Nearly Global Results for Time-Varying, Directed Graphs

33. Achieving Nearly Global Results for Time-Varying, Directed Graphs

34. Parallel and Circular Formations: Time-Invariant Case

35. Parallel and Circular Formations: Time-Varying Case

36. Further Results • Resonant patterns.

37. Non-constant Curvature

38. Planar Particle Model: Oscillatory Speed Model Swain, Leonard, Couzin, Kao, Sepulchre, submitted Proc. IEEE CDC, 2007

39. Two Sets of Coupled Oscillator Dynamics

40. Steady State Circular Patterns

41. Steady State Circular Patterns for Individual

42. Stabilization of Circular Patterns

43. Circular Patterns with Prescribed Relative Phasing

44. Stabilization of Circular Patterns with Noise

45. Convergence with Limited Communication Definition of blind spot angle Simulation with blind spot