Cooperative Control and Mobile Sensor Networks

Transcription

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 Slide 1 Collective Motion Stabilization Problem Achieve synchrony of many, individually controlled dynamical systems. How to interconnect for desired synchrony? with Rodolphe Sepulchre (University of Liege), Derek Paley (Princeton) Use simplified models for individuals. Example: phase models for synchrony of coupled oscillators. Kuramoto (1984), Strogatz (2000), Watanabe and Strogatz (1994) 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). (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). Slide 2 1

2 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. Slide 3 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), [5] Scardovi, Sepulchre, Collective optimization over average quantities, Proc. IEEE CDC, [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. Slide 4 2

3 Planar Unit-Mass Particle Model Steering control Speed control Slide 5 Planar Particle Model: Constant (Unit) Speed [Justh and Krishnaprasad, 2002] Shape variables: Slide 6 3

4 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] Slide 7 Phase Model If steering control only a function of relative phases: Then reduced model corresponds to phase dynamics: Slide 8 4

5 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). Slide 9 Key Ideas Centroid of phases of group: Average linear momentum of group: [Kuramoto 1975, Strogatz, 2000] is phase coherence, a measure of synchrony, and it is equal to magnitude of average linear momentum of group. Slide 10 5

6 Synchronized state Balanced state Slide 11 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: Slide 12 6

7 Phase Potential Slide 13 Phase Potential: Stabilized Solutions Slide 14 7

8 Stabilization of Circular Formations: Spacing Potential Slide 15 Stabilization of Circular Formations: Spacing Potential Slide 16 8

9 Stabilization of Circular Formations: Spacing Potential Slide 17 Composition of Phasing and Spacing Potentials Can also prove local exponential stability of isolated local minima. Slide 18 9

10 Phase + Spacing Gradient Control Slide 19 Stabilization of Higher Momenta Slide 20 10

11 Stabilization of Higher Momenta Slide 21 Symmetric Balanced Patterns Slide 22 11

12 Symmetric Balanced Patterns Slide 23 Symmetric Patterns, N=12 M=1,2,3 M=4,6,12 Slide 24 12

13 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. Slide 25 Graph Representation of Communication Particle = node Edge from k to j = comm link from particle k to j (Jadbabaie, Lin, Morse 2003, Moreau 2005) Slide 26 13

14 Circulant Graphs (undirected) P.J. Davis, Circulant Matrices. John Wiley & Sons, Inc., Slide 27 Time-Varying Graphs Moreau, 2004 Slide 28 14

15 Phase Synchronization and Balancing: Time Invariant Communication Slide 29 Phase Synchronization and Balancing: Time Invariant Communication Slide 30 15

16 Well-Studied Result in Euclidean Space See also Moreau 2005, Jadbabaie et al 2004 for local results. Slide 31 Achieving Nearly Global Results for Time-Varying, Directed Graphs Slide 32 16

17 Achieving Nearly Global Results for Time-Varying, Directed Graphs Slide 33 Parallel and Circular Formations: Time-Invariant Case Slide 34 17

18 Parallel and Circular Formations: Time-Varying Case Slide 35 Further Results Resonant patterns. Slide 36 18

19 Non-constant Curvature Slide 37 Planar Particle Model: Oscillatory Speed Model Swain, Leonard, Couzin, Kao, Sepulchre, submitted Proc. IEEE CDC, 2007 Slide 38 19

20 Two Sets of Coupled Oscillator Dynamics Slide 39 Steady State Circular Patterns Slide 40 20

21 Steady State Circular Patterns for Individual Slide 41 Stabilization of Circular Patterns Slide 42 21

22 Circular Patterns with Prescribed Relative Phasing Slide 43 Stabilization of Circular Patterns with Noise Slide 44 22

23 Convergence with Limited Communication Definition of blind spot angle Simulation with blind spot Slide 45 23