Cooperative Control and Mobile Sensor Networks

Transcription

1 Cooperative Control and Mobile Sensor Networks Cooperative Control, Part I, D-F Naomi Ehrich Leonard Mechanical and Aerospace Engineering Princeton University and Electrical Systems and Automation University of Pisa Slide 1 Outline and Key References A. 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, B. 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 , C. 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, Slide 2 1

2 D. Virtual Tensegrity Structures: Outline and Key References 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, 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, T.R. Smith, H. Hanssmann and N.E. Leonard. Orientation control of multiple underwater vehicles. In Proc. 40th IEEE CDC, pages , S. Nair and N.E. Leonard. Stabilization of a coordinated network of rotating rigid bodies. In Proc. 43rd IEEE CDC, pages , 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, Slide 3 D. Virtual Tensegrity Structures with Ben Nabet Slide 4 2

3 Linear Model (see papers by R. Connelly) Real cables do not increase in length and real struts do not decrease in length. Slide 5 Potential Slide 6 3

4 Equilibria This fixes the shape of the equilibria but not the size. Slide 7 Nonlinear Model Slide 8 4

5 Potential Slide 9 Equilibria Slide 10 5

6 Slide 11 Shape Change Choose a path from initial to final configuration that consists of a path of stable tensegrity structures. Can then prove boundedness of transient and convergence to final structure. Slide 12 6

7 Cable Strut Initial shape Final shape Slide 13 Multi-Scale Shape Change Slide 14 7

8 E. Networks of Mechanical Systems/Rigid Bodies Geometric framework: Method of Controlled Lagrangians with A.M. Bloch and J.E. Marsden - Energy shaping for stabilization of (otherwise unstable) underactuated mechanical systems. - Restrict to control dynamics that derive from a Lagrangian. - Theory is constructive for certain classes: Synthesis! also D.E. Chang and C.A. Woolsey, P.S. Krishnaprasad, G. Sanchez de Alvarez, see also IDA-PBC method Blankenstein, Ortega, Spong, van der Schaft et al Slide 15 Method of Controlled Lagrangians Given a mechanical system, possibly underactuated and possibly with unstable dynamics. Design L c so corresponding Euler-Lagrange equations match original equations with control law. Matching conditions are PDE s. For certain classes of systems, use structured modification L c of L - Q=S x G. L invariant to G. Shape kinetic energy metric. - Modify potential energy to break symmetry (if desired). Yields parametrized family of L c that satisfy matching conditions. Theory provides conditions on (control) parameters for stability: g φ m M l u - Construct energy function. - Consider dissipation and asymptotic stability s Bloch, Leonard, Marsden, IEEE TAC, 2000, 2001 Slide 16 8

9 Coordination Design artificial potentials to couple N individual systems. - Relative position/orientation of vehicle pairs. - Potential well = desired group configuration. Treat coupled multi-body system with same approach as for individual. - Symmetry group G for Hamiltonian + potentials. - Reduce action of G on phase space. - Construct energy function to prove: Individual dynamics are stabilized and group is stably coordinated. Nair and Leonard; Smith, Hanssmann and Leonard Slide 17 Role of Symmetry Potentials will break symmetry: E.g., consider N vehicles and Q = SE(3) x... x SE(3) Suppose Q is original symmetry group. N times - Break N-1 copies of SO(3) to align orientations. - Break N-1 copies of SE(3) to align and distribute. - Break N copies of SO(3) to align and to orient whole group, etc. Break symmetry for coordination and group cohesion. Preserve symmetries when control authority is limited. Discrete symmetries in homogeneous group with no ordering. Slide 18 9

10 Same Features as in Particle Systems Distributed control. Neighborhood of each vehicle can be prescribed. (Global info not required) No ordering of vehicles is necessary. Provides robustness to failure. Vehicles are interchangeable. Illustrations: A. Two (underwater) vehicles in SE(3) B. N inverted-pendulum-on-cart systems. Slide 19 A: Coordinated Orientation of 2 Vehicles in SE(3) with Troy Smith and Heinz Hanssmann A B A B Slide 20 10

11 Introduce Artificial Potential Slide 21 Reduced System Slide 22 11

12 Underwater Vehicles Slide 23 B: Coordination of Mechanical Systems with Unstable Dynamics with Sujit Nair Extend controlled Lagrangians to collection of unstable mechanical systems with controlled coupling. Class of systems includes inverted pendulum on a cart. Goal: Stabilize each pendulum in the upright position while synchronizing the motion of the carts. Slide 24 12

13 Extension to Network of Systems M g φ l u m s c sin φ y Slide 25 Curvature Control and Level Set Tracking Generating a contour plot with three clusters: Exploring Scalar Fields Fumin Zhang and N.E. Leonard, Proc. IEEE Swarm Symposium, 2005 Slide 26 13

14 Four moving sensor platforms, each takes one measurement a time: r 1 r 2 Filter Design Taylor Series: r c r 4 r 3 Slide 27 Filtering problem: From a series of measurements find and at the center. Filter Design Step k-1: Step k: Slide 28 14

15 Prediction: Step k-1: Step k: Filter Design Slide 29 Update: Find that minimizes Filter Design error covariance of prediction error covariance of measurements We get: Slide 30 15

16 Estimate: y r 3 Filter Design r 1 a r c D cy r 4 b D cx D c r 2 x A special arrangement to simplify the estimators Slide 31 Estimate: Filter Design How to estimate the Hessian We have a prediction Slide 32 16

17 Estimate: r 2 r 3 r F D F r K Filter Design r 1 r E D E r c D c r J D K r 4 D J Assuming formation is small enough Slide 33 We now know the Hessian: Filter Design Slide 34 17

18 Estimation Error: Error in estimate of field value at center. Error in estimate of gradient at center. Formation Design Optimization: Find and that minimize the mean square error L. Slide 35 Optimization: Formation Design Covariance matrix of updated measurements Error in estimate of first diag. el. of Hessian General solutions are numerical. We found analytical solutions when B is diagonal. [Ögren, Fiorelli and Leonard 04],[FZ, Leonard SIS05] [FZ, Leonard CDC06]. Slide 36 18

19 Goals for cooperative controllers: Cooperative Control 1. Achieve the cross formation with optimal shape and. * a 2. Align the horizontal axis of the formation with the tangent vector to the level curve at the center. 3. Control the motion of the center to go along the desired level curve. We get a contour plot with gradient estimates along the level curve. * b Slide 37 Jacobi Vectors: r 3 q 2 Cooperative Control q 3 q 1 r1 2 r 4 r Slide 38 19

20 Decoupled Dynamics: Formation Center Cooperative Control where and Slide 39 Formation Control: Cooperative Control r 1 * a y 1 r 3 * b x 1 r 2 r 4 Slide 40 20

21 Reduced center dynamics: y 1 x 1 Tracking Level Curves r(s) Boundary tracking is a special case. Slide 41 Control Lyapunov Function: Tracking Level Curves Steering Control: which achieves Convergence proved using LaSalle s Invariance Principle. [FZ, Leonard CDC06, SIS05] Slide 42 21