Distributed Consensus in Multivehicle Cooperative Control: Theory and Applications

Transcription

1 Distributed Consensus in Multivehicle Cooperative Control: Theory and Applications Wei Ren and Randal W. Beard Springer ISBN: Tutorial Slides Prepared by Wei Ren Department of Electrical and Computer Engineering Utah State University

2 Potential Applications for Autonomous Vehicles Civil and Commercial: Automated Mining Monitoring environment Monitoring disaster areas Communications relays Law enforcement Precision agriculture Military: Special Operations: Situational Awareness Intelligence, surveillance, and reconnaissance Communication node Battle damage assessment Homeland Security: Border patrol Surveillance Rural/Urban search and rescue 2 Intel Epson

3 Cooperative/Coordinated Control Motivation: While single vehicles performing solo missions will yield some benefits, greater benefits will come from the cooperation of teams of vehicles. Common Theme: Coordinate the movement of multiple vehicles in a certain way to accomplish an objective. - e.g. many small, inexpensive vehicles acting together can achieve more than one monolithic vehicle. e.g., networked computers Shifts cost and complexity from hardware platform to software and algorithms. courtesy: SRI, courtesy: Airforce Technology, Multi-vehicle Applications: Space-based interferometers, future combat systems, surveillance and reconnaissance, hazardous material handling, distributed reconfigurable sensor networks courtesy: NASA, 3

4 Cooperative Control Categorization Formation Control - Approaches: leader-follower, behavioral, virtual structure/leader, artificial potential function, graph-rigidity - Applications: mobile robots, unmanned air vehicles, autonomous underwater vehicles, satellites, spacecraft, automated highways Task Assignment, cooperative transport, cooperative role assignment, air traffic control, cooperative timing - Cooperative search, reconnaissance, surveillance (military, homeland security, border patrol, etc.) - Cooperative monitoring of forest fires, oil spills, wildlife, etc. - Rural search and rescue. 4

5 Cooperative Control: Inherent Challenges Complexity: Systems of systems. Communication: Limited bandwidth and connectivity. What? When? To whom? Arbitration: Team vs. Individual goals. Computational resources: Will always be limited 5

6 Cooperative Control: Centralized vs Distributed Schemes Centralized Schemes Assumptions: availability of global team knowledge, centralized planning and coordination, fully connected network Practical Issues: sparse & intermittent interaction topologies (limited communication/sensing range, environmental factors) Distributed Schemes Features: Local neighbor-to-neighbor interaction, evolve in a parallel manner Strengths: reduced communication/sensing requirement; improved scalability, flexibility, reliability, and robustness 6

7 Distributed Consensus Algorithms Basic Idea Each vehicle updates its information state based on the information states of its local (possibly time-varying) neighbors in such a way that the final information state of each vehicle converges to a common value. Extensions Relative state t deviations, incorporation of other group behaviors (e.g., collision avoidance) Feature Only local neighbor-to-neighbor interaction required R Vicsek s Model Boids: 7

8 Consensus Algorithms Literature Review Historical Perspective biology, physics, computer science, economics, load balancing in industry, complex networks Theoretic Aspects algebraic graph theory, nonlinear tools, random network, optimality and synthesis, communication delay, asynchronous communication, Applications rendezvous, formation control, flocking, attitude synchronization, sensor fusion, WiR Wei Ren, Randal dlwb W. Beard, Distributed ib t Consensus in Multi-vehicle l Cooperative Control, Communications and Control Engineering Series, Springer-Verlag, London, 2008 (ISBN: ) 8

9 Modeling of Vehicle Interactions A directed spanning tree A undirected graph that is connected (i) Separated groups A directed graph that is strongly connected A graph that has a spanning tree but not strongly connected 9 (ii) Multiple leaders

10 Modeling of Vehicle Interactions (cont.) adjacency matrix 10 Laplacian matrix

11 Outline Part 1: Consensus for Single-integrator integrator Kinematics Theory and Applications Part 2: Consensus for Double-integrator Dynamics Theory and Applications Part 3: Consensus for Rigid Body Attitude Dynamics Theory and Applications Part 4: Synchronization of Networked Euler- Lagrange Systems Theory and Applications 11

12 Consensus Algorithm for 1 st -order Kinematics 12

13 Convergence Result (Fixed Graph) The Laplacian matrix has a simple zero eigenvalue and all the others have positive real parts. Directed graph has a directed spanning tree. Consensus is reached. 13

14 Sketch of the Proof An inductive approach Step 1: find a spanning tree that is a subset of the graph Step 2: show that consensus can be achieved with the spanning tree (renumber each agent) Step 3: show that if consensus can be achieved for a graph, then adding more links will still guarantee consensus (1) (2) (3) 14

15 Consensus Equilibrium (Fixed Topology) The initial condition of a node contributes to the equilibrium value if and only if the node has a directed path to all the other nodes. 15

16 Convergence Result (Switching Graphs) 16

17 Sketch of the Proof 17

18 Examples - Consensus and Directed Spanning Trees Consensus can be achieved: Equilibrium determined by vehicles 1 and 2 Cases when consensus cannot be achieved: Consensus can be achieved: (i) Separated groups (ii) Multiple leaders 18 Union of (i) and (ii)

19 Rendezvous Experiments Experiment was performed in CSOIS (joint work with Chao, Bougeous, Sorensen, and Chen) Switching Topologies Union Topology 19

20 Rendezvous Demos Switching Topologies Switching Topologies Union Topology 20

21 Axial Alignment 21

22 Consensus with a Virtual Leader 22

23 Sketch of the Proof 23

24 Examples Consensus Tracking Subcase (a) reference ξ i t (sec) Subcase (b) reference 1 ξ i t (sec) 24

25 Example - Virtual Leader/Structure Based Formation Control (Centralized) r jf r j rj d (x vc,y vc, vc ) C F C o 25

26 Example - Virtual Leader/Structure Based Formation Control (decentralized) r d j r jf C Fj C o 26

27 A Unified Scheme for Distributed Formation Control Communication Network Consensus-based Formation State Estimator Module #i Group Leader Follower Consensus-based Formation Control Module #i Vehicle #i 27

28 Formation State Estimation Level 28

29 Vehicle Control Level 29

30 Experimental Platform at USU 30

31 Mobile Robot Kinematic Model y d (x,y) θ v (x h,y h ) o 31 x

32 Experimental Specification 32

33 Formation Control with a Simple Group Leader 33

34 Experimental Demonstration (formation control) Four robots maintaining a square shape Three robots in line formation 34

35 Outline Part 1: Consensus for Single-integrator Kinematics Theory and Applications Part 2: Consensus for Double-integrator Dynamics Theory and Applications Part 3: Consensus for Rigid Body Attitude Dynamics Theory and Applications Part 4: Synchronization of Networked Euler- Lagrange Systems Theory and Applications 35

36 Consensus Algorithm for Double-integrator Dynamics 36

37 Convergence Analysis (Relative Damping) 37

38 Example with Relative Damping 1.4 ξ i ξ 1 ξ 2 ξ 3 ξ t (s) ζ 1 ζ 2 ζ 3 ζ 4 ζ i t (s) 38

39 Switching Topologies Directed Case 39

40 Sketch of the Proof 40

41 Application - Cooperative Monitoring with UAVs Single UAV Experiment 41

42 Application UAV Formation Flying 42 3D postflight analysis of the flight on July 4th, (Leader and follower are set at different altitudes to avoid possible collision. The blue dots represent the leader's positions and the red dots represent the follower's positions.)

43 Actuator Saturation 43

44 Coupled Second-order Harmonic Oscillators 44

45 Convergence Analysis (Leadless Case) 45

46 Application - Synchronized Motion Coordination 46

47 Outline Part 1: Consensus for Single-integrator Kinematics Theory and Applications Part 2: Consensus for Double-integrator Dynamics Theory and Applications Part 3: Consensus for Rigid Body Attitude Dynamics Theory and Applications Part 4: Synchronization of Networked Euler- Lagrange Systems Theory and Applications 47

48 Consensus for Rigid Body Attitude Dynamics 48

49 Sketch of the Proof 49

50 Unavailability of Angular Velocity Measurements 50

51 Sketch of the Proof 51

52 Application - Spacecraft Attitude Synchronization Synchronized spacecraft rotations Courtesy: NASA JPL 52

53 Experimental Demonstration (attitude control) 53

54 Reference Attitude Tracking 54

55 Reference Attitude Tracking Control Law 55

56 Sketch of the Proof 56

57 Application - Spacecraft Reference Attitude Tracking 57

58 Outline Part 1: Consensus for Single-integrator Kinematics Theory and Applications Part 2: Consensus for Double-integrator Dynamics Theory and Applications Part 3: Consensus for Rigid Body Attitude Dynamics Theory and Applications Part 4: Synchronization of Networked Euler- Lagrange Systems Theory and Applications 58

59 Synchronization of Networked Euler-Lagrange Systems 59