1. Information Architecture and Control Design for Rigid Formations Brian DO Anderson, Changbin (Brad) Yu, Baris Fidan Australian National University and National ICT Australia 2007 Chinese Control Conference

2. Chinese Control Conference 中国控制会议 2 Thanks Coauthors: J Hendrickx J Lin A Morse I Shames D v d Walle W Whiteley R Yang To the conference organisers for inviting me To my colleagues for helping me: Contributions of all these individuals appear somewhere in this talk P Belhumeur V Blondel M Cao S Dasgupta T Eren J Fang D Goldenberg Baris Fidan Changbin (Brad) Yu Collaborators

3. Chinese Control Conference 中国控制会议 3 Aim of Presentation To expose current problems involving swarms To indicate a typical messy application problem To indicate some of the tools (building blocks) being developed to tackle general application problems  We shall describe a number of ‘standardised’ swarm problems and partial solutions  Solutions to real swarm problems depend on many of these standardised problems

4. Chinese Control Conference 中国控制会议 4 Outline Swarm Problems Rendezvous Consensus and Flocking Station Keeping, Rigidity and Persistence Merging, Splitting and Closing Ranks Conclusions

5. Chinese Control Conference 中国控制会议 5 Swarms A number of individual agents The agents exhibit a spatial pattern, which implies some sort of interaction between the agents What is a swarm?

6. Chinese Control Conference 中国控制会议 6 A particular swarm problem Scenario Three or more UAVs overfly an area, which includes no-fly zones There are some objects of interest at unknown locations in the area The UAVs take bearing measurements on perceived objects of interest, and they wish to localize the objects Non-motion constraints They have intermittent GPS connection They cannot ‘look’ straight down, i.e. they have a blind spot. Constraints on motion Groups of three must stay within 5 km of one another They must stay as spread out as possible, and at same height They must fly at different constant average airspeeds all about 80 km/hour They must operate in windy conditions They have a minimum turning radius, say 1.5 km

7. Chinese Control Conference 中国控制会议 7 A particular swarm problem How do they:  Search the area?  Modify their search strategy if lots of objects turn up in one area?  Avoid collisions?  Avoid obstacles and no-fly zones?  Deal with moving objects?  Complete the task in minimum time?  Modify their strategy if they lose GPS?  Cope with a loss of a communications link? How do we:  Decide whether having more agents would or would not be worthwhile?

8. Chinese Control Conference 中国控制会议 8 Generic Operational Problems Certain problems apply to most artificial swarms as well as the particular scenario described: Dealing with failures of agents and/or communication links Achieving a self-repair capability to a swarm Reconfigurable computing Capacity constrained communication Environmental hazards: smoke, heat, … Practical problem solutions need theoretical building Blocks...

9. Chinese Control Conference 中国控制会议 9 Classes of considered problems Rendezvous (not part of written paper) Consensus and flocking (not part of written paper) Station keeping (maintaining formation shape) Moving formation from A to B while maintaining shape Splitting, merging and repairing formations A number of ‘simpler’ idealized swarm problems have been tackled. This talk describes some.

10. Chinese Control Conference 中国控制会议 10 Meta Problem Virtually all swarm problems require answers to a meta problem: What are the ARCHITECTURES for each of: SENSING, COMMUNICATIONS, CONTROL?

11. Chinese Control Conference 中国控制会议 11 Outline Swarm Problems Rendezvous Consensus and Flocking Station Keeping, Rigidity and Persistence Merging, Splitting and Closing Ranks Conclusions

12. Chinese Control Conference 中国控制会议 12 Rendezvous Consider N agents  In the plane  Agents are point agents  Agents have same sensing radius of r  Agents all have their own local coordinate basis, and no compass  Each agent knows difference between its x coordinate and that of each sensed agent, and its y coordinate and that of each sensed agent  Each agent has its own clock Rendezvous control task:  Using local calculations at each agent, and only the information available at that agent,  determine a motion strategy for each agent that will promote the assembly of all agents to the one point.

13. Chinese Control Conference 中国控制会议 13 Rendezvous Consider N agents  In the plane  Agents are point agents  Agents have same sensing radius of r  Agents all have their own local coordinate basis, and no compass  Each agent knows difference between its x coordinate and that of each sensed agent, and its y coordinate and each sensed agent  Each agent has its own clock Rendezvous control task:  Using local calculations at each agent, and only the information available at that agent,  determine a motion strategy for each agent that will promote the assembly of all agents to the one point. No centralized controller! No global information!

14. Chinese Control Conference 中国控制会议 14 Rendezvous Consider N agents  In the plane  Agents are point agents  Agents have same sensing radius of r  Each agent knows difference between its x coordinate and that of each sensed agent, and its y coordinate and each sensed agent (ie each agent knows the distance vector to each neighbour) Rendezvous control task:  Using local calculations at each agent, and using the information available at that agent,  determine a motion strategy for each agent that will promote the assembly of all agents to the one point. Could be R 3 Restrictive Could have angles, distances, location on hyperbola (time difference of arrival), etc Over-simplified No centralized controller! No global information!

15. Chinese Control Conference 中国控制会议 15 Using a Graph Agents represented by vertices of the graph When two agents are within sensing distance of r, an edge joins the corresponding vertices.

16. Chinese Control Conference 中国控制会议 16 Using a Graph Agents represented by vertices of the graph When two agents are within distance of r, an edge joins the corresponding vertices. Assume that each agent listens/senses in a certain interval, and moves in another interval Synchronous case much easier but less practical than asynchronous case r

17. Chinese Control Conference 中国控制会议 17 Using a Graph Control law for agent J is continuous function of offsets from ‘neighbours’. Once a neighbour always a neighbour. As time evolves, edge set increases. When initial graph is not connected, may get one rendezvous point or several. Important Result: Rendezvous is always possible with initially connected graph!

18. Chinese Control Conference 中国控制会议 18 Connected graph RV Graph initially connected Neighbors are never lost Each node progressively acquires more neigbors

19. Chinese Control Conference 中国控制会议 19 Rendezvous with leader Leader here Can always assign a leader--which does not move Everybody goes to him/her

20. Chinese Control Conference 中国控制会议 20 Disconnected Graph RV Graph is not initially connected Unconnected interior agents are ‘captured’

21. Chinese Control Conference 中国控制会议 21 Outline Swarm Problems Rendezvous Consensus and Flocking Station Keeping, Rigidity and Persistence Merging, Splitting and Closing Ranks Conclusions

22. Chinese Control Conference 中国控制会议 22 Consensus and Flocking Consider a group of agents collecting data, e.g. air temperature, particle concentration, etc. Suppose each agent can only communicate with designated neighbours. Can they share information to all learn the average value?

23. Chinese Control Conference 中国控制会议 23 Another motivation

24. Chinese Control Conference 中国控制会议 24 Vicsek et al problem A collection of agents moves with the same speed but different headings Each agent can sense the heading in which its neighbours are moving Agents update their headings at the same time: new heading of an agent = average of headings of itself and all neighbours No centralized controller/coordinator but may have a leader. Neighbour sets may be time-varying. Observation: Agents align, within one or more flocks Vicsek simulation explained by Jadbabaie, Lin, Morse

25. Chinese Control Conference 中国控制会议 25 Vicsek et al problem 2 Intuitive picture: averaging headings (or temperatures or air pollution measurements) is like a discrete time and space version of heat flow equation Idea works with communication delays Extensions have been done to cope with dynamics in agents Vicsek simulated effect of noise Algorithm was known in another form in computer science and ‘flocking’ goes back at least to 1989 Tools for analysis include graph theory and properties of matrices with nonnegative entries, mainly from inhomogeneous Markov chain literature (decades old)

26. Chinese Control Conference 中国控制会议 26 Normal flocking Agents start with random orientations --but get aligned. Alignment direction not easily predictable

27. Chinese Control Conference 中国控制会议 27 Flocking with a fast leader Red is leader Yellow is neighbor of leader Blue is nonneighbor of leader Agents can follow a leader Agents may lose connection through not turning fast enough

28. Chinese Control Conference 中国控制会议 28 Outline Swarm Problems Rendezvous Consensus and Flocking Station Keeping, Rigidity and Persistence Merging, Splitting and Closing Ranks Conclusions

29. Chinese Control Conference 中国控制会议 29 Station Keeping Suppose a collection of agents in R 2 or R 3 is supposed to maintain a cohesive formation shape. They may or may not be moving. Suppose they can sense their neighbours. The key questions: What needs to be sensed and what needs to be controlled to maintain the formation shape?

30. Chinese Control Conference 中国控制会议 30 Formations

31. Chinese Control Conference 中国控制会议 31 Formations

32. Chinese Control Conference 中国控制会议 32 Formations A formation is a collection of agents (point agents for us) in two or three dimensional space A formation is rigid if the distance between each pair of agents does not change over time In a rigid formation, normally only some distances are explicitly maintained, with the rest being consequentially maintained. The distances ab,bc,cd,ad and ac are explicitly maintained and the distance bd is maintained as a result of the topology. a b c d

33. Chinese Control Conference 中国控制会议 33 Rigid and Nonrigid Formations MINIMALLY RIGID NONRIGID RIGID, BUT NOT MINIMALLY SO a b a b c d a d c c b a d c d

34. Chinese Control Conference 中国控制会议 34 Undirected/Directed Graphs Maintaining a formation shape is done by maintaining certain inter-agent distances Angles may sometimes be usable--not considered here. If the distance between agents X and Y is maintained, this may be:  A task jointly shared by X and Y, or  Something that X does and Y is unconscious about, or conversely (leader/follower concept)--may be easier/cheaper Undirected graphs model the first situation. Rigid graph theory is applicable. Directed graphs model the second situation. All the rigidity type questions and theories have to be validated and/or modified with new results for directed graphs. X Y X Y

35. Chinese Control Conference 中国控制会议 35 Formation Rigidity What undirected graphs give rise to rigid formations? What directed graphs give rise to formations which can maintain their structure? Answers to these questions have been provided using:  Linear algebra for formations in R 2 and R 3  Graph theory for formations in R 2 Some results are known using graph theory for R 3 formations.

36. Chinese Control Conference 中国控制会议 36 Formation Rigidity Let’s look at undirected graphs first…. X Y X and Y are jointly responsible for maintaining the distance.

37. Chinese Control Conference 中国控制会议 37 Each edge can remove a single degree of freedom For a whole rigid formation, just rotations and translations will be possible (three degrees of freedom), so at least 2n-3 edges are necessary for a graph to be rigid. Total degrees of freedom:2n given n point agents in R 2 Rigidity Characterization This edge does not eliminate any degree of freedom but may not!

38. Chinese Control Conference 中国控制会议 38 Rigidity Characterization (cont’d) Necessary and sufficient condition for rigidity: At least 2n-3 well-distributed edges n = 3, 2n-3 = 3 yes n = 4, 2n-3 = 5 yes n = 5, 2n-3 = 7 no Are these graphs rigid?

39. Chinese Control Conference 中国控制会议 39 Rigidity Characterization (c’td) Notion of ‘well-distributed’ can be formalized, resulting in graphical test for rigidity in R 2 (necessary and sufficient condition--known as Laman’s theorem) Only differing necessity and sufficiency conditions are known in R 3. A linear algebra test is available in R 2 and R 3 :  When graph has |V| vertices and |E| edges and is in R d, an |E| by d|V| matrix is formed.  Rigidity corresponds to kernel of matrix having dimension (1/2)d(d+1).  Smallest nonzero singular value appears to measure closeness to nonrigidity.

40. Chinese Control Conference 中国控制会议 40 Rigid Formations v1v1 v2v2 v3v3 v4v4 (1,2) x 1 - x 2 y 1 - y 2 x 2 - x 1 y 2 - y 1 00 (1,3) x 1 - x 3 y 1 - y 3 0x 3 - x 1 y 3 - y 1 0 (1,4) x 1 - x 4 y 1 - y 4 00x 4 - x 1 y 4 - y 1 (2,3) 0x 2 - x 3 y 2 - y 3 x 3 - x 2 y 3 - y 2 0 (2,4) 0x 2 - x 4 y 2 - y 4 0x 4 - x 2 y 4 - y 2 (3,4) 00x 3 - x 4 y 3 - y 4 x 4 - x 3 y 4 - y 3 Sample two dimensional Rigidity Matrix--a Matrix Net ∑ x i M i +y i N i in coordinates of points.

41. Chinese Control Conference 中国控制会议 41 Formation Persistence To distinguish directed case from undirected, we use the term Formation Persistence instead of formation rigidity. And now we look at directed graphs. X Y X is responsible for maintaining the correct distance from Y, and Y is unconscious of X

42. Chinese Control Conference 中国控制会议 42 Rigidity is not enough! Agent 1 is unconstrained (leader), and agent 2 must follow agent 1, and agent 3 must follow agent 2. Agent 3 can move on a circle, even if agent 1 and agent 2 are stationary. 1 2 3 4 Undirected graph is rigid!

43. Chinese Control Conference 中国控制会议 43 Rigidity is not enough! Agent 1 is unconstrained (leader), and agent 2 must follow agent 1, and agent 3 must follow agent 2. Agent 3 can move on a circle, even if agent 1 and agent 2 are stationary. Agent 4 can no longer maintain all three distances 1 2 3 4 Undirected graph is rigid! Set-up is not constraint consistent. 3’

44. Chinese Control Conference 中国控制会议 44 Directed graph generalization Rigidity says shape maintained if certain distances are maintained; constraint consistence says these distances can be maintained Formation maintenance requires a directed graph to be both rigid and constraint consistent. We call this persistence In R 2 persistence can be checked by running multiple rigidity tests R 3 is more complicated.

45. Chinese Control Conference 中国控制会议 45 Persistence Characterization Result: graph is persistent iff rigidity holds for certain subgraphs. They are obtained by removing outgoing edges in excess of 2 at each vertex. Persistent Rigid in each of three cases (neglecting directions)

46. Chinese Control Conference 中国控制会议 46 Persistence Characterization Result: graph is persistent iff rigidity holds for certain subgraphs. They are obtained by removing outgoing edges in excess of 2 at each vertex. Just one vertex has out degree 3 Persistent Rigid in each of three cases (neglecting directions)

47. Chinese Control Conference 中国控制会议 47 Leader Cycle-Free Graphs Persistence is preserved after addition/deletion of vertex with no incoming edges and at least two outgoing edges. Follower Every cycle-free persistent graph can be obtained by a succession of such additions to initial Leader-Follower seed One starts with a leader-follower pair. Control for station keeping is straightforward, due to the one way (triangular) coupling

48. Chinese Control Conference 中国控制会议 48 Leader First-follower There is feedback around the loop. Linearized analysis for station keeping is possible Natural closed-loop system is of form  is adjustable and almost diagonal, and A is fixed from geometry Graphs with Cycles

49. Chinese Control Conference 中国控制会议 49 Cohesive Motion Problem Control Task: Move a persistent formation whose initial position and orientation are specified to a new desired position and orientation maintaining shape Specifications: Use a decentralized scheme Each agent can sense its position and the positions of the agents it follows Satisfying distance constraints has higher priority: Guidance is from positive DOF agents Continuous-time domain Simplifications: Planar motion Point-agent model Perfect measurement Simple integrator model for agent kinematics:

50. Chinese Control Conference 中国控制会议 50 Cohesive Motion Movie

51. Chinese Control Conference 中国控制会议 51 Cohesive Motion Movie 2 Formation maintenance with two approaches to obstacle avoidance (based on path planning concepts) Some distortion occurs

52. Chinese Control Conference 中国控制会议 52 Outline Swarm Problems Rendezvous Consensus and Flocking Station Keeping, Rigidity and Persistence Merging, Splitting and Closing Ranks Conclusions

53. Chinese Control Conference 中国控制会议 53 Formation Merging How many links will be needed? Where should we put the links? Can the establishing of new links be done in a decentralized way?

54. Chinese Control Conference 中国控制会议 54 Formation Splitting How many links will be needed? Where should we put the links? Can the establishing of new links be done in a decentralized way?

55. Chinese Control Conference 中国控制会议 55 One (or more) agents is removed in 3D formation, generally destroying rigidity Right hand diagram depicts losing one agent and its 7 links Remaining links kept and 4 new ones added restoring rigidity Closing Ranks

56. Chinese Control Conference 中国控制会议 56 One (or more) agents is removed, generally destroying rigidity Diagram depicts 3D formation losing one agent and its 7 links Remaining links kept and 4 new ones added giving rigidity Closing Ranks Same questions: Where, how many and decentralized possible?

57. Chinese Control Conference 中国控制会议 57 Common issues Splitting is a special case of closing ranks with multiple agent loss (and conversely):  The agents in subformation 2 are like lost agents as far as subformation 1 is concerned Subformation 2 Subformation 1

58. Chinese Control Conference 中国控制会议 58 Closing Ranks Key Conclusion 1: Closing ranks can always be achieved when one vertex with its incident edges is lost by making connections among neighbours of the lost vertex Key Conclusion 2 (consequence of 1): Closing ranks can always be achieved when several vertices with their incident edges are lost by making connections among the neighbours of the lost vertices Note that all edges remaining after the vertex or vertices loss are retained for use. Number of possibilities to check is not massive.

59. Chinese Control Conference 中国控制会议 59 Closing Ranks One (or more) agents is removed, generally destroying rigidity Diagram depicts three-dimensional formation losing one agent and its 7 links Remaining links kept and 4 new ones added giving rigidity

60. Chinese Control Conference 中国控制会议 60 Closing Ranks Neighbours of lost vertex

61. Chinese Control Conference 中国控制会议 61 Formation Splitting Requirement to add two single links only is consequence of number of lost links and R 3 problem character Requirement to join vertices of former neighbors means only possibility is new links at 3-5 and 6-10 Limited communications between agents will figure this out.

62. Chinese Control Conference 中国控制会议 62 Common issues All problems, splitting, merging and closing ranks, deal with finding extra edges to establish or re-establish rigidity in a formation that already has some edges An algorithm can be found for systematically adding further edges to a nonrigid formation already including some edges to provide rigidity There is no real decentralized version of the algorithm currently. But there are some key insights, as e.g. for closing ranks and formation splitting. Merging is really a matter of making a rigid meta formation out of two formations:  Three new edges (with careful choice of associated agents) are needed for merging two rigid formations in R 2 in order that the merged formation be again rigid  Six new edges are required for R 3

63. Chinese Control Conference 中国控制会议 63 Outline Swarm Problems Rendezvous Consensus and Flocking Station Keeping, Rigidity and Persistence Merging, Splitting and Closing Ranks Conclusions

64. Chinese Control Conference 中国控制会议 64 Conclusions Flocking and formations are presented by nature, and have civilian and military applications Architectures for sensing, communication and control are important Practical formation problems are hard: solutions will probably use building blocks that are currently the subject of much effort These include: rendezvous, consensus and flocking, station keeping and rigid/persistent motion, formation change maneuvers

65. Chinese Control Conference 中国控制会议 65 Conclusions Challenging current basic problems include:  Doing three-dimensional problems well  Understanding what formations are easy to control, what are hard to control  Designing formations to be tolerant of link loss or agent loss  Dealing with conflicting objectives retaining the autonomy and architecture constraints Applications and theory are nevertheless in their infancy.

66. Chinese Control Conference 中国控制会议 66 Questions?