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Analysis of Autonomous Multiagent Formations

From Rigidity to Persistence DSTO-NICTA Meeting

22 July 2005

Brad C. YUNational ICT Australia Limited

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Outline

• Problem Description• Recap on Rigidity • Persistence• Structural Persistence• Summary and Future Work

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Problem

What’s the right information and control structure and the right analysis model to maintain formation?

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Outline

• Problem Description• Recap on Rigidity • Persistence• Structural Persistence• Summary and Future Work

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Rigid Formations

• A formation is a collection of agents (point agents initially) in two or three dimensional space

• A formation is rigid if the distance between each pair of agents does not change over time

• Steady motion of a formation normally needs rigidity

• 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 andthe distance bd is maintained as

a result of the topology.

a b

cd

Rigidity is an Undirected Notion!

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Scenarios of Control

• Undirected notion of rigidity requires peer-wise control, it’s a joint effort of both agents

• How do we deal with scenarios with leader-follower structure?

LeaderFollower

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Rigidity notion is insufficient in directed case

1

3

2

4

A

B

C

1

3

2

4

1

3

2

4NOT RIGID

So, need to take direction constraints into account in addition to distance constraints

B is rigid. But, if 3 moves, 4 is unable to react

??

Rigidity insufficient because

•Essentially undirected notion (although definition OK for directed graphs)

Directed distance constraints

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Outline

• Problem Description• Recap on Rigidity • Persistence• Structural Persistence• Summary and Future Work

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Constraint Consistence (C. C.)

This leads to new notion of “persistence”

•Intuitively, an agent CANNOT have too many directed distance constraints , i.e., constraints on the distances to be maintained from other agents.•An agent is fitting if it satisfies all constraints in all cases•A formation is C. C. if every agent is fitting

Example:p(1)

p(3)

p(2)

p(4)

p’ fitting w.r.t. 1,2,3But 4 is non-fitting

p not C.C.(although p rigid)

p’(3)

=p’(2)

p’(4)=

=p’(1)

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Rigidity to Persistence

• Rigidity: “All constraints satisfied structure preserved”

• Constraint Consistence: “Every agent tries to satisfy all its constraints all the constraints are satisfied”

• Persistence: “Every agent tries to satisfy all its constraints structure preserved”

Persistence ||

Rigidity + C. Consistence

1

3

2

4

1

3

2

4

1

3

2

4

Rig. NO C.C. YES

Rig. YESC.C. NO

Rig. YESC.C. YES

A

B

C

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Half-way summary

• Rigidity has been studied extensively with aid of undirected graphs (distance constraints)

• Persistence is a recently developed notion, principally due to ourselves and collaborators, for analysis of formations with leader-follower type of control structure

• (directed distance constraints)

• We have developed systematic ways of checking persistence which involves repetitive tests for rigidity.

• What if we look at the formation structure as a whole???

(is it structurally persistent? ---- the new question)

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Outline

• Problem Description• Recap on Rigidity • Persistence• Structural Persistence• Summary and Future Work

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Two leaders’ problem in 3D formation

• A persistent formation is NOT structurally persistent if it has two leaders.

• 2D persistent formation is always structurally persistent.

1

2

34

5

????

??

A leader is an agent that has NO constraints on its

movement.

1 and 2 are the leaders

in this 3D formation

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• Collections of autonomous vehicles (Point agents)

• Rigid

• Persistent, but two leader agents may create problem

• Persistent, and only one leader agent

Persistence to Structural Persistence

Rigid Persistent Struc Pers. Structural Persistence

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Analysis of Structural Persistence

Our studies show that in contrast to the fact that

• Rigidity and persistence only deal with individual agent’s constraints collectively

• Structural persistence deals with a formation as a whole.It puts constraints on the allocation of “degrees of freedom”

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Allocation of DOF

• DOF is an abstraction of agent’s autonomy in its movement

• An agent’s DOF defines its “role” in the formation• In 3D formation,

no-constraint = 3 DOF = move freely in 3D = a leader

one constraint = 2 DOF = move in spheretwo constraints = 1 DOF = move in circlethree constraints = no DOF = fixed (w.r.t

formation)four or more constraints ??

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Transfer of DOF

•Change of agents’ “role”(esp. leadership) of a formation may be required as part of mission plan, new agent carrying new mission maybe added as leader

•Transfer of DOF can be made via a technique we developed for formation in arbitrary dimension (s)

2

3

1

4

5

Ok!

Join us

Sorry, I havetoo many

guys to follow!

Alright,I can give

one DOF to you.

Same here!Let’s lead together

Let me follow 5

then

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Outline

• Problem Description• Recap on Rigidity• Persistence• Structural Persistence• Summary and Future Work

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A summary of results

Notion Handles Works inRigidity Distance Constraints 2D , 3DPersistence Directed Distance Const. 2D Stru. Pers. (as above) 2D , 3D

How to construct a structurally persistent

formation?Tip: Henneberg

Sequence

What about the robustness of the

formation??Tip: Redundancy

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Future Work (cont’d)

Practical• 1> Formation health and quality measurement/assurance

auto discovering, decentralized detection algorithms/procedures

• 2> Formation robustness, self-curing (closing ranks), splitting, merging formations,

• 3> Implementation/deployment related control issues(e.g., cycles in the graph = a loop of L-F, who is the leader?)

Theoretical4> Link graphical results to that of Linear Algebra5> Extend/generalize results to arbitrary dimensions6> Still many open problems around

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Future work

The Grand Challenge: • A rigorous analysis framework based on Graph

Theory and Linear Algebra (Matroid) that works for both 2D and 3D formations in practical applications.

• With comprehensive results based on the framework in formation construction, manipulation, closing-ranks with guaranteed level of robustness, and at any time, the health and quality of such formation can be measured and determined at low computational complexity.

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Main References:• [1] J. M. Hendrickx, B.D.O. Anderson, V. Blondel & J. –C. Delvenne , “ Directed

graphs for the analysis of rigidity and persistence in autonomous agent systems”, submitted to International Journal of Robust Nonlinear Control, 2005

• [2] J. M. Hendrickx, B. Fidan, C. YU, B.D.O. Anderson & V. Blondel , “ Rigidity and Persistence of Three and Higher Dimensional Directed Formations”, to appear in MARS05, Barcelona, Sep 2005

• [3] C. YU, J. M. Hendrickx, B. Fidan & B.D.O Anderson, “Structural Persistence of Three Dimensional Autonomous Formations”, to appear in MARS05, Barcelona, Sep 2005

• [4] C. YU, B. Fidan & B.D.O. Anderson, “Persistence Acquisition and Maintenance for Autonomous Formations”, Submitted to the 2nd International Conference on Intelligent Sensors, Sensor Networks and Information Processing, Melbourne, Dec 2005.

Thank you