Information Architecture and Control Design for Rigid Formations Brian DO Anderson, Changbin (Brad) Yu, Baris Fidan Australian National University and

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

Chinese Control Conference 中国控制会议 2

Thanks

Coauthors:

J HendrickxJ LinA MorseI ShamesD v d WalleW WhiteleyR 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 FidanChangbin (Brad) Yu

Collaborators


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


Outline

• Swarm Problems

• Rendezvous

• Consensus and Flocking

• Station Keeping, Rigidity and Persistence

• Merging, Splitting and Closing Ranks

• Conclusions


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?


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


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?


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...


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.


Meta Problem

• Virtually all swarm problems require answers to a meta problem:

What are the ARCHITECTURESfor each of:SENSING,

COMMUNICATIONS,CONTROL?


Outline

• Swarm Problems

• Rendezvous




• Conclusions


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.


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!


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 R3

Restrictive

Could have angles, distances, location on hyperbola (time difference of arrival), etc

Over-simplified

No centralized controller! No global information!


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.


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


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!


Connected graph RV

QuickTime™ and aCinepak decompressor

are needed to see this picture.

Graph initially connected

Neighbors are never lost

Each node progressively acquires more neigbors


Rendezvous with leader



Leader here

Can always assign a leader--which does not move

Everybody goes to him/her


Disconnected Graph RV



Graph is not initially connected

Unconnected interior agents are ‘captured’


Outline

• Swarm Problems

• Rendezvous




• Conclusions


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?


Another motivation


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


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)


Normal flocking

QuickTime™ and aVideo decompressor


Agents start with random orientations --but get aligned.

Alignment direction not easily predictable


Flocking with a fast leader

Red is leader

Yellow is neighbor of leader

Blue is nonneighbor of leader

QuickTime™ and aVideo decompressor


Agents can follow a leader

Agents may lose connection through not turning fast enough


Outline

• Swarm Problems

• Rendezvous




• Conclusions


Station Keeping

• Suppose a collection of agents in R2 or R3 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?


Formations


Formations


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 andthe distance bd is maintained as

a result of the topology.

a b

cd


Rigid and Nonrigid Formations

MINIMALLY RIGID NONRIGID

NONRIGIDRIGID, BUT NOT MINIMALLY SO

a b a b

cd

a

d cc

ba

d

cd


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


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 R2 and R3

Graph theory for formations in R2

• Some results are known using graph theory for R3 formations.


Formation Rigidity

Let’s look at undirected graphs first….

X Y

X and Y are jointly responsible for maintaining the distance.


Each edge can remove a single degree of freedomFor 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 R2

Rigidity Characterization

This edge does not eliminate any degree of freedom

but may not!


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?


Rigidity Characterization (c’td)

• Notion of ‘well-distributed’ can be formalized, resulting in graphical test for rigidity in R2 (necessary and sufficient condition--known as Laman’s theorem)

• Only differing necessity and sufficiency conditions are known in R3.

• A linear algebra test is available in R2 and R3: When graph has |V| vertices and |E| edges and is in Rd, 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.


Rigid Formations

v1 v2 v3 v4

(1,2) x1 - x2 y1 - y2 x2 -x1 y2 - y1 0 0

(1,3) x1 - x3 y1 - y3 0 x3 - x1 y3 - y1 0

(1,4) x1 - x4 y1 - y4 0 0 x4 - x1 y4 - y1

(2,3) 0 x2 - x3 y2 - y3 x3 - x2 y3 - y2 0

(2,4) 0 x2 - x4 y2 - y4 0 x4 - x2 y4 - y2

(3,4) 0 0 x3 - x4 y3 - y4 x4 - x3 y4 - y3

Sample two dimensional Rigidity Matrix--a Matrix Net ∑ xi Mi +yi Ni in

coordinates of points.


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


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!


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’


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 R2 persistence can be checked by running multiple rigidity tests

• R3 is more complicated.


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)


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)


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


LeaderFirst-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


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:


Cohesive Motion Movie




Cohesive Motion Movie 2

Formation maintenance with two approaches to obstacle avoidance (based on path planning concepts)

Some distortion occurs


Outline

• Swarm Problems

• Rendezvous




• Conclusions


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?


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?


• 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


• 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?


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


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.


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


Closing Ranks

Neighbours of lost vertex


Formation Splitting

Requirement to add two single links only is consequence of number of lost links and R3 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.


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 R2 in order that the merged formation be again rigid

Six new edges are required for R3


Outline

• Swarm Problems

• Rendezvous




• Conclusions


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


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.


Questions?

Documents

Information Architecture and Control Design for Rigid Formations Brian DO Anderson, Changbin (Brad) Yu, Baris Fidan Australian National University and