Information Architecture and Control Design for Rigid Formations

1
Information Architecture and Control Design for
Rigid Formations
2007 Chinese Control Conference

• Brian DO Anderson, Changbin (Brad) Yu, Baris
  Fidan
• Australian National University and National ICT
  Australia

2
Thanks

• To the conference organisers for inviting me
• To my colleagues for helping me

Contributions of all these individuals appear
somewhere in this talk

•  Coauthors

Collaborators
P Belhumeur V Blondel M Cao S Dasgupta T Eren J
Fang D Goldenberg

• J Hendrickx
• J Lin
• A Morse
• I Shames
• D v d Walle
• W Whiteley
• R Yang

Changbin (Brad) Yu
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
Outline

• Swarm Problems
• Rendezvous
• Consensus and Flocking
• Station Keeping, Rigidity and Persistence
• Merging, Splitting and Closing Ranks
• Conclusions

5
Swarms
What is a swarm?

• A number of       individual agents
• The agents exhibit a     spatial pattern,
  which     implies  some sort of     interaction
  between     the agents

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

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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
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...

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Classes of considered problems
A number of simpler idealized swarm problems
have been tackled. This talk describes some.

• 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

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Meta Problem

• Virtually all swarm problems require answers to a
  meta problem

What are the ARCHITECTURES for each
of SENSING, COMMUNICATIONS,CONTROL?
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Outline

• Swarm Problems
• Rendezvous
• Consensus and Flocking
• Station Keeping, Rigidity and Persistence
• Merging, Splitting and Closing Ranks
• Conclusions

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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.

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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!
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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.

No centralized controller! No global information!
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.

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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
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Using a Graph
Important Result Rendezvous is always possible
with initially connected 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.

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Connected graph RV
Graph initially connected Neighbors are never
lost Each node progressively acquires more
neigbors
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Rendezvous with leader
Can always assign a leader--which does not move
Everybody goes to him/her
Leader here
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Disconnected Graph RV
Graph is not initially connected Unconnected
interior agents are captured
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Outline

• Swarm Problems
• Rendezvous
• Consensus and Flocking
• Station Keeping, Rigidity and Persistence
• Merging, Splitting and Closing Ranks
• Conclusions

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?

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Another motivation
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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

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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)

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Normal flocking
Agents start with random orientations --but get
aligned. Alignment direction not easily
predictable
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Flocking with a fast leader
Agents can follow a leader Agents may lose
connection through not turning fast enough
Red is leader Yellow is neighbor of leader Blue
is nonneighbor of leader
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Outline

• Swarm Problems
• Rendezvous
• Consensus and Flocking
• Station Keeping, Rigidity and Persistence
• Merging, Splitting and Closing Ranks
• Conclusions

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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?
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Formations
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Formations
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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.

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Rigid and Nonrigid Formations
a
b
b
a
d
c
d
c
MINIMALLY RIGID
NONRIGID
a
a
b
d
d
c
c
RIGID, BUT NOT MINIMALLY SO
NONRIGID
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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.

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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.

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Formation Rigidity

• Lets look at undirected graphs first.

X and Y are jointly responsible for maintaining
the distance.
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Rigidity Characterization
Total degrees of freedom
2n given n point agents in R2
Each edge can remove a single degree of freedom
but may not!
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.
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Rigidity Characterization (contd)
Are these graphs rigid?
Necessary and sufficient condition for
rigidity At least 2n-3 well-distributed edges
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Rigidity Characterization (ctd)

• Notion of well-distributed can be formalized,
  resulting in graphical test for rigidity in R2
  (necessary and sufficient condition--known as
  Lamans 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 dV matrix is formed.
• Rigidity corresponds to kernel of matrix having
  dimension (1/2)d(d1).
• Smallest nonzero singular value appears to
  measure closeness to nonrigidity.

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Rigid Formations
Sample two dimensional Rigidity Matrix--a Matrix
Net ? xi Mi yi Ni in coordinates of points.
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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 is responsible for maintaining the correct
distance from Y, and Y is unconscious of X
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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
4
2
3
Undirected graph is rigid!
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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
4
2
3
3
Undirected graph is rigid!
Set-up is not constraint consistent.
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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.

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

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Cohesive Motion Movie
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Cohesive Motion Movie 2
Formation maintenance with two approaches to
obstacle avoidance (based on path planning
concepts) Some distortion occurs
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Outline

• Swarm Problems
• Rendezvous
• Consensus and Flocking
• Station Keeping, Rigidity and Persistence
• Merging, Splitting and Closing Ranks
• Conclusions

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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?
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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?
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Closing Ranks

• 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

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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

Same questions Where, how many and decentralized
possible?
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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
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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.

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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

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Closing Ranks
Neighbours of lost vertex
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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.
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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

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Outline

• Swarm Problems
• Rendezvous
• Consensus and Flocking
• Station Keeping, Rigidity and Persistence
• Merging, Splitting and Closing Ranks
• Conclusions

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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

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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.

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Questions?