Distributed Cooperative Control of Multiple Vehicle Formations Using Structural Potential Functions Reza Olfati-Saber Postdoctoral Scholar Control and.

Distributed Cooperative Control of Multiple Vehicle Formations Using

Structural Potential Functions

Reza Olfati-Saber

Postdoctoral Scholar Control and Dynamical Systems

California Institute of [email protected]

UCLA, March 2nd, 2002

Outline• Introduction• Multi-vehicle Formations• Past Research• Coordinated Tasks

– Stabilization/Tracking– Rejoin/Split/Reconfiguration Maneuvers

• Why Distributed Control?• Formation Graphs

– Rigidity/Foldability of Graphs• Potential Functions• Distributed Control Laws• Simulation Results• Conclusions

Introduction

Definition: Multi-agent Systems are systems that consist of multiple agents or vehicles with several sensors/actuators and the capability to communicate with one another to perform coordinated tasks.

Applications:– Automated highways– Air traffic control– Satellite formations– Search and rescue operations– Robots capable of playing games (e.g. soccer/capture the flag)– Formation flight of UAV’s (Unmanned Aerial Vehicles)

Multi-Vehicle Formations

A group of vehicles with a specific set of inter-vehicle distances is called a Multi-Vehicle Formation.

Formation Stabilization

Dynamics: 2

1,2, ,

, ,i i

i i i i i

q p i n

p u q p u

Past Research

• Robotics: navigation using artificial potential functions (Rimon and Koditschek, 1992)

• Multi-vehicle Systems: – Coordinated control of groups using artificial

potentials (Leonard and Fiorelli, 2001)

– Information flow on graphs associated with multi-vehicle systems (Fax and Murray, 2001)

Why Distributed Control?• No vehicle knows the state/control of all other vehicles

• No vehicle knows its relative configuration/velocity w.r.t. all other vehicles unless n = 2,3

• The control law for each vehicle must be distributed so that the overall computational complexity of the problem is acceptable for large number of vehicles

• A system controlled via a centeralized computer does not function if that computer breaks.

What is a Formation?

Formation Representation

2( 2)3 3( , , , , , )

1 2( 2) 2 3, 2

nn nl x y x y

f n n n

Coordinated Tasks

attitude

Tracking

Trajectory

Rejoin

Split

Reconfiguration

Diamond Formation Delta Formation

TwoFormations

One Formation

Switching

Split/Rejoin Maneuvers

Operational Graph

Formation Graphs1

2 3

4 ii) measures ij j ix q q

iii) knows its desired distance to must be jviv ijd

an Edge means

i) is a neighbor of

Formation Graph:

Connectivity Matrix

Distance Matrix

Set of Vertices

, ,G I C D

1,2,3,4 ,0I I =

2 3 0

1 3 0

1 0 0

1 2 3

C

3

a a

a aD

a

a a a

i j

jv iv

iv

Rigidity

2 3

4

1

Definition: A planar formation graph with n nodes and 2n-3 critical links is called a rigid formation graph.

Definition: A critical link is a link that eliminates a mobility degree of freedom of a multi-body system.

a a

b b

c

dRemark: c (or d) is called a single mobility degree of freedom of the formation graph.

c

Foldability

2 3

4

1

Definition: A rigid formation graph is foldable iff the set of structural constraints associated with the formation graph does not have a unique solution.

Definition: The following non-redundant set of equations are called structural constraints of a formation graph.

a a

b b

c

d

c

: 0ij j i ijq q d

4

Deviation Variable:

Node Orientation

1 2

3

1 2

3

z

yx

1 2

1 2

1 2

1

( , , ) det 1 ( ) ( )

1

0 1,

1 0T

x x

h x y z y y x z y z

z z

a b a Qb Q

0h

0h

Unambiguous FG’s

Definition: A formation graph is called unambiguous if it is both rigid and unfoldable.

1

2

3

4

5

6

1

2 3

4 5

67

Potential Functions

x

x

( )x

( )f x

Potential Function: 2( ) 1 1x x

Force: 2

( )( )

1

d x xf x

dx x

iq jq

( ) j i ij

i j i ij iji

j j i ij jii

j iij

j i

V q q q d

Vu f q q d

q

Vu f q q d

q

q q

q q

n

n

n 0i ju u

ijn

Distributed Control Laws

Potential Function :

2

max

1( , ) ( )

2

1 1( )

2

1,2, , , 0 1,i

ii I

j i iji I j Ji

i

H q p p V q

V q u q q dJ

I n J I

Hamiltonian :

indices of the neighbors of iv

Theorem(ROS-RMM-IFAC’02): The following state feedback

is a gradient-based bounded and distributed control law that achieves collision-free local asymptotic stabilization of any unambiguous desired formation graph .

maxmax

2 2max2

( 1)

, : , :1

i

i j i ij ij ij Ji

i

uu f q q d u p

J

xx i u u

x

n

I :

( , , )G I C D

Operational Graph

: ( , , ) ( )

( , ), 1,2, ,5

ii i i j J j is G I C D u K p q q

x q p i I

Split Maneuvers

Rejoin Maneuver

Reconfiguration I

Reconfiguration II

Tracking

Conclusions

• Introducing a framework for formal specification of unambiguous formation graphs of multi-vehicle systems that is compatible with formation control.

• Providing a Lyapunov function and a bounded and distributed state feedback that performs coordinated tasks such as formation stabilization/tracking, split/rejoin, and reconfiguration maneuvers.

• Introducing a Hybrid System that represents split, rejoin, and reconfiguration maneuvers in a unified framework as a discrete-state transition where each discrete-state is an unambiguous formation graph.