A Control Lyapunov Function Approach to Multi Agent Coordination

Cen

tre f

or

Auto

nom

ou

s S

yst

em

s

Petter Ögren CAS talk 1

A Control Lyapunov Function Approach to

Multi Agent Coordination

A Control Lyapunov Function Approach to

Multi Agent Coordination

P. Ögren, M. Egerstedt* and X. HuRoyal Institute of Technology (KTH), Stockholm

and Georgia Institute of Technology*

IEEE Transactions on Robotics and Automation, Oct 2002

Cen

tre f

or

Auto

nom

ou

s S

yst

em

s


Multi Agent RoboticsMulti Agent Robotics

Motivation:

Flexibility

Robustness

Price

Efficiency

Feasibility

Applications:

Search and rescue missions

Spacecraft inferometry

Reconfigurable sensor array

Carry large/awkward objects

Formation flying

Cen

tre f

or

Auto

nom

ou

s S

yst

em

s


Problem and Proposed Solution Problem and Proposed Solution

Problem: How to make set-point controlled robots moving along trajectories in a formation ”wait” for eachother?Idea: Combine Control Lyapunov Functions (CLF) with the Egerstedt&Hu virtual vehicle approach.Under assumptions this will result in:

Bounded formation error (waiting)Approx. of given formation velocity (if no waiting is nessesary).Finite completion time (no 1-waiting).

Cen

tre f

or

Auto

nom

ou

s S

yst

em

s


Quantifying Formation KeepingQuantifying Formation Keeping

Will add Lyapunov like assumption satisfied by individual set-point controllers. =>

Think of as parameterized Lyapunov function.

Definition: Formation Function

Cen

tre f

or

Auto

nom

ou

s S

yst

em

s


Examples of Formation FunctionExamples of Formation Function

• Simple linear example !• A CLF for the combined

higher dimensional system:

Note that a,b, are design parameters.

• The approach applies to any parameterized formation scheme with lyapunov stability results.

Cen

tre f

or

Auto

nom

ou

s S

yst

em

s


Main AssumptionMain Assumption

We can find a class K function such that the given set-point controllers satisfy:

This can be done when -dV/dt is lpd, V is lpd and decrescent. It allows us to prove:

Bounded V (error): V(x,s) < VU

Bounded completion time.

Keeping formation velocity v0, if V ¿ VU.

Cen

tre f

or

Auto

nom

ou

s S

yst

em

s


Speed along trajectory: How Do We Update s?

Speed along trajectory: How Do We Update s?

Suggestion: s=v0 t

Problems: Bounded ctrl or local ass stability

We want:

V to be small

Slowdown if V is large

Speed v0 if V is small

Suggestion:

Let s evolve with feedback from V.

Cen

tre f

or

Auto

nom

ou

s S

yst

em

s


Evolution of sEvolution of s

Choosing to be:

We can prove:

Bounded V (error): V(x,s) < VU

Bounded completion time.

Keeping formation velocity v0, if V ¿ VU.

Cen

tre f

or

Auto

nom

ou

s S

yst

em

s


Proof sketch: Formation error Proof sketch: Formation error

Cen

tre f

or

Auto

nom

ou

s S

yst

em

s


Proof sketch: Finite Completion Time Proof sketch: Finite Completion Time

Find lower bound on ds/dt

Cen

tre f

or

Auto

nom

ou

s S

yst

em

s


The Unicycle Model, Dynamic and Kinematic

The Unicycle Model, Dynamic and Kinematic

Beard (2001) showed that the position of an off axis point x can be feedback linearized to:

Cen

tre f

or

Auto

nom

ou

s S

yst

em

s


Example: FormationExample: Formation

Three unicycle robots along trajectory.

VU=1, v0=0.1, then v0=0.3 ! 0.27

Stochastic measurement error in top robot at 12m mark.

Cen

tre f

or

Auto

nom

ou

s S

yst

em

s


Extending Work by Beard et. al. Extending Work by Beard et. al.

”Satisficing Control for Multi-Agent Formation Maneuvers”, in proc. CDC ’02

It is shown how to find an explicit parameterization of the stabilizing controllers that fulfills the assumption

These controllers are also inverse optimal and have robustness properties to input disturbances

Implementation

Cen

tre f

or

Auto

nom

ou

s S

yst

em

s


What if dV/dt <= 0 ?What if dV/dt <= 0 ?

If we have semidefinite and stability by La Salle’s principle we choose as:

By a renewed La Salle argument we can still show: V<=VU , s! sf and x! xf.

But not: Completion time and v0.

Cen

tre f

or

Auto

nom

ou

s S

yst

em

s


Formations with a Mission: Stable Coordination of Vehicle Group Maneuvers

Mathematical Theory of Networks and Systems (MTNS ‘02)

Visit: http://graham.princeton.edu/ for related information

Edward Fiorelli and Naomi Ehrich Leonard [email protected], [email protected]

Mechanical and Aerospace Engineering

Princeton University, USA

Optimization and Systems Theory

Royal Institute of Technology, Sweden

Petter [email protected]

Another extension:Another extension:

Cen

tre f

or

Auto

nom

ou

s S

yst

em

s


•Configuration space of virtual body is for orientation, position and expansion factor:

• Because of artificial potentials, vehicles in formation will translate, rotate, expand and contract with virtual body.

• To ensure stability and convergence, prescribe virtual body dynamics so that its speed is driven by a formation error.

• Define direction of virtual body dynamics to satisfy mission.

• Partial decoupling: Formation guaranteed independent of mission.

• Prove convergence of gradient climbing.

Approach: Use artificial potentials and virtual body with dynamics.

Approach: Use artificial potentials and virtual body with dynamics.

Cen

tre f

or

Auto

nom

ou

s S

yst

em

s


ConclusionsConclusions

Moving formations by using Control Lyapunov Functions.Theoretical Properties:

V <= VU, error

T < TU, time

v ¼ v0 velocity

Extension used for translation, rotation and expansion in gradient climbing mission

Cen

tre f

or

Auto

nom

ou

s S

yst

em

s


Related PublicationsRelated Publications

A Convergent DWA approach to Obstacle Avoidance

Formally validatedMerge of previous methods using new mathematical framework

Obstacle Avoidance in FormationFormally validatedExtending concept of Configuration Space Obstacle to formation case, thus decoupling formation keeping from obstacle avoidance

A Control Lyapunov Function Approach to Multi Agent Coordination

Documents