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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

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Petter Ögren CAS talk 2

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

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Petter Ögren CAS talk 3

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).

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Petter Ögren CAS talk 4

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

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Petter Ögren CAS talk 5

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.

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Petter Ögren CAS talk 6

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.

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Petter Ögren CAS talk 7

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.

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Petter Ögren CAS talk 8

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.

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Petter Ögren CAS talk 9

Proof sketch: Formation error Proof sketch: Formation error

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Petter Ögren CAS talk 10

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

Find lower bound on ds/dt

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Petter Ögren CAS talk 11

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:

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Petter Ögren CAS talk 12

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.

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Petter Ögren CAS talk 13

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

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Petter Ögren CAS talk 14

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.

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Petter Ögren CAS talk 15

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 eddie@princeton.edu, naomi@princeton.edu

Mechanical and Aerospace Engineering

Princeton University, USA

Optimization and Systems Theory

Royal Institute of Technology, Sweden

Petter Ogrenpetter@math.kth.se

Another extension:Another extension:

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Petter Ögren CAS talk 16

•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.

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Petter Ögren CAS talk 17

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

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Petter Ögren CAS talk 18

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