a control lyapunov function approach to multi agent coordinationclfcas03

8/10/2019 A Control Lyapunov Function Approach to Multi Agent CoordinationclfCas03

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Petter gren CAS talk 1

A Control Lyapunov Function

Approach to

Multi Agent Coordination

P. gren, 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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Multi Agent Robotics

Motivation:

Flexibility

Robustness

Price

Efficiency

Feasibility

Applications:

Search and rescuemissions

Spacecraft inferometry

Reconfigurable sensorarray

Carry large/awkwardobjects

Formation flying


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Problem and Proposed Solution

Problem:How to make set-point controlledrobots moving along trajectories in a formationwait for eachother?

Idea:Combine Control Lyapunov Functions(CLF) with the Egerstedt&Hu virtual vehicleapproach.Under assumptions this will resultin:

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


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Quantifying Formation Keeping

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

Think of as parameterized Lyapunov function.

Definition: Formation Function


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Examples of Formation Function

Simple linear example !

A CLF for the combinedhigher dimensional

system:

Note that a,b, are designparameters.

The approach applies toany parameterizedformation scheme withlyapunov stability results.


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Main Assumption

We can find a class K function ssuch that thegiven set-point controllers satisfy:

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

Bounded V (error): V(x,s) < VUBounded completion time.

Keeping formation velocity v0, ifV VU.


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Speed along trajectory:

How Do We Update s?

Suggestion: s=v0t

Problems: Boundedctrlor localass stability

We want:V to be small

Slowdown if V is large

Speed v0if V is small

Suggestion:Let s evolve withfeedbackfrom V.


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Evolution of s

Choosing to be:

We can prove:

Bounded V (error): V(x,s) < VUBounded completion time.Keeping formation velocity v0, ifV VU.


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Proof sketch: Formation error


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Proof sketch: Finite Completion Time

Find lower bound on ds/dt


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The Unicycle Model,

Dynamic and Kinematic

Beard (2001) showedthat the position of an

off axis point xcan befeedback linearized to:


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Example: Formation

Three unicyclerobots alongtrajectory.

VU=1, v0=0.1, thenv0=0.3 ! 0.27

Stochasticmeasurement error

in top robot at 12mmark.


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Extending Work by Beard et. al.

Satisficing Control for Multi-Agent FormationManeuvers, in proc. CDC 02

It is shown how to find an explicit

parameterization of the stabilizing controllersthat fulfills the assumption

These controllers are also inverse optimal andhave robustness properties to inputdisturbances

Implementation


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What if dV/dt


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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 Ogren

[email protected]

Another extension:


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Configuration space of virtual body isfor orientation, position and expansionfactor:

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

To ensure stability and convergence, prescribe virtual bodydynamics so that its speedis driven by a formation error.

Define directionof virtual body dynamics to satisfy mission.

Partial decoupling: Formation guaranteed independent of mission.

Prove convergence of gradient climbing.

Approach: Use artificial potentials and virtual bodywith dynamics.


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Conclusions

Moving formations by using ControlLyapunov Functions.Theoretical Properties:

V


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Related Publications

A Convergent DWA approach to ObstacleAvoidance

Formally validated

Merge of previous methods using newmathematical framework

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

a control lyapunov function approach to multi agent coordinationclfcas03

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