a control lyapunov function approach to multi agent coordinationclfcas03
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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
eddie@princeton.edu, naomi@princeton.edu
Mechanical and Aerospace Engineering
Princeton University, USA
Optimization and Systems Theory
Royal Institute of Technology, Sweden
Petter Ogren
petter@math.kth.se
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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