Transcript Document

Centre for Autonomous Systems
A Control Lyapunov Function
Approach to
Multi Agent Coordination
P. Ögren, M. Egerstedt* and X. Hu
Royal Institute of Technology (KTH), Stockholm
and Georgia Institute of Technology*
IEEE Transactions on Robotics and Automation, Oct 2002
Petter Ögren
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Multi Agent Robotics
Applications:
Search and rescue
missions
Spacecraft inferometry
Reconfigurable sensor
array
Carry large/awkward
objects
Formation flying
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Motivation:
Flexibility
Robustness
Price
Efficiency
Feasibility
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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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Quantifying Formation Keeping
Definition: Formation Function
Will add Lyapunov like assumption satisfied by
individual set-point controllers. =>
Think of as parameterized Lyapunov function.
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Examples 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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Main Assumption
We can find a class K function s 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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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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Evolution 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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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) showed
that the position of an
off axis point x can be
feedback linearized to:
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Example: 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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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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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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Another extension:
Formations with a Mission: Stable
Coordination of Vehicle Group Maneuvers
Petter Ogren
Edward Fiorelli and Naomi Ehrich Leonard
[email protected]
[email protected], [email protected]
Optimization and Systems Theory
Mechanical and Aerospace Engineering
Royal Institute of Technology, Sweden
Princeton University, USA
Mathematical Theory of Networks and Systems (MTNS ‘02)
Visit: http://graham.princeton.edu/ for related information
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Approach: Use artificial potentials and virtual body
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with dynamics.
•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.
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Conclusions
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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Related Publications
A Convergent DWA approach to Obstacle
Avoidance
Formally validated
Merge of previous methods using new
mathematical framework
Obstacle Avoidance in Formation
Formally validated
Extending concept of Configuration Space
Obstacle to formation case, thus decoupling
formation keeping from obstacle avoidance
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