A Control Lyapunov Function Approach to Multi Agent Coordination

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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
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Multi 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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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 EgerstedtHu 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. gt
• 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) lt VU
• Bounded completion time.
• Keeping formation velocity v0, if V VU.

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Speed along trajectory How Do We Update s?

• Suggestion sv0 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) lt 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.
• VU1, v00.1, then v00.3 ! 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 lt 0 ?

• If we have semidefinite and stability by La
  Salles principle we choose as
• By a renewed La Salle argument we can still show
  VltVU , 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
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 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 lt VU, error
• T lt 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