Implementable

1
Implementable Efficient Trajectory Design for
an Autonomous Underwater Vehicle.

• Monique Chyba
• Thomas Haberkorn
• Ryan N. Smith
• George R. Wilkens
• Song K. Choi
• University of Hawaii at Manoa

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Research Goal

• Given two configurations at rest, we wish to
  design controllers which produce implementable
  trajectories for autonomous underwater vehicles.

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Motivation

• Merge theory and application
• Interesting problem from any viewpoint
• AUVs are becoming more prevalent in research
• Optimal control of underwater vehicles is
  presently incomplete
• Efficient and versatile underwater vehicle
• Different from torpedo shape AUV
• Exploit all degrees of freedom

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Interdiciplinary Research Team

• Monique Chyba
• Optimal Control Theory
• Thomas Haberkorn
• Numerical Analysis
• George Wilkens
• Differential Geometry
• Song Choi
• Mechanical Engineering
• Ryan Smith
• Ocean Engineering
• ODIN
• Omni-Directional Intelligent Navigator
• Test-bed AUV

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Project Components

• Geometric Control Theory
• Vehicle Model
• Control Strategy
• Numerical Computations Modeling
• Experiments
• Repeat...Update...Repeat...Update...

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Model

• Complex and highly nonlinear
• Model based on ODIN
• Good understanding of forces and geometry
• Experiments keep updating the model
• Viewed from a differential geometry point of view

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Equations of Motion
Configuration space is the Lie Group
SE(3). Velocities belong to the lie algebra se(3).
Kinetic Energy
Translation
Rotation
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• Second order forced affine-connection control
  system on Q SE(3).
• ? is the Levi-Civita affine connection
• Unique connection for the Riemannian metric (G)
  induced by the kinetic energy of the system
• View below as Acceleration Force/mass

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Ideal Fluid
Real Fluid
Forced Affine-Connection Control System on SE(3)
Affine-Connection Control System on SE(3)
2nd order
First order system on TSE(3)
First order system on TSE(3)
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Control Strategy

• Efficiency
• In computation time and duration along the
  trajectory
• Implementability
• Both trajectory and algorithm implementable on an
  AUV
• Understand physical constraints
• Thrusters have limited power
• Optimize cost(s) along trajectories
• Time or energy consumption or BOTH
• Begin with time minimization
• Extend previous results
• Real fluid characterization

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Time Optimal Trajectory

• Apply Pontryagins Maximum Principle
• Optimal strategy is a concatenation of bang-bang
  control and singular arcs
• Leads to chattering
• Complicated control strategy
• No optimal synthesis analytically
• Must consider numerical methods

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Numerical Techniques

• Direct methods
• Full discretization of state and control
• Very fine discretization and long solving time
• Always works
• Indirect methods
• Very sensitive to initialization
• Require a priori knowledge of the control
  structure
• Fast and accurate IF it converges

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Time Optimal Strategy
?f (6,4,1,0,0,0) and tf 23.21s
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Numerical Algorithm

• Use actuator switching times as additional
  unknowns of the optimal control problem
• Concatenation of constant thrust arcs
• Connected by a linear junction to avoid
  instantaneous switching of actuators
• Eliminates singular arcs and controls the number
  of switching times

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Omni-Directional Intelligent Navigator (ODIN)

• 65 cm spherical body
• 8 thrusters
• 4 horizontal
• 4 vertical
• Weight 127 kg
• bouyant by 0.3 kg
• Full 6 DOF motion
• Max depth 100m
• Max speed 3 m/s
• Tuned to our needs
• CG location
• Buoyancy
• Robust
• Remedied many issues
• Thrusters
• Grounding
• Batteries
• Tether
• Positioning

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Detailed Parameters

• CB (0,0,-1) cm
• Mif 70 kg
• Jif 0 kg.m2
• Ix 5.46, Iy 5.29, Iz 5.72 kg.m2
• D?11 35.8, D?21 35.8, D?31 31.2
• D?12 36.8, D?22 36.8, D?32 34.8
• DO11 18.9, DO21 18.9, DO31 16.9
• DO12 35.4, DO22 34.4, DO32 31.5
• All parameters were first empirically tested and
  are continually updated

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Experiments

• Weekly testing
• Test-bed AUV is ODIN
• Test control strategies and trajectory design
• Update the model
• Observe practical constraints not visible in the
  theory
• Thrusters
• Environment

ODIN from the 10m platform
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Pure Motion Thrust Strategy
?f (6,4,1,0,0,0) and tf 72.77s
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Pure Motion Evolution
b2 (m)
?f (6,4,1,0,0,0) and tf 72.77s
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STPP1 Strategy Evolution
?f (6,4,1,0,0,0) and tf 35.9s
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STPP2 Thrust Strategy
?f (6,4,1,0,0,0) and tf 25.29
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STPP2 Evolution
?f (6,4,1,0,0,0) and tf 25.29
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Conclusion

• We are able to design a control strategy which
  gives implementable trajectories which are both
  efficient to compute and are efficient with
  respect to time and/or energy.
• Demonstrated time efficiency. Energy consumption
  will be approached through similar analysis and
  is under investigation.

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Kinematic Reduction

• Ideal fluid case - no external forces
• .
• System on TQ
• .
• Driftless system on Q
• ?kin is a KINEMATIC REDUCTION of ?dyn if
• X is of locally constant rank
• For every controlled trajectory (?,ukin) for ?kin
  there exists dynamic controls such that (?,udyn)
  is a controlled trajectory for ?dyn

X (resp. Y) denotes the distributions generated
by X (resp. Y )
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Decoupling Vector Fields

• Kinematic reductions of rank one
• Vector field whose integral curves can be
  followed by a trajectory of the dynamic system
• X is decoupling for ?dyn iff X, ?XX ? G8(Y)

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Further Research

• Characterize a kinematic reduction for the forced
  affine-connection control system
• Identify decoupling vector fields under this
  characterization
• Find the simple motions for the forced system
• Use these motions in place of the piecewise
  continuous arcs in the numerical algorithm