Velocity Control of a Wheeled Inverted Pendulum by Partial Feedback Linearization

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Velocity Control of a Wheeled Inverted Pendulum
by Partial Feedback Linearization
43rd IEEE Conference on Decision and
Control Thursday December 16, 2004 1700-1720,
Paper ThC09.4

• Kaustubh Pathak, Grad. Student, Univ. of
  Delaware,
• Jaume Franch, Ph. D., Univ. Pol. de Catalunya,
• Sunil K. Agrawal, Ph D. , Professor
• Mechanical Systems Lab,
• University of Delaware,
• Newark, DE 19716, USA

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Introduction
Inverted Wheeled Pendulum
Segway 2002
Joe Grasser et al, 2002
Segway RMP 2003

• Dynamic stabilization needed for vehicle
  operation.
• Existing Controllers have mostly been derived
  using Linearization.
• Our Aims
• Mathematical modeling for autonomous operation.
• Design a Nonlinear Controller to follow
• Velocity(direction, magnitude) set-points.

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Overview

• Introduction
• Dynamic Modeling
• Obtaining the Dimension of the Largest Feedback
  Linearizable System
• Model Reformulation by Partial Feedback
  Linearization
• Velocity Controller Design
• Lower-level fast controller
• Higher-level slow controller
• Simulation Results
• Conclusions

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Modeling

• Two wheels with torque-input.
• Lagrangian Formulation
• of Equations of Motion.

FRONT
The CM of the pendulum is at (0,0,Cz) in
Body(blue) coordinates
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Model Reduction
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Strong Accessibility Largest Feedback
Linearizable Set

• The following results are derived in the paper
• The system is strongly accessible
• The largest feedback linearizable
• sub-system has dimension 4
• (Using Marino, Sys. Cont. Lett. 1986)

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Partial Feedback Linearization
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Transformed Dynamic System
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Two Time-scale Controller Faster Controller LLC
Lower-level Controller LLC
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Slower Higher Level Controller (HLC)
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HLC Design
Consider Lyapunov Function
Using the facts that
By LaSalles Thm.
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Simulation Results (1)
Stop command (zero velocity set-point) for two
different cz
HLC sampling-rate should Be about 0.2 sec.
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Simulation Results (2)
Snap-shot of input behavior
am
Non-zero velocity set-point for two different
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Position Controller Brief Discussion
Aim Make r converge to zero, at rest, Starting
from arbitrary state. Control Strtaegy LLC
input w2 same as in Velocity Controller
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Position Controller Animation
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Conclusions

• Detailed mathematical modeling and
  controllability analysis of the under-actuated
  inverted wheeled pendulum system.
• A velocity set-point controller design based on
  time-scale separation. The controller also bounds
  the pitch angle of the system explicitly.

Velocity and Position Control of a Wheeled
Inverted Pendulum by Partial Feedback
Linearization, Kaustubh Pathak, Jaume Franch,
and Sunil Agrawal, IEEE Transactions on Robotics.
(accepted for publication)