Designing a Working Physical Model to Demonstrate Global Stabilization of the Classical Inverted Pendulum Problem; Step One

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Designing a Working Physical Modelto Demonstrate
Global Stabilizationof the Classical Inverted
Pendulum ProblemStep One Motor Specification

• Brian Ganus
• EE 691/692 - Spring 2006

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BASIC INVERTED PENDULUM
http//www.eas.asu.edu/aar/pix/cart_pend_anim1.gi
f
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PROBLEM DEFINITION

• Classic Inverted Pendulum Problem
• Global Stabilization Using Dual Mode Control as
  Presented in 1971 Masters Thesis 1
• Unable to Construct Working Model Due to
  Technology Limitations
• Want to Investigate Building a Working Model
  Using Current Technology
• Motor Choice Drives the Design

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ORIGINAL 1971 CART DESIGN
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METHOD

1. Use Simulink to build and model the open loop
   system from the equation given in 1.
2. Verify that the simulation correctly models the
   expected behavior of the desired open loop
   system.
3. Use Simulink to build and model the closed loop
   system using the dual-mode control described.
4. Verify that the simulation correctly models the
   expected behavior of the desired closed loop
   system.
5. Run simulations to determine the maximum
   acceleration required to stabilize the system.
6. Use the maximum acceleration value found to
   define the motor choice parameters.
7. Choose motor.

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STEP ONE

• Use Simulink to build and model the open loop
  system from the equation given in 1.
• Define Parameters
• Build a Simulink Model

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PLANT DEFINITION

• Placement, measured from the desired vertical (T
  gt 0 ? clockwise rotation)
• Angular velocity
• Non-negative normalized damping coefficient
  associated with the angular velocity.
• Horizontal acceleration y of the cart used to
  control the motion of the pendulum

http//members.cox.net/srice1/pendulum/page0.htm
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BASIC ASSUMPTIONS

• 0 is defined as the vertical position.
• Pendulum freely rotates from 0.
• Continued rotation is unrestricted.
• Can make infinite number of turns.
• Any multiple of 2 n ? will be valid.
• Angular rate is measurable.
• Pendulum position can be determined and fed back
  into the controller to tell the cart which
  direction and speed to move.
• Forward/reverse motion of the cart used to
  control the pendulums position.

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MATHEMATICAL MODEL

• Mathematical Model of the Dynamics of Pendulum
  Angle and Cart

• Phase Variables

• Phase-Variable State Equations

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• State Model

• Math Flow Diagram of the Open loop System

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STEP TWO

• Verify that the simulation correctly models the
  expected behavior of the desired open loop system.

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TESTING OPEN LOOP SYSTEM

• Initial Open Loop System Model
• Position ? 10
• Angular Velocity ? 0 rad/sec
• Damping Factor b 0.7

• Final Values
• Position ? 180
• Angular Velocity ? 0 rad/sec

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TESTING OPEN LOOP SYSTEM

• Initial Open Loop System Model
• Position ? 180
• Angular Velocity ? 0 rad/sec
• Damping Factor b 0.7

• Final Values
• Position ? 180
• Angular Velocity ? 0 rad/sec

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TESTING OPEN LOOP SYSTEM

• Initial Open Loop System Model
• Position ? 350
• Angular Velocity ? 0 rad/sec
• Damping Factor b 0.7

• Final Values
• Position ? 180
• Angular Velocity ? 0 rad/sec

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TESTING OPEN LOOP SYSTEM

• Initial Open Loop System Model
• Position ? 10
• Angular Velocity ? 1 rad/sec
• Damping Factor b 0.7

• Final Values
• Position ? 180
• Angular Velocity ? 0 rad/sec

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TESTING OPEN LOOP SYSTEM

• Initial Open Loop System Model
• Position ? 10
• Angular Velocity ? 1-10 Rad/sec
• Damping Factor b 0.7

• Final Values
• Position ? 180,540,900
• Angular Velocity ? 0 rad/sec

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STEP THREE

• Use Simulink to build and model the closed loop
  system using the dual-mode control described in
  1.

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HYBRIDDUAL-MODE CONTROL

• Bang-Bang Control
• Motor causes the cart to move forward and back to
  pump the pendulum to a desired angle.
• Linear Control
• Fine tunes the position until vertically stable.
• Uses a threshold to compare the position to
  determine when to switch modes.

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• Dual Mode Control

• Math Flow Diagram of the Closed loop System

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STEP FOUR

• Verify that the simulation correctly models the
  expected behavior of the desired closed loop
  system.

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TESTING CLOSED LOOP SYSTEM

• Initial Open Loop System Model
• Position ? 10
• Angular Velocity ?1 rad/sec
• Damping Factor b 0.325

• Final Values
• Position ? 0
• Angular Velocity ? 0 rad/sec

• Required fine tuning Damping Factor to stabilize.
• Dual Mode Control utilized backwards.

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TESTING CLOSED LOOP SYSTEM

• Initial Open Loop System Model
• Position ? 50
• Angular Velocity ?1 rad/sec
• Damping Factor b 0.325

• Final Values
• Position ? NOT 0
• Angular Velocity ?NOT0 rad/sec

• If initial position changed, then needed to be
  retuned or it would not stabilize.

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ERROR IN PAPER

• Through Simulations, found that the paper had an
  error in the threshold definition of the
  Dual-Mode Control.

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• Altered Dual Mode Control

• Altered Math Flow Diagram of the Closed loop
  System

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TESTING ALTERED CLOSED LOOP SYSTEM

• Initial Open Loop System Model
• Position ? 10
• Angular Velocity ?0 rad/sec
• Damping Factor b 0.016

• Final Values
• Position ? 0
• Angular Velocity ? 0 rad/sec

• Stabilized, but only used linear control.

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TESTING ALTERED CLOSED LOOP SYSTEM

• Initial Open Loop System Model
• Position ? 80
• Angular Velocity ?0 rad/sec
• Damping Factor b 0.016

• Final Values
• Position ? 0
• Angular Velocity ? 0 rad/sec

• Stabilized and used full Dual-Mode Control.

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STEP FIVE

• Run simulations to determine the maximum
  acceleration required to stabilize the system.

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DETERMINING MAX ACCELERATION REQUIRED

• Initial Open Loop System Model
• Position ? 180
• Angular Velocity ?0 rad/sec
• Damping Factor b 0.016

• Final Values
• Position ? 0
• Angular Velocity ? 0 rad/sec

• Used worst case scenario for initial position to
  determine max acceleration required to stabilize.

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STEP SIX

• Use the maximum acceleration value found to
  define the motor choice parameters.

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DETERMINING MAX ACCELERATION REQUIRED

• 4 Basic parameters allow specification of motor
  characteristics
• Derived the relationship between
• Experimentally determined maximum acceleration
• Maximum torque of the motor
• Maximum mass of the cart
• Maximum length of the pendulum

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Basic Derivation
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Results

• Maximum value of horizontal acceleration required
  to stabilize the system

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REVIEW

• Found Error in Original Paper
• Given
• Maximum Acceleration (y)
• Motor Torque (t)
• Able to Determine Maximum Value of Product
• Mass of the Cart (M)
• Radius (r) of the pendulum
• Define Design Parameter
• If Mr lt t / y
• Then Model Will Demonstrate the Global
  Stabilization of the pendulum as described in 1
  and as seen in the simulations.

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STEP SEVEN

• Choose motor.
• Now can compare the motors
• We wanted an RC off-the-shelf motor
• But what type?

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BRUSHED MOTOR
http//electronics.howstuffworks.com/motor1.htm
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BRUSHED MOTOR
http//electronics.howstuffworks.com/motor6.htm
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BRUSHED MOTORS
http//www.rcboataholic.com/motors/motor_brushless
.htm
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BRUSHED MOTOR

• Advantages
• Cheaper
• Disadvantages
• Less Torque
• Brushes Wear Out
• Change direction slower
• Less Accurate

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BRUSHLESS MOTORS
http//www.rcboataholic.com/motors/motor_brushless
.htm
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BRUSHLESS MOTOR

• Advantages
• More Torque
• No Brushes to Wear Out
• Changes Directions Quicker
• More Accurate
• Disadvantages
• More Expensive
• Requires a controller

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BRUSHLESS MOTORS
http//www.rcboataholic.com/motors/motor_brushless
.htm
http//www.hobbyworks.com/default.cfm/Content/full
product/hs/RC/ID/502204
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OUTRUNNER STYLE BRUSHLESS MOTOR

• Outrunner style motor provides the most torque of
  any brushless style motor because case and rotors
  spin.
• Allows a single shaft which simplifies the design
  and makes it more durable.

http//www.hobby-lobby.com/brushless-axi4130.htm
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FACTORS TO CONSIDER

• Most motors are 80 efficient or less.
• Motor turns electric input power into mechanical
  output power.
• Output Power is in terms of Torque and
  Revolutions Per Minute (RPM).
• Few manufacturers provide information on torque
  produced.
• Need to define variables to compare products.
• Voltage Constant (Kv)
• Torque Constant (Kt)
• Torque
• Amperage range

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DEFINE UNITS
Used System International (SI) Units Mass
Kilograms (kg) Force Newtons (kg?m/s2) Time Sec
ond (s) Current Ampere Angular Velocity
Radian/Second (rad/s) Acceleration
Meter/Second/Second (m/ s2) Radian
Degrees?p/180 Torque Newton Meter
(kg?m2/s2) Radius Meter (m)
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RESULTS

• AXI Outrunner AXI-2815-10
• Given Torque Constant (kt) of 3.782 m/s2
• Given Range of 20-30 Amps
• For Calculations used 25 Amps

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RESULTS

• Maximum Torque of this motor
• Now can determine the other values.

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CONCLUSION
Specification Specification
Nominal Voltage 7.0 V
Operating Voltage 6-9 V
No Load Motor Speed 11,130 RPM
RPM per Volt 1,590 RPM/V
Max. Efficiency with ESC 81
Max. Efficiency Current 20-30 A
Max. Loading 40 A/60 s
No load Current / 8V 2,4 A
Ri 37 ohm
Dimensions 35x37 mm
Shaft Diameter 4 mm
Weight w/cables 106 g
Propeller range (direct drive) 9,5x6" - 11x6,5"240/150 - 300/165
http//www.modelflight.com.au/rc_model_electronics
/axi_brushless_2814-10.htm
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REFERENCES

1. Baumann, James L. and C. D. Johnson,
   Global-Stabilization of the Inverted-Pendulum
   An Early Solution and Hardware Implementation,
   Proceedings of the 2004 Southeastern Symposium on
   System Theory, Georgia Institute of Technology
   Atlanta, GA, March, 2004.
2. Boucher, Robert J. Electric Motor Handbook,
   Astro Flight Inc., Marina Del Ray, California,
   1994.
3. Halliday, David and Robert Resnick and Jearl
   Walker Fundamentals of Physics Third Sixth
   Edition, John Wiley Sons, Inc., New York, NY,
   2001.
4. Cannon, Robert H. Dynamics of Physical Systems,
   McGraw-Hill Book Company, New York, NY, 1967.
5. Brogan, William L. Modern Control Theory Third
   Edition, Prentice Hall, Upper Saddle River, NJ,
   1991.
6. Baumann, William T. and Wilson J. Rugh, Feedback
   Control of Nonlinear Systems by Extended
   Linearization, IEEE Transactions on Automatic
   Control, Vol. AC-31, No. 1, January 1986, pp
   40-46.