Transcript Control of a Ball & Plate System

Control of a Ball &
Plate System
Andrew Percy
Academic Supervisor – James Welsh
Summary
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Introduction to the Ball & Plate System
Background & Project Motivation
System Modeling
Actuator Design & Modeling
Sensor Design & Integration
Control System Design
Remaining Work
The Ball & Plate System
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Objective
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Control Characteristics
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Balance a Ball on a
Moveable Plate
Unstable
Non-Linear
Examples
Plate
Ball
qy
y
qx
x
Project Motivation
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Applications
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Control Research
Stabilized Platforms
Rolling Contact
Manipulation
Existing Apparatus
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Actuator Backlash
Measurement Errors
Rolling with Slipping
System Model
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z'
A Plate Fixed Reference
Frame

cos q y

   sin q x sin q y
  cos q sin q
x
y

sin q y
0
cos q x
sin q x


 sin q x cos q y 
cos q x cos q y 
z''
qx
Plate
y''
ry
Ball
x
R
H
qx
y'
O
z
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Plate Angular Velocity
z'
qy
 0

        z

y

 z
0
x
y 

 x 
0 
Plate
x'
rx
Ball
R
y
H
qy
O
x
System Model
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Modeling Assumptions
The Ball is a Perfect Sphere
 Rolling Without Slipping
 Neglect Plate Potential Energy
 Small Plate Angles
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Lagrange's Equation
L d L

 0, k  1,2,..., n
 q k dt  q k
System Model
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Complete & Coupled Non-Linear Model
rx 
ry 


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5
rx qx2 sin 2 q y  qy2  3R 5  H qy  sin q y g cos q x  R  H qx2 cos q y  ryqx  2ryqx
7
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
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5 2
ryq x  g sin q x  2qxqy sin q y 6 R 5  H   qx cos q y 7 R 5  H   sin q y 2rxqx  rxqx  2rxqxqy cos q y
7
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Decoupled Linear Approximation
rx 
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5
53

gq y   R  H qy
7
75

5
57

ry   gq x   R  H qx
7
75
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System Transfer Functions
5 g  s 2 3R  5H 

qy
7s 2
rx
s 2 7 R  5H   5 g

qx
7s 2
ry
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Actuation
Actuator Selection
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Servo Motor
Stepper Motor
Linear Solenoid
Linear Induction Motor
Magnetic Force
20
10
0
0
1    l l   
Fm    0 1 3
2 
lg
2
 2 2
N I

1
2
3
4
5
-10
Resultant Force (N)
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-20
-30
0At
400At
-40
800At
1200At
-50
-60
-70
-80
Displacement (mm)
6
7
8
9
10
Sensors
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Plate Angle
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Ball Position
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Potentiometers
Tilt Sensor
Image Processing
Pressure Sensitive Plate
Sensor Grid
Camera
Buffer
Sync
Separation
Ball
rx
ry
Current Sense Resistor
Micro
Controller
Plate
I
Coil
Solenoid Current
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Sample
& Hold
I
+
Current
Sense
Out
Control Design
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Ball Position Control
State Feedback
 Cascade Control
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Desired Ball
Position
+
-
Ball Position
Controller
+
-
Plate Angle
Controller
Plate
Dynamics
Plate Angle
Sensor
Ball Position
Sensor
Ball
Dynamics
Actual Ball
Position
Remaining Work
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Continue Development of Solenoid Actuator
Rebuild the Existing Ball & Plate Apparatus
Design Sensor Interface & Control Circuits
Continue Development of System Models
Design & Simulation of Controllers
Develop Control Software
Implement & Test Control System