Transcript Introduction - City University of New York

Capstone Design -- Robotics
Motors and Control
Jizhong Xiao
Department of Electrical Engineering
City College of New York
[email protected]
Robot Actuators
Stepper motors
DC motors
AC motors
Physics review:
Nature is lazy.
Things seek lowest energy states.
• iron core vs. magnet
• magnetic fields tend to line up
N
N
Electric fields and magnetic
fields are the same thing.
S
+ v -
+ v -
S
Stepper Motor Basics
stator
N
S
N
S
S
rotor
N
Stator: made out of coils of
Electromagnet
wire called “winding”
Current switch in winding
==>Magnetic force
Rotor: magnet rotates on
==>hold the rotor in a position
bearings inside the stator
• Direct control of rotor position (no sensing needed)
printers
computer drives
• May oscillate around a desired orientation (resonance at low speeds)
• Low resolution
Increased Resolution
S
torque
N
S
N
Half stepping
angle
Increased Resolution
S
N
S
N
Half stepping
More teeth on rotor or stator
Increased Resolution
S
N
S
N
Half stepping
More teeth on rotor or stator
How to Control?
4 Lead Wire Configuration
Step Table
Step Red Blue Yellow White
0
+
+
1
+
+
2
+
+
3
+
+
4
+
+
-
A+
Red
4 lead
motor
ABlue
Yellow
B+
White
B-
Clockwise Facing Mounting End
Each step, like the second hand of a clock => tick, tick
Increase the frequency of the steps => continuous motion
Motoring along...
• direct control of position
• precise positioning (The amount of
rotational movement per step depends
on the construction of the motor)
• Easy to Control
• under-damping leads to oscillation at low speeds
• torque is lower at high speeds than the primary alternative…
DC motors -- exposed !
DC motor basics
permanent
magnets
N
N
rotor
S
S
stator
brush
+
V
-
commutator
attached to shaft
DC motor basics
permanent
magnets
N
N
rotor
S
N
S
stator
+
+
V
-
V
-
S
DC motor basics
permanent
magnets
N
N
rotor
S
N
S
N
N
S
stator
+
+
V
-
+
V
-
V
-
S
S
Position Sensors
 Optical Encoders
 Relative position
 Absolute position
 Other Sensors
 Resolver
 Potentiometer
Optical Encoders
• Relative position
light sensor
light emitter
grating
decode
circuitry
- direction
- resolution
Optical Encoders
mask/diffuser
• Relative position
light sensor
decode
circuitry
light emitter
grating
A diffuser tends to
smooth these signals
Ideal
Real
Optical Encoders
• Relative position
light sensor
light emitter
grating
decode
circuitry
- direction
- resolution
Optical Encoders
• Relative position
light sensor
light emitter
- direction
- resolution
decode
circuitry
grating
A
A
A lags B
B
B
Optical Encoders
• Relative position
light sensor
- direction
- resolution
decode
circuitry
light emitter
grating
Phase lag between A and B is 90 degree
A
B
A leads B
Optical Encoders
• Detecting absolute position
something simpler ?
Optical Encoders
• Detecting absolute position
wires ?
Gray Code
#
0
1
2
3
4
5
6
7
8
9
Binary
0
000
1
001
10
011
11
010
100
110
101
111
110
101
111
100
1000
1001
among others...
Other Sensors
• Resolver
= driving a
stepper motor
• Potentiometer
= varying
resistance
Control
Control: getting motors to do what you want them to
What you want to control
For DC
motors:
=
what you can control
speed
voltage
windings’
resistance
N
V
N
R
w
V
S
e
back
emf
S
e
is a voltage generated by the rotor
windings cutting the magnetic field
emf: electromagnetic force
Controlling speed with voltage
• The back emf depends only on the motor speed.
e = ke w
• The motor’s torque depends only on the current, I.
t = kt I
R
V
e
DC motor model
Controlling speed with voltage
• The back emf depends only on the motor speed.
e = ke w
• The motor’s torque depends only on the current, I.
t = kt I
Istall = V/R
current when motor
is stalled
speed = 0
torque = max
V = IR + e
How is V related to w ?
tR
V=
+ ke w
kt
R
V
• Consider this circuit’s V:
e
- or -
V
w=- R t+
ke
kt ke
DC motor model
Speed is proportional to voltage.
speed vs. torque at a fixed voltage
speed w
V
ke
no torque at max speed
max torque when stalled
torque t
ktV
R
speed vs. torque at a fixed voltage
speed w
V
ke
no torque at max speed
Linear mechanical power Pm = F  v
Rotational version of Pm = t  w
torque t
ktV
R
stall torque
speed vs. torque at a fixed voltage
speed w
V
ke
Linear mechanical power Pm = F  v
Rotational version of Pm = t  w
max speed
power output
speed vs.
torque
torque t
ktV
R
stall torque
speed vs. torque
speed w
V
ke
gasoline engine
max speed
power output
speed vs.
torque
torque t
ktV
R
stall torque
Motor specs
ke
Electrical Specifications (@22°C)
For motor type 1624
003S
006S
012S
024
--------------------------
--------
--------
--------
---------
-------
nominal supply voltage
armature resistance
maximum power output
maximum efficiency
no-load speed
no-load current
friction torque
stall torque
velocity constant
back EMF constant
torque constant
armature inductance
(Volts)
(Ohms)
(Watts)
(%)
(rpm)
(mA)
(oz-in)
(oz-in)
(rpm/v)
(mV/rpm)
(oz-in/A)
(mH)
3
1.6
1.41
76
12,000
30
.010
.613
4065
.246
.333
.085
6
8.6
1.05
72
10,600
16
.011
.510
1808
.553
.748
.200
12
24
1.50
74
13,000
10
.013
.600
1105
.905
1.223
.750
24
75
1.92
74
14,400
6
.013
.694
611
1.635
2.212
3.00
kt
Back to control
Basic input / output relationship:
tR
V=
+ ke w
kt
We can control the
voltage applied V.
We want a particular
motor speed w .
How to change the voltage?
V is usually controlled via PWM -- “pulse width modulation”
PWM
 PWM -- “pulse width modulation
 Duty cycle:
 The ratio of the “On time” and the “Off time” in one cycle
 Determines the fractional amount of full power delivered to the
motor
Open-loop vs. Close-loop Control
Open-loop Control:
V(t)
desired speed w
Controller
solving for V(t)
Motor
w
actual speed
If desired speed wd  actual speed wa, So what?
Closed-loop Control: using feedback
wd - wa
desired wd
-
compute V from
the current error
V
PID controller
actual speed wa
wa
Motor
PID Controller
PID control: Proportional / Integral / Derivative control
V = Kp (wd - w) + Ki ∫ (wd - w) dt + Kd ddte
V = Kp • ( e + Ki ∫ e + Kd ddte )
Error signal e
wd - wa
desired wd
-
compute V using
PID feedback
V
actual speed w
actual w
Motor
Evaluating the response
overshoot
settling time
steady-state error
ss error -- difference from the
system’s desired value
overshoot -- % of final value
exceeded at first oscillation
rise time -- time to span from
10% to 90% of the final value
settling time -- time to reach
within 2% of the final value
rise time
How can we eliminate the steady-state error?
Control Performance, P-type
Kp = 20
Kp = 50
Kp = 200
Kp = 500
Steady-state Errors, P-type
Kp = 50
Kp = 200
Control Performance, PI - type
Kp = 100
Ki = 50
Ki = 200
You’ve been integrated...
Kp = 100
instability &
oscillation
Control Performance, PID-type
Kd = 2
Kd = 5
Kd = 10
Kd = 20
Kp = 100
Ki = 200
PID final control
PID Tuning
How to get the PID parameter values ?
(1) If the system has a known mathematical model (i.e., the
transfer function), analytical methods can be used (e.g., rootlocus method) to meet the transient and steady-state specs.
(2) When the system dynamics are not precisely known, we
must resort to experimental approaches.
Ziegler-Nichols Rules for Tuning PID Controller:
Using only Proportional control, turn up the gain until the system
oscillates w/o dying down, i.e., is marginally stable. Assume that K
and P are the resulting gain and oscillation period, respectively.
Then, use
for P control
for PI control
for PID control
Kp = 0.5 K
Kp = 0.45 K
Kp = 0.6 K
Ki = 1.2 / P
Ki = 2.0 / P
Kd = P / 8.0
Ziegler-Nichols Tuning
for second or higher
order systems
Implementing PID
Use discrete approximations to the I and D terms:
• Proportional term:
ei = wdesired - wactual
at time i
i=now
• Integral term:
S
e
i=0 i
• Derivative term:
ei - 2ei-1 + ei-2
How could this discretization affect the performance of a system?
Sampling time is critical!!
What is proper sampling
 Proper sampling:
 Can reconstruct the analog signal
from the samples
 Aliasing:
 The higher frequency component
that appears to be a lower one is
called an alias for the lower
frequency
 Aliasing: the frequency of the
sampled data is different from the
frequency of the continuous signal
Aliasing
b. 0.09 of sampling rate might represent, a 90 cycle/second sine wave being sampled at 1000
samples/second; in another word, there are 11.1 samples taken over each complete cycle of the sinusoid
d. Aliasing occurs when the frequency of the analog sine wave is greater than the Nyquist frequency (onehalf of the sampling rate); in other word, the sampling frequency is not fast enough. Aliasing
misrepresents the information, so the original signal cannot be reconstructed properly from the samples.
Shannon’s Sampling Theorem
 An analog signal x(t) is completely specified by the samples
if x(t) is bandlimited to wBL  ws / 2 , where ws  2 / Ts
 In other word, a continuous signal can be properly sampled,
only if it does not contain frequency components above onehalf of the sampling rate.
 Definitions:
 Given a signal bandlimited to f BL , must sample at greater than 2 f BL
to preserve information. The value 2 f BL is called Nyquist rate (of
sampling for a given f BL )
 Given sampling rate f s , the highest frequency in the signal must be
less than f s / 2 if samples are to preserve all the information. The
value f NYQ  f s / 2 is called the Nyquist frequency (associated with a
fixed sample frequency).
Rule of Thumb
 For a closed-loop control system, a typical
choice for the sampling interval T based on
rise time is 1/5 th or 1/10 th of the rise time.
(i.e., 5 to 10 samples for rise time)
Motor Drive
 Micro-controller
 Logic Level
 Motor Drive Components
 Power transistors
 H-Bridge Drivers
 etc ...
Useful Links
 6.270 MIT’s Autonomous Robot Design Competition,
http://web.mit.edu/6.270/www/home.html
 Acroname Inc. for Easy robotics, sensors, kits, etc,
http://www.acroname.com/
 Interactive C User’s Guide, etc., http://www.newtonlabs.com/ic/
 Handy board, http://www.handyboard.com/
 Pitsco Lego Dacta, lego components, http://www.pitscolegodacta.com/intro.htm
 The Electronic Goldmine: cheep motors, electronics components,
http://www.goldmine-elec.com
 Applied Motion Products: Step/DC motors and drives,
http://www.applied-motion.com
 Jameco Electronics: http://www.jameco.com