1. Capstone Design -- Robotics
Motors and Control
Jizhong Xiao
Department of Electrical Engineering
City College of New York

2. Robot Actuators Stepper motors DC motors AC motors Physics review:
Things seek lowest energy states.
Nature is lazy.
iron core vs. magnet N S
magnetic fields tend to line up v + - N S
Electric fields and magnetic fields are the same thing. v + -

3. Stepper Motor Basics
stator S N S N N S
rotor
Stator: made out of coils of wire called “winding”
Rotor: magnet rotates on bearings inside the stator
Current switch in winding ==>Magnetic force
==>hold the rotor in a position
Electromagnet
Direct control of rotor position (no sensing needed)
May oscillate around a desired orientation (resonance at low speeds)
Low resolution
printers
computer drives

4. Increased Resolution S
torque N S
angle N
Half stepping

5. Increased Resolution More teeth on rotor or stator Half stepping S N S

6. Increased Resolution More teeth on rotor or stator Half stepping S N S

7. Step Red Blue Yellow White
How to Control?
4 Lead Wire Configuration
Step Table
Step Red Blue Yellow White
Clockwise Facing Mounting End
Each step, like the second hand of a clock => tick, tick
Increase the frequency of the steps => continuous motion

8. 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…

9. DC motors -- exposed !

10. DC motor basics brush permanent magnets N S N rotor S stator+ V -
commutator attached to shaft

11. DC motor basics permanent magnets N S N rotor S N S N S stator + + - -V V - -

12. DC motor basics permanent magnets N S N rotor S N S N S N N S S stator+ + + V V V - - -

13. Position Sensors Optical Encoders Other Sensors Relative position
Absolute position
Other Sensors
Resolver
Potentiometer

14. Optical Encoders Relative position light sensor decode circuitry
- direction
light sensor
- resolution
decode circuitry
light emitter
grating

15. Optical Encoders Relative position light sensor decode circuitry
mask/diffuser
Relative position
light sensor
decode circuitry
light emitter
grating
A diffuser tends to smooth these signals
Ideal
Real

16. Optical Encoders Relative position light sensor decode circuitry
- direction
light sensor
- resolution
decode circuitry
light emitter
grating

17. Optical Encoders Relative position light sensor decode circuitry
- direction
light sensor
- resolution
decode circuitry
light emitter
grating A A
A lags B B B

18. Optical Encoders Relative position
- direction
light sensor
- resolution
decode circuitry
light emitter
grating
Phase lag between A and B is 90 degree A B
A leads B

19. Optical Encoders
Detecting absolute position
something simpler ?

20. Optical Encoders
Detecting absolute position
wires ?

21. Gray Code #
Binary 1 2 3 4 5 6 7 8 9 1 10 11
100
101
110
111
1000
1001
000
001
011
010
110
111
101
100
among others...

22. Other Sensors Resolver = driving a stepper motor Potentiometer
= varying resistance

23. Control Control: getting motors to do what you want them to
What you want to control = what you can control
For DC motors:
speed
voltage
windings’ resistance R N S w e V V
back emf N S e
is a voltage generated by the rotor windings cutting the magnetic field
emf: electromagnetic force

24. Controlling speed with voltage
e = ke w
The back emf depends only on the motor speed.
The motor’s torque depends only on the current, I.
t = kt I R V e
DC motor model

25. Controlling speed with voltage
e = ke w
The back emf depends only on the motor speed.
The motor’s torque depends only on the current, I.
t = kt I
Istall = V/R
Consider this circuit’s V:
V = IR + e
current when motor is stalled
How is V related to w ?
speed = 0
torque = max
V = ke w
t R kt R V e
- or - R V
w = t +
kt ke ke
DC motor model
Speed is proportional to voltage.

26. speed vs. torque V ke ktV R at a fixed voltage speed w torque t
no torque at max speed ke
max torque when stalled
ktV
torque t R

27. speed vs. torque Linear mechanical power Pm = F  v
at a fixed voltage
Linear mechanical power Pm = F  v
speed w
Rotational version of Pm = t  w V
no torque at max speed ke
ktV
stall torque
torque t R

28. speed vs. torque Linear mechanical power Pm = F  v
at a fixed voltage
Linear mechanical power Pm = F  v
speed w
Rotational version of Pm = t  w V ke
max speed
power output
speed vs. torque
ktV
stall torque
torque t R

29. speed vs. torque V ke ktV R speed w gasoline engine torque t max speed
power output
speed vs. torque
ktV
stall torque
torque t R

30. Motor specs
Electrical Specifications
For motor type 1624   003S 006S 012S 024
nominal supply voltage (Volts)
armature resistance (Ohms)
maximum power output (Watts)
maximum efficiency (%)
no-load speed (rpm) 12,000 10,600 13,000 14,400
no-load current (mA)
friction torque (oz-in)
stall torque (oz-in)
velocity constant (rpm/v)
back EMF constant (mV/rpm)
torque constant (oz-in/A)
armature inductance (mH) kt ke

31. Back to control Basic input / output relationship: t R V = + ke w kt
We can control the voltage applied V.
V = ke w
t R kt
We want a particular motor speed w .
How to change the voltage?
V is usually controlled via PWM -- “pulse width modulation”

32. 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

33. Open-loop vs. Close-loop Control
Open-loop Control:
V(t)
Controller solving for V(t)
desired speed w
Motor w
actual speed
If desired speed wd  actual speed wa,
So what?
Closed-loop Control: using feedback
desired wd V
Motor wa
actual speed wa -
compute V from the current error
wd - wa
PID controller

34. compute V using PID feedback
PID Controller
PID control: Proportional / Integral / Derivative control
V = Kp (wd - w) + Ki ∫ (wd - w) dt + Kd
V = Kp • ( e + Ki ∫ e + Kd )
d e dt
d e dt
Error signal e
wd - wa V
actual w
desired wd
compute V using PID feedback -
Motor
actual speed w

35. Evaluating the response
overshoot
steady-state error
ss error -- difference from the system’s desired value
settling time
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?

36. Control Performance, P-type
Kp = 20
Kp = 50
Kp = 500
Kp = 200

37. Steady-state Errors, P-type
Kp = 50
Kp = 200

38. Control Performance, PI - type
Kp = 100
Ki = 50
Ki = 200

39. You’ve been integrated...
Kp = 100
instability & oscillation

40. Control Performance, PID-type
Kp = 100
Kd = 5
Kd = 2
Ki = 200
Kd = 10
Kd = 20

41. PID final control

42. 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., root-locus 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 = K
Kp = 0.6 K
Ziegler-Nichols Tuning for second or higher order systems
Ki = 1.2 / P
Ki = 2.0 / P
Kd = P / 8.0

43. Implementing PID Use discrete approximations to the I and D terms:
Proportional term: ei = wdesired - wactual
at time i
i=now
Integral term: S ei
i=0
Derivative term: ei - 2ei-1 + ei-2
How could this discretization affect the performance of a system?
Sampling time is critical!!

44. What is 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 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 (one-half 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.

45. Shannon’s Sampling Theorem
An analog signal x(t) is completely specified by the samples if x(t) is bandlimited to , where
In other word, a continuous signal can be properly sampled, only if it does not contain frequency components above one-half of the sampling rate.
Definitions:
Given a signal bandlimited to , must sample at greater than to preserve information. The value is called Nyquist rate (of sampling for a given )
Given sampling rate , the highest frequency in the signal must be less than if samples are to preserve all the information. The value is called the Nyquist frequency (associated with a fixed sample frequency).

46. 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)

47. Motor Drive Micro-controller Motor Drive Components Logic Level
Power transistors
H-Bridge Drivers
etc ...

48. Useful Links
6.270 MIT’s Autonomous Robot Design Competition,
Acroname Inc. for Easy robotics, sensors, kits, etc,
Interactive C User’s Guide, etc.,
Handy board,
Pitsco Lego Dacta, lego components,
The Electronic Goldmine: cheep motors, electronics components,
Applied Motion Products: Step/DC motors and drives,
Jameco Electronics:

49. Assignment Refresh you memory Laboratory Control Theory
(Text book: K. Ogata, Modern Control Engineering, Prentice Hall)
Electronics (OP-amp, motor drive)
Laboratory
Specs of Motors
Motor Drive Circuit
Looking for Drive Components