Motors and Control

Jizhong Xiao

Department of Electrical Engineering

City College of New York

jxiao@ccny.cuny.edu

Capstone Design -- Robotics

Robot Actuators

Stepper motors

DC motors

AC motors

Physics review:

Electric fields and magnetic fields are the same thing.

Nature is lazy.Things seek lowest energy states.• iron core vs. magnet• magnetic fields tend to line up

v+ - v+ -

N

S

N S

Stepper Motor Basics

S

N

Stator: made out of coils of wire called “winding”

Rotor: magnet rotates on bearings inside the stator

• Direct control of rotor position (no sensing needed)

• May oscillate around a desired orientation (resonance at low speeds)

• Low resolution

printerscomputer drives

SN

Electromagnet

stator

rotor

N S

Current switch in winding ==>Magnetic force

==>hold the rotor in a position

Increased Resolution

Half stepping

S

S

N

N

angle

torque

Increased Resolution

More teeth on rotor or stator

Half stepping

S

S

N

N

Increased Resolution

More teeth on rotor or stator

Half stepping

S

S

N

N

How to Control?

Step TableStep Red Blue Yellow White

0 + - + -1 - + + -2 - + - +3 + - - +4 + - + -

4 leadm otor

Red

B lue

A+

A-

B+ B-

Yellow W hite

4 Lead Wire Configuration

Clockwise Facing Mounting End

Increase the frequency of the steps => continuous motion

Each step, like the second hand of a clock => tick, tick

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

N

S

N S

stator

rotor

permanent magnets

commutator attached to shaft

V

+

-

brush

DC motor basics

N

S

N S

stator

rotor

permanent magnets

N SS N

V

+

-V

+

-

DC motor basics

N

S

N S

stator

rotor

permanent magnets

N SS N N SN S

V

+

-V

+

-

V

+

-

Position Sensors

Optical Encoders Relative position Absolute position

Other Sensors Resolver Potentiometer

Optical Encoders

• Relative position - direction

- resolution

grating

light emitter

light sensor

decode circuitry

Ideal

Optical Encoders

• Relative position mask/diffuser

grating

light emitter

light sensor

decode circuitry

Real

A diffuser tends to smooth these signals

Optical Encoders

• Relative position - direction

- resolution

grating

light emitter

light sensor

decode circuitry

Optical Encoders

• Relative position - direction

- resolution

grating

light emitter

light sensor

decode circuitry

A

B

A

B

A lags B

Optical Encoders

• Relative position - direction

- resolution

grating

light emitter

light sensor

decode circuitry

A

B A leads B

Phase lag between A and B is 90 degree

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

1

10

11

100

101

110

111

1000

1001

000

001

011

010

110

111

101

100

among others...

Other Sensors

• Resolver

= driving a stepper motor

• Potentiometer

= varying resistance

Control

What you want to control = what you can control

For DC motors:

speed voltage

N

S

N SV

V e back emf

R

windings’ resistance

e is a voltage generated by the rotor windings cutting the magnetic field

emf: electromagnetic force

Control: getting motors to do what you want them to

Controlling speed with voltage

DC motor model

V e

R

• The back emf depends only on the motor speed.

• The motor’s torque depends only on the current, I.

e = ke

= k I

kke

Controlling speed with voltage

DC motor model

V e

R

• The back emf depends only on the motor speed.

• The motor’s torque depends only on the current, I.

e = ke

= k I

• Consider this circuit’s V: V = IR + eIstall = V/Rcurrent when

motor is stalledspeed = 0

torque = max

How is V related to

V = + ke R k

- or -

= - + R ke V

Speed is proportional to voltage.

speed vs. torque

torque

speed

ke V

at a fixed voltage

R kV

max torque when stalled

no torque at max speed

speed vs. torque

torque

speed

ke V

at a fixed voltage

R kV stall torque

no torque at max speed

Linear mechanical power Pm = F v

Rotational version of Pm =

speed vs. torque

torque

speed

ke V

at a fixed voltage

R kV stall torque

max speed

Linear mechanical power Pm = F v

Rotational version of Pm =

power output

speed vs. torque

speed vs. torque

torque

speed

ke V

R kV

power output

speed vs. torque

gasoline enginemax speed

stall torque

Motor specs

Electrical Specifications (@22°C)For motor type 1624   003S 006S 012S 024

-------------------------- -------- -------- -------- --------- -------nominal supply voltage (Volts) 3 6 12 24armature resistance (Ohms) 1.6 8.6 24 75maximum power output (Watts) 1.41 1.05 1.50 1.92maximum efficiency (%) 76 72 74 74no-load speed (rpm) 12,000 10,600 13,000 14,400no-load current (mA) 30 16 10 6friction torque (oz-in) .010 .011 .013 .013stall torque (oz-in) .613 .510 .600 .694velocity constant (rpm/v) 4065 1808 1105 611back EMF constant (mV/rpm) .246 .553 .905 1.635torque constant (oz-in/A) .333 .748 1.223 2.212armature inductance (mH) .085 .200 .750 3.00

ke

k

Back to control

Basic input / output relationship:

How to change the voltage?

We want a particular motor speed .

We can control the voltage applied V.

V = + ke R k

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:

actual speed

desired dV

Motor

a

actual speed a

- compute V from the current error

d a

Closed-loop Control: using feedback

desired speed Controller solving for V(t)

V(t)

Motor

If desired speed d actual speed a,

So what?

PID controller

PID Controller

desired dV

Motoractual

actual speed

- compute V using PID feedback

d a

Error signal e

PID control: Proportional / Integral / Derivative control

V = Kp (d ) + Ki ∫ (d ) dt + Kd

V = Kp • ( e + Ki ∫ e + Kd )d e dt

d e dt

Evaluating the response

How can we eliminate the steady-state error?

steady-state error

settling time

rise time

overshoot

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

ss error -- difference from the system’s desired value

Control Performance, P-type

Kp = 20

Kp = 200

Kp = 50

Kp = 500

Steady-state Errors, P-type

Kp = 200 Kp = 50

Control Performance, PI - type

Kp = 100

Ki = 50 Ki = 200

You’ve been integrated...

Kp = 100

instability & oscillation

Control Performance, PID-typeKp = 100 Ki = 200 Kd = 2

Kd = 10 Kd = 20

Kd = 5

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., 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.

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

Ziegler-Nichols Tuning for second or higher order systems

for P control for PI control for PID control

Kp = 0.6 K

Ki = 2.0 / P

Kd = P / 8.0

Kp = 0.45 K

Ki = 1.2 / P

Kp = 0.5 K

Ziegler-Nichols Rules for Tuning PID Controller:

Implementing PID

Use discrete approximations to the I and D terms:

• Proportional term: ei = desired - actual

• Integral term: ei

at time i

i=0

i=now

• 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

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 (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.

Aliasing

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).

2/sBL ss T/2

BLfBLf2

BLf2

BLf

sf

2/sf2/sNYQ ff

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.pitsco-

legodacta.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

Assignment Refresh you memory

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