9- 104/19/23

©T.C. Chang

Chapter 9. FUNDAMENTALS OFNUMERICAL CONTROL

HARDWARE AND INTERPOLATORS

Dr. T. C. ChangSchool of Industrial EngineeringPurdue University

9- 204/19/23

©T.C. Chang

HISTORICAL DEVELOPMENT

• 15th century - machining metal.

• 18th century - industrialization, production-type machine tools.

• 20th century - F.W. Taylor - tool metal - HSS

Automated production equipment -

Screw machines

Transfer lines

Assembly lines

...

using cams and preset stops

Programmable automation -

NC

PLC

Robots

9- 304/19/23

©T.C. Chang

A 3-AXIS MACHINING CENTER

9- 404/19/23

©T.C. Chang

HEXAPOD 6-AXIS MACHINES

A Giddings & Lewis Hexapod machine

Another hexapod configuration

9- 504/19/23

©T.C. Chang

NEW NCS

high speed spindle (> 20,000 rpm)

high feed rate drive ( > 600 ipm)

high precision ( < 0.0001" accuracy)

9- 604/19/23

©T.C. Chang

NC MACHINES• Computer control

• Servo axis control

• Tool changers

• Pallet changers

• On-machine programming

• Data communication

• Graphical interface

MCUMachineTool

CLUDPU

MCU - Machine control unit

CLU - Control-loops unit

DPU - Data processing unit

9- 704/19/23

©T.C. Chang

NC MOTION-CONTROLN C Pr o g r am

Ex e c u t io nSy s t e m

In t e rp o la t o r &S e r v o - c o n t r o lM e c h a n is m

Co n t ro l L o g ic

L in e ar M o t io n

P o w e r

T r a n s l a t o r

Re la y

S o le n o id

Co m m an d sD im e n s i o n s

9- 804/19/23

©T.C. Chang

NC MACHINE CLASSIFICATIONS

1. Motion control: point to point (PTP) and continuous (contouring) path

2. Control loops: open loop and closed loop

3. Power drives: hydraulic, electric, or pneumatic

4. Positioning systems: incremental and absolute positioning

5. Hardwired NC and softwired Computer Numerical Control (CNC)

9- 904/19/23

©T.C. Chang

POINT TO POINT• Moving at maximum rate from point to point.

• Accuracy of the destination is important but not the path.

• Drilling is a good application.

9- 1004/19/23

©T.C. Chang

CONTINUOUS PATH• Controls both the displacement and the velocity.

• Machining profiles.

• Precise control.

• Use linear and circular interpolators.

9- 1104/19/23

©T.C. Chang

MAJOR COMPONENTS OF AN NC MACHINE TOOL

Magnetics control cabinet

Controller

Servo drive

Machine tablePosition transducer

Leadscrew

Gear box

Tachometer

Motor

9- 1204/19/23

©T.C. Chang

BALL-BEARING-NUT LEADSCREW

Precision ground screw

Preloaded ball-bearing nut

9- 1304/19/23

©T.C. Chang

MACHINE BED

Linear ways

Leadscrew

Bearing

9- 1404/19/23

©T.C. Chang

TOOL CHANGE

Tool changearm

ToolSpindle with a tool

9- 1504/19/23

©T.C. Chang

5-AXIS MACHINE

Tool

Workpiece

9- 1604/19/23

©T.C. Chang

NC MACHINE RATING

Accuracy

Repeatability

Spindle and axis motor horsepower

Number of controlled axes

Dimension of workspace

Features of the machine and the controller.

9- 1704/19/23

©T.C. Chang

NC ACCURACY AND REPEATABILITY

• Accuracy = control instrumentation resolution and hardware accuracy.

• Control resolution: the minimum length distinguishable by the control unit (BLU).

• Hardware inaccuracies are caused by physical machine errors.

9- 1804/19/23

©T.C. Chang

HARDWARE INACCURACIES

Component tolerances: inaccuracies in the machine elements, machine-tool assembly errors, spindle runout, and leadscrew backlash.

Machine operation: Tool deflection (a function of the cutting force), produces dimensional error and chatter marks on the finished part.

Thermal error: heat generated by the motor operation, cutting process, friction on the ways and bearings, etc. Use cutting fluids, locating drive motors away from the center of a machine, and reducing friction from the ways and bearings

9- 1904/19/23

©T.C. Chang

REPEATABILITY

Avg. error

Programmed position

Test result

Repeatability

9- 2004/19/23

©T.C. Chang

PRIME MOVERS

Hydraulic power drive. Advantages:

Large torques and fast responses.

Large power/size ratio.

Use for large and heavy-duty machines.

Disadvantages:

High cost.

Additional peripherals.

Noise.

Response lag due to hydraulic fluid viscosity.

Contamination from leaking fluid.

9- 2104/19/23

©T.C. Chang

A HYDRAULIC POWER DRIVE

DC Motor

Hydraulic pump

Hydraulic Motor

Sump

High pressure line

Low pressure lineReturn

to leadscrewServo Valve

Signal from NC controller

9- 2204/19/23

©T.C. Chang

HYDRAULIC POWER DRIVE

The servo valve output flow rate: q = k V where: q : output flow rate, in3/s k : valve constant V : signal voltage, voltThe power of the motor: p q = T where: p : input pressure, psi q : input flow rate, in3/s T : output torque, in lb : angular speed, rad/s

v

m

m

v

9- 2304/19/23

©T.C. Chang

HYDRAULIC DRIVE (CONTINUE)

The steady state rotational speed of the motor:

= K q

where:

K: motor constant

9- 2404/19/23

©T.C. Chang

ELECTRIC POWER DRIVESStepper motors:

Rotates in angular increments.

Used in NC, robots, printers, plotters, VCRs, cameras, etc.

Rating: torque, from 1 oz-in to several hp.

step angle: 0.72° to 90°; 1.8°, 7.5°, and 15° are most popular.

For each input pulse (signal), the motor shaft advance one step.

9- 2504/19/23

©T.C. Chang

STEPPER MOTOR PRINCIPLES• A simple stepper motor with two pairs of stator

electrical magnets and a single permanent magnet rotor. The motor has four steps per revolution.

Coil set 2 is energized, rotor is attracted to the vertical position.

Turn off coil set 2 and turn on coil set 1. The rotor rotate to a horizontal position.

9- 2604/19/23

©T.C. Chang

HALF STEPPING AND MICRO-STEPPING

• The control shown in the previous slide is for full step control. With smart control, the same motor can have higher step resolution. Following are two strategies used in motor control.

Half stepping: turn both sets of coil on. Rotor rotates at 45 degree.

Micro stepping: turn the coil on at different power level. The rotor rotates proportional to the strength of the fields.

9- 2704/19/23

©T.C. Chang

A PRACTICAL STEPPER MOTR

• To increase the resolution of a stepper motor, more poles are added to the rotor. By doing so, smaller stepping steps can be made.

9- 2804/19/23

©T.C. Chang

BIPOLAR AND UNIPOLAR MOTORS

The coils take bi-directional current to control the motor rotation.

Each coil only takes current in the fixed direction. Often there is a center power lead for each set of the coils (see the motor drawing on the next page.)

9- 2904/19/23

©T.C. Chang

STEPPER MOTOR AND DRIVER

Stepper Motor

Allegro MicroSystems

UCN5804B BiMOS II Unipolar Stepper-motor Translator/Driver

Unipolar winding, 6 leads. Center leads are for power.

9- 3004/19/23

©T.C. Chang

STEPPER MOTOR CONTROL

Step S1 S2 S3 S4

1 1 0 1 0

2 1 0 0 1

3 0 1 0 1

4 0 1 1 0

1 1 0 1 0

Stepping motorSignal leads

s1 s2

s4s3

power ground

Clockwise stepping

Counterclockwisestepping

Rotation controlled by pulse sequence on the signal leads (connect to ground).

9- 3104/19/23

©T.C. Chang

EXAMPLETurn a 1.8° step angle motor 2000 steps at 360 rpm,what is the number of pulses and pulse rateto be sent to the motor?

The number of pulses should be the same as thedesired steps. It, therefore, is 2000 pulses.

360 rpm = 360 (rotation/min) / 60 (second/min) = 6 rotation/second

Number of steps per rotation, N:

N = 360°/1.8° = 200 steps/rotation

Pulse rate = 6 (rotation/second) x 200 (steps/revolution) = 1,200 pulses/second

9- 3204/19/23

©T.C. Chang

STEPPING MOTORS (continue)Advantages:

Digital input.

Accurate positioning with non- cumulative errors.

simple and rugged construction.

Bi-directional rotation and control with no additional control complexity.

Disadvantages:

Loss of synchronization at certain operating range.

9- 3304/19/23

©T.C. Chang

TORQUE VS STEPPING SPEED

Each motor has its own torque characteristics. However, for all motors, torque drops at higher speed.

Torque

Stepping speed (steps/second)

9- 3404/19/23

©T.C. Chang

MOTOR SIZING

• Select motor torque requirement• Torque = Fxr where F: load force, r: radius of arm

r

F

IWr

2

2 For a disc, I: moment of Inertia lb-in2 , w: weight

For a cylinder

Torque required:

T It

2180

1

240

: step angle (degree)

: step rate (steps/second)

Io: total inertia (motor+load) lb.in2

T: oz. in

IW

r r 2 1

222( )

TI

24: angular acceleration, rad/sec2

9- 3504/19/23

©T.C. Chang

A 3-AXIS NC SYSTEM USING STEPPER MOTORS

9- 3604/19/23

©T.C. Chang

SERVO MOTORSMotor controlled using a feedback mechanism. A transducer feedback and a speed control forms a servo loop.

DC & AC servos.

DC: speed controlled by voltage

AC: speed controlled by frequency

+

_

Commutation bars

Carbon brushes

Rotor

A DC permanent magnet motor

+

ShaftDC Motor

Tacho- meter

Differential amplifier

Feedback

Velocity command

A DC servo motor system

9- 3704/19/23

©T.C. Chang

DC MOTORS

• DC brush motor and DC brushless motor.• DC brush motor is most popular and easy to

control. – Three coil windings on the rotor and two permanent magnets

on the stator.– Coil c1 is energized. The magnetic field pull it toward the

permanent magnet pole.– Immediate after it lined up with the permanent magnet, the

power is disconnected and coil c2 is energized. – Continue these steps, the rotor will keep turning.– The connection of power to the coil at the right time is called

“commutation”.– Commutation is achieved by commutation bars and two carbon

brushes. Rotor S

9- 3804/19/23

©T.C. Chang

COMMUTATORS

Coils

Six commutator bars for three sets of coils.

Brush

9- 3904/19/23

©T.C. Chang

BUILD YOUR OWN MOTORFollowing is a motor with one coil winding (Figure 1). It needs a gentle push to start the rotation.

Use coated transformer wire to loop around a AA battery about 10 times. Leave 2” leads on either end. Use scotch tape to keep the loop together. It is important that the leads come out of the center of the loop (see figure 2), so the rotor will be balanced.

The coil is turned on for half of the rotation (see figure 3)

AA battery

Connector (paper clip)

Permanent magnet

Rotor coil

Rotor coil

Remove coating completely on this end of the lead.

Sand away only the top half of the coating.

Figure 1

Figure 2

Figure 3

Commutation is done half the rotation when the exposed wire has contact with the connector.

9- 4004/19/23

©T.C. Chang

BRUSHLESS DC MOTOR

• A brushless motor has windings on the stators and the rotor is made of permanent magnet.

• It requires electronic commutation.

• The position of the rotor is detected by sensors. The stator is activated in sequence by the controller based on the rotor position.

• Unlike stepper motor, brushless motor makes continuous rotation instead of stepping.

• More complex control, smooth rotation.

9- 4104/19/23

©T.C. Chang

AC MOTORS

• Induction motor– Squirrel cage induction motor

– Wire-wound induction motor

• Magnetic field from the stator is induced into a conductor on the rotor. This induced current establishes a magnetic field around the rotor's conductor in the opposite polarity.

• The stator does not move. However, the alternative current generate rotating magnetic field.

9- 4204/19/23

©T.C. Chang

SQUIRREL CAGE ROTOR

AC current induced into the rotor by the fields on the stator. The current in the rotor generates a magnetic field which pulls the rotor to follow the changing stator field (due to AC).

9- 4304/19/23

©T.C. Chang

AC MOTOR PRINCIPLES

Two-phase voltage for developing a magnetic field in a stator

A single phase motor

(use a capacity to provide the starting power)

Three-phase motor is omitted from this page.

9- 4404/19/23

©T.C. Chang

HOW DOES AC MOTOR WORK

9- 4504/19/23

©T.C. Chang

OTHER AC MOTOR TYPES

• Wound rotor motors: Instead of using induction, the rotor consists of electric magnets. Slip rings are used to pass the AC current to the rotor windings. For high power and low rpm motors.

• Synchronous motors: designed to run at a constant speed. Modified squirrel cage design to enable the rotor to lock onto the rotating magnetic field of the stator. Use for very large industrial applications, or low power applications in clocks and timing devices.

• Universal motors: Use commutator. Can use both AC and DC power. Vacuum cleaners, food mixers, portable drills, etc. Low power.

9- 4604/19/23

©T.C. Chang

AC MOTOR CONTROL

• AC motor is controlled by the input voltage frequency. The rotor is usually lagging behind the changing magnetic field in the stator (slew). Based on the design (number of poles) each motor is rotated at a multiple of the power frequency.

• Motor speed = AC frequency/# of poles in the stators * 120

• It is more difficult to change the frequency of high current power than changing voltage as it is in DC motor control. AC motor control thus costs more.

9- 4704/19/23

©T.C. Chang

TRANSDUCERS

A transducer is a device which transforms one physical phenomenon to another.

Speed transducer

Tachometer, which is a DC generator.

where v = output voltage, volts

= shaft angular speed, rad/s

= tachometer constant, volt/rad

v = k t

k t

9- 4804/19/23

©T.C. Chang

TRANSDUCERS (continue)

f = K

where

f = output pulse frequency, pulse/s = input angular speed, rpm K = encoder gain

e

e

Position transducer:

Encoders - digital

Encoder disk

Output

Photoelectric sensor

Schmitt-triggerPhoto diode

time

Incremental encoder: angle rotated (in pulses)

Absolute encoder: shaft rotation angle (in a binary number)

9- 4904/19/23

©T.C. Chang

TRANSDUCERS (continue)

The output of a resolver is:

v = V sin( t + )

where

V = input voltage, volt = shaft angle t = input signal phase

Resolvers

Input

Output

Phase difference

Rotation angle is measuredby the phase change.

9- 5004/19/23

©T.C. Chang

TRANSDUCERS (continue)LVDTs. Linear Variable Differential Transformer

Input

output

ACoutput

ACExcitation

armature(moveable magnetic core)

sine wave

The output shows the differencebetween two coils. When the armatureis in the central position, the outputis zero.

Range from micro inch to inches.Repeatability in a few micro inches.

9- 5104/19/23

©T.C. Chang

TRANSDUCERS (continue)Inductosyns. Trade name by Farrand Controls.

Attached to the machine bed and the table, provides direct reading of the position, either linear or rotational (two different configurations).

9- 5204/19/23

©T.C. Chang

LEADSCREW CALCULATION

• Axial force (lb)

p: leadscrew pitch

Eff: screw efficiency, Ball screw: 85-95%, ACME screw: 35-45%

• Inertia

F T p eff 2

16

I D lengthscrew 4 0 028. For steel

9- 5304/19/23

©T.C. Chang

LEADSCREWS

Leadscrew

Pitch

Nut

Converting the rotational motion of the motors to a linear motion.

• pitch (p): the distance between adjacent screw threads

• the number of teeth per inch (n):

• n = 1 / p

• BLU: Basic Length Unit (machine resolution)

• BLU = p / N

• e.g. an NC machine uses a 0.1" pitch leadscrew and a 100 pulse/rev encoder.

• BLU = p / N = 0.1 (in/rev) /100 (pulses/rev) = 0.001"

9- 5404/19/23

©T.C. Chang

CONTROL LOOPSOpen loop - No position feedback.

Use stepping motor.

A machine has 1 BLU = 0.001".To move the table 5" on X axis at a speed (feed rate) of 6 ipm.

pulse rate = speed/BLU = 6 ipm/0.001 ipp = 6,000 pulse/min

pulse count = distance/BLU = 5/0.001 = 5,000 pulses

motor

table

pulses

9- 5504/19/23

©T.C. Chang

CLOSED LOOP

Reference pulses

+ DC Motor

Tacho- meter

Differential amplifier

EncoderUp-down counter AmpDAC

_

+

Shaft

Closed-loop control mechanism

9- 5604/19/23

©T.C. Chang

INTERPOLATION

Control multiple axes simultaneously to move on a line, a circle, or a curve.

(3,2)

(10,5)

X

Y

Point-to-point control path

(3,2)

(10,5)

X

Y

Linear path

Vy =6

(5-2)

(10-3)2+ (5-2)2= 6

3

49+ 9= 2.3635

Vx =6

(10-3)

(10-3)2+ (5-2)2= 6

7

49+ 9= 5.5149

9- 5704/19/23

©T.C. Chang

INTERPOLATORS

Most common interpolators are: linear and circular

Since interpolation is right above the servo level, speed is critical, and the process must not involve excessive computation.

Traditional NC interpolators: Digital Differential Analyzer (DDA)

Higher order curves, such as Bezier's curve, use off-line approximation algorithms to break the curves into linear or circular segments.

9- 5804/19/23

©T.C. Chang

A DDA

Accumulator

Adder

Register (a)

f

output

²t

fr

Each time a pulse is received, the value of the register(a value) is added to the accumulator. The overflowbit of the accumulator is output to the motor control.

fr = a f2

N N: accumulator width, bit

9- 5904/19/23

©T.C. Chang

LINEAR INTERPOLATORUp- down counter

DDA Xfr

Up- down counter

DDA Yfr

f

X axis

Y axis

DDA ffc

A two axis control

f = af fc2

Nf

fr = af fc2

Nf a

2 N

= a af 2

(Nf + N) fc

Feedrate control

Output to axis control

x

y

9- 6004/19/23

©T.C. Chang

LINEAR INTERPOLATOR (continue)

Since feedrate is the linear speed, how to convert itin Vx and Vy without using a computer?

frx = Vfx

x2 + y2

ax af 2

(Nf + N) fc = Vf

x

x2 + y2

af 2

(Nf + N) fc = Vf

x2 + y2

Set ax to x (ay = y)

af = Vf

x2 + y 2 2

( Nf + N)

fc

is a constant based on the hardware design

2 (Nf + N)

fc

af =

AVf

x2+y2

This is called inversed timecode.

A value is usually 10.

9- 6104/19/23

©T.C. Chang

EXAMPLE

X

Y

BLU

0 000 0 000 01 100 0 011 02 000 1 110 03 100 1 001 14 000 2 100 15 100 2 111 16 000 3 010 27 100 3 101 28 000 4 000 3

clock X X counter Y Y counter

N = 3dX = 4 BLUdY = 3 BLU

Speed controlledby the clock rate.

9- 6204/19/23

©T.C. Chang

CIRCULAR INTERPOLATOR

Figure 9.28. A circular arc

P0P1

P2

x - xo

X

Y

Vf

R

Vf =R

ddt

x = R cos + x 0

y = R sin + y0

R cos = x – x 0

R sin = y – y 0

dxdt

= – R sinddt

= – (y -y0)ddt

dydt

= R cosddt

= (x - x0)ddt

9- 6304/19/23

©T.C. Chang

CIRCULAR INTERPOLATOR (continue)

to X counter

to Y counter

+

+

+

-

fDDA X

DDA Y

d2x

dt2= – R cos

ddt

ddt

= –dydt

ddt

d2y

dt2= – R sin

ddt

ddt

=dxdt

ddt

af =Vf

R

2(Nf + N)

fc

=10 Vf

R

fc

fx

fy

dx/dt

dy/dt

9- 6404/19/23

©T.C. Chang

Next Generation Controller

• US Air Force program

• Open architecture controller for all manufacturing applications.

workstation

machine tool task

robottask

conveyortask

machinetool motion

m/c toolapplication

axis servo

Workstation

Task

E-move

Primitive

Servo

NML

NML

NML

NML

conveyorservo

NML: Neutral manufacturing language

9- 6504/19/23

©T.C. Chang

Future Controllers

• Open architecture– Standard hardware platform, plug-and-play

– Modular software, custom features

9- 6604/19/23

©T.C. Chang

5-AXIS MACHINE CONFIGURATIONS

Rotational axes on the spindle

Rotational axes on spindle and the table

Rotational axes on the table

9- 6704/19/23

©T.C. Chang

MACHINE KINEMATIC MODELS

A three axis machine. Set three coordinate systems on the machine.

1. Spindle (Z)

2. Machine bed (Y),

3.machine table (X),

4. Workpiece

Tip to spindle (l: tool length)

T l

l

tz ( )

L

N

MMMM

O

Q

PPPP

1 0 0 0

0 1 0 0

0 0 1 0

0 0 1

Spindle to bed (z: z axis motion)

T z

a b c z

zy

y y y

( )

L

N

MMMM

O

Q

PPPP

1 0 0 0

0 1 0 0

0 0 1 0

1

Bed to table (y: y axis motion)

T y

a b y c

yx

x x x

( )

L

N

MMMM

O

Q

PPPP

1 0 0 0

0 1 0 0

0 0 1 0

1

Table to workpiece (x: x axis motion)

T x

a x b c

xw

z z z

( )

L

N

MMMM

O

Q

PPPP

1 0 0 0

0 1 0 0

0 0 1 0

1

Please pay attention to the sign of a, b and c. Depending on the design, some of them are positive and some negative.

9- 6804/19/23

©T.C. Chang

3-AXIS MACHINE MODELTool tip to workpiece

p x y z l T T T T

a a a x b b b y c c c l z

w tz zy yx xw

x y z x y z x y z

( , , , )

L

N

MMMM

O

Q

PPPP

0 0 0 1

1 0 0 0

0 1 0 0

0 0 1 0

1

Where ax, ay, az, …, cz are design variables (relative position between each reference points at the home position), l is the tool length. In a floating zero machine, when the set zero button is pushed, these values get cancelled out (compensated). Therefore, the final equation is:

p x y z l

x y z

w ( , , , )

L

N

MMMM

O

Q

PPPP

1 0 0 0

0 1 0 0

0 0 1 0

1

The tool tip position in the workpiece coordinate system is the result of X,Y,Z motion only.

9- 6904/19/23

©T.C. Chang

5-AXIS MACHINE

Two rotational axes on the spindle. The rest of the machine is the same as a 3-axis machine

Tip in the B axis: [0 0 -l 1]

B to C

Tbc ( )

cos sin

sin cos

L

N

MMMM

O

Q

PPPP

0 0

0 1 0 0

0 0

0 0 0 1

C to spindle

T

z z z

cz

a b c

( )

cos sin

sin cos

L

N

MMMM

O

Q

PPPP

0 0

0 0

0 0 1 0

1

Tool tip in the workpiece coordinate system

p x y z l l T T T T Tw bc cz zy yx xw( , , , , , ) 0 0 1

9- 7004/19/23

©T.C. Chang

5-AXIS MACHINE

• The equation presented in the previous slide need to be changed based on the machine configuration.

• Given the same programmed x,y,z position, when the rotational angles are changed, the tool tip will change its location and orientation.