Motion Control Theory. servomotor

Motion Control Theory

Kimberly-Clark GNW & HC Electrical Forum

Topics

• Closed Loop System

• Servo Motors

• Servo Drives

• Mechanical Gearing

• Feedback Types

• Motion Controllers

• Torque

• Inertia

• Design Considerations

Closed Loop

System

• Accuracy – This term defines the ability of the controlled

axes to position an object in a spatial domain. (how close you can come to the bulls-eye). Another definition is the ability to reach a predetermined point in space.

• Resolution – Resolution is defined as resolving by the

breaking into parts. • The action of a rotary or linear feedback device

used for control purposes.

• Repeatability – The ability to exactly replicate or reproduce a

motion profile as a continuous operation.

– This term defines the ability of the controlled

axes to position the controlled object after

several moves

Definitions

• Position Error – The dynamic difference between the position commanded by the controller

and the actual position of the object being moved.

• Position Loop – The controller algorithm correcting the difference between the controller

commanded position and the feedback from the controlled axes used to determine actual position

• Speed or Velocity Loop – This loop is typically located in a drive. The function of the loop is to output

the motor torque required to maintain the speed/velocity commanded to the drive amplifier.

• Torque (current) Loop – The loop in the drive amplifier that is responsible for controlling the torque

producing current by comparing the motor actual shaft position’ (incremental shaft position is integrated within the controller to determine the shaft speed/velocity) to the actual shaft position.

Definitions

When is Motion Control Required?

• Precise Control of Position

– Rule of Thumb: Accuracy of motor shaft less than

5 degrees or linear movement less than .0001”.

• Precise Control of Speed

– Rule of Thumb: Speed regulation of .05% or better

• Rapid Acceleration and Deceleration Requirements

– Rule of Thumb: Motor acceleration from 0 to 2000 rpm in

less than .5 sec.

• Control of Torque

– Ability to provide full torque at 0 rpm.

Speed Feedback

Example – Closed Loop Control System

Movement

Controller AC/DC

Servo Amp

0 to ±10V DC

Position Loop Velocity

Encoder

Ballscrew

Reducer

Motor Power

What is a Servo?

• A Servo is a control system which, through the use of feedback or internal control,

has the capability for accurate and repeatable control of one or more of the following

dynamic parameters…

– Position

– Velocity

– Torque

• A Servo Axis is a principal direction along which motion occurs. The machine

hardware (mechanics) that make up that movement.

ROTARY

LINEAR

Ballscrew

Ballnut

Gearbox Pulleys

Components of a Motion Control System

Software

Motion

Controller

Drive/

Amplifier

Actuator/

Mechanical

Linkages

And Load

Feedback

Transducer

Servo Axis

Motors

windings

Air gap

Magnet

DC Motor Limitations

• Limitation - Heat in a brush type DC motor must be conducted from

the rotor windings to the case of the motor.

• Limitation to how fast a motor can be run using mechanical

commutation...

Commutator

Arcing

Permanent

Magnets

Windings

Servo Motors - Theory

• A brushless servo motor locates the permanent magnets on

the rotor and the windings on the stator...

Heat generated in

windings is directly

conducted to motor

Winding is electronically

commutated.

• Brushless motor design reduces the problems of heat

dissipation and commutation induced speed limits.

Magnet material

on rotor is lighter

than copper wire

windings.

• Also, the rotor inertia is reduced because heavy copper wire

windings are not required.

Performance Limitations

• Windings in the stator

– Better heat dissipation

– Higher continuous torque

– Magnets on the rotor

• Mechanical commutator

replaced by electronic

commutator in the amplifier

– No Commutation Limit

– No speed limit due to arc-

over Continuous Duty Zone

Intermittent Duty Zone

Magnets

2 4 6 8 10 12

De-magnetizing force - “H” (Kilo Oersteds)

Flux Density - “B”

(Kilo Gauss)

Servo Motors - Properties

• Magnet material properties

Rare Earth vs. Ferrite Motors

• Servo motor magnets primarily fall into two classes:

– Rare Earth Magnets (Samarium Cobalt or Neodymium-Iron-Boron)

– Ceramic Magnets (Ferrite)

• Rare Earth Motors fall into two categories:

– Medium Inertia Motors

– Low Inertia Motors

• Rare Earth Magnets considerations.

– Lower inertia means higher theoretical acceleration • Stability and Bandwidth become an important consideration

– Proper System inertia sizing becomes very critical • Do not exceed 6 to 1 system to rotor inertia (3 to 1 in critical/contouring applications)

when using standard encoder or resolver feedback

• Rare Earth Advantages:

– Excellent magnetic properties (up to 30 MGOe).

– Lower rotor inertia.

– Smaller motor sizes for a given torque

– Effective where high response speed, quick acceleration, high efficiency and small size are required.

• Rare Earth Disadvantages:

– Magnetic material is expensive.

– Samarium Cobalt and Neodymium material has a limited supply (only available in a few areas of the world).

– The system inertia must also be low (within at least 6 to 1 of motor inertia). BEWARE! BEWARE! Inertia mismatch means stability and control problems. System sizing is very critical. (High resoultion feedback can help expand the range)

• Ferrite Motor Advantages:

– Low cost magnetic material

– Virtually unlimited availability

– May be easier to match the motor to a system due to inertia.

• Ferrite Motor Disadvantages:

– Inferior magnetic properties (~4 MGOe). More material required to

provide the same flux.

– Larger frame size and higher inertia per given torque.

• Torque to Rotor Inertia Comparison (Rule of Thumb)

– The Rare-Earths’ ratio of Torque to Rotor Inertia is about 4 times

higher than the standard Ferrite motor.

Servo Drives

Power from

DC Bus

200 Volts

20 Volts

Transistor is pulsed

on and off - low

power dissipation.

10 Amps

Transistor Power Dissipation = Much Lower vs Linear supply!!

Servo Drives - Theory

• PWM drive operation - transistor is pulsed on and off

Positive

Negative

Period Period

50% 50% Positive

Negative

Motor Current PWM Voltage

• Pulse width modulation - zero current:

Positive

Negative

Period Period

75% 25% Positive

Negative

• Pulse width modulation - positive current:

Positive

Negative

Period Period

25% 75% Positive

Negative

• Pulse width modulation - negative current:

Reference

Saw Tooth

Carrier

Reference

PWM Output

Comparator

• Pulse width modulation - pulse generation:

Voltage

Current

• Pulse width modulation is possible because the

inductance of the motor has a smoothing effect on

the pulses...

What makes a Servo Drive Unique?

• Operation without cogging to zero rpm

• Controls output Torque to 0 rpm

• Always use Motor feedback (velocity and commutation)

• Speed Regulation < 0.1% standard with 100% torque disturbance

• Extreme stiffness to Transient Loading

• Velocity Loop Bandwidth > 100 Rads/sec (16Hz)

– Motor not included

– Digital drive has 40+ Hz typical

• Use special permanent magnet servomotors

• Current limit and bandwidth control standard

• External control/sequencing circuits required

Trapezoidal

Sinusoidal

• There are two common methods of servo drive commutation:

Trapezoidal

Sinusoidal

• There are two common methods of servo drive commutation:

– Trapezoidal

– Sinsoidal

• “Trapezoidal” and “Sinusoidal” refer to the shape of the voltage

waveforms that the amplifier generates...

Trapezoidal Commutation-

Requires only hall effect

feedback of rotor position

Sinusoidal Commutation–

Requires encoder or

resolver feedback of rotor

position

• “Trapezoidal” and “Sinusoidal” commutation

Torque

Trapezoidal commutation

- Torque ripple

- Only hall effect feedback

required

Sinusoidal commutation

- Minimal torque ripple

- Encoder or resolver

feedback required

10-15%

Mechanical

Gearing

Motion

Dictionary

Definitions

• Backlash

– The relative movement between interacting mechanical

parts resulting from looseness.

• Preload:

– The process of forcing interacting mechanical parts

together to eliminate backlash.

• Angular measurement:

– 60 Arc-minutes = 1 degree of rotation

– 3600 Arc-seconds = 1 degree of rotation

Backlash

• A relative movement between interacting mechanical parts,

resulting from looseness when motion is reversed.

Backlash in

tooth profiles

Gearing

• Types

– Gearbox, Belt and Pulley, Gear Mechanism

• Why?

– Torque Increaser (Ratio), Tout = Tin x Z

– Speed Reducer (Ratio), nout = nin x Z

– Inertia Reducer (Ratio2) , jref = jload / Z2

Output

(Application)

(Servomotor)

Gear Reduction

Gearing Advantages

• Most Motion Control Applications require some form of

reduction - Approx. 70%

• Torque Multiplication

• Inertia Matching

• Increased Stiffness

• Increased Resolution

• Speed Reduction

• Utilize Full Motor Characteristics

• What is the machine controlling? – Is the machine moving material in space where there is no

concern about path accuracy? • In this case torque is the primary concern.

• Why am I concerned about inertia mismatch?

– What is the customers primary concern? • An application that resists external disturbances during a move

– select a motor with higher inertia

• Not paying for KW that is simply used to get the motor moving

– This was one of the first criteria in the development of rare earth motors

• (high torque and very low inertia)

Application Considerations

Gearing Drawbacks

• Can add backlash to the system

• Reduces system efficiency

• Reduces output speed

– requires higher motor speed

• Can increase audible noise

• Increases cost

Speed Reduction

• Output speed equals input speed divided by the gearbox

ratio: Vo = VM/Z

– Where: • VM = Motor Speed(RPM) = 500 RPM

• VO = Output Speed(RPM) = 50 RPM

• Z = Gearbox Ratio = 10:1

Torque Multiplication

• Output Torque equals motor torque times the gearbox ratio

times the gearbox efficiency: TO = TM x Z x e

– Where: • TO = Output Torque(In-Lb) = 90 In-Lb

• TM = Motor Torque(In-Lb) = 10 In-Lb

• e = Gearbox Efficiency(%) = 90 %

• You cannot exceed the gearbox output torque specification!

Motor - coupled to

input of speed

reducer.

Input gear

Output gear - twice

the size of input gear

so torque output is

doubled and speed

is halved.

Torque Multiplication

• 2:1 Speed Reducer

Inertia Matching

• The reflected inertia as seen by the motor is equal to the total

system inertia divided by the square of the gearbox ratio:

• JREF =JL/Z2

– Where: • JL = System inertia(In-Lb-Sec2) = 2 In-Lb-Sec2

• JREF = Reflected inertia(In-Lb-Sec2) = 0.5 In-Lb-Sec2

Typical Gearing Technologies

• Parallel Axis Gearing

– Spur/Helical

– Planetary

• Non-Parallel Axis Gearing

– Worm

– Bevel

• Other

– Belt Driven

Spur/Helical Gear Technology

• Moderate Backlash

• High Stiffness

• Very Smooth Operation

• 96-98% Efficient Per Pass

• High Input Speeds

• Low Inertia

• Excellent back-drive capability

Helical

Planetary Gear Technology

• Low Backlash

• Very High Stiffness

• Smooth Output Torque

• 90-95% Efficient Per Pass

• Input Speeds < 3000 Rpm

• Low Inertia

• Limit to gear ratios without

staging planetary

Worm/Bevel Technology

• Worm Gear

– Low Backlash

– Very High Stiffness

– 40% Efficient, Lower At Start-Up

– Moderate Input Speeds, Frictional

– Low Inertia

– Poor Back-Drive Capability

• Bevel Gear

– Spur gear technology but right angle

– Right Angle Torque Transmission

– Moderate Backlash

– High Stiffness

– 96-98% Efficient

– High Input Speed

– Excellent Back Drive Capability

Belt Driven Technology

• Belt and Pulley for Gear Ratio

• High Efficiency - 90%

• Minimal mechanical noise

• Long life

• Potential for stretch

• Potential for slip if positive drive

isn’t used

Gearhead Sizing

• When selecting or sizing the correct gearhead for an

application,

– It is important to first determine whether the application is

of a cyclic or continuous operation.

– Whether the application is cyclic or continuous

determines which of a gearhead’s speed and torque

ratings should be used for proper sizing and selection of

a gearhead (maximum vs nominal ratings).

Feedback

Devices

Servo Axis

Feedback Position/ Velocity

Motion Control Drive

Position/ Velocity Control

Feedback Devices

• Basic feedback loops:

– Current / Commutation Loop

• Motor Based Feedback Provides

– Commutation Feedback (motor rotor position for AC control)

– Can also be used for Velocity & Position Loop feedback

– Velocity / Position Loop

• Velocity Loop controls speed of motor

• Position Loop controls linear or rotary machine position

• Velocity Loop could be performed by the Drive

Current/ Commutation

Velocity/Position Loop

Feedback Devices

• Sensor devices mounted to the actuator (motor) or

load that detect speed and/or position.

Pulses Indicate Position

Pulses/Sec Indicate Speed

Voltage

TimeEncoder

Output

Feedback Devices

• Encoder

Incremental Encoder Output

• Electrical Characteristics

– TTL (5V) or CMOS (12v)

– Single Ended

– Complementary

– Differential Line Driver

• Differential Line Driver

(A+Noise)-(-A+Noise)=A+Noise+A-Noise=2A

• Detecting Rotation Direction

Channel A

Channel B

Marker

Channel A

Channel B

Marker

Clockwise

Rotation

Counterclockwise

Rotation

• Line Count Multiplication

Channel A

Channel B

1X Multiply

2X Multiply

4X Multiply

Resolvers

• What Are They?

• Rotary transformers

– Single rotating winding (rotor)

– Two stationary windings (stators)

– Coupling between rotor and stators varies with shaft

• Brushless transformer couples signal to rotor

• Provide absolute position over one revolution

• There are 2 types of Resolvers

– Transmitter • Transmitter is excited at the rotor signal.

• Position information is read at the SIN and COSINE windings

– Receiver • Receiver is excited at the SIN and COSINE windings and the absolute

position is read from the rotor winding

• Functional Differences

– Transmitter is more noise immune and requires less

hardware to support.

Resolvers

Circular Transformer

Two Phase Stator

Sinusoidal

Output

Sinusoidal

Output

Resolvers

• Resolver - (Transmitter style)

Winding A

(Sine Winding) Winding B

(Cosine Winding)

Resolvers

• Resolver Operation - Concepts

Signal sent to rotor:

Signal on Cosine

winding:

Signal on

Sine winding:

0 Degree Position

Resolvers

Signal on Cosine

winding:

Signal on

Sine winding:

20 Degree Position

Resolvers

Signal on Cosine

winding:

Signal on

Sine winding:

45 Degree Position

Resolvers

Signal on Cosine

winding:

Signal on

Sine winding:

90 Degree Position

Resolvers

Rotor Signal

Cosine Output

Sine Output

0o 0o 90o 180o 180o 270o 360o

Electronics looks

at portion of signal

indicated by green

lines.

Resolvers

• Resolver Operation Concept - 360o Rotation

Sine value

Cosine value

Sine Output

Cosine Output

Values are measured by a

comparator and compared to

the reference signal.

0o 90o 180o 270o 360o

Resolvers

Sine value

Cosine value

Sine Output

Cosine Output

0o 90o 180o 270o 360o

Resolvers tell you where you are within

1 shaft rotation (Even immediately

after power up).

Resolvers

Excitation

Oscillator

Tracking

Converter

Binary Angle

Output

Resolver

Resolver - Interface

• Tracking Resolver-to-Digital (R/D)

Converter

• Digital counter tracks the Resolver

position

• Counter provides parallel binary

output representing absolute position

• Up to 16 bit resolution typical

Resolvers vs. Incremental Encoders

• Resolvers

– Lower device cost

– Absolute within one rev

– Higher noise immunity

– Lower maintenance

– Passive device

– Smaller Package

– Less Wires

– Higher environmental specs.

• Encoders

– Lower interface cost

– Variable resolutions

– Easier to debug

– Easier to ratio together

Magnet

Hall Effect

Device

Feedback Devices

• Hall Effect Switches – When a magnet is passed by a Hall effect device, current flows

through it.

Magnetic disk attached

to motor shaft.

Hall Effect

Devices

Hall A

Hall B

Hall C

0o 60o 120o 180o 240o 300o 360o

Feedback Devices

• Hall Effect Switches – Hall effect switches are often used as a way to roughly determine

the position of the rotor on a brushless motor.

Signal

0o 60o 120o 180o 240o 300o 360o

“ABS” Signal

Feedback Devices

• Hall Effect Switches – Rockwell-Automation products combine the A, B, and C signals

into one output called the “ABS” signal to reduce wire count.

Channel A+

Channel A-

Channel B+

Channel B-

Channel Z+

Channel Z-

Feedback Devices

• Hall Effect Switches

– Hall effect switch functionality (including the ABS signal)

is now being built into standard optical encoders.

Feedback Trends

• Devices:

– Moving towards Motor Based Absolute Encoders

– Encoders are approaching resolvers in robustness

• Resolution:

– Applications demanding 1M to 4M counts/motor rev in

precision applications (CNC, converting, etc.)

• Motor based Absolute Encoders:

– Want to eliminate homing on power up

– Smart encoders that store motor parameters

Vos2 V

Hi-Res Feedback

• Hiperface

• SRS, SRM Optical Encoder

• ST & MT Absolute Versions

• 1024 Sine/Cosine Per Rev. (1Vpp)

• >2 Million Counts per rev

• RS485 Parameter Channel

• Low Voltage & Frequency Signal

• 8 Wires

• Internal Voltage Regulator

– Allows long cable lengths (to 300m)

– Excellent Power Supply Rejection (PSRR)

• On Board Temperature Sensor

• Plug and Play, on board E2PROM

7 to 12 VDC

Diagnostics Position Offset Motor Rating Label

Event Counter Absolute Position User defined Data 42H

Process

Channel

Parameter

Channel

SRS/SRM Drive

RS 485

Driver/Receiver

Controller

Counter

• Smart Sensor Functions

Stegmann Hiperface

optical pick up

system

customized

integrated

circuit

vector

controlled

LED-current

mechanical

gearbox with

magnetic (hall)

pick up system

for the multiturn

function

controller

EEprom

RS 485

driver

operational

amplifier

Sin/Cos

parameter

power supply Linear

Regulator

Block Diagram of the Stegmann Multi-Turn Encoder

Accuracy Resolution Shock Temp Comments

Tachometers N/A N/A 25 G -10o to 120o C Velocity feedback

Incremental

Encoders

~1.5 Arc-Min ~32,000 counts 25 G 0o to 100o C

Resolvers ~10 Arc-Min ~16,000 counts 50 G -55o to 175o C

Hi-Res

Encoders

~5 Arc-Sec ~4 million

counts

25 G 0o to 125o C

Hall Effects ~1 Degree ~1/6 Revolution 25 G -10o to 120o C Course rotor

position for

commutation

*Typical values

Feedback Devices

• Comparison of Feedback Devices

Motion

Controllers

Motion Controllers

• The “brain” of the system which typically use

microprocessors to accumulate an input command, compare

it to a feedback and make appropriate corrections

• Usually one of the following types….

– PLC Based

– Bus-Based

– Integrated Drive/Controller

– Stand-Alone

– Open Architecture

Machine

Controller

Sensors

Gauges

Meters

Data Acquisition

Proportional Valves

Displays

Keyboards

Touch Screens

Servos

Steppers

Hydraulics

VFD’s

Mainframes

Peer to Peer

Switches

Indicators

Readout

Actuators

Motion Controller Elements

Motion Controllers Provide

• Precise programmed positions of a load

• Precise speed regulation & high acceleration rate control

• Precise control of servo motors, stepper motors, hydraulic actuators,

VFD’s, Linear Motors

• Feedback is often used for position and speed control

• Networking to host or peer computer/controllers

• Synchronization of multiple moving machine members (axes)

• Processing Inputs & Outputs (discrete or analog)

Apply PID to

Position Error

Determine

Position Error

Calculate

Desired Position

Check Inputs and

Set Outputs

Check for Serial

Commands

Check for

Faults

P (Proportional) - For Speed Response.

I (Integral) - For Accuracy - Slow response.

D (Derivative) - For stability and Damping.

Motion Control

• Motion Controller Firmware Operation

Motion Control Profiles

What are the critical parameters that must be controlled or

important for a successful completion of a process? • Distance

• Velocity

• Acceleration

• Deceleration

• Torque

• Inertia

Other Motion profile items • Index

• Incremental Move

• Absolute Move

• Home

Accel Decel

Triangular Profile Accel to speed and decel back to original speed or zero, rest and repeat the process as needed.Ex. Pick & Place

When to use Triangular Velocity Profiles?

S-Curve Profile Accel to speed at a variable rate (slower first, then faster, then slower), travel constant speed, decel to zero at a variable rate (slower first, then faster, then slower).Ex. Bottling; Train ride at Airports.

Trapezoidal Profile Accel to constant speed, travel at constant speed, and decel to zero. Ex.Cut to Length Accel Decel

Constant Speed

time (s)

time (s) 1 2 0

When to use Trap / S-Curve Velocity Profiles?

An entire operation can be plotted as multiple velocity profiles, including time at rest.

time (s) 1 4 3 6 5 8 7 10 9

Rest or Dwell Cycle

Profiles

An “index”:

Constant

Index Profile

• Incremental Move

• Absolute Move

0 1 2 3 4 5 -5 -4 -3 -2 -1

Go in the direction indicated (+/-) from where you are at the time the command is issued. The # of units is specified.

A B = +2 2 units positive B C = +1 1 unit positive

C D = -7 7 units negative

A B = 2 go to a position of 2 B C = 3 go to a position of 3

D = -4 go to a position of -4 C

0 1 2 3 4 5 -5 -4 -3 -2 -1

0 1 2 3 4 5 6 7 8 9 10 11

Home Position

Home Position = all absolute moves are positive (+)

Home position is the base (zero) reference for all absolute moves.

It can be defined anywhere in the travel.

All moves are CW. No CCW rotation from zero

Overtravel

• Overtravel = Going beyond the physical limits of the machine

0 1 2 3 4 5 -5 -4 -3 -2 -1

Overtravel limit switches shut down the drive before damage occurs from crashing into machine limits.

Software overtravel limits are established inside of the hardware overtravel limits (eg. +4.5, -4.5)

Move to 4 = Travel to 4 units = OK = Move Move to 10 = Beyond SW OT = No Move

Resolution

• The smallest increment into which a quantity can be divided

• In motion...

– Defined by the feedback counts/rev & the smallest

programmable distance.

-0.0002 -0.0001

Commanded Position

+0.0001 +0.0002

If position A and B look the same to the controller but position C does not, the positioning resolution of the system is ± 0.0001

A B B C C

Application Accuracy Requirements

• Linear Accuracy is relatively easy to understand

– Terms are in linear measurements of fractions of inches or millimeters

• Rotary Accuracy can be addressed in various terms

– Radians

– Units of linear measurements about the circumference of a roll

– Degrees • Minutes 1/60 of a degree

• Seconds 1/60 of a Minute

• Arc Seconds

– 1 arc second = 1/3600 of one degree of an arc

– Length of arc for a center angle of nº= 0.008727d (where d is the diameter)

Bandwidth

• The limiting frequency of commands to which an

actuator can respond

• The higher the Bandwidth, the more commands / Unit of

time the system can respond to

We are most interested in the torque and speed

required by each application.

Triangular profiles are limited by the maximum speed of

the system.

Trapezoidal profiles can be used when maximum speed

is a limitation.

Trapezoidal profiles are limited by the maximum

acceleration of the system.

S-Curve and Parabolic profiles have smoother speed

transitions but require greater acceleration and

deceleration rates.

Motion Profiles

• Things to Remember

Torque

Motion

Dictionary

Torque

• The tendency of a force to produce rotation about an axis.

• The turning force applied to a shaft tending to cause rotation

• Torque is defined by these two equations:

– Torque = Force ´ Radius • = F ´ X

– Torque = Inertia x rotational acceleration • = J ´ a

– Unit = in.- lb; N m;

Torque:

Radius

Inertia

If radius is 16 inches, and

the force applied is 1

pound:

Torque = Force x Radius

Torque = 1 lbs x 16 inches

Torque = 16 inch-lbs

Torque

Produced

Torque

Torque - Example:

Velocity

83p rad/sec

0 Seconds

67p rad/sec

50p rad/sec

33p rad/sec

0 rad/sec

17p rad/sec

1 2 4 3

100p rad/sec

a = 50p rad/sec2 = 157.1 rad/sec2

Jcyl = .933 in-lbs-sec2

T = Jcyl x a = 146.6 in-lbs

Torque

• From the Torque/Speed

profile you must

determine:

– Peak Intermittent Torque

is within Servo System

capabilities

– RMS Torque is within the

continuous operating

region

Continuous Duty Zone

Intermittent Duty Zone

Velocity/Torque vs. Time Profile

Desired Load

Velocity

Required Motor Torque

t1 t2 t3 t4

Trms T1

2 t1 + T2

2 t2 + T32 t3 + T4

t1 + t2 + t3 + t4 =

Inertia

• The product of the weight of an object (W) and the square of the radius of gyration (K) (how the weight is distributed around the axis of rotation).

• Result = WK2 = Lb - Ft2

• The magnitude of inertia is a function fourth power of its radial dimension. Therefore a small diameter cylindrical rotor inherently has a much lower inertia than a large diameter motor.

• A smaller Radius part has much less inertia than a larger radius part. – Double radius - 24 = 16 times the inertia

– Triple radius - 34 = 81 times the inertia

• Servo System inertias are generally defined as IN-LB-SEC2

Rotating a disk Rotating a cube

The inertia (how hard it is to rotate an object) is

determined by the length, diameter,and density of

the object.

Inertia

• Inertia - Examples:

Inertia - Examples:

The inertia of a cylinder

rotating on the axis shown is:

Jcyl = pLrr4 = Wr2

p = 3.14

r = Density (lbs/in3)

W = weight (lbs)

g = 386 in/sec2

L = Length (in)

r = Radius (in)

Inertia

Inertia - Examples:

Jcyl = pLrr4 = (3.14)(10 in)(.283 lbs/in3)(3 in)4 =

2g (2)(386 in/sec2)

L = 10 inches r = 3 inches

.933 in-lbs-sec2

Steel cylinder (r = .283 lbs/in3)

Inertia Calculation using Diameter

Inertia - Examples:

Jcyl = Wr2 = (80 lbs)(3 in)2 =

2g (2)(386 in/sec2)

r = 3 inches

.933 in-lbs-sec2

Steel cylinder (W = 80 lbs)

Inertia Calculation using Mass

Torque, Inertia, and Time

Inertia Torque Directly

Proportional

Directly Proportional

Inversely Proportional

Torque = Acceleration x Inertia

Design

Considerations

When is Motion Control Required?

• Precise Control of Position

– Rule of Thumb: Accuracy of motor shaft less than

5 degrees or linear movement less than .0001”.

• Precise Control of Speed

– Rule of Thumb: Speed regulation of .05% or better

• Rapid Acceleration and Deceleration Requirements

– Rule of Thumb: Motor acceleration from 0 to 2000 rpm in

less than .5 sec.

• Control of Torque

– Ability to provide full torque at 0 rpm.

Selecting the Correct Motor

• What are the application requirements

– Continuous process

– Pick and Place

– Motion Intensive • Feed to Length

• Rotary Shear

• Flying Cutoff

• Is high Inertia a help or a hindrance

– Is the application subject to disturbances

– Do axes need to follow a precise path

Relevant Application Data

• Load / Motor Inertia Ratio

• Motor / Load Coupling Type

– Rigid

– Backlash (i.e. Gearbox)

– Compliant (i.e. Resonance)

• Profile

– Fast acceleration / deceleration

– Slow acceleration / deceleration

• Dynamic Performance required

Backlash Influences

• Any backlash will have a severe effect on the maximum

achievable gain

• From a previous slide :- ‘If the inertia doubles, the gain can

also be doubled thus restoring the bandwidth’

• But in the middle of the backlash range, the motor and load

are effectively disconnected, so the gain at this point will be

too high. The effect is severe ‘hammering’ across the

backlash.

• The maximum usable gain is that which would be appropriate

without the load.

Inertia Ratio and Backlash

• This helps to answer the perennial questions :-

– ‘What is the maximum allowable inertia mismatch ?’

– ‘What is the optimum inertia match ?’

• The best situation for performance when backlash is present

is a motor that dominates the load :-

Jmotor >> Jload (A ‘high’ inertia motor helps here)

• Even a 1:1 inertia ratio reduces bandwidth by 50%

• With a rigidly coupled motor and high resolution feedback

even extreme inertia ratios may perform satisfactorily

Coupling and Resonance

• If the load and motor are coupled by a compliant (springy)

coupling, the effect on limiting gain is very similar to that of

backlash so, again, a ‘high’ inertia motor helps .

• At the moment the motor starts to accelerate from rest the

‘spring’ is unloaded so the load is effectively disconnected

from the motor.

• As the motor moves and ‘winds up the spring’ the load

inertia is felt by the motor.

• Note that compliance may exist in the rotating parts or the

motor mounting.

Acceleration Requirements

• Slow acceleration -- tens of seconds to ramp to speed

– Typically used in process lines such as calenders, extruders, • Normally select induction motors since high acceleration is not a requirement

• Normal Acceleration

– several seconds to attain maximum speed • These applications may require more investigation to determine the correct solution

• High acceleration -- complete cycle occurs in milliseconds

– requires PID or zero following error system • Includes feed forward and integral terms to keep the proportional error near zero.

There is always a small error

• sizing based on motor inertia and gearing

Selecting The System

• Carefully identifying the application requirements will allow

you to identify the appropriate motion control components

and a successful application.

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