Motion Control Theory. servomotor

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Motion Control Theory
Kimberly-Clark GNW & HC Electrical Forum

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Topics
• Closed Loop System
• Servo Motors
• Servo Drives
• Mechanical Gearing
• Feedback Types
• Motion Controllers
• Torque
• Inertia
• Design Considerations

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Closed Loop
System

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• 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

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• 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

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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.

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Speed Feedback
Example – Closed Loop Control System
Load
Table
Movement
Controller
AC/DC
Servo
Amp
0 to ±10V DC
Position
LoopVelocity
Loop
Encoder
Ballscrew
Reducer
Motor
Power

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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
Load
Motor
LINEAR
Ballscrew
Ballnut
Motor
Gearbox
Pulleys

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Components of a Motion Control System
Software
Motion
Controller
Drive/
Amplifier
Actuator/
Motor
Mechanical
Linkages
And Load
Feedback
Transducer
Servo Axis

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Servo
Motors

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Rotor
windings
Air gap
Magnet
Case
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...
Brush
Commutator
Arcing

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Permanent
Magnets
Windings
Servo Motors - Theory
• A brushless servo motor locates the permanent magnets on
the rotor and the windings on the stator...

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Heat generated in
windings is directly
conducted to motor
case.
Winding is electronically
commutated.
• Brushless motor design reduces the problems of heat
dissipation and commutation induced speed limits.

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Magnet material
on rotor is lighter
than copper wire
windings.
• Also, the rotor inertia is reduced because heavy copper wire
windings are not required.

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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

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Rotor
Magnets
24681012
2
4
6
8
10
12
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De-magnetizing force - “H” (Kilo Oersteds)
Flux Density - “B”
(Kilo Gauss)
Servo Motors - Properties
• Magnet material properties

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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

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• 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)

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• 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.

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Servo Drives

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Power from
DC Bus
Motor
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

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Positive
Zero
Negative
Period Period
50% 50%
Positive
Negative
Motor CurrentPWM Voltage
• Pulse width modulation - zero current:

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Positive
Zero
Negative
Period Period
75% 25%
Positive
Negative
• Pulse width modulation - positive current:

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Positive
Zero
Negative
Period Period
25% 75% Positive
Negative
• Pulse width modulation - negative current:

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+
+
Reference
Input
Saw Tooth
Carrier
Carrier
and
Reference
PWM Output
Comparator
• Pulse width modulation - pulse generation:

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Motor
Voltage
Motor
Current
• Pulse width modulation is possible because the
inductance of the motor has a smoothing effect on
the pulses...

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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

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Trapezoidal
Sinusoidal
• There are two common methods of servo drive commutation:

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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...

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 Trapezoidal Commutation-
 Requires only hall effect
feedback of rotor position
 Sinusoidal Commutation–
 Requires encoder or
resolver feedback of rotor
position
• “Trapezoidal” and “Sinusoidal” commutation

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Torque
Torque
Trapezoidal commutation
- Torque ripple
- Only hall effect feedback
required
Sinusoidal commutation
- Minimal torque ripple
- Encoder or resolver
feedback required
10-15%

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Mechanical
Gearing

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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

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Backlash
• A relative movement between interacting mechanical parts,
resulting from looseness when motion is reversed.
Backlash in
tooth profiles

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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)
Input
(Servomotor)
Gear Reduction

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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

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• 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

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Gearing Drawbacks
• Can add backlash to the system
• Reduces system efficiency
• Reduces output speed
– requires higher motor speed
• Can increase audible noise
• Increases cost

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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

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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!

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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

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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

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Out
In
In
Out
Typical Gearing Technologies
• Parallel Axis Gearing
– Spur/Helical
– Planetary
• Non-Parallel Axis Gearing
– Worm
– Bevel
• Other
– Belt Driven

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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
Spur
Helical

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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

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Worm/Bevel Technology
• Worm Gear
– Low Backlash
– Very High Stiffness
– 40% Efficient, Lower At Start-Up
– Moderate Input Speeds, Frictional
Heat
– 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
Worm
Bevel

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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

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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).

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Feedback
Devices

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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
Loop
Velocity/Position Loop

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Feedback Devices
• Sensor devices mounted to the actuator (motor) or
load that detect speed and/or position.

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Pulses Indicate Position
Pulses/Sec Indicate Speed
Voltage
TimeEncoder
Output
Feedback Devices
• Encoder

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Incremental Encoder Output
• Electrical Characteristics
– TTL (5V) or CMOS (12v)
– Single Ended
– Complementary
– Differential Line Driver

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• Differential Line Driver
(A+Noise)-(-A+Noise)=A+Noise+A-Noise=2A

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• Detecting Rotation Direction
Channel A
Channel B
Marker
Channel A
Channel B
Marker
Clockwise
Rotation
Counterclockwise
Rotation

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• Line Count Multiplication
Channel A
Channel B
1X Multiply
2X Multiply
4X Multiply

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Resolvers
• What Are They?
• Rotary transformers
– Single rotating winding (rotor)
– Two stationary windings (stators)
– Coupling between rotor and stators varies with shaft
angle
• Brushless transformer couples signal to rotor
• Provide absolute position over one revolution

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• 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

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Circular Transformer
Two Phase Stator
Sinusoidal
Input
Sinusoidal
Output
Sinusoidal
Output
Resolvers
• Resolver - (Transmitter style)

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Winding A
(Sine Winding)
Winding B
(Cosine Winding)
Rotor
Resolvers
• Resolver Operation - Concepts

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Signal sent to rotor:
Signal on Cosine
winding:
Signal on
Sine winding:
0 Degree Position
Resolvers

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Signal on Cosine
winding:
Signal on
Sine winding:
20 Degree Position
Resolvers

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Signal on Cosine
winding:
Signal on
Sine winding:
45 Degree Position
Resolvers

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Signal on Cosine
winding:
Signal on
Sine winding:
90 Degree Position
Resolvers

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Rotor Signal
Cosine Output
Sine Output
0o0o 90o 180o180o 270o 360o
Electronics looks
at portion of signal
indicated by green
lines.
Resolvers
• Resolver Operation Concept - 360o Rotation

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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

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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

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Excitation
Oscillator
Tracking
R/D
Converter
Binary Angle
Output
Resolver
R2
R4
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

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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

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NS
Magnet
Hall Effect
Device
Feedback Devices
• Hall Effect Switches
– When a magnet is passed by a Hall effect device, current flows
through it.

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N
S
Magnetic disk attached
to motor shaft.
Hall Effect
Devices
A
B
C
Hall A
Hall B
Hall C
0o 60o 120o 180o 240o 300o 360o
Feedback Devices
– Hall effect switches are often used as a way to roughly determine
the position of the rotor on a brushless motor.

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ABS
Signal
0o 60o 120o 180o 240o 300o 360o
“ABS” Signal
N
S
A
B
C
Feedback Devices
– Rockwell-Automation products combine the A, B, and C signals
into one output called the “ABS” signal to reduce wire count.

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Channel A+
Channel A-
Channel B+
Channel B-
Channel Z+
Channel Z-
ABS
Feedback Devices
– Hall effect switch functionality (including the ABS signal)
is now being built into standard optical encoders.

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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

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Vpps2
Vos2 Vo_s2
Sin
_Sin
Cos
_Cos
Voc2
Vpps2
Vo_c2
F2
sc2
Data
_Data
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

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7 to 12 VDC
Diagnostics Position OffsetMotor Rating Label
Event CounterAbsolute Position User defined Data42H
40H
Process
Data
Channel
Parameter
Channel
Power
SRS/SRM Drive
RS 485
Driver/Receiver
Power
Controller
2 n+2
Counter
• Smart Sensor Functions
Stegmann Hiperface

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1 8
1 8
1 8
1 8
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

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Accuracy Resolution Shock Temp Comments
Tachometers N/A N/A 25 G -10o
to 120o
C Velocity feedback
only
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

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Motion
Controllers

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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

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Machine
Controller
Sensors
Gauges
Meters
Data Acquisition
Proportional Valves
Displays
Keyboards
Touch Screens
Servos
Steppers
Hydraulics
VFD’s
Mainframes
MIS
SPC
Peer to Peer
Switches
Indicators
Readout
Actuators
Motion Controller Elements

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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)

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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

87
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

88
1 20
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?

89
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.
1
20
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 20
When to use Trap / S-Curve Velocity Profiles?

90
20
An entire operation can be plotted as multiple velocity profiles,
including time at rest.
time (s)1 43 65 87 109
Rest or Dwell
Cycle
Profiles

91
An “index”:
Time
Speed
Constant
Speed
Dwell
Batch
Index Profile

92
Move
• 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
A B = +2 2 units positive
B C = +1 1 unit positive
B C
D C
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 -4C

93
Home
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.
Load
Home
All moves are CW. No CCW rotation from zero

94
Overtravel
• Overtravel = Going beyond the physical limits of the machine
0 1 2 3 4 5-5 -4 -3 -2 -1
+5-5
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

96
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 BBC C

97
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)

98
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

100
 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

110
Torque

111
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;

112
Torque:
Force
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

113
Torque - Example:
Time
Velocity
83p rad/sec
0 Seconds
67p rad/sec
50p rad/sec
33p rad/sec
0 rad/sec
17p rad/sec
1 2 43
100p rad/sec
qa
wmax
a = 50p rad/sec2 = 157.1 rad/sec2
Jcyl = .933 in-lbs-sec2
T = Jcyl x a = 146.6 in-lbs
Torque

114
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

115
Velocity/Torque vs. Time Profile
time
time
Desired
Load
Velocity
Required
Motor
Torque
Vpeak
t1 t2 t3 t4
T1
T2
T3
T4
Tpeak
Trms
T1
2 t1 + T2
2 t2 + T3
2 t3 + T4
2 t4
t1 + t2 + t3 + t4
=

116
Inertia

117
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

118
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:

119
Inertia - Examples:
The inertia of a cylinder
rotating on the axis shown is:
Jcyl = pLrr4 = Wr2
2g 2g
p = 3.14
r = Density (lbs/in3)
W = weight (lbs)
g = 386 in/sec2
L = Length (in)
L
r = Radius (in)
r
Inertia

120
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

121
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

122
Torque, Inertia, and Time
Time
Inertia TorqueDirectly
Proportional
Directly
Proportional
Inversely
Proportional
Torque = Acceleration x Inertia

123
Design
Considerations

124
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.

125
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

126
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

135
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.

137
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

138
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.

140
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

141
Selecting The System
• Carefully identifying the application requirements will allow
you to identify the appropriate motion control components
and a successful application.

Motion Control Theory. servomotor

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