Electrical feed drives in automation : basics, computation, dimensioning 3895781487, 9783895781483

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GroB/Hamann/Wiegartner Electrical Feed Drives in Automation

Electrical Feed Drives in Automation

Basics, Computation, Dimensioning

By Hans GroB, Jens Hamann and Georg Wiegartner

Publicis MCD Corporate Publishing

Pie Deutsche Bibliothek- CIP-Cataloguing-in-Publication-Data ....

A catalogue record for this publication is available from Die Deutsche Bibliothek

This book was carefully produced. Nevertheless, authors and publisher do not warrant the information contained therein to be free of errors. Neither the authors nor the publisher can assume any liability or legal responsibility for omissions or errors. Terms reproduced in this book may be registered trademarks, the use of which by third parties for their own purposes my violate the rights of the owners of those trademarks.

ISBN 3-89578-148-7 Editor: Siemens Aktiengesellschaft, Berlin and Munich Publisher: Publicis MCD Corporate Publishing, Erlangen and Munich Translated by Hema Desai, transXpert, Illinois, USA © 200 I by Publics MCD Werbeagentur GmbH, Munich This publication and all parts thereof are protected by copyright. All rights reserved. Any use of it outside the strict provisions of the copyright law without the consent of the publisher is forbidden and wi11 incur penalties. This applies particularly to reproduction. translation, microfilming or other processing, and to storage or processing in electronic systems. It also applies to the use of extracts from the text. Printed in Germany

Preface

In the last 20 years numerous changes and developments have markedly impacted the field of drives technology. The book "Electrical Feed Drives for Machine Tools" which appeared in 1981 therefore demanded a complete revision and is now replaced with this present edition in the ever growing application field. Due to continuously increasing personnel and manufacturing costs, automation technology has become an integral part of all manufacturing processes. Advances in drive technology and control engineering originating in the metal processing industry are now being applied to the wood processing, glass and ceramics industries, the packaging machine industry, to robot and handling technology and to many other automation tasks on processing machines. Regardless of the fact that these applications may require features different from the feed drives of machine tools, the position controlled electrical feed drive, which has been standard in the metal processing industry for almost 40 years, still provides the basis for positioning and low-inertia drives. This entire field of drive technology is called Motion control. This book contains a comprehensive summary of the currently used fundamental dimensioning and calculation methods. From the beginning, Siemens has been actively supporting the field of mechanical engineering, and with its many inventions and products ensured that efficient drives meeting the applicable requirements are provided. These efforts were supported by institutes of technology, with special mention of the Institute of Control Theory for Machine Tools and Manufacturing Equipment at the University of Stuttgart. The basics and fundamental methods described herein also apply to the field of high speed processing and to the transition to linear motors. In many applications, the electrical feed drive has proven to be superior to other drive systems. In addition, it has the potential for further improvements and the ability to adapt to the continually growing requirements in regard to dynamics and costs, the indicators of productivity and efficiency. Special emphasis has been placed on practical applications. The theoretical fundamentals have only been included to the extent necessary for the understanding of the applied methods, and always refer to practical examples. The objective is to provide engineers and technicians with a basic understanding of the subject matter to enable them to design and select components appropriately.

This book does also attempt to promote inter-disciplinary understanding between different fields of study, because a well designed feed drive is the result of a joint effort of control engineers, electrical engineers and mechanical engineers. Mechanical engineering students who plan to major in drive technology will find a comprehensive listing of requirements from different fields of technology. At this point we also want to express our thanks to the many associates who lent their knowledge and expertise to this book. Their contributions ensure that practice-relevant key items are included, and that the reader is able to benefit from the expertise of a leading supplier of drive technology for automation. We are convinced that in the case of future developments from the Siemens company these fundamentals will contribute to the creation of products enabling mechanical engineering to maintain its leading role. Erlangen, May 2001

Aubert Martin President Motion Control Systems Division

Siemens AG, Automation and Drive Technology Motion Control Systeme

Contents

1

Fundamentals of Control Theory . . . . . . . . . . . . . . 12

1.1 1.1.1 1.1.2 1.1.3 1.1.4 1.1.5 1.1.6

Terminology. . . . . . . . . . . . . . . . . . Open Control Chain and Closed Control Loop Closed Control Loop Signals . . . . . . . . . Sampling . . . . . . . . . . . . . . . . . Linearization . . . . . . . Normalizing . . . . . . . . . . Referenced Variables .

1.2 1.2.1 1.2.2 1.2.3

Response Characteristics . . . . . Steady-state transfer characteristic Dynamic Response . . . . . . Basic Transfer Elements . . . . .

1.3 1.3.1 1.3.2

Representation in the Time Domain Differential Equation . . . . . . . . . . Step Response. . . . . . . . . . . .

1.4 1.4.1 1.4.2 1.4.3 1.4.4

Representation in the Frequency Domain. . . . . . . . . . . . Frequency Response . . . . . . . . . . Bode Diagram . . . . . . . . . . . . . . P-T 1- and P-T2-Response . . . . . . Frequency Response Measurement .

. . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

1.5 Root Loci . . . . . 1.5.1 Representation. . . . . . . . . 1.5 .2 Poles . . . . . . . . . . 1.5.2 Zeros. . . . . . . . . . . . . . . . 1.5.4 Root Loci of the P-T2-Element .. 1.5 .5 Root Loci of 1- and 2-Mass Oscillators . 1.6

Linking of Transfer Elements . . . . . .

1.7

Comparison of the Description Methods .

. . . . 12 . . . . 12 . . 13 . . . . 16 . 19 . . . . . . . . 19 . 20

. . . .

. . . .

. . . .

. . . . . . 21 . . . . 21 . . . . . . 21 . . . . 23

. . . . . . . . 35 . . . . . . . . 35 . . . 42 . . . . . . 47

. . . . . . 47 . . . . . . 51 . . . . . . 52 . 61

. . . . . . . . . . . . . . . . .. .. . .

61 61 62 63 64 66

. . . . 76

.. 79

1.8 . . . . . . .. 80 Appendix . . . . . . . . . . . . . . . . . . 1.8.1 Fundamentals of Control Theory . . . . . . . . . . . . . . . 80 1.8.2 Analogies between electrical and mechanical oscillators. . . . . 89

7

2

Control Loops for Feed Drives . . . . . . . . . . . . . . . 92

2.1 2.1. l 2.1.2 2.1.3 2.1.4 2.1.5

Terminology. . . . . . . . . . . Block Diagram Step Function . . . . . . Accuracy . . . . . . . Frequency Responses . . . . . Characteristic Stability Values

2.2 2.1.6 2.2.2 2.2.3 2.2.4 2.2.5

Controller Design Options Feed Drive Controllers . Analog PI Controller Digital PI Controller . . . Sampling Controller. . . PI Controller with Reference Model

2.3 2.3.1 2.3.2

Optimization Rules . . . . . . . . . Double Ratios . . . . . . . . . . . . Symmetrical Optimum, Absolute Value Optimum ..

113 113 114

2.4 2.4.1

115

. 92 . 92 . . . . 94

. . . . 95 . . . . 96 . . . . 97 .. 99 . 99 102 103 105 110

2.4.3 2.4.4 2.4.5 2.4.6

Dynamic Response Characteristics . . . . . . . . . . Command Frequency Response of the PI Controller with Reference Model. . . . . . . . . . . . . . . . . Command Frequency Response of the Conventional PI Controller . . . . . . . . . . . . . . . . . . . . . . . . Interference Frequency Response of the PI Controller . Determining the Controller Parameters . . . . . . . . . Standard 2 nd and 3 rd Order Drive Frequency Responses Summary . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

116 117 117 118 121

2.5

Command and Disturbance Response in the Time Domain

121

2.5.1 2.5 .2

Command Response. . . . . . . . . . . . . . . . . . . . Disturbance Response. . . . . . . . . . . . . . . . . . .

121 124

2.6 2.6. l 2.6.2

Command and Disturbance Response in the Frequency Domain . . . . . . . . . . . . . Command Response. . . . . . . . . . . . . . . . Disturbance Response. . . . . . . . . . . . . . .

125 127 127

2. 7

Modeling an example of the Speed Control Loop

129

3

Feed Drive Position Control . . . . . . . . . . . . . .

141

2.4.2

3.1 Terminology, Basics . . . . . 3. I . 1 Configuration and Function . . 3.1.2 Position Measurement. . . . .

115

141 141 143

3.1.3 Parameters and Characteristics of Position Control Loops . 3.1.4 Modeling Linear Position Control Loops . . . . . . . .

146 149

3.2

Dynamic Response Characteristics of Linear Position Control Loops . . . . . . . . . . . . . . . . . . . 154 3.2.1 Command Response of Linear Position Control Loops with Indirect Measuring System. . . . . . . . . . . . . 154 3.2.2 Command Response of Linear Position Control Loops with Direct Measuring System. . . . . . . . . . . . 160 3.2.3 Disturbance Response of Linear Position Control Loops. . 166

3.3 Circular Contour Errors . . . . . . . . . . . 3.3.1 Causes, Influencing Variables . . . . . . 3.3.2 Errors caused by the Command Response

177 177

at the Indirect Measurement System . . . . . . . . . . 3.3.3 Errors caused by the Command Response at the Direct Measuring System. . . . . . . . . . . . . . . . . . . . 3.3.4 Circular Contour Distortions by the interaction of both feed axes . . . . . . . . . . . . . . . . 3.3.5 Summary of the Errors and Distortions on the Circular Contour

178

189 194

3 .4

Feed Forward Control . . . . . .

200

3.5

Command Variable Control . . .

203

188

Position Control Limits for Oscillatory Mechanical Systems . 206 3.6.1 Natural Frequencies . . . . . . . . . . . . . . . . 206 3.6.2 Frequency Response of the Mechanical System . 208 3.6.3 Feed Axis Modeling. . . . . . . . . . . . . . . . 210 3.6.4 Disturbance Optimum and Damping Optimum. . 213 3.6.5 Influence of the Sampling Periods . . . . . . 225 3 .6

3.7

Final Conclusions regarding the K v Factor . .

4

Steady-State Layout and Calculation . . . . . . . . . . . 231

4.1

Calculation Methods

4.2 4.2.1 4.2.2 4.2.3 4.2.4

Steady-state Layout . . Requirements . . . . . . . Feed Drive with Lead Screw . Rack-and-Pinion Feed Drive Linear Motor Drive . . . . .

4.3 4.3.1

Steps for the Dynamic Layout Requirements . . . . . . . . . .

226

231 . . . . 232

. ... . . 232 . . . . . . . . . 233 ..... 241 . . . . . . . . . 245 . . . . . . . . .

247 247

9

4.3.2 Rough Sizing Estimate . . . . . . . . . . . . . . . 4.3.3 Requirements based on Speed and Acceleration . 4.3.4 Examples . . . . . .

. . . . 247

248 . . . . 252

. 260

4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.4.5 4.4.6

. . . . . . . Motion Diagrams . . . .. . . . .. .. Jerk limitation . . . . . . Running Travel Distances. . . . . . Estimating the Positioning Time . . Energy Content of the Moved Masses . . . . . . . . . . . . . Attainable Acceleration . . Periodic Duty Cycle . . . . . . . . . . . ... ....

4.5 4.5.1

. . . . . . . . Summary . . . . . . . . . . . . . . . . Feed Drive Layout based on a Flow Diagram . . . . . . . .

5

Technical Appendix . .

5.1

Formula Symbols . .

5.2

SI Units . .

5.3

Conversion Tables. .

. . . . 322

5.4

Equations . . . .

..

. . . . . . . . . 326

5.5

Technical Data of Motors and Power Converters. .

IO

.....

.

. . . .

. . ..

. 296 . 296

.................. .

... ..

260 266 282 285 286 290

306

. . . 306 319

.

. . . 327

Introduction

In order to understand the characteristics of electrical feed drives for automation some basic knowledge of control engineering is required. These fundamentals are summarized in the first Section of this book. Selected control engineering processes are being presented in a manner that also enables readers without previous training to access and understand the criteria developed using these methods. Referenced literature gives the interested reader the option to gain a deeper understanding of the subject matter. The remaining chapters contain demonstrations on how to apply the fundamentals to speed and position control loops. They are explained with the help of numerous practical examples. Several specific methods used for control devices and numerical controls are introduced. The study of these methods is based on the frequency response, which, in the form of a Bode diagram permits the practical description of complex structures. Bode dia-

grams may be generated metrologically, or as a result of simulated calculations, they still represent sound documents. Such simulations in the time domain and the frequency domain permit the response of a machine to be evaluated while still in the design phase, and can detect potential weaknesses. The feed drives are represented in the form of coupled two-mass oscillator systems, which cannot be accessed anymore for some simple calculations. In this case, empirical values may have to be used, since especially the correct damping ratio cannot be determined mathematically. The concluding chapter contains the calculation methods and formulas required for the stationary layout of feed motors for different mechanical assemblies. Even though here too, computer calculations may be used to simplify the task of the person dimensioning the system, it remains essential that the engineer and technician maintain an understanding of the general scheme of things. Here, you will find the relevant formulas establishing the link between mechanical and electrical engineering in the field of feed drives. A planned follow-up volume will describe the individual state-of-the-art feed drive components. The performance of feed drives is determined by the characteristics and features of motors, electronic power converters and mechanical transfer elements. The volume will also include a listing of test results from numerous applications.

11

1 Fundamentals of Control Theory

1.1 Terminology 1.1.1 Open Control Chain and Closed Control Loop Numerical controls and drives for machines are based on the principle of intervention into the energy and/or material flow through information. A system is defined as an arrangement of devices related to each other as required by the automation task. The information is contained in the value or the characteristic of the physical variables of a signal. These variables are called information parameters. An alternating voltage, for example, is a signal, the physical variables voltage, frequency and phase angle are the information parameters, which will be utilized by the system according to their definition. Variables must al ways be indicated as the product of a nu-

merical value and a unit of measures. Open loop control is a process within a system, where one or several input variables affect one or several output variables. This type of control is characterized by an open path of action, the open control chain. Closed loop control is a process within a system, where a controlled variable x is continuously detected, compared to a command variable w, and adjusted according to the command variable. This type of control is characterized by a closed path of action, the closed control loop. During its operation the direction of the control action must be reversed in order to generate the error variable e. Controlled variable x does continuously adjust itself. Fig. 1.1 shows the block diagrams for an open control chain and a closed control loop. It lists the major terms [1.1], [1.2]. Note: For the control theory symbols used in this document we will use from here on the reserve symbols pursuant to IEC 60 0027-2 Table 11.2 of November 2000, which are being used throughout the German speaking areas. Terms appearing in italic are specified in DIN 19226 of February 1994, and explained in Section 1.8. The functional interrelationships within a system are indicated by signal flow paths and by arrows pointing in the variables' direction of action. Signal flow paths, summation points and branch-off points, together with rect-

12

1.1 Terminology

..... u

Generation ofthe command variable

w

Controlling equipment

Y__

w e Controlling .: ·0,.- element

y __

----

--

X

Controlled system

----

Generation of the final XA,V controlled variable

-

Generation of the final controlled variable

----

a) Open control chain

..... u

Generation ofthe command variable

Comparing element, comparator I

~ ~

--

Controlled system

X

I

-

---

...

XA,, V

,

I

I

r I

I

Measuring device

b) Closed control loop

---·

I

I

I

u v w y z

Input variable Output variable Command variable Manipulated variable Disturbance variable

Fig. 1.1

x xA r e

Controlled variable Final controlled variable Feedback variable Error variable (e = w- r)

Block diagram of an open control chain a) and a closed control loop b)

angular blocks, form the block diagram of a system. Inside each block is indicated how the output variable v is affected by the input variable( s) u. This dependency is the response characteristic of the respective transfer element. There is a distinction between the steady-state and the dynamic transfer characteristic. The steady-state transfer characteristic may, for example, be described by characteristic curves. It describes the steady-state behavior, after all transient conditions have settled. The dynamic behavior describes the time related response of an element, and is mostly applied to transient conditions. This behavior is described by the unit step response or by a differential equation.

1.1.2 Closed Control Loop Signals The signals used for the control of open or closed loops may be continuous or discretely timed signals. In addition, the information parameter may have a continuous or a discrete range of values. We distinguish between analog, digital and binary signals.

13

I Fundamentals of Control Theory

Analog signals are characterized by a continuous range of values, and may be applied continuously or at discrete time intervals. Their characteristic may be steady or non-steady. Figure 1.2 shows different examples of analog signals. A transfer element or system operating in analog mode is characterized by an analog input signal resulting in an analog output signal. It is not necessary for one signal to have a linear dependency on the other. Examples of analog signaling devices are potentiometers, resolvers or tacho-generators. Examples of analog transfer elements are, for example, transistors or operational amplifiers. In Figure 1.2 b) the analog value is scanned only at certain time intervals. The depicted periodic sampling with a sampling period TAT is a method often used on digital systems (see additional information in Section 1.1.3

r

a)

I

0

r

0

TAT 2TAT 3 TAT 4TAT 5TAT STAT

t

...

b)

''

I

TAT 2TAT 3TAT 4TAT STAT STAT

t

r

...

I

....

I I

c)

I

I

I I I

d)

I

I

0

0

TAT 2TAT 3TAT 4TAT STAT &TAT

t

...

TAT 2TAT 3TAT 4TAT STAT STAT

t

•

a) Steady, time-continuous analog signal u. (Information parameter: value of u (t)) b) Non-steady, time-discrete analog signal u2 , generated through periodic sampling. (Information parameter: value of ½ (k TArH c) Non-steady, time-discrete analog signal v, (0 < 8 < TAr) generated through periodic sampling and conversion into a width-modulated pulse signal. (Information parameter: value of BITAr of the previous sampling period) d) Non-steady, time continuous analog signal v2 , generated with a holding element from ½ in b). (Information parameter: value of v2 (t)) Fig. 1.2

14

Examples of analog signals

1.1 Terminology

with explanations to Figure 1.2 d) ). An analog signal may also be represented as a pulse width. Figure 1.2 c) illustrates the conversion of the timediscretely sampled signal u2 into such a pulse signal. Digital signals have a finite number of discrete value ranges. They may be time-continuous or time-discrete. Their values do not always change steadily. If, for example, an analog voltage is converted into a digital signal, the information parameter (e.g. the voltage u) is broken down into a number of finite value ranges k, and each value range is assigned a specific information. Figure 1.3 a) shows the variable u divided into 6 value ranges k = 0 ... 5. The numerical value of k is in this case a voltage level. In this example only the discrete values of the voltage, i.e. 0 V, 1 V, 2 V, 3 V, 4 V and 5 V can be represented as a digital signal. The corresponding digital information does only change when the information parameter u exceeds the limits of the value range, i.e. in the value range k = 1, the voltage u may change between 0.5 V and 1.5 V without causing a change in the digital signal. Only when the limits are exceeded the information content of the digital signal will change. This analog-to-digital conversion is also called "quantization" of an analog variable. Digital signals may therefore be numbers. This makes them well suited for digital systems, which only use the binary states O and 1 when processing infonnation internally. Commonly used are the binary system and the hexadecimal system.

4 3 a)

2

4.5-+-----,----~-------3.5-t-----,........,___ _ _~,,..___ _ _ _ __

2.5-+--------.il,-......+----------__.....-----1.5+--~~""--+-.&-.-------L----+----'-~lr------

1 0.5+--,------4~+-------t-----r---+-------ir---__,,;::i~--

0

o

~t b)

I

I I I I I I I I I

I I I

I I I

lI tI lI 1,:

lI tI I : ,..............,

I I I I I I

I I

I I

t

.

I I

' Fl ,----, ...... R ' I I

I

I

I

I '

,

I I

'

a) Digital signal represented by discrete value ranges k b) Representation as binary number with a 3-bit signal width Fig. 1.3 Curve of an analog signal u (-) and its generated digital signal (= ) with the value ranges k. Information parameter: values of k(t)

15

I Fundamentals of Control Theory

In order to be utilized in digital systems, the digital signal shown in Figure 1.3 a) is represented in b) as a series of binary numbers. Each value range k of the source signal u corresponds to one binary number. The number of parallel paths (the number of bits) depends on the degree of resolution desired for the analog source signal. For example: 8 bits can be used to represent 256 individual values. The resolution is 1/256, i.e. approx. 0.4%. In digital systems, a decrease in resolution has a negative effect on the quality of the control. According to the type of input signal, a signal width of 8, 16 or 32 bits is commonly used. Binary numbers can be transferred and processed in parallel or in series (see Figure 1.5 and Section 1.1.3). While parallel processing requires separate paths for each partial value of the signal, it has the advantage that the complete information is available at any time. The parallel digital signal is thus a continuous timed signal. Serial processing requires only one transmission path for all values. This carries the disadvantage that the complete information is only available after all data have been transmitted, i.e. after the transfer period has been completed. Depending on the width of the signal (the resolution), and the transfer rate (Bps or baud rate), this may take more or less time. The serial digital signal is thus a discretely timed signal. Another digital signal is the binary signal. It is a signal with only two value ranges, 0 and 1, as information parameters. Binary signals are used in binary switching systems.

1.1.3 Sampling Digital signal processing systems as well as open control chain and closed control loop systems operate with individual time cycles. If these time cycles are equidistant (like clock pulses), these systems can be mathematically analyzed just like continuously operating systems [1.3]. These so called data sampling systems are characterized by at least one sample-and-hold action. Figure 1.4 shows the principle of sampling for the analysis of analog and digital signals. The sampling of an analog input signal has already been shown in Figure 1.2 b ). In order to keep the time-discrete sampled signal u2 defined between the sampling times, a storage element is connected in series with the sampler. During the sampling period, this memory cell assigns the output the value of the sampled signal at the start of this period, generating the sampled signal v2 shown in Figure 1.2 d). Figure 1.4 a) shows the sampler and the storage element for an analog input signal.

16

1.1 Terminology

a)

u

b)

TAT

.....1'--. Sampler

U2

~ T Storage element

V2

TAT

TAT

~*' k►lillll ►1~,V2►~

Quantification (analog-to-digital converter)

Sampler

Storage element

Converter

a) Sampling of an analog input signal b) Analog-to-Digital conversion with sampling Fig. 1.4

Principle of sampling using a sampler and storage element

Figure 1.4 b) shows a sampler and a storage element processing a digital signal. In the example, the digital signal resulting in Figure 1.3 a) is to be sampled at the times TAT, 2TAT, 3TAT, 4TAT, .... This generates the digital signal v 2 • Figure 1.5 a) illustrates this conversion. Using a converter connected in series, the generated signal v2 is converted into a binary number (Figure 1.5 b ). A comparison of this bit string with the one in Figure 1.3 b) shows that the sampled signal contains less information than the digital signal, even though both were generated from the same analog input signal. The length of the sampling time affects the degree of distortion. If the signal is represented and processed in series rather than in parallel, the complete transfer of a binary number requires a certain amount of time (see Figure 1.5 c)). This transfer period n · ~tis effectively contained as an integer multiple within the sampling period TAT. (n is the number of parallel signal paths; i.e. it is equal to the number of bits in the binary number. In this example, the transfer period equals the sampling period). The curve indicated by a broken line in Figure 1.5 a) is the steady average signal equivalent reproduced out of the sampled signal. Only this information is available by the evaluation of the signal. If we compare this information with the information content of the analog source signal u (thin solid line), we notice a time shift and a change in amplitude: t> Sample-and-hold elements cause on average a delay of one-half of the sampling period and a deviation in the amplitude ~ v of one-half of the resolution [1.4].

When digital control chain and control loop systems use sampling devices, the resulting change in amplitude and the delay must be taken into consideration. In order to achieve results comparable to analog closed control loops, the sampling period should be as short as possible. Short means that in comparison to the delay times (time constants) inherent to the control circuit, the sampling period may only be half as long as the shortest delay time being processed. Considering the quality of the control, it should be

17

l Fundamentals of Control Theory

-

4

I

3 2

l

1

0

I

I I

'

I

'

I

I I

I

I

I

I

I

I

I

l

I I

I I

I I

!

I I

I

-=-

a:::a I

-==a

.

l

I

I c:::m I

. l

I

I

I I

I

I

I

I

I

I

I

I

I

I

4 +----'-____;_,..,..___......_ .............,.....

.

.

'

'

I

I

~

I

Mean value of ~

3

a)

v2

2 + - -..........._.__----'-_ ____._ ___,...,....,__---L-_

I

-r-

___.__ 1

1 +,~"t-----t----t-----1r---t---...,...~...---+-

0

I

TAT I

i2 b)

21

2°

I

2 TAT

I

3 TAT

I

4TAT

I

I

STAT I

I

I

STAT

7TAT f

I

I I I

I I I

I I I

I I I

I

I

I

I

I

I

I

I

8TAT -~~•

~-pii-....!-......1--+---+--_..,,_.....,.!_.....,.______ ~i-+--..'~-..' _...,.,_....,.,__._,____., _. . ,.,__~ ~+1---1,.'. _.,.,_......,.j_ __,.j.............i--,.'. .-➔1-. . . .I

I

I

3~

4~

5~

I

7~ t

6~

8~ T.

~tS AT

I I

I

I

I

17TAr!I 18TAT I

I

n n = Number of parallel signal paths

t--1~ ►

a) Sampled digital signal v2 , non-steady and time-continuous b) Parallel representation of v2 as binary number with 3 bits c) Serial representation of v2 as binary number (time-discrete) Fig. 1.5

Characteristic of a sampled digital signal

reduced even further. A serial signal transmission causes an additional delay. Serial signal transmission paths therefore should not be located within control loops, since the delay causes destabilization. There is no fundamental difference between the behavior of continuously operating analog controls and discretely timed digital controls with a suffi-

18

1.1 Terminology

ciently short equidistant sampling period TAT· The fundamentals of control theory apply to both systems. There remains, however, the difference that an analog system can respond at any time while a digital system has to wait for the next sampling period. The result is a lag time.

1.1.4 Linearization The theory of linear systems can also be applied to non-linear systems with a continuous characteristic curve under the condition that the operating range is small, i.e. only small deviations from a fixed operating point are allowed. For this purpose, we replace the continuous but non-linear characteristic curve of a transfer element with the tangent in the operating point. The resulting operating-point dependent ratio between the output and the input variable is the proportional coefficient Kp. Starting point for the linearization is the steady-state transfer characteristic of the transfer element, found either by measurements or expressed by a mathematical relation. Linearization includes the following steps: (see Figure 1.6 a)) 1. Determination of the operating point A on the characteristic curve with the coordinates uA and vA. 2. Introduction of a new system of coordinates whose zero reference point is located in the operating point, and which is only used to indicate the applicable deviations. In this example, we use a system with the coordinates L1u and L1v. 3. Linearization at the operating point, i.e. approximating the characteristic curve with the tangent. The gradient of the tangent indicates the operating-point dependent proportional coefficient. When the characteristic curve is described by a known mathematical equation, or the change of the output signal v as a function of a change at the input signal u is determined empirically, the operating point-dependent proportional coefficient is calculated using equation ( 1.1 ): (1.1)

1.1.5 Normalizing Normalizing means to relate a variable to a certain characterizing value of the same dimension in order to render the variable dimensionless, and thereby to obtain manageable numerical values (DIN 5485). This method simplifies the computation of control related problems significantly. Rated and maximum values with the same dimension are well suited for this nor-

19

I Fundamentals of Control Theory

r

i~ t:

a)

I Nominal or I maximum point

~~-• ~uA ~u

1

Uo

-

~VA To ____

A - - - ~~-+---+

I Operating point I I I I I I I I I 0

-

b)

I No~inal or . I maximum point

~uA ~u I

Operating uo point

I I I U

...

Uo

I I I I I ,

0

u Uo

a) linearize

b) normalize

Transfer element with plotted non-linear characteristic curve.

Linearized and normalized transfer element with plotted unit step response.

Fig. 1.6

...

Transfer element with non-linear characteristic curve

malizing or relating. In the example, this is demonstrated on the linearized values lluA and llvA in Figure 1.6 b). The normalized or related proportional coefficient becomes: llVA

vo

KpA No == fl

UA

uo

== KpA-

(1.2)

Vo

Uo

1.1.6 Referenced Variables Referenced values are quotients resulting from different types of variables characterizing a physical condition. Examples are the referenced rpm deviation in Figure 2.14 and the referenced velocity gain Kvlmd min (see Figure 3.6). Referenced values are used in diagrams in order to reduce the number of parameters.

20

1.2 Response Characteristics

1.2 Response Characteristics 1.2.1 Steady-state transfer characteristic The steady-state transfer characteristic takes place after all transient responses in a stable system or transfer element have settled. It constitutes the dynamic borderline response for t ➔ 00 • The transfer elements of a system can be broken down according to their behavior in the steady-state condition. Figure 1. 7 shows different characteristic curves of transfer elements. If the input and output variables of an element correspond to a continuous function, the element is called a continuous element; if they correspond to a discontinuous function, the element is called a discontinuous element.

1.2.2 Dynamic Response The dynamic response of a system indicates the time response of the output signal to a changing input signal. This dynamic response can be described in different possible ways:

t> The representation in the time domain describes the system response as a function of the time v(t). It can be used to describe linear and nonlinear systems. [>

The representation in the frequency domain indicates, for example, the response as a function of the imaginary angular frequency v(jm). The

V

V

u

u

Linear, steady

V

Non-linear, steady

u

Non-linear, non-steady

V

Dead band

Fig. 1.7

Limit

Hysteresis

Two-point behavior

Three-point behavior

Characteristic curves of transfer elements

21

l Fundamentals of Control Theory

imaginary angular frequency jm would be selected as the reference value, since many variables in control circuits occur in the form of imaginary or complex numbers. Another method of representation is as a function of the complex frequency v(p) or v(s). Frequency domain methods can only be used to describe linear systems. Those methods which have the greatest practical significance, and which will be explained in the following chapters, are shown in Figure 1.8. In these descriptions we refer as often as possible to practical applications. Readers who wish to further their studies of the theory may refer to the listed publications, especially [1.5], on which many of the derivations in this book are based. Table 1.1 on pages 24 to 27 contains a summarized overview of these descriptions for the so-called basic or standard transfer elements. The first column lists the differential equations; the remaining columns contain the describing options for step-response and frequency response F(jm ). With the exception of the proportional and the proportional-integral element (first and last row) a proportional coefficient of I is assumed for these transfer elements. In order to solve complex open and closed loop control problems, we can convert the system to be analyzed into a simulated system, which only contains these basic transfer elements. In the block diagram, these individual elements are connected as required by the function of the system to

I

Description methods for the dynamic response characteristic

Time domain

Description by differential equation

-

-,

Description by the step response

Unit step response (for linear and non-linear systems)

Fig 1.8

22

Frequency domain

Description by the frequency response and the transfer function

'

Frequency response equation (algebraic)

,-1 __ Bode Diagram (graphic)

I

I I

Transfer function

1

I I

L _ _ _ _ .J

Root Loci, Laplace transformation (for linear systems)

Selected description methods for the dynamic response as used in this book

1.2 Response Characteristics

summary points and branch-off points using series, parallel, and feedback circuits. For the time domain and the frequency domain, certain mathematical rules apply, which can be used to determine the total system response [1.6]. In the case of time domain representation, empirical analyses can be obtained with the step response or ramp response, in frequency domain represented systems with the sinusoidal response. The characteristic values and the type of the measured transfer element can be determined from the characteristics of the measured output variables [ 1. 7].

1.2.3 Basic Transfer Elements Table 1.6 in Section 1.8.2 establishes the relation between mechanical and electrical systems. Common characteristics of control related basic transfer elements are pointed out, and the oscillatory characteristics in mechanical and electrical systems are made comparable. (Remark: The equations (1.3) to (1.32) are in Table 1. 1.)

Electrical Proportional Element (P-Element) This is the ideal amplifier with a resistor network (see Figure 1.9). The output voltage is

U2

R2 R1

= --U1

=Kp· U1

( 1.33)

with the proportional coefficient (gain)

R2

Kp=--

R1

( 1.34)

(The negative sign in equation (1.33) is caused by reversing of the signal in the shown circuit. The proportional coefficient is in this case negative.) A sudden change at the input voltage U 1 results in a sudden change at the output voltage U2 •

Mechanical Proportional Control Element with 1st Order Delay (P-T 1-Element) If a spring with speed-dependent friction is released, the equilibrium of the forces - assuming the components have zero mass - results from the following equation: ( 1.35)

23

1 Fundamentals of Control Theory

Table 1.1 Listing of differential equation, step response and frequency response of basic transfer elements (summary) Response Type

Proportional

Element

P-element

Proportional P-T1with I st order element delay

Proportional with 2nd order delay

2 P-T 1elements . . 1n senes

Differential Equation

Step Response

System Equation

Equation for Solution

v=Kp ·U

( 1.3) v = Kp · us

dv T-+v=u dt

( 1.4) v = us ( I - e -

( 1.12)

i)

( 1.13)

:sKt=.t

:s1k: I

Kp= I

I

dv

+ T2)dt

V

=

+v=u (1.5) Kp=l

+

Us

(1-

T2 Ti -T2

Ti · e - _!__ r, + Ti -T2 (1.14) _!__) . e T2

:s1fz=. P-Tr

[

v = us I -

element D>l

P-T2selement Oscillatory element Ol

Unit Step response

2 dv 2d v T - 2 +2DT d + V = u dt t ( 1.6) Kr= I

e-D!T

JD2-1

1

· sin h

(; J DL I+ arc cos h D)] ( 1.15) [

v = us I -

e -D!r JI

-D2

· sin

:s11£::

1

(; ✓1 -D2 +arc cos D)] (1.16)

Oscillatory element D=O

24

v = us (

I- cos ; )

( 1.17)

:SIV!st.l

1.2 Response Characteristics

Frequency Response Equation Bode Diagramm Poles ( x) - - - - - - - - . - - - - - - - - - - Z e r o s (0) Amplitude Response

FUm)

( 1.23)

=Kp

.

FUm)=

.

1

l +ja>T

(1.24)

1

T=-

mo

FUm)=

.

F(jro) ' d920 Kp--0 -20 2 1 a2=- ro/mo 10· 10· 1 10 1

F(jro} ' dif20 0 -20 ~ :- ro/ro0 -40 10·210· 1 1 10 102

"

Phase Response

10·2 10·1 1 10 102 t :- ro o 90 0 -90 - q, (ro)I° 1• 10·2 10·1 1 10 1a2 : ro/roo 90 -45--0 -- _'J l-:. --90 -180 - q, (ro)I° 1•

jOJ h -.~

,,

k-lr-

-a

1

(1.25) F(jro) , • dif 20 +--,,..........---.-_, O-t--1i--.t--t--1

-20

FUa>)=

\.

-40 2 1 ' :: ro/ro0 10· 10· 1 10 1a2

1

I Um)2 · 21 l+jm2D-+ mo a>o

a>o

(1.26)

1 =T

j.

F(jro) d920 0 \ -20 -40 ' :- ro/ro0 10·2 10· 1 1 10 102

.

FUm)=

1

F(jwJ .~ dB 20 ~-J-l.....--. 0~~f-+----1 -20 \

I

1 +Uml·w2 0

( 1.27)

1 T

-40 2 1 ' :: ro/ro0 10· 10- 1 10 102

10-2 10· 1 1 10 1a2=- w/roo 90 0 ' ..,, -90 -180 -cp(w)I°••

,,

jro, •

¥

D - - - - - - - - - - - - - - ' JM

--

UA

l diA UA-eM - A dt_ - - ~

l ~ I

~.

;A RA

6M

ML Jl

1/LA

cM, Kr

~

jA -

-

I

RA

Ml

- ,,

t=

M~

1/Jiot

-

~

Me

~

-(OM

I

t= -

cM, Ke

t= I~ -

Block dia gram

uA eM

Time variable armature voltage Time variable induced back EMF in the motor. (Also called ·electromotive Force". When carrying the opposite sign, this voltage is also referred to in the literature as

'source voltage', DIN 1323).

,A RA

LA

"'M nM MM

Mt. M8 JM JL CM

Ke K1

Time variable armature current Effective armature circuit resistance Effective armature circuit inductance Motor angular velocity Motor speed Motor torque Load torque reflected on the motor shaft Acceleration torque Motor moment of inertia Load moment of inertia reflected on the motor shaft Motor constant Voltage constant Torque constant

Fig. 1.15 Equivalent circuit diagram and block diagram of an unregulated electrical drive (permanent magnet excitation, DC motor with/without brushes, self-regulated synchronous motor)

Motor torque

MM

=CM .;A

( 1.58)

Acceleration torque Me

dWM =ltot

dt

( 1.59)

37

I Fundamentals of Control Theory

cM

is the motor constant. The selection tables for the motors show two values:

KE Voltage constant with the unit Vs/rad (Another common form often used is V/min- 1 or V/1000 min-I; the conversion factor is l V/1000 min-I =9.549 • 10-3 Vs/rad)

KT Torque constant with the unit Nm/A Both variables have the identical numerical value where the units are used as Vs/rad and Nm/A: KE

V s/rad or Nm/ A

Vs/rad

( 1.60)

(Note: For multi-phase, self-regulated synchronous motors the values indicated for KE and KT often differ. In these cases equation ( 1.60) does not apply.) Equations ( 1.55) through ( 1.59) can now be used to develop the block diagram. In accordance with the sample to equation ( 1.35. l) in Section 1.2.3, equation ( 1.55) will be represented as an integral element with a proportional negative feedback. The equivalent is a P-T I-element with the proportional coefficient 1/RA and the time constant Tei A=LA/RA (see Figure 2.2). The motor torque equation ( 1.58) is represented by the subsequent proportional element with the coefficient cM =KT.At the output, the motor torque MM is available, which according to equation ( 1.57), together with the load torque ML will be supplemented to the acceleration torque M 8 in the subsequent summation point. The following I-element represents equation (1.59); the output represents the angular velocity ~- This speed is fed back to the input via an additional proportional element with the coefficient cM=KE to produce the back EMF eM per equation (1.56). As shown on the left side of equation (1.55), -eM will be added to uA. The block diagram is always developed from the physical interrelationships using the basic transfer elements per Table 1.1. These block diagrams are the basis for the calculation of the response as will be shown on later examples. From equations ( 1.55) through ( 1.59) we derive the differential equation for the unregulated drive. By inserting eM from equation ( 1.56) and ;A from equations ( 1.58) and ( 1.57) into equation ( 1.55) we get

( 1.61)

38

1.3 Representation in the Time Domain

If we cancel RA on both sides and introduce the electrical time constant Te 1 A of the drive,we get with

LA

( 1.62)

TetA=-

RA

the equation 2

UA CM ML ltot dmM Tei A dML Tel A · ltot d mM RA-RA·WM= CM+ CM. dt + CM dt + CM . dt 2

( 1.63)

For practical reasons we normalize the terms of the equation to the following values: The motor angular velocity to the maximum no-load motor angular velocity lOmax M,

the armature voltage to the maximum no-load armature voltage UmaxA

= EmaxM =CM· lOmaxM

( 1.64)

and the load torque to the maximum drive torque at full armature voltage U max A and locked shaft. We will call this torque the short-circuit torque of

the drive: ( 1.65) By multiplying the equation with RAIUmaxA, these normalized variables are introduced to the differential equation ( 1.63 ). After applying a few conversions, we get T. elA

1101

.MSt A

2

d mM 1101 dmM dt 2 + M St A dt

WM

+ lOmax M

If we now define the mechanical time constant of the drive as Ti mechA

ltot · RA

_ ltot · lOmax M M

( 1.66)

2 CM

StA

we obtain the 2 nd order differential equation for the unregulated electrical drive.

d2 lOM Tel A· TmechA ·

d WM

+ TmechA ·

~m;"M

t

d uA

ML

UmaxA

MstA

-

T.

elA.

W;axM t

+ lOmax WM M

( 1.67)

ML MstA

dt

The left side of this equation indicates the motor angular velocity mM as the output variable, and the right side describes the two input variables ar-

39

I Fundamentals of Control Theory

mature voltage uA and load torque ML. The latter acts as a disturbance variable to the drive. If we put ML= 0, we can determine the command response for the input voltage by solving the simplified differential equation. The differential equation has now the form of equation ( 1.6) in Table 1.1. Thus, the unregulated drive responds to command value changes like a P-T2-element. By comparing the coefficients, we obtain the characteristic value T~

= JTelA · TmechA

T;. ( 1.68)

and the damping ratio o;. * _

DA-

!

TmechA

2

Tel A

( 1.69)

( • indicate the values of the unregulated drive) The solution equations ( 1.15) through ( 1.17) in Table 1.1 indicate the transient response to a sudden input voltage change. Figure 1.16 shows this response to command value changes. Depending on the values of Tei A and Tmech A, the unregulated drive might already exhibit an oscillatory characteristic. The damping ratio D;. determines the overshoot and the setting time. The damping ratio of a drive can be increased by continuous friction torque, e.g. slideway friction. In the case of feed drives, this is a desired effect to be achieved through the appropriate layout of the mechanical transfer

COM

2)

The output variable does not exhibit an initial response to a sudden input change; the initial tangent is horizontal.

(n is the number of derivations of the output variable, m is the number of derivations of the input variable) In our case, n = m + 1 according to equation ( 1.67); which means that the angular velocity ~ at a load torque change with a finite rate of rise changes from its previous steady-state value to the new steady-state value. The transient response is thus a function of the damping ratio D l- Figure 1.17 shows the disturbance response of the unregulated drive at a load impulse of 10% of the short-circuit torque. · As we will see later, a controlled feed drive may also be described by an approximate P-T2 response. The response of a speed-controlled feed drive to command values changes is then also the same as in Figure 1.16. In the disturbance response the initial deviation will be the same as in Figure 1.17, however, the integral portion in the speed controller will reduce the permanent deviation again to 0. Using the controller parameters, the damping ratio DA at the controlled feed drive will be set to >0.5 in order to quickly compensate load pulses (see Figures 2.13, 2.14).

41

1 Fundamentals of Control Theory

i~\ ~,ro_:""'T":_M_r---~----r--~-r---~----r--~---, 0.9

0.8 - - - DA*=0.3

-~- - - - - - - - - - - - - - ---- ------

ML/Ms1A

0.1

t ~ML Ms1A ♦ I I

0

2

3

4

5

6

7

10

Fig. 1.17 Disturbance response of the unregulated drive with constant voltage (load torque pulse liML/Mst A =0.1)

In addition to equation ( 1.16) in Table 1.1 the literature contains one other form for O ro0

Amplitude response

20 ...........--......,....---...----....-------. dB

------.tu·

r

COo

Sinusoidal response

-20-+-----+------1-........--+------t

-40-+------+------1---...,_-----t 10·2 10· 1 10° 101 102

t

Phase response

m/m0

►

cp

r -180°-+-----+----+ 10•2

I

102

♦-q>(ro)

Fig. 1.21

Frequency response of a P-T2 -element as a Bode diagram

Characteristic Angular Frequency One defining characteristic of an oscillation is its frequency f Better suited for calculations is the angular frequency w, with the mathematical relation:

2n m==21t·f==r

( 1.97)

r == I/f is the period of an oscillation. For frequency analyses of P-T 1- and P-T2-elements we will, as introduced earlier, replace the delay time and the characteristic value T with the characteristic angular frequency

56

1.4 Representation in the Frequency Domain

1 mo=T

( 1.89)

and use the characteristic or the natural frequency when calculating practical examples. The absolute value of the frequency response of a P-T 1-element at the point m0 is pursuant to equation ( 1. 91)

P-T 1: IF(im) lt»o =

~=

0,7071 or correspondingly - 3,01 dB

and of a P-T2-element pursuant to equation ( 1. 95)

P-T2: IF(im)lt»o= ~ 2

( 1.98)

The absolute value of the frequency response of a P-T2-element at m = mo is therefore inversely proportional to the damping ratio times 2.

Resonant Angular Frequency Besides by the characteristic angular frequency, oscillating elements are described by additional angular frequencies. To start with, there is the resonant angular frequency Wr, at which the frequency response reaches its maximum. The resonant angular frequency of the P-T25 -element is:

mr = moJl - 2D2

( 1.99)

where the absolute value of the frequency response reaches a maximum with {1.100) Based on equation ( 1.99), a real value would only result for D < 1/ \/2, i.e. at D > 1/ \/2 = 0, 7071 there doesn't exist any resonant rise in the frequency response anymore. The step function, however, does still overshoot. Aperiodic settling into the steady-state value does only take place at D > 1 (see Figure 1.21 and Figures 1.16 and 1.17). The resonant angular frequency mr and the absolute value at the resonance point IFUm)lwr can be obtained from the measured amplitude response. Both values can be used in equation ( 1. 99) and ( 1.100) to determine the damping ratio D and the characteristic angular frequency m0 • Resonance does occur when a sinewave signal carrying the resonant frequency is applied to the transfer element or the system. In higher order systems several resonant frequencies can occur. In the field of drive engineering, the excitation frequencies are usually speed-dependent, meaning that existing resonance points are very often subject to excitation. If necessary, the ampli-

57

I Fundamentals of Control Theory

tude response and the phase response of the control loop may have to be adjusted accordingly.

Example: For a typical damping ratio of D=0.5 in a speed control loop with P-T25 response, the resonant angular frequency is

mr

=

0.707

mo,

and the absolute value of the frequency response at this point is

IFUm) l"'r =

1.15, or correspondingly = 1.25 dB.

As the damping ratio decreases, mr continues to approach m0 , and the resonant rise in the amplitude response increases. (see dotted line in Fig. 1.2 I).

Natural Angular Frequency While the resonant frequency is a characteristic variable for the frequency response, the natural angular frequency md of the P-T2 -transfer element or system characterizes the transient response in the time domain. The natural angular frequency is: (1.101) A system oscillates at its natural angular frequency after having been excited by a disturbance and then being left on its own without any further external intervention ( see Section 3 .6.1). For a better understanding, let's examine the transient response in the form of the unit step response in the time domain as shown in Figure 1.22. (In order to be applicable to any characteristic angular frequency, the abscissa is labeled with the time related to Tor 1/m0 .) Indicated are the transient functions at D = 0.1 and D = 0.5. The decaying oscillation moves within an envelope outlined by two e-functions, which has the decay time constant TA· The following applies: v

- = 1 ±A• e us

__, rA

where

T 1 d 1 TA== - == - - an A== _1 D mo· D v I - D2

( 1.102)

(The functions can be derived from the solution ( 1.16.1) of the differential equation (1.6) [1.7].) At damping ratios of D < 0.3 is A : : : 1, so that in this case the starting points of these envelopes can be assumed on the referenced ordinate with sufficient accuracy at O and + 2. We obtain the decay time constant TA and the period r from a measured transfer response. The magnitude of the overshoots and undershoots during the transient response are a function of the damping ratio D. In the case of the referenced step response (the unit step

58

1.4 Representation in the Frequency Domain

tI ;s 2 ---,.-----r----.--1-r----,.--~--------,.....--------~----~ ~kt'l+Ae TA

,' wo~:~·~n ~---7~- --\

, 0.8

:

I Wo

Vm+1

w0 r=-2rc

I

;'

~~/

'. /-~

~~

l -_ ,-i-.._11--- - -.......,-++---#,-... __.., -0.6 -+--~j'- -11.------------t---+-\t-t-.,y------#---------r-~~""-----+----+---+----t---------t-----i Wo

i

I

/~

Q_4 -t-....---:-+/--+----+/---7c;__f-----+----+----+---+-----+----+----1

:,,\/ 2

4

6

8

10

Fig. 1.22 Transient response of a P-T 25 -element

12

14

16

18

t Wo, -t T

20

.

response) of a P-T 2 s-element, the following applies to the ratio of two subsequent amplitudes Vm

Vm+l

V _m_+_2

}

Vm

Vm+l

~D

== . . . . . == e - ~

{ l.103)

Equations (1. I), (1.101 ), ( 1.102), ( 1.103) and the period r can be used to determine the characteristic variables md, m0 , D, and Kr of a P-T 25 -element from a step response (see [1.7]). The three variables characteristic angular frequency w0 , resonant angular frequency Wr. and the natural angular frequency Wct begin to approach each

59

l Fundamentals of Control Theory

other as the damping ratio decreases, and are at D = 0 identical. Due to the fact that in practical applications the damping ratio Dmech for mechanical oscillators in most cases is very small, and that these oscillating elements, due to additional non-linearities, often do not represent true P-T25 -elements, we will from now on use the term natural angular frequency or natural frequency as the characteristic variable for mechanical transfer elements, and we will use the term characteristic angular frequency only for true P-T25-elements.

Cut-Off Angular Frequency, Gain Cro~ver Angular Frequency, Phase Crossover Angular Frequency Some publications mention one additional angular frequency. This is the cut-off angular frequency lOg, a point at which the amplitude response has dropped by - 3 dB. A conversion per equation (l.88) results in IFUlO)lw = 10- 3/ 20 =0.7079. The frequency range from Oto lOg is called the band~idth. For stability analyses using the Bode diagram (see Section 2.1.5), we will also introduce the gain crossover angular frequency lOo and the phase crossover angular frequency OJ,c. Since in most practical applications the frequency instead of the angular frequency is used, it should be pointed out that each of the listed angular frequencies is associated with a corresponding frequency, which can easily be determined from equation (1.97). In Bode diagrams, either OJ or f can be entered on the abscissa. Watch for this difference when comparing or taking measurements!

Frequency Response of the Delay Elements The amplitude responses in Figures 1.20 and 1.21 indicate that in the case of low-frequency oscillations, P-T 1- and P-T2-elements respond like proportional elements. High-frequency oscillations, on the other hand, will be heavily damped. For a P-T 1-element, the amplitude drop approximates asymptotically a straight line with a gradient of -20 dB/decade through the point lO/lOo = 1, IFUlO) I = 0 dB. The phase shift ranges from 0° for low frequency oscillations to a maximum of - 90° for high frequency oscillations. At the characteristic angular frequency lOo == 1/T, the absolute value of the frequency response is IFUlO)I == - 3.01 dB, or correspondingly, ==0.7071 and the phase shift is - 45°. For the P-T2 -element, the amplitude drop approximates asymptotically a straight line with a gradient of -40 dB/decade through the point l0/0J0 = 1, IF Um) I == 0 dB. The phase shift ranges from 0° for low frequencies to a maximum of - 180° for higher frequencies. At the angular frequency lOo

60

1.5 Root Loci

the absolute value of the frequency response depends on the damping ratio D. The phase shift is here - 90°. Proportional elements with higher order delays respond accordingly: For third order delays, the amplitude drop is -60 dB/decade. According to the lengths of the delay times within the system, 2-3 comer angular frequencies with possible resonant rises at the resonant frequencies are the result. At higher frequencies, the phase shift approaches - 270°. The P-T0 -elements cannot be accessed anymore for a complete calculation. We therefore replace such transfer elements with known elements connected in series. The solutions are shown in [l. 7].

1.4.4 Frequency Response Measurement In addition to the oscilloscope or the chart recorder, the signal analyzer provides a practical method to measure and record the frequency response of unknown transfer elements or systems. These types of signal generators operate with the "Fast Fourier Transformation", abbreviated FFf. This transformation provides a method where manageable measurement instruments can be used to perform frequency analyses and frequency response measurements with high-performance signal processors. Such measurement methods are also directly integrated into the numerical control. They permit an analysis of the controlled system as well as of open and closed control loops. An electrical input signal containing a broad spectrum of frequencies (band-limited white noise), is applied to the system to be examined, and the resulting output signal is measured and analyzed. Quick mathematical algorithms permit the direct representation of the frequency response in the Bode diagram. Auxiliary functions allow the conversions between the open and closed control loop, and as a result the stability criteria are read. The poles and the zeros (see Section 1.5) can also be directly determined. Such devices can also be used to perform simulations, which may indicate weaknesses in the design of the control structure and the machines. At this point, characteristic parameters can be entered to optimize the response.

1.5 Root Loci 1.5.1 Representation The last column in Table 1.1 shows another possible way to regard the behavior of linear, time continuous systems without lag time elements under the heading "Poles, Zeros". This approach is known as the root locus dia-

61

I Fundamentals of Control Theory

gram, and is described for example in (1.8]. It is used for stability analyses in closed control loops. Root loci indicate the geometrical locations of the poles in the complex p-plane of the closed, i.e. feedback system, as a function of a parameter, e.g. the proportional coefficient or the integral-action time of a controller. In order to create them we need the poles and the zeros of the entire open system, i.e. of all the transfer elements in a control loop. At this point, we want to limit our discussion to the explanation of the poles and zeros of transfer elements, and only list the important conditions required to understand this method. From equation ( 1.86) we know the transfer function of a linear element or system. Prerequisite was m < n, so that G(p) is a rational function. The numerator and the denominator each are a polynomial in p (i.e. a function of the complex angular frequency p ), where we define the complex pplane more accurately as:

p = o+jm.

{l.104)

Variables depending on p are thus complex variables, which may be represented as vectors in a Gaussian numerical plane with o as the real and jw as the imaginary axis (see example in Fig. 1.23). The solutions to these characteristic equations can be found when a polynomial is set to equal 0. These solutions, also called 'roots', are generally complex variables. When they are entered into the polynomial, it becomes zero. Thus, the following statement applies to the transfer function G(p): t> The zeros of the numerator polynomial are the zeros of the transfer function. The zeros of the denominator polynomial are infinite points, which are called the poles of the transfer function. t> The poles and zeros of a transfer element are entered into a complex numerical plane as x or O . Their positions provide important information about the system characteristics.

1.5.2 Poles The poles of a system determine the stability and the damping, i.e. the transient response (see Figure 1.22). As shown in the example below, the characteristic angular frequency and the natural angular frequency of a system decrease the closer a pole is located to the coordinate origin. The damping ratio becomes smaller as the angle between the vector of a pole and the imaginary axis decreases. A pole causes a negative phase rotation in the phase response (thus it should be noted that poles with complex coordinates always occur in pairs as conjugate complex poles).

62

1.5 Root Loci

The following statements about the response characteristic can be derived from the position of the poles:

t> A system is stable, when all poles have only negative real components, i.e. they are located to the left of the jro axis (left half-plane).

t> Poles with positive real components (right half-plane) indicate unstable transfer elements.

t> If the poles do not have an imaginary component, i.e. if they are located on the negative a-axis, the transfer function has components representing decaying e-functions with the decay time constants T = 1/a.

t> If poles have negative real components as well as imaginary components, the transfer function will be characterized by dampened oscillations. The characteristic angular frequency ro0 , the natural frequency rod, the decay time constant TA, and the damping ratio D can be obtained from the coordinates of the poles. These poles located symmetrically to the a-axis each represent an oscillating element: • The length of the vector from the coordinate origin to the pole equals the characteristic angular frequency ro0 of the oscillating element. • The damping ratio D equals the cosine of the angle pole vector and the negative a-axis.

/3

between

• The imaginary component of the pole vector on the jro-axis equals the natural angular frequency rod of the oscillating element (see equation 1.101 ). • The reciprocal of the real component of the pole vector on the aaxis equals the decay time constant TA of the transient response (see equation (1.102) and Figure 1.22).

t> If the poles are located on the imaginary axis, undamped oscillations occur.

t> If a pole is located at the coordinate origin of the complex numerical plane, it represents an integral response.

1.5.2 Zeros The zeros of a system determine the characteristic of the response function. Their position and distribution within the complex numerical plane affect the amplitude response. In negative feedback systems the zeros are important for the stability of the closed system, e.g. for a control loop. A zero introduces a positive phase rotation into the system. Non-dominant poles can be compensated with zeros, stabilizing the control loop response. Compensated poles in the left half-plane improve the response of the control loop to setpoint changes as long as no limits are being reached. This compensation does not affect the disturbance response.

63

1 Fundamentals of Control Theory

1.5.4 Root Loci of the P-T2-Element The frequency response of a P-T2-element with the proportional coefficient Kp = 1 is according to equation (1.94):

FUm)=vUm)= uUm)

I

. 2D U )2 1 1 +Jl.t) - + l.t) -

mo

( 1.94)

CtJ5

Using the transformation according to equations ( 1.85) from Table 1.3, we obtain the transfer function: 1 2D

G(p)=

1

( 1.105)

l+p-+p2-

mo

m5

We obtain the poles of the transfer function by setting the denominator to 0 and determining the roots: 1

2

2D

2P +-p+l=0.

mo

mo

A quadratic equation must be solved, where the following applies: _2D±

P1,2

==

mo

✓4D2 _~ aif mo 2

~

and simplified P1,2

= -mo D ± moJD2 - 1 == -mo D ±jmoJl - D2

(1.106)

Depending on the value of the damping ratio D there are 3 cases:

t> Case 1: D > 1: There are 2 values for p; both are real and negative. For high values of D, one pole approaches the coordinate origin, the other pole approaches -2m0 D.

t> Case 2: D = 1: This is the borderline case for 2 real poles coinciding in -mo. t> Case 3: 1 > D > 0: 2 conjugate complex poles occur in the left p-halfplane. As D decreases, they approach the jm-axis, and for D == 0 are located directly on it. Figure 1.23 shows these three cases for the position of the poles. Since the numerator in equation ( 1.105) equals 1, the P-T 2-element doesn't have any zeros. The poles for the transient responses shown in Figure 1.22 with the damping ratios D == 0.1 and D == 0.5 are represented in a complex p-plane refer-

64

1.5 Root Loci

+--1

Tendency for D >> 1

I I P2

i-+

I I P1

---- - - -I - - - - 0 . 1111 -q I 1 _.!_=-mo (D-,j 0 2-1·) I T1 2 --=-m 1 0 (D+1D -1) I+- T2 - - - - - - - - - a) Casel:0>1 -jm 1 I

...

-(1

i -jro

b) Case 2: D = 1

P1.2

Tendency for D

Eo

u2

..

□..fq_,ntrol for

C: '-

cu

l

-

Conditioning Wv

0

.... "'C Q)

C: (1)

Q) Q) (!)-

W3 ""'

j.

83 .....

- -

R3

W2l •e2 ...: -.:

-

j

I

-

R2

Controlled system 2

I

w2

2

I

Controlled system 3

I

I

.g ~

w1

1

Conditioning Feed forward

§~

~~

□Feed ..!_orward c~ ntrol for

W1

--

1Controlled system 1

Wvi

:

14

I I I I

j I

-RA

I ~ I

-

-

R,

~

~

Ml

I

1

I

,e1

2

.. I

!JA

--

,l_-

~IA!

~

IA ~~

Kr

MM Ms - ,, - -

C

-

v-

-

I l I

l

JIOI

~ ... ~

1:

WM

Controller 3 Controller 2 Controller 1 Actuating ( In regard to the compensation of disturbance variables and the avoidance of a permanent system deviation, both controllers respond the same: the settling time is a function of the proportional coefficient KPn and the reset time T00 , and, when set-up according to the symmetrical optimum SO, is approximately:

Tsett Ref = Tsett PI ~ 20 · Tan = constant

{2.39)

t> The temporary system error is proportional to the load torque change, the sum total of the minor delay times Tan, and inversely proportional to the total moment of inertia 1 101 • If K Pn is increased, the settling time and the system error decrease. In contrast, a longer reset time T00 , as is effective at a set-up of 2 • SO, increases the settling time. The setting 2 · KPn, T00 = 4 Tan results therefore in a better disturbance response. (Compare Section 3.2.3, Figure 3.12.) By utilizing the computer options, the reference model can be used to achieve a speed controller configuration with improved command response, since only a P controller is active. A time delay caused by the integral component within the controller will be avoided, and the equivalent delay time of the speed control loop will only be 50% that of a conventional PI controller. From measurements at machines and drive facilities it becomes apparent that by setting a smaller setpoint delay and a higher proportional coefficient, conventional PI controllers can also achieve equivalent delay times of < 4 Tan. Also apparent is the fact that a P-T 1-element does not simulate a speed control loop sufficiently enough to reflect reality. The position control examples in Sections 3.2 and 3.6 will also show that not only the speed controller determines the dynamic limits, but that the characteristics of the mechanical transfer elements also strongly affect the response of the speed controller and the position control.

2.6 Command and Disturbance Response in the Frequency Domain Based on the simplified control loop shown in Figure 2.12, we will use a simulation to compute the frequency responses for the PI controller with reference model, and for the conventional PI controller. Regarding the control loop responses in the frequency domain, the command frequency response according to equations (2.29) and (2.31) is represented in Figure 2.15, and the interference frequency response per equation (2.32) in Figure 2.16. Here too, the frequency coordinates are related to Tan, so they can be used for any Tan values.

125

2 Control Loops for Feed Drives

i

lfwnl f,Tan .. dB 3 20 30 50 70 100 200 400 ,1()" 1000 10 10...---.,_---................________..........-.._ for 0,._=0.707: I for 0,._=0.5: I 3 3 1 _fo,..• Tan =113,1()" fo,..• Tan=160,1CT :

0

,- ........ """' -- ... "' "' ~

. I

' I I I

I I

.1011

.

I

-45 I

20 30 50 70 100 200

10

-90

'

I I I

I

I .I

'

I

I I I

' \J-oA:Q.5

I

~

I I

I

f,Tan .. 400 ,1cr31000

I

~A=Q.707

I I I

I

-135

-180

1-225 +-< ~

3

-

"'O

< m

Integral-action coefficient of the speed controller (Pl)

lKin. ·s/

:::0

(J)Q

~~: 0

8531

291

581

456

912

-

228

(Tl

0

~

g.

----

-

-

Speed setpoint delay tin1e (Pl) [TG 0 /ms)

Cl!

-

I]

-

:::J

-

-

(ii

-

-

0.274

1.10

-

-

C/J

"'O

Response ti1ne [Tresp/ms]

0.69

I 1.10

(ii

0.9

1.29

0.93

1.08

1.37

1.15

("D

Q.

(j )>

Charact. frequency, (at (5), (6)

z

-

vJ vJ

C( lfner

frequency/EA)

Charact. angular frequency. (at (5). (6) corner angular frequency WEA) [cooA ~ 1]

[/0 A/Hz]

547

412

372

263

584

292

273

292

3440

2 591

2338

165 l

3669

1835

1720

I 8J5

0 :::::,

ar

0 0

"'O

2 Control Loops for Feed Drives ➔

(2)

PI controller with reference model: Set to a damping ratio of DA = 0. 707. KPn is calculated based on the data in Table 2.3, equation (2.24). The command response is only affected by the P component. T00 will be set according to equation (2.26), it only affects the disturbance response. The response time Tresp in the command response is according to Figure 2.13: Tresp (2) ~ 4 · Tcm = 4 · 0. 274 · l 0- 3 s = l. 10 · l 0- 3 s is according to the optimization rules theoretically 4. 7 Tcm, however, the non-exact reference model does have an effect (see note to equation (2.38)). We obtain the characteristic frequency foA and the characteristic angular frequency ro0A from Figure 2.15: Tresp

foA (2) ~

and (3) ➔

113-10- 3 Tcm

rooA( 2)

113-10- 3 412 0.274 · 10- 3 s = Hz

= 2591 s- 1•

Conventional PI controller without setpoint delay: Set to double the proportional coefficient KPn and double the reset time T00 in relation to the symmetrical optimum (2 • SO). The response time Tresp based on Figure 2.13 is: Tresp(3)

~ 2.1 · Tan = 2.1 · 0.43 · 10- 3 S = 0.9 · 10- 3 S

and the characteristic frequency foA and the characteristic angular frequency ro0 A based on Figure 2.15 is: foA(3) ~

160 · 10- 3 T.

on

and ➔

(4)

rooA())

160- 10-3 = 372 Hz 0.43 · 10- 3 s

== 2338 s- 1

Conventional PI controller without setpoint delay: Kp0 and T00 are set based on the symmetrical optimum (SO) according to Table 2.3. The response time Tresp is according to Figure 2.13: Tresp(4)

~ 3 ·Tan= 3 · 0.43 · 10- 3 S == 1.29 · 10- 3 S

The characteristic frequency foA and the characteristic angular frequency ro0 A are according to Figure 2.15: foA(4)

and (5)

➔

134

~

113- lo- 3

rooA(4)

T. on

==

I 651

113-10- 3 - - - - == 263 Hz 0.43 · 10- 3 s s-

1

Conventional PI controller with setpoint delay: Set to double the value of Krn and double the value of Tnn in relation to the symme-

2. 7 Mcxleling an example of the Speed Control Loop

trical optimum (2 · SO). The setpoint delay is TGn sponse time Tresp is according to Figure 2.13: Tresp(5)

==

Tan· The re-

~ 3.4 · Tcm== 3.4 · 0.274 • 10- 3 S == 0.93 · 10- 3 S

The comer frequency !EA and the comer angular frequency will be calculated based on Figure 2.15:

~

fEA(5)

and

(6)

➔

160 · 10- 3

Tan

Wf:A( 5 )

mEA

160 · 10- 3 0.274 • 10-3 s == 584 Hz

== 3 669 s- 1

Conventional PI controller with setpoint delay: KPn and T00 are set pursuant to the symmetrical optimum (SO), Table 2.3. The setpoint delay is TGn == 4Tan· The response time Tresp ( which is also theoretically correct), follows from Figure 2.13: Tresp(6)

~ 7.6 ·Tan= 7.6 · 0.274 · 10- 3 S = 2.08 · 10- 3 S

and the comer frequency f EA and the comer angular frequency mEA from Figure 2.15 is:

~

fEA(6)

and

80 · 10- 3

Tan

~A( 6 )

80- 10- 3 0.274- 10-3 s == 292 Hz

== 1835 s- 1

The last two columns with the setting criteria ( 1) and (3) have been calculated in accordance with the previously named cases. The affect of the greater Tan on the attainable values of Tresp, foA and moA is apparent; they are independent from the moment of inertia. The step responses in Figure 2.13 apply until limits within the controlling device and the controlled system have been reached. The first limit value is the current limit given by the power converter. Short-time overloads of 2-3 times the current rating are common. For our example with the I FT6064-1 AF motor, at the current limit a torque of about 28.5 Nm is possible with the assigned power modul. At a set proportional coefficient in option A, criterion (2) with KPn == 5.66 Nm s/ rad the current limit will be reached with a normalized step in the speed setpoint of: i\WM

i\nM

Wrated

nrated -

MM max

28.5 Nm. 60 s. min-). 100

- - = - - > - - - - == - - - - - - - - - - - - -1 == KPn · Wratcd

5.66 Nm s/rad · 2 7t rad· 3 000 min-

C-1

1.6 /(

This value appears to be relatively small, it depends, however, on the great proportional coefficient K Pn , which, in tum is determined by the sum total of the minor delay times Tan. The limit for a conventional PI controller can be calculated in the same way~ when the step is applied without a set-

135

2 Control Loops for Feed Ori ves

point delay. When a setpoint delay exists, the current limit will be reached only at increased step levels, since the step function takes a delayed effect on the speed controller. Regarding the disturbance step response, Figure 2.14 indicates an referenced maximum value for the speed deviation of approx. - 0.28. For option A and setting criterion (2) (symmetrical optimum, 1Ff6064-1AF motor, including load moment of inertia), the short-term speed deviation at a load pulse of half the stall torque is Tan

AnM == -0.28-AML · - = ltot

== _

.

0 ·28 4 ·75

N

0.274 · 10- 3 s • 60 s • minm 0.0031 Nm s2

1

== _

. _1 7 · 3 min

The same method can be used to calculate the speed deviation for other motors and/or other load moments of inertia for a correspondingly calculated total moment of inertia J 101 and other sum totals of the minor delay times T00 in the speed control loop. When calculating the deviation of the angular velocity AmM, the ordinate value must be multiplied with 21t. Table 2.7 lists the computed characteristic and comer frequencies moA and mEA for the command response of the Pl controllers. The previous examples illustrated how to use the diagram (Figure 2.15). When we compare the characteristic angular frequency of the ideal P-T2 response obtainable from equation (2.34) for the PI controller with reference model, option A, setting criterion (2), we obtain with moA == 2580 s- 1 a value, which well matches the value in Table 2. 7. If we also compare the damping ratio, we see that with equation ( 1.98) and the value of approx. - 2.3 dB at the location moA obtained from the diagram in Figure 2.15, the damping ratio is 1

DA=---

2-IFI

WoA

I 2 · 0.767

== 0.652

while equation (2.34) results in a value of DA=

1

v'2 = 0.707

The small difference is caused by the inexact simulation of the reference model, due to which the control loop does not respond exactly like a P-T 2 element with a damping ratio of DA== 0.707 (at m0 A, the frequency response shouldn't have a resonant rise anymore). Now let's have a look at the utilization of the diagram in Figure 2.16. The logarithmic representation makes this more difficult. We'll use an example to explain its use. We obtain for the comer frequency fEz of a control loop

136

2. 7 Modeling an example of the Speed Control Loop

with Tan= 0.274 ms, which has been optimized based on the symmetrical optimum (SO) the value /Ez

~

78 · 10-3

Tan

78 · 10- 3 == 0.274 · l0- 3 s == 285 Hz

For this 285 Hz the associated referenced amplitude ratio is IFzl · lt0 JTan == - 5.86 dB. In order to obtain IFzl, the value must be multiplied by Tani1101 • To ensure that the physical units associated with the variables are correct, and to avoid difficulties with the mathematical rules for logarithms, we convert the dB value IFzl •J10JTan back into the linear value:

IFzl ltot __T._a_n = -5.86 dB

------>

IFzl

ltot

Tan

= 10-\: = 0.509

Accordingly, we obtain for drive option A with Tan== 0.274 ms and a total load moment of inertia of 1101 = 0.0031 kg m 2 for the amplitude ratio (lkgm2 =1Nms 2)

IFzl =

~M

ML

°·0.0031 Nms!

274 10 3 = 0.509 Ton = 0.509 · - = 0.045 ltot

I

Nms

If we want to obtain the absolute value of the amplitude ratio for the output variable motor angular velocity mM, we must calculate:

WM Tan IFzl ==... =27t·0.509-== ML

ltot

== 2 7t rad• 0.509 0.274 · 10-3s == 0.283 rad

0.0031 Nm s2

Nm s

When we apply a sinusoidal disturbance torque with an amplitude of ML== 4,7 Nm and a frequency of 285 Hz to the drive, the amplitude of the speed deviation is: A

nM

= IFzl -ML= 0.045

I -4.7 Nm= 0.21 s- 1 = 12.7 min- 1 Nms

The result is: A periodic disturbance torque with an amplitude of 4. 7 Nm results at the

corner frequency of the disturbance response offEz = 285 Hz in a speed amplitude of nM = 12.7 min- 1 overlaying the actual speed.

In another simulation, let's examine the stability criteria at the open control loop pursuant to Figure 2.5. If we use the values in options A and B and the setting criteria ( 1) and (2) as well as (3) and (4) in Table 2. 7 to calculate the frequency responses of the open control loop, we obtain Figure 2.19:

137

2 Control Loops for Feed Drives

i';1

i';I

40-

40

20

20

0

0

•

-20

-20

90

90

0

0

-90 . -180

-90

-------

-------------

i-270 -((JI°

Setting criterion

Kpn T00ls Kps

Tanis 1IK15

DA

= = = = = =

-180

1 9.35000

2 (= SO)

5.66000

00

00

1.00000 0.00027 0.00310 0.55000

1.00000 0.00027 0.00310 0.70700

Results

I -1 coos The innermost control loop, in this case the current control loop, generates the system time constant with its equivalent delay time TEI· The equivalent delay time of the speed control loop TEn is greater than the system time constant by a factor of approximately 2. TE 1 determines the dynamic response of all higher-level control loops. I> The command response of the closed speed control loop can be approximated by a P-T 1-element with a delay time of TEn = (l ... 2) T00 • A P-T2-element with the characteristic angular frequency of the drive moA and the damping ratio DA provides a better approximation. TEn and the attainable characteristic angular frequency moA are dependent on the sum total of the minor delay times Tan and the damping ratio DA.

139

2 Control Loops for Feed Drives

The setting takes place for use with a higher-level position control loop with a damping ratio of DA = 0.5 ... 0.6. If a matched setpoint delay is applied at the input of the speed control loop, the equivalent delay time increases up to approximately 4 Tan. The response of the speed control loop then approximates the response of a P-T 3-element.

t> The effect of disturbances on the speed in the closed speed control loop caused by a load torque change llML or by a periodic change in load with the amplitude ML is proportional to the sum total of the minor delay times Tan, and inversely proportional to the total moment of inertia 1101 •

t> The magnitude of the proportional coefficient KPn (the gain of the speed controller) is proportional to the total moment of inertia 1101 • Its set value, however, is limited, due to interference signals and due to the fact that often there is no stiff coupling between the load and the motor. This means that the value of the total moment of inertia is subject to restrictions. It must be kept low by a small load moment of inertia.

t> Low mechanical natural frequencies with low damping only permit a greater reset time Tnn of the speed controller than would be possible based on the optimization rules. When setting feed drives with low mechanical natural frequencies to the damping optimum (see Section 3.6) the proportional coefficient KPn has to be reduced additionally.

140

3 Feed Drive Position Control

3.1 Terminology, Basics 3.1.1 Configuration and Function The machining of a workpiece using a machine tool to create a desired shape requires that the tool and the workpiece move in relation to each other. In order to accomplish this, the individual feed drives must be adjustable in their position. Position controls with speed controlled electrical drives are predominantly deployed for this purpose. Position control based on a purely open loop, e.g. using stepping motors, is limited to situations where small drive powers are required. This type of control gains in importance for simple machines nowadays. The components of the closed-loop-controlled electrical drive are the control device or the converter system, which usually contains closed loops for speed and current, a clocked transistor actuator, and the servo motor. Used are DC drives, brushless DC drives, self-regulated synchronous drives and asynchronous drives being built for rotary as well as linear motion. Position control loops for numerically controlled machine tools as well as for tracer milling machines and tracer lathes are required to move the associated feed drives according to stored geometrical and velocity data as free of delays and distortions as possible by following the position command variables generated in the numerical control unit (CNC). Position controls are characterized by a circular system configuration. This provides a closed signal action path between the position command signal at the input of the position control loop, and the position feedback or the position controlled signal at the output of the position control loop. The closed action path of the position control ensures the position controlled variable to be detected by a suitable measuring device, and fed back to be compared to the position command variable. The response characteristic of the position command variable can also be modified by a command variable control, and may be supplemented by feed forward control values (see Sections 3 .4 and 3 .5). The position control deviation between the position command variable and the position feedback variable modulates the position controller, which, in the case of a cascaded configuration, consti-

141

3 Feed Drive Position Control

tutes the command variable for the speed control loop of the electrical drive (compare Figure 2.2). A position control loop, like any other control system, consists of a controlling device, a controlled system and a measuring device. The controlling device is comprised of the position controller, consisting of the comparator and the controlling element, as well as the speed and current control loops, which are components of the power converter. The controlled system includes the transistor actuator of the electrical drive with the servo motor, and the sum total of the mechanical transfer elements of the machine tool. The position controller, the closed-loop-controlled electrical drive and the mechanical transfer elements will hereafter be called the feed drive, where the position controller may also be integrated into the numerical control. Figure 3 .1 shows the basic configuration of a position controlled feed drive with the defined terms and variables. The individual functional components of the position control perform the fallowing tasks: The position controller compares the position command variable wx (position setpoint) to the position feedback variable rx detected and delivered by the position measuring system. Using the position control deviation ax (following error), the controller generates the setpoint value for the velocity Vs of the motion unit. Feed force fv

-------7

---------------r-- Ret~oacti;-----, r---------------, ,-----L_ __t._ _______ l :~ I

Position controller

- -

- -

n-, I-control Actuator loops

~

Servomotor

·---------------Electrical drive

__ Controlling device

___

Fig. 3.1

142

I

~

Mechanical transfer elements

_l __ __ _ Controlled system Measuring device

xi Position controlled variable (Actual position value) rx Position feedback variable wx Position command variable (Position setpoint value) ~x Position control deviation (following error, error variable)

•

I

l'i Actual velocity value v5 Velocity setpoint value fv Feed force (disturbance variable) vv Feed rate

Basic configuration of a position controlled feed drive

I

__

__

!

3.1 Terminology, Basics

The position measuring system uses the position controlled variable xi ( actual position value) of the machine part it is connected to, to generate the position feedback variable rx. The electrical drive converts the velocity setpoint value in the case of a linear motion unit into a corresponding actual velocity, and in the case of a rotary motion unit into a corresponding actual angular velocity. The mechanical system integrates the actual velocity into a travel distance or an angle. The mechanical transfer elements extend up to the point where the cutting forces attack the workpiece. They are, for example, configured as a lead screw drive consisting of a coupling, a transmission and a lead screw. The feed force F v resulting from the external load is the disturbance variable z. It is introduced into the position control loop via the mechanical transfer elements. Depending on the design of the mechanical transfer elements (distribution of masses and stiffness) an additional retroaction affecting the electrical drive does exist. This retroaction must be taken into account in the parameter settings of the speed control loop.

3.1.2 Position Measurement Measured Data Acquisition As stated in Section 1.1.2, information between the individual elements of the control loop can be transmitted using analog or digital signals. Depending on the type of measured data acquisition we thus differentiate between analog and digital position measuring systems. In analog measuring systems, the geometrical variable (travel distance or angle) is proportionally converted into another corresponding physical variable, e.g. into the phase angle or the amplitude of an electrical signal, which can be processed by the measuring system. This mode has the advantage that the actual position values can be exactly assigned to the measured variable. Disadvantageous is the amount of hardware required. In digital systems, the analog travel distance is broken down into a number of equally long travel distance increments A\'. The value of A\' determines the position resolution. The advantage of the digital mode is its simple hardware configuration and the secure transmission of measured values. There are special types of digital measuring systems, which detect the actual value in the form of an encoded numerical value, allowing the absolute position value to be measured.

Measuring Methods In addition to the type of measured data acquisition, position measuring systems are differentiated by their measuring methods. They are: absolute, incremental and cyclically absolute methods.

143

3 Feed Drive Position Control

Absolute position measurements can be used for analog as well as digital measured data acquisition. Each measured value is assigned a unique signal value. This has the advantage that the actual position value is available immediately after the machine has been powered up, and no home position has to be approached. Examples of a digital absolute measuring method are the absolute value encoders available for rotary as well as linear position pickup. They can be configured with several analog tracks being sampled in parallel, whose evaluation results in a binary encoded numerical value. Using a specific protocol, this numerical value will be transmitted in a parallel or serial manner to the controller. Incremental position measurements are only possible in digital measured data acquisition. There is no fixed zero position; any position can be defined as the zero position by using the zero mark in the encoder to set the measuring device accordingly, accordingly allowing a machine specific zero position to be set. This method has the advantage that only little effort is required for the measured data acquisition and transmission, and that the zero position can be easily selected and shifted. It has the disadvantage that, after the supply voltages have been cut off, the measuring device is not able to reconstruct the correct actual position. It is therefore necessary that after the machine has been powered up, to approach a mechanical reference marker, or to provide an appropriate storage device to ensure that the measured actual position values will be saved. Examples are the incremental pulse encoder or the incremental linear scale.

The cyclic/absolute measuring method is possible for analog as well as digital measured data acquisition. An absolute range of values is measured repeatedly and in cycles, while the number of repeats is counted incrementally. This method combines the advantage of a high resolution with the advantage of a low transmit frequency between encoder and the evaluating electronics. On self-regulated synchronous drives, this I-encoder measuring method is combined with the detection of the absolute rotor position. Generating the average value of two subsequent samples of the absolutely detected and summarized small travel distance increments allows the speed to be computed in such a way, that the position, speed and absolute angular position of the motor can be detected with one and the same measuring system. One of these methods is described in [2.2]. The following applies to the motor angular velocity (see Figure 3 .21): W

,...__, ~'PM M"""

~t

'PM(k) - 'PM(k-1)

TATn

(3.1)

The delay generated by this detection will be included with 0.5 • T ATn into the sum total of the minor delay times of the speed control loop. Additional information about measuring methods and the corresponding measured value transmitters can be found in [3.1] and [3.2].

144

3.1 Terminology. Basics

Measuring Points Depending on the location at the machine tool where the measured data acquisition takes place, we differentiate between direct and indirect position measurernents. Figure 3.2 illustrates these two options on a schematic diagram of the position control for a lead screw drive. In the case of the indirect measuring system, which is fixed to the servo motor, the I-encoder principle mentioned above can be applied (A). Linear motor drives operate with a direct measuring system. ·o ue to the fact that there aren't any rotating mechanical transfer elements, the difficulties caused by backlash described below do not apply. For rotary as well as linear drives, where the actual speed value is obtained from the position measurement, the actual value encoder must be installed free of backlash and tilt. Elasticities in the encoder and in the add-on parts have a very negative effect on the speed control loop. The resulting mechanical natural frequencies interfere with the frequency response of the drive and can result in a reduction of the characteristic frequency of the drive. When taking indirect position measurements, the position of the table is detected indirectly via the angular position of the lead screw, of a gear or of the servo motor. Backlash occurring past the measurement point takes effect outside the position control loop. It takes effect as permanent position error, and does accordingly influence the dimensional accuracy of the workpieces to be processed. This permanent position error can be partially eliminated by compensation via the position command variable. A back-

Current, and speed (resp. Position velocity) controller controller

Workpiece Feed force Fv ' - - - r - , Direct position measuring system Feed rate vv ____... Transistor Servoactuator motor Gearbox

~.__~~~~:GD-

M

For analog control

~ ~ ~ ~ - + - - L __

_j-f-+-f-+--~

For digital control 11-encoder system)

IT achoge nerator) Measurement conditioning

-, □

® (~

X

~

Indirect position measuring system

,...,__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ___,

II

I ◄ -------------------------------------- - ---j -----x1 d1r Direct position measuring system

xiin Indirectly measured actual

position value

W

with 1 -encoder system at the motor

@

with actual position va lue encoder at the lead screw or on the gearbox

xi dir Directly measured actual position value

Fig. 3.2

Indirect and direct position n1easurernent in the position control loop

145

3 Feed Drive Position Control

lash within the position control loop (in the case of configuration (B) e.g. at the couplings between motor and gearbox or lead screw, or measuring system and lead screw) has the same consequences for the settings of the position and speed control loops as described for the direct position measurement. For direct position measurements the measuring system is installed directly between the fixed and the moving machine part. Existing backlash talces effects inside the position control loop. Permanent position errors do not occur. However, in order to malce the layout and the optimization of the position control loop easier, the backlash must be small. This requires an accordingly constructive layout of the mechanical transfer elements. Backlash in the position control loop causes hunting (oscillations) around the controlled position or, with appropriate damping, an inexact position approach. The direct measuring system should be located as close as possible to the processing location (Abbe's Principle). Tilting errors at the feed slides result in errors at the workpiece.

3.1.3 Parameters and Characteristics of Position Control Loops In order to compute the response of position controlled feed drives in good approximation, the response characteristics of the individual elements and of the entire system must be known. In addition to the characteristics of the mechanical transfer elements, their steady-state and dynamic response is responsible for how well the machine performs its processes. Position controllers and, depending on its design, the position measuring device as well, exhibit a proportional response, i.e. they respond to changes in the corresponding input variables without delay. In dynamic analyses, the measuring device can in this case be neglected. For the cyclic/absolute encoders in digital control circuits mentioned above, however, the actual values must be computed first. It is therefore necessary to include a lag time element with its associated delay time in the block diagram. The proportional response of the controlling device is indicated by the proportional coefficient of the controller. This value, the velocity gain Kv, also called the Kv factor, is defined as the ratio between the setpoint velocity Vs and the position control deviation ax: V s Kv---

~x

(3.2)

For a straight-line motion with constant velocity the position control deviation, the so-called following error is based on this correlation: Vs

~x==-·

Kv

146

(3.3)

3.1 Terminology, Basics

The following error indicates how much the actual position value lags behind the position setpoint value. In straight-line motions the following error doesn't cause any contouring errors, provided the velocity gains in all position control loops generating the contouring path are identical. If the velocity gains are not equal in the case of straight-line motion, a deviation thus already arises due to a parallel offset of the actual path compared to the commanded path. Curved contours are always subject to dimensional errors, whose value is inversely proportional to the K v factor. Different K v factors and/or different dynamic characteristics of the axes involved in the motion cause additional errors, which increase as the contouring rate increases. The unit of the velocity gain Kv is s- 1 • Another commonly used unit is m/(min · mm) with the corresponding correlation m/min ls _ 1 = 6 • 10_2 -mm

( or mm/min) µm

and 1 m/min = 16.67 s- 1 mm

Common to all position control loops is the integral response of the controlled system. This I-element corresponds to the physical correlation between the actual value of the velocity controlled actual position variable xi.

vi

of a feed slide or table and the

The following applies to the linear motion of the feed slide: t

dxi v·--1 dt

and

Xi

=/

vi

(3.4)

dt

0

or to the rotary motion of the servo motor:

and

0.5 /Tax. Thus DL < I/ v'2. If the Kv factor continues to increase, the overshoot increases as well and the damping ratio within the position control loop decreases. Also apparent is that as the K v factor increases, the foil owing error decreases. On the right side of the figure, the K v factor is kept constant and the sum total of the delay times within the position control loop has been changed. In this case, it becomes apparent that for a certain K v factor the sum total of the delay times in the position control loop must be limited to a specific value. If this value grows larger, overshoots will occur. Based on this analysis of a simplified position control loop without any influence of the mechanical transfer elements, we can conclude: t> For a specific K v factor a specific delay time within the position control loop might not to be exceeded. t> With a fixed K v factor, decreasing this delay time within the position control loop does not improve the response.

In the following sections we will see that in addition to the sampling period within the position control loop, and the equivalent delay time of the drive, the dynamic characteristic values of the mechanical transfer elements do also limit the K v factor.

3.2 Dynamic Response Characteristics of Linear Position Control Loops 3.2.1 Command Response of Linear Position Control Loops with Indirect Measuring System For an indirect position measuring system, the mechanical transfer elements are located outside the control loop. Even though they therefore do not affect the stability of the position control loop, the attainable K v factor is limited by the condition, that we do not want any overshoot to occur at the end of the mechanical transfer system during the positioning. To accomplish this, we need to know the response of the mechanical transfer elements.

The Determining Mechanical Transfer Element Initially, we are assuming that there is no retroaction of the mechanical system present in the electrical drive. In Section 1.5.5 we derived the block diagram and the response of a 2-mass oscillator. Equation ( 1.113.1) showed us the response characteristics between the motor angular velocity and the motor torque. For a position control loop block diagram derived

154

3 .2 Dynamic Response Characteristics of Linear Position Control Loops

from Figure 3.3 however, we need the response characteristics between the rotation angle at the load Errors on the circular contour, which are dependent from the contour-

ing rate and the circle radius, and C> Distortions of the circular contour, which result, when the participating

axes do not respond equally. Let's first examine the errors in position control loops with indirect measuring systems, and let's assume that the axes have the same response characteristics. We base our analysis on Figure 3.3, which we simplified in Section 3.2.1 into Figure 3.5. In place of the P-T 1-element of the speed setpoint delay we now represent two versions of the electrical drive, as they were already introduced in Section 3.1.4:

(1) The drive as P-element without additional time delay in the position control loop (from here on referred to as 1st order position control loop),

(2) The drive as P-T 1-element with additional time delays in the position

control loop (from here on referred to as 2 nd order position control loop). We combine all delays, including that of the drive, into a P-T 1-element with the delay time Tax•

We will furthermore assume that the contouring process shall take place in the plane of both the X and the Y axis.

3.3.2 Errors caused by the Command Response at the Indirect Measurement System Here, the position control loop and the determining mechanical transfer element can be separately calculated. In position control loops of the 1st order, the command response of the position control loop follows the response of a P-T 1-element; in the position control loops of the 2nd order, it follows the response of a P-T 2-element. The command frequency responses of the position control loops are according to equations (3.6) and (3.7) in Section 3.1.4:

178

3.3 Circular Contour Errors

FwL 1.0

=

1 1 l+jro-

Kv

and FwL2.0

(3.22)

1

=

1

l+jm-+Um)

2

Kv

T. ax

Kv

The command frequency response of the determining mechanical transfer elements located outside the position control loops with their respective lowest mechanical natural frequency rod min is with equation (3 .13): 1 F mech ~ - - - - - - - - - l + jm 2Dmech 1 + (jm)2 1

(3.23)

roa min

(l)d min

(In this case we can neglect the numerator, because it only takes effect at angular frequencies far above rod min). With theses equations we can determine the circular contour errors resulting from the amplitude and the phase response.

Amplitude Response For a circular motion with a contouring rate v8 and the radius ri, an angular velocity equal to the angular frequency ro of the command variables of the position controllers can be defined. (The reciprocal of the time t per rotation is equal to the rotational frequency f). We obtain m from

21t · Ti VB==--=21t·f •Tj t and subsequently

Va 21t·f=m=ri

When we determine the absolute values of the position control loop command frequency responses with equations (3.22), we obtain according to equations (1.91) and (1.95) 1

IFwL 1. 0 I = -;:::===== 1

l+ro2 -K2 V and

(3.24) 1

IFwL 2. 0 I = ----;:::=================================2

(1-m ;j\(w ;J 2

179

3 Feed Drive Position Control

or, when entering the angular velocity determined above

1

IFwL I. 0 I = -----;::=====2 l

+

(

Ve

ri · Kv

)

(3.24.1)

and

1

IFwL 2.ol = ---;:::========= 2 2

Tax) +(Ve __I ) ( l - ~. ,f Kv ri Kv

For the 2 nd order response of the position control loops, the maximum Kv factor is known from equation (3.12). If we set the sum total of the minor delay times to T0 xl/µm +/-0...t

Pha~e ctTor

I

3.24) 2nd order

3 .24) 2nd order

3. l\kchankal Transf()r Elementc,

only Radiu\ error JO Hl. fdnun .h.1 min

Dml!ch

I

= 0.2

~ 1.7

'1rv nm:1/µm

52.3

= 50 Hz. Dnm. h I = 0. l

13.0 5.0

19.9

Distortion through: .h.j min X = 30 Hz. Dmei.:h I X = 0.2 .h.1 nun y = 20 Hz. Drm.·,:h 1 y = 0.2 - ---- Amplitude error l~r1- mci.:11l/µm 66.8

0.0

0.7

0.0

t27)

3.25) or (3.33. I)

I

1~,.~1

Pha~e error

-

16.2

--

Pha-.;c error

1~,.~

-

l+/-

-

s.1

--

I +/-8.0

-198

OA

2.8

11.3 ~

µm

0.1

(3.39) (3.25)

+/-8.8

f3.45) 3.30)

0.0

?3.39l 3.25

I

I

-

2.2

I +/- 5.\.0

+/- 5J.t

-

Distortion through: .fu min x = 50 Hz. Dme-~h I x = 0.1 0.1 f,1 Ullll y == ..io Hz. Dn11:1. h 1 'I -Amplitude etTor l-1 ,.F mc1.h l pm -

j

-

+/- 55.S

µm

-

--+/- 1.3

+/-8.0

-- --

i

(3.45) (3.30)

3.3 Circular Contour Errors

Table 3.3

(continued)

4. Position Control Loops, incl. the Mechanical Transfer Elements, Indirect measuring system Radius error ArvJµm Kv = 3 m/(min · mm) /d nun = 50 Hz, Dmech I = 0.1 Distortion through: Kv x = 3 m/(min-mm) Kv v = 3.15 m/(min·mm) Ten x = Ten v = 0 ms Amplitude error IArFl/µm

Phase error

IArcpl/µm

- 339.1 (Tm= 0 ms) -94.3 - 173.1 (T.,. = 5 ms) -44.9 + 4.0 (Tm= 10 ms) +4.7

-12.7 -6.0 + 0.7

-0.4 -0.2 +0.0

(3.28) (3.24) (3.25)

30.1

9.2

1.2

0.0

(3.40) (3.24~ J"orde r (3.25

+/-41.5

+/- 47.2

+/-47.6

+/-7.9

(3.45l st (3.29 1 orde r (3.30)

Distortion through: = 50 Hz, Dmech I X = 0.1 /d min Y = 40 Hz, Dmech I Y = 0.1 Kv x = 3 m/(min-mm) Kv v = 3.15 m/(min·mm) Tax x = Tax v = 0 ms /d min X

Amplitude error

IArFl/µm

40.7

11.9

1.6

0.0

(3.40) (3.24l I" orde r (3.25

Phase error

IArcpl/µm

+/-33.3

+/- 39.2

+/- 39.7

+/-6.6

(3.45l st ( 3.29 1 orde r (3.30)

-12.5 - 6.0 + 0.8

- 0.3 - 0.2 +0.0

(3.26) (3.36)

5.1 Position Control Loops with Direct Measuring Systems Radius error ArvJµm Kv = 3 m/(min·mm) /d min = 50 Hz, Dmech I = 0.1 Distortion through: /d min X = 50 Hz, Dmech I x = 0.1 /d min Y = 40 Hz, Dmech l Y = 0.1 Tax. x = Tax v = IO ms Kv x = Kv v = 3 m/(min·mm) Amplitude error !ArFwd/µm

Phase error

IArcpl/µm

- 334.5 (T.,, = 0 ms) -92.9 - 167.3 (T.,, ·- 5 ms) -43.5 + 11.2 (T.,, -. IO msJ + 6.1

see Figure 3. I 6 c)

1.6

0.2

0.0

+/- 1.6

+/- 0.1

+/-0.0

+/-0.0

-·- -

5.2 Position Control Loops with Direct Measuring Systems Radius error ArvJµm Kv = 10 m/(min-mm) /d min= 150 Hz, Dmech J = 0.1 Distortion through: /d min X = 150 Hz, Dmech t X = 0.1 /d min y = 135 Hz, Dmech 1 y = 0. I Tax x = Tax v = 3 ms Kv x = Kv v = 10 m/(min·mm) Amplitude error IA r..-"-d/µm

Phase error

1.1,q>l/µm

-----

7.4

-

- - --

-

_ _ ___

...._ _

--

(3.38) (3.36) (3.45) (3.37)

-

+ 2.4 (T.,, = 3 ms)

- 8.4 +0.6

- 1.1 + 0.1

- 0.0 + 0.0

(3.26) (3.36)

0.3

0.1

0.0

0.0

( 3.38) (3.36)

+/-0.0

+/-0.0

+/-0.0

+/-0.0

- 33.1

(T,,,,.., 0 ms)

-- --

~ ~

--

-

-

-- -

- -- - - - - - -

l -

- -

-

(3.45)

(3.37) -

- --

- -

199

3 Feed Drive Position Control

5.2 Position Control Loops with Direct Measuring Systems

In this example, the K v factors and the mechanical natural frequencies are significantly higher as they are in example 5.1. The errors are accordingly smaller. For comparison purposes, they have also been calculated for Tax= 0. In this case, their negative sign indicates the dominance of the position control loops. For the distortions we again assume a difference in the mechanical natural frequencies. Only small errors are generated. This example can be used for high-quality linear motor drives. However, rotary drives with small masses to be moved and good stiffness values can also achieve these results. In summary we can say that for the indirect position measurement the amplitude errors increase as the rotational frequency increases, while the phase errors increase as the contouring rate increases. For the direct measurement system in comparison, the phase error is also dependent on the rotational frequency. Here, the position controllers can counter the distortions caused by the mechanical systems and reduce them accordingly. All errors and distortions are, to a great extent, dependent on the K v factor and the lowest mechanical natural frequencies, as is shown in the previously derived equations. Careful, equal parameter settings for the axes involved in the circular motion are very important to ensure that the remaining errors will not be increased by the normally not influencible mechanical natural frequencies. The additional distortions caused by the shifting and tilting motions of the feed axes, including axes not involved in the contouring travel, cannot be calculated in advance. They are not contained in these examples.

3.4 Feed Forward Control From the previous comments about errors on contoured paths it is obvious that the factor v8 /(ri · Kv) significantly affects the errors and distortions. Equation (3.3) indicates that the quotient v8 /Kv constitutes the following error Lit, i.e. it is important for the accuracy on the workpiece to keep the following error as small as possible. The feed forward control shown in Section 2.1.1 can be used to generate a small following error with a cascaded configuration of the control loops and to increase the accuracy accordingly. As shown in Figure 2.2, this improvement of the command response is based on the fact, that the higher-level control of the motion guidance for the generation of the position setpoints, also generates the values for the speed setpoints and, if applicable, for the current setpoints which correspond to the acceleration. These values will be applied via matching elements directly as additive command variable wv to the respective control-

200

3.4 Feed Forward Control

lers. A correct match leads to the result that the bypassed controller must only correct the disturbance variables in its own control loop, which it achieves with relatively small manipulated variable changes. Command variable changes will be routed past the controller and are therefore processed quicker. In the case of the position controller being bypassed by a speed feed forward control, the following error will be substantially smaller. The speed dependent errors will be reduced, the distortions caused by different command and disturbance responses of the axes, in comparison, continue to exist with the same magnitude. The difficulty consists in generating exact feed forward control variables and determining a balancing filter in the setpoint channel of the bypassed controller. The sampling periods of the cascaded control loops increase towards the outside. The higher-level interpolator has in comparison to the position controller a sampling period, which is greater by an additional factor of 2-4. The generated feed forward control variables for the speed and, if applicable, for the current and the acceleration, require therefore intermediate interpolation, otherwise they would be applied in a too rough clock pattern. The balancing filters can only conditionally simulate the non-linearities and lag times that are present in the real controlled system. This is the reason the real-life applications are limited to a speed feed forward control and, if applicable, a current setpoint injection as acceleration feed forward control without complete compensation of the associated manipulated variables. Figure 3 .17 shows a simplified configuration of a speed feed forward control derived from Figure 3.3. From a command variable model with two integrators in the motion guidance of the CNC a speed feed forward control variable wv 0 is obtained, which, multiplied within the clock pattern of the speed control loop with the feed forward control factor K0 v will be added to the output of the position controller. The feed forward control factor allows this signal to be matched. In order to prevent the position controller to counter this applied signal, a simulated model of the position controlled system must be inserted into the path of the position command variable Wx. This is done with the balancing filter, which simulates the equivalent delay time TEn of the speed control loop, including the speed setpoint delay T0 0 • Initially, we'll look at an indirect measuring system with a stiff mechanical system. Depending on how accurately the controlled system can be simulated, the position will be controlled by the speed feed forward control variable wv 0 without intervention by the position controller. The purpose of the balancing filter is to ensure that the position command variable Wx always remains equal to the position feedback variable rx, i.e. that the following error Llx approaches 0. As we have seen, a simple filter configuration (e.g. a P-T 1-element) is not sufficient to simulate the speed control loop, so that the following error cannot be completely compensated. More complex filter configurations are required, which

201

3 Feed Drive Position Control

Command variable model in the motion guidance 1

Fine inter-

1 jro

I

1

1

polation

jro

I

~' Ten+ TGn

~

C

Balancing

Feedforward

-- - - control factor

- - - filter

omech ,. rod 1 r.---,

DA, WQA,

Kv

w. - -11.X )(

~

~

r

.- -

TGn

Wvn

t=

1 ,

- .. ~

-

-

~

-

L

1

Kv

FA

1+jro TGn Position controller

Speed setpoint delay

1

Ten ..._

-

~

Xiil ..._ I I ,,-• 1 Xi dir

-1

Fmech

- I

I/

I

__ I I_Ii. _ _ _ .J

jro

Electrical I-element drive Stiff mechanical syste,m

T• I I I I I I

Compliant mechanical : system ....1

-

_____

Fig. 3.17 Simplified block diagram of a position control loop with speed feed forward control

cannot be easily matched to the actual controlled system. (Compare Section 2. 7, Figure 2.18). The speed feed forward control reduces the following error Ax according to how well it is matched by the balancing filter and the feed forward control factor K0 v, and generates pursuant to equation (3.3) a higher Kv factor for the command response. This means that the position command variable affects the actual position value with the equivalent delay time of the speed control loop (including the speed setpoint delay, if applicable), and this translates, for example for TEn = 1 ms into an effective K v factor of 1000 s- 1 or 60 m/(min • mm). This can be derived from the effective frequency response F:x for the position control chain between Xs and xi. Without speed setpoint delay the frequency response of the system is

From the frequency response of the I-element in the reference model of the motion guidance we obtain Xs ==

202

I ns . -. JW

and accordingly

.

ns == X s · JW

3.5 Command Variable Control

This results in an effective frequency response of: F

*

Xi

•

I

=~ - = Jm · Knv ·FA·-.== Knv ·FA ~ J{iJ

This corresponds to the frequency response of the drive multiplied by the feed forward control factor. In Section 2.4.5 we identified the factor a 1 in the denominator of the drive frequency responses as the equivalent delay time TEn. From the frequency response equations of the position control loop (equations (3.6), (3.7) and (3.9)) we know that there the factor is a1 = 1/Kv. The fictitious effective Kv factor is accordingly Kv == 1/TEn. When the speed setpoint delay Tan is included, it must be added to TEn. The disturbance response, however, is only affected by the Kv factor, which will be set in the position controller. Regarding information about the K v factor of control loops with feed forward control it is therefore important to be aware of these differences in order to prevent misinterpretations. The analysis above was based on the assumption of stiffly coupled mechanical transfer elements without any retroaction. In the case of a compliant coupling of the mechanical elements, an additional oscillator with the frequency response F mech exists, whereby the elastic coupling prevents direct feedthrough of the feed forward controlled position to the workpiece. At the direct position measuring system, a delayed and distorted feedback variable exists, to which the balancing filter has to be adapted. The simple motion guidance model now doesn't meet the requirements of the existing system either anymore, and provides inaccurate feed forward control values. For these cases, [3.3] suggests to generate a four-fold integral-action chain, and to derive four feed forward control values. This method requires exact knowledge of the controlled system in order to dimension the three balancing filters. Due to the fact that the mechanical transfer elements usually are multi-mass oscillators with non-linearities, which cannot easily be simulated with simple models, this method not being widely used in real-life applications.

3.5 Command Variable Control The generation of command variables in the CNC links the geometrical data of the workpiece contour with the technical specifications for the machining. It is important to ensure that the resulting acceleration and velocity characteristics can be followed by the feed drives of all axes contributing to the motion. This can be achieved by control of the command variables. This control reduces the dynamic contour errors on numerically controlled machine tools generated by distortions in the signal transfer characteristic of the position control loops. The time characteristics of the posi-

203

3 Feed Drive Position Control

tion setpoints generated to create the smallest travel distance increments will be matched to the response characteristic of the position controlled system. This prevents the feed drives from reaching current and voltage limits. The reversal errors generated by the elasticities of the mechanical transfer elements will be reduced. In addition to improving the contouring accuracy, the command variable control decreases the stress on the drive systems, including the mechanical transfer elements. The improvement of the contouring accuracy and the stress reduction at the feed drives through the control of the command variables does, however, result in an extended motion. The degree of this lengthening depends on the parameters of the command variable control. When the controllable variables jerk, acceleration and velocity are limited, each step of the motion runs optimally with regard to time if any one of the three variables is at its limit value at any given time. A jerk limitation ensures a steady transition of the velocity and contouring characteristics of the sequential geometric data from one to the next. In this case, their time characteristics are not broken at any point, but show instead continuous transitions. This control of the command variables does therefore supplement the linear, circular and spline interpolation performed in the interpolator between the geometrical interpolation points of the workpiece contour. The travel distance-time diagram in Figure 3.18 compares the uncontrolled position setpoint of one axis to the position setpoint with contouring control for a straight-line positioning step. It indicates that the kinks at the beginning and at the end of the motion are replaced by steady transitions. For acceleration steps, this position setpoint control means that the linear time function of the setpoint value Xs ( t) will be replaced by a time function of a higher order. This results in tangential transitions at the beginning and at the end of the travel distance-time function and a lengthening of the time by t 3-t2 • For the velocity controlled position setpoint generation, the first derivative of the position setpoint, i.e. the command variable of the contouring rate

►

►

Without command variable control

Fig. 3.18

204

With command variable control

Position setpoints Xs for a straight-line positioning step

3.5 Command Variable Control

is partially constant. Restricting the acceleration has the effect that the 2 derivative of the position setpoint, i.e. the command variable of the acceleration as, is during a velocity change limited to the constant reference value aso. This is called acceleration controlled position setpoint generation. If in addition the 3rd derivative of the position setpoint, i.e. the command variable of the acceleration change (jerk), is limited to the reference value rso, it is called jerk controlled position setpoint generation . Figure 3.19 shows the time characteristic of the position setpoint and the command variables for the two methods. The time lengthening increases with each limited variable. Vs,

nd

For numerically controlled machine tools, an acceleration controlled position command variable control with a restriction of the command variables for the acceleration and the velocity is usually sufficient to achieve a good dynamic response. When contouring comers, the command variable of the velocity must then temporarily be brought back to zero in all axes simultaneously, otherwise the demand for an acceleration limitation is not guaranteed. The maximum acceleration alimit can be calculated based on the drive data. It depends on the acceleration torque and the total moment of inertia. The maximum jerk limitation is determined by the damping ratio Dmech and the determining minimum natural frequency of the mechanical elements (see Section 4.4.1 ).

tx•

-----------

.,,,,,-

------------_;;i--~-..--/

/ /

/

/

/

/ / _,L

0

/

/ / /

.

•

r

t1

t2

t3

t

t1

t2

t3

t

t2

t3

0 0

/

/

/

/

/

0

/

/

/

f8·

.

/

o~---------,.---.----0

.

850

0

I

I

0 t1 -as0,-----------

t

t'·

.

. o•

t1

t2

t3

t

.

I

I I I I

0

----

I I I I

t1

I I I I

-

I I I I I

t2

. t3

I I I I

t

►

r5 not limited

Acceleration limited

Jerk limited

Fig. 3.19 Position setpoint characteristic with acceleration and jerk limited contouring control

205

3 Feed Drive Position Control

The position setpoint control has advantages for starting and holding steps as well as directional changes within the position control loop. It offers the possibility to reduce the dynamic contour errors occurring on numerically contour controlled machines [3 .3]. Regarding the errors caused by the compliances of the mechanical transfer elements, there are also advantages. When the mechanical stiffness of an axis cannot be increased, the overshoot response can be improved by restricting the acceleration and the jerk at the expense of the positioning times.

3.6 Position Control Limits for Oscillatory Mechanical Systems In the previous sections, the need for a specific K v factor has repeatedly been pointed out. As the proportional coefficient of the position controller in the outer loop, it is dependent on the sum total of the minor delay times Tax and accordingly also dependent on the equivalent delay times TE of the lower level control loops. TEn and TEI of the speed and current control loops in digital controllers are due to small sampling periods, and in analog control loops due to the avoidance and reduction of circuit delay times sufficiently dynamic to reach Kv factors of 50 m/(min • mm) (equation (3.10) and Table 2.7). In digital position controllers, the sampling period TATx constitutes a first restriction of the Kv factor. If, for example, TATx = 1 ms, equation (3.12) results without speed setpoint delay and with an effective equivalent delay time TEn = 0.5 ms of the speed control loop in an attainable Kv factor of< 15 m/(min • mm). The mechanical natural frequencies of the mechanical transfer elements, which are elastically coupled together in the form of 1 and 2 mass oscillators, create another restriction. The lowest of these natural frequencies limits the value of the possible K v factor in the position control loop, as we have already seen in Sections 3.2.1 and 3.2.2. Let's take another look at this limit and examine it more closely.

3.6.1 Natural Frequencies We used equation (1.113.1) to calculate the transfer function G(p) = mMf MM of the 2-mass oscillator. The graphic representation of this response is shown in the Bode diagram in Figure 1.26. The mechanical transfer elements of a feed axis can be represented by several such 2-mass oscillators connected together. The Bode diagram does then indicate a series of zeros and poles (see Figure 3.21 ). The influence of the individual oscillators on the characteristic of the frequency response (i.e. on the absolute values of

206

3.6 Position Control Limits for Oscillatory Mechanical Systems

the amplitude and the angular changes at zero and pole) depends on the damping ratio Dmech, by which the physical damping coefficient i}co, the torsion spring constant cTo, and the moments of inertia J of the masses concerned are considered (see equations ( 1.112) and ( 1.118)). A zero causes in this frequency response a lowering of the amplitude response, and a positive phase shift in the phase response. A pole, in contrast, causes an amplitude rise and a negative phase shift. The pole frequency is higher than the zero frequency. The frequency spacing between the pole and the zero depends on the ratio of the two masses. In Section 1.5 we established the connection between zeros and poles of 2-mass and 1-mass oscillators. A 2-mass oscillator turns into a 1-mass oscillator when we lock one of the two masses. This simulates approximately the response occurring when the two masses are greatly different. In the block diagram, we can then more simply represent a 2-mass oscillator as a 1-mass oscillator with the smaller mass. (This correlation is shown on the derivation of the frequency response of the mechanical system, Figure 3.20). In Section 1.4.3 we defined that for mechanical transfer elements with Dmech < 0.3, the characteristic frequency m0 shall be equal to the natural angular frequency md . Therefore the following statements apply: • The natural angular frequency md 1 is approximately equal to the characteristic angular frequency m0 mech 1, equal to the zero angular frequency ~ of the 2-mass oscillator pursuant to equation ( 1.127): (3.46) At this frequency id 1 the 2-mass oscillator in Figure 1.24 oscillates, when the mass JM is locked. It corresponds to the natural frequency of a 1-mass oscillator with cTo and JL ( also called locked rotor frequency). This frequency determines the attainable dynamic of the transfer system and accordingly also the possible Kv factor. • The natural angular frequency md 2 is approximately equal to the characteristic angular frequency m0 mech 2 , equal to the pole angular frequency lOp of the 2-mass oscillator pursuant to equation ( 1.123): lOd 2 ~ lOo mech 2

==

(3.47)

At this frequency id 2 the coupled masses of the 2-mass oscillator oscillate when both masses are freely moving. It is the natural frequency of the 2-mass oscillator.

207

1 feed JJn·. e P 17.55 ms The proportional coefficient and the reset time for a P or PI controller, are according! y Nm s (0.01 + 0.03) kg m 2 KPn < - - - - - 3- - = 2 .28 17. 55 · 1o- s rad T 2(0.01 + 0.03) kg m 2 l nn > N == 35 . l ms 2.28 ms rad

Frequency Responses, Damping Optimum With a help of a simulation with the parameters calculated above we will illustrate the Bode diagrams of the frequency responses of the drive FA, and of the drive including the mechanical system F w. The frequency responses of the controlled system of the speed control loop F sn and of the mechanical system Fmech in Figure 3.23 a) and c) do not change when the proportional coefficient of the speed controller is changed. Figure 3.25 shows the frequency responses of the damping optimum.

221

3 Feed Drive Position Control

a)

b)

i

IFAI dB

fd min=12.8Hz

20------------------.

...

o~~

lii.:..

i

IFwl dB

fd min= 12.8Hz

20.....------..--------r-----, ______________ +3d8 -~:-:-:-_- - - .:~3d8 0 ========

-20....,___ _____..,_____--+-_...;::a,,,,--4

--40+-----+-----+-----t

--40 + - - - - - - - - + - - - . I W - - - - - - - 4

-60-----------

~+------+-----+-~-----4

0

0

-90 -180

-90 -180 -270

-270

!

1~

-q,r

101

102

103 .. f/Hz

!

10°

101

102

-q>I°

1a3 .. f/Hz

a) Frquency response FA=roM*/ro 5 , drive b) Frquency response Fw =CJh/CJJ 5 , drive+ mechanical system

Fig. 3.25 Bode diagram for the damping optimum of oscillating mechanical elements

When comparing Figure 3.25 a) to Figure 3.23 b), we can see now a significantly less dynamic drive, whose frequency response already starts to drop off at approx. 30 Hz. The amplitude collapse at the lowest mechanical natural frequency fd min is much more distinct, because the speed controller with the lower proportional coefficient KPn = 2.28 Nms/rad is less able to compensate it. The effect on the command frequency response Fw of the drive and the mechanical elements is that no resonant rise occurs at the point fd min anymore, and that the characteristic indicates approximately a 3rd order response (see Figure 3.25 b) in comparison to Figure 3.23 d)). The retroaction from the mechanical oscillator to the position control loop is now suppressed. Position Control Loop with Damping Optimum

In the position control loop with direct measuring system, the speed control loop, including the mechanical elements with the command frequency response F w per Figure 3.25 b) is in effect. Added are the computing cycle time and a sample-and-hold element with a total delay time of 1.5 · T ATx = 6 ms. The equivalent delay time of the speed control loop was T En > 17 .55 ms, so that the sum total of the minor delay times in the position control loop results in: Tax== TEn + 1.5 · TATx == 17.55 ms+ 1.5 · 4 ms== 23.55 ms

222

3.6 Position Control Limits for Oscillatory Mechanical Systems

Accordingly, we can use equation (3.12) to calculate the maximum Kv factor: 1

1 -1 2 2 2 • 23.55 ms= 1. s

Kvmax < 2Tax

= 1. 27 m/min mm

or

In comparison to the calculated values for the disturbance optimum without and with speed setpoint delay, this K v factor is clearly higher. The advantage of using a damping optimized setting is also apparent when we compare the three command frequency responses in Figure 3.26. Compared to the disturbance optimized setting in Figure 3.26 a) and b ), the frequency response with the damping optimum in the speed control loop indicates no resonant rise at the point /d min (Figure 3.26c) ). In spite of the higher Kv factor the damping is significantly higher, and the actual contour is less distorted. The difference between the two optimization methods is also apparent in the graphic representation of the step response of the speed control loop. The step responses with disturbance optimum are shown in parts a) and b) of Figure 3.27, and in part c) with damping optimum. At the damping optimum, the load angular velocity ~ is marked by a small, one-time overshoot. The motor angular velocity increases in steps. This corresponds to only a two-time load change with possible noise in the mechanical transfer

i

l~I fd min= 12.BHz 20....-------...___ _.....,

i

l~I uu

fd min=12.8Hz

20-----------.. . . +3dB

+3dB

+3dB

0

-20--------

-20---------

-20--------

-40--------

-40+---~-~-~

-40-+----+-~-~

-60---------

-60---------

-60-+----+--~~

0 -90 P-s.;~~

-90

0

0 F-=-...,_,. -90

-

-180 -180 -180 - 2 7 0 - - - - - - - -270-+------+---""""I -270+-------+---~

+1

10° cpl°

101

102

----1..,► f/Hz

a) Disturbance optimum, Kv= 0.43 . m without mm-mm speed setpoint delay

I

f-

10° cp/ 0

101

102

-----1..,~f /Hz

b) Disturbance optimum, Kv= 1 . m with speed mm-mm setpoint delay TG0 =23ms

I

f-

10° cpl°

102 -----1..,~ f /Hz

c) Damping optimum, Kv= 1.27 . m without mm-mm speed setpoint delay

Fig. 3.26 Command responses F wL == q,Jwx of the position control loop with direct measuring system and oscillating mechanical elements. Settings for the disturbance optimum and the damping optimum

223

3 Feed Drive Position Control

o.......,_______

O• - ...__---- ~ - - 50 100 150 200 0

0

50

150

200

----1-.-.tfms

---.--t/ms a) Disturbance optimum, /G,,,= 16.2 ~ without speed setpoint delay

100

I

b) Disturbance optimum, KPn=16.2~ with speed setpoint delay TGn= 23 ms

o----------0

50

100

150

200

--.,~ttms

c) Damping optimum,

/G,,,=2.~ without speed setpoint delay

Fig. 3.27 Step response of the speed control loop with oscillating mechanical system, settings for the disturbance optimum and the damping optimum

elements. The rise characteristic is significantly more damped than with the disturbance optimum. The rise time increases. The settling time is only a fraction of the settling time occurring at the disturbance optimum. Drives adjusted in this fashion can be used when the load provides little damping during the machining process. Their areas of application include robot axes, manipulating systems, and processes without load retroaction, e.g. laser machining and water jet cutting.

Additional Optimization Possibilities In the previous numerical example, the first term in the maximum condition according to equation ( 3 .51) is applicable in the calculation of the equivalent delay time TEn. At higher natural frequencies or at a greater load moment of inertia, the second term may become the determining variable. If the first term has the greater value and does therefore determine TEn, it is possible to increase the sum total of the minor delay times Tcm· However, this method has its limitations for the control of additional oscillators in the mechanical system based on the previously described correlations where Tan can be found in the denominator, as well as for the disturbance response according to Figure 3 .12 or equation ( 3 .17 .1). This greater value of Tan can be calculated with equations (3.49), (3.50)

T.

1 JM 2 JM + q · JL

~

1, since in most applications gears are used to reduce the speed. When the lead screw is coupled directly to the motor, i must be set to 1. The torque Mv at a 10 mm lead screw pitch can be roughly calculated as FvL/N 500

Mv Nm

-~---

(4.3.1)

(The product of the efficiencies 1Ja · 1JsM is calculated with approx. 0.8.)

fv

--+ fcov ----+

I I I

JGt2

I

I I. I

I I

I,

a

17G, J Gt n 1 J)_nM

--0-JG,1

234

Fig. 4.1 Machining forces and torques at a feed drive with gearbox and lead screw drive

4.2 Steady-state Layout

Table 4.1

Efficiency coefficients of feed drive gearboxes

Type --

- .

--

77G

Toothed belt, single-stage

0.95 ... 0.97

Gear drive, single-stage

0.9 ... 0.95 - --

Planetary gear drive (2-/single-stage)

0.88 ... 0.94

Gear drive, multi-stage

0.8 ... 0.88 - -

---

Cycloidal gear drive (single-stage)

0.85 ... 0.9

Harmonic drive

0.8 ... 0.9

Wormgear drive

ca. 0.7

In addition to toothed belt or gear drives, planetary gear drives, cycloidal gear drives, or the harmonic drive are also being used. They have a more or less large reversal error due to backlash and/or elastic deformation, a fact that must be taken into consideration when using them for accurate feed drives.

Torque for Friction and Losses The torque for friction and losses I,MR is comprised of several variables:

Guideway Friction The required torque reflected on the lead screw for the lead screw drive is

MRF = µF ·

hsp 2 7t • 1JsM

[(mw + mT) · g · cos a + FVT)

(4.5)

µp is the friction factor of the way. It depends on the pairing of the work

materials, and can be proportional to the velocity. Tables 4.2 and 4.3 contain references. The weight force (mw + mT) • g includes all linearly moved masses. Multiplied by the cosine of the angle of inclination a yields the component of the weight force vertical to the way that is added to the component FvT of the feed force. For using roller slideways, an additional constant force component Fway must be considered in equation (4.5). This component depends on the design and model, and must be obtained from catalog information provided by the roller slideway manufacturer. Roller slideways have a significantly smaller friction factor µp than simple slideways. Equation (4.5) is then: MRF

=

hsp 2 7t • 1JsM

{ Fway + µF[(,nw

+ mT) · g · cos a+ Fvr] }

(4.5.1)

235

4 Steady-State Layout and Calculation

Table 4.2

Friction factors of different guideway systems Roller slideways

Slideways with material combination Gray cast iron-Gray cast iron

0.18

Roller slides

0.005-0.01

Gray cast iron-Epoxy

0.1

Linear ball bearing

0.002-0.003

Gray cast iron-Teflon

0.06

Note: The force Fway obtainable from the manufacturer must be added (see examples in Section 4.2.3)

Table 4.3

Shifting forces

Aerostatic guideway

~0N/m

Hydrostatic guideway Forces are dependent from the width of the guideway and from the velocity. Information in reference to the length of the guideway.

4114N

With these specifications and some margins, the lFNl 186 motor with winding F meets the requirements. Its force/velocity diagram is shown in Figure 4.5. At v = 200 m/min we obtain at the voltage limit curve the force FM limit= 5 000 N. The technical data sheet provides the attractive force Fa = 22 500 N. That's why we must take a first iterative step back and calculate the friction forces again. By including the motor masses, including cables of approx. 2 • 65 kg = 130 kg, we obtain for the friction force:

FRF = 60 N + 0.007 [(600 + 130) kg· 9.81 m/s

2

+ 2 · 22500 N] = 425 N

tFM/N

11000 10000 9000

8000 7000

6000 hAlimil

FMN

4000

3000

0

50

100

1501 200 160 m/min Vmax V

m/min

258

250

...

Fig. 4.S Force/velocity diagram of the 1FN1186-5AF71 linear motor

4.3 Steps for the Dynamic Layout

When adding the force for the way cover we obtain for the sum total of the friction forces and losses

LFR =425 N +200N = 625 N The available acceleration force is then (equation (4.37)) Fa= 2 · 5000N -625 N = 9375 N

We can now calculate the ramp-up time with equation (4.36): ttt=

730-200 -3 _ s=260-10 s=260ms 60 9375

(The actual linear acceleration is aw= 9375 N/730 kg= 12.84 m/s 2 • It is thus slightly smaller than the specified value of 13 m/s 2.) The acceleration time is according to equation (4.29): ta =

-3 200 . s = 256 · 10 s = 256 ms 60 13

Calculating with the actual accelereration of 12.84 m/s 2, we see that the condition (4.30) tH < t 8 is fulfilled. The motor selection is permitted. As we will see in Section 4.4.3, as the maximum velocity is being reached, a steady characteristic in the velocity curve is generated by a jerk limitation or by the increase of the following error, and the acceleration limit value will not be reached. When braking at maximum velocity, the sum total of the friction forces is added to the deceleration force, so that neither during acceleration nor during braking the danger of an acceleration limitation does exist caused by the voltage limit. Nevertheless, this example illustrates that small differences between the mass and/or friction forces have a strong effect on the sizing of fast linear motor drives. These differences must therefore be carefully determined. (Example continued in Section 4.4.2.) It is necessary here as well to calculate the duty cycle related thermal stress, and to determine the power converter. The current at the limit force is according to the data sheet /M == 36.2 A. For the motors operating in parallel, the power module 56/112 A is most likely sufficient, which has reserve at the peak current to allow a ramp-up time of 260 ms. The thermal stress caused by the RMS value of the force can be calculated similarly to the example in Section 4.4.6. For linear motor drives, the increase of the friction forces at high speeds must be observed, and this force requirement must be calculated into the effective load.

259

4 Steady-State Layout and Calculation

4.4 Motion Diagrams 4.4.1 Jerk limitation In Section 3 .5 it was pointed out, that steady transitions in the motion of the feed axes during directional changes brought about by an acceleration and jerk limitation reduces the dynamic contouring errors. The excitation of oscillating mechanical transfer elements can also be limited by these restrictions to a tolerable value. The lower the minimum mechanical natural frequency /d min, and the smaller the damping ratio Dmech of the mechanical elements to be moved, the greater the overshoot past the specified setpoint position during acceleration and deceleration (compare Section 3.6). Overshoots during machining processes where tool and workpiece are in contact, are only allowed within the specified dimensional tolerances. They can be limited by decreasing the K v factor as well as by restricting the acceleration and the jerk to acceptable levels. In the case of contour controlled feed axes, the surface quality, e.g. for milling and turning processes, depends on the overshoot characteristic. Correct limitation settings are therefore very important. On positioning-only axes, e.g. on manipulators, these limitations can be used to achieve gentle positioning with low stress on the mechanical system. Since the productivity requires a specific acceleration, and since the jerk limitation time enters into the machining and non-machining times, we must match the jerk limitation in the command variable control to the specified acceleration and the existing mechanical system. In order to derive the proper adjustment criteria, the ramp response of a P-T 25 -element is measured in dependency on the ramp time of the input signal. The overshoot in reference to the steady-state value 1 as a function of the ramp time will be analyzed. We obtain the diagrams shown in Figure 4.6 for two defined natural frequencies at a damping ratio D = 0.1. If we set the mechanical system of a feed axis equal to a P-T25 -element with the lowest natural frequency fd min and the damping ratio Dmcch 1 , we can transfer the results of this measurement in approximation onto the effect of a jerk limitation. The ramp time then corresponds to the jerk limitation time, which we also call tR. The diagrams in Figure 4.6 indicate that the overshoot Vm initially drops sharply as the jerk limitation time tR increases, and then, starting with values of about IR> 1//d min asymptotically approaches 0, with minimums occurring at the points IR== n//d min· A practical setting for the jerk limitation is at values of tR > l //d min. For smaller jerk limitation times, i.e. at higher jerk limitation setpoint values rso, the overshoot increases noticeably.

l()(}

4.4 Motion Diagrams

t

..

t

..

►

t

- - - Measured values Vm ( tR) - - - - - Equivalent hyperbola for tR ~ 1/fd min - - - - - Equivalent straight line for tR < 1/fd min

fd = 10Hz

D=0.1

0.8 0.6

0.6 -

0.4

0.4 I

0.2

0.2

I

0 -t---r--r,- - - - - - - - - - - - - - - - -0 40

...........

0 .., _ ......................-................._..........,...._......................................0 40t 80 120 160 200 1/fd min

tR/ms

Fig. 4.6 Overshoot Vm of a P-T25 -element as a function of the input signal ramp time the natural frequencies /d = 10 Hz and 20 Hz, and a damping ratio D = 0.1

tR

►

at

A jerk is defined as

da r=dt

(4.38)

For the constant jerk rso and a final acceleration value aso, we obtain for the jerk limitation time (4.39) whereby a50 is the reference value for the acceleration limitation. At a constant jerk limitation setpoint value the jerk limitation time tR changes proportionally to the final acceleration value. This condition must be considered for short travel distances, where the set acceleration limit will not be reached. tR will then decrease, and the overshoot will increase due to excitation of the mechanical natural frequency (compare Figure 4.13 c)). For the range of the jerk limitation time tR > 1/fd min in which we are interested, we approximate the correlation empirically determined above with a hyperbolic curve. The overshoot will be accordingly

261

4 Steady-State Layout and Calculation

1 rso Vm~0.15---=0.15--IR ·/d min as0 ·/d min

(4.40)

For the sake of completeness, we define in the range of tR < overshoot will be approximated by a straight line as follows:

0.9as0 ·/dmin) ~0.9(1-0.9tR ·/dmin) =0.9 ( 1 - - - - - rso

Vm

1//d min

the

(4.40.1)

For the P-T25 -element we found based on equation ( 1.16) the characteristic of the overshoot as a function of the damping ratio with equation (1.103). (Compare Section 1.4.3 with Figure 1.22). The following applies at a unit step response to the overshoot of a mechanical system with the response of a P-T25-element: X·Dmech I

= e ✓•-o2mcchl

v*

m

(4.41)

v~ is the overshoot as a function of the damping ratio Dmech 1 at a sudden input change. This dependency will also be used to approximate the overshoot as a function of the ramp time in the restricted range 0.05 < Dmech 1 1//d min Comment

7.7

Dmech 1

50

50 0.2

125

0.1

0.2

325

450

2.5 5.6

no Approx. per equation (4.40) is invalid

2.5 7.7

20 yes

5.6 no

With approximation per equation

With approximation per equation

(4.40)

(4.40.1)

'1sm/µm

3.8

for

-

0.1

2.5

I

-

2.8

'1sm/µm

15

12

= 0.2: 2

2

( 2.5m/s 0.9-2.5m/s -5os- 1) 0.316- x-1 00-222 L\sm~-----·0.9 1 - - - - - - 3- - - ·e ✓ - - ~12µm 2 2 (21t • 50) s450 m/s

(Derivation from equation (4.44) with Vm from equation (4.40.1).) The results of these examples are summarized in Table 4.5. The effects of the jerk limitation time on the travel diagram are represented in Sections 4.4.2 and 4.4.3.

4.4.2 Running Travel Distances In practical applications of servo drives, there are two versions of the travel diagram for a specific travel distance s: • triangular velocity characteristic: the limit value of the velocity will either not be reached at all, or just at the peak, • trapezoidal velocity characteristic: the drive runs for some time at maximum velocity. Other factors are the influence of a jerk limitation, the effect of the existing delay and lag times, and the influence of the following error build-up and decay. Together, they cause deviations from the ideal response. The theoretical correlations to idealized curve characteristics are listed below.

266

4.4 Motion Diagrams

Acceleration limited Operation According to Figure 4.8 a) the specification of a constant acceleration + a 80 followed by -a80 having the same time span t 1 each results in a triangular velocity characteristic. For a linear motion, the maximum velocity is Vmax

= aso · t1 = 2 ~ = ✓2 · aso · St

(4.46)

ti

and the acceleration distance

I

St

~2

I

v~ax

= 2 aso. fi = 2 Vmax. t1 = 2aso

(4.47)

(The complete travel distance is 2s 1.) For a rotary motion, we obtain for the maximum angular velocity mmax

= aso . t1 = 2 "' 1 = ✓2 . aso . ( 80A) A · 0.52 s = 0.99 s 58

and we obtain 60 s/0.99 s < 61 steps/min. In this example, the power module does therefore limit the number of steps per minute. When the drive is operated with smaller steps, it must be noticed that the limitation setpoint values for jerk and acceleration remain fixed. During jerk limited operation, the travel time will increase in comparison to acceleration limited operation only. The term t 1 = 2tR + t3 does then not apply anymore but, similar to the 2nd case (equal travel distance, but greater acceleration setpoint value as Tr = 1.41 as Re), the time t 3 must be calculated similar to equation (4.64). Generally, the following applies to this time at a defined ratio of as Tr/as Re:

tf 1 3 - - - - + - t 2 --tR asTr/asRe 4 R 2

(4.86)

Figure 4.13 b) shows an example. For t 1 = 0.15 s the travel time at jerk limited operation is approx. 10% longer than 2 t 1• As the steps decrease, t3 equals 0, and the acceleration trapeze changes into a triangle. The increased acceleration as Tr will not be reached anymore. A similarity analysis between the resulting acceleration triangles provides a maximum acceleration of asor

=

2 tR · as Tr

(4.87)

2tR

We obtain the even further increased time 2t~ at jerk limited operation compared with acceleration limited operation only from *

2tR ==

3

4

2

- - - - · t 1 • lR as Tr/as Re

(4.88)

(Terms in Figure 4.13 c). * stands here for the decreasing jerk limitation time. It is derived from equating the travel distances for the rectangular and the triangular acceleration profile.) Figure 4.13 c) shows an example for t 1 = 0.08 s, where the time 2t~ for the jerk limited operation takes approx. 38% longer than t 1• The jerk limitation time t~ will be shorter than the time tR calculated from the limitation setpoint values aso and rso with equation (4.39). This means that the overshoot L'.\ sm from equation (4.44) increases, and the contouring accuracy decreases. L\sm rises significantly at t~ < 1/fct min. When t~➔ 0, then the acceleration will also approach 0 (as or ➔ 0). The overshoots dsm

?.96

4.5 Summary

do therefore again decrease at t~ < 1/(2 · /d min). The straight-line approximation provided by equation (4.40.1) cannot be used anymore ( see Sections 4.4.1 and Figure 4.6). The RMS loading on the motor and the power module will for these longer steps always be lower than in the borderline case calculated above. In this example of laser cutting the machining forces are very small. For a duty cycle similar to Figure 4.12, it is therefore sufficient to only consider the acceleration and the braking phases when determining the RMS value per equation (4.82). The continuous force of the linear motor is not dependent on the velocity. This simplifies the calculation of the RMS value. On feed axes installed on top of each other, where the base axis is designed with 2 motors operating in parallel according to the gantry principle, the load distribution on these motors of the base axis must be observed. Depending on the position of the upper axes, different acceleration forces are generated, and therefore different loads for the two base axis motors. (A calculation example can be found in the project planning guide of the 1FN ... linear motors series).

4.5 Summary 4.5.1 Feed Drive Layout based on a Flow Diagram The complex correlations involved in the selection of a feed drive can be represented in a schematic. We obtain the flow diagram in Figure 4.14. As the previous calculation examples have shown, additional optimization criteria can be pursued. Depending on the application, it is therefore practical to include different steps into the selection process. It should once again be pointed out that the natural frequencies of the mechanical transfer elements significantly influence the layout of a feed drive based on the steadystate requirements. It is therefore very important to perform the mechanical analysis listed at the end of the layout process. This analysis, however, requires appropriate equipment and knowledge [4.3], [4.4]. Of the iterative steps starting with the comparison of tH and t8 , the reduction of the moments of inertia and the selection of a different gear ratio promise the best results. Especially when the load moment of inertia is greater than 2 · JM, it is worth checking the mechanical system in regard to where reductions can be introduced without affecting a sufficient overall stiffness. Selecting a larger motor is usually of no use, since its own greater moment of inertia will consume what is gained by the torque. Improvements are possible, however, when the voltage limit characteristic of the modified motor yields a significantly higher acceleration torque.

297

4 Steady-State Layout and Calculation 17

Determine type of drive, forces, velocity, acceleration

•

_.

-

Calculate friction force, feed force, speed n2

Without · eor o · Decide With gearbox how to mount motor . · I

·~,

.,, r

,.

.

Calculate load moment of inertia, motor speed

,,,

.

'""!

,'

-

Pre-select motor based on technical documentation

,,

,,

'

I

I

Calculate moments of inertia J4 kt

,r

~

Calculate ramp-up time ~

~

and acceleration time ta

,

-...

tH > ts I

~, tH

~

'

ta

• • • •

Select different gearbox layout Change moments of inertia Select different stator winding Select larger size motor ~

Calculate RMS torque

MM RMS>

.

MIMs

MM 1ll0

~ MM RMS~ MM,no

'U

,--

Select power converter

,,

--

Calculate RMS current IMRMs I

IM RMS > IGRMS

,

IM RMS~ IG RMS

Drive OK -

~, Frequency analysis of the mechanical system Simulate frequency responses fsn, Fwn, f wt

Figure 4.14

298

Flow diagram for the layout of feed drives

-

-

1

Increase power module current

I

I

4.5 Summary

In the case of linear motor drives, the diagram is somewhat simpler, since the gearbox and other mechanical components do not exist. The moments of inertia are replaced by the masses, and operating in place of the torques are the forces. Here, it is important that the masses of the motor, the cable power track, and the measurement system are taken into account, and that the friction factors increase as the velocity increases. Equations (4.11) and (4.13) should be used as important quality criteria, which in the case of linear motor drives apply to the forces. In the case of roller slideways, the conditions for equation (4.11) often can not be met. This means that the mechanical system is only slightly damped, and the natural frequencies cause more interference. The result is a reduction of the K v factor as shown in Sections 3.2 and 3.6. The sizing equations for the three types of drives are summarized in Table 4.9. They are explained on the corresponding examples in the previous Sections 4.2 through 4.4. (Equation (4.89) through (4.101) are only listed in this table and do not appear in the running text.)

299

4 Steady-State Layout and Calculation

Table 4.9 Summary of equations for the sizing of feed drives Lead Screw Drive

Motor, without Gears F

\.1lap, Vy --~~ tI

---+

F

V

---+ FvL MRSL SL

mr

Workpiece mass

mw/kg

Table or slide ma~s

mT/kg *)

Guideway friction factor

µF

Limit torque at the voltage limit curve

Feed force

Fv/N

Motor torque

MM/Nm

Machining force in axis direction

FvL/N

Motor speed

nMfmin-•

Component perpendicular to way

FvT/N

Motor moment of inertia

JMf'kgm2

Coupling moment of inertia

JKfkgm

MMlimiJNm

Feed rate

vv/m/min

Rapid traverse rate

VRap/m/min

Lead screw pitch

hsp/m

Lead screw length

lsp/m

Lead screw diameter

dsp

Average lead screw bearing diameter

dmL/m

Lead screw moment of inertia

lsp/kg m **)

Lead screw bearing friction factor

µSL

Axial force in the lead screw direction

Fasp/N

Ballscrew nut efficiency

1JSM

Angle of inclination

a/o

Acceleration torque

Force for load independent friction on roller slideways

Fway/N

Load moment of inertia ( reflected on the motor shaft)

Force for way cover

Fcov/N

*)

nM

= n1

2

==

n2

lat ==0

i == 1 1JG == 1

2

Load torque

Plus all moved masses (Accessories, cable power track, linear measuring system, cover, counter balance) **) Plus bearings and half of the coupling ***) Including tensioning element, reflected on the respective shaft

300

4.5 Summary

Table 4.9

(continued)

Motor, with Gears

Gear ratio

j Rack-and-Pinion Drive

i==n 1/n 2 (4.4) n /min- 1

Input speed (shaft 1)

1

mw/kg

Table or slide mass

mT/kg

Guideway friction factor

µF

Feed force

Fv/N

1

Output speed (shaft 2)

n 2/min-

Moment of inertia gearbox (refl. on shaft 1)

JGJkgm

2

Machining force in axis direction

FvJN

Component perpendicular to way

FvT/N

Feed rate 2

***) Rapid traverse rate

Moment of inertia gear 1

JG1 1/kgm

Moment of inertia gear 2

lat 2/kg m ***)

Gear efficiency

TJG

MJNm

Workpiece mass

2

g == 9.81 m/s

2

*)

vv/m/min VRap/m/min

Pinion radius

rR/m

Pinion moment of inertia

JRjkgm2

Pinion mesh efficiency

1JRi

Angle of inclination

a/o

Force for load independent friction on roller slideways

Fway/N

Force for way cover

Fcov/N

Me/Nm J1Jkgm

2

301

4 Steady-State Layout and Calculation

Table 4.9

(continued)

Lead Screw Drive: Torques: Torque for the machining force:

Mv

=

- - - ~ Load torque:

FvL ·hsp 2 7t • i · 11a · 11sM

(4.3) Torque balance:

Way friction: Available acceleration torque: (without machining torque Mv)

hsp {Fway+µF[(mw+mT)· 2 7t · 1JsM •g • cos a + F VT]}

MRF=

(4.5.1)

Lead screw bearing friction (or manufacturer's info):

1 MRsL =µSL· drnL · Fasp 2

(4.7)

Ramp-up time:

(4.6)

Acceleration time:

4.1 o)

Linear acceleration setpoint value:

Way cover friction losses: Mcov

=2

hsp 7t • TJsM

· Fcov

Torque for friction and losses:

'""'MR = MRF + ~Cov + MR SL ~ l • TJa

-----1 (

Torque for the weight force: MG

==

Requirement:

(mw + mT ) · g · sin . a _ __. (4 . l 2) . hsp 2 7t · 1 • 11a · 11sM

Moments of inertia: Linearly moved mass: 2

lw+T = (mw +mT)G~)

(4.89)

Cylindrical body (e.g. steel made lead screw):

)4· lsp mm mm

lsp == 0.77. 10-12 (dsp kgm

2

Load moment of inertia: With single-stage gears:

With multi-stage gears:

(4.90)

For other materials, multiply with the factor 3 p/7.85 kg/dm •

Direct coupling:

Sum total reflected on the lead screw:

Total moment of inertia: (reflected on the motor shaft)

12 == lw+ T + lsp

302

(4.9 I)

4.5 Summary

Table 4.9

(continued) j Rack-and-Pinion Drive

(4.1)

Torque for the machining force: FvL-rRi

( 1.57)

(4.15)

Mv=-.--1 "f7G . 11Ri

Way friction:

MRF - Acceleration

= 'Ri { Fway + µF[(mw +mT) · g · cos a+ FVT]}

i + Upward traverse

i

+ Deceleration

(4.16.1)

- Downward traverse

Way cover friction losses: tH

ltot L\nM kg m2 • min- 1

s

9.55. Me

-------

Mcov

Torque for friction and losses:

~ MR

VRap

ts

m/min

s

_ aso m/s

(4.17)

(4.28)

Nm

60

= 'Ri · Fcov

(4.29)

~

= M_Cov + MRF l .

11G . 11Ri

(4.18)

2

Torque for the weight force: Ma

=. l

'Ri

"11G •11Ri

(mw + mT) · g · sin a

(4.19)

(4.30)

l -.l L -

Gtl

la12 +12 .2

+

(4.95)

l

12 lL=lat+~

Linearly moved mass: lw+T

(4.96)

l

= (mw +mT)r~i

(4.92)

Pinion (aluminium made): JRi

kg m

2

2.7 kg/dm3 . 0.77. 10-12. 7 .85 kg/ dm 3

(4.93)

4

lL == JK +12

(4.97)

(4.98)

. ( -dRi) ·/Ri -

mm

mm

Sum total, reflected on the pinion 12 = lw+T +lRi

(4.94)

303

4 Steady-State Layout and Calculation

Table 4.9

(continued)

Lead Screw Drive: Actual acceleration values: Angular acceleration:

Linear acceleration: aw== aM.

hsp . 21t · l

(4.78)

Speed values: Lead Screw speed: Feed traverse:

Motor speed: Rapid traverse:

(4.99)

Linear Motor Drive: Forces: Machining force:

F vL

Fw,v __. Few

~

Friction force of the way:

FRF== Fway+ µF [(mw + mT) · g · cos a+ +FvT + Fa]

(4.21.1)

Workpiece mass

Sum total of friction forces and losses:

Table or slide mass

LFR=FRF+Fcov

Guideway friction factor

µp

Feed force

Fv/N

Weight force:

Machining force in axis direction

FvL/N

Fa == (mw

Component perpendicular to way

FvT/N

Feed rate

vv/m/min

Maximum velocity

Vmax/m/min

Angle of inclination

a./0

Force for load independent friction on roller slideways

Fway/N

Force for way cover

Fcov/N

Attractive force

FJN

+ mT) · g · sin a

(4.22)

(4.23)

*) Plus all moved masses (Motor, accessories, cable power track, linear measuring system, cover, counter balance)

304

4.5 Summary

Table 4.9

(continued)

I Rack-and-Pinion Drive:

Linear acceleration: (4.77)

(4.79)

Pinion speed: (4.101)---Feed traverse:

n2

=

vv

21t · 'R.i

Rapid traverse:

VRap

n2=---

2 7t • rRi

(4.100)

Ramp-up and acceleration time: ~Loadforce: FL=FvL+LFR+FG (4.2)

Force balance: FM =FL+ Fe

(4.33)

Ramp-uptime:

ftt

s

mw+mT ~v kg m/min

(4.36)

60· Fe

N Available acceleration force (without machining force F vL):

Acceleration time: 437 ( )

- Acceleration

i

+ Deceleration

i + Upward traverse - Downward traverse

Limit force at the voltage limit curve:

Vmax

te

m/min

s

60. aso 2 m/s

-----

Requirement:

(4.29)

ttt