solution manual of cnc technology

8/18/2019 solution manual of cnc technology

1/17

Virtual CNC system. Part I. System architecture

Chi-Ho Yeunga, Yusuf Altintasa, Kaan Erkorkmazb,*

a Manufacturing Automation Laboratory University of British Columbia, Department of Mechanical Engineering,

2324 Main Mall, Vancouver, BC, Canada V6T 1Z4b Department of Mechanical Engineering, University of Waterloo, 200 University Avenue West, Waterloo, Ont., Canada N2L 3G1

Received 8 July 2005; accepted 4 August 2005

Available online 29 September 2005

Abstract

The paper presents a comprehensive virtual simulation model of a realistic and modular CNC system. The Virtual CNC architecture

represents an actual CNC, but with modular feed drives, sensors, motors, and amplifiers. The CNC software library includes a variety of

trajectory interpolation and axis control laws. Constant, trapezoidal and cubic acceleration profiles can be selected as a trajectory generation

module. The control laws can be selected ranging from a simple PID to complex Pole Placement, Generalized Predictive Control or Sliding

Mode Controller with friction compensation. When the Virtual CNC is assembled, its performance can be tested using frequency and time

domain response analyses, which are automated. The Virtual CNC includes both analytical tuning methods for linear controllers, as well as

Fuzzy Logic based expert auto-tuning system for Adaptive Sliding Mode Control. The paper includes detailed experimental verification of

the Virtual CNC.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: Virtual machining; CNC; Modeling; Controls; Interpolation

1. Introduction

The objective of the Virtual Manufacturing technology is

to design a completely digital factory where the part is

modeled, machined with optimized process parameters, and

resulting errors are predicted with corrective actions being

taken in a computer simulation environment. This paper

presents a Virtual model of a CNC system for machine

tools.

The CNC system consists of mechanical feed drives,

motors, amplifiers, position-velocity-acceleration sensors,

and real-time computer algorithms which generate time

stamped position commands through trajectory generation

and close the axis servo loops [1]. The Virtual CNC requires

a realistic mathematical model of each CNC component and

their logical interconnection.

There has been significant research reported in modeling

various trajectory generation algorithms [2–4], control laws

[5–9], and physical components of the drives such as

motors, amplifiers, ball-screw and linear drives with various

friction characteristics [10–13]. The majority of the past

research focused on trajectory generation, modeling and

identification of feed drive dynamics, and control law

development. When the trajectory generation leads to

discontinuous position commands, their first (velocity),

second (acceleration) and third (jerk) derivatives contain

fluctuation and sudden changes. High fluctuation in the

velocity results in poor feeds as well as high frequency

content beyond the bandwidth of the closed loop system,

which leads to poor tracking. High frequency content in

acceleration and large jerk commands may excite structural

modes of the machine, which are not desirable. To deal with

these problems, Pritschow [14] recommended jerk-limited

trajectories with trapezoidal or sine square acceleration

profiles, favoring the latter for better continuity. Chen and

Tlusty [2] also used the jerk-limited trajectories with

trapezoidal acceleration profiles in their feed drive

International Journal of Machine Tools & Manufacture 46 (2006) 1107–1123

www.elsevier.com/locate/ijmactool

0890-6955/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijmachtools.2005.08.002

* Corresponding author. Tel.: C1 519 885 1211 5214; fax: 1 519 885

5862.

E-mail addresses: [email protected] (C.-H. Yeung), altintas@

mech.ubc.ca (Y. Altintas), [email protected]

(K. Erkorkmaz).

http://www.elsevier.com/locate/ijmactoolhttp://www.elsevier.com/locate/ijmactool


2/17

development work. Later, Erkorkmaz and Altintas [4]

developed a trajectory generation with cubic acceleration

profiles for high speed milling operations.

Typically, the position errors originating from the CNC

are dependent on the robustness and tracking ability of the

axis control laws. Pritschow [8] noted the importance of

achieving high servo bandwidth, in order to be able to track sudden changes in the motion commands accurately, while

rejecting disturbances originating from axis friction and

cutting forces. Tomizuka [9] developed the Zero Phase

Error Tracking Controller (ZPETC), which provides a wide

bandwidth with zero phase delay. This design philosophy is

based on the cancellation of stable components of the closed

loop servo dynamics in a feedforward fashion. Boucher et

al. [6] used cascaded Generalized Predictive Control (GPC)

for accurate tracking of velocity and position commands.

Erkorkmaz and Altintas [5,12] applied the Sliding Mode

Control (SMC) technique for the control of high speed feed

drives. They first designed a SMC based on the rigid bodydynamics alone. Later, they improved their control

methodology to actively compensate the effect of the

structural flexibilities of ball screw drives.

Numerous publications have dealt with the modeling and

identification of the dynamics of the machine tool’s feed

drive system. Koren [15] used a simple first order linear

model for identifying the dynamics of classical feed drives.

In a lead screw driven mechanism, non-linear guideway

friction becomes a significant factor in degrading the

tracking and contouring accuracy, especially at motion

reversals in sharp corners and circular arc quadrants.

Armstrong et al. [10] presented an excellent survey on the

physics behind the friction phenomenon. They pointed out

that the typical friction characteristics for lubricated

metallic surfaces in contact could be described by the

Stribeck curve. Backlash is also responsible for the motion

accuracy limitation of CNC machine tools. Kao et al. [13]

reported an analytical method to observe and model the

backlash behavior on the motion accuracy of CNC lathes.

Tuning plays an important role in controller design. Over

the last two decades, the use of fuzzy logic control has been

widely proposed in literature for auto-tuning control

parameters. de Silva and MacFarlane [16] conducted

research on auto-tuning servo controllers through a

hierarchical, multi-level system that could be applied inthe control structure of industrial robotic manipulators.

Wickramarachchi and de Silva [17,18] applied a knowl-

edge-based hierarchical control system for fish processing

applications. Later, de Silva [19] investigated an analytical

framework for knowledge-based tuning of servo controllers.

He pointed out that a substantial reduction in computational

effort could be achieved through the application of the

analytical framework. Goulet et al. [20] has also used a

hierarchical control structure for a deployable orbiting

manipulator. They addressed the advantages of combining

crisp conventional control with knowledge-based fuzzy

logic control for stabilizing a space-based manipulator with

flexible deployable and slewing links.

Henceforth, the paper is organized as follows. The

architecture of the Virtual CNC system is presented in

Section 2, which includes the mathematical modeling of

actuators, friction, trajectory generation and control laws.

Fuzzy logic auto tuning of the axis control law is shown inSection 3. The Virtual CNC is experimentally verified in

contouring standard diamond and circular paths in Section 4,

and the paper is concluded in Section 5.

2. Virtual CNC architecture

The Virtual CNC system must be able to predict the

performance of a real CNC system during part machining.

The mathematical models of all working modules of a real

CNC must be identified and integrated as in a real CNC

system for accurate prediction of part machining perform-

ance. The architecture of the developed Virtual CNC system

is illustrated in Fig. 1, which resembles the real,

reconfigurable and open CNC developed in the Manufactur-

ing Automation Laboratory, UBC [21]. The Virtual CNC

system accepts reference toolpath generated on CAD/CAM

systems in the form of industry standard Cutter-Location

(CL) format. Each block in the CL file contains NC block

numbers, tool paths in the form of linear, circular and spline

segments, the cutter dimensions, tool center coordinates,

and feed speed for machining a particular part on a CNC

machine tool. Each tool path segment, such as linear,

circular or spline path, is passed through the trajectorygeneration algorithm which creates displacement, feed,

acceleration and jerk expression at divided segments [1].

Trajectory generation sets up the real time interpolation

parameters such as discrete displacement along the path and

its frequency. The interpolator generates discrete displace-

ment commands for each axis at every control interval. The

axis commands are passed on to the control law, which

shapes the overall response of the feed drive transfer

function, consisting of Digital to Analog (D/A) converter,

amplifier, servo motor, inertia, viscous damping, guideway

friction and lead screw backlash. The axis can be configured

to have acceleration, velocity and position sensors with

defined accuracy and noise parameters. The position error of

each axis is evaluated in the feedback loop and combined to

predict the contouring error at each control interval. The

user can select any control law, lead screw or linear drive

parameters, as well as amplifier, motor, friction field and

sensor so that most machine tools can be reconfigured

automatically by the user, who can add new modules or

modify the existing algorithms which are created in

MATLAB environment.

The Virtual CNC system consists of three main modules:

Feed drive, trajectory generation, and axis tracking control

modules.

C.-H. Yeung et al. / International Journal of Machine Tools & Manufacture 46 (2006) 1107–11231108


3/17

2.1. Dynamic model of feed drive

A typical ball screw drive system consists of current

amplifier, servo-motor, lead screw coupling mechanism,

ball screw with preloaded nut, table carrying the workpiece,

guideway friction and feedback sensors, as shown in Fig. 2.

The ball screw drive and rotary servo-motor are replaced by

a linear motor and bearings for direct drive feed

mechanisms. Both drive mechanisms are considered in the

Virtual CNC system.

(i) Rigid body motion: The linear dynamic model of a

classical feed drive is shown in Fig. 3. The control signal

ua [V] generated by the axis controller is applied to the

current amplifier which has a gain of K a [A/V]. In the

motor armature, the motor dynamic torque T m [Nm]

is produced, which is assumed to be linearly proportional

to the motor current with the motor torque constant K t[Nm/A]. The total dynamic torque delivered by the motor

is spent in overcoming the external disturbance torque T d[Nm] which is due to nonlinear static and Coulomb

friction in the drive and cutting forces, and in

accelerating the inertia ( J [kg/m2]) of the rigid body

motion of the table, and overcoming the system’s viscous

damping ( B [kg/m2 /s]). In return, the angular velocity u

[rad/s] of the motor shaft is determined, and then

transferred to the angular position q [rad] through

integration. The angular position of the motor shaft is

then transmitted as a linear displacement of the table by

the ball screw and nut mechanism, which has a

transmission gain of r g [mm/rad]. Further basics of feed

drive design and modeling are given in [1].

Considering Fig. 3, the angular velocity of the motor

shaft and actual position of the table can be expressed in

Laplace domain as:

uðsÞZ

1

JsC B K tK auaðsÞK

T dðsÞ

xlðsÞZr g

suðsÞZ

r g

s

1

JsC BK tK auaðsÞKT dðsÞ (1)

Fig. 3. Linear dynamics of a feed drive.

Fig. 2. Feed drive mechanism with a lead screw drive.

Fig. 1. Architecture of Virtual CNC system.

C.-H. Yeung et al. / International Journal of Machine Tools & Manufacture 46 (2006) 1107–1123 1109


4/17

or in state space as:

_uðt Þ

_ xðt Þ

( )Z Ac

uðt Þ

xðt Þ

( )C Bc

uaðt Þ

T dðt Þ

( )

Where A

cZ

K B = J 0

r g 0" #

and B

cZ

K aK t = J K1 = J

0 0" #

(2)

The discrete time equivalent of Eq. (2) is,

uðk C1Þ

xlðk C1Þ

( )ZFcðT sÞ

uðk Þ

xlðk Þ

( )C H cðT sÞ

uaðk Þ

T dðk Þ

( ) (3)

where FcðT sÞZe AcT s , H cðT sÞZ

Ð T s0

e Act dt

! Bc, k is the

sample counter, and T s [s] is the digital control interval.

(ii) Saturation of actuation system: the saturation of the

actuation system is considered at the input as,

uaðk ÞZ

umin; usðk Þ!umin

usðk Þ; umin%usðk Þ%umax

umax; usðk ÞOumax

8><>: (4)

where us is the current (i.e. torque) command generated by

the control law and ua is the actual control signal passed to

the amplifier. umin and umax are the lower and upper

saturation limits, which are identified from torque limits of

the motor T min/max as,

umin = max Z1

K aK t T min = max (5)

(iii) Quantization error and measurement noise: the

digital control command is converted into voltage for

analog drives through the Digital-to-Analog converter

circuit (DAC). The DAC quantization error ~u is assumed

to be present in the control signal after being quantized by

the DAC, which has a resolution of du:

usðk ÞZucðk ÞC ~uðk Þ0 ~uðk ÞZusðk ÞKucðk Þ (6)

where uc is the command control signal generated by the

motion controller. The quantization error ~u has a uniform

probability distribution with zero mean and is bounded by

Kdu /2 and Cdu /2 as shown in Fig. 4,

pð ~uÞZ1 = du; Kdu = 2% ~u%Cdu = 2

0; otherwise

( (7)

The variance R ~u of ~u is computed as,

R ~uZE ½ð ~uKE ð ~uÞÞ20as E ð ~uÞZ 0

r R ~u ZE ½ð ~uÞ2Z

ð CNKN

pð ~uÞ ~u2d ~uZ

ð Cdu = 2Kdu = 2

1

du~u

2d ~u

ZðduÞ2

12

(8)

The position measurement noise ~ x is defined as,

xmðk ÞZ xaðk ÞC ~ xðk Þ0 ~ xðk ÞZ xmðk ÞK xaðk Þ (9)

where xa and xm are the actual and measured axis position

respectively. Similarly, ~ x is assumed to have a uniformdistribution between Kd x /2 and Cd x /2 with a zero mean

and a variance of,

R ~ xZðd xÞ2

12 (10)

where d x is encoder resolution. The tachometer measure-

ment noise ~u is defined as:

umðk ÞZuðk ÞC ~uðk Þ0 ~uðk Þ Z ~uðk Þ Zumðk ÞKuðk Þ

(11)

where u and um are the actual and measured angular velocity

respectively. The profile of the tachometer measurementnoise is reasonably assumed to have a normal distribution

with zero mean and variance of R ~u. The variance of the

tachometer measurement noise is obtained either from the

catalog, or by monitoring the noise in the tachometer signal

when CNC is powered, but the axis is at rest.

(iv) Guideway friction: the disturbance torque T d consists

of non-linear guideway friction T f and cutting forces T creflected to the motor shaft as,

T dðk ÞZT f ðk ÞCT cðk Þ (12)

The dominant non-linear axis friction reflected on the

motor shaft is considered as the typical friction characteristic

for lubricated metallic surfaces in contact, which can be

Fig. 4. Quantization error of DAC: (a) Block diagram of DAC quantization, (b) Uniform probability distribution function.



5/17

described by the Stribeck curve, as shown in Fig. 5. The

Stribeck curve consists of four different regions: static

friction zone, boundary lubrication zone, partial lubrication

zone, and full fluid lubrication zone [10]. T stribeck can be

characterized analytically with sufficient closeness as:

T Gstribeck ðuÞZT

Gstate

Ku = uG1 CT

Gcounlð1Ke

Ku = uG2 ÞCT viseu

(13)

where T stat and T coul are the static friction and the Coulomb

friction torque respectively. T visc is the viscous damping

coefficient that corresponds to B in Eq. (1). u1 determines the

spacing between the boundary lubrication and partial fluidlubrication zones. u2 determines the spacing between the

partial lubrication and full fluid lubrication regions. The

superscript ‘G“ corresponds to positive and negative

directions of motion. The parameters of the Stribeck friction

curve can be entered in the drive model. The characteristics

of guideway friction torque T f in discrete time domain can besummarized as follows,

T f ðk ÞZ

0; uðk ÞZ0 and T aðk ÞZ0

T aðk Þ; uðk ÞZ0 and T Kstat!T aðk Þ!T

Cstat

T Cstat; uðk ÞZ0 and T aðk ÞRT CstatO0

T Kstat; uðk ÞZ0 and T aðk Þ!T Kstat!0

T Cstribeck ðuðk ÞÞ; uðk ÞO0

T Kstribeck ðuðk ÞÞ; uðk Þ!0

8>>>>>>>>>><>>>>>>>>>>:

(14)

where

T aðk ÞZT mðk ÞKT cðk Þ*T mðk ÞZK aK tuaðk Þ (15)

Since the effect of viscous damping is included in the

model of rigid body dynamics, the friction torque expression

in Eq. (13) is re-written, neglecting viscous damping

component as,

T Gstribeck ðuðk ÞÞZT

Gstate

Kuðk Þ = uG1 CT

Gcounlð1Ke

Kuðk Þ = uG2 Þ (16)

(v) Backlash: The spacing or dead-zone between the

screw and nut creates the backlash, which is modeled by: a

disengaged zone, engagement in the positive direction, andengagement in the negative direction, as shown in Fig. 6.

The analytical expressions for the backlash model are

summarized as follows,

where Db denotes the backlash of the feed drive

mechanism. xa and xl are the actual axis position due

to backlash and the axis position obtained from Eq. (3)respectively. d Ke and d Ce are the positions of the negative

and positive ends of the dead-band. The axis position is

assumed to be located at the middle of the dead-zone at

initial conditions. Therefore, d Ke and d Ce are initially set

to be:

initial0d Ke ZK Db = 2 and d Ce ZC Db = 2 (18)

The parametric model of the overall feed drive is

shown in Fig. 7.

Fig. 5. Stribeck friction curve for two lubricated metallic surfaces in contact.

xaðk ÞZ

xaðk K1Þ; d Ke% xlðk Þ%d

Ce

xlðk ÞK Db = 2; xlðk ÞRd Ce 0set d

Ke Z xlðk ÞK Db and d

Ce Z xlðk Þ

xlðk ÞC Db = 2; xlðk Þ%d Ke0set d

Ke Z xlðk Þ and d

Ce Z xlðk ÞC Db

8>><>>: (17)



6/17

2.2. Trajectory generation mechanism

The trajectory generation mechanism implemented in

the Virtual CNC is illustrated in Fig. 8. After interpreting

the CL file, the start and end coordinates of the toolpath,

the types of the tool movement and the feedrate are

recognized and stored into a buffer. By executing the

buffer block by block, the descriptions for each toolpath

segment are obtained and then passed to the trajectory

generation process sequentially. The trajectory generation

algorithm identifies the distance to be traveled, and

divides it into acceleration, constant velocity and

deceleration sub-segments depending on kinematic pro-

files selected in the CNC design. The Virtual CNC

system presented here gives a choice of constant,

trapezoidal, or cubic acceleration profiles, as shown in

Fig. 7. Overall block diagram of feed drive model.

Fig. 6. Classical dead-zone backlash model: (a) disengaged zone, (b) system engaged in positive direction, (c) system engaged in negative direction,

(d) backlash plot.



7/17

Fig. 9. Step changes in the constant acceleration profile

lead to infinite jerk and severe acceleration disconti-

nuities, which contain high frequencies in the generated

position commands. Such high frequency content may

excite structural vibrations and cause severe tracking

errors in high-speed machines. Cubic acceleration leads

to much smoother position commands with continuity in

velocity, acceleration and jerk, which is better suited for

high speed and precision drives, but at the expense of

increased computational load on the CNC computer.

The details of each algorithm can be found in [22] and

are not repeated here.

2.3. Axis control module

There are a significant number of control laws, which can

be implemented in CNC systems. Typically, any axis

control law has two components: the feedforward part

which processes the reference position commands, and the

feedback part that shapes the measured states such as

Fig. 8. Mechanism of trajectory generation.



8/17

position, velocity and acceleration to stabilize the closed

loop dynamics, as shown in Fig. 10. The control law

generates the control signal command (i.e. uc(k ))) which is

sent to the physical drives as:

ucðk ÞZ DTð zÞ X rðk ÞK Dsð zÞ X mðk Þ

where

DTð zÞZ d T1ð zÞ d T2ð zÞ d T2ð zÞ

1!3

DSð zÞZ d S1ð zÞ d S2ð zÞ d S2ð zÞ

1!3

X rðk ÞZ

xrðk Þ

_ xrðk Þ

€ xrðk Þ

264

375

3!1

and X mð zÞZ

xk ðk Þ

_ xk ðk Þ

€ xk ðk Þ

264

375

3!1

8>>>>>>><>>>>>>>:

(19)

DT(z) and Ds( z) are the matrices corresponding to the

feedforward and feedback transfer functions respectively, in

the discrete time domain. X r(k ) is the reference axis

command state vector and X m(k ) is the axis measurement

state vector.

The presented Virtual CNC system has a number of

user reconfigurable control laws, which have all been

experimentally proven on our open CNC system [21].

The conventional control laws include P-PI control, PID

control and Lead-Lag control [1]. More sophisticated

control laws include Pole Placement Control (PPC) [23],

Zero-Phase Tracking Error Control (ZPTEC) [9], Gener-

alized Predictive Control (GPC) [6], and Sliding Mode

Control (SMC) [5]. While the conventional control laws

can be found in standard control texts, details of the moreadvanced control laws can be found in the referenced

articles.

3. Auto-tuning of axis control laws in virtual CNC

It is desirable to tune the feed drives in a simulated,

virtual environment, provided that the mathematical model

of the system is sufficiently accurate. The virtual tuning

allows modification to the machine tool drive mechanism as

well as proper selection of motors and sensors during the

initial design. Once the control law is well tuned and

optimized, further adjustments can be made on the real

machine. Depending on the structure of the control laws,

they are either analytically tuned according to desired

performance criteria, or manually tuned with trial-and-error.Pole placement and model based control laws can be tuned

by defining the rise time, overshoot and bandwidth of the

closed loop position control system [1]. Since the analytical

tuning techniques can be found in literature, a newly

developed fuzzy logic based tuning of an adaptive Sliding

Mode Controller is presented here.

3.1. Sliding mode controller

The control law of rigid body-based Sliding Mode

Controller (SMC) for high speed feed drives is given as

follows [5]:

uSMCc ðk ÞZ J e lð _ xrðk ÞK _ xmðk ÞÞC € xrðk Þ

C Be _ xmðk Þ

CK SS ðk ÞCd h

ðk Þ (20)

where

S ðk ÞZl½ xr ðk ÞK xmðk ÞC½_ xr ðk ÞK_ xmðk Þ

d ^ðk Þzr l T s z

zK1 xr ðk ÞK

T s z

zK1 xmðk Þ

24

35C xr ðk ÞK xmðk Þ

8<

:

9=

; J eZ

J

K aK t r g ; and BeZ

B

K aK t r g ;

8>>>>>>>>><

>>>>>>>>>: (21)

Fig. 9. Feed motion profiles: (a) constant acceleration, (b) trapezoidal acceleration, (c) cubic acceleration.

Fig. 10. Axis control law in a standard form.



9/17

Je and Be are the equivalent inertia and viscous

damping reflected on the motor shaft respectively, S is a

stable sliding surface function and d̂ is the axis

disturbance estimated using a simple observer for

adaptation. The control parameters that need to be

tuned are: sliding surface bandwidth l [rad/s], feedback

gain K S [V/(mm/s)], and disturbance adaptation gain r

[V/mm]. l is assumed to be fixed and is determined

according to the achievable bandwidth of the drive, and

the two control parameters (K S and r) are considered inthe auto-tuning of SMC.

Considering Fig. 11, the auto-tuning process has a three-

level hierarchical structure: bottom layer for SMC which

needs to be tuned, intermediate layer for system perform-

ance evaluation, and upper layer for decision making using

fuzzy logic tuning. A smooth back-and-forth motion

command is applied to the closed loop. The system

performance is observed and characterized in the inter-

mediate layer, where descriptors of the system response

such as oscillation of the control signal, stability of the loop,

and tracking error are evaluated. If the system response isnot found to be satisfactory, then new control parameters are

Fig. 12. Fourier spectrum analysis of control signal: (a) clean signal, (b) noisy signal.

Fig. 11. Hierarchical structure of auto-tuning strategy.



10/17

assigned to the SMC through fuzzy logic tuning in the upperlayer. The control parameters are iteratively adjusted until

the overall system performance becomes satisfactory.

3.2. System performance evaluation and specification

System performance attributes are evaluated and classi-

fied into linguistic statements such as acceptable, over-

qualified, unsatisfactory, and so on. The oscillation level of

the control signal, loop stability, and tracking accuracy are

evaluated as follows.

(i) Oscillation level of control signal (OSC): a noisy

control command contains high frequency harmonics whichdamages the actuators and degrades the tracking accuracy;

hence, it needs to be constrained. The frequency content and

strength of the unwanted jitter can be evaluated by

considering the Fourier spectrum of the control commands,

see Fig. 12. The index of deviation corresponding to the

oscillation level of the control signal, OSCindex, is defined as,

OSCindex ZS cð H maxK H meanÞ

ð L maxK L meanÞ

wherelow frequency range00% f %F thresh

high frequency range0F thresh% f %F nyquist

9>>>>>=>>>>>;

(22)

F thresh is introduced as a threshold value to distinguish

between low and high frequency regions. F Nyquist corre-

sponds to the nyquist frequency, or the half of the control

sampling frequency. L max and H max are the maximum power

spectrum magnitudes of the dominant frequencies in the low

and high frequency ranges, respectively. Similarly, L mean and

H mean are the averages of magnitudes corresponding to thelow and high frequency ranges in the power spectrum

respectively. S c is a scaling factor to adjust the sensitivity of

the control signal oscillation level, as needed.

(ii) Stability margin analysis (PHM): The system

stability can be analytically defined by evaluating the

phase margin (fm) of the loop frequency response as,

PHMindex Zfm Z1808Carg b Glð jugÞ c (23)

where PHMindex is the index corresponding to the system

stability. G( ju) is the loop transfer function corresponding

to the linear system model in the frequency domain. The

gain cross-over frequency ug is the frequency where themagnitude of the loop transfer function Gl( ju) is equal to

unity.

Table 2

Mapping from index of deviation to discrete performance index

Discrete performance index Index of deviation for OSC Index of deviation for PHM Index of deviation for TRE

1 1.00%OSCindex PHMindex!308 20 mm%TREindex2 0.75%OSCindex!1.00 308%PHMindex!408 5 mm%TREindex!20 mm

3 0.50%OSCindex!0.75 408%PHMindex!458 1 mm%TREindex!5 mm

4 0.25%OSCindex!0.50 458%PHMindex!608 0.1 mm%TREindex!1 mm

5 0%OSCindex!0.25 608%PHMindex!1808 0%TREindex!0.1 mm

Table 3

Membership functions of fuzzy input variables for OSC, PHM, and TRE

Fuzzy input variables—OSC, PHM, and TRE

5 Primary fuzzy set Discrete performance index

For OSC and PHM For TRE 1 2 3 4 5

HIUN EL 1 0.7 0.3 0 0

LIUN LA 0.7 1 0.7 0.3 0

ACCP ME 0.3 0.7 1 0.7 0.3

LIOQ SM 0 0.3 0.7 1 0.7

HIOQ OK 0 0 0.3 0.7 1

Table 1

Primary fuzzy sets for performance attributes of OSC, PHM, and TRE

Performance attributes Discrete

performance

index

Index of deviation

(IOD) for OSC, PHM,

TREOSC and

PHM

TRE

HIUN EL 1 Threshold(1)%IOD

LIUN LA 2 Threshold(2)%IOD!Threshold(1)

ACCP ME 3 Threshold(3)%IOD!

Threshold(2)

LIOQ SM 4 Threshold(4)%IOD!

Threshold(3)

HIOQ OK 5 Threshold(5)%IOD!

Threshold(4)

For OSC and PHM : HIOQ, over-qualified; LIOQ, little-qualified; ACCP,

acceptable; LIUN, little unsatisfactory; HIUN, highly unsatisfactory

For TRE : OK, okay; SM, small; ME, medium; LA, large; EL, extremely

large



11/17

(iii) Tracking accuracy (TRE): During the back-and-forth

motion cycle, the tracking error is obtained by comparing

the reference ( xr ) and the measured ( xm) axis position at

each time step. The maximum absolute value of tracking

error (i.e. TREindex) is evaluated as,

TREindex Zmaxðj xrðt ÞK xmðt ÞjÞ; 0% t %T end (24)

where T end is the total duration needed for one cycle of

back-and-forth motion. Once the system performance

attributes corresponding to the oscillation level of control

signal (OSC ), the system stability (PHM ), and tracking

accuracy (TRE ) are evaluated, they are mapped into discrete

performance indices corresponding to selected primary

fuzzy sets as shown in Tables 1 and 2.

3.3. Development of fuzzy decision table

A Fuzzy Decision Table, which is used to tune the

control law parameters in real time, is first prepared off-line

as follows.

(i) Membership functions: In development of the fuzzy

decision table, the system performance attributes and the

corresponding tuning actions must be fuzzified into pre-

defined membership functions. In auto-tuning of SMC, the

fuzzy input variables are the system performance attributes

based on the evaluation of OSC, PHM, and TRE. The fuzzy

output variables correspond to the tuning actions that are

defined as follows: DK S and Dr denote the changes in K Sand r, respectively. The membership function for each

fuzzy input or output set is designed in a discrete tabular

form, by applying a triangular shaped function with a full

Table 5

Heuristic manipulation of SMC parameters

Performance attribute Action for performance improvement

K S r

Oscillation of control signal (OSC) Decrease No change (or slightly decrease)

System stability (PHM) Decrease DecreaseTracking accuracy (TRE) No change (or slightly decrease) Increase (decrease if system is unstable)

Table 4

Membership functions of fuzzy output variables for DK S and Dr

Fuzzy output variables —DK S and Dr

7 selected fuzzy set Representative numerical value (ci)

K3 K2 K1 0 C1 C2 C3

NL (negative large) 1 0.9 0.8 0.7 0.6 0.5 0.4

NM (negative medium) 0.9 1 0.9 0.8 0.7 0.6 0.5NS (negative small) 0.8 0.9 1 0.9 0.8 0.7 0.6

ZR (no change) 0.7 0.8 0.9 1 0.9 0.8 0.7

PS (positive small) 0.6 0.7 0.8 0.9 1 0.9 0.8

PM (positive medium) 0.5 0.6 0.7 0.8 0.9 1 0.9

PL (positive large) 0.4 0.5 0.6 0.7 0.8 0.9 1

Table 6

Linguistic rules for tuning SMC

Performance attributes Tuning action

If OSCZ HIUN then DKSZ NL and DrZ NS

Or If OSCZ LIUN then DKSZ NM and DrZ ZROr If OSCZ ACCP then DKSZ ZR and DrZ ZR

Or If OSCZ LIOQ then DKSZ PS and DrZ ZR

Or If OSCZ HIOQ then DKSZ PM and DrZ PS

If PHMZ HIUN then DKSZ NM and DrZ NM

Or If PHMZ LIUN then DKSZ NS and DrZ NM

Or If PHMZ ACCP then DKSZ ZR and DrZ ZR

Or If PHMZ LIOQ then DKSZ ZR and DrZ PS

Or If PHMZ HIOQ then DKSZ PS and DrZ PM

If TREZ EL then DKSZ NS and DrZ NL

Or If TREZ LA then DKSZ ZR and DrZ NM

Or If TREZ ME then DKSZ PS and DrZ PL

Or If TREZ SM then DKSZ PS and DrZ PS

Or If TREZ OK then DKSZ ZR and DrZ ZR



12/17

membership grade (i.e. unity) assigned at its normalized

representative value (i.e. diagonal elements in the member-

ship function tables). The off diagonal terms represent the

reducing weight to model the degree of fuzziness as

indicated in Tables 3 and 4.

(ii) Fuzzy linguistic rules: Expert knowledge on tuning

the sliding mode controller is expressed as a set of linguistic

rules containing reference to the defined fuzzy sets.

According to the expert knowledge and experience, the

SMC tuning rules developed are summarized in Tables 5

and 6. For example, the first row of the Table 6 is interpreted

as follows: If the oscillation level of the control signal is

highly unsatisfactory (if OSCZHIUN), then the feedback

gain (K S) must be largely decreased and disturbance esti-

mator gain (r) must be slightly decreased (then DK SZNL

and DrZNS).

(iii) Fuzzy composite relation table: The performanceattributes of the controller, the tuning actions, and the rules

of tuning must be related to each other using fuzzy

composite relationship functions [24]. Through fuzzy

implication (IF-THEN), the fuzzy relation (mRi) for the ith

individual rule ( Ri) defined in Table 6 is developed as,

mRiOSC; PHM; TRE

DK s;Dr

Zmin minputi ðOSC; PHM; TREÞmoutputi ðDK s;DrÞ

(25)

where minput, formed as a 5!1 column vector, is amembership function of the measured fuzzy state for the

input variables (i.e. OSC, PHM, and TRE) listed in Table 3.

moutput denotes a 1!7 row vector which relates to

membership functions of the corresponding fuzzy state for

the output variables (i.e. DK S and Dr) given in Table 4.

½minputi ðOSC; PHM; TREÞ+moutputi ðDK S;DrÞ leads to the

comparison of each corresponding element and the min

operation enforces the selection of the element which has

the minimum value. Each resulting rule m Ri becomes a 5!7

matrix, and there are five rules mapping from each input

Table 7

Fuzzy decision table for SMC tuning

Performance attributes (input) Tuning action (output)

DK S Dr

OSC HIUN K0.2545 K0.1552

LIUN K0.2456 0

ACCP 0 0LIOQ 0.1552 0

HIOQ 0.2456 0.1552

PHM HIUN K0.2456 K0.2456

LIUN K0.1552 K0.2456

ACCP 0 0

LIOQ 0 0.1552

HIOQ 0.1552 0.2456

TRE EL K0.1552 K0.2545

LA 0 K0.2456

ME 0.1552 0.2545

SM 0.1552 0.1552

OK 0 0

Table 8

Summary of axis feed drive parameters

Feed drive parameters X -axis Y -axis

Feed drive rigid body dynamics

Current amplifier gain, K a [A/V] 6.4898 7.5768

Motor gain, K t [Nm/A] 0.4769 0.4769

Total reflected inertia, J [kg m2] 0.0077736 0.0098109

Viscous damping, B [kg m2 /s] 0.019811 0.028438

Pitch length h p [mm] 10 10

Transmission gain r g [mm/rad] 1.5915 1.5915

Quantization and saturation limits

D/A Converter bit [ ] 16 16

Voltage range of D/A chip [V] G10 G10

D/A resolution, du [V] 3.0518!10K4 3.0518!10K4

Quantization variance, R ~u [V2

] 7.761!10K9

7.761!10K9

Saturation limits, umin / umax [V] K5,C5 K5,C5

Measurement noise

Linear encoder resolution, d x [mm] 1.22!10K3 1.22!10K3

Linear encoder noise variance, R ~ x [mm2] 1.2418!10K7 1.2418!10K7

Tachometer noise variance, R ~u [(rad/s)2] 50!10K3 50!10K3

Non-linear friction characteristics

Static friction, T Gstat [Nm] 2.6256, K1.8672 2.7658, K2.4520

Coulomb friction, T Gcoul [Nm] 2.1529, K1.4730 2.5228, K2.3887

Velocity threshold 1, uG1 [rad/s] 3.88, K3.51 2.37, K4.41

Velocity threshold 2, uG2 [rad/s] 4.20, K3.52 3.01, K4.20

Backlash

Backlash, Db [mm] 0.003 0.003



13/17

variable to each output variable, totaling 15 sets of matrices

for each of DK S and Dr. As the individual rules are joined

through the fuzzy connective (OR) operator, the resulting

fuzzy relation matrix between each of the fuzzy input and

output variables is formed as,

mROSC; PHM; TRE

DK S;DrZmax

5

iZ1mRi

OSC; PHM; TRE

DK S;Dr

#" (26)

By comparing each element of the fuzzy relation table for

individual rules, the maximum of the membership grades

corresponding to each pair of input-output states is chosen.

Using a similar fashion, the composite fuzzy relation tables

between all condition and action variables are constructed.

Since there are 3 fuzzy input variables (i.e. OSC, PHM, and

TRE) and 2 fuzzy output variables (i.e. DK S and Dr) that are

considered in SMC tuning, 6 resulting fuzzy composite

relation matrices or tables are developed in total.

(iv) Fuzzy decision table: The fuzzy decision table

is calculated by matching the system performance

attributes to the composite fuzzy relation tables, through

the use of the compositional rule of inference (i.e. sup of

min operation) as [24],

mC Zsupmin minputðOSC;PHM;TREÞ;m ROSC;PHM;TRE

DKs;Dr

!

(27)

where minput, m R, and mC are the membership functions of the

fuzzy input variables, the resulting fuzzy composite relation

calculated using Eq. (26), and the corresponding output

variables, respectively. In the use of a sup of min operation,

the membership function vector of the input variables is first

compared with each column of the fuzzy relation table. Then,

the lower value in each pair of compared elements is taken.For each column, the largest value in the vector of elements is

selected, thus resulting in a row vector. In order to obtain a

numerical value, ĉ, that determines the degree of tuning

action, a de-fuzzification is applied using the discrete

centroid method as,

ĉZ

P7i¼1 cimC ðciÞ

mC ðciÞ (28)

Table 9

Summary of fuzzy tuning for three different scenarios

Results of fuzzy

tuning for SMC

Case 1 Case 2 Case 3

Pre-selected l [rad/s] 200 200 200

Initial/Tuned K S[V/(mm/s)]

1/0.363 0.01/0.315 10/0.269

Initial/tuned r

[V/mm]

100/47.16 1/39.12 10,000/37.16

Initial/final maximum

tracking error [mm]

0.009/0.009 0.136/0.011 Unstable/0.012

Fig. 13. System performance before and after fuzzy tuning: (a) initially noisy system, (b) initially highly stable and sluggish system, (c) initially unstable

system.



14/17

where ci corresponds to the representative numerical value

for the tuning parameters DK S and Dr, given in Table 4. The

fuzzy decision table, shown in Table 7, is obtained off-line

which allows efficient real-time tuning of the control

parameters. Through this fuzzy decision table, the new

tuning actions are assigned according to system performance

attributes. The formulation used for updating the control

parameters is given in Eq. (29),

K SnewZK SoldCX3ðOSC;PHM;TREÞiZ1

DK S

!!K Sold ;

rnewZroldCX3ðOSC;PHM;TREÞiZ1

Dr

!!rold

(29)

where K Snew and rnew correspond to the new SMC

parameters after fuzzy decision making. K Sold and rold are

the current SMC parameters that need to be tuned. DK S and

Dr are the degree of the tuning actions obtained from the

fuzzy decision table.

4. Experimental validation of virtual cnc

The Virtual CNC system is experimentally verified

on a three-axis vertical machining center with ball

screw drives. The identified feed drive parameters of

the machine are summarized in Table 8. The machine

tool is controlled by an in house developed open CNCsystem [21], which allows modular integration of any

trajectory generation, control law, and compensation

strategies.

4.1. Fuzzy tuning of axis control law

The sliding mode control law for the drives is

automatically tuned in the Virtual CNC environment

using Fuzzy Logic. The desired sliding surface bandwidth

(l) is pre-selected to be 200 rad/s. A back-and-forth test

motion with a travel distance of 100 mm and a command

feedrate of 200 mm/s is used as the reference input forsystem performance diagnosis. The noise in the control

law and the tracking error are identified from Virtual

CNC simulation for given feedback (K S) and disturbance

observer (r) gains. The phase margin of the closed loop

controller is also calculated. The auto-tuning starts with

initial values of K S and r, and the fuzzy logic tuner

determines the optimal gains automatically, regardless of

the initial guess. The initial and tuned control law

parameters are listed along with their performance results

in Table 9. The corresponding control performances

before and after tuning with experimental validation are

shown in Fig. 13.

Table 10

Summary of control parameters for PID and SMC

PID Control X -Axis Y -Axis

Proportional gain, K p [V/mm] 70 70

Integral gain, K i [V/(mm$s)] 800 800

Differentiation gain, K d [V/(mm/s)] 0.30 0.30

Sliding mode control (SMC) X

-axis Y

-axisSliding surface bandwidth, l [rad/s] 200 200

Feedback gain, K S [V/(mm/s)] 0.30 0.30

Disturbance adaptation gain, r [V/mm] 30 30

Fig. 14. Diamond-shaped contouring tests with PID control.



15/17

4.2. Prediction of CNC performance in machining

circular and diamond profiles

The Virtual CNC system has been experimentally

verified by trying out various tracking control schemes,

such as P-PI, PID, SMC, GPC, and Pole Placement Control

with feedforward friction compensation. PID and SMC

results are reported here as examples. The control interval

was 1 ms. The control parameters of the developed PID

and SMC are summarized in Table 10. Standard circular

Fig. 15. Diamond-shaped contouring tests with sliding mode control (SMC).

Fig. 16. Circle-shaped contouring tests with PID control.



16/17

and diamond toolpath contouring tests were conducted: The

diamond had a side length of 50 mm and the circular path

radius was 50 mm. The reference trajectories have been

generated to achieve a feedrate of 200 mm/s with maximum

acceleration and jerk values of 2000 mm/s2 and 50,000 mm/

s3, respectively. The selected acceleration profile was

trapezoidal, resulting in piecewise constant jerk commandsalong the toolpath. The experiments were conducted under

air-cutting conditions in order to avoid structural defor-

mations of the machine under cutting load.

The measured and predicted tracking and contour error

profiles are shown in Figs. 14–17, and summarized in

Tables 11 and 12. It can be seen that the most critical parts

on the diamond paths are located at the corners where

transients in reference trajectory occur. The most significant

deviations on the circular paths are located at the quadrants

where the direction of the axis motion changes, and the

errors are mostly due to friction, which holds the slide until

sufficient torque is accumulated to overcome the staticfriction disturbance. The experimental results clearly

indicate the accurate prediction capability of the Virtual

CNC system, also indicating that the sliding mode controller

(SMC) yields significantly less contouring and tracking

error than the standard PID controller.

Fig. 17. Circle-shaped contouring tests with sliding mode control (SMC).

Table 12

Summary of simulation and experimental results for air-cutting of circle

profile

Circle contouring tests

Axis control

law

Tracking and contouring

error

Comparison analysis

Simulation

(mm)

Experiment

(mm)

Abs. Differ-

ence (mm)

Deviation

(%)

X-axis tracking analysis

PID 0.0471 0.0436 0.0035 8.0275

SMC 0.0117 0.0109 0.0008 7.3394

Y-axis tracking analysis

PID 0.0223 0.0220 0.0003 1.3636

SMC 0.0077 0.0072 0.0005 6.9444

Contouring error analysis

PID 0.0215 0.0226 0.0011 4.8673

SMC 0.0117 0.0108 0.0009 8.3333

Table 11

Summary of simulation and experimental results for air-cutting of diamond

profile

Diamond contouring tests

Axis control

law

Tracking and contouring

error

Comparison analysis

Simulation

(mm)

Experiment

(mm)

Abs. differ-

ence (mm)

Deviation

(%)

X -axis tracking analysis

PID 0.0375 0.0376 0.0001 0.2659

SMC 0.0122 0.0115 0.0007 6.0870

Y-axis tracking analysis

PID 0.0412 0.0414 0.0002 0.4831

SMC 0.0074 0.0070 0.0004 5.7143

Contouring error analysis

PID 0.0126 0.0138 0.0012 8.6957

SMC 0.0084 0.0081 0.0003 3.7037



17/17

5. Conclusion

A comprehensive virtual model of a Modular CNC is

presented. The Virtual CNC allows modular integration of

trajectory planning and interpolation routines, mathematical

models of ball screw and linear drives, friction, feedback

sensors, amplifiers, D/A converters and flexible motioncontrol laws. The system allows the designer to try out

various feed drive design alternatives, control laws, and

sensors with different resolutions. The accurate model of the

drives allows realistic simulation of the drives’ high speed

contouring capability, and auto-tuning of sophisticated axis

control laws in a virtual environment, as has been

demonstrated for the presented Sliding Mode Controller.

Acknowledgements

This research is sponsored by NSERC and Pratt &

Whitney Canada under research chair and strategic grant

agreements.

References

[1] Y. Altintas, Manufacturing automation: metal cutting mechanics

Machine Tool Vibrations, and CNC Design, Cambridge University

Press, Cambridge, 2000.

[2] Y.C. Chen, J. Tlusty, Effect of low-friction guideways and lead-screw

flexibility on dynamics of high-speed machines, Annals of CIRP 44

(1) (1995) 353–356.

[3] K. Erkorkmaz, Y. Altintas, High speed CNC system design. Part I.

Jerk limited trajectory generation and quintic spline interpolation,International Journal of Machine Tools and Manufacture 41 (9)

(2001) 1323–1345.

[4] K. Erkorkmaz, Y. Altintas, Trajectory generation for high speed

milling of molds and dies, Proceedings of the Second International

Conference and Exhibition on Design and Production of Dies and

Molds, Kusadasi, Turkey, DM_46 2001.

[5] Y. Altintas, K. Erkorkmaz, W.-H. Zhu, Sliding mode controller design

for high speed drives, Annals of CIRP 49 (1) (2000) 265–270.

[6] P. Boucher, D. Dumur, K.F. Rahmani, Generalized predictive cascade

control (GPCC) for machine tool drives, Annals of CIRP 39 (1) (1990)

357–360.

[7] K. Erkorkmaz, Y. Altintas, High speed CNC system design. Part III.

High speed tracking and contouring control of feed drives,

International Journal of Machine Tools and Manufacture 41 (11)

(2001) 1637–1658.

[8] G. Pritschow, On the influence of the velocity gain factor on the path

deviation, Annals of CIRP 45 (1) (1996) 367–371.

[9] M. Tomizuka, Zero phase error tracking algorithm for digital control,

ASME Journal of Dynamic Systems, Measurement, and Control 109

(1987) 65–68.

[10] H.B. Armstrong, P. Dupont, C. Canudas De Wit, A survey of models,

analysis tools and compensation methods for the control of machines

with friction, Automatica 30 (7) (1994) 1083–1138.[11] K. Erkorkmaz, Y. Altintas, High speed CNC system design. Part II.

Modeling and identification of feed drives, International Journal of

Machine Tools and Manufacture 41 (10) (2001) 1487–1509.

[12] K. Erkorkmaz, PhD Thesis: Optimal Trajectory Generation and

Precision Tracking Control for Multi-Axis Machines, University of

British Columbia, Department of Mechanical Engineering, Vancou-

ver, BC, Canada, 2004.

[13] J.Y. Kao, Z.M. Yeh, Y.S. Tarng, Y.S. Lin, A study of backlash on the

motion accuracy of CNC lathes, International Journal of Machine

Tools and Manufacture 36 (5) (1996) 539–550.

[14] G. Pritschow, Course notes: control techniques of machine tools and

industrial robots, Institute of Control Technology for Machine Tools

and Manufacturing Units, Stuttgart University, Germany, 1997.

[15] Y. Koren, Computer Control of Manufacturing Systems, McGraw-

Hill, New York, 1983.

[16] C.W. de Silva, A.G.J. MacFarlane, Knowledge-based control

approach for robotic manipulators, International Journal of Control

50 (1) (1989) 249–273.

[17] N. Wickramarachchi, C.W. de Silva, Knowledge-based supervisory

control system of a fish processing workcell. Part I. System

development, Engineering Applications of Artificial Intelligence 11

(1) (1998) 97–118.

[18] N. Wickramarachchi, C.W. de Silva, Knowledge-based supervisory

control system of a fish processing workcell. Part II. Implementation

and evaluation, Engineering Applications of Artificial Intelligence 11

(1) (1998) 119–134.

[19] C.W. de Silva, An analytical framework for knowledge-based tuning

of a servo controller, Engineering Applications in Artificial

Intelligence 4 (3) (1991) 177–189.[20] J.-F. Goulet, C.W. de Silva, V.J. Modi, Hierarchical knowledge-based

control of a deployable orbiting manipulator, Acta Astronautica 50 (3)

(2002) 139–148.

[21] Y. Altintas, N.A. Erol, Open architecture modular tool kit for motion

and machining process control, Annals of CIRP 47 (1) (1998) 295–

300.

[22] C.H. Yeung, MASc Thesis: A Three-Axis Virtual Computer

Numerical Control (CNC) System, University of British Columbia,

Department of Mechanical Engineering, Vancouver, BC, Canada,

2004.

[23] K.J. Astrom, B. Wittenmark, Computer-Controlled Systems: Theory

and Design, third ed., Prentice-Hall Inc., New Jersey, 1997.

[24] C.W. de Silva, Intelligent Control—Fuzzy Logic Applications, CRC

Press LLC, Boca Raton, FL, 1995.


solution manual of cnc technology

Documents

Transcript of solution manual of cnc technology