Evaluation of thermal models on a machining center


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Evaluation of thermal models on a machining center
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vii, 147 leaves : ill. ; 29 cm.
Mize, Christopher Daryl, 1965-
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Mechanical Engineering thesis, Ph. D   ( lcsh )
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Thesis (Ph. D.)--University of Florida, 1998.
Includes bibliographical references (leaves 142-146).
Statement of Responsibility:
by Christopher D. Mize.
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An undertaking of this scale requires the assistance of many people. I would like

to thank my chairperson Dr. Scott Smith for his assistance and persistence for remaining

on my committee after his departure to the University of North Carolina at Charlotte. I

also thank the remaining members of my committee for their input and time spent

reviewing this document. Special thanks go to Dr. John Ziegert for his availability to

discuss the many topics encountered in this research.

I would like to thank Mike Niemotka at Tetra Precision for his invaluable

assistance programing the machine tool and helping with the measurements. Thanks go

to Tony Schmitz in the Machine Tool Research Center for writing the CMM program and

providing his simulation software for the tool dynamics. I would also like to thank

Narayan Srinivasa for providing the neural network algorithm and his discussions about

the network.

Thanks go to the personnel at Cincinnati Milacron for providing the necessary

support for the special software required for compensation, and to Joe Godschalk and Joan

Staubach for programming the compensation algorithm in the controller.

I would also like to thank the USAF, without whose funding this research would

not have been possible.


ACKNOWLEDGMENTS ...................................... ii

Abstract ........ .............................. ........... vi

INTRODUCTION ......... ............................. 1
Precalibrated Compensation ................. ............... 7
Research Objectives ......... ..... ...................... 8

LITERATURE REVIEW ............................... 12
Background ................. .. ...................... 12
Geometric Compensation ................. ............... 12
Thermal Modeling and Compensation ........................ 15
Motivation for the Research .............................. 22
Scope of the Research ................. ................. 23

GEOMETRIC MODEL ................. .............. 25
Geometric Compensation ................................ 25
Kinematic Model for the Machining Center ..................... 27
Implementation in the Controller ........................... 32

THERMAL ERROR MODELS ............................ 34
Introduction .......... .. ................ ............. 34
First Order Thermal Model ................. .............. 35
Implementation of the Model ................. ......... .... 36
Thermal Drift ............................. ........... 37
The Neural Network Model ............................... 38
Incorporation of the Network in the Model ..................... 41
Implementation on the Machine ............................ 44
Thermal Sensor Placement ................ .............. 49

Geometric Error Measurement with the Laser Ball Bar .............. 52
Data Collection Procedure ................................ 53
Data Collection for Model Verification ........................ 58
Body Diagonal Measurement Procedure ...................... 60
B5.54 Part Machining Procedures ........ ................. 60

TEST RESULTS ........ .............................. 64
Model Evaluations ........ ...... ...................... 64
Model Stability Evaluation ................. .............. 64
Diagonal Measurement Evaluation of the Models ................. 69
M machine Part Evaluation ................................. 74

Success of Implementing Fuzzy ART-MAP on a Milling Machine ....... 81
Geometric vs. Thermal Modeling ................ ..... 87
Deterministic vs. Non-deterministic Modeling ................... 87
Training with Machining vs. Non-machining .................... 89
Metal Removal Testing vs. Non-machining Testing ................ 91
Surface Distance Error ....... .......... .............. 92
Future W ork ................. .............. .......... 93

THE LASER BALL BAR .............................. 95
Background ...................... .......... .......... 95
The Instrument ......... .... ....................... 96
Trilateration with the LBB ................................ 98
Determining LBB to Machine Coordinate Transformation .......... 102
Parametric Error Reduction from Coordinate Data ................ 104
Coordinate Error Sensitivity to Tetrahedron Geometry ............. 105
Instrument Accuracy .......... ....... ................. 108
Parametrics from Six Measurements ....................... 109

KINEMATIC MODEL IN CONTROLLER ..................... 112


LIST OF REFERENCES ................ .................... 142

BIOGRAPHICAL SKETCH ................. .................. 147

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



Christopher D. Mize

August 1998

Chairperson: Kevin S. Smith
Major Department: Mechanical Engineering

Machine tool positioning accuracy varies with the thermal state of the machine.

As electrical energy is input into the servomotors, hydraulic pumps, and other machine

systems, energy is transferred into the part, atmosphere, and most importantly the

machine's structure. This transfer of energy throughout the machine results in temperature

changes and thus structural deformations that change the machine's accuracy. In order to

mitigate these detrimental accuracy variations, models have been employed that try to

predict and correct for these variations based on discrete temperature readings from the

machine. In this study three thermal models and one geometric model are evaluated by

comparing their accuracy improvement on a Cincinnati Milacron Maxim 500 machining

center. One thermal model is the simple geometric model with first order correction of

the scale errors. The two remaining thermal models utilize a new implementation of a


neural network. One of the neural network models is trained from error measurements

taken as the thermal state is varied with non-machining actuation, while the other utilizes

actual machining. The laser ball bar is utilized to collect the training data for the models

in a timely manner and to allow machining between measurements. The models are

evaluated by measuring body diagonals with the laser ball bar and by comparing the

accuracy of machined parts at different thermal states of the machine.

The body diagonal and part machining tests reveal that the thermal models are

capable of 2-4X error reduction at several thermal states. The completely deterministic

first order thermal model performed as well or better than the neural network models.

Durability tests showed that the models were capable of error reduction over a 9 month

period. No clear preference was found for training with or without machining, rather the

use of coolant appeared to be a more important factor. Thermal compensation is a viable

technique that should be embraced by industry.



Machine tools, assembly robots, and computer controlled positioning machines in

general, have as their primary function to place an end effector relative to a work piece

according to a controller's commanded position and orientation. In machine tools, this is

manifested in the machine's ability to position the cutting tool relative to the part raw

material. A machined part's accuracy is directly, one to one, dependent on this

positioning capability. Other factors such as tool wear and dynamic effects (chatter) can

also contribute, but unlike positioning error, their effects can be mitigated by proper

planning at the time of production.

The accuracy with which positioning can be accomplished depends on many

factors: accuracy of the axis feedback sensors, Abbe' effects, thermal stability, structural

integrity, and dynamic stability, to name a few. Much effort has been spent in this century

on improving machine tool accuracy, and understandably so, since machine tools are

important to a nation's economy. They are a critical early link in a chain that has at one

end a nation's natural resources and at the other its salable products. In 1991, the

machining processes of turning, milling, drilling, and grinding, accounted for

approximately $115 billion annually in the U.S. discrete part industry, or about 2% of the

GDP [Soons, 1995].


To reduce a machine's errors it is best to understand their source first. This can

be accomplished by examining the individual systems making up the machine and how

their non-ideal behavior effects part accuracy. A modern machine tool is comprised of a

control system (digital computer), an actuation system (rotary or linear servomotors), a

feedback system (linear or rotary encoders), a cutting system (spindle motor, and cutting

tools), a part fixturing system (pallet or chuck), and a mechanical system (prismatic and

rotary joints). Each of these systems can contribute to a machined part's inaccuracies.

Traditional machine tools achieve multi-degree positioning through a chain of

prismatic and rotary joints combined in series, with the cutting tool at one end of the chain

and the work piece at the other. Each joint is intended to provide a single degree of

freedom. Recently machine tools have been developed that are based on parallel kinematic

structures [Aronson, 1996, 1997]. While promising, these machines are still in the

developmental phase. The focus of this research will be upon the industry dominant serial

link machine.

Since direct volumetric positioning feedback between the cutting tool and the work

table has not been perfected, machine tools rely upon position feedback from each joint

for determining tool placement relative to the work table. This arrangement requires,

infinitely stiff components, a perfect control system, perfect scales, a single degree of

freedom at each joint, and isothermal behavior to achieve perfect positioning.

Unfortunately this ideal scenario is not achievable. Errors are correspondingly classified

into four major categories: control, elastic, geometric, and thermal.


Elastic deformations can result from cutting load induced deflections in the tool and

machine structure as well as dynamic effects such as vibration (chatter). Quasi-static

deflections in the structure caused by cutting forces and gravity can be reliably predicted

through finite element modeling and adequately minimized by proper stiffening at the

design stage. Elastic deformations caused by vibrations in the tool and spindle can be

significant, but their discussion is beyond the scope of this research and necessary

precautions can be taken to avoid their domain of existence. The control system can also

be reliably modeled and optimized, minimizing this source of error. Geometric and

thermal errors have reached a cost/design tradeoff limit that leaves them as the largest

error contributors [Venugopal and Barash, 1986; Hocken, 1980; Bryan, 1990]. The

geometric errors are limited by the precision and assembly of the machine's components.

Cost to reduce these errors at the manufacturing stage can rise exponentially. Thermal

errors, like elastic errors, can be reasonably modeled; however, they are not as easily

mitigated at the design stage because the unavoidable heat sources can not be sufficiently

insulated from the machine structure. The geometric and thermal errors remain as the

primary obstacle to improved accuracy in machining and are the focus of this research.

Geometric errors can be attributed to the mechanical component inaccuracies that

result in unwanted motions at each joint. Each joint is intended to have a single degree

of freedom, but in reality has six degrees of error motion. For prismatic joints there is

an intended displacement freedom which can be in error, two straightness errors

perpendicular to the displacement, and three angular motions about the Cartesian axes

(these errors are often referred to as roll, pitch, and yaw), as shown in Figure 1-1.


Figure 1-1 Six degrees of error freedom.


Rotary joints also have six degrees of error motion. There is the angular positioning

error, an axial error motion, two perpendicular translational motions, and the two

remaining angular error motions that are referred to as tilt to distinguish them from the

intended angular freedom. These fundamental error motions are often referred to as

parametric errors in the literature. Each of these parametric errors propagate through a

kinematic chain of joints and bodies, producing a resultant positioning error between the

tool and work piece.

Thermal error is the change that occurs in the geometric errors as a result of

structural deformation caused by temperature gradients induced from the heat sources. It

can be separated into two classifications: drift and kinematic. Drift is the change in

position of the tool relative to the work piece at some nominal reference point in the

machine and can be seen as the DC offset that occurs in the parametric errors. The

kinematic portion is seen as the change in shape of the parametric error motions. Drift can

be minimized by the industry accepted practice of periodically probing a reference feature

to compensate for the effect in the part program. Minimizing the kinematic changes

requires more elaborate techniques, which have not been adopted by industry

Geometric and thermal errors can be mitigated in two ways: error avoidance and

error compensation [Blaedel, 1980]. Error avoidance involves eliminating the source of

the error, e.g., making perfect machine components, and temperature controlling the

machine [Bryan, 1979]. This obviously has physical and fiscal limitations and is not

usually a viable solution for commercial machine tools. Error compensation involves

canceling the effect of the error by appropriately modifying the machine's commanded


motions. This can be achieved through active compensation or precalibrated

compensation. Active compensation involves accurately monitoring the machines motions

against a metrology frame and modifying the commanded motions in real time [Estler and

Magrab, 1985]. Precalibrated compensation involves pre-measuring the machine's error

motions with reference to some independent variables such as the nominal axes positions

and then compensating at a later time based on the errors computed from the independent


Obviously the success of precalibrated compensation depends on and is limited by

the machine's repeatability. Precalibrated compensation usually involves making a

mathematical model of the machine that is fit to some premeasured data set. The model

can be as simple as a look-up table of errors mapped to the machine's scale readings. The

errors are predicted based on inputs such as the machine's commanded position and

machine temperatures.

Most research has focused on reducing machine tool errors through precalibrated

compensation. It has the advantage that it is not prohibitively expensive to install and can

be retrofitted to existing machines. Active compensation can be expensive due to the

multitude and cost of the dedicated sensors required (laser interferometers). This research

focuses on precalibrated compensation, since it has the greatest probability of being

adopted by industry.

Precalibrated Compensation

During the past 20 years, the majority of research involving machine tool accuracy

enhancement has focused on precalibrated compensation. Geometric errors are normally

addressed by fitting the parametric errors with polynomial equations or placing them in

a look-up table for interpolation and then including them in a kinematic model for tool

point error prediction. The kinematic model is a mathematical representation used to

propagate the parametric errors through the kinematic chain, usually via homogenous

transformation matrices. Concatenation of the transformations from one end of the chain

to the other results in the positional error of the tool relative to the work table. Correction

of geometric errors is only effective at the thermal state in which the errors are measured.

Since commercial machine tools do not operate at a single thermal state, geometric

compensation alone is not a panacea. Thermal errors must be addressed if significant

accuracy enhancement is desired.

Thermal errors are normally addressed by incorporating models that correlate the

changing geometric errors to selected temperature inputs around the machine. Models can

be as simple as one dimensional parametric fitting of errors to temperature or as complex

as the backward propagation neural network [Chen, 1991].

In each of these models, the machine is exercised through a duty cycle with

intermittent measurement of the changing geometric errors and temperature sensors. The

temperatures and nominal machine positions are used as inputs to the models and the


measured errors are used to determine the terms for the input/output transfer function.

After the training set of data is collected, the models predict errors based on nominal

machine position and temperature inputs. Temperatures are valid inputs for predicting

changing machine geometry because the displacement-strain field in a linear elastic

medium is completely defined by the temperature field and the physical boundary


Previous research involved with thermal error compensation has experienced two

major limitations from the available metrology equipment. First, most equipment can not

be removed from the work zone during the thermal duty cycle without loss of the

measurement reference. Second, most instruments can only measure a single component

of error at a time. There have been some recent equipment developments that allow up

to five error measurements simultaneously per axis [Ma et al., 1996; Huang and Li, 1996],

but these still require multiple set-ups to measure all of the parametric errors.

The first limitation requires a thermal duty cycle free of machining, and the

second limitation necessitates a repeat of the cycle for each measurement. This deviates

from real machine operation and requires several days of machine down time for

measurement. With the introduction of the Laser Ball Bar (LBB), these two limitations

can be overcome [Mize, 1993; Ziegert, 1994].

Research Objectives

Geometric and thermal compensation, while proven in the laboratory, has not been

implemented on commercial machine tools with the level of proliferation that geometric


compensation has been incorporated on Coordinate Measuring Machines (CMMs).

Perhaps geometric compensation has been dismissed because thermal errors can

overshadow any improvements. Additionally thermal compensation may appear too

complex or appear to be a maintenance nightmare with the multitude of thermal sensors

employed. This study attempts to evaluate and compare the amount of benefit obtained

from geometric, elementary thermal, and complex thermal compensation models on a 3-

1/2 axis machining center. Measurements were made with the Laser Ball Bar (See

Appendix 1) developed at the University of Florida and refined by Tetra Precision

Incorporated. The geometric model is based on the traditional method of deriving

positional error equations using homogeneous transformation matrices. The independent

variables for the position error equations are the parametric errors that are measured with

the LBB while the machine is in its cold state. The first order thermal model is the

geometric model with the addition of linear scale corrections based on scale temperature

readings. The complex thermal models comprise a new and unique combination of the

deterministic geometric model and a neural network for correlating 31 machine

temperatures to changes in the parametric errors. Two neural network models were

evaluated. These models are identical with the exception of their training cycles. One

of the models was trained with the traditional method of warming the machine by actuation

free of machining. The other model was trained by warming the machine with actual

machining, since the LBB makes this possible. Comparing these two models will help

determine the importance of training with real machining, which until now has been

difficult or impossible.

Evaluating accuracy improvement in the context of the complexity of model

implementation should provide useful information for a commercial builder in deciding

what type of compensation, if any, is most viable for their product.

Tests were conducted on a 3-1/2 axis Cincinnati Milacron Maxim 500 machining

center, Figure 1-2. The Maxim has axes travels of 750x700x750 mm (29.5x27.6x29.5 in)

in X, Y, and Z respectively. The cinematic structure of the machine has the pallet carried

on a moving Z axis carriage running on recirculating linear bearing/ways supported on a

cast iron 'T' bed. The spindle is mounted to a Y axis carriage running on recirculating

linear bearing/ways on a vertical column. The spindle is rated at 20 KW (33.5 HP)

continuous with a maximum speed of 7000 RPM and has 50 V-flange tool holder. The

machine is also equipped with a Renishaw MP8 touch trigger probe.

The column runs on X axis recirculating linear bearing/ways mounted to the bed.

The pallet is mounted on a B axis indexing table (12 axis). A rotating pallet shuttle allows

part loading and removal at the front of the machine. A 40 socket tool chain and changer

is located opposite the operator station. The controller is a model Acramatic 950

manufactured by Cincinnati Milacron. It has a custom passive backplane computer with

multiple i386 based processing boards.

Figure 1-2 Maxim 500 machining center (graphic courtesy of Cincinnati



Software compensation of machine tools became possible with the introduction of

computer controlled servo systems in the 1950s. Error correction itself predates numerical

control [SIP, 1952; Schlesinger, 1927]. The amount of compensation possible was limited

to lead screw correction through custom manufactured cams. With the introduction of

numerical actuator control, cross compensation of axes became possible. With this

advancement, not only could error motions along an axis be corrected, but errors

perpendicular to the axis could be corrected through actuation of the other axes as a

function of the commanded axis.

Geometric Compensation

Compensating for the errors of the machine at some fixed thermal state, such as

the cold state, is often referred to as geometric compensation. Research in this area for

CNC machine tools began to take place in the 1960s. Leete first proposed a method to

compensate for errors by breaking the feedback loop in the servo system to introduce

corrective signals [Leete, 1961]. While he never implemented such a system, several


researchers have since used this technique [Okushima et al., 1975; Donmez 1985, Sumanth


One of the first documented software corrections of a machine tool was performed

by French and Humphries [1967; NIST-60NANB2D1214, 1993]. Backlash and alignment

errors were compensated by modifying the part program based on a machine model

derived using Euclidian geometry. In the same year, a paper was published detailing the

implementation of an online compensation system for a large boring mill [Schede, 1967].

Autocollimators were used for angle measurement and an automatic alignment

interferometer provided signals for compensation of machine geometry errors. Similarly,

Wong and Koenigsberger also implemented an active compensation system using an optical

error detection system [Wong and Koenigsberger, 1967].

Much of the foundation of our modern understanding of measuring and modeling

machine tool errors can be attributed to Tlusty [1971]. While no systematic modeling

approach was presented, he did mathematically detail the effects each of the angular errors

contribute to corresponding linear errors at the tool tip. This is essentially what HTMs

accomplish in a more systematic manner. He also introduced a very descriptive

nomenclature, that is widely accepted [ASME B5.54], to describe the six degrees of error

motion for an axis: [6x(x), 6b(x), 68x), E (x) E (f), e (I)]. The Greek delta, 6,

represents the translational errors and the subscript denotes the direction of the error. The

Greek epsilon, e, represents the angular errors and the subscript denotes the axis about

which it rotates. The independent variable inside the parentheses represents the motion

axis, x in this example.


One of the first papers dealing with our modern use of software correction was by

Wasiukiwicz [1974]. The concept of the machine as an information storage device was

introduced. That is, a machine's unique kinematic arrangement and structure serve as an

error storage device at least as well as machines repeat their error motions. This is a

simple but profound observation. This "memory storage" capability makes precalibrated

compensation possible, allowing for instance, cheaper less accurate scales to perform like

more accurate and expensive master scales. Wasiukiewicz discusses measuring and storing

a three dimensional lattice of errors to be used for correction in the same way scales can

be corrected from lookup tables.

The first documented study to incorporate the complete trio of measurement,

kinematic modeling, and software correction was presented in a paper by Hocken, et al.

at NIST (then NBS) [1977]. The work was performed on a Moore 5-7 coordinate

measuring machine. A combination of preprocess gaging and active compensation was

utilized as well as intermittent reference probing to mitigate thermal drifts. Parametric

measurements were taken over a cubic lattice of two inches and stored in an auxiliary

computer for retrieval into a kinematic model made up of 3x3 rotation matrices (a small

step away for HTMs). Measuring these errors over a cubic grid allowed non-rigid effects

to be included. Schultshik presented a similar paper at the same conference [1977].

Measurements were made on a three-axis jig borer and combined with a rigid body model

via matrix mathematics, though not as elegant as Homogeneous Transformation Matrices

(HTMs). Unfortunately, he did not use his measurements and model for correction, only

verifying the prediction of the model against a ball gage standard, with favorable results.

In the mid 1970s, work continued at NIST on software correction on CMMs and

machine tools [NIST-60NANB2D1214, 1993]. Software correction was implemented on

a Brown & Sharpe machining center with geometric compensation and temperature

correction of the scales (1st order thermal model). Much of this research was applied by

commercial CMM manufacturers to correct for geometric errors in their machines.

In 1980 the Machine Tool Task Force at Lawrence Livermore National Laboratory

(LLNL) completed its survey of the state of the art [Hocken, 1980; Tlusty 1980]. Five

volumes were dedicated to accuracy, mechanics, controls, and system management and

utilization. A chapter written by Ken Blaedel at LLNL detailed the state of error reduction

through avoidance and software compensation. Blaedel makes an interesting analogy

regarding predicting a machine's thermal error behavior with that of predicting the


if we had a mathematical model of sufficient sophistication, enough sensors
located in the right places, and a large enough high-speed computer to process the
data, then one can imagine an NC machine whose control compensates for all
thermal deformations. This is a problem of such complexity as to rival that of
accurately predicting the weather with the aid of giant computers
[Blaedel, 1980, pg. 70]

Thermal Modeling and Compensation

The studies mentioned above demonstrate that software compensation can greatly

reduce the effects of geometric error in CMMs and machine tools. Unlike CMMs,

geometric error is not the most significant contributor of error for a machine tool. Internal

heat sources (e.g. servo motors, guide way friction, cutting energy dissipation), and


changing ambient conditions can cause the geometric errors to change. This problem is

not so severe for CMMs which normally have good environmental control and servo

systems that operate with very little load. For this reason, machine tool research had to

focus on understanding and correcting for thermally induced errors.

A comprehensive survey and in-depth look at thermal errors was conducted by

James Bryan [1968]. Bryan surveyed many researchers in the field and presented a

heuristic analysis of heat sources, heat transfer mechanisms, and their effects on individual

components. He summarized this in a now historic flow chart duplicated in Figure 2-1.

He conducted a follow up survey in 1990 to assess the progress since 1967 [Bryan, 1990].

Unfortunately he concluded that little had changed in industry since 1967. However he

felt much progress was on the near horizon and that more would be accomplished in the

next five years than occurred in the past twenty-three. It appears that this prophesy has

yet to be fulfilled.

One of the first documented predictive software thermal models was tested by Ray

McClure for his Ph.D. dissertation at Lawrence Livermore laboratories in 1969 [McClure,

1969]. McClure presents an in-depth analysis of thermally induced errors. Error

prediction was implemented using lumped parameter models to estimate drift caused by

spindle growth on a vertical milling machine. Tests were also conducted to measure the

effects of work piece and tool expansion due to the cutting process. Turning tests revealed

that using coolant reduced tool expansion by over 80% and reduced part expansion about


Thermal effects diagram

Figure 2-1 Reproduction of heat flow chart [Bryan, 1990].


An important observation was made by Okushima, et al. [1975], that only a few

key temperature locations need to be monitored to obtain useful compensation. In this

study a vertical machining center was run under no load conditions while temperatures and

displacements were measured. Relationships were formulated between temperature and

displacement error, and significant error reduction was reported. Tlusty and Mutch

[1973], also made a similar observation regarding key temperature locations and gave an

explanation for their existence. They observed that machine tools typically reach

repeatable thermal mode shapes. These mode shapes can be predicted from temperatures

at a few key locations.

Many researchers in the early 1980s began to tackle the lowest order thermal

effects of drift and scale expansion [Koda and Yoshiro, 1981; Zhang et al., 1985]. Drift

is the motion measured between the tool and table at a fixed nominal machine position

during changing thermal conditions. It can be caused by thermal growth between the

machine's scale reference points, thermally induced bending of machine structure, thermal

growth of the end bodies in the kinematic chain, and scale expansion. The predominant

solution found in the literature and industry is to probe the tool with a tool setting station

or probe a fixed reference artifact to measure the drift [Koda and Yoshiro, 1981, Hocken

et al., 1977, Bryan 1990]. Once measured, appropriate compensation moves are

undertaken to eliminate the drift.

One of the first complete thermal compensations of a machine tool was performed

at NIST by Alkan Donmez on a two-axis turning center [Donmez, 1985]. Parametric

errors and selected machine temperatures were measured at discrete intervals as the


machine was warmed up under loadless conditions to steady state and then allowed to cool

down to room temperature. The parametric errors were fit with polynomials in both space

and temperature. The parametric errors could then be predicted from temperature and

nominal machine position inputs. The errors were then input into a kinematic model built

using HTMs to compute the tool point error. Correction was achieved by breaking the

feedback loop from the rotary encoders and injecting or suppressing pulses. Up to 20

times error reduction was reported.

Donmez's research proved that through a completely deterministic approach,

thermal errors could be successfully predicted and compensated. A few commercial

applications using thermal modeling began to surface after this research was published

[Janeczko, 1988], but only a few rudimentary systems remain commercially available

today. Other researchers have followed Donmez's approach of combining polynomial

parametric error fitting with HTM modeling [Teeuwsen et al., 1989, Balsamo et al.,


Contemporaneously at Purdue University other researchers were investigating

thermal modeling on three-axis machining centers [Venugopal and Barash, 1986]. In this

paper, thermoelastic equations were combined with heat transfer equations to show

analytically that deformation is instantaneously dependent on the temperature. A finite

difference model for temperature prediction was combined with a finite element model to

estimate deformations. Like Donmez's work, parametric errors were predicted from a few

key temperatures. However, no machining tests were mentioned.

During the early 1990s a new modeling strategy appeared in the literature based


on parallel learning neural networks. In an attempt to extend Donmez's work from a two

dimensional to a three dimensional machine, Jenq-Shyong Chen [1991] implemented

compensation on a machining center using an artificial neural network model. He used

a backward propagating (BP) network which attempts to mimic human synapse/neuron

interaction through three layers of nodes with weighted connections between node layers.

He also compared the neural network against a multiple regression empirical model by

assessing their ability to predict spindle drift. He concluded that the neural network model

compared favorably to the multiple regression model. His final test model incorporated

both active and predictive compensation. While he concluded that the back propagation

ANN was satisfactory, he felt an adaptive resonance theory (ART) network might be more

tolerant of noisy input data.

A year later, a BP network was evaluated for use on a two-axis turning center

through simulation by Ziegert and Kalle [1994]. They assumed realistic functions for the

parametric errors and computed volumetric errors with a kinematic model to train the

network. The simulation indicated accurate prediction of error might be possible and this

was verified experimentally by Srinivasa, Ziegert, and Smith on a two-axis turning center

[1993]. Like Chen, Srinivasa noted drawbacks to the BP network which included, long

training times, trial and error selection of the network architecture, and a vagueness of

how the network parameters relate to real world parameters. Because of these

shortcomings, Srinivasa adapted a fuzzy ART map network to predict the compensation

values for the same two-axis turning center used in his previous work at the University of

Florida [Srinivasa, 1994]. Srinivasa trained the ART map by intermittently measuring


volumetric errors with the LBB as the machine was warmed through non-machining

actuation and allowed to cool. Only a single thermal cycle was needed because all the

necessary errors for training could be collected simultaneously with the LBB. Cutting

tests revealed that the ART map correction system improved feature accuracies by 2.0 to

15.3 times.

A interesting use of thermal imaging was used to investigate the temperature

gradients of a machine tool during a thermal duty cycle [Allen et al., 1996]. Thermal

imaging revealed unexpected heat sources on a three-axis machining center. The imaging

was also used to locate key locations that were subsequently monitored to predict thermal

drift. A reduction in drift from 70Mm to 10 pm over a three hour duty cycle was


A recent study at the Department of Precision Instrument Engineering, Tianjin

University China, modeled the effects of thermal errors by correlating them to spindle

speed as opposed to temperatures [Shuhe et al., 1997]. Measurements were made with a

1-D ball array for errors in the Z direction. The errors were correlated to the spindle

speed at four locations on the Z axis using least squares fit. The errors were compensated

by premodifying the part program based on its spindle speed requirements. A simple 1-D

depth cutting test was performed with and without compensation after 1 hour of operation.

An error reduction from 7 pIm to 2 im was reported.

While no thermal modeling was conducted, it is appropriate to mention recent

research that occurred at Lawrence Livermore National Labs [Krulewich et al., 1995].

Krulewich utilized the LBB for measurement and error compensation on the same


Cincinnati Milacron Maxim 500 used in this research. Krulewhich introduced a new

measurement technique she termed the projection method. It consists of measuring lengths

between two magnetic sockets and projecting the length along the nominal direction

between the two sockets. The nominal direction, or vector, is obtained from knowledge

of the approximate location of the sockets in the machine's coordinate frame. The error

of the projected length is used to fit the parameters of a rigid body kinematic model with

assumed polynomial fits to the parametric errors. The projection method allows for an

extended work volume over trilateration with the LBB since for a given position the length

measurement between each base socket is not required. The method relies on the

assumption of knowing the nominal position of the sockets to within about 1.5 mm (0.062


Three thousand lengths were taken over a 3 1V hour period to fit the model with

linear regression techniques. The model was tested by measuring face and body diagonals

with a laser interferometer. The predicted errors were within about 2.0 Am for

approximate 800mm diagonals. Compensation was also carried out using part program

modification with nearly equal results.

Motivation for the Research

Based on the above literature survey it is evident that significant geometric and

thermal error reduction is possible. CMM manufacturers have realized this and

incorporated geometric compensation in almost every machine sold today. Machine tool

builders have not followed their lead. While dramatic improvements have been


demonstrated for at least the last 15 years, most machine tools sold today have only

geometric scale compensation systems incorporated. Perhaps builders have not taken the

next step of full geometric compensation because the thermal errors appear so

overwhelming. As mentioned earlier, thermal errors are not an issue for CMMs. It is

understandable that the thermal compensation schemes demonstrated by researchers might

make a builder reluctant to implement such systems. The many temperature sensors

normally used would surely increase the maintenance level on a machine. The perceived

benefits apparently do not outweigh the perceived complexity in the machine tool builder's

and user's minds.

Scope of the Research

In an attempt to assess the benefit/complexity trade off, the performance

enhancement of three thermal compensation models was evaluated against geometric

compensation and no compensation on a three-axis machining center. The geometric

compensation model is included to further reference how much thermal effects contribute

to the overall error.

The first thermal model evaluated was the simple first order scale compensation

model utilized by earlier researchers[ NIST-60NANB2D1214, 1993; Koda and Yoshiro,

1981; Zhang et al., 1985]. It is based on elementary physics and requires only four

thermal sensors. Nowhere in the literature search has a comparison been made to see how

much is really gained over this simple completely deterministic model.

The remaining two thermal models utilize the fuzzy ART map refined by Srinivasa


in a new implementation that combines it with kinematic modeling. One network was

trained using a traditional no load actuation thermal duty cycle and the other was trained

with a machining thermal duty cycle. This serves to quantify what errors are being missed

when the machine is measured by the common practice of excluding actual machining.

These models were evaluated at different thermal states by measuring body

diagonals with the Laser Ball Bar and by machining the B5.54 precision positioning test

part [ASME B5.54, 1992]. Thermal drift was addressed by using the industry accepted

practice of probing a reference feature.

All of the models utilized rigid body kinematics via homogeneous transformation

matrices. This type of model has proven successful in the research, but no testing of its

durability was found. Machine tool users will want to minimize the frequency of the

measurements for their models as much as possible. To address this, a durability test will

be conducted over several months to see how a model degrades.

From the tests mentioned above, the data was examined to shed light on the five

following areas: 1. success of implementing the Fuzzy ART-MAP to a 3D milling

machine, 2. model durability, 3. geometric modeling vs. thermal modeling, 4.

deterministic vs. non-deterministic modeling, 5. training with machining and without.

Chapter 3 describes the geometric model that was used and incorporated into the

thermal models. Chapter 4 describes the three thermal models employed. Chapter 5

details the measurement and machining procedures. Chapter 6 presents the results of the

durability tests, body diagonal measurements, and machined part tests. Chapter 7 draws

appropriate conclusions and makes suggestions for future work in this area.


Geometric Compensation

A machine tool's thermal error can be considered as geometric error with an

additional dependance on the machine's thermal state. In this definition, geometric error

is the machine's error at some given thermal state, usually the power on, machine idle,

steady state. Each thermal model evaluated in this study is built from the same geometric

model. The models only differ in how the variables to the geometric model are modified

with respect to temperature. The geometric model is a function of the machine's

parametric error motions:

GEOMETRIC= F(5 (X), 6 (Y) 5 (Z), (X), E (Y), ei(Z)}

i index representing the X, Y, and Z axes

and the machine's parametric errors gain a dependence on the thermal state in the thermal


"HERMAL=F{6,(X, T), 6i(Y,T), (Z,T), Ei(X,T), Ei(Y,T), i(Z, T)

T temperature field of the machine.


The geometric model utilized is the successfully demonstrated kinematic joint chain

model mathematically represented with homogeneous transformation matrices [Donmez,

1985; Chen, 1991; Mize, 1993] and will hence be referred to as the Homogenous

Transformation Matrix model, HTM model.

In an HTM model, the machine is represented as an open loop chain of rigid bodies

connected by rotary or prismatic joints. The chain is open between the work table and the

tool. The work piece fills this opening and closes the chain. HTMs mathematically relate

the position and orientation of one body relative to the next body in the kinematic chain

and can be used to transform vectors between coordinate systems. This matrix

representation was introduced by Denavit and Hartenberg [1955], as a systematic method

to compute the position of a robot end effector based on its joint positions.

In the notation representing the transformation, 'T, the superscript, I, denotes the

reference frame and the subscript, j, denotes the current frame. Premultiplying a vector

described in the current frame by this transformation matrix will produce the vector

described in the base frame. The 3x3 matrix bounded by the first three rows and the first

three columns contains the orientation information between frames and is called the

rotation matrix. The columns of this 3x3 matrix are the unit vectors of the current frame's

axes as described in the reference frame. The first three terms of the last column are the

components of the position vector from the origin of the reference frame to the origin of

the current frame, as described in the base reference. The last row is set to (0 0 0 1) for

our purposes. Other choices for the last row result in non-homogeneous transformations

which do not, in general, preserve lengths and angles. A convenient feature of HTMs is


that they can be multiplied in series to obtain a resultant HTM between two frames (e.g.

OT2 = Tl'T2).

A fundamental assumption implied in the model is that the bodies are rigid, i.e.,

that deflections of the bodies due to gravity and loading are negligible. This assumption

generally holds true in machine tools since positioning accuracy and good cutting dynamics

require very stiff machine components. However, if non-rigid effects are significant,

their effects will only be modeled if they are present at the machine positions selected for

parametric error measurements.

The HTM model requires measurements of all six degrees of error motion for each

moving body in the kinematic chain. For a 3 axis machine this results in 18 error

measurements. Traditionally the three orthogonalities between the axes are also included

in the total, resulting in 21 errors. However for modeling, it is convenient to include the

squareness errors into the appropriate straightness errors since squareness is merely

straightness with a linear dependence on position. When straightness has squareness

included it is has been appropriately referred to as lateral error to distinguish between the

two [Tlusty, 1980].

Kinematic Model for the Machining Center

The construction of the model begins by assigning coordinate systems to each body

in the kinematic chain. For the 3-1/ axis machine, ignoring the 'B' axis, we begin by

placing a coordinate system on the work table. This will be the frame of reference, since

this is the frame where the work piece is machined, see Figure 3-1. Working through the


chain, a frame is attached to the bed of the machine, then the 'X' carriage, and finally to

the tool itself. It is important to note that frame placement is arbitrary. A frame does not

even have to be within the physical bounds of the body. A frame can be placed outside

the bounds of the body and an assumed imaginary rigid attachment back to the body. The

goal of the HTM model is to know the error of the tool in the reference frame. For error-

free motion and assuming that all of the frames are coincident at some initial start position,

the three transformation matrices between the bodies on the machine are as follows:

0100 010 0 1 0 0 0
T, = 0 1 T2 = 0 1 23 = 0 1
0 0 0010 0 010
0001 0001 0 0 0 1

Since each body's motion is not free of error, the error motions need to be included

These are easily included by including an additional transformation for each body's perfect

motion that is due to the error motion. The true transformations between bodies then

becomes [T] =[T][E], where

1 -Ez(i) ey() 5x(i)
EZ(i) 1 -PX(i) 6y(i)
-ey(i) E(i) 1 65(i)
0 0 0 1

i index representing the prismatic axes

Figure 3-1 Machine coordinate frame assignment.


The error matrix is valid for small error motions where higher order terms can be ignored.

The actual transformations then become

1 -e(Z) E(Z) 5x(Z)
E (Z) 1 -ex(Z) 6r(Z)
-E (Z) ex(Z) 1 Z+68(Z)
0 0 0 1

1 -Ez(X) EC(X) 6x(X) +X
s. (X) 1 -Ex(X) 6y(X)
T, M
S -e (X) Ex(X) 1 56(X)

0 0 0 1

1 -Ez(Y) Ey(Y) 5x(Y)

2 E ) 1 -x(Y) 5+Y
-E (Y) ex(Y) 1 5(Y)
0 0 0 1

Concatenating the matrices from the work table frame to the tool frame, OT, = OT,

'T2 2T3, a transformation is obtained that transforms a vector in the tool frame into the

work table frame. For a given point in the tool frame, premultiplying it by the oT3 frame

will describe the point in the work table frame. Realizing that any point described in the

tool system not its origin is a tool offset, we can determine the actual position of the tool

for a given offset as


With the actual position determined, the error can be computed as:


XA Actual position
Xc Commanded position
XNOMNAL Commanded position of gage point
XT Tool offset from gage point

Carrying out the matrix manipulations, neglecting higher order terms, and substituting into

the above equation, the error equations are obtained:

Px = 6(X) +56(Y) +5x(Z) -Y{e,(X) +EZ(Z) } -Y{f (X) +Ez(Y) +Cz(Z) }
+Z{TEy(X) +Ey(Y) +Ey(Z) }

Py = 6y(X) +56y(Y) +6,(Z) +XEz(Z) +X T{E,(X) +E,(Z) }
-ZT{EX(X) +EX(Y) +ex(Z) )

P, = Z(X) +6 (Y) +5-(Z) -XE (Z) -XT E (X) + y(Y) +Ey(Z) }
+Y{(E(X) +Ex(Z) } +Y{(E(X) +Ex(Y) +Ex(Z)

The values X, Y and Z in the equations are the nominal machine coordinates from

the machine point where all of the errors are initially assumed to be zero. In this


application, this occurs at the intersection of the three measurement lines (0, 280, 340)mm.

This is the basic geometric model. Thermal effects are included by modeling each

parametric error with temperature as well as position, e.g., 6x(X,TO,Tl,...,Tn).

Implementation in the Controller

The geometric model is implemented inside the A950 controller. Each of the 18

parametric errors are least squares fit with 4th order polynomials with respect to axis

position. Reversal errors for each of the 18 parametric errors are stored as a single

coefficient computed as the average reversal error from the measurements. Squareness has

been included in the appropriate straightness coefficients, 1st order terms of straightness

errors. The coefficients for the polynomials are stored in a table that can be updated

through the parallel port on the controller. The polynomials and three error equations

shown above, are executed in a peripheral board running inside the control computer

autonomous from the servo control loop, and the program is shown in Appendix B. This

i386 based board monitors the axes positions and computes the positioning error to modify

the nominal position at every closure of the feedback loop (approximately 5 ms). A block

diagram of the servo system with the compensation is shown in Figure 3-2. The thermal

modeling was implemented by changing the appropriate coefficients in the table based on

temperatures monitored around the machine, and is discussed in the next chapter.




Figure 3-2 Block diagram of compensation system.



As mentioned previously, thermally induced error was compensated by modifying

the parametric error inputs in the kinematic model as a function of temperature as opposed

to directly correlating temperature to the volumetric errors. This is a valid deterministic

approach, since the parametric errors are known to change with temperature and the

volumetric errors can be predicted via the parametric errors and a kinematic model. Three

thermal models were evaluated that follow this general approach. Only the method by

which the parametric errors are correlated to temperature will differ.

The simplest model to be evaluated deals with only the linear variation of the

displacement errors with temperature. It is based on the elementary physics of material

expansion and contraction. The remaining two models utilize the fuzzy ART network

developed by Srinivasa in a new implementation to predict all 18 of the parametric errors

based on temperature inputs from the machine [Srinivasa, 1994]. These two models differ

only in their training cycle. One is trained with the traditional method of machine warm

up using axis and spindle actuation free of machining. The other uses actual machining

in its warm up cycle to more closely model realistic use.

First Order Thermal Model

The first level of thermal compensation is naturally to compensate for the thermal

expansion and contraction of the scales. This error results in a one to one error at the

tool, e.g., an 'X' scale error of 1 pm will contribute 1 /jm of tool 'X' coordinate error.

The scale error can be represented by the formula

6x(X, Txs.le) 5X (X) + fJa (T(X)) [T (X) xsca- TEF] dX

6x(X), Scale error at some reference temperature
a(T(X)) Temperature dependent coefficient of expansion
TR, Reference temperature scale is measured at

Assuming a uniform temperature distribution and a constant coefficient of expansion, the

equation reduces to
6x(XTxscale) x (X) + [Txscae- TR] X

The thermal portion of this error manifests itself in lead screw expansion for rotary

encoder feed back systems, and in scale expansion for glass scales and other solid linear

metrics. Laser interferometer scales have a similar error based on atmospheric wavelength

variation with temperature, although the dependency is typically an order of magnitude


This model has been chosen for comparison, because it is thought to be the most

readily accepted by industry. It is understandable that industry might shy away from

more complex models that require many thermal sensors (>10) and their associated


maintenance. This model utilizes 4 sensors, one for each axis and one for the work piece.

The literature review did not reveal any comparisons between more elaborate thermal

models and this simple model based on elementary physics.

Implementation of the Model

A temperature sensor, in this case a 'T' type thermocouple, has been placed on the

metallic housing surrounding each glass scale at its mid-span. The effectiveness of this

placement depends on the homogeneity of the scale temperature and the similarity of the

housing temperature to the actual scale temperature. The scales on the Maxim are

contained in an aluminum housing anchored at their midpoint to the machine structure.

Heat transfer is primarily from natural convection, so the homogeneity assumption should

be reasonable. The 'Y' axis scale is most likely to deviate from this assumption since it

has a vertical orientation that is more susceptible to vertical temperature gradients.

Multiple sensors could be placed along the length of the scale and included in a piecewise

integration to handle the gradients, but this digresses from the simple model objective of

as few sensors as possible. A sensor is also placed on the work piece to get an estimate

of the part's bulk temperature. This is used to compensate the scale to expand and

contract like the work piece material at its bulk temperature. This should help to minimize

errors caused by differing material/scale expansions when either the scale's or part's

temperature deviates from the international temperature standard of 200C. Thermal

gradients will surely exist in the part as a result of local heat transfer from the cutting

process, however coolant fluid will be used to minimize the effect. The system as stated,


is only capable of compensating for that portion of the part thermal expansion that is

equivalent to the temperature as read by the work piece sensor.

The data collection procedure requires the scale temperature, T,,, be recorded as

the displacement errors are measured. The displacement correction can then be modified

to make the scale expand and contract like the part material.

The error of the scale to be corrected is

SCALE ERROR = 5x(X) + [ (Tscale TRF)scae (Tpart 200C)Cpart]X

By correcting the scale's displacement error, 6x(X), at a reference temperature,

TREF, the temperature sensor's absolute temperature accuracy is not important, only its

relative accuracy. Thermocouples are well suited for relative accuracy over limited

temperature ranges. However, the material correction to 200C does require accurate

absolute temperature measurement, and for this reason a thermistor is used. The term

[(Tscale Tref)ac (Tpart 200C )ap~, is added to the 1st order coefficient in the 4th

order curve fit of the displacement errors.

Thermal Drift

This model does not have the capability to compensate for all sources of thermal

drift. Drift can be caused by thermal expansion/contraction of the material between the

scale anchor points. It can be caused by thermal expansion/contraction of the material

between the scale read head and the tool or work piece. Thermal drift can also be caused

by the scale expansion/contraction, which is the only source this model is capable of

addressing. Instead drift will be addressed by the industry accepted practice of probing


a reference feature on the part or machine prior to machining which mitigates all drift


The Neural Network Model

Because the relationship between machine tool error and the thermal state of the

machine is complex and beyond all but the most complex deterministic approaches,

Artificial Neural Networks (ANN) have been employed to mimic the thermal/error

relationship [Chen, 1991; Ziegert and Kalle, 1994; Srinivasa, 1994]. The network mimics

the relationship, because ANN's do not express this relationship based on the physics of

heat transfer and solid mechanics like finite element modeling. Rather, they develop a

relationship between empirically generated inputs and outputs based on a set of rules. The

ANN evaluated in this study is based on the ART MAP developed by Carpenter et al.

[1991], and refined for machine tools by Srinivasa [1994].

In simplest terms, the ART MAP categorizes a set of input vectors into discrete

classes and maps these input classes to corresponding categorized output vector classes.

In our case, the components of the input vectors are composed of the temperature sensor

readouts, the output vectors are the coefficients of the polynomial fits to the parametric

errors. Neural networks have two operating modes, training, and prediction. Neural

networks must first be trained on empirical data before prediction can take place.

During training, a new input vector is introduced and is compared to the existing

classes to see if a match exists or if the vector is a new unique class. A class is defined

as a unique input vector, where uniqueness is judged by a fuzzy logic comparison to the

existing classes. This comparison is made via a fuzzy logic equation whose output is a

similarity factor called vigilance, represented by the Greek letter rho.

XA w. min (Xi, W)
P --XA- = Xi

A fuzzy AND operator.
i index for the vector component.
J index for the existing classes being compared.
X vector to be compared.
W class vector.

Vigilance can range between zero and one. A vigilance of one is achieved if a

vector matches a class exactly. If the vigilance between a vector and an existing class is

larger than a user defined threshold, it is considered to be equivalent to the class. If the

input vector fails to meet this threshold similarity with any of the existing classes, it is

considered to be a new and unique class.

Simultaneously the same procedure is performed on the output vectors. Additional

logic insures that the input classes map to only one output class. However, an output class

can be mapped from more than one input class. If an output vector is found to be unique,

but its corresponding input is not, the input is forced, by changing the vigilance threshold,

to become a unique class. Figure 4-1, shows a graphical representation of the network.

The fuzzy ART-MAP can be thought of as an algorithm that identifies similarity

in input and output data sets to minimize redundancy, and maps the remaining input data

to the remaining output data sets. After training, the network is used to predict machine

0 1 0 0*-L
MAP 0 -0 1 ""1 RTo

. /L

CLASS I = + + W + ... .


cuSS A,= %,+ w.+ ... //





YIt c'A.

Figure 4-1 Schematic representation of fuzzy ART-MAP.


tool parametric errors by comparing new input vectors to the existing classes using another

fuzzy logic comparison. The fuzzy logic equation for comparison is called the activity


TXA A WJAI min(xi,Wj,
T (X A)
I wAl WA

A fuzzy AND operator.
i index for the vector component.
J index representing class number.
X Input vector to be compared.
W Existing class to be compared

The class that exhibits the most similarity based on this fuzzy logic test, is

selected and its corresponding output vector is the predicted error. The FAM's ability to

predict accurately is limited by the similarity of the training set to that of inputs presented

to it. Therefore, it is critical to present diverse inputs during training, to assure all real-

world thermal states are included.

Incorporation of the Network in the Model

The neural network can be utilized for mapping a variety of input/output feature

sets. The selection of where a network or networks is included in the model will be an

important factor in the model's performance. For instance, a network could be the model

itself if it were used to predict all three components of the volumetric error based on inputs

of temperature and position. This is similar to the approach Srinivasa used for

compensating the two dimensional errors of a lathe [Srinivasa, 1994]. However, such an

approach discards any deterministic solutions to the problem.

At the other end of the spectrum, networks could be used to predict each of the

parametric errors for input into a rigid body kinematic model. This would retain the

proven deterministic technique of computing volumetric error from the parametric errors,

and dedicate the network to mimicking the complex relationship of the parametric errors

with temperature and position. This architecture was considered for our model, but was

excluded for fear of attenuation of the positional inputs due to the more numerous

temperature inputs. Because all inputs in an ART-MAP inherently have equal weight, the

large number of temperatures, 31, could overshadow the 3 positional inputs. Srinivasa

experienced a similar problem with directional inputs in his model [1994]. He used an

ART-MAP to predict a single component of the 2D error vector on a lathe from the 2D

position input and selected temperatures. Originally his networks had an input for axis

direction. He found that the networks could not successfully predict errors of reversal

with this architecture and he opted to use separate networks for axis direction.

With this in mind, the approach is to avoid overworking the network and allow it

to handle only that part of the problem for which the deterministic approach has failed or

been overly complex to implement. Considering that the deformation field of an elastic

medium is uniquely determined from its temperature field and its kinematic boundary

conditions, a machine's deformations might be reasonably dependent on its temperature

field since the kinematic boundary conditions are relatively fixed. Realizing that the

variation of the parametric errors can be attributed to changes in the machine's geometry,

networks will be used to predict the parametric errors from selected temperature inputs.


Axes position will be eliminated as a direct input by using the network to predict the

coefficients of a 4th order least squares polynomial. The coefficients to the polynomial

inherently describe the error with respect to position, eliminating it as a direct input.

At each measurement cycle, the parametric errors are fit to 4th order polynomial

equations. The average reversal error for each parametric error is also computed and

included as an extra coefficient.

The coefficients are predicted from the temperatures using seven networks. The

input/output relationship of each network is shown in Table 4-1. Each of the scale error

coefficients are predicted from a single network using the scale's temperature sensor as

an input. This retains the deterministic part of the solution in which thermal scale error

is known to be purely a function of its temperature distribution. The remaining parametric

error coefficients are mapped to the temperature sensors on the body supporting their

prismatic axis. For example, the X-Axis angular and straightness parametric error

coefficients are mapped to the temperature sensors on the bed where the X-Axis ways are

mounted. This assumes that these errors are a function of their way's shape. Finally a

single network is used to map the parametric error drifts to all 31 temperature sensors,

since the deformation of each body in the kinematic chain can contribute to drift.

This type of architecture was selected to retain the proven deterministic technique

of computing volumetric errors from the parametric errors, minimizing the task of the

networks to determining the thermal dependence of the parametric errors. This is thought

to be a good application for an ART-MAP network, since they are good at selecting the

closest match to a finite number of selections. When measured parametric errors are









* See Figure 4-4 for temperature sensor location

plotted with their drift removed, curves tend to maintain their form and at most change

slope, Figure 4-2. Therefore, over a thermal duty cycle, selecting a curve from the finite

choices, is unlikely to be in great error since adjacent curves do not differ greatly.

Implementation on the Machine

In this study, two different training sets are evaluated: 1. warm-up without

machining, and 2. warm-up with machining. The literature review revealed no instance

where actual machining was utilized in the training cycle. The reason probably involves

the delicacy and incremental nature of the measurement transducers. Most measurement

transducers can only measure changes in their measurement parameter, not absolute

values. Laser interferometers, one of the more common metrology instruments, can only

measure changes in length (displacement), and can therefore not measure thermal drift if


E 0.010
"r 0.00oo5 --, -----------

-400 -300 -200 -100 0 100 200 300
X Axis (mm)
-- 1 2 --3 4 -5 --- 6 -.-7 8 9 -10 -*-11 12

Z -0.010
w -0.015
. -0.020
-0.025 -
-400 -300 -200 -100 0 100 200 300
X Axis (mm)


-0.020 .---
-400 -300 -200 -100 0 100 200 300
X Axis (mm)

-+-1 --2---3 -- 4 --5 --6 -+ 7 --8 9 10 -11 --12

Figure 4-2 Typical parametric errors at different thermal states.


their signal is interrupted by machining operations. Also their operation is restricted to

linear motion, further neglecting any useful machining.

The LBB overcomes the incremental measurement problem because it is initialized

to an absolute distance and trilateration produces absolute coordinates. It can therefore be

removed from the work zone during machining and reintroduced after, without loss of

reference. No loss of reference occurs as long as the magnetic sockets are not disturbed

from their original placement. They can move due to thermal contraction and expansion

of the machine elements, but this is merely the thermal drifting of the machine we intend

to measure. Also, any change in distance between the base sockets will be accounted for

by their measurement before each cycle. We are able to machine parts without disturbing

the socket placements, because the sockets on the spindle are mounted on the periphery

of the spindle housing allowing the simultaneous use of a cutting tool, Figure 4-3. Also

the pallet with the base sockets can be shuttled out while a part is shuttled in for

machining. By shuttling the pallet with the sockets in and out, we are depending on the

repeatability of the pallet relocation with respect to the Z axis carriage. Repeatability tests

were conducted by shuttling the pallet in and out several times and measuring the location

of a socket mounted on the spindle each time. The results are shown in Table 4-2. This

repeatability will limit the accuracy of the translational drift measurement.


Pallet Change Repeatability (Max variation of 10 runs)
X Axis Y Axis Z Axis

3.3,um 4.04m 1.64m

Figure 4-3 Socket placement on the machine.


A small compromise was also utilized in placing the sockets on the periphery of

the spindle. This was done because fixturing the sockets on a tool holder was found to be

too compliant. When the LBB is in a socket that is offset from the spindle center line,

enough torque is produced to cause unwanted rotation of the spindle. The servo locking

mechanism also tends to hunt. By not fixturing to the spindle, a small part of the

kinematic chain is bypassed. Any growth or drifting of the rotating portion of the spindle

relative to the housing will be missed. However, the socket placement on the housing is

very near the bearing connection to the spindle and any lost drift motion is thought to be

outweighed by the gain in socket rigidity.

The non-machining duty cycle is similar to that implemented by other researchers

[Donmez, 1985; Chen, 1991; Srinivasa, 1994]. It involves actuating the axes and spindle

at predetermined speeds and feeds, followed by an idle "machine on" cool down period

with intermittent pauses taken for measurement. Two thermal cycles were utilized over

two days for training. The first thermal cycle consisted of the spindle operating at 2500

RPM and the three axes actuated at 5 m/min for 30 minutes, followed by a measurement.

This routine was repeated continuously over an eight hour period. After this warm-up

period, the machine was left in the idle power on state for four hours. Measurements were

taken after 30 minutes and then every hour after that. The second thermal cycle was

identical except that the spindle was operated at 5000 RPM and the axes were operated at

10 m/min.

For the machining duty cycle, a production machine run was emulated, in which

an identical part or part features is/are repeatedly machined over a single shift day.


During this shift, several parts will typically be loaded, machined, and removed. This

cycle was emulated by machining for 30 minutes, cooling down for 10 minutes, and

measuring for 20 minutes. The machining consisted of a facing operation with an insert

type face mill in 6061 aluminum. Training occurred over two days so that different metal

removal rates could be used for the warm-up cycle. The first training cycle used a metal

removal rate of 76,200 mm'/min, for an approximate power consumption from the spindle

of 1 kW. The second training cycle used a metal removal rate of 381,000 mm'/min, for

an approximate power consumption from the spindle of 5 kW. These 60 minute cycles

were repeated over an 8 hour period. The cool down portion of the duty cycle was the

same as for the non-machining duty cycle, idle machine state for 4 hours, with

measurement after the first half hour followed by measurements approximately every hour.

Thermal Sensor Placement

The prediction capability of the neural network is dependent on the placement of

the thermal sensors on the machine. For this study, 'T' type thermocouples were utilized

as the temperature sensors. They were interfaced to a i486-33Mhz computer through a

National Instruments multiplexer board with cold junction compensation and a 16 bit

analog to digital converter with programmable gains. The multiplexer allows 31

thermocouples to be used. Each thermocouple input into the analog to digital converter

was conditioned with a low pass filter providing a cutoff frequency of 4 Hz with -3 db

attenuation and then amplified with a gain of 100. The filtering was necessary to reduce

noise pick up from the servo motors and transformers. This system provides 0.04 C of


resolution, but noise limits the useful resolution to about 0.1 'C. Placement of the

thermocouples is shown in Figure 4-4.

Locations were selected based on engineering judgement. Preference was given

to the Y column due to its proximity to the primary heat source, spindle motor, and its

relatively small mass. Sensors were placed along the front and back of the Y column on

both sides to allow the network to recognize any significant gradients that might cause

bending. Similarly, the spindle housing was instrumented around its relatively

unconstrained periphery. Both the X and Z beds were instrumented along their top

surfaces. Deformation of their bottom surfaces is constrained by 13 mounting pads

epoxied to a 12" thick concrete floor. Each glass scale was also instrumented with a

thermocouple on its housing at mid-span.

Figure 4-4 Thermal sensor placement on the machine.

i ~rill"


Geometric Error Measurement with the Laser Ball Bar

Because evaluation of thermal error is accomplished by measuring the machine's

geometric error at different thermal states, there are only parametric error measurement

techniques to be discussed. As was mentioned in Chapter 3, the geometric (or volumetric)

errors for a machine can be reliably represented by the machine's parametric errors and

a rigid body kinematic model, the HTM model. Utilizing the LBB's volumetric

measurement capability, the parametric errors can be decomposed from a series of

volumetric measurements.

Trilateration of a single point on a body as it moves along an axis, 'I', directly

provides the translational parametric errors [6x(I), 6y(I), 6z(I)] for that body. To measure

angular parametric errors [Ex(I),ey(I),ez(I)] with the LBB, the volumetric position of at

least three points on the moving body must be known; three non-collinear points on a rigid

body uniquely define its orientation and position. This can be accomplished by

trilaterating to three points on the moving body. This requires measuring nine variable


leg lengths. However, this actually provides redundant information. Only six distances

between two bodies are needed to define the position and orientation between the bodies.

This is the kinematic arrangement utilized in Stewart Platforms which are commonly

employed for flight simulators.

To minimize collection time, a six leg measurement procedure was utilized that was

developed by John Ziegert [Kulkarni, 1996], see Appendix A. The only draw back to this

procedure is the increased likelihood of bad measurement sensitivities in some coordinate

directions. With care in the set-up, adequate sensitivity can be maintained. A sensitivity

problem was encountered during early testing and was remedied by adding a 75 mm

extension, in the -Z direction, to one of the tool sockets, Figure 5-1. To verify the

accuracy of this set-up, angular parametrics were measured with both the nine and six leg

techniques for comparison. The results were favorable, with only slight deviation, 3 arc-

sees, between the methods for a few of the angular parametrics. The comparisons for the

X axis angular parametrics are shown in Figure 5-2.

Keeping the collection time to a minimum is important to insure the errors are

collected before significant thermal change can occur. The six leg measurement technique

reduces the 21 parametric error data collection time from 20 to 13 minutes over the

standard nine leg technique.

Data Collection Procedure

Data collection for a given thermal state begins with initialization of the LBB.

Reinitializing before each measurement cycle minimizes thermal errors that can occur in

Figure 5-1 Measurement set-up for LBB error measurement on the machine.


UP -2.0

fi -6.0 ---------------------" --

-10.0 --
-300 -250 -200 -150 -100 -50 0 50 100 150 200 250
X Axis (mm)

S-9LEG --86LEG


4.0 ---------
S3.0 --
g 2.0c----------- _----- --_---- --11- "

-300 -250 -200 -150 -100 -50 0 50 100 150 200 250
X Axis (mm)

$-9LEG -s-6LEG

4.0 -
3.0 ..

0.0 --1
i" 0.0- -----a---....
1.0 -

-4.0- ---
-300 -250 -200 -150 -100 -50 0 50 100 150 200 250
X Axis (mm)

1.0 5- -

g -1.0 1 .. -'--- "- "

-3.0 -

-300 -250 -200 -150 -100 -50 0 50 100 150 200 250
X Axis (mm)

-- 9LEG --.--6LEG

Figure 5-2 Error comparison between 6 and 9 leg technique.


in the LBB, see Appendix A. After initialization, the base lengths between the three

sockets on the pallet are measured, Figure 5-1. Next the distances between the three

sockets on the tool body are computed by trilaterating to each tool socket. The distance

between each of the tool sockets is required for computing their location with the six leg

measurement technique. Trilaterating to each tool socket is required to determine the

distances between the tool sockets, since the LBB is too large to directly measure between

them. The sockets are measured while the machine is at the commanded coordinate

reference location, (0 280 340)mm. This coordinate reference is the position where all of

the parametric errors are set to zero for the first measurement cycle. Comparing the

position and orientation at this commanded location to that of the first measurement cycle

provides the thermal displacement and orientation drifts.

Following these measurements, the six variable lengths are collected by running

the machine through a measurement sequence six times. While the base lengths and tool

positions can change over time, their variation is assumed to be negligible during a

measurement cycle. Base length measurements were taken before and after a measurement

cycle to verify this. Originally excessive growth was found in the I-Beam portion of the

fixture, but was remedied by insulating it with 6 mm thick foam. Table 5-1 shows the base

length measurement variations before and after a measurement cycle with and without




NO INSULATION 2.354m 4.174m 4.43gm

INSULATED 0.35gm 1.35g/m 1.524m

The CNC program was repeatedly executed for each leg. Recording data for a

single leg during a complete measurement sequence is referred to as sequential

trilateration, see Appendix 1. The CNC program's commanded moves are as follows: The

tool was moved from its start position (0 280 340)mm in the negative X direction -341.000

mm and then preloaded in the positive direction 1.000 mm. After a dwell of 0.5 seconds

a relay was closed and then reopened to trigger a measurement from the LBB. One

thousand laser readings were sampled at approximately 10 kHz and averaged to reduce

vibrational noise. The machine was dwelled for 0.7 seconds to insure the data collection

was complete. Next the machine was moved 67.000 mm in the positive X direction and

the collection cycle (dwell-trigger-dwell-move) repeated. A total of nine moves in the

positive direction were commanded. After the last positive move the axis was moved an

additional 1.000 mm and then reversed 1.000 mm to preload the axis in the negative X

direction. The data collection in the reverse direction was identical to that of the forward

direction described above. At the completion of the X reversal measurements the machine

was returned to its start position. Immediately following, a similar cycle was repeated for

the Y axis with 56.000 mm steps and then for the Z axis with 45.000 mm steps. Each of

the six variable legs were measured with this machine program. Measurement of the six

legs took approximately 10 minutes. Parametric measurements over these ranges


combined with the HTM model resulted in a compensation volume of 603 x 504 x 405

mm^3, Figure 5-3.

Data Collection for Model Verification

The part machining tests evaluated the complete compensation system, but only

over a limited amount of the compensated work space. To evaluate the models over a

larger volume and verify their functionality before material is machined, body diagonals

were measured with the LBB. Normally diagonal measurements are performed with a

linear interferometer. However, alignment and set-up can be very time consuming. In

order to further reduce the data collection time, only five points were measured

bidirectionally along the diagonals. Diagonal measurements are a good evaluation of the

whole model's functionality since all 21 parametric errors can contribute to their

inaccuracy. Four body diagonals were measured at four different thermal states for each

model: cold state-measure diagonals-30 minutes at 3500 RPM and 7 m/min all axes-

measure diagonals-1 hour at 3500 RPM and 7 m/min all axes-measure diagonals-1 hour

at 3500 RPM and 7 m/min all axes-1 hour cool down- measure diagonals. A spindle speed

of 3500 RPM and 7 m/min axes actuation was chosen because it falls between the non-

machining training cycles of 2500 RPM with 5 m/min and 5000 RPM with 10 m/min.

Using the LBB did limit the diagonal measurement lines to 499 mm, 609 mm, 422 mm,

and 394 mm. Figure 5-3 shows the body diagonals relative to the compensated work




#1 (-187.165, 430.257, 267.650) to (144.945, 152.673, 490.709)= 498.607 mm
#2 (-231.296, 144.857, 496.990) to (183.855, 491.849, 218.157) = 608.688 mm
#3 (-177.698, 422.344, 460.991) to (144.945, 152.673, 490.709)= 421.550 mm
#4 (-192.002, 177.700, 264.402) to ( 76.459, 402.085, 444.711)= 393.613 mm

Figure 5-3 Body diagonal's location relative to the compensation zone.


It should be noted that the compensation system implemented did not update the error

coefficients in real time. The controller utilized did not permit real time updates. The

coefficients were updated via an external computer using the controller's parallel port just

before a CNC part program execution. For short programs such as the diagonal

measurements and the B5.54 parts, this should not be a problem

Body Diagonal Measurement Procedure

First the coefficients were downloaded from the appropriate compensation model

based on the temperature sensor readings taken from the external computer. Next the LBB

was initialized and the three base lengths were measured. Then the LBB was placed

between tool socket #1 and each of the three base sockets as the CNC program was

sequentially executed three times for each length measurement, see Appendix A.

Measurement of body diagonals 2, 3, and 4 followed, each requiring about 3.5 minutes

to collect. The LBB was not reinitialized and the base lengths were not remeasured for

the other body diagonals since the whole procedure could be completed in about 15


B5.54 Part Machining Procedures

The ultimate test of any compensation system is to improve the accuracy of

machined parts. Each of the compensation systems for this study were compared by

machining the B5.54 precision positioning test part with compensation active at different

thermal states, Figure 5-4. The part is intended to provide a statistically significant


number of features to test bidirectional positioning accuracy, repeatability, squareness, and

circular profiling capability [ASME B5.54, 1992].

The test began by machining a part while the machine was in its cold state. Next

the machine was warmed by machining scrap 6061 aluminum at a metal removal rate

requiring 5 kW of energy for 8 minutes. Another B5.54 part was then machined at this

new thermal state followed by a 1 hour cycle of 5000 RPM and 10 m/min spindle and axis

actuation and then additional scrap machining requiring 10 KW for 4 minutes. A third

B5.54 part was then machined. The machine was then allowed to cool, with all systems

powered without actuation, for one hour. A final and fourth part was then machined at

this cool down thermal state. This test cycle was performed with the machine tool control

in its normal non-compensated mode and with the each of the 4 compensation systems

active. Each model was evaluated with this test sequence on separate days with the

ambient conditions held at nearly 24 "C before each test. Each part took approximately

30 minutes to machine including 10 tool changes. Approximately 285,120 mm3 of

aluminum was removed from the part. Figure 5-5 and 5-6 show the scrap material and

B5.54 part just prior to machining.

Y17.500 P 36 PL
13 000


305 000

m -J -

B .000.0 .( 000 0

Figure 5-4 B5.54 precision positioning test part.

Figure 5-5 Scrap material just prior to machining.

Figure 5-6 B5.54 test part just prior to machining.


Model Evaluations

The compensation models were evaluated for stability, accuracy, and post machine

crash durability. Translational parametric errors (displacement and straightness) were

measured over a four month period with 1st order thermal compensation active to evaluate

stability. Body diagonals were measured with the LBB to evaluate accuracy improvements

at different thermal states for all of the models. Test parts were also machined and

inspected to evaluate all of the model's accuracy improvements at different thermal states.

As the result of an accident, the durability of the models were tested after the tool was

crashed into the tombstone.

Model Stability Evaluation

The durability of the geometric and thermal models is dependent on the stability

of the machine's parametric errors. It is important that these errors remain constant over

a long enough period so that remeasurement cycles can be minimized. Thermal duty cycle

measurements with the LBB require a day to complete and manufacturers will certainly

wish to minimize this non-machining use of their machines.


To evaluate the machine's geometric stability, parametric data were measured over

a 4 month period. Correction data were collected on 8-20-97 and input into the

controller. Translational parametric measurements were then taken over a 4 month period

with the 1st order thermal created on 8-20-97 running inside the controller. Only the

translational parametrics were examined because the angular parametrics can not be

directly corrected without any rotary actuation capability. The translational parametrics

were evaluated on the same lines in space along which compensation data was collected.

During this 4 month period, the machine was used to machine a few miscellaneous parts,

the B5.54 precision positioning part included, and some trial warm up tests involving

spindle and axis actuation without machining. The amount of machine use would be

considered light by industry standards. Figure 6-1, 6-2, and 6-3 compare the X, Y, and

Z axis parametrics without compensation and with compensation over the 4 month period.

The compensation is shown to be significant and stable over this time period. For

instance, the 6y(X) error is reduced from about 24 pm to 3 4tm and remains in this range

throughout the 4 month period. Table 6-1 shows the axes squareness before compensation

and after compensation for the 4 month period.


Squareness Errors over 4 Month Period with Compensation (arc-secs)
No Comp 8-20-97 9-18-97 10-14-97 12-19-97

XY Squareness 7.5 -0.7 -0.1 0.4 2.2

XZ Squareness -10.9 1.7 2.9 1.7 3.6

YZ Squareness 12.6 1.1 0.2 0.1 3.3


-400 -300 -200 -100 0 100 200 300 400
X Axis (mm)

----20-Aug 18-Sep -14Oct -- 19-Dec --No Comp

E 0.005 -

0.000 -
m -0.005
-0.015 ------
-400 -300 -200 -100 0 100 200 300 400
X Axis (mm)

-- 20-Aug -i 18-Sep -14-Oct -X- 19-Dec -- No Comp


E 0.002
w -0.002 .
-400 -300 -200 -100 0 100 200 300 400
X Axis (mm)

I --"-20-Aug X 18-Sep ---- 14-Oct 19-Dec -1--No Comp

Figure 6-1 Compensated X-Axis translational parametrics over a 4 month period.

m -0.002

E 0.003
x -0.001


100 200 300 400 500

Y Axis (mm)
---20-Aug 18-Sep -14-Oct -X 19-Dec -W-- No Comp


100 200 300 400 500

Y Axis (mm)

----20-Aug ---- 18Sep -14-Oct -X 19-Dec --- No Comp

E 0.001
-0001 /
uw -0001
N -0 002
0 100 200 300 400 500 600

Y Axis (mm)

----20-Aug 18-Sep --- 14-Oct -(- 19-Dec ---NoComp

Figure 6-2 Compensated Y-Axis translational parametrics over a 4 month period.


0.008 I

0o00 ---8 ----

N -0002

0 100 200 300 400 500 600
Z Axis (mm)

--- 20-Aug 1- 14-Oct 9-Dec --No Comp


100 200 300
Z Axis (mm)

400 500 600

--0 20-Aug -- 18-Sep r -14-Oct -(- 19-Dec --- No Comp


100 200 300 400 500 600
100 200 300 400 500 600

Z Axis (mm)

- 4- 20-Aug ---- 18-Sep -14-Oct 19-Dec --- No Comp

Figure 6-3 Compensated Z-Axis translational parametrics over a 4 month period.

6 0.001

E 0.001
W -0.001
-0 001



Diagonal Measurement Evaluation of the Models

To evaluate the models over a larger volume than encompassed by the test parts

and check for programming mistakes before wasting material, body diagonals were

measured in the compensated work volume. The four body diagonals were measured at

four different thermal states for each of the compensation systems with the LBB. Graphs

for the first body diagonal at the four thermal states are shown in Figures 6-4 and 6-5.

Table 6-2 shows the total ranges of the errors over all four thermal states and the

improvement ratio over the uncompensated machine for each body diagonal.


Diagonal Error Ranges over 4 Thermal States [microns][improve ratio]

No Comp Geo Comp 1st Therm ANN #1 ANN #2

Diag #1 33.5 16.8 2.0 5.8 5.8 12.8 2.6 11.8 2.8

Diag #2 34.0 21.0 1.6 8.3 4.1 12.1 2.8 14.7 2.3

Diag #3 19.8 16.8 1.2 8.6 1.7 11.3 1.8 11.7 1.7

Diag #4 23.6 21.7 1.1 12.1 1.9 8.3 2.8 8.7 2.7

Diagonal #1 No Compensation


0 100 200 300 400 500
Diagonal length (mm)

---State 1 State 2 -- -State 3 --- State 4

Diagonal #1 Geometric Compensation

E 0.02
0.01 -----.-- --.


0 100 200 300
Diagonal length (mm)

400 500

- State 1 State 2 State 3 -x State 4

Diagonal #1 1st Order Thermal Compensation

0 100 200 300
Diagonal length (mm)

400 500

Figure 6-4 Body diagonal #1 for four thermal states: no compensation, geometric, and
1st order thermal.

E 0.02
t 0.01

-0.01 :

-+- State 1 + State 2 -A State 3 State 4

Diagonal #1 ANN Nonmachining Training

E 0.02


-0.01- --
0 100 200 300 400 50
Diagonal length (mm)

--- State 1 -- State 2 -- -- State 3 x State 4

Diagonal #1 ANN Machining Training

0.03 ---
E 0.02

o .....-..^SS
0 100 200 300 400 500
Diagonal length (mm)

-- State 1 i- -State 2 -State 3 --x- State 4

Figure 6-5 Body diagonal #1 for four thermal states: ANN #1, and ANN #2.



Post Machine Crash Data

During data collection of the 5 kW machining training the face mill was

accidentally crashed into the face of the tombstone between the 2nd and 3rd data collection

runs. The crash occurred while the tool was approximately 50 mm from the tombstone

as it was erroneously commanded into the tombstone via a Z axis rapid move. Five of

the six carbide inserts were damaged absorbing much of the impact. Subsequent

measurements showed a significant change in the linear portion of the straightness errors

of the X and Z axis, specifically the 6x(Z) and 6z(X) errors, Figure 6-6. However the

squareness between the X and Z axes did not significantly change, indicating the axes

themselves remained unchanged. Observing the change in the first order terms of these

two errors between the 2nd and 3rd runs revealed they changed by approximately the same

magnitude, indicating that a rigid body rotation of the pallet locating interface must have

occurred; change in 6z(X) term: -5.3 to -56.6 microns/meter, change in 6x(Z) term: 39.8

to 87.3 microns/meter. Subsequent measurements of these straightness errors showed that

their coefficients were approximately the same as before the crash. Since this test required

the removal of significant amounts of expensive aluminum, approximately $800, it was not

feasible to repeat it. Instead the appropriate first order coefficients were changed by 50

microns/meter to remove the rigid body rotation. This type of rigid body rotation would

be significant and non removable if it occurred during the cutting of a part or if any part

feature needed to be referenced to some feature on the pallet, such as a T-slot. However,

for this research no features are machined that are dependent on any pallet reference.

0.020 -----
E 0.010 i----------- -- ^ '*
- 0.0050----

-200 -150 -100 -50 0 50 100 150 200 250 300
Z Axis (mm)
-- 2 --3



E 0.040

0.030 -- -

: 0.020 ----
0.010 .

0.000 4--
-400 -300 -200 -100 0
-F-2 -- 3 X Axis (mm)

100 200 300

Figure 6-6 Change in straightness parametrics due to machine crash.


It is interesting to note that the parametric error coefficients lend themselves well for

correction or modification of the model. This could be valuable if a machine crash

revealed a change in alignment, because only a limited number of coefficients would

require modification instead of remeasuring over an entire thermal state.

Machine Part Evaluation

The precision positioning test parts were inspected on a Brown and Sharpe

Microval PFx CMM with an uncertainty of 0.002 ,m in the plane of the measurements.

The CMM was compensated in the plane of the part measurements and checked by

measuring a fixed length (300 mm) ball bar at 6 different orientations in the plane.

Per the B5.54 standard, a coordinate system was defined relative to holes 1A, 2A,

and 3A [ASME B5.54, 1992]. The bores and counter bores at locations IB-36B were

located by probing 6 points on their periphery to find a best fit circle, see Figure 5-4. The

peripheral planes were probed at 54 and 11 locations for the X and Y surfaces,

respectively. Table 6-3 through 6-6 compares the feature errors for the different models

at the four thermal states. The features compared are: the X and Y positioning error for

the bores and counter bores machined from the forward X and Y directions (locations 1B-

18B) and reverse X and Y directions (locations 19B-36B); the XY squareness between

surfaces -J- and -E-; and the X and Y dimensions between the peripheral surfaces. The

errors reported for the bore and counter bores are given as the maximum, minimum, and

span (max-min) of the 36 features. Circularity evaluations of the contouring cuts were

inconclusive. A low pressure problem existed with the counter balance mechanism on the


B5.54 Part Errors-Thermal State 1

Feature No Comp Geo Comp 1st Thermal ANN #1 ANN #2

X (jm) max 12.0 13.0 6.0 8.0 8.0
Positioning -
FWD Bores min -22.5 -23.8 -10.1 -17.8 -18.5
& C-Bores
1B-18B span 34.5 36.8 0.9 16.1 2.1 25.8 1.3 26.5 1.3

Y (gm) max -1.1 1.8 -2.3 -1.3 2.4
FWD Bores min -11.8 -13.7 -11.4 10.6 -6.0
& C-Bores
IB-18b span 10.7 11.9 0.9 9.1 1.2 9.3 1.2 8.4 1.3

X (4m) max 17.0 14.0 16.0 16.0 17.0
REV Bores min -23.5 -24.2 -8.1 20.9 -14.3
& C-Bores
19B-36B span 40.5 38.2 1A1 24.1 1.7 36.9 1.1 31.3 1-3

Y (m) max -2.2 -3.3 0.4 11.6 0.5
REV Bores min -12.3 -12.9 -5.5 21.0 --7.2
& C-Bores
19B-36B span 10.1 9.6 1.1 5.9 1.7 7.8* 1.3 7.7 1.3

XY Square -10.4 -13.3 0.8 7.2 1.S 19.8 0.5 32.4 0.3

X-DIM (1m) 18.1 16.2 1.1 56.8 0.3 35.9 0.5 43.3 0.4

Y-DIM 36.0 33.0 1.1 47.2 0.8 22.1 16 26.7 1,4
* Three outlier points excluded.


B5.54 Part Errors-Thermal State 2

Feature No Geo Comp 1st Thermal ANN #1 ANN #2
Error Comp

X (m) max 13.0 80 0.0 16.0 14.0
ng FWD min -23.8 -27.9 13.4 -22.3 -25.1
Bores &
C-Bores span 36.8 35.9 1.0 13.4 2.8 38.3 0.9 39.1 0.9

Y (fm) max 1.8 0.2 4.3 5.9 2.3
ng min -13.7 -13.5 -18.6 11.9 -11.4
Bores &
C-Bores span 11.9 13.7 0.9 22.9 0.5 17.8 0.7 13.7 0.9

X (Cm) max 14.0 21.0 15.0 22.0 21.0
ng min -24.2 -20.6 -6.8 -19.0 -23.0
Bores &
C-Bores span 38.2 41.6 0.9 21.8 1.8 41.0 0.9 44.0 0.9

Y (m) max -3.3 -4.3 6.6 3.1 -3.7
ng min -12.9 15.4 -10.3 -12.7 -13.9
Bores &
C-Bores span 9.6 11.1 0.9 16.9 0.6 15.8 0.6 10.2 0.9

XY Square -13.3 13.3 1.0 7.6 1.8 24.5 1.8 10.4 1.3

X-DIM (pm) 16.2 18.5 0.9 66.6 0.2 23.1 0.7 27.1 0.6
Y-DIM 33.0 33.2 1.0 51.8 0.6 29.3 1.1 21.3 1-6
L (m)


B5.54 Part Errors-Thermal State 3

Feature No Comp Geo Comp 1st Thermal ANN #1 ANN #2

X (tm) ma 16.0 16.0 1.0 19.0 20.0
Positioning x
FWD mi -22.7 -25.5 -11.9 20.9 24.6
Bores &C- n
Bores spa 38.7 41.5 0 12.9 3.0 39.9 0.9 44.6 1.1
1B-18B n

Y (Lm) ma -4.4 1.0 -3.1 0.1 0.4
Positioning x
FWD mi -14.3 11.4 19.1 -13.6 11.9
Bores & n
C-Bores pa 9.9 12.4 0.8 16.0 0.6 13.7 0.7 15.9 06
1B-18B n

X (Um) ma 16.0 33.0 14.0 29.0 32.0
Positioning x
REV mi -21.8 -21.7 4.2 -17.2 20.2
Bores & n
C-Bores spa 37.8 54.7 0.7 18.2 2.1 46.2 0.8 52.2 0.7
19B-36B n

Y ma -3.6 0.76 6.5 0.0 0.0
(Cm) mi -11.5 -10.9 -8.2 -13.2 -14.2
Bores & n
C-Bores -
C-Bores spa 7.9 11.6 0.7 14.7 0.5 13.2 0.6 14.2 0.6
19B-36B n

XY Square -6.8 10.4 0.7 5.7 1.2 9.0 0.8 10.1 0.7

X-DIM (zm) 13.2 14.4 0.9 73.4 0.2 16.5 0.8 17.2 0.8

Y-DIM 37.6 32.1 1.2 56.2 0.7 28.1 1.3 19.1 2.0


B5.54 Part Errors-Thermal State 4

Feature No Geo 1st ANN #1 ANN #2
Error Comp Comp Thermal

X (Cm) max 13.0 16.0 1.0 23.0 15.0
FWD Bores min -24.4 25.8 -13.8 -28.3 31.1
& C-Bores
1B-18B span 37.4 41.8 0.9 14.8 2.5 51.3 0.7 46.1 0.8

Y (m) max 2.3 3.6 -0.2 -0.2 4.7
FWD Bores min -11.6 18.1 -18.6 -13.9 11.4
& C-Bores
1B-18B span 13.9 21.7 0.7 18.4 0.8 13.7 1.0 16.1 0.9

X (an) max 15.0 26.0 12.4 33.0 27.0
REV Bores min -22.8 19.8 -8.8 26.2 -27.1
& C-Bores
19B-36B span 37.8 45.8 0.8 21.2 1.8 59.2 0.6 54.1 0.7

Y(rm) max -2.3 1.6 5.8 -3.8 -1.3
REV Bores min -11.5 16.9 -8.2 -15.5 -15.5
& C-Bores
19B-36B span 9.2 15.3 0.6 14.0 0.7 11.7 o08 14.2 0.7

XY Square -14.4 10.4 1.4 16.9 0.9 11.5 1.3 14.8 0.9
X-DIM (/m) 15.2 11.6 1.3 70.1 0.2 4.2 3-6 12.1 1,3

Y-DIM 34.2 34.0 1.0 54.9 0.6 22.7 15 17.1 2.0


machine that caused unwanted vibrations at feed rates below 500 mm/min in the negative

Y axis direction. This resulted in poor surface finish at the top and bottom of the milled

reliefs in the center of the part, greatly overshadowing any potential improvements.

Examining the data reveals that only the 1st order thermal model showed

significant improvement. The X positioning accuracy improved between 1.7 and 3.0 times

over the non compensated part, similar to the diagonal test results. The Y axis positioning

degraded as the machine was warmed, but the magnitude of the error span never exceeded

18.9 microns. The squareness error in general stayed the same magnitude or improved

while changing sign from acute to obtuse. It should be noted that the accuracy of the

squareness measurements is limited to about 4 arc-secs for a positional accuracy of

1.0 /m over the short Y axis surface of 50 mm. The only features to get substantially

worse were the dimensions between the surfaces. The surfaces were very long (78 uum

max), and this is unusual since all of the hole positions were slightly short (10 pzm). This

occurred, to a lesser extent, for all of the models. It is possible that the surface error was

caused by either static or dynamic deflection of the tool. Down milling can be especially

prone to undercutting since the forces tend to move the cutter away from the work piece.

The hole locations would not be sensitive to tool deflection, only their size could be

affected. The surface error was investigated by measuring the transfer function of the tool

and inputting the cutting parameters into a simulation program to check for deflection, this

is discussed in the next chapter.

The two neural network models only showed improvements at the cold state,

improvement ratios of approximately 1.3. There appeared to be little difference between


them. Their accuracy deteriorated just below the accuracy of the non-compensated

machine at the later thermal states. Their error was short, which is to be expected since

the parts were machined at a higher temperature than 20 C and this was not compensated

for as in the 1st order thermal model. Additional cutting tests were conducted while the

part temperature was monitored to correct for the part expansion. These tests are

discussed in the next chapter.


The diagonal tests indicate that all of the models improved the machine's

positioning accuracy. Unfortunately they all did not show significant improvement in the

machining tests. Metal removal does add additional uncertainties that must be controlled

to achieve similar results as in non-machining tests. However, machining accurate parts

is the ultimate goal and function of a machine tool. Fortunately, the discrepancy is

explainable, as subsequent testing proved.

In the following paragraphs, relevant conclusions are made from the data regarding

the questions this research intended to investigate.

Success of Implementing Fuzzy ART-MAP on a Milling Machine

Previous research implemented the Fuzzy ART-MAP neural network on a 2-Axis

lathe [Srinivasa, 1994]. The lathe network was trained using the volumetric errors at a

grid of points in the 2D work zone as inputs. Implementing the network on 3-Axis

machine, excluded measuring the volumetric errors as direct inputs into the network due

to the order of magnitude increase in the number of points. Instead, as much of the

deterministic solution was retained as possible, by using the kinematic model and allowing

the network to predict its coefficients. This freed the network from having to determine

the positional dependence of the errors, and focus on only their thermal dependence.



The diagonal tests showed that the network did a good job of reducing the errors

at four different thermal states. Error reduction was on the order of 2-3X, maintaining the

errors between about + 5.5gm regardless of temperature. The 1st order thermal model

performed slightly better, maintaining errors between about 4.0im. From translational

parametric error measurements made after some of the diagonal tests, it appears the

network's weakness was in its ability to accurately predict the displacement error variation

with temperature. Given that only 24 thermal states were provided for training, the

network performance was reasonable. This is an area where the deterministic approach

is probably superior and adequate. Future researchers might consider combining the 1st

order thermal compensation of the displacement errors with the neural network

compensation of the remaining parametric errors.

At first glance, the data from the machining tests does not appear as favorable as

the diagonal tests. But the apparent degradation in accuracy of the parts cut using the

neural networks is explainable and expected. During the diagonal tests, the models are

measured against the laser interferometer, which is compensated to be independent of

temperature. During machining tests, the part material, unless cut at 200C will affect the

machined dimensions. The networks were designed to compensate the machine to be

accurate, regardless of temperature, as the diagonal tests proved. The network produced

parts improved the accuracy in the cold state, but appeared to degrade the accuracy at the

later thermal states when compared to the non-compensated machined parts. This occurs,

because as the machine warms, the atmosphere and part temperatures increase. The non-

compensated scales expand with the part, helping to reduce the error, while the networks


attempt to maintain the scales at the same correct length. This has the effect of making

the accuracy of the networks appear to degrade. Relative to a part's accuracy they do, but

they are correctly positioning the machine. To be a more useful system, part temperature

compensation should be added to the system as demonstrated in the 1st order thermal

model. To verify what would be achievable with such a system, two additional parts were

machined with network #2 and their part temperatures recorded to compensate out the part

expansion error in the data. The results are shown in Table 7-1 and 7-2. As expected, the

neural network compensated parts performed about as well as the 1st order thermal model.

This is similar to the results from the diagonal tests. The surface distances are still too

large, but this was found to be unrelated to the compensation models and is discussed

below. In future work, performance might be improved if the 1st order and neural

network models are combined.

Model Durability

The four month durability tests indicated that the 1st order thermal model

maintained the machine's accuracy very well over this period. Squareness errors remained

below 3.6 arc-sees and the translational parametric errors never exceeded a band of 5

pm. During this four month period, a limited amount of machining occurred. As a final

evaluation, the translational parametric errors were measured on 5-19-98 with the

compensation data collected on 8-20-97, about a nine month period. During this period,

all of the B5.54 parts and trial parts (approximately 30) were machined as well as about

67,000,000 mm3 of scrap material. The spindle unit was also replaced after the oil-mist


cooling and lubrication system quit, causing damage to the bearings. Also all 31

thermocouples were replaced prior to the final testing. The squareness errors and error

ranges for the translational parametric errors


B5.54 Part Errors-Thermal State 1

Feature No Comp Geo Comp 1st Thermal ANN #2
Error Temp

X (um) max 12.0 13.0 6.0 0.9
FWD Bores & mm -22.5 -23.8 10.1 -11.5
1B-18B span 34.5 36.8 0.9 16.1 2.1 12.4 2.8

Y (.m) max -1.1 -1.8 -2.3 0.2
FWD Bores & min -11.8 13.7 -11.4 -12.5
IB-18B span 10.7 11.9 0.9 9.1 1.2 12.7 0.8

X (Mm) max 17.0 14.0 16.0 18.9
REV Bores & min -23.5 24.2 -8.1 -3.3
19B-36B span 40.5 38.2 1.1 24.1 1.7 23.3 1-7

Y (0m) max -2.2 -3.3 0.4 13.1
REV Bores & min -12.3 -12.9 -5.5 -7.3
19B-36B span 10.1 9.6 11 5.9 1.7 20.3 0.5

XY Square -10.4 -13.3 0.8 7.2 1.5 28.1 0.4

X-DIM (um) 18.1 16.2 1.1 56.8 0.3 28.8 0.6

Y-DIM 36.0 33.0 1.1 47.2 0.8 16.9 2.1
(Im) _


B5.54 Part Errors-Thermal State 3

Feature No Comp Geo Comp 1st Thermal ANN #2
Error Temp

X (im) max 16.0 16.0 1.0 1.2
FWD Bores & min -22.7 -25.5 -11.9 13.0
1B-18B span 38.7 41.5 0.9 12.9 3.0 14.1 2.7

Y (Um) max -4.4 1.0 -3.1 -1.1
FWD Bores & min -14.3 -11.4 19.1 19.1
IB-18B span 9.9 12.4 0.8 16.0 0.6 18.0 0.6

X (/m) max 16.0 33.0 14.0 15.2
REV Bores & min -21.8 -21.7 -4.2 0.1
19B-36B span 37.8 54.7 0.7 18.2 2.1 15.1 2.5

Y (Am) max -3.6 0.76 6.5 7.6
REV Bores & min -11.5 10.9 -8.2 -8.0
19B-36B span 7.9 11.6 0.7 14.7 0.5 15.6 0.5

XY Square -6.8 10.4 0.7 5.7 1.2 15.8 0.4

X-DIM (ym) 13.2 14.4 0.9 73.4 0.2 48.5 0.3

Y-DIM 37.6 32.1 1.2 56.2 0.7 40.9 0.9


are shown in Table 7-3. The XZ and YZ squareness errors have degraded to 6.4 and 5.2

arc-secs, respectively, but the XY squareness is still quite good, -0.7 arc-secs. All of the

translational parametric errors remained at about the original compensation levels except

for the X and Y displacement errors (10.5 and 10.9 /m). This could be caused by the

replaced thermocouples. The 'T' type thermocouples only have an accuracy and

repeatability of about 1.0 'C. This would account for about 7.2 ym error in the X scale

and about 6.0 ym in the Y scale measurements.


Model Durability Follow-up Test
No Comp 8-20-97 9-18-97 10-14-97 12-19-97 5-19-98

XY (arc-sec) 7.5 -0.7 -0.1 0.4 2.2 -0.7
XZ (arc-sec) -10.9 1.7 2.9 1.7 3.6 6.5
YZ (arc-sec) 12.6 1.1 0.2 0.1 3.3 5.8

8x(X) (0m) 11.9 7.1 7.0 4.9 7.7 10.6

8y(Y) (Cm) 9.7 6.1 5.7 7.5 4.6 11.0
6z(Z) (am) 11.0 4.0 3.1 3.2 3.7 2.8

8y(X) (Gm) 25.3 1.9 2.1 2.0 4.1 5.6
6z(X) (am) 8.8 3.1 4.3 3.7 2.5 4.7
6x(Y) (Gm) 8.5 1.8 2.5 3.1 3.3 2.5
8z(Y) (Um) 3.7 1.4 1.4 1.3 2.7 3.4
6x(Z) (Um) 3.0 4.4 3.8 3.6 3.8 4.4

by(Z) (Om) 2.4 2.1 1.5 1.7 2.0 1.5

The technique of inputting the parametric errors into a rigid body model appears

to be useful for error reduction over a 9 month period for this particular machine. How


well compensation durability translates to other machines will depend on the quality of

their construction. It is reasonable to assume a machine with similar components would

perform in a similar fashion.

Geometric vs. Thermal Modeling

Clearly the results of the diagonal and part tests reveal that geometric compensation

only proves helpful at a single thermal state. The diagonal tests indicate that the geometric

compensation reduced the errors at the cold state, but as soon as the machine was warmed,

the accuracy deteriorated at the same rate as the uncompensated machine. In fact, due to

the sign of the errors in the uncompensated cold state, as the machine warmed the

uncompensated machine's accuracy improved for some of the diagonals, while the

geometric compensated machine degraded. Geometric modeling might be adequate for

scenarios in which the machine will be operated at or near a single thermal state such as

the steady state reached in a long production run. However, thermal modeling must be

considered if machines are to be used for flexible manufacturing, in which different

numbers and types of parts are machined from day to day or week to week.

Deterministic vs. Non-deterministic Modeling

The first order thermal model utilized only deterministic methods. The volumetric

error was computed from the parametric errors with a rigid body kinematic model. The

displacement errors were modified according to scale temperature readings and published

expansion coefficients. Only the thermal dependence of the remaining errors was not

modeled. The neural network models also utilized the deterministic method of parametric


errors and rigid body modeling, but employed the non-deterministic ART-MAP's to model

the parametric error variation with temperature.

From the diagonal measurements, the methods proved comparable, with the 1st

order model fairing slightly better (improvement ratios of 4.1 vs. 2.8 were typical). From

parametric measurements taken after each diagonal measurement, the neural network's

weakness appeared to be in its inability to accurately model the displacement error

variation with temperature. All of the other errors were reduced significantly. It is

understandable that this occurred, since the displacement errors exhibited the largest

change in shape with temperature. Each of the neural network models were only presented

with 24 different thermal states for training. Assuming a typical variation in scale

temperature over a thermal cycle to be on the order of 10C, and an even distribution of

these temperatures when measurements were taken, would at best produce 0.4 C

resolution. This is clearly inferior to a deterministic system capable of 0.1 "C resolution

and it is unlikely the scale temperatures were evenly distributed..

An important feature provided by the 1st order thermal model was the ability to

compensate the scales to the part material's temperature. This was not readily applicable

to the neural networks since they would require additional training with different materials

at different temperatures, greatly increasing the training time. The benefit of this feature

is apparent in the machined part tests. The parts were machined at temperatures between

23-27"C and inspected at the international temperature standard of 20 "C. In the parts,

this would be manifested in a shortness error of 26 zm at the maximum hole distance of

234 mm. In fact, the distance between the far holes were significantly short (approx. 25


4m) on all of the compensated parts except those machined with the 1st order thermal

compensation (approx. 10ym). This attest to the importance of considering all sources of

error, since any one might overshadow the mitigation of the others.

Thermal scale compensation is an area where deterministic methods are adequate

and the non-deterministic methods do not appear to offer any substantial improvement.

Training with Machining vs. Non-machining

The two neural network models were identical in their architecture, but differed

in their training cycles: one was trained with non-machining actuation for warm-up, and

the other was trained with actual machining for warm-up. The diagonal tests did not

reveal any clear preference. Over all the improvement ratios for the non-machining

trained network were slightly better (2.6 to 2.8, 2.8 to 2.3, 1.8 to 1.7, and 2.8 to 2.7),

however the diagonal tests were conducted with non-machine warm up.

During testing it was observed that the spindle housing reached much higher

temperatures when not actually machining. Analysis revealed that this was not directly

related to machining, but rather a related issue of whether coolant was or was not being

used. The coolant is circulated through the spindle housing to jets directed at the tool.

This forced convection cooling is most likely more significant to thermal distortion than

the cutting process itself. Most of the cutting energy is transferred to the chip

(approximately 80% at high cutting speeds) [Schey, 1987] which spends a short time in

contact with the bed of the machine before being conveyed to a bin. The heat transfer from

the chips to the bed will affect the bed temperature, but the lower mass spindle housing


combined with the large convective heat transfer is probably more significant contributor

to thermal deformation. The greatest contribution of the chip removal is probably from

its heat transfer to the coolant.

A secondary issue related to the coolant effect on the spindle housing and the LBB

measurement procedure was observed. The X rotational errors were noticed to be larger

for the non-machining tests. The spindle housing was suspected of large drifts during the

measurement procedure as a result of the housing temperature rise. Drifting needs to be

minimal during the measurement cycle for the measurements to be accurate with the LBB.

The sequential measurement procedure relies on the assumption that the machine is in a

quasi-stable state. To investigate this, the location of the tool sockets #1 and #2 were

measured just after a 30 minute run of the spindle at 5000 RPM with the coolant off,

followed by measurements 20 minutes later at the same commanded position. This is to

simulate the conditions under which the non-machining measurements were taken. Table

7-4 shows the delta movement of the two sockets from just after warm-up to 20 minutes



Socket Movement after 20 min dwell, following 30 min at 5000 RPM
AX(Cm) AY(um) AZ(Um)

Socket #1 -15.2 0.0 33.0

Socket #2 -1.0 -20.0 34.0

The movements indicate that due to the rapid rise in temperature and low thermal

mass, the spindle housing is contracting radially about 15 to 20 pm, and axially about 34


lm during the measurement cycle. This will distort the measurements taken with the

LBB, especially the rotational errors since they rely on data taken over about a 15 minute

cycle. The translational parametrics are collected in about 7 minutes and should be less

affected. This contraction is less severe when the coolant is used, because it keeps the

spindle from rising so much higher than the ambient temperature. Mounting to a fixture

held in the spindle may have reduced the radial contraction, but not the axial. This was

a surprising and unfortunate effect on the LBB measurement procedure. Fortunately the

effect was not large enough to prevent satisfactory improvement of the machine's accuracy

using the neural network models.

Metal Removal Testine vs. Non-machining Testing

This was not an intended topic of this research, but due to the discrepancy in the

diagonal tests and the part machining tests for the neural network models, it seemed

appropriate to discuss here. The diagonal measurement tests indicated that all three

thermal models investigated improved the machines accuracy by a minimum improvement

of about 2X. The B5.54 precision positioning tests only showed this kind of improvement

for the 1st order thermal model. Also in all of the models, the surface to surface distance

measurements were long while the bored hole spacings were short. Machining the parts

at a higher temperature than 20'C explains the short dimensions, but not the large surface

distances. This discrepancy was found to be partly caused by an incorrect tool diameter

entered into the tool offset table. The nominal diameter of 12.000 mm was input into the

table, subsequent measurements revealed that the actual diameter was 11.977 mm. This


error of 0.023 mm would cause the dimension across the surfaces to be larger by this

amount. Cutting tests were conducted using the corrected tool diameter, see Table 7-1 and

7-2, but the surface distance still remained large by about 20-50 pm. Examination of the

milling parameters revealed that the finishing cuts for these surfaces consisted of: 8.3%

radial immersion, 11 mm depth of cut, down milling, 7000 RPM, and at a feed rate of

1310 mm/min. In down milling finishing cuts, forces can be present to move the cutter

away from the surface of the part causing larger dimensions. The transfer function for this

tool was measured and input into a simulation program to determine how much dynamic

deflection would be expected based on the cutting parameters. The results are discussed


Surface Distance Error

Simulation tests on the tool deflection were inconclusive. The simulation was for

a straight tooth cutter and showed excessively large deflections. To get a feel for the

potential deflection magnitudes the static deflection can be computed. The static stiffness

for a carbide cylindrical bar of 12 mm diameter at a tool length of 45 mm is 17,760

N/mm. The maximum cutting force per tooth obtained by integrating over the contact

length of the helix is 256 N. This force would cause a deflection of 0.014 mm in the

direction of the force. This force would not be normal to the cut surface, reducing the

error effect. However the stiffness will probably be less than calculated due to imperfect

and flexible clamping. With these considerations in mind, this calculation can still provide

an estimate of the deflection. Such a deflection normal to the surface would account for


0.028 mm of the large dimension. This falls within the range of errors that were

measured. Depending on the phasing of the tooth passing frequency and the tool's

vibration this number could easily be larger since the dynamic stiffness is always smaller

than the static stiffness. Regardless, the error most likely was caused by cutting

deflection since the deflection insensitive boring operations did not reveal this type of

positioning error.

As is evident from the testing, when conducting machining tests or machining

actual parts, care must also be taken in controlling the machining parameters that affect

accuracy. At these reduced levels of error, machine positioning improvement can be

overshadowed by things such as cutter tolerance. Since precautions can be taken at the

time of production to avoid these problems, positioning evaluation free of machining is

still a satisfactory analysis tool. It provides valuable information about the machine itself,

independent of user induced errors and dynamic effects.

Future Work

Further accuracy improvement might be achieved if the 1st order and neural

network models would be combined. The 1st order model does a superior and satisfactory

job of correcting the scale errors, but misses error variation such as changing squareness,

which was observed in this study. Such a hybrid model might reduce the errors to within

the 2.5 gm range. Also the use of the part material temperature compensation, which

was not feasible on the neural network model, should be incorporated. The part material

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