Production Engineering & Management M

Production Engineering & Management

Master Thesis

Assessment of Capability Variation due to Repeatability of Machine Tools

Author:

Mahsa Sorbi

Supervisor:

Dr. Andreas Archenti Co-Supervisor:

Karoly Szipka

July 27, 2017

A B S T R A C T

In recent years, several studies have been dedicated to the accuracy prediction of machine tools, leading to an enhancement in the accu- racy and precision predictability. Prediction is of great importance in performing more controllable operations and it additionally opti- mizes/reduces the measuring operations that can be both financially expensive and time consuming.

This thesis commence with an overview of measurements and ma- chine tools diagnostics to pinpoint the error sources (e.g. geometric error, motion error, quasi static behavior, dynamic behavior, thermal behavior which can be categorized as systematic or random error).

Next a novel evaluation method is presented to study the role of re- peatability (precision) of a machine tool, based on the measured kine- matic and geometric errors under quasi static load condition. Linear axis of machine tools with a significant effect on repeatability is un- der consideration in this project.

Data management of the experimental results is the first essential step following by analysing the variation of random errors along travel axis and on specific points, using statistical approaches. In this step probability of the positioning errors in 3D space is calculated and visualized.

Finally by numerical investigation, using homogeneous transforma- tion method, repeatability on an arbitrary defined path is calculated and visualized.

3

A C K N O W L E D G M E N T S

I have had the privilege of working under the supervision of Dr. An- dreas Archenti, who has always shown constant support and trust to- wards me and this study. I would like to express my sincere gratitude to him for his patient guidance, enthusiasm, his immense knowledge and basically for giving me this golden opportunity to do this inter- esting project.

I am very grateful to Kroly Szipka, who his willingness to help has been very much appreciated and essential in the project success. I would like to thank Theodoros Laspas for his guidance and contribu- tion in DBB measurements for the validation.

I would also thank my dear friends specially Eva, Maria and Mina for being always there for me and the nice time we have spent to- gether in and out of the office.

Last but not least, I wish to thank my dear husband for his encour- agement and endless patience and my beloved family specially my parents for their support throughout my study. With his own brand of cuteness, my little sweet nephew, Arman, has always been a source of energy.

5

C O N T E N T S

i c h a p t e r s 9

1 i n t r o d u c t i o n 11

1.1 Purpose and motivation . . . 12

1.2 Scope and Assumptions . . . 15

2 s tat e o f t h e a r t 17 2.1 Introduction to performance of Machining Systems . . 17

2.1.1 Machine Tools. . . 17

2.2 Machine Tool Errors . . . 18

2.2.1 Error Definition . . . 18

2.2.2 Error Types . . . 18

2.2.3 Error Sources in Machine Tools (Diagnostics) . 19 2.3 Fundamental Terms . . . 20

2.3.1 Accuracy and Precision . . . 20

2.3.2 Repeatability . . . 20

2.3.3 Capability . . . 21

3 k i n e m at i c e r r o r m o d e l i n g 23 3.1 Measurement of Kinematic Errors . . . 23

3.1.1 Input data for modeling . . . 24

3.2 Modeling of the effect of machine tool repeatability . . 24

3.2.1 Probability Density Function of Kinematic Errors 24 3.2.2 1Dimensional PDF. . . 25

3.2.3 2Dimensional PDF. . . 28

3.2.4 3Dimensional PDF. . . 30

3.2.5 Modeling on Volumetric Error Level. . . 32

3.3 Statistical Prediction of Kinematic Errors for Arbitrary Defined Tool Path . . . 35

3.3.1 Simple Trajectory . . . 35

3.3.2 Complex Trajectories . . . 37

3.4 Validation . . . 39

4 s u m m a r y & conclusions 45

5 f u r t h e r e f f o r t s 47

ii a p p e n d i c e s 49

a s p e c i f i c at i o n s 51

b i n p u t d ata 53

c m a c h i n e t o o l d i a g n o s t i c s 55

d 1 d p d f r e s u lt s 61

e p r o b l e m s o lv i n g t o o l s 65

7

iii b i b l i o g r a p h y 67

Bibliography 69

Part I C H A P T E R S

1

I N T R O D U C T I O N

Quality of the produced parts is tightly connected to the machine’s performance.Traditionally, inspection process to check the accuracy of the produced components mainly take place after production. Dimen- sional error or tolerance has been investigated using various statisti- cal techniques after a product is fabricated. However, the difficulty of these conventional approaches is because the errors must be inferred after a part is made, since machine performance is unknow[27].Therefore in order to avoid or minimize the inspection and scrap parts , it is nec- essary to study the behavior of the machine tool carefully before the process get involve.

Productive machining operations require maximizing material re- moval and at the same time satisfying the quality based on the speci- fications to reach or get closer to ideal machining unit.

As time goes on role of precision engineering particularly in manu- facturing new products and in process development is increasing[2].

“Even a stable process produces products with small deviations from a defined target value; this phenomenon is called natural deviation”[1].

Therefore researches move toward of this idea to have more stable ma- chining systems that produce products with small deviations from the desired target value and having a mean value equal to target value.This raises the question of whether and to what extent these deviations are measurable or controllable.

Machine tool itself (apart from cutting process or other modules of a machining system) generates errors associated with imperfec- tions in machine tool components,assembly or due to frictions,errors of controlling system and etc. Incompetence of machine tools might cause economical forfeiture of tens of billions of US dollars per year[5].

Thus, maintenance of machine tools and their availability is signifi- cantly important for cost-effective production[6].

The concept of this project is mainly proposing and evaluation of a new approach that is closely related to the capability of a machining system.It is expected that this study will provide a base for further investigations/conclusions regarding the effect of probability and re-

11

Figure 1.1.: Schematic illustration of mchine tool and process natural deviations[1]

peatability of machine tools in capability of machining systems.

Error measurement, computational analysis and validation are es- sential steps to build a robust model for capability assessment of ma- chine tools. Concolusions from such studies can be used for finding preferred machining system in respect to a given specification [3] or in order to analyse and control of the evaluation aid through analysis and control of the accuracy loss and act [4].

1.1 p u r p o s e a n d m o t i vat i o n

Interaction between the machine tool, process and the control system has a significant effect on the part accuracy. By focusing on machine tools, prediction of kinematic errors on a nominal toolpath has been a major interest in recent years. However, the role of repeatability and its effect on the accuracy prediction is still unexplored. The aim of this project is therefore to fill this gap in the literature by investigat- ing this issue which is closely related to the capability of a machining system.

Since repeatability is one of the fundamental problems in precision of machine tools, this project is motivated due to lack of a clear analy- sis about the behavior of random errors and repeatability of machine tools. “Correct assessment of geometric and kinematic errors is neces- sary if one is to make error compensation in the machine tool process instructions for tool path” [7]. Therefore final aim is to propose a procedure to model the probability of positioning errors for a known

1.1 purpose and motivation 13

toolpath based of the measurement results.

A summary of the necessary steps that can be considered as mile- stones in the current project is given below:

1. Getting a comprehensive vision of fundamental terms such as capability, variation, variability, accuracy, precision and repeata- bility.

2. Overview of measurements and diagnostics of machine tools in order to pinpoint the error sources (e.g. geometric error, motion error, quasi static behavior, dynamic behavior, thermal behavior which can be categorized as systematic or random error).

3. Numerical investigation of a machine tool accuracy, using a pre- defined tool path. (calculations will be performed to study the role of repeatability on arbitrary trajectories.)

4. Proposal for a comprehensive framework for executing Root Cause Analysis (RCA) of machine tool errors: i.e. a flowchart that provides root causes of failure and how to correct/prevent them.

Since the error sources of machine tools are under the focus of this study, another use of such evaluations can be used for a more in- depth cause and effect analysis in manufacturing which is a very challenging topic in industry. Within the current work an attempt is made on managemental problem solving tools in order to have a more comprehensive perspective on root cause analysis in produc- tion specially when an unwanted event or an errors appear.(A table including hundred of management problem tools can be found in ap- pendix E).

”Root cause analysis (RCA) is a process designed for use in investi- gating and categorizing the root causes of events with safety, health, environmental, quality,reliability and production impacts”[8]. Based on this definition and different examples of root-cause analysis charts for different levels of causes in industrial cases by AvBjrn et al. [9] and also by inspiring from the problem solving tools such as 5M and Ishiwaka(fish bone) diagram, a RCA flowchart is proposed in figure 1.2. 5M is a problem solving tool used in risk management, trouble shooting and root/cause analysis that these five Ms stands for man, machine, manufacturing(process), measurement, material. Ishikawa diagrams are causal diagrams that shows different causes of a special effect or event. The root-cause analysis flowchart below represents the importance of the current project and such approaches by illus- trating a more general perspective.

This provided flow chart illustrates the process of eliminating un- wanted events from a symptoms recognition including various stages to identify the root cause by means of 5M together with the Ishikawa diagrams. This cycle provides a feedback after implementing the nec- essary steps in order to eliminating the cause. Feedback supplied to- wards the earlier stage (when the unwanted event reappear), allows modifications throughout this process.

Figure 1.2.: Proposal of a root cause analysis (RCA) flowchart

This flowchart illustrates the process that can be used to find the root cause of an inefficiency or evaluate the result of a process. Cur- rent project result is an effort to make insight of the first cause and effect diagram, representing machine. Here are more details of the factors effecting machine tool accuracy:

Figure 1.3.: The effecting errors in the accuracy of a machining system Material[10]

1.2 scope and assumptions 15

1.2 s c o p e a n d a s s u m p t i o n s

This thesis commence with a summery of relevant state of the art and introducing error sources of a machine tool(diagnostics) and then more specifically focus on certain kinematic errors that will be imple- mented in forthcoming analytic model(chapter 3).In this section scope of the project will be clearly explained as well as the major assump- tions.

As mentioned previously, aim of this project is mainly evaluating a new approach that would be used to make conclusions on repeatabil- ity of the machine tools. Since machine tool itslef is a very complex system ,therefore it is necessary to clearly define the scope of the project and boarder of the analytic model.

Machine tool errors further described later on in the section 2.2.3.

However since the focus of this project is on kinematic errors , here after in figure 1.4 scope of the project from the errors perspective is shown. These triangles illustrates a categorization of error budgets of kinematic errors into positioning error, straightness errors and angu- lar errors. Total error components of a 3 axis machine tool includes 3positioning errors, 6 straightness errors and 9 angular errors. Posi- tioning error and straightness error might classed into a group called translational error which is the scope of this project.

In this project angular errors are excluded from the model and the reason is because of the final goal of the project. As mentioned pre- viously, charechterizing the machine tool and making conclusions on repeatability of the machine tool is the final objective of this study.

Since the angular errors comprises measurement uncertainty more significantly in comparison to machine tool uncertainty.However mea- surement uncertainty of the translational errors are inconsiderable in comparison to machine tool uncertainty. Therefore merely position- ing and straightness errors(two of the linear components of kinematic errors)will be evaluated in this model.

Uncertainty analysis for separating the role of the measurement un- certainty and the machine tool uncertainty is not in the scope of cur- rent study. However since the input data comes from the measure- ments, the results of this modeling are affected by both the measure- ment and machine tool. Therefore it is necessary to mention that in this step due to this condition that the measurement uncertainty (for transnational errors) is much less than the machine tool uncertainty, then it is possible to relate the results of the computational modeling (chapter 3) to behavior of random errors of the machine tool(machine tool repeatability).

Figure 1.4.: Scope of the project

Another required simplification is rigid body assumption in order to simplify the model by involving less parameters in analysis. In this case it is assumed that the machine tool components might not deform. According to this fact that the deformation of the shape and the body of the components is several magnitudes smaller than the movement of the components, this simplification is reasonable.

The purpose of this study is to focus on random errors and eval- uate their behavior by means of statistical approaches and mathe- matical formulas. In this step random errors are assumed normally distributed. This assumption also makes it possible to use mathemat- ical tools such as probability density function in order to study the randomness of translational errors. More details will be explained within the modeling steps in chapter 3.

2

S TAT E O F T H E A R T

2.1 i n t r o d u c t i o n t o p e r f o r m a n c e o f m a c h i n i n g s y s t e m s Machining system(MS) can be classed into five main modules: ma- chine tool, cutting tool, cutting process, work piece and fixture. Ev- ery module as a sub-system has its own contribution and effect in ultimate results. Interaction of these sub-systems also result in errors that are more complex to split and study.

Figure 2.1.: Machining system(MS) modules

2.1.1 Machine Tools

A machine tool as a complex mechatronic system consists of abun- dant subsystems with interaction with each other. Machine tools in general include three main sub systems: the elastic structure, the drives and the control system, which includes the measuring system[11].

Servo system provides the power for linear and rotary movements in machine tools. The elastic structure mainly is the frame and other mechanical components of the machine tool.

Machine tool as the main physical structure in machining systems is under focus of current project. Accuracy and precision of machine tools are tightly interlinked with the produced part accuracy.

17

2.2 m a c h i n e t o o l e r r o r s

2.2.1 Error Definition

Error (of measurement)The measurement result minus the true value of the measurand is defined as the error of the measurement. How- ever since literally there is not a value as a real or true value, in practice a conventional true value is ment [31].

2.2.2 Error Types

According to the characteristics of errors, they can be categorized into two broad groups as random and systematic errors.

Random error(of a result)“A component of the error which, in the course of a number of test results for the same characteristic, varies in an unpredictable way.”[28]

Random errors are difficult to promote and impossible to correct or fully control.

Outside interference error such as changes in operation conditions or environment instability (such as temperature) are the cause of ran- dom errors.Random Errors may also referred as precision errors.

Systematic error“A component of the error which, in the course of a number of test results for the same characteristic, remains constant or varies in a predictable way” [28]. By correcting the NC codes based on the measurements analysis it is possible to compensate for the systematic errors.

Figure 2.2.: A schematic illustaration of random and systematic errors[13]

2.2 machine tool errors 19

2.2.3 Error Sources in Machine Tools (Diagnostics)

All the components and nodes of a machine tool have contribution to the machine tool accuracy and precision. After studying different key note papers regarding machine tool diagnostics and their error sources point to this fact that the topic is enormously wide. Given also the fact that each study provides a new way of classification based on different needs, having an overall view is almost impossible due to the existing limitations. Therefore most of the studies in the literature partly refer to another work in order to cover the desired subject, which can be time consuming for the readers in the industry where time is a critical factor. Therefore in this step of the work a comprehensive study is done and The aim of the work is to give a comprehensive overview, categorization and evaluation of different machine tool characterization approaches.

“Diagnostics of technical systems can be defined as a process of functional faults and their causes, on the basis of data obtained by control, supervising or monitoring”[14].

Since studying of machine tool diagnostics is fundamental in posi- tioning precision researches, an attempt is made within the current work in order to have a comprehensive categorization on machine tool error sources, their error budget, relevant indicator signals, mea- surement instruments based on 60 references. This table and more details about this categorization and the list of references that are used for this summary can be found in appendix C.

Error sources of a machining system are main error sources that affect machining system accuracy and might classed into four general categories:

• Kinematic errors(geometric errors & motion control errors)

• Static load behavior

• Dynamic behavior

• Thermal behavior

Interaction between the components of these error sources affect the total accuracy of a machine tool.

Since the focus of this thesis is on kinematic errors, it will be de- scribed hereafter. Definitions of other error sources (static load behav- ior, dynamic behavior, thermal behavior) can be found in [4].

2.2.3.1 Kinematic Errors

Kinematic errors have a significant contribution to machine tool ac- curacy which is connected to the capability variation of a machining system.

In this thesis geometric and motion errors are considered as kinematic errors. As such, motion errors are functions of the carrying axis and their effect on the positioning that might mainly occure while execut- ing algorithms[4].

Kinematic errors of a CNC machine tool in general in dealing with imperfection of machine tool components and nodes. Following as- pects gives a better understanding of geometric root causes: [29][30]

manufacturing errors of the machine tool itself: This is the machine’s original construction error that refers to imperfection and faults in the machine tool components or surface quality.

Machine control system error: Errors that are caused by the CNC interpolation algorithm and the servo system, including the machine shaft and ball screw.

Thermal deformation error: Thermal deformation in machine tool components which occur due to the thermal disturbances in both internal sources in the machine and its surrounding environment.

Detection system test error: Manufacturing errors in measurement instruments and their installation; e.g. error in the feedback to the system, measured by various sensors.

2.3 f u n d a m e n ta l t e r m s

2.3.1 Accuracy and Precision

Accuracy of measuremt: “The accuracy is defined as the closeness of agreement between a test and the accepted reference value”[31].

Precision of measurement: “closeness of agreement between indications or measured quantity values obtained by replicate measurements on the same or similar objects under specified conditions”[31].

2.3.2 Repeatability

Repeatability is an important component in precision concept.Following terms would clarify this term in the scope of Machining systems and measurements.

2.3 fundamental terms 21

Figure 2.3.: Kinematic Errors

Repeatability of measurements can be defined as “measurement precision under a set of repeatability conditions of measurement”[31].

Repeatability of Machine tool can be defined as the “maximum value of spread in positioning error at any target point along the range of motion when the system is moved in both the directions multiple times under similar pre-specified conditions”[19].

Repeatability conditions “Conditions where independent test re- sults are obtained with the same method on identical test items in the same laboratory by the same operator using the same equipment within short intervals of time”[28].

2.3.3 Capability

Due to Iso definition capability is the ability of a manufacturing unit to produce a given part within specified tolerances at a specified con- fidence level, a concept mainly applied to batch production[20]. For a machine tool, capability index C_m showsshows how well the ma-

chine produce details in comparison to defined tolerances.It can be formulized based on upper limit, lower limit and the standard devi- ation. It must be noted that the calculation of capability indices is based on normal distribution assumption. Six sigma in the equation, points to the normal distribution condition of the process[25].

Cm = ^Tu−T_l 6σ

Corrected capability index for machine tools represented by C_mk and mathematically can be calculated by:

C_mk =_min| ^Tu−µ

6σ ,µ−T_l 6σ |

This will give a more accurate result about capability and perfor- mance of the machine tool. According to this equation instead of tol- erance interval, standard deviation is evaluated in comparison with each tolerance level. Standard deviation in this formula is the point that connects this study - on repeatability of machine tools - to the concept of capability(the equation and definition of standard devia- tion will be explained in section 3.2.2.).

3

K I N E M AT I C E R R O R M O D E L I N G

This chapter represents the results of the numerical simulation and statistical analysis on collected data from measurements in order to build a model to predict and evaluate probability of the linear er- rors(positioning and straightness errors of each axis) in 3D space for a single point and developing the model for arbitrary trajectories.

It must be noted that randomness, repeatability or precision are the concepts that is dealing with variance and a probability function is needed to describe the behavior of random errors while accuracy is dealing with average value of measured data. Both accuracy and precision are essential in order to have a to have a more comprehen- sive view on machine tool characteristics.

A summary of the necessary steps that can be considered as mile- stones in the current project is given below:

1. Evaluation of the kinematic errors in one dimensional space.

(on the points in measurement range of one axis)

2. Evaluating combination effect of two kinematic errors in two dimensional space.(on the points in measurement range of two axis)

3. Evaluating combination effect of three kinematic errors in three dimensional space(volumetric).(on the points in measurement range of three axis)

4. Building a computational model by means of HTM theory (Ho- mogeneous Transformation Matrices) to analyze total volumet- ric errors for arbitrary tool path.

5. Developing the model for more complex trajectories.

6. Proposing a testing method to validate the simulation

3.1 m e a s u r e m e n t o f k i n e m at i c e r r o r s

The current work analyze the measurements data which the results as well as the measurement methodologu can be found in [15] and [4].

23

The measurement is done on a 5 axis machine tool by laser interferom- etry. Characteristics of the machine tool and the laser interferometry can be found in Appendix 1.

The input data to current model is measurement data including the transnational errors (positioning error and straightness error for X,Y and Z axis). As mentioned in section 1.2, respect to the scope of this project, angular errors(roll, pitch, yaw, tilt) will not be included in the analysis.

3.1.1 Input data for modeling

Entire results of the measurements can be found in appendix C. Table below is the summery of the input data:

Table 3.1.: Input data X axis travel range [0-980]mm

Measured points [0,20,40,...,980] mm

50Points (average of 5 measurements per point) Errors Positioning Straigtness

in Y and Z direction Roll Pitch Yaw Y axis travel range [-30,-490]mm

Measured points [-30,-50,-70,...,-490]mm

23Points (average of 5 measurements per point) Errors Positioning Straigtness

in X and Z direction Roll Pitch Yaw Z axis travel range [-30,-450]mm

Measured points [-30,-50,-70,...,-450]mm

21Points (average of 5 measurements per point) Errors Positioning Straigtness

in X and Y direction Roll Tilt Yaw

3.2 m o d e l i n g o f t h e e f f e c t o f m a c h i n e t o o l r e p e ata b i l- i t y

By adopting the measurement data and considering rigid body kine- matics along with homogeneous transformation, this section seeks to propose an analytic model to characterize the repeatability of the machine tools.

3.2.1 Probability Density Function of Kinematic Errors

Probability density function is a mathematical tool helping to model random variables by a continuous function. In this case studying ran-

3.2 modeling of the effect of machine tool repeatability 25

dom variables are identical to the collected data from measurements.

Calculations will be performed to study the role of repeatability on specific measured points on a defined trajectory and furthermore to make conclusions on arbitrary points based on their predicted be- havior.Applying this function is based on the normal distribution as- sumption.

3.2.2 1 Dimensional PDF

For continuous random variables of X, the normal distribution is a function of the mean value and the standard deviation :

X∼ F (µ, σ²) (3.1)

µ is the mean value and σ is the standard deviation that can be calculated by :

σ= v u u t

1 N−₁

∑

(x_i−µ)² (3.2)

That x_i is ith measured value and N is the number of measure- ments.

Forµ =0, probability density function can be expressed as :

P(x) = ¹ σ_x

√

2πe^−(x)

2. 2σ²_x

(3.3)

For each points on each axis of X,Y and Z of the machine tool, average of 5 measurements is repeated in both forward and reverse directions. By calculating the average values and standard deviations of the measured values, probability density functions are calculated and plotted for each of these points. In this step positioning errors of each axis is targeted.

Here PDFs for positioning errors of Y axis for measured points and for both directions(forward and reverse)is plotted separately. The plots illustrate the difference in the behavior of the machine tool in forward and reverse directions. It also shows the probability of errors along the transitional axis of Y. Such plots can help to make conclu- sions about the most precise travel ranges of a machine tool.

(a) Positioning Error

(b) Straightness in X direction

(c) Straightness in Z direction

(d) Pitch Error

Figure 3.1.: Probability density function of geometric errors in Y axis

Rest of the plots for various error components of three axis of X, Y and Z can be found in Appendix 3.(In this step angular errors are also calculated and plotted).

3.2 modeling of the effect of machine tool repeatability 27

3.2.2.1 Unidirectional Repeatability

According to the ISO [41] unidirectional repeatability of positioning at each of the target(measured) points is defined as:

R↑=4×σ↑ (Forward) (3.4)

R↓=4×σ↓ (Revearse) (3.5)

The results for positioning error in Y axis is plotted in figure 3.2.

Figure 3.2.: Repeatability and standard deviation for both directions along the travel range in Y axis

Another way to illustrate the behavior of Y axis respect to position- ing error is shown in figure 3.3.

Figure 3.3.: Repeatability along the travel range of Y axis for both forward and re- verse directions.Stars shows the average values for each of the measured points along the Y axis and triangles shows the standard deviation of measurement data.

3.2.2.2 Maximum Value of the Repeatability

“Maximum value of the repeatability of positioning at any position Pi along or around the axis can be calculated by” [41]:

RR↑= Max[RR_i ↑](Forward) (3.6)

RR↓= Max[_RRi ↓](Reverse) _(3.7) As it is visible from the graph 3.2, maximum value of repeatability in case of positioning error in Y axis is:

Max Repeatability Y axis Positioning /Forward = 8.0329 Max Repeatability Y axis Positioning /Reverse = 5.5828

3.2.3 2 Dimensional PDF

Table 3.2 demonstrates all possible conditions in order to combine two set of variables (errors) in probability density calculations:

3.2 modeling of the effect of machine tool repeatability 29

Table 3.2.: 2D Possibilities Axis DOF Error Source Possibilities

1 2 Positioning Error Straightness Error

Positioning X & Straightness(Y or Z) Positioning Y & Straightness(X or Z) Positioning Z & Straightness(X or Y)

As mentioned in previous section, for each set of random variables of X and Y, normal distribution can be expressed as:

P(x) = ¹ σ

√2πe^−(x)

2. 2σ²

(3.8)

P(y) = ¹ σ

√2πe^−(Y)

2. 2σ²

(3.9) Respect to variable independency assumption, following condition holds[26]:

P(x, y) =P(x).P(y) (3.10) Thus two dimensional probability density function of the pair (x,y) can be obtained as:

P(x, y) = ¹

2πσ_xσ_yExp[−( ^x

2

2σ_x² + ^y

2

2σ_y²)] (3.11) The probability density function of this distribution has the follow- ing form[26]:

P(x, y) =







1

πσxσy f or _2σ^x22

x + ^y2

2σ_y² ≤1 , 0 f or _2σ^x22

x +_2σ^y2₂

y >1 . (3.12) Probability density function formula needs to be normalized in or- der to make right conclusions of the plots. In the other words the area under the plot (line or surface depends on the dimension) should be equal to one so that we can use the word ” probability ” in decla- rations and interpretation of diagrams. Constant coefficient in PDF formula (2D & 3D ) is considered in calculations for the aim of nor- malized plots.

This step seeks to visualize the distribution of errors for each points on each axis by considering two degree of freedom (DOF). Even be- fore visualization, from the formula 3.10, it is predictable that the plots should be ellipse (and not circular) due to the inequality of stan- dard deviations for X and Y variables.

Therefore the plots show the random deviation of positioning er- rors for 2 degree of freedom around the mean value. Red rings repre- sents the area for the confidence level of 95.45% and 99.7% equivalent to expansion factor of K=2 and K=3 respectively.

(a) Top view (b) Iso view

Figure 3.4.: PDF for 2DOF of Errors: Positioning Error in X axis and straightness error of X axis in Y direction

3.2.4 3 Dimensional PDF

Table 3.3 includes all possible conditions in order to combine three set of variables (errors) in probability density calculations:

Table 3.3.: 3 Degree of freedoms

Axis DOF Error Source Possibilities

1 3

Positioning Error Straightness Error I Straightness Error II

Positioning X & Straightness(Y and Z) Positioning Y & Straightness(X and Z) Positioning Z & Straightness(X and Y) By following the same logic from previous section, 3D probability

density function formula can be derived as:

P(x, y, z) = ¹

2πσ_xσ_yσ_zExp[−( ^x

2

2σ_x² + ^y

2

2σ_y² + ^z

2

2σ_z²)] (3.13) therefore x, y and z in this formula represents the variables that are 3.2.4.1 Visualization

Visualizing the results for three set of errors or 3 DOF can be challeng- ing due to demonstration of the fourth dimension. As in previous steps for 1DOF and 2DOF, plots were 2d and 3d respectively. Same

3.2 modeling of the effect of machine tool repeatability 31

here, plotting three variables in 3D plot needs an extra dimension to demonstrate quantity of the probability density function. Therefore, by mean of colors this dimension is illustrated (figure 3.5).

As shown in Fig. 3.5 (a) four bars are designed as follows:

• X Slice Controller: Enables the the user to manualy fit an arbi- trary Y-Z plane in 3D PDF domain.

• Y Slice Controller: Enables the the user to manualy fit an arbi- trary X-Z plane in 3D PDF domain.

• Z Slice Controller: Enables the the user to manualy fit an arbi- trary X-Y plane in 3D PDF domain.

• Iso Surface Bar: Shows the Iso surface which means it repre- sents the surface that have a constant probability. Most sig- nificant benefit of this bar is the ability to set different values respect to arbitrary coverage factors (ex: K=2 or K=3) and com- pare results.

(a) Probability density function for 3DOF

(b) 3D PDF at K=1,K=2,k=3 Figure 3.5.: 3D probability Density Function

The blue surface in figure 3.5(a) represents the possible domain of translational errors for one point along Y axis. It is plotted around the mean value for a probability adequate to K=2 which is calculated and as shown Iso surface bar, is equal to 0.0123. There fore the iso bar contour represents the probability domain and the center of this ellipse is the average value of the measurements for a single point along the axis.

As shown in figure 3.5(b) the red color in the iso bar contour repre- sents higher probability (proportional to smaller K values) illustrates smaller ellipse in comparison to the other end of the iso bar contour with blue color, lower probability, larger K values and repectively big- ger ellipses.

As shown in figure 3.6 this is also possible to visualize the rela- tionship between four variables which means that the user can fit arbitrary planes to observe the probability contours there as well.

Figure 3.6.: Visualization of two slices in X and Z planes and the iso surface with approximately 57% probability

3.2.5 Modeling on Volumetric Error Level

This section seeks to introduce a transformation methodology to pro- vide the possibility to predict the random error distribution around arbitrary points and in an advanced phased around arbitrary trajecto- ries. As shown in figure 3.7 the final aim is to analyze and visualize the probability density functions on an arbitrary tool path in 3D do- main. In the other word by means of this modeling it will possible to observe the behavior of random errors or the repeatability of the machine tool on an arbitrary trajectory in 3D domain/space of the machine tool.

3.2 modeling of the effect of machine tool repeatability 33

Figure 3.7.: Schematic illustration of modeling the probability density function on volumetric error level for three different K values

3.2.5.1 Analytic description of machine tool errors

For the purpose of machine tool error prediction a valid analytical de- scription of errors is needed. A comprehensive literature survey on modeling techniques can be found in[34]. In this section two models of the recent approaches will be introduced. Selected approach for this thesis will be explained in more details.

1) Error Matrix Model[35,36]:

In the model that Dufour and Groppetti [33] proposed, the mea- surement result of errors at various positions within the working space of the machine tool stored for various loading and thermal con- ditions. These information were interpolated at each location in order to model the errors in volumetric level[27]. This method requires a huge number of measurements and data storage.

2) Homogeneous Transformation Matrices(HTM):

A homogeneous transformation matrix (HTM) describes position of two coordinate systems releatively. In case of machine tools these two coordinates are rigidly connected to the components of the ma- chine tool[15]. HTM is also able to include components’ imperfec- tion in the model by applying three rotational and three transnational components[39]. By measuring individual error components(section 3.1.2) and analyzing them(section 3.1.3) total volumetric error can be calculated[38].

According to the assumptions (section 1.2) for a rigid body kine- matic modeling and for perfect link and joints, an arbitrary point in space can be described by:

~r₀ =

∏

A_i

!

~rp (3.14)

Figure 3.8.: General transformation between two coordinates[27]

A_iis transformation matrix from a coordinate to another.As shown in figure 3.6 for small rotational errors of α, β, γ and transnational er- rors of ∆x, ∆y, ∆z, positional error (both angular and transnational) between two coordinates can be expressed as :

∆A=









1 −α β ∆x

α 1 −γ ∆y

−β γ 1 ∆z

0 0 0 1









(3.15)

As it mentioned before due to the rigid body assumption, all the components and nodes of a machine tool can be modeled by means of HTM. For a kinematic modeling of translational errors (and not angular errors) the equation will be:

∆A=









1 0 0 ∆x 0 1 0 ∆y 0 0 1 ∆z

0 0 0 1









(3.16)

Positional Error in each of the axis X,Y and Z can be expressed as :

∆x=µx+δx (3.17)

∆y=µy+δy (3.18)

∆z=µz+δz (3.19)

where µ is the systematic portion of error that is equal to the the average value of translational errors for each measured points and δ is the random component of the total error.

3.3 statistical prediction of kinematic errors for arbitrary defined tool path 35

These equations are simplified due to the focus of this project. Com- plete form of the equations can be found in [15, ?, ?,27].

Following the normal distribution assumption (1.2) probability den- sity functions of f1(δ_x)_{, f2}(δ_y)_{, f3}(δ_z)_{will be:}

f₁(δ_x) = √ ¹ 2πσx

e^−(δx)

2. 2σ_x²

(3.20)

f₂(δy) = √ ¹ 2πσy

e⁻(^δy)^2.2σ_y²

(3.21)

f₃(δ_z) = √ ¹ 2πσz

e^−(δz)

2. 2σ_z²

(3.22)

Translational error for the arbitrary point of E can be expressed as:

∆E =µ_E+δ_E (3.23)

It is possible to calculate µE and compensate it for CNC machines [27]. This portion is assigned to systematic errors and can be calcu- lated by averaging the measured data for each point. This part of the equation is evaluated and visualized in [15,4]. However it is not possible to define the exact deviation for the random portion due to characteristics of random errors and their stochastic behavior.

Due to the independency assumption for variables/errors (section 1.2), for f₁(δ_x), f₂(δ_y), f₃(δ_z), the density function of δ_E is a joint func- tion of f₁, f₂, f₃ or δ_E = f₃(δ_z)f₂(δ_y)f₃(δ_z).

3.3 s tat i s t i c a l p r e d i c t i o n o f k i n e m at i c e r r o r s f o r a r- b i t r a r y d e f i n e d t o o l pat h

In this section by means of Matlab, translational errors will be ana- lyzed and visualized for different trajectories.

3.3.1 Simple Trajectory

First step is to define the trajectory. In this step a linear tool path with the following coordinates is confined (Table 3.4).

Table 3.4.: Coordinates of the line

X1 X2

450mm 550mm

Y1 Y2

-400mm -490mm

Z1 Z2

-400mm -400mm

(a) Black line is the nominal path, Red line is the systematic (actual) path, ellipses illistrate the domain of random errors with confidence interval K=1 and Step size=2 mm

(b) Black line is the nominal path, Step size for random error calculation decreased to 0.2 mm .(Colors of the contoured line represent the deviation level in Z axis) Figure 3.9.: Probability density function for a defined line. for the random error calculation is K=1. Magnification coeficient for systematic errors is set to 500.

Defined line that might be called the nominal trajectory is shown by black color in figure 3.9. Actual trajectory is the path under ef- fect of systematic errors and is illustrated by red color. As its shown the nominal tool path is defined in X-Y plane (constant Z coordinate) however the deviations in Z axis for the actual tool path is visible.

Consider that the magnification coefficient for systematic errors is set

3.3 statistical prediction of kinematic errors for arbitrary defined tool path 37

to 500 for the aim of visualizing.

The ellipses around the actual path represent the domain of the random errors in discrete points. The step size controls the density of these calculating points along an axis. For example in Figure 3.9 (a) the step size is set to 2 mm. It means that the evaluation (based on the measured data, probability density function and HTM) is calculated and plotted with 2 mm interval between consecutive defined line. As shown in figure 3.9 (b), when using smaller step sizes ellipses collapse to a tunnel shape space around the actual trajectory. In the other words regarding the assumptions and measured data on this specific machine tool, the predicted actual tool path on the real machine will fall in this simulated tunnel. The probability of this prediction is based on the K value.

3.3.2 Complex Trajectories

This section seeks to develope the model for more complex trajecto- ries. Following the smae steps for the simple trajectories ,in the first place the arbitrary (nominal) tool path, circular and rectangular trajec- tories will be defined. Consider that dimension and location of these trajectories is arbitrary however it should be within the measurement range of the machine tool. It is also possible to set arbitrary step sizes depending on the aim of study. As it shown in previous section, de- creasing step size may help to have a better look at the behavior of the machine tool along the sensitive cutting direction although may cause lose of information in the direction along the tunnel (non-sensitive di- rection).

3.3.2.1 Circular trajectory

First step is to define the arbitrary trajectory. Coordinates of the circle should be within the measured travel range of the three axis of the machine tool. Table 3.7 illustrates the coordinates of the nominal circular tool path.

Table 3.5.: Coordinate of the circular trajectory X coordinate of center point 600.45 mm Y coordinate of center point -158.71 mm Z coordinate of center point -331.3 mm

Radius 50mm

(a) Black trajectory is the nominal path, Red line is the systematic (actual) path.

Magnification coefficient for systematic errors is set to 500. Ellipses illustrate the domain of random errors that is magnified to 3 times bigger than the actual value for the purpose of visualization, Step size=3 mm

(b) Black trajectory is the nominal path, Step size for random error calculation decreased to 0.5 mm at K=3. (Colors of the contoured trajectory represent the deviation level in Z axis)

Figure 3.10.: Probability density function for a defined circular trajectory.

3.3.2.2 Rectangular Trajectory

Coordinate of the rectangle is defined as:

Table 3.6.: Coordinate of the rectangular trajectory

X1 X2 X3 X4

100mm 300mm 300mm 100mm

Y1 Y2 Y3 Y4

-350mm -350mm -150mm -150mm

Z1 Z2 Z3 Z4

-100mm -100mm -100mm -100mm

3.4 validation 39

(a) Black trajectory is the nominal path, Red line is the systematic (actual) path.

Magnification coefficient for systematic errors is set to 500. Ellipses illustrate the domain of random errors that is magnified to 9 times bigger than the actual value for the purpose of visualization, Step size=3 mm

(b) Black trajectory is the nominal path, Step size for random error calculation decreased to 0.2 mm at K=3. (Colors of the contoured trajectory represent the deviation level in Z axis)

Figure 3.11.: Probability density function for a defined rectangular trajectory.

Figure 3.12 represents the same trajectory in section 3.3.2.1, how- ever the simulation is run for three differant K values at K=1, K=2, K=3.

3.4 va l i d at i o n

Verification of computer simulation model is conducted during the development of the model with the ultimate goal of producing a cred- ible model[42,43]. The proposed model in this thesis, as explained in previous chapters is based on measurement data, mathematical equa- tions and assumptions. Despite of verification of the steps, it will be a big value to add a validation test method to this numerical simula- tion. Therefor it will confirm that this volumetric error analysis can be confirmed in a test condition.

Figure 3.12.: Demonstartion of random errors at different confidence intervals of K=1, K=2, K=3. Due to visualization purposes random errors are plotted 3 times bigger than their actual size.

It must be noted that the focus of this project is on the machine tool itself therefore the experiments for the aim of validation have to just involves the machine’s effect and not the cutting process. Thus the challenge would be designing a close prototype to the assumptions and conditions of the problem.

For the circular trajectories, Double Ball Bar (DBB) would be a proper device. DBB measurement is a popular direct measurement method, used to identify geometric errors of machine tools. The ad- vantage of this method is the simplicity and quickness of the set up and the test.

Figure 3.13.: Double Ball Bar (DBB)[44]

3.4 validation 41

The actual circular path coincides with the nominal trajectory, given an ideal performance for the machine tool in theory, However vari- ous factors including kinematic errors introduce deviations from the nominal trajectory. The aim of this test is to find out whether the repeatability of the machine tool, measured directly for a given trajec- tory, confirms the prediction from the numerical model.

In the test, several circular trajectories are measured around a fixed point with a certain radius. The average deviation on the test data can then be calculated and compared to the result from the model (the tunnel shaped path). In other words, the measured trajectories should lie within the tunnel path, calculated by the model, with the same probability, reported for a specific tunnel path. However it should be noted that the numerical model considers only the translational er- rors, while more errors (angular, environmental, control system, etc.) are present in the test condition. Few points should be taken into account in order to minimize the effect of other factors, highlighting the kinematic errors which are the focus of the numerical model:

1. The double ball bar test should be performed with the lowest possible feed rate.

2. Large radius circles are better to highlight the machine geometry errors since the smaller circles are more sensitive to servo errors.

There are three extension bars with 50 mm, 150 mm and 250 mm length that reflects the radius of the arbitrary circular trajectory. Due to this fact that the larger radius is more appropriate to highlight the geometric errors and also respecting to the measured data range, 150 mm radius is chosen for this test set up.(250 mm radius will be out of the range of measurement data that is used for the numerical simulation for the Y axis). Calibration shows that the precise radius is 150.075 mm. The test is carried out with the feed rate of 1000 mm/min and radius of 150 mm and then it is repeated 10 times in the same condition.

Table 3.7 represents the coordinate of the nominal trajectory that is given to the CNC machine and also to the software.

Figure 3.14 shows the result of the first test. There are different format of reports that the software offers. However for the purpose of this section the raw data is needed in order to analyze it. This report includes thousands of the points for each run (CW and CCW) that the probe captured data. This report of raw captured data is shown in figure 3.15. These values are the deviations from the calibrated radius.

Table 3.7.: Circular trajectory coordinate and test conditions X Center 600,45 mm

Y Center 250mm Z Center 350,739 mm Radius 150.075 mm Feed Rate 1000mm/min Temperature 20.0^◦

Figure 3.14.: Renishaw ball bar software analyze and plot the captured data [44]

Figure 3.15.: Raw data captured by DBB system

3.4 validation 43

To be able to compared these results of 10 repeated tests with the simulated model these steps carried out :

Firstly, It is necessary to have both data sets (from the test and the simulation) in the same coordination. Therefore in this step by using a Matlab code the data from the DBB test converted to X-Y plane.

Figure 3.16

Figure 3.16.: A plot of DBB test in X-Y coordinate. In this fig- ure, due to the scale of the trajectory (mm) and errors (µm), theradiusisreduced f orthevisualizationpurpose

By means of Matlab nine points at every 45 degree of the trajecto- ries is captured. The aim is to compare the random errors in these points. Figure 3.17 shows the arbitrary points on the simulated trajec- tory.

Figure 3.17.: Selected points

Next step is to calculate the repeatability of the machine tool based on the DBB test. Therefore the average value and the standard de-

viations of the value for each point is calculated based on 10 data sets. The standard deviations from the simulated model also for cor- responding points carried out.

Results show that the random errors in the machine tool in average is 2.6 times bigger than the random errors from the simulation. Since the simulation represents a simplified model of the real machine this difference seems reasonable. Here are some of the most important factors that helps to interpret the difference of the simulation and the test result:

1. Age of the measurement data can play an important role since the simulation is based on the measurements that is done two years ago and the DBB test is done recently.

2. The simulation is not including other kinematic errors such as angular errors.

3. Thermal behavior of the machine is not included in the simula- tion.

4. Uncertainty of the DBB is not investigated for the test results.

5. Another factor is that the data for the simulation is obtained two years while this test is done recently.

4

S U M M A R Y & C O N C L U S I O N S

In this project a new volumetric error modeling technique introduced that provides this ability to evaluate and visualize the probability of random errors around an arbitrary trajectory based on the machine tool’s measurearment.

In the first chapter the aim of this thesis explained as well as the scope of the project and the fundamental assumptions that the model is built on them.

The outcome of the second chapter is a comprehensive study on ma- chine tool diagnostics based on 50 papers. This table including can be found in appendix C. In this table error sources of a machine tool as well as the error budget and the relevant instruments for the in- spection of these errors presented. Then with a focus on kinematic errors the fundamental terms and aspects of a machine tool for the analytic model fulfilled.

In chapter 3 the computational model proposed. This model com- mence with one dimensional evaluation of kinematic errors and then developed for two and three dimensional space. By means of HTM theory and in conjunction with statistical approaches the model reached the final goal that was to predict the probability of random errors for arbitrary trajectories. At the end of this chapter the validation tests described and the results ensures that the simulation does what is in- tended to. However the differences can be justified by the discussion mentioned earlier.

The result of this thesis is a systematic tool that can be used to predict the tolerances. The importance of such studies in industrial environments is to avoid going through costly and time consuming inspections.

Finally, this study provides a probability based view for further in- vestigations regarding the repeatability performance of machine tools and by fulfilling all the necessary steps,the final aim of the research work is accomplished.

45

5

F U R T H E R E F F O R T S

Suggestions for further research can be :

1)Dependency or independency of variables :

PDF can be calculated whether by type (a) or (b) in figure below. In case (a) major axis of ellipses has an angle is it is not same as case (b) in which the major axis lies right on x2 axis. In case (b) variables are assumed independent since in case (a)they are more realistic there- fore variables are dependent.Due to limitation of the data (too few measurements per point) formula for case b is used in this thesis.In the other word it means that for example positioning errors in X di- rection has no effect on the straightness error in Y or Z directions. To be able to use (a) we need to know the correlation between our values.

Figure 5.1.: Dependency/independence of variables[32]

2)Including uncertainty of measurement instruments:

In practice even with accurate and precise measurements still some uncertainty will remain therefore uncertainty analysis for the laser in- terferometer and the LDBB measurement results is a must in further steps.

3)Applying validation tests or methods for current model

47

4)Extending the model for five-axis machine tool

As mentioned before in this thesis rotational errors (roll, pitch, yaw, tilt) are excluded.

5)Extending the model by involving the thermal effects:

Ferreira et all. due to experiments observed that as the machine heated, the positioning error could be as big as 0.15 mm to 0.2 mm[37].

It would be important to add They proposed to use GDMH pro- grams(group method of data handling) in order to relate positioning errors to variables such as the positions on the axis and the data from thermocouples.

Part II A P P E N D I C E S

A

S P E C I F I C AT I O N S

Effective travel range of the machine tool can be seen in table below.

(a) Machine Tool

(b) Laser

Figure A.1.: Charachterisations

51

B

I N P U T D ATA

Figure B.1.: The input data sheet[15]

53

C

M A C H I N E T O O L D I A G N O S T I C S

Error sources Error budget Indicator signal Instruments (Sensors) References

Dynamic stiffness Vibration & Force

Thermal Behaviour Temprature Deformation Temprature / distance

Temperature sensors[40][43]

Thermal deformation sensors IR-camera

Pyrometer

Telescopic double ball bar [40][E26]

Thermistors + touch-fire probe+LVDT (a type of analogue position sensors)[44][45]

Temperature Detector, RTD [45]

Thermocouple [51]

[2][40][37]

[43][44][45]

[51]

[19][48][10]

[3][11][12]

[17][18][20]

[21][22][23]

[24][25][26]

[27][28][29]

[30][31][32]

[33][34][35]

[36][42][39]

[40][41][45]

[47]

Accelerometers (piezoelectric) Vibrometer[49]

Ultrasonic Sensor

[19][49][7]

[52][54][53]

[57][58][55]

[56][49][2]

[59][60][53]

Kinematic Errors

Distance / angle

Noise induced by motor and servo errors

Vibrations caused by hydraulic and pneumatic systems

Position error affected by control stability and encoder performance Feedrate accuracy, stability, and interpolation accuracy [51]

Positioning errors

Straightness errors of each axis in its perpendicular axes Pitch , Roll, Yaw angular errors Abbé error Reversal errors

Backlash errors

Straightness error of each axis in its perpendicular axes Squareness error between two axes Contouring error of each axis

Hysteresis errors Nonperpendicularity Spindle radial drift

Spindle axial deviation Acceleration of axes Spindle inclination (Tilt or wobble error) Spindle radial play (run-out error) Friction and stick slip motion errors Inertia force errors while braking/accelerating Machine assembly errors

Parasitic movements Servo errors

Interpolation algorithmic errors

Mismatch of position loop gain Instrumentation errors Noise

laser interferometer [11][18][42][45]

Angular interferometer [11][23][45]

Ball Bar [11][13][42][1][39]

3D ball plate [11] [12][13]

Dial gauge [19][11]

Capacitance gauges[11]

Electronic gauges[11]

Gage blocks[11]

Step gauges [11][E6]

Lasertracer [11]

PSD (position sensitive devices) [11][20][21][22][23]

Autocollimator [11][E13]

Capacitive or inductive sensors [11][25][26]

Displacement sensor [19]

Ball array plate [19][41]

Sweeping alignment laser [19]

Angle lever /artefact [19]

Carriage [40]

Gyroscope[40]

Accelerometer[40]

Inclinometer [40]

Quasi Static Behaviour/analysis Static stiffness Force

[9][2][3]

[7][46][51]

Dynamic Behaviour

Natural Frequency(Vibration) Vibration

Force gauge Force Indicator Micrometer Linear displacement sensor Loaded double ball bar (LDBB) [46]

Figure C.1.: Diagnostics of Machine Tools.

Error sources are main sources of errors in a machine tool that affect machining system accuracy. Error budget is the components of the main error and the focus area of applied diagnostic method. Indicator signal is the signal type used to describe the focus area. Instruments shows the list of devices to measure and monitor the errors.

55

[1] Analysis of Dynamic Characteristics and Evaluation of Dynamic Stiffness of a 5-Axis Multi-tasking Machine Tool by using F.E.M and Exciter Test. Shil-Geun Kim, Sung-Hyun Jang, Hyun-Young Hwang, Young-Hyu Choi and Jong-Sik Ha.

[2] http://www.rcmt.cvut.cz/zkuslab/en/text/36

[3] http://www.planlauf.com/en/measurement/foundation-measurement/

[4] Machining Dynamics Fundamentals Kai Cheng.

[5] Measurement techniques for determining the static stiffness of foundations for machine tools

A Myers, S M Barrans and D G Ford.

[6] Static and dynamic stiffness. J. Tlusty.

[7] Optimizing static and dynamic stiffness of machine tools spin- dle shaft, for improving machining product quality. Tri Prakosa, Agung Wibowo and Rizky Ilhamsyah.

[8] Development of an energy consumption monitoring procedure for machine tools. Thomas Behrendt, Andre Zein, Sangkee Min.

[9] Stiffness Estimation of a Tripod-Based Parallel Kinematic Ma- chine. Tian Huang, Xingyu Zhao, and David J. Whitehouse .

[10] Integrated geometric error modeling, identification and com- pensation of CNC machine tools. Shaowei Zhua, Guofu Ding, Shengfeng Qin, Jiang Lei, Li Zhuanga, Kaiyin Yana.

[11] Geometric error measurement and compensation of machine- sAn update. H. Schwenkea, W. Knappb, H. Haitjemac, A. Weckenmannd, R. Schmitte, F. Delbressinef.

[12] A Measuring Artifact for True 3D Machine Testing and Cali- bration. Bringmann,Knapp(2005) .

[13] ISO 230:2005, Test Code for Machine Tools.

[14] Self-Calibration: Reversal, Redundancy, Error Separation and Absolute Testing. Evans RJ, Hocken R, Estler Wt.

[15] Error Mapping of CMMs and Machine Tools by a Single Track- ing Interferometer. Schwenke H, Franke M, Hannaford J (2005)