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Modeling and compensation of machine tool volumetric errors for virtual CNC environment Bal, Evren 2003

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M O D E L I N G A N D C O M P E N S A T I O N OF M A C H I N E T O O L V O L U M E T R I C ERRORS FOR V I R T U A L C N C E N V I R O N M E N T by EVREN BAL  B . S c , Istanbul Technical University, 2002  A THESIS SUBMITTED IN P A R T I A L F U L F I L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF  M A S T E R OF APPLIED SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES M E C H A N I C A L ENGINEERING  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH C O L U M B I A October 2004 © Evren Bal, 2004  [UBCL  THE UNIVERSITY OF BRITISH C O L U M B I A  Library  FACULTY OF G R A D U A T E STUDIES  Authorization  In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  EVRBN  A2//o/<2ooy  BAL  Name of Author (please print)  Title of Thesis:  Degree: Department of  Date (dd/mm/yyyy)  A/0£>El/A/<£,  MASTER.  Of?  MBCHANICAL  /MO  COM  PENS A T) OA/  APPLIED EN&tkf£eX.I  SC/EA/CE N  mACHl^S  Year:  2.0  TOOL  0<-^  &  The University of British Columbia Vancouver, B C  Canada  grad .u bc.ca/forms/?form I D=TH S  page 1 of 1  last updated:  20-M-04  Abstract Machine tools produce positioning errors due to thermal and structural deformations of the structural elements, geometric errors due to inaccuracies in the manufacturing and assembly of their components, and computer numerical control errors caused by the bandwidth of their servo drives. This thesis presents the measurement, modeling and compensation of geometric errors of three axis machine tools in order to simulate and improve their volumetric accuracy in virtual environment.  The geometric accuracy of each axis is measured using a laser interferometer. The displacement, pitch, yaw and backlash errors of each axis element are measured and mapped to the machine coordinates using rigid body kinematic transformations of the system. The identified errors are curve fitted to the position of each drive in the machine coordinates. The algorithm allows prediction of relative positioning error between the tool and workpiece within the working volume of the machine tool, or pre-compensates the errors by adding the estimated positioning errors to Numerical Control (NC) program in Virtual Environment before the machining takes place.  The prediction and compensation of geometric errors are experimentally demonstrated on a three axis, vertical C N C machining center. Standard ISO geometric profiles, a circle and a diamond, slot milled on the machine. The machined profiles are measured using a coordinate measuring machine, and the lengths and offsets from the command profiles'are estimated from the measurements. The measurement results and the predicted geometric errors are compared. The contributions of C N C and tool deflections to the total errors are estimated, and the remaining errors are correlated to geometric errors of the machine tool. Although, it was not possible to account all the machine tool errors, the proposed prediction and correction method in virtual environment improved the compensation of geometric errors significantly. The overall model is integrated to U B C Manufacturing Automation Laboratory's Virtual C N C system for use in industry.  ii  Table of Contents Abstract  ii  Table of Contents  iii  List of Tables  v  List of Figures  vi  Acknowledgement  :  :  .-  x  Chapter 1 Introduction  1  v.-  1.1 Problem Statement. 1.2 Research Objective  '.  :  1.3 Organization of the Thesis  1 5 6  Chapter 2 Literature Review  7  2.1 Error Analysis  7  2.2 Error Measurement  15  2.3 Error Compensation  18  2.4 Summary  22  Chapter 3 Mathematical Modeling of the Machine Tool  3.1 Introduction  :  3.2 Modeling approach - Homogeneous Transformation Matrices  24  24 25  3.2.1 Homogenous Transformation Matrices - Background  28  3.2.2 Homogeneous Coordinate Representation of Machine Slides  36  3.2.3 Homogeneous Coordinate Representation of Rotational Axes  39  3.2.4 The Mathematical Model for Machine Tools 3.3 Modeling Approach -Vector Representation Method  41 51  3.3.1 Background of Vector Representation Method  51  3.3.2 Vector Representation ofFADAL VMC-2216 3.4 Summary  :  54 60  Chapter 4 E r r o r Measurements by Laser Interferometry  61  4.1 Overview  61  4.2 Laser Interferometers and Their Working Principle  61  4.2.1 General Set-up for Measuring Linear Positioning Errors  iii  62  4.2.2 General Set-up for Measuring Angular Errors  63  4.3 Machine Tool Testing Parameters  67  4.4 Measurement Results of Three-Axis FADAL VMC-2216 Machining Center  68  4.4.1 Linear Positioning Errors  68  4.4.2 Pitch Errors  87  4.4.3 Yaw Errors  92  4.4.4 Backlash Errors of FADAL VMC-2216  96  4.5 Summary  99  Chapter 5 E r r o r Compensation and Experimental Results  102  5.1 Overview  102  5.2 Geometric Error Compensation  103  5.2.1 Compensation when moving to a target point  ....  104  5.2.2 Compensation for motion along a straight line 5.2.3 Compensation for motion along a circle  106 108  5.3 Backlash Compensation  110  5.4 Automation of Geometric Error Compensation Algorithms  115  5.5 Experimental Results  120  5.5.7 Cutting Tests  120  5.5.2 Linear Interpolation Test Results  122  5.5.3 Results for the Circular Interpolation  127  5.5.4 CNC and Deformation Error Sources  130  5.5.5 Discussion of Results  136  Chapter 6 Conclusions  139  6.1 Conclusions  139  6.2 Future Research Directions  140  Bibliography  142  Appendix A : The Resulting Equations of the Mathematical Model  148  Appendix B : Principles of Laser Interferometry Systems  163  iv  List of Tables Chapter 4: Error Measurements by Laser Interferometry  Table 4.1: Rotational errors on a VMC  64  Table 4.2: The travel lengths of three axes in the FADAL VMC-2216  68  Table 4.3: The periodic errors of Y axis  86  Table 4.4: The periodic errors of Z axis  86  Table 4.5: The equations of the pitch errors of Y axis  90  Table 4.6: The equations of pitch errors of Z axis  90  Table 4.7: The yaw errors of Y axis  94  Table 4.8: The yaw errors of Z axis  96  Table 4.9: Constant backlash zones and mean values of reversal errors of X axis  97  Chapter 5: Error Compensation and Experimental Results  Table 5.1: Identification of the direction changes  115  Table 5.2: The comparison of errors  138  Appendix B: Principles of Laser Interferometry Systems  Table B . l : Some specifications of the Renishaw laser system  v  163  List of Figures Chapter 1: Introduction Figure 1.1: Contour error in 2-axis machining  ....4  Chapter 2: Literature Review Figure 2.1: Accurate (dotted line) and inaccurate (solid line) motion in X axis  11  Figure 2.2: Block diagram for error compensated CNC axis drive  21  Chapter 3: Mathematical Modeling of the Machine Too Figure 3.1: Model of a sliding joint (translational axis)  25  Figure 3.2: Reference, ideal and actual coordinate frames  28  Figure 3.3: Coordinate systems  32  Figure 3.4: Definition of angles  33  Figure 3.5: Rotation about X axis  35  Figure 3.6: Linear and angular errors on a sliding joint  37  Figure 3.7: Model of a rotational axis  .  40  Figure 3.8: Common configurations of three-axis machining centers. The letter(s) before F shows the possible motion of the workpiece whereas the letters after F shows the possible motions of the tool  41  Figure 3.9: FADAL VMC-2216 3-axis milling machine  42  Figure 3.10: Coordinate Frames on the machine tool  48  Figure 3.11: Sketch of a 3-axis XYFZ type machine tool  49  Figure 3.12: Mathematical Model of a 3-axis XYFZ type machine tool  49  Figure 3.13: General transform of a vector  53  Figure 3.14: Sketch of a XYFZ type 3-axis machine tool  55  Figure 3.15: Coordinate systems on FADALVMC-2216  55  Chapter 4: Error Measurements by Laser Interferometry Figure 4.1: General set-up for linear measurements  63  Figure 4.2: The set-up for linear positioning measurement on the X axis of FADAL  63  vi  Figure 4.3:  Angular errors for motion in X axis  64  Figure 4.4:  General set-up for angular measurements  66  Figure 4.5:  The set-up for yaw measurement of Z axis on FADAL  66  Figure 4.6:  The set-up for pitch measurement of Z axis on FADAL  67  Figure 4.7:  Linear positioning errors along X-axis  69  Figure 4.8:  Reversal errors along X-axis  Figure 4.9:  Periodic errors in the forward and backward direction of motion of X-axis  :  70 71  Figure 4.10:  Mean of periodic errors in the backward direction of X-axis and the fitted curve  72  Figure 4.11:  Mean of periodic errors in forward direction of motion of X-axis and the fitted curve  73  Figure 4.12:  Periodic errors of X-axis.„  75  Figure 4.13:  Set-up with Abbe error (Source: Renishaw [42])  76  Figure 4.14:  Set-up for X-axis on FADAL VMC-2216 with Abbe error  77  Figure 4.15:  Linear positioning errors of X-axis in forward direction with pitch compensation  78  Figure 4.16:  Linear positioning errors of X-axis in backward direction with pitch compensation  78  Figure 4.17:  Linear positioning errors of X-axis without pitch errors and the fitted curves  80  Figure 4.18:  Linear positioning errors of X-axis in the forward direction  81  Figure 4.19:  Linear positioning errors of X-axis without pitch errors in the backward direction  81  Figure 4.20:  Linear measurement set-up for negative Y axis  83  Figure 4.21:  Linear measurement set-up for positive Y axis  83  Figure 4.22:  Linear positioning errors of Y axis in forward and backward directions  84  Figure 4.23:  Linear positioning errors of Z axis  85  Figure 4.24:  Sign convention for pitch errors of X axis  88  Figure 4.25:  Pitch errors of X-axis, raw data  88  Figure 4.26:  The difference between forward and backward pitch errors of X-axis..'  89  Figure 4.27:  Pitch errors of X-axis for both directions of motion  89  Figure 4.28:  Pitch errors of Y axis  91  Figure 4.29:  Pitch errors of Z axis, fitted curves to raw data for both directions  91  Figure 4.30:  Yaw errors of X-axis, raw data  92  Figure 4.31:  Yaw errors of X-axis in forward motion of direction  93  Figure 4.32:  Yaw errors of X-axis in backward motion of direction  95  Figure 4.33:  Yaw errors of Y axis  95  vii  Figure 4.34: Yaw errors of Z axis in forward direction of motion, fitted curve to raw data  96  Figure 4.35: Backlash errors of X axis and sub-regions with constant backlash errors  98  Figure 4.36: Backlash errors of Y axis and sub-regions with constant backlash errors  98  Figure 4.37: Backlash errors of Z axis and sub-regions with constant backlash errors  99  Chapter 5: Error Compensation and Experimental Results  Figure 5.1 : Flowchart of recursive scheme of compensation  106  Figure 5.2: Determination of point at which original straight line is to be broken in the compensation scheme along a line  107  Figure 5.3: (a) Influence of long movement of one axis, (b).Influence of non-perpendicularity of axes on circular paths  108  Figure 5.4: Compensation for motion along a circle Figure 5.5: Backlash errors and compensations  109 .':  '.  113  Figure 5.6: Machine tool configuration dialog box  117  Figure 5.7: Loading the errors measured for X axis  118  Figure 5.8: NC-tape modification for linear and circular interpolations  119  Figure 5.9: Top view of the second and third workpiece (dimensions given in mm)  121  Figure 5.10: Finished workpieces  122  Figure 5.11: Results of measurements along the diamond for the uncompensated case  123  Figure 5.12: Results of measurements along the diamond for the compensated case  126  Figure 5.13: Results of measurements along the circle for the uncompensated case  128  Figure 5.14: Static stiffness test set-up on FADAL  131  Figure 5.15: Forces generated during cutting in X and Y directions  133  Figure 5.16: Forces generated during cutting in X and Y directions in a short range  134  Figure 5.17: Form errors left on the surface of the workpiece  134  Figure 5.18: Contouring errors in X and Y axes during linear and circular interpolations  136  Appendix B: Principles of Laser Interferometry Systems  Figure B . l : Set-up of the laser interferometer system for displacement measurements  164  Figure B.2: Linear and angular optics of Renishaw laser system  165  Figure B.3: Linear interferometer and retroreflector arranged for displacement measurements  166  viii  Figure B.4: Angular interferometer and angular reflector arranged for angular measurements  167  Figure B.5: Set-up with cosine error  169  Figure B.6: Set-up with Abbe error (Source: Renishaw [42])  170  ix  Acknowledgement  I would like to express my sincere appreciation to my research supervisor Dr. Yusuf Altintas for his perpetual guidance, support and encouragement throughout my graduate studies at the University of British Columbia. I thank my co-supervisor Dr. Alkan Donmez for his diligence and counsel during my research. M y thanks are also due to Nesrin Altintas and Petek Donmez for the hospitality they have shown to me.  I am grateful for my colleagues in the Manufacturing Automation Laboratory. They have shared their knowledge, experience and culture with me, which have made my life at U B C easier and more enjoyable. I would like to thank Megi Senmenek and Fuat Atabey for their friendship and endless support. I was also very lucky to have my roommates, Bahar Orhan and Nimet Kardes, whom I spent two years with. It was very enjoyable to share every aspect of my life with them.  Finally, I am deeply grateful to my beloved parents, Cevahir and Halil Ibrahim Bal; my dear sister and brother-in-law; Elen and Ozgur Zafer for their constant support, patience and encouragement during my graduate study abroad. M y special thanks are due to my soul mate, Tolga Yasa, for always sharing my loneliness even from such a far distance. I dedicate this work to my family and Tolga.  Chapter 1 Introduction The main objective of the machining process is to produce a workpiece within the imposed tolerances of shape, dimension and surface finish. With the requirement of higher productivity and increased automation, the demand for greater accuracy and better reliability became more important in industry. Many key factors such as cutting tools and cutting conditions, resolution of the machine tool controller, the shape and material of workpiece play important roles in machining accuracy. However, once these are determined before the operation, the consistent performance of the machine tool depends upon its ability to move the tool tip to the commanded positions in the work volume with the required accuracy, which is greatly constrained by errors either built into the machine or occurring on a periodic basis due to temperature changes or variation in cutting forces along the path.  Basically, there are two approaches for the improvement of a machine's accuracy. The first approach is to improve the structure of the machine at the design, manufacturing and assembly stage. The second is to enhance the accuracy of the machine tool by error compensation methods. Although the improvement of the structure leads to a better performance of the machine, in many cases the physical limitations on the achievable accuracy can not be overcome only by production and design techniques. Therefore, no matter how well a machine tool may be manufactured, there is a limit to the accuracy that could be achieved. It is impossible to completely account for some of the errors like thermal deformations by detailed design. Thus, effective error compensation schemes become very important for precision manufacturing. In the case of periodic machine errors which can be measured and stored, compensation methods can be a good solution to increase the accuracy of finished workpiece [41]. 1.1 Problem Statement As the tight tolerances in accuracy become one of the major requirements in machining, the precision machines become essential elements of an industrial society. Indeed, fhodern industry is significantly dependent on precision machines ranging from those for the  Chapter 1. Introduction  2  manufacture of integrated circuits and optical components to dies and molds. In space and optics industry, the tolerances are indicated with submicron accuracy. To be able to satisfy tight tolerances in many industries such as aerospace industry, all the errors sources affecting the positioning of axes should be identified and compensated for. Therefore, precision and low cost machining is vital for the survival and competitive manufacturing of automotive, aerospace and electronics industry. Current machining practice is mostly based past experience and narrowly applicable analytical techniques. The produced part is inspected and compared against design drawings and tolerances. The discrepancy between the measured and design dimensions, the errors, are accepted if they are within the tolerance of the part; otherwise the part is scrapped.  In machining operations, the precision of the workpiece depends on the accuracy of the relative position between the cutting tool and the workpiece. The factors which affect the accuracy of this relative position can be classified as four major groups:  1. Relative structural deformations between the tool and the workpiece 2. Thermal effects on the machine tool, workpiece and cutting tool 3. Tracking errors of axis servo 4. Volumetric errors of the machine tool  The relative structural deformations between the tool and workpiece, where the cutting forces are applied, are passed as errors. If the deformation produced by cutting forces remained constant throughout the machining operation a workpiece of accurate form could be obtained. However, if it varies, form errors are left on the workpiece. The dynamic stiffness of all the components of the machine tool that are within the force-flux flow of the machine is responsible for errors caused as a result of the cutting action. As a result, the position of the tool tip with respect to the workpiece varies due to static and dynamic loading of the structure by cutting forces.  As reported in Bryan's paper, errors due to thermal factors account for 40-70% of the total dimensional and shape errors of a workpiece in precision engineering [8]. Six sources of  Chapter 1. Introduction  3  thermal influence are identified in machine tools, which are (i) heat generated during cutting operations, (ii) heat generated by the machine, (iii) heating or cooling provided by the systems, (iv) heating or cooling influence of the room, (v) the effect of people and (vi) thermal memory from any previous environment. The most critical thermal source is the heat generated by the machine. The change of temperature causes relative expansion or contraction of the various elements of the machine tool resulting in inaccurate positioning of the cutting tool tip.  Tracking errors of axis servo also influence the accuracy of the workpiece. The goal of control systems is to keep the actual tool position on the desired tool path at all times, which will lead to the part being machined with the desired part geometry. The limited bandwidth of the servo drives, as well as the mismatch of position loop gain, will cause the tracking errors.  The last group of errors that influence the overall accuracy of the workpiece is the volumetric errors of the machine tool. The term 'volumetric error' is used for geometric and kinematic errors. Geometric errors in a machine are due to its design, the inaccuracies created during assembly and as a result of the components used on the machine. For instance when one axis is built on top of another with an error in their orthogonality, a squareness geometric error is produced between the two axes. This squareness error is known as a geometric error. The geometric errors are concerned with the quasi-static accuracy of surfaces moving relative to one another. Kinematic error is the error appearing in the ability of the machine tool to reach the exact specified position by the controller. The geometric errors are particularly significant during the combined motion of different axes.  •-  *  Besides the errors mentioned above, there are also other error sources due to tool wear, weight deformations and fixturing errors. Errors in fixturing are caused by the set-up and geometric inaccuracies of the locating elements. Also, various methods are employed to compensate for weight deformations, to counter-balance the weights, or to minimize their effects. It is generally necessary to design the responsible parts of structure with sufficient  Chapter 1. Introduction  4  stiffness so as to keep the errors due to weight deformations within limits arising from the rather tight tolerances.  In order to machine a part, the tool is commanded to follow a desired tool path as shown with dotted line in Figure 1.1 which the tool is supposed to follow. However due to the mentioned error sources, the tool may follow an actual tool path shown in Figure 1.1 with a solid line. The difference between the actual and reference tool paths gives the contour error, which is caused by the summation of all the errors. Hence, reducing the individual error sources of the machine tool is a key requirement for improving overall workpiece accuracy.  Depending upon the severity of the cutting action or the thermal condition of the machine tool, the error components change continuously. As mentioned earlier, the strategies to reduce the errors that adversely affect the accuracy of machine tool can be grouped into two categories which are 'error avoidance' and 'error compensation' categories. Error avoidance approach involves a high degree of investment as machine costs rise exponentially with the level of accuracy involved. In the contrary, the error compensation is implemented on the machine by experimentally identifying and correcting them during machining. This thesis focuses on the measurement and compensation of geometric machine tool errors.  velocity  actual toolpath  Figure 1 . 1 : Contour error in 2-axis machining  Error compensation could further be divided into two sub-categories depending on the repeatability of the system. One method is 'pre-calibrated error compensation', or passive error compensation, in which the error is measured either before or after a machining process  Chapter 1. Introduction  5  and used to calibrate the process during the subsequent operation. This method is more preferable especially when the errors of the system can be considered as highly repeatable. The second method is called 'active error compensation' where the error monitored during the machining process is used to alter the process simultaneously. The advantage of the active error compensation is that a higher workpiece accuracy could be achieved in a relatively lower grade machine tool with error compensation techniques.  In this thesis, volumetric error compensation with pre-calibration is used, which basically involves a study of the various sources of volumetric error in the machine tool and a method to compensate them. The error sources can be summarized as the linear positioning errors of each axis and angular deviations, which are generally considered as systematic and repeatable errors in machine tools. The aim of this research is to identify the errors, left on the part prior to cutting operations, and improve the accuracy by using off-line compensation schemes. By designing virtual machining algorithms, it is possible to simulate structural deformations, servo errors and volumetric errors of the machine tool in virtual environment. The system allows the prediction of the magnitude and source of each error component and optimizes feeds, speeds, machine tool correction offsets so that the part is produced in an optimal fashion without trial and error cuts.  1.2 Research Objective The thesis has the following objectives:  -  To identify the machine tool linkage structure and to derive error synthesis model using homogeneous transformation matrices and vector representation method.  To measure error components at target positions on each axis  To create the error component model (curve fitting) and combine all the individual errors into one.'Error Synthesis' model, and  Chapter 1. Introduction  6  To develop an error compensation system and integrate it to the virtual Computer Numerical Control (CNC) system for automatic geometric error compensation of tool path.  1.3 Organization of the Thesis The thesis is organized as follows: The review of related literature is presented in Chapter 2, followed by the mathematical modeling of three-axis machine tools in Chapter 3. The rigid body kinematics and small angle assumption are used in order to derive the equations representing the error components in X , Y and Z directions as a combination of individual link and squareness errors. The individual link errors include linear positioning error, angular deviations (roll, pitch and yaw) and straightness errors (horizontal and vertical). In addition to link errors, the squareness errors are also considered in the mathematical modeling of the machine tool. The methods developed by Schultschik [43] and Donmez [14] are applied to a three axis machine tool ( F A D A L VMC-2216) available at the Manufacturing Automation Laboratory at U B C . Two methods were used in order to check the accuracy of the resulting equations. After the equations representing the error components in three orthogonal directions are derived, the individual error components of the machine tool are measured in its total work volume. The measurement techniques and results of the linear and angular measurements are given in Chapter 4 while the compensation strategies are explained in Chapter 5. The compensation  schemes  for linear and circular interpolations  are  experimentally tested on the machine tool, and the results are analyzed in Chapter 5. The thesis is concluded with the summary of contributions in Chapter 6. Clarifications of some of the equations and explaining the working principle of laser interferometer system are provided in the Appendices.  1 Chapter 2 Literature Review Due to the need for higher accuracy in many manufacturing industries, error analysis and compensation methods have become a major consideration in high precision applications. In the past years, many researchers have done much work in this area.  Geometric accuracy is a concept that was historically established in the 1930s. Professor Schlesinger in Berlin undertook the task of preparing accuracy specifications and tests for the acceptance of machine tools for both machine tool builders and their end users. Schlesinger's tests, which have been used;as a basis of most national and international .standards, were conceived as non-machining tests; instead, they mostly concentrated on measuring the accuracy of parts of the machine-tool structure by means of levels and dial gages. Typical measurements are for the flatness of a table, straightness of a guideway, and squareness of X and Y guideways [47].  In further development of machine tool metrology, Tlusty (1970) proposed a different approach that is based on the measurement of errors of the machining motions and relating them to the errors on the workpiece [48]. The introduction of a well-developed and easy-touse laser interferometer was a very important event in the development of the kinematic methods. The laser interferometer system is an accurate and reliable tool for measuring errors of motions. However, it is much less preferable for measuring errors of surfaces due to the difficulties in the set-up procedure, and useless for the static measurement of distances.  Since then a lot of work has been devoted to the development of machine tool metrology. The following sections introduce the essence of the present state of the machine tool metrology art. 2.1 Error Analysis Errors in position and orientation in a multi-axis machine are the result of the individual links of machine and of the interactions between them.  Chapter 2. Literature Review  8  Schultschik (1977) models the machine as a closed vector chain to obtain expressions for the volumetric error vector [43]. Most commonly coordinate transformation methods are used to derive mathematical representations of machine volumetric errors based on rigid body assumption. Schultschik's method consists of the following steps; defining the axes of motion, measuring the location of these axes in relation to each other, then obtaining the single error effects and combining all of the error components. He demonstrated this method using a jig boring machine. The geometrical model in this approach does not take either squareness or straightness errors into account. It only deals with linear positioning and angular errors (roll, pitch and yaw).  Duffie and Yang (1985) used the vector representation method for the generation of parametric kinematic error-correction functions from volumetric error measurements [16]. A kinematic model for a proportional tactile probe was developed and the errors found in the probe were identified using this model. A n automatic volumetric error measurement technique was developed which utilized a special fixture into which the probe tip was inserted. The average error of the probe was reduced by a factor of ten by the error compensation technique.  Dufour and Groppetti (1981) used the error matrix method and stored the error vector components at different locations in the machine workspace for various loads and thermal deformations [17]. Error predictions are obtained by interpolation between the stored values. They suggested several correction schemes in the machine control circuit once the error vector is known. The error matrix method, however, requires a large amount of data which changes over time due to the heating of the machine tool. In their work, they assumed that the angular and linear positioning errors along an axis of a machine are constant.  Zhang, et al, (1985) proposed that a correct geometric model, a correct thermal model and careful machine calibration can increase the accuracy of a coordinate measuring machine (CMM) at least by ten times [50]. Based on the vector representation method developed by Schultschik, he managed to get the geometric model of a three-axis C M M . Furthermore, he added straightness and squareness errors. A l l the errors of the C M M other than squareness  9  Chapter 2. Literature Review  and roll errors were measured using a HP laser interferometer system. In addition, a thermal model with a simple compensation sensitive to ± 1°C change in the temperature was used. The squareness was determined by measuring along machine diagonals, which proved to be a very sensitive and accurate method for squareness determination.  Donmez, et al., (1985) introduced a general methodology for increasing the accuracy of machine tools by compensating for the inherent systematic errors [14]. This methodology includes a general mathematical model, relating the error in the position of the cutting tool with respect to the workpiece to the errors of the individual elements of the machine tool structure. Both thermal and geometric errors are considered in the model, generated by utilizing homogenous transformation matrix manipulations, with the assumption of rigid body kinematics. Each axis of motion is represented by a coordinate frame and by using a homogenous transformation matrix for each slide, it is possible to describe the motion of the slide in the reference coordinate system. To validate this methodology, a turning centre was used as an example and the results showed that accuracy enhancement of up to 20 times was achievable.  Eman and Wu (1987) used rigid body kinematics and Denavit-Hartenburg approach [13] to build a generalized error model of a multi-axis machine of arbitrary configuration [18]. As Donmez used in his methodology, they utilized homogenous transformation matrices. The model can include first, second and higher order error contributions as required. However, the approximation of the actual position of the tool point by first order influences.may be sufficiently accurate for most multi-axis machine applications.  Using the same methodology Donmez developed, Okafor and Ertekin (1999) modeled the Cincinnati  Milacron  Sabre  750  three-axis  C N C V M C by  applying' homogenous  transformation matrix manipulations to each slide [37] They proved that the resultant volumetric error increases with increasing machine operation time (temperature) and axis nominal position. This is not a general fact because the change in the errors due to the increase in the nominal position is dependent on the mechanical structure of the machine.  10  Chapter 2. Literature Review  Based on Donmez's approach, Jha and Kumar (2002) developed a generalized error model for the effects of geometric errors of the components of the kinematic chain of a machine in the workspace [26]. The results obtained by this model were verified experimentally. A significant improvement was obtained in the machining accuracy of a specific cam profile.  Ferreira and L i u (1986) proposed an analytical quadratic model for the prediction of geometric errors of a machine tool using rigid body kinematics [19]. The model allows for the variation of angular and positioning errors of the machine tool and relates the error vector at a point in the machine tool's workspace to the coordinates of that point by the dimensional and form errors of the individual links and joints of the machine's kinematics scheme. A quadratic expression for the geometrical error at a point in a three-axis machine tool's work space was developed for the case where the individual joint errors vary linearly along the axis. In Ferreira and Liu's work, only first order errors are considered using small angle approximations in utilizing the homogenous transformation matrices. The angular and position errors of each joint, oc(x), (3(x), y(x) and A(x), are expressed as a function of their own link variable (See Figure 2.1). The angular errors are given by; a(x) = x ^ , dx  P{x) = x ^ , dx  Y(x) = x^dx  (2.1)  The variation of the angular errors along the joint's axis will cause displacement errors, Ayi and Azu in the other two directions perpendicular to the joint's axis. Ay =ja(x)dx, i  o  Az =]-fi(x)dx :  (2.2)  o  where Xi is the displacement of the joint.  According to the equations Eq. (2.2), the machine's position errors along the Y and Z axes (horizontal and vertical straightness errors of X) depend only on the angular errors. That is to say, there are no translational errors either without angular errors. In fact, a machine could have arbitrary position errors due to imperfect manufacturing, assembly and  11  Chapter 2. Literature Review  misalignment between the joint elements. Therefore, this model will not be true for most cases.  Y  F i g u r e 2.1: Accurate (dotted line) and inaccurate (solid line) motion in X axis  Later, based on the work of Ferreira and L i u who used the rigid body kinematics assumption, Kiridena and Ferreira (1994) developed an nth order quasi-static error model which is a function of error components of each link [28, 29]. They assumed that the error components for each link are a function of their own link variable only and used nth order polynomials to represent the angular and positional errors without loss of generality; 'a,u,) = 3 X * / , 7=1  A(* ) = £ / V / > j=l , ;  .  r,W =E r W 7=1  = !>«,*/  (2-3)  (-> 2 4  7=1 where /0,•(*,•) is the accumulation of positioning error along the ith axis.  The equations (Eq. (2.3) and Eq. (2.4)) are basically nth order polynomials in x„ x,- for /=1,3.being the ith axis where ty, /%  and p,y are the coefficients of the jth power in the  polynomial representing the angular (roll, pitch and yaw) and positioning error characteristics along the ith axis. After the parameters in the error model are estimated, the quasi-static errors for the three-axis machine were predicted. This research also addresses error  12  Chapter 2. Literature Review  compensation strategies in circular and linear interpolations not explored in the work done by Ferreira and Liu [27].  Zhang, et al., (1988) have proposed a displacement method to determine the machine's geometric errors [51]. Based on the rigid body assumption, the error terms of each link can be determined by a straightforward measurement of the displacement. This method requires measurements along a minimum of 22 lines: three parallel lines for each axis, three diagonal or half diagonal lines on the X Y , Y Z and X Z planes, three similar diagonals on the plane Z=Zi parallel to the X Y plane, an additional Y Z diagonal. The actual motion of the tool tip of a three axis machine with respect to the base coordinate frame is expressed as: X  p  = X+S (Y) x  + S (Z)-Ya x  -Za^  YX  Y =Y P  z  z  + 6 (X) y  z  Z =Z p  + S (X) z  Y,(e (X)  YZ  x  x  z  +  e (Y))Y  ¥  x  x  ( Z ) ) - Z, (e (X) + e (Y) + e (Z))+ Y x  + S (Y) + S (Z)-Ye (X)-XXe (X) + e (Y)  x  Y  y  z  z  +  6 (Y)-Za -Z(e (X)-e (Y))+  y  z  Y  y  + 6 (Z) +  X, (e (X) + e (Y) + e z  + z(e (X)  z  (Z))+ Z, (e (X) + e (Y) + e (Z))+ X,  Y (e (X) + e (Y) + e t  -Y£ (X)  x  Y  x  t  + e (Y) + Y  e (Z))+Z x  e (Z))Y  (2.7)  t  where X , Y and Z are the nominal values of the carriage positions, X , Y and Z are the X , Y and Z offsets of the tool tip, t  t  t  8 (v) is the translational error of v axis in u direction, u  e (v) is the angular error about u axis under v motion, u  u and v are arbitrary arguments, which may be X , Y and Z, otxY, ocyz and ocxz are out-of -squareness in the X Y , Y Z andX Z planes respectively.  In the displacement measurements for machine calibration, only the positioning errors along the three axes are measured directly. A l l other error terms are derived from these displacement measurements. The pitch and yaw errors are derived by measuring the displacements errors along two lines parallel to the axis of motion but separated by a distance in the appropriate orthogonal direction. For example, if it is desired to measure the yaw error  Chapter 2. Literature Review  13  of the x-axis, first the displacement, S (x), along the line Yi = Z, = X, = Y, = Z, = 0 is x  measured and denoted by d(X, 0, 0, 0, 0, 0). Next the same displacement along the line Y = F, is measured and d(X, F/, 0, 0, 0, 0) = 6 (x)-Y £ (x)is x  i  l  obtained from Eq. (2.5). Thus, the  yaw error of X-axis can be obtained as; e (x) = [-d(X,y„0,0,0,0)+d(X,0,0,0,0,0)]/r z  i  (2.8)  The other two angular error, e (x) and e (y), can be determined in a similar fashion. y  x  Furthermore, by measuring along diagonal lines, the roll errors and squareness and straightness can also be determined. Zhang's method [51] requires only the measurement of displacement errors along a set of lines. Therefore it is simpler and less time-consuming to perform the measurements along a set of lines than conducting a full parametric calibration. Zhang's method also has the advantage of reducing the amount of instrumentation needed. However, since the measurements of the errors other than linear positioning errors are not directly made, the uncertainties in the results may be higher.  The displacement method of Zhang et al. [51] was improved by Chen, et al., (2000) [11]. They proposed that by measuring the positioning errors along the 15 lines in the machine work zone, a total of 21 geometric errors can be determined quickly. To evaluate the performance of the proposed method, experimental tests were conducted on a three-axis C N C horizontal machining center. The calculated values for pitch, yaw and straightness errors were compared against the values measured by using the optics of the laser interferometer system. A good match between the errors measured directly and the errors derived using the displacement method was obtained in terms of the trends and magnitudes of the errors.  A l l of the research done in this area is, in general, limited to the study of three axis C N C machine tools or C M M s . When this investigation of error is extended to five-axis machine tools, the complexity of the problem increases. Extra sources of error are introduced by the added degrees of freedom. The interaction of all the axes complicates the relationship between the sources of error and the final error at the tool tip. Srivastava, et al., (1994)  Chapter 2. Literature Review  14  described a systematic approach for the development of geometrical and thermal errors based on the kinematic analysis of the machine structure on five-axis machine tools [46]. The shape and joint transformations for inaccurate links and joints (sliding and rotary) are obtained using small angle approximations. These transformations are used to set up the kinematic equations of the machine. The resultant matrix is solved to express the position and orientation errors in terms of individual error components of links and joints. The model was developed using rigid body kinematics, proper transformations for inaccurate links, as well as sliding and rotary joints of an existing RRTTT (R: Rotational axis, T: Translational axis) type machine tool. Besides, a compensation scheme was also presented to improve the total position and orientation accuracy at the tool tip.  In 1999, Barakat, et al., used the same systematic approach for kinematic modeling of a coordinate measuring machine [4]. The C M M is considered as an open chain mechanism where Denavit-Hartenburg modeling methodology can be applied. The links and joints of the C M M are modeled as homogenous transformation matrices. Two methods are used and compared in this research to measure the errors; (1) standard artifact measurements and, (2) laser interferometer measurements.  Chen, et al. (1997) proposed a new method using the meshing concepts developed in finite element method literature in order to predict the quasi-static errors at any point in the workspace [10]. Firstly, the geometric errors at some positions in the workspace are measured. For error prediction at any point in the workspace, the workspace is sub-divided into a finite number of smaller three-dimensional elements. Once the elements have been constructed and the errors at the nodal points are determined, the errors at any position in every element can be interpolated by introducing suitable interpolation functions. This methid is similar with Dufour and Gropetti's error matrix method.  Generally the non-rigid case is not included in many past studies in error modeling area. In practice, because of the limited stiffness of the machine's components, deformations due to dynamic behavior and load conditions can not be avoided. The use of heavier components is one of the methods to increase the stiffness, but this lead to larger moving masses which  Chapter 2. Literature Review  15  are, in general, not desirable. Therefore, in order to precisely predict the errors in the operations that demand extra-high accuracy, the consideration of the non-rigid body case is necessary. However, no study has included the effect of non-rigidity in the error model.  2.2 Error Measurement The use of error compensation techniques has been recognized as an effective way in the improvement of the performance of multi-axis machines. A n essential part of the error compensation is the ability to measure the errors of the machine tool accurately. Studies on machine accuracy tests have been carried out by many authors. In general, two approaches appear to exist, including: (1) the parametric error measurement approach, and (2) the master part tracing approach. The parametric error measurement approach emphasizes the measurement of individual geometric error components and the use of a kinematic model to combine these individual error components to estimate the volumetric errors. In the master tracing approach, the machine probe is commanded to trace a master part instead of measuring each individual error component.  In the parametric error measurement method, error components commonly measured as a function of axis position using laser interferometers, electronic levels, straightedges, and square block as stated in the ISO standards [23]. Week (1986) developed a method using laser beam and a four-quadrant photodiode to measure the radial run-out and tilt-motion of a rotating table of a gear hobbing machine, as well as the deviation of parallelism between the axis of rotation and a linear slide [49]. Although the complete mapping of all geometric error components is very time consuming, the disadvantages of using standard specimens can be avoided. The physical form errors in the standards as well as adjustment errors in the standards against the machine part to be measured are the main disadvantages of using a standard specimen.  The most commonly used method in order to get the individual geometric errors of machine tool slides is using a laser interferometer. This technique is used in the researches of Donmez (1985) [15], Zhang, et al. (1985) [50], Okafor, et al. (1999) [35, 36], Chen et al.  Chapter 2. Literature Review  16  (2001) [11], Barakat (2000) [4], etc. A S M E B5.57-1998 Standard defines the procedure that shall be used in measuring the geometric errors of a machine tool using a laser interferometer [3]. The standard specifies each step in the measurement, such as the alignment, setting premeasurement parameters, measurements and data analysis. According to this standard, measuring intervals shall be no larger than 1/10 of the axis length. Besides, the environmental compensation of the laser beam is very important for linear measurements in this technique because the measurements depend on the wavelength of laser, which is the resolution of the laser interferometer. The wavelength of laser beam changes according to the changes in the velocity of the beam, which is dependent on the temperature, pressure and humidity of the ambient medium in which the laser beam travels. However, roll errors can not be measured with a laser interferometer system. Instead electronic levels are employed.  In 2002, Castro and Burdekin described a method for evaluating the positioning accuracy of machine tools and C M M s under dynamic conditions, using a HP laser interferometer which is capable of performing dynamic calibration instead of static calibration [9]. In static calibration technique, one measurement is recorded each time the slide stops at a prespecified target position along the axis under test. A n electronic interface was employed in order to permit the "on-the-fly" data acquisition. This consists of reading instantaneously frozen displays as the probing point is passing through the targets.  Instead of measuring each individual error component, some of the works done in the field of error measurement were focused in tracing a master part such as a circular disc, a ball-bar, or any other metrological artifact. Bryan (1982) developed the magnetic ball-bar consisting of precision balls and L V D T to obtain the position error of the machine at various points [6, 7]. The test, though not complete, is quick and easy to perform and gives good estimates of some of the error components.  Pahk, et al., (1997) proposed a technique for assessing the volumetric errors in multiaxis machine tools using a kinematic double ball bar [38]. The parametric errors such as positional, straightness, angular and backlash errors are modeled as polynomial functions of the position along each axis. The squareness and errors due to servo gain mismatch are also  Chapter 2. Literature Review  17  modeled between the neighboring axes. Then the constructed volumetric error model is applied to the ball bar measurement data. The developed system was applied to two practical cases of machine tools, demonstrating high efficiency for the assessment of the threedimensional error components in a relatively short time when compared with conventional length and angle measuring equipment.  Lei and Hsu (2003) proposed a new measurement device for an accuracy test of five axis C N C machines [33]. This device is named probe-ball and consists of a 3D probe, an extension bar and a base plate with a measuring ball on one side. A permanent magnet is integrated in the socket of the extension bar so that the extension bar and the measuring ball can be connected together with magnetic force. After installing the probe-ball device, the kinematic chain of the five axis machine tool is closed. The tool paths are defined as the curves on a spherical test surface to check the accuracy of the machine tool. The overall positioning errors of the relative motion are measured by the 3D probe and are used to justify the volumetric accuracy of five axis machine tool.  Knapp (1983) developed circular test technique, which is a fast way of testing the geometrical accuracies of three-axis machines, i.e. positioning, straightness, roll, pitch, yaw and perpendicularity errors [30]. In the circular test a standard disc is used and it is measured in different positions in the working area of the machine. The calculated mean square fit diameters, standard deviations and Fourier analysis result in an analysis of the error sources and of the geometric error components. Computer simulations show that each of 21 error components yields an excessive (or short) axial movement, a non-perpendicular movement, out-of straightness or a combination of these three basic effects. Thus, the circular test is said to be sensitive to 21 error components.  ISO Standard 230-4, Part 4 gives the details of the procedure of circular tests for numerically controlled machine tools [24]. The parameters of the test are diameter of the nominal path, contouring feed, contouring direction, machine axes moved to produce in the actual path, location of measuring instrument in the machine tool work zone, temperature, positions of slides or moving elements on the axes which are not being tested and data  Chapter 2. Literature Review  18  acquisition method. A graphical method of presenting results should include circular hysteresis, circular deviations for clockwise and counter clockwise contouring and radial deviations for clockwise and counter clockwise contouring, corrected to 20°C. The standard also explains the influences of each error source, including the geometric deviations and deviations caused by the numerical control and its drives on the circular path.  Some other researches were also based on measuring standards to be able to derive the geometric errors of machine components and volumetric accuracy. Lee and Burdekin (2001) showed the possible use of a hole-plate artifact method to measure the volumetric error of C M M s and 21 parametric error components [32].  2.3 E r r o r Compensation Compensation techniques are increasingly used in precision machine tools and coordinate measuring machines. After various design efforts are exhausted, compensation methods can be integrated to achieve further cost-effective improvement of the machine's accuracy. The major task in the compensation is how to collect necessary information for the compensation of the machine's inaccuracy. In general, passive and active error compensation techniques are employed for machine tools.  With passive compensation, errors are measured or identified offline either before or after the machining process, then the known errors are compensated in the subsequent operations. Two basic approaches may be used to introduce quasi-static error compensations, when the errors are known, which are (1) hardware approaches, and (2) software (preprocessing) approaches. For a hardware compensation scheme, a micro-processor is used to store the error model with its current parameters. It also keeps a track of the system's parameters such as the coordinates in the fixed reference frame and the error compensation given to the machine. Then the compensatory motion can be obtained by deleting / adding reference pulses in the forward loop or from the feedback circuit of the N C system. However, making hardware modifications to an existing machine may be costly and limited by the technical limitations of existing machine controllers. Therefore, the software approach becomes a more  Chapter 2. Literature Review  19  feasible tool. In this approach, the N C program is modified or preprocessed based on an error model to achieve the desired machining process performance [41].  For active error compensation, the identification of error and its compensation are simultaneously executed on-line. Most current compensation methods, utilize error maps collected off-line before the machine starts to cut or to inspect a part. When the machine is considered to be highly non-repeatable or the disturbances generated during machining as well as the effects of the machine's time-varying characteristics are significant, then the error mapping during processing become highly desirable.  A real time compensation method for static position errors and thermal errors of a C N C turning center is proposed by Donmez, et al. (1986) [14]. A modular, flexible and structured software system was generated to compensate for the predicted errors in real-time and was implemented in a low-cost, single-board micro-computer. A typical C N C controller calculates the position command signals, compares the command value to the position feedback signal and performs the speed control either by monitoring the velocity feedback or by deriving the feedback signal from a position feedback signal. A real-time compensation system is an attachment to the controller which injects the error compensation signals into the position servo loop as shown in. Based on the current nominal position, the direction of motion, the temperature data and the tool-setting station data, the components of the resultant error vector were calculated and the corresponding servo counts for each axis of the machine were sent to the machine tool controller.  The real time compensation technique was also used by Zhang et al. in 1985 [50]. A software was added as a single subroutine in the control computer for the measuring machine, which, was a small minicomputer. During operation, the subroutine read the nominal machine coordinates, performed a linear interpolation to calculate the expected value for the error terms and calculated the compensated coordinates.  Chen, et al. (1997) proposed computational approaches to compensating errors of the basic motions, such as linear interpolation and helical interpolation taking the non-linearity  Chapter 2. Literature Review  20  of quasi-static errors into account [10]. For linear interpolation, a recursive scheme of compensation is developed and used to iteratively calculate the error-compensated position until the difference between the actual position and the desired position is smaller than a predefined tolerance value.  Based on a general nth order quasi-static error model and the method for estimating the parameters in the model, Kiridena and Ferreira (1994) proposed a method of compensating errors when locating the tool at a point, moving it along a straight line trajectory and along circular arcs [27]. The general approach has been to compensate a trajectory by connecting its end point locations. For straight line trajectories, this involves two points. For circular trajectories, it involves three points because an arc is defined in terms of its end points and the center of the circle which it is a part of. Then the trajectory is divived into sub-segments and end point compensations are applied to this segments. The drawback of this approach is that it results in longer N C programs.  Another method of passive compensation was applied by Okafor and Ertekin in 2000 [37]. Calculated error compensation values were downloaded as error compensation tables to the machine controller to apply software based error compensation. This is a cost-effective and easy way to apply volumetric error compensation. Another similar method was used by L i (2001) which involved in modification of the N C programme for finish turning [34].  Postletwaite and Ford (1997) produced a Windows based software package that uses proven techniques to simulate the effect of axis geometric errors on volumetric accuracy at the University of Huddersfield [39]. The program can calculate the effects of the measured geometric error components throughout the machine volume in order to determine a measure of the volumetric accuracy of the machine. The program is designed to work with standard geometric error files produced by the Renishaw laser measurement system. The logical sequence of operations can be defined such as (1) definition of the machine tool configuration and machine specific details, (2) selection of geometric error components for use in simulation and creating the error map, (3) setting the simulation parameters such as the  Chapter 2. Literature Review  21  step size, direction of travel and length of axis travel. (4) running the volumetric simulation, and (5) analysis and display of the results.  Pi ID ' 3o •in oa.  3  w  calc  a o 4*o • fa ~3  o o  o ID o  O o  &  Figure 2.2: Block diagram for error compensated C N C axis drive  22  Chapter 2. Literature Review  After this work, Fletcher, et al. (1999) introduced an expert system for addressing the machine tool accuracy systems, named as Machine Tool Error Identification and Compensation Advice System (MTEICAS) [20]. This system is not only able to display the measured errors and calculate the volumetric error at a given position, but also capable of estimating the possible errors caused by thermal distortions.  2.4 Summary In this chapter, an outline of the literature in volumetric error modeling, error measurement, and compensation schemes has been presented. Even though a great deal of research on error analysis, error measurement and error compensation has been accompished in the past, there still remains much work to be done to develop a feasible and quick way to improve the machining accuracy of machine tools and coordinate measuring machines, which can also be used in machine shop environment.  Many researchers worked on the modeling of geometric and kinematic errors, which is the very first step in increasing the machining accuracy. The methods, which are capable of handling all 21 errors for a three-axis machine tool including the straightness and squareness errors, improve the machining accuracy. The vector representation method can be regarded as a simplified version of the methodology which uses the homogenous transformation matrix manipulations. The assumptions made in both methods are rigid body kinematics and small angle approximation. In this research, both of the.modeling techniques are explained and applied to F A D A L VMC-2216, which is subject to this thesis.  The results of two  methods, which are the geometric error components in three orthogonal directions, are compared.  '  - . ;  ,  Error measurement is always the most direct and accurate way to determine the errors of a machine. Direct measurements of the errors are usually used to provide the necessary knowledge for error compensation and also the capability of verifying the error analysis system. The laser interferometers are the best devices with high accuracy and resolution for this task. That's why the measurements in this research was done with a Renishaw laser  Chapter 2. Literature Review  23  interferometer system. However, the laser system is always expensive and requires a complex set-up, especially with straightness and squareness optics.  Hardware and software approaches can be applied in order to compensate for the volumetric errors. Hardware approaches are, however, more difficult to implement on existing machine-tools due to technical limitations and warranty problems. A complete software compensation scheme which performs as a preprocessor is an alternative. The compensation scheme uses the uncompensated NC-program to generate a compensated N C program which produces the anticipated higher degree of accuracy as long as the mathematical model and measurements are derived correctly. In this research, the compensation scheme which is involved in NC-tape modification is used. It works as a preprocessor which modifies the uncompensated N C code according to the error values at those locations specified in the NC code.  Although the volumteric model of the machines are studied extensively, their off line automatic correction by modifying the N C program or the visualization of volumetric errors left on the part prior to machining have not been studied. This thesis aims to develop a methodology to enhance the accuracy of three-axis machine tools by developing the mathematical model of the machine which explains the relationship between the individual slide errors and the geometric error components of any point in the workzone of the machine tool. Laser interferometry is the chosen method to measure the individual slide errors. Subsituting the individual slide errors in the mathematical model provides the compensation offsets that should be commanded to the machine in order to get a better workpiece accuracy without any geometric errors. Compensation schemes for linear and circular interpolations are applied in the virtual environment prior to cutting operations on the machine.  Chapter 3 Mathematical Modeling of the Machine Tool  3.1 Introduction This chapter presents two generalized methods to determine the total volumetric error of a machine tool at any position in the workspace in terms of geometric errors of individual machine tool components. The two methods, developed by Schultschik [43] and Donmez [14], are used to derive the mathematical model of an existing three-axis vertical machining center. The objective of this thesis is to machine a part by compensating volumetric errors based on building the mathematical model of the machine. Both of the generalized mathematical models presented in this chapter split the machine tool into its basic components and relate each component to the others. In addition, the models are modular and structured so that they can be easily modified to be applicable to three-axis machine tool structures and coordinate measuring machines with different configurations.  In this chapter, the methods developed by Schultschik and Donmez are applied to F A D A L VMC-2216, a XYFZ-type three-axis machining center, to derive the expressions explaining the relationship between the individual slide errors and the geometric error components that need to be compensated for an enhanced accuracy. Following that, the slide errors.are measured and fed back into the mathematical model to get the geometric error components in X , Y , and Z directions at any position in the machine tool work zone. The calculated geometric error components are compensated; thus the produced workpiece will not be influenced by the geometric errors of the machine tool.  Schultschik developed a  method of vector representation of the machine tool using rigid body dynamics. He models the machine as a closed vector chain to obtain expressions for the volumetric error. In Donmez's  method,  the  mathematical  model is derived by utilizing  homogenous  transformation matrix manipulations to describe the spatial relationships between the machine tool structural elements. Donmez's method gives the tool compensation offsets similar to those in Schultschik's method, but it also gives the orientation of the tool with respect to the workpiece. The two modeling approaches explained above assume that the  Chapter 3. Mathematical Modeling of the Machine Tool  25  errors are so small that small angle assumption can be used and the multiplication of two errors converges to zero. The models are also based on rigid body dynamics, which means every component of the machine tool is assumed to be a rigid link. The principles of the techniques are explained in detail in the following sections of this chapter. Firstly, the machine tool subject to this research is modeled using Donmez's method. Then the mathematical model of the machine tool is derived using Schultschik's method. The tool compensation offsets received from the two modeling approaches are the same for F A D A L VMC-2216. 3.2 Modeling approach - Homogeneous Transformation Matrices  Figure 3.1: Model of a sliding joint (translational axis)  Multi-axis machines are typically composed of a sequence of linkages connected by joints that provide either rotational or translational motion. Using rigid body kinematics, each axis of a machine tool relative to each other and to the reference frame can be modeled using a homogenous transformation matrix.  In Figure 3.1, a machine slide is shown. The desired motion of this slide is only in X direction. For an ideal case, the slide moves in X direction with an amount of desired motion, which is equal to x. However, in reality, the slide does not move in only X direction, but also has angular and translational deviations. Since the motion in X axis is not straight with  26  Chapter 3. Mathematical Modeling of the Machine Tool  respect to Y and Z axes, there exist two straightness errors, called horizontal and vertical straightness errors, respectively. Also, the motion in X axis does not come equal to the desired amount of motion, which adds a translational or linear positioning error in the direction of motion. The slide also has angular errors about three orthogonal axes, X , Y and Z axes, which are named as roll, pitch, and yaw errors respectively.  An ideal slide, which moves along its X axis, can be described by a homogenous transformation matrix in the following form; 1 0 r 1  0 = Ri 0 0  0  x+X  1 0  Y  0  1  z  0  0  1  (3.1)  where X, Y and Z are defined as offsets of the origin of the slide coordinate system (Fi) with respect to the reference coordinate system [FR] as shown in Figure 3.1 and x is the variable position of the slide coordinate system origin with respect to the reference system. The transformation matrix given in Eq. (3.1) is used to transform coordinates of any point in the ideal frame [F;] to the coordinates in reference frame [FR]. x  R  y  =T Ri Z.i 1 _1_ R  1  Frame-1 shown in Figure 3.1 is supposed to be attached to the ideal slide. It moves with the slide and the reference frame stays stationary.  However, in reality in addition to its intended motion, any single degree of freedom slide has an error in six degrees of freedom. Due to unwanted motions, the actual position and also the orientation of the slide are different from those of the ideal slide. The dotted lines of the slide show the actual case in Figure 3.1. The total error matrix consists of three rotational and three translational errors. The homogenous transformation matrix, E, which represents the individual errors can be written as;  Chapter 3. Mathematical Modeling of the Machine Tool 1  dx  y  £  1  z  £  £ x  0  0  ~  27  by  £ x  1  Sz  0  1  (3.3)  where e is rotational error about X axis and defined as roll for motion in X axis, x  £ is rotational error about Y axis and defined as pitch for motion in X axis, y  s is rotational error about Z axis and defined as yaw for motion in X axis, z  5 is the linear positioning error of X axis while the slide is moving in X direction, X  8 is the horizontal straightness error of X axis (Y straightness of X ) , X  8 is the vertical straightness error of X axis (Z straightness of X). X  This error matrix shows the transformation between the two coordinate frames, ideal frame [Fj] and actual frame [F ]. a  l  X  y  t  y  = T.  a  1  I  =>  - E 1  y  (3.4)  a  1  With this general error matrix, the actual position and orientation of the slide in reference coordinate frame is found by the following equation; T 1  (3.5)  =T T Ra Ri - ia 1  where T  Ri  l  is the transformation from the reference to the ideal case,  T = E is the transformation from the ideal case to the actual case or error matrix, ia  T  Ra  is the transformation from the reference to the actual case.  The homogenous transformation matrix representation is used for all slides of the machine tool in the mathematical model. The ideal and actual cases are illustrated in Figure 3.2.  '  Chapter 3. Mathematical Modeling of the Machine Tool  28  Y, a ..• / Actual Carriage  F, Z,j *  Frame Ideal Carriage Frame  Reference Frame  Figure 3.2: Reference, ideal and actual coordinate frames  Since a machine tool consists of several supposedly rigid links connected to each other by sliding joints, it is possible to describe the motion of a slide in a reference frame by assigning a coordinate frame to a slide and using the homogenous transformation matrix. Before •building the mathematical model, the background of homogenous transformations matrices is given in the following section in order to explain how the error matrix is derived.  3.2.1 Homogenous Transformation Matrices - Background In general, a homogenous coordinate representation is defined as the representation of an N-dimensional position vector with (N+l) components. The physical N-dimensional vector is obtained by dividing the homogenous coordinates by the (N+l) coordinate. Thus in threeth  dimensional space, a position vector of a point defined in its coordinates; P = \P,  Py  0.6)  PY t  is represented in the homogenous coordinate representation by the augmented vector: p = kVp  x  W  Py  W  w]  T  Pz  (3.7)  The forth coordinate W is a non-zero scaling factor. The concept of homogenous coordinate representation of points in three-dimensional Euclidean space is useful in developing a matrix transformation including rotation, translation (positioning), scaling, and perspective.  Chapter 3. Mathematical Modeling of the Machine Tool  29  A homogenous transformation matrix in 3-dimensional space is a 4 x 4 matrix that is defined for the purpose of mapping a homogenous position vector from one coordinate system to another. The homogenous transformation matrix can be partitioned into four submatrices R, P, W, and F of dimensions 3 x 3 , 3 x 1 , l x l , and 1 x 3 respectively. It has the form:  T=  R,  rotation matrix perspective transformation  lxl  1*3  ]  position vector scaling factor  (3.8)  The upper left 3 x 3 sub-matrix denotes the orientation of one frame with respect to the other whereas the upper right 3 x 1 sub-matrix denotes the position of the origin of one frame to the other frame.  For kinematics of mechanisms and robot manipulators, the scaling factor is taken to be 1 and the elements of the perspective transformation are three zeros; hence the homogenous transformation matrix is given by: T=  r 0  i p~  0  (3.9)  01 1  The homogenous transformation can be used to explain the geometric relationship between the two coordinate systems Ouvw and Oxyz such that: [P  Py  P  Z  l] =T[p  [Pu  P  P  W  l l ^ ^ k  x  V  T  u  P  P  v  w  Py  P  Z  (3-10)  The inverse homogenous transformation matrix could be computed in the following manner. Let's assume that the inverse of the homogenous transformation matrix has the following form: T~ =  R'  l  0  0  \P'\ 0 11  (3.11)  30  Chapter 3. Mathematical Modeling of the Machine Tool  R' and P' are the unknowns in Eq. (3.11). They should be expressed in terms of known parameters in the homogenous transformation matrix such as R and P.  The product of the homogenous transformation matrix and its inverse gives a 4 x 4 unity matrix: RR'  1  0  0  1 0  0  0 .1 0  0 0" 0  0  0  1 0  0  0  0  1  From Eq. (3.12); flP'+P = [0 0  Oj  (3.13)  P'=-R P T  Besides, from Eq. (3.12), we can write; RR'= I => R'= R~  (3.14)  l  Rotation matrices are orthogonal matrices. A n n x n matrixRis an orthogonal matrix if RR =I  (3.15)  T  where R is the transpose of' R, and / is the identity matrix. In particular, an orthogonal T  matrix is always invertible, and; R~ =R l  \  T  "  -  -•: ,  ;  :.; "  > • (3.16)  Therefore: R = R~ =R l  T  Hence the inverse homogenous transformation matrix becomes:  .  (3.17)  31  Chapter 3. Mathematical Modeling of the Machine Tool  R'  \ - R  T  P  (3.18)  With the homogenous transformation matrix it is possible to describe the relative rotation and translation between any two coordinate frames. Besides, an important feature of the homogenous transformation matrix is that they can be multiplied in series to describe one object with respect to several different Coordinate frames. This feature is very useful to describe structures which consist of several components that are positioned with respect to each other such as machine tools. In robotics applications, this is the common way to describe the end effector with respect to the base of the robot [44]. In machine tool applications, the machine tool is considered as a chain of linkages and a mathematical model of the machine is generated utilizing homogenous transformation matrix manipulations for the spatial relationships between the machine tool structural elements. The methodology used to derive the homogenous transformation matrices is based on the properties of homogenous transformation  matrices  explained in  the  previous  paragraphs.  The  homogenous  transformation matrices are obtained by using kinematic transformations which can be represented by rotation matrices and position vectors.  Rotation Matrices: When pure rotation of one coordinate frame with respect to another is the only consideration, the origins of the coordinate frames may be assumed to be coincident at o, as shown in Figure 3.3. The coordinate system Oxyz is fixed in three-dimensional space and considered to be the reference frame. The coordinate system Ouvw is rotating with respect to that reference frame, Oxyz. Let (i ,j ,k ) x  y  z  and (i ,j ,k ) u  v  w  be the unit vectors  along the coordinate axes of Oxyz and Ouvy systems respectively. A point P in the space can be represented by its coordinates with respect to both coordinate systems. Assuming that the point is fixed with respect to the Ouvw coordinate frame, then; —>  —>  Puvw = Pu K + P v j  P. xyz = Px  h  + Py  v  Jy  + Pw  k  + Pz  k  w  z  (3.19)  Chapter 3. Mathematical Modeling of the Machine Tool  .  .  32  or in homogenous coordinates; p l = k  P  P  Pxn=[Px  Py  Pz  p  uvw  v  and p  I]'  w  (3-20)  Y  l  represent the same point P in the space with reference to the Ouvw and  m  Oxyz coordinate systems respectively. The transformation matrix R will transform the coordinates of p  uvw  to the coordinates of p  in such a way that;  Pxyz = Pu  (3-21)  R  VW  The projections of the vector oPalong the axes ox, oy, oz are p ,p , x  y  and  p  z  respectively. Px x-P=  hiPu K + Pv Jv+ PwK)=  =i  =  Px  cos a .  x Pv  where a ,fi ,y x  V  x  x Pw  are the angular rotations between the unit vector i and each of the unit x  vectors i , j , and k respectively (See Figure 3.4). u  v  (3.22)  +cos/3 . +cosy .  x Pu  x  h •»'„ -Pu + -h J Pv + h -K P«  w  Figure 3.3: Coordinate systems  33  Chapter 3. Mathematical Modeling of the Machine Tool By applying the same transformation (p = j .p and p = k .p ), y  Pz.  i .i X u j .i Jy u _k -i  Px  cos a  cos /3  cos y  x  Pu  Py - cos a cos a .Pz.  y  cos P  cos y  y  Pv  z  cos /3  cos y  z  -P».  ~Px' Py  -  z  y  z  z  i.j i .k ~P ~ x Jv X w j .i i .k P JyJv Jy w k .j k .k _ -P». U  (3.23)  v  u  z  x  v  z  w  X  y  Z  (3.24)  or \Pxyz\= W\Puv } w  where a ,/5 ,y y  y  y  are angles between the unit vector j and each of the unit vectors i , j y  u  v  and k respectively, w  a ,fo ,y z  z  a  r  angles between the unit vector k and each of the unit vectors i , j  e  z  z  u  v  and k respectively. w  The rotation matrix defines the rotations of the axes of the coordinate system Ouvw about each of the principal axes of the reference coordinate system Oxyz. The following special cases are important for deriving the general rotational error matrix [44].  (a)  (b)  (c)  Figure 3 . 4 : Definition of angles (a)a ,fi ,y , x  x  x  (b) a ,fi ,y , y  y  y  (c)a ,fi ,y z  z  z  34  Chapter 3. Mathematical Modeling of the Machine Tool  (a) The coordinate system Ouvw rotates at an angle a about the ox axis: Pxyz  x,a Puvw  R  i  o  o  0  cos a  - sin a  0  sin a  cos a  (3.25)  (b) The coordinate system Ouvw rotates at an angle /J about the Oy axis: Pxyz  Ry.sPuvw  =  cos ft 0 sin /5 y,P  R  ~  0  1  0  (3.26)  - sin /5 0 cos /?  (c) The coordinate system Ouvw rotates at an angle y about the Oz axis: . Pxyz  ^z,yPuvw  cos y  - sin Y 0  ,r = s i n / 0  cos/ 0  R  0  (3.27)  1  These special cases can also be derived using geometry. As shown in Figure 3.5, when the coordinate system rotates about Ox axis with an amount of a; y w v - —-— + . sin a => y = v cos a - z sin a cos a cos a  (3.28)  z = wcosa + v s i n a  or in matrix form; xl  [ 1 0  y = 0 coscc z  0 sin a  0 -sinor cos a  (3.29)  Chapter 3. Mathematical Modeling of the Machine Tool  35  which is the same rotation matrix derived using the direction cosines given with Eq. (3.25).  The rotations about Y and Z axes, which are given in the previous paragraphs (b) and (c) can also be derived using the same manner.  U  y  Figure 3.5: Rotation about X axis  Position Vectors: Pure Translation along X axis: The position vector for a pure translation along X axis is p - [x 0 x  Of where x is the amount of desired motion.  The homogeneous transformation matrix representing the intended translation, x, along X axis with unity rotation matrix is as follows; "1 0  0  0  1 0  0  0  0  0  JC"  0 1 0  0  (3.30)  1  Pure Translation along Y axis: The position vector for a pure translation along Y axis is P  = v  [0  y  Of where y is the amount of desired motion.  The homogeneous transformation matrix with this position vector is as follows;  Chapter 3. Mathematical Modeling of the Machine Tool 1 0  0  .,  ;  .  36  0  0  1 0  0  0  y 1 0  0  0  0  (3.31)  I  Pure Translation along Z axis: The position vector for a pure translation along Z axis is p - [0 0 z  zf  where z is the amount of desired motion.  The homogeneous transformation matrix with this position vector is as follows; 1 0  0  0  0  1 0  0  0  0  1 z  0  0  0  (3.32)  1  Any translational motion can be expressed as a combination of three pure translational vectors by taking the product of the homogenous transformation matrices.  3.2.2 Homogeneous Coordinate Representation of Machine Slides The generalized mathematical model developed by Donmez [14] splits the machine tool into its basic components such as rigid links connected to each other by sliding joints. In this way, it becomes possible to describe the motion of a slide in a reference frame by assigning a coordinate frame to each slide and using homogenous transformation matrices. The total error matrix for a machine slide, given in Eq. (3.38), consists of three rotational and three translational errors. A machine slide is shown in Figure 3.6. The intended motion is in X direction. The general error matrix can be written as a product of the rotational error matrix and translational error matrix. The rotational error matrix,  TROT, represents  the errors  coming from the angular deviations. Each angular deviation can be represented with a homogenous transformation matrix that is derived in Eq. (3.25), Eq. (3.26) and Eq. (3.27).  Chapter 3. Mathematical Modeling of the Machine Tool  37  Figure 3.6: Linear and angular errors on a sliding joint  TROT ~ Te Tg x  y  T  6  B  6  0  0  sin0  cosd  0  0  0  o  0  1  0  1  0  0  0  1  1  0  0  0"  cos0  0  0  COS0,  -sine?,  0  0  1  0  0  0  sin0,  COS 6^-  0  -sin  0  cos^  0  0  0  1  0  y  6  y  sin  0  y  d  0" cosf?..  - sin  z  Z  z  .  The resulting rotational error matrix becomes:  cosf? cos# t 1  =  ROT  - cos 6 sin 6  7  y  sin 6 sin 6 cos 6 + cos 6 sin 6 X  y  Z  X  Z  - cos 0, sin 6„ cos 6, + sin 6„ sin 6. 0  Z  - sin 0 sin 6 sin 6 + cos d cos 6 X  y  Z  cos 0 sin 0 sin 0 + sin X  Z  x  Z  cos 0  Z  0  sin (9^ -sinf? cos#„ r  x  y  (3.34)  COS0 COS0 r  V  0  Under the assumption of small angles; cosine of an angular error is assumed to be 1 and sine of an angle is equal to its tangent or itself. Besides, the multiplication of two errors is assumed to converge to zero. Therefore, the rotational error matrix becomes;  Chapter 3. Mathematical Modeling of the Machine Tool 1  ~z 1 £  z  £  y  0  - *  0  1  0  0  1  £  £  £ x  0  0  38  (3.35)  where e is rotational error about X axis and defined as roll for motion in X axis, x  8 is rotational error about Y axis and defined as pitch for motion in X axis, y  8 is rotational error about Z axis and defined as yaw for motion in X axis. Z  Besides, the total translational error matrix, T  , consists of three components which are  tram  translations along the three orthogonal axes. The total translational error matrix is derived in a similar way to that used to derive the total rotational error matrix;  1 0 r 1  0 =T T T TRANS XYZ 0 0 1  1  1  0  0 1  1  Sx "1  0  1 0  0  0  0  1  0  0  0  0  1  0  0  0" "1  0  0  0  1 0  Sy  0  1 0  0  0  1  0  0  0  1 &  0  0  1  0  0  0  0  1  0 dx 1 0 ^  TRANS 0  0  1  &  0  0  0  1  (3.36)  where S is the linear positioning error of X axis while the slide is moving in X direction x  8 is the horizontal straightness error of X axis (Y straightness of X) X  8 is the vertical straightness error of X axis (Z straightness of X ) X  The total error matrix E representing the errors of a slide (neglecting the second-order terms) becomes; E = iT .iT i^ R 0 T  TRANS  39  Chapter 3. Mathematical Modeling of the Machine Tool ~  ~z £  E =  y  1  z  - e.  0  £ x  Sx  $  x  I  &  0  o  1  £  (3.37)  Each individual error term in the error matrix for X slide is a function of the position on the X axis. Hence, Eq. (3.37) can be re-written as; 1 E =  -sAx)  £ (x) v  -£ (x)  5y(x)  £ (x) x  1  Sz(x)  0  0  1  x  -£ (x) y  0  Sx(x)  (3.38)  With this general error matrix, the actual position and orientation of the slide in reference coordinate frame can be found. The homogenous transformation representation is used for all slides of the machine tool to build the general mathematical model of the machine tool.  3.2.3 Homogeneous Coordinate Representation of Rotational Axes The axes not only provide linear motions, but also rotational motions in a five-axis machine tool. Generally, three axes provide three linear motions in three orthogonal directions and the other two axes provide two rotational motions.  It is not possible to use the same transformation matrix derived for linear slides in modeling the motion of a rotational axis. For rotational axes, it cannot be assumed that all the angles are small enough to neglect their multiplications because the intended motion is rotation about one axis. Therefore, the rotation which is the intended motion around the axis cannot be assumed to be small, hence it needs to be considered. For example, if the rotation axis is around Z direction, i.e. the C axis in machine tools, then 6 is not small enough to Z  make a small angle assumption (See Figure 3.7). The model is simplified by assuming that the angular deviations about the axes other than the axis of rotation are small. After the simplification, the transformation matrix representing the rotational axes can be represented in the following equation;  Chapter 3. Mathematical Modeling of the Machine Tool  E=  40  P  Sx  -a 1  oy  0  1  (3.39)  Sz  where y(6 ) is the error about the rotation axis, Z in the Figure 3.7, z  Ax, A8y are the radial errors along X and Y axes respectively, Az is the axial error along the rotation axis, 6 , 6 are tilt errors about X and Y axes respectively. X  y  There are three translational motions which are orthogonal to each other in the three axis machine tools. Therefore, the homogenous coordinate representation of rotational axes is not used in this thesis. This type of transformation is crucial for a five axis machine tool which is not available in our laboratory.  Z  •Y~  xr  Figure 3.7: Model of a rotational axis  2  Chapter 3. Mathematical Modeling of the Machine Tool  41  3.2.4 The Mathematical Model for Machine Tools The total error matrices for a machine slide and a rotational axis are derived in the previous sections. This section presents a mathematical model of the complete machine which leads to the evaluation of errors between the tool tip and slide positions when they move simultaneously. As mentioned earlier in the chapter, a machine tool can be considered as a chain of linkages. Denavit-Hartenberg's approach of describing the spatial geometry of linkages with respect to a reference frame by matrix multiplications [13], which has roots in the literature of robotics field, can be used to determine the spatial relationship between the cutting tool and the workpiece.  Generally, three-axis machine tools and C M M s can be classified into four groups. These groups are called F X Y Z , X F Y Z , X Y F Z , and X Y Z F . In the name of each group, the letters before F show available directions of motion of the work piece with respect to the reference frame, and the letters after F show the available directions of motion of the tool or probe with respect to the same frame. For example, in group X Y Z F , the tool is fixed, whereas in the group F X Y Z , the workpiece is fixed.  (a) XYFZ  (b) XFYZ  (c) FXYZ  Figure 3 . 8 : Common configurations of three-axis machining centers. The letter(s) before F shows the possible motion of the workpiece whereas the letters after F shows the possible motions of the tool  Chapter 3. Mathematical Modeling of the Machine Tool  42  Hence, the mathematical model is not the same for all three-axis machining centers because it also depends on the configuration of the machine tool. Figure 3.8 depicts some configurations that are common in three-axis machine tools.  Fadal VMC-2216 is an X Y F Z type machine, which means the table has motions in X and Y axes with respect to a fixed frame and the tool has motion only in Z direction with respect to the same fixed frame. A F A D A L Vertical Machining Center is shown in Figure 3.9.  Since this type of machine does not have any rotational axes, the transformation matrix derived in Eq. (3.39) for the rotational axes is not used. Only the homogenous transformation representation derived for linear sliding joints (Eq. 3.37) is needed to build the mathematical model. Three slides of the machine tool shall be modeled using the same principle.  The procedure for developing the mathematical model is as follows: First of all, one coordinate frame is assigned to each slide. Then, the transformations for the ideal case are considered. Considering 6 degrees of freedom and unwanted six motions for each axis, the error matrices are derived. The product of the ideal transformation and error matrix gives the actual transformation. Hence, the transformations from the reference frame to actual cases are derived.  Figure 3.9: F A D A L VMC-2216 3-axis milling machine  Chapter 3. Mathematical Modeling of the Machine Tool  43  Machine structures can be decomposed into a series of coordinate transformation matrices using the rigid motion assumption. The matrices describe the relative position of each axis and any intermediate coordinate frames that may assist in the modeling process. If N rigid bodies are connected in series and the corresponding homogenous transformation matrices between connecting axes are known, then the position of the tip ( N axis) in terms th  of the reference coordinate system is the sequential product of all of the homogenous transformation matrices: N'T 0 1  N 1 I"  mrji  \rri'2rp'irpArji Nrji I 2 3 •••N-l  = 1 1 m-l =0 1  1  1  1  l  (3.40)  1  m=l  By multiplying homogenous transformation matrices corresponding to a series of elements such as saddle, table, cutting tool, and spindle, one can describe the tip of the cutting tool and the point on the workpiece which has to be in contact with the cutting tool. Ideally, the resultant homogenous transformation matrices TWORK  TTOOL  for the cutting tool and  for the point on the workpiece are identical. However, due to errors involved in  machining operations, two matrices describe the two separate coordinate systems. This causes an erroneous cutting operation and degrades the accuracy of the workpiece. The general error matrix can be represented by the following equation: (3-41)  TTOOL ~T ORKE W  Pulling the general error matrix from Eq. (3.41), the general error matrix becomes;  E  =  (3-42)  TWORRTTOOL  Thus the position vector of the resultant error matrix gives the actual coordinates of the tool tip in the table frame.  -  ••'  •'  For the case of a three axis machining center which is X Y F Z type, the coordinate frames should be assigned first such that the transformations can be kept to their simplest forms. The structure of the machine consists of a tool, a spindle, a base, a saddle, and a table. The spindle is connected to the column of the machine with a prismatic joint and the base is  Chapter 3. Mathematical Modeling of the Machine Tool  44  rigidly connected to the column as well. The saddle is connected to the base and the table is connected to the saddle with prismatic joints. Finally, the cutting tool is rigidly connected to the spindle. Figure 3.10 shows the locations of the coordinate frames, which are assigned to each element of the machine-tool-workpiece system from two views (top and left). In this figure, the reference frame is chosen to coincide with the saddle's frame at the position where Y axis is equal to machine zero at the start position. ^T is the transformation between the reference frame and the saddle frame. T is the transformation between the saddle and the 2  {  table. T is the transformation between the spindle and the reference frame whereas \T is the 3  0  transformation to go from the spindle's frame to the tool's frame. This transformation shows only the tool offsets. We again assume that any rotational or linear errors of the tool do not exist. The ideal homogenous transformation matrices for these elements are given in the following:  For the saddle the only degree of freedom is in the Y direction; 1 0  0  0  0  1 0  0  0  1  y 0  0  0  0  I  (3.43)  where y is the motion along the Y axis (variable).  For the table, the only degree of freedom is in the X direction; 1 0  0  X  0  1 0  0  0  0  1  z,  0  0  0  1  where x is the intended motion along the X axis, Z/ is a constant offset.  For the spindle, the only degree of freedom is in Z direction;  (3.44)  Chapter 3. Mathematical Modeling of the Machine Tool  0  0  10  0  10  0 T  =  45  . (3.45)  0 0 1 z+Z  2  0 0 0  1  where z is the intended motion in Z direction, . Z2 is a constant.  The homogenous transformation matrices representing the actual position and orientation of each element can be obtained using the error matrices derived in the previous section.  For the saddle, by applying the transformation matrix;  0 0  10  0  1 0  0 0  1  y  £ (y)  10  -£ iy)  1 -e,{y)  0  e (y) x  0  y  S (y)  -e (y)  s (y)  x  £ (y)  1  x  0  z  T =  1  y  -£ (y)  £ (y)  z  z  0 0 0 1  1  -£ (y)  0  0  S (y)  -e (y)  y + S (y)  x  1 0  y  s (y) t  1  £ (y) y  x  x  y  S (y)  (3.46)  z  1  where e (y) is the pitch of the saddle (Y) motion, x  e (y) is the roll of the saddle (Y) motion, y  £ (y)is the yaw of the saddle (Y) motion, z  S (y) is the horizontal straightness of the saddle (Y) motion (X straightness of Y), x  S (y) is the linear positioning error of the saddle (Y) motion, y  S (y)is z  the vertical straightness of the saddle (Y) motion (Z straightness of Y),  y is the intended motion of the saddle.  For the table which has the intended motion in X direction;  Chapter 3. Mathematical Modeling of the Machine Tool x  1  •eAx)  eAx)  0  eAx)  1  - £ (J)  1 z,  •e(x)  eAx)  1  0  0  0  1  0  0  0  1 0  0  0  0  0 0 1  1 -e (x) z  S (x)-x.a  1  •e (x)  •e (x) y 0  e (x) x  1  0  0  v  1  x + S (x)  eAx)  (x)-x.a XY S (x) z  e (x) y  S  x  x  y  x  (3.47)  Z S (x) l+  z  1  where e (x) is the roll of the table (X) motion, x  e (x) is the pitch of the table (X) motion, y  £ (x) is the yaw of the table (X) motion, z  S (x) is the linear positioning error of the table (X) motion, x  S (x) is the horizontal straightness of the table (X) motion (Y straightness of X), y  6 (x) is the vertical straightness of the table (X) motion (Z straightness of X), Z  a  is the squareness error between X and Y axes,  XY  x is the intended motion of the table, and  , .. -  Zy is the constant offset between the reference frame and the table's frame.  For the spindle, "10 T  =  0  0  0  1 0  0  0  0  0  0 0 1  T  =  £ (z) z  -eAz)  eJz)  s (z)  1  -eAz)  -£y(z)  e (z)  1  0  0  z  1 z+Z  2  0  1 -e (z) 1 t  x  £ (z)  d (z)-z.a  -e (z)  S (z)-z.a  y  x  x  y  YZ  1  z + Z +S (z)  0  0  1  2  z  where £ (z) is the pitch of the spindle (Z) motion, x  £ (z) is the yaw of the spindle (Z) motion, y  S (z)-z.a XZ d(z)-z.a. YZ SAz) x  xz  e (z) x  0  1  (3.48)  Chapter 3. Mathematical Modeling of the Machine Tool  47  £ (z) is the roll of the spindle (Z) motion, z  S (z) is the horizontal straightness of the spindle (Z) motion (X straightness of Z), x  S (z) is the vertical straightness of the spindle (Z) motion (Y straightness of Z), y  S (z) is the linear positioning error of the spindle (Z) motion, z  a  is the squareness error between X and Z axes,  xz  a  is the squareness error between Y and Z axes,  n  z is the intended motion of the spindle, Z2 is the constant offset between the reference frame and the spindle's frame.  For the tool, "10  0  0  1 0  Y,  0  0  1  0  0  0  z, 1  (3.49)  where X , Y and Z, are tool offsets in X , Y and Z directions respectively. t  t  The elements of the matrices may be mathematical functions of the position on the axis and the entire representation is called the form shaping function for the machine. Representing the errors as mathematical functions of the position allows a closed form mathematical representation of the machine tool. The homogenous transformation matrix representation method described here can readily accommodate any number of coordinate frames.  Using the transformation matrices given in Eq. (3.48) and Eq. (3.49), the cutting tool with respect to the reference frame can be represented by the following matrix: =lT*T  TOOL  ' (3.50)  Similarly, the workpiece with respect to the reference frame can be represented by: T  = T^T l  WORK 0 l  1  1  x  (3 511 v.-'—'V  Chapter 3. Mathematical Modeling of the Machine Tool Ideally,  T  WORK  and  T  49  should give the same location in the work volume with respect  T 0 0 L  to the reference coordinate frame. Using the Eq. (3.51), the components of TWORK are given as follows:  l-£ (y)e (x)-e (y)e (x) z  z  y  £ (y) + £ (x) + z  WORK  L  -e (y) + y  -£ {x)-e (y)  y  z  £ (y)£ (x)  z  x  -£ (y)£ (x)  y  z  e (y)e (x)-e (x) x  z  +  z  +  z  £ (y)£ W  y  y  +  z  0  £ (y)£ (x) y  x  l-£ (y)£ (x) x  x  Ay) + £A )  £  x  0 £ (x) + £ (y)£ (x) y  z  x  + £ (y) y  £ {y)£ {x)-£ (x)-£ (y) z  y  x  y  y  x  x  0  Pjw (x)  P-rwiy)  x  -£ (y)£ (x)-£ (y)£ (x)  (3.52)  +l  PTW(Z)  1  T (tool)  Figure 3.11: Sketch of a 3-axis X Y F Z  Figure 3.12: Mathematical Model of a 3-  type machine tool  axis X Y F Z type machine tool  Chapter 3. Mathematical Modeling of the Machine Tool P (x) = x + d (x) -e (y)(S (x) m  x  z  50  - a x) + £ ( y ) ( Z + C5;(J0) - S (y)-  y  XY  y  1  (3.53)  x  Pm (y) = (y)(* + t (*)) + S (x)-xa -  e (y)(Z, + 6 (xj+ y + 6 (y)  Pm (z) = -£ (y)(* + & (*)) + £ (y)(-a x  + 6 (x)) + Z +S (x)  £  l  x  y  y  x  xr  x  x  XY  Z  y  y  l  + S (y)  z  z  (3.54) (3.55)  From Eq. (3.50), 1  -e (z)  x,-£ (z)y,+s (z)z,+S (z)-a z  e <z)  t  z  y  e (z) t  z  TOOL  l  (z)  1  0  0  £  -£y(Z)  x  0  y  x  xz  (z)x +y,  £  -e (z)z,+S (z)-a z  t  x  y  YZ  - e (z)x, + e (z)y, +z,+z + Z +S y  x  2  z  (z)  (3.56)  1  Eq. (3.42) gives the relation between the tool and work transformations to get the error matrix. This method assumes that the errors are very small and the multiplication of any two errors converges to zero, hence it is negligible. Under these assumptions, the simplified components of the position vector and rotation matrix of the general error matrix are given as follows; R  i WORK  1  TOOL  1  0  E  P  E  0 0  1  (3.57)  P (x) E  (y)  p  P. =  (3.58)  E  P (z) 1 E  P (x) = [-x -d (x)-6 (y) E  x  + 6 (z) + x,+y (£ (x)  x  x  l  + s (y)-e  z  z  z  (z))+  z, (- e (x) - e (y) + £ (z))- (z + Z ie (x) + e ( y ) ) - y{s (x) + e (y)) y  -a z  y  y  2  y  z  z  (3-59)  + Z e (x)]  xz  l  y  (y) = [-y-S (y)-S (x)  p  y  E  y  + 6 (z) + y,+x (-e (x)-£ (y)  y  y  + x£ (x) + z, (e (x) + £ (y)-£ z  x  x  x  t  z  + e (z))  z  z  (z))+ (z + Z \e (x) + e (yj)-Z e 2  x  x  x  x  (x)  (3.60)  51  Chapter 3. Mathematical Modeling of the Machine Tool P (z) = [z + 6 (z) - <5 (*) - <5 (y) + y(£, (y) + e (x))-xe (x) + z,-Z +Z Z  E  Z  x,{e (y) + e (x)-e  Z  x  (z))+ y,(-e (y)-e (x) x  y  x  +  2  (3.61)  + e (z))  x  x  The general rotational error matrix is; 1 R* =\-£ (y)-£ (x) z  y  z  + £ (x) + £ (y) z  + £ (z)  z  -£ (z)  -£ (z)  1  z  + £ (x) + £ (y) y  y  £ {z)-£ {x)-£ (y)  z  y  -£ (z)  £ (z)-£ (x)-£ (y) x  x  x  x  y  y  + £ (x) + £ (y) (3.62) x  x  1  The components of the position vector of the error matrix give the amount of motions that are essential for the actual transformation. They are the actual magnitude of motion that the tool moves when it is commanded to move to (x,y,z) coordinates. The compensation offsets can be calculated using the position vector.  The components,of the rotation matrix and position vector before the simplifications are given in Appendix A .  3.3 Modeling Approach -Vector Representation Method 3.3.1 Background of Vector Representation Method The geometric error components are derived using the homogenous transformation matrices given in the previous section which used the method developed by Donmez et al. [14]. A different method, which is like a simplified version of Donmez's method, is also explained and applied to F A D A L VMC-2216 to compare whether the same error components are derived. The only advantage of the vector representation method over Donmez's approach is the simplicity of the resulting equations. Thus, it is more efficient to use this method in terms of the computational time and effort to simplify the resulting equations. The disadvantage of this method is that it does not give any information about the orientation of the tool. The orientation of the tool tip is not taken into consideration because there is no possible way to  Chapter 3. Mathematical Modeling of the Machine Tool  52  compensate for the orientation of the tool tip on a three-axis machine tool controller. However, the vector representation is able to calculate the error components in three directions for a three-axis machine tool as derived in Donmez's method.  The model developed by Schultschik [43] does not include any straightness or squareness errors of the slides. A model without straightness and squareness errors is not complete. However, these errors can easily be added to the model as explained in the following sections.  Schultschik's model is designed to compensate for the systematic geometric errors in the machine and the geometric errors are pre-measured and stored for off-line correction. Also, since it is necessary to keep the machine model as simple as possible, the rigid-body assumption is used to compute the machine geometry.  In the vector representation method the aim is to represent the position of the tool tip in the reference coordinate system in two different approaches. The tool tip is assumed to be a point in space and it is located with a 3 x 1 position vector in the reference frame. The position vector can be written either with respect to the spindle coordinate frame or with respect to the table coordinate frame. The resulting equations from the two relations help to derive the compensation offsets.  As shown in Figure 3.13, if body j has been assigned a coordinate system with origin Oj and moves with respect to another body i, which has been assigned a coordinate system with origin Ou then point K on body j can be represented by the vector equation;  -»  where p  :j  -»  .  ->  is the vector 0 0 • . Similarly p {  ik  -»  is the vector O O . Rtj is a matrix which t  k  defines the rotation of the coordinate system with origin Oj with respect to the coordinate system with origin 0,. For example, if the rotation is specified first about the Z axis, followed  Chapter 3. Mathematical Modeling of the Machine Tool  53  by the Y axis and X axis of the coordinate system at 0 „ then the rotation matrix can be expressed as;  ij  R  *j  R  (3.64)  - xij yij uj R  =  R  R  1  0  0  cos r..  0  sin r  cos r  xij  xij  cos r  0  sinr^  1  0  0  - sm r  (3.66)  cos r.yj  cosr  -sm r^  sin r. 0  cos r. UJ 0  zij  (3.65)  - sin r  A.IJ  0' 0  (3.67)  1  where r , r and r are rotation angles about X , Y and Z axes respectively. In machine xjj  xij  xij  tools, the rotation angles can be assumed to be small and the cosine and sine terms can be approximated with only the lowest terms. Using this approximation the rotation matrix given in Eq. (3.64) becomes; 1  — Z'J r..  ryj.  rZ'J  1  — XI] r..  CXIJ  1  — r .. yu  (3.68)  The error associated with this approximation is on the order of the neglected 2" order terms. K  Figure 3.13: General transform of a vector  Chapter 3. Mathematical Modeling of the Machine Tool  54  3.3.2 Vector Representation of FADAL VMC-2216 The first step is to locate the coordinate systems on the machine tool. The model requires five independent coordinate systems. They are the X , Y , and Z carriage systems, tool system and a reference stationary system. In order to compare the results with Donmez's method, the locations of the coordinate frames are chosen in the same manner as shown in Figure 3.10 where X axis is the starting axis for squareness compensation. Vector representation method of analysis is based on an equivalent diagram in which every structural and variable magnitude is depicted as a vector, either in a workpiece chain or in a tool chain. The following statement is true for the unloaded faultless machine; Z+T = Y+X+P  (3.69)  where X,Y,Z are the vectors representing the translations of X , Y , and Z , —»  T is the vector representing the tool offsets, —»  P is the vector representing the ideal motion of the tool tip with respect to workpiece in the workpiece coordinate system. When the table moves a nominal distance, y, the ideal position of the Y-carriage in the reference system is given by the vector: "0" Y=\y  (3.70) 0  The displacement errors occur in the motion of Y-carriage. The actual position of the Y carriage in the reference system is given by the sum of the displacement error vector and ideal Y-carriage vector;  Ey = S (y) y  Y =Y E 0  i+  Y  = y + $ (y) y  S (y) z  (3.71)  Chapter  3. Mathematical  Modeling  of the Machine  55  Tool  where d (y) is the horizontal straightness of motion in Y direction, x  S (y) is the vertical straightness of motion in Y direction, z  S (y) is the linear positioning accuracy of motion in Y direction. y  .Z  T  0  T  ;z  v  x  ZR  'OR  yR  -yy  Figure 3.14: Sketch of a X Y F Z type 3-  Figure 3.15: Coordinate systems on  axis machine tool  FADALVMC-2216  Similarly, when the X-carriage moves a nominal distance, x, and the Z-carriage moves a nominal distance, z, we have two additional ideal vectors; x  (3.72)  X, = 0  0 Z. =  (3.73)  0 z + Z  2  where Z i and Z2 are constant offsets along Z-direction.  Chapter 3. Mathematical  Modeling of the Machine  56  Tool  In an ideal case, P can be calculated using the ideal transformations and the following results for the components of P vector are derived: Z,. + ? = r;+X,.+ ^ 0 0  -  + y,  z+ Z  z,  2  P (x) = x i  0  X  P(x)  y +  0  + P(y)  0  (3.74)  P{z)  -x  t  P (y) = y,-y  (3.75)  i  P,'z) = z + z,+Z -Z 2  l  However, due to displacement and squareness errors, the actual positions of the X and Z carriages in the reference system are defined by the following vectors; S (x)  X y  z  y  z  -  0  -  ~S (z)-za ~ x  +  0  8 {£)-za y  n  S (y)  2  t  x  y  y  S (x),d (y),S (z) x  y  Z S (x) l+  _  z  d (z)-za <z x  /  (3.77)  S (z)-za y  _z +  Z +S (y)_ 2  z  are squareness errors,  ZY  ^ {y)^ {z),6 (x),6 (z),6 (x),5 (y)are x  -  (3.76)  XY  '  xz  _z + Z _  ,a  = 8 {x)-xa  XY  S (x)  _ i.  XY  x  + 5 (x)-xa  -- 0  where a ,  x + S (x)  x  z  z  z  straightness errors,  are linear displacement errors, and  x, y, z are nominal values of the carriage positions specified by the controller.  The whole Y carriage system has rotational deviations with respect to the reference system due to angular error motions. This rotation can be expressed by the infinitesimal rotation matrix:  Chapter 3. Mathematical Modeling of the Machine Tool  1  ~£ (y)  £ (y)  z  £ (y)  R(y) =  y  1  z  57  -£ (y) x  (3.78)  Similarly, the X-carriage and Z-carriage have rotational errors; 1  £ (X)  £ {x)  1  -e 'x)  •£ {x)  £ (x)  1  1  -£ (Z)  Z  £fx)  R(x) =  y  x  x  y  1  R(z) =  x  y  x (y)>  6  £ (x) z  e  y  X  (3.80)  1  x  (y)' z (y)  £  -£ (Z)  £ (z)  -£y(z)  (3.79)  £ {z)  Z  where e (x), £ {x),  y  are roll, pitch and yaw of X-carriage respectively, a r e  p i » roll and yaw of Y-carriage respectively, tcn  £ (z), £ (z), £ (z) are pitch, yaw and roll of Z-carriage respectively. x  y  z  The tool vector only consists of tool offsets. The tool compensation, which can generally be done by machine tool controllers, should be off when using the error compensation offsets derived using the mathematical modeling. The tool compensation is only the offsets of the tool in three axes and it does not take the angular orientation of the tool into account. The tool offsets are affected by the angular errors of Z-carriage, therefore the angular orientation of the tool should be taken into account: <  x  (3.81)  T = y, Z,\  i  where x , y , z are tool offsets in X , Y , and Z directions, respectively. t  t  t  Chapter 3. Mathematical Modeling of the Machine Tool  58  According to Eq. (3.63), the tool tip in the reference coordinate system can be written using two approaches; one from the spindle coordinate system (tool chain) and another using the saddle and table coordinate systems (workpiece chain).  From the spindle coordinate system: —»  —>  PRT=  PR  —»  (3-82)  RZPZT  +  R  Z  From the saddle and table coordinate systems; —>  — >  —>  PRT=  PRy+ Ry  ( -83) 3  PyT  R  (3-84)  PyT^Py + y P T R  X  X  X  — >  When p in Eq. (3.82) is substituted in Eq. (3.81), then; T  PRT  ~ PRy  + R (p +R ) Ry  yx  (3.85)  yxPxT  By equating Eq. (3.83) to Eq. (3.80): - »  — >  —>  —»  —>  PRT = PRy + Ry(Py y P T) R  +R  X  —»  X  ( -86) 3  = PRZ+ RZPZT R  X  — » — » - * — >  Knowing that p -Z,p =Y, Rz  —>  p  —>  Ry  yx  —»  —»  = X , p =T, zT  the vector representation of F A D A L  VMC-2216 can be written as: Y+R(y)(X+R(x)P)  = Z+R(z)T  (3.87)  — >  The vector  P , representing the actual motion of the tool tip with respect to the  workpiece coordinate system, can be expressed as: P = R- (x)R~ ( y ) ( Z - Y) - R- (x)X+ TT (x)R- (y)R(z)T 1  l  1  1  ]  (3.88)  Chapter 3. Mathematical Modeling of the Machine Tool  59  When the rotation matrices and vectors developed from Eq. (3.71) to Eq. (3.79) are substituted into Eq. (3.86), the resultant P vector is obtained as detailed in Appendix-A. Under small angle assumption, the components of the P vector are simplified as follows:  P (x) = [~x-6 (x)-d (y) a  x  + 6 (z) + x,+y,(£  x  x  z, (- e (x) - e (y) + e ( z ) ) - (z + Z y  -a z  y  y  (z))+  (x) + £ (y)-£  z  z  z  (x) + £ (y))- y(£ (x) + e (y))  2  y  z  (3.89)  z  + Z e (x)]  xz  l  y  P (y) = [-y-S (y)-S (x) a  y  + S (z) + y,+x {-e (x)-£ (y)  y  y  l  + x£ (x) + z, (e (x) + e (y) - e (z))+ (z + Z z  x  x  P (z) = [z + S (z)S a  z  x  + £ (z))  z  z  (*).+ e (y))-Z e  2  x  x  (x) -S (y) + y(£ (y) + £ (x))-x£  z  z  x  x, (e (y) + £ (x) - e (z))+ y, (- £ (y)y  z  y  y  x  (x)  x  (x) + z,-Z +Z  y  1  2  (3.90)  +  e (x)+e (z))  x  x  x  Hence, the difference between the ideal and actual P vector components gives the compensation offsets for each direction: AP(x) = P (x)-P (x) a  i  AP(x) = [S (x) -S (y) + S (z) + y, (e (x) + £ (y) -e (z)) x  x  + z, (-S (x) -£ (y) y  -a z xz  x  z  z  z  + £ (z))- (z + Z %£ (x) + £ (y))-  y  y  2  y  y(£ (x) + £ (y))  y  z  z  + Z e (x)] l  y  AP(y) =  P (y)-P (y) a  AP(y) = [S  i  (y)-S  y  y  (x) + 5 (z) + x, (- e (x) -£ (y) y  z,{E (x) + £ (y)-e (z))+ x  x  AP(z) =  z  (z + Z \e (x)  x  2  x  + £ (z))+ xe (x) +  z  z  + e (y))~Z e (x) x  x  x  + xa  XY  (3.93)  - za^]  P (z)- (z) p  a  i  AP(z) = [S (z) - S (x) -S (y) + y{e (y) + e (x))-xe (x) z  z  z  x  x  y  + x, [e (y) + e (x) - e (z))+ y, (- e (y)- £ (x) + £ (z)) y  t  y  y  x  x  x  (3.94)  Chapter 3. Mathematical  Modeling of the Machine Tool  60  The results derived using the vector representation method are consistent with the results derived using the homogenous transformation matrix approach.  3.4 Summary In this chapter, the mathematical model of an X Y F Z type three-axis machine tool is derived in two ways and the results of two modeling techniques are exactly the same. The inputs to the mathematical model are the individual error components of three slides. Therefore, the next step is to measure the angular (pitch and yaw) and linear error components using laser interferometry. The other errors of F A D A L VMC-2216 are not measured due to lack of equipment. The measurement results are fed back into the mathematical model, which can be expressed by Eqs. (3.91), (3.92), and (3.93). Since the individual error components are measured along the complete travel length of each axis, the geometric error mapping of the machine tool can be derived.  (pt  Chapter 4 Error Measurements by Laser Interferometry  4.1 Overview  The methodology of axis error prediction and its mapping on machine tool coordinates presented in Chapter 3 is applied to the measurements of an experimental vertical machine tool in this chapter. The laser interferometer based axis error measurement procedures and their modeling to predict the total positioning error within the work volume of the machine tools are presented.  4.2 L a s e r Interferometers and T h e i r W o r k i n g Principle  Laser interferometer position measurement systems provide accurate displacement information for dimensional measurements or motion control for many applications. Since the 1950s, laser interferometers are used for high precision measurements of distances from nanometers up to 80 m, but also for measuring angles, flatness, straightness, and squareness.  The laser interferometer, the most widely used instrument for machine tool calibration because of its high accuracy and resolution, is based on Michelson's interferometer developed in the 1880s. In the basic Michelson interferometer a monochromatic light is directed at a half mirror, which passes half the beam to a movable mirror, while the rest of the light is reflected to a stationary mirror. The reflections from the fixed and movable mirrors are recombined at the beam splitter, where their interaction is observed. Although today's laser interferometers are more sophisticated and more accurate, the underlying principle is the same as Michelson's interferometer. However, several improvements converted the basic Michelson interferometer into a reliable position measuring device. First, the source is selected as the laser light because of its long range due to its high purity of light. This property of the laser makes the measurements highly accurate. Secondly, the flat mirrors are replaced with cube-corner retro-reflectors which reflect the incident light parallel to the incoming direction regardless of the angle of incidence. Flat mirrors reflect the light in the  Chapter 4. Error Measurements by Laser Interferometry  62  same direction of the incoming beam. The replacement of the flat mirrors reduces the difficulty level of alignment in the measurements. Finally, photocells are used in the laser head to convert the monitored fringes into voltage pulses that are processed to provide both the amount and direction of displacement changes [31]. Detailed information on the working principles of laser interferometers is given in Appendix B .  4.2.1 General Set-up for Measuring Linear Positioning Errors Linear positioning or linear displacement error, is defined as the translational error movement of a machine element along its axis of motion. In general, this type of error is caused by the geometric inaccuracies of the drive mechanism and the feedback unit. In the case of ball screw driven slides, linear displacement errors are caused by erroneous pitch of the ball screw, misalignment between its axis of rotation and its centerline, irregularities in its geometry, defective coupling between the feedback unit (a rotary encoder in F A D A L Vertical Machining Center), and the ball screw.  To set up for a linear measurement, one of the linear reflectors should be attached to the beam-splitter to form the reference path for the laser beam. This combination element is called the linear interferometer. The linear Interferometer is positioned in the beam path between the laser head and the linear reflector, as shown in Figure 4.1. In general, the laser interferometer is kept stationary, whereas the measurement retro-reflector is located on the moving part of the machine tool to avoid the vibrations during the beam splitting. A n :  example on the F A D A L Vertical Machine Tool is given in Figure 4.2. The figure depicts a set-up for linear positioning errors of the X axis.  The working principle of the laser system for linear measurements can be explained as follows: First, the beam from the laser head enters the linear interferometer, where it is split into two beams. One of the beams is directed to the reflector, which is attached to the beamsplitter, forming the reference beam, while the second beam passes through the beam-splitter to the second reflector mounted on the moving part of the machine, to form the measurement beam. Both beams are then reflected back from the reflectors to the beam-splitter where they are re-combined and directed back to the laser head where the interference between the two  Chapter 4. Error Measurements by Laser Interferometry  63  beams is monitored by a detector and a differential positional measurement between the two optical components is produced by monitoring the change in optical path difference between the measurement and reference beams [42]. The working principle is very similar to Michelson's interferometer. LINEAR INTERFEROMETER LASER HEAD  LASER BEAM-SPLITTER  LINEAR REFLECTOR  Figure 4.1: General set-up for linear measurements  Figure 4.2: The set-up for linear positioning measurement on the X axis of F A D A L  4.2.2 General Set-up for Measuring Angular Errors Angular errors are caused by geometric inaccuracies of the slideways and the misalignment in the assembly of structural elements of the machine tool. The three angular errors, which are about the three orthogonal axes of the machine slide, are defined as roll,  Chapter 4. Error Measurements by Laser Interferometry  64  pitch, and yaw (See Figure 4.3). Yaw error is the rotational error of the slide around the axis perpendicular to the plane in which the axis of motion lies. Roll error is the rotational error of the slide around the axis of motion, and the pitch error is the angular error of the slide about the axis which is in the plane of motion and perpendicular to the axis of motion. Z  Direction of motion Figure 4.3: Angular errors for motion in X axis  The rotational errors can be best defined using the notation for rotational axes in machine tools. The rotational axis which is about X axis is known as A whereas the rotational axis about Y and Z are known as B and C axes of machine tools, respectively. The definitions of angular errors are given in Table 4.1.  Table 4.1 states that when the direction of motion is along the X axis, the rotation about X axis (A) is known as roll angle. The first letter shows the rotational axis whereas the second letter shows the direction of motion. Also, the rotation about Y axis (B) is named as pitch error and the rotation about Z axis (C) is known as yaw error.  Table 4.1: Rotational errors on a V M C Direction of motion  Pitch  Yaw  Roll  X  B  of X  Cof X  A of X  Y  A of Y  Cof Y  B  Z  A of Z  B  Cof Z  of Z  of Y  65  Chapter 4. Error Measurements by Laser Interferometry  The error component is generally measured in terms of arc seconds or pm /m. The evaluation of the angular error is complicated because the positioning inaccuracy produced by an angular error is influenced by three variables; (1) the magnitude of the angular error at the target position, (2) the travel length of the axis, and (3) the machine tool configuration.  The size of an angular error affects the machine tool positioning accuracy. The greater the magnitude of the error, the bigger is its effect on the positioning inaccuracy. However, the angular error of an axis only creates a positioning error in the machine if a movement in a second orthogonal machine axis is existent. For instance, in F A D A L VMC-2216, consider two axes; X and Y , where X is mounted on top of Y . Positioning errors are produced in the direction of Y when the X axis starts to move. The greater the movement of the X axis the greater the positioning error produced. Generally, angular errors are more significant and dominant on larger machine tools [39]. The last factor which influences the contribution of angular errors to positioning inaccuracy is the machine-configuration, which defines the arrangement of its axes and their relationship to the tool and workpiece. It does not influence the size of the resulting positioning errors caused by angular errors but defines which axes experience the position error's.  =  '  -  There must be a rotation of one optical component with respect to the other for angular measurements to take place. The rotation of the angular reflector causes a change in the path difference between the two measurement beams, which is determined by the fringe-counting circuitry in the laser head and is converted to an angular measurement.  The angular interferometer is placed in the beam path between the laser head and the angular reflector, as shown in Figure 4.4. The beam coming from the laser head is split into two beams by the beam-splitter of the angular interferometer. One part of the beam passes through the interferometer and is reflected from one half of the angular reflector, which is a combination of two linear interferometers, back to the laser head. The other beam is reflected by 90° in the angular interferometer and sent to the other half of the angular reflector, which returns it through the interferometer to the laser head [42].  Chapter 4. Error Measurements by Laser Interferometry  ANGULAR LASER HEAD  INTERFEROMETER  AXIS O F M O V E M E N T  ANGULAR REFLECTOR  Figure 4.4: General set-up for angular measurements  Figure 4.5 depicts the set-up for yaw measurements of Z axis whereas Figure 4. illustrates the set-up for pitch measurements of the same axis.  Figure 4.5: The set-up for yaw measurement of Z axis on F A D A L  Chapter 4. Error Measurements  by Laser Interferometry  67  4 . 3 Machine Tool Testing Parameters Since only one variable, which is the nominal position, is considered to affect the geometric errors in this research, measurements should be carried out to find the relationship between this variable and the individual error components over the axes' possible length of travel. For each axis, the measurement starts when the slide under study is at one end of its travel range. Then the slide is moved toward the other end of its travel range, and measurements are collected when the machine is stopped for a short duration (i.e. to-five seconds) at the pre-specified target points. In the A S M E standard about the methods for performance evaluation of C N C machine tools, it is recommended that the laser readings be averaged for about 0.25 seconds [3]. After one run is completed, the motion is reversed at the end of the travel, and the slide is sent back to its starting position, again with readings taken at every measuring interval. With this procedure, it is possible to derive the reversal error of each axis from the difference between the values of errors in forward and backward directions. In selecting the measuring interval, one has to consider the periodic error components such as the one caused by the leadscrew misalignment. In order to eliminate the effect of the periodic errors, the measuring interval should be selected at even multiples of  Chapter 4. Error Measurements by Laser Interferometry  68  the leadscrew lead. However, in order to determine the periodic error, the positioning errors should be taken separately over a short region with a very fine resolution with the assumption of uniformity over the whole travel range.  Table 4.2: The travel lengths of three axes in the F A D A L VMC-2216 Name of the axis  Length of travel  X axis  530 mm  Y axis  380 mm  Z axis  350 mm  According to A S M E standards, measuring intervals should be no longer than 25 mm for axes of 250 mm length or less. For longer axes, the interval should be no more than 1/10 of the axis length [3]. In our case, the lengths of the three axes of F A D A L VMC-2216 are given in Table 4.2. The lead of the ballscrew is 10 mm for three axes and the measuring interval is chosen to be 5 mm in linear and angular measurements to get more detailed information about the errors along an axis. Besides, 1 mm is used as the measuring interval in linear periodic errors. The periodic error measurements have not been taken along the whole travel length but in a shorter range such as 40 mm assuming that periodic errors of one axis are uniform along it.  4.4 Measurement Results of Three-Axis F A D A L VMC-2216 Machining Center The results of the measurements conducted on three axes of the F A D A L vertical machining centre are given in this section of the thesis. First the results and the procedure to model the geometric errors are explained for X axis. Then, Y and Z axes are explained and the results are plotted against the nominal position on the axis. 4.4.1 Linear Positioning Errors  The displacement error data were taken along the X , Y , and Z axes in the F A D A L Vertical Machining Center. In this section of the thesis, the error measurement and analysis procedure is explained for X axis. Then the results of error data taken along Y and Z axes are presented.  69  Chapter 4. Error Measurements by Laser Interferometry  Before the measurements were taken, firstly the direction of sense of the laser system was set to be the same as the machine under test. The machine coordinate system is shown in Figure 3.15. The displacement error data taken along the X axis are shown in Figure 4.7. The crosses represent the runs in forward (positive) direction whereas the dots represent the runs in backward (negative) direction of the axis. The mean values for forward and backward data are shown in the same plot.  Linear Positioning Errors of X axis 30  20 Forward direction of motion  " 1 0 1 —  •4—<—  O u  oh 5ft  r-  2-101  . _i  o  x  x x  LU  -20 h  ....  •  -30 h -40  Backward direction! of motion  -250 -200  -150 -100 -50 0 50 Position on X axis  • • •  100  mean forward forward - run1 forward - run2 forward - run3 forward - run4 forward - run5 mean backward backward - run1 backward - run2 backward - run3 backward - run4 backward - run5  150  200 250  Figure 4.7: Linear positioning errors along X-axis The initial observation from Figure 4.7 indicates that there are about 3 micrometers of backlash error in spite of the use of a preloaded ball nut/leadscrew assembly. The reversal errors vary along the length of the leadscrew due to the nonlinearities of the leadscrew and complex dynamics of ball nut friction mechanisms. The variation of the reversal error along X axis is given in Figure 4.8. Therefore, the fitted curves for forward and backward  Chapter 4. Error Measurements by Laser Interferometry  70  directions will not be the same curve. Different error calibration for forward and reverse directions will be established to compensate for the backlash.  Reversal Errors on X axis 5  ITL  ^4.5  ...4*.  CO  ^r - -  ^  | 3.5 £2.5  i . iui  1  ^  +ff  _T* _ +  + _  r* * • +  — .  . ,  *  o 2 w 1.5 1 r"* 0.5-250 -200 -150 -100  -50 0 50 Position on X axis  100  150  200  250  Figure 4.8: Reversal errors along X-axis  In order to predict the errors between the target points along the X axis, the errors are fitted into a curve using the least squares method. The curve fitting requires the prediction of the periodic errors of X axis.  Periodic Errors: The periodic errors of X axis are measured at 1 mm interval along a range of 40 mm. The combination of the ball screw-nut drive system and the rotary encoder feedback unit is expected to create a periodic type of displacement error. Figure 4.9 shows the data obtained from the measurements for forward and backward directions. The difference between the incremental motions as measured by the laser interferometer and given by the machine tool controller is the net periodic error motion.  The Fourier transforms are used in curve fitting for periodic errors. The errors are periodic and can be represented by their harmonic components. As it can be seen in Figure 4.9, the errors are periodic with a period of 10 mm, which is the leadscrew lead. The periodic errors can be expanded into discrete Fourier series such as;  71  Chapter 4. Error Measurements by Laser Interferometry a °° °° E(x) = — + ^ a cos nwx + ^ b sin nwx 2 „ i „ \ n  (4.1)  n  =  =  where n is the harmonic of the fundamental frequency w and the coefficients of; 2  N  a„ = — >_,£, cos Nt( ' NT  (4.2)  L  n  o„ = — / £; sin " NT  L  Periodic errors of linear positioning in X axis Forward- mean Backward - mean  10  15  20  25  30  35  40  Position on X axis, [mm]  Figure 4.9: Periodic errors in the forward and backward direction of motion of X-axis  As it can be seen from Figure 4.9, the periodic errors of forward and reverse directions of motion are not the same, therefore, different curve fittings are applied for each direction of motion. The difference between the backlash and forward directions is coming from the backlash error of X axis which is about 3 micrometers. Besides, the trends in backward and  Chapter 4. Error Measurements by Laser Interferometry  7 2  forward directions are not exactly the same at some specific locations. If the difference were pure backlash, then the periodic errors in forward and backlash directions could be represented by the same a„ and b coefficients with different ao. n  Figure 4.10 depicts the measurements in the backward direction of motion in X axis. The errors between 10 mm and 19 mm are chosen in order to apply Fourier transforms. After the equation is derived, it is plotted over the whole range of the periodic error measurement as shown in Figure 4.12.  P e r i o d i c e r r o r s o f linear p o s i t i o n i n g in X a x i s - - B a c k w a r d O • 0 • + — —  w 4 d>  12  13  14  backward-run 1 backward-run2 backward-run3 backward-run4 backward-run5 Periodic error curve fit to data Mean of measured values  15  Position on X axis,  16  17  [mm]  Figure 4.10: Mean of periodic errors in the backward direction of X-axis and the fitted curve  The values measured and used in this Fourier transforms are; x = [l0  11 12  y =[1.6668 b  -2.3096  13 14  -3.2424 -0.2310  15 16 -1.8498 2.0096]  17  18 19] mm  2.2162  0.3638  0.5570  -0.6174  pm  where y is the mean value of five runs for the backward direction of motion, and b  Chapter 4. Error Measurements by Laser Interferometry  73  x is the nominal position on X axis. T = x(2) -  x(l)  =  -1 mm  TV = 10  (4.4)  w = 2nl(NT) = -0.6283 where T is the interval, N is the number of samples, w is the fundamental frequency.  P e r i o d i c e r r o r s o f linear p o s i t i o n i n g in X a x i s - - F o r w a r d  121  1  1  r  1  O + 0 • x — —  10 v>  forward-runt forward-run2 forward-run3 forward-run4 forward-run5 Periodic error curve fit to data Mean of measured values  n  <D  10  11  12  13  14  15  Position on X axis,  16  17  18  19  [mm]  Figure 4.11: Mean of periodic errors in forward direction of motion of X-axis and the fitted curve  As shown in Figure 4.10, the fitted curve matches with the mean of measured values. The first four harmonic components are enough to express the mean error curve accurately. Thus, n is chosen as four. The Fourier coefficients for four harmonic components are calculated from Eq. (4.2) as follows: 0.0509  0.0593 0.7047 a,  0.6040 0.5509  pm  -1.9080 -1.1541 0.4360  pm  (4.5)  Chapter 4. Error Measurements by Laser Interferometry  74  a ll = -0.1437 um 0  From Eq. (4.1), the periodic error function for the backward direction of motion can be expressed as: S (x) = -0.144 - 0.059 cos(wx) + 0.705 cos(2wx) + 0.604 cos(3wx) xb  + 0.55 lcos(4wx) +0.05 lsin(wx)-l.908 sin(2wx)-1.154 sin(3wx)  (4.6)  + 0.436 sin(4wx)  The same procedure is followed for the periodic errors in the forward direction of motion. The same X axis range and number of samples are used. The mean values of measured errors in this range are; JC  = [10  11 12  yf = [4.4188 0.6202  13  0.5628 3.4468  14  15  16  17  18  1.2148 5.3946 5.2452]  19]  4.3920  mm 3.6808  2.5434  ( 4  '  7 )  pm  where y is the mean value of five runs for the backward direction of motion. f  The interval length and the fundamental frequency are same as the backward direction given in Eq. (4.4), respectively.  As shown in Figure 4.11, the fitted curve for the forward direction of motion matches the mean of measured values well, As in the backward direction of motion, the first four harmonic components are enough to express the mean error curve accurately. The Fourier coefficients for forward direction components are as follows:  a  -0.1184  •0.1505  0.6800  -2.0659  0.4362  -0.8379  0.2178  0.5649  a =6.3038 pm Q  pm  (4.8)  Chapter 4. Error Measurements by Laser Interferometry  75  Therefore, the periodic error function for the forward direction of motion can be expressed as; & (x) = 3.152-0.118cos(w;c) + 0.68cos(2wx) + 0.436 cos(3wx) 4  [pun]  + 0.218cos(4wx) + 0.151sin(H'jc)-2.066sin(2wx)  (4.9)  - 0.838 sin(3wx) + 0.565 sin(4wx)  Periodic errors of linear positioning in X axis Forward - mean Backward - mean Fitted curve-forward Fitted curve-forward  20  25 30 35 40 Position on X axis, [mm]  50  Figure 4.12: Periodic errors of X-axis  Abbe  Offset  Correction:  The errors shown on Figure 4.7 were measured when the  retroreflector was located on the pillar at a height of 16.5 cm. Because of the Abbe offset, these errors are not the same as the ones that could be measured at the level of the feedback device of X axis. The linear positioning errors defined in the mathematical model are supposedly measured at the level of the feedback device. Therefore, the Abbe error correction is required. Figure 4.13 depicts a general set-up for linear measurements and the effect of the Abbe error on the measurements.  Chapter 4. Error Measurements by Laser Interferometry  76  Where a measurement is made with the beam aligned parallel to, but with an offset from the defined axis of calibration, machine angular errors (e.g. pitch or yaw) can produce an Abbe offset measurement error. For each arc second of angular motion the error introduced is approximately 0.005 pm/mm of offset. Hence, the magnitude of the Abbe error at any point depends on the location on the pillar and the magnitude and direction of the pitch error of the axis under study at that point.  M o v i n g Optic  A b b e Error  A x i s of L a s e r B e a m  Extension  Laser Head Offset  Y a w or Pitch Angle  Pivot Point  A x i s of Motion  Figure 4.13: Set-up with Abbe error (Source: Renishaw [42])  The direction of pitch error should be analyzed carefully. In our case, because of the mechanical structure of the machine tool, it was an expected result that as the location on the pillar increases, the Abbe error increases in X direction. The sign of the linear positioning error and the sign of pitch error should be examined carefully to see if the Abbe error is destructive or constructive to linear positioning errors.  The magnitude of the pitch error at any location on the X axis is multiplied with the location on the pillar and this Abbe error is subtracted from or added to the value of the linear positioning error at that position depending on the sign of the angular deviation. As shown in  Chapter 4. Error Measurements by Laser Interferometry  11  Figure 4.14, the result will give the errors measured on the surface of the table. If one wants the value of the linear positioning errors at the feedback device of the X axis, then the distance from the feedback device to the top of the table should be known or measured precisely. The corrections in this research are made according to the level of the feedback devices.  Abbe error  •  ZA  4  Location on the pillar  Figure 4.14: Set-up for X-axis on F A D A L VMC-2216 with Abbe error The results of pitch of X measurements are given in Section 4.4.2. The only angular error that affects the linear positioning accuracy of X axis is its pitch because the effect of its yaw is eliminated by taking the measurements along the axis where the nominal position on Y axis is equal to zero. The comparison between the linear positioning errors with Abbe errors and linear positioning errors with the correction according to the level of the leadscrew is given in Figure 4.15 and Figure 4.16 for forward and backward directions respectively. The linear positioning errors of X axis were measured for five times to see the consistency of the errors. The measured values are shown with crosses and diamonds for forward and backward directions, respectively. The dotted lines show the linear positioning errors at the level of the feedback device with the Abbe correction, whereas the solid lines show the mean of measured values at the top of the pillar. For a travel of 265 mm along X axis, the maximum Abbe errors calculated in forward and backward directions are about 19 micrometers as evident in Figure 4.15 and Figure 4.16.  Chapter 4. Error Measurements by Laser Interferometry Linear Positioning Errors of X axis - Forward  —  mean forward run1 run2 run3 x run4 run5 •— with pitch comp. x  3:  -100 0 100 Position on X axis,[mm]  200  Figure 4.15: Linear positioning errors of X-axis in forward direction with pitch compensation Linear Positioning Errors of X axis - - Backward  • o o  -200  mean backward run1 run2 run3 run4 run5 with pitch comp.  -100 0 100 Position on X axis  200  Figure 4.16: Linear positioning errors of X-axis in backward direction with pitch compensation  Chapter 4. Error Measurements by Laser Interferometry  79  Results for the Linear Positioning Errors of X axis: After the Abbe errors are taken away, the equations for periodic errors are derived. The linear positioning errors of X axis can be plotted versus the nominal position on X axis as shown in Figure 4.17. The errors at target points are fitted to a curve which is a 5 order polynomial equation as shown in the figure. th  The fitted polynomials for forward and backward directions of motion are given as follows;  For forward direction: S (x) = 9 . 0 9 6 . 1 0 " x 12  + 2 . 3 3 . 1 0 " x - 6.856.10"  5  12  xf f  4  7  x  3  [pm]  (4.10)  - 3.51.10" x + 0.087x + 2.041 5  2  For backward direction: S.'x) = + 1 . 4 1 9 . 1 0 " * +1.3236.10" x 11  5  9  4  xb  -1.339.10 x _ 6  3  [pirn]  (4.11)  - 9.436.10" x +0.10419*-1.0468 5  2  where x is the nominal position along X axis [mm]. The periodic errors are added on top of the error functions given in Eq. (4.6) and Eq. (4.9), the following polynomials which represent the final form of the equations for the linear positioning errors of X axis are derived as shown in Figure 4.18 and Figure 4.19. The period of the oscillations are equal to the lead of the ball screw because the nonlinearities of the lead are being repeated in every lead of the ballscrew, which cause the periodicity in the linear displacement errors. The combination of the ball screw-nut drive system and the rotary encoder feedback unit is expected to influence the periodicity in the linear displacement error.  For forward direction, S (x) = - 0.1184 cos(wx) + 0.6800 cos(2wx) + 0.4362 cos(3wx) xf  + 0.2178cos(4wx) + 0.1505sin(wx) - 2.0659 sin(2wx) - 0.8379sin(3wx) +0.5649sin(4iyx) + 9 . 0 9 5 5 . 1 0 " x 12  + 2.3295.10 " x - 6 . 8 5 5 9 . 1 0 ~ x - 3 . 5 0 6 . 1 0 ~ x -1  2  4  + 0.08707x +2.0411  7  3  5  [pm]  5  2  •  (4.12)  Chapter 4. Error Measurements by Laser Interferometry  80  Linear positioning errors of X axis with Abbe error correction  -250  -200  -150  -100  -50  0  50  100  150  200  250  Position on X axis [mm] Figure 4.17: Linear positioning errors of X-axis without pitch errors and the fitted curves  For backward direction: S ( ) =- 0.0593 COS(WJC) + 0.7047 cos(2wx) + 0.6040 cos(3wx) xb  x  + 0.5509 cos(4wx) + 0.0509 sin( wx) -1.9080 sin(2w;c) -1.1541sin(3wx) + 0.4360sin(4vyx) + 1 . 4 1 9 . 1 0 +1.3236.10" x - 1 . 3 3 9 . 1 0 " * 9  4  6  3  _11  x  -9.436.10 x - 5  5  [pm\  (- ) 4  13  2  + 0.10419*-1.0468  where x is the nominal position along X axis [mm].  The curves shown in Figure 4.18 and Figure 4.19 have the final form of the equations which are used in the compensation scheme for linear positioning errors of X axis.  Chapter 4. Error Measurements by Laser Interferometry  L i n . p o s . errors o f X with periodic errors a n d A b b e E r r o r c o r r e c t i o n  -200  -100  0  100  200  Position on X axis,[mm] Figure 4.18: Linear positioning errors of X-axis in the forward direction  L i n . p o s . errors of X with periodic errors a n d A b b e Error correction  201  -200  -100  0  100  200  Position on X axis,[mm] Figure 4.19: Linear positioning errors of X-axis without pitch errors in the backward direction  Chapter 4. Error Measurements by Laser Interferometry  82  The procedure for deriving the linear positioning errors along the whole travel length of an axis can be summarized as follows: Firstly the measurements are taken along any line parallel to the target axis with a pre-specified interval, which was 5 mm in the measurements. Secondly, the periodic errors are measured with a finer interval along a shorter range. If the measurements are not taken along the axis of calibration, the Abbe error correction is required. Then the general trend of errors with Abbe correction is combined with the periodic errors of the axis, which gives the final form of the linear positioning errors.  The linear  positioning errors of Y and Z axes are modeled in a similar fashion. Before the results for the axes are given, the specific arrangements done on Y axis are explained.  On the tested machine, it is impossible to position the spindle in X and Y directions due to its configuration type. The motions in X and Y are provided by the table. Therefore it was not possible to take measurements over the whole travel length of Y axis in one shot. As shown in the Figure 4.20 and Figure 4.21, when one of the optics is mounted on the table while the other one is mounted on the spindle, only one side of Y axis can be subjected to the measurements. For the other, side of Y axis, the optic should be removed from the table and relocated on the table looking in the reverse direction. Also, the linear interferometer should be mounted on the spindle such that one of the beams can go to the reference reflector whereas the other is targeted to the measuring reflector. Therefore, Y axis is presented in two sections; (1) positive side of Y axis, and (2) negative side of Y axis. Finally the equations representing the errors of Y axis in each side are combined and plotted together.  The results of the curve fitting for the periodic errors in Y axis are summarized in Table 4.3. The following equations represent the general trend of the errors in Y axis after the Abbe correction is applied to the raw measurement data.  In positive side, for forward direction of motion: S  ( y ) = 0.00091814y -0.13467y-2.550 2  yf  p  [pm]  (4.14)  [jLtm]  (4.15)  In positive side, for backward direction of motion: S  (y) = 0.00048878y -0.05429y + 4.8387 2  yb  p  Chapter 4. Error Measurements by Laser Interferometry  83  Spindle  Linear Retroreflector Linear Interferometer  Table -200 m m  200 m m  0  z •> Y  Figure 4.20: Linear measurement set-up for negative Y axis  I Spindle  Linear Retroreflector  Linear Interferometer  Table -200 mm  200 m m  u ZA  ->• Y  Figure 4.21: Linear measurement set-up for positive Y axis  In negative side, for forward direction of motion: 5  ( ) = 0.00031134y +0.0043608y-2.7657 2  n  y  [pm]  (4.16)  Chapter 4. Error Measurements by Laser Interferometry  84  In negative side, for backward direction of motion: S  (y) = 0.00057837y -0.0013779y + 6.7838  [pm]  2  yb  n  (4.17)  where y is the nominal position along Y axis [mm]. The functions given in Eq. (4.14) to Eq. (4.17) represent the general trend of the linear positioning errors of Y axis with Abbe correction. The periodic errors are added and the final forms representing the linear positioning accuracy along Y axis are plotted against nominal position in Figure 4.22.  Linear Positioning Errors of Y axis 2  0  f  i  -150  !  •  -100  !  !  !  -50  0  50  •  i  >  1  1  —  Backward  100  il  150  Position in Y axis [mm] Figure 4.22: Linear positioning errors of Y axis in forward and backward directions The measurement results of the periodic linear positioning errors of Z axis are summarized in Table 4.4. The general trends of the errors in Z axis are represented with the following equations:  For forward direction:  85  Chapter 4. Error Measurements by Laser Interferometry  b (z) 4  = -0.956z + 84.294 [pm]  (4.18)  S (z)= -0.927z + 86.342 [pm] zb  (4.19)  where z is the nominal position along Z axis [mm].  As a result, the linear positioning errors of Z axis can be expressed as the combination of the periodic errors and general trends.  Linear Positioning Errors of Z axis  Position on Z axis [mm]  Figure 4.23: Linear positioning errors of Z axis  The Abbe error correction is not applied to measurements in Z axis because the axis that the measurements were taken along was very close to the axis of the spindle. The effect of the Abbe errors on the linear positioning errors is so small (i.e. under the resolution of the machine) that it can be considered as negligible. The final forms of linear positioning errors in Z axis are plotted in Figure 4.23.  Chapter 4. Error Measurements by Laser Interferometry  86  Table 4.3: The periodic errors of Y axis Fourier Series [pm], y[mm]  Periodic error  Forward motion in  S  yfp  the positive direction  (y) = -7.514 + 0.355cos(wy) + 3.801cos(2wy)  p  - 0.246cos(3wy) + 0.757 cos(4wy) + 0.26sin(wy)  of Y axis  +1.93 sin(2wy) +1.06 sin(3wy) + 0.048 sin(4wy)  Backward motion in  S  ybp  the positive direction  (y) = 3.318-0.294cos(vvy) + 3.474cos(2>vy)  p  - 0.174cos(3wy) + 0.035 cos(4wy) - 0.255 sin(wy)  of Y axis  + 1.731 sin(2wy) + 0.975 sin(3wy) + 0.155 sin(4wy)  Forward motion in &  the negative  n(y)=  yfp  -2-2958 + 0.848 cos(wy) + 3.0752cos(2wy)  + 0.1927 cos(3wy) - 0.6368 cos(4wy) - 0.4029 sin( wy)  direction of Y axis  -1.9891 sin(2wy) -1.162 sin(3wy) - 0.2469 sin(4wy)  Backward motion in  <W„ (y) = 8-59 - 0.0618cos(wy) + 3.0714cos(2wy)  the negative  + 0.1579cos(3wy) -0.7234cos(4wy) + 0.0499sin(wy)  direction of Y axis  -2.013sin(2wy)-1.1033sin(3wy)-0.264 sin(4wy)  Table 4.4: The periodic errors of Z axis Fourier Series [pm] , z[mm]  Periodic error  6^ (z) = 90.0621 + 2.762cos(wz) + 2.202 cos(2wz) Forward direction of  -0.357cos(3wz)-0.552cos(4wz)-0.853cos(5wz)  motion  + 2.391sin(wz) + 0.417 sin(2vvz) - 0.46sin(3wz) + 0.093 sin(4wz) s  Backward direction of motion  6 (z) = 88.0391+ 3.4738cos(wz) +2/7975cos(2wz ) zbp  -0.5969cos(3wz)-0.2783cos(4wz)  • *  - 0.3484 cos(5wz) + 1.1869sin(wz) + 0.7559 sin(2wz) + 0.2707 sin(3wz) + 0.2600 sin(4wz)  .  87  Chapter 4. Error Measurements by Laser Interferometry  4.4.2 Pitch Errors  The pitch error of one axis is defined as the angular error about the axis that lies in the plane and perpendicular to the direction of motion. The sign convention used for angular errors can be best explained with the right hand rule. The right hand rule indicates that when a thumb is aimed in the positive direction of the axis of rotation and the fingers are curled around the same axis, the curl direction is the positive angle. For example, when the positive direction of pitch angle of X axis is to be found, the thumb is aimed in the positive direction of Y axis and the finger curl direction is the positive pitch angle. This case is illustrated in Figure 4.24 for the F A D A L Vertical Machining Center. The mean values of the pitch errors of X axis are shown in Figure 4.25.  Since there is a very small difference between the errors in forward and backward directions, only one analysis is used for both of motion directions. The average of the difference between forward and backward directions at each target position is 0.0226 arcsecs and the maximum difference is 0.2250 arcsecs. The variation in the difference is plotted against the position on X axis in Figure 4.26. The magnitudes of the pitch errors of X axis are relatively large compared to the other angular errors of the machine because of the mechanical structure of the table. When the table starts to move towards its ends, it starts to bend causing pitch error because the nut support lies only in the middle.  The pitch error model is found using a regression analysis with a single variable (nominal position) as shown in Figure 4.27 and given by the following expression: e (JC) = -4.6567. I O * +1.1723.10" V -10  V  y V  '  4  + 0.065278*- 0.55347  + 9.8653.10"V  [arcsec]  (4.20)  where x is the nominal X reading from the machine controller [mm].  The pitch error models of Y and Z axes are derived in a similar manner. The pitch errors of Y axis are angular deviations about X axis when the intended motion is in Y axis.  The pitch errors of Y axis were measured in two sections (in positive and negative sides of Y axis) as done in linear positioning errors. Since there is a difference of about 1 arcsec  Chapter 4. Error Measurements by Laser Interferometry  88  (i.e. 4.8 |im/m) between the errors in forward and backward directions, different calibrations are used for backward and forward directions of motions. The analysis results are given in Table 4.5. The fitted curves for the pitch errors of Y axis in forward and backward directions of motion are plotted against the nominal position on Y axis in Figure 4.28.  Positive direction for pitch of X axis  Figure 4.24: Sign convention for pitch errors of X axis  89  Chapter 4. Error Measurements by Laser Interferometry The difference between forward and backward pitch errors  A  0.2  0.15  ^  0.1  £  t "V ...i.i..v  0.05  H  <0  III • J  0  o  != -0.05  UJ  -0.1 -0.15 -0.2 -300  i.LL  i A !\ , | I , r1 V  IV  V  I.J.  1  -200  -100  0  100  200  300  Position in X axis [mm]  Figure 4.26: The difference between forward and backward pitch errors of X-axis  Pitch Errors of X axis  -200  -100  0  100  200  Position in X axis [mm]  Figure 4.27: Pitch errors of X-axis for both directions of motion  The pitch errors of Z axis are the angular deviations about X axis when the motion is in Z axis. Although the measured errors for forward and backward directions show almost the  Chapter 4. Error Measurements by Laser Interferometry  90  same trend, different calibrations are applied for each direction due to the increasing difference in the negative end of Z axis. The summary of the single variable regression for the pitch errors of Z axis is given in Table 4.6. The mean values of pitch errors of Z axis and fitted curves are plotted in Figure 4.29 for both directions.  T a b l e 4.5: The equations of the pitch errors of Y axis Pitch errors of  Curve Fitted Functions [arcsec], y[mm]  Y axis  Forward e _„(y) = -5.2114.10- y -1.2848.10- y +5.2715.10- y  direction in the  u  5  8  4  8  xf  negative side  + 6.0417.10 y + 0.019885y + 0.5980 -5  2  Backward direction in the  E _„(y)  = -5.1209.10- y - 1 . 3 9 1 2 . 1 0 - V - 2 . 9 2 7 4 . 1 0 - V 11  XB  negative side  5  + 5.6832.10~ y +0.025278y+ 0.2493 5  2  Forward direction in the  { ) = 2.47.10- y - 6 . 6 4 . 1 0 - V - 6 . 4 1 . 1 0 " V 9  E x f  4  y  positive side  + 0.0295y +0.598  Backward direction in the positive side  e ( ) = 2.71.10- y - 6 . 6 5 . 1 0 ' V - 5 . 4 4 . 1 0 " V 9  xb  4  y  + 0.0254y +0.253  Table 4.6: The equations of pitch errors of Z axis Pitch Error of Z axis  Forward  Curve Fitted Error Functions [arcsec], z[mm]  e (z) = -3.16.10" z -0.142.10" z 7  direction Backward direction  3  3  xf  -0.087z +10.985  ^ ( ) = -3.7.10- z -0.175.KrV 7  3  z  -0.091z + 11.488  2  3  er 4. Error Measurements by Laser Interferometry Pitch Errors of Y axis 3  r  T  2 1 W  0  u  « 1-2 LU  Forward Backward  -200  -150  -100  -50 0 50 100 Position in Y axis [mm]  150  200  Figure 4.28: Pitch errors of Y axis  Pitch Errors of Z axis 30  i  i  i i  x  •  O  ii  25 •  [  Forward - mean Fitted curve-forward Backward - mean Fitted curve-backward  »  20 o  w 15  t  5 LU  1 i  10 i  i i i  i  -200  -150  -100  •  -50  0  50  100  Position in Z axis [mm] Figure 4.29: Pitch errors of Z axis, fitted curves to raw data for both directions  92  Chapter 4. Error Measurements by Laser Interferometry  4.4.3 Yaw Errors  The yaw errors of one axis are the angular deviations about the axis that is perpendicular to the plane where the direction of motion lies. The sign convention is taken as explained in Pitch Errors. To find the positive direction of the yaw angle of X axis, the thumb is aimed in the positive direction of Z axis and the curl direction is the positive yaw angle of X axis. The yaw errors of X axis, which are angular deviations of X about Z, are shown in Figure 4.30.  Yaw Errors of X axis - Raw data  5I -250  I  -200  I  -150  I  I  -100  -50  I  0  I  50  I  100  I  150  I  200  I 250  Position in X axis [mm]  Figure 4.30: Yaw errors of X-axis, raw data Since there is a difference in forward and backward directions, they are analyzed separately. The yaw errors in forward direction for the complete measurement range is shown in Figure 4.31, whereas Figure 4.32 shows the same for the reverse direction with the fitted curves obtained with a regression and given as follows:  For forward direction of motion:  93  Chapter 4. Error Measurements by Laser Interferometry  (x) = -1.1035.10" x - 4.4512. I O x + 2.2649. IO 16  £ r f  1  -15  6  + 3.5651.10" JC -1.4819.10 X - 2 . 1 2 2 4 . 1 0 X 10  4  _6  3  JC  -11  _5  5  2  [arcsec]  (4.21)  [arcsec]  (4.22)  + 0.021469* - 0.84994  For backward direction of motion: £ {x) zb  = -A.\2.\Q'  xl  x + 2.3647. IO" * +1.2869.10"" x 1  15  6  5  -8.1077.10" * -1.0595. IO" * +3.4861. I O * 10  4  6  3  -5  2  + 0.016075.*-1.29 where x is the nominal X reading from the machine controller [mm].  Yaw Errors of X axis - Forward 1  1  1  1 [  1  +  measured data • fit curve  +* +/  <> / p  j  -250  J  J  J  -200 -150 -100  J  J  -50  0  J  50  100  150  200 250  Position in X axis [mm] Figure 4.31: Yaw errors of X-axis in forward motion of direction The yaw errors of Y and Z axes are analyzed in a similar manner. The yaw errors of Y axis, which are angular deviations of Y about Z axis, are analyzed in positive and negative sides of Y axis. Since the trend and magnitudes in forward and backward directions are totally different, they are analyzed separately. Due to the nonlinearity in the yaw error of Y axis at the position of -50 mm, the yaw errors were divided into two sections in the negative side of Y such as from -200 mm to -55 mm and from -55 mm to 0 mm. This approach can  Chapter 4. Error Measurements by Laser Interferometry  94  decrease the degree of the curve representing the yaw errors of Y in the negative direction and increase the computational efficiency in the compensation schemes. The errors in the positive side were analyzed in a similar fashion and the results are summarized in Table 4.7 and plotted in Figure 4.33.  Table 4.7: The yaw errors of Y axis Yaw error of Y axis  Equation of the fitted curve [arcsec], y [mm]  Forward errors of in the range of  (y) = 1.03.10" y -0.00022y + 0.021y + 2.8764 5  [-55 0] mm. Backward errors in the range of  200 -55] mm. Backward errors of Y axis in the  9  zb  side  -7  4  _5  3  -0.00015y + 0.01 ly-1.3644 2  ^ ( y ) = -4.92.10" y -0.0021y -0.28y-10.145 6  3  2  e { ) = -1.08.10 y -6.264.10~ y - 0 . 0 0 0 1 4 y -9  zb  5  7  4  3  y  -0.016y -0.859y-20.483 2  E^(y)  side Backward errors in the positive  5  y  range of [-200 -55] mm. Forward errors in the positive  2  e ( ) = 7.26.10~ y +1.548.10 y -1.002.10 y  [-55 0] mm. Forward errors in the range of [-  3  = 2.21.10" y -7.96.10~ y + 6 . 5 . I O y 12  6  10  5  -8  4  + 4.84.10 y -0.00028y +0.015y + 1.26 _6  3  2  £ ( y ) = 2.52.10- y -9.73.10- y +1.52.10 y 12  6  10  5  _7  4  z6  -1.078.10~ y +5.83.10~ y 5  3  5  2  +0.02y-1.65  The yaw errors of Z axis are the angular deviations about Y axis while moving in Z direction. The equations of the curves fitted to the yaw errors of Z axis are given in Table 4.8.  Chapter 4. Error Measurements by Laser Interferometry  Figure 4.33: Yaw errors of Y axis  95  Chapter 4. Error Measurements by Laser Interferometry  96  Table 4.8: The yaw errors of Z axis Y a w error of Z a x i s  C u r v e fitted y a w error f u n c t i o n s [arcsec], z[mm]  f ( z ) = 1.863.10" z + 8 . 9 1 . 1 0 V + 1 . 0 2 . K T V 13  -  6  y /  Forward direction  - 1 . 0 5 . l C r V - 0 . 2 6 . 1 0 " V - 0 . 0 1 2 z + 2.973 £ (z) = 5.209.10~ z + 3.23.10" z + 4.744.10~ z 14  Backward direction  6  11  5  9  4  yb  - 5 . 7 5 6 . 1 0 " V - 0 . 1 8 3 . 1 0 " z - 0 . 0 1 2 z + 2.633 3  2  The yaw errors summarized in Table 4.8 are plotted against the nominal position on Z axis in Figure 4.34.  Yaw Errors of Z axis  -150  -100  -50  0  Position in Z axis [mm] Figure 4.34: Yaw errors of Z axis in forward direction of motion, fitted curve to raw data  4.4.4 Backlash Errors of FADAL VMC-2216 In general, the backlash errors depend only on the axis and they generally do not vary along the travel length. In our case, the leadscrews of F A D A L are worn in some specific positions along their lengths, therefore the reversal errors are not constant along the travel  Chapter 4. Error Measurements by Laser Interferometry  97  length. The leadscrews had to be divided into zones in which backlash errors can be assumed as constant. In each zone, the backlash error is calculated as the mean value of the measured values as tabulated in Table 4.9. In Figure 4.35, the backlash errors of X axis are illustrated. Similarly Figure 4.36 shows the reversal errors along the Y axis and Figure 4.37 depicts the backlash errors along Z axis and the zones specified with the constant backlash errors.  Table 4.9: Constant backlash zones and mean values of reversal errors of X axis Position [mm]  X  < X  SIX < >  (/) • x <-, N  [-265 -250] [-245 -230] [-225 -220] [-215 -210] [-165 -105] [-100 -55] [-50 -15] [-10 90] [95 265] [-150 -135] [-130 -115] [-110 -85] [-80 -55] [-50 30] [30 150] • [-140 -120] [-115 -80] [-75 -30] . [-25 70] [75 90]  Error [um]  0.56 1.28 1.95 2 . 62 3 .45 4.96 3 .37 2 .42 3.09 -15.78 -14.46 -12 .70 -10.82 -9.65 -12.24. -4.09 -1.53 1.80 .3.78 0.54  Chapter 4. Error Measurements by Laser Interferometry Reversal Errors on X axis  w o 4  i+i  (D  £ o  • 4 - J  4**  £.  o  44 o UJ +  measured reversal errors  — — sub-regions with constant reversal errors _J_  0 — -300 1  -200  0  -100  100  200  300  Position on X axis  Figure 4.35: Backlash errors of X axis and sub-regions with constant backlash errors  Reversal Errors of Positive Y axis 1  •*•  measured reversal errors  — — sub-regions with constant reversal errors  i  c  2  -s  u L  .-*._:fcj  i  * - /  .  E UJ •12 o  *— y  +  +  1 + 1+,  *  fc* r * ' : r f c  *4-i  + * +  +'*•:.  5-14  +  CC -16 -18 -150  +  •  i -  -.100  -50  .0  50  100  150  Position in Y axis [mm]  Figure 4.36: Backlash errors of Y axis and sub-regions with constant backlash errors  99  Chapter 4. Error Measurements by Laser Interferometry R e v e r s a l Error on Z axis (  0  O w  0)  4 0  2  o  r; 1  - - t >0  r  J,  s  0  0*  </)  r—-  I_  o  0  10 0  ^..1..°..  £ 0  - -2 HI ^  r - H r— o - ^ < 10  0  0  •4—»  1  O  1°  >  -4 0 „  1  1  -100  O measured reversal errors — — sub-regions with constant reversal errors 1 1 1  -50  0  50  Position on Z axis [mm]  Figure 4.37: Backlash errors of Z axis and sub-regions with constant backlash errors 4.5 Summary In this chapter, the data obtained from the measurements which were carried out on a three-axis vertical machining center and the models for the individual error components created by the application of least squares curve fitting on the data are presented. Using the error models, it is possible to predict each geometric error individually at any position of the tool within the work volume of the machine tool. The individual error models are fed back into the mathematical model which calculates the total geometric error between the tool and the workpiece at any position. The total geometric error coming from linear positioning errors and angular deviations is dependent on the position in the work volume of the machine, the axes that the motion takes place and the direction of motion along the axes. A n angular deviation with a magnitude of 1 arcsecond produces a linear positioning error of 2.4 micrometers over a travel of 0.5 meter. For example, when the machine moves from 0 to 100 mm along X axis, the actual position that the tool reaches can be calculated as following:  Coordinates of the initial and end points: X i = 0 mm  X = 100 mm  Y i = 0 mm  Y2  2  = 0 mm  (4.23)  Chapter 4. Error Measurements by Laser Interferometry Z2 = -200 mm  100  Z2 = -200 mm  In order to move the tool to the specified position, the positions in X and Y should be given to the machine with a negative sign because X and Y motions are provided by the table. Therefore the coordinates that should be commanded to the machine are (-100, 0, -200) mm. The actual position that the' machine moves to are calculated as the sum of the desired position and error component: X=  -100-0.016 = -100.016 mm  a  Y=  0+0.0001 =0,0001 mm  Z =  200 - 0.0042 = 199.9958 mm  a  a  v  '  "... .  ,-'(4.24)  The total error in X direction is calculated as given in Eq. (4.25). The sum of the linear positioning error and effect of the pitch error of X axis on the linear positioning accuracy is the total positioning error of the tool. The linear positioning error in X direction calculated at 100 mm is 16.211 pm while the pitch of X is 6.14 arcsecs and the yaw of X is -0.15 arcsecs. E = (-Z, +Z +Z,+z x  2  e  )e {x) + (5,0:) - £ 0 ) y , = 16.211 pm y  z  (4.25)  where Z i , Z are the machine constants (given in Figure 3.10), 2  x y, and Zt are tool dimensions, ti  x , y , Ze are the coordinates of the destination point. e  e  The first term in Eq. (4.25) represents the influence of the pitch of X on the linear positioning accuracy along X axis while the second is the linear positioning error of X axis. The last term is the effect of yaw of X . Out of 16.211 pm, 5.1603 pm is coming from the pitch deviation of the X axis while 0.03 pm error is coming from the yaw error of X . The biggest contribution is the linear positioning error of X axis, which is equal to 11.0215 pm.  If the intended motion is not only in X axis, but also in Z axis such as the tool goes from (0, 0, 0) mm to (100, 0, -200) mm, the coordinates of the actual end position can be given as:  Coordinates of the initial and end points:  Chapter 4. Error Measurements by Laser Interferometry X i = 0 mm  X = 100 mm  Yi =0mm  Y = 0 mm  Z2 = 0 mm  Z = -200 mm  101  2  (4.26)  2  2  X = -100-0.012 = -100.012 mm a  Y=  0 + 0.0126 = 0.00126 mm  Z =  200 + 0.266 = 200.266 mm  a  a  (4.27)  Since the motion is not only in X axis, it is not possible to use Eq. (4.25) to derive the total error component in X direction. Eq. (4.28) is used to calculate the error component for the motion in X and Z axes: E = 6 (x) -£ (x)y,+ x  x  z  (-Z, + Z + Z, + z )£,(*) + y z, = 12.0733 pan 2  e  z  (4.28)  The contributions to the total positioning error for the motion from (0, 0, 0) mm to (100, 0, -200) mm can be given as following: out of 12.1315 um, 11.0215 pm comes from the linear positioning error while the effect of the pitch of X is 2.5 uxn, the effect of yaw of Z is 1.42 pm and the effect of yaw of X is 0.03 pm on the geometric error component in X axis.  The errors in Z axis are much larger than the other two components due to linear positioning errors along Z axis. According to the mathematical model developed in Chapter 3, when there is not'-any motion in Z axis, the error component in Z is due to the pitch error of X axis for the case in this research which only deals with the angular (pitch and yaw) deviations and linear positioning errors of the machine tool.  As evident from the examples given in the section, the total positioning error is affected by the magnitudes of the individual errors, travel distance, direction of motion and the number of axes that the motion takes place in. The calculated error components in three directions are the inputs to the compensation schemes that are explained in the following chapter.  Chapter 5 Error Compensation and Experimental Results  5.1 Overview The compensation schemes used for linear and circular interpolations for geometric errors in C N C machine tools are explained in this chapter. The influence of backlash on geometric accuracy along a path is presented, and its measurement on a vertical C N C machining center (FADAL) is demonstrated. The compensation algorithms are integrated to a virtual C N C software through a user friendly interface, which is presented by relating the input parameters to the measurement and mathematical models given in earlier chapters.  There are hardware and software approaches to compensate the geometric errors of the machine tools. In a hardware approach, a microprocessor is generally used to store the error model with its current parameters. The coordinates of the tool tip in a fixed reference system are tracked and the error compensation offsets given to the machine in real time, where the compensatory motion can be obtained by deleting or adding reference pulses in the forward loop or from the feedback circuit of the N C system [14]. The hardware approach is more useful in the development of a new machine controller. In most production environments, making hardware modifications to an existing machine might be very difficult due to the technical limitations of existing machine controllers. In such cases, software approach becomes essential for error compensation of machine tool motions. The N C program to be executed is preprocessed or modified based on an error model of the machine. In this research, the method of offline error compensation (preprocessing) is investigated in order to predict and compensate geometric errors of the machine tool in a virtual C N C environments.  Compensations given to a machine tool may be divided into three categories [27];  (1) Tool Dimensions: A variety of tools, are used according to the aim of the cutting operation. That is why the tool length and tool diameter should be considered during the planning of the tool path. By compensating for the tool diameter and length a 102  Chapter 5. Error Compensation and Experimental Results  103  proper cutting tool and workpiece relationship is maintained. Changing the tool diameter and length is actually modifying the kinematic constants of the machine as presented in Chapter 3.  (2) Geometric Error Compensation: The second category is the compensation of the geometric errors of the machine tool in the workspace. Compensations that should be given are dependent on the position in the workspace, and the trajectory of motion. In this chapter, the computation of compensations for three different situations are addressed: •  Compensation when moving to a target point in the workspace,  •  Compensation for motion along a straight line,  •  Compensation for motion along a circle.  (3) Backlash Compensation: The compensations for backlash generally depend only on the axis and the direction of motion along the axis. It will be independent of the position along the axes and, consequently, in the workspace. This might not be true for machines in which a leadscrew is worn in specific positions along its length. The leadscrew should be divided into constant backlash error zones in such a case. For F A D A L VMC-2216, the backlash errors along the axes are not constant, especially towards the ends of the axes. Therefore, the leadscrew of three axes are divided into sub-regions where the backlash errors can be considered as constant. 5.2 Geometric Error Compensation The first section in this part presents the scheme for computing the error-compensated position in order to move the tool to a desired location in the presence of errors. In the second and third sections, the results of compensation at a point are generalized to compensate errors along the two kinds of basic motions (linear interpolation motion and circular motion) of a C N C machine tool. The geometric compensation is based on point to point motion and giving the necessary error corrections to these points in three directions. The geometric errors are analyzed in  104  Chapter 5. Error Compensation and Experimental Results  three different sub-categories  to clarify the error compensation schemes in different  interpolations.  5.2.1 Compensation when moving to a target point One of the simplest compensations when moving to a target point is to move the machine axes to the position which is the desired position subtracted by the error as;  Actual position of the tool  _  Error-compensated position of the tool  (5.1)  X =X +e a  d  X =X - e c  d  (5.2)  where Xd is the desired position and e is the error at the desired position.  If the machine is commanded to go to 100 mm and the calculated error at 100 mm is 5 microns in X direction, then the machine should be commanded to 99.995 mm, whereas the actual position of the tool is 100.005 mm without any error correction. This compensation scheme, which is trivial to implement in real-time applications, is also used in the work done by Rahman et al. [40]. Instead of using this basic algorithm, another method was developed by Chen et al. to decrease the amount of residual errors at the target point [10]. In their work, they developed a recursive algorithm to calculate the residual errors at the error-compensated positions. They compensate for the errors until the residual errors remain under a prespecified tolerance value, which can be the resolution of the machine tool controller. The recursive method is explained below.  Supposing that Xd represents the desired tool position whereas the error-compensated position is shown with X , the error at the desired tool position can be expressed by: c  e(X ) d  = X -X d  (5.3)  c  Thus, the error compensated position is then equal to: X = e  X -e(X ) d  d  (5.4)  105  Chapter 5. Error Compensation and Experimental Results  When the tool is commanded to move to the compensated position, X , the actual position c  reached by the tool is given by the following equation; X = X +e(X ) a  c  = X -e(X )  c  d  + e(X -e(X ))  d  d  (5.5)  d  where X is the actual position that the tool will reach after compensation, a  e(X ) is the error at the error-compensated position. c  Consequendy, the residual error of the tool after compensation, which is the difference between the actual position and the desired position, can be obtained by the following: e = X - X = e(X - e(X )) - e(X ) a  d  d  d  (5.6)  d  Based on the first error-compensated position X and the first residual error e, the second c  error-compensated location can be established as follows; X (2) = X -e c  c  = X -e(X d  d  -e(X ))  (5.7)  d  where X (2) is the second error compensated position. c  The actual position reached by the tool after introducing the second error compensation is given by the following: X (2) = X (2) + e{X (2)) = X B  c  c  d  e(X - e(X )) + e{X - e(X - e{X ))) d  d  d  d  d  (5.8)  The residual error can be given by the following: e(2) = e(X -e(X -e(X )))-e(X -e(X )) d  d  d  d  d  (5.9)  Similarly, it is possible to establish the third error-compensated position. Thus a recursive scheme of compensation is developed by Chen et al. [10] and used to iteratively calculate the error-compensated position until the difference between the actual and the desired positions is kept smaller than a predefined tolerance value. Figure 5.1 depicts a flowchart of the recursive scheme of compensation.  Chapter 5. Error Compensation and Experimental Results  X -^e(i-l)  w  d  Predict error  X (n) r  106  e(X (n) c  t  Error-compensated position  Figure 5.1: Flowchart of recursive scheme of compensation 5.2.2 Compensation for motion along a straight line During a linear interpolation, the controller automatically calculates a series of tiny single departures to move from its current position to a specified destination with a linear requirement on the cutter path. In this section, the procedure for compensation at a point is generalized to compensation along a straight line.  Every point on the linear motion can be expressed using the start point (Xj) and end point (X2) such as; ££[0,1]  X = X +k(X -X ) l  2  l  (5.10)  Firstly, the compensation is applied to the start (k =0) and end (k = 1) points of the line. Using the compensation scheme at a point, presented in the previous section, the errorcompensated positions at the start and end points can be computed using the compensation scheme at a point, which is presented in the previous section. The new linear trajectory, X , c  can be derived using the two new error-compensated positions as: X  r  = X .+k(X -X .) u  2c  H  ke[Q,\]  (5.11)  Since the parameter k is only a scaled and displaced image of X and consequently a oneto-one correspondence remains between the original and modified trajectories, it can also be  Chapter 5. Error Compensation and Experimental Results  107  used for the new line, X , given in Eq. (5.11). The residual errors that remain uncompensated c  by only introducing corrections at the starting and ending points can be expressed as: e=  X +e(X )-X c  c  = e(X +k(X -X )) lc  2c  lc  + (X -X ) lc  l  + k[(X -X )-(X -X )] 2c  2  ic  i  "  (5  12)  k e [0,1] After the compensation is applied to the start and end points of the linear trajectory, the residual errors of the points that lie along the line can be checked. If the square of the magnitudes of the residual errors at all points of the trajectory are kept less than or equal to the square of the predefined tolerance value, the positions of the tool will be kept within a sphere with a radius equal to the tolerance value, from each point along the trajectory. The errors are generally greater than the predefined tolerance value due to the nonlinear variation of the geometric errors in the workspace. In this case, the same procedure is repeated by breaking the original straight line up into two parts as shown in Figure 5.2. The breaking point of the line is determined by the position at which the square of the residual error has its maximum [10].  Figure 5.2: Determination of point at which original straight line is to be broken in the compensation scheme along a line  Chapter 5. Error Compensation and Experimental Results  108  5.2.3 Compensation for motion along a circle Circular motion in machine tools is probably the most commonly specified motion in N C programs after linear motion [27]. Circular paths produced by two linear axes are influenced by geometric deviations of the two axes and by deviations caused by the numerical control and its drives. The geometric inaccuracy of a machine tool can influence the overall accuracy of a circular path in several ways such as the influence of a progressive linear positioning deviation, influence of non-perpendicularity, and influence of periodic deviations. The radius should be kept constant along a circle. However, if one of the movements in one axis is longer than the movement in the other axis due to a scale deviation, then the circular path is changed to an ellipse with its major diameter parallel to the longer axis as shown in Figure 5.3.  Compensations for circular interpolations are performed only in planes parallel to the coordinate axis because three-axis C N C machines are not capable of performing circular interpolation in oblique planes. In the following compensation scheme, it is assumed that circular interpolation is performed parallel to the X Y plane and the tool is mounted in Z direction as it is the case in the F A D A L Vertical Machining Center [27].  (1) nominal path , (2) actual path Figure 5.3: (a) Influence of long movement of one axis, (b) Influence of nonperpendicularity of axes on circular paths  Chapter 5. Error Compensation and Experimental Results  109  The approach in circular motion compensation is very similar to the compensation for motion along a straight line, which is based in point-to-point compensation [27]. The procedure can be explained as the following: First, the compensation is applied to the end points of the trajectory. In this case, instead of two end points, three points, two end points, and the mid-point of the arc are required to define a circular motion. Figure 5.4 depicts the overall approach behind geometric error compensation along an arc or a circle.  -•X Figure 5.4: Compensation for motion along a circle  Considering that the tool is commanded to move along an arc PQ with its center at point C and the mid-point of the arc as the point S, the coordinates of points P, Q, S, and C can be specified as: P =  >  -  :  '  (P ,Py) X  (5.13) C = (c ,c ) x  y  P', Q', and S', which are the error-compensated positions, can be found by the following equations; P>=P-e(P) = (p ',p ') x  y  Q'=Q-e(Q) = {q ',q ') x  y  S'=S-e'S) = (s ',s ') x  y  where e(P), e(Q), and e(S) are the errors at P, Q and S.  (5.14)  Chapter 5. Error Compensation and Experimental Results  110  If the center and the radius of the arc passing through P', Q', and S' are C = (c ',c ') and x  y  r' then it should satisfy the equation of circle; (x-c ') +'y-c *) =r 2  2  x  (5.15)  2  y  for points P', Q' and S'. Therefore;  ov-o'+ov-o ^ =» 2  2  (P, ' ) + (Py l = r - {c ') - (c, ') + 2(p >c '+ 'c ') 2  2  2  2  2  x  x  (^-c l +(q '-c/) =r 2  2  x  ' ) + ( 9 , ' ) = r - (c ') - (c, ') + 2 ( 2  2  2  2  x  (^-^') +(^-c/) =, 2  Py  y  =>  2  y  2  x  2  9 x  'c '+9, 7:,') x  =>  2  ') + (J, ') = r - ( O - (c, ') + 2(,, -c,'+ 'c,') 2  2  2  2  2  Jy  This can be rewritten in matrix form as:  W ) (0  + Q>/) '  2  2  2  +(0  2  2P '  2Py'  1  29/  2q '  1  2s '  25,.' 1 r ' - ( c / ) - ( c / )  X  y  (5.17) 2  2  2  By solving the above equations, the coordinates of the center C'= (c ',c ') and the radius x  y  of the arc P ' S ' Q ' , r' can be calculated.  Because the geometric errors cause a non-linear distortion of the work volume of the machine, one can also check if the intermediate points on the corrected arc lie within a prespecified error limit of the desired circular trajectory if needed. The checking algorithm of mid-points is not applied in this research.  -  5.3 Backlash Compensation In order to develop a compensation scheme for the effects of backlash on the volumetric accuracy of a machine tool, it is necessary to know the important characteristics of backlash  Chapter 5. Error Compensation and Experimental Results  111  errors. For a motion system the backlash is defined as the lost motion after reversing direction. It may be existent on any axis and the magnitude of this error does not usually depend on the position along the axis and along the other axes of the machine. Generally, it is reasonable to assume that backlash remains almost constant along each axis. According the experimental studies done by Ferreira and Kiridena [27], the backlash errors are not significantly affected by the thermal effects on the machine. The C N C machine used in this research (FADAL) indicated various backlash errors along each axis of its axes. Two backlash compensation methods can be employed. One of them is to use different calibrations for forward and backward directions of motion as explained in Chapter 4. The following backlash compensation developed by Kiridena and Ferreira [27] is easier to implement and more effective in terms of computational time because the machine is calibrated for one direction and the motions in the other direction are modified depending on the backlash errors of the axis.  The nature of backlash can be explained using a two dimensional example shown in Figure 5.5 [27]. In the work developed by Kiridena and Ferreira, the tool is programmed to move from point Pn to point P5 in the X Y plane. It is assumed that the motion along X is considered to change in direction but the motion along Y is considered to increase and there is no motion in Z direction. x ,y are the coordinates of the point n, eb is the value of the n  n  x  backlash error on X axis whereas e\, . is defined as the value of the backlash error on Y axis. y  Initially the cutter is programmed to move along [P  0  P  X  P  P  2  P  3  4  P ]. However, 5  due to backlash errors in X axis, it does not move along its commanded path, but instead it moves along \P  O  along the  P  X  P  P  2  P  2  compensated  movement will be along [P  path 0  P  3  X  4  P \. If the cutter is programmed to move 5  [P  P  X  P  2  P  P  P  3  P  A  P ] which is the intended motion. The four  Q  P  P  A  2  2C  P  3C  P  4C  P  4  P\ 5  then  the  5  possible cases of backlash compensation developed in [27] are illustrated in Figure 5.5.  Case 1:  (e.g.  n=l)  actual  Chapter  5. Error  Compensation  and Experimental  (e.g. n = 2)  Case 2:  Case 3:  Case 4:  112  Results  X  n+l <x  X<X  (5.18)  (e.g. n = 3)  n-l  (e.g. n = 4)  If the machine is considered to be calibrated for forward motion, the backlash error is zero. However when the machine reverses direction of motion, the backlash error is Motion from point Pi to P2 is characterized by Case 1. The motion is in the forward direction without any reversal motion in X axis. Therefore no compensation is required at point P2 to move the tool in the programmed path. Motion from P2 to P3 is an example of the second case, which involves a direction change in X axis. The direction of motion changes, from forward to backward in X axis at point Pi. The error due to backlash can be corrected in two steps. First the tool has to be shifted in the negative direction by a distance equal to the backlash, e^ and then the target point should also be moved in the negative direction by et,. The third case is illustrated by the motion of the tool from P to P . It requires point P4 to be 3  4  compensated for backlash. Motion from P4 to P5 has to be carried out in two steps. First, the tool should be moved in the positive X direction by a distance equal to backlash eb and then it should be moved to P5. This is an example for Case 4.  The explained algorithm corrects the motion so that the motion in the negative direction along the axes is always corrected by the magnitude of backlash. A correction is only made when the machine reverses itself from motion in the negative direction to motion in the forward direction. When one wants to integrate this backlash compensation algorithm with the compensating algorithms for geometric inaccuracies, the backlash compensation will have to be performed after the other compensations have been introduced.  An implementation of the scheme is shown in an example on F A D A L V M C . Assuming that the tool is commanded to perform a series of linear interpolations in X axis, the geometric error and backlash error compensation are applied to the given specified positions.  113  Chapter 5. Error Compensation and Experimental Results  Case 4  Case3  Case 2  Casel  '  • X Figure 5.5: Backlash errors and compensations  The desired coordinates of points [mm] that the tool should follow are: P ,= (0,0,0) 0(  P = (50,0,0) ld  P  = (70,0,0)  P  = (40,0,0)  P  = (20,0,0)  2d  3d  Ad  P  5d  = (60,0,0)  The calculated geometric errors for the motion only along X axis at the positions specified in Eq. (5.19) are: E, 00 = 9.7527  £ , ( y ) = 0.0025  £ , ( z ) = -1.1985  E (x) = 12.3855  E (y) = 0.0458  E (z) = -2.1830  E (x) = 8.4151  £ ( y ) = -0.0453  £ ( z ) =-0.8066  E (x) = 5.7057  E ( y ) =-0.1282  £ ( z ) =-0.2202  £ ( y ) = 0.0307  £ ( z ) =-1.6571  2  3  A  £ (JC) S  = 11.0767  2  3  4  5  2  3  4  5  [pm]  (5.20)  114  Chapter 5. Error Compensation and Experimental Results  The  coordinates  of actual  positions  that  tool  would  follow  without  any error  compensation are: P  =(50.010,0,-0.001)  P  =(70.012, 0,-0.002)  P  =(40.008,0,-0.001)  P  = (20.006,0,0)  la  2a  3a  Aa  P  [mm]  (5.21)  =(60.011,0,-0.002)  5a  Using the recursive compensation algorithm given in Section 5.2.1, the coordinates of the compensated positions are calculated as: p =(49.9902,0, 0.0005) ic  P  = (69.9876, 0,0.0008)  P  3c  =(39.9916,0,0.0004)  4 c  = (19.9943, 0,0.0001)  5c  = (59.9889,0,0.0007)  2c  P P  [mm]  (5.22)  Since the resolution of F A D A L V M C - 2 2 1 6 is 1 micrometer, it is not possible to command the machine to go to a position specified by less than a micrometer. Instead, the machine is commanded to go to the following positions: P = (49.990,0,0.001) ic  P  = (69.988,0,0.001)  P  = (39.992,0,0)  2c  3c  P = 4c  P  5c  [mm]  (5.23)  (19.994,0,0) =(59.989,0,0.001)  The backlash value of X axis in the range of -10 mm to 90 m m is "constant and equal to 2.42 pm. The backlash compensation is applied according to the change direction in X axis, since there is not'any motion in Y OF Z axes. Table 5.1 shows the identification of the direction changes in the motion.  Chapter 5. Error Compensation  and Experimental  Results  115  Table 5.1: Identification of the direction changes Origin  Destination,  Case #  (0, 0, 0)  (49.990, 0, 0.001)  Case 1  (49.990, 0, 0.001)  (69.988, 0, 0.001)  Case 1  (69.988, 0, 0.001)  (39.992, 0, 0)  Case 2  (39.992, 0, 0)  (19.994, 0, 0)  Case 3  (19.994, 0, 0)  (59.989, 0, 0.001)  Case 4  Using the algorithm explained in the section, the following coordinates should be given to the tool in order to compensate for the backlash as well as the geometric errors,: P= 0c  P  lc  (0,0,0)  =(49.990,0,0.001)  P = (69.988,0, 0.001) 2c  7V= (69.986,0,0.001) [mm]  (5.24)  P = (39.990, 0, 0) 3c  7V= (19.992, 0,0) P = (19.994,0,0) 4c  P  5c  =(59.989,0,0.001)  It is possible to handle the backlash in three axes using the backlash compensation developed by Kiridena and Ferreira.  5.4 Automation of Geometric Error Compensation Algorithms A Visual Basic program is developed and interfaced to the laboratory's Virtual C N C software system. The software runs on PCs and computations are carried out in a M A T L A B environment which is linked to a Virtual C N C system.  The program can calculate the effects of the measured geometric error components throughout the machine volume in order to determine a measure of the volumetric accuracy of the machine.  Chapter 5. Error Compensation and Experimental Results  116  The compensation program is designed to work with text files which store the necessary information for the calibration of measured errors. The output files of the laser interferometry software can be easily converted to text format without any loss of any information. Thus, the file with text format includes the nominal position along the axis and error readings for forward and backward directions. Using the information provided by the laser interferometry system, the program assists the user in the generation of individual error models by employing the Basic Curve Fitting Toolbox in M A T L A B . After the measurement data for each individual error is analyzed and fitted to a curve, the user is able to enter the necessary information for the NC-program modification.  The  specified NC-program is imported from a C A D / C A M system to display the  commanded cutter motions and locations. However, the real machine tool deviates from the command positions due to-geometric errors. The virtual C N C . system with integrated geometric error correction algorithms can predict the errors and modify the command tool positions for compensation. The corrected N C programs can be sent to an actual machine which leads to improved accuracy. The software integrates the machine tool error compensation by guiding the user interactively as follows:  (1) Define the machine tool configuration and provide machine specific details such as the travel lengths of the axes (See Figure 5.6).  (2) Select geometric error components and enter the stored measurement result for each type of error for each axis.  '  (3) Plot the selected error and choose the curve which fits to the measured data best using the M A T L A B Basic Curve Fitting Toolbox. Copy the coefficients of the curve and paste it into the dialog box in the program.  (4) View the accuracy, repeatability, and systematic deviations for the linear error type if required.  Chapter 5. Error Compensation and Experimental Results  117  (5) Enter the constant offsets of the machine tool such as the tool offsets.  Volumetric Erros M a c h i n e Information; f ~  Data Files  J  Results  NC-Tape Correction  P l e a s e c h o o s e the m a c h i n e configuration from the g i v e n options b e l o w for a three axis machine tool. The letters coming before F show available motion directions of the work piece with respect to the reference frame and the letters aftei F show the available motion directions of the tool (probe) with respect to the reference frame.  XYFZ  FXYZ  r  XFYZ  Machine Information  Enter the name of the machine X Axis Information  Travel Length:  - J~  / +f  -  /+  Y Axis Information  Travel Length:  [  Z Axis Information  Travel Length:  Machine Configuration  MACHINE XYFZ : X and Y axes motions are associated with the movement of the table whereas the tool provides the motion in Z axis.  Help  Cancel  »  Next  Figure 5.6: Machine tool configuration dialog box  (6) Choose the type of interpolation and enter its parameters. In linear interpolation, only the coordinates of the start and end points are required whereas the coordinates of the center are additionally required in circular interpolations. Instead of modifying the tool path line by line, another option is to load the NC-program. The output of a loaded NC-program is the modified trajectory stored in a new file.  (7) View the summary of the machine information and the results, including the error components and the error-compensated positions.  Chapter 5. Error Compensation  Machine Information  J  and Experimental  Data Files  jf  118  Results  NC-Tape Correction  Results  f~  Error Types / Raw Dala PlottingC  Lineal  Browse  Plot  C  Peiiodic  Browse  Plot  rr  Roll of X [X ot X]  Browse  Plot  r  Pitch of X (Y of X)  Browse  Plot  r  YawofX(ZofX)  Browse  Plot  P  Y Straightness of X  Browse  Plot  C  Z Straightness of X  Browse  Plot  • Curve fitting result for the selected error type ——  •—  ^—  Enter (he coefficients of the equation that best fits the entered data for the selected error type:  ACCEPT FORWARD COEFF.  |  ACCEPT BACKWARD COEFF.  Statistical Analysis for the linear positioning errors Accuracy  I  Repeatability  Help  Systematic Deviations  Cancel  Back <<  >> Next  Figure 5.7: Loading the errors measured for X axis  Defining a Machine Configuration: This operation allows the user to define a configuration appropriate to a specific machine tool. The software is designed for use with any machine with X Y F Z type configuration but it is easy to improve the structure of the program so that it can handle any type of three-axis machine tool. Initially, the user is required to select the type of the machine and enter its axes travel lengths. After the configuration is selected, the next step is to load the errors measured.  Loading of the Measured Errors: For each axis, 7 types of error are defined in this section; linear positioning errors with the periodic terms, straightness errors in horizontal and vertical directions, and angular errors such as roll, pitch and yaw deviations. The text file can be browsed and the errors can be  Chapter 5. Error Compensation and Experimental  Results  119  plotted. The linear positioning error is assumed to be entered in micrometer units whereas the angular deviations are assumed to be entered in arcsecond units. After the plot button is pressed for any type of error, the user can display the measured errors both in'forward and backward directions of motion. At the same time, the basic curve fitting toolbox can be used to choose the best curve representing the fneasured data. The user shall copy the coefficients of the curve he/she chooses and paste it into the following dialog box so that the equation representing that error is defined in the main program (See Figure 5.7). The program displays the accuracy, repeatability and systematic deviations for linear positioning errors, which assist the user to compare two machine tools.  'PS Yqlumettji_c IK^CS, Machine Information  J  Data Files  Results  J NC-Tape Correction |f~  Constant Offsets for the XYFZ-type machine tool  —  Enter the constant offset between the table and zero position of the Z axis: Enter the constant offset between the saddle feedback device and the top of the table: Enter the constant offset between the table feedback device and the top of the table: Tool Offsets:  CREATE FILE  X"  r-NC-Program Data File Import Browse  Please browse the NC-Program  Circular Interpolation—'•  -Linear Interpolation  Enter the coordinates of the initial point:  To locate the tool to a target position in the workspace:  (X1.Y1.Z1)| Enter the coordinates of the final point:  Enter the coordinates of the initial point:  X1 Y1 Z1  (X2. Y2. Z2)j  | Enter the coordinates of the center of the circle:  Enter the coordinates of the final point:  X2 Y2 Z2  I  »  Next I  (Xc. Yc. Zc)| Enter the direction of the circle: C  Clockwise  C Counter-clockwise  Help  Cancel  Back <<  Figure 5.8: NC-tape modification for linear and circular interpolations  Chapter 5. Error Compensation and Experimental Results  120  NC-Program Modification: The program does either predictions of geometric errors along the tool path or compensates them automatically based on the error and correction methods presented in the thesis. The program requires some information from the user such as the cutting tool geometry, the distance from the feedback device to the top of the table, the distance from the table, and the distance from the table to the zero position of the spindle. The user can modify the NC-program line by line or a new file with the modified NC-program can be automatically generated by specifying the file which consists of the original NC-program that needs to be modified. The backlash errors are also taken into consideration in N C modification. For circular interpolation, the coordinates of the initial and the final points are entered in addition to the coordinates of the center of the circle (See Figure 5.8). The software calculates the errors as well as the error-compensated positions along the tool path. 5.5 Experimental Results  .  A set of milling operations were carried out on the F A D A L V M C - 4 0 C N C Milling Centre in order to verify the geometric error model of the machine tool and the error compensation strategies presented  earlier. Two parts were machined, one without  compensation to compared the predicted and measured errors, and the second was machined with the compensated N C program within a Virtual C N C environment. The part geometry reflects machine tool testing standards defined by ISO 10791-7 [25]. The dimensional accuracy of the part is discussed, while presenting the influence of non-geometric errors such as C N C contouring and structural deformation errors caused by cutting forces. 5.5.1 Cutting Tests  The cutting tests have been performed on the F A D A L VMC-40 C N C Milling Centre to verify the overall accuracy of the workpiece improvement during linear and circular interpolations.  The test part contains a diamond with four linear interpolations and one full circle, which allows the testing of geometric errors with linear and circular interpolations. The three-axis machine tool ( F A D A L VMC-2216) is controlled with an in-house developed open C N C , ORTS [2]. ORTS allows the testing of real algorithms such as interpolations, control and  Chapter 5. Error Compensation and Experimental  Results  121  trajectory generation laws, in the machine tool. In total, two parts were machined: the first part was machined without any error compensation, whereas the second was machined with error compensation offsets calculated using the mathematical model of the machine tool as presented in Chapter 3. The toolpath of the workpiece and the nominal dimensions are shown in Figure 5.9. The error compensation was tested only in the X - Y plane due to limitations on the Z axis controller. The photograph of machined parts is shown in Figure 5.10.  The coordinates of the points A , B, C, and D are calculated in the ( X , Y ) coordinate P  P  system (part coordinates) for the linear interpolations as shown in Figure 5.9. Then, they are transferred to the machine coordinates. The error offsets for linear interpolation are derived at the calculated locations in the machine coordinates and the coordinates of the corresponding points are changed by the error compensation offsets in the second workpiece as explained in the chapter.  ido  70  Figure 5.9: Top view of the second and third workpiece (dimensions given in mm)  Chapter 5. Error Compensation and Experimental Results  122  Since it is not possible to make any change in Z direction during motion, the axial depth of cut, i.e. Z axis, is set at the desired depth. Also, due to the same reason, the diamond and the circle are combined with a linear interpolation to avoid the motion in Z axis during cutting as shown in Figure 5.10.  The accuracy of both parts, with and without geometric error compensation, was measured using Brown and Sharpe 50 Validator, which is a manual three-axis coordinate measuring machine (CMM).  Figure 5.10: Finished workpieces  5.5.2 Linear Interpolation Test Results Since we do not have a stationary reference which is valid for the machine tool and the coordinate measuring machine, it is impossible to detect the absolute errors at points A , B, C, and D. Instead, the errors of the length of each arm of the diamond can be determined. The error correction offsets are calculated at the tool tip. Therefore it is necessary to track and  Chapter 5. Error Compensation and Experimental Results  123  measure the path that the end mill follows. However, due to the diameter of the end mill, it is not possible to follow the cutter center on the coordinate measuring machine. Therefore, first the side walls of the arms were measured. The center points of the cutter along the tool can be determined from the mean line passing in between the edges. This method eliminates the effect of the diameter of the probe that,was used in the measurements on the C M M as well.  Part - 1 Without compensation i  1  -100 -80  1  -60  1  -40  1  1  -20  0  TI  20  1  1  r  40  60  80  100  Position on X axis [mm] Figure 5.11: Results of measurements along the diamond for the uncompensated case  The arms are named beginning from point A to point B as the 1 arm, from B and C as st  the 2 , C to D as the 3 , and D to A as the 4 arm. nd  rd  th  Any point on the workpiece can be chosen as the origin of the workpiece coordinate system and all calculations and measurements are done with respect to this origin. As illustrated in Figure 5.11, random points are measured along the sidewalls of each arm. The measured points along the lines are curve fitted using least squares fitting technique and the mean line passing between the two sidewalls is identified as the path that the cutter center  Chapter 5. Error Compensation and Experimental Results  124  followed. The intersection points of four arms provide the coordinates of points A, B, C, and D, which are used in the calculations of the length of each arm. The calculations are as follows:  For the external sidewall of the first arm, [AB], the following data are derived from the measurements: x = [13.755 -1.679  -7.300  -21.926 -38.758 -53.013  y = [12.168 3.220 0.042 -8.499 -18.087  -26.365  -70.366]  -36.390]  (5.25)  For the internal sidewall: * = [-60.473 -53.099  -40.790..-29.025  y = [-39.870 -35.621 -28.494  -21.653  -15.388 -0.496 13.936] -13.752 -5.202  3.126] •  (5.26)  The lines passing through the external and internal edges can be derived from the measurement data as: > W « =0.60925* -6.2954 3 W - = 0.60867* + 3.0463 y x =0.6090* -1.6245 Arm  (5.28)  The same procedure is followed for the remaining three arms: y rm2 A  = -1 -64785* -174.51 for [BC]  yArmi ~  y ,4 Am  0.6076*-130.255 for [CD]  (5.29)  =-1-14885*+ 37.429 for [DA]  Using the equations (5.28 and 5.29), the coordinates of the intersection points of the arms can be given as follows: A - (20.3418,11.4049) B = (75.0562, -83.7572) (5.30) C = (-20.1996,-138.8408) D = (-75.0659, -43.6944)  v  '  125  Chapter 5. Error Compensation and Experimental Results  Hence, the lengths of the arms are calculated as: AB X  ~xj + (y -yA!J = 109.7702  h_ o  =  h_  = Vlfc - B J+(y  ^ _  = Vita  --^c)  =^\L A  - DJ +(y  mC  m  B  J \=  3  c  M  4_ com  l  X  un  2  1  - y B  X  uncom  iO-0358  mm mm  (5.31)  +(v -y C ) J= 109.8324 mm 2  D  -y J\ = -1751 mm 110  X  A  D  The nominal value for the four edges is given as: / = ^1(20.13 + 75.13) + (75.13 - 20.13) J = 109.9976 mm 2  2  (5.32)  The difference between the nominal length and length of the arm gives the errors for the geometrically uncompensated linear motions: = -227.4071 £V2_ )  = 38.1889 pm  uncom  £(h_  )  uncom  .  =-165.2154 pm  ^•(/ ,„,,„„,) = 177.5628 pni 4  Results of measurements along the diamond where the geometric errors of the machine are compensated using the algorithms presented in the chapter are shown in Figure 5.12. The lengths of the edges and the errors of the compensated  arms are calculated using the  same procedure as presented for the uncompensated case.  For the compensated case, the lengths of the edges and the errors are calculated using the same procedure: 109.9479 mm  for [AB]  = 109.9146 mm  for [BC]  h_cam =  l_ 2  com  = 109.8408 mm  for [CD]  h com = 109.9443 mm  for [DA]  h_co  m  (5.34)  Chapter 5. Error Compensation and Experimental Results  126  = -49.6408 pm -82.9865 / ^  £(!i_coJ  =  e(h_«J  = "156.7529 pm  = "53.3179  ^  Part - 2 With compensation i  -100  1  1  1  1  1  71  -80  -60  -40  -20  0  20  1  1  40  60  1  80  r  100  Position on X [mm] Figure 5.12: Results of measurements along the diamond for the compensated case  Although the error in the second length increases with the compensation, the sum of absolute errors decreases by almost 50%. 4  Uncompensated case:  = 6O8.3742/0n  (5.36)  4  Compensated case:  = 342.698Ipm  (5.37)  i=l  The reason why the errors are large even in the compensated case may be the other error sources which were not considered in compensation. The factors that influence the accuracy  Chapter 5. Error Compensation and Experimental Results  127  of the workpiece can be static deflections, C N C contouring errors, and thermal behavior of the machine tool.  5.5.3 Results for the Circular Interpolation In order to obtain the errors-of the circular interpolation, firstly the sidewalls of the circular trajectory were measured. Then the measured points are fitted to a circle. For the :  circular regression, the Levenberg-Marquardt algorithm was employed.  The Levenberg-Marquardt algorithm finds the vector, p, that minimizes an objective function, J{p) = ^ d (p), 2  /  j  where d (p) t  is the distance from the point to the geometry  defined by the vector of parameters, p [45]. For a two-dimensional circle fit, if the coordinates of the measured points are given, it is possible to calculate the coordinates of the center and the radius when the coordinates of the measured points are given as follows:  y = [y,  y  2  y  3  y  4  (5.38)  y] 5  The unknown parameters (x ,y ) c  of the center and radius ( r ) of the circle need to be  c  evaluated from the measured points. Because of the symmetry of a circle, the minimum distance of a data point (x,y) from the boundary of the circle is given by: (5.39)  We wish to find the values of x , y , and r that minimize the total squared error for the c  c  data set. (5.40)  Chapter 5. Error Compensation and Experimental  128  Results  Eq. (5.34) gives the objective function to be minimized in the algorithm. This technique is applied both to the external and internal sidewalls of the circular interpolation as shown in Figure 5.13. The mean circle lies between the internal and external circles is the circle that the tool tip follows during the machining.  Circular Interpolation  -50  -40  -30  -20  -10  0  10  20  30  40  50  Position on X [mm] Figure 5.13: Results of measurements along the circle for the uncompensated case  The points used in the algorithm for the external and internal circles are as follows; x  Extemal  =[43.585  38.261  -42.608  -41.524  24.612  4.619  -28.560  -14.359  0.570  18.787  -32.674 33.334  43.156] (5.41) y  External  =  [-70.027  -85.656  -93.072  -74.787  -34.955  -55.234]  -100.080 -49.166  -107.418  30.332  -105.280  -19.717  -23:893  Chapter 5. Error Compensation and Experimental Results  in,er i  x  na  =[20.888  -23.539  29.717  28.099  1.552 17.224  32.624]  -3.333  -29.442  129 -33.775 (5.42)  y i«, ai =[-34.414 en  -51.531  -43.558 -36.433  -86.204 -27.797  -99.542 -32.107  -84.452 -48.711]  The unknown parameters for the external circle are calculated as: *c_£«««,/ =0.0274 mm y a w =-63.6962 mm  (5.43)  c  ^ , = 4 3 . 9 9 0 7 mm  Hence, the equation of the external circle is: (jc-0.0274) +(y + 63.6962) =1935.1817 2  (5.44)  2  Similarly, the internal circle parameters are derived as: x _i«,en«d =-0.0167 mm  •  c  r  '  y _ „ = - 6 3 . 7 2 5 0 mm  .  .  .  (5.45)  c  rhaemal =35.9716 mm  Hence, the equation of the internal circle is: {x + 0.0167) + (y + 63.7250) = 1293.956 2  2  (5.46)  Although the centers of internal and external circles should coincide with each other, there are differences both in X and Y directions such as: A =0.0274 + 0.0167 = 0.0441mm = 44. ljum  (5.47)  A = -63.6962 + 63.725 = 0.0288mm = 2S.8pm  (5.48)  x  y  The uncompensated mean circle of the internal and external circles can be expressed by its parameters as follows:  130  Chapter 5. Error Compensation and Experimental Results  x = 0.0054 mm c  y = -63.7106 mm  (5.49)  c  r = 39.9812 mm  (x-0.0054) +(y + 63.7106) =1598.4964 2  2  (5.50)  Therefore the error in locating the circle center is calculated as the difference between the nominal coordinates of the center of the circle and the calculated coordinates. The error in the radius can also be calculated: e(x ) = 0.033125-0.0054 = 0.0277 mm = 21.1pm c  (y )  £  £  c  = -63.721875 + 63.7106 = -0.0113 mm = -U3pm  (5.51)  ( r ) = 40-39.9812 = 0.0188 mm = 18.8/zm  The predicted error components at the center of the circle can be calculated as: e(x ) = 3.5pm c  £(y ) c  =-4.6pm  Although the predicted errors of the center of the circle from the geometric error model of the machine are smaller than the actual errors measured on the machined circle, the sign of the predicted errors are consistent with the actual errors left on the workpiece. The reason why there is a difference between the predicted and actual errors of the center of the circle may be the other error sources as mentioned in the Linear Interpolation Test Results. The circle was not machined with error compensation offsets because the C N C was not able to do this in real-time, except the circle center.  5.5.4 CNC and Deformation Error Sources There is a discrepancy between the predicted and measured geometric error left on the part. The error models, including their compensation, use only the geometric errors of the machine in this thesis. However, thermal and structural deformations of the machine tool, as well as the C N C dynamics leave additional errors on the part. The actual measurements on  131  Chapter 5. Error Compensation and Experimental Results  the part reflect the summation of all errors imprinted on the part. The modeling of all error sources are beyond the scope of this thesis. However, sample investigations about the static cutter deflection and C N C errors are given here as examples in order to provide quantitative comparisons of error sources in machining.  Static Deflection Errors: The end mill, which is cantilevered to the spindle, deflects under periodic milling forces during machining. The static stiffness of the end mill is experimentally evaluated from impact modal tests as well as static loading of the end mill while measuring the tool deflections with a precision, laser displacement sensor as shown in Figure 5.14. The stiffness of the end mill is identified as: k =5.814  Nlpm (5 53)  k =12.5 Nlpm  ' '  K  y  Figure 5.14: Static stiffness test set-up on F A D A L  The static deformation (8) of the end mill at its tip is; S~j-  ( 5  -  5 4 )  Chapter 5. Error Compensation and Experimental Results  132  where F, k are the cutting forces and equivalent stiffness at the tool tip. a  However, the deflection marks left on the finish surface are never equal to the tool deflection due to the helix angle and surface generation mechanism in milling. The cutting edge generates the finish surface only when its edge is perpendicular and in contact with the finish wall. The kinematics and surface generation mechanism are explained by Altintas in [1], and the application of the model is presented here.  A slot milling operation was performed with a 12 mm diameter end mill with 2 helical flutes having 30 degree of uniform helix angle. The spindle speed was 5000 rev/min with a feed rate of 1000 mm/min (chip load of 0.1 mm/tooth) and an axial depth of 5 mm, which was a chatter free cutting condition. The measured cutting forces in X and Y directions of the machine tool axes are shown in Figure 5.15. The zoomed view of the cutting forces between 22 and 22.4 seconds are shown in Figure 5.16 to illustrate the periodicity of the generated forces. The forces in X and Y axes are periodic with tooth passing frequency due to the nature of the milling, which is an intermittent cutting operation. The tooth passing period is T=  ^5000 j  12 = 0.0060 [seclwhich is also evident from the period of the measured forces  shown in Figure 5.16.  The stiffness of the tool can be calculated using the expression k - ( 3 E I ) l l where E is 3  the Young Modulus, 7 = (mi )164 is the inertia and 1 is the tool length [1]. The stiffness of 4  the tool used in the tests is calculated as 3.98 N/um. The difference between the calculated and measured stiffness values may be due to the spindle-chuck flexibility.  Using the static stiffness of the end mill and surface generation mechanism explained by Altintas [1], the maximum error left on the part surface at the tool tip must be about 0.06 mm as shown in Figure 5.17 during circular and linear paths.  er 5. Error Compensation and Experimental  Results  Forces Generated During Cutting 800  25  30  35  Time [sec] Figure 5.15: Forces generated during cutting in X and Y directions  Chapter 5. Error Compensation and Experimental Results Forces in X and Y Directions  0.p058 sep 22  22.005  22.01  22.015  22.02  22.025  22.03  22.035  22.04  Time [sec] Figure 5.16: Forces generated during cutting in X and Y directions in a short rang  Form errors left on the surface 0.06  i » H  0.05  tt tt  0.04  tt X  tt  .2 u  tt X  003  I  K X  !*  0.02  tt  tt  0.01  0  0.5  1  1.5  2  2.5 ',  3  tt  tt tt  3.5  X  1  4  tt  tt tt  X  4.5  Axial depth.of cut [mm] Figure 5.17: Form errors left on the surface of the workpiece  5  Chapter 5. Error Compensation and Experimental Results  135  The deflection marks are left in the start and exit angles of immersion due to the number of teeth of the tool. The tool leaves form errors at the two sides of the slot with the same magnitude but with a reverse sign. Therefore, the mean line that the cutter follows is not shifted resulting in the conclusion that the form errors left on the finished surface do not have a significant influence on the positioning of the tool. However, since the C M M readings were taken along the finished surfaces, they are affected by the deflection marks in a great extent. The C M M measurements were taken with the probe at a half o f the depth of cut. The magnitude of the errors at that depth is about 22 micrometers as evident from Figure 5.17.  C N C Contouring Errors: The digital control law is periodically executed at the control sampling frequency to maintain the tool position at the programmed rates in the axis servo control closed-loop. Typically, the control law decides how much control voltage should be generated by comparing the actual output measured by the feedback devices against the reference input at the same state in order to keep the error to a minimum. The tracking performance of a controller is dependent on the tracking abilities of the control laws used in the system [12]. Using the Virtual C N C System, which was developed in the Manufacturing Automation Laboratory at the University of British Columbia, the contouring errors in X and Y and the controller performance can be simulated using different control laws and trajectory generation profiles. The trajectory was generated with simple trapezoidal velocity using Sliding Mode Control during the cutting tests. Therefore, the following error in X and Y axes are simulated as shown in Figure 5.18 using the same conditions used in the cutting tests. As observed from Figure 5.18, the maximum contouring error is derived to be about 0.009 mm in X direction and 0.012 mm in Y direction during the cutting tests.  136  Chapter 5. Error Compensation and Experimental Results  E jE  o UJ  0.015 0.01  ii  i  i  .......  i i i  0.005 E  o  o  IJJ -0.005 -0.01 -0.015  10  i 15  20  25  T i m e [sec]  30  i 35  •  40  45  50  Figure 5.18: Contouring errors in X and Y axes during linear and circular interpolations  5.5.5 Discussion of Results  In order to verify the methodology, cutting tests with and without the error compensation were carried out on the machine. While doing the tests, the forces generated during cutting were measured and the structural errors left on the finished surfaces were calculated using the forces and measured static stiffness of the tool. The maximum magnitude of the form errors left on the machined surface was about 60 micrometers. The form errors do not have any significant effect on the positioning accuracy of the tool tip, but they influence the C M M readings which were taken along the sidewalls of the slot.  The C N C contouring errors of the machined parts were simulated in Virtual C N C Program, which was developed in the Manufacturing Automation Laboratory. The maximum  Chapter 5. Error Compensation and Experimental  Results  137  contouring error is derived to be 9 micrometers in X direction and 12 micrometers in Y direction. The effect of contouring errors on the lengths is shown in Table 5.2 with the magnitudes of the other error sources. As seen in the table, there is a discrepancy between the total measured error for the uncompensated case and the sum of calculated errors coming from the controller and geometric error sources. This result brings the fact that there may be other error sources other than the geometric and contouring errors, which could have affected the positioning accuracy of the machine. One of the important error sources during metal cutting is the thermal condition of the machine tool. The ambient temperatures that the workpiece was machined during the cutting test and measured with the C M M were not same. Considering that the expansion coefficient of aluminum and its alloys is around 22 ppm/C, a minor change in temperature can affect the overall accuracy of the workpiece essentially. If there was a 1 °C difference in measuring and cutting the part, then, in a travel of 100 mm, there will be 10 microns of error coming from the expansion of the workpiece. In addition, while doing the interferometry tests, the material temperature sensor was located on the table in order to estimate the accuracy of the machine if it was operated in an environment of 20 °C. So the individual errors of the machine tool were taken as if it was operated in an environment of 20°C. Then the cutting was performed in a higher temperature, such as 26°C. Finally, the errors were measured in a dower temperature such as about 23°C. The thermal changes on the part will apparently bring some extra errors. In addition to workpiece and machine thermal expansion or contraction, the geometric errors are also influenced by the change of temperature.  Besides, although the measurements on the C M M were taken very precisely and repeatedly, using a manual C M M may also bring some extra errors. As many as possible points were taken into consideration to be more accurate. Also, the measurements were repeated more than once. The structural deformations left on the workpiece to the forces generated during machining influence the measurements done with the C M M as explained in the previous section.  In addition, all the geometric errors were not measured on the machine, such as straightness, squareness and roll errors due to the lack of equipment. However, the resulting  Chapter 5. Error Compensation and Experimental Results  138  equations of the mathematical model prove that the geometric accuracy is certainly affected by all geometric error types.  Table 5.2: The comparison of errors Total Measured Length Error for the uncompensate d case (TE) [urn]  Predicted length error from Geometric Accuracy of the Machine (A) [u-m]  Predicted Error from CNC (B) [um]  Unaccounted Error TE-(A+B) [nm]  Measured Length Error for the compensate d case [u.m]  Li  -227.41  -46.1  -15.2  -166.11  -49.64  L  2  38.19  27.8  44  -33.61  -82.99  L  3  -165.22  -16.5  12.1  -160.82  -156.75  177.56  36.5  40  101.06  -53.31  608.38  126.9  111.3  461.6  342.69  L-4 Absolute s u m of length errors  /3J Chapter 6 Conclusions  6.1 Conclusions The prediction and off-line compensation of geometric errors of the machine are important in improving the, accuracy of machining parts in industry. The aim is to design an accurate machine within the physical and cost constraints, and compensate the remaining errors through modeling the error map of the machine tool within its working volume. This thesis contributes to measurement, modeling and compensation of machine tool errors in virtual environment before the part is machined on actual machine.  The volumetric errors of a three axis machine tool are measured using a laser interferometer. The linear positioning and angular deviations due to pitch and yaw errors of each axis are measured, and correlated to the coupled axes by considering rigid body motion of the machine tool. It is assumed that the angular deviations are small, and each axis acts as a rigid link where the motion errors are only due to geometric errors of the individual components of the machine tool. The geometric errors include backlash on the ball screw drive, inconsistent pitch of the ball screw which leads to linear displacement error, and pitch and roll action of each axis around the remaining axes which lead to angular deviations form the commanded direction. The angular deviations are transmitted as a positioning error of the tool in the machine tool coordinate system. The errors contributed by each axis are fitted to a polynomial as a function of axis position in the machine tool coordinate system. The kinematic model of the machine tool and errors is developed mathematically, and used to predict the total geometric error between the tool and workpiece within the working volume of the machine tool.  The volumetric error model of the machine is experimentally tested on the modeled three axis machine tool. Since the model considers only the volumetric errors, there was a significant difference between the measured and predicted errors on the machined part. Out of 608.38 micrometer error which is reflected on the dimension of the diamond profile, 111.3  Chapter 6. Conclusions  140  micrometer is contributed by the tracking errors of the C N C system. The unaccounted error of 461.6 micrometer is assumed to originate from the measurement errors on the finished part, and thermal deformation and approximations made during the mathematical model of the machine tool.  Although, the volumetric accuracy of the machine tools is studied before, the errors are mainly either reported to assess the accuracy of the machine tool or to compensate them within the C N C system which controls the machine. Most C N C systems do not allow the compensation of angular errors by allowing only tool offsets at discrete locations of the machine tool.  This thesis presents a new method of correcting the volumetric errors within a Virtual C N C Environment. The commanded tool path is imported from the commercial C A D system through ISO standard A P T file format. The machine tool motion is simulated by including the dynamics of the servo drives, amplifiers and drives. The geometric positioning error at any tool location is predicted by using the proposed volumetric error model of the machine tool, and compensated automatically by altering the N C program. The system can handle any linear or spline paths which are approximated by a series of small line segments. The circular paths are interpolated in real time on CNC, and more accurate compensation is viable only by integrating the algorithms to the real time trajectory generation of C N C systems. The virtual compensation of the linear and backlash positioning errors are also experimentally evaluated, and the contributions of geometric errors are reduced significantly while noting the influence of structural and thermal deformations of the machine tool.  6.2 Future Research Directions Although it is a challenge to measure, model and compensate all positioning errors in the machine tool, it is viable to compensate some of them in virtual environment as presented here, and others in real time with the aid of sensors.  Chapter 6. Conclusions  141  The thermal compensation of the machine tool can be conducted in a similar way. The thermocouples can be added to the machine, and the geometric errors due to thermal deformation can be correlated to the temperature readings. There have already been activities in this direction as reported in the literature review. Since the compensation scheme is similar, the combination of volumetric and thermal errors can be extended to five axis machine tools where the accuracy is crucial when machining dies, molds and aerospace parts.  Due to lack of instrumentation, some of the geometric errors, such as roll, straightness and squareness errors of the machine, are not considered in this thesis, although they may contribute to the positioning error of the machine significantly depending on the individual machine. Since such a measurement was not possible, it is difficult to asses the contribution of unconsidered errors to the overall accuracy of the machine tool presented in the thesis. Future study may be able to include all errors to the overall geometric error model of the machine tool presented in the thesis.  Again, the experimental verification of the proposed virtual compensation method could have been done better if the laboratory had an access to a modern Coordinate Measuring Machine located in a temperature controlled room. The part inspection was done on a worn, manual machine with varying room temperature. It is possible that some of the errors are contributed by the poor measurement equipment.  Bibliography  1. Altintas Y . , 2000, Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design, Cambridge University Press.  2. Altintas Y . , Erol N . A . , 1998, "Open Architecture Modular Kit for Motion and Machining Process Control", Annals ofCIRP, Vol. 47, N o . l , pp. 295-300.  3. A S M E B5.57, Methods for performance  evaluation of computer numerically  controlled lathes and turning centers, American Society of Mechanical Society, New York, 1998.  4. Barakat N . A . , Elbestawi M . A . , Spence A . 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N . , 2000, "Error compensation in machine tools - a review Part I: geometric, cutting-force induced and fixture-dependent  errors",  International Journal of Machine Tools and Manufacture, Vol. 40, pp. 1235-1256.  42. Renishaw's Laser System Manual, Renishaw Co., 2002.  Bibliography  147  43. Schultschik R, 1977, "The components of volumetric accuracy", Annals of CIRP, Vol. 25, No. 1, pp. 223-228.  44. Sciavicco L . and Siciliano B., 2000, Modeling and Control of Robot Manipulators, Springer-Verlag.  45. Shakarji, C M . , 1998, "Least-squares fitting algorithms of the NIST algorithm testing system", Journal of Research of the National Institute of Standards and Technology, Vol. 102, No. 6, pp. 633-641.  46. Srivastava A . K . , Veldhuis S.C., Elbestawi M . A . , 1994, "Modelling geometric and thermal erros in a five-axis C N C machine tool", International Journal of Machine Tools and Manufacture, Vol. 35, pp. 1321-1336.  47. Tlusty J., 1999, Manufacturing Processes and Equipment, Prentice Hall.  48. Tlusty J., 1970, Specifications and Tests of Metal Cutting Machine Tools, Revell and George Limited.  49. Week M . , Schmidt M . , 1986, " A new method for determining geometric accuracy in the axis of movement of machine tools", Precision Engineering, Vol.8, No.2, pp. 97103.  50. Zhang G., Veale R., Charlton T., Borchardt B., Hocken R., 1985, "Error compensation of coordinate measuring machines", Annals of CIRP, Vol. 34, N o : l , pp. 445-448.  51. Zhang G., Ouyang R., Lu B., Hocken R., Veale R., Donmez A . , 1988, " A displacement method for machine geometry calibration", Annals of CIRP, Vol. 37, N o . l , pp. 515-518.  Appendix A : The Resulting Equations of the Mathematical Model  In Chapter 3, The Mathematical Modeling of the Machine Tool, the general error matrix is derived as given in Eq. (3.57). £ = 7;WORK  1  where  T OOL T  TOOL  0  0  0  (A.1)  1  is the resultant homogenous transformation matrices for the cutting tool with  respect to the reference frame and  TWORK  is the homogenous transformation matrix for the  workpiece with respect to the reference frame.  Eq.s (3.59), (3.60) and (3.61) are the components of the position vector of the error matrix and they are the simplified according to the assumption that the errors are so small that small angle assumption can be made and the product of two errors is considered to converge to zero. However, the resulting equations showing the components of the position vector in three directions are derived without any simplifications as follows;  p (y) E  P (z) E  1  (A.2)  Appendix A. The Resulting Equations of the Mathematical Model P (x) = )[x -y £ (z) E  l  l  + z £ (z)  z  l  +  y  6 (z)-za X£ (y)£ (x)£ (y)e (x) x  xz  z  z  x  x  - £ (y)£ (x) - £ (y)£ (x) +1 + £ (y)£ (x)£ (y)£ (x) + e (y) £  (xf  2  z  z  y  y  x  x  y  y  x  x  + e (y)e (x)£ (x) + £ {y)£ (x)£ (y) - e (y)£ (y)£ (x) + e (yf y  z  x  y  -£ (y)£ (x)e (x) z  y  z  x  x  + £ (xf)+(e (z)x  x  x  z  z  y  + y, -z,£ (z)  t  x  + S (z)-za^)  x  -  y  (- £ (x)e (y)e (x) + £ (x) - £ (y)e (y)£ (x) + £ (y) z  x  x  z  z  y  y  z  + £ (y) £  (x)e (x) + £ (y)£ {xf £ (y) + e (y)£ (x)e (y)e (x)  + £ (y) £  (x) + £ (y)£  2  y  x  y  2  y  z  y  x  x  x  x  x  x  (x) -£ (y?£  z  (-£ (z)x,  x  z  y  + £ (z)y,  y  z  y  z  x  x  x  y  z  z  (x) £  y  y  (y))  2  x  z  z  + z, + z + Z + S (z))}{l + £ (y) £ (xf  + £ (yfe  2  x  2  +  (xf)  x  (x) + £ (y)£ (x)£ (y) - £ (x)  y  - £ (y) + £ (y)£ (x)£ (x) + £ (y)£ y  y  (x) + £ (y)£ (y) + £ (x)£ (x) + £ (y)£  y  + (£ (x)£ (x) + £ (y)£ z  x  z  y  y  z  (xf  z  £ (yf+£ (yfe (xf+e (yf+£ (yf£ (xf+£ (yfe (xf+£ (xf z  +  x  z  y  y  x  x  y  z  £ (yfe (xf+£ (yfe (x) +e (xf+£ (y) £ (xf+e (yfe (xf 2  y  + e (yf  + £* (xf)-  x  z  z  2  x  y  [e (x)£ (x)S (y) + £ (yfe z  x  z  y  2  x  z  + £ (y) £  y  y  - £ (y) £ (x)a x£ y  x  £ (yfS  XY  y  w  + x £ (yf w  z  x  z  x  z  z  (x)£ (yf  x  +  z  +S  x  (y)£ (yf  x  x  (xf e (y)S (y)  ( )£y ( ) y CO + £ (y)£  (x)Z e (x)  x  x  y  2  x  2  z  + x£ (x) + 8 (x)£ (x) + S (y)£ (xf + £ (y)£  x  x  x  y  2  + £ (y) £  (x)  x  y  (x) + x £ (yf+x£ (yf+S  x  (xfS  x  (x) + £ (x)£ (x)y + £ (y)y + £ (yfx  2  X  x  x  x  x  x  x  z  (x)e (x)S (y) + e (y) £  s  x  2  x  y  z  y  z  x  + £ (y)£ (x)£ (x)S (y) + £ (y)£ (x)e (y)8 (y) + £ (y) £ z  x  - £ (y) £  y  (x)a x - £ (y)£  2  y  x  z  XY  x  z  y  XY  + £ (y)£ (y) z  x  x  y  z  x  y  z  x  x  z  y  (x)S (x) + £ (x)S (y) + £ (x)S (x)  y  y  z  y  z  y  (x)£ (x)  2  y  x  z  y  + £ (x)y + x £ (xf+£ (y)S (y) z  w  z  z  z  x  + £ (y)£ (x)£ (y)£ (x) - e (y)e {xf + £ (y) £  x  x  (y)£ (x)S (y) - e (x)e (y)e (x)8 (y)  z  - £ {x)£ (x)a x + e (x)£ (yfe x  x  z  (x)£ (x)S (x)  2  y  x  (x) + £ (y)£ (x)£ {y)£ (x)S (y)  2  z  z  x  (x)S (x) + y.£ (y) £  2  z  x  x  y  2  x  y  (x)S (x)  z  + e (y)e (y)S (y) + x.£ (y) e (xf + £ (y) £ z  z  y  x  z  + x £ (y) £ (xf  - £ (yf  2  y  w  x  z  z  x  £ (x)8 (y)  z  y  z  - ^ ( y ) ^ W Z , - ^ ( y ) ^ W ^ U ) + e (y) ^U) x , + ^ ( y ) ^ ( x ) ^ 2  2  2  2  2  + £ (yf£ (xfS (x) y  x  +  x  + £ (x)£ (yf8 z  x  e (yf£ (xfx -£ (y)£ (xfS (y) y  x  w  -e (x)e (yf y  x  z  z  x  z  + £ (y)£  x  x  y  z  z  z  x  Z, -e (x)e (yfS (x) y  + £ (y)£ (xfS z  y  x  z  (x) + £ (x)E (y)S (y) + £ {x)£ (x)Z  y  x  2  >l  x  + e (x)£ (x)8 (x) - £ (y)£ (x)8 (y) + e (yfe z  2  y  x  z  (y) + £ (yf£ (xfx z  (x)S (y) + £ (y)£  x  y  x  y  y  z  {  (x)8 (y) y  + £ (yfx e (xf x  w  + £ (y) £ 2  z  + e (y)e (xf  y  x  z  (xf S (x) + e (y)e (x)y x  (y)y + £ (x)£ (x)8 (y) x  y  y  x  y  x  y  Appendix A. The Resulting Equations of the Mathematical Model + e {x)e (x)e (y) 8 (x) + e (x)e (x)e (y) y + £ (y)£ (y)e (x) y 2  x  y  2  y  y  y  x  2  y  y  + e (x)e (y) S (y)e (x) + e (y)e (x) S (y)£ (y) 2  x  y  y  - e (x)£ {y) a x x  x  y  2  x  z  x  (x)  Z£ X  X  + e {x)e (y) e (x)S (x) - e (x)ye (y)e (x)  2  z  y  x  + e (x)e {y)  2  y  x  2  XY  z  x  x  z  z  x  x  - e (x)e {y)e (x)S (y) - e (x)a x + x e (x)  2  z  x  x  y  z  - £ (x)e (y) a xe z  XY  y  x  y  + £ (y)£ (x)£ (y)£ (x)S (y) y  z  z  + £ (y) x  x  (y)  ()y x  £  y  y  +  y  w  y  x  (x)  2  x  £ (y) £ (x) x 2  2  z  x  (x)S (y) -£ (y?£  £  £  x  w  (x) + s (y)e (y)S (y) + x £  2  x  XY  x  w  (x)e  x  (x)a x  y  XY  + e (y)^(x)e (y)^(x)y + ^(y)e U)^(y)£ (y)e (x) y  z  z  ;c  + £ (y)£ (x)£ (y)S (y) + £ (y) £ (x) x 2  z  z  y  z  - y£ (y)£  2  z  z  w  . ^  ;i  z  (x)£ (y)  y  y  - £ (y)£ (x)£ (y)8 (y) + £ (y) £ (X)E (X)Z + £ (y) £ 2  z  y  y  + £ (y) £  z  Z  X  (x)£ (x)S (x) - £ (y) £  2  z  y  x  z  z  x  (x)a x-  2  z  (x)£ (x)S (y)  2  z  z  z  £  XY  x  (x)£ (y)S (y)  y  y  x  + £ (y) £ (x) x +£ (y) £ (x) x +£ (y) £ (x) 5 (y) 2  2  y  z  2  w  + x + x +£ w  2  x  x  2  w  x  x  y  y  + £ (x) x £ (y) )l^ y  + £ (y)  2  y  z  y  (y) -Z £ (y) £ 2  y  y  x  y  (x) -£ (y?8  2  x  2  y  y  +£ (y) £ (x) 2  y  y  y  y  X  (x)£ (x)  z  y  + £ (y) £ (x) +£ (y) £ (x) +£ (y) +£ (y) £ (xf  2  w  w  (x) - £ (x)Z  2  y  z  x  £ (y)S (y) + £ (y) S  2  y  z  2  x  (x)£ (x)S (x) + £ (y) x-  x  - £ (x)S (x) - £ (x)S (y) + x £ y  + S (x) + S (y)  2  x  z  2  2  y  2  z  z  +£ (y?£ {x)  2  x  z  x  z  2  y  (A.3)  2  +£ (x) + £ (y) £  2  x  2  2  z  y  z  (xf  + £ (y) £ W +^(x) +e (y) ^(*) +£ (y) e W +^(y) +e (x) )] 2  z  2  ;c  2  2  z  2  2  ;c  2  ;t  2  2  ;c  Appendix A. The Resulting Equations of the Mathematical Model  P (y) = i(£ (y)£ (y) x  (*) -  e  + e (y)e (x) £ (y) + £ (y)  2  £  E  z  x  x  y  y  + £ (y)£ x  ("0 + £ ( * ) £ ( y ) £ , W - « (*)  z  2  z  z  y  x  x  y  (x)- e (y)e (x) + e (x)e (x))  z  2  x  y  )+ (-  xz  (x)  2  z  y  y  + £,(y)£ (x)£ x  z  z  x  y  (y)e,(x) +1 +  £ (y)£ (x)e (y)£ (x) - £ (y)£ (x) + £ (y) £ z  y  2  (x, - y,e (z) + z,£ (z) + £ (z) - za z  z  (x)£, (x) + f, (y)£, (*) + e (y)£ (y)  x  (X)E (y)e (x) - £ (y?£  z  y  y  y  2  (y)£ (x) + £ (y)£ (y)£ (x) + £ (y) - £ ( y ) £  (x) -  2  y  y  y  z  x  y  z  y  £, (y)£ (x)e (y) + e (x) + E (y)e (x)e (x)) 2  z  x  (E (Z)X,  y  z  y  x  + y, - z,£ (z) + ^ ( z ) - za )+  Z  x  (x) + £ ( y )  n  x  + £ (y) £ (x)£ (x) - £ {y)£ (x)£ (y)  + £ {y)£  2  z  z  y  z  -£ (y)£ (x)£ (x) y  y  z  x  + £ (y) £ (x) z  x  y  + £ (y)£ (x) + £ (y)£ (x) + £ (y)£ z  x  (-£ (z)x,  y  +£ (z)y,  y  x  2  z  +  z  £ (x)£ (x) y  z  (x)e (y)e (x))  2  y  (x) £ (y)  y  + £ (y)£ (y)  2  x  y  y  t  x  +z, +z + Z +S (z))}(l + £ (y) £ (x) 2  x  2  z  +  2  y  y  £ (y) £ (x) 2  2  z  z  + £ (y) + £ (y) £ (x) + £ (y) + e (y) £ (x) + £, (y) ^ (x) + ^ (x) 2  2  z  2  x  + £ (y) £ (x) 2  2  z  +£ (x)  2  y  2  x  2  w  2  +£ (y) £ (x)  2  z  2  y  2  y  2  +£ (y) £ (x)  2  x  2  x  ( x ) ) - [(£ (y) y e (x) + ^ (y) £  2  x  2  y  2  z  + £ (y) +£  2  y  +£ (y) £ (x)  2  y  2  z  2  x  (x)S (y)  2  x  x  x  z  + ^ ( y ) ^ W y + ^ ( y ) Z ^ ( x ) + f (y) ^(x)^(x)-^(y)e (x)y 2  2  2  2  w  1  z  - £ (y)£ (x)8 (y) + £ (y?£ z  z  y  z  z  (x)£ (x)S (y) + £ (y) £ y  z  z  + £ (y) £  (x)£ (x)S (x) - £ (y)£ (x)£ (y)S (y)  + £ (x) £  (y)£ (y)S (y) -£ (y) £  2  z  z  y  2  y  z  z  z  x  z  z  y  £ (x)£ (x)x  2  y  y  + £ (y) £ y  x  (x)£ (x)S (x)  2  XY  y  x  x  + £ (y) Z e (x) - £ (x)£ (y) x + £ (x)E (y)S (y) + £ (y)£ (x)y£ 2  2  x  l  x  z  y  z  + S (y) + £ (y)£ (x)£ (x)£ (y)8 y  Z  z  y  y  Y  z  z  (y) + e (y) £ t  x  + £ (y)£ (x)y£ (y)£ (x) + £ (y)e (x)S (y)£ (y)£ y  y  x  x  y  y  y  - £ {y)£ (x)£ (x)8 (y) + £ (y) £ y  x  z  - £ (y) £ (x) a 2  2  x  y  x  -  XY  x  +  x  x  x  z  (y)S (y) x  x  x  y  y£ (y)£ (x) x  x  w  z  x  z  2  z  2  2  z  y  (y)S (y) + y ^ (x) + £ (x)Z, + 2  z  w  z  x  (x)x + e (x)£ (y?£  x  x  x  y  y  z  y  w  (x)£ (y)y - £ (y)£ y  x  x  (y) + e (y) S (x) + y ^ (y) +  2  + £ (x)£ (y)£ (x)8 (y) + y £ - £ (y)£  y  2  + £ (y)£ (y)£ (x)y + £ (y)£ (y)£ z  y  y  z  w  (x) + y £  z  y  y  x  x  ( x ) ^ (y) + £ (x)£ {y) £ z  x  + £ (y) £ (x)S (x)-  XY  2  y  x  2  2  (x) +  y  (x)  x  (x)x£ (x) + £ (y) £ (x)£ (x)S (x) + S (x)  y  x  w  £ (y?S  y  (y) + £ (x) y ^ (y) - £ (x)S (x) - £ (x)S (y)  - e (x)x + y £ z  x  w  (y)£,  £ j t  x + £ (y) £  x  (y)£ (x)  2  y + y -a x  2  x  2  y  2  XY  -£ (y) a x+  x  y  (x)£ (x)8 {y) + £ (y)£ {x) £  2  y  z  x  z  (x)8 (x) x  (x)S (y) + e (x)£ (y)£ (x)y  x  y  (y) £ 2  z  z  (x)5 (x)  y  z  (x) + £ {y)£ (x)£ (y)£ (x)S (y) x  z  (x)£ (y)S (y) - £ (y) a y  x  2  y  2  z  :  (x)Z,  2  y  .  y  + £ (y)  2  y  z  z  (x) a x  2  y  (x)£ (x)Z,  2  z  y  z  z  y  XY £y(X) X  x  Appendix A. The Resulting Equations of the Mathematical Model  + £ (y) £  (x)£ (x)Z, + e (yf e (x)e (x)8 (x) - e (y)£  2  x  z  y  - £ (y)£ x  x  z  y  z  x  (x)e (x)y  z  y  (x)£ (x)8 (y) + £ (x)£ (y)£ (x)8 (y) + e (x)£ (y) Z e 2  z  y  y  + £ (x)£ (y) £ y  y  y  x  (x)S (x) + £ (y)£ (y)e  2  z  z  y  z  z  x  152  z  y  x  y  (x)  (x)S ( y )  x  x  + £ (y)£ (x)£ (y)£ (x)S ( y ) + £ ( y ) y + £ (y) S (x) 2  x  y  l  + £ (y) y  y  y  y  y  y  z  £y  x  y  z  +  w  z  £ (yf£ (xfS (x) y  + £ (yfS (y)£ (xf  y  £ (x) y 2  y  + £ (yf£ (x)y  y  2  -£ (y)S (y)-£ (y) £ (x)x  w  + £ (yf (xfy  x  y  +  y  y  y  -£ (yf£ (x)S (x) z  y  + £ (xf  2  w  2  y  + £ (x) S (y)  2  y  + £ (x) S (x)  2  y  z  z  + £ (y) S (y)  2  y  x  y  y  -e (yfa x  y  z  XY  + £ (y) £  (xf y .+ y e (yfe  + £ (yf£  (xfS (x) + e (y)e (xfd (y) + £ (y)e (y)8 (y)  2  z  x  x  w  y  w  y  y  - £ (x)£ (yfS z  y  x  x  z  z  -£ (yfe x  y  x  z  w  z  y  z  x  y  z  z  y  x  y  z  +  XY  x  £ (xf y z  w  (x)S (y)  2  x  + £ (yf  XY  (x)S (x) -£ (y) £  -e (xfa x  y  z  (x)8 (x) - £ (yfa x  x  z  -8 (y)£ (y)£ (xf x  z  y  (x)x -£ (yf£  z  y  (x) + £ (x)£ (x)Z + £ (x)£ (x)5 (x)  + £ (x)£ (x)5 (y) + £ (y) £ y  x  y  2  z  (xf + e (yf e (xf y  z  x  £ (y) £ (xfS (x) 2  z  y  y  + £ (y)£ (x)8 (y) + £ (y)£ (y)8 (y) + e (x)xe (x) y  x  x  y  x  x  y  x  + £ (x)£ (x)8 (x) + £ (x)£ (x)8 (y)) I y  x  x  y  ({ + £ (yf£ (xf y  x  +£ (yfe (xf  y  z  +  +£ (yf  z  z  +£ (yf£ (xf x  +£ (yf  z  y  £ (yf£ (xf+£ (yf£ (xf+£ (xf+£ (yf£ (xf y  + £ (yf£ z  x  x  x  (xf + £ (xf + £ (yf y  £ (y) £Axf+£ (yf+e (xf)] 2  x  x  x  x  z  £y  y  (xf +  z  y  z  Appendix A. The Resulting Equations of the Mathematical Model (z) = ( (y) * (30 + z (y) x (x) + £ (y)£ (xf + £ (x)£ (x) £  E  P  £  £  £  z  + £ (y)£ x  y  (x)£ (y)£ (x) + £ (y) £ y  z  + £ (y)£ (x) £ z  z  (x,-y (z)  x  + z,£ (z)  £  1 z  y  y  x  (y)e  y  (x)e (y)  x  x  (y) + £ (yf £ (x)£ (x)- £ (x)£ (y)£ (x) + £ (x))  2  x  x  z  (x) + £ (y)-£  2  y  z  z  +  y  x  z  z  y  y  S (z)-za )+. x  xz  (- £ (y) - e (x) +£ (y)£ (xf £ (y) + £ (y)£ (x)£ (x) + £ (yf £ (x)£ (x) x  x  z  z  y  z  z  + £ (y)£ (x)£ (y) + £ (y)£ (y) - £ (y) £ y  x  y  z  y  x  + £ (y)£ (x)£ (y)£ (x)+£ (x)£ (x) x  z  y  x  y  (e (z)x, + y, -z,£ (z) z  y  + £ (y)£ (x)£ (y)£ z  ~  z  x  £  y  x  y  y  y  y  y  z  + £ (yf£ (xf + £ (xf  x  z  z  x  2  (xf + £ (yf  z  y  +£ (yf£ (xf  z  x  (z))  z  2  +e (xf  y  +£ (yf£ (xf  z  z  +£ (yf  x  z  (x)  y  +z,+z + Z +S  (xf +£ (yf£ (xf  y  z  x  £ (yf  y  + £ (x)£ (y)£  z  £y  +£ (yf£  y  2  z  y  (xf + £ (yf + £ (y) £  z  +£ (yf (xf  x  y  X  x  /(l + £ (yf £ (xf + e (yf£ y  £ (y) £ (xf  x  (X)£ (X)  z  (y)£ (x)\- £ (z)x, + £ (z)y,  z  y  x  z  z  x  + £ (y)£  y  (l-e (y)£ (x) +  + £ (xf - £ (y)£  £  z  z  n  z  z  (xf  z  (x) + £ (y)e (x)s (y)£ (x) - £ (y)£ (x) +  x  (y) (y) (X)  £  y  £ (y)£ (x))  + S (z)-za )+  x  x  +  z  y  (x) - £ (y)£  2  y  x  x  +£ (xf)-  x  x  [(-£ (yf£ (x)S (x)+£ (xf£ (y)5 (y)-£ (yf£ (x)y y  x  y  z  y  x  y  x  - £ (y)£ (x)S (y) + £ (x)£ (y)y-£ y  y  + (yf £  y  x  (y)  £  z  y  (y) £  y  z  x  + £ (yf  y  z  1+  y  z £ (xf + £ (yfZ e  y  w  z  y  x  y  + £ (y) £  (xf z + £ (y)S (y)£ (x) + £  x  2  x  y  w  x  z  x  x  x  x  - £ (y)£ x  z  x  x  (y)S (y)£ (y)  z  x  (xfy + £ (y) £  x  z  + e (xfe z  z  x  z  z  z  x  + £ (y) £  y  z  + £ (y) £  x  w  + £ (y) £ (xfz £  z  (yf  £  z  z  z  w  z  (xf8  z  y  y  z  y  x  (x)8 (y) + £ (y) £ z  z  z  x  x  y  x  y  y  y  z  + £ (xf z + £ (x)8 (x) y  w  y  x  z  y  (xf z  w  +  z  XY  z  (x)  (x)£ (y)8 (y)  y  2  x  (x)  (x)y£  y  + £ (y)e  x  x  l  x  y  (y) + e (yf £ (x)a x - e (x)£ (yf £  z  + £ (y)£ (x)£ (x)£ (y)y x  z  + £ (yf£ (xfZ +£ (yf£ (xf8 (x)  2  z  y  z  x  z  z  (xf z + £ (y)£ (x)£ (y)£  2  z  y  (y)  y  £ (x)£ (x)8 (x) + £ (yfe  y  y  x  y  (x)e (x)8 (y) + £ (y)£ (x)y£ (y)  2  y  y  (xfS  z  (x)£ (x)8 (x)  z  y  + £ (y)£ (x)£ (x)8 (y) + £ (yf z  x  (y)e (y)8 (y) + £ (x)£ (y)y£  2  y  x  (x) - £ (y)£  x  z  (y)£ (y)y + £ (x) £  z  x  (x)x£ (x) + e (yfe  2  z  z  x  (x)5 (y)  y  + £ (x)x£ (x) + e (x)8 (x)£ (x) + £ (x)5 (y)£ z  (x)5 (x)  y  z  2  y  w  (x)£ (xf  z  (x)S (x) + £ (y) £  2  y  z  y  (xf + £ (yfS  z  y  + S (y) + z  z  (x)x + £ (y) £  y  (x)x + £ (y) £  2  z  2  + £ (y) £ x  y  (x)S (y) + Z S (x)  x  z  (x)8 (x) + £ (yf e (x)x  2  z  (x)£ (y)e (x)S (y) + £ (yfe  £  y  y  (x)S (x) -£ (yf£  £  z  +  z  (x)a x XY  Appendix A. The Resulting Equations of the Mathematical Model  + e (x)S (y) + £ (x)x + e {y)e y  x  y  (x)e (y)e (x)8 (y) + e (y) e  z  (x) z + e (y) e  2  z  z  x  (x) Z  2  y  (x) 8 (x) + e (y) e  + e (y) e  2  2  x  z  x  154  x  y  2  (x)  2  w  z  l  z  2  y  2  z  x  w  + £ (y)£ (y)y + £ (y)£ (y)$ (y) + £ (x)e (x)S (y) y  z  y  z  y  y  y  t  + £ (x)£ (x)S (x) + £ (x)£ (x)y + £ (x)£ (y)S (y) y  z  y  y  z  + £ (y)£ (x)£ (y)S (y)£ x  y  y  + £ (y) z  x  y  XY  + E (y) S (y)  2  z  (x)a x  z  +£ (y) S (x)  z  y  (x)£  2  2  w  z  (x) -£ (y) £  z  + £ (y) Z,  2  z  x  y  + £ (x) z  2  z  z  z  + e (x) Z, + e (x) 8 (x) + e (x) 8 (y) + £ (y) £ 2  2  2  z  Z  z  -£ (y) £ (x)S (x)  + £ (y) £ (x) z  2  x  2  y  x  x  2  w  x  w  z  x  - e ( y ) £ (x)a x£  + £ (y) Z '  x  y  z  x  x  y  x + £ (y)  £ (x)£ (x)5 (x)  2  XY  :  l  - £ (x)S (y)  x  (x)a  2  z  2  x  x  y  t  y  z  x  y  z  - £ (x)£ (y)£ (x)S (y) + £ (y) £ y  x  + £ (x) £ (y)£  (y)S (y) + £ (y) £  y  (x)a x  + £ (y) £  (x)£ (x)x + £ (y) £  2  x  2  x  x  x  x  + £ (y) £ 2  z  y  x  x  (x)£ (x)8 (x)  z  (x)E  + £ (y) S (y)  y  x  x  y  z  y  z  x  x  2  x  y  2  l  y  x  (x)8 (y)  x  y  x  w  y  XY  +  2  z  (x)a x  z  + £ (y) z )/{l  + £ (y) Z +£ (y) S (x)  2  (x)S (y)  X  (x)£ (x)S (x) - £ {x)£ (y)£ (x)S (y) - £ (x)£ z  x  x  {x)£ (x)5 (x) - £ (y)£ (y)£  z  z  z  y  2  z  2  z  y  z  x  2  x  2  XY  (y)£ (x)S (y)  y  (x)£ (x)x + £ (y) £  z  + £ (y) £ y  z  z  2  z  z  y  z  (x)e (x)5 (y) + £ (y)£ (x)£ (y)S (y) - £ (y)£  + £ (y)£ x  XY  w  z  x  x  (x) + £ (y) £  2  y  y  y£ (x) - S (y)£ (y)  2  y  x  2  x  + £ ( y ) S (x) - y£ (y)x  w  + £ (x) z +£ (y) z -S (x)£ (x)  x  z  2  x  -£ (y)£ (x)8 (y)  2  y  + a x£ (x) XY  x  w  (x)  2  z  z  ,  2  z  £ (y) £ (x) 2  y  +  £ (y) £ (x) +£ (y) +£ (y) £ (x) +£ (y) +£ (y) £ (x)  +  £ (y) £ (x) +£ (x) +£ (y) £ (x) +£ (y) £ (x) +£ (x)  2  z  2  z  2  x  + £ (y) £ 2  z  2  y  (x) +£ (y?£ (x) +£ (y) + 2  y  2  x  x  2  x  2  z  2  x  2  z  2  z  2  y  2  2  y  2  z  y  2  z  2  x  2  x  2  y  2  y  £ (x) )] 2  x  Also the simplified components of the rotation matrix of the general error matrix are given with Eq.4.62. The components of the rotation matrix before the simplifications are given in the following equations.  R=  11  R  R  21  R  R  31  l2  22  32  R  n  23 33  R  (A.6)  Appendix A. The Resulting Equations of the Mathematical Model  i i = l£ (y) M) (y) (x)  R  £  £  z  - Sy) z  £  x  £  x  + e (y)e (x)£ (y)£ (x) + e (yfe x  x  y  y  (*)  £  x  - (y) (x)+1 £  £  y  y  (xf + £ (y)£  x  y  (x)e (x)  z  x  + e (y)e (x)e (y) - e (y)e (y)e (x) + e (yf y  -  z  (y)  £  x  z  y  x  (x)e (x) + £ (xf]+ e (z)[- £ (x)£ (y)£ (x) +  £  z  x  y  x  x  z  z  x  £ (x) - £ (y)£ (y)£ (x) + £ (y) + £ (yfe z  z  y  y  + £ (y)£ (x) £ (y) x  + £ (y)£ x  x  y  z  (x) + e (y)£ (y)  y  y  x  z  x  z  x  x  z  z  z  y  z  y  (xf]  x  y  (xf +e (yfe  x  z  y  x  x  (x)£  x  £ (yf  z  y  y  + (yf (xf+£ (yf+ £  z  £ (y) £  £  x  z  z  (xf +  y  z  (xf  2  y  x  + £ (yf£ (xf+£ (xf+£ (yf£ (xf+£ (yf£ (xf x  y  z  £ (xf  y  z  z  +£ (yf£ (xf  y  z  +  x  +£ (yf£ (xf  y  x  (x)  (x)  y  2  x  x  (x)£  z  + £ (y)£ (xfe (y)])/[l + £ (yf £ (xf +£ (y) £ x  (y)£ (y)  z  (x) - e (y) + £ (y)£  y  z  (x) + £  2  z  y  y  y  (x) - £ (y) £  z  + £ (y)£ (x)£ (y)-£  x  x  + £ (y)£ (x)£ (y)£ (x) - £ (y)£ y  (x)  y  + e (x)e (x) + £ (y)£  y  - £ (z)[£ (x)£ (x) + £ (y)£ y  (x)£  x  + £ (y)£ (x)£ (y)£ (x) + £ (yf £ (x)  2  y  z  x  +£ (yf  x  +£ (xf]  x  x  n={- (z)i (y) (x)£ (y)£ (x)-£ (y)£ (x)  R  £  £  z  £  z  z  x  z  z  + l + £ (y)£ (x)£ (y)£ (x)  -£ (y)£ (x) y  x  y  x  x  y  + £ (y)£ (x)£ (x) + £ (y)£ y  z  x  y  + £ (yf-£ (y)£ x  z  x  x  .  x  (y)£ (x)  z  y  (x)£ (x) + £ (x) ]+ [- £ (x)£ (y)£ (x) 2  y  z  £ (xf  x  (x)e (y) - £ (y)£  z  x  x  + £ (x)-£ (y)£ (y)£ (x) z  + £ (yf  y  y  z  + £ (y) +  y  x  x  £ (y) £ (x)£ (x) 2  z  y  x  y  + £ (y)£ (xf £ (y) + £ (y)£ (x)£ (y)£ (x) + £ (y) £  (x)  2  y  x  x  y  z  z  x  y  z  + £ (y)£  (x) + e (y)£ (y) + £ (x)£ (x) + £ (y)£ (xf ]  +£ (z)[e  (X)£ (X)  x  y  x  x  z  y  x  + £ (y)£ x  X  y  z  (x) - £ (y) £  (x) + £  2  z  z  x  y  + £ (y)£ (x)£ (y)£ (x) - £ (y)£  (xf + £ (y) £  + £ (y)£  + £ (y)£  y  z  x  z  y  y  (x)£ (y)-£  z  y  (x) -£ (y)  y  y  x  x  +  + £ (y)£ (xf£ (y)^[l x  z  z  z  x  (x)£  x  x  (x)£  2  x  (y)£ (y)  z  (x)  (x)  y  £ (yf£ (xf+£ (yf£ (xf y  +  y  z  z  £ (yf+£ (yf£ (xf+£ (yf+£ (yf£ (xf+£ (yf£ (xf z  + £ (xf +£ (yf£ z  + £ (yf£  y  z  y  (xf +£ (yf£  z  z  x  (xf +£ (yf£ (xf+£ (yf+ x  x  x  x  z  y  y  (xf + £ (xf y  £ (xf ] x  x  x  y  Appendix A. The Resulting Equations of the Mathematical Model n = ^ (z)[e (y)£ (x)e (y)£ (x)  - £ {y)£  R  y  t  z  x  x  z  (x)  z  - £ (y)£ (x) + l + £ ( y)£ (x)£ {y)£ (x) + £ (y) £  (x)  2  y  y  x  •+ £ (y)£ {x)e y  z  y  (y) y  2  y  z  (*) +  £  z  y  x  (x) + £ (y)£ (x)£ (y)  x  Ay) -£  £  x  2  x  - £ (y)£  x  x  (y)£ (x) +  z  y  (x) ]- £ (z)[- £ (x)€ (y)£ (x)  £  2  x  x  z  x  x  + £ (x) - £ (y)£ (y)£ (x) + £ (y) + £ (y) £ (x)£  (x)  2  z  z  y  y  z  y  x  y  + £ (y)e (x) £ (y) + £ (y)£ {x)e {y)e (x) + £ (y) £ 2  y  + £ (y)  x  y  M + £ (y)£  £  x  y  x  X  y  x  z  i  x  y  x  y  z  (x) -s (y) s z  y  z  y  z  z  (x) - £ (y) + £ (y)£  + £ (y)£ (x) £ (y)§l\  +  y  y  y  2  x  x  z  + £ (y) +£ (y) £ (x) 2  2  z  x  (x)  y  2  2  y  y  +  2  y  2  z  +  2  z  (x)£  x  (x)  x  £ (y) £ (x) +£ (y) £ (x)  + £ (y)  2  x  (x)£  2  x  +  x  (x) - £ (y)£ (x) + £ (y) £ 2  y  z  x  (x) + £ (y)£ (y)  2  z  + £ (y)£ (x)£ (y)-£ z  z  2  x  £ (y)£ (x)£ (y)£  z  (y) + £ (x)£ (x)+ £ (y)£ (x) ]  y  + [E (X)£ (X) + £ (y)£ Z  (x)  2  x  z  £ (y) £ (x) +£ (y) £ (x) 2  2  y  2  x  2  x  y  £ (x) +£ (y) £ (x) +£ (y) £ (x) +£ (x) +£ (y) £ (x) 2  2  z  +  2  y  2  z  z  2  2  x  2  y  2  z  y  £ (y) £ (x) +£ (y) +£ (x) ] 2  2  x  n =i  R  (y) Ay)  £  £  z  + £ (y)£ x  x  (x)-£ (y)  x  + £ (x)£ (y)£  z  z  x  (y) + £ (y)£ x  y  z  y  x  y  y  y  z  y  x  x  x  y  z  x  y  y  z  y  x  £ (y)£ (x) £ (y)~ 2  y  z  z  y  y  (y)£ (x)  y  y  y  y  +  z  2  y  z  (x)£ (x) - £ (y)£ (x)£ (y)  z  y  x  £ (x) +£ (y)£ (x)£ (.  z  z  y  x  +  x  £ (y)£ (x)£ (x) + £ (y) £ (x) + £ (y)£ (y) 2  y  y  x  z  x  y  z  x  y  x  z  (x)£ (y)£ (*)]}  2  z  x  z  + £ (x)£ (x) + £ (y)£ (x) + £ (y)£ (x) + £ (y)£ y  z  (x)£ (y)£ (x) + £ (y)£ (y)£ (x)  x  x  2  x  y  z  x  - £ (z)[£ (x) + £ (y) + £ (y) £ y  y  (x) + l + £ (y)£ (x)£  x  2  y  y  -£ (x)£ (y)£ (x)-£ (y)£ (x)£ (y)  2  y  y  (x) - £ (y)£ (x)£ (y)  y  (x) + £ (y)£  2  z  x  (x)£ (y)£  z  - £ (y)£ (x) + £ (y) £  y  z  x  2  + £ (y)  y  2  + £ (x) + £ (y)£ (x)£ (x) - £ (y)£ z  2  z  £ (y) £ (x) +£ (y)£ (x)£ (y)£ (x) x  y  (x) - E (y)£ (x)  z  z  2  z  x  2  + £ (y)£ (y)£ (x) + £ (y) -£ y  z  (x) + l + £ (y)£ (x)£ (y)£ (x)  x  +  z  -£ (x)  x  2  z  z  -£ (y)£ (x)  y  (x)£ (y)£ (x) -£ (y) £  z  y  (x)  y  x  + £ (x)£ (x)]+ £ (z)[- £ (y)£ x  y  (x)£ (x) + £ (y)£ (x) +  2  y  y  x  (y) + £ (y) £  2  £ (y)£  2  x  £  (x) £  y  2  x  y  z  x  l\l +  £ (y) £ (x) +£ (y) £ (x) +£ (y) +£ (y) £ (x) +£ (y)  +  £ (y) £ (x) +£ (y) £ (x) +£ (x) +£ (y) £ (x) +£ (y) £ (x)  2  2  y  z  2  2  y  + £ (x) +£ (y) £ 2  y  2  z  2  x  x  2  x  2  x  2  z  2  x  (x) +£ (y) £ (x) +£ (y) + 2  y  2  y  2  2  y  2  z  2  x  2  z  2  z  y  2  z  £ (x) ] 2  x  2  y  2  z  2  x  Appendix A. The Resulting Equations of the Mathematical Model  R  =  22  {-£ (z)[£ (y)£ (y) x( )- z(y) zM (y) y( ) £  z  - s (x) + e (y)£ (x) £ x  y  x  (y) (x) + £ (y)£ (y)  £  x  x  -£ (y) £ x  z  z  [-£ (y)£ (x) x  +  (y)  x  y  + £ (yf  z  (x)£ (x)]+  x  y  £ (y)£ (x)£ (y)£ (x)-£ (y)£ (x) z  x  z  z  y  (x)£ (y)£  x  x  y  y  + £ (y)£ (x)£ (x)]+£z  y  x  y  z  z  (x) + £ (y)£ (y)£ (x)  y  y  z  + £ (y) £  x  x  (x) + £ (y) +£ (y) £  y  (x)£ (x)  2  x  y  y  + £ (y)£ (xf x  £ (xf  z  z  -  z  y  £ (y)£ (x)£ (x) y  y  (x) + £ (y)£ (y) + £ (x)£ (x) + £ (y)£  2  z  y  x  +  x  + £ (y)£ (xfs (y)  x  z  y  (Z)[E  x  - £ (y)£ (x)£ (y) z  y  z  y  x  (x)  z  £ (y)£ (x)£ (y)£ (x)$l x  \l + £ (y) £ (xf  y  z  x  +£ (y) £ (xf  2  y  z  +  y  x  y  -e (x)e (y)e (x)-e (y)e-(x)e (y)  y  £  x  z  (x) +£  y  (xf + £ (y)£  l£  y  £  (x)£ (x) +  y  2  +l+  x  £  y  (x) - £ (y)£  2  +£  + £ (y)£ (x)£ (y)£ (x)  £  y  £  y  2  y  x  x  (y) + £ (y) £  2  z  z  +£ (yf  2  y  z  z  +  +£ (y) £ (xf 2  z  x  z  £ (yf+£ (yf£ (xf+£ (yf£ (xf+£ (xf+£ (y) £ (xf 2  y  + £ (y) £  2  x  +  x  (x) + £ (x) + £ (y) £  2  z  y  2  z  y  z  y  (x) +£ (y) £  2  y  x  2  y  (xf  2  x  z  x  £ (yf+£ (xf] x  x  i3 = ^ (z)[e (y)£  (y)£ (x)-e (y) + e (x)e (y)e (x)  R  y  z  x  x  - £ (x) + £ (y)£ z  x  + £ (y)£ x  z  (x) £ (y) + £ (y) £ 2  y  x  x  y  y  x  y  (*)£ (y)£ (*) - £ (yf  z  z  y  y  + £ (y)£ z  x  y  x  z  y  y  z  + £ (y)£ (x)£ {y)£ (x) x  y  x  z  y  y  + £ (y)£ (y)£ (x)  y  £ (x)£ (y)£ z  (x) +1  x  {x)£ (y)£ (x) - £ (y)£ (x) + £ (yf £ {xf  z  x  x  x  £ (*)  x  - £ (y)£ (xf + £ (x)£ (x)]- £ (z)[- £ {y)£ z  y  (x)£ (x) + £ (y)£ (x)  2  y  (y) + £ (y)£  y  z  y  z  + £ (yf  x  -  y  (x) - £ (y)e (x)£ (y) + £ (xf +  y  y  z  x  y  £ (y)£ (x)£ (x)]+ [£ (x) + £ (y) + £ (yf £ (x)£ (x) z  y  x  x  x  z  - £ (y)£ (x)£ (y) + £ (y)£ (xf £ (y)z  z  x  y  y  z  z  y  £ (y)£ (x)e (x) y  y  x  + £ (yf £ (x) + e (y)£ (y) + £ (x)£ (x) + £ (y)£ (x) z  x  + £ (y)£ x  y  y  z  (xf + £ (y)£ x  y  z  y  z  (x)£ (y)e (x)}[l + £ (y) £  (xf  2  y  z  x  y  y  +  £ (yf£ (xf+£ (yf+£ (yf£ (xf+£ (yf+£ (yf£ (xf  +  £ (yf (xf+£ (xf+£ (yf£ (xf+£ (yf£ (xf+£ (xf  z  z  x  + £ (yf£ z  y  (xf +£ (yf£ (xf x  x  z  £y  z  + £ (yf+ x  x  z  y  z  £ (xf ] x  y  z  y  x  x  y  Appendix A. The Resulting Equations of the Mathematical Model  158  K I = (y) (y) + (y) * (•") + (y) W + Me w + £ {y)£ (x)e {y)e (x) + £ (y) £ (x) + £ (y)£  £  x  3  £  £  z  £  y  £  2  z  z  x  2  x  y  y  z  x  y  y  £ (y)£ (x)e (y) + £ (y)£ (xf £ (y) + £ (y) £ (x)e (x) - £ (x)£ (y)£ (x) + £ (x)]+ £ (z)[- £ {y) -£ (x) + 2  y  x  x  y  y  x  z  z  z  z  x  z  z  x  x  x  £ (y)£ (x) £ (y) + £ (y)£ (x)e (x) + £ (y) £ (x)£ (x) + £ (y)£ (x)£ (y) + £ (y)£ (y) - £ (y) £ (x) - £ (y)£ (x) + £ (y)£ (x)e (y)e (x) + e (x)e (x)+ e (y)e (x)]2  z  2  z  y  z  z  x  y  y  z  2  y  y  x  x  z  y  y  z  2  y  x  y  x  x  z  z  z  y  £ (z)[l - £ (y)£ (x) + £ (y) £ (x) + £ (y)£ {x)£ {y)£ (x) 2  y  x  x  2  z  z  z  z  x  x  + £ (y)£ (x)£ (y)£ (x)-£ (y)£ (x) + £ (y) 2  z  z  y  y  y  y  z  £ (y)£ (y) (x) + £ (x) -£ (y)£ (x)£ (x) + £ (x)£ (y)£ (x) £  y  z  2  x  z  y  z  x  z  x  y  + £ (y)£ (y)£ U)M + £ (y) £ (x) + £ (y) £ W + 2  x  z  y  2  y  2  y  z  £ (y) +£ (y) £ (x) +£ (y) +£ (y) £ (x) 2  2  z  2  x  +  2  z  2  y  2  +£ (y) £ (x)  2  y  z  2  x  2  x  y  £ (x) +£ (y) £ (x) +£ (y) £ (x) +£ (x) +£ (y) £ (x) 2  2  z  +  y  2  2  z  2  z  2  x  2  y  2  z  y  £ (y) £Ax) +£ (y) +£ (x) ] 2  2  2  x  2  x  x  (y)£ (y) + £ (y)£ (x) + £ (y)£ (x) + £ (x)£ (x) + £ (y)£ (x)£ (y)£ (x) + £ (y) £ (x) + £ (y) - £ (y)£ (x)£ (y) + £ (y)£ (x) £ (y) + £ (y) £ (X)E (X) - £ (x)£ (y)£ (x)+ £ (x)]+ [- £ (y)- £ (x) +£ (y)£ (x) £ (y) + £ (y)£ (x)£ (x) + £ (y) £ (x)£ (x) + £ (y)£ (x)£ (y) n = {-  R  £  z  2  (z)[£  x  z  z  x  y  z  2  z  x  x  y  y  y  x  x  x  z  z  y  y  z 2  x  z  y 2  z  y  x  z  X  2  x  x  z  z  y  2  z  z  x  y  y  z  y  y  x  + £ (y)£ (y)-£y(y) £ W - (y) W + (y) (x)£ (y)£ 00 2  y  z  £  x  £  x  £  2  z  £  x  z  y  x  + £ (x)£ (x) + £ (y)£ (x)]+ £ (z)[l - £ (y)£ (x) + £ (y) £ (x) + £ (y)£ (x)£ (y)£ (x) + £ (y)£ (x)£ (y)£ (x) - £ (y)£ (x) + £ (y (y)£ (y)£ (x) + £ (x) - £ (y)£ (x)£ (x) + £ {x)£ (y)£ (x) + £ (y)£ (y)£ (x)M + £ (y) £ (x) +£ (y) £ (x) +£ (y) + £ (y) £ (x) +£ (y) +£ (y) E (x) +£ (y) £ (x) + £ (x) + £ (y) £ (x) +£ (y) £ (x) +£ (x) +£ (y) £ (x) +£ (y) £ (x) + £ (y) +£ (x) ] (A.14) 2  y  z  z  z  - £y  z  x  z  y  x  x  z 2  x  z  z  y  2  x  z  y  2  2  y  2  z  z  2  x  y  2  x  2  x  2  x  z  2  z  y  2  y  2  y  x  z  2  x  z  2  z  2  y  z  2  z  2  z  y  x  2  y  2  y  z  2  2  z  y  y  y  2  x  z  2  x  x  2  y  2  x  2  x  Appendix A. The Resulting Equations of the Mathematical Model  #33 =  (y)e (y) + (y)  fc  £  x  (*)+£  £  z  x  (y)e (xf  y  z  + £ (x)£ (x) + £ (y)£ (x)£ (y)£ (x) + £ (y) £  (x)  2  z  x  x  y  + e (y) - £ (y)£ y  y  Ay)  e  2  x  £  £  (yf  £  y  £ (y) £ y  +  x  £  y  £  z  (y)  y  x  z  z  y  z  z  x  y  2  y  z  y  + £ (x) 2  x  z  (x)+ £ (y)£ (y)£ (*)]}  y  x  z  z  y  +  2  z  z  +  £ (y) £ (xf 2  x  z  £ (yf+£ (yf£ (xf+£ (yf£ (xf+£ (xf+e (yf£ (xf y  + £ (y) £  y  (xf + e (xf +£ (y) £  2  z  +  z  (x) + £ (y) £ (xf+£ (yf  2  y  (x) + £ (y)e (x)e (y)e (x)  x  x  /\[ + £ (y) £  x  (xf  2  z  2  z  z  y  (x) + £ (y) £  x  z  y  x  (x)e (y)e (x) + e (x)e (x)  (x) + £ (y) - £ (y)£ (y)£ (x)  y  z  y  z  £ (y)£ (x)£ (x) + £ (x)£ (y)e y  z  x  x  x  £  y  (y) + £ (y)£ (x)£ (x)  y  (xf + £ (y)£  z  + £ (y)£ (x)£ (y)£ -  y  z  (x)]+ [l - e {y)£  y  z  y  z  x  (y)  z  (x) +£ (y)£ (x) £  z  £  z  (xf e (y) +  z  z  (x) - £ (y)£  2  y  (x)£ (x) + £ (y)£ (x)£ (y) + £ (y)£ (y) -  £  y  x  2  x  x  +  x  + £ (y)£  x  z  (z)[- Ay)~  £  z  (*) - £ (x)£ (y)£ (x)+ £ (x)]-  £  z  (x)£ (y)  x  M  £  y  159  y  z  x  (xf + £ (yf  2  x  x  y  x  y  z  y  z  £ (xf x  ( A - 1 5 )  £ (yf+£ (xf] x  x  The same assumption was made in the Vector Representation method. The vector representing the actual motion of the tool tip with respect to the ..workpiece coordinate system is expressed with Eq.3.88 in Chapter 3. The equation is;  P= R'\x)R' (y)(Z-Y)-R-'(x)X+R- (x)R- (y)R(z)T l  1  l  Without any simplification, the results of vector, P, can be given as follows;  (A.16)  160  Appendix A. The Resulting Equations of the Mathematical Model  P (x) = {l-£ (y)£ (x) P  z  + -e (y)£ (x)l6 (z)-a z-6 (y))  z  y  y  x  xz  (e (y) + £ (x) + e (y)e (x)\S (z)-a z-y-S z  z  x  y  y  YZ  y  x  -x-S (x)-  z  y  (y)) +  y  (- e (y) + e (y) £ (x) - £ (x)\z + Z +S 2  +  JC  (z) - S (y))  Z  z  £ (x)(S (x) - a x) + £ (x)(Z + S (x) +  x  z  y  XY  y  l  "(1 + e ( z ) )(1 - £ (y)£ (x) + - £ (y)£ (x))  z  2  x  z  z  [l +  y  y  |  £ (z) +£ (z) +£ (z) ] 2  2  x  2  z  y  (£ (y) + e (x) + £ (y)£ (x))(e ( z ) +£ (z)£ z  z  x  y  [l +  z  y  (z)  x  £ (z) +£ (z) +£ (z) ] 2  2  x  2  z  y  ( - £ (y) + £ (y) £ (x) - £ (x))(e (z)£ ( z ) - £ ( z ) ) 1 y  x  z  y  z  x  y  [l + £ (z) +£ (z) +£ (z) ] 2  2  x  U\-£  z  z  [  y  y  z  y  x  l + e (z) +e (z) +£ (z) 2  x  (£ (y)  2  z  x  z  l+ y  2  +  y  +£ (z) )  £ (y)£ (*))(!  + e (x) +  z  (- £  y  (y)£ (x) + -e (y)g (x))(e (z) - £ (z)e (z)  z  +  ]'  2  2  |  y  y  £ (z) +£ (z) +£ (z) 2  2  x  (y) + £ (y) x  2  z  £ (x) - £ z  y  ( Z ) + £ (z)£  (X)\E  Y  X  z  ( z ) ) l  y  l + £ ( ) +£ (z) +£ (z) 2  x Z  2  \ '  2  z  y  y  z  z  y  l+  y  y  x  t  ^(z) +^(z) +^(z) 2  + £ (z))  (z)£ (z)  ' (l - e (y)£ (x) + -e (y)£ (x)\e 2  +  2  (e (y) + £ (x) + £ (y)£ (x)\-e (z) + £ (z)£ ( z ) z  z  x  y  l+  x  z  £ (z) +£ (z) +£ (z) 2  x  2  2  z  y  x  z  y  1+ ^(Z) +£ (Z) +£,(Z) 2  2  Z  z  2  (z) ) 2  (- £ (y) + £ (y) £ (x) - £ (x)\l + e y  y  (A. 17)  Appendix A. The Resulting Equations of the Mathematical Model  + £ (y)e (x)\S (z) - a  P (y) = (" £ (*) -£ (y) P  z  z  y  x  x  x z  z - 8 (y)) + X  (- £ (y)e (x) +1 + -£ (y)£ (x)\8 (z)-a z-y-8 z  z  x  y  x  (y)£ (x) + £ (y)£ (x)\z + Z +8 (z)-  (  £y  z  x  2  x  8 (y)) + £ (x)(x + 8 (x)) Z  z  8 (x) + a x-£ (x)(Z +8 (x))  +  '(-£ (x)-£ (y)  + £ (z) )  y  XY  z  l  x  z  + £ (y)£ (x))l  z  y  x  2  2  x  l + e,(z) +^(z) +^(z) 2  z  |  2  +  ( ~ £ (y)£ (x) +1 + -£ (y)£ (X)XE ( Z ) + £ z  z  x  x  l+ J£  (y)£  y  Z  (z)£ \z)  y  x  £ (z) +£ (z) +£ (z) 2  2  x  (x) + £ (y)£  z  (y)) +  y  YZ  x  z  y  (Z)£ ( Z ) - £ (z)) 1  (X)\E  x  2  z  X  y  l + £ (z) +£ (z) +£ (z) 2  2  x  y  2  z  y  I" (-£ (x) - £ (y) + £ (y)£ (x)\£ z  z  [  +  y  x  l+  (z) - £  z  (z)£ (z))  y  x  £ (z) +£ (z) +£ (z) 2  2  x  2  z  y  (- £ (y)£ (x) +1 + -£ (y)£ (x)Xl + £ (z) ) 2  z  z  x  x  y  l + ^(z) +e (z) +^(z)  +  2  |  2  z  2  (e (y)£ (x) + £ (y)£ {x)\e (z) + £ (z)£ (z)) 1 y  z  x  x  x  z  i+^(z) +^(z) + ^(z) 2  2  I" (- £ (x) - £ (y) + £ (y)£ (x))j£ z  +  z  [  y  z  y  2  (z)£ (z) + £ (z))  2  z  2  (*) + ! + -£ (y)£  z  x  x  (x)X-£  X  2  (z) + £ (z)£ (z))  l+ ^(z) +£ (z) +^(z)  +  2  ( £ (y)£ ix) + £ (y)£ y  z  x  x  z  z  2  z  2  (x))l + £ (z) ) I z  i+^(z) +e (z) +^(z) 2  y  x  l + ^(z) +^(z) +^(z)  (- £ ty)£  +  x  y  2  2  2  y  y  w  y  X  Appendix A. The Resulting Equations of the Mathematical Model  P (z) = (£,(*) + e (y) P  W + £ , ( y ) K ( z ) - a z -(5,(y)) +  z  xz  k(y)s,(*) - e (x) - e (y)\S (z)-a x  + (-£ (y)£ (x) y  x  y  - £ (y)e (x)  y  x  z-  y-S (y))  YZ  y  + l\z + Z +S (z)-S (y))  x  2  z  z  £ (x)(x + S (x)) + £ (x)(S (x) - a x) -Z -5 y  x  x  y  XY  X  { (x) + £ (y) £ (x) + £ (y)\l £y  +  z  x  l+  (x)  Z  2  x  2  x  x  y  z  (£ (z) + £ (z)£ (z)  x  l+  z  y  x  £ (z) +£ (z) +£ (z) 2  2  x  2  z  y  (- £ (y)£ (*) ~ £ (y)£ O) + l\e (z)e (z) - £ y (z)) y  y  x  x  z  x  l + g (z) +g,(z) +g (z) 2  2  v  y  x.  2  v  (g(x) + £ (y) £ (x) +  +  -  + £ (z) ) •+ £ (zf+£ (z) +£Azy y  (e (y)£ (x) - e Q) - £ (y))+ z  162  z  £ (y)\e (z)-£ (z)£ (z))  x  l+  y  t  y  x  £ (z) +£ (z) +£ (z) 2  2  x  2  z  y  (g (y)g W-g U)-g,(y)):i + g (z) ) l + g (z) +^(z) +^(z) 2  ;  ;  +  y  ;c  y  2  2  2  | +  x  (- £ (y)£ (x) - £ {y)£ y  y  x  (x) + l\e (z) + £ (z)£ (z)  x  x  l+e  x  +  z  2  2  7  (£ (x) + £ (y)£ (x) + £ (y))(g (z)£ (z) + £ (z)) l + g,(z) +g (z) +g (z) y  z  x  y  z  2  2  y  x  x  y  y  (£ (y)g (*) - £ (x) - £ (y))(-£ z  x  2  z  +  y  (z) +£ (z) +£Az) 2  (z) + £ (Z)£ (z))  x  z  l + g,(z) +g (z) +g (z) 2  2  z  y  2  y  (- £ (y)g (*) - g, (y)g, (*) + + £ (z) )' z, z„ i+grz) +g (z) +g (z)  •+  2  y  y  z  2  2  z  2  v  (A. 19)  Appendix B : Principles of Laser Interferometry Systems  A Renishaw M L 1 0 X Laser Interferometer system is used for linear and angular error measurements in this research. The laser interferometer system is capable of measuring displacements up to 80 meters with 1 nanometer resolution and 0.05 ppm (part per million) frequency accuracy and 0.7 ppm (part per million) linear measurement accuracy, which means there may be an error of 0.7 micrometers when the laser beam travels 1 meter because of the inherent inaccuracy of the laser system [42].  The laser source used M L 1 0 X laser head is a homodyne HeNe laser tube. The interferometer optics splits the laser light into two paths; a reference path and a measurement path, then recombines the light returning from the two paths and directs it to a photodiode in the laser head where it produces an interference signal. The measuring electronics measures and accumulates the phase and provides a position output for the application. Some specifications of the Renishaw M L 1 0 X laser system are given in Table B . l .  Table B . l : Some specifications of the Renishaw laser system Laser S o u r c e  HeNe laser tube (Class II)  Laser Power  <1 Mw  V a c u u m wavelength Laser frequency a c c u r a c y Resolution Operating temperature Operating humidity Linear A c c u r a c y  632.9906 nm ±0.05 ppm 1 nm 0 - 40 °C 0 - 95% (non-condensing) ±0.7 ppm  A general set-up of the laser interferometer system is shown in Figure B . l . The main components of the system are the laser head, which is the laser source, linear and angular  Appendix B. Principles of Laser Interferometry Systems  164  optics, environmental compensation unit, sensors (air temperature sensor and material temperature sensors) and data acquisition system. Linear Beam Splitter  Linear Reflectors  Figure B . l : Set-up of the laser interferometer system for displacement measurements (Source: Renishaw Inc. [42])  Four types of optics are used for displacement and angular measurements. They are linear retroreflector, linear interferometer, angular interferometer and angular reflector which are illustrated in Figure B.2. A linear retroreflector is basically a trihedral prism which reflects an incoming beam parallel to coming direction but with an offset of the incoming beam's distance from the corner apex [42]. The linear interferometer is the combination of a linear retroreflector and a beam splitter which separates the incoming beam into two components and sends them in two perpendicular directions. The angular interferometer is a linear interferometer with a beam bender attached to it while the angular reflector is an optic consisting of two linear retroreflectors. The linear retroreflectors inside the angular reflector are spaced at a precisely known distance apart.  Appendix B. Principles of Laser Interferometry Systems  165  Figure B.3 shows a linear interferometer and a retroreflector combination used for displacement measurements. The beam from the laser head enters the linear interferometer, where it is split into two beams. The reference beam is directed to the reflector attached to the beam-splitter, while the second beam, which is the measurement beam, passes through the beam-splitter to the second reflector. Both beams are then reflected back to the beamsplitter where they are re-combined and directed back to the laser head where a detector within the head monitors the interference between the two beams.  During linear measurements, one of the optical components remains stationary, while the other moves along the linear axis. A displacement measurement is produced by monitoring the change in optical path difference between the measurement and reference beams. It is a differential measurement between the two optical components and is independent of the position of the laser head. This measurement can be compared to the read-out from the scale of the machine under test to establish any errors in the machine's accuracy.  166  Appendix B. Principles of Laser Interferometry  Angular measurements are made from a sine measurement with the optics arranged as illustrated in Figure B.4. The optics create two parallel beams between the interferometer and the reflector. For angular measurement to take place, there must be a rotation of one optical component with respect to the other. This causes a change in the path difference between the two measurement beams. This change in path difference is determined by the fringe-counting circuitry in the laser head and is converted to an angular measurement, or angular error, by the software. RETROREFLECTOR  RETROREFLECTOR F r o m L a s e r Head  To M e a s u r e m e n t Receiver LINEAR INTERFEROMETER  Figure B . 3 : Linear interferometer and retroreflector arranged for displacement measurements  The angular interferometer is placed in the beam path between the laser head and the angular reflector, as shown in Figure B.4. The side of the angular interferometer with two optical faces must face away from the laser head, towards the reflector. For pitch measurements along a horizontal axis, both optical components are mounted vertically; for yaw measurements, they are both mounted horizontally.  The laser beam is split into two by the beam-splitter contained within the angular interferometer. One part of the beam passes straight through the interferometer and is reflected from one half of the angular reflector back to the laser head. The other beam passes through the periscope of the angular interferometer to the other half of the angular reflector, which returns it through the interferometer to the laser head.  167  Appendix B. Principles of Laser Interferometry Systems  Angular measurements are achieved by comparing the path difference between the beams. The measurements are independent of the distance between the laser and the interferometer and also of the distance between the interferometer and the reflector.  ANGULAR INTERFEROMETER  ANGULAR REFLECTOR  Figure B.4: Angular interferometer and angular reflector arranged for angular measurements  The factors that affect the accuracy of the measurements can be categorized as environmental factors and alignment errors. Errors caused by environmental factors include change in velocity of light changes, incorrect positioning of temperature sensors, localized heat sources and deadpath error. Hence, the use of an environmental compensation unit is essential especially in linear displacement measurements.  In the laser head the measured displacement is calculated using the half wavelength of the laser beam. Therefore the accuracy of the laser interferometry system depends highly on the wavelength of the laser beam, which is dependant on the refractive index of the medium through which it is passing. Since the refractive index of air varies with temperature, pressure and relative humidity, the wavelength value used to compute the measured values may need to be compensated for changes in these environmental parameters. The absolute accuracy of the measurement is affected by one part per million for any of: air temperature change of 1 °C, air pressure change of 3.3 mbar, or humidity change of 30%. If the variation in  Appendix B. Principles of Laser Interferometry Systems  168  wavelength is not compensated for, then linear laser measurement errors can reach 50 ppm. When the measurements are done for angular motions, it is not necessary to compensate for the environmental factors since the angular measurements are based on a comparison of two lengths.  The air temperature sensor and material temperature sensors are used in the laser systems to get some information for the wavelength compensation and the expansion of the structural components of the machine tool. Therefore, it is important to place these sensors to appropriate locations. It. should be avoided to place the sensors close to localized hear sources or in cold draughts. The air temperature sensor should be placed as close as possible to the laser beam's measurement path. The material temperature sensors are used to avoid the calibration error of the machine tools. The international reference temperature is assumed to be 20 °C and C M M s and machine tools are normally calibrated with reference to this temperature. In a factory environment, it is hard to maintain this temperature every time. Because most machines expand or contract with temperature, the change in the temperature can cause an error in the calibration. To avoid the calibration error, the software normalizes measurements using the coefficient of expansion, which must be entered manually, and a mean machine temperature measured using the environmental compensation unit. The objective of the correction is to estimate the laser calibration results that would have been obtained if the machine calibration had been performed at 20 °C.  Deadpath error is an error associated with changes in the environmental factor during a linear measurement. Under normal conditions, deadpath error is insignificant and only occurs if the environment changes after a datum and during measurement. To avoid the deadpath error, the laser software should be datumed when two optics are close to each other to decrease the effect of the changes in the environmental changes.  Appendix B. Principles of Laser Interferometry Systems  169  The alignment errors include the cosine error, Abbe error and optics not positioned correctly or optics not fixed rigidly. Any misalignment of the laser beam path relative to the axis of motion results in a discrepancy between the measured distance and the actual distance traveled. This error is known as cosine error. As shown in Figure B.5, the size of this error is related to the angle of misalignment between the laser beam and the axis of travel.  Where a measurement is made with the beam aligned parallel to, but offset from, the defined axis of calibration, machine angular errors (e.g. pitch or yaw) can introduce an Abbe offset measurement error (see Figure B.6).  Appendix B. Principles of Laser Interferometry Systems  Moving Optic  170  A b b e Error  A x i s of _Laser B e a m  |  Extension.  Laser Head Offset  Y a w or  J  Pitch  it  A x i s of M o t i o n  Figure B . 6 : Set-up with Abbe error (Source: Renishaw [42])  To minimize the effect of Abbe offset error, the laser measurement beam should be coincident (or as close as possible) to the line along which calibration is required. For example, to calibrate the linear positioning accuracy of the Z axis of a lathe, the laser measurement beam should aligned close to the spindle centre line. This will minimize the contamination of the linear measurement from machine pitch or yaw errors. The effect of Abbe errors in linear measurements should be taken into account.  

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