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                                    Publicat de
                     Universitatea Tehnica “Gh. Asachi”, Iasi
                       Tomul XLVII (LI), Supliment, 2001
                         CONSTRUCTII DE MASINI



                   DUMITRU ZETU and IONEL SOFRON

           Abstract. Geometric errors belong to the systematic errors category of
     machine tools. The measurement and compensation of this errors is an effective
     method to enhance the machining operations. This paper proposes a review of
     measurement and compensation problems geometrical errors due to the motion
     the axis of machine tool.

          Keywords: geometric      errors,    compensation,   measuring   instrument,
     optoelectronic sensor.

                                1. Introduction

     The manufacturing accuracy on machine tools is determined by two
categories of errors [1]:
     - Systematic errors, like the tools wear, thermal and elastic deformations of
machine tool, geometric errors and measuring error, outlined errors and dynamic
     - Random errors, due to the tool breaking, errors caused by non-
homogeneus manufactured material, errors caused by programing of machine
tools, eroror of fastening of piece.
     With respect to ensure that the processed pieces are in tolerance limit
request by designer, the necessary to make the systematic errors compensation
34                        Dumitru Zetu and Ionel Sofron             Constructii

and to prevent random errors appearance [2]. Geometric errors belong to the
systematic errors category of machine tools.

        2. General Comments on Geometric Error of Machine Tool

     Murphy observe in paper [3], geometric errors effects can contribute more
than 40% to the entire sources of machine errors, accordingly to Figure 1.

                        Fig. 1 - Machine sources of inaccuracies.

                                        Table 1
                                      Key to errors
             Alg     Linear axis alignments
             Ang     Spindle angular runout
             Ax      Spindle axis runout
             Beta    Rotary axis (table)
             C       Beta axis centrality
             Clnt    Coolant thermal errors (machining)
             En      All other environment thermal errors
de Masini            Bul. Inst. Polit. Iasi, t. XLVII (LI), Supliment, 2001      35

                                        Table 1
              Ge       Machine geometry
              Ma       Machine thermal errors
              O        All other machine thermal errors
              Pch      Beta axis pitch
              Pos      Linear and Beta axes positioning errors
              Rad      Spindle radial runout
              R&F      Beta axis rise & fall
              Sp       Spindle geometry and thermal error
              Str      Linear axis straightness
              Th       Machine termal errors
              Tlt      Linear axis tilts (roll, pitch, yaw)
              Tr       Spindle axis tram
              XYZ      Linear X, Y, Z axes

     There are 21 volumetric error components in machine tool. Linear,
horizontal, vertical errors and roll, pitch, yaw motion errors in each x, y, z axis,
and three squareness error, accordingly to Figure 2.

                    Fig. 2 – Volumetric error components in machine tool.
36                        Dumitru Zetu and Ionel Sofron                Constructii

           3. Geometric Errors Measurement and Compensation

                     3.1. Conceptual Design of the System

     The conceptual design of the real time on-axis volumetric error extraction
system is shown as part of the compensation system illustrated in Figure 3, [4].
It consists of two functional subsystems, the displacement detection subsystem
and an error extraction and compensation subsystem.

                           Fig. 3 – Compensation system.

     The displacement detection subsystem detects the dinamic displacements
of the linear moving element such as a table. The error extraction subsystem will
calculate the three linear and three angular errors from these measured

                          3.2. Single Encoder System

      The optical encoder system is based on an interferential cross grid encoder
such as Heidenhain`s VM 182. The grating consists of squares of a width of 4
µm [5]. However, the measurement resolution can be increased to 0.05 µm with
a 1024-fold interpolation. In the proposed system, two diffrent sensing units
were used as shown in Figure 4, [4]. A single sensor unit that includes two
sensing elements for detection of two perpendicular linear displacements (Figure
4, a). The other sensor unit as shown in Figure 4,b can detect small angular and
one linear displacement.
de Masini           Bul. Inst. Polit. Iasi, t. XLVII (LI), Supliment, 2001   37

                   3.3. Three Encoder System Configuration

     The proposed three dimensional encoder system consists of two-direction
sensing units (Figure 4, a) and one single directional and angular sensing unit
(Figure 4, b) as shown in Figure 5, so that six displacements can be measured.
All scan units are attached to a bracket so that they act as a rigid body.

                     Fig. 4 – Configuration of the sensing units

                       Fig. 5 – The encoder system schematic

                           3.4. Design Considerations

     The following requirements have to be taken into consideration:
     - Resolution: 1 µm for linear displacement and 10 µrad for angular
     - Measurement range: 0 – 1000 mm;
38                         Dumitru Zetu and Ionel Sofron                 Constructii

     - Traversing speeds of up to 120 m/min;
     - Simultaneous detection of three linear and three angular motions;
     - Real-time and in-process environment: motion, vibration, and temperature
     - Compactness and simplicity for mounting and setup.
     From the six measured displacements six error components (three linear
components and three angular components) can be extracted and simultaneously
detected. This gives a complete and accurate description of the volumetric error
on a single axis. To get a resolution of the system that fulfills the requirements
the distance between adjacent encoder scales has to be more that 75 mm. The
interdependence between the beam width W and the angular resolution δ for an
encoder with a resolution of 0.05 µm is shown in Figure 6.

                 Fig. 6 – Angular encoder resolution vs. encoder width

                        3.5. Error Extraction Procedure

     Erroneous motion of a machine slide can be detected by the relative
position of the encoder carriage to the encoder mounting structure. In Figure 7,
the sensor displacements are indicated by arrows according to their
measurement direction.
     From these displacements, the linear and angular displacements of the
carriage with respect to its coordinate frame x-y-z can be calculated. The system
consists of three coordinate systems. The reference coordinate (R) system XYZ,
the encoder center (EC) coordinate system x', y', z' which travels along the Z-
axis of the reference coordinate system and the encoder (S) coordinate system
which describes the motion of the encoder bracket. The encoder head locations
are described by the vectors s1 to s6 in the EC coordinate system. With a
coordinate transformation of these vectors in the EC system a relationship
among the angles (pitch (α), yaw (β), roll (θ)), translation (dx, dy, dz) and the
displacements (ds1, ds2, ds3, ds4, ds5, ds6) can be obtained. Using
de Masini               Bul. Inst. Polit. Iasi, t. XLVII (LI), Supliment, 2001               39

homogeneous coordinates transformation of the EC to R coordinate system is
described by the matrix A and the transformation from encoder (S) to encoder
center (EC) is described by matrix B.

                     1         0 0 dx     1                     −α        β      dx 
                     0         1 0 dy    α                       1       −θ      dy 
(1)                A=                  B=                                           
                     0         0 1 dz    − β                     θ        1      dz 
                                                                                    
                     0         0 0 1      0                       0       0      1

                          Fig. 7 – Placement of the sensing elements

       The position vectors of the sensor scan heads in the S-coordinate system

                 s1 := [x1 , y1 , z1 ], s2 := [x2 , y 2 , z 2 ], s3 := [x3 , y3 , z3 ],
                 s 4 := [x 4 , y 4 , z 4 ], s5 := [x5 , y5 , z5 ], s6 := [x6 , y6 , z 6 ].

     When multiplying these vectors by the matrix B the position of the vectors
with respect to the EC coordinate system can be obtained. By finding the
Jacobian and its inverse, the position can be extracted.

                                          4. Conclusions

     This method suppose the real time geometric errors measurement and
compensation of machine tools. Through this method application, resulting a
series of advantages:
     - Can provide direct error detections;
     - Large measurement range;
     - Simultaneous detection of three liniar and three angular motions;
40                             Dumitru Zetu and Ionel Sofron                        Constructii

      - Compactness and simplicity for monting and setup.

                                                Technical University “Gh. Asachi”, Jassy,
                                                     Department of Machine-Tools


1. Z e t u D., C a r a t a E., Sisteme flexibile de fabricatie. Ed. Junimea, Iasi, 1998.
2. S a r t o r i S., Z h a n g G X., Geometric Error Measurement and Compensation of
      Machines. Annals of the CIRP, 44, 2, pp. 599-609 (1995).
3. M u r p h y S D., In-Process measurement and Control. Textron, Inc. Danville,
      Pennsylvania, 1990.
4. Y a m a z a k i K. et al., A study on the Developement of Three Dimensional Linear
      Encoder System for In-Process Motion Error Calibration and Compensation of
      Machine Tool Axes. Annals of the CIRP, 49, 1, pp. 403-406 (2000).
5. S p i e s A., Längen in der Ultrapräzisionstechnik messen. Feinwerktechnik &
      Messtechnik, 98, 10, pp. 406-410 (1990).

                       ALE MASINILOR-UNELTE


       Lucrarea prezinta, în prima parte, unele din problemele cu caracter general ale masurarii
si compensarii erorilor geometrice ale masinilor-unelte. În partea a doua, se prezinta un studiu
asupra unei metode de masurare si compensare a acestor erori geometrice ale masinilor-unelte
si în special ale erorilor datorate axelor de miscare ale saniilor masinilor-unelte.