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BULETINUL INSTITUTULUI POLITEHNIC DIN IASI Publicat de Universitatea Tehnica “Gh. Asachi”, Iasi Tomul XLVII (LI), Supliment, 2001 Sectia CONSTRUCTII DE MASINI A STUDY ON THE GEOMETRIC ERRORS MEASUREMENT AND COMPENSATION IN MACHINE TOOLS BY 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 : - Systematic errors, like the tools wear, thermal and elastic deformations of machine tool, geometric errors and measuring error, outlined errors and dynamic errors; - 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 . Geometric errors belong to the systematic errors category of machine tools. 2. General Comments on Geometric Error of Machine Tool Murphy observe in paper , 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 (continuation) 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, . 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 displacements. 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 . 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, . 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 displacements; - 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 disturbances; - 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 are: s1 := [x1 , y1 , z1 ], s2 := [x2 , y 2 , z 2 ], s3 := [x3 , y3 , z3 ], (2) 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 REFERENCES 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). UN STUDIU AL MASURARII SI COMPENSARII ERORILOR GEOMETRICE ALE MASINILOR-UNELTE (Rezumat) 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.
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