BULETINUL INSTITUTULUI POLITEHNIC DIN IASI
Universitatea Tehnica “Gh. Asachi”, Iasi
Tomul XLVII (LI), Supliment, 2001
CONSTRUCTII DE MASINI
A STUDY ON THE GEOMETRIC ERRORS MEASUREMENT
AND COMPENSATION IN MACHINE TOOLS
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,
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
- 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.
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
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
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
- 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
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.
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,
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
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.