Geometrical Errors

Figure 6.3 Plot of F vs C, (a) theoretical curve (b) with geometrical error,...

The fractional light absorption can be measured in a separate experiment. Knowing and estimating the extent of decomposition, f , the incident intensity can be calculated in the units of einstein cm-3s-1 falling on the reaction cell. To avoid geometrical errors due to differences in absorptivity of the actinometer solution and the sample, the same cell is used for actinometry and for the reaction, under conditions of equal optical densities. There are a number of photochemical reactions which have been found suitable as actinometers. They are useful within their specific wavelength ranges. 

Another reason for the preference of up-numbering is actually due to a manufacturing problem. Tolerances in silicon etching are absolute tolerances which means that with decreasing size of the channels the geometric errors increase. This influence is balanced if a large number of channels are combined. 

Geometrical errors associated with manual measurements include height error, peak width error, half-height error, and baseline position error. 

In a diffractometer measurement by the standard focusing technique, the chief geometrical error is a displacement of the specimen surface from the diffractometer axis. If the specimen is displaced by an amount x along the reflecting-plane normal, then a line shift will occur even for a stress-free specimen, as it is rotated from j/ = 0 to = ij/. This shift, in radians, is given by [16.29]... 

The geometrical errors shown in Table 1.3 include 6 N-H- 0 bond lengths and angles and 11 N-H N bond lengths and angles. fix(-H) Y and y denote the bond length between X-Y and H-Y,... 

For machine tools, we group errors in error motions of spindles and rotary axes (e.g., axial error motion, radial error motion), error motions of linear axes (e.g., linear positioning error motion, straightness error motion, pitch error motion), contouring errors (e.g., circular error) (ISO 230-1 2012). Errors of machine tools may be also grouped according to the error source geometric errors, errors of the numerical control, dynamic errors, thermal errors, load errors, etc. 

The positioning error (sometimes called positioning and orientation error) can be defined as the deviation of the actual values of the parameters from the nominal ones. This error is due to geometrical inaccuracies in the mating parts of the device (machine tool or robot) and clearances. Geometrical errors can be compensated by proper calibration techniques. 

Thermal errors are the geometric errors that a piece suffered during its manufacturing processes as a consequence of temperature changes. 

A strategy for geometric error characterization in muiti-axis machine... 

Keywords Laser tracker, volumetric verification, geometrical errors, regression functions, kinematic model. 

Generally machine tool accuracy is obtained by reducing the influence of qausi-static errors, especially geometric error. Since these source of error is the main contributor to overall error of the machine tool. The different verification techniques used in order to improve the accuracy of the machine reducing the influence of geometric errors are divided into direct measurement techniques [3-4] and indirect measurement techniques of errors [5-8]. Verification by direct measurement of errors is based on the calculated independently to each one geometrical error of the machine tool using measurement system with known dimensions. Errors obtained in a position of the MT... 

Approximation functions of machine toll geometric errors... 

Machine tool geometric error identification strategy depends on the type and configuration of the machine as well as the purpose of verification [5, 6].The difference between theoretical and real point represents the influence combined of machine errors for each point, volumetric error (ve) Eq, 1. Minimizing the machine tool volumetric error using non-lineal optimization techniques, the approximation functions of each error can be obtained, if there are enough measured points. 

The development of kinematic models based on machine structure is one of the most important steps for an error compensation strategy. The MT kinematic model is used to understand and mathematically describe the motion of the machine [5,6,9], The sequence of movements that describes the kinematic model is determined by the t5q)e of machine, the geometrical structure and the number of axes of the same. The position of a tool tip relative to a measurement system in cartesian coordinates (LT) is determined by the following the programmed nominal position, the position of the tip of the tool with respect to the reference machine (offsets T) and the geometric errors of the axes. 

Volumetric error reduction depends on polynomial regression functions used to characterize the geometric errors To realise a proper volumetric verification of any machine tool with rotational axes, such as XCFZ, it is necessary to obtain the approximation functions of each of the errors of a rotational axis, as well as approximation functions of lineal ones. 

Equation 13 presents the approximation function of a geometric error in a lineal axes, where a,... 

The physical behaviour of the geometric errors on rotary axis makes it impossible to characterize them by a simple polynomial of order three. This is due to the periodic behaviour of these geometric errors. To realize a better characterization of the errors, periodic functions must be used. 

In order to realise the characterization of all geometric errors of a machine regardless of the measurement strategy to used, it is necessary to place the LT on a profile attached to the turntable. Consequently, the LT is available to measure the range of axes x and z in different positions of c between 0° and 360°. This measurement method is limited either by protective housings of the machine or by the structure of it, therefore the range of the axes to measure depends on the strategy selected (Fig. 5). 

Independent optimization of linear and rotational axes. This strategy requires two independent measurements. In the first one, a set of points forming a XZ plane is measured in a position of C (Fig.5 B). This measurement is used to characterize only error of the linear axes. The second measurement is formed by a point XZ which is measured in different positions of C. From which, the geometrical errors of the rotary axis are characterized (Fig. 5 C). 

Combined optimization of linear and rotational axes. This method requires of two tests. In the first test, a plane of points in C=0 is measured. It is used to realise an independently geometrical error characterization of lineal axis (Fig.5 B). Once these errors are characterized, a new plane XZ which is capable of being measured at different position of C is measured, test 2. This test is influenced by rotational and lineal axes errors (Fig. 5 C). 

The approximation functions obtained by intensive process of parameter identification provide a mathematical compensation of the combined effect of all geometrical errors. 

N.A.Barakat, M.A. Elbestawi, A.D. Spence. Kinematic and geometric error compensation of a coordinate measuring machine. Machine Tool Manufacture International Jumal of Machine Tools and Manufacture. 2000, vol 40, pp. 833-850. 

Chen, J.S., Kou, T.W, Chiou, S. H.Geometric error calibration of multi-axis machines using an autoalignment laser interferometer. Precision Engineering, 1999, Vol.23, pp. 243-252. 

H.Schwenke, W.Knapp, H.Haitjema, A Weckenmann, KSchmitt, F.Delbressine. Geometric error measurement and compensation of machines. CIRP Annals-Manufacturing Technology. 2008,... 

Machine Tool Calibrator MTC for measuring some major geometrical errors of the machine tool ... 

Abstract. Calibration of Laser Tracker systems is based most times in the determination of its geometrical errors. Some standards as the ASME B89.4.19 [1] and the VDI 2617-10 [2] describe different tests to calculate the geometric misalignments that cause systematic errors in Laser Tracker measurements. 

Laser Tracker with beam source in the rotating head. This model is typical of API and FARO and determination of their geometric errors has been systematized by Muralikrishnan et al. [3] and Hughes et al. [4]. According to the proposed model, calibration corrections are based on 15 parameters, each representing the influence of a particular geometric error in the overall error of the equipment ... 

Laser Tracker with beam source column. LT model used by Leica and whose geometric error model was set by Loser et al. (1999) [5]. Geometric errors are similar to those from the previous model but with the particularity that the laser beam that emerges from the column in the vertical direction LT, is directed to the reflector by a mirror whose center of rotation coincides with the nominal theoretical intersection of the azimuth and tilt axes. The errors described are ... 

It has been shown the beginning of a process which will lead to with the development of a simple procedure for the calibration of laser tracker systems. The work done so far include the tests on SMR and planning of them with Active Target. Also the definition of LT models to study and geometric errors is at an advanced stage along with the definition of an automatic generator of synthetic data with simulated errors. 

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