Error compensation in machine tools — a review

International Journal of Machine Tools & Manufacture 40 (2000) 1257–1284

Error compensation in machine tools — a review Part II: thermal errors R. Ramesh, M.A. Mannan *, A.N. Poo Department of Mechanical and Production Engineering, The National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 Received 7 July 1999; received in revised form 22 December 1999; accepted 14 January 2000

Abstract One of the major errors in machine tools namely geometric/kinematic errors was discussed at length in Part I of this paper. Here, in Part II, another major source of inaccuracy, namely thermal error that occurs due to extended usage of the machine is analysed. Continuous usage of a machine tool causes heat generation at the moving elements and this heat causes expansion of the various structural elements of the machine tool. It is this expansion of the structural linkages of the machine that leads to inaccuracy in the positioning of the tool. Such errors are called thermal errors and constitute a significant portion of the total error in a machine tool. Thus the overall volumetric error of a machine tool is not only dependent on errors due to the assembly and the specific kinematic structure of the machine but also on the thermal errors. In Part II of this paper, an attempt is made to review the work carried out over the last decade in the estimation and compensation of temperature dependent errors.  2000 Elsevier Science Ltd. All rights reserved. Keywords: Thermal error; Real-time thermal error compensation; Thermocouple; Laser interferometer; Thermal deformation

1. Introduction The accuracy of the workpiece produced on metal-cutting machine tools is largely influenced by deviations from the planned relative movement between the tool and the workpiece, the elastic deformation of the tool and tool wear and the elastic deformation of the work holding fixture. Deviation from the defined relative motion between the tool and the workpiece may be classified as geometric and kinematic errors. When errors due to the increase in the temperature of the * Corresponding author. Fax: +65-779-1459.

0890-6955/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 0 0 ) 0 0 0 1 0 - 9

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machine elements (thermal errors) need to be appraised, only those thermal deformations that lead to a relative displacement at the cutting point and thus have an influence on the accuracy of the work being produced, are considered. However, during a machine acceptance test, it is rare for the thermal behaviour of a machine to be separately appraised like the introduction of a heat source in the machine etc. The effect of the temperature in the change in shape of the machine components may be determined by measuring the geometric/kinematic behaviour whereby the temperature distribution over the whole machine is a parameter [44]. One of the techniques employed in solving the problem of errors caused by temperature variation is to use materials like cement concrete, fibre reinforced plastics etc. in the construction of the machine tool as seen in the work carried out by various researchers [21,22,35,37,38]. Although these methods do reduce the deformation of the structure on account of temperature changes, this technique of error elimination by careful design tends to be very expensive. It has been found that it is more cost effective to compensate for these thermal errors. Consequently, the common procedure that is employed to this effect is to measure the temperatures of critical points on the machine and also the error induced in the machine at these temperature states. By analysis of these two sets of data, a predictive model is arrived at that could map the error data with the recorded temperature readings. This model is thus capable of predicting the error in the machine tool for any specific temperature condition that might be encountered. Based on the predicted data, the necessary compensation values are calculated and these are incorporated in the respective axes to effect the compensation.

2. Thermal errors As explained earlier, thermal errors are those that cause a relative displacement between the workpiece and the tool on account of deformation or expansion of the machine elements due to an increase in their temperature. Relative movement between the various elements of the machine causes heat to be generated at the contact zones and it is this heat that leads to the deformation of the machine elements. Some of the possible heat sources are: a) Bearings b) Gear and hydraulic oil c) Drives and clutches d) Pumps and motors e) Guideways f) Cutting action and swarf g) External heat sources The effect of these sources of heat is as shown in Figs. 1 and 2. The effect of these thermal variations on the machining accuracy may be determined by measuring the geometric and kinematic behaviour (as explained in the previous section) of the machine considering the temperature distribution pattern over the whole machine. In addition to the errors as mentioned in the previous section, a few other errors arise on account of the temperature variations namely spindle and origin drift errors and spindle inclination errors [9,25,44].

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Fig. 1. Thermal effects diagram [9].

Fig. 2. Examples of thermally induced displacements on a milling machine [44].

In general, the thermal errors could be divided into two categories. In the first category are those errors that change as a function of the temperature but not the axis position. These errors effectively change the machine offsets and are known as position independent thermal errors (PITE). The second category of errors deals with those that change as a function of axis position

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as well as temperature. They effectively alter the linear positioning of the machine and are known as position dependent thermal errors (PDTE). In simplifying the problem and determining the most suitable compensation technique, it is useful to define these categories of thermal error. The effect of a PITE on component accuracy is strongly dependent on the rate of change of the PITE relative to the time taken to produce a component. Even large changes in a PITE may result in little error if there is no significant change during the time taken to machine a component. A PDTE will produce component errors if the change in linear positioning of the machine does not match the change in linear positioning required by the thermal expansion of the component [1]. In analysing the behaviour of the machine tool structure under various thermal profiles, the theory of thermoelastic behaviour of structural joints has been applied [2–6]. This theory is based on the recognition of the fact that the distribution of contact pressure along the joint controls the transfer of heat from one structural element to another [4,6]. In general, thermal deformation is calculated based on the assumption of either perfect thermal contact or perfect thermal insulation. Attia et al. [4,5] showed that the thermal deformation of structural elements in contact is affected significantly by the non-uniformity in the distribution of the thermal contact resistance. Heat transfer across the joints in machine tool structures exhibits non-linear thermoelastic behaviour. Two groups of factors affect the behaviour of the joint. The first group consists of factors like external loading, surface texture and material properties of structural elements while the second group contains contact pressure, thermal contact resistance, thermal field, mechanical constraints and thermal deformation of structural elements. A joint in a machine tool represents the contact between elements with their contacting surfaces machined and characterised by a certain roughness and waviness. The convergence of the heat flow lines towards these micro-contact points results in a thermal constriction resistance (refer to Fig. 3). The thermal contact resistance is due to this constriction resistance. The mechanism of heat transfer across the machine tool joint shows that conduction is the only significant mode of heat transfer. The contact resistance of the joint is thus controlled by the contact configuration, the thickness of the interfacial gap and the thickness of the surface film. For particular mechanical properties of contacting elements, all these factors depend on the local contact pressure and hence, the thermal contact resistance should be expressed as position dependent and not an average value.

Fig. 3. Heat transfer across a joint [5].

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When there is a change in the local contact configuration, there is a change in the interfacial gap that causes a change in the surface film. Due to the nature of the mechanical and thermal loading on the structure, the distribution of the contact pressure is inherently non-uniform. This nonuniformity in the contact pressure causes variation in the contact resistance that in turn causes variation in the heat flux. The resulting heat flux generates thermal contact stresses that alter the existing pressure distribution. Thus a closed loop interaction is activated which remains in effect until a state of equilibrium is achieved [2,5]. Thermal contact resistance should thus be defined as a distribution and not as a single value on account of the non-uniformity in the distribution of the contact pressure along the machine tool joint.

3. Progress of thermal error research It has long been understood that thermal errors are a major source of inaccuracy in machine tools and significant improvements in accuracy stand to be achieved by the effective elimination of the same. The high cost involved in dealing with the problem can be greatly offset by the gains that could be achieved upon minimising this problem. Elimination of thermal errors obviates routine activities like statistical quality control measures and ensures reduced scrap rates and better repeatability of the machine. One of the most important initial considerations seems to be the need for temperature control. It is imperative to keep the machine tool in a controlled ambient temperature rather than have it exposed to the vagaries of the atmosphere. At the meeting of the International Committee of Weights and Measures in Paris (1931), it was agreed upon that, ‘length of an object’ would mean its length at 20°C. It could thus be understood that some errors would exist in the machine if measuring or machining were to be carried out at temperatures other than 20°C with the degree of error depending on its thermal distance from this value. A generalised approach that has been proposed by many researchers towards handling the problem of non-uniform temperatures is that all solutions fall into one of the following three categories namely: a) Control of heat flow into the machine tool environment. b) Redesign of the machine tool system to reduce sensitivity to heat flow. c) Compensation through controlled movement. Design should be optimised before compensation. It is quite deceptive to believe that real-time correction for thermal deformations could eliminate design defects connected with thermal phenomena [9]. The thermoelastic behaviour of a machine tool is one of the most important factors in determining the accuracy capability. Due to inaccurate knowledge of the heat source, thermal boundary conditions, mechanism of heat transfer etc., precise prediction of the behaviour of a machine at the design stage is very difficult. Where materials with a low expansion coefficient including well designed temperature control systems are used, it is possible to achieve minimum thermal distortion [45]. Heat from the cutting operation is a major source of error as regards the final accuracy of the job. Though this accuracy depends on the rate of stock removal, which is less in a finish-cut, the intense heat generated as a result of the roughing operation affects the accuracy obtained. This

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factor takes precedence over all other sources of heat generation in the case of high volume production. The principle solution to the problem continues to be the high volume of coolant that is used for heat dissipation. Of late, new solutions that are gaining ground are high speed machining and grinding techniques that divert the heat onto the chip instead of the workpiece. One important suggestion in arriving at a solution to the problem of thermal error is the use of temperature-controlled boxes. These enclosures are designed to contain the machine and provide a controlled atmosphere. They seem to be the better option than the design and construction of temperature controlled rooms that are costlier and difficult to maintain. Boxes have the advantage of being moved along with the machine. As far as instrumentation is concerned, researchers have used thermocouples, platinum resistance thermometers and thermistors for the measurement of the temperature variation of different elements of the machine tool. An innovative technique that was used in measuring the temperature of a rotating workpiece was floating a thermocouple bead on the hydrodynamic oil film adhering to the workpiece [9]. 4. Modelling of the thermal error Thermally induced error is a time-varying non-linear process caused by non-uniform temperature variation in the machine structure. The interaction between the heat source location, its intensity, thermal expansion coefficient and the machine system configuration creates complex thermal behaviour [28]. Researchers have employed various techniques namely finite element analysis, coordinate transformation methods, neural networks etc., in modelling the thermal behaviour. Some of the work carried out in each of these areas respectively is listed herein. Jedrzejewski et al. [18] presented a new method of modelling thermal behaviour of a machine tool based on the determination of the power losses in the kinematic system components. The factors affecting the thermal state of a machine tool are shown in Fig. 4. This method was based on the assumption that the amount of energy dissipated in particular components of the kinematic system is a function of the operational conditions. Once the heat sources are identified and the operational parameters like spindle speed, cutting power, ambient temperature, operational time etc. are known, the power loss values in all components of the drive system as well as temperatures and thermal displacements could be determined automatically. The advantage of modelling the complete structure (shown in Fig. 5) was that actual conditions of heat flow could be adequately accounted for. The 3-dimensional thermal model of the machine tool was built using finite element methods. In somewhat related work, the technique of using 3D finite elements was also employed [19] to simulate the thermal behaviour of a machine tool based on energy losses, temperature distribution and thermal displacements. A predictive model of the thermal displacements was built using regression analysis describing the relationship between temperature increases and displacement. Jedrzejewski et al. [17] also used finite element techniques to determine the temperature distribution and then to determine the displacement or deformation as a result of the same. Optimisation was conducted both in terms of the elements that contribute to heat generation and with those elements of the support structure that contain heat sources. Thus parametric optimisation as well as structural optimisation were conducted. The main goal of optimisation was to minimise the machining errors due to thermal distortion of the machine tool structure.

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Fig. 4. Links between factors affecting the thermal state of a machine tool [18].

Kim et al. [20] analysed the temperature distribution along a ballscrew system using finite element methods (FEM) with bilinear (quadrilateral, four nodes) type of elements. Heat induced on account of friction is the main source of deformation in a ballscrew system, the heat generated being dependent on the preload, the lubrication of the nut and the assembly conditions. The proposed FEM model was based on the assumption that the screw shaft and nut are a solid and hollow shaft respectively. The problem was defined as transient heat conduction in a nondeforming media without radiation. The gap elements that were considered to generate heat were inserted between the screw shaft and nut. Venugopal et al. [42] identified that deformations at any particular point in time are dependent on the temperature of the body at that particular instant. The procedure adopted was to calculate the thermal profile in the domain of interest and then determine the resultant deformation by solving the associated thermoelastic equations using finite difference methods. Finite element techniques were employed to obtain the deformations that were used later for analysis. Thus using the finite element model, the accuracy of the machine tool could be predicted by measuring the temperature at a few points on the structure. It is well known that there are 21 error components in a three-axis machine. Chen et al. [14] however proposed a new model with 32 error components synthesising the geometric as well as the thermal errors. The 11 additional errors were used to represent the thermal effects of the machining centre. Even without any nominal movement of the spindle, the position of the tool

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Fig. 5. Finite element model of the geometrical structure of a machine tool [18].

tip would gradually change on account of the thermal distortion of the spindle and column. This would theoretically produce six error components at the tool tip but since the rotation error of the spindle axis has no effect on the position of the tool tip, only five errors were considered. This formed the first group of the thermal errors. The second group consisted of thermal drifts at the origins of the axes. This would create nine additional error components but only six were considered as one of the three axes was taken as a reference. The volumetric model was developed using homogeneous transformation matrices (HTM). The position vectors of the tool tip were expressed in the spindle co-ordinate system and later transformed into the machine co-ordinate system. Similarly, the workpiece was represented in the cross slide co-ordinate system and later transformed into the reference system. Work was done on a turning centre in the compensation of thermal errors. Two types of data were obtained namely, error and related data such as temperature, co-ordinates etc. A separate model was created for each of the error components except squareness and parallelism errors. The PDTE components were mathematically modelled and synthesised using error synthesis models for the purpose of real-time error compensation [25]. Srivastava et al. [36] employed HTMs in modelling the errors on a five-axis CNC machine. The model considered shape and joint transformations for inaccurate links and joints using small angle approximations and arrived at the total volumetric error as a function of all the possible errors. Consequently, errors due to inaccurate links, errors due to inaccurate motion of the slides and errors in axis rotation were considered in the model. The five-axis vertical machining centre (VMC) considered was modelled as a kinematic chain with several links connected in series by prismatic joints and rotational axes. One end of the chain was attached to the work table while

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the other end was attached to a tool on the spindle. A point in the work table co-ordinate system was expressed in the spindle co-ordinate system in order to obtain a complete error matrix. Wang et al. [43] worked on the assumption that there are always only six errors for each axis: one positioning error, two linearity errors, one pitch error, one roll error and one yaw error. A three-axis machine tool was studied for the 21 error sources inclusive of the three squareness errors between each axis. The homogeneous transformation matrix was utilised in analysing the volumetric movement of the machine considering it to be a rigid body. Lo et al. [27] described a 2–D error synthesis model of a four-axis dual-spindle vertical turning centre to explain the positioning errors on the machine. With the realisation of the machine linkage system, four co-ordinate frames were introduced and transformation matrices arrived at for each linkage. The tool position was transformed with respect to the part co-ordinate frame as the desired tool position represents the part dimension. 11 error components were included in this model. Error transformation was obtained as the difference between the actual and desired tool positions. Mou et al. [29] used feature-based analysis techniques for improving the accuracy of CNC machine tools. With the help of assumptions based on rigid body kinematics, the geometricthermal error model was generated. Each error component was characterised as a parametric function of tool position and the machine tool’s temperature profile. Statistical regression methods were used. Yuan et al. [48] charted a general procedure for the compensation of geometric, thermal and cutting force-induced errors. The error synthesis model for the geometric and thermal errors was based on rigid body and small error assumptions. Using HTMs the co-ordinates of the tool were transformed to the part co-ordinate system. The error components were estimated using an inverse kinematic algorithm. The volumetric error vector at any position was then formulated using the error synthesis model. It is found that when the sensor locations were optimised, the thermal error model tends to be linear and has good interpolation and extrapolation capabilities. Yoshida et al. [47] analysed the effect of thermal deformation on the accuracy of a cylindrical grinding machine. As it is very difficult to measure the exact distortion of slideways, the same was calculated using simple equations constructed on the basis of temperature distribution and thermal displacement of the machine. The bed of the cylindrical grinding machine was assumed to be a simple beam. Thermal deflection of the beam was calculated under various kinds of temperature gradients. Krulewich et al. [23] worked on the assumption that temperature distribution in a particular region can be estimated by a polynomial. The same approach, though followed by others previously, involved the use of as many as 100 sensors to capture the full thermal distribution. Here, the fact that the thermal error is related to the integral of the temperature profile rather than the entire profile was made use of. The need to capture the entire profile was thus eliminated. The Gaussian integration method was used and from the data obtained for a minimum number of points the exact solution was generated. This method, though applicable only to position independent errors, was also extended to position dependent errors using a similar integration procedure. Huang et al. [16] used multiple variable regression analysis in order to develop the mathematical foundation for analysing a ballscrew feed drive system. Herein the thermal error was expressed as a function of temperature rise. The coefficients for the model were estimated from sampling data. Balsamo et al. [8] expressed the thermal model as a mathematical equation relating the tempera-

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ture at each machine point as a function of point co-ordinates. The model was a second order polynomial in three co-ordinates. The machine was conceived as consisting of four independent volumes at least: the three guide bearings, on which the three co-ordinate axes were located, and the piece. The deformation model was a link between the deformations and the temperature distribution. Spur et al. [34] analysed the theoretical foundations for describing the thermal behaviour of machine tools. Three different types of machines namely axis-oriented (turret lathe), area-oriented (planning machine) and mixed type (knee-type milling machine) were studied. The different modes of thermal deformation namely linear thermal expansion, thermal bending in the case of single-sided clamping and thermal bending in the case of free support were theoretically analysed and methods of improving the thermal behaviour by cooling arrangements were evaluated. Most current research is focused on the prediction of thermal errors from discrete temperature data. Analytical techniques attempt to capture the entire temperature profile from a finite number of temperature measurements. Empirical models are also used to relate temperature measurements to thermal deformation. ‘Neural Networks’ is a popular method to develop empirical models between discrete temperature measurements and thermal errors. Chen et al. [11] used the homogeneous co-ordinate transformation method to develop the kinematic model of a three axis horizontal machining centre. Since the error map method is time-consuming, the geometric and thermal errors on machine linkages were measured and these errors were synthesised by the model built based on the specific machine configuration. The position vectors of the tool and the workpiece were transformed to the global co-ordinate system of the machine using these HTMs. The difference between the two vectors was taken as a measure of the relative positioning error. The kinematic model of the machining centre was thus developed. A multiple-layer feedforward artificial neural network (ANN) was used to map the measured temperature values with the thermal error. Mou et al. [28] worked out a simplified approach to the problem of thermal error modelling and analysis using neural networks. As the first part, a generalised error model was derived using HTMs that related the positioning errors to the structural and kinematic errors. Each link and joint was modelled as a link transformation matrix and a joint constraint matrix, respectively. The predicted machine error was then used to generate the compensation signal. Temperature measurements were matched with the measured positioning errors. The output of the neural network was used to iteratively modify the error model coefficients. Yang et al. [46] used the generalised delta rule model in order to predict the scalar and volumetric thermal errors of machine tools according to the temperature change occurring at each point. A two layer neural network model, consisting of eight input nodes corresponding to the mounted temperature sensors and six output nodes corresponding to the six measured thermal errors, was used. Veldhuis et al. [40] outlined a strategy for error compensation on a five-axis machine based on neural networks. The initial configuration of the network was explored with data based on the simulation results of the machine. HTMs and small angle approximations were employed in the simulation. The neural network was trained for 20 000 iterations with the simulation data. Two networks were used: one for the z position error and the other for angular error. The hidden layers of both the networks used sigmoidal thresholding functions while the output layers used linear functions. Each network had three hidden layers, the first with 30, the second with 15 and the

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third with seven nodes respectively. The neural network model was later optimised with measured data. Five different tests were run over a period of ten hours. Of these, four were used in training with 500 iterations each and the fifth was reserved for testing. As real-time estimation of thermal errors requires accurate on-line monitoring of thermal errors during machining, this is usually achieved through pre-established analytical or empirical models, correlating thermal error with the temperature at critical points on the machine. Chen et al. [13] also used an artificial neural network based model to map the relationship between the thermal errors and the temperature measurements. A three-layer feedforward ANN using the sigmoidal function was employed. The network was trained by a training set consisting of 540 training pairs of the input vector (temperature) with the output vector (thermal error) collected from several cutting conditions. Chen et al. [12] used a three-layer ANN with a supervised backpropagation algorithm to map the thermal errors to temperature measurements. A multiple output hybrid ANN model was arrived upon after carefully considering the need to use a single-output ANN (one for each thermal error). While the later was effective in both interpolation and extrapolation prediction, it was found to be ineffective when the temperature patterns were totally different from those used in model estimation. It was therefore decided to use the hybrid ANN model with multiple-output. Chen et al. [10] studied both a multi-variant regression analysis (MRA)-based model and an ANN-based model built from air and real cutting conditions. A three-layer feed-forward ANN with ‘sigmoid’ activation function was used to map the temperature measurements with the calibrated thermal errors. The MRA model had the advantage that the physical meaning of the model could be easily interpreted. The model was also robust against conditions that differed from those used in model estimation. The advantage of the ANN model over the MRA model was that multiple thermal errors could be easily modelled and thus complex models could be automatically learned. But the physical meaning was difficult to interpret from the error model. In general, homogeneous transformation matrices have been used to model the errors on the machine. Just as in the case of the geometric errors, a kinematic linkage chain of the machine tool is established between the various linkages (axes) and the tool. In some cases, however, finite element analysis techniques have been used to generate a three-dimensional model of the heat source, or the machine at large, in order to map the heat flow in the system. From this model the temperature distribution and the thermoelastic deformation of the structure could be evaluated. Through this process, both the heat-generating source and the structure itself could be studied. ‘Neural Networks’ is another popular method that has been adopted in the modelling of thermal errors. Temperature measurements of critical points of the machine tool are made and these are then mapped with the measured thermal errors. This gives a correlation between the thermal state of the system and the expected accuracy of the machine. Neural networks have also been used in conjunction with error models using HTMs in order to iteratively modify the error model coefficients. 5. Thermal error measurement and compensation In pursuance of the objective of creating an error compensation system, apart from the activity of modelling described above, the two major tasks that need to be carried out are temperature

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and error measurement and error compensation. Extensive research work has gone into the identification of the various critical points on machine tools where the temperature needs to be monitored on a periodic basis. Different methods have been employed to analyse the data obtained in order to arrive at a suitable method of compensation. In the measurement of temperature, both external factors like the environmental conditions and internal factors like heat generated by motors, pumps, heat due to friction etc. (usually measured by thermocouples), need to be considered [41]. However, measurement of the temperature and error components is only the first step towards the goal of improving the accuracy of machine tools [31]. The other important step to take is the correction of these errors in real-time such that the effect of the errors on the accuracy is minimised. A lot of the research work mentioned below has already been cited before from the point of view of evaluating the modelling techniques that have been employed. This section however examines this aspect, namely of error measurement and compensation, of their research work. Measurement of temperatures has generally been carried out through the use of thermocouples. These are mostly of foil type construction, either T–type or J–type thermocouples. The sensors are pasted onto the surface of the heat source and the data monitored periodically. Lo et al. [27] identified the need to mount 80 thermocouples at various locations on a turning centre as indicated in Fig. 6 based on the knowledge of the major heat sources and possible thermal distortions. Of the 11 identified error components, eight were measured using the laser interferometer while the three spindle errors were measured using non-contact capacitance sensors. The locations of the

Fig. 6. Machine co-ordinate frames and thermocouple locations [27].

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Table 1 Location of thermocouples on the machine [27] Sensor number

Location of thermocouple

13 15, 16, 17, 18 24 27 82, 83 84, 85 88 90 94 96

End of x guide way x and z ball screw bearings and nuts Spindle bearings (front) Rear corner of spindle housing Right of the column front face Middle of the column front face Left of column front face (in the middle) Middle of dual spindle fixture Top of the column rear face Environment

sensors that were actually required are as shown in Table 1. Several combinations of air cutting sequences were conducted to cover most of the thermal conditions commonly encountered. From the experiments carried out, it was observed that among the error components, the largest transitional error was the linear displacement error along the z–axis while the biggest angular error was the yaw error along the z–axis. All the 11 error components were identified through empirical measurements under different thermal conditions. The compensation system consisted of an IBM computer, and interface module with four different interface boards and error model software. Data from the machine regarding axis positions, temperature values, tool length and number etc. were fed into the various interface boards. The real-time error compensation control system is shown in Fig. 7.

Fig. 7. Real-time error compensation control system [27].

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Two hardware systems were developed to perform the compensation action. The Quadrature Correction was used to interrupt the encoder feedback and add or subtract quadrature pulses to accomplish the necessary compensation. The compensated error values were then sent to a CNC I/O address and the tool origin was shifted during cutting. These values are sent to the controller of the machine based on the compensation values calculated by the error model. Once the compensation was achieved, laser measurements as well as a series of actual cutting tests were performed to verify the effectiveness of the proposed system. The results showed that 90% of the machine errors were corrected. Kim et al. [20] worked on the estimation of the temperature distribution on a ball screw system. Six T type thermocouples with a 0.1 s response time were used in measuring temperatures at specific points. Since the main source of heat is the frictional resistance due to the moving nut and rotating bearings, these sensors were specifically mounted to record temperature variations of the same. Sensors were thus mounted on the driving and driven side bearing surfaces, three different surfaces of the nut and one deep inside the nut (5 mm away from the inner diameter of the nut along the radial direction). The one deep inside was used to measure the temperature of the contact area between the screw and the nut. From the data obtained it was found that the temperature distribution and thermal deformation of the ball screw shaft could be considered to vary only in the axial direction. Experiments were performed with the above set-up in order to estimate the heat transfer and deformation. The heating pattern was generated by repeatedly moving the nut of the ballscrew from one end to the other with a certain stopping time at each end. Point temperatures were measured using the sensors mounted while the temperature distribution was measured using an infrared radiation thermocouple. It was observed that stop time as well as the moving velocity affected the temperature rise. Similar work done by Huang et al. [16] involved mounting of thermocouples to the end bearings and nut of the ballscrew. A displacement gauge located at the end edge of the ball screw was used to measure its thermal error. A PCL–818 data acquisition system and PCLD–889 AMP/MUX board were used to collect the data samples. It was observed that there was a distinct relation between the thermal error and the measured temperature increase at the said positions. These results corroborate the conclusions arrived at by Kim et al. Chen et al. [11] identified 23 thermocouple locations on the machine structure and the data from these sensors was scanned by a 16 bit A/D converter updated at a sampling rate of every 2 s. Capacitance sensors were used to measure the thermal drift of the spindle. Thermal errors were measured using a HP laser interferometer and an electronic differential level. A PC-based compensation controller (as shown in Fig. 8) was used for real-time error compensation. The position of the slides of the three-axis horizontal machining centre was accessed through a Q/D board. The data from the thermocouples and the encoder feedback were fed into the A/D and Q/D boards of the PC respectively. The kinematic model arrived at the compensation to be incorporated into the three axes by accepting data from the Q/D board and the ANN thermal error model. The compensation controller then sends the compensation values to the programmable logic controller (PLC) board of the CNC controller through a digital I/O port. The effectiveness of the compensation was verified using the HP laser interferometer. It was found that the accuracy was increased by one order of magnitude after compensation. Veldhuis et al. [40] worked on the identification of the optimum number and location of thermo-

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Fig. 8. Block diagram of the error compensation scheme [11].

couples for significant improvement in the accuracy of five-axis machining. Excess thermocouples were initially mounted on the machine and those that were not found to contribute significantly in the calculation of the final compensation values were eliminated. The region of the spindle slide and the column was identified as the major generator of heat and where the largest structural components were located. 17 thermocouples were mounted on the surface of the spindle support arm and the Z column of the machine. Position and orientation errors between the tool and the workpiece were measured using a laser interferometer. The output of the trained neural network provided the compensation values that could be directly applied to the machine’s controller. As a result of the compensation, the error was reduced from 0.06 mm to 0.02 mm in tests conducted on P20 tool steel during a finishing operation. Chen et al. [13] worked on a vertical machining centre wherein 15 E type thermocouples were used and six thermal errors were measured. The locations and the monitored thermal error of each thermocouple are as given in Table 2. A quick set-up and multiple-error measurement system consisting of on-line probes was developed. Seven measurement points were selected over the entire working zone of which one point was on the centre of the artefact. The various error components were measured at the different points on the artefacts. The thermal effects at any Table 2 Location and monitored thermal error of each thermocouple [13] Sensor Nunber

Locations

Monitored thermal error

T1 T2, T3

Spindle housing Top and bottom sides of cantilever arm Nut and bearing of the z-axis Four sides of the column Environment and the machine base Nuts and bearing of the x-axis Nuts and bearing of the y-axis

Thermal growth of the spindle Thermal bending of the cantilever arm Lead screw thermal expansion Thermal bending of the column Temperature references Lead screw thermal expansion Lead screw thermal expansion

T4, T5 T6 - T9 T10, T11 T12, T13 T14, T15

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location of the working zone were then linearly interpolated between these seven points. Thus, error measurement was carried out using an on-machine probe and with the set of artefacts. The artefacts used are shown in Fig. 9. A PC based compensation controller was employed to achieve the necessary correction. The machine co-ordinates were accessed through a Q/D board while the temperature values were accessed through an A/D board. These errors, along with the static errors and the machine co-ordinates, were synthesised to yield the overall error, which was fed into the CNC controller through the digital I/O board. Contour errors were measured by the circular test using the ball bar. Radial spikes as well as spikes at the direction changing quadrants were observed. The feedforward control was used to eliminate the radial error while the feedforward torque compensation was used to reduce the protrusion at quadrant changes. As a result of these techniques, the contouring errors were reduced. The position accuracy was also enhanced with the help of the thermal error model. Tseng et al. [39] measured all possible temperature distributions on a VMC, such as the temperature of the spindle, bearings on each axis and room temperature. During a series of cutting operations, the temperature records showed significant heat generation in sliding surfaces. Consequently, thirteen major thermal measuring positions were identified. These included the body, the x, y and z bearings, the guideway (two locations) and six different locations on the spindle. The room temperature was also monitored as it serves as the reference. A zigzag cutting process was executed for thirty minutes with a cutting tool. Thereafter, the cutting tool was replaced with a

Fig. 9.

Layout of the artefacts [13].

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RENISHAW optical probe and the thermal displacements were measured. Errors were successfully reduced to 4 µm for general cutting. Yang et al. [46] worked on the prediction of thermal errors using two spherical balls. Thirteen thermocouples were installed on the machine as shown in Fig. 10. The measurement system consisted of a touch probe, an artefact with two spherical balls, a thermal data logger and a personal computer (refer to Fig. 11). The measurement values and the temperature data were uploaded into the PC through the NC controller and the data logger respectively. Thermal errors were divided into scalar and volumetric thermal errors. The volumetric errors were represented by the difference between the measured values of the two spherical balls at time t and the reference values of the co-ordinate measuring machine (CMM). The total errors in the x, y and z directions were thus described using the thermal errors of the reference ball and the volumetric thermal errors. Mou et al. [30] used feature based analysis techniques for improvement of the machine performance. The machine in question was a two-axis turning centre. Nine types of errors were examined on this machine tool and incorporated into the geometric-thermal error model. The errors investigated were: x linear displacement, z linear displacement, x yaw, z yaw, x straightness of z motion, z straightness of x motion, squareness of x axis with respect to spindle axis, parallelism of z axis with respect to spindle axis and spindle drift. The temperature variations associated with these errors were monitored by recording the output of the 39 thermocouples attached to the machine (as shown in Fig. 12). For each error characterisation, sensors were set up to detect both the geometric and thermally induced deviations in the actual machine position and the position

Fig. 10. Set-up positions of the thermocouple [46].

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Fig. 11.

Configuration of the inspection system [46].

Fig. 12. Schematic diagram of a turning centre showing locations of thermocouples [30].

requested by the machine controller. A laser interferometer system was used to measure the various error components. A capacitance probe was used to measure the squareness of the x–axis with respect to the spindle axis. Two optical probes were used to measure the parallelism of the z–axis in respect to the spindle axis. The capacitance probe and the optical probes were used to sense the spindle drift. The errors were monitored and recorded over time as the machine was put through a series of warm-up and cool-down periods. Mou et al. [28] conducted experiments in order to study the time-varying thermal effect on the accuracy of a machine tool. Forty thermocouples were installed at various locations on the vertical machining centre. A NAS979 standard part was first measured on a CMM and the data was used

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as reference. The part was then mounted on the table of the machine for on-machine measurement using the canned probing cycles available on the machine. This on-machine probing was repeated at various temperature profiles in order to study the time-varying thermal effect on the positioning accuracy of the machine. The neural network model used the temperature values and the positioning errors in order to arrive at an error prediction system. Once the network was trained, it could predict the errors based on the temperature values of the thermocouples. Using the principle of inverse kinematics, the compensation values were generated from the errors obtained. Yuan et al. [48] identified seven error components with column thermal deformation namely: two straightness components, pitch, yaw, roll errors of the y–axis and the squareness errors between the x–y and y–z pairs of axes. Using the telescopic ball bar system the volumetric errors were measured at some points in the working volume and the error components estimated using an inverse kinematic algorithm. A total of nineteen sensors were needed (as shown in Fig. 13) for the seven error components mounted. The method of origin shift was used in this research in order to compensate for the error. Data regarding temperature, slide position etc., were collected in real-time and the error components were estimated by the error models. The resultant errors between the tool tip and the workpiece were calculated using the error synthesis model. The compensation signals were then sent to the CNC controller that effected the origin shift. The block diagram of the real-time error compensation system used is as shown in Fig. 14. In the case of the hexapod [33], the thermal deformation was found to be a major source of errors on account of the thermal expansion of the struts due to heat generated by friction in the telescope and the ballscrew drive. Temperature elevations were measured using 34 thermocouples located mainly on the struts. Only a fraction of the observed drift of the tool could be predicted owing to the difficulties in the determination of the effective temperature distribution of the leadscrew using sensors on the outer surface of the strut. Allen et al. [1] developed various techniques to assess a machine’s thermal behaviour. One of the techniques adopted was the recording of a sequence of thermal images whilst the machine ran through a simple cycle. As some structural elements of the machine are likely to undergo greater temperature changes than others, they are likely to produce larger thermal errors. These recorded thermal images give a complete picture of the temperature distribution in the structure.

Fig. 13. Optimal sensor locations on the machine column [48].

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Fig. 14.

Block diagram of the real-time error compensation system [48].

Non-contact sensors and a laser interferometer were also used to directly measure the thermal distortion between the tool and the workpiece. An IBM/PC computer was integrated with the CNC system of a turning centre in order to achieve a comprehensive Geometric–Thermal–Cutting force (G–T–F) error compensation system. The temperature information was collected from thermistors in real-time. The current positions of the slides were read from the controller. The PC also collected data from the laser interferometer and the capacitance sensors for the purpose of calibration. The detailed layout of the error compensation system is shown in Fig. 15. The error synthesis model evaluated the planar error in two directions. The error values were then transferred to the CNC controller in order to implement the error compensation on real-time basis [25]. Experiments conducted at the Purdue University [42] have shown that thermal effects on the accuracy of NC machine tools could be predicted by monitoring the temperature of a few selected points on the machine. A laser interferometer was used in the measurement of the error components. Four thermometers mounted at points around the machine were used to monitor room temperature. Both the data acquisition system and the interferometer were interfaced to a microcomputer. From the data obtained, it was found that positioning errors were the largest and these could easily be compensated using an external reference such as a magnetic scale. The other errors were predictable as a function of the temperature of the machine tool. In work conducted on the estimation of the cylindrical accuracy of workpieces that are longitudinally ground on a cylindrical grinding machine, Yoshida et al. [47] arrived at equations that could be used to calculate the temperature induced cylindrical error. The temperature distribution of the grinding machine was determined at about 50 points by means of thermistors and the output recorded. In order to measure the thermal displacements, test stands were built around the machine and strain gauges were mounted at the 30 measuring points. The points were mainly around the upper section of the bed in the horizontal plane. The thermal distribution within the machine after six hours of running without load indicated a rise in temperature at the rear side of the bed that

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Fig. 15. The hardware configuration of the error compensation system [25].

was higher than that at the front side. From the experiments conducted, it was concluded that the variation of the cylindrical accuracy of the workpiece was caused by the thermal deformation of the bed of the grinding machine and this error could be determined by measuring the temperature distribution or the thermal deformation of the machine. Jedrzejewski et al. [19] studied the problem of correcting thermal displacements in a three-axis machine tool. In this machine, the temperature measuring points were primarily located within the headstock near the bearings. The thermal characteristics of the machining centre were determined assuming some operational parameters. Temperature values and thermal displacements were measured and using the collected data, the prediction of the thermal errors was achieved. Chen et al. [14] mounted 17 thermocouples on a machining centre (as shown in Fig. 16) after conducting preliminary experiments. One extra sensor was used to measure the ambient temperature. The temperature fields under selected conditions are shown in Fig. 17. A compensation controller based on an IBM/PC was implemented on the horizontal machining centre to compensate for the error. Compensation in the volumetric error was effected every 10 ms based on the compensation signals received from the compensation controller. Thermally induced positioning errors of the cutting edge for a vertical machining centre result from a combination of spindle growth, cantilever-arm bending, z–axis expansion etc. Consequently, the volumetric positioning error of the cutting edge is spatial-variant in the working zone and time-variant in the cutting time span. Chen et al. [12] arrived at a quick set-up measurement system consisting of on-machine probes and artefacts in order to calibrate these thermal errors. Spindle drift was calibrated using a spindle-mounted MP7 probe and gauge block fixed on the work table. Thermal expansion of the horizontal linear axis was determined by the variation of the calibrated length of a quartz tube put in parallel with the axis while that of the vertical axis was achieved by using a granite height gauge. Column bending was calibrated by measuring the

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Fig. 16.

The mounting of thermocouples on the machining centre [14].

Fig. 17. Temperature fields under different conditions [14].

co-ordinates of a gauge block at two z levels. Four tests were carried out in order to study the characteristics of thermal errors under different cutting conditions. The friction of the spindle bearings is one of the main sources of heat in a machine tool. Li et al. [24] analysed the same by measuring the thermal errors using a simple 1–D ball array. This array was used to measure the 21 geometric errors of the three-axis vertical machining centre as well as the thermal errors. The thermal drifts of the machine under different co-ordinates could

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be evaluated very quickly as this measurement system is very fast. As the spindle was the major source of heat, only the influence of the spindle was considered. Compensation was achieved by correcting the NC part program constantly. Every 10 min the thermal errors in the z direction were calculated using the pre-established model. Once the ⌬z values were obtained, the program was corrected by adding ⌬z in the program. The thermal errors were thus compensated before machining. Temperature measurement is an essential part of studying the thermal deformation response of machine tools. Attia et al. [7] addressed the factors like distance between the thermocouple and the heat source, the effect of heat flow along the thermocouple wires etc., that affect the measurement of temperature. The accuracy of surface temperature measurements is greatly reduced by the following factors: a) The distortion of the temperature field at the surface area, b) The presence of a surface coating (eg. paints) which act as additional thermal barriers, c) Imperfect contact between the thermocouple and the surface, d) The uncertainty associated with the exact location of the effective junction of the thermocouple. General guidelines were presented in order to minimise the errors. Experimental detection of thermal deformations could be carried out in two ways namely measurement during actual machining or measurement under conditions simulating the machining process. Novotny et al. [32] conducted measurements on a machine for grinding gears up to a diameter of 300 mm, by a generating process using worm-shaped grinding wheels. The size and shape of the workpieces produced from the first piece machined on the cold machine tool up to the final piece when the machine tool has attained its steady state were monitored. Measurements were also made under simulated machining conditions. A very precise simulation of the machining conditions was obtained by hydraulic braking of the spindle of a hobbing machine. The thermal deformations of the machine after seven hours of testing are as shown in Fig. 18. It was observed that the bed deflected upwards (convexly), the main column inclined longitudinally and deflected to one side (away from the table) while the table and the supporting column were inclined to the opposite side. The temperature increases of the different points of the machine and of the cutting oil (shown in Fig. 19) were helpful in finding the causes of the thermal deformations. Spur et al. [35] investigated the heat transfer on the working side of a spindle housing (headstock). Two techniques for the compensation of spindle displacement were outlined. In the first technique, the axial displacement of the front portion of the spindle was maintained at a constant value through the use of a radiant heating system. A heat balance was undertaken for the spindle system in order to determine the heating requirements for the spindle displacement compensation. With the heat balance and the displacement measured at constant spindle speed, the quantity of heat and the time constant were determined using non-linear regression techniques. Heating requirements for compensation were evaluated by changing the speed of revolution. The expansion of a longitudinal element is proportional to its length, the increase in temperature and the coefficient of thermal expansion. In the temperature range of 0°C to 100°C, resistance as a function of temperature could be taken to be linear. If a resistance thermometer is brought into thermal contact with the component, the expansion of which needs to be measured, the increase

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Fig. 18.

Thermal deformation of the hobbing machine after seven hours of the test duration [32].

Fig. 19.

Increase of the temperature after seven hours of the test duration [32].

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in the resistance of the thermometer directly corresponds to the linear expansion of the component. The second technique used this very concept in the determination of the spindle displacement. The signal obtained from the resistance thermometer could directly be used in the CNC machine tool in order to correct the relative positioning of the workpiece to the cutting tool. As one of the key phases in the elimination of thermal error in machine tools, extensive work, as described above, has gone into the measurement and compensation of these errors. In order to simulate the thermal distortion, the machine is put into a continuous cycle at different spindle speeds and feeds. Since it is not possible to exactly simulate the cutting conditions, the machine is generally made to run at considerably higher parameters than normal so that the heat generated would be comparable to that generated in an actual production environment. One of the primary activities in the measurement phase is the measurement of temperatures at critical points on the machine structure. In general, thermocouples have been used to carry out this task. Data from these sensors is then transmitted to a PC for further processing. In the meantime, the different error components of the machine are also measured using a laser interferometer. In some cases, non-contact capacitance sensors have also been used to measure the thermal distortion of the machine structure particularly at the tool. The thermal error data is then matched with the temperature readings in order to arrive at a generalised error model that could predict the error depending on the measured temperature values during actual machining. These predicted error values are then converted, depending on the number of axes extant on the machine, into individual error components with the help of homogeneous transformation matrices. All these activities, namely collection of temperature data, collection of encoder feedback signals from the machine, generation of the temperature-thermal error map, evaluation of the compensation to be carried out at each axis and interfacing of this data to the machine, are all carried out by an externally situated PC. The compensated data is then transferred to the CNC controller that carries out the actual compensation. Compensation is done either by interrupting the encoder feedback and adding/subtracting quadrature pulses or by shifting the origin of the axes during cutting. Thus real-time compensation of thermal errors could be achieved. 6. Conclusion Real-time thermal error compensation (RTEC) in general can be broken down into three stages namely modelling, measurement and compensation [26]. In order to achieve the final objective of minimising error in the machine tool, the behaviour of the machine structure is first of all modelled through the use of FEM (thermo-elastic modelling) techniques or empirical models. Measurement of the temperature variation at critical elements on the machine is carried out using a variety of sensors. The corresponding error components are usually measured directly using the laser interferometer or the ball–bar or evaluated with the help of extremely precise artefacts. Once the temperature and error data are obtained, a distribution pattern is then arrived at that correlates the temperature variation with the overall error of the system. Such a system would then be able to predict the response of the machine under actual working conditions. Nonetheless, many of the developed models assume that the component errors are already available when actually they have to be obtained by time-consuming and tedious experiments that invariably prove to be a costly exercise. In addition, most of the geometric and master part tests

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are very time-consuming and hardly any of the analytical methods of modelling describe how these models could be updated. The error matrix and multiple redundancy methods used for prediction of the error require far too much time to be applicable to a regular production environment [15]. Moreover, most of the modelling techniques employed in the analysis of thermal behaviour employ an overall machine model to arrive at the volumetric error. One way to address the complex system of the machine tool is to develop individual models for each of the major machine elements and then superpose these individual models to get the overall error model of the system. In addition, most of the compensation systems available at present utilise an externally situated computer to perform the task of data acquisition and error modelling/compensation and the compensation values are fed back into the CNC system of the machine tool through the interface. Further improvements in this system of compensation could be incorporated through the use of PC-based CNC systems wherein the PC itself directly controls the machine. Error modelling and compensation could then be carried out on the same PC thereby achieving the same result in one single system. However, despite such research, the industrial implementation of these error compensation systems, in particular that for thermal error, is still to be achieved primarily due the unavailability of a viable commercial system. With the rapid advancement in computer technology, it is expected that, in the future, such systems would be a standard feature of advanced machine tools employed in precision component manufacture.

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