Keywords

1 Introduction: Motivation and State of the Art

Thermal errors in machine tools, meaning positioning inaccuracies due to thermal influences, are the most significant factors in determining the machining precision. They outweigh geometric, static and dynamic error components and are generally assumed to account for around two thirds of the overall positioning inaccuracies [1].

There are numerous strategies for dealing with thermal errors in machine tools. They can generally be grouped into:

  • methods for cooling or stabilizing the temperature field (e.g. by reducing the waste heat influx, cooling, air conditioning, controlling the heat flow, increasing the thermal inertia, etc.),

  • methods for reducing the impact of temperature changes (e.g. thermo-symmetric design, materials with low thermal expansion, decoupling of heat sources, etc.),

  • methods for directly measuring the thermal error with subsequent compensation,

  • methods for estimating the thermal error indirectly using model and/or sensor based prediction algorithms with subsequent compensation.

The first two groups are typically combined into error avoidance and the latter two into error compensation [2]. The thermal error compensation based on measurements or predictions is performed in the machine tool control via position offsets. The optimal strategy typically depends on the machine tool (type, kinematics, workspace, heat sources and sinks, support systems, available sensors, etc.) and the use case (small/large batch size, wet/dry cutting, workpiece, CAM strategy, etc.). Other factors such as experienced personnel, cost of installation and model training, maintenance effort, achievable accuracy improvement, reliability, transferability to other machines, etc. also play an important role.

With this multitude of factors, it is inevitable that all methods have their individual strengths and weaknesses. While cooling and thermal machine tool design are a necessary and ubiquitous element of modern machine tools, by themselves they are rarely enough to guarantee high-precision machining. Therefore, error compensation is likewise an important part of thermal error management. Error compensation also has the advantage that it is usually energy-efficient and quite effective at dealing with thermal errors. Measurement based compensation, while effective and straightforward in its usage, is often disruptive to the machining process, as it requires, e.g. production halts for intermittent recalibration. Model based compensation is less intrusive and usually only requires software and sensor installation. Popular model types for thermal error compensation include multiple regression models, transfer functions, artificial neural networks, support vector machines, fuzzy models, finite element models, lumped capacitance models, geometric elasticity models and control oriented models.

While all of these compensation models have their uses, they each have weaknesses as well. Regression analysis, while simple and time-independent, is also highly dependent on both the training data selection and the temperature sensor placement. Simulation based approaches, such as using FEM, are generally quite accurate in a wide range of load cases but they require a large modelling and parametrization effort. Compensation based on transfer functions presents a good compromise between the aforementioned but has often trouble if too many heat sources and sinks affect the same components. Compensation using integrated deformation sensors (IDS) is simple and effective but IDS cannot always be installed in all relevant components. To solve this dilemma, more and more researchers are turning to hybrid compensation schemes combining two or more methods to eliminate individual weaknesses and combine the strengths of each method [6].

One way to achieve this, is by combining sensor-based and sensorless methods. Jedrzejewski and Kwasny have used temperature sensors to approximate the error of the machine tool structure and combined this with a spindle shift compensation based on control-internal data [7]. In another work by Zhou and Harrison, a hybrid compensation model is constructed using a fuzzy controller enhanced by a supervised neural network to estimate the time-varying positioning error from in-cycle measurements via a touch trigger probe [8]. Xiang et al. combine a multivariable linear regression model, a natural exponential model and a finite element method model to a vector-angle-cosine prediction vector. The weights of the constituent models within the hybrid model are then optimized by maximizing the cosine between the prediction vector and the measured deformation vector [9]. Thiem et al. use an FEM simulation based compensation model and eliminate the problem of unknown initial conditions and thermal drift over time by temperature sensor based monitoring and recalibration of their thermal model [10]. Finally, a hybrid compensation model combining regression analysis and IDS based compensation has previously been suggested in [11] but had not been fully developed and tested, yet.

This contribution presents the combination of an IDS measurement based with a characteristic diagram based compensation method. Section 2 briefly describes the demonstrator and the FE model used to simulate the thermo-mechanical behavior of this machine tool. This model is the basis for the compensation methods in the upcoming sections. Section 3 describes the use of the FEM simulations to construct the characteristic diagrams. In Sect. 4, the IDS will be explained and an experience-based and a mathematical placement strategy will be described. The validation in Sect. 5 shows the efficiency of the compensation approach for a test load case. Finally, a summary and outlook will be given in Sect. 6.

2 FE Model of the Demonstrator DMU 80 eVo

For the investigation and later the validation of the individual as well as the hybrid compensation strategies, the DMU 80 eVo of DMG Mori was used. It is a five-axis machining center with a workspace of \(850 \times 650 \times 550 \, \text{mm}^{3}\). The DMU has eight temperature sensors installed, see Fig. 1. There are also various cooling systems, e.g. for the spindle, the axis bearings, the motors, the guides and the machining table.

For the training of the characteristic diagram based compensation, an FEM simulation model of the DMU was created in ANSYS. A detailed description of the modeling approach can be found in [3]. This model has been validated using measurements inside a climate chamber. This ensures, that the initial conditions of simulation and measurement match well and that the machine tool’s behavior under known ambient temperature changes also matches the simulation model. In Sect. 4, this same FEM simulation model is used for both the manual and optimal sensor placement.

Fig. 1.
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Temperature sensors of the DMU 80 eVo

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3 Characteristic Diagram Based Compensation

Characteristic diagram based compensation is a type of regression analysis using high-dimensional characteristic diagrams. Usually, a set of temperature sensors along with the axis positions of the machine tool are mapped directly onto the TCP displacement with a piecewise multilinear model. The details on how to construct these models and how to select suitable temperature sensor locations can be found in [12]. Some publications on the subject speak of “correction” instead of the more commonly used term “compensation” and may be viewed as synonymous in the context of thermal error reduction. Correction is in some ways a more clear description of this type of method since it implies the existence of the thermal error, which is to be estimated or measured, and thus clearly sets it apart from error avoidance methods.

In the context of the DMU 80 eVo, characteristic diagram based compensation is only partially suitable due to the predetermined temperature sensor placement. For the column, by which we refer to the assembly carrying the X, Y and Z axis, the single temperature sensor per assembly is not enough to accurately identify different thermal states. For the machine table, using characteristic diagrams based on the three temperature sensors installed there, is a good way to predict the thermal deformation. Moreover, installing IDS in the table is not possible and simulation based approaches have had difficulties in modelling and correctly parametrizing the table due to its complex design.

The initial compensation model proposed for the table in [13] used only five parameters to describe the thermal displacement of the table. These parameters are dx, dy, dz, rx and ry, referring to the displacement and tilting of the table as a whole. In this model, the table cannot expand in x or y direction and it cannot assume convex or concave shapes. These limitations were made because the limited reliability of the simulated table deformation made determining the exact deformed shape of the table nearly impossible and it is unclear just how much of the table expansion would even affect a workpiece fixed on that table. In order to determine how well characteristic diagram based compensation could theoretically predict the thermal error of the table, a more detailed model will now be presented using the assumption that the simulated table deformation correctly matches that of the actual machine tool. The improved modelling technique will be useful for similarly designed machine tools as well as for the DMU 80 eVo once more accurate FEM models have been developed.

In order to provide full flexibility regarding the table deformation, the first step was to approximate the deformation using a 2D characteristic diagram of the table surface, where in the case of the DMU 80 eVo a 5-by-5 grid was sufficient for good results. For each grid vertex, the table deformation as calculated by FEM can be mapped in the second step using a thermal compensation model dependent on the temperature sensor readings. Since for many of the table nodes, the thermal deformation is highly non-linear and cannot be accurately predicted with regression analysis alone, a hybrid model using first order time delay functions (PT1) and temperature sensors was used for the thermal models. For each table node, individual compensation models have to be created and parametrized depending on their nonlinear thermal behavior. This process was automated by providing a set of function types and parameter ranges followed by a brute-force trial-and-error model optimization, which simply tested all possible models and compared them by maximum and mean residual errors. In the future, a neural network or genetic algorithm could be used to speed up the training process by preselecting suitable model types or refining the parameter ranges. Figure 2 shows the old table model on the left and the new, improved model on the right.

Fig. 2.
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New vs old table model for thermal error prediction

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One last aspect, which proved to be important was that separate compensation models were needed for the dry heating, heating with coolant and cool-down phases of the machine tool operation. Due to the completely changing thermo-elastic behavior of most table vertices for these situations, separate modelling was the only way to obtain good approximations. More complex thermo-elastic models instead of the mostly regression based models could eliminate this need in the future, however.

4 Integrated Deformation Sensors

4.1 Description of the IDS

To compensate the thermal deviation of the TCP, the Fraunhofer IPT developed integrated deformation sensors, abbreviated as IDS [15]. These IDS consist of CFRP rods which, in theory, are perfectly thermally stable, though in reality, the thermal expansion coefficient (TEC) can fluctuate between −0.12 and 1.9 \(\frac{\mu m}{{m \cdot K}}\) [4]. Compared to materials that are often used in machine tools, like grey cast iron (9 \(\frac{\mu m}{{m \cdot K}}\)) or steel (ca. 11.5 \(\frac{\mu m}{{m \cdot K}}\)) in a range of temperature that is suitable for production halls, this low value enables the CFRP rods to be used in thermal deformation sensors, as was shown in [5].

To measure the thermally induced length change of the machine’s structural parts, the CFRP rod is mounted with a fixed bearing on the right-hand side and a loose bearing on the left-hand side. A displacement sensor is then added on the left and can measure the change in length via tactile measurement (Fig. 3).

Fig. 3.
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Operating principle of the integrated deformation sensor

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According to this principle, the thermo-elastic behavior of the large structural components of the machine can be measured. We apply the Euler-Bernoulli beam theory to obtain a model, which gives the relationship between the measured IDS values and the TCP displacement. Depending on the shape of the machine components, other theories might also be applied, for example shell theory for thin components. Expert placement of the IDS using this model was compared to mathematical approaches in [16].

4.2 Experience Based Sensor Placement

The placement of the IDS is the basis for a good prediction of the TCP deviation. In order to determine a suitable sensor placement, the machine’s structural parts are investigated in an FEM simulation. Heat sources like motors, friction in axis guides or bearings are simulated as well as fluctuations in the ambient temperature. The simulations calculate the resulting temperature fields and based on this, the thermally induced deformation. An example result of such a simulation is shown in Fig. 4.

Fig. 4.
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Simulation results, left: temperature field, right: deformation field

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Based on this displacement, the sensor placement shown in Fig. 5 was proposed.

Fig. 5.
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Sensor placement in two views

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Three sensors are placed at the headstock. Two additional sensors are placed at the cross slide and five sensors are placed at the moving column for the y-direction. Although the biggest deformation can be found at the table, no sensors are placed there. This is because the measuring unit itself is not suitable for the workspace due to its lack of resistance to cooling lubricants. Furthermore, there are no sensors placed at the machine bed. This can be justified by the fact that there is no main direction of deformation, which prevents a reasonable placement.

4.3 Mathematical Sensor Placement Using an FE Model

The basis for the mathematical sensor placement is the estimation of the uncertain model parameters using measurements. Given the estimation problem, one has to place the sensors such that the uncertainty of the predicted TCP displacement is as small as possible. The FE model described above has the structure

$$ \begin{aligned} & \quad E\dot{T}\left( {t, p} \right) = A\left( {t,p} \right) T\left( {t, p} \right) + Bu\left( {t, p} \right) \\ & y_{mech} \left( {t, p} \right) = C_{mech} T\left( {t, p} \right) \\ & d_{CFK} \left( {t, p} \right) = H\left( {y_{mech} \left( {t, p} \right)} \right) \\ & \quad \quad \ \ \ y_{TCP} = C_{TCP} T\left( {t,p} \right) \\ \end{aligned} $$

where the coefficient matrices \(E\), \( A\left( {t,p} \right)\) and \(B\) are the heat capacity matrix, the heat conduction (and heat exchange) matrix and the input matrix, respectively.

The uncertain heat exchange coefficients enter the matrix \( A\), whereas the heat loads are part of the inputs \( u\). The output matrix \(C\) contains the stationary equations of linear thermo-elasticity reduced to the desired displacements in the sensor locations.

The output operator \( H\) provides the distances between the mounting points from the displacements \( y_{mech}\), which correspond to the measurements. The change in length is nonlinear in the displacement, but the relative change is very small. Therefore, it is customary to linearize the change in length in the undeformed state.

The parameter estimation corresponds to solving the least square problem

$$ \mathop {\min }\limits_{p} \,\sum\limits_{i,m} {\left| {d_{{CFK_{m} }} \left( {t_{i} , p} \right) - \hat{d}_{{CFK_{m} }} \left( {t_{i} } \right)} \right|_{{\sigma_{m}^{ - 2} }}^{2} + \left| { p - p^{bg} } \right|_{{\alpha^{ - 2} }} } $$

where \(\hat{d}_{CFK} \left( {t_{i} } \right)\) are the measured rod lengths. All measurement errors are independent and normally-distributed with standard deviation \( \sigma_{m}^{2}\). Due to the length-dependent measurement errors, there are in general different standard deviations for every rod. The parameter \(\alpha\) is a small regularization parameter, which penalizes too large deviations from the background parameters.

In the optimal sensor placement, the measurement carries the most information about the parameters. At the same time, the parameters are weighted according to their impact on the TCP displacement. One measure for the accuracy of the TCP displacement is the D-criterion [14], which is related to the volume of the covariance ellipsoid of the TCP displacement \( y_{TCP} \left( {t, p} \right)\). The optimal sensor placement leads to a problem of the form

$$ \begin{aligned} & \mathop {\min }\limits_{w} \,\log \;\det \;{\text{Cov}}_{{{\text{TCP}}}} \,\left( w \right) \\ & {\text{Cov}}_{{{\text{TCP}}}} \left( {\text{w}} \right) = { }J_{{y_{mech} }} \left( {t_{f} } \right)Cov^{ - 1 } \left( w \right)J_{{y_{mech} }}^{T} \left( {t_{f} } \right) \\ & \quad \quad {\text{Cov}}\left( {\text{w}} \right) = \sum\limits_{{{\text{i}},{\text{m}}}} {\frac{{w_{m} }}{{\sigma_{d,m}^{2} }}} {\text{ J}}_{{CFK_{m} }}^{{\text{T}}} { }\left( {{\text{t}}_{i} } \right){\text{J}}_{{CFK_{m} }} { }\left( {{\text{t}}_{i} } \right) + \upalpha ^{ - 2} {\text{ I}} \\ \end{aligned} $$

The symbols \(J_{{y_{mech} }}\) and \({\text{J}}_{{CFK_{m} }}\) denote the Jacobian of the displacement and the length with respect to the parameters p, respectively. The placement-dependent matrix \({\text{Cov}}_{{{\text{TCP}}}}\) represents the covariance matrix of the TCP displacement at the final time step.

Thus, the optimal sensor placement depends on the derivative of the observation and of the quantity of interest (QOI, here the TCP location) with respect to the parameters. This observation is key for an efficient implementation as well as for the offline computation of optimal sensor positions during the machine design. Parameters which enter only the sources \(u\) have temperature-independent sensitivities. Their sensitivity, however, might depend on other parameters. Examples for those are the amplitude of friction losses and ambient temperatures. In contrast, if a parameter influences the coefficients, the derivatives are temperature-dependent and thus the optimal placement also depends on the corresponding loads.

Figure 6 depicts both sensor placements. The mathematical placement adds two sensors next to the rails on the bed and one additional rod inside the x-sledge near the motor, which is not visible. The right image shows only the optimized sensor locations for a fixed number of five IDS (Table 1).

Fig. 6.
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Sensor placement manual and optimized (left) and optimized sensor placement for five IDS (right)

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Table 1. Optimization criteria and highest uncertainty of the TCP relative to expert placement
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Table 1 compares the optimization criteria relative to the expert placement. The expert placement always consists of ten rods, whereas the optimized placement was computed for smaller numbers of rods, too. A difference of about -2.3 corresponds to approximately one tenth of the volume due to the natural logarithm. The second column relates the largest standard deviations µ of the different placements to each other. Thus, the optimal placement reduced the uncertainty to about 2/3. Furthermore, one can expect a reduction to about 3/4 of the uncertainty with eight sensors.

5 Validation

For the validation of the new hybrid compensation method, an independent test load case was simulated, which could then be compared to the estimated relative displacement. The estimate is calculated from the characteristic diagram based estimation of the table error minus the IDS based estimation of the column’s thermal error.

The test load case consists of four parts: a short 2h heating phase with X 90%, Y 34%, Z 56% and B 38% of maximum axis speed; followed by a 1h cool-down phase (all 0%); followed by a longer 4h heating phase with X 45%, Y 60%, Z 30%, B 90% and C 45% of maximum axis speed; and finally another 4h cool-down. Since the simulation does not actually contain motion, the 90% in this example refers to 90% of the maximum waste heat, which is applied to the guides, bearings and motors. 0% is a predefined low standby waste heat.

The resulting temperatures at the sensor locations can be seen in Fig. 7.

Fig. 7.
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Temperatures of the test load case containing four sections

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Fig. 8.
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Displacements of tool and table: simulated vs. estimation

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Figure 8 shows the simulated error of the test load case as compared to the estimation. It shows a good agreement of the estimation with the simulated values, reducing the error by about 2/3. For the characteristic diagram based estimation of the table error, more training data would result in better approximations, as the test data was not part of the limited set of training data. Since some other parts of the table have a very different thermo-elastic behavior, an additional validation for all table nodes would be recommended. The IDS based estimation of the tool error resulting from the thermal column deformation also shows a high achievable accuracy. This result is based on the optimized IDS placement in Fig. 6, right. The x-displacement is calculated from the elongation of IDS2 and the difference in elongation of IDS3 and IDS4, although IDS2 has little effect, since the x-axis position is in the middle. The y-displacement is calculated from IDS2 and the average of IDS1 and IDS5, where the latter again has little effect due to the position being the middle of the y-axis. The z-displacement is calculated from IDS2 and the average of IDS3 and IDS4. Thus the bending of the y-sledge as detected by IDS2 contributes strongly to the thermal error of the column. It is also clear, that along with the amplitude of the IDS elongation, a geometric understanding of the elongation and bending behavior of the components is required for good estimations using IDS. Testing further machine poses and different load cases will also be required to optimize and fully validate the approach.

6 Summary and Outlook

The thermal behavior of a machine tool is one of the most significant factors in determining its positioning accuracy. Various methods from thermal error avoidance (e.g. cooling or thermo-symmetric design) to thermal error compensation (e.g. model based error estimation) can be employed to improve this thermal behavior. Compensation strategies are especially important in this regard, since they are effective, energy-efficient and are often non-invasive and thus suitable for both new and existing machine tools. Since each thermal error compensation strategy has its own strengths and weaknesses depending on the situation, hybrid strategies combining several different methods can be used to eliminate the weaknesses of individual methods.

The DMU 80 eVo is a five-axis machining center which is popular in the industry due to its small space requirements compared to its workspace. Since it has, however, a complex thermal behavior, a hybrid compensation approach combining IDS based compensation for the machine tool column and characteristic diagram based compensation for the machining table was developed and described in this paper.

In a first step, the machine’s thermal behavior was analyzed via FEM simulations. Based on the results of this analysis, the placement of the IDS on the main structural parts was determined. Based on the expected thermal behavior, a model of the machine was developed to be able to calculate the TCP deviation from the IDS measurement.

To show the functionality of the IDS and the model, a theoretical experiment was performed. As the simulation model was experimentally validated, it can be used as a substitute for an experiment actually carried out on the physical machine. This eliminates the need to physically install the IDS in the machine tool. An example load case for the DMU 80 eVo was simulated, which computed the length change of the IDS and the corresponding TCP deviation. The length change of the IDS could then be calculated into a predicted TCP deviation via the model. In order to be able to make a statement about the quality of the IDS as the basis of a compensation method, these TCP deviations can be compared.

The estimation of the table error was calculated from the three temperature sensors installed in the table using regression analysis combined with first order time delay elements (PT1). This model has shown good agreement with the simulated displacement for the investigated test load case. One crucial part of this table model was the subdivision into dry milling, wet milling and cool-down (stand-by). To further validate this new table compensation model, more test load cases will be created with several sudden shifts between these three production phases in order to test the composite model under more realistic conditions. Additional tests will also be performed to evaluate the accuracy of the table model at different table locations within the workspace. Finally, the entire hybrid compensation model will be tested and compared to existing thermal compensation methods.