This paper proposes a new method for predicting spindle deformation based on temperature data. The method introduces the adaptive neurofuzzy inference system (ANFIS), which is a neurofuzzy modeling approach that integrates the kernel and geometrical transformations. By utilizing data transformation, the number of ANFIS rules can be effectively reduced and the predictive model structure can be simplified. To build the predictive model, we first map the original temperature data to a feature space with Gaussian kernels. We then process the mapped data with the geometrical transformation and make the data gather in the square region. Finally, the transformed data are used as input to train the ANFIS. A verification experiment is conducted to evaluate the performance of the proposed method. Six Pt100 thermal resistances are used to monitor the spindle temperature, and a laser displacement sensor is used to detect the spindle deformation. Experimental results show that the proposed method can precisely predict the spindle deformation and greatly improve the thermal performance of the spindle. Compared with back propagation (BP) networks, the proposed method is more suitable for complex working conditions in practical applications.
Manufacturers should promote the performance of the components of machine tools to improve the quality and accuracy of their workpieces. Among all of the components of machine tools, the spindle plays the most crucial role in machining operations because it provides the cutting speed of the tool and is a part of the force chain between the machine tool structure and the tool or the workpiece [
Thermal deformations are the main sources of errors in machine tools and of geometrical errors of machined workpieces [
Weck et al. [
The majority of existing studies have focused on using artificial neural networks (ANNs) to build thermal error compensation models based on discrete temperature data. Unlike other mathematics models, ANNs have the advantages of information distribution saving, parallel processing, and self-learning ability. In recent years, different kinds of ANNs have been developed for thermal error compensation models, including back propagation (BP) [
In this paper, we propose a new thermal error predictive model for spindle deformation. The model integrates the kernel and geometrical transformation and the adaptive neurofuzzy inference system (ANFIS). The kernel and geometrical transformation is used for transforming temperature variables into the input space of ANFIS. The ANFIS model is trained by using two transformed temperature variables as input and the spindle deformation as output. After the training, the output of the model is used for predicting spindle deformation. A thermal error experiment for the spindle is implemented to evaluate the performance of the model. The results show that the model has high prediction accuracy and robustness, and it has a better machining performance than BP networks in practical applications.
ANFIS combines neural network adaptive capabilities and fuzzy logic qualitative characteristics. Therefore, it is more flexible in terms of network structure and it can approximate a highly nonlinear surface more effectively than traditional ANNs. As a neurofuzzy modeling approach, it has been widely applied for different purposes, including prediction [
The remaining of this paper is organized as follows. Section
ANFIS [
ANFIS architecture using the first-order Sugeno fuzzy model.
Using the first-order Sugeno fuzzy model for the output of each rule, the rule set with four fuzzy if-then rules is as follows: Rule 1: if Rule 2: if Rule 3: if Rule 4: if
where
ANFIS consists of five layers that perform different actions. We define the output of the layer
Figure
ANFIS training employs the hybrid learning algorithm combining the least-square and gradient descent methods, which consist of the forward and backward passes. In the forward pass, the least-squares method is used to optimize consequent parameters with premise parameters fixed. When the optimal consequent parameters are obtained, the backward pass begins immediately. In the backward pass, the gradient descent method is employed to optimize premise parameters with consequent parameters fixed. When the output error is less than a specified value or the maximum number of iterations is reached, the iteration stops.
As illustrated in Figure
Schematic illustration of experimental equipment.
In the spindle, because of the friction between the rolling elements and the inner and outer raceways, the bearings generate a large amount of heat and it is the main thermal error source of the spindle. The heat is dependent on the speed, preload, and lubrication. Faster speeds lead to higher contact forces, hence, higher friction and more heat.
A spindle has two groups of bearings to support the mandrel. They are arranged as follows: face-to-face angular contact bearing at the spindle nose and double row roller bearing at the rear of the spindle. As shown in Figure
As illustrated in Figure
Schematic illustration of data process system.
To investigate the spindle’s thermal behavior, the experiment was performed throughout the entire machining process. The spindle speed used for training the proposed ANFIS model is shown in Figure
According to the installed position of the Pt100 thermal resistances, the six measurement points were divided into two groups: the front and the rear. The temperatures for each group are illustrated in Figure
In this paper, 4500 training data pairs used in the proposed model contain the input vector
We calculate the average curve of the temperatures in Figure
As can be seen from Figure
Block diagram of the ANFIS model.
Using the Gaussian kernels below, it maps the input vector
Discovering linear relations among the images
Transforming the original data to the input space of the ANFIS model performs the geometrical transformation and interval normalization for the dataset
Interval normalization includes the translation and scaling operations for dataset
We train the ANFIS with two inputs, four rules, and one output. The inputs
Data transformation maps the training data pair to the square region
Figure
((a), (b)) MFs before training; ((c), (d)) MFs after training.
Initial MFs on
Initial MFs on
Final MFs on
Final MFs on
Training dataset: (a) the cutting condition; (b) the temperature of front bearings and the rear bearing; (c) the prediction of the ANFIS model; (d) the prediction of BP network.
We obtain the testing dataset from a new cutting condition used for validating the ANFIS model, and the cutting condition is shown in Figure
For the comparison between different models, we construct the BP network with the 2-5-1 topology structure. As shown in Figure
BP network architecture.
Testing dataset: (a) the cutting condition; (b) the temperature of front bearings and the rear bearing; (c) the prediction of the ANFIS model; (d) the prediction of BP network.
To compare the performances of the ANFIS model and BP network, we employ the following evaluation criteria.
Root mean squared error (RMSE):
Mean absolute percentage error (MAPE):
Correlation coefficient (
Performances of the ANFIS model and BP network.
Model | RMSE | MAPE (%) |
|
---|---|---|---|
Training dataset | |||
ANFIS model | 0.56 | 0.29 | 0.9994 |
BP network | 1.53 | 3.42 | 0.9983 |
|
|||
Testing dataset | |||
ANFIS model | 0.89 | 0.67 | 0.9991 |
BP network | 1.62 | 3.53 | 0.9974 |
Using the BP network based on the training and testing datasets, we predict the spindle deformation along the
Because of the change in working conditions, the measured values of the spindle deformation in intervals
This paper proposes a new method for establishing the relationship between temperature data and spindle deformation along the With the use of data transformation, the temperature data is gathered in the square region. The ANFIS model can conveniently partition the input space and effectively reduce the number of rules. In addition, the transformation procedure can reduce the randomness of the temperature data and the influence of unpredictable noises. Experimental validation was implemented. The experimental results indicate that the proposed method could precisely predict the spindle deformation and greatly improve the thermal performance of the spindle. Under the new condition, the prediction error of the spindle deformation along the According to the evaluation criteria, the ANFIS model outperforms the BP network. Under the new cutting condition, the MAPE of the ANFIS model is less than 0.7%, whereas the MAPE of the BP network is greater than 3.5%. The ANFIS model can respond more quickly than the BP network, and it can produce smaller prediction error. Unlike the BP network, ANFIS is transparent rather than a black box. Its if-then rules are easy to understand and interpret. By nature, BP network is a black box, and the relationships between its inputs and outputs are difficult to interpret. By contrast, ANFIS is transparent, and its if-then rules are very easy to understand and interpret. However, the ANFIS model has the imperfection of only modelling a single output, whereas the BP network can have several outputs. Overall, the ANFIS model has better performances than the BP network, and it is more suitable for modelling the spindle deformation caused by heat.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The work here is supported by Collaborative Innovation Center of High-End Manufacturing Equipment, the State Key Basic Research Program of China (no. 2011CB706803).