Abstract
Considering the detrimental impact of thermal phenomena on the geometric precision of machine tools, a machine tool ball screw’s omnidirectional error model is created using the LSTM neural network algorithm. Subsequently, the machine tool ball screw's omnidirectional error compensation module is devised by combining the core functions of the Huazhong numerical control system with the visual programming environment of QT and the numerical computation capability of Matlab. To enhance the practicality and accuracy of the compensation model, this study has employed the Whale Optimization Algorithm (WOA) to optimize the parameters of the LSTM model. This has resulted in an improvement in the model's generalization ability and prediction performance, making it more effective. During the experimental validation phase, the $Z$axis error of the machine tool was practically operated and analyzed using the compensation method. Results manifestly show that, after employing the compensation method, the peak amplitude of the $Z$axis error fluctuations have been notably curtailed to ±0.006 mm – a considerable reduction compared to the initial error bandwidth of ±0.0145 mm. These empirical findings substantiate the efficacy of the proposed compensation strategy in substantially boosting the machining precision of products, thus furnishing a substantial and instructive benchmark for future inquiries into CNC machine tool error compensation technologies.
1. Introduction
Ball screws play a critical role in the manufacturing industry, but their reciprocating motion in CNC machine tools generates frictional heat, leading to thermal expansion [1]. This expansion phenomenon subsequently induces shifts in the relative positioning of the cutting point, ultimately impacting the precision of workpiece machining processes [2]. While conventional screw error compensation technology in the axial direction is effective, radial compensation still needs improvement, especially for the two ends of the fixed screw structure [3]. Against this backdrop, the present study focuses its investigative efforts on the development and resolution of the multifaceted challenge posed by omnidirectional error compensation necessitated by thermal effects in dynamically functioning screws.
In recent times, research on thermal error modeling methods based on the principle model (PBM) and empirical model (EBM) [4] has been limited due to their low prediction accuracy. Consequently, a range of thermal error modeling techniques have been proposed by scholars both domestically and internationally; among these, the application of deep learning to error compensation has progressively grown to become a research hotspot in the field. For instance, Li Bin et al. [5] harnessed a wavelet neural network optimized via genetic algorithm to build a thermal error model for CNC machine tools, which they experimentally validated for its superior attributes, including accuracy, resistance to interference, and robustness. Similarly, the fourlayer neural network architecture proposed by PuLing Liu et al. [6] showcased commendable precision and resilience in predicting thermal errors during machining operations. Moreover, various scholars – such as Hu Shi et al. [7], Guo Qianjian et al. [8], Li Guangbao et al. [9], and Li et al. [10] – have independently enhanced the predictive and compensatory capabilities of error models by implementing different optimization algorithms and neural network architectures.
Amidst the ongoing evolution of error prediction technologies, a deeper understanding of thermal error characteristics becomes increasingly crucial. Li S. S. et al. [11] employed the finite element method to scrutinize the thermal attributes, vibrational modes, and natural frequencies of a motor spindle in both steadystate and static conditions. Li Yi et al. [12] conceptualized and experimentally substantiated an efficient watercooling system tailored for motor spindles. H. Qu et al. [13] systematically reviewed the principal sources of thermal errors and prevailing control strategies while envisioning prospective research directions. Parallel to this, advancements in error compensation technology significantly contribute to enhancing machining precision. Lu Chengwei et al. [14] presented a feature decompositionbased key geometric error analysis and compensation strategy applicable to fiveaxis CNC machine tools, offering detailed analyses and efficacious remedies for critical geometric errors. Li et al. [15] developed a thermal error modeling and compensation technique for heavy gantry machine tools, demonstrating marked effectiveness, with the mean error diminishing from 101 microns to 13.5 microns. Chen GuoHua et al. [16] constructed a volumetric error model for machine tools grounded in Abbe error theory, incorporating it into the system to realize error compensation, which they confirmed by contrasting tool accuracy pre and postcompensation.
Synthesizing these studies, preventative and compensatory measures constitute the dual mainstream strategies for coping with thermal error problems [17]. Nevertheless, preventive measures are often costly and are limited by technological constraints. Coupled with the lessexplored issue of errors emanating from the fixed state of CNC machine tool screws’ ends, this study innovatively introduces an omnidirectional error compensation method targeting the screw structure's interaction with heat sources. This novel approach addresses a gap in the literature and promises significant improvements in coping with thermal error challenges.
2. Principle and methods
2.1. Principle analysis of LSTM
Deep learning is a widely used machine learning technique that combines underlying features to reveal complex patterns [18]. Harnessing highcapacity training algorithms on extensive datasets, this technique excels at discerning sophisticated relationships.
Among its variants, Long ShortTerm Memory (LSTM) [19] represents a refined rendition of recurrent neural networks, explicitly designed to mitigate the notorious challenges of gradient disappearance and explosion inherent in conventional recurrent architectures. The LSTM network state is regarded as a dynamic system, where a memory cell and three gate structures (input gate, output gate, and forget gate) constitute a long and shortterm memory unit. A graphical depiction of these gates is provided in Fig. 1, elucidating their operational dynamics within the LSTM structure.
In LSTM, the update of each state variable follows the following formula:
The oblivion gate determines the degree of oblivion of the hidden cell state at the previous moment by using Eq. (1), where ${f}_{t}$ denotes the output of a sigmoid function that proportionally “obliviates” the previous hidden state based on the current input ${x}_{t}$:
The filtering of valid information is indicated by the ${i}_{t}$ in Eq. (2), and each component is given a weight ranging from 0 to 1. The likelihood that a component will be chosen and added to the unit state increases with weight:
Eq. (3) delineates the formulation of a candidate state vector ${u}_{t}$, wherein the utilization of a hyperbolic tangent activation function ($\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}$) plays a pivotal role. This function restricts the output spectrum to the interval [–1, 1], a characteristic that fortifies the LSTM's resilience against the common issues of vanishing and exploding gradients, thereby preserving the inherent expressiveness of the encoded information.
Fig. 1Schematic diagram of LSTM unit structure
The activation function ($\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}$) behaves as:
In addition, Eqs. (2) and (3) together form the mechanism for the operation of the input gate (memory gate), which determines whether the data at time step $t$ is integrated into the cell state.
At time step $t$, the cell state ${c}_{t}$ undergoes an update governed by Eq. (5), where the retention or forgetting of past information is initially modulated by a sigmoid function, utilizing the output ${h}_{t1}$ from the previous time step and the input ${x}_{t}$ at the current instant. A sigmoid value of 1 indicates a strong retention of the previous state ${c}_{t1}$, while a value of 0 indicates complete forgetfulness of the previous state:
Eqs. (6) and (7) expound upon the operative mechanism of the output gate. First, a sigmoid function is engaged to sieve out pertinent information from a composite vector $\left[{h}_{t1},{x}_{t}\right]$, which consolidates the current input and the output from the immediately preceding time point. Subsequent to this, the current cell state is transformed onto the interval (–1, 1) through the application of a hyperbolic tangent function $\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}$, thereby shaping the ultimate output value that encapsulates the processed temporal information.
In the above equation, at time $t$, ${x}_{t}$ represents the input data, ${h}_{t}$ represents the state of the hidden layer, + represents the superposition operation, $U/W$ represents the matrix weight, $b$ represents the offset, $\sigma $ represents the sigmoid function, and the symbol $*$ represents the vector outer product. The usual LSTM network structure is shown in Fig. 2.
Fig. 2Schematic diagram of long shortterm memory network
2.2. WOA optimization algorithm
The WOA algorithm is designed to optimize a problem by imitating biological strategies used in the hunting behaviour of humpback whales, such as encircling predation and bubble net predation. Within this framework, each artificial humpback whale embodies a potential solution, and its position iteratively evolves to approach the global optimum:
The strategic essence of the encircling predation strategy is instantiated in Eq. (8) and (9), wherein, once a prevailing global optimum ${X}_{\left(t\right)}^{*}$ is identified, the remaining members of the population progressively converge towards it in accordance with predefined rules. Herein, $A$ and $B$ denote dynamically tuned coefficient vectors contingent upon the iteration count $t$, while ${X}_{\left(t\right)}$ stands for the present initial position vector of an individual whale agent. The position adjustment process is judiciously guided by the utilization of the absolute value operator $\left\right$ and the Hadamard product (elementwise multiplication) operator $\u2022$, ensuring a systematic movement toward the most promising areas of the search space:
Throughout the iteration process, the computation of vectors $A$ and $B$ follows Eqs. (10) and (11), where $a$ decreases linearly to zero with the number of iterations and ${r}_{1}$ and ${r}_{2}$ are random numbers in the interval [0, 1]:
Eqs. (12) and (13) describe the equation of a logarithmic spiral that is used to calculate the position update of whales and their prey during bubble net predation. In the equation, $T\mathrm{\text{'}}$ represents the distance between the current individual and the current optimal solution, $b$ is the spiral shape parameter, and $l$ is a random number uniformly distributed between [–1, 1]:
Eq. (14) shows the process of WOA selecting different predation strategies based on the probability P, which is a random probability in the range of [0, 1]:
In Eqs. (15) and (16), when $\leftA\right\ge $1, the exploratory agents not only gravitate towards the identified optimal solution but also recalibrate their positions in relation to the distances from randomly chosen peers, thereby instigating a broader scope of random search. Here, $T\text{'}\text{'}$ denotes the distance between the current individual and the random individual, and ${{X}_{rand}}_{\left(t\right)}$ is the position of the random individual.
To conclude, the WOA algorithm ingeniously amalgamates a biomimetic heuristic with a pliable probabilistic decisionmaking process, which guides the searching individuals to effectively traverse the solution space in the iterative process to find the globally optimal solution. The comprehensive workflow of the WOA algorithm is visually depicted in Fig. 3, which lucidly illustrates the algorithm’s entire operational sequence.
Fig. 3Flowchart of WOA algorithm optimization
2.3. Model framework
Since the ball screw error data manifest temporal correlations and continuity, and are notably influenced by the thermal dynamics of the machine tool, this study adopts an LSTM network to examine the data within a dynamic time series framework, aiming to capture their intricate spatiotemporal properties and deliver precise thermal error predictions [20].
The construction of the threedimensional error prediction model for the ball screw encompasses several stages: data preprocessing, model architectural setup, hyperparameter tuning, and model testing and evaluation.
During the preprocessing phase, a variety of multivariate data are gathered and meticulously organized, concatenated, and normalized, encompassing the running distance, servomotor temperature, slider temperature, screw nut temperature, and the threedirectional errors ($\mathrm{\Delta}x$, $\mathrm{\Delta}y$, $\mathrm{\Delta}z$). Among them, the preset temperature data includes motor mount temperature, ball screw nut temperature, and guideway slider temperature, with weights of 70 %, 20 %, and 10 %, respectively. Meanwhile, the running distance, sampling position, and preset temperature data are taken as model input parameters, and the threeway error is taken as output parameters. The dataset is divided into 70 % as the training set, 10 % as the validation set, and 20 % as the test set.
The model architecture is depicted in Fig. 4. Its input layer contains the input sequence at time t and its previous period, while the output layer represents the value of the threeway error of the ball screw predicted by the model at time $t+1$. Multiple hidden layers are set up inside the model, including the FNN layer, LSTM layer, and dropout layer, with a serial connection of two LSTM layerdropout layer pairs culminating in a fully connected FNN layer. Among them, the LSTM layer mainly undertakes the task of extracting key features from the input sequence and maintaining the internal state.
Fig. 4Framework diagram of predictive modeling
In order to obtain optimal performance from the LSTM model during training, it is necessary to adjust its configuration and parameter tuning. This involves finding the best combination of hyperparameters, such as the learning rate, number of hidden neurons in the LSTM layer, regularization coefficient, and number of training rounds, through several iterations. However, traditional optimization algorithms may be computationally expensive and inefficient, so this study uses the WOA algorithm. It is chosen for its fast convergence and excellent global optimal solution search capability. Moreover, to guarantee the gradient stability of the screw’s threeway error prediction model, the uniform initialization method [21] is employed to initialize model parameters. The LSTM model adopts the tanh and sigmoid functions as activation functions, complemented by the Adam optimization algorithm [22] for parameter adjustment, aimed at minimizing the loss function.
In the final stage of model assessment, this study primarily utilizes the root mean square error (RMSE) as the primary metric for gauging prediction accuracy, complemented by the mean absolute error (MAE) to quantify the magnitude of disparity between the predicted and actual values:
2.4. Training of algorithm
In this study, a ball screw threeway error prediction model is shown in Fig. 5. During the training process, the dataset is divided into small batches. In each batch, the model receives a set of input sequences and calculates the gradient of the loss function through a backpropagation algorithm. These gradients are then used to update the model parameters to minimize the loss function. On this basis, the optimizer exploits the computed gradients to refine the model parameters, and this iterative training process persists until a convergence criterion is met. Of particular note, the Dropout layer is used to prevent overfitting by implementing an early stopping strategy.
Fig. 5Flowchart of Threeway error prediction of ball screws
As per the previously discussed Section 2.3, the WOA algorithm is used to optimize the best hyperparameter combination for the experiment. Since the training sets are different, the set hyperparameters are also different. Ultimately, following rigorous training and meticulous parameter optimization, the highperforming LSTM neural network model was successfully integrated into a custombuilt screw error compensation module. This integration enables the accurate prediction and efficacious compensation of the ball screw's threedirectional errors in practical application contexts.
2.5. Training of algorithm
The error compensation function of the CNC machine tool is a result of integrating various technologies such as Fast Ethernet data interaction technology, sensor technology, realtime control system, and adaptive control technology. The principle of the error compensation system is illustrated in Fig. 6. The error prediction value, computed by the threeway error prediction model of the ball screw, is quickly stored in the nonvolatile memory of the programmable machine controller (PMC) through data interaction. When the machine tool error compensation system interacts with the PMC, the data is transferred to the builtin interpolation system. The data is then adjusted by an internal interpolation algorithm that converts the compensation value into an interpolation step, which is transmitted to the interpolator to correct the interpolation vector.
Fig. 6Schematic diagram of ball screw error compensation
This study aims to address the issues of cumbersome testing and low efficiency of CNC systems by developing a ball screw error compensation module. The module is designed to be highly stable, personalized, and userfriendly, with a high degree of humancomputer interaction. It is developed using the secondary development of QT software and is designed, developed and debugged in a Win environment, which includes: module function definition, interface design, operation logic processing, parameter configuration, motion control, etc. The module interface is shown in Fig. 7.
QT introduces the signal and slot mechanism to enhance the communication function between the system and interpolator, making the interaction between system components more convenient and flexible. The signal and slot mechanism allows communication and interaction between components through a reactionslot (slot) approach, making it easy to communicate between different threads and achieve realtime data transfer and synchronization. This improves the response speed and efficiency of the system. In the secondary development of the ball screw error compensation module, the sensor data can be sent as a signal and then connected to the slot function of the interpolator to realize realtime data interaction and control.
2.6. Construction of Omnidirectional error compensation method
The omnidirectional error compensation method is to address the issue of thermal error resulting from thermal deformation when the ball screw’s two ends are fixed. The detailed implementation process is illustrated in Fig. 8. To accomplish this, the method employs both QT and MATLAB software for secondary development of the CNC system, as well as to establish a threeway error prediction model for the ball screw.
Fig. 7Ball screws error compensation module interface
During this process, data about the ball screw's movement is collected, including its realtime position, feed speed, running time, running distance, and temperature at the sensitive point. These data are then fed into a WOALSTM prediction model, which has already been established. After performing mathematical calculations and fitting the model, the system obtains the threeway errors ($\mathrm{\Delta}x$, $\mathrm{\Delta}y$, $\mathrm{\Delta}z$) at the realtime position of the machining platform. These errors are then sent to the interpolator, which uses the values to generate control signals and adjust the servo motors accordingly. The main goal of this method is to use feedback control to ensure that the tool or workpiece moves as close as possible to the desired trajectory, thereby improving machining quality and accuracy. By using realtime error data, interpolation is achieved through the interpolators in the compensation system. This system ensures that the machine is stable when handling complex workpieces.
For data acquisition, it is necessary to preset an experimental program for the acquisition process and use the control variable method. The experimental program must include adjustments to motor speed and machining platform feed speed, presetting segmented measurement points, and developing a temperature rise test program. The segmented measurement points are preset by dividing the feed ball screw of the CNC machine tool into multiple parts according to the same spacing ($\mathrm{\Delta}L$), and dividing them into different threeway error measurement areas. In each region, the point at $\mathrm{\Delta}L$/2 is set as the measurement point. The temperature rise test program involves writing Gcode to control the reciprocating motion of the measuring platform feed system within a given distance (S) to mimic the actual machining state of the CNC machine tool and generate localized high temperatures. During this process, the feed axis ball screw of the CNC machine tool deforms due to thermal expansion.
During data acquisition, the CNC machine's three axes were run without any load and stopped for 3 seconds at 1/2 of each pitch, $\mathrm{\Delta}L$/2, in accordance with the ISO international standard [23]. During the halt period, a laser interferometer was used to measure the threeway error data of the current segmented position of the ball screw, while a temperature sensor collected data on the ball screw's temperature during operation, including the temperature of the servo motor, slide, and ball screw nut. After collecting the data, they were preprocessed and integrated into the dataset required for the model.
Fig. 8Flowchart of Omnidirectional error compensation method
3. Thermal error measurement experiment
3.1. Construction of experimental environment
This study zeroes in on the thermal error issue pertaining to the feed screw within the S7H composite machining center, employing highprecision instruments such as PT100 temperature sensors, an HSP9ER temperature acquisition device, and a Renishaw XL80 laser interferometer for experimental execution. Notably, the detailed layout arrangement of the $Y$axis is graphically depicted in Fig. 9, whereas the positioning of the laser interferometer for the $X$ and $Z$axis screws is tailored according to the experimental requisites.
Concurrently, in pursuit of obtaining a representative ambient temperature, two additional temperature measurement points, T4 and T5, are symmetrically installed on both the inner and outer surfaces of the machine, with their mean value serving as the ambient temperature reference. The initial positioning error of the machine is accurately measured by a laser interferometer [24]. Given that the total length of the screw measures $L=$ 600 mm, it is uniformly divided into thirty sections, each segment of length $\mathrm{\Delta}L=$ 20 mm, with an error detection point established at the midpoint of every segment for meticulous error analysis.
In light of the heat dissipation patterns intrinsic to the ball screw, temperature sensors are strategically positioned in thermally sensitive regions such as the screw nut, guideway slide, and servo motor, designated as monitoring points T1, T2, and T3, respectively. These PT100 resistance temperature sensors [25] boast a ±0.3 °C highprecision measurement capability and operate across a wide temperature range from –200 °C to 300 °C.
The configuration of the laser interferometer system at the ball screw is elucidated in Fig. 10, comprising elements such as the interference head, refractor, reflector, stable tripod, and other components. Initially, the robust tripod is stationed atop a dedicated support pedestal, facilitating the mounting of the laser interferometer's main apparatus to ensure that the interference head is aligned in the zerocoordinate horizontal plane of the machine tool's working platform, thus guaranteeing the precision of the measurement reference point. Subsequently, an adjustable support rod is affixed to the $Z$axis column using a magnetic base, allowing it to ascend and descend in unison with the $Z$axis motion. Upon this rod, a reflector is securely attached, which tracks the vertical displacement of the $Z$axis and fundamentally serves to receive and redirect the laser beam horizontally from the laser interferometer.
Fig. 9Schematic diagram of thermal error measurement setup
Simultaneously, a refractor mirror is prescriptively positioned on a support bar, its exact placement lying at the geometric intersection of the laser interferometer’s $X$axis extension line and the principal axis of the machine tool – this position remains invariant with the worktable’s movements. The refractor's function is to collect the horizontally emitted laser beam from the interferometer, refract it towards the reflector, and thereafter intercept the laser beam returning from the reflector, thereby ensuring it is vertically impinged upon the receiver aperture for meticulous optical analysis and measurement.
Fig. 10Arrangement of laser interferometer
3.2. Protocol for measurement experiment
The experimental protocol unfolds in the following sequential phases:
1) Precise temperature sensors are strategically placed in zones of the machine tool apparatus that are susceptible to significant temperature fluctuations. At the same time, key error detection points are marked, and the distribution of positioning errors at each point is accurately quantified using a laser interferometer.
2) Proceeding to the temperature rise experiment, the feed system of the machining center was operated with its cooling mechanism deactivated, while the worktable executes reciprocating motions along the feed axes at a steady speed of 10 meters per minute. The temperature data from the sensors are recorded in realtime. When the temperature approaches a predetermined threshold, the feed rate is reduced, and there is a mandatory pause of 5 seconds at each error detection point to collect relevant data.
3) Throughout each pause interval, the threedimensional error data were meticulously captured at each segment along the ball screw's length using a laser interferometer. Simultaneously, a continuous surveillance and logging of the comprehensive temperature data took place, encompassing the servo motor, slider, and screw nut temperatures.
4) To examine the impact of varying feed speeds on error magnitudes and temperature changes, the experiment includes a stepped increase in feed rates, starting from 5 meters per minute up to 25 meters per minute. The previous experimental procedures are repeated rigorously at each feed speed to ensure comprehensive data collection and credibility.
3.3. Data analysis
Under ambient room temperature conditions, the ball screw is driven to reach thermal equilibrium status at a feeding velocity of 10 m/min, which is considered achieved when the temperature increase approximates around 95 %. The perusal of experimental data reveals that the internal and external ambient temperatures of the machine tool stabilize at approximately 20 °C. The $Y$axis feeding system exhibits a characteristic pattern of rapid initial temperature escalation followed by a gradual slowdown and eventual stabilization. Specifically, the temperature escalates swiftly during the initial 70 minutes before plateauing around 110 minutes later. The temperature variation curves for the three temperaturesensitive points and the measuring points on the inner and outer surfaces of the machine tool are presented in Fig. 11. Owing to the semienclosed nature of the motor end, which leads to relatively inferior heat dissipation conditions, the temperature at measurement point T1 (the motor) peaks and stabilizes at around 36 °C. Following this, the screw nut at measurement point T2 attains a steadystate temperature of about 27 °C, and the slider at measurement point T3 records the lowest temperature, stabilizing at approximately 24 °C. This disparity is attributed to the slider's mobility with the feeding system and larger air contact surface area, resulting in more efficient heat dissipation. The elevated temperature of the screw nut compared to the slider is mainly due to the heat generation from the nutpair interaction. Ultimately, all temperature data were collated, and the temperatures at T1, T2, and T3 are preset to the temperature record value according to the weights of 70 %, 20 %, and 10 %, respectively.
Once the temperature within the monitoring system reaches the preestablished threshold, a laser interferometer is utilized to quantify the thermal error, which is calculated as the difference between the measured values at each time interval and the initial positioning error.
The radial offsets and expansion deformations caused by mechanical forces and thermal influences manifest positively along the positive direction of the system coordinates. Fig. 12 illustrates the variations in thermal error at a representative error detection point under different feed speeds, effectively demonstrating the substantial correlation between feed speed and the thermal error of the ball screw – specifically, an increase in feed speed results in a commensurate rise in thermal error. Hence, a judicious decrease in feed speed can be instrumental in mitigating thermal error.
Table 1 encapsulates select experimental error data, indicating that the threeway error at the ball screw measurement points closer to the motor end is substantial, particularly beyond point P26, where the maximum error can reach up to ±14.59 μm. It is noteworthy that the initial positioning error registers negatively and veers in both positive and negative directions, which can be attributed to the fixed mounting of the screw’s two ends.
Fig. 11Temperature variation curves at various positions under a feed rate of 10 m/min
Fig. 12Thermal error variation curve under different feed speeds at the same position
Table 1Threeaxis errors at varying inspection points (mm)
Feed speed  Points of error identification  $X$Axis  $Y$Axis  $Z$Axis 
$X$, $Y$, $Z$  $X$, $Y$, $Z$  $X$, $Y$, $Z$  
5 m/min  P1  –0.0026, –0.0018, –0.0025  0.0041, –0.0029, –0.0036  –0.0044, –0.0035, –0.0015 
P2  0.0018, –0.0038, –0.0042  0.0013, 0.0005, 0.0048  –0.0045, –0.0021, –0.0012  
…  …  …  …  
P30  0.0038, –0.0068, 0.0063  –0.0054, 0.0029, 0.0064  0.0024, 0.0038, –0.0021  
…  …  …  …  … 
20 m/min  P1  –0.0016, –0.0029, 0.0043  0.0023, –0.0006, –0.0016  0.0031, –0.0043, –0.0025 
P2  0.0020, –0.0028, –0.0053  0.0023, 0.0011, 0.0023  0.0008, –0.0040, –0.0026  
…  …  …  …  
P30  0.0061, –0.0108, 0.0086  –0.0086, 0.0056, 0.0075  0.0091, –0.0084, –0.0036  
25 m/min  P1  0.0005, –0.0051, 0.0059  0.0017, 0.0016, 0.0020  0.0055, –0.0050, –0.0029 
P2  0.0035, 0.0012, –0.0048  –0.0013, 0.0021, 0.0023  0.0062, –0.0036, –0.0020  
…  …  …  …  
P30  0.0083, –0.0141, 0.0124  –0.0118, 0.0076, 0.0124  0.0145, –0.0104, –0.0050 
4. Experimental results analysis and model validation
4.1. Experimental analysis of predictive modeling
In the present study, a series of error measurement experiments were systematically conducted on an S7H composite machining center under diverse preestablished temperature and feed rate scenarios, thereby generating datasets essential for model training and validation purposes.
Subsequent to this, the determination of model parameters was executed by identifying the most optimal loss function on the validation dataset. The efficacy of the model was further substantiated through its testing phase, wherein the performance of the proposed WOALSTM model was juxtaposed against conventional LSTM and GRU models. As evidenced in Table 2, the WOALSTM model consistently excels in all predictive metrics, demonstrating superior performance compared to its counterparts. Remarkably, the loss function values yielded by the WOALSTM model on both the validation and test sets meet the expected thresholds, indicative of its commendable accuracy in predicting the threedirectional error of the screw.
Table 2Evaluation result of each model
Model  Different axes  RMSE  MAE / mm 
LSTM  $X$  0.0062, 0.0064, 0.0079  0.0044, 0.0045, 0.0055 
$Y$  0.0057, 0.0051, 0.0069  0.0038, 0.0035, 0.0048  
$Z$  0.0062, 0.0053, 0.0064  0.0044, 0.0039, 0.0044  
GRU  $X$  0.0100, 0.0087, 0.0119  0.0079, 0.0060, 0.0089 
$Y$  0.0115, 0.0098, 0.0134  0.0090, 0.0068, 0.0105  
$Z$  0.0105, 0.0139, 0.0114  0.0078, 0.0112, 0.0083  
WOALSTM  $X$  0.0030, 0.0033, 0.0041  0.0017, 0.0016, 0.0020 
$Y$  0.0040, 0.0051, 0.0045  0.0013, 0.0021, 0.023  
$Z$  0.0033, 0.0041, 0.0043  0.0020, 0.0026, 0.0025 
To validate the efficacy of the Dropout technique in mitigating overfitting, an LSTM network model bereft of the Dropout layer was trained on the identical dataset. As illustrated in Fig. 13, the validation and training set's loss function trajectories exhibit a discernible divergence throughout the model training progression, clearly suggesting that the absence of the Dropout layer engenders overfitting tendencies within the model structure. In contrast, when incorporating the Dropout technique, the loss function values display a convergent behavior as the number of iterations increases, settling in the range between 0.2 to 0.4. This outcome underscores the significance of Dropout in maintaining a balance between model complexity and generalizability.
Fig. 13Effect of model training without dropout layer
a) With dropout layer
b) No dropout layer
The prediction outcomes for the threeaxis errors, namely the $X$axis, $Y$axis, and $Z$axis, are graphically depicted in Figs. 14, 15, and 16 respectively. A comparative analysis reveals that the WOALSTM model yields improved predictions over the LSTM and GRU models. Specifically, concerning the $X$axis error, the mean root mean square error (RMSE) of the WOALSTM model decreases by 49.19 % and 65.98 % relative to the LSTM and GRU models. Similarly, for the $Y$axis error prediction, the WOALSTM model achieves a decrease in average RMSE of 23.17 % and 60.82 %, while in predicting the $Z$axis error, the average RMSE drops by 34.63 % and 67.31 %. In order to authenticate the accuracy and robustness of the WOALSTM model, an extensive examination of the ball screw error measurement and prediction under a variety of experimental conditions was undertaken. As Table 2 manifests, under equivalent sample sizes, the WOALSTM model exhibits notably enhanced prediction accuracy and efficiency visàvis the LSTM and GRU models, with the maximum absolute residuals being 0.0194 mm, 0.0213 mm, and 0.0202 mm for the $X$, $Y$, and $Z$ axes respectively.
Fig. 14Comparison of Xaxis threeway error prediction
Fig. 15Comparison of Yaxis threeway error prediction
Fig. 16Comparison of Zaxis threeway error prediction
Based on Figs. 1416, it is evident that the WOALSTM model provides the most accurate predictions for the $X$axis on various axes. Therefore, the prediction is fitted for the $X$axis. Additionally, Fig. 17 shows that the WOALSTM model has a higher Rsquared value, indicating a more significant and statistically relevant fitting effect compared to the LSTM model.
Fig. 17Fitting impact of the Xaxis forecasts
a) LSTM
b) WOALSTM
Additionally, considering the superior prediction performance on the $X$axis, the KDE method was employed to fit the error distribution based on the $X$axis data. Fig. 18 shows that the normalized average width of the prediction intervals becomes narrower with the change in sample data. However, at peak and valley error values, these intervals widen considerably, possibly due to the influence of uncertainty factors on the error data at these extremes.
In an effort to appraise the efficacy of model interval prediction, this paper incorporates two key evaluation indices: Prediction Interval Coverage Probability (PICP) and Normalized Prediction Interval Average Width (PINAW), with their respective calculation formulas given below:
Fig. 18Xaxis interval forecasting effectiveness
a)$X$direction
b)$Y$direction
c)$Z$direction
Table 3 consolidates the findings related to the $X$axis prediction interval coverage (PICP) and normalized prediction interval average width (PINAW) for the three models across varied confidence levels.
The PINAW values of the WOALSTM model show a decrease between 53.81 % to 73.56 % compared to the LSTM and GRU models at the 95% confidence level. This trend continues at 90 %, 75 %, and 50 % confidence levels, where the WOALSTM model consistently outperforms the other models in both metrics.
To summarize, the WOALSTM model not only improves prediction accuracy but also significantly reduces the width of the prediction interval, indicating its superior predictive capabilities. This highlights the significant impact of the Whale Optimization Algorithm (WOA) in optimizing the LSTM model, making the WOALSTM model the best choice for precise and efficient prediction of the threeaxis screw errors.
Table 3Comparative evaluation of interval estimates under different levels of confidence
Confidence level  Model  PICP (%)  PINAW 
$X$, $Y$, $Z$  $X$, $Y$, $Z$  
95 %  LSTM  93.15 %, 87.67 %, 93.15 %  0.9498, 0.9958, 1.1712 
GRU  93.15 %, 90.41 %, 86.30 %  1.4428, 1.3645, 1.5889  
WOALSTM  93.15 %, 93.15 %, 93.15 %  0.4387, 0.4508, 0.4201  
90 %  LSTM  87.67 %, 84.93 %, 86.30 %  0.8172, 0.8066, 1.0228 
GRU  90.41 %, 83.56 %, 80.82 %  1.2759, 0.9862, 1.4157  
WOALSTM  89.04 %, 87.67 %, 90.41 %  0.3077, 0.1996, 0.2922  
75 %  LSTM  71.23 %, 73.97 %, 73.97 %  0.5424, 0.3881, 0.6709 
GRU  72.60 %, 75.34 %, 69.86 %  0.9222, 0.4608, 0.9873  
WOALSTM  72.60 %, 79.45 %, 76.71 %  0.0873, 0.0941, 0.1028  
50 %  LSTM  47.95 %, 54.79 %, 52.05 %  0.1634, 0.1916, 0.1540 
GRU  52.05 %, 49.31 %, 50.68 %  0.5173, 0.2582, 0.4302  
WOALSTM  49.31 %, 49.31 %, 49.32 %  0.0573, 0.0650, 0.0690 
4.2. Experimental verification of error compensation
The $Z$axis of the S7H composite machining center was studied to compensate for machining errors. The threeway error of the $Z$axis before and after the compensation was compared. As shown in Fig. 19, the maximum error of the threeway error of the $Z$axis was significantly reduced from ±0.0145 mm to ±0.006 mm after the compensation. At the same time, the average threeway error was also significantly improved from –0.00486, –0.00325, –0.00209 to –0.00197, –0.00049, –0.00038, –0.00197, –0.00049, –0.00038, and –0.00209, respectively. This means that the average threeway error was improved from –0.00209 to –0.00197, –0.00049 and 0.00038 after the compensation.
Fig. 19Comparison of threeway error before and after compensation
5. Conclusions
The objective of this study is to investigate the implications of dualend constraints on the precision of ball screw calibration during error compensation procedures and to propose a comprehensive approach for screw error mitigation. Focusing on the feed screws of the $XYZ$ axes within the S7H composite machining center, the following salient points emerge from the conducted research:
1) This investigation employs the Whale Optimization Algorithm (WOA) to finetune the hyperparameters of the Long ShortTerm Memory (LSTM) model, thereby addressing the issues surrounding substantial computational overhead and low search efficiency typically associated with the adjustment of hyperparameters in the threeaxis error prediction model for screws.
2) Comparative analysis of the three modeling techniques visàvis experimental measurements of screw errors reveals that the threeaxis error prediction model constructed by integrating the WOA with the LSTM neural network demonstrates superior fitting performance. This validates the efficacy of the proposed screw's threeaxis error prediction model in accurately forecasting the actual screw errors and enhancing the machining precision of the machine tool.
3) Using QT software and Matlab software for the secondary development of the CNC system, the ball screw error compensation module communicates more effectively with the interpolator to achieve realtime 3axis error compensation.
4) The empirical outcomes highlight that the radial screw error exerts a significantly greater influence on the machine's practical machining accuracy relative to the axial screw error. Thus, an allencompassing screw error compensation methodology is devised to cater to different mounting configurations. The effectiveness of this holistic approach is corroborated through error compensation machining experiments, demonstrating an improvement in machining accuracy by approximately 59 %.
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About this article
This work was supported by Hubei Provincial Natural Science Foundation Joint Fund for Innovation and Development Project (No.: 2022CFD081), Major Science and Technology Projects of Hubei Province (No.: 2021AAA003), Hubei Province Intellectual Property Application Demonstration Project in 2020 and Special Project for Supporting Technological Innovation and Development of Enterprises (No.: 2021BAB011), XiangYang Science and Technology Project in 2022 (No.: 2022ABH006436).
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
Bo Zhou: conceptualization, methodology, software, investigation, data curation, writingoriginal draft. Guo Hua Chen: methodology, investigation, resources, data curation, writingreview and editing, project administration. Jie Mao: validation, formal analysis, resources. Yi Li: resources, visualization. Shuai Wei Zhang: investigation, data curation.
The authors declare that they have no conflict of interest.