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The geometric precision of machined parts is greatly reduced due to thermal errors, thereby the thermal errors should be controlled and compensated timely to improve the machining accuracy. But the previously established thermal error models in related studies are not robust, and the predictive accuracy is not high, and the convergence is weak. Moreover, the error control delay is serious due to the network bandwidth limitation. To overcome the above challenges, the theoretical and data-driven methods are combined. The intrinsic long-term memory characteristic of thermal errors is revealed by the theoretical method and the finite element analysis, providing a theoretical guidance for the data-driven method. Then a residual-based one-dimensional convolution-minimum gate unit model is designed based on the residual connection. The designed one-dimensional convolution layer extracts the data features, and the residual connection with multiple pooling layers is used to compress the dimension of the error data. The minimal gate unit is used to improve the convergence rate. Then a new physical-data-edge-cloud terminal architecture is proposed by combining the advantages of the cloud computing and edge computing. Finally, the experiment was conducted to verify the designed physical-data-edge-cloud terminal architecture. The results indicate that the predictive accuracy is 98.18% and that the convergence time is shortened to 13.719 s compared with traditional thermal error models. Furthermore, the geometric error is reduced by more than 80%, and the total time consumption is 166 s, and the data transmission time is 25 s. The system efficiency of the designed physical-data-edge-cloud terminal system is far higher than that of physical-edge-cloud, physical-data-cloud, and physical-cloud systems. [ABSTRACT FROM AUTHOR]

The thermal errors in ball screws present complex spatial-temporal characteristics and exhibit persistent long-term memory, significantly affecting the machining accuracy of the whole machine tool in applications. Deep learning has emerged as a promising approach to predict these errors using thermal data. However, the scarcity of extensive thermal datasets, typically due to the prohibitive costs and time required for experimental acquisition, hampers the comprehensive exploitation of spatial-temporal and long-term memory attributes. Moreover, the current thermal error compensation system does not support the deployment of deep learning-based thermal error models due to its weak real-time performance. These limitations lead to suboptimal prediction accuracy and inadequate error compensation. Addressing this challenge, a novel spatial-temporal interactive integration network is proposed, and this is a sophisticated model that synergistically blends a time memory gate and a spatial-temporal fusion gate within an advanced attention-based spatial-temporal graph convolutional framework. Our network ingeniously leverages spatial data to anchor long-term memory while employing temporal data for selective memory feature extraction, facilitating a refined integration of spatial-temporal features. Concurrently, the integration of gated recurrent units and a time attention layer meticulously extracts and enhances temporal features, bolstered by our innovative time memory gate, which is adept at handling small-sample scenarios and refining thermal error predictions. Critically, the digital twin system for thermal error compensation is constructed to improve the system's real-time performance. The integration of this network into the digital twin system marks a pivotal advancement in thermal error compensation. Our empirical results underscore the system's remarkable robustness and superior predictive accuracy, demonstrating a significant reduction in positioning and machining errors (over 90% and 80% respectively) even with constrained thermal data inputs. These advancements not only delineate a substantial leap in predictive accuracy but also underscore our contribution to reducing operational costs and enhancing the efficacy of machine tools. [Display omitted] • Spatial-temporal fusion gate is proposed to extract spatiotemporal features. • STIIN is proposed to extract spatiotemporal features of thermal error. • Digital twin system for thermal error compensation is established. • STIIN model is embedded into digital twin system. • Robust compensation is realized with small-volume thermal information data. [ABSTRACT FROM AUTHOR]

The precision machine tool is essential for machining parts with high-precision requirements, and are widely used in the aviation, aerospace, and other fields. The thermal error is an important factor affecting the geometric precision of machined parts, and then its effective control is important. To enhance the control accuracy and execution efficiency, an edge–cloud system is designed to predict and control thermal errors. To obtain the thermal error data, a sensor network, which is composed of the temperature and displacement sensors, is designed for the thermal behavior measurement. Then the temporal and spatial behaviors of thermal errors are revealed from the heat transfer perspective, and a novel sequence-to-sequence model based LSTM network with attention mechanism (SQ-LSTMA) is designed with the full exploration of the long-term (LT) and short-term (ST) memory information of thermal errors. For the designed edge–cloud system framework, the data collection is conducted by the user layer. The edge layer performs the data processing, data storage, and error prediction with the real-time data, and the SQ-LSTMA model training is conducted by the cloud layer with the historical data. The results show that the execution time is shorten effectively and that the geometric precision of the machined part is increased. Additionally, the SQ-LSTMA model is able to reflect the dependence of the current thermal error on the historical thermal errors, and should have the ability to utilize the LT and ST memory information, and the robustness and prediction accuracy of SQ-LSTMA is superior to that of the traditional time-series models. [Display omitted] • Memorizing characteristics of thermal errors are revealed. • Novel prediction architecture is designed based on seq2seq and LSTM. • SQ-LSTMA is proposed to improve predictive accuracy and convergence rate. • Attention mechanism is used to enhance long- and short- term information. • An edge–cloud system is proposed to realize error control. [ABSTRACT FROM AUTHOR]

[Display omitted] • Four-terminal-architecture CEDTS is proposed for thermal error control. • Hyper-parameters of ILSTM network are optimized by ISOA. • Comprehensive machining error model is constructed according to HCT theory. • Data services, including clustering analysis, status monitoring, and data funsion, are provided. • GPU-based cloud computing layer is used to expedite executing process. Production lines are important for the high-accuracy and efficient machining of parts. The thermal error of key machining equipment in production lines has a significant effect on the geometric accuracy of machined parts. To improve the geometric accuracy of machined parts, the thermal error of key machining equipment in a production line should be controlled. Then the collection, storage, analysis, and calculation of the large-volume manufacturing data are essential. But the processing involving the large-volume manufacturing data is time-consuming and challenging, which leads to low executing efficiency. To solve the problem that the system is inefficient in the processing of the large-volume manufacturing data, a four-terminal-architecture cloud-edge-based digital twin system (CEDTS) is proposed with a reasonable functional division of four terminals, and thus the executing efficiency of CEDTS is expedited. Then the error mechanism is studied to prove the long-term memorizing behavior, and an improved seagull optimization algorithm (ISOA) is proposed based on the chaos thought to optimize the weights, thresholds, and the number of iterations of an improved long short term memory (ILSTM) network with the attention mechanism. The ISOA-ILSTM error model is embedded into the intelligent decision-making terminal of CEDTS to predict the thermal error. Moreover, a comprehensive machining error model is proposed and embedded into the intelligent decision-making terminal of CEDTS to control the thermal error. Finally, the effectiveness of CEDTS is verified on a production line. The results show that the reduction of the large-volume manufacturing data for the collection, storage, analysis, and calculation is significant. With the implementation of CEDTS, the fluctuation range of geometric errors of machined parts is reduced significantly. The executing time is reduced by more than half by CEDTS with the GPU acceleration. [ABSTRACT FROM AUTHOR]

• Mapping between tooth surface error and machine errors is constructed. • Internal attention gate is designed to propose AGBi-LSTMN. • MVWOA is proposed to optimize hyper-parameters of AGBi-LSTMN. • Relative position errors are proportional to tooth profile tilt deviations. • Simultaneous geometric and thermal error control is realized. The gear profile grinder is important to ensure the geometric accuracy of tooth surfaces. However, its machining accuracy is significantly reduced by the geometric and thermal errors. So the geometric and thermal errors should be controlled simultaneously. The geometric error modeling and decoupling of the gear profile grinder are studied, and the mapping relationship between the geometric accuracy of tooth surfaces and machine errors is constructed. To establish a robust thermal error model, a physically-based model is established to prove the feasibility of using the bidirectional long short term memory network (Bi-LSTMN) to establish a self-learning error model. The high-frequency noises are filtered and singular values are reduced by the Savitzky–Golay (SG) filtering algorithm. An attention-gate-based Bi-LSTMN (AGBi-LSTMN) is proposed to reduce the time complexity. Then the mixed-variants whale optimization algorithm (MVWOA) is proposed to optimize the hyper-parameters, and SG-MVWOA-AGBi-LSTMN is proposed for the thermal error prediction. Finally, the simultaneous geometric and thermal error control is implemented. The results show that the total tooth profile precision is increased by ISO level 3, and the tooth profile tilt precision is increased by ISO level 5 with the simultaneous geometric and thermal error control. [Display omitted] [ABSTRACT FROM AUTHOR]

Thermal error has become a key reason hindering machine tool's thermal stability improvement. The error compensation is carried out from the view of error mechanism of spindle systems to increase the thermal stability of machine tools. The hysteresis effect of the thermal expansion is revealed with theoretical modeling of error mechanism, and long-term memory characteristics of thermal error on historical thermal information are demonstrated. Then the applicability of long short term memory (LSTM) neural networks for the training of the error model is proved. The variational mode decomposition (VMD) decomposes error data into several inherent modal function (IMF) components to remove the coupling effect of high- and low-frequency data, improving the robustness and generalization capability of the error model. Moreover, the hyper-parameters of LSTM neural networks are optimized with grey wolf (GW) algorithms to remove the sensitivity of the predictive performance to its hyper-parameters. Finally, error models are trained with VMD-GW-LSTM neural networks, VMD-LSTM neural networks, and recurrent neural networks (RNNs). To verify the effectiveness of compensation methods, the error compensation and machining were performed at different working conditions. The results show that compensation rates of the VMD-GW-LSTM network model are 77.78%, 75.00%, and 77.78% for Sizes 1, 2, and 3, respectively. Moreover, the predictive performance and compensation performance of the VMD-GW-LSTM network model is far better than that of VMD-LSTM network and RNN models. [Display omitted] • Thermal hysteresis effect and long-term memorizing behavior are revealed explicitly. • The applicability of LSTM neural networks to train error model is proved. • The compensation is realized with VMD-GW-LSTM network model. • The compensation effect of VMD-GW-LSTM network model is the best. • The direct estimate of thermal characteristic is inaccurate. [ABSTRACT FROM AUTHOR]

The thermal error stands as a pivotal determinant, significantly impacting both the precision and its stability of gear hobbing processes. Thus far, conspicuous delays in compensation and insufficiencies in precision have been duly noted. To achieve a proficient and exceedingly precise thermal error compensation, an intricate mathematical correlation is expounded, delineating the connection between the spatial pose error of the hob and the inherent thermal error in the gear hobbing machine. In addressing the thermal error linked to the B-axis, a bidirectional-minimal gated unit network is harnessed to characterize long-term memory attributes. An advanced crystal structure algorithm is introduced, encompassing components such as the population initialization mechanism, the nonlinear convergence factor, the Gaussian perturbation mechanism, and an enhanced optimization equation. The weights of the bidirectional-minimal gated unit network are judiciously assigned via the self-attention mechanism. This innovation, recognized as the improved crystal structure algorithm-self-attention-bidirectional-minimal gated unit network, is introduced for the training of the thermal error model for the B-axis. Subsequently, thermal analysis models are formulated, yielding the thermal errors of the X- and Z- axes. An error control model and an error decoupling model are established, employing the thermal errors of the B-, X-, and Z- axes as their input. Ultimately, an intelligent error control system is crafted, in which the error control model, error decoupling model, and thermal error model are seamlessly integrated. The aim is to mitigate tooth flank errors. With the integration of the intelligent error control system, the levels of precision pertaining to gear helix deviation, maximum deviation of a single pitch, and cumulative total pitch deviation are elevated by one, one, and two levels, respectively. More significantly, the geometric error reduction of the tooth flank achieved by the meticulously designed intelligent error control system surpasses that of traditional pitch compensation by a substantial margin. Furthermore, the execution efficiency of the intelligent error control system, when operating within the machine-edge-fog-cloud framework, exhibits a remarkable improvement of 24.18%, 16.20%, and 57.84% when compared to the intelligent error control system operating within the machine-edge-cloud, machine-fog-cloud, and machine-cloud frameworks, respectively. [ABSTRACT FROM AUTHOR]