Analysis of Dual-Driven Feed System Vibration Characteristics Based on Computer Numerical Control Machine Tools: A Systematic Review

: Vibration in state-of-the-art machining impacts accuracy by diminishing the machine’s dynamic precision and the workpiece surface quality. The dependability of the cutters and productivity becomes a severe problem for optimizing the computer numerical control machine tools’ (CNCMT) efﬁciency. Therefore, investigating the twin ball screw drive system vibration properties as well as its corresponding control measures is vital. This paper thoroughly reviews the recent works on methods of analyzing and controlling vibration for dual-driven feed systems (DDFS). The research on vibration control technologies, parameter identiﬁcation, and system modeling are identiﬁed and summarized; the merits and drawbacks of various methods are discussed for comparative purposes. Furthermore, the asymmetrical relation between DDFS and single-driven feed systems are thoroughly discussed based on their dynamic properties. Finally, based on existing studies, related research prospects are described systematically, and these research directions are sure to markedly contribute to developing methods for dampening vibrations on DDFS of CNCMT.


Introduction
Since the world is transitioning from the third to the fourth industrial revolution, manufacturing organizations need more open and interoperable systems to be confident with increasingly frequent and unpredictable market changes [1,2].The automobile, aerospace, shipbuilding, mold, and energy industries have increasingly adopted high-speed machining technology as the modern manufacturing sector has emerged [3][4][5][6].Vibration in machining is an essential factor affecting its accuracy for high-speed CNCMT.It decreases cutting tool life, lowers workpiece surface quality, and compromises the machine's dynamic precision.In this regard, CNCMT vibration control is a critical point under consideration in machining [7][8][9][10][11][12].
A unidirectional dual-feed servosystem of DDFS, shown in Figure 1 [13,14], features excellent precision and stability due to the dual-drive structure's powerful force, low inertia, substantial transmission stiffness, and broad control frequency.DDFS is extensively applied in high-precision manufacturing machinery, including gantry-type tools and rectangular coordination robotics [15][16][17].Due to its potential to considerably affect the control performance, machining precision, and stability of massive gantry-type equipment, the twin ball screw feed system's (TBSFS) dynamic features occupy an enormous part of Symmetry 2023, 15, 2012 2 of 21 increasing the positioning and manufacturing consistency [18][19][20][21][22][23].Therefore, to enhance CNCMT's machining capabilities, it is necessary to investigate the vibration features of the drive system's ball screws and the techniques used in dampening.In order to effectively mitigate system vibration and enhance its bandwidth, while simultaneously dampening it, it is essential to thoroughly assess the resulting feed inside a precise model and conduct system parameter identification using relevant control methods [24].Prior reviews have concentrated on trajectory-tracking control approaches [21,[25][26][27][28]. Haojin Yang et al. [25] studied the vibration of single-drive feed systems by elaborating on their controlling and modeling methods.Tao Huang et al. [9] proposed a comprehensive review of vibration management and modeling in the feed-drive system.
Symmetry 2023, 15, x FOR PEER REVIEW 2 of 23 control performance, machining precision, and stability of massive gantry-type equipment, the twin ball screw feed system's (TBSFS) dynamic features occupy an enormous part of increasing the positioning and manufacturing consistency [18][19][20][21][22][23].Therefore, to enhance CNCMT's machining capabilities, it is necessary to investigate the vibration features of the drive system's ball screws and the techniques used in dampening.In order to effectively mitigate system vibration and enhance its bandwidth, while simultaneously dampening it, it is essential to thoroughly assess the resulting feed inside a precise model and conduct system parameter identification using relevant control methods [24].Prior reviews have concentrated on trajectory-tracking control approaches [21,[25][26][27][28]. Haojin Yang et al. [25] studied the vibration of single-drive feed systems by elaborating on their controlling and modeling methods.Tao Huang et al. [9] proposed a comprehensive review of vibration management and modeling in the feed-drive system.
Nevertheless, all researchers have focused on analyzing and controlling vibration using a single-drive feed system.This paper promises a deeper understanding of the methods for analyzing and controlling vibration of the TBSFS.In contrast to prior research, this study focuses on the application of the machine DDFS modeling process, identification of different parameters, and vibration management methods.Furthermore, the asymmetrical relation between dual-driven and single-driven feed systems are thoroughly discussed based on their dynamic properties.Finally, the work mainly analyzed representative methods of vibration analysis and control.

Comparing Perspectives between Dual-and Single-Drive Feed Systems
Dual-driven and single-driven feed systems are the mechanisms various machines apply to transport materials or components through a processing or assembly line.Several studies have been offered to provide a comprehensive understanding of DDFS [16,17,[29][30][31].While comparing these two different feeding systems, as shown in Table 1, this research considers the number of motors, the control, the speed, the structural complexity, and the load distribution.The DDFS can be conceptualized as a parallel positioning system generated by two motors moving in sync within a single orientation.In other words, instead of adopting one engine and unidirectional screw, the suggested dual-driven feed mechanism uses twin permanent magnet synchronous motors (PMSMs) with two screws.Herein, the feed system's stiffness can be improved by reducing its inertia with this structure [14].In addition, the drive-by-center-of-gravity method centered on a DDFS reduces vibration and increases control system bandwidth [32].Moreover, the DDFS can provide high driving forces, which are widely used in gantry-type machines [17].
Based on the following sections, this study has confidently demonstrated that most single-driven feed system analysis and vibration control methods can be applied to DDFS.However, due to the fundamental inherent complexity of DDFS, its modeling approach is Nevertheless, all researchers have focused on analyzing and controlling vibration using a single-drive feed system.This paper promises a deeper understanding of the methods for analyzing and controlling vibration of the TBSFS.In contrast to prior research, this study focuses on the application of the machine DDFS modeling process, identification of different parameters, and vibration management methods.Furthermore, the asymmetrical relation between dual-driven and single-driven feed systems are thoroughly discussed based on their dynamic properties.Finally, the work mainly analyzed representative methods of vibration analysis and control.

Comparing Perspectives between Dual-and Single-Drive Feed Systems
Dual-driven and single-driven feed systems are the mechanisms various machines apply to transport materials or components through a processing or assembly line.Several studies have been offered to provide a comprehensive understanding of DDFS [16,17,[29][30][31].While comparing these two different feeding systems, as shown in Table 1, this research considers the number of motors, the control, the speed, the structural complexity, and the load distribution.The DDFS can be conceptualized as a parallel positioning system generated by two motors moving in sync within a single orientation.In other words, instead of adopting one engine and unidirectional screw, the suggested dual-driven feed mechanism uses twin permanent magnet synchronous motors (PMSMs) with two screws.Herein, the feed system's stiffness can be improved by reducing its inertia with this structure [14].In addition, the drive-by-center-of-gravity method centered on a DDFS reduces vibration and increases control system bandwidth [32].Moreover, the DDFS can provide high driving forces, which are widely used in gantry-type machines [17].
Based on the following sections, this study has confidently demonstrated that most single-driven feed system analysis and vibration control methods can be applied to DDFS.However, due to the fundamental inherent complexity of DDFS, its modeling approach is considerably more complicated than that of a single-drive feed system; hence, parameter identification is strongly recommended to reduce structural complexity.

Description Advantages Disadvantages
Single-drive feed system [8,9,19,23] A single-drive feed system is a positioning system for motorized carriages traveling in the same direction.
Its modeling approach is simple.It is easy to control dynamic errors since it has one drive system.No synchronization errors Features low precision over workspace ratio and low dynamic performance.It covers low precision and common feed, which affect the machining accuracy.
Dual-driven feed system [13,18,33,34] The dual-driven feed system represents a unidirectional paralleled X-Y positioning system generated by two synchronized motors.
It has high rigidity, stability, and control bandwidth inertia of the feed system.DDFS features high precision and stability over workspace ratio and good dynamic performance due to its unique mechanical construction and advancements in electrical linear drives.
Its modeling approach is significantly more complex.Notwithstanding the benefits of DDFS, engineers, and researchers still have significant difficulty, especially in analyzing natural frequencies, monitoring errors, positional preciseness, and manipulating the desynchronized errors.

TBSFS Modeling Methods
System modeling is adopted to assess and control the twin ball screw's (TBS's) vibration.Most DDFS comprises two servo motors, TBS, two linear guide rails, bearings, a worktable, and other components [35][36][37].As illustrated in Figure 1, the mechanical structure components of the TBSFS have been described as intricate elastic structures [14,38].Establishing the vibration mode of a system, which essentially requires obtaining the displacement of every point in the system, is a complicated operation.Typically, the lumped parameter approach or the finite element method transforms an infinite-degree of freedom (DOF) continuously operating system into a finite-DOF independent system.From the recent progress in modeling control, the research has confirmed that lumped parameters, finite element (FE), and hybrid modeling approaches are the most prevalent methods for modeling the TBSFS [14,39].Therefore, the following section thoroughly emphasizes reviewing the aforementioned procedures for comparative purposes.

Finite Element Approaches of TBFDS
Generally, to acquire the FE equations of the core system, the TBSFS must be divided into finite elements, and each piece must be interpolated to produce the finite element equations.The proper finite element approximation method must be selected, and the global stiffness matrix must be generated.This approach's computational speed is symmetrical to the number of discrete components and the complexity of the method approximation.The main disadvantages of the FE approach cover the computation time due to the model complexity, which limits its applicability in that model scaling and process transformation are essential for applying methods in resolving engineering problems.As illustrated in Figure 2, Meng Duan et al. [13] adopted FE approaches to evaluate the TBSFS's dynamic properties of joint rigidity.Zaeh et al. [40] adopted FE methods to present the detailed modeling of ball screw drives and to enable direct FE modeling of transmission systems by discrete geometrical computation of the transmission shafts, thereby assuring a natural torque or force overloading in the transmission system.Moreover, Okwudire et al. [41] utilized the FE method to establish the model by incorporating axial, bending, and torsional coupling components in the stiffness matrix.Their research concluded that model scaling and process transformation are essential for applying FE methods.Table 2 covers the work performed on DDFS's dynamics analysis using FE approaches with their advantages and disadvantages.
assuring a natural torque or force overloading in the transmission system.Moreover, Okwudire et al. [41] utilized the FE method to establish the model by incorporating axial, bending, and torsional coupling components in the stiffness matrix.Their research concluded that model scaling and process transformation are essential for applying FE methods.Table 2 covers the work performed on DDFS's dynamics analysis using FE approaches with their advantages and disadvantages.

Application Benefits Drawback
Meng et al. [13] adopted the FE approach to construct a stiffness model for the transmission chain, taking into account the rigidity of screwnut and bearing couplings.
Due to its ability to accurately describe DDFS's complicated geometries, material properties, and boundary conditions, this approach has shown to be among the most effective for accurately modeling systems with variable properties.
Within the analysis of their model, DDFS has been characterized as one of the largest and most complex systems, it may necessitate considerable computational capacity to meet the demands of fine meshing and solving enormous matrices.
To enhance the dynamic features of the dual differential feed system, Wang et al. [42] employed the FE method by integrating orthogonal experiments and gray relational The FE approach is capable of effectively addressing a diverse set of boundary conditions, encompassing different constraints, loads, and contact interactions.The results indicate that the modal flexibility of the first mode experienced Despite the fact that this approach has the potential to enhance DDFS dynamics analysis, as a consequence of the sensitivity of its obtained results to the size and quality of the used finite elements, it still faces formidable Table 2.The applications, benefits and drawbacks of FE methods.

Application Benefits Drawback
Meng et al. [13] adopted the FE approach to construct a stiffness model for the transmission chain, taking into account the rigidity of screw-nut and bearing couplings.
Due to its ability to accurately describe DDFS's complicated geometries, material properties, and boundary conditions, this approach has shown to be among the most effective for accurately modeling systems with variable properties.
Within the analysis of their model, DDFS has been characterized as one of the largest and most complex systems, it may necessitate considerable computational capacity to meet the demands of fine meshing and solving enormous matrices.
To enhance the dynamic features of the dual differential feed system, Wang et al. [42] employed the FE method by integrating orthogonal experiments and gray relational analysis, to optimize the joint's stiffness configuration.
The FE approach is capable of effectively addressing a diverse set of boundary conditions, encompassing different constraints, loads, and contact interactions.The results indicate that the modal flexibility of the first mode experienced a reduction of 20.6%, 5.6%, and 17.04% in the X, Y, and Z directions, respectively, following the optimization process, as compared to the values prior to optimization.
Despite the fact that this approach has the potential to enhance DDFS dynamics analysis, as a consequence of the sensitivity of its obtained results to the size and quality of the used finite elements, it still faces formidable challenges.However, coarser meshes might result in decreased precision of the outcomes.
Chan et al. [43] applied The FE Methods to analyze the structure optimization of a gantry-type high-precision machine tool.
The FE approach could potentially be employed to analyze the static and dynamic properties of a gantry-type CNC machine, since it takes into account nonlinear factors including material nonlinearity, contact, and significant deformations, it can be used for a wide variety of dynamic problems.
This method involves the adoption of precise material characteristics and boundary conditions.If the parameters in concern are not adequately described, the obtained results could lack reliability.

Lumped Parameter Modeling of TBSFS
Lumped parameter modeling is a widely used method for simulating the dynamic behavior of DDFS.This approach divides a complex system of TBSFS into smaller and more manageable pieces called "lumps," each model with its own set of simplified mathematical equations.Moreover, The TBSFS's screw, nut, and bearing joint are all dynamically represented as spring-damping elements in a corresponding model wherein the worktable correlates to the lumped mass model.As shown in Figure 3 in this method, a finite number of masses blocks stands for the mass elements of the TBSFS, and some massless springs represent the elastic structures.Equivalent damping elements represent the structural damping elements.The associated concept offers a simple model that may represent the fundamental dynamic features of the low-frequency domain.Unfortunately, the lumped parameter model has poor solution accuracy because it fails to describe the screw's elastic characteristics.In this regard, Lagrange's energy approach was proposed to formulate the dynamical modeling of TBSFS [8,15,44].Equation (2) expresses the potential energy of the feed system as indicated below.
The variables  and  represent the correspondingly equivalent  and  's axial stiffness.The value of  ,  and  are well described in [14].
Assume that L = T − V.The following is a well-known representation of the Lagrange equation for extended coordinates and forces: where,  and  represent the standardized coordinate matrix and force matrix, respectively.Generating the value in Equation (3) into Lagrange equation matrix form, we will get It is noted that C, K, and M stand for damping, stiffness, and mass matrixes, respectively.Thus, the dynamic system model is represented by considering or ignoring the damping matrix.In this regard, to learn more about the connection between the ball screw preload and its vibration, Feng et al. [45] proposed the lumped parameter approach to develop a dynamic model of the pre-load-adjustable feed drive system.While Garcia-Herreros I et al. [46] employ Lagrange's equation to create a simple model of the ETEL(Établissements Techniques Électriques) gantry-type dual-driven worktable and assume the lumped parameter to understand further how a ball screw's vibration is related to its preload.Chen et al. [47] adopted the same method to establish an analytical model for investigating the nonlinear dynamic characteristics of a ball screw with preload.Zhang et al. [21] also adopted these methods to develop an equivalent dynamic model of transmission components, considering the impact of high acceleration on the dynamic proper- The Equation (1) [14] listing T as the energy of total kinetics for all components, Therefore, T can be represented by the following express: where m e = J e ( P 2π )2 , P and J e stand for total axial mass, the ball screw's lead, and rotary inertia, respectively, the expression m w = m 1 + m 2 where m 1 , m 2 stand for cross beam's and moving parts' mass, respectively.J e represent the ball screw rotary inertia, J e stands for rotary inertial for the coupled part between the center of gravity O c ; while x 1 , x 2 are the equivalent displacement of the X 1 and X 2 axis, respectively.Equation ( 2) expresses the potential energy of the feed system as indicated below. (2) The variables K e1 and K e2 represent the correspondingly equivalent X 1 and X 2 's axial stiffness.The value of l 1 , l 2 and θ z are well described in [14].
Assume that L = T − V.The following is a well-known representation of the Lagrange equation for extended coordinates and forces: where, q i and Q i represent the standardized coordinate matrix and force matrix, respectively.Generating the value in Equation ( 3) into Lagrange equation matrix form, we will get It is noted that C, K, and M stand for damping, stiffness, and mass matrixes, respectively.Thus, the dynamic system model is represented by considering or ignoring the damping matrix.In this regard, to learn more about the connection between the ball screw preload and its vibration, Feng et al. [45] proposed the lumped parameter approach to develop a dynamic model of the pre-load-adjustable feed drive system.While Garcia-Herreros I et al. [46] employ Lagrange's equation to create a simple model of the ETEL(Établissements Techniques Électriques) gantry-type dual-driven worktable and assume the lumped parameter to understand further how a ball screw's vibration is related to its preload.Chen et al. [47] adopted the same method to establish an analytical model for investigating the nonlinear dynamic characteristics of a ball screw with preload.Zhang et al. [21] also adopted these methods to develop an equivalent dynamic model of transmission components, considering the impact of high acceleration on the dynamic properties of feed systems.While Zhang HJ et al. [48] also used the same approaches to establish a dynamic model of the high-speed feed system while considering joint stiffness, they also looked at how factors like axial force, friction force, and preload affected the transmission chain's rigidity.Lu Hong et al. [14] proposed lumped parameter approaches to provide the overall dynamic model of the DDFS accounting for the contact stiffness of kinematic joints.

Hybrid Modeling Method on TBSFS
The hybrid model method refers to the combination of FE approaches with lumped parameter approaches to perform the dynamic structure of the given design.The TBS in the TBSFS is modeled by finite elements as a flexible structure, whereas the lumped mass method applies to analyzing the rest of the system.This method primarily establishes the TBS distributed parameter model and the nut/screw joint surface modeling.The modeling techniques for a screw are categorized in two ways: the straightforward procedure that treats the screw like a full beam, and models that use the beam's analytical equation.This approach provides reliable predictions of the DDFS's open-loop dynamic features when juxtaposed with the results collected from experiments [49][50][51].
The second approach uses a beam-element-based finite element model to simulate the screw.This new approach uses a more efficient and concise model description than the previous one [52][53][54].In addition, the joint surface of the transmitting coupling is designed to characterize the screw-to-table motion, vibration, and force transfer process, which makes it a crucial component of the ball screw's hybrid modeling method.Mostly in modern technology, the stiffness matrix is used to explain the transport properties of the joint's surface.Research has confirmed three stages of development: the screw's axial and torsional are considered in the first stage, its axial, rotational torque, and horizontal permanent deformations in the second, and their interactive coupling re-patroonships in the third.In this regard, by viewing a ball screw as Timoshenko's beam, Dong et al. [55] adopted the hybrid model method to investigate the effect of table parameters like mass, position, and preload on the ball screw dynamic properties.Zhang et al. [49] utilized a hybrid strategy to develop a thorough model of the ball screw drive spindle system for investigating its rigid-flexible coupling vibration.They obtained the continuous deformation and motion equation through the Lagrange approach.This hybrid method promises a beneficial numerical simulation result for the controller and the matching stiffness design of ball screw drives.Table 3 covers the comparatives aspects of TBSFS modeling methods.
The main objective of this section consists of comprehensively examining the modeling techniques employed in the investigation of dynamic behaviors of DDFS.These techniques comprise lumped parameter modeling, the finite element approach, and hybrid methods.The investigation yielded a thorough understanding of the three methodologies by detailing their reported applications along with their respective advantages and draw-backs.Briefly, this research revealed that: lumped parameter models are appropriate for expeditious evaluations, the FE method is for comprehensive and precise analyses, and hybrid approaches are for situations that necessitate a compromise between efficiency and accuracy.Nevertheless, it is imperative for engineers and researchers to thoroughly assess the merits and constraints associated with each methodology, taking into consideration the intricacy and specific demands of their respective DDFS systems.

Method Benefits Drawbacks
Lumped parameter method [7,25,26,35,36] The system's natural frequency and vibration modes are shown using this method.It decreases model DOF and function transfer order.Integration with the control system is straightforward.
A few crucial elements are neglected, including joint rigidity, structural adaptability, and the slender rod distinctive of the TBS.Therefore, when the model is constructed using this method, there will be some inaccuracies in the computation results.
The finite element modeling method [45][46][47] The specifics of system dynamics are assured by this method.Its dynamic variations while the table moves can also be described, in addition to its general static properties.
The control system cannot easily integrate the complex model with several DOFs.Each reduced-order method has necessities; many parameters must be considered in actual implementations.
Once compared to the other two methods, this one is superior.However, structural adaptability's impact on dynamic performance is underestimated when applying the model to the control system.

Parameter Identification of the TBSFS
The model-based control of TBSFS requires a precise model for the controller design.Therefore, as the above-mentioned typical modeling techniques simplify the actual model, they result in errors between the theoretical and practical models due to inaccuracies in dynamic parameters like the system's damping and stiffness.To assess and identify the dynamic parameters of the large CNCMT, a mathematical model is created based on the linear relationship between the input and output signals of the TBSFS [56].Figure 4 depicts the description of all methods needed to analyze any dynamics models.These are distributed into three main parts: major identification elements, parameter extraction methods, and identification technologies.The main objective of this section consists of comprehensively examining the modeling techniques employed in the investigation of dynamic behaviors of DDFS.These techniques comprise lumped parameter modeling, the finite element approach, and hybrid methods.The investigation yielded a thorough understanding of the three methodologies by detailing their reported applications along with their respective advantages and drawbacks.Briefly, this research revealed that: lumped parameter models are appropriate for expeditious evaluations, the FE method is for comprehensive and precise analyses, and hybrid approaches are for situations that necessitate a compromise between efficiency and accuracy.Nevertheless, it is imperative for engineers and researchers to thoroughly assess the merits and constraints associated with each methodology, taking into consideration the intricacy and specific demands of their respective DDFS systems.

Parameter Identification of the TBSFS
The model-based control of TBSFS requires a precise model for the controller design.Therefore, as the above-mentioned typical modeling techniques simplify the actual model, they result in errors between the theoretical and practical models due to inaccuracies in dynamic parameters like the system's damping and stiffness.To assess and identify the dynamic parameters of the large CNCMT, a mathematical model is created based on the linear relationship between the input and output signals of the TBSFS [56].Figure 4 depicts the description of all methods needed to analyze any dynamics models.These are distributed into three main parts: major identification elements, parameter extraction methods, and identification technologies.

Major Identification Element
Each axis among two synchronized axes of a TBSFS consists of a table, bearings, ball screw, rolling guide, and servo motor coupled via various joints.Typical rolling joints

Major Identification Element
Each axis among two synchronized axes of a TBSFS consists of a table, bearings, ball screw, rolling guide, and servo motor coupled via various joints.Typical rolling joints comprise the rolling direction, ball screw, and bearing.When subjected to complicated external dynamic loads, joint component substructures are prone to micro-vibration with multi-directional degrees of freedom (DOFs).The joints will be able to show off the crucial characteristic of flexible connections by storing and dissipating energy and demonstrating elasticity and damping [57][58][59].Statistics indicated that joint factors like damping and stiffness account for 30% to 50% of stiffness and 90% of the CNCMT's dynamic performance damping.It has also proved that more than 60% of vibration issues are directly linked to the joint itself [60,61].
Consequently, the joint dynamic properties are among the most influential aspects determining dynamic performance.In the study of the dynamics of CNCMT, the emphasis is on precisely identifying the characteristic dynamic parameters of joints and analyzing the ball screw drive system's dynamic characteristics.Moreover, the TBSFS incorporates both fixed and rotating joints.In this regard, the method for identifying the dynamic properties of joint parameters is described in the following section.

Rolling Joint Dynamic Parameters Identification
So far, most rolling joints comprise bearing, rolling guide, and ball screw joints that transmit motion from the motor to the feed mechanism.It is challenging to improve the TBSFS dynamic modeling precision and the rolling joints' dynamic parameters identification accuracy since it involves the infrequent assessment of the coupling between each of the rolling joints with their unidirectional dynamic factors, which is one of the traditional modeling techniques' limitations.Some approaches were proposed by researchers to identify rolling joint dynamic parameters to acquire a deeper understanding of the vibration characteristics of the CNCMT, which can be used to improve their performance and reduce the likelihood of failure.Wei Zhang et al. [62] proposed innovative approaches to identify the rolling joint's dynamic parameters accounting for the assembled BSFS's digital twin dynamic model.The geometric model was constructed by the physical entity's synchronizing information.Additionally, they introduced the FE-analyzed model, which may simultaneously consider numerous rolling joints and their dynamic parameters in different orientations.Wherein they had demonstrated that the mean inaccuracy of identifying parameters was under 3%, implying that the suggested technique is practicable, efficient, and more accurate.Jidong Zhang et al. [63] proposed a multi-criteria integrated optimal control design method to identify the dynamic parameter and obtain the optimal excitation signal taking the multi-axis movable coupling features into account.Peng fei Yan et al. [64] analyzed the influence of the vibration amplitude in the faulty bearing considering the five DOF dynamic models of the rolling bearing with the local defect.Harris et al. [65] defined the rolling bearing joint as a spring unit with two degrees of freedom and derived the analytic equation in the calculation of bearing stiffness.To obtain the ball screw joints' stiffness matrices, Yongjiang Chen et al. [66] investigated the impact of nut rotational speed, axial load, as well as pitch on the stiffness by utilizing a quasi-static analysis.

Fixed Joint Dynamic Parameters Identification
The stiffness and damping characteristics can be calculated using the experimental data supplied by Yoshimura in his integration method.To obtain stiffness and damping element graphs beneath the unit positive pressure, Yoshimura measured the total contact stiffness as well as the damping parameters through various mixed situations while studying fixed bolted joints of CNCMT [67].Kashani et al. [22] established the theory of the identification method on the optimum equivalent linear frequency response function (OELF).Herein, authors proposed nonlinear joint identification approaches which did not require sophisticated monitoring hardware and techniques.These approaches can also be utilized without any joint pre-assumption.To evaluate the constructed mode and component substructure without noting the masses of bolted joints, Tsai et al. [68] solved each joint's stiffness and damping characteristics using the theoretical model's integration.Herein, they have presented a novel approach aimed at identifying the characteristics of a newly developed bolt joint.The dynamic properties of this single bolt joint are obtained by directly extracting the observed frequency response functions of the substructures and the assembled structure, as they represent the combined features of the joint.According to the Hertzian contact theory, Duan et al. [32] provided the mathematical algorithm for TBS virtual material characteristic parameters of joints.Their results confirmed the virtual material modeling method's viability and correctness.Damjan et al. [69] applied a substructure synthesis approach to developing the joints' analytical model in each screw.They applied modal measurement data to determine the joints' stiffness and damping parameters.Ren et al. [70] created a more generic version of the frequency response function (FRF) standard identification method for systems with several rigid and flexible joints.They determined the joint characteristics from the experimental data and developed a theoretical model.Dynamic joint stiffness was determined by integrating mass, stiffness, viscosity, and structural damping matrices.One approach to mitigate the impact of measurement mistakes was establishing an appropriate condition for the optimal solution.

TBS Modal Parameters' Extraction Method
So far, the research has confirmed two standard methods for extracting modal parameters, including operational modal analysis (OMA) as well as experimental modal analysis (EMA) [71,72].When comparing the two methods, each has benefits and drawbacks.EMA is more precise, but it can be costly in time and resources because it needs to be excited in a controlled manner.OMA is not as exact but can be performed on active structures without external excitation [73][74][75].The requirements of the modal analysis and the resources at hand will determine the adopted methods.According to EMA, modal parameters can be retrieved from a structure by evaluating its response to known excitation, such as a hammer blow, shaker input, or natural vibrations.Accelerometers and strain gauges can record the structure's response, which can then be processed to generate the frequency response functions (FRFs).
However, curve-fitting algorithms like the poly reference least-squares complex exponential (PLSCF) method and the eigensystem realization algorithm (ERA) extract the modal parameters from the FRFs.Bertolaso et al. [76] adopted the ball screw drive systems modal analysis using experimental and numerical techniques.Two ball screw encoders were set in motion to ascertain the axial and torsional modal characteristics by a stimulus applied by Xiangsheng et al. [77] using the measured dynamic stiffness and the specified beginning range of stiffness and damping, and they calculated the joint's dynamic characteristics utilizing the particle swarm optimization (PSO) method.The analysis shows that the recognized and estimated modal parameters generated are consistent.Casquero et al. [78] determined the dynamic characteristics by planning motor action stimulated by the feed system.

Modal Parameters Identification Method
Dual-driven CNCMTs are complex, multi-degrees-of-freedom systems with various dynamic behaviors, such as vibration, resonance, and instability.To understand and control these behaviors, it is necessary to determine the machine's modal parameters; such characteristics include natural frequencies, damping ratios, and mode analysis.This method's modal parameter identification (PI) methods are classified according to the various signal domains as the time domain, frequency domain, and time-frequency domain identification.The following is a discussion of the numerous identification techniques, organized following the classification of signal domains: (1) Frequency domain identification (ID) method: This involves exciting the machine tool at different frequencies and measuring the corresponding response to varying points on the machine using accelerometers or other sensors.The most prevalent frequency domain identification techniques are admittance circle analysis and orthogonal polynomial techniques [79,80], optimization, and total least squares techniques [81,82].
(2) Time domain ID method: this is a powerful tool for identifying the structure's modal parameters and understanding its dynamic behavior, which requires careful measurement and analysis to obtain accurate results.This method corresponds to the parameter identification of time series information gathered via the system's vibration response.Time domain methods feature the time series model that combines an autoregressive (AR) with a moving average (MA) model (ARMA method) [83], random sub-space process, and cross-correlation method [84].
(3) Time-frequency domain ID method: This is highly effective for identifying modal parameters of vibrating structures, particularly for complex systems with non-stationary responses.It integrates the system's information in the time and frequency domains to adequately characterize and express the vibration properties of a mechanical system.Consequently, the vibration signal time-varying laws may be accurately uncovered to ensure helpful direction in analyzing the system's dynamic properties.While using these methods, the response signal is initially transformed into the time-frequency domain using a technique like the short-time Fourier transform (STFT) or the continuous wavelet transform (CWT) [85,86].The time-frequency representation must be analyzed to identify the modal parameters of the structure by using various algorithms, such as the S-method or the Hilbert-Huang Transform (HHT), for modal frequency extraction and damping ratios from the time-frequency spectrum [87].Typically, the choice of method is determined by the nature of the application and the data provided.When high-frequency data is available, time domain identification is performed, whereas frequency domain identification is applied when the system's reaction to specific frequencies must be evaluated.The time-frequency domain is favorable when both time and frequency information are required to comprehend the system's behavior.The time-frequency techniques yield promising positive outcomes over nonlinear and non-stationary signals, which caused it to be extensively investigated.
This section focuses on the parameter's identification (PI) that facilitates the analysis of the dynamic properties of DDFS, which is crucial for assuring the accuracy and efficiency of the system in diverse industrial and automation applications.PI in DDFS is typically engaged in the task of identifying and optimizing the diverse parameters that dictate the system's behavior and performance.This study examined the primary components of identification, such as dynamics PI in rolling or stationary elements, as well as the TBS modal parameter extraction method and model PI method.Based on prior research, it was concluded that, the process of PI in a DDFS is commonly achieved using a combination of empirical investigation, quantitative measurement, and mathematical modeling.The process may also entail the utilization of calibration equipment, analysis of data, and software tools to optimize the system's characteristics in order to achieve superior performance, accuracy, and dependability.

Vibration Controlling Methods for DDFS
Due to the asymmetry of the two motors, a DDFS is susceptible to vibrations, resulting in decreased accuracy, productivity, and premature wear and tear.Therefore, its vibration control is necessary to ensure dependable and accurate performance, increase productivity, and lower maintenance costs.So far, the fundamental needs of its management are trajectory tracking and vibration suppression.Position tracking of the workpiece via trajectory tracking necessitates high processing speed and precision, while vibration suppression dampens the tremors caused by the fluctuating cutting force and feed rate.The drive system's performance is hindered by several factors, including the following: the ball screw's limited structural flexibility restricts the maximum tracking speed; applying cutting force near the resonant frequency causes the mechanical structure to vibrate; and the machining process causes constant changes in the system's dynamic parameters.
Moreover, even as the DDFS performs complex trajectories, it is noted that synchronous control is just as important as system capacities and precise tracking in each axis when it comes to motion precision.Several TBSFS control strategies were presented to address the previously-mentioned issues.Therefore, as described in the following sections, the proposed methods can be categorized into two categories, including traditional vibration control or modern vibration control methods.

Traditional Vibration Control Methods
Traditional vibration controllers typically use simple control strategies, such as proportional-integral-derivative (PID) control, to reduce vibrations in mechanical systems.These controllers are widely used due to their adaptability, user-friendliness, and simple parameter adjustment.Nevertheless, many are exclusively constructed following the TBSFS rigid body dynamics, disregarding the flexible modes of structural resonance, which restricts the possible bandwidth [88][89][90][91].As discussed in the previous sections, the TBSFS propels the load by two ball screws.The control system may employ crosscoupling to ensure that the two ball screws are synchronized and that the load is distributed evenly between them [92].Figure 5 displays the dual-driving control system configuration.The research has confirmed that the main control methods include the notch filter, the input shaping, the pole placement, the active damping, and the feed-forward control methods [25].sections, the proposed methods can be categorized into two categories, including traditional vibration control or modern vibration control methods.

Traditional Vibration Control Methods
Traditional vibration controllers typically use simple control strategies, such as proportional-integral-derivative (PID) control, to reduce vibrations in mechanical systems.These controllers are widely used due to their adaptability, user-friendliness, and simple parameter adjustment.Nevertheless, many are exclusively constructed following the TBSFS rigid body dynamics, disregarding the flexible modes of structural resonance, which restricts the possible bandwidth [88][89][90][91].As discussed in the previous sections, the TBSFS propels the load by two ball screws.The control system may employ cross-coupling to ensure that the two ball screws are synchronized and that the load is distributed evenly between them [92].Figure 5 displays the dual-driving control system configuration.The research has confirmed that the main control methods include the notch filter, the input shaping, the pole placement, the active damping, and the feed-forward control methods [25].Notch filters can be incorporated into the control loop to stop the control signals from exciting the flexible modes during machining.In this regard, the dual-stage notch filter control techniques were introduced by C. Peng et al. [94] for damping the synchronous vibration of the magnetically suspended rotor system with solid gyroscopic impacts.Their works confirmed that the system could maintain a stable suspension across the whole speed range while incorporating the proposed algorithm for vibration suppression.Their vibration control process is summarized in Figure 6, wherein they considered the control in the high/low parts.Smith et al. [95] applied notch filters to suppress the resonance effect effectively; however, these filters have minimal influence on phase margin recovery and limit the attainable bandwidth.
Further measures to prevent structural vibrations are input command pre-shaping (ICP) or input shaping (IS) procedures [96,97].The central principle of the IS approach is based on the idea that vibrations can be mitigated by carefully planning the system's control input parameters to allow the intended controlled input command to be pre-shaped in advance.For the closed-loop control system, the modified reference input signal is the new input and is compared to the measured output.Tsai et al. [98] developed a full-order modified input shaping with a zero vibration (FMISZV) algorithm integrated into the CNC interpolator to relieve vibrations of the servo-feed drive system.They compared the Notch filters can be incorporated into the control loop to stop the control signals from exciting the flexible modes during machining.In this regard, the dual-stage notch filter control techniques were introduced by C. Peng et al. [94] for damping the synchronous vibration of the magnetically suspended rotor system with solid gyroscopic impacts.Their works confirmed that the system could maintain a stable suspension across the whole speed range while incorporating the proposed algorithm for vibration suppression.Their vibration control process is summarized in Figure 6, wherein they considered the control in the high/low parts.Smith et al. [95] applied notch filters to suppress the resonance effect effectively; however, these filters have minimal influence on phase margin recovery and limit the attainable bandwidth.
Further measures to prevent structural vibrations are input command pre-shaping (ICP) or input shaping (IS) procedures [96,97].The central principle of the IS approach is based on the idea that vibrations can be mitigated by carefully planning the system's control input parameters to allow the intended controlled input command to be pre-shaped in advance.For the closed-loop control system, the modified reference input signal is the new input and is compared to the measured output.Tsai et al. [98] developed a full-order modified input shaping with a zero vibration (FMISZV) algorithm integrated into the CNC interpolator to relieve vibrations of the servo-feed drive system.They compared the effectiveness of FMISZV vibration dampening to that of conventional input shapers, including zero vibration with derivation (ZVD) as well as traditional linear deceleration or acceleration interpolators (CLAI).FMISZV was preferred compared to the ZVD and CLAI techniques in their studies. and  : rotor displacements in the radial, and rotational directions, g : transfer function of the power amplifier.g (s) and g (s): the baseline controllers,  : proportional gain of the sensor system, J and J : moments of inertia of the suspended rotor in the radial direction, J is the moment of inertia in the axial direction.In addition to counteracting the stable dynamics of a closed-loop system, the feedforward control techniques enable the generation of complete control highly similar to the uniform transmission function over a broad frequency range.García et al. [46] developed a decoupling feedforward/feedback control method using a model-based decoupling control.The primary purpose was to solve the asymmetry and mechanical coupling between dual-drive gantry stage machine tool actuators.Their algorithmic structure is depicted in Figure 7. Wan et al. [29] implemented model-based composite adaptive feedforward and PID with adaptive RISE feedback with a 2-DOF control technique.The model concurrently suppresses the disturbance caused by structural and unstructured model uncertainties without excitation of flexible actual dual-drive checkerboard gantry modes.Pritschow et al. [99] demonstrated the performance of feedforward controllers highly depends on the model's accuracy.Even though the feed system's dynamic properties are precisely determined, the parameters change over time.Thus, the open-loop design of this method is one of its most significant disadvantages.
Moreover, active damping was proposed by Trusty et al. [100,101] to eradicate the excitation of the system's flexible mode by the cutting force.Their controller was created with accelerated feedback and incorporated feedforward compensation.In contrast, the simulated results demonstrated the effectiveness of vibration dampening.The research has proved that classical vibration control approaches have advanced recently in DDFS; On the basis of a dynamic model of a cross-arm bearing an unsteady payload, Giams et al. [102] suggested an adaptive tuning strategy for the PID controllers of the master/master design This technique is intended to adjust for unstable load distribution among crossarm actuators caused by an altering load.Altintas et al. [103] added an input shaping filter to the reference commands of a CNCMT to prevent the retained vibrations.The revised position directives are subsequently supplied to the actual axis control loop to decrease the drive structure's residual vibrations and contour inaccuracies.In the TBSFS, ring shaping, notch filters, and active damping pole placement were introduced [104,105].
The system's uncertainties and nonlinear variables still constrain the classic controllers.However, while emphasizing these problems in the controller design, it is well recommended to implement the adaptive controllers [106].In addition to counteracting the stable dynamics of a closed-loop system, the feedforward control techniques enable the generation of complete control highly similar to the uniform transmission function over a broad frequency range.García et al. [46] developed a decoupling feedforward/feedback control method using a model-based decoupling control.The primary purpose was to solve the asymmetry and mechanical coupling between dual-drive gantry stage machine tool actuators.Their algorithmic structure is depicted in Figure 7. Wan et al. [29] implemented model-based composite adaptive feedforward and PID with adaptive RISE feedback with a 2-DOF control technique.The model concurrently suppresses the disturbance caused by structural and unstructured model uncertainties without excitation of flexible actual dual-drive checkerboard gantry modes.Pritschow et al. [99] demonstrated the performance of feedforward controllers highly depends on the model's accuracy.Even though the feed system's dynamic properties are precisely determined, the parameters change over time.Thus, the open-loop design of this method is one of its most significant disadvantages.
Moreover, active damping was proposed by Trusty et al. [100,101] to eradicate the excitation of the system's flexible mode by the cutting force.Their controller was created with accelerated feedback and incorporated feedforward compensation.In contrast, the simulated results demonstrated the effectiveness of vibration dampening.The research has proved that classical vibration control approaches have advanced recently in DDFS; On the basis of a dynamic model of a cross-arm bearing an unsteady payload, Giams et al. [102] suggested an adaptive tuning strategy for the PID controllers of the master/master design This technique is intended to adjust for unstable load distribution among cross-arm actuators caused by an altering load.Altintas et al. [103] added an input shaping filter to the reference commands of a CNCMT to prevent the retained vibrations.The revised position directives are subsequently supplied to the actual axis control loop to decrease the drive structure's residual vibrations and contour inaccuracies.In the TBSFS, ring shaping, notch filters, and active damping pole placement were introduced [

Modern Vibration Control Methods
Modern vibration control methods for DDFS often employ sophisticated control techniques, sensors, and actuators.In this technology, the controller design considers the flexible modes of the TBS and uses active feedback to ensure robustness.In addition, the closed-loop system's susceptibility to parameter change might be diminished while its bandwidth may be boosted.Many researchers have recently presented several trajectory tracking and vibration control methods for the TBSFS.When taking into account the dynamic variations of the TBS, one of the controllers displayed in DDFS is a multi-input multi-output (MIMO) controller, which allows for adequate results in trajectory tracking and vibration dampening.
The research has proposed the principal method, including predictive model control, sliding mode control, H∞ control, and LPV control methods.
The sliding mode controller featuring vibration compensation was recently developed in DDFS to address the unpredictable synchronous inaccuracy.It implies nonlinear elements such as servo properties' variation and mechanical connection [91,102,107,108].Le-Bao Li et al. [109] integrated adaptive sliding mode control (ASMC) into a ring coupling synchronization control to harmonize the motion of multiple motors so that their speed tracking errors converging to zero requires stabilizing the speed tracking of a single motor.
Po-Huan Chou et al. [110] proposed the cross-coupled intelligent complementary sliding mode control (ICSMC) system for the synchronous control of a twin linear motors servo system based on a digital signal processor (DSP).Yung Chen et al. [111] developed an ASMC controller that uses unknown parameters to manage the 12 DOFs in a dual-axis maglev positioning system; the proposed approaches aim to accomplish precise guiding and positioning.Figure 8 describes the design of the n-motor system's multiple-motor control strategy proposed by Kaiyu Da et al. [112] while developing the universal sliding mode surface in the second layer to stabilize both position tracking and synchronization; The system's uncertainties and nonlinear variables still constrain the classic controllers.However, while emphasizing these problems in the controller design, it is well recommended to implement the adaptive controllers [106].

Modern Vibration Control Methods
Modern vibration control methods for DDFS often employ sophisticated control techniques, sensors, and actuators.In this technology, the controller design considers the flexible modes of the TBS and uses active feedback to ensure robustness.In addition, the closed-loop system's susceptibility to parameter change might be diminished while its bandwidth may be boosted.Many researchers have recently presented several trajectory tracking and vibration control methods for the TBSFS.When taking into account the dynamic variations of the TBS, one of the controllers displayed in DDFS is a multi-input multi-output (MIMO) controller, which allows for adequate results in trajectory tracking and vibration dampening.
The research has proposed the principal method, including predictive model control, sliding mode control, H∞ control, and LPV control methods.
The sliding mode controller featuring vibration compensation was recently developed in DDFS to address the unpredictable synchronous inaccuracy.It implies nonlinear elements such as servo properties' variation and mechanical connection [91,102,107,108].Le-Bao Li et al. [109] integrated adaptive sliding mode control (ASMC) into a ring coupling synchronization control to harmonize the motion of multiple motors so that their speed tracking errors converging to zero requires stabilizing the speed tracking of a single motor.
Po-Huan Chou et al. [110] proposed the cross-coupled intelligent complementary sliding mode control (ICSMC) system for the synchronous control of a twin linear motors servo system based on a digital signal processor (DSP).Yung Chen et al. [111] developed an ASMC controller that uses unknown parameters to manage the 12 DOFs in a dual-axis maglev positioning system; the proposed approaches aim to accomplish precise guiding and positioning.Figure 8 describes the design of the n-motor system's multiple-motor control strategy proposed by Kaiyu Da et al. [112] while developing the universal sliding mode surface in the second layer to stabilize both position tracking and synchronization; The authors also suggested an adaptive synchronization tracking controller based on n hierarchical sliding mode control (HSMC) for multi-axis drive systems with uncertain parameters and external load.This is based on its ability to predict the system's future behavior over a specified time horizon while considering its current state and the expected disturbances.Several researchers have proposed model predictive control (MPC) as a feedback control method; to improve tracking accuracy, smoothness of motion, and energy consumption in DDFS.The authors also suggested an adaptive synchronization tracking controller based on n hierarchical sliding mode control (HSMC) for multi-axis drive systems with uncertain parameters and external load.This is based on its ability to predict the system's future behavior over a specified time horizon while considering its current state and the expected disturbances.Several researchers have proposed model predictive control (MPC) as a feedback control method; to improve tracking accuracy, smoothness of motion, and energy consumption in DDFS.
e , ( ) : the synchronization error of i engine with the external disturbance e : tracking errors, θ , : angular position of the motors ,  , :angular speed, p , ∶ servo valve's load pressure drop, x , : the real angular position of the motor.By addressing MPC online, Kayacan et al. [113] tackle the robustness of model predictive control in mismatched uncertainty, such as disturbance, noise, and parameter fluctuations.For minimizing the ball screw axial vibrations, Dumur et al. [106] adopted a generalized predictive controller and an adaptive loop to fine-tune the control parameters.Nevertheless, the table position effect did not dampen the structure's trembling, and the influence of the variable inertia of the drive system has been examined.By integrating the minimum tracking error filters (MTEFs) for monitoring error in BSFS with structural flexibility, Okwudire et al. [114] effectively dampened vibrations in the ball screw feed system utilizing a model-compensating disturbance adaptive discrete-time sliding mode controller (MCDADSC).The proposed techniques enhance the system's tracking precision and noise immunity.MPC approaches have been demonstrated to enhance the system's monitoring precision and perturbation avoidance capacity.Due to the intricate construction of the filters, their performance could be influenced by dynamic fluctuations induced by the table's motion.
While modeling the BSFS and developing controllers, Symens et al. [115] initially accounted for variations in stiffness due to the time-dependent features and model By addressing MPC online, Kayacan et al. [113] tackle the robustness of model predictive control in mismatched uncertainty, such as disturbance, noise, and parameter fluctuations.For minimizing the ball screw axial vibrations, Dumur et al. [106] adopted a generalized predictive controller and an adaptive loop to fine-tune the control parameters.Nevertheless, the table position effect did not dampen the structure's trembling, and the influence of the variable inertia of the drive system has been examined.By integrating the minimum tracking error filters (MTEFs) for monitoring error in BSFS with structural flexibility, Okwudire et al. [114] effectively dampened vibrations in the ball screw feed system utilizing a model-compensating disturbance adaptive discrete-time sliding mode controller (MCDADSC).The proposed techniques enhance the system's tracking precision and noise immunity.MPC approaches have been demonstrated to enhance the system's monitoring precision and perturbation avoidance capacity.Due to the intricate construction of the filters, their performance could be influenced by dynamic fluctuations induced by the table's motion.
While modeling the BSFS and developing controllers, Symens et al. [115] initially accounted for variations in stiffness due to the time-dependent features and model parameter uncertainty.Rather than using a traditional fixed-parameter controller, which cannot account for the drive system time-varying structural vibration characteristics, they implemented the H∞ robust control approach based on gain scheduling.While utilizing the linear interpolation method and the LPV approach and implementing the HAC-LAC structure in the control, their experimental results demonstrated that linear interpolation yields superior results due to its conservatism and extremely tight assumptions in their controller.The high authority motion controller (HAC) is constructed around the low authority vibration controller (LAC).Van de Wal et al. [116] obtained an improved model from their proposed four-block H∞ control technique, to robust electromechanical moving devices by choosing the proper weighting function.The reliability of such a plan is highly questionable due to factors like the possibility of a mismatched resonance frequency.As a result, it has the potential to be used for positioning tools that generate vibrations.

Other Vibration Control Methods
Recently, the research has identified master/slave (M/S), master/master (M/M), and cross-coupled control designs for DDFS [102].In the M/S design, the cross-arm-impacting actuators are separately managed, and the final product of the master control loop functions as a benchmark for the slave control loop.These described methods are mainly applied to gantry stages like a gantry crane.Path-tracking accuracy is supplementary because the slave motor is perpetually delayed and desynchronized compared to the master motor.The cross-arm actuators are individually controlled in the M/M control scheme and share the same set-point reference.Every autonomous controlling loop is computationally optimized to deliver the highest feasible positioning efficiency.This method of control is utilized in numerous commercial gantry stages to compensate for variations in the engines' dynamics and load [117].Eventually, most academic studies have proposed the crosscoupled control scheme as the optimal control due to the fact it employs the master/master control technique to transport the cross-arm.And the desynchronization within the motors bearing the cross-arm can be tracked and adjusted.It is possible to utilize another controller placed across the two separate control systems to accelerate the slowest and fastest motors, respectively [118].
Since the DDFS follows complex trajectories, motion precision depends on system ability, reliability of tracking across every autonomous axis, and synchronous control [119].Therefore, different servo properties, asymmetrical forces, and mechanical coupling will negatively impact synchronous performance, especially concerning high-speed as well as high-precision machining.In this regard, the cross-coupled control was proposed as the first synchronous approach by Koren et al. [120] to eradicate contour inaccuracy in a 2DOF's feed system.In addition, several researchers presented a cross-coupled synchronous control technique to regulate a surface mounted machine's biaxial gantry structure.Moreover, to enhance the dynamic efficiency of cross-coupled control within DDFS by two PMSMs, the optimal control was investigated through high-speed, high-precision machines [121,122].Nian et al. [123] applied an advanced algorithm, such as robust control for synchronous control.Their research reveals that the proposed controller has excellent dynamic, static, and synchronization performance, can handle load uncertainty, and has robustness for disturbance.
It has been demonstrated that the chattering phenomena combined with mechanical coupling may worsen synchronous error under reversing and abrupt acceleration changes in a DDFS with cross-coupled control.Several studies have advocated the incorporation of the adaptive fuzzy SMC that adopts the tan h function in place of the signum process to overcome these challenges and obtain higher performance.So far, the research has successfully suggested that fuzzy logic control (FLC) is an efficient solution for parametervarying plants [124].In addition, a digital signal processor (DSP), and the control system, were used to build an adaptive and robust control approach with a disturbance observer for a motor tracking control application [125,126].Numerous researchers have been motivated to combine these control strategies to highlight the evident benefits of sliding mode control (SMC) and FLC in nonlinear systems [127][128][129].The control method incorporating FLC and SMC is intended to mitigate the drawbacks of the sliding mode yet preserve the guarantees of uniform global stability.Furthermore, research has demonstrated that increasing the bandwidth and enhancing the single axis disturbance rejection is insufficient for achieving perfect synchronization in a DDFS, as shown in Figure 9, to overcome synchronous errors produced by mechanically connected disturbances between two axes.Finally, Lu et al. [93] suggested the new cross-coupled fuzzy logic sliding mode control for control, which can be applied in DDFS.
Symmetry 2023, 15, x FOR PEER REVIEW 17 of 23 motivated to combine these control strategies to highlight the evident benefits of sliding mode control (SMC) and FLC in nonlinear systems [127][128][129].The control method incorporating FLC and SMC is intended to mitigate the drawbacks of the sliding mode yet preserve the guarantees of uniform global stability.Furthermore, research has demonstrated that increasing the bandwidth and enhancing the single axis disturbance rejection is insufficient for achieving perfect synchronization in a DDFS, as shown in Figure 9, to overcome synchronous errors produced by mechanically connected disturbances between two axes.Finally, Lu et al. [93] suggested the new cross-coupled fuzzy logic sliding mode control for control, which can be applied in DDFS.

Conclusions
Since vibration affects machining accuracy, high-precision, and high-speed CNCMT has stricter requirements for the feed drive system.This study investigates the vibration analysis and control methods of DDFS for CNCMT applications.The findings of this research can be summarized as follows: (1) Following the number of motors and the dynamic behaviors of dual-and singlydriven feed systems, their asymmetric characteristics are extensively discussed throughout section two of this work.Herein, it examines the positive aspects of DDFS in terms of machining accuracy and manufacturing productivity in contrast with single-drive feed systems.However, it was found that DDFS features high precision and stability over workspace ratio and good dynamic performance compared to the single drive feed system.
(2) Various modelling techniques, like the lumped parameter, finite element, and hybrid approaches have been reviewed for comparison purposes.However, the findings of this study indicated that lumped parameter models are suitable for quick assessments, the FE method is adequate for thorough and accurate analyses, and hybrid approaches are optimal for circumstances requiring a trade-off between speed and accuracy.
(3) By describing the primary steps of the TBS parameter identification processes to assess and pinpoint the essential CNCMT dynamic characteristics, it turns out that the process of PI in a DDFS is often accomplished by a combination of empirical inquiry, quantitative measurement, and mathematical modeling.The procedure may further involve the utilization of calibration equipment, the analysis of data, and the utilization of software tools to enhance the system's features with the aim of attaining greater performance, accuracy, and reliability.

Conclusions
Since vibration affects machining accuracy, high-precision, and high-speed CNCMT has stricter requirements for the feed drive system.This study investigates the vibration analysis and control methods of DDFS for CNCMT applications.The findings of this research can be summarized as follows: (1) Following the number of motors and the dynamic behaviors of dual-and singlydriven feed systems, their asymmetric characteristics are extensively discussed throughout section two of this work.Herein, it examines the positive aspects of DDFS in terms of machining accuracy and manufacturing productivity in contrast with single-drive feed systems.However, it was found that DDFS features high precision and stability over workspace ratio and good dynamic performance compared to the single drive feed system.
(2) Various modelling techniques, like the lumped parameter, finite element, and hybrid approaches have been reviewed for comparison purposes.However, the findings of this study indicated that lumped parameter models are suitable for quick assessments, the FE method is adequate for thorough and accurate analyses, and hybrid approaches are optimal for circumstances requiring a trade-off between speed and accuracy.
(3) By describing the primary steps of the TBS parameter identification processes to assess and pinpoint the essential CNCMT dynamic characteristics, it turns out that the process of PI in a DDFS is often accomplished by a combination of empirical inquiry, quantitative measurement, and mathematical modeling.The procedure may further involve the utilization of calibration equipment, the analysis of data, and the utilization of software tools to enhance the system's features with the aim of attaining greater performance, accuracy, and reliability.
(4) Based on previous research, the associated studies' prospects of the vibration control method for TBS were comprehensively described in Section 5 through three essential validation methods, including traditional, modern, and other techniques.From the presented methods, it was revealed that even though a great deal of effort has been put into finding ways to reduce the vibration of DDFS, some critical points, particularly synchronizing error (the most significant obstacles in the DDFS' structure), still need to be developed in future work.
Furthermore, since synchronizing error remains a significant challenge in DDFS, the future work of this article will emphasize developing the methods, technologies or innovative techniques with the objective of minimizing the synchronizing errors.In other words, we are targeting the development of control algorithms involving advanced technologies and dynamic characteristics analysis to enhance the mechanical design towards precision and stability of DDFS systems, thus, mitigating vibrations and maintaining machining accuracy.

Symmetry 2023 ,
15, x FOR PEER REVIEW 13 of 23 effectiveness of FMISZV vibration dampening to that of conventional input shapers, including zero vibration with derivation (ZVD) as well as traditional linear deceleration or acceleration interpolators (CLAI).FMISZV was preferred compared to the ZVD and CLAI techniques in their studies.(a) (b)

Figure 6 .
Figure 6.Synchronous vibration control diagram (a) in the high-speed region (b) in the low-speed region [94].

Figure 8 .
Figure 8.Control strategy design for the multi-axis drive system [112].

Figure 8 .
Figure 8.Control strategy design for the multi-axis drive system [112].

Table 2 .
The applications, benefits and drawbacks of FE methods.
,   :   shaft's nut left and light side equivalent stiffness.  ,   : X1 total damping, as mentioned above.  ,   and   ,   stand for total stiffness and damping on the left and right sides of the   shaft's nut, successively.  ,   and   ,   represent the joint stiffness between screw and nut with the shaft's damping along the   /  axes, successively.