Research on Equivalent Simplified Modeling and Simulation of Digital Hydraulic Cylinder

Research on Equivalent Simplified Modeling and Simulation of Digital Hydraulic Cylinder | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Research on Equivalent Simplified Modeling and Simulation of Digital Hydraulic Cylinder Changlin Ma, Lin Hao, Yunguang Gao This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7215125/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 18 Dec, 2025 Read the published version in Scientific Reports → Version 1 posted 11 You are reading this latest preprint version Abstract To address the challenges in traditional modeling methods for highly integrated electro-mechanical-hydraulic digital hydraulic cylinders, such as complex nonlinear factors, high parameter uncertainty, and low simulation efficiency, this study proposes two equivalent simplified modeling and simulation methods based on the negative feedback mechanism. Firstly, a valve port control model based on equivalent feedback of displacement signals is established by constructing a closed-loop feedback mechanism between piston displacement and spool displacement. Secondly, a mathematical mapping between the valve opening and the relative displacement of the spool and sleeve is formed by utilizing the linkage following relationship between the valve sleeve and spool, resulting in a modeling method based on equivalent feedback of sleeve following. Two equivalent models of the digital hydraulic cylinder are developed using the AMESim platform, and the correctness and basic equivalence of the model were verified through simulation and performance testing. Based on the equivalent model, the influence of key parameters on the dynamic characteristics of the system can be analyzed. This can provide a relatively simplified modeling method for the complex engineering application system analysis of digital hydraulic cylinders. The proposed negative feedback equivalent modeling approach offers a referable approach for facilitating modeling and simulation implementation in other systems. Physical sciences/Engineering Physical sciences/Mathematics and computing Digital Hydraulic Cylinder Simplify Equivalent Model Modeling and Simulation Negative Feedback Mechanism AMESim Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 Figure 21 Figure 22 Figure 23 0. Preface A digital hydraulic cylinder combines a hydraulic cylinder, a hydraulic valve, a feedback mechanism, and control components. It is a type of single hydraulic component of the system engineering level with built-in closed- and open-loops. It can convert a pulse into a precise power drive and achieve micron-level control precision. It successfully combines hydraulic technology with digital technology to realize digital, long-distance, and intelligent control of hydraulic transmission. It has the advantages of high control precision, a simple structure, and convenient control. This is a typical example of digital hydraulic technology [ 1 ], and its engineering applications are becoming increasingly extensive. The digital hydraulic cylinder specializes in fluid collection, machinery, electrical, automation, and so on. It exhibits the characteristics of high electromechanical and hydraulic integration. Its performance is affected by the response lag of the hydraulic system, the dead zone, mechanical clearance, oil leakage, temperature, load, and other factors. Therefore, in the performance analysis of a digital hydraulic cylinder and its engineering application system, modeling and simulation are necessary auxiliary methods. By establishing a reasonable mathematical model, the simulation results were obtained, and the structural parameters were optimized to improve the working performance of the digital hydraulic cylinder. Several scholars have conducted extensive research in this regard. Jia [ 2 ] and Changlin [ 3 ] deduced the transfer function of a valve-controlled asymmetric cylinder and established a general transfer function model for a digital hydraulic cylinder. Li [ 4 ], Jia [ 5 ], Zhiquan [ 6 ], and Shouling [ 7 ] analyzed all types of nonlinear factors and obtained a complete nonlinear state-space model of a digital hydraulic cylinder. Wang [ 8 ] and Jiang [ 9 ] established the AMESim model of an internal indirect feedback digital hydraulic cylinder, including a ball screw and feedback nut, and conducted simulation research. Qing [ 10 ], XIA [ 11 ], and Panguo [ 12 ] used the HCD library model and screw nut mechanism model in AMESim to establish the AMESim simulation model. These models were established based on the specific needs of this research. The models are relatively complex, and there are some difficulties in parameter setting and simulation solutions. For example, it is difficult to determine the comprehensive moment of inertia, comprehensive viscous damping coefficient, frequency, and other parameters of the slide valve in the transfer-function model. The nonlinear state equation contains a set of differential equations for 11 state variables, and some boundary conditions must be considered when solving it. However, this is difficult to achieve. It is relatively easy to model relying on the professional simulation software AMESim; however, the model contains a screw nut or ball screw model, which contains nonlinear characteristics such as friction and damping. For complex systems, the solution time is too long. Based on the above analysis, although extensive research has been conducted on the modeling and simulation of digital hydraulic cylinders, existing models generally suffer from high complexity, difficulty in parameter determination, and low solving efficiency, making it challenging to meet the demands of rapid and efficient analysis in practical engineering applications. Particularly for performance optimization and real-time control of digital hydraulic cylinder application systems, there is an urgent need for a simplified modeling approach that can accurately reflect system characteristics while being easy to implement. Therefore, this paper aims to propose an equivalent simplified modeling method. Based on the working mechanism of servo feedback in digital hydraulic cylinders, a model incorporating signal negative feedback and valve sleeve follow-up equivalence is established. This model is implemented using AMESim software, and the results are validated through a digital hydraulic cylinder performance test bench. The proposed method aims to provide a relatively simplified modeling approach for the design optimization, performance enhancement, and engineering application of digital hydraulic cylinders. 1. Typical structure and principle of digital hydraulic cylinder As shown in Figure 1, it is a typical structure of a digital hydraulic cylinder that is mainly composed of a stepper motor, three-way slide valve, feedback nut, screw, piston, piston rod, and cylinder [1]. The piston rod is a hollow structure that provides space for screw movement. The feedback nut and piston are fixed together. Generally, screws and spools are integrated. The output shaft end of the stepper motor was connected to the hydraulic spool through the connecting sleeve. The inner wall of the connecting sleeve has an axial chute that cooperates with the spool and the shaft end. In this manner, the spool and stepper motor output shaft can rotate synchronously and slide axially. The working principle of this digital hydraulic cylinder is as follows: After inputting a certain number of pulses to the stepper motor, the motor shaft outputs an angular displacement and drives the spool of the three-way slide valve to rotate at the same time. Owing to the action of the screw and feedback nut, the spool moves axially while rotating, so that the valve port is opened so that the oil inlet or oil return path is connected to the rodless cavity. The positive and negative steering of the stepper motor determines whether the rodless cavity is connected to the oil inlet or return path. Specifically, when the spool moved to the right, the oil inlet was connected to the rodless cavity, and the piston rod extended outward. Simultaneously, the feedback nut drives the spool to return to the equilibrium position after the motor stops rotating. When the spool moves to the left, the return oil circuit is connected to the rodless cavity and the piston rod retracts. Similarly, the feedback nut drives the spool to return to the equilibrium position after the motor stops rotating [13]. Therefore, the number of input stepper motor pulses determines the displacement of the hydraulic cylinder piston rod and the frequency of the pulse corresponds to the speed of the piston rod. From the above principle analysis, it can be seen that the digital hydraulic cylinder of this structure realizes internal mechanical feedback through the screw pair between the screw and feedback nut. The control block diagram is shown in Figure 2 It can be seen from the Figure that the working principle of the digital hydraulic cylinder can be summarized as a three-way slide valve to control a differential cylinder. The screw was used to provide precise feedback on the piston position, and the two hydraulic and screw technologies were skillfully used to achieve both a large output force and precise position accuracy. 2. Analysis of equivalent simplified modeling ideas for digital hydraulic cylinder Based on the above analysis, it is clear that when the digital hydraulic cylinder is working, the stepper motor drives the spool of the hydraulic valve to move, and the hydraulic valve outputs hydraulic oil to drive the piston and load of the hydraulic cylinder to move, which is the forward channel of the hydraulic feedback control. The load, executive hydraulic cylinder, and control slide valve constitute the control structure unit, which is called the valve-controlled cylinder hydraulic power unit. Each tiny valve opening generated by the movement of the control valve core causes the hydraulic cylinder to close the valve port to the corresponding reaction movement. Therefore, the control valve mainly works near the median or its working valve port is relatively small. In the structure of an actual digital hydraulic cylinder, the opening of the valve port is the result of the compound motion of the valve core and the mechanical feedback mechanism in space. As shown in Figure 3, under the action of the stepper motor drive and screw/nut pair, the valve core rotates to produce axial displacement, the valve port opens, and the pressure oil drives the piston to move. The piston moves in the opposite direction to follow the action of the valve core, weakening or counteracting the movement effect of the valve core, thus forming negative feedback. It can be said that the generation of negative feedback is established by the screw pair between the piston and the valve core and the screw. In other words, the stepper motor drove the rotation of the valve core. This rotation was converted into the input displacement x m of the slide valve using the screw pair. Once the valve port opens, the displacement xp of the oil cylinder piston is fed back through the large lead screw and screw nut, which becomes the feedback displacement x f of the slide valve core. Ultimately, the input displacement xm and feedback displacement x f of the valve core are combined to form the absolute displacement (i.e., the valve port opening) x v . So there are where: P the nut pitch, σ the feedback coefficient for the single-stage spiral feedback structure σ =1 for the double-stage spiral feedback structure σ = P / S ( S is the lead of the lead screw). When the actual digital hydraulic cylinder is operating, the opening amount of the valve port is the difference between the active displacement of the valve core and the feedback displacement of the piston/screw. When using professional software for modeling and simulation, if the software is convenient for building a detailed simulation model based on the actual physical model, the opening amount of the valve port can also be equivalently processed when simplifying the modeling. There are two equivalent treatment ideas. First, the real-time signal of the piston displacement is compared with the displacement signal of the valve core driven by the stepping motor using signal feedback. This difference controls the action of the valve core, that is, the opening amount of the valve. The control block diagram is shown in Figure 4, which shows the modeling and simulation method of a digital hydraulic cylinder based on the equivalent feedback of the displacement signal. Second, the piston displacement is directly applied to the valve sleeve, and a position follow-up relationship is formed between the valve sleeve and the valve core, so that the valve opening amount is equal to the displacement difference between the valve core and valve sleeve, so that it is equivalent to the original structure. The control block diagram shown in Figure 5 can be used to simulate the digital cylinder mechanism. In fact, it is similar to the composition of the common mechanical and hydraulic servo mechanism, which is called the digital hydraulic cylinder modeling and simulation method, based on the equivalent feedback of the valve sleeve follow-up. 3. Equivalent simplified model of digital hydraulic cylinder 3.1 Model of main components of digital hydraulic cylinder The hydraulic cylinder, control valve, stepper motor, and other components of the digital hydraulic cylinder can be directly modeled by the hydraulic module in the AMESim software. For the modules that are not in the hydraulic library or existing modules but cannot meet the requirements of accurate modeling, the hydraulic component design library ( hereinafter referred to as the HCD library ), mechanical library ( Mechanical ) and signal library ( Signal, Control ) can be used to model these components. 3.1.1 Control Valve Model The HCD library of AMESim software is a basic element composed of basic geometric structural units. It adopts a modular modeling method based on the geometric structure of the hydraulic components and can establish a detailed component model considering the dynamic performance of moving bodies, fluid compressibility, friction, leakage, hydraulic power, and other factors. It is used to construct various hydraulic components according to geometric shapes and physical characteristics. The library is suitable for modeling and analyzing the dynamic characteristics of nonstandard hydraulic components [14]. The HCD library provides a variety of types of spool valve components, including components with ring grooves, round hole grooves, grooves, and custom slotted spool valves. Table 1 lists information on some spool valve components. In actual modeling, the appropriate component sub-model is selected according to the structure of the valve. The control valve was the core component of the digital hydraulic cylinder. According to the actual structure, it can be composed of BAO011 and BAO012 modules in the HCD library and a mass block in the mechanical library, as shown in Figure 6. BAO012 differs from BAO011 only because the variables associated with ports 3 and 4 are interchanged. The established control valve model is shown in Figure 7, where the right mass block, MECMAS21, represents the spool. The stroke limit was set at both ends of the mass block such that the maximum displacement of the valve core could be controlled. The valve core adopted a cylindrical section, according to the actual situation. The diameter parameters of the piston and piston rod could be modified according to the three-way valve in the BAO012 and BAO011 modules. MECMAS21 represents the one-dimensional motion of a two-port mass under the action of two external forces in terms of N, weight, and frictional forces. 3.1.2 Hydraulic cylinder model The 'hydraulic cylinder' in the digital hydraulic cylinder is generally a single-rod piston cylinder. The HJ020 and HJ000 models listed in Table 2 can be used in the AEMSim software hydraulic library. The main difference between the two models was the mass load. In addition, the HCD library components can also be used to build hydraulic cylinder models. The optional models are BAP1 and BAP2, as shown in Table 3, each of which contains the BAP11 and BAP12 submodels. The difference between the two sub-models is that the variables defined by ports 2 and 3 are in opposite directions, as shown in Figure 8. The hydraulic cylinder model constructed using the HCD module library is illustrated in Figure 9. The rod cavity and rodless cavity of the hydraulic cylinder composed of BAP12, BAF01, and BAP11 were the same by default. If the action areas of the two cavities are not the same, they can be realized by setting different piston-rod diameters. 3.1.3 Stepping Motor Model For the modeling and simulation of stepping motor in digital hydraulic cylinder, different models or equivalent models can be selected according to the simulation purpose, and there are many ways. In general, the focus of modeling is the analysis of mechanical and fluid parts. The stepper motor can use the signal to drive the spool directly or select the transfer function module in the signal library. This model is illustrated in Figure 10. The left module is the input pulse, and the pulse frequency can be modified by setting the slope such that the pulse quantity is determined when the simulation time is determined. 3.2 Construction of a Digital Hydraulic Cylinder System Model Based on Equivalent Signal Feedback According to the principle block diagram shown in Figure 4, the model of each component of the digital hydraulic cylinder was selected and combined according to the mechanism. The feedback mechanism adopts the displacement signal feedback method, and the sub-model in the signal library is applied to the negative feedback equivalent modeling. The AMESim model of the digital hydraulic cylinder system ( AMESim16.0 ), based on the equivalent feedback of the displacement signal, was obtained. As shown in Figure 11, the real line represents the connected hydraulic pipeline, where the rough and real lines represent the characteristics of the length, diameter, and complex flow state of the hydraulic pipeline, and the imaginary line represents the connection relationship between the signals. For the rotary motion of the valve core driven by the stepping motor, linear displacement of the valve core occurs under the side effect of the screw nut. The feedback signal can be selected as the displacement or speed signal. The displacement signal feedback is shown in Figure 11, and the displacement signal was collected by the displacement sensor at the piston output end. The signal conversion module and its interface definitions used in Figure 11 are listed in Table 4. XVLC01 converts the dimensionless signal input at port 1 into a linear displacement of the same value in meters, and the linear displacement is the output at port 2 in meters per second. The speed is obtained by approximate differentiation of the angle using the first-order lag method of the time constant provided. The initial speed can be set by the user or zero by default. The VELC02 model accepts the dimensionless signal input at port 1 and converts it into velocity (m/s) and displacement (m) at port 2. Displacement is a state variable obtained by velocity integration. If the speed feedback signal is used, the piston output end is replaced by a speed sensor and the drive input of the control valve is changed to a speed signal. As shown in Figure 12, the corresponding signal source can be set at a constant speed. 3.3 Construction of a Digital Hydraulic Cylinder System Model Based on Valve Sleeve Servo Feedback Equivalence According to the principle block diagram of the digital hydraulic cylinder modeling method based on the valve sleeve servo feedback equivalent shown in Figure 5, the modeling was performed on the AMESim software platform. The movable spool valve parts of the valve sleeve are provided in the HCD library. Table 5 presents several typical models. The main difference is that the valve-port shapes differ. For example, the BRO011 / BRO012 model was used. The difference between the two model interface definitions is that ports 3 and 6 are reciprocal, as shown in Figure 13. When using the valve sleeve movable spool valve component to establish the spool valve model, attention was paid to the interface connection between the components. In general, it is necessary to increase the MAS011RT module to simulate the force between the spool and the valve sleeve. The connections between BRO011 and BRO012, BRO011, BRO012, MAS011RT, BRO012, and VELC02 conversion sub-models, and FVXSG1 are shown in Figure 14. The model of the hydraulic oil source and hydraulic cylinder was the same as that shown in Figure 11. By adding the connection between the piston output and valve sleeve, a complete model of the digital hydraulic cylinder system based on the equivalent feedback of the valve sleeve can be established, as shown in Figure 15. Two conversion sub-modules are added to the diagram to increase the scale factor k at ports v and x. When k is 1, the model is equivalent to a digital hydraulic cylinder with single-stage spiral feedback. When the k value is ta/tb ( ta is the pitch of the spool screw, tb is the pitch of the ball screw ), the model is equivalent to a digital hydraulic cylinder structure with double-stage spiral feedback. 3.4 Model Parameter Settings The relevant parameters of the model were set. The structural parameters, such as the hydraulic cylinder and control valve, were set according to the actual parameters of the research object. The initial balance parameters were determined using static calculations. Other parameters can be determined based on the technical manual or empirical data of the component. Table 6 lists the values of the simulation parameters determined by the system. Table 6. Relevant parameters of digital hydraulic cylinder simulation Component name Parameter name Numerical value Unit Hydraulic cylinder Cylinder inner diameter 160 mm Piston rod outer diameter 100 mm Cylinder stroke 220 mm Screw External diameter 20 mm Pitch 3 mm Slide valve Valve core diameter 18 mm Groove diameter 16 mm Shoulder width 5 mm 4. Simulation and test result analysis For the AMESim simulation model of the digital hydraulic cylinder constructed above, the input signal and simulation parameters are set, and the simulation can be performed to obtain the dynamic characteristics of the digital hydraulic cylinder. By setting different values for the same variable of the digital hydraulic cylinder, such as different working pressures, external loads, and stepper motor frequencies, the values of other variables were not changed, and the influence of this variable on the dynamic characteristics was analyzed. The simulation results were compared with the test results to verify the accuracy and equivalence of the simulation model. 4.1 Performance test device The performance test device for the digital hydraulic cylinder is shown in Figure 16. It is mainly composed of a hydraulic pump station, digital hydraulic cylinder, loading hydraulic cylinder, constant pressure valve block, loading valve block, sensors, electrical control cabinet,measurement and control host,and software. The constant pressure valve block provides constant oil supply pressure to the digital hydraulic cylinder, and the loading valve block controls the output tension or pressure of the loading hydraulic cylinder to load the digital hydraulic cylinder. The measurement and control signals communicate with an industrial computer through an electric control cabinet. Based on LabVIEW, a human-computer interaction interface was developed to realize the functions of the hydraulic system state display, loading force setting and control, constant pressure source setting and control, digital hydraulic cylinder action setting and control, and data display and storage. 4.2 Working characteristics of digital cylinder under different working pressures The pressure of the loading cylinder was set to 3Mpa as the external load, the frequency of the stepping motor was 400 Hz, and the number of input pulses was 240, which corresponded to a theoretical displacement of 3 mm. The working pressures of the digital cylinder were 5MPa, 7MPa and 9MPa respectively, and the simulation was carried out based on the equivalent model. The expected piston displacement increased from 0 mm to 3 mm and then decreased from 3 mm to 0 mm. The corresponding input-signal curves are shown in Figure 17. The simulation curves of the digital cylinder displacement under different working pressures are shown in Figure 18 and 19, respectively. It can be seen from the diagram that under the same external load, the piston extension process is less affected by the working pressure, whereas the piston retraction process is obviously affected by the working pressure. The higher the working pressure, the faster is the response speed of the digital hydraulic cylinder. Under the same conditions, the test bench was used to test the positioning error of the digital hydraulic cylinder under different working pressures, and the results were compared with the simulation results of the two models. As shown in Table 7, 'Simulation 1' represents the simulation results based on the signal feedback equivalent model, 'Simulation 2' represents the simulation results based on the valve sleeve servo equivalent model, and 'Simulation 3' represents the simulation results based on the model in Reference [12]. It can be seen that the simulation results of the two methods proposed in this paper are consistent, the motion displacement errors were all smaller than those obtained from the simulation results based on the model in Reference [12], and all showed close agreement with the experimental results. With an increase in the working pressure, the positioning accuracy of the digital cylinder increased. In general, the positioning accuracy of the piston rod was higher than that of retraction. Table 7. Simulation and experimental results of displacement under different working pressures Working pressure 3mm retraction displacement error (mm) 3mm extension displacement error (mm) Simulation 1 Simulation 2 Simulation 3 Test Simulation 1 Simulation 2 Simulation 3 Test 5 MPa 0.00034 0.00034 0.0014 0.001 0.0039 0.0039 0.0041 0.004 7 MPa 0.0007 0.0007 0.0009 0.001 0.0033 0.0033 0.0031 0.003 9 MPa 0.0005 0.0005 0.0002 0.001 0.002 0.002 0.0025 0.002 4.3 Working characteristics of digital cylinder under different external loads The working pressure of the digital hydraulic cylinder was set to 9 Mpa, and the frequency of the stepping motor was 400 Hz. The pressures of the loading chamber of the given loading cylinder were 3, 5, and 7 MPa. The number of input pulses was 240, which corresponded to a theoretical displacement of 3 mm. The expected piston displacement increased from 0 mm to 3 mm and then decreased from 3 mm to 0 mm. The corresponding input-signal curves are shown in Figure 17. The simulation results show that the displacement curves of different models of digital cylinders under different external loads are as shown in Figure 20 and 21. It can be observed that under the same working pressure, the piston extension process is less affected by the load pressure. The piston retraction process is significantly affected by the load. The smaller the load, the faster is the response speed of the digital hydraulic cylinder. Under the same conditions, the test bench was used to test and extract the positioning error of the digital cylinder under different external loads, and the results were compared with the simulation results. As shown in Table 8, it can be seen that the simulation results of the two methods proposed in this paper are consistent. The motion displacement errors were all smaller than those obtained from the simulation results based on the model in Reference [12], and all showed close agreement with the experimental results. At the same working pressure, the smaller the load, the higher the positioning accuracy of the digital cylinder. The positioning accuracy when the piston rod is extended is higher than that when the piston rod is retracted. Table 8. Simulation and experimental results of displacement under different load pressures Load Pressure 3mm retraction displacement error (mm) 3mm extension displacement error (mm) Simulation 1 Simulation 2 Simulation 3 Test Simulation 1 Simulation 2 Simulation 3 Test 3MPa 0.0005 0.0005 0.0002 0.001 0.002 0.002 0.0025 0.003 5MPa 0.0002 0.0002 0.0012 0.002 0.003 0.003 0.0048 0.005 7MPa 0.0003 0.0003 0.0015 0.003 0.005 0.005 0.0053 0.006 4.4 Working characteristics of the digital hydraulic cylinder under different stepper motor frequencies The working pressure of the digital cylinder was set at 5 Mpa. The pressure in the loading chamber of the loading cylinder was 3 MPa. The frequencies of the stepping motor were 200 Hz, 400 Hz, and 600 Hz. The number of input pulses was 240, which corresponds to a theoretical displacement of 3 mm. The expected piston displacement increased from 0 mm to 3 mm and then decreased from 3 mm to 0 mm. The corresponding input-signal curves are shown in Figure 17. The displacement curves of the different digital cylinder models under different pulse frequencies are shown in Figure 22 and 23. It can be observed that under the same load and working pressure, the piston extension process is less affected by the stepping motor frequency. The piston retraction process was affected by the frequency of the stepping motor. The higher the frequency of the stepping motor, the faster is the response speed of the digital hydraulic cylinder. Under the same conditions, the test bench was used to test and extract the positioning error of the digital cylinder under different pulse frequencies, and the results were compared with the simulation results. As shown in Table 9, it can be seen that the simulation results of the two methods proposed in this paper are consistent. The motion displacement errors were all smaller than those obtained from the simulation results based on the model in Reference [12], and all showed close agreement with the experimental results. The lower the pulse frequency of the stepper motor, the higher the positioning accuracy of the digital cylinder. The positioning accuracy when the piston rod was extended was higher than that when it was retracted. Table 9. Simulation and experimental results of displacement at different pulse frequencies Pulse frequency 3mm retraction displacement error (mm) 3mm extension displacement error (mm) Simulation 1 Simulation 2 Simulation 3 Test Simulation 1 Simulation 2 Simulation 3 Test 200Hz 0.0016 0.0016 0.0009 0.001 0.0024 0.0024 0.0027 0.003 400Hz 0.00034 0.00034 0.0014 0.001 0.0039 0.0039 0.0041 0.004 600Hz 0.0004 0.0004 0.0016 0.002 0.004 0.004 0.0045 0.005 5. Conclusion and Discussion Based on the mechanical feedback mechanism of digital hydraulic cylinders, two methods for constructing simplified equivalent models have been proposed. One was based on the equivalent feedback of the displacement signals, and the other was based on the equivalent feedback of the valve sleeve follow-up. Two equivalent models of digital hydraulic cylinders were established based on AMESim and were verified by simulation and performance tests. The following conclusions were drawn: The simulation and experimental results verify the accuracy of the two models. These two ideas are based on the mechanism of negative feedback and are essentially equivalent. Under the same external load and stepping motor frequency, the higher the working pressure, the faster the response speed of the digital hydraulic cylinder and the higher the positioning accuracy, and the piston extension process is less affected by the working pressure. Under the same working pressure and stepping motor frequency, the smaller the load, the faster the response speed of the digital hydraulic cylinder and the higher the positioning accuracy, and the piston extension process is less affected by the load pressure. Under the same load and working pressure, the higher the frequency of the stepper motor, the faster is the response speed of the digital hydraulic cylinder. The piston extension process is less affected by the frequency of the stepper motor, while the piston retraction process is significantly affected by the frequency of the stepper motor. The positioning accuracy of the digital hydraulic cylinder piston rod extension action exceeded that of the retraction action. Compared with traditional modeling methods, the approach proposed in this paper achieves a certain degree of simplification in modeling digital hydraulic cylinders. It provides an efficient and practical modeling solution for digital hydraulic cylinder application systems, especially for complex engineering systems involving multiple hydraulic cylinders, thereby greatly reducing the implementation difficulty of simulating complex systems. Furthermore, the simplified modeling approach based on negative feedback equivalence adopted in this paper is not only applicable to digital hydraulic cylinders but also provides reference value for the modeling and simulation of other complex electromechanical-hydraulic systems. The research presented in this paper primarily proposes two equivalent simplified modeling approaches based on negative feedback mechanisms, while theoretical analysis remains relatively limited. In the experimental testing section, the measurement accuracy is constrained by sensor precision and requires further improvement.Several aspects warrant additional investigation: （1）The influence of mechanical friction and hydraulic fluid viscosity on positioning accuracy in the model requires further study. （2）Theoretical analysis of positioning accuracy variations under different loads, working pressures, and stepper motor frequencies for digital hydraulic cylinders needs more comprehensive discussion. These identified areas present valuable directions for future research on digital hydraulic cylinders and their engineering applications: Enhanced theoretical modeling of multi-factor interactions Improved experimental measurement methodologies Systematic investigation of performance under various operational conditions Declarations Funding This work is supported by the Natural Science Basic Research Program of Shaanxi Province, China (Grant No. 2025JC-YBQN-686). Author Contribution Principle analysis of digital hydraulic cylinder, Changlin Ma and Yunguang Gao, Mod-eling Method and Implementation of Digital Hydraulic Cylinder, Changlin Ma and Lin Hao, Digital hydraulic cylinder simulation, Changlin Ma. Data Availability The data that support the findings of this study are available from the corresponding author upon reasonable request. References Ma Changlin, Li Feng, Gao Yunguang, et al. Modeling, Simulation and Test of Digital Hydraulic Cylinder [M]. Xi'an: Xi'an University of Electronic Science and Technology Press, 2020.09. Chen Jia, Xing Jifeng, Peng Likun. Modeling and Analysis of Digital Hydraulic Cylinder Based on Transfer Function [J]. China Mechanical Engineering，2014,25(1):65-70. 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Mechanism Analysis and Control System Research of Digital Hydraulic Cylinder [J].Machine Tool and Hydraulics,2020,48(24):8-12. Liang Quan, Su Qiying. AMESim computer simulation guide for hydraulic system [M]. Beijing: China Machine Press, 2014. Tables Tables 1 to 5 are available in the Supplementary Files section. Additional Declarations No competing interests reported. Supplementary Files Table1to5.docx Cite Share Download PDF Status: Published Journal Publication published 18 Dec, 2025 Read the published version in Scientific Reports → Version 1 posted Editorial decision: Revision requested 29 Aug, 2025 Reviews received at journal 28 Aug, 2025 Reviews received at journal 25 Aug, 2025 Reviewers agreed at journal 17 Aug, 2025 Reviewers agreed at journal 16 Aug, 2025 Reviewers agreed at journal 11 Aug, 2025 Reviewers invited by journal 11 Aug, 2025 Editor assigned by journal 11 Aug, 2025 Editor invited by journal 07 Aug, 2025 Submission checks completed at journal 04 Aug, 2025 First submitted to journal 03 Aug, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7215125","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":501653419,"identity":"a8fc978b-5934-412c-813c-ad7255224af1","order_by":0,"name":"Changlin Ma","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA4ElEQVRIiWNgGAWjYHACxgMJDAxybMwMiQ8SKmyI0wPSYszP3vDY4MGZNCK1AHHizJ6DzyQfth0irFy3/fCDAw931DJuuJGcVpHAdoCBv707Aa8WszNpBgcSzxxnNriRlnYjgecOg8SZsxvwazmQw3Agse0Ym8GNHKAWiWcMBhK5BLScfwPWwmNwI/9bQYLBYSK03ADbUiMh2XMgjSEhgSgtz4B+aTtgAAzkZImEA2k8hP1yPvnhw59tdfVtwKj8+POfjRx/ey9+LVBwGM7iIUY5CNQRq3AUjIJRMApGIgAAR4ZWf0AXp7wAAAAASUVORK5CYII=","orcid":"","institution":"Rocket Force Uinversity of Engineering","correspondingAuthor":true,"prefix":"","firstName":"Changlin","middleName":"","lastName":"Ma","suffix":""},{"id":501653420,"identity":"97d0da93-79dd-4008-b473-b7131045efd4","order_by":1,"name":"Lin Hao","email":"","orcid":"","institution":"Rocket Force Uinversity of 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1","display":"","copyAsset":false,"role":"figure","size":145320,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of digital hydraulic cylinder\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/d17c7bfc17c6814d729f1cec.png"},{"id":89371737,"identity":"33f80698-63b8-438a-a06a-088dfbe8ccf0","added_by":"auto","created_at":"2025-08-19 10:17:24","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":52889,"visible":true,"origin":"","legend":"\u003cp\u003eClosed loop control principle block diagram of digital hydraulic cylinder\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/f6391a5ab5ee03191c4bb43e.png"},{"id":89372638,"identity":"13c3d159-c17d-4ef6-8a48-62c7f6797088","added_by":"auto","created_at":"2025-08-19 10:25:24","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":107921,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of mechanical feedback principle of digital hydraulic cylinder\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/6c24a15be5d31d86f41d9179.png"},{"id":89371270,"identity":"6e704e8e-8f53-4882-af70-ef8d87a26dc2","added_by":"auto","created_at":"2025-08-19 10:09:24","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":56842,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic block diagram of digital hydraulic cylinder modeling based on displacement signal feedback equivalence\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/bcc69d2de46f0b5ecfa7bc34.png"},{"id":89371739,"identity":"51770dc3-dad2-49d9-b43e-ca02b95f3613","added_by":"auto","created_at":"2025-08-19 10:17:24","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":52186,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic block diagram of digital hydraulic cylinder modeling based on valve sleeve servo feedback equivalence\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/c916cf2cd782af0fbd65fcb0.png"},{"id":89371743,"identity":"29425df1-d4f2-4d20-887b-d60a9224e879","added_by":"auto","created_at":"2025-08-19 10:17:24","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":77061,"visible":true,"origin":"","legend":"\u003cp\u003eThe External variables of BAO011and BAO012\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/9c36ee8a4bad7e144b8b5248.png"},{"id":89372649,"identity":"1ae0c323-7dc6-4406-a539-18e04e360c2d","added_by":"auto","created_at":"2025-08-19 10:25:25","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":91390,"visible":true,"origin":"","legend":"\u003cp\u003eControl valve model\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/0eb5de7665a377e0f08c23ec.png"},{"id":89372993,"identity":"6c0ffa90-9cfb-4d8f-a176-85c18435a064","added_by":"auto","created_at":"2025-08-19 10:33:24","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":71900,"visible":true,"origin":"","legend":"\u003cp\u003eDifferences between two models of hydraulic cylinder components\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/57f4355b338ef725f506d613.png"},{"id":89372639,"identity":"5d82305c-d9bb-4bfc-9067-e56aa24aa94e","added_by":"auto","created_at":"2025-08-19 10:25:24","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":73652,"visible":true,"origin":"","legend":"\u003cp\u003eHCD model of hydraulic cylinder\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/7d179201fc81ded767f56385.png"},{"id":89371742,"identity":"3e3101d8-2025-45a8-90db-83b2c3ddabf1","added_by":"auto","created_at":"2025-08-19 10:17:24","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":45351,"visible":true,"origin":"","legend":"\u003cp\u003eSimplified model of stepping motor\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/7ddc70a7cf315de2666fb6d3.png"},{"id":89371305,"identity":"eea2bbbf-a311-4f08-909c-be25a660c8ed","added_by":"auto","created_at":"2025-08-19 10:09:25","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":100125,"visible":true,"origin":"","legend":"\u003cp\u003eDigital hydraulic cylinder simulation model based on displacement signal equivalent feedback\u003c/p\u003e","description":"","filename":"11.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/edf105727676bbf077d9572a.png"},{"id":89371753,"identity":"104fb45a-c699-499e-8bf6-e5413b9fd1c7","added_by":"auto","created_at":"2025-08-19 10:17:25","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":107438,"visible":true,"origin":"","legend":"\u003cp\u003eSimulation model of digital hydraulic cylinder based on equivalent feedback of speed signal\u003c/p\u003e","description":"","filename":"12.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/9806f2d3630e39f68ff031f0.png"},{"id":89371744,"identity":"1614a9d1-73d3-4cc3-b505-981020150dab","added_by":"auto","created_at":"2025-08-19 10:17:24","extension":"png","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":108393,"visible":true,"origin":"","legend":"\u003cp\u003eDefinition diagram of interface parameters of bro011/12 valve sleeve movable slide valve components\u003c/p\u003e","description":"","filename":"13.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/80ebb578a3abce9f77a28cf2.png"},{"id":89372994,"identity":"84ce767c-f19d-4937-a169-7b019dc42470","added_by":"auto","created_at":"2025-08-19 10:33:25","extension":"png","order_by":14,"title":"Figure 14","display":"","copyAsset":false,"role":"figure","size":75105,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of HCD model construction of valve sleeve servo slide valve\u003c/p\u003e","description":"","filename":"14.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/a417c0027fe286261745e307.png"},{"id":89371284,"identity":"a027a5c0-08a8-42a2-96e8-09f74c47af17","added_by":"auto","created_at":"2025-08-19 10:09:25","extension":"png","order_by":15,"title":"Figure 15","display":"","copyAsset":false,"role":"figure","size":136054,"visible":true,"origin":"","legend":"\u003cp\u003eSimulation modeling of digital hydraulic cylinder based on valve sleeve servo equivalent feedback\u003c/p\u003e","description":"","filename":"15.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/7f3561549d15ba7b82fd7017.png"},{"id":89372644,"identity":"b59d0402-2025-4ba2-9ee6-795ae4e7dad0","added_by":"auto","created_at":"2025-08-19 10:25:25","extension":"png","order_by":16,"title":"Figure 16","display":"","copyAsset":false,"role":"figure","size":1089100,"visible":true,"origin":"","legend":"\u003cp\u003ePhysical diagram of digital hydraulic cylinder performance test device\u003c/p\u003e","description":"","filename":"16.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/bc75b2d93b6f2b8e9a4eae89.png"},{"id":89372646,"identity":"b8cf7fa7-64bd-46d8-8382-9f6d75067c77","added_by":"auto","created_at":"2025-08-19 10:25:25","extension":"png","order_by":17,"title":"Figure 17","display":"","copyAsset":false,"role":"figure","size":117512,"visible":true,"origin":"","legend":"\u003cp\u003eAngle signal curve corresponding to input pulse\u003c/p\u003e","description":"","filename":"17.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/e20589f261d3d94a20cfa616.png"},{"id":89371752,"identity":"9e6f0980-6ace-4b4f-b8d1-a3fc2bdcb4f3","added_by":"auto","created_at":"2025-08-19 10:17:25","extension":"png","order_by":18,"title":"Figure 18","display":"","copyAsset":false,"role":"figure","size":232077,"visible":true,"origin":"","legend":"\u003cp\u003eDisplacement simulation curve of digital hydraulic cylinder based on equivalent feedback model under different working pressures\u003c/p\u003e","description":"","filename":"18.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/589734a39b77033a5125aba4.png"},{"id":89372648,"identity":"1acc9768-6c3a-4209-8345-d0dea7cc7f90","added_by":"auto","created_at":"2025-08-19 10:25:25","extension":"png","order_by":19,"title":"Figure 19","display":"","copyAsset":false,"role":"figure","size":237854,"visible":true,"origin":"","legend":"\u003cp\u003eDisplacement simulation curve of digital hydraulic cylinder based on valve sleeve follow-up model under different working pressures\u003c/p\u003e","description":"","filename":"19.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/d4c1706d3108c646e0c566a2.png"},{"id":89371288,"identity":"2a12ddee-347a-4489-a971-12b4cbd61119","added_by":"auto","created_at":"2025-08-19 10:09:25","extension":"png","order_by":20,"title":"Figure 20","display":"","copyAsset":false,"role":"figure","size":204025,"visible":true,"origin":"","legend":"\u003cp\u003eDisplacement curve of digital hydraulic cylinder based on equivalent feedback model under different external loads\u003c/p\u003e","description":"","filename":"20.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/f19d196b97d51fde3c8e22f6.png"},{"id":89374043,"identity":"eb739d81-056c-4357-ae95-27c317ff7bac","added_by":"auto","created_at":"2025-08-19 10:41:26","extension":"png","order_by":21,"title":"Figure 21","display":"","copyAsset":false,"role":"figure","size":198333,"visible":true,"origin":"","legend":"\u003cp\u003eDisplacement curve of digital hydraulic cylinder based on valve sleeve follow-up model under different external loads\u003c/p\u003e","description":"","filename":"21.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/b17bc5d229dadd7477a06a22.png"},{"id":89371755,"identity":"9dcdba97-d6b3-46d2-b151-5ef0e3cf8e05","added_by":"auto","created_at":"2025-08-19 10:17:25","extension":"png","order_by":22,"title":"Figure 22","display":"","copyAsset":false,"role":"figure","size":180849,"visible":true,"origin":"","legend":"\u003cp\u003eAngle signal curves corresponding to different pulse frequencies based on equivalent feedback model\u003c/p\u003e","description":"","filename":"22.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/f754e5ade25607f1200e25b2.png"},{"id":89371783,"identity":"62d611de-aef9-44f2-92f2-500ab72fb066","added_by":"auto","created_at":"2025-08-19 10:17:26","extension":"png","order_by":23,"title":"Figure 23","display":"","copyAsset":false,"role":"figure","size":181944,"visible":true,"origin":"","legend":"\u003cp\u003eSimulation curves of digital cylinder displacement at different pulse frequencies based on valve sleeve follow-up model\u003c/p\u003e","description":"","filename":"23.png","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/e052e4f442f130d0ae75a635.png"},{"id":98813919,"identity":"022399fd-a791-4814-b356-cfe9844cd11c","added_by":"auto","created_at":"2025-12-22 16:07:44","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":4098661,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/6b121db5-8ed7-455c-b7a1-26d8e987d5d6.pdf"},{"id":89371263,"identity":"264f78ae-370f-4de4-8a53-8a45798d40c1","added_by":"auto","created_at":"2025-08-19 10:09:24","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":36308,"visible":true,"origin":"","legend":"","description":"","filename":"Table1to5.docx","url":"https://assets-eu.researchsquare.com/files/rs-7215125/v1/08982f752a65532e956c5443.docx"}],"financialInterests":"No competing interests reported.","formattedTitle":"Research on Equivalent Simplified Modeling and Simulation of Digital Hydraulic Cylinder","fulltext":[{"header":"0. Preface","content":"\u003cp\u003e\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003eA digital hydraulic cylinder combines a hydraulic cylinder, a hydraulic valve, a feedback mechanism, and control components. It is a type of single hydraulic component of the system engineering level with built-in closed- and open-loops. It can convert a pulse into a precise power drive and achieve micron-level control precision. It successfully combines hydraulic technology with digital technology to realize digital, long-distance, and intelligent control of hydraulic transmission. It has the advantages of high control precision, a simple structure, and convenient control. This is a typical example of digital hydraulic technology [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e], and its engineering applications are becoming increasingly extensive. The digital hydraulic cylinder specializes in fluid collection, machinery, electrical, automation, and so on. It exhibits the characteristics of high electromechanical and hydraulic integration. Its performance is affected by the response lag of the hydraulic system, the dead zone, mechanical clearance, oil leakage, temperature, load, and other factors. Therefore, in the performance analysis of a digital hydraulic cylinder and its engineering application system, modeling and simulation are necessary auxiliary methods. By establishing a reasonable mathematical model, the simulation results were obtained, and the structural parameters were optimized to improve the working performance of the digital hydraulic cylinder. Several scholars have conducted extensive research in this regard. Jia [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e] and Changlin [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e] deduced the transfer function of a valve-controlled asymmetric cylinder and established a general transfer function model for a digital hydraulic cylinder. Li [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e], Jia [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e], Zhiquan [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e], and Shouling [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] analyzed all types of nonlinear factors and obtained a complete nonlinear state-space model of a digital hydraulic cylinder. Wang [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e] and Jiang [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e] established the AMESim model of an internal indirect feedback digital hydraulic cylinder, including a ball screw and feedback nut, and conducted simulation research. Qing [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e], XIA [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e], and Panguo [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e] used the HCD library model and screw nut mechanism model in AMESim to establish the AMESim simulation model. These models were established based on the specific needs of this research. The models are relatively complex, and there are some difficulties in parameter setting and simulation solutions. For example, it is difficult to determine the comprehensive moment of inertia, comprehensive viscous damping coefficient, frequency, and other parameters of the slide valve in the transfer-function model. The nonlinear state equation contains a set of differential equations for 11 state variables, and some boundary conditions must be considered when solving it. However, this is difficult to achieve. It is relatively easy to model relying on the professional simulation software AMESim; however, the model contains a screw nut or ball screw model, which contains nonlinear characteristics such as friction and damping. For complex systems, the solution time is too long.\u003c/p\u003e\u003cp\u003eBased on the above analysis, although extensive research has been conducted on the modeling and simulation of digital hydraulic cylinders, existing models generally suffer from high complexity, difficulty in parameter determination, and low solving efficiency, making it challenging to meet the demands of rapid and efficient analysis in practical engineering applications. Particularly for performance optimization and real-time control of digital hydraulic cylinder application systems, there is an urgent need for a simplified modeling approach that can accurately reflect system characteristics while being easy to implement. Therefore, this paper aims to propose an equivalent simplified modeling method. Based on the working mechanism of servo feedback in digital hydraulic cylinders, a model incorporating signal negative feedback and valve sleeve follow-up equivalence is established. This model is implemented using AMESim software, and the results are validated through a digital hydraulic cylinder performance test bench. The proposed method aims to provide a relatively simplified modeling approach for the design optimization, performance enhancement, and engineering application of digital hydraulic cylinders.\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e"},{"header":"1. Typical structure and principle of digital hydraulic cylinder","content":"\u003cp\u003eAs shown in Figure 1, it is a typical structure of a digital hydraulic cylinder that is mainly composed of a stepper motor, three-way slide valve, feedback nut, screw, piston, piston rod, and cylinder [1]. The piston rod is a hollow structure that provides space for screw movement. The feedback nut and piston are fixed together. Generally, screws and spools are integrated. The output shaft end of the stepper motor was connected to the hydraulic spool through the connecting sleeve. The inner wall of the connecting sleeve has an axial chute that cooperates with the spool and the shaft end. In this manner, the spool and stepper motor output shaft can rotate synchronously and slide axially.\u003c/p\u003e\n\u003cp id=\"_Toc528070774\"\u003eThe working principle of this digital hydraulic cylinder is as follows: After inputting a certain number of pulses to the stepper motor, the motor shaft outputs an angular displacement and drives the spool of the three-way slide valve to rotate at the same time. Owing to the action of the screw and feedback nut, the spool moves axially while rotating, so that the valve port is opened so that the oil inlet or oil return path is connected to the rodless cavity. The positive and negative steering of the stepper motor determines whether the rodless cavity is connected to the oil inlet or return path. Specifically, when the spool moved to the right, the oil inlet was connected to the rodless cavity, and the piston rod extended outward. Simultaneously, the feedback nut drives the spool to return to the equilibrium position after the motor stops rotating. When the spool moves to the left, the return oil circuit is connected to the rodless cavity and the piston rod retracts. Similarly, the feedback nut drives the spool to return to the equilibrium position after the motor stops rotating [13]. Therefore, the number of input stepper motor pulses determines the displacement of the hydraulic cylinder piston rod and the frequency of the pulse corresponds to the speed of the piston rod.\u003c/p\u003e\n\u003cp\u003eFrom the above principle analysis, it can be seen that the digital hydraulic cylinder of this structure realizes internal mechanical feedback through the screw pair between the screw and feedback nut. The control block diagram is shown in Figure 2 It can be seen from the Figure that the working principle of the digital hydraulic cylinder can be summarized as a three-way slide valve to control a differential cylinder. The screw was used to provide precise feedback on the piston position, and the two hydraulic and screw technologies were skillfully used to achieve both a large output force and precise position accuracy.\u003c/p\u003e"},{"header":"2. Analysis of equivalent simplified modeling ideas for digital hydraulic cylinder","content":"\u003cp\u003eBased on the above analysis, it is clear that when the digital hydraulic cylinder is working, the stepper motor drives the spool of the hydraulic valve to move, and the hydraulic valve outputs hydraulic oil to drive the piston and load of the hydraulic cylinder to move, which is the forward channel of the hydraulic feedback control. The load, executive hydraulic cylinder, and control slide valve constitute the control structure unit, which is called the valve-controlled cylinder hydraulic power unit. Each tiny valve opening generated by the movement of the control valve core causes the hydraulic cylinder to close the valve port to the corresponding reaction movement. Therefore, the control valve mainly works near the median or its working valve port is relatively small.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eIn the structure of an actual digital hydraulic cylinder, the opening of the valve port is the result of the compound motion of the valve core and the mechanical feedback mechanism in space. As shown in Figure 3, under the action of the stepper motor drive and screw/nut pair, the valve core rotates to produce axial displacement, the valve port opens, and the pressure oil drives the piston to move. The piston moves in the opposite direction to follow the action of the valve core, weakening or counteracting the movement effect of the valve core, thus forming negative feedback. It can be said that the generation of negative feedback is established by the screw pair between the piston and the valve core and the screw.\u003c/p\u003e\n\u003cp\u003eIn other words, the stepper motor drove the rotation of the valve core. This rotation was converted into the input displacement \u003cem\u003ex\u003csub\u003em\u003c/sub\u003e\u003c/em\u003e of the slide valve using the screw pair. Once the valve port opens, the displacement xp of the oil cylinder piston is fed back through the large lead screw and screw nut, which becomes the feedback displacement \u003cem\u003ex\u003csub\u003ef\u003c/sub\u003e\u003c/em\u003e\u003csub\u003e\u0026nbsp;\u003c/sub\u003eof the slide valve core. Ultimately, the input displacement xm and feedback displacement \u003cem\u003ex\u003csub\u003ef\u003c/sub\u003e\u003c/em\u003e of the valve core are combined to form the absolute displacement (i.e., the valve port opening) \u003cem\u003ex\u003csub\u003ev\u003c/sub\u003e\u003c/em\u003e. So there are\u003c/p\u003e\n\u003cp\u003e\u003cimg 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\" width=\"596\" height=\"266\"\u003e\u003c/p\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003ewhere:\u003cem\u003eP\u003c/em\u003e the nut pitch, \u0026sigma; the feedback coefficient for the single-stage spiral feedback structure \u003cem\u003e\u0026sigma;\u003c/em\u003e=1 for the double-stage spiral feedback structure \u003cem\u003e\u0026sigma;\u003c/em\u003e=\u003cem\u003eP\u003c/em\u003e/\u003cem\u003eS\u003c/em\u003e (\u003cem\u003eS\u003c/em\u003e is the lead of the lead screw).\u003c/p\u003e\n\u003cp\u003eWhen the actual digital hydraulic cylinder is operating, the opening amount of the valve port is the difference between the active displacement of the valve core and the feedback displacement of the piston/screw. When using professional software for modeling and simulation, if the software is convenient for building a detailed simulation model based on the actual physical model, the opening amount of the valve port can also be equivalently processed when simplifying the modeling. There are two equivalent treatment ideas.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eFirst, the real-time signal of the piston displacement is compared with the displacement signal of the valve core driven by the stepping motor using signal feedback. This difference controls the action of the valve core, that is, the opening amount of the valve. The control block diagram is shown in Figure 4, which shows the modeling and simulation method of a digital hydraulic cylinder based on the equivalent feedback of the displacement signal.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eSecond, the piston displacement is directly applied to the valve sleeve, and a position follow-up relationship is formed between the valve sleeve and the valve core, so that the valve opening amount is equal to the displacement difference between the valve core and valve sleeve, so that it is equivalent to the original structure. The control block diagram shown in Figure 5 can be used to simulate the digital cylinder mechanism. In fact, it is similar to the composition of the common mechanical and hydraulic servo mechanism, which is called the digital hydraulic cylinder modeling and simulation method, based on the equivalent feedback of the valve sleeve follow-up.\u003c/p\u003e"},{"header":"3. Equivalent simplified model of digital hydraulic cylinder","content":"\u003cp\u003e3.1 Model of main components of digital hydraulic cylinder\u003c/p\u003e\n\u003cp\u003eThe hydraulic cylinder, control valve, stepper motor, and other components of the digital hydraulic cylinder can be directly modeled by the hydraulic module in the AMESim software. For the modules that are not in the hydraulic library or existing modules but cannot meet the requirements of accurate modeling, the hydraulic component design library ( hereinafter referred to as the HCD library ), mechanical library ( Mechanical ) and signal library ( Signal, Control ) can be used to model these components.\u003c/p\u003e\n\u003cp\u003e3.1.1 Control Valve Model\u003c/p\u003e\n\u003cp\u003eThe HCD library of AMESim software is a basic element composed of basic geometric structural units. It adopts a modular modeling method based on the geometric structure of the hydraulic components and can establish a detailed component model considering the dynamic performance of moving bodies, fluid compressibility, friction, leakage, hydraulic power, and other factors. It is used to construct various hydraulic components according to geometric shapes and physical characteristics. The library is suitable for modeling and analyzing the dynamic characteristics of nonstandard hydraulic components [14].\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe HCD library provides a variety of types of spool valve components, including components with ring grooves, round hole grooves, grooves, and custom slotted spool valves. Table 1 lists information on some spool valve components. In actual modeling, the appropriate component sub-model is selected according to the structure of the valve.\u003c/p\u003e\n\u003cp\u003eThe control valve was the core component of the digital hydraulic cylinder. According to the actual structure, it can be composed of BAO011 and BAO012 modules in the HCD library and a mass block in the mechanical library, as shown in Figure 6. BAO012 differs from BAO011 only because the variables associated with ports 3 and 4 are interchanged. The established control valve model is shown in Figure 7, where the right mass block, MECMAS21, represents the spool. The stroke limit was set at both ends of the mass block such that the maximum displacement of the valve core could be controlled. The valve core adopted a cylindrical section, according to the actual situation. The diameter parameters of the piston and piston rod could be modified according to the three-way valve in the BAO012 and BAO011 modules. MECMAS21 represents the one-dimensional motion of a two-port mass under the action of two external forces in terms of N, weight, and frictional forces.\u003c/p\u003e\n\u003cp\u003e3.1.2 Hydraulic cylinder model\u003c/p\u003e\n\u003cp\u003eThe \u0026apos;hydraulic cylinder\u0026apos; in the digital hydraulic cylinder is generally a single-rod piston cylinder. The HJ020 and HJ000 models listed in Table 2 can be used in the AEMSim software hydraulic library. The main difference between the two models was the mass load. In addition, the HCD library components can also be used to build hydraulic cylinder models. The optional models are BAP1 and BAP2, as shown in Table 3, each of which contains the BAP11 and BAP12 submodels. The difference between the two sub-models is that the variables defined by ports 2 and 3 are in opposite directions, as shown in Figure 8.\u003c/p\u003e\n\u003cp\u003eThe hydraulic cylinder model constructed using the HCD module library is illustrated in Figure 9. The rod cavity and rodless cavity of the hydraulic cylinder composed of BAP12, BAF01, and BAP11 were the same by default. If the action areas of the two cavities are not the same, they can be realized by setting different piston-rod diameters.\u003c/p\u003e\n\u003cp\u003e3.1.3 Stepping Motor Model\u003c/p\u003e\n\u003cp\u003eFor the modeling and simulation of stepping motor in digital hydraulic cylinder, different models or equivalent models can be selected according to the simulation purpose, and there are many ways.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eIn general, the focus of modeling is the analysis of mechanical and fluid parts. The stepper motor can use the signal to drive the spool directly or select the transfer function module in the signal library. This \u0026nbsp;model is illustrated in Figure 10. The left module is the input pulse, and the pulse frequency can be modified by setting the slope such that the pulse quantity is determined when the simulation time is determined.\u003c/p\u003e\n\u003cp\u003e3.2 Construction of a Digital Hydraulic Cylinder System Model Based on Equivalent Signal Feedback\u003c/p\u003e\n\u003cp\u003eAccording to the principle block diagram shown in Figure 4, the model of each component of the digital hydraulic cylinder was selected and combined according to the mechanism. The feedback mechanism adopts the displacement signal feedback method, and the sub-model in the signal library is applied to the negative feedback equivalent modeling. The AMESim model of the digital hydraulic cylinder system ( AMESim16.0 ), based on the equivalent feedback of the displacement signal, was obtained. As shown in Figure 11, the real line represents the connected hydraulic pipeline, where the rough and real lines represent the characteristics of the length, diameter, and complex flow state of the hydraulic pipeline, and the imaginary line represents the connection relationship between the signals.\u003c/p\u003e\n\u003cp\u003eFor the rotary motion of the valve core driven by the stepping motor, linear displacement of the valve core occurs under the side effect of the screw nut. The feedback signal can be selected as the displacement or speed signal. The displacement signal feedback is shown in Figure 11, and the displacement signal was collected by the displacement sensor at the piston output end.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe signal conversion module and its interface definitions used in Figure 11 are listed in Table 4. XVLC01 converts the dimensionless signal input at port 1 into a linear displacement of the same value in meters, and the linear displacement is the output at port 2 in meters per second. The speed is obtained by approximate differentiation of the angle using the first-order lag method of the time constant provided. The initial speed can be set by the user or zero by default. The VELC02 model accepts the dimensionless signal input at port 1 and converts it into velocity (m/s) and displacement (m) at port 2. Displacement is a state variable obtained by velocity integration.\u003c/p\u003e\n\u003cp\u003eIf the speed feedback signal is used, the piston output end is replaced by a speed sensor and the drive input of the control valve is changed to a speed signal. As shown in Figure 12, the corresponding signal source can be set at a constant speed.\u003c/p\u003e\n\u003cp\u003e3.3 Construction of a Digital Hydraulic Cylinder System Model Based on Valve Sleeve Servo Feedback Equivalence\u003c/p\u003e\n\u003cp\u003eAccording to the principle block diagram of the digital hydraulic cylinder modeling method based on the valve sleeve servo feedback equivalent shown in Figure 5, the modeling was performed on the AMESim software platform.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe movable spool valve parts of the valve sleeve are provided in the HCD library. Table 5 presents several typical models. The main difference is that the valve-port shapes differ. For example, the BRO011 / BRO012 model was used. The difference between the two model interface definitions is that ports 3 and 6 are reciprocal, as shown in Figure 13.\u003c/p\u003e\n\u003cp\u003eWhen using the valve sleeve movable spool valve component to establish the spool valve model, attention was paid to the interface connection between the components. In general, it is necessary to increase the MAS011RT module to simulate the force between the spool and the valve sleeve. The connections between BRO011 and BRO012, BRO011, BRO012, MAS011RT, BRO012, and VELC02 conversion sub-models, and FVXSG1 are shown in Figure 14.\u003c/p\u003e\n\u003cp\u003eThe model of the hydraulic oil source and hydraulic cylinder was the same as that shown in Figure 11. By adding the connection between the piston output and valve sleeve, a complete model of the digital hydraulic cylinder system based on the equivalent feedback of the valve sleeve can be established, as shown in Figure 15. Two conversion sub-modules are added to the diagram to increase the scale factor k at ports v and x. When k is 1, the model is equivalent to a digital hydraulic cylinder with single-stage spiral feedback. When the k value is ta/tb ( ta is the pitch of the spool screw, tb is the pitch of the ball screw ), the model is equivalent to a digital hydraulic cylinder structure with double-stage spiral feedback.\u003c/p\u003e\n\u003cp\u003e3.4 Model Parameter Settings\u003c/p\u003e\n\u003cp\u003eThe relevant parameters of the model were set. The structural parameters, such as the hydraulic cylinder and control valve, were set according to the actual parameters of the research object. The initial balance parameters were determined using static calculations. Other parameters can be determined based on the technical manual or empirical data of the component. Table 6 lists the values of the simulation parameters determined by the system.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 6.\u003c/strong\u003e Relevant parameters of digital hydraulic cylinder simulation\u003c/p\u003e\n\u003cdiv align=\"\"\u003e\n \u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"520\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 130px;\"\u003e\n \u003cp\u003eComponent name\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 172px;\"\u003e\n \u003cp\u003eParameter name\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 123px;\"\u003e\n \u003cp\u003eNumerical value\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 94px;\"\u003e\n \u003cp\u003eUnit\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd rowspan=\"3\" style=\"width: 130px;\"\u003e\n \u003cp\u003eHydraulic cylinder\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 172px;\"\u003e\n \u003cp\u003eCylinder inner diameter\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 123px;\"\u003e\n \u003cp\u003e160\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 94px;\"\u003e\n \u003cp\u003emm\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 172px;\"\u003e\n \u003cp\u003ePiston rod outer diameter\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 123px;\"\u003e\n \u003cp\u003e100\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 94px;\"\u003e\n \u003cp\u003emm\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 172px;\"\u003e\n \u003cp\u003eCylinder stroke\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 123px;\"\u003e\n \u003cp\u003e220\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 94px;\"\u003e\n \u003cp\u003emm\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd rowspan=\"2\" style=\"width: 130px;\"\u003e\n \u003cp\u003eScrew\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 172px;\"\u003e\n \u003cp\u003eExternal diameter\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 123px;\"\u003e\n \u003cp\u003e20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 94px;\"\u003e\n \u003cp\u003emm\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 172px;\"\u003e\n \u003cp\u003ePitch\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 123px;\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 94px;\"\u003e\n \u003cp\u003emm\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd rowspan=\"3\" style=\"width: 130px;\"\u003e\n \u003cp\u003eSlide valve\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 172px;\"\u003e\n \u003cp\u003eValve core diameter\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 123px;\"\u003e\n \u003cp\u003e18\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 94px;\"\u003e\n \u003cp\u003emm\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 172px;\"\u003e\n \u003cp\u003eGroove diameter\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 123px;\"\u003e\n \u003cp\u003e16\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 94px;\"\u003e\n \u003cp\u003emm\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 172px;\"\u003e\n \u003cp\u003eShoulder width\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 123px;\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 94px;\"\u003e\n \u003cp\u003emm\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e"},{"header":"4. Simulation and test result analysis","content":"\u003cp\u003eFor the AMESim simulation model of the digital hydraulic cylinder constructed above, the input signal and simulation parameters are set, and the simulation can be performed to obtain the dynamic characteristics of the digital hydraulic cylinder. By setting different values for the same variable of the digital hydraulic cylinder, such as different working pressures, external loads, and stepper motor frequencies, the values of other variables were not changed, and the influence of this variable on the dynamic characteristics was analyzed. The simulation results were compared with the test results to verify the accuracy and equivalence of the simulation model.\u003c/p\u003e\n\u003cp\u003e4.1 Performance test device\u003c/p\u003e\n\u003cp\u003eThe performance test device for the digital hydraulic cylinder is shown in Figure 16. It is mainly composed of a hydraulic pump station, digital hydraulic cylinder, loading hydraulic cylinder, constant pressure valve block, loading valve block, sensors, electrical control cabinet,measurement and control host,and software. The constant pressure valve block provides constant oil supply pressure to the digital hydraulic cylinder, and the loading valve block controls the output tension or pressure of the loading hydraulic cylinder to load the digital hydraulic cylinder. The measurement and control signals communicate with an industrial computer through an electric control cabinet. Based on LabVIEW, a human-computer interaction interface was developed to realize the functions of the hydraulic system state display, loading force setting and control, constant pressure source setting and control, digital hydraulic cylinder action setting and control, and data display and storage.\u003c/p\u003e\n\u003cp\u003e4.2 Working characteristics of digital cylinder under different working pressures\u003c/p\u003e\n\u003cp\u003eThe pressure of the loading cylinder was set to 3Mpa as the external load, the frequency of the stepping motor was 400 Hz, and the number of input pulses was 240, which corresponded to a theoretical displacement of 3 mm. The working pressures of the digital cylinder were 5MPa, 7MPa and 9MPa respectively, and the simulation was carried out based on the equivalent model. The expected piston displacement increased from 0 mm to 3 mm and then decreased from 3 mm to 0 mm. The corresponding input-signal curves are shown in Figure 17. The simulation curves of the digital cylinder displacement under different working pressures are shown in Figure 18 and 19, respectively. It can be seen from the diagram that under the same external load, the piston extension process is less affected by the working pressure, whereas the piston retraction process is obviously affected by the working pressure. The higher the working pressure, the faster is the response speed of the digital hydraulic cylinder.\u003c/p\u003e\n\u003cp\u003eUnder the same conditions, the test bench was used to test the positioning error of the digital hydraulic cylinder under different working pressures, and the results were compared with the simulation results of the two models. As shown in Table 7, \u0026apos;Simulation 1\u0026apos; represents the simulation results based on the signal feedback equivalent model, \u0026apos;Simulation 2\u0026apos; represents the simulation results based on the valve sleeve servo equivalent model, and \u0026apos;Simulation 3\u0026apos; represents the simulation results based on the model in Reference [12]. It can be seen that the simulation results of the two methods proposed in this paper are consistent, the motion displacement errors were all smaller than those obtained from the simulation results based on the model in Reference [12], and all showed close agreement with the experimental results. With an increase in the working pressure, the positioning accuracy of the digital cylinder increased. In general, the positioning accuracy of the piston rod was higher than that of retraction.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 7.\u0026nbsp;\u003c/strong\u003eSimulation and experimental results of displacement under different working pressures\u003c/p\u003e\n\u003cdiv align=\"\"\u003e\n \u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"605\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd rowspan=\"2\" style=\"width: 47px;\"\u003e\n \u003cp\u003eWorking pressure\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"4\" style=\"width: 265px;\"\u003e\n \u003cp\u003e3mm retraction displacement error (mm)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"4\" style=\"width: 293px;\"\u003e\n \u003cp\u003e3mm extension displacement error (mm)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003eSimulation 1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003eSimulation 2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003eSimulation 3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003eTest\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003eSimulation 1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003eSimulation 2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003eSimulation 3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003eTest\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003e5 MPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.00034\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0.00034\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0.0014\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003e0.001\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0.0039\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e0.0039\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e0.0041\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003e0.004\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003e7 MPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.0007\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0.0007\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0.0009\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003e0.001\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0.0033\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e0.0033\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e0.0031\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003e0.003\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003e9 MPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.0005\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0.0005\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0.0002\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003e0.001\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0.002\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e0.002\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e0.0025\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003e0.002\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003e4.3 Working characteristics of digital cylinder under different external loads\u003c/p\u003e\n\u003cp\u003eThe working pressure of the digital hydraulic cylinder was set to 9 Mpa, and the frequency of the stepping motor was 400 Hz. The pressures of the loading chamber of the given loading cylinder were 3, 5, and 7 MPa. The number of input pulses was 240, which corresponded to a theoretical displacement of 3 mm. The expected piston displacement increased from 0 mm to 3 mm and then decreased from 3 mm to 0 mm. The corresponding input-signal curves are shown in Figure 17. The simulation results show that the displacement curves of different models of digital cylinders under different external loads are as shown in Figure 20 and 21. It can be observed that under the same working pressure, the piston extension process is less affected by the load pressure. The piston retraction process is significantly affected by the load. The smaller the load, the faster is the response speed of the digital hydraulic cylinder.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eUnder the same conditions, the test bench was used to test and extract the positioning error of the digital cylinder under different external loads, and the results were compared with the simulation results. As shown in Table 8, it can be seen that the simulation results of the two methods proposed in this paper are consistent. The motion displacement errors were all smaller than those obtained from the simulation results based on the model in Reference [12], and all showed close agreement with the experimental results. At the same working pressure, the smaller the load, the higher the positioning accuracy of the digital cylinder. The positioning accuracy when the piston rod is extended is higher than that when the piston rod is retracted.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 8.\u003c/strong\u003e Simulation and experimental results of displacement under different load pressures\u003c/p\u003e\n\u003cdiv align=\"\"\u003e\n \u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"597\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd rowspan=\"2\" style=\"width: 46px;\"\u003e\n \u003cp\u003eLoad Pressure\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"4\" style=\"width: 268px;\"\u003e\n \u003cp\u003e3mm retraction displacement error (mm)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"4\" valign=\"top\" style=\"width: 283px;\"\u003e\n \u003cp\u003e3mm extension displacement error (mm)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 69px;\"\u003e\n \u003cp\u003eSimulation 1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003eSimulation 2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003eSimulation 3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003eTest\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003eSimulation 1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003eSimulation 2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003eSimulation 3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003eTest\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 46px;\"\u003e\n \u003cp\u003e3MPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 69px;\"\u003e\n \u003cp\u003e0.0005\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e0.0005\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.0002\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003e0.001\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003e\n \u003cp\u003e0.002\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 66px;\"\u003e\n \u003cp\u003e0.002\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0.0025\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.003\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 46px;\"\u003e\n \u003cp\u003e5MPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 69px;\"\u003e\n \u003cp\u003e0.0002\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e0.0002\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.0012\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003e0.002\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003e\n \u003cp\u003e0.003\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 66px;\"\u003e\n \u003cp\u003e0.003\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0.0048\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.005\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 46px;\"\u003e\n \u003cp\u003e7MPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 69px;\"\u003e\n \u003cp\u003e0.0003\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 85px;\"\u003e\n \u003cp\u003e0.0003\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.0015\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003e0.003\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003e\n \u003cp\u003e0.005\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 66px;\"\u003e\n \u003cp\u003e0.005\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0.0053\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e0.006\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003e4.4 Working characteristics of the digital hydraulic cylinder under different stepper motor frequencies\u003c/p\u003e\n\u003cp\u003eThe working pressure of the digital cylinder was set at 5 Mpa. The pressure in the loading chamber of the loading cylinder was 3 MPa. The frequencies of the stepping motor were 200 Hz, 400 Hz, and 600 Hz. The number of input pulses was 240, which corresponds to a theoretical displacement of 3 mm. The expected piston displacement increased from 0 mm to 3 mm and then decreased from 3 mm to 0 mm. The corresponding input-signal curves are shown in Figure 17. The displacement curves of the different digital cylinder models under different pulse frequencies are shown in Figure 22 and 23. It can be observed that under the same load and working pressure, the piston extension process is less affected by the stepping motor frequency. The piston retraction process was affected by the frequency of the stepping motor. The higher the frequency of the stepping motor, the faster is the response speed of the digital hydraulic cylinder.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eUnder the same conditions, the test bench was used to test and extract the positioning error of the digital cylinder under different pulse frequencies, and the results were compared with the simulation results. As shown in Table 9, it can be seen that the simulation results of the two methods proposed in this paper are consistent.\u0026nbsp;The motion displacement errors were all smaller than those obtained from the simulation results based on the model in Reference [12], and all showed close agreement with the experimental results.\u0026nbsp;The lower the pulse frequency of the stepper motor, the higher the positioning accuracy of the digital cylinder. The positioning accuracy when the piston rod was extended was higher than that when it was retracted.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 9.\u0026nbsp;\u003c/strong\u003eSimulation and experimental results of displacement at different pulse frequencies\u003c/p\u003e\n\u003cdiv align=\"\"\u003e\n \u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"605\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd rowspan=\"2\" style=\"width: 66px;\"\u003e\n \u003cp\u003ePulse frequency\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"4\" style=\"width: 274px;\"\u003e\n \u003cp\u003e3mm retraction displacement error (mm)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"4\" valign=\"top\" style=\"width: 264px;\"\u003e\n \u003cp\u003e3mm extension displacement error (mm)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003eSimulation 1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 79px;\"\u003e\n \u003cp\u003eSimulation 2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 73px;\"\u003e\n \u003cp\u003eSimulation 3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003eTest\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003eSimulation 1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003eSimulation 2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003eSimulation 3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003eTest\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e200Hz\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0.0016\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 79px;\"\u003e\n \u003cp\u003e0.0016\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 73px;\"\u003e\n \u003cp\u003e0.0009\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003e0.001\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 66px;\"\u003e\n \u003cp\u003e0.0024\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003e\n \u003cp\u003e0.0024\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0.0027\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003e0.003\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e400Hz\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0.00034\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 79px;\"\u003e\n \u003cp\u003e0.00034\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 73px;\"\u003e\n \u003cp\u003e0.0014\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003e0.001\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 66px;\"\u003e\n \u003cp\u003e0.0039\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003e\n \u003cp\u003e0.0039\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0.0041\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003e0.004\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 66px;\"\u003e\n \u003cp\u003e600Hz\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0.0004\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 79px;\"\u003e\n \u003cp\u003e0.0004\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 73px;\"\u003e\n \u003cp\u003e0.0016\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003e0.002\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 66px;\"\u003e\n \u003cp\u003e0.004\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003e\n \u003cp\u003e0.004\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e0.0045\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47px;\"\u003e\n \u003cp\u003e0.005\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e"},{"header":"5. Conclusion and Discussion","content":"\u003cp\u003eBased on the mechanical feedback mechanism of digital hydraulic cylinders, two methods for constructing simplified equivalent models have been proposed. One was based on the equivalent feedback of the displacement signals, and the other was based on the equivalent feedback of the valve sleeve follow-up. Two equivalent models of digital hydraulic cylinders were established based on AMESim and were verified by simulation and performance tests. The following conclusions were drawn:\u0026nbsp;\u003c/p\u003e\n\u003cul\u003e\n \u003cli\u003eThe simulation and experimental results verify the accuracy of the two models. These two ideas are based on the mechanism of negative feedback and are essentially equivalent.\u0026nbsp;\u003c/li\u003e\n \u003cli\u003eUnder the same external load and stepping motor frequency, the higher the working pressure, the faster the response speed of the digital hydraulic cylinder and the higher the positioning accuracy, and the piston extension process is less affected by the working pressure.\u0026nbsp;\u003c/li\u003e\n \u003cli\u003eUnder the same working pressure and stepping motor frequency, the smaller the load, the faster the response speed of the digital hydraulic cylinder and the higher the positioning accuracy, and the piston extension process is less affected by the load pressure.\u0026nbsp;\u003c/li\u003e\n \u003cli\u003eUnder the same load and working pressure, the higher the frequency of the stepper motor, the faster is the response speed of the digital hydraulic cylinder. The piston extension process is less affected by the frequency of the stepper motor, while the piston retraction process is significantly affected by the frequency of the stepper motor.\u0026nbsp;\u003c/li\u003e\n \u003cli\u003eThe positioning accuracy of the digital hydraulic cylinder piston rod extension action exceeded that of the retraction action.\u0026nbsp;\u003c/li\u003e\n \u003cli\u003eCompared with traditional modeling methods, the approach proposed in this paper\u0026nbsp;achieves a certain degree of simplification in modeling digital hydraulic cylinders.\u0026nbsp;It provides an efficient and practical modeling solution for digital hydraulic cylinder application systems, especially for complex engineering systems involving multiple hydraulic cylinders, thereby greatly reducing the implementation difficulty of simulating complex systems. Furthermore, the simplified modeling approach based on negative feedback equivalence adopted in this paper is not only applicable to digital hydraulic cylinders but also\u0026nbsp;provides reference value\u0026nbsp;for the modeling and simulation of other complex electromechanical-hydraulic systems.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eThe research presented in this paper primarily proposes two equivalent simplified modeling approaches based on negative feedback mechanisms, while theoretical analysis remains relatively limited. In the experimental testing section, the measurement accuracy is constrained by sensor precision and requires further improvement.Several aspects warrant additional investigation:\u003c/p\u003e\n\u003cp\u003e（1）The influence of mechanical friction and hydraulic fluid viscosity on positioning accuracy in the model requires further study.\u003c/p\u003e\n\u003cp\u003e（2）Theoretical analysis of positioning accuracy variations under different loads, working pressures, and stepper motor frequencies for digital hydraulic cylinders needs more comprehensive discussion.\u003c/p\u003e\n\u003cp\u003eThese identified areas present valuable directions for future research on digital hydraulic cylinders and their engineering applications:\u003c/p\u003e\n\u003cul\u003e\n \u003cli\u003eEnhanced theoretical modeling of multi-factor interactions\u003c/li\u003e\n \u003cli\u003eImproved experimental measurement methodologies\u003c/li\u003e\n \u003cli\u003eSystematic investigation of performance under various operational conditions\u003c/li\u003e\n\u003c/ul\u003e"},{"header":"Declarations","content":"\u003ch2\u003eFunding\u003c/h2\u003e\u003cp\u003eThis work is supported by the Natural Science Basic Research Program of Shaanxi Province, China (Grant No. 2025JC-YBQN-686).\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003ePrinciple analysis of digital hydraulic cylinder, Changlin Ma and Yunguang Gao, Mod-eling Method and Implementation of Digital Hydraulic Cylinder, Changlin Ma and Lin Hao, Digital hydraulic cylinder simulation, Changlin Ma.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eThe data that support the findings of this study are available from the corresponding author upon reasonable request.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eMa Changlin, Li Feng, Gao Yunguang, et al. Modeling, Simulation and Test of Digital Hydraulic Cylinder [M]. Xi\u0026apos;an: Xi\u0026apos;an University of Electronic Science and Technology Press, 2020.09.\u003c/li\u003e\n \u003cli\u003eChen Jia, Xing Jifeng, Peng Likun. Modeling and Analysis of Digital Hydraulic Cylinder Based on Transfer Function [J]. China Mechanical Engineering，2014,25(1):65-70.\u003c/li\u003e\n \u003cli\u003eMa C ,Li F ,Hao L , et al.Mechanism Modeling and Simulink Simulation Analysis of Digital Hydraulic Cylinder[C]//Advanced Science and Industry Research Center.Proceedings of 2020 2nd International Conference on Computer Modeling,Simulation and Algorithm(CMSA2020).\u003c/li\u003e\n \u003cli\u003eLiu Youli, Ma Changlin, Li Feng. Research on Nonlinear Modeling, Simulation and Experiment of Digital Hydraulic Cylinder [J]. Hydraulic and Pneumatic,2018(10):118-124.\u003c/li\u003e\n \u003cli\u003eChen Jia, Xing Jifeng, Peng Likun. Nonlinear Dynamic Characteristic Analysis and Experiment of Digital Hydraulic Cylinder [J]. Mechanical Science and Technolog, 2016,35(7):1035-1042.\u003c/li\u003e\n \u003cli\u003eJiang S L, Zhang K, Wang H, et al.Research on adaptive friction compensation of digital hydraulic cylinder based on LuGre friction model[J].Shock and Vibration,2021,(2):1-10.\u003c/li\u003e\n \u003cli\u003eXiao Zhiquan, Peng Likun, Xing Jifeng, et al. Modeling Analysis of Digital Servo Stepping Hydraulic Cylinder [J]. China Mechanical Engineering，2007,18(16):1935-1938.\u003c/li\u003e\n \u003cli\u003eJiang Shouling. Research on control strategy of digital hydraulic cylinder position system [D]. Liaoning Technical University, 2023.\u003c/li\u003e\n \u003cli\u003eWang Hui, Jiang Shouling, Qi Panguo,et al. Stiffness Analysis and AMESim Simulation of Digital Hydraulic Cylinder [J]. Control Engineering of China,2018,25(10):1849-1853.\u003c/li\u003e\n \u003cli\u003eTan Qing,Yao ZhiWei,Xia YiMin, et al. Research on the Influence of Different Valve Ports on the Performance of Digital Hydraulic Cylinder [J]. Modern Manufacturing Engineering, 2020(2):131-137.\u003c/li\u003e\n \u003cli\u003eQi PanGuo,Liu ZhengQi,Chen HongYue, et al. Modeling and Analysis of 120 t Digital Hydraulic Cylinder for Shearer Rocker Arm [J]. Control Engineering of China, 2022,29(12):2184-2193,2203.\u003c/li\u003e\n \u003cli\u003eXia Yimin, Shi Yupeng, Yuan Ye,et al. Analyzing of influencing factors on dynamic response characteristics of double closed-loop control digital hydraulic cylinder[J].JOURNAL OF ADVANCED MECHANICAL DESIGN SYSTEMS AND MANUFACTURING,2019,13(3).\u003c/li\u003e\n \u003cli\u003eZhu Haoyu, Pei Zhongcai, Liu Honglin, et al. Mechanism Analysis and Control System Research of Digital Hydraulic Cylinder [J].Machine Tool and Hydraulics,2020,48(24):8-12.\u003c/li\u003e\n \u003cli\u003eLiang Quan, Su Qiying. AMESim computer simulation guide for hydraulic system [M]. Beijing: China Machine Press, 2014.\u003c/li\u003e\n\u003c/ol\u003e"},{"header":"Tables","content":"\u003cp\u003eTables 1 to 5 are available in the Supplementary Files section.\u003c/p\u003e\n"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Digital Hydraulic Cylinder, Simplify Equivalent Model, Modeling and Simulation, Negative Feedback Mechanism, AMESim","lastPublishedDoi":"10.21203/rs.3.rs-7215125/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7215125/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eTo address the challenges in traditional modeling methods for highly integrated electro-mechanical-hydraulic digital hydraulic cylinders, such as complex nonlinear factors, high parameter uncertainty, and low simulation efficiency, this study proposes two equivalent simplified modeling and simulation methods based on the negative feedback mechanism. Firstly, a valve port control model based on equivalent feedback of displacement signals is established by constructing a closed-loop feedback mechanism between piston displacement and spool displacement. Secondly, a mathematical mapping between the valve opening and the relative displacement of the spool and sleeve is formed by utilizing the linkage following relationship between the valve sleeve and spool, resulting in a modeling method based on equivalent feedback of sleeve following. Two equivalent models of the digital hydraulic cylinder are developed using the AMESim platform, and the correctness and basic equivalence of the model were verified through simulation and performance testing. Based on the equivalent model, the influence of key parameters on the dynamic characteristics of the system can be analyzed. This can provide a relatively simplified modeling method for the complex engineering application system analysis of digital hydraulic cylinders. The proposed negative feedback equivalent modeling approach offers a referable approach for facilitating modeling and simulation implementation in other systems.\u003c/p\u003e","manuscriptTitle":"Research on Equivalent Simplified Modeling and Simulation of Digital Hydraulic Cylinder","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-08-19 10:09:19","doi":"10.21203/rs.3.rs-7215125/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-08-29T13:39:24+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-08-28T11:04:59+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-08-25T13:43:01+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"181124342674201364161091062958774344446","date":"2025-08-18T03:18:09+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"1676788519705935958185328744513792899","date":"2025-08-16T09:59:50+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"60760143845703230435571677419069686916","date":"2025-08-11T11:29:05+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-08-11T09:55:17+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-08-11T05:17:38+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2025-08-07T06:39:47+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-08-04T08:21:17+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2025-08-04T01:13:21+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"c65c61ca-8a29-4e01-b624-0b60ef3ae1cc","owner":[],"postedDate":"August 19th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":53277330,"name":"Physical sciences/Engineering"},{"id":53277331,"name":"Physical sciences/Mathematics and computing"}],"tags":[],"updatedAt":"2025-12-22T16:00:39+00:00","versionOfRecord":{"articleIdentity":"rs-7215125","link":"https://doi.org/10.1038/s41598-025-27686-3","journal":{"identity":"scientific-reports","isVorOnly":false,"title":"Scientific Reports"},"publishedOn":"2025-12-18 15:57:04","publishedOnDateReadable":"December 18th, 2025"},"versionCreatedAt":"2025-08-19 10:09:19","video":"","vorDoi":"10.1038/s41598-025-27686-3","vorDoiUrl":"https://doi.org/10.1038/s41598-025-27686-3","workflowStages":[]},"version":"v1","identity":"rs-7215125","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"identity":"rs-7215125","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}