# Adaptive Friction Compensation for a Novel Two-Axis Differential Microfeed System

Authors: Anning Wang; Xianying Feng; Dong Wang; Peigang Li; Zhenyu Xu
Journal: Shock and Vibration
 (2022)

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## Abstract

The conventional drive feed system inevitably enters the nonlinear creep region at very low speed. This seriously affects its low-speed performance and tracking accuracy. The new two-axis differential microfeed system (TDMS) can overcome the low-speed crawling and creep phenomenon caused by the inherent characteristics of the traditional electromechanical servo system structure. In this study, the dynamic model of TDMS is first established, and the LuGre friction model is improved according to its motion characteristics. The friction model of the screw nut whole assembly of TDMS is established. On this basis, two nonlinear observers are designed to estimate the internal state of the friction model, and the TDMS adaptive friction compensation controller is designed to estimate the uncertain parameters in TDMS through the parameter adaptive law. The global asymptotic stability of the TDMS controller is proved by Lyapunov theorem. Finally, the experimental verification is carried out by using dSPACE system. The results showed that the proposed TDMS adaptive friction compensation controller has good control accuracy. And it can effectively compensate the friction torque of lead screw nut differential drive. Compared with the application of traditional PID controller in TDMS, the tracking accuracy is improved by an order of magnitude.


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## Body

## 1. Introduction

The large-stroke and high-precision microfeed system plays a key role in high-end manufacturing equipment, and it is becoming increasingly significant in the fields of machining, assembly measurement, and so on. However, under the condition of low-speed motion, the conventional drive feed system (CDFS) based on linear motor and ball screw is inevitably affected by nonlinear friction due to the inherent characteristics of the system structure, resulting in low-speed creep, hysteresis creep, and limit cycle oscillation. It has a significant negative impact on the positioning accuracy and tracking error of the feed system [1–4]. In order to suppress the influence of friction, the friction compensation of drive system is needed.There are two main types of friction compensation methods. The first is compensation not based on the friction model. The friction is regarded as a disturbance, and the influence of friction is suppressed by improving the anti-interference ability of the system [5]. Li et al. designed an active disturbance rejection controller with robust item to compensate the nonlinear friction and external disturbance of the system [6]. Tomizuka proposed a zero phase error tracking controller to eliminate the instability of zero dynamics and greatly reduce the tracking error caused by friction [7]. He et al. proposed an active disturbance rejection control strategy for macro-micro-composite motion platform, which can effectively suppress the influence of nonlinear friction and elastic vibration [8]. Hu et al. designed the sliding membrane variable structure controller for friction compensation based on the disturbance observer [9]. The second kind is compensation based on the friction model. This kind of compensation method depends on establishing an accurate friction model. More mature friction models include the Coulomb model, the Stribeck model, the Dahl model, and the LuGre friction model. The LuGre friction model [10, 11] is a more perfect dynamic friction model, which can accurately describe the complex friction characteristics of the mechanical system, such as presliding displacement, variable static friction, friction hysteresis, and Stribeck effect. Zhang et al. [12] proposed a friction observer for compensation based on the LuGre model. Jiang et al. designed a double observer friction compensation method to effectively reduce the nonlinear friction interference in the performance of the digital hydraulic cylinder [13]. Wei et al. proposed a friction compensation control method based on double observers for pneumatic position servo system, which suppressed the crawling phenomenon at low speed and stick-slip oscillation at high speed [14]. The authors of [15, 16] also proposed an adaptive control method with friction compensation based on the accurate friction model to reduce the adverse impact of nonlinear friction on the tracking accuracy of the servo drive system. Yaolong Tan and Ioannis Kanellakopoulos took the three friction parameters of the system as uncertain parameters for adaptive law design and reflected the uneven change of each friction parameter through the parameter estimation value [17].However, the above methods do not fundamentally solve the adverse effects of nonlinear friction. Therefore, we designed a novel two-axis differential microfeed system (TDMS) [18–21]. The lead screw and nut are driven by servo motor and run at a speed higher than the critical crawling speed. Their speed is close and the synthetic speed is low. Compared with CDFS, the nonlinear friction interference is significantly reduced. In the past studies, it was found that the critical speed of crawling phenomenon caused by TDMS synthetic motion was much lower than that of CDFS. However, under the condition of low rotational speed of two axes, the lead screw drive shaft and nut drive shaft were affected by nonlinear friction to various degrees. A friction feedforward compensation method is proposed in [19], but it depends on an accurate friction model. It is necessary to accurately identify the friction parameters of the system and cannot reflect the change of friction state with wear, temperature, and other factors. In order to further suppress the friction interference, based on the LuGre friction model, this study improves the friction model for the new two-axis differential microfeed system and proposes an adaptive friction compensation control method. The experimental results show that the control strategy proposed in this study obviously suppresses the adverse effect of nonlinear friction of the lead screw drive shaft and nut drive shaft. The tracking accuracy of the new two-axis differential microfeed system is improved.


## 2. Structure of Two-Axis Differential Microfeed System

The structure of the two-axis differential microfeed system is shown in Figure1. The traditional ball screw nut pair driven by the lead screw rotation is changed into a new transmission mechanism in which the lead screw nut can rotate. The servo motor of screw drive shaft is directly connected with the ball screw through the coupling, and the servo motor of nut drive shaft drives the nut to rotate through the synchronous belt. The motion mechanism is as follows. Based on the differential composite principle of screw drive, two servo motors drive the screw and the nut, respectively. The working speed area of the drive shaft is shown in Figure 2, which is higher than the critical speed for crawling. The rotation direction of the two drive shafts is the same and the speed is close to the same. The worktable moves evenly at a very low speed. In this way, the micro-nanofeed of TDMS worktable is realized. In addition to the differential movement mode, the system can also control the movement direction of the lead screw and nut to be opposite, in order to achieve fast feed. It can also be driven separately by the lead screw drive shaft, that is, the traditional conventional drive mode. Therefore, TDMS with multiple working modes has the advantages of low critical crawling speed, large movement stroke, wide speed change range, strong load capacity, large stiffness, and so on.Figure 1
Schematic diagram of mechanical structure of the two-axis differential microfeed system.Figure 2
Drive shaft speed distribution.


## 3. Mathematical Model of Two-Axis Differential Microfeed System

### 3.1. Dynamic Model of TDMS

Based on the screw transmission principle, differential synthesis principle, and servo drive technology, the two-axis differential microfeed system obtains microfeed on the same axis through differential speed. Figure3 shows the dynamic model of TDMS.Figure 3
Dynamic model of the two-axis differential microfeed system.Its mathematical expression is described as(1)Jθ¨=Ktu−Mf−Tl,(2)Mf=Msf−Mnf,(3)Tl=Tsl−Tnl,(4)θ=θms−θmn,(5)u=us−un,(6)Tsl=r⋅Fds,(7)Tnl=r⋅Fdn,(8)Fds=Mtx¨tθ˙ms>θ˙mn0θ˙ms≤θ˙mn,(9)Fdn=0θ˙ms>θ˙mnMtx¨tθ˙ms≤θ˙mn,where J is the equivalent moment of inertia of the drive shaft. In the predesign process, the equivalent moment of inertia of screw drive shaft Js and the equivalent moment of inertia of nut drive shaft Jn are equal by adjusting the quality of flange and coupling, and Js=Jn=J; Kt is motor torque constant; us is output control quantity for lead screw drive shaft; un is output control quantity for lead nut drive shaft; Msf is the friction torque of the lead screw drive shaft, and it also includes the equivalent friction torque at the guide rail; Mnf is the equivalent friction torque of the nut drive shaft; Tsl is the load torque equivalent to the screw drive shaft, and it consists of rated load torque and external disturbance load torque; Tnl is the load torque equivalent to the nut drive shaft; Fd is the thrust on the table; Mt is the total mass of the table; r is transmission ratio between linear motion and rotary motion.


### 3.2. Friction Modeling of TDMS

The friction modeling of the conventional drive feed system is usually regarded as a whole, and the friction is equivalent to the drive shaft for modeling. For the two-axis differential microfeed system, based on the LuGre friction model, the friction modeling of lead screw drive shaft and nut drive shaft in TDMS is carried out, respectively:(10)Mif=σi0zi+σi1z˙i+σi2θ˙mi,(11)z˙i=θ˙mi−θ˙migθ˙mizi,(12)σi0gθ˙mi=Tci+Tsi−Tcie−θ˙mi/θ˙i2,where i=s,n represents the screw drive shaft and nut drive shaft, respectively, Mif is the friction torque of the drive shaft, Tci is the Coulomb friction torque of the drive shaft, Tsi is the maximum static friction torque of the drive shaft, zi is the average offset between sideburns, σi0 is the rigidity coefficient of the drive shaft, σi1 is the damping coefficient of the drive shaft, σi1 is viscous friction coefficient of drive shaft, θ˙i is the Stribeck speed of the drive shaft, and gθ˙mi>0.When the two-axis differential microfeed system works, the screw drive shaft and nut drive shaft are in rotational motion. And their motion coupling will produce a certain vibration, resulting in the sideburns in the model being affected by stress. Furthermore, the elastic deformation will change and the displacement of sideburns will change, thus leading to the change of the friction torque. In addition, the differential speed, lubrication, wear, and temperature rise of the two shafts will further lead to changes in the friction torque [15]. These factors affect these friction parameters in a very complex way, which leads to changes in both static and dynamic friction parameters. Except parameter z, the other six parameters of the friction model can be determined through complex identification. A single unknown parameter may be used to link the physical reason of friction change with the parameters of friction model [22, 23]. However, the friction changes may not be uniform. Using only one unknown parameter can only reflect the uniform change of friction, which is just a special case. Therefore, the friction variation adaptive coefficient λsj and λnj are introduced to reflect the change of friction torque of lead screw drive shaft and nut drive shaft, respectively(j = 0,1,2). In this case, (10) can be expressed as(13)Mif=λi0σi0zi+λi1σi1z˙i+λi2σi2θ˙mi.When the TDMS is in the working state of single axle drive and the external environment remains unchanged,λsj=λnj=1. In this case, the TDMS friction torque model is the same as the LuGre basic model.


## 3.1. Dynamic Model of TDMS

Based on the screw transmission principle, differential synthesis principle, and servo drive technology, the two-axis differential microfeed system obtains microfeed on the same axis through differential speed. Figure3 shows the dynamic model of TDMS.Figure 3
Dynamic model of the two-axis differential microfeed system.Its mathematical expression is described as(1)Jθ¨=Ktu−Mf−Tl,(2)Mf=Msf−Mnf,(3)Tl=Tsl−Tnl,(4)θ=θms−θmn,(5)u=us−un,(6)Tsl=r⋅Fds,(7)Tnl=r⋅Fdn,(8)Fds=Mtx¨tθ˙ms>θ˙mn0θ˙ms≤θ˙mn,(9)Fdn=0θ˙ms>θ˙mnMtx¨tθ˙ms≤θ˙mn,where J is the equivalent moment of inertia of the drive shaft. In the predesign process, the equivalent moment of inertia of screw drive shaft Js and the equivalent moment of inertia of nut drive shaft Jn are equal by adjusting the quality of flange and coupling, and Js=Jn=J; Kt is motor torque constant; us is output control quantity for lead screw drive shaft; un is output control quantity for lead nut drive shaft; Msf is the friction torque of the lead screw drive shaft, and it also includes the equivalent friction torque at the guide rail; Mnf is the equivalent friction torque of the nut drive shaft; Tsl is the load torque equivalent to the screw drive shaft, and it consists of rated load torque and external disturbance load torque; Tnl is the load torque equivalent to the nut drive shaft; Fd is the thrust on the table; Mt is the total mass of the table; r is transmission ratio between linear motion and rotary motion.


## 3.2. Friction Modeling of TDMS

The friction modeling of the conventional drive feed system is usually regarded as a whole, and the friction is equivalent to the drive shaft for modeling. For the two-axis differential microfeed system, based on the LuGre friction model, the friction modeling of lead screw drive shaft and nut drive shaft in TDMS is carried out, respectively:(10)Mif=σi0zi+σi1z˙i+σi2θ˙mi,(11)z˙i=θ˙mi−θ˙migθ˙mizi,(12)σi0gθ˙mi=Tci+Tsi−Tcie−θ˙mi/θ˙i2,where i=s,n represents the screw drive shaft and nut drive shaft, respectively, Mif is the friction torque of the drive shaft, Tci is the Coulomb friction torque of the drive shaft, Tsi is the maximum static friction torque of the drive shaft, zi is the average offset between sideburns, σi0 is the rigidity coefficient of the drive shaft, σi1 is the damping coefficient of the drive shaft, σi1 is viscous friction coefficient of drive shaft, θ˙i is the Stribeck speed of the drive shaft, and gθ˙mi>0.When the two-axis differential microfeed system works, the screw drive shaft and nut drive shaft are in rotational motion. And their motion coupling will produce a certain vibration, resulting in the sideburns in the model being affected by stress. Furthermore, the elastic deformation will change and the displacement of sideburns will change, thus leading to the change of the friction torque. In addition, the differential speed, lubrication, wear, and temperature rise of the two shafts will further lead to changes in the friction torque [15]. These factors affect these friction parameters in a very complex way, which leads to changes in both static and dynamic friction parameters. Except parameter z, the other six parameters of the friction model can be determined through complex identification. A single unknown parameter may be used to link the physical reason of friction change with the parameters of friction model [22, 23]. However, the friction changes may not be uniform. Using only one unknown parameter can only reflect the uniform change of friction, which is just a special case. Therefore, the friction variation adaptive coefficient λsj and λnj are introduced to reflect the change of friction torque of lead screw drive shaft and nut drive shaft, respectively(j = 0,1,2). In this case, (10) can be expressed as(13)Mif=λi0σi0zi+λi1σi1z˙i+λi2σi2θ˙mi.When the TDMS is in the working state of single axle drive and the external environment remains unchanged,λsj=λnj=1. In this case, the TDMS friction torque model is the same as the LuGre basic model.


## 4. Design of Adaptive Friction Compensation Controller

Based on the dynamic model of the two-axis differential microfeed system, the output control quantity of the two axis is designed by using the integral backstepping method.First, from equations (1), (9), and (13), we can get the following equation:(14)Jθ¨ms−θ¨mn=Ktus−un−Tsl−λs0σs0zs−λs1σs1z˙s−λs2σs2θ˙ms+λn0σn0zn+λn1σn1z˙n+λn2σn2θ˙mn.Bring (11) into (14) to obtain(15)Jθ¨ms−θ¨mn=Ktus−un−Tsl−λs0σs0zs+λs1σs1φs1zs−λs1σs1θ˙ms−λs2σs2θ˙ms+λn0σn0zn−λn1σn1φn1zn+λn1σn1θ˙mn+λn2σn2θ˙mn,where φs1=θ˙ms/gθ˙ms; φn1=θ˙mn/gθ˙mn.By (15), we can get the control quantity of the lead screw drive shaft servo motor us and the control quantity of the nut drive shaft servo motor un. They are shown as(16)Ktus=Jθ¨ms+Tsl+λs0σs0zs−λs1σs1φs1zs+λs1σs1θ˙ms+λs2σs2θ˙ms,(17)Ktun=Jθ¨mn+λn0σn0zn−λn1σn1φn1zn+λn1σn1θ˙mn+λn2σn2θ˙mn.The tracking error of position loop of screw drive shaft servo systemes1 is defined as the first error variable:(18)es1=θs−θms,e˙s1=θ˙s−θ˙ms,where θs is expected rotation angle of lead screw drive shaft.The first Lyapunov function is preliminarily designed:(19)Vs1=12es12.Take the derivative of (19) to obtain(20)V˙s1=es1e˙s1.The designed reference speed control signal is as follows.(21)ωs=θ˙s+k1es1+k2χs,(22)χs=∫0tes1dt,where ωs is output control quantity of lead screw drive shaft and χs is the integral term; its function is to ensure that the tracking error of the system can approach zero under the interference of external load and the uncertainty of the model; k1 and k2 are the positive real number.The second error variablees2 describing the speed fluctuation in the lead screw drive shaft servo system is designed:(23)es2=ωs−θ˙ms.Bringing (21) into equation (24), we can obtain(24)es2=θ˙s−θ˙ms+k1es1+k2χs.Differentiate (24) to obtain(25)e˙s2=θ¨s−θ¨ms+k1e˙s1+k2es1.It can be seen from (25)that(26)θ¨ms=θ¨s+k1e˙s1+k2es1−e˙s2.It can be obtained from (16) and (26)that(27)Je˙s2=Jθ¨s+Jk1e˙s1+Jk2es1−Ktus+Tsl+φs1λszs+φs2λsθ˙ms,where λs, zs, and Tsl are unknown, and they are expressed as estimated values λ^s, z^s, and T^sl.Sideburns’ offsetzi in the LuGre friction model is not measurable. Two nonlinear observers are designed to estimate the friction state zi. The mathematical expression is as follows:(28)z^˙s0=θ˙ms−θ˙msgθ˙msz^s0+Ls0,(29)z^˙s1=θ˙ms−θ˙msgθ˙msz^s1+Ls1,where Ls0 and Ls1 is the error compensation term of the nonlinear observer of the lead screw drive shaft. The error compensation term can reduce the error between the estimated value and the actual value of sideburns offset. Its expression will be further determined in subsequent design.The second Lyapunov function is defined as(30)Vs=Vs1+12es22+12k2χs2+12γ1λ˜s02+12γ2λ˜s12+12γ3λ˜s22+12γ4T˜sl2+12λs0z˜s02+12λs1z˜s12,where γ1>0, γ2>0, γ3>0, and γ4>0, and they are the positive real number. Their values are determined by the designer. λ˜s0, λ˜s1, λ˜s2, T˜sl, z˜s0, and z˜s1 represent the errors of λs0, λs1, λs2, Tsl, zs0, and zs1, respectively:(31)λ˜s0=λs0−λ^s0λ˜s1=λs1−λ^s1λ˜s2=λs2−λ^s2,(32)T˜sl=Tsl−T^sl,(33)z˜s0=zs0−z^s0z˜s1=zs1−z^s1,The dynamics of the state estimation error can be derived from (28) and (33):(34)z˜˙s0=−θ˙msgθ˙msz˜s0−Ls0z˜˙s1=−θ˙msgθ˙msz˜s1−Ls1.The control rate of the servo motor of the lead screw drive shaft is designed according to (27):(35)Ktus=Jθ¨s+Jk1e˙s1+Jk2es1−Je˙s2+T^sl+σs0λ^s0z^s0−σs1φs1λ^s1z^s1+σs1λ^s1θ˙ms+σs2λ^s2θ˙ms.The control rateus can be obtained by introducing (18) and (24) into (35):(36)us=JKtθ¨s+1+k2−k12es1+k1+k3es2−k1k2χs+T^slJ+1Jσs0λ^s0z^s0−1Jσs1φs1λ^s1z^s1+1Jσs1λ^s1θ˙ms+1Jσs2λ^s2θ˙ms,where k3 is the positive real number. Their values are determined by the designer.Under a given working condition, the fluctuation of load torque is very small. So, the load torque can be assumed as a constant, which means that its derivative is equal to zero. However, the load torque has some a little variation in fact, and this assumption will reduce the control accuracy to some extent. Besides, in the actual working conditions, the nonconstant load torque changes slowly in the control cycle of the servo motor, so the time-variable load torque can still be assumed as a constant in the control cycle [24]. However, in the case of the large load torque change rate, it is necessary to design a new torque observer and redefine the Lyapunov function to complete the stability proof. The stability proof in this study is completed on the basis of the above assumption. The following proof method is not suitable for the situation where the load torque changes greatly and rapidly.It can be obtained from equation (28), equation (32)–(34), and equation (37) that(37)e˙s2=−es1−k3es2+1JT˜sl+σs0Jλs0z˜s0+λ˜s0z^s0−σs0φs1Jλs1z˜s1+λ˜s1z^s1+σs1λ˜s1θ˙ms+σs2λ˜s2θ˙ms.Take equation (20) and equations (35) and (37) into (30):(38)V˙s=−k1es12−k3es22+T˜sles2J−T^˙slγ4−λs0φs1z˜s02−λs1φs1z˜s12+λs0z˜s0σs0es2J−Ls0−λs1z˜s1Ls1+σs1φs1es2J+λ˜s0σs0es2Jz^s0−λ^˙s0γ1−λ˜s1σs1φs1es2Jz^s1+λ^˙s1γ2−es2σs1θ˙ms+λs2es2σs2θ˙ms−λ^˙s2γ3.The design of adaptive law and other parameters is further completed, and the controller can obtain asymptotic tracking performance under this parameter. And the system is globally asymptotically stable. The following adaptive laws and error compensation terms are preliminarily selected:(39)dλ^s0dt=γ1σs0es2Jz^s0,dλ^s1dt=γ2es2σs1θ˙ms−σs1φs1Jz^s1dλ^s2dt=γ3es2σs2θ˙ms,,(40)dT^sldt=γ4es2J,(41)Ls0=σs0es2JLs1=−σs1φs1es2J,Bring equations (40)–(42) into (38) to obtain(42)V˙s=−k1es12−k3es22−λs0θ˙msgθ˙msz˜s02−λs1θ˙msgθ˙msz˜s12.It can be seen from the above design and the properties of the LuGre friction model,k1>0, k3>0, λs0>0, λs1>0, and gθ˙ms>0. So, V˙s≤0.Combined with (30), when t⟶∞, Vs has a limit. Differentiate (42) to obtain(43)V¨s=−2k1es1θ˙s−θ˙ms−2k3es2θ¨s−θ¨ms+k1θ˙s−k1θ˙ms+k2es1−2λs0θ˙msgθ˙msz˜s0−θ˙msgθ˙msz˜s0−σs0es2J−2λs1θ˙msgθ˙msz˜s1−θ˙msgθ˙msz˜s1+σs1φs1es2J.Sinceθs, θms, θ˙s, θ˙ms, θ¨s, θ¨ms, es1, es2, z˜s0, and z˜s1 are all bounded in formula (43), therefore, V¨s is bounded. So, V˙s is uniformly continuous. According to Barbalat lemma, when t⟶∞, it can be known that es1⟶0 and es2⟶0.Therefore, errorses1 and es2 gradually converge to zero and meet the requirements of system stability.For the control quantityun of nut drive shaft, the same design idea and stability proof method are adopted according to (17), and the design of control rate is as follows:(44)un=JKtθ¨n+1+k5−k42en1+k5+k6en2−k4k5χn+1Jσn0λ^n0z^n0−1Jσn1φn1λ^n1z^n1+1Jσn1λ^n1θ˙mn+1Jσn2λ^n2θ˙mn.The adaptive law and error compensation term are designed as follows:(45)dλ^n0dt=γ5σn0en2Jz^n0,dλ^n1dt=γ6en2σn1θ˙mn−σn1φn1Jz^n1dλ^n2dt=γ7en2σn2θ˙mn,,(46)Ln0=σn0en2JLn1=−σn1φn1en2J.Therefore, in the TDMS system, the errors of the lead screw drive shaft servo system and the nut drive shaft servo system are close to zero, that is, the two-axis differential microfeed system meets the requirements of global asymptotic stability. The overall control block diagram is shown in Figure4.Figure 4
Control block diagram of two-axis differential microfeed system.


## 5. Real-Time Simulation Experiment and Result Analysis

### 5.1. Experimental Platform

In order to verify the effect of the designed adaptive friction compensation controller on the two-axis differential microfeed system, dSPACE controller and Panasonic servo driver are selected to realize the motion control of TDMS. As shown in Figure5, the test-bed of two-axis differential microfeed system has a maximum stroke of 280 mm, a platform mass of 20 kg, a ball screw model of DIR1605-THK, a diameter of 16 mm, a lead of 5 mm, and a linear guide rail of EGH15CA-HIWIN. Two permanent magnet synchronous servo motors MHMF042LJV2M are used to drive the lead screw shaft and nut shaft, respectively. Both drive shafts are equipped with MBDLN25SG driver. The controller model used in this study is MicroLabBox(ds1202). It depends on the Simulink mathematical model in MATLAB. It is required to solve it in a fixed step size and save it as an SDF file. The algorithm is directly imported into MicroLabBox through ControlDesk to realize the user-defined algorithm control of the motor.Figure 5
Test bench of two-axis differential microfeed system.The data acquisition and measurement system adopts a color laser coaxial displacement measuring instrument with model CL-3000, as shown in Figure6. The sampling period of the displacement measuring instrument is 0.001s, and the accuracy can reach 10 nm.Figure 6
Color laser coaxial displacement measuring instrument.The friction parameters of TDMS are identified. Combined with the method in [18], the friction parameters of the whole assembly are identified first to obtain the friction parameters of the nut drive shaft, and then, the friction at the guide rail is equivalent to that at the lead screw drive shaft for identification. The results are shown in Table 1. The friction torques of forward and reverse rotation of ball screw are different. The corresponding friction parameters are selected under different motion directions. In the follow-up experiments of this study, tests were carried out in the forward rotation mode.Table 1
Friction model parameters of two-axis differential microfeed system.
ParameterLead screw drive shaftLead nut drive shaftForward rotationReverse rotationForward rotationReverse rotationFc/N92.3−86.576.6−72.1Fs/N114.8−105.1108.9−101.3vs/m/s1.13 × 10–3−1.16 × 10–39.31 × 10–4−9.46 × 10–4σ2/N⋅s/m3.05 × 1032.64 × 1033.12 × 1032.99 × 103σ0/N/m3.56 × 1073.62 × 107σ1/N⋅s/m5.12 × 1035.70 × 103


### 5.2. Experimental Verification

The adjustment of controller parameters is a very complex process; this paper adopts a relatively simple engineering tuning method. The controller parameterk affects overshoot and response speed. Smaller k will lead to low control precision and poor tracking performance. Larger k can lead to system instability and deterioration of control effect. The parameter γ mainly affects the speed and effect of the adaptive process of the system. Usually, a smaller parameter γ is conducive to faster and more stable convergence of the estimated value. Adjust the parameters by dSPACE, and the controller parameters increase from zero. Through a large number of experimental tests, the optimal parameter combination is selected. After repeated debugging, the controller parameters are as follows: k1=55, k2=10, k3=12, γ1=0.3, γ2=0.3, γ3=0.5, γ4=0.01, k4=70, k5=2, k6=20, γ5=0.2, γ6=0.2, and γ7=0.3.Firstly, the TDMS input slope signal is compared with the traditional PID controller to analyze the effectiveness of the TDMS adaptive friction compensation controller proposed in this study. The PID controller is tuned by the critical proportion method, and the appropriate parameters are obtained by continuous fine-tuning. When the parameters of the two controllers are optimal, the experimental test and comparative analysis are completed. Set the speed of the lead screw motor equivalent to linear motion asvs  = 9 mm/s and the speed of the lead nut motor equivalent to linear motion as vn  = 6 mm/s. That is, the actual speed of the workbench is vt  = 3 mm/s. The tracking error under different control modes is shown in Figure 7. Under the PID control without friction compensation, the tracking error of the system is stable at 40 μm. When adaptive friction compensation control is adopted, the tracking accuracy is significantly improved, and the tracking error is about 5 μm.Figure 7
Ramp signal input tracking error curve. (a) Nonfriction compensated PID control tracking error curve. (b) Adaptive friction compensation control tracks the error curve.
(a)(b)In order to further verify the effectiveness of the proposed controller for two-axis friction compensation, the sine wave input is considered. The lead screw shaft position command isxs=9sin0.4πt. The lead nut shaft position command is xs=6sin0.4πt. The workbench position command after motion synthesis is xt=3sin0.4πt. Figure 8 shows the sinusoidal response curve of the system under PID control, and its tracking error is about 50 μm. Figure 9 shows the response curve of adaptive friction compensation control. When vt  = 0, it is most seriously affected by nonlinear friction, and the tracking error reaches the maximum value of 5.5 μm. In addition, the tracking error is stable at ± 4 μm.Figure 8
Sinusoidal input response curve of PID control.Figure 9
Sinusoidal response curve of adaptive friction compensation control. (a) Sinusoidal input response curve of the table. (b) Table sinusoidal input tracking error curve.
(a)(b)Under the adaptive friction compensation control, the sine wave frequency is changed to make the position command of the worktable after motion synthesis bext=3sin0.6πt. The tracking error curve is shown in Figure 10. The maximum tracking error is 6.5 μm when the worktable is reversing. In addition, the tracking error is stable at ± 5 μm. It can be seen that the tracking accuracy is significantly improved compared with the PID control without friction compensation, which proves the effectiveness of the adaptive friction compensation controller proposed in this study.Figure 10
Sinusoidal input tracking error curve of adaptive friction compensation control.Change the combination mode of two-axis input position command curve to give a constant speed component to the nut drive shaft. And ensure that the desired position curve of the synthesized workbench isxt=3sin0.4πt. Take the lead screw drive shaft position command as xs=3sin0.4πt+6t, and take the lead screw drive shaft position command as xn=6t. This ensures that both shafts work in a high-speed motion range greater than the creeping speed of the drive shaft. As can be seen from Figure 11, under this motion combination mode, the phenomenon of significant increase in tracking error caused by zero crossing of sinusoidal motion speed is well improved, and the tracking accuracy is stable at ± 4 μm. This proves the superiority of the two-axis differential microfeed system in which both the lead screw drive shaft and the nut drive shaft work in the high-speed area.Figure 11
Tracking error curve.In order to explore the influence of constant speed value of nut drive shaft on the accuracy of worktable, the position command of the composite motion is maintained asxt=3sin0.4πt, and nut drive shaft speed vn  = 0∼15 mm/s. So, the lead screw drive shaft position command is xs=3sin0.4πt+vnt. The experiment is carried out under the same controller parameters, working environment, and other conditions. And the maximum tracking error is taken for analysis. The results are shown in Figure 12. When the speed of nut drive shaft is low, it is greatly affected by nonlinear friction, and the error is about 11 μm. With the increase of rotating speed, the two shafts gradually work in the high-speed area, leaving the presliding friction area and the Stribeck effect area. The tracking accuracy is gradually improved, and the optimal value is reached at 6 mm/s accessories. Then, the tracking error gradually increases with the increase of nut drive shaft speed. Therefore, when using the adaptive friction compensation controller proposed in this study, the constant speed of nut drive shaft should be in the range of 4 ∼ 10 mm/s to ensure that TDMS is in the optimal working state.Figure 12
Influence of nut drive shaft speed on tracking error.


## 5.1. Experimental Platform

In order to verify the effect of the designed adaptive friction compensation controller on the two-axis differential microfeed system, dSPACE controller and Panasonic servo driver are selected to realize the motion control of TDMS. As shown in Figure5, the test-bed of two-axis differential microfeed system has a maximum stroke of 280 mm, a platform mass of 20 kg, a ball screw model of DIR1605-THK, a diameter of 16 mm, a lead of 5 mm, and a linear guide rail of EGH15CA-HIWIN. Two permanent magnet synchronous servo motors MHMF042LJV2M are used to drive the lead screw shaft and nut shaft, respectively. Both drive shafts are equipped with MBDLN25SG driver. The controller model used in this study is MicroLabBox(ds1202). It depends on the Simulink mathematical model in MATLAB. It is required to solve it in a fixed step size and save it as an SDF file. The algorithm is directly imported into MicroLabBox through ControlDesk to realize the user-defined algorithm control of the motor.Figure 5
Test bench of two-axis differential microfeed system.The data acquisition and measurement system adopts a color laser coaxial displacement measuring instrument with model CL-3000, as shown in Figure6. The sampling period of the displacement measuring instrument is 0.001s, and the accuracy can reach 10 nm.Figure 6
Color laser coaxial displacement measuring instrument.The friction parameters of TDMS are identified. Combined with the method in [18], the friction parameters of the whole assembly are identified first to obtain the friction parameters of the nut drive shaft, and then, the friction at the guide rail is equivalent to that at the lead screw drive shaft for identification. The results are shown in Table 1. The friction torques of forward and reverse rotation of ball screw are different. The corresponding friction parameters are selected under different motion directions. In the follow-up experiments of this study, tests were carried out in the forward rotation mode.Table 1
Friction model parameters of two-axis differential microfeed system.
ParameterLead screw drive shaftLead nut drive shaftForward rotationReverse rotationForward rotationReverse rotationFc/N92.3−86.576.6−72.1Fs/N114.8−105.1108.9−101.3vs/m/s1.13 × 10–3−1.16 × 10–39.31 × 10–4−9.46 × 10–4σ2/N⋅s/m3.05 × 1032.64 × 1033.12 × 1032.99 × 103σ0/N/m3.56 × 1073.62 × 107σ1/N⋅s/m5.12 × 1035.70 × 103


## 5.2. Experimental Verification

The adjustment of controller parameters is a very complex process; this paper adopts a relatively simple engineering tuning method. The controller parameterk affects overshoot and response speed. Smaller k will lead to low control precision and poor tracking performance. Larger k can lead to system instability and deterioration of control effect. The parameter γ mainly affects the speed and effect of the adaptive process of the system. Usually, a smaller parameter γ is conducive to faster and more stable convergence of the estimated value. Adjust the parameters by dSPACE, and the controller parameters increase from zero. Through a large number of experimental tests, the optimal parameter combination is selected. After repeated debugging, the controller parameters are as follows: k1=55, k2=10, k3=12, γ1=0.3, γ2=0.3, γ3=0.5, γ4=0.01, k4=70, k5=2, k6=20, γ5=0.2, γ6=0.2, and γ7=0.3.Firstly, the TDMS input slope signal is compared with the traditional PID controller to analyze the effectiveness of the TDMS adaptive friction compensation controller proposed in this study. The PID controller is tuned by the critical proportion method, and the appropriate parameters are obtained by continuous fine-tuning. When the parameters of the two controllers are optimal, the experimental test and comparative analysis are completed. Set the speed of the lead screw motor equivalent to linear motion asvs  = 9 mm/s and the speed of the lead nut motor equivalent to linear motion as vn  = 6 mm/s. That is, the actual speed of the workbench is vt  = 3 mm/s. The tracking error under different control modes is shown in Figure 7. Under the PID control without friction compensation, the tracking error of the system is stable at 40 μm. When adaptive friction compensation control is adopted, the tracking accuracy is significantly improved, and the tracking error is about 5 μm.Figure 7
Ramp signal input tracking error curve. (a) Nonfriction compensated PID control tracking error curve. (b) Adaptive friction compensation control tracks the error curve.
(a)(b)In order to further verify the effectiveness of the proposed controller for two-axis friction compensation, the sine wave input is considered. The lead screw shaft position command isxs=9sin0.4πt. The lead nut shaft position command is xs=6sin0.4πt. The workbench position command after motion synthesis is xt=3sin0.4πt. Figure 8 shows the sinusoidal response curve of the system under PID control, and its tracking error is about 50 μm. Figure 9 shows the response curve of adaptive friction compensation control. When vt  = 0, it is most seriously affected by nonlinear friction, and the tracking error reaches the maximum value of 5.5 μm. In addition, the tracking error is stable at ± 4 μm.Figure 8
Sinusoidal input response curve of PID control.Figure 9
Sinusoidal response curve of adaptive friction compensation control. (a) Sinusoidal input response curve of the table. (b) Table sinusoidal input tracking error curve.
(a)(b)Under the adaptive friction compensation control, the sine wave frequency is changed to make the position command of the worktable after motion synthesis bext=3sin0.6πt. The tracking error curve is shown in Figure 10. The maximum tracking error is 6.5 μm when the worktable is reversing. In addition, the tracking error is stable at ± 5 μm. It can be seen that the tracking accuracy is significantly improved compared with the PID control without friction compensation, which proves the effectiveness of the adaptive friction compensation controller proposed in this study.Figure 10
Sinusoidal input tracking error curve of adaptive friction compensation control.Change the combination mode of two-axis input position command curve to give a constant speed component to the nut drive shaft. And ensure that the desired position curve of the synthesized workbench isxt=3sin0.4πt. Take the lead screw drive shaft position command as xs=3sin0.4πt+6t, and take the lead screw drive shaft position command as xn=6t. This ensures that both shafts work in a high-speed motion range greater than the creeping speed of the drive shaft. As can be seen from Figure 11, under this motion combination mode, the phenomenon of significant increase in tracking error caused by zero crossing of sinusoidal motion speed is well improved, and the tracking accuracy is stable at ± 4 μm. This proves the superiority of the two-axis differential microfeed system in which both the lead screw drive shaft and the nut drive shaft work in the high-speed area.Figure 11
Tracking error curve.In order to explore the influence of constant speed value of nut drive shaft on the accuracy of worktable, the position command of the composite motion is maintained asxt=3sin0.4πt, and nut drive shaft speed vn  = 0∼15 mm/s. So, the lead screw drive shaft position command is xs=3sin0.4πt+vnt. The experiment is carried out under the same controller parameters, working environment, and other conditions. And the maximum tracking error is taken for analysis. The results are shown in Figure 12. When the speed of nut drive shaft is low, it is greatly affected by nonlinear friction, and the error is about 11 μm. With the increase of rotating speed, the two shafts gradually work in the high-speed area, leaving the presliding friction area and the Stribeck effect area. The tracking accuracy is gradually improved, and the optimal value is reached at 6 mm/s accessories. Then, the tracking error gradually increases with the increase of nut drive shaft speed. Therefore, when using the adaptive friction compensation controller proposed in this study, the constant speed of nut drive shaft should be in the range of 4 ∼ 10 mm/s to ensure that TDMS is in the optimal working state.Figure 12
Influence of nut drive shaft speed on tracking error.


## 6. Conclusions

In this study, for the two-axis differential microfeed system, considering the influence of its motion synthesis on the sideburns offset and other friction states of the LuGre friction model, the friction change adaptive coefficientsλsj and λnj are introduced. The adaptive friction compensation controller of TDMS is designed based on the dynamic model. In the specific condition of operation for which the load torque can be considered varying slowly, through the real-time online simulation experiment, the following conclusions are obtained:(1)
Through comparative experiments, it is proved that the adaptive friction compensation controller proposed in this study is more effective than the traditional PID controller for friction compensation of the two-axis differential microfeed system. And the tracking accuracy is stable at ± 5μm. The problem of tracking accuracy degradation caused by friction is effectively improved.(2)
Through the different position command combination of lead screw drive shaft and nut drive shaft, it is verified that the two-axis differential microfeed system effectively solves the problem of sudden increase of tracking error caused by zero crossing of sinusoidal motion speed. Both axes can work in high-speed area and avoid low-speed crawling and nonlinear creep area, which improves the control performance of the system.



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*Source: 3915600-2022-05-02.xml*