# Dynamic Characteristics Analysis and Test of Dual-Driving Feed System Driven by Center of Gravity

Authors: Hong Lu; Wei Fan; Xinbao Zhang; Yongquan Zhang; Shaojun Wang; Meng Duan
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Dual-driving feed system (DDFS) driven by center of gravity (DCG) has been widely used in advanced manufacturing machine for its high rigidity and precision. However, the DCG technology requires that the joint force coincides with the center of gravity of the sliding stage. The dual-driving synchronization and tracking performance will be affected by the change of center of gravity of the sliding stage. Therefore, this paper proposes dynamic characteristics modeling, identification, and control scheme for DDFS driven by center of gravity (DCG). Firstly, a redundancy dynamic model including rotation and pitch vibration caused by the change of the position of center of gravity is presented for DDFS DCG based on the Lagrange method. The model parameters are identified by system identification experiment, and the predictive natural frequencies and vibration modes by the proposed dynamic model are compared by modal experiment. Moreover, the dynamic model-based cross-coupled sliding mode control (CCSMC) is proposed for DDFS DCG. Then, the proposed dynamic model-based CCSMC has been compared with normal cross-coupled sliding mode control (NCCSMC). Both the simulation and experimental results show that the proposed dynamic characteristics analysis and test scheme of DDFS DCG are validated effectively by comparisons.


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## Body

## 1. Introduction

Traditional drive mechanism used single servo motor and ball screw in gantry-moving machine. However, in this single drive structure, the deformation and vibration of screw are inevitably caused by the deviation between the drive force and center of gravity. Furthermore, the load capacity is restricted owing to the limit of the power and torque of single servo motor. The DDFS is used on high-end industrial applications such as surface chip mounting, precision metrology, and high-speed and heavy load manufacturing [1]. Since the joint force is provided by twin motors and screws, the system rigidity would be significantly increased, and double overall thrust would afford to drive the stage.Moreover, based on driving by center of gravity (DCG), DDFS can improve the system dynamics with high acceleration and low vibration. The DCG technology is innovatively investigated by Mori Seiki. This twin ball screws structure could ensure the direction of drive forces close to the center of stage gravity, so that the vibration of machine tools could be decreased dramatically [2].Numerous researches have conducted ball screw system and DDFS in CNC machine tools [3–6]. Most of them focused on the synchronous control algorithm based on master-slave or cross-coupled control structure and simplified the mechanical system as transmission ratio for simply research [7, 8]. The synchronous control strategies have certain effects to improve the synchronization performance for common DDFS. However, the mechanical characteristics diversity and coupling factor of DDFS are ignored in these researches. In fact, the performance of DDFS suffered from the diversity of mechanism characteristics, deflection of joint thrust, and mechanical coupling [9–12]. To improve the synchronous performance, some scholars established the mechanical coupling models by identification technique [13–15], yet the models are established as first-order inertia links, and the parameters are identified without actual physical meaning [16].Furthermore, as to the dynamic modeling based on DCG technology, not only the differences of servo and mechanical characteristics between the two axis but also the change of the center of gravity of the sliding stage should be considered. Nevertheless, most researches seldom consider the effect of position of center of gravity. Most of the researches assume that the DCG feed drive is implemented on a horizontal plane, and the position of center of gravity always remains invariant. Actually, if the sliding stage was mounted with different loads, or the position of the sliding stage along the directionY, perpendicular to dual-driving feed direction, was changed, the synchronization of the DDFS will be inevitably affected. Since the height of center of gravity does not equal zero, both torsion vibration and pitch mode should be taken into consideration in dynamic research of dual-driving system.In addition, the synchronization and tracking accuracy of DDFS are determined by the bandwidth of servo control loop [17, 18]. The bandwidth of the DDFS is still limited by the structural resonances arising from the mechanical structure [19]. Especially in high acceleration motion, the bandwidth is limited by a specific vibration mode in which the dual-driving stage oscillates as the double ball screws undergo torsion and pitch vibrations. Some researches built the dynamic model by considering vibration characteristics of feed system in CNC machine tool [20–22]. These are helpful for us to analyze the dynamic characteristics of DDFS DCG.As to the synchronous control, many scholars have developed the motion control strategy for DDFS from the perspective of tandem control and cross-coupled control [23]. In the former case, the tandem control is the most widely used method in industry and the DDFS is distinguished as master and slave axis. The output of the master motor is considered as the input of the slave motor. The slave motor has to follow the exact position of the master motor to maintain synchronization. However, the disturbance of the slave axis cannot affect the controller of the master axis, and the synchronization accuracy needs to be improved. In the second one, the DDFS is considered as two coupled axes, and the parameters variation is considered as disturbances. The cross-coupled controller between the dual axes can compensate the synchronous error. However, to avoid the large overshoot of dual axis following each other [24], the dynamic characteristics caused by parameters variation and coupling factor of DDFS need to be modeled and analyzed.Hence, in order to help to analyze the natural frequency and dynamic characteristics of DDFS DCG and also benefit the synchronous control, the redundancy dynamic model including rotation and pitch errors for DDFS DCG is proposed in this paper. The contributions of this work can be listed as follows:(i)
The redundancy dynamic model including rotation and pitch errors of the DDFS DCG is developed. This model is different from the previous mathematical model which considering the center of gravity is in the geometric center. In this way, the influence because of the time-varying position change of center of gravity of the sliding stage is theoretically discussed and experimentally validated. Moreover, the variation parameters of dual axis are obtained by practical identification experiment.(ii)
The dynamic model-based CCSMC has been proposed for DDFS DCG. The maximum and average synchronous errors of the proposed control scheme are compared with NCCSMC. The comparison results show the effectiveness of the proposed model and control scheme.(iii)
This work focuses on the practical application of high-precision feed system in machine tools. The proposed model of used dual-driving stage is verified by natural frequency with the increasing of height of center of gravity; the value and changing trend of simulated natural frequency is tested by experimental results.The structure is organized as follows. In Section2, the redundancy dynamic model including rotation and pitch errors for DDFS DCG is developed, along with the DCG mechanism analysis. Section 3 concerns the validation and comparison of the proposed model by practical DDFS DCG. The dynamic model-based CCSMC is presented in Section 4. The experimental results show the effectiveness of the proposed dynamic analysis and test method. Conclusions are drawn in Section 5.


## 2. Dynamic Modeling and Parameters Identification of DDFS DCG

The DDFS DCG used in this paper is composed of twin AC200-v permanent magnet synchronous motor (PMSM) and ball screws; namely, on each feed direction, the table is driven by two motors and two screws joint together. A simplified sketch of the proposed typical gantry stage with DCG DDFS is illustrated in Figure1. In the DDFS DCG, the one-axis movement is achieved by parallel twin motors and screws. Two ball screws aligned in parallel guide to move the table in y-axis, and this moving table serves as the base of the sliding stage which moves along x-axis.Figure 1
Basic structure and mechanism of DCG DDFS.The dual-driving stage is a high-speed, high-precision positioning system. It consists of a dual-driving stage that is moved by two parallel ball screws (axesy1 and y2). The stage is mounted on the geometry center so that the initial coordinate of center of gravity x0,y0,z0 locates on the resultant forces. At this case, the DCG dual-driving can be achieved if the driving forces are equal. However, as can be seen in Figure 1, the feed accuracy of DDFS is suffered from the diversity of servo and mechanical characteristics, deflection of joint thrust, frictions, and so on. Moreover, the stage moved along x-axis, and different workpiece which mounted on the stage will increase the height of center of gravity. The dynamics of DDFS should be considered based on transmission coordinate xg,yg,zg. The travel length along x-axis can vary from the middle position to the left-hand (or right-hand) side. And the height of center of gravity is increased depending on the application and work required.


### 2.1. Redundancy Dynamic Model of DCG DDFS

The proposed redundancy dynamic model of DCG DDFS is developed in this section. Based on the dual-driving structure and mechanism shown in Figure1, the lumped-parameter model is described in Figures 2 and 3. The lumped model is described based on the change of position of center of gravity. In this model, the new coordinate system considers not only the linear positions of the sliding stage, but also the rotation and pitch around center of gravity of the sliding stage.Figure 2
Lumped-parameter model considers rotation aroundz-axis.Figure 3
Lumped-parameter model considers pitch aroundx-axis.As can be seen in Figures2 and 3, the parameters are defined as follows:myi is the equivalent mass of each axis.ms is the mass of the sliding stage.yi is the actual linear displacement of each axis.Fei is the equivalent force of each axis.Xg, Yg, and Zg are the displacements between xg,yg,zg and x0,y0,z0 in x-axis, Y-axis, and z-axis, respectively.θgx is the pitch angle around x-axis;  θgz is the rotation angle around z-axis.khz is the support stiffness of slider; kbx is the later stiffness of slider.ke is the equivalent axial stiffness of screw systems, which would be identified by actual feed system.l1 and l2 are the displacement from force points of screws to center of mass in the sliding stage.L is the distance of dual screws.a is the axial distance of sliders.The drive force can be obtained from equivalent axial stiffness:(1)F1=key1-ys1,F2=key2-ys2,where ys1 and ys2 are the equivalent linear displacements of double servo motors.The total drive force of table can be deduced as(2)F=Fe1+Fe2,Fe1=F1-cy1sign⁡y˙1,Fe2=F2-cy2sign⁡y˙2.c
y
1 and cy2 are forces corresponding to equivalent total friction of axis y1 and y2.When the workpiece is not clamped on the sliding stage, the drive torque of DDFS DCG structure around center of gravity is zero, and the pitch error will not be produced. After clamping the workpiece, according to the geometric relationship, the driving torque can be converted from the driving forces, that is(3)Tgx=Fe1+Fe2Zg.The driving torque of DCG dual-driving structure is smaller than normal single driving structure for its shorter arm of force. However, once the workpiece is installed on the stage, the driving torque is generated for the height of center of gravity is not zero. The pitch errors are also generated and cannot be compensated by adjusting the driving forces of dual axis.The lumped-parameter model of DCG dual-driving stage is shown in Figure2. The fundamental coordinate is established based on the original center of gravity Oox0,y0,z0. Oo is center of gravity of dual-driving stage without clamping the workpiece, which is designed to be on the horizon plane of axes of double screws. However, when the workpiece is clamped and moved along x-axis, the translation coordinate system is built based on the actual center of gravity Ogxg,yg,zg. During the dual-driving process in y-axis, the stage is driven by dynamic joint forces. Thus, the rotation around z-axis and pitch vibration around x-axis will be induced and impact on the feed accuracy.Since the yaw angle of cross-arm is limited to nearly 0.1 rad, thesin⁡θgz can be considered as θgz, and cos⁡θgz is simplified to be 1. Considering the torsion and pitch vibration, the displacements of dual axis are given by(4)Rotx,θgx=10000cos⁡θgxsin⁡θgx00-sin⁡θgxcos⁡θgx00001,Rotz,θgz=cos⁡θgzsin⁡θgz00-sin⁡θgzcos⁡θgz0000100001.Then, the displacements of each axis can be expressed as(5)y1=y-L2-Xgθgz+Zgθgx=y-l1θgz+Zgθgx,y2=y+L2+Xgθgz+Zgθgx=y+l2θgz+Zgθgx,where y1 and y2 are the displacements of the nuts of each axis and y is the displacement of the stage in Y direction. Since the actuator is mounted on the stage, y also represented the actual displacement of actuator. From (5) we can see that y is affected by the position of the center of gravity, the pitch errors θgx, and yaw errors θgz, which means that the research of position of center of gravity is necessary to improve the feed accuracy. The yaw errors θgz are reflected in the synchronous errors. In order to improve the tracking accuracy, the pitch errors θgx caused by the increase of center of gravity are required to be noticed and compensated.The microrotation angle can be expressed as(6)θgz=y2-y1L.Then the motion equations of the DCG dual-driving stage will be derived using Lagrange-Euler formalism. Based on the conservation of energy, Lagrange-Euler equation can be used in the dynamics modeling and analysis of multidegree of freedom system. Through the eigenvalue decomposition, the natural frequencies and vibration modes of DDFS DCG can be obtained. To this purpose, the kinetic energy of the DCG dual-driving system can be expressed as(7)T=12my1y˙12+my2y˙22+12msy˙2+12Jzθ˙gz2+12Jxθ˙gx2.Thus, two sets of geometric relationships in x-axis and z-axis can be used to specify the displacements of slide blocks. xbi are the linear displacements of slide blocks in the direction of x-axis. The relationship between xbi and torsion angle θgz is given by(8)xb1,3=xg±a2-Ygθgz,xb2,4=xg∓a2+Ygθgz.And the relation between displacements of slide blocks inz-axis and pitch angle θgx is given by(9)zb1,3=a2-Ygθgx,zb2,4=a2+Ygθgx.The dual-driving stage is designed to be symmetrical, and the center of gravity is designed to be located on the geometric center as mentioned. Compared to the deflections of center of gravity inx-axis and z-axis, the deflection in y-axis is negligible when clamping workpiece of different weights.The potential elastic energy can be obtained(10)U=12ke1ys1-y12+12ke2ys2-y22+12kbx∑i=14xi2+12kbz∑i=14zi2.According to Lagrange method, the system dynamics can be defined as(11)ddt∂L∂q˙j-∂L∂qj=Qj,where(12)q1=y1,q2=y2,q3=x,q4=θgz,q5=θgx,L=T-U.Normally, the center of gravity can be designed to be on the centerline ofy-axis. The changing of Yg is always in the same direction of driving force and would not cause the deviation between center of gravity and equivalent forces. Therefore, to simplify the dynamic model, Yg=0 is assumed here.The motion equations of the DCG dual-driving stage can be deduced as(13)Mq¨+Kq=F+U,where M and K are the simplified inertia (14) and stiffness (15) matrices; F is the force vector (16).(14)M=my100000my200000ms00000Jgz00000Jgx,(15)K=ke0-kel1ke-Zgke0ke-ke-l2ke-Zgke-ke-ke2kel2-l1ke2Zgkel1ke-l2kel2-l1kel12+l22ke+a2kbxl1-l2Zgke-Zgke-Zgke2Zgkel2-l1Zgke2Zg2ke+a2kbz,(16)F=Fe1Fe20TgzTgxFe1=ke1y1-ys1-cy1sign⁡y˙1Fe2=ke2y2-ys2-cy2sign⁡y˙2Tgz=Fe2l2-Fe1l1Tgx=Fe1+Fe2Zg.The stiffness matrixK points out to the mechanical coupling between the dual-driving axes y1 and y2 in the multidegree of freedom, which is affected by equivalent stiffness ke, kbx, kbz, and different positions of the center of gravity l2-l1, Zg. The simplified inertia and damping matrices show the nonuniform load and friction distribution. The first two lines of F are the force vector. Assume that l1=l2, my1=my2, cy1-cy2=0, and y1-ys1=y2-ys2, which means the mass, inertia, and friction of dual axes are equal and the workpiece is located on the middle of the stage. In this case, dual-driving synchronous feed can be ensured by Fe1=Fe2, which represent that the equivalent forces are equal and the joint force coincides with the center of gravity in X-Y plane. However, due to the different mechanical and servo characteristics of dual axis, and the center of gravity moving along x- and z-axis, the perfect synchronous dual-driving is hard to be achieved.Moreover, the last two lines of the force vectorF are torques around z-axis and x-axis, respectively. Traditionally, the dynamic model of DDFS is established based on two degrees of freedom, the linear displacement in y-axis and rotation angle around z-axis; the pitch errors caused by increase of position of center of gravity are totally ignored [5, 12]. Tgx is the torque around x-axis associated with the deviation of center of gravity in height. Only if the center of gravity is located on the X, Y plane, which means Zg=0, the torque around x-axis approaches zero. However, when the workpiece with different weight is located, it is impossible to ensure Zg=0. The joint force will not go through the center of gravity and pitch error would be generated. To achieve satisfactory DCG feed accuracy, the dynamic of pitch and torsion vibrations should be measured and analyzed in the dual-driving process.The determinant of the Lagrange equation can be obtained as(17)D=A110-kel1ke-Zgke0A22-ke-l2ke-Zgke-ke-keA33l2-l1ke2Zgkel1ke-l2kel2-l1keA44l1-l2Zgke-Zgke-Zgke2Zgkel2-l1ZgkeA55,(18)A11=-my12w2+ke,A22=-my22w2+ke,A33=-2ms2w2+ke,A44=-Jgzw2+l12+l22ke+a2kbx,A55=-Jgxw2+2Zg2ke+a2kbz.The determinant (17) expresses the axial, pitch, and torsion resonant modes. Solve the determinant D=0; wii=1⋯5 represent the natural frequencies.


### 2.2. Model Parameters Identification

As to (17) and (18), the dynamic parameters can be divided into three parts: inertia parameters (the inertia matrix M), jointed parameters (the stiffness matrix K and damping matrix C), and structure parameters (the constant parameters such as a, L). For the determined dual-driving DCG feed system, the structure parameters are assumed to be constant during the machining process. Hence, only the inertia and jointed parameters are analyzed and identified in this section.


#### 2.2.1. Test Platform Setup

As shown in Figures4 and 5, DDFS and National Instruments (NI) test platform are set up for system parameters identification. The DDFS is constructed by Beckhoff servo system and controller, which will also be employed in the feed verification experiment later. The sweep sine signal is generated as identification input by the dynamic signal analyzer (DSA) from NI. The frequency response data is also calculated and analyzed in DSA. A data acquisition (DAQ) card inserted in a PC is used to receive the analog signal. All the AD/DA conversion, system memory allocation, and accessing module are executed by controller of NI. The detection data is analyzed by LabVIEW frequency response analyzer code.Figure 4
Frequency response experiment diagram.Figure 5
Experiment setup of identification.The sweep sine input signal applied to the drive is generated by DSA module. And the same signal is also linked to the Beckhoff controller or amplifier. The velocity and acceleration limits of double motors are 1.5 m/s and 5 m/s2. The desired motion trajectories of dual-driving axis are given exactly the same to avoid desynchronization.


#### 2.2.2. Parameters Identification

As a major component of a dual-driving NC machine tool, the stiffness of the ball screws and linear guides are crucial to affect the positioning accuracy directly. Normally, the axial stiffnesske of screw system is estimated as the related components in series and parallel. The screw, nut, and support bearing influence the stiffness most. The dynamic model is shown in Figure 6.Figure 6
Simplified axial stiffness of screw.The integral axial stiffness of the screw transmission system is made up of the stiffness of screw, nut joint, and support bearing joints series-parallel connections. The equivalent axial stiffness can be given by(19)ke=ks1yb+kb1/ks1ybkb1+ks2yb+kb2/ks2ybkb2+knutks1yb+kb1/ks1ybkb1+ks2yb+kb2/ks2ybkb2knut.The equivalent axial stiffness can be obtained from the tension and torsion stiffness when the structure is determined, and this axial stiffness function is only affected by the located position of nut.Different frictional characteristics will generate the unbalanced forces in double axes. So the frictions model should be built accurately. The nonlinearity friction model is described as [11](20)cyv=FC+FS-FCe-v/vσδσ+Fvvv≠0,where vσ is the Stribeck velocity, FC is Coulomb friction, and FS is the maximum static friction. According to Newton’s Second Law, when the table moves in uniform motion, the Coulomb and viscous frictions of the y1 and y2 axes are equal to drive forces of motors. The velocities of dual axes are given at various constant velocities within 5 mm/s~14 m/min and the frictions can be easily obtained by the feedback drive forces of dual motors. Then, the measurement data can be processed with MATLAB nonlinear fitted function. The identification frictions of dual axis are fitted by practical detection results, as shown in Figure 7. As the velocity increases, friction starts decreasing, and as the velocity exceeds a certain critical value, which is called Stribeck velocity, friction starts to rise gradually again. The fitted parameters of friction model are shown in Table 1.Table 1
Physical parameters of the dual-driving system.
Name
Value
Description
m
y
1
7.53 kg
Mass of the axisy1
m
y
2
7.61 kg
Mass of the axisy2
m
s
510 kg
Mass of the sliding stage
J
g
z
78 kg·m2
Rotary inertia of the stage
J
g
x
51 kg·m2
Pitch inertia of the stage
k
e
1631.8 N·m/rad
Equivalent stiffness of screw
k
h
z
1572.5 N·m/rad
Support stiffness of slider
k
b
x
2.6734e6 N·m/rad
Lateral stiffness of slider
L
0.8 m
Distance of dual screws
a
0.45 m
Axial distance of the sliders
r
g
5 mm
Screw lead
F
s
1
117.7352 N
Maximum static friction ofy1
F
s
2
102.1365 N
Maximum static friction ofy2
F
c
1
30.2017 N
Coulomb friction of axisy1
F
c
2
26.8520 N
Coulomb friction of axisy2
F
v
1
0.0052 Ns/m
Viscosity coefficient of axisy1
F
v
2
0.0060 Ns/m
Viscosity coefficient of axisy2
v
σ
1
73.2533 mm/min
Stribeck velocity of axisy1
v
σ
2
109.3258 mm/min
Stribeck velocity of axisy2Figure 7
(a) Friction measuring and fitting of axisy1; (b) Friction measuring and fitting of axis y2.
(a)
(b)The equivalent mass of each axis can be estimated by moving they1 and y2 at constant acceleration cycles and making the center of gravity of the sliding stage in the geometric center. In this case, the sliding stage is located on the middle of dual axis, and there is no workpiece mounted on the stage, which means y¨1=y¨2, θgz≐0, and θgx=0. Then, the variation mass can be derived as (21)my1=key1-ys1-cy1sign⁡y˙1y¨1-1,my2=key2-ys2-cy2sign⁡y˙2y¨2-1.The displacements ofysi and yi can be obtained by the feedback of encoders and grating scales in each axis.The rotary inertia in the inertia matrix is estimated by the formula of moment of inertia.(22)Jgz=my1+my2+msL212,Jgx=my1+my2+msa212.The identified parameters are given in Table1.


## 2.1. Redundancy Dynamic Model of DCG DDFS

The proposed redundancy dynamic model of DCG DDFS is developed in this section. Based on the dual-driving structure and mechanism shown in Figure1, the lumped-parameter model is described in Figures 2 and 3. The lumped model is described based on the change of position of center of gravity. In this model, the new coordinate system considers not only the linear positions of the sliding stage, but also the rotation and pitch around center of gravity of the sliding stage.Figure 2
Lumped-parameter model considers rotation aroundz-axis.Figure 3
Lumped-parameter model considers pitch aroundx-axis.As can be seen in Figures2 and 3, the parameters are defined as follows:myi is the equivalent mass of each axis.ms is the mass of the sliding stage.yi is the actual linear displacement of each axis.Fei is the equivalent force of each axis.Xg, Yg, and Zg are the displacements between xg,yg,zg and x0,y0,z0 in x-axis, Y-axis, and z-axis, respectively.θgx is the pitch angle around x-axis;  θgz is the rotation angle around z-axis.khz is the support stiffness of slider; kbx is the later stiffness of slider.ke is the equivalent axial stiffness of screw systems, which would be identified by actual feed system.l1 and l2 are the displacement from force points of screws to center of mass in the sliding stage.L is the distance of dual screws.a is the axial distance of sliders.The drive force can be obtained from equivalent axial stiffness:(1)F1=key1-ys1,F2=key2-ys2,where ys1 and ys2 are the equivalent linear displacements of double servo motors.The total drive force of table can be deduced as(2)F=Fe1+Fe2,Fe1=F1-cy1sign⁡y˙1,Fe2=F2-cy2sign⁡y˙2.c
y
1 and cy2 are forces corresponding to equivalent total friction of axis y1 and y2.When the workpiece is not clamped on the sliding stage, the drive torque of DDFS DCG structure around center of gravity is zero, and the pitch error will not be produced. After clamping the workpiece, according to the geometric relationship, the driving torque can be converted from the driving forces, that is(3)Tgx=Fe1+Fe2Zg.The driving torque of DCG dual-driving structure is smaller than normal single driving structure for its shorter arm of force. However, once the workpiece is installed on the stage, the driving torque is generated for the height of center of gravity is not zero. The pitch errors are also generated and cannot be compensated by adjusting the driving forces of dual axis.The lumped-parameter model of DCG dual-driving stage is shown in Figure2. The fundamental coordinate is established based on the original center of gravity Oox0,y0,z0. Oo is center of gravity of dual-driving stage without clamping the workpiece, which is designed to be on the horizon plane of axes of double screws. However, when the workpiece is clamped and moved along x-axis, the translation coordinate system is built based on the actual center of gravity Ogxg,yg,zg. During the dual-driving process in y-axis, the stage is driven by dynamic joint forces. Thus, the rotation around z-axis and pitch vibration around x-axis will be induced and impact on the feed accuracy.Since the yaw angle of cross-arm is limited to nearly 0.1 rad, thesin⁡θgz can be considered as θgz, and cos⁡θgz is simplified to be 1. Considering the torsion and pitch vibration, the displacements of dual axis are given by(4)Rotx,θgx=10000cos⁡θgxsin⁡θgx00-sin⁡θgxcos⁡θgx00001,Rotz,θgz=cos⁡θgzsin⁡θgz00-sin⁡θgzcos⁡θgz0000100001.Then, the displacements of each axis can be expressed as(5)y1=y-L2-Xgθgz+Zgθgx=y-l1θgz+Zgθgx,y2=y+L2+Xgθgz+Zgθgx=y+l2θgz+Zgθgx,where y1 and y2 are the displacements of the nuts of each axis and y is the displacement of the stage in Y direction. Since the actuator is mounted on the stage, y also represented the actual displacement of actuator. From (5) we can see that y is affected by the position of the center of gravity, the pitch errors θgx, and yaw errors θgz, which means that the research of position of center of gravity is necessary to improve the feed accuracy. The yaw errors θgz are reflected in the synchronous errors. In order to improve the tracking accuracy, the pitch errors θgx caused by the increase of center of gravity are required to be noticed and compensated.The microrotation angle can be expressed as(6)θgz=y2-y1L.Then the motion equations of the DCG dual-driving stage will be derived using Lagrange-Euler formalism. Based on the conservation of energy, Lagrange-Euler equation can be used in the dynamics modeling and analysis of multidegree of freedom system. Through the eigenvalue decomposition, the natural frequencies and vibration modes of DDFS DCG can be obtained. To this purpose, the kinetic energy of the DCG dual-driving system can be expressed as(7)T=12my1y˙12+my2y˙22+12msy˙2+12Jzθ˙gz2+12Jxθ˙gx2.Thus, two sets of geometric relationships in x-axis and z-axis can be used to specify the displacements of slide blocks. xbi are the linear displacements of slide blocks in the direction of x-axis. The relationship between xbi and torsion angle θgz is given by(8)xb1,3=xg±a2-Ygθgz,xb2,4=xg∓a2+Ygθgz.And the relation between displacements of slide blocks inz-axis and pitch angle θgx is given by(9)zb1,3=a2-Ygθgx,zb2,4=a2+Ygθgx.The dual-driving stage is designed to be symmetrical, and the center of gravity is designed to be located on the geometric center as mentioned. Compared to the deflections of center of gravity inx-axis and z-axis, the deflection in y-axis is negligible when clamping workpiece of different weights.The potential elastic energy can be obtained(10)U=12ke1ys1-y12+12ke2ys2-y22+12kbx∑i=14xi2+12kbz∑i=14zi2.According to Lagrange method, the system dynamics can be defined as(11)ddt∂L∂q˙j-∂L∂qj=Qj,where(12)q1=y1,q2=y2,q3=x,q4=θgz,q5=θgx,L=T-U.Normally, the center of gravity can be designed to be on the centerline ofy-axis. The changing of Yg is always in the same direction of driving force and would not cause the deviation between center of gravity and equivalent forces. Therefore, to simplify the dynamic model, Yg=0 is assumed here.The motion equations of the DCG dual-driving stage can be deduced as(13)Mq¨+Kq=F+U,where M and K are the simplified inertia (14) and stiffness (15) matrices; F is the force vector (16).(14)M=my100000my200000ms00000Jgz00000Jgx,(15)K=ke0-kel1ke-Zgke0ke-ke-l2ke-Zgke-ke-ke2kel2-l1ke2Zgkel1ke-l2kel2-l1kel12+l22ke+a2kbxl1-l2Zgke-Zgke-Zgke2Zgkel2-l1Zgke2Zg2ke+a2kbz,(16)F=Fe1Fe20TgzTgxFe1=ke1y1-ys1-cy1sign⁡y˙1Fe2=ke2y2-ys2-cy2sign⁡y˙2Tgz=Fe2l2-Fe1l1Tgx=Fe1+Fe2Zg.The stiffness matrixK points out to the mechanical coupling between the dual-driving axes y1 and y2 in the multidegree of freedom, which is affected by equivalent stiffness ke, kbx, kbz, and different positions of the center of gravity l2-l1, Zg. The simplified inertia and damping matrices show the nonuniform load and friction distribution. The first two lines of F are the force vector. Assume that l1=l2, my1=my2, cy1-cy2=0, and y1-ys1=y2-ys2, which means the mass, inertia, and friction of dual axes are equal and the workpiece is located on the middle of the stage. In this case, dual-driving synchronous feed can be ensured by Fe1=Fe2, which represent that the equivalent forces are equal and the joint force coincides with the center of gravity in X-Y plane. However, due to the different mechanical and servo characteristics of dual axis, and the center of gravity moving along x- and z-axis, the perfect synchronous dual-driving is hard to be achieved.Moreover, the last two lines of the force vectorF are torques around z-axis and x-axis, respectively. Traditionally, the dynamic model of DDFS is established based on two degrees of freedom, the linear displacement in y-axis and rotation angle around z-axis; the pitch errors caused by increase of position of center of gravity are totally ignored [5, 12]. Tgx is the torque around x-axis associated with the deviation of center of gravity in height. Only if the center of gravity is located on the X, Y plane, which means Zg=0, the torque around x-axis approaches zero. However, when the workpiece with different weight is located, it is impossible to ensure Zg=0. The joint force will not go through the center of gravity and pitch error would be generated. To achieve satisfactory DCG feed accuracy, the dynamic of pitch and torsion vibrations should be measured and analyzed in the dual-driving process.The determinant of the Lagrange equation can be obtained as(17)D=A110-kel1ke-Zgke0A22-ke-l2ke-Zgke-ke-keA33l2-l1ke2Zgkel1ke-l2kel2-l1keA44l1-l2Zgke-Zgke-Zgke2Zgkel2-l1ZgkeA55,(18)A11=-my12w2+ke,A22=-my22w2+ke,A33=-2ms2w2+ke,A44=-Jgzw2+l12+l22ke+a2kbx,A55=-Jgxw2+2Zg2ke+a2kbz.The determinant (17) expresses the axial, pitch, and torsion resonant modes. Solve the determinant D=0; wii=1⋯5 represent the natural frequencies.


## 2.2. Model Parameters Identification

As to (17) and (18), the dynamic parameters can be divided into three parts: inertia parameters (the inertia matrix M), jointed parameters (the stiffness matrix K and damping matrix C), and structure parameters (the constant parameters such as a, L). For the determined dual-driving DCG feed system, the structure parameters are assumed to be constant during the machining process. Hence, only the inertia and jointed parameters are analyzed and identified in this section.


### 2.2.1. Test Platform Setup

As shown in Figures4 and 5, DDFS and National Instruments (NI) test platform are set up for system parameters identification. The DDFS is constructed by Beckhoff servo system and controller, which will also be employed in the feed verification experiment later. The sweep sine signal is generated as identification input by the dynamic signal analyzer (DSA) from NI. The frequency response data is also calculated and analyzed in DSA. A data acquisition (DAQ) card inserted in a PC is used to receive the analog signal. All the AD/DA conversion, system memory allocation, and accessing module are executed by controller of NI. The detection data is analyzed by LabVIEW frequency response analyzer code.Figure 4
Frequency response experiment diagram.Figure 5
Experiment setup of identification.The sweep sine input signal applied to the drive is generated by DSA module. And the same signal is also linked to the Beckhoff controller or amplifier. The velocity and acceleration limits of double motors are 1.5 m/s and 5 m/s2. The desired motion trajectories of dual-driving axis are given exactly the same to avoid desynchronization.


### 2.2.2. Parameters Identification

As a major component of a dual-driving NC machine tool, the stiffness of the ball screws and linear guides are crucial to affect the positioning accuracy directly. Normally, the axial stiffnesske of screw system is estimated as the related components in series and parallel. The screw, nut, and support bearing influence the stiffness most. The dynamic model is shown in Figure 6.Figure 6
Simplified axial stiffness of screw.The integral axial stiffness of the screw transmission system is made up of the stiffness of screw, nut joint, and support bearing joints series-parallel connections. The equivalent axial stiffness can be given by(19)ke=ks1yb+kb1/ks1ybkb1+ks2yb+kb2/ks2ybkb2+knutks1yb+kb1/ks1ybkb1+ks2yb+kb2/ks2ybkb2knut.The equivalent axial stiffness can be obtained from the tension and torsion stiffness when the structure is determined, and this axial stiffness function is only affected by the located position of nut.Different frictional characteristics will generate the unbalanced forces in double axes. So the frictions model should be built accurately. The nonlinearity friction model is described as [11](20)cyv=FC+FS-FCe-v/vσδσ+Fvvv≠0,where vσ is the Stribeck velocity, FC is Coulomb friction, and FS is the maximum static friction. According to Newton’s Second Law, when the table moves in uniform motion, the Coulomb and viscous frictions of the y1 and y2 axes are equal to drive forces of motors. The velocities of dual axes are given at various constant velocities within 5 mm/s~14 m/min and the frictions can be easily obtained by the feedback drive forces of dual motors. Then, the measurement data can be processed with MATLAB nonlinear fitted function. The identification frictions of dual axis are fitted by practical detection results, as shown in Figure 7. As the velocity increases, friction starts decreasing, and as the velocity exceeds a certain critical value, which is called Stribeck velocity, friction starts to rise gradually again. The fitted parameters of friction model are shown in Table 1.Table 1
Physical parameters of the dual-driving system.
Name
Value
Description
m
y
1
7.53 kg
Mass of the axisy1
m
y
2
7.61 kg
Mass of the axisy2
m
s
510 kg
Mass of the sliding stage
J
g
z
78 kg·m2
Rotary inertia of the stage
J
g
x
51 kg·m2
Pitch inertia of the stage
k
e
1631.8 N·m/rad
Equivalent stiffness of screw
k
h
z
1572.5 N·m/rad
Support stiffness of slider
k
b
x
2.6734e6 N·m/rad
Lateral stiffness of slider
L
0.8 m
Distance of dual screws
a
0.45 m
Axial distance of the sliders
r
g
5 mm
Screw lead
F
s
1
117.7352 N
Maximum static friction ofy1
F
s
2
102.1365 N
Maximum static friction ofy2
F
c
1
30.2017 N
Coulomb friction of axisy1
F
c
2
26.8520 N
Coulomb friction of axisy2
F
v
1
0.0052 Ns/m
Viscosity coefficient of axisy1
F
v
2
0.0060 Ns/m
Viscosity coefficient of axisy2
v
σ
1
73.2533 mm/min
Stribeck velocity of axisy1
v
σ
2
109.3258 mm/min
Stribeck velocity of axisy2Figure 7
(a) Friction measuring and fitting of axisy1; (b) Friction measuring and fitting of axis y2.
(a)
(b)The equivalent mass of each axis can be estimated by moving they1 and y2 at constant acceleration cycles and making the center of gravity of the sliding stage in the geometric center. In this case, the sliding stage is located on the middle of dual axis, and there is no workpiece mounted on the stage, which means y¨1=y¨2, θgz≐0, and θgx=0. Then, the variation mass can be derived as (21)my1=key1-ys1-cy1sign⁡y˙1y¨1-1,my2=key2-ys2-cy2sign⁡y˙2y¨2-1.The displacements ofysi and yi can be obtained by the feedback of encoders and grating scales in each axis.The rotary inertia in the inertia matrix is estimated by the formula of moment of inertia.(22)Jgz=my1+my2+msL212,Jgx=my1+my2+msa212.The identified parameters are given in Table1.


## 2.2.1. Test Platform Setup

As shown in Figures4 and 5, DDFS and National Instruments (NI) test platform are set up for system parameters identification. The DDFS is constructed by Beckhoff servo system and controller, which will also be employed in the feed verification experiment later. The sweep sine signal is generated as identification input by the dynamic signal analyzer (DSA) from NI. The frequency response data is also calculated and analyzed in DSA. A data acquisition (DAQ) card inserted in a PC is used to receive the analog signal. All the AD/DA conversion, system memory allocation, and accessing module are executed by controller of NI. The detection data is analyzed by LabVIEW frequency response analyzer code.Figure 4
Frequency response experiment diagram.Figure 5
Experiment setup of identification.The sweep sine input signal applied to the drive is generated by DSA module. And the same signal is also linked to the Beckhoff controller or amplifier. The velocity and acceleration limits of double motors are 1.5 m/s and 5 m/s2. The desired motion trajectories of dual-driving axis are given exactly the same to avoid desynchronization.


## 2.2.2. Parameters Identification

As a major component of a dual-driving NC machine tool, the stiffness of the ball screws and linear guides are crucial to affect the positioning accuracy directly. Normally, the axial stiffnesske of screw system is estimated as the related components in series and parallel. The screw, nut, and support bearing influence the stiffness most. The dynamic model is shown in Figure 6.Figure 6
Simplified axial stiffness of screw.The integral axial stiffness of the screw transmission system is made up of the stiffness of screw, nut joint, and support bearing joints series-parallel connections. The equivalent axial stiffness can be given by(19)ke=ks1yb+kb1/ks1ybkb1+ks2yb+kb2/ks2ybkb2+knutks1yb+kb1/ks1ybkb1+ks2yb+kb2/ks2ybkb2knut.The equivalent axial stiffness can be obtained from the tension and torsion stiffness when the structure is determined, and this axial stiffness function is only affected by the located position of nut.Different frictional characteristics will generate the unbalanced forces in double axes. So the frictions model should be built accurately. The nonlinearity friction model is described as [11](20)cyv=FC+FS-FCe-v/vσδσ+Fvvv≠0,where vσ is the Stribeck velocity, FC is Coulomb friction, and FS is the maximum static friction. According to Newton’s Second Law, when the table moves in uniform motion, the Coulomb and viscous frictions of the y1 and y2 axes are equal to drive forces of motors. The velocities of dual axes are given at various constant velocities within 5 mm/s~14 m/min and the frictions can be easily obtained by the feedback drive forces of dual motors. Then, the measurement data can be processed with MATLAB nonlinear fitted function. The identification frictions of dual axis are fitted by practical detection results, as shown in Figure 7. As the velocity increases, friction starts decreasing, and as the velocity exceeds a certain critical value, which is called Stribeck velocity, friction starts to rise gradually again. The fitted parameters of friction model are shown in Table 1.Table 1
Physical parameters of the dual-driving system.
Name
Value
Description
m
y
1
7.53 kg
Mass of the axisy1
m
y
2
7.61 kg
Mass of the axisy2
m
s
510 kg
Mass of the sliding stage
J
g
z
78 kg·m2
Rotary inertia of the stage
J
g
x
51 kg·m2
Pitch inertia of the stage
k
e
1631.8 N·m/rad
Equivalent stiffness of screw
k
h
z
1572.5 N·m/rad
Support stiffness of slider
k
b
x
2.6734e6 N·m/rad
Lateral stiffness of slider
L
0.8 m
Distance of dual screws
a
0.45 m
Axial distance of the sliders
r
g
5 mm
Screw lead
F
s
1
117.7352 N
Maximum static friction ofy1
F
s
2
102.1365 N
Maximum static friction ofy2
F
c
1
30.2017 N
Coulomb friction of axisy1
F
c
2
26.8520 N
Coulomb friction of axisy2
F
v
1
0.0052 Ns/m
Viscosity coefficient of axisy1
F
v
2
0.0060 Ns/m
Viscosity coefficient of axisy2
v
σ
1
73.2533 mm/min
Stribeck velocity of axisy1
v
σ
2
109.3258 mm/min
Stribeck velocity of axisy2Figure 7
(a) Friction measuring and fitting of axisy1; (b) Friction measuring and fitting of axis y2.
(a)
(b)The equivalent mass of each axis can be estimated by moving they1 and y2 at constant acceleration cycles and making the center of gravity of the sliding stage in the geometric center. In this case, the sliding stage is located on the middle of dual axis, and there is no workpiece mounted on the stage, which means y¨1=y¨2, θgz≐0, and θgx=0. Then, the variation mass can be derived as (21)my1=key1-ys1-cy1sign⁡y˙1y¨1-1,my2=key2-ys2-cy2sign⁡y˙2y¨2-1.The displacements ofysi and yi can be obtained by the feedback of encoders and grating scales in each axis.The rotary inertia in the inertia matrix is estimated by the formula of moment of inertia.(22)Jgz=my1+my2+msL212,Jgx=my1+my2+msa212.The identified parameters are given in Table1.


## 3. Test of Proposed Dynamics Model for DDFS DCG

### 3.1. Model Parameters

Once the parameters are identified by NI vibration test platform, the modal test and dynamic measurement can be compared and analyzed by simulation and experiments. The experiments are carried out on an industry milling machine based on DCG DDFS. The milling machine is shown in Figure1. The physical parameters of DCG DDFS are identified in Table 1. The servo parameters of adopted PMSMs are listed in Table 2. The body is subject to mechanical friction with a viscous friction coefficient of Fv1 = 0.0072 Ns/m and a Coulomb friction level of FC1 = 50 N.Table 2
Motor parameters.
Name
Motory1
Motory2
Description
J
0.000167
0.000165
Inertia kg·m2
B
0.00023
0.00019
Damping kg·m2/sec
k
t
1.37
1.37
Torque coefficient NmA
k
a
8.4355
7.7686
Amplifier A/VTo investigate the comparison between simulation and experiment results, different parameters are separated from the influence of natural resonant and feed performance. The height of center of gravityZg and the position along x-axis Xg are discussed, respectively. The proposed typical gantry stage with DCG DDFS is illustrated in Figure 8.Figure 8
Photograph of DDFS DCG.


### 3.2. Modal Test and Comparison

To investigate the accuracy of the proposed mechanical-coupled model, the natural frequency is tested by modal experiment. The variation of the acceleration vibration response of the dual-driving DCG feed system is measured by NI vibration measurement instrument. The hammer is used to simulate with a 3 Hz test bandwidth. Four three-dimensional acceleration sensors are assembled on the four corners of the sliding stage, so that the torsion and pitch vibration can be measured. The DDFS DCG is shown in Figure8 and the modal experiment diagram is shown in Figure 9. When the height of center of gravity is zero, the simulation and experiment results of natural frequencies are shown in Tables 3 and 4.Table 3
Natural frequencies and modes of vibration.
Eigenvalue
Natural frequency
Vector
Mode of vibration
w
1
0 Hz
1,1
,
0,0
,
0
T
Axial
w
2
48.4 Hz
1
,
-
1,0
,
-
0.55,0
T
Low-order torsion
w
3
147.5 Hz
0.35,0.35,1
,
-
0.05,0
T
Axial and torsion
w
4
317.2 Hz
0.03,0.03,0
,
1,0
T
High-order torsion
w
5
341.6 Hz
[
0,0
,
0,0.42,1
]
Pitch and torsionTable 4
Comparison of simulation and modal experiment.
Mode of vibration
Prediction natural frequency
Experiment natural frequency
Relative error
Axial
-
-
-
Low-order torsion
48.4 Hz
51.7 Hz
6.3%
Axial and torsion
147.5 Hz
136.2 Hz
8.3%
Pitch
317.2 Hz
284.4 Hz
11.5%
Pitch and torsion
341.6 Hz
302.2 Hz
13.0%Figure 9
Diagram of modal test.In order to study the influence of height of center of gravity on natural frequencies, the stage is carried on different workpiece and the height of center of gravity is changed from 0 to 300 mm.The height of center of gravity is a structure parameter defined above. To facilitate the analysis of height of center of gravity, the changing of inertia and jointed parameters is assumed to have no related effect.The simulated and experimental results of variation heights of center of gravity effect on each natural frequency are shown in Figure10. As can be seen in Figure 10, the increase of height of center of gravity has relatively large effect on 3rd (axial and torsion) and 5th (pitch and torsion) vibration. As the height of center of gravity increases, the axial and torsion vibration is reduced from 147.5 Hz to 123.7 Hz. The pitch and torsion vibration is increased from 341.6 Hz to 397.9 Hz. In addition, the low-order torsion and high-order torsion vibrations are constant as the height of center of gravity increased.Figure 10
Simulation results of height of center of gravity on natural frequency.From Figures11 and 12, the comparison results show the similarity of the proposed model and actual experiment. The axial and torsion vibration of experiment is reduced from 136.2 Hz to 112.9 Hz with the increasing of height of center of gravity. Meanwhile, the experimental pitch and torsion vibration are increased from 302.2 Hz to 361.7 Hz. The simulation errors confirm that the trend of experiment and the range is also approximate, which illustrates that the model can reflect the actual operation of DDFS.Figure 11
Comparison of pitch and torsion vibration.Figure 12
Comparison of axial and torsion vibration.


## 3.1. Model Parameters

Once the parameters are identified by NI vibration test platform, the modal test and dynamic measurement can be compared and analyzed by simulation and experiments. The experiments are carried out on an industry milling machine based on DCG DDFS. The milling machine is shown in Figure1. The physical parameters of DCG DDFS are identified in Table 1. The servo parameters of adopted PMSMs are listed in Table 2. The body is subject to mechanical friction with a viscous friction coefficient of Fv1 = 0.0072 Ns/m and a Coulomb friction level of FC1 = 50 N.Table 2
Motor parameters.
Name
Motory1
Motory2
Description
J
0.000167
0.000165
Inertia kg·m2
B
0.00023
0.00019
Damping kg·m2/sec
k
t
1.37
1.37
Torque coefficient NmA
k
a
8.4355
7.7686
Amplifier A/VTo investigate the comparison between simulation and experiment results, different parameters are separated from the influence of natural resonant and feed performance. The height of center of gravityZg and the position along x-axis Xg are discussed, respectively. The proposed typical gantry stage with DCG DDFS is illustrated in Figure 8.Figure 8
Photograph of DDFS DCG.


## 3.2. Modal Test and Comparison

To investigate the accuracy of the proposed mechanical-coupled model, the natural frequency is tested by modal experiment. The variation of the acceleration vibration response of the dual-driving DCG feed system is measured by NI vibration measurement instrument. The hammer is used to simulate with a 3 Hz test bandwidth. Four three-dimensional acceleration sensors are assembled on the four corners of the sliding stage, so that the torsion and pitch vibration can be measured. The DDFS DCG is shown in Figure8 and the modal experiment diagram is shown in Figure 9. When the height of center of gravity is zero, the simulation and experiment results of natural frequencies are shown in Tables 3 and 4.Table 3
Natural frequencies and modes of vibration.
Eigenvalue
Natural frequency
Vector
Mode of vibration
w
1
0 Hz
1,1
,
0,0
,
0
T
Axial
w
2
48.4 Hz
1
,
-
1,0
,
-
0.55,0
T
Low-order torsion
w
3
147.5 Hz
0.35,0.35,1
,
-
0.05,0
T
Axial and torsion
w
4
317.2 Hz
0.03,0.03,0
,
1,0
T
High-order torsion
w
5
341.6 Hz
[
0,0
,
0,0.42,1
]
Pitch and torsionTable 4
Comparison of simulation and modal experiment.
Mode of vibration
Prediction natural frequency
Experiment natural frequency
Relative error
Axial
-
-
-
Low-order torsion
48.4 Hz
51.7 Hz
6.3%
Axial and torsion
147.5 Hz
136.2 Hz
8.3%
Pitch
317.2 Hz
284.4 Hz
11.5%
Pitch and torsion
341.6 Hz
302.2 Hz
13.0%Figure 9
Diagram of modal test.In order to study the influence of height of center of gravity on natural frequencies, the stage is carried on different workpiece and the height of center of gravity is changed from 0 to 300 mm.The height of center of gravity is a structure parameter defined above. To facilitate the analysis of height of center of gravity, the changing of inertia and jointed parameters is assumed to have no related effect.The simulated and experimental results of variation heights of center of gravity effect on each natural frequency are shown in Figure10. As can be seen in Figure 10, the increase of height of center of gravity has relatively large effect on 3rd (axial and torsion) and 5th (pitch and torsion) vibration. As the height of center of gravity increases, the axial and torsion vibration is reduced from 147.5 Hz to 123.7 Hz. The pitch and torsion vibration is increased from 341.6 Hz to 397.9 Hz. In addition, the low-order torsion and high-order torsion vibrations are constant as the height of center of gravity increased.Figure 10
Simulation results of height of center of gravity on natural frequency.From Figures11 and 12, the comparison results show the similarity of the proposed model and actual experiment. The axial and torsion vibration of experiment is reduced from 136.2 Hz to 112.9 Hz with the increasing of height of center of gravity. Meanwhile, the experimental pitch and torsion vibration are increased from 302.2 Hz to 361.7 Hz. The simulation errors confirm that the trend of experiment and the range is also approximate, which illustrates that the model can reflect the actual operation of DDFS.Figure 11
Comparison of pitch and torsion vibration.Figure 12
Comparison of axial and torsion vibration.


## 4. DCG Dual-Driving Synchronous Feed Experiment

In this section, the dynamic model-based CCSMC method which considers the synchronous errors is proposed. The synchronous errors and pitch vibrations are measured by laser interferometer and grating scale. The experimental and simulated results are compared to validate the proposed model. Figure13 is the overall diagram of dynamic model-based CCSMC.Figure 13
Diagram of dynamic model-based CCSMC.


### 4.1. Dynamic Model-Based Sliding Mode Control System

For its advantages of completely insensitive to modeling inaccuracies, SMC is particularly suited into nonlinear system. The control objective is to generate a robust sliding mode controller to force the actual motion position to track the given bounded desired reference trajectory and then guarantee that the tracking errorE(t)=Yt-Yref converges asymptotically to zero.In the practical scenario, the displacements of each axisY=y1,y2,x,θgz,θgxT are measurable. In order to assist with the designing of synchronous controller, the redundancy dynamic model can be simplified as(23)Y¨=M5×5-1U+F5-K5×5Y,where M5×5-1, K5×5, and F5 are the simplified inertia (14), stiffness (15), and the force (16) matrices.In the DDFS DCG, the control problem is to establish a control law to guarantee the convergence of both the tracking error and synchronous error to zero simultaneously. Hence, the sliding surface is defined as follows:(24)St=λiEt+E˙t,where E(t)=Yt-Yref is defined as the error between actual measurements and reference value; λi is the sliding surface parameter, which should be a positive constant like a transmission gain to determine the convergence speed of the dynamic error on the sliding surface; i depends on the order of E.Taking the derivative of (24) and using (23), the general sliding surface can be obtained.(25)S˙=λiE˙t+E¨t=λiE˙t+Y¨ref-M5×5-1Ut+F5-K5×5Y.It follows fromS˙i(t=0)=0 that the equivalent control law can be derived:(26)Ueq=λiM5×5E˙t+M5×5Y¨ref-F5+K5×5Y.A switching control law makes it possible to guarantee the stability of the system, which is defined as(27)U=Ueq+Usc,where Usc is the switching control function and the traditional SMC adopted signum switching function as switching function, which will improve the robustness and uncertainty disturbances; however, the discontinuous switching can also injects chattering phenomenon into the system, and this chattering introduces undesirable high frequency dynamics in the dual-driving feed system. According to the signum function switch between −1 and +1, a continuous saturation function is adopted to reduce chattering phenomena with the same function domain. (28)Usc=κitanh⁡Sφi.Based on the sliding surface function in the conventional SMC, the switching control law of proposed integral SMC is given as(29)Ui=λiM5×5E˙t+M5×5Y¨ref-F5+K5×5Y+κitanh⁡Sφi.The Lyapunov function can be expanded by substituting (16) in(30)V˙=SiλiE˙t+Y¨ref-M5×5-1Ut+F5-K5×5Y.To ensure the derivative of Lyapunov function to be negative, the following control equation should be satisfied:(31)λiE˙t+Y¨ref-M5×5-1Ut+F5-K5×5Y<0forS>0=0forS=0>0forS<0.By substituting (29) into (30),(32)V˙=Siκitanh⁡Siφi.As tanhSi/φi ranges from -1 to +1 and is only located in the first and third quadrant. Apparently, the sign of Si and tanh⁡Si/φi is always sameness, which implies that V˙ is negative semidefinite.


#### 4.1.1. Pitch Errors Measurement and Analysis

It is known that the dual-driving performance is expressed by the tracking and synchronization errors. However, the pitch vibrations are also crucial to analyze the feed accuracy of DDFS. In particular, when different workpiece is carried on the stage and the center of gravity is changed in height direction, the pitch errors are generated and would intensely reduce the feed precision.The pitch errors are evaluated by using both the developed dynamic model and laser interferometer. The rotation angle aroundz-axis θgz can be deduced by measuring the axial displacements of y1 and y2. Nevertheless, the pitch error around x-axis and roll error around y-axis are hardly to be obtained by the normal measuring method which using double grating scales mounted on the both sides of the stage.A novel pitch angle measurement method of DCG DDFS is proposed in this section. Figures14 and 15 show the outline and photograph of the measurement scheme. The whole measurement system is integrated on the DCG DDFS. In the setup, a laser interferometer is employed as the light source. Each interference mirror is mounted on the fixed table. The reflective mirrors are mounted on the sliding stage to make the measurement axis of the interferometer coincide with the mirrors.Figure 14
Schematic diagram of pitch error measurement.Figure 15
Measurement of laser interference.The original pitch errorθgx of each axis is determined as the assembly accuracy. Although the initial deviations of heights of dual axis θgx0 are calibrated to be same, the angle errors are inevitably existed. The pitch vibration errors of each axis at any position of the travel can be expressed as(33)θgx=θgx0+Δθgx.The pitch errors are obtained by the simulation and practical measurement. The results are changed from variation located positions of sliding stage alongx-axis, from the middle position to left-hand (or right-hand side). On the other hand, the change of height of center of gravity can directly increase the pitch errors. According to the practical processing requirements, the center of gravity is raised from 0 mm to 300 mm. In order to make specific analysis of the position of center of gravity, the influences of the mass and inertia are not discussed here. The maximum pitch errors are measured at velocity of 100 mm/s and acceleration of 1 m/s2.The comparison of simulation and practical measurement is shown in Table5. And the pitch error change with different position of center of gravity is shown in Figure 16. From the results we can notice the following:Table 5
Comparison between simulation results and practical measurement results.
Displacement inx-axis (l2-l1)
Height of center of gravityZg (mm)
Maximum simulation resultsθgx (°)
Maximum experiment pitch errorθgx (°)
Relative error
0
0
1.0
6
E
-
3
1.1
E
-
3
3.6%
0.2
5
L
0
1.2
9
E
-
3
1.3
3
E
-
3
3.0%
L
0
1.5
8
E
-
3
1.6
9
E
-
3
6.5%
0
150
2.0
7
E
-
3
2.2
E
-
3
5.9%
0
300
4.3
5
E
-
3
5.1
E
-
3
14.7%
0.2
5
L
150
2.4
4
E
-
3
2.6
E
-
3
6.2%
0.2
5
L
300
4.6
7
E
-
3
5.4
E
-
3
13.5%
L
150
2.5
6
E
-
3
2.7
E
-
3
5.2%
L
300
5.0
7
E
-
3
5.9
E
-
3
14.1%
Maximum variation by displacement inx-axis
—
0.5
2
E
-
3
0.5
9
E
-
3
11.9%
Maximum variation by height of center of gravity inz-axis
—
3.4
9
E
-
3
4.2
1
E
-
3
17.1%Figure 16
Pitch errors change with different position of center of gravity. Feed speed from 0 to 500 mm/s.(1) The simulated pitch errors are close to the experiment ones, and the relative error is about 3% to 15%. When the height of center of gravity is constant, the maximum pitch errors of simulation and experiment are increased when the sliding stage is moved from middle position to the left-hand (or right-hand side). When the position of sliding stage alongx-axis is constant, the pitch errors are increased with the increase of the height of center of gravity.(2) From Figure15, the pitch errors are mainly caused by the increase of center of gravity, which is inevitable when different workpieces are mounted on the stage. The pitch errors are generated by the joint forces and can be eliminated by the compensation of position tracking control.


#### 4.1.2. Synchronous Errors Measurement and Analysis

A dual-driving feed experiment is developed based on the computer-controlled dual-driving gantry stage as shown in Figure17. The Beckhoff control card is the core of the control computer. The control card is integrated in PC, so that the control algorithm, motion control, and other NC tasks can be executed at different levels of CPU, which provide a much better real-time communication property. Each amplifier runs in current control mode which has a circle time 0.1 ms.Figure 17
Overall hardware architecture.The feed conditions are conventional where the sliding stage is fed at a speed of 100 mm/s by dual axis. Furthermore, the parameters and test conditions are designed to be the same as in the simulation. The resolution of the grating scales is 0.1μm and measuring accuracy is ±1 um. After the input command is set, the feedback loop is realized by the encoder sampling at 10 [kHz]; then the synchronous errors are calculated by the algorithm. Finally, the dual motors are driven by the current commands disposed in the inherent current loop. To describe and clarify the comparison results clearly, the parallel uncoupled control strategy is adopted.The experimental and simulated synchronous errors of dual axis with an oscillating behavior are studied. While the dual-driving system is feed alongy-axis, the sliding stage is moved along x-axis, and the positions are changed from the middle position l2=l1, to left-hand (or right-hand side) l2-l1≈L.To validate the effectiveness of the proposed model, a normal 2-DOF dynamic model has been developed as (24). In this 2-DOF model, the effects of center of gravity and identified friction are ignored. (34)D·y¨+key=F1+F2,F1·l2-F2·l2-kd·θgz=Jgz·θ¨gz,Fi=2πhTMi.Then the equivalent displacement of dual-driving system can be yielded as(35)y¨1y¨2=1LD00Jt+kds2l2l21-1-111l2-l2F1F2,where D=my1+my2+ms is the total mass; Fi=F1F2 is the drive force, respectively; kd is rotational stiffness of the sliding stage, which points out the rotation error caused by parameters variation and mechanical coupling. The performance tests of the proposed redundancy dynamic model and normal 2-DOF model for the comparison by a dual-driving experiment are shown in Table 6. Comparing the normal 2-DOF model, the relative errors of synchronous error simulated by the proposed model are much reduced. Although the accuracy is slightly increased when the sliding stage is located on the middle, in other two cases, when the sliding stage is located on the 1/4 and one end of the table, the average and max of relative error simulated by the proposed dynamic model are significantly reduced, respectively.Table 6
Performance comparison with variation location inx-axis of the sliding stage.
Experimental results (um)
Proposed DCG model (um)
Relative error
Normal 2-DOF model (um)
Relative error
(
l
2
-
l
1
)
/
L
=
0
Max
9.8
9.1
7.1%
8.9
9.2%
Avg
3.6
3.2
11.1%
3.1
13.9%
(l2-l1)/L=0.25
Max
12.5
11.6
7.2%
9.7
22.4%
Avg
4.1
3.7
7.3%
3.3
19.5%
(
l
2
-
l
1
)
/
L
=
1
Max
14.7
12.9
12.2%
10.3
29.9%
Avg
4.3
4.0
6.9%
3.1
27.9%Therefore, the proposed dynamic model can provide significant improvement of the accuracy of dual-driving stage driven by center of gravity.Then, the practical control performance and synchronization accuracy of proposed dynamic model-based cross-coupled control are testified in the experiment. The proposed control algorithm is applied to the dual-driving servo system by using “C” and MATLAB/SIMULINK language, which will be called by module in the control card. Based on the actual require of manufacturing, two PMSMS are driven by the desired trajectory with 200 mm peak value, at the speed of 100 mm/s, as shown in Figure18. For comparison, the normal cross-coupled control is applied to the DDFS with the same process condition. Figures 19 and 20 depict the experimental results of synchronous control based on the proposed dynamic model-based cross-coupled control and normal cross-coupled control in DDFS DCG.Figure 18
Experiment and simulation results of synchronous errors in dual-driving feed process. (a) Reference trajectory ofy1 and y2; (b) synchronous errors of dual axes when stage is located on (l2-l1)/L=0; (c) synchronous errors of dual axis when stage is located on (l2-l1)/L=0.25; (d) synchronous errors of dual axis when stage is located on (l2-l1)/L=1.
(a)
(b)
(c)
(d)Figure 19
Reference trajectory of dual-driving feed experiment.Figure 20
Experimental results of NCCSMC: (a) tracking responses of axisy1 and y2 and (b) synchronous errors between axis y1 and y2.
(a)
(b)In comparison to the synchronous performance between proposed dynamic model-based CCSMC and NCCSMC, some phenomena should be noticed in control of dual-driving system:(1) As can be seen in Figures19, 20, and 21, the proposed dynamic model-based CCSMC has a favorable tracking response and synchronous performance benefits from the dynamics in handling parameters variation and disturbances. Table 7 shows the details of the comparison results to the NCCSMC, and the proposed dynamic model-based CCSMC has better performance with respect to the precise synchronization and tracking as well as offering an acceptable performance to the sudden changing of velocity.Table 7
Performance comparison of dual-driving feed experimental results.
Errors/control schemes
Sinusoidal trajectory
NCCSMC
Dynamic model-based CCSMC
Maximum (um)
Tracking error
53.2
31.7
Synchronous error
12.4
6.2
Average (um)
Tracking error
15.1
7.6
Synchronous error
3.8
2.1Figure 21
Experimental results of the proposed dynamic model-based CCSMC: (a) tracking responses of axisy1 and y2 and (b) synchronous errors between axis y1 and y2.
(a)
(b)(2) The synchronous errorsy2-y1 are influenced by the position of center of gravity, as shown in Figure 18. The peak value of synchronous errors is increased with the sliding stage moving from the middle position to the left-hand (right-hand side).


## 4.1. Dynamic Model-Based Sliding Mode Control System

For its advantages of completely insensitive to modeling inaccuracies, SMC is particularly suited into nonlinear system. The control objective is to generate a robust sliding mode controller to force the actual motion position to track the given bounded desired reference trajectory and then guarantee that the tracking errorE(t)=Yt-Yref converges asymptotically to zero.In the practical scenario, the displacements of each axisY=y1,y2,x,θgz,θgxT are measurable. In order to assist with the designing of synchronous controller, the redundancy dynamic model can be simplified as(23)Y¨=M5×5-1U+F5-K5×5Y,where M5×5-1, K5×5, and F5 are the simplified inertia (14), stiffness (15), and the force (16) matrices.In the DDFS DCG, the control problem is to establish a control law to guarantee the convergence of both the tracking error and synchronous error to zero simultaneously. Hence, the sliding surface is defined as follows:(24)St=λiEt+E˙t,where E(t)=Yt-Yref is defined as the error between actual measurements and reference value; λi is the sliding surface parameter, which should be a positive constant like a transmission gain to determine the convergence speed of the dynamic error on the sliding surface; i depends on the order of E.Taking the derivative of (24) and using (23), the general sliding surface can be obtained.(25)S˙=λiE˙t+E¨t=λiE˙t+Y¨ref-M5×5-1Ut+F5-K5×5Y.It follows fromS˙i(t=0)=0 that the equivalent control law can be derived:(26)Ueq=λiM5×5E˙t+M5×5Y¨ref-F5+K5×5Y.A switching control law makes it possible to guarantee the stability of the system, which is defined as(27)U=Ueq+Usc,where Usc is the switching control function and the traditional SMC adopted signum switching function as switching function, which will improve the robustness and uncertainty disturbances; however, the discontinuous switching can also injects chattering phenomenon into the system, and this chattering introduces undesirable high frequency dynamics in the dual-driving feed system. According to the signum function switch between −1 and +1, a continuous saturation function is adopted to reduce chattering phenomena with the same function domain. (28)Usc=κitanh⁡Sφi.Based on the sliding surface function in the conventional SMC, the switching control law of proposed integral SMC is given as(29)Ui=λiM5×5E˙t+M5×5Y¨ref-F5+K5×5Y+κitanh⁡Sφi.The Lyapunov function can be expanded by substituting (16) in(30)V˙=SiλiE˙t+Y¨ref-M5×5-1Ut+F5-K5×5Y.To ensure the derivative of Lyapunov function to be negative, the following control equation should be satisfied:(31)λiE˙t+Y¨ref-M5×5-1Ut+F5-K5×5Y<0forS>0=0forS=0>0forS<0.By substituting (29) into (30),(32)V˙=Siκitanh⁡Siφi.As tanhSi/φi ranges from -1 to +1 and is only located in the first and third quadrant. Apparently, the sign of Si and tanh⁡Si/φi is always sameness, which implies that V˙ is negative semidefinite.


### 4.1.1. Pitch Errors Measurement and Analysis

It is known that the dual-driving performance is expressed by the tracking and synchronization errors. However, the pitch vibrations are also crucial to analyze the feed accuracy of DDFS. In particular, when different workpiece is carried on the stage and the center of gravity is changed in height direction, the pitch errors are generated and would intensely reduce the feed precision.The pitch errors are evaluated by using both the developed dynamic model and laser interferometer. The rotation angle aroundz-axis θgz can be deduced by measuring the axial displacements of y1 and y2. Nevertheless, the pitch error around x-axis and roll error around y-axis are hardly to be obtained by the normal measuring method which using double grating scales mounted on the both sides of the stage.A novel pitch angle measurement method of DCG DDFS is proposed in this section. Figures14 and 15 show the outline and photograph of the measurement scheme. The whole measurement system is integrated on the DCG DDFS. In the setup, a laser interferometer is employed as the light source. Each interference mirror is mounted on the fixed table. The reflective mirrors are mounted on the sliding stage to make the measurement axis of the interferometer coincide with the mirrors.Figure 14
Schematic diagram of pitch error measurement.Figure 15
Measurement of laser interference.The original pitch errorθgx of each axis is determined as the assembly accuracy. Although the initial deviations of heights of dual axis θgx0 are calibrated to be same, the angle errors are inevitably existed. The pitch vibration errors of each axis at any position of the travel can be expressed as(33)θgx=θgx0+Δθgx.The pitch errors are obtained by the simulation and practical measurement. The results are changed from variation located positions of sliding stage alongx-axis, from the middle position to left-hand (or right-hand side). On the other hand, the change of height of center of gravity can directly increase the pitch errors. According to the practical processing requirements, the center of gravity is raised from 0 mm to 300 mm. In order to make specific analysis of the position of center of gravity, the influences of the mass and inertia are not discussed here. The maximum pitch errors are measured at velocity of 100 mm/s and acceleration of 1 m/s2.The comparison of simulation and practical measurement is shown in Table5. And the pitch error change with different position of center of gravity is shown in Figure 16. From the results we can notice the following:Table 5
Comparison between simulation results and practical measurement results.
Displacement inx-axis (l2-l1)
Height of center of gravityZg (mm)
Maximum simulation resultsθgx (°)
Maximum experiment pitch errorθgx (°)
Relative error
0
0
1.0
6
E
-
3
1.1
E
-
3
3.6%
0.2
5
L
0
1.2
9
E
-
3
1.3
3
E
-
3
3.0%
L
0
1.5
8
E
-
3
1.6
9
E
-
3
6.5%
0
150
2.0
7
E
-
3
2.2
E
-
3
5.9%
0
300
4.3
5
E
-
3
5.1
E
-
3
14.7%
0.2
5
L
150
2.4
4
E
-
3
2.6
E
-
3
6.2%
0.2
5
L
300
4.6
7
E
-
3
5.4
E
-
3
13.5%
L
150
2.5
6
E
-
3
2.7
E
-
3
5.2%
L
300
5.0
7
E
-
3
5.9
E
-
3
14.1%
Maximum variation by displacement inx-axis
—
0.5
2
E
-
3
0.5
9
E
-
3
11.9%
Maximum variation by height of center of gravity inz-axis
—
3.4
9
E
-
3
4.2
1
E
-
3
17.1%Figure 16
Pitch errors change with different position of center of gravity. Feed speed from 0 to 500 mm/s.(1) The simulated pitch errors are close to the experiment ones, and the relative error is about 3% to 15%. When the height of center of gravity is constant, the maximum pitch errors of simulation and experiment are increased when the sliding stage is moved from middle position to the left-hand (or right-hand side). When the position of sliding stage alongx-axis is constant, the pitch errors are increased with the increase of the height of center of gravity.(2) From Figure15, the pitch errors are mainly caused by the increase of center of gravity, which is inevitable when different workpieces are mounted on the stage. The pitch errors are generated by the joint forces and can be eliminated by the compensation of position tracking control.


### 4.1.2. Synchronous Errors Measurement and Analysis

A dual-driving feed experiment is developed based on the computer-controlled dual-driving gantry stage as shown in Figure17. The Beckhoff control card is the core of the control computer. The control card is integrated in PC, so that the control algorithm, motion control, and other NC tasks can be executed at different levels of CPU, which provide a much better real-time communication property. Each amplifier runs in current control mode which has a circle time 0.1 ms.Figure 17
Overall hardware architecture.The feed conditions are conventional where the sliding stage is fed at a speed of 100 mm/s by dual axis. Furthermore, the parameters and test conditions are designed to be the same as in the simulation. The resolution of the grating scales is 0.1μm and measuring accuracy is ±1 um. After the input command is set, the feedback loop is realized by the encoder sampling at 10 [kHz]; then the synchronous errors are calculated by the algorithm. Finally, the dual motors are driven by the current commands disposed in the inherent current loop. To describe and clarify the comparison results clearly, the parallel uncoupled control strategy is adopted.The experimental and simulated synchronous errors of dual axis with an oscillating behavior are studied. While the dual-driving system is feed alongy-axis, the sliding stage is moved along x-axis, and the positions are changed from the middle position l2=l1, to left-hand (or right-hand side) l2-l1≈L.To validate the effectiveness of the proposed model, a normal 2-DOF dynamic model has been developed as (24). In this 2-DOF model, the effects of center of gravity and identified friction are ignored. (34)D·y¨+key=F1+F2,F1·l2-F2·l2-kd·θgz=Jgz·θ¨gz,Fi=2πhTMi.Then the equivalent displacement of dual-driving system can be yielded as(35)y¨1y¨2=1LD00Jt+kds2l2l21-1-111l2-l2F1F2,where D=my1+my2+ms is the total mass; Fi=F1F2 is the drive force, respectively; kd is rotational stiffness of the sliding stage, which points out the rotation error caused by parameters variation and mechanical coupling. The performance tests of the proposed redundancy dynamic model and normal 2-DOF model for the comparison by a dual-driving experiment are shown in Table 6. Comparing the normal 2-DOF model, the relative errors of synchronous error simulated by the proposed model are much reduced. Although the accuracy is slightly increased when the sliding stage is located on the middle, in other two cases, when the sliding stage is located on the 1/4 and one end of the table, the average and max of relative error simulated by the proposed dynamic model are significantly reduced, respectively.Table 6
Performance comparison with variation location inx-axis of the sliding stage.
Experimental results (um)
Proposed DCG model (um)
Relative error
Normal 2-DOF model (um)
Relative error
(
l
2
-
l
1
)
/
L
=
0
Max
9.8
9.1
7.1%
8.9
9.2%
Avg
3.6
3.2
11.1%
3.1
13.9%
(l2-l1)/L=0.25
Max
12.5
11.6
7.2%
9.7
22.4%
Avg
4.1
3.7
7.3%
3.3
19.5%
(
l
2
-
l
1
)
/
L
=
1
Max
14.7
12.9
12.2%
10.3
29.9%
Avg
4.3
4.0
6.9%
3.1
27.9%Therefore, the proposed dynamic model can provide significant improvement of the accuracy of dual-driving stage driven by center of gravity.Then, the practical control performance and synchronization accuracy of proposed dynamic model-based cross-coupled control are testified in the experiment. The proposed control algorithm is applied to the dual-driving servo system by using “C” and MATLAB/SIMULINK language, which will be called by module in the control card. Based on the actual require of manufacturing, two PMSMS are driven by the desired trajectory with 200 mm peak value, at the speed of 100 mm/s, as shown in Figure18. For comparison, the normal cross-coupled control is applied to the DDFS with the same process condition. Figures 19 and 20 depict the experimental results of synchronous control based on the proposed dynamic model-based cross-coupled control and normal cross-coupled control in DDFS DCG.Figure 18
Experiment and simulation results of synchronous errors in dual-driving feed process. (a) Reference trajectory ofy1 and y2; (b) synchronous errors of dual axes when stage is located on (l2-l1)/L=0; (c) synchronous errors of dual axis when stage is located on (l2-l1)/L=0.25; (d) synchronous errors of dual axis when stage is located on (l2-l1)/L=1.
(a)
(b)
(c)
(d)Figure 19
Reference trajectory of dual-driving feed experiment.Figure 20
Experimental results of NCCSMC: (a) tracking responses of axisy1 and y2 and (b) synchronous errors between axis y1 and y2.
(a)
(b)In comparison to the synchronous performance between proposed dynamic model-based CCSMC and NCCSMC, some phenomena should be noticed in control of dual-driving system:(1) As can be seen in Figures19, 20, and 21, the proposed dynamic model-based CCSMC has a favorable tracking response and synchronous performance benefits from the dynamics in handling parameters variation and disturbances. Table 7 shows the details of the comparison results to the NCCSMC, and the proposed dynamic model-based CCSMC has better performance with respect to the precise synchronization and tracking as well as offering an acceptable performance to the sudden changing of velocity.Table 7
Performance comparison of dual-driving feed experimental results.
Errors/control schemes
Sinusoidal trajectory
NCCSMC
Dynamic model-based CCSMC
Maximum (um)
Tracking error
53.2
31.7
Synchronous error
12.4
6.2
Average (um)
Tracking error
15.1
7.6
Synchronous error
3.8
2.1Figure 21
Experimental results of the proposed dynamic model-based CCSMC: (a) tracking responses of axisy1 and y2 and (b) synchronous errors between axis y1 and y2.
(a)
(b)(2) The synchronous errorsy2-y1 are influenced by the position of center of gravity, as shown in Figure 18. The peak value of synchronous errors is increased with the sliding stage moving from the middle position to the left-hand (right-hand side).


## 4.1.1. Pitch Errors Measurement and Analysis

It is known that the dual-driving performance is expressed by the tracking and synchronization errors. However, the pitch vibrations are also crucial to analyze the feed accuracy of DDFS. In particular, when different workpiece is carried on the stage and the center of gravity is changed in height direction, the pitch errors are generated and would intensely reduce the feed precision.The pitch errors are evaluated by using both the developed dynamic model and laser interferometer. The rotation angle aroundz-axis θgz can be deduced by measuring the axial displacements of y1 and y2. Nevertheless, the pitch error around x-axis and roll error around y-axis are hardly to be obtained by the normal measuring method which using double grating scales mounted on the both sides of the stage.A novel pitch angle measurement method of DCG DDFS is proposed in this section. Figures14 and 15 show the outline and photograph of the measurement scheme. The whole measurement system is integrated on the DCG DDFS. In the setup, a laser interferometer is employed as the light source. Each interference mirror is mounted on the fixed table. The reflective mirrors are mounted on the sliding stage to make the measurement axis of the interferometer coincide with the mirrors.Figure 14
Schematic diagram of pitch error measurement.Figure 15
Measurement of laser interference.The original pitch errorθgx of each axis is determined as the assembly accuracy. Although the initial deviations of heights of dual axis θgx0 are calibrated to be same, the angle errors are inevitably existed. The pitch vibration errors of each axis at any position of the travel can be expressed as(33)θgx=θgx0+Δθgx.The pitch errors are obtained by the simulation and practical measurement. The results are changed from variation located positions of sliding stage alongx-axis, from the middle position to left-hand (or right-hand side). On the other hand, the change of height of center of gravity can directly increase the pitch errors. According to the practical processing requirements, the center of gravity is raised from 0 mm to 300 mm. In order to make specific analysis of the position of center of gravity, the influences of the mass and inertia are not discussed here. The maximum pitch errors are measured at velocity of 100 mm/s and acceleration of 1 m/s2.The comparison of simulation and practical measurement is shown in Table5. And the pitch error change with different position of center of gravity is shown in Figure 16. From the results we can notice the following:Table 5
Comparison between simulation results and practical measurement results.
Displacement inx-axis (l2-l1)
Height of center of gravityZg (mm)
Maximum simulation resultsθgx (°)
Maximum experiment pitch errorθgx (°)
Relative error
0
0
1.0
6
E
-
3
1.1
E
-
3
3.6%
0.2
5
L
0
1.2
9
E
-
3
1.3
3
E
-
3
3.0%
L
0
1.5
8
E
-
3
1.6
9
E
-
3
6.5%
0
150
2.0
7
E
-
3
2.2
E
-
3
5.9%
0
300
4.3
5
E
-
3
5.1
E
-
3
14.7%
0.2
5
L
150
2.4
4
E
-
3
2.6
E
-
3
6.2%
0.2
5
L
300
4.6
7
E
-
3
5.4
E
-
3
13.5%
L
150
2.5
6
E
-
3
2.7
E
-
3
5.2%
L
300
5.0
7
E
-
3
5.9
E
-
3
14.1%
Maximum variation by displacement inx-axis
—
0.5
2
E
-
3
0.5
9
E
-
3
11.9%
Maximum variation by height of center of gravity inz-axis
—
3.4
9
E
-
3
4.2
1
E
-
3
17.1%Figure 16
Pitch errors change with different position of center of gravity. Feed speed from 0 to 500 mm/s.(1) The simulated pitch errors are close to the experiment ones, and the relative error is about 3% to 15%. When the height of center of gravity is constant, the maximum pitch errors of simulation and experiment are increased when the sliding stage is moved from middle position to the left-hand (or right-hand side). When the position of sliding stage alongx-axis is constant, the pitch errors are increased with the increase of the height of center of gravity.(2) From Figure15, the pitch errors are mainly caused by the increase of center of gravity, which is inevitable when different workpieces are mounted on the stage. The pitch errors are generated by the joint forces and can be eliminated by the compensation of position tracking control.


## 4.1.2. Synchronous Errors Measurement and Analysis

A dual-driving feed experiment is developed based on the computer-controlled dual-driving gantry stage as shown in Figure17. The Beckhoff control card is the core of the control computer. The control card is integrated in PC, so that the control algorithm, motion control, and other NC tasks can be executed at different levels of CPU, which provide a much better real-time communication property. Each amplifier runs in current control mode which has a circle time 0.1 ms.Figure 17
Overall hardware architecture.The feed conditions are conventional where the sliding stage is fed at a speed of 100 mm/s by dual axis. Furthermore, the parameters and test conditions are designed to be the same as in the simulation. The resolution of the grating scales is 0.1μm and measuring accuracy is ±1 um. After the input command is set, the feedback loop is realized by the encoder sampling at 10 [kHz]; then the synchronous errors are calculated by the algorithm. Finally, the dual motors are driven by the current commands disposed in the inherent current loop. To describe and clarify the comparison results clearly, the parallel uncoupled control strategy is adopted.The experimental and simulated synchronous errors of dual axis with an oscillating behavior are studied. While the dual-driving system is feed alongy-axis, the sliding stage is moved along x-axis, and the positions are changed from the middle position l2=l1, to left-hand (or right-hand side) l2-l1≈L.To validate the effectiveness of the proposed model, a normal 2-DOF dynamic model has been developed as (24). In this 2-DOF model, the effects of center of gravity and identified friction are ignored. (34)D·y¨+key=F1+F2,F1·l2-F2·l2-kd·θgz=Jgz·θ¨gz,Fi=2πhTMi.Then the equivalent displacement of dual-driving system can be yielded as(35)y¨1y¨2=1LD00Jt+kds2l2l21-1-111l2-l2F1F2,where D=my1+my2+ms is the total mass; Fi=F1F2 is the drive force, respectively; kd is rotational stiffness of the sliding stage, which points out the rotation error caused by parameters variation and mechanical coupling. The performance tests of the proposed redundancy dynamic model and normal 2-DOF model for the comparison by a dual-driving experiment are shown in Table 6. Comparing the normal 2-DOF model, the relative errors of synchronous error simulated by the proposed model are much reduced. Although the accuracy is slightly increased when the sliding stage is located on the middle, in other two cases, when the sliding stage is located on the 1/4 and one end of the table, the average and max of relative error simulated by the proposed dynamic model are significantly reduced, respectively.Table 6
Performance comparison with variation location inx-axis of the sliding stage.
Experimental results (um)
Proposed DCG model (um)
Relative error
Normal 2-DOF model (um)
Relative error
(
l
2
-
l
1
)
/
L
=
0
Max
9.8
9.1
7.1%
8.9
9.2%
Avg
3.6
3.2
11.1%
3.1
13.9%
(l2-l1)/L=0.25
Max
12.5
11.6
7.2%
9.7
22.4%
Avg
4.1
3.7
7.3%
3.3
19.5%
(
l
2
-
l
1
)
/
L
=
1
Max
14.7
12.9
12.2%
10.3
29.9%
Avg
4.3
4.0
6.9%
3.1
27.9%Therefore, the proposed dynamic model can provide significant improvement of the accuracy of dual-driving stage driven by center of gravity.Then, the practical control performance and synchronization accuracy of proposed dynamic model-based cross-coupled control are testified in the experiment. The proposed control algorithm is applied to the dual-driving servo system by using “C” and MATLAB/SIMULINK language, which will be called by module in the control card. Based on the actual require of manufacturing, two PMSMS are driven by the desired trajectory with 200 mm peak value, at the speed of 100 mm/s, as shown in Figure18. For comparison, the normal cross-coupled control is applied to the DDFS with the same process condition. Figures 19 and 20 depict the experimental results of synchronous control based on the proposed dynamic model-based cross-coupled control and normal cross-coupled control in DDFS DCG.Figure 18
Experiment and simulation results of synchronous errors in dual-driving feed process. (a) Reference trajectory ofy1 and y2; (b) synchronous errors of dual axes when stage is located on (l2-l1)/L=0; (c) synchronous errors of dual axis when stage is located on (l2-l1)/L=0.25; (d) synchronous errors of dual axis when stage is located on (l2-l1)/L=1.
(a)
(b)
(c)
(d)Figure 19
Reference trajectory of dual-driving feed experiment.Figure 20
Experimental results of NCCSMC: (a) tracking responses of axisy1 and y2 and (b) synchronous errors between axis y1 and y2.
(a)
(b)In comparison to the synchronous performance between proposed dynamic model-based CCSMC and NCCSMC, some phenomena should be noticed in control of dual-driving system:(1) As can be seen in Figures19, 20, and 21, the proposed dynamic model-based CCSMC has a favorable tracking response and synchronous performance benefits from the dynamics in handling parameters variation and disturbances. Table 7 shows the details of the comparison results to the NCCSMC, and the proposed dynamic model-based CCSMC has better performance with respect to the precise synchronization and tracking as well as offering an acceptable performance to the sudden changing of velocity.Table 7
Performance comparison of dual-driving feed experimental results.
Errors/control schemes
Sinusoidal trajectory
NCCSMC
Dynamic model-based CCSMC
Maximum (um)
Tracking error
53.2
31.7
Synchronous error
12.4
6.2
Average (um)
Tracking error
15.1
7.6
Synchronous error
3.8
2.1Figure 21
Experimental results of the proposed dynamic model-based CCSMC: (a) tracking responses of axisy1 and y2 and (b) synchronous errors between axis y1 and y2.
(a)
(b)(2) The synchronous errorsy2-y1 are influenced by the position of center of gravity, as shown in Figure 18. The peak value of synchronous errors is increased with the sliding stage moving from the middle position to the left-hand (right-hand side).


## 5. Conclusion

A modeling, identification, and control scheme for dynamic characteristics analysis of DDFS DCG is presented in this paper.First, a redundancy dynamic model including rotation and pitch errors is developed based on the Lagrange method. The influence of center of gravity is considered in terms of the translation from basic coordinate (x0,y0,z0) to actual coordinate (xg,yg,zg). This model accounts for the coupling effects associated with the mechanical coupling between the dual axis and includes the pitch modes caused by joint driving forces. Then the primary parameters (such as inertia and jointed parameters) are identified by the vibration detection system which can be used in the similar DDFS DCG.The second part of this paper is devoted to the dynamic characteristics test based on a practical DDFS DCG. The relative errors of natural frequencies between simulation and experiment are about 6% to 13%. With the increase of height of center of gravity, the changing trend of natural frequencies is analyzed. Then, the influence of variation of center of gravity on the dynamic characteristics is discussed with reference to the dual-driving feed experiment. The maximum pitch errors aroundx-axis are obtained by the proposed laser interferometer measurement method. The pitch errors are close between the simulation and the experiment measurement, and the relative error is about 3% to 15%.Finally, the dynamic model-based CCSMC is presented for the dual-driving synchronous control. The dynamic model precision is further tested by dual-driving feed experiment. Both the experimental and simulated results show the effectiveness of the proposed model and control scheme.The main implication of this paper is the dynamic characteristics analysis and control scheme for DDFS DCG. The dynamic model which considers the effect of position of center of gravity has been presented, and the dynamic model-based CCSMC has been developed for DDFS DCG. The predictive natural frequencies and modes can be the reference for filter design and high bandwidth control. Moreover, the dynamic characteristics including pitch errors and synchronous errors are helpful for the tracking and synchronization control and compensation.



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*Source: 9490826-2018-01-31.xml*