Dualdriving 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 dualdriving 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 modelbased crosscoupled sliding mode control (CCSMC) is proposed for DDFS DCG. Then, the proposed dynamic modelbased CCSMC has been compared with normal crosscoupled 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.
Traditional drive mechanism used single servo motor and ball screw in gantrymoving 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 highend industrial applications such as surface chip mounting, precision metrology, and highspeed and heavy load manufacturing [
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 [
Numerous researches have conducted ball screw system and DDFS in CNC machine tools [
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 direction
In addition, the synchronization and tracking accuracy of DDFS are determined by the bandwidth of servo control loop [
As to the synchronous control, many scholars have developed the motion control strategy for DDFS from the perspective of tandem control and crosscoupled control [
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:
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 timevarying 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.
The dynamic modelbased 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.
This work focuses on the practical application of highprecision feed system in machine tools. The proposed model of used dualdriving 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 Section
The DDFS DCG used in this paper is composed of twin AC200v 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 Figure
Basic structure and mechanism of DCG DDFS.
The dualdriving stage is a highspeed, highprecision positioning system. It consists of a dualdriving stage that is moved by two parallel ball screws (axes
The proposed redundancy dynamic model of DCG DDFS is developed in this section. Based on the dualdriving structure and mechanism shown in Figure
Lumpedparameter model considers rotation around
Lumpedparameter model considers pitch around
As can be seen in Figures
The drive force can be obtained from equivalent axial stiffness:
The total drive force of table can be deduced as
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
The driving torque of DCG dualdriving 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 lumpedparameter model of DCG dualdriving stage is shown in Figure
Since the yaw angle of crossarm is limited to nearly 0.1 rad, the
Then, the displacements of each axis can be expressed as
The microrotation angle can be expressed as
Then the motion equations of the DCG dualdriving stage will be derived using LagrangeEuler formalism. Based on the conservation of energy, LagrangeEuler 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 dualdriving system can be expressed as
And the relation between displacements of slide blocks in
The dualdriving 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 in
The potential elastic energy can be obtained
According to Lagrange method, the system dynamics can be defined as
Normally, the center of gravity can be designed to be on the centerline of
The motion equations of the DCG dualdriving stage can be deduced as
The stiffness matrix
Moreover, the last two lines of the force vector
The determinant of the Lagrange equation can be obtained as
The determinant (
As to (
As shown in Figures
Frequency response experiment diagram.
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/s^{2}. The desired motion trajectories of dualdriving axis are given exactly the same to avoid desynchronization.
As a major component of a dualdriving NC machine tool, the stiffness of the ball screws and linear guides are crucial to affect the positioning accuracy directly. Normally, the axial stiffness
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 seriesparallel connections. The equivalent axial stiffness can be given by
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 [
Physical parameters of the dualdriving system.
Name  Value  Description 


7.53 kg  Mass of the axis 

7.61 kg  Mass of the axis 

510 kg  Mass of the sliding stage 

78 kg·m^{2}  Rotary inertia of the stage 

51 kg·m^{2}  Pitch inertia of the stage 

1631.8 N·m/rad  Equivalent stiffness of screw 

1572.5 N·m/rad  Support stiffness of slider 

2.6734 
Lateral stiffness of slider 

0.8 m  Distance of dual screws 

0.45 m  Axial distance of the sliders 

5 mm  Screw lead 

117.7352 N  Maximum static friction of 

102.1365 N  Maximum static friction of 

30.2017 N  Coulomb friction of axis 

26.8520 N  Coulomb friction of axis 

0.0052 Ns/m  Viscosity coefficient of axis 

0.0060 Ns/m  Viscosity coefficient of axis 

73.2533 mm/min  Stribeck velocity of axis 

109.3258 mm/min  Stribeck velocity of axis 
(a) Friction measuring and fitting of axis
The equivalent mass of each axis can be estimated by moving the
The displacements of
The rotary inertia in the inertia matrix is estimated by the formula of moment of inertia.
The identified parameters are given in Table
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 Figure
Motor parameters.
Name  Motor 
Motor 
Description 


0.000167  0.000165  Inertia kg·m^{2} 

0.00023  0.00019  Damping kg·m^{2}/sec 

1.37  1.37  Torque coefficient NmA 

8.4355  7.7686  Amplifier A/V 
To 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 gravity
Photograph of DDFS DCG.
To investigate the accuracy of the proposed mechanicalcoupled model, the natural frequency is tested by modal experiment. The variation of the acceleration vibration response of the dualdriving DCG feed system is measured by NI vibration measurement instrument. The hammer is used to simulate with a 3 Hz test bandwidth. Four threedimensional 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 Figure
Natural frequencies and modes of vibration.
Eigenvalue  Natural frequency  Vector  Mode of vibration 


0 Hz 

Axial 

48.4 Hz 

Loworder torsion 

147.5 Hz 

Axial and torsion 

317.2 Hz 

Highorder torsion 

341.6 Hz 

Pitch and torsion 
Comparison of simulation and modal experiment.
Mode of vibration  Prediction natural frequency  Experiment natural frequency  Relative error 

Axial       
Loworder 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% 
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 Figure
Simulation results of height of center of gravity on natural frequency.
From Figures
Comparison of pitch and torsion vibration.
Comparison of axial and torsion vibration.
In this section, the dynamic modelbased 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. Figure
Diagram of dynamic modelbased CCSMC.
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 error
In the practical scenario, the displacements of each axis
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:
Taking the derivative of (
It follows from
A switching control law makes it possible to guarantee the stability of the system, which is defined as
Based on the sliding surface function in the conventional SMC, the switching control law of proposed integral SMC is given as
The Lyapunov function can be expanded by substituting (
To ensure the derivative of Lyapunov function to be negative, the following control equation should be satisfied:
By substituting (
It is known that the dualdriving 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 around
A novel pitch angle measurement method of DCG DDFS is proposed in this section. Figures
Schematic diagram of pitch error measurement.
Measurement of laser interference.
The original pitch error
The pitch errors are obtained by the simulation and practical measurement. The results are changed from variation located positions of sliding stage along
The comparison of simulation and practical measurement is shown in Table
Comparison between simulation results and practical measurement results.
Displacement in 
Height of center of gravity 
Maximum simulation results 
Maximum experiment pitch error 
Relative error 

0  0 


3.6% 

0 


3.0% 

0 


6.5% 
0  150 


5.9% 
0  300 


14.7% 

150 


6.2% 

300 


13.5% 

150 


5.2% 

300 


14.1% 
Maximum variation by displacement in 
— 


11.9% 
Maximum variation by height of center of gravity in 
— 


17.1% 
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 lefthand (or righthand side). When the position of sliding stage along
(2) From Figure
A dualdriving feed experiment is developed based on the computercontrolled dualdriving gantry stage as shown in Figure
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 0.1
The experimental and simulated synchronous errors of dual axis with an oscillating behavior are studied. While the dualdriving system is feed along
To validate the effectiveness of the proposed model, a normal 2DOF dynamic model has been developed as (
Then the equivalent displacement of dualdriving system can be yielded as
Performance comparison with variation location in
Experimental results (um)  Proposed DCG model (um)  Relative error  Normal 2DOF model (um)  Relative error  



Max  9.8  9.1  7.1%  8.9  9.2% 
Avg  3.6  3.2  11.1%  3.1  13.9% 
( 

Max  12.5  11.6  7.2%  9.7  22.4% 
Avg  4.1  3.7  7.3%  3.3  19.5% 


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 dualdriving stage driven by center of gravity.
Then, the practical control performance and synchronization accuracy of proposed dynamic modelbased crosscoupled control are testified in the experiment. The proposed control algorithm is applied to the dualdriving 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 Figure
Experiment and simulation results of synchronous errors in dualdriving feed process. (a) Reference trajectory of
Reference trajectory of dualdriving feed experiment.
Experimental results of NCCSMC: (a) tracking responses of axis
In comparison to the synchronous performance between proposed dynamic modelbased CCSMC and NCCSMC, some phenomena should be noticed in control of dualdriving system:
(1) As can be seen in Figures
Performance comparison of dualdriving feed experimental results.
Errors/control schemes  Sinusoidal trajectory  

NCCSMC  Dynamic modelbased 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  2.1 
Experimental results of the proposed dynamic modelbased CCSMC: (a) tracking responses of axis
(2) The synchronous errors
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 (
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 dualdriving feed experiment. The maximum pitch errors around
Finally, the dynamic modelbased CCSMC is presented for the dualdriving synchronous control. The dynamic model precision is further tested by dualdriving 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 modelbased 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.
The authors declare that they have no conflicts of interest.
The work was supported by the National Natural Science Foundation of China (no. 51675393) and National Natural Science Foundation of China (no. 51375197).