design and implementation of speed ﬂuctuation reduction ... · ddfs structure and its control...

Mech. Sci., 11, 163–172, 2020https://doi.org/10.5194/ms-11-163-2020© Author(s) 2020. This work is distributed underthe Creative Commons Attribution 4.0 License.

Design and implementation of speedfluctuation reduction for a ball screwfeed system at low-speed operation

Zhaoguo Wang1,2, Xianying Feng1,2, Fuxin Du1,2, Hui Li1,2, and Zhe Su1,2

1School of Mechanical Engineering, Shandong University, Jinan, Shandong, China2Key Laboratory of High Efficiency and Clean Mechanical Manufacture of Ministry of Education,

Jinan, Shandong, China

Correspondence: Xianying Feng ([email protected])

Received: 7 July 2019 – Revised: 7 March 2020 – Accepted: 11 April 2020 – Published: 14 May 2020

Abstract. In the high-precision servo feed system, when the permanent magnet synchronous motor (PMSM)is operated at low speed in the classical drive feed system (CDFS), the speed fluctuation caused by the motortorque harmonics seriously affects the speed smoothness of the servo system. In this paper, a novel double-drivedifferential feed system (DDFS) is proposed to effectively suppress the effect of torque harmonics of PMSM onspeed fluctuation of the linear feed system at low-speed operation. Firstly, the effect of motor torque harmonicson motor speed for the DDFS is analyzed by the sensitivity function of the servo system, which indicates that thetorque harmonics have little effect on the motor speed at high-speed operation. Then, in the DDFS, we make twomotors rotate in the same direction at high speed and differentially synthesize at the ball screw to obtain low-velocity linear motion. Compared with the CDFS, the DDFS can suppress the effect of motor torque harmonicson speed fluctuation of the table and improve speed smoothness at low-speed operation.

1 Introduction

In the high-precision manufacturing industry, higher require-ments are placed on the dynamic response speed and speedsmoothness of the servo feed system at low speed. Perma-nent magnet synchronous motors (PMSMs) are widely usedin high-precision servo feed systems since they have high re-liability, efficiency, and a torque inertia ratio with a fast dy-namic response (Pillay, 1990; Jahns and Soong, 1996). How-ever, when the motor is rotating at low speed, the speed fluc-tuation caused by motor torque harmonics will seriously af-fect the performance of the servo system (Houari et al., 2018;Qian et al., 2004). According to the difference of frequencyvalues, the torque harmonics of the PMSM can be dividedinto the low-frequency components and the high-frequencycomponents. The high-frequency components can be effec-tively suppressed by controlling loop bandwidths, and thelow-frequency components within the controlling loop band-widths need special handling (Houari et al., 2018; Chan et al.,1994). Therefore, in order to increase the speed smoothness

of the servo system, when the motor operates at low speed, itis necessary to suppress the low-frequency components thatcause the motor torque fluctuation.

The speed fluctuation of a PMSM is affected by many fac-tors at low speed, e.g., cogging torque, current measurementerrors, and flux harmonics (Guemes et al., 2011; Gebreger-gis et al., 2015). To effectively reduce the speed fluctuation,many valuable methods have been proposed and verified tosuppress motor torque ripple. Generally, these methods areusually divided into two groups. The first group focuses onmotor design improvement, such as skewing the slot or mag-net (Binns et al., 1993; Carlson et al., 1989), using a frac-tional number of slots per pole (Chung et al., 2012; Senol andUstun, 2011; Donato et al., 2010; Cavagnino et al., 2013), oroptimizing the winding distribution (Nakao et al., 2014; Ge-bregergis et al., 2015). The torque ripple can be effectivelyreduced by optimizing the motor design. However, the opti-mization of motor design is limited by the development ofnew technologies, and they can result in higher manufac-turing cost and a further complicated realization. The sec-

Published by Copernicus Publications.

164 Z. Wang et al.: Design and implementation of speed fluctuation reduction for a ball screw feed system

ond group uses advanced control methods, which regulatethe input current or voltage to reduce torque ripple (Pandaet al., 2008). Houari et al. (2018, 2015) proposed an effec-tive method to reduce speed fluctuation of PMSM at lowspeed, which improved the conventional PMSM controllerby superposing an appropriate compensation signal on thequadratic-current reference. Xia et al. (2015) proposed aspeed and current proportional–integral–resonant (PIR) con-trol strategy to reduce speed fluctuation for the low-speedhigh-torque PMSM drive system. Qian et al. (2005) proposedtwo iterative learning control (ILC) schemes implemented inthe time domain and frequency domain, respectively, to sup-press speed ripples due to torque ripple at low speed. Us-ing a robust iterative learning control algorithm (ILC) basedon an adaptive sliding mode control (SMC) technique, themotor torque ripple was effectively suppressed, and the anti-interference ability of the servo system was improved (Liu etal., 2018). Tan et al. (2011) proposed the identification andcompensation method of torque ripple based on steady-stateerror analysis, in which the function of torque ripple was de-termined by the least-squares method and the compensationof torque ripple was achieved by a programmable multi-axiscontroller (PMAC). The advantage of the above method isthat the controller is a component of the servo system anddoes not require additional hardware. However, this methodrequires a special control algorithm to be written, which iscomplicated to implement.

In this paper, a novel DDFS is proposed to reduce the influ-ence of torque harmonics of PMSM on velocity fluctuationsof a table for a linear feed system at low-velocity operation.In the DDFS, the screw and nut are both driven by PMSMs,which rotate in the same direction at high speed and differen-tially synthesize at the ball screw pair to obtain low-velocitylinear motion. This design ensures the driven table travelsat low velocity while guaranteeing the two motors rotate athigh speed, in which the high-frequency components of thetorque ripples limited by the bandwidth of the servo systemsuppress the influence of the motor torque ripple on the ve-locity fluctuation of the table. Compared with the CDFS, theDDFS can reduce the effect of motor torque harmonics on thespeed fluctuation of the table and improve speed smoothnessat low-speed operation.

The remainder of this paper is arranged as follows. TheDDFS structure and its control strategy are described inSect. 2. The speed fluctuation caused by motor torque har-monics is analyzed in Sect. 3. The influence of motor torqueharmonics on motor speed for the DDFS is analyzed by thesensitivity function of the servo system in Sect. 4. The ve-locity fluctuations of the table for the DDFS compared withthe CDFS are simulated in Sect. 5. The experimental resultsvalidate the effectiveness of the proposed method in Sect. 6.The paper is concluded in Sect. 7.

2 The DDFS description

2.1 The DDFS structure

In the CDFS, the screw shaft is driven by a servo motorthrough a coupling and the rotary motion of a motor is con-verted into linear motion of the table. Compared with theCDFS, the DDFS structure based on the nut-driven ball screwpair is illustrated in Fig. 1.

In the DDFS, the screw shaft and screw nut are both drivenby PMSMs, which rotate in the same direction at high speedand differentially synthesize though the ball screw pair to ob-tain low velocity of the table. This design ensures the driventable travels at low velocity while guaranteeing the two mo-tors rotate at high speed. Superposing two rotating speeds ofPMSMs at high speed by the ball screw pair, the driven tablecould obtain low velocity and effectively avoid the crawlingzone of the motors at low speed (Du et al., 2018a, b, 2017;Yu and Feng, 2015). In the DDFS, when the nut servo motordoes not rotate, the DDFS changes to the CDFS.

2.2 PMSM model

Assuming that the PMSM iron loss, eddy current loss, andhysteresis loss are not considered, the d–q axis current equa-tion in the stator d–q axis rotation reference coordinate sys-tem can be described as follows:

ddtid =−Rs

Ldid +

ωeLq

Ldiq +

1Ldud , (1)

ddtiq =−Rs

Lqiq −

ωeLd

Lqid −

ωeψf

Lq+

1Lquq , (2)

ddtωe =

p

JM

(Te− TL−

B

pωe

), (3)

ddtθe = ωe, (4)

where p is the number of pole pairs, ωe is the electrical an-gular speed, Rs is the stator resistance, ud and uq are thevoltages of the stator along the d and q axes, respectively,Ld and Lq are the inductances of the stator along the d andthe q axes, respectively, id and iq are the currents of the sta-tor along the d and q axes, respectively, TL is the load torque,Te is the electromagnetic torque, B is the viscous friction co-efficient equivalent to the motor shaft, and JM is the momentof inertia equivalent to the motor shaft, including the mo-tor shaft moment of inertia, the screw moment of inertia, thecoupling moment of inertia, and the mass of the table.

When the id = 0 vector control is performed on thePMSM, the electromagnetic torque of the motor is largest,and the electromagnetic torque can be shown as follows:

Te = p[ψfiq +

(Ld −Lq

)id iq

]. (5)

Mech. Sci., 11, 163–172, 2020 https://doi.org/10.5194/ms-11-163-2020

Z. Wang et al.: Design and implementation of speed fluctuation reduction for a ball screw feed system 165

Figure 1. The DDFS structure.

Table 1. Parameters of the servo system for the DDFS.

Parameters Value

Stator resistance Rs 1.7 �Stator inductances Ls 5.9 mHNumber of pole pairs p 5Magnet flux ψf 0.11 WbEquivalent inertia JM 7.2× 10−4 kg m2

Equivalent frictional coefficient B 0.002 N m s−1

Rated speed 3000 r min−1

Rated torque 1.3 N mCutoff frequency of current loop ωc 3000 rad s−1

Cutoff frequency of speed loop ωω 200 rad s−1

Helical pitch of ball screw l 5 mm

For the surface-mounted PMSM used in this paper, Ld andLq are equal and the electromagnetic torque is simplified asfollows:

Te = pψfiq . (6)

The parameters of the servo system for the DDFS are shownin Table 1.

2.3 Control strategy of the DDFS

The speed fluctuation of the DDFS is mainly studied in thispaper; therefore, the PMSMs are set to work under the speedloop. The conventional PI control strategy is adopted for thePMSMs with the same parameters selected. The control strat-egy of the DDFS is shown in Fig. 2.

When the velocity of the table v = vrefe1−vrefe2 is required,the velocity signals vrefe1 = ω refe1 · r of the screw motor andvrefe2 = ω refe2·r of the nut servo motor are given by the servodrivers of the two motors, respectively. The velocity of thetable can be obtained by the ball screw pair.

v = (ωe1−ωe2)r (7)

r =l

2πp, (8)

where v is the velocity of the table, ω refe1 and ω refe2 are theelectrical angular speed reference value of the screw servomotor and the electrical angular speed reference value of thenut servo motor, respectively, r is the transmission coeffi-cient, l is the helical pitch of the ball screw, and p is thenumber of poles of the PMSM.

The current loop controller and speed loop controller aredesigned by a simple pole placement strategy. When the cur-

https://doi.org/10.5194/ms-11-163-2020 Mech. Sci., 11, 163–172, 2020


Figure 2. Control strategy of the DDFS.

rent loop bandwidth is selected, the current loop adjustmentresponse time is required to be as small as possible and thecurrent loop is required to have good disturbance perfor-mance. When selecting the speed loop bandwidth, it is nec-essary to ensure a better control target while avoiding mutualinterference between the current loop and the speed loop.

The control parameters of the current loop and speed loopare as follows:

Kpc = 2ξLsωc−Rs, (9)

Kic = Lsω2c , (10)

Kpω =2ξJMωω

p2ψf, (11)

Kiω =JMω

2ω

p2ψf, (12)

where Kpc is the current loop proportional gain, Kic is thecurrent loop integral gain, ξ is the damping coefficient, ξ =0.7, ωc is the current loop cutoff frequency, Kpω is the speedloop proportional gain, Kiω is the speed loop integral gain,and ωω is the speed loop cutoff frequency.

3 Speed fluctuation analysis of a PMSM at lowspeed

When the PMSM is controlled at low speed, the main influ-ence factors for the speed fluctuation of a motor are currentsensor offset error, gain mismatch, and flux harmonics.

3.1 Current sensor offset error

Due to the dc offsets present during phase current detectionand controller input conversion, the phase current measuredby the sensor is deviated. The torque ripple due to the sta-tor current measurement error can be expressed as follows(Houari et al., 2018):

Tem1 = β cos(θe+α)√1i2a +1ia1ib+1i

2b , (13)

where1ia and1ib are the offset errors, β is a constant, θe isthe electrical angle, and α is a constant displacement of theangle.

3.2 Gain mismatch

The frequency of the torque ripple caused by the current sam-pling error is twice the fundamental frequency, and the torquefluctuation caused by the current sampling error can be ex-pressed as follows (Houari et al., 2018):

Tem2 = γ cos(

2θe−2π3

), (14)

where γ is the sensor gain deviation.

3.3 Flux harmonics

The non-sinusoidal flux linkage in the air gap is another fac-tor that causes torque ripple. The torque harmonics of the 6thand multiples of the 6th caused by the non-sinusoidal fluxlinkage can be expressed as (Houari et al., 2018)



TFH =

n∑i=1

T6i cos(6iθe) , (15)

where T6i is the amplitude of the multiple of six harmonics.

3.4 Speed fluctuation analysis

The motor electromagnetic torque Te can be expressed as thesum of the main dc component (T0) and the harmonic com-ponents (1T ).

Te = T0+1T (16)1T = Tem1+ TFH+ Tem2 (17)

The motor electromagnetic torque Te contains the 1st, 2nd,6th, and 12th harmonics. The presence of these harmonicscan cause speed fluctuations, especially when the motor ro-tates at low speed.

4 The influence of torque harmonics on motorspeed for the DDFS

The relationship between the motor output speed ωe,i , themotor input speed ω refe,i (i = 1,2), and the disturbance TLcan be obtained by the Laplace transform as shown in thefollowing equation.

ωe,i =H (s)ωrefei − S (s)TL, (18)

where H (s) and S(s) are the closed loop transfer functionand the sensitivity function, respectively. The expressions ofH (s) and S(s) are listed in Appendix A.

The Bode magnitude plot of S(s) is shown in Fig. 3, inwhich the horizontal axis represents the electrical angularspeed of the PMSM. The torque harmonics are regarded asthe disturbance signals. Figure 3 shows that as the electricalangular speed increases, the influence of the torque harmon-ics on the torque ripple increases first and then decreases.When the DDFS is used, according to the Bode magnitudeplot of the S(s), the high-speed operating conditions with lessinfluence of torque harmonics on motor speed fluctuationsare selected for the both PMSMs to achieve low velocity ofthe table through ball screw pair differential synthesis, whichcan effectively improve velocity smoothness of the table atlow velocity compared with the CDFS.

The analysis was carried out with an example of the ta-ble velocity 2.5 mm s−1. When the CDFS is used, the cor-responding motor speed is 30 r min−1 (the electrical angu-lar speed corresponding to the 6th harmonic of the motor is900 rpm and is marked A6; the electrical angular speed cor-responding to the 12th harmonic of the motor is 1800 rpmand is marked A12). In the DDFS, when the nut servo motorspeed is 300, 500, and 800 rpm, the electrical angular speedof the 6th harmonic of the nut servo motor is marked B6, C6,

Figure 3. Bode magnitude plot of the S(s) for the DDFS.

and D6, and the electrical angular speed of the 12th harmonicof the nut servo motor is marked B12, C12, and D12, respec-tively.

The electrical angular frequency corresponding to theabove torque harmonics is marked in Fig. 3. Because thescrew servo motor speed is only 30 rpm higher than the nutservo motor in the DDFS, the screw servo motor torque har-monics are located near the corresponding torque harmonicsof the nut servo motor. It can be seen from Fig. 3 that theBode magnitude plot of S(s) has better suppression of theharmonic torque of the PMSM for the DDFS at a high speed,and the low velocity of the table is obtained by both motors’synthesis. It can effectively reduce the velocity fluctuation ofthe table compared with the CDFS.

5 Simulation of the velocity fluctuation for the table

To simulate the velocity fluctuation of the table at low veloc-ity for the DDFS compared with the CDFS, the torque har-monics caused by a non-sinusoidal flux density distribution issimulated by using Eq. (15), in which the 6th and 12th torquecomponents are taken as 6 % and 2 % of the rated torque, re-spectively (Houari et al., 2015). The velocity ripple factor(VRF) is used to evaluate the velocity fluctuation of the ta-ble, which indicates the percentage of the velocity deviatingfrom the reference velocity.

VRF=Vpk-pk

V refe× 100%, (19)

where Vpk-pk is the peak-to-peak value of the velocity fluc-tuation for the table and V ref

e is the reference velocity of thetable.

The reference velocity 2.5 mm s−1 of the table is selectedfor analysis under the load of 1 Nm. Figure 4 shows the ve-locity fluctuation of the table for the DDFS at different syn-thesis speeds of both PMSMs compared with the CDFS (thespeed of the nut servo motor is 0 mm s−1) when the reference



Figure 4. Comparison of the table velocity fluctuations of CDFSand DDFS at 2.5 mm s−1 under 1 N m load torque.

velocity of the table is 2.5 mm s−1. Figure 4a shows the com-parison of the table velocity fluctuations of CDFS and DDFSwhen the nut servo motor speed is 300 rpm in the DDFS.Figure 4b shows the comparison of the table velocity fluctu-ations of CDFS and DDFS when the nut servo motor speedis 500 rpm in the DDFS. Figure 4c shows the comparison ofthe table velocity fluctuations of CDFS and DDFS when thenut servo motor speed is 800 rpm in the DDFS.

Figure 4 indicates that the two motors can be synthesizedat a high speed to obtain a low velocity of the table in theDDFS, which can effectively reduce the velocity fluctuationof the table compared with the CDFS.

Figure 5. Experimental setup device.

Figure 6. Comparison of the table velocity fluctuations of CDFSand DDFS when the nut servo motor speed is 300 rpm in the DDFS.




6 Experiment results validate

In order to compare the velocity fluctuations at low velocityof the table for the CDFS and the DDFS, the experimentalsetup device is shown in Fig. 5. The experimental equipmentincludes the DDFS, Renishaw RLE 10 fiber laser scale (RLD10 detector head, RLU 10 laser unit, RCU 10 compensationsystem), and data acquisition system. In the experiments, thesampling frequency of the fiber laser scale is set to 10 kHz.

When the CDFS is used, the speed of the nut servo mo-tor is set to 0 rpm and the table is driven only by the screwservo motor. When the DDFS is used, we make both motorsrotate in the same direction at high speed and differentiallysynthesize by the ball screw pair to obtain low velocity of thetable.

The velocity fluctuations of the table for the DDFS at dif-ferent synthesis speeds of both PMSMs compared with theCDFS are studied when the reference velocity of the tableis 2.5 mm s−1. Figures 6, 7, and 8 show a comparison of thetime and harmonic spectrum for the table velocity fluctua-tions of CDFS and DDFS when the nut servo motor speedsare 300, 500, and 800 rpm in the DDFS, respectively. FromFig. 6 it can be seen that the VRF rates of the table aredecreased from 20.1 % with the CDFS to 13.4 % with theDDFS, when the speed of the nut servo motor is 300 rpm in


Figure 9. The VRF rate of the table varies with the speed of the nutservo motor.

the DDFS. Figure 7 indicates that the VRF rates of the tableare decreased from 20.1 % with the CDFS to 10.6 % with theDDFS, when the speed of the nut servo motor is 500 rpm inthe DDFS. Figure 8 illustrates that the VRF rates of the tableare decreased from 20.1 % with the CDFS to 8.2 % with theDDFS, when the speed of the nut servo motor is 800 rpm inthe DDFS.

When the velocity of the table is 2.5 mm s−1 in the DDFS,the VRF rate of the table varies with the speed of the nutservo motor as shown in Fig. 9, which indicates that the



VRF rate of the table gradually decreases with the increasein the speed of the nut servo motor in the DDFS. Comparedwith the CDFS, the DDFS can significantly improve the low-velocity smoothness of the table.

7 Conclusion

In this paper, a novel DDFS is developed to reduce the influ-ence of torque harmonics of PMSM on speed fluctuations ofa linear feed system at low-speed operation.

Some conclusions can be drawn as follows.

1. The law of the influence of motor torque harmonics onmotor speed for the DDFS is analyzed by the sensitivityfunction of the servo system, which indicates that thetorque harmonics have little effect on the motor speedat high-speed operation.

2. According to the law of the influence of motor harmon-ics on motor speed in the DDFS, we make two motorsrotate in the same direction at high speed and differen-tially synthesize at the ball screw to obtain low-velocitysmoothness linear motion.

3. The analysis is carried out with an example of the tablevelocity 2.5 mm s−1, which indicates that the VRF rateof the table gradually decreases with the increase in thespeed of the nut servo motor in the DDFS. Comparedwith the CDFS, the DDFS can significantly improve thelow-velocity smoothness of the table.



Appendix A

The closed loop transfer function H (s) and the sensitivityfunction S(s) can be expressed as follows:

H (s)= NumH (s)/Den(s) , (A1)S (s)= NumS(s)/Den(s) , (A2)

NumH (s) =(KpcKpωψfp

2)s2

+

(KicKpωψfp

2+KiωKpcψfp

2)s

+KicKiωψfp2, (A3)

NumS(s) = (Lsp)s3+(Kpcp+Rsp

)s2+ (Kicp)s, (A4)

Den(s)= (JMLs)s4+(BLs+ JMKpc+ JMRs

)s3

+

(KpcKpωψfp

2+BKpc+BRs+ JMKic

)s2

+

(BKic+KicKpωψfp

2+KiωKpcψfp

2)s

+KicKiωψfp2. (A5)



Data availability. All data included in this study are availableupon request by contact with the corresponding author.

Author contributions. ZW made substantial contributions to theconception and design, the acquisition, the analysis, and the inter-pretation of data for the work. He also drafted the work or revisedit critically for important intellectual content. ZW made a contri-bution to the acquisition of simulation experimental data and datacollation. XF supervised and structured the process of the paper.FD helped in making figures. HL checked the writing language. ZShelped in the writing language.

Competing interests. The authors declare that they have no con-flict of interest.

Acknowledgements. The authors wish to thank the supportof the Science and Technology Development Plan of ShandongProvince (grant no. 2015GGX103036).

Financial support. This work is fully supported by the Scienceand Technology Development Plan of Shandong Province (grantno. 2015GGX103036), the National Natural Science Foundation ofChina (grant no. 51375266), the Young Science Foundation Projectof China (grant no. 51705289), the Key Research & DevelopmentProgram of Shandong Province (grant no. 2019GGX104101), andthe Natural Science Foundation of Shandong Province, China (grantno. ZR2017PEE005).

Review statement. This paper was edited by Zi Bin and reviewedby Assylbek Jomartov and one anonymous referee.

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design and implementation of speed ﬂuctuation reduction ... · ddfs structure and its control...

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