資源目錄里展示的全都有，所見即所得。下載后全都有,請放心下載。原稿可自行編輯修改=【QQ：401339828 或11970985 有疑問可加】 Int J Adv Manuf Technol (2003) 21:604611Ownership and Copyright 2003 Springer-Verlag London LimitedBacklash Estimation of a Seeker Gimbal with Two-Stage GearReducersJ. H. Baek, Y. K. Kwak and S. H. KimDepartment of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1 Gusung-dong Yusung-gu Daejon,KoreaA novel technique for estimating the magnitude or contributionratio of each stage backlash in a system with a two-stage gearreducer is proposed. The concept is based on the change offrequency response characteristic, in particular, the change ofanti-resonant frequency and resonant frequency, due to thechange of the magnitude of the backlash of each stage, eventhough the total magnitude of the backlash of a system witha two-stage gear reducer is constant. The validity of thetechnique is verified in a seeker gimbal and satisfactory resultsare obtained. It is thought that the diagnosis and maintenanceof manufacturing machines and systems with two-stage gearreducers will become more efficient and economical by virtueof proposed technique.Keywords: Anti-resonantfrequency;Backlashestimation;Contribution ratio; Frequency response characteristic; Resonantfrequency; Seeker gimbal1.IntroductionThe automation of manufacturing machines and the frequentuse of robots and servo systems have greatly increased thedemand for servo systems with servomotors. With the advanceof motor manufacturing techniques, servo systems have beendeveloped with direct drive type motors that do not requiregear reducers. However, thus far, servo systems with gearreducers have been used extensively in manufacturing machinesin many fields, because the servo system volume and weightis larger than that of the gear reducer, while its torque isrelatively small in comparison.Servo systems with gear reducers have had problems relatedto gear backlash since their inception. Accordingly, manystudies have been performed in order to deal with the problems.Correspondence and offprint requests to: J. H. Baek, Research andDevelopment 7 Group, LG Innotek Co., Ltd., 1481 Mabuk-ri Gusung-eup Yongin-city Kyonggi-do, 449910, Korea.E-mail: Received 5 February 2002Accepted 29 March 2002In order to diagnose and maintain the performance of therobots and servo systems, a method of monitoring and detectingthe magnitude and change of backlash has been developed.Dagalakis and Myers used a coherence function and the magni-tude of resonant peak in the frequency response between themotor voltage and the acceleration of a robot link as measures1. Stein and Wang developed a technique based on momen-tum transfer analysis in order to detect and estimate thebacklash of a servo system with a gear reducer. They foundthat the speed change of the primary gear due to impact withthe secondary gear is related to the magnitude of the backlash2. Saker et al. developed a technique to complement the workof Stein and Wang using the impulsive torque due to impact,instead of the speed change of the primary gear 3. Pan et al.developed a technique for detecting and classifying the backlashof a robot by using WignerVille distributions combined witha two-dimensional correlation of the relationship between thesinusoidal joint motion and the acceleration of the robot link4. However, there is no technique for estimating the magni-tude or contribution ratio of each stage of the backlash in aservo system with a multistage gear reducer, which is oftenused in manufacturing machines and robots. It is very importantto know the magnitude of each stage backlash of system inorder to obtain the desired magnitude of backlash and tomaintain that magnitude in a correct range. The purpose ofthis paper is, therefore, to present a technique for estimatingthe magnitude or contribution ratio of each stage backlash ofa servo system with a two-stage gear reducer. The contributionratio is defined as the ratio of the magnitude of the first stagebacklash to that of the total backlash. The concept for estimat-ing the magnitude of each stage backlash is based on thechange of anti-resonant frequency (ARF) and resonant fre-quency (RF) in the frequency response characteristic of a servosystem, according to the change of the magnitude of eachstage of backlash, even though the total backlash of the servosystem is constant. In order to verify the validity of theproposed technique, two driving servo systems of a seekergimbal, which are used in order to stabilise the orientation ofan object, are considered. One is an azimuth driving servosystem (ADSS); the other is an elevation driving servo system(EDSS). Both servo systems have two-stage gear reducers.Backlash Estimation of a Seeker Gimbal6052.Model of Seeker Gimbal2.1Model of the ADSS in the Seeker GimbalA photograph of the seeker gimbal with two-stage gear reducerswhich is considered in this paper is presented in Fig. 1(a).The ADSS and EDSS correspond to the two driving parts ofthe seeker gimbal. In the case of the ADSS, the hatchedcomponents, pinion 2, shaft 1, gear 1, pinion 1, motor andbearings rotate with respect to the AA? axis except that gear2 is attached on a fixed shaft as shown in Fig. 1(b). It isassumed that bearings support each shaft without any clearance,due to the preload. Also, the influence of the damping charac-teristic is neglected. The model of the ADSS obtained underthese assumptions is presented in Fig. 1(c). The moment ofFig. 1. (a) Seeker gimbal; (b) structure of ADSS; (c) model of ADSS; (d) structure of EDSS; (e) model of EDSS.inertia of pinion 1 is included with that of the motor. Thetorsion spring represented at the right side of gear 1 indicatesthe torsion stiffness due to tooth stiffness between pinion 1and gear 1. In the case of shaft 1, the moment of inertia islumped at the centre of the distance between gear 1 and pinion2 and torsion springs, with twice the value of torsion stiffnessof shaft 1, are connected with gear 1 and pinion 2. Becausethey are fixed, gear 2 and the fixed shaft are modelled so thatthey have only torsion springs without moment of inertia. Eachbacklash is represented as the angles of rotation of the gearswhen the pinions are fixed. Components enclosed by a phantom(double dot) line in Fig. 1(c) indicate the load of the ADSS.The ADSS considered consists of a tachometer filter, a motoramplifier and the aforementioned structure. The motor amplifieris used to amplify the input voltage of motor. A permanent606J. H. Baek et al.magnetic field type d.c. motor with a tachometer is used asan actuator. In order to filter the output voltage of the tach-ometer, a second-order low-pass filter is used. The governingelectric Eq. of these components are as follows 5:Vm= kaVi(1)Ladiadt+ Rmia+ kb?m= Vm(2a)Tm= ktia(2b)Vt= kts?m(3)Vo(s) = Gf(s)Vt(s)(4)The Eq. of motion for the motor is as follows:Jm?m+ Bm?m= Tm?Tg1N1? Tf,msign (?m)(5)The torque transmitted to gear 1 is represented as a nonlinearEq., presented in Eq. (6), due to the backlash between pinionFig. 2. The bode diagram (Vo/Vi) of ADSS according to contribution ratio: (a) case 1; (b) case 2; (c) case 3; (d) case 4; (e) case 5. (Sim:simulation; Exp: experiment.).1 and gear 1. The model of the dead zone is used as themodel of the backlash 6.Tg1=?kg1(?d1?1),?d1?10,?d1?1kg1(?d1+?1),?d1?1?(6)where?d1=?m/N1?g1(7)The Eq. of motion for gear 1 is as follows:Jg1?g1= Tg1? 2ks1(?g1?s1)(8)The Eq. of motion for shaft 1 is as follows:Js1?s1= 2ks1(?g1+?p2) ? 4ks1?s1(9)Besides, the equation of motion for pinion 2 is as follows:Jp2?p2= 2ks1(?s1?p2) ?1NrTL(10)Backlash Estimation of a Seeker Gimbal607The torque of the load is represented in Eq. (11), like Eq. (6).TL=?k2(?d2?2),?d2?20,|?d2|?2k2(?d2+?2),?d2?2?(11)where?d2=?p2/Nr?L(12)Here, the equivalent torsion stiffness between gear 2 and shaft2 is as follows 7:k2=kg2ks2kg2+ ks2(13)Finally, the equation of motion for the load is as follows:JL?L= TL? Tf,Lsign (?L)(14)The response of the output voltage of the tachometer filterwith respect to the input voltage of the motor amplifier isobtained from these Eq. In addition, the relation between thetotal backlash and each stage backlash is as follows:bt= b2+1Nrb1(15)wherebi= 360 ?i/?(i = 1,2)(16)2.2Model of the EDSS in the Seeker GimbalIn this subsection, the EDSS models and Eq. of motion arederived. The structure of the EDSS is presented in Fig. 1(d).Because gear 2 is directly attached to the load, the momentof inertia of gear 2 is included with that of the load and gear2 has only a torsion spring model, as shown in Fig. 1(e). TheEq. of motion for the EDSS between the motor amplifier andtachometer filter are the same as those of the ADSS, exceptfor replacing Eqs (10)(13) and Eq. (15) with Eqs (17)(20)as follows:Jp2?p2= 2ks1(?s1?p2) ?1N2TL(17)TL=?kg2(?d2?2),?d2?20,|?d2|?2kg2(?d2+?2),?d2?2?(18)where?d2=?p2/N2?L(19)bt= b2+1N2b1(20)From Eqs (1)(9), Eq. (14) and Eqs (17)(20), the responseof the output voltage of the tachometer filter with respect tothe input voltage of the motor amplifier is obtained.3.SimulationIt is well known that an increase in the total backlash in asystem causes the frequency response characteristic, of theoutput voltage of the tachometer filter with respect to the inputvoltage of the motor amplifier, to change because it reducesthe effective equivalent torsional stiffness of the system 8.However, it has not been reported yet that although the totalbacklash magnitude is constant, a servo system with a differentbacklash magnitude at each stage has different frequencyresponse characteristic. In this work, each stage of backlashof a servo system is examined by this phenomenon and hypoth-esis. In order to verify this hypothesis, the frequency responsecharacteristic of ADSS is investigated according to the contri-bution ratio. The bode diagrams of ADSS obtained from thesimulation are represented in Fig. 2. The specifications usedfor the simulation are presented in Table 1. The combinationsof the magnitude of backlash of each stage obtained accordingto the change of contribution ratio are listed in Table 2. Theyare obtained from Eqs (15) and (20). In order to obtain thesimulation results of Fig. 2, the equation of motion outlinedin the previous section are converted into a block diagram.The simulation is then performed using MATLAB SimulinkV. 6.1 software. The peak amplitude of the sinusoidal voltagesupplied to the motor amplifier is 2.5 V and the sampling timeused is 10 ?sec. Bode diagrams of Fig. 2 are made fromthe frequency analysis to extract only the excited frequencycomponent from the output voltage of the tachometer filterwith respect to the sinusoidal voltage supplied to the motoramplifier. The ARF and RF obtained are summarised in Table2 and are represented in Fig. 3(a). The difference between theARF and RF is shown in Fig. 3(b). From Fig. 3(a) and (b),it is found that the frequency response characteristic of a servoTable 1. Specifications for ADSS and EDSS.ParameterADSSEDSSGear ratio 1, N15.946.41Torsion stiffness, kg1(m ? N/rad)3.40E4 4.74E4Moment of inertia of gear 1, Jg1(kg ? m2)2.34E-5 3.69E-5Torsion stiffness of shaft 1, ks1(m ? N/rad)22.81.54E2Moment of inertia of shaft 1, Js1(kg ? m2)8.30E-8 2.04E-7Moment of inertia of pinion 2, Jp2(kg ? m2)2.21E-7 4.84E-7Gear ratio, Nr,N210.57.75Equivalent torsion stiffness, k2,kg2(m ? N/rad)7.74E4 2.54E5Moment of inertia of load, JL(kg ? m2)2.75E-3 1.44E-2Static friction torque of load, Tf,L(m ? N)7.0E-37.1E-3Total backlash, bt(deg.)0.0660.276Motor inductance, La(H)8.50E-4Motor resistance, Rm(?)4.10Back-EMF const., kb(V ? s/rad)3.44E-2Torque sensitivity, kt(m ? N/A)3.49E-2Moment of inertia of motor, Jm(kg ? m2)8.60E-6Static friction torque of motor, Tf,m(m ? N)1.40E-2Gain of motor amplifier, ka4.11Tachometer sensitivity, kts(V ? s/rad)8.60E-2Transfer function of low-pass filter, Gf(s)723439s2+ 1710s + 723439Viscous damping coeff. of motor, Bm(m ?1.6E-4N/(rad/s)608J. H. Baek et al.Table 2. The simulation result and experiment result of ADSS andEDSS according to the contribution ratio (Exp: experiment).Case Contribution b1b2Anti-Resonantratio (%)resonant(dB/Hz)(dB/Hz)ADSS1000.066?33.6/125?12.8/1272250.1730.0495?33.5/131?14.3/1353500.3470.0330?33.3/134?14.0/1454750.5190.0166?32.2/137?9.6/14951000.6930?30.8/1410.2/153Exp. 230.1610.051?22.3/128?18.6/137EDSS1000.276?24.7/50?3.4/792250.5350.207?23.7/51?15.1/843501.070.138?27.5/52?3.2/974751.600.069?20.8/52?5.9/9251002.140?22.4/51?3.9/89Exp. 40.08560.265?14.6/40?1.8/75Fig. 3. The simulation results according to contribution ratio: (a) ARF and RF of ADSS; (b) difference between ARF and RF of ADSS; (c)error index of ADSS; (d) ARF and RF of EDSS; (e) difference between ARF and RF of EDSS; (f) error index of EDSS.system is changed according to the change of the magnitudeof the backlash of each stage in spite of having the same totalbacklash. In order to investigate this phenomenon once more,the EDSS of the seeker gimbal is simulated in same manneras the ADSS. The results obtained are presented in Fig. 3(d)and (e), and listed in Table 2. From Fig. 3(a), (b), (d), and(e), it is confirmed that although the magnitude of the totalbacklash is constant, a servo system with a two-stage gearreducer has a different frequency response characteristic accord-ing to the change of the magnitude of the backlash of eachstage.4.ExperimentsTo obtain experimental bode diagrams of the ADSS and EDSS,a dynamic analyser (HP35670A) is used and the bode diagramsobtained are represented in Fig. 4(a) and (b). The ARF andBacklash Estimation of a Seeker Gimbal609RF of the ADSS and EDSS obtained from the experimentsare presented in Table 2. In order to verify the accuracy andvalidity of the proposed technique, the backlash of each stageof the ADSS and EDSS is measured using an optical micro-scope, after disassembly of each gear reducer from the systems.Measurement examples of the backlash of each stage arerepresented in Fig. 4(c) and (d) and the measured data arelisted in Table 2.5.Results and DiscussionBecause the simulation results are obtained under the assump-tions that ignore damping effects and bearing clearances, it isdifficult to obtain exactly consistent results between the experi-Fig. 4. (a) Experiment result of ADSS; (b) experiment result of EDSS; (c) backlash measurement of ADSS; (d) backlash measurement of EDSS;(e) the comparison of the estimated contribution ratio with the measured contribution ratio.ment and the simulation. Thus, the error index between thesimulation results and the experiment results is defined as Eq.(21), and the minimum contribution ratio is found.error index =|fAR,S? fAR,E|+|fR,S? fR,E|+|fD,S(21)? fD,E|The error indices of the ADSS and EDSS, according to thecontribution ratio, are represented in Fig. 3(c) and (f). It isshown that the contribution ratio having the minimum errorindex for the ADSS is 25% and that for the EDSS is 0%.The contribution ratios of the ADSS and EDSS obtained fromthe measurement of each stage backlash are 23% and 4%,respectively. From Fig. 4(e), it is also found that the proposedtechnique is sufficiently accurate to estimate the magnitude orcontribution ratio of the backlash of each stage of a seekergimbal with two-stage gear reducers.610J. H. Baek et al.Comparing Fig. 3(c) with Fig. 3(f), the EDSS has a muchhigher minimum error index than the ADSS (EDSS: 20 Hz,ADSS: 10 Hz). It is thought that the dominant error originatesfrom the assumption of neglecting the damping characteristic.The exact transfer function analysis of the model in Fig. 1(c)and (e) is very complex and complicated. Therefore, in orderto simplify the analysis of the damping characteristic, eachservo system is considered simply as a linear system with twomasses and one spring model 9. From Fig. 4(a) and (b), theapproximated damping factors are obtained and the frequencyreduction ratios of the ARF and RF are calculated using thefollowing Eq. 9,10?AR=12QAR=(f2,E? f1,E)2fAR,E(22)?R=fR,EfAR,E?AR(23)RAR= 1 ?1 ? 2?2AR(when 0?AR?0.707)(24a)RR= 1 ?1 ? 2?2R(when 0?R?0.707)(24b)The damping factors and frequency reduction ratios obtainedare represented in Fig. 5(a) and (b). The damping factors ofthe ADSS are 0.075 at the ARF and 0.083 at the RF, whilethose of the EDSS are 0.135 at the ARF and 0.246 at the RF,Fig. 5. (a) Damping factor of ADSS and EDSS. (b) The frequencyreduction ratio of ADSS and EDSS due to damping factor.respectively. The frequency reduction ratios of the ADSS are0.56% at the ARF and 0.69% at the RF, while those of theEDSS are 1.8% at the ARF and 6.2% at the RF, respectively.From Fig. 5(a) and (b), it is thought that the error of theEDSS is larger than that of the ADSS mainly because of thedamping factor, as the former has a more complicated structurethan the latter in terms of load. It is also thought that theremainder of the error arises from the uncertainty of the loadof the EDSS. Finally, it is thought that the ARF and RF inthe frequency response characteristic can be used to estimatethe magnitude or contribution ratio of the backlash of eachstage of a seeker gimbal with two-stage gear reducers if itsload has a small damping coefficient and small uncertainty.6.ConclusionsThe ARF and RF of the frequency response characteristic areconsidered as measures in order to estimate the magnitude orthe contribution ratio of the backlash of each stage of aseeker gimbal with two-stage gear reducers. The concept ofthe proposed technique is based on changes of the ARF andRF according to the change of the magnitude of the backlashof each stage, even though the total magnitude of the backlashis constant. It is verified that the technique can estimate eachstage backlash of the ADSS and EDSS with two-stage gearreducers, respectively, if the servo system, in particular, theservo system load, has a small damping coefficient and smalluncertainty. The technique has several advantages as follows:first, it is a novel method in that it estimates the backlash ofeach stage if the total magnitude of the backlash of servosystem is available. Second, the technique does not require anadditional sensor such as an accelerometer or torque sensor,because it measures the angular velocity of the motor usingthe tachometer. Third, it is efficient and economical becauseonly a loose or an excessively loose gear stage needs to beadjusted or replaced rather than having to replace the wholegear reducer. Fourth, it can be applied to nonrobotic servosystems such as NC machines because it is unnecessary toattach a sensor on the link of robot or the output shaft of aservo system 2. It is thought that using the proposed tech-nique, the diagnosis and maintenance of various manufacturingmachines and many servo systems will become more efficientand economical.AcknowledgementsWe would like to thank LG Innotek Co. for supporting thisstudy and Sung Min Hong, Ho Young Kim and Byung HoLee for their assistance.References1. N. G. Dagalakis and D. R. Myers, “A Technique for the detectionof robot joint gear tightness”, Journal of Robotic Systems, 2(4),pp. 414423, 1985.2. J. L. Stein and C. H. Wang, “Estimation of gear backlash:theory and simulation”, ASME Journal of Dynamic Systems,Measurement and Control, 120, pp. 7482, 1998.Backlash Estimation of a Seeker Gimbal6113. N. Sakar, R. E. Ellis and T. N. Moore, “Backlash detection ingeared mechanisms: modeling, simulation, and experimentation”,Mechanical Systems and Signal Processing, 11(3), pp. 391408,1997.4. M. C. Pan, H. V. Brussel, P. Sas and B. Verbeure, “Fault diagnosisof joint backlash”, ASME Journal of Vibration and Acoustics,120, pp. 1324, 1998.5. M. Clifford, Modern Electronic Motors, Prentice Hall, EnglewoodCliffs, NJ, 1990.6. M. Nordin, J. Galic and P. O. Gutman, “New models for backlashand gear play”, International Journal of Adaptive Control andSignal Processing, 11, pp. 4963, 1997.7. B. A. Chubb, Modern Analytical Design of Instrument Servo-mechanisms, Addison-Wesley, Reading, MA, 1967.8. R. Dhaouadi, K. Kubo and M. Tobise, “Analysis and compensationof speed drive systems with torsional loads