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Int J Adv Manuf Technol 2003 21 604 611 Ownership and Copyright 2003 Springer Verlag London Limited Backlash Estimation of a Seeker Gimbal with Two Stage Gear Reducers J H Baek Y K Kwak and S H Kim Department of Mechanical Engineering Korea Advanced Institute of Science and Technology 373 1 Gusung dong Yusung gu Daejon Korea A novel technique for estimating the magnitude or contribution ratio of each stage backlash in a system with a two stage gear reducer is proposed The concept is based on the change of frequency response characteristic in particular the change of anti resonant frequency and resonant frequency due to the change of the magnitude of the backlash of each stage even though the total magnitude of the backlash of a system with a two stage gear reducer is constant The validity of the technique is verifi ed in a seeker gimbal and satisfactory results are obtained It is thought that the diagnosis and maintenance of manufacturing machines and systems with two stage gear reducers will become more effi cient and economical by virtue of proposed technique Keywords Anti resonantfrequency Backlashestimation Contribution ratio Frequency response characteristic Resonant frequency Seeker gimbal 1 Introduction The automation of manufacturing machines and the frequent use of robots and servo systems have greatly increased the demand for servo systems with servomotors With the advance of motor manufacturing techniques servo systems have been developed with direct drive type motors that do not require gear reducers However thus far servo systems with gear reducers have been used extensively in manufacturing machines in many fi elds because the servo system volume and weight is larger than that of the gear reducer while its torque is relatively small in comparison Servo systems with gear reducers have had problems related to gear backlash since their inception Accordingly many studies have been performed in order to deal with the problems Correspondence and offprint requests to J H Baek Research and Development 7 Group LG Innotek Co Ltd 148 1 Mabuk ri Gusung eup Yongin city Kyonggi do 449 910 Korea E mail jhbaekb Received 5 February 2002 Accepted 29 March 2002 In order to diagnose and maintain the performance of the robots and servo systems a method of monitoring and detecting the 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 the motor voltage and the acceleration of a robot link as measures 1 Stein and Wang developed a technique based on momen tum transfer analysis in order to detect and estimate the backlash of a servo system with a gear reducer They found that the speed change of the primary gear due to impact with the secondary gear is related to the magnitude of the backlash 2 Saker et al developed a technique to complement the work of 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 backlash of a robot by using Wigner Ville distributions combined with a two dimensional correlation of the relationship between the sinusoidal joint motion and the acceleration of the robot link 4 However there is no technique for estimating the magni tude or contribution ratio of each stage of the backlash in a servo system with a multistage gear reducer which is often used in manufacturing machines and robots It is very important to know the magnitude of each stage backlash of system in order to obtain the desired magnitude of backlash and to maintain that magnitude in a correct range The purpose of this paper is therefore to present a technique for estimating the magnitude or contribution ratio of each stage backlash of a servo system with a two stage gear reducer The contribution ratio is defi ned as the ratio of the magnitude of the fi rst stage backlash to that of the total backlash The concept for estimat ing the magnitude of each stage backlash is based on the change of anti resonant frequency ARF and resonant fre quency RF in the frequency response characteristic of a servo system according to the change of the magnitude of each stage of backlash even though the total backlash of the servo system is constant In order to verify the validity of the proposed technique two driving servo systems of a seeker gimbal which are used in order to stabilise the orientation of an object are considered One is an azimuth driving servo system ADSS the other is an elevation driving servo system EDSS Both servo systems have two stage gear reducers Backlash Estimation of a Seeker Gimbal605 2 Model of Seeker Gimbal 2 1Model of the ADSS in the Seeker Gimbal A photograph of the seeker gimbal with two stage gear reducers which is considered in this paper is presented in Fig 1 a The ADSS and EDSS correspond to the two driving parts of the seeker gimbal In the case of the ADSS the hatched components pinion 2 shaft 1 gear 1 pinion 1 motor and bearings rotate with respect to the AA axis except that gear 2 is attached on a fi xed shaft as shown in Fig 1 b It is assumed that bearings support each shaft without any clearance due to the preload Also the infl uence of the damping charac teristic is neglected The model of the ADSS obtained under these assumptions is presented in Fig 1 c The moment of Fig 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 The torsion spring represented at the right side of gear 1 indicates the torsion stiffness due to tooth stiffness between pinion 1 and gear 1 In the case of shaft 1 the moment of inertia is lumped at the centre of the distance between gear 1 and pinion 2 and torsion springs with twice the value of torsion stiffness of shaft 1 are connected with gear 1 and pinion 2 Because they are fi xed gear 2 and the fi xed shaft are modelled so that they have only torsion springs without moment of inertia Each backlash is represented as the angles of rotation of the gears when the pinions are fi xed 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 fi lter a motor amplifi er and the aforementioned structure The motor amplifi er is used to amplify the input voltage of motor A permanent 606J H Baek et al magnetic fi eld type d c motor with a tachometer is used as an actuator In order to fi lter the output voltage of the tach ometer a second order low pass fi lter is used The governing electric Eq of these components are as follows 5 Vm kaVi 1 Ladia dt 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 Tg1 N1 Tf msign m 5 The torque transmitted to gear 1 is represented as a nonlinear Eq presented in Eq 6 due to the backlash between pinion Fig 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 the model of the backlash 6 Tg1 kg1 d1 1 d1 1 0 d1 1 kg1 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 1 Nr TL 10 Backlash Estimation of a Seeker Gimbal607 The torque of the load is represented in Eq 11 like Eq 6 TL k2 d2 2 d2 2 0 d2 2 k2 d2 2 d2 2 11 where d2 p2 Nr L 12 Here the equivalent torsion stiffness between gear 2 and shaft 2 is as follows 7 k2 kg2ks2 kg2 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 fi lter with respect to the input voltage of the motor amplifi er is obtained from these Eq In addition the relation between the total backlash and each stage backlash is as follows bt b2 1 Nr b1 15 where bi 360 i i 1 2 16 2 2Model of the EDSS in the Seeker Gimbal In this subsection the EDSS models and Eq of motion are derived The structure of the EDSS is presented in Fig 1 d Because gear 2 is directly attached to the load the moment of inertia of gear 2 is included with that of the load and gear 2 has only a torsion spring model as shown in Fig 1 e The Eq of motion for the EDSS between the motor amplifi er and tachometer fi lter are the same as those of the ADSS except for replacing Eqs 10 13 and Eq 15 with Eqs 17 20 as follows Jp2 p2 2ks1 s1 p2 1 N2TL 17 TL kg2 d2 2 d2 2 0 d2 2 kg2 d2 2 d2 2 18 where d2 p2 N2 L 19 bt b2 1 N2 b1 20 From Eqs 1 9 Eq 14 and Eqs 17 20 the response of the output voltage of the tachometer fi lter with respect to the input voltage of the motor amplifi er is obtained 3 Simulation It is well known that an increase in the total backlash in a system causes the frequency response characteristic of the output voltage of the tachometer fi lter with respect to the input voltage of the motor amplifi er to change because it reduces the effective equivalent torsional stiffness of the system 8 However it has not been reported yet that although the total backlash magnitude is constant a servo system with a different backlash magnitude at each stage has different frequency response characteristic In this work each stage of backlash of a servo system is examined by this phenomenon and hypoth esis In order to verify this hypothesis the frequency response characteristic of ADSS is investigated according to the contri bution ratio The bode diagrams of ADSS obtained from the simulation are represented in Fig 2 The specifi cations used for the simulation are presented in Table 1 The combinations of the magnitude of backlash of each stage obtained according to the change of contribution ratio are listed in Table 2 They are obtained from Eqs 15 and 20 In order to obtain the simulation results of Fig 2 the equation of motion outlined in the previous section are converted into a block diagram The simulation is then performed using MATLAB Simulink V 6 1 software The peak amplitude of the sinusoidal voltage supplied to the motor amplifi er is 2 5 V and the sampling time used is 10 sec Bode diagrams of Fig 2 are made from the frequency analysis to extract only the excited frequency component from the output voltage of the tachometer fi lter with respect to the sinusoidal voltage supplied to the motor amplifi er The ARF and RF obtained are summarised in Table 2 and are represented in Fig 3 a The difference between the ARF and RF is shown in Fig 3 b From Fig 3 a and b it is found that the frequency response characteristic of a servo Table 1 Specifi cations for ADSS and EDSS ParameterADSSEDSS Gear ratio 1 N15 946 41 Torsion stiffness kg1 m N rad 3 40E4 4 74E4 Moment of inertia of gear 1 Jg1 kg m2 2 34E 5 3 69E 5 Torsion stiffness of shaft 1 ks1 m N rad 22 81 54E2 Moment of inertia of shaft 1 Js1 kg m2 8 30E 8 2 04E 7 Moment of inertia of pinion 2 Jp2 kg m2 2 21E 7 4 84E 7 Gear ratio Nr N210 57 75 Equivalent torsion stiffness k2 kg2 m N rad 7 74E4 2 54E5 Moment of inertia of load JL kg m2 2 75E 3 1 44E 2 Static friction torque of load Tf L m N 7 0E 37 1E 3 Total backlash bt deg 0 0660 276 Motor inductance La H 8 50E 4 Motor resistance Rm 4 10 Back EMF const kb V s rad 3 44E 2 Torque sensitivity kt m N A 3 49E 2 Moment of inertia of motor Jm kg m2 8 60E 6 Static friction torque of motor Tf m m N 1 40E 2 Gain of motor amplifi er ka4 11 Tachometer sensitivity kts V s rad 8 60E 2 Transfer function of low pass fi lter Gf s 723439 s2 1710s 723439 Viscous damping coeff of motor Bm m 1 6E 4 N rad s 608J H Baek et al Table 2 The simulation result and experiment result of ADSS and EDSS according to the contribution ratio Exp experiment Case Contribution b1b2Anti Resonant ratio resonant dB Hz dB Hz ADSS100 0 066 33 6 125 12 8 127 2250 173 0 0495 33 5 131 14 3 135 3500 347 0 0330 33 3 134 14 0 145 4750 519 0 0166 32 2 137 9 6 149 51000 693 0 30 8 1410 2 153 Exp 230 161 0 051 22 3 128 18 6 137 EDSS100 0 276 24 7 50 3 4 79 2250 535 0 207 23 7 51 15 1 84 3501 07 0 138 27 5 52 3 2 97 4751 60 0 069 20 8 52 5 9 92 51002 14 0 22 4 51 3 9 89 Exp 40 0856 0 265 14 6 40 1 8 75 Fig 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 magnitude of the backlash of each stage in spite of having the same total backlash In order to investigate this phenomenon once more the EDSS of the seeker gimbal is simulated in same manner as 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 confi rmed that although the magnitude of the total backlash is constant a servo system with a two stage gear reducer has a different frequency response characteristic accord ing to the change of the magnitude of the backlash of each stage 4 Experiments To obtain experimental bode diagrams of the ADSS and EDSS a dynamic analyser HP35670A is used and the bode diagrams obtained are represented in Fig 4 a and b The ARF and Backlash Estimation of a Seeker Gimbal609 RF of the ADSS and EDSS obtained from the experiments are presented in Table 2 In order to verify the accuracy and validity of the proposed technique the backlash of each stage of 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 are represented in Fig 4 c and d and the measured data are listed in Table 2 5 Results and Discussion Because the simulation results are obtained under the assump tions that ignore damping effects and bearing clearances it is diffi cult 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 the simulation results and the experiment results is defi ned 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 the contribution ratio are represented in Fig 3 c and f It is shown that the contribution ratio having the minimum error index for the ADSS is 25 and that for the EDSS is 0 The contribution ratios of the ADSS and EDSS obtained from the measurement of each stage backlash are 23 and 4 respectively From Fig 4 e it is also found that the proposed technique is suffi ciently accurate to estimate the magnitude or contribution ratio of the backlash of each stage of a seeker gimbal with two stage gear reducers 610J H Baek et al Comparing Fig 3 c with Fig 3 f the EDSS has a much higher minimum error index than the ADSS EDSS 20 Hz ADSS 10 Hz It is thought that the dominant error originates from 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 order to simplify the analysis of the damping characteristic each servo system is considered simply as a linear system with two masses and one spring model 9 From Fig 4 a and b the approximated damping factors are obtained and the frequency reduction ratios of the ARF and RF are calculated using the following Eq 9 10 AR 1 2QAR f2 E f1 E 2fAR E 22 R fR E fAR E AR 23 RAR 1 1 2 2 AR when 0 AR 0 707 24a RR 1 1 2 2 R when 0 R 0 707 24b The damping factors and frequency reduction ratios obtained are represented in Fig 5 a and b The damping factors of the ADSS are 0 075 at the ARF and 0 083 at the RF while those 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 frequency reduction ratio of ADSS and EDSS due to damping factor respectively The frequency reduction ratios of the ADSS are 0 56 at the ARF and 0 69 at the RF while those of the EDSS 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 the EDSS is larger than that of the ADSS mainly because of the damping factor as the former has a more complicated structure than the latter in terms of load It is also thought that the remainder of the error arises from the uncertainty of the load of the EDSS Finally it is thought that the ARF and RF in the frequency response characteristic can be used to estimate the magnitude or contribution ratio of the backlash of each stage of a seeker gimbal with two stage gear reducers if its load has a small damping coeffi cient and small uncertainty 6 Conclusions The ARF and RF of the frequency response characteristic are considered as measures in order to estimate the magnitude or the contribution ratio of the backlash of each stage of a seeker gimbal with two stage gear reducers The concept of the proposed technique is based on changes of the ARF and RF according to the change of the magnitude of the backlash of each stage even though the total magnitude of the backlash is constant It is verifi ed that the technique can estimate each stage backlash of the ADSS and EDSS with two stage gear reducers respectively if the servo system in particular the servo system load has a small damping coeffi cient and small uncertainty The technique has several advantages as follows fi rst it is a novel method in that it estimates the backlash of each stage if the total magnitude of the backlash of servo system is available Second the technique does not require an additional sensor such as an accelerometer or torque sensor because it measures the angular velocity of the motor using the tachometer Third it is effi cient and economical because only a loose or an excessively loose gear stage needs t

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