Backlash Estimation of a Seeker Gimbal with Two-Stage Gear Reducers

Transcription

1 Int J Adv Manuf Technol (2003) 21: 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, 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 verified 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 efficient and economical by virtue of proposed technique. Keywords: Anti-resonant frequency; Backlash estimation; 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 fields, 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., Mabuk-ri Gusungeup Yongin-city Kyonggi-do, , Korea. jhbaekb@lginnotek.com 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 magnitude 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 momentum 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 magnitude 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 defined as the ratio of the magnitude of the first stage backlash to that of the total backlash. The concept for estimating the magnitude of each stage backlash is based on the change of anti-resonant frequency (ARF) and resonant frequency (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.

2 Backlash Estimation of a Seeker Gimbal Model of Seeker Gimbal 2.1 Model 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 fixed shaft as shown in Fig. 1(b). It is assumed that bearings support each shaft without any clearance, due to the preload. Also, the influence of the damping characteristic is neglected. The model of the ADSS obtained under these assumptions is presented in Fig. 1(c). The moment of 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 fixed, gear 2 and the fixed 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 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 motor amplifier and the aforementioned structure. The motor amplifier is used to amplify the input voltage of motor. A permanent Fig. 1. (a) Seeker gimbal; (b) structure of ADSS; (c) model of ADSS; (d) structure of EDSS; (e) model of EDSS.

3 606 J. H. Baek et al. magnetic field type d.c. motor with a tachometer is used as an actuator. In order to filter the output voltage of the tachometer, a second-order low-pass filter is used. The governing electric Eq. of these components are as follows [5]: V m = k a V i (1) di a L a dt + R mi a + k b m = V m (2a) T m = k t i a (2b) V t = k ts m (3) V o (s) = G f (s)v t (s) (4) The Eq. of motion for the motor is as follows: J m m + B m m = T m T g1 T N f,m sign ( m) (5) 1 The torque transmitted to gear 1 is represented as a nonlinear Eq., presented in Eq. (6), due to the backlash between pinion 1 and gear 1. The model of the dead zone is used as the model of the backlash [6]. T g1 = k g1( d1 1 ), d1 1 0, d1 1 (6) k g1 ( d1 + 1 ), d1 1 where d1 = m /N 1 g1 (7) The Eq. of motion for gear 1 is as follows: J g1 g1 = T g1 2k s1 ( g1 s1 ) (8) The Eq. of motion for shaft 1 is as follows: J s1 s1 = 2k s1 ( g1 + p2 ) 4k s1 s1 (9) Besides, the equation of motion for pinion 2 is as follows: J p2 p2 = 2k s1 ( s1 p2 ) 1 N r T L (10) Fig. 2. The bode diagram (V o /V i ) 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.).

4 Backlash Estimation of a Seeker Gimbal 607 The torque of the load is represented in Eq. (11), like Eq. (6). T L = k 2( d2 2 ), d2 2 0, d2 2 (11) k 2 ( d2 + 2 ), d2 2 where d2 = p2 /N r L (12) Here, the equivalent torsion stiffness between gear 2 and shaft 2 is as follows [7]: k 2 = k g2k s2 (13) k g2 + k s2 Finally, the equation of motion for the load is as follows: J L L = T L T f,l sign ( L) (14) The response of the output voltage of the tachometer filter with respect to the input voltage of the motor amplifier is obtained from these Eq.. In addition, the relation between the total backlash and each stage backlash is as follows: where b t = b N r b 1 (15) b i = 360 i / (i = 1,2) (16) 2.2 Model 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 amplifier and tachometer filter are the same as those of the ADSS, except for replacing Eqs (10) (13) and Eq. (15) with Eqs (17) (20) as follows: where J p2 p2 = 2k s1 ( s1 p2 ) 1 T N L 2 (17) T L = k g2( d2 2 ), d2 2 0, d2 2 k g2 ( d2 + 2 ), d2 2 (18) d2 = p2 /N 2 L (19) b t = b N 2 b 1 (20) From Eqs (1) (9), Eq. (14) and Eqs (17) (20), the response of the output voltage of the tachometer filter with respect to the input voltage of the motor amplifier 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 filter with respect to the input voltage of the motor amplifier, 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 hypothesis. In order to verify this hypothesis, the frequency response characteristic of ADSS is investigated according to the contribution ratio. The bode diagrams of ADSS obtained from the simulation are represented in Fig. 2. The specifications 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 amplifier 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 filter with respect to the sinusoidal voltage supplied to the motor amplifier. 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. Specifications for ADSS and EDSS. Parameter ADSS EDSS Gear ratio 1, N Torsion stiffness, k g1 (m N/rad) 3.40E4 4.74E4 Moment of inertia of gear 1, J g1 (kg m 2 ) 2.34E E-5 Torsion stiffness of shaft 1, k s1 (m N/rad) E2 Moment of inertia of shaft 1, J s1 (kg m 2 ) 8.30E E-7 Moment of inertia of pinion 2, J p2 (kg m 2 ) 2.21E E-7 Gear ratio, N r,n Equivalent torsion stiffness, k 2,k g2 (m N/rad) 7.74E4 2.54E5 Moment of inertia of load, J L (kg m 2 ) 2.75E E-2 Static friction torque of load, T f,l (m N) 7.0E-3 7.1E-3 Total backlash, b t (deg.) Motor inductance, L a (H) 8.50E-4 Motor resistance, R m ( ) 4.10 Back-EMF const., k b (V s/rad) 3.44E-2 Torque sensitivity, k t (m N/A) 3.49E-2 Moment of inertia of motor, J m (kg m 2 ) 8.60E-6 Static friction torque of motor, T f,m (m N) 1.40E-2 Gain of motor amplifier, k a 4.11 Tachometer sensitivity, k ts (V s/rad) 8.60E-2 Transfer function of low-pass filter, G f (s) s s Viscous damping coeff. of motor, B m (m 1.6E-4 N/(rad/s))

5 608 J. 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 b 1 b 2 Anti- Resonant ratio (%) resonant (db/hz) (db/hz) ADSS / / / / / / / / / /153 Exp / /137 EDSS /50 3.4/ / / /52 3.2/ /52 5.9/ /51 3.9/89 Exp /40 1.8/75 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 confirmed 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 according 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 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.

6 Backlash Estimation of a Seeker Gimbal 609 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 microscope, 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 Results and Discussion Because the simulation results are obtained under the assumptions that ignore damping effects and bearing clearances, it is difficult to obtain exactly consistent results between the experi- ment and the simulation. Thus, the error index between the simulation results and the experiment results is defined as Eq. (21), and the minimum contribution ratio is found. error index = f AR,S f AR,E + f R,S f R,E + f D,S (21) f D,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 sufficiently accurate to estimate the magnitude or contribution ratio of the backlash of each stage of a seeker gimbal with two-stage gear reducers. 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.

7 610 J. 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 = (f 2,E f 1,E) (22) 2Q AR 2f AR,E R = f R,E f AR AR,E (23) R AR = AR (when 0 AR 0.707) (24a) R R = 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 at the ARF and at the RF, while those of the EDSS are at the ARF and at the RF, 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 coefficient 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 verified 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 coefficient and small uncertainty. The technique has several advantages as follows: first, 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 efficient and economical because only a loose or an excessively loose gear stage needs to be adjusted or replaced rather than having to replace the whole gear reducer. Fourth, it can be applied to nonrobotic servo systems such as NC machines because it is unnecessary to attach a sensor on the link of robot or the output shaft of a servo system [2]. It is thought that using the proposed technique, the diagnosis and maintenance of various manufacturing machines and many servo systems will become more efficient and economical. Acknowledgements We would like to thank LG Innotek Co. for supporting this study and Sung Min Hong, Ho Young Kim and Byung Ho Lee for their assistance. References Fig. 5. (a) Damping factor of ADSS and EDSS. (b) The frequency reduction ratio of ADSS and EDSS due to damping factor. 1. N. G. Dagalakis and D. R. Myers, A Technique for the detection of robot joint gear tightness, Journal of Robotic Systems, 2(4), pp , J. L. Stein and C. H. Wang, Estimation of gear backlash: theory and simulation, ASME Journal of Dynamic Systems, Measurement and Control, 120, pp , 1998.

8 Backlash Estimation of a Seeker Gimbal N. Sakar, R. E. Ellis and T. N. Moore, Backlash detection in geared mechanisms: modeling, simulation, and experimentation, Mechanical Systems and Signal Processing, 11(3), pp , M. C. Pan, H. V. Brussel, P. Sas and B. Verbeure, Fault diagnosis of joint backlash, ASME Journal of Vibration and Acoustics, 120, pp , M. Clifford, Modern Electronic Motors, Prentice Hall, Englewood Cliffs, NJ, M. Nordin, J. Galic and P. O. Gutman, New models for backlash and gear play, International Journal of Adaptive Control and Signal Processing, 11, pp , B. A. Chubb, Modern Analytical Design of Instrument Servomechanisms, Addison-Wesley, Reading, MA, R. Dhaouadi, K. Kubo and M. Tobise, Analysis and compensation of speed drive systems with torsional loads, IEEE International Workshop on Advanced Motion Control, Yokohama, Japan, pp , W. J. Bigley, Wideband base motion isolation control via the state equalization technique, Optical Engineering, 32(11), pp , L. Meirovitch, Principles and Techniques of Vibrations, Prentice- Hall, Upper Saddle River, NJ, Notation B m viscous damping coefficient of motor (m N/(rad/sec)) b i angular backlash to be measured at the gear i ( ) (i = 1, 2) b t total backlash to be measured at the output stage ( ) f 1,E, f 2,E frequencies of half-power points at the near f AR,E obtained from experiment (Hz) f AR,E, f R,E ARF and RF obtained from experiment (Hz) f AR,S, f R,S ARF and RF obtained from simulation (Hz) f D,S, f D,E difference between ARF and RF in the simulation and experiment (Hz) G f (s) transfer function of tachometer filter motor current (A) i a J g1,j L,J m,j s1 moment of inertia of gear 1, load, motor, and shaft 1 (kg m 2 ) J pi moment of inertia of pinion i (kg m 2 )(i = 1, 2) k 2 equivalent torsion stiffness between gear 2 and pinion 2 (m N/rad) gain of motor amplifier k a k b k gi back e.m.f. constant (V s/rad) torsion stiffness between pinion i and gear i (m N/rad) (i = 1, 2) k si torsion stiffness of ith shaft (m N/rad) (i = 1, 2) k t k ts L a L bi torque sensitivity of motor (m N/A) tachometer sensitivity (V s/rad) inductance of motor (H) arc length due to the angular backlash between ith gear and ith pinion (m) (i = 1, 2) N i gear ratio of pinion i and gear i (i = 1, 2) N r revolution gear ratio between pinion 2 and gear 2 (N r = N 2 + 1) Q AR factor to define the damping factor R AR,R R frequency reduction ratio of ARF and RF due to damping effect (%) R m resistance of motor ( ) T f,l,t f,m static friction torque of load and motor (m N) T g1 transmitted torque of gear 1 (m N) T L load torque (m N) T m motor torque (m N) V i input voltage of motor amplifier (V) V m input voltage of motor (V) V t output voltage of tachometer (V) V o output voltage of tachometer filter (V) i half value of angular backlash to be measured at the side of gear i (rad) (i = 1, 2) di angular transmission error of ith stage (rad) (i = 1, 2) g1, L, m, p2, s1 AR, R sign( ) rotation angle of gear 1, load, motor, pinion 2, and shaft 1 (rad) anti-resonant and resonant damping factor sign of value in the parenthesis