Error Analysis for Gyroscopic Force Measuring System

Transcription

1 Proceedings of the 7 th International Conference on Force, Mass, Torque and Pressure Measurements, IMEKO TC3, 7- Set., Istanbul, Turkey Error Analysis for Gyroscoic Force Measuring System Shigeru Kurosu *, Kazuhiro Kodama **,Motoyuki Adachi ***, Kazuyuki Kamimura **** * Oyama National Collage of Technology, Oyama, Jaan ** Kyushu Institute of Technology, Iizuka, Jaan *** Yamamoto Scale Co., Ltd., Akashi, Jaan **** Yamatake Building Systems Co., Ltd, Tokyo, Jaan Abstract This aer concerns the develoment of a new tye of force sensor called Gyroscoic Force Measuring System, (simly called GFMS) for measuring a force vectorialy. The GFMS realized a highly accurate measurement using an estimator for the unknown angles of incidence of a force vector. The estimator, however, assumed an exact knowledge on the dynamical equations in the steady-state. This caused much degradation in the measurement accuracy unless the steady-state of the GFMS could be attained. The estimation errors of the angles of incidence directly affect the accurate measuring of the force. We can get the better erformance by choosing the adequate daming coefficient of gyroscoe (for nutation mode). From the error analysis, some essential information for designing the GFMS is highlighted.. Introduction The early art of the twenties century saw gyroscoes being alied to a variety of fields. It has been used to navigate shis and rockets, to stabilize the rolling of shis, to counter vibration and to oerate innumerable control mechanisms. The small directional unit of the gyro-comass oerates by the same rinciles as the massive rotor of the shi stabilizer. sinning raidly about its axis would be difficult to find in industrial alication fields. This is because secial devices for the sensor using entirely different rinciles have been emerged. For examle, fiber-otic gyros, quartz vibrating gyros and the like have been con- ceived in stead of traditional mechanical gyros. Today, this mechanical gyros consist of a rotor 6

2 Proceedings of the 7 th International Conference on Force, Mass, Torque and Pressure Measurements, IMEKO TC3, 7- Set., Istanbul, Turkey In this aer, of articular interest to us is other end of this lever is mounted on the G at how to aly the unique features of mechanical one end of the sin-axis of the rotor. The gyros for a force sensor. Three decade ago, a new tye of mechanical gyro for measuring a force was invented and roduced in Germany. The original Gyroscoic Force Measuring System, (simly called GFMS) has earned much interest as a weighing scale due to its inner and outer turntables G 3 and G 4, which suort the gyroscoe, are free to rotate relative to the G 4 about OZ and the frame about OX, resectively. The force to be measured acts vectorially on the center of the lever through a swivel and therefore alies a advantages of high resolution []~[3]. This f o r c e o n o n e e n d rincile can easily be extended to the measurement of a vectorial force in sace. This aer exlores the ossibility of the GFMS to the vectorial measurement of a force. An analysis in this study allows us to design the GFMS with suitable arameters to offer a reasonable accuracy. The total erformance of the GFMS is demonstrated by numerical Force detector F Swivel & otentiometer orizontal gimbal G Vertical gimbal G Outer turntable G Z 4 X gyro rotor Controller Torque motor Inner turntable G 3 Torque motor (fixed to the frame) Torque motor & revolution counter Y simulations. Some systematic errors associated with the GFMS are analyzed and discussed.. Basic Princile and Equations of Motion. Construction and Princile Figure. shows a construction model of the Figure. Construction model of GFMS of the sin-axis of the rotor at the unknown angles of incidence and. The best esti- mated angles and can easily be defined as estimates for which the force F will be at the GFMS schematically. The vertical gimbal G, maximum value. The servomechanisms of which suorts the rotor, is free to rotate relative to the horizontal gimbal G about OX, the G 3 and G 4 serve this urose even better because the algorithm can be easily ro- while the G is free to rotate relative to the grammed. This rincile of the GFMS is inner turntable G 3 about OY. The G suorts somewhat similar to that of a well-known one end of a ivoted lever vertically. The weathercock. The reason for this is that to 63

3 Proceedings of the 7 th International Conference on Force, Mass, Torque and Pressure Measurements, IMEKO TC3, 7- Set., Istanbul, Turkey control the force detector s asect erendicular to the direction of the force in sace is of dominant imortance. The torque generated by the force alied causes the rotor in its gimbals to turn (called recession) about OY and the recession rate is directly roortional to the force F alied. When the rate can be measured accurately, the GFMS oerates as a recise linear force transducer as follows: a F () where F is the magnitude of the force, is the sin angular momentum of the gyro-rotor and a is the length of a lever arm... Analytical Descrition To consider now the oeration of the GFMS as shown in Figure with the vertical sin-axis OZ and to oint the outut axis OY toward the direction of the force F, we assume that the force vector F which acts on the force detector can be defined by two rotations, namely about OX and about OZ. The equations of motion in this system can be described as follows: A c T B c T x 3 ( EF cosf { c 4 x ( F sin T 3z3 x y ( CDcos Dcos) { c ( D D) sin cos} T T sin) F) sin cos} cos cos T y sin T 4x4 () where A, B, C and E are the moments of inertia about OX and OY of gimbals and turntables, D and D are the moments of inertia including G about OZ and G about OX, F and F are the moments of inertia of the G 3 about OX and OY, T x, T y, T 3z3 and T 4x4 are the torques alied about OX, OY, OZ and OX, and c,c,c 3 and c 4 are the viscous friction coefficients about OX, OY, OZ and OX, resectively. The feedback torques exerted on the G, G 3 and G 4 by the torque motors are given by T T T y 3z3 4x4 k k K K P3 P4 where k,k i and i t dt k ( ) K r ( ) K r d D3 D4 (3) k d are the roortional, integral and derivative gains of the torque motor, K P3, K I3 and K D3 are the roortional, integral and derivative gains of the torque motor, and r and r are the reference inuts for the servo mechanisms. The con- trolled oututs and are relative rota- tions of the turntables G 3 and G 4. 64

4 Proceedings of the 7 th International Conference on Force, Mass, Torque and Pressure Measurements, IMEKO TC3, 7- Set., Istanbul, Turkey The external torque T x sulied by the force F to the gimbal G can be written as T x Fa{cos cos cos( ) sin sin cos cos sin( ) sin {cos sin cos( ) sin cos }..3. Estimations of and The steady-state value of the outut can be written as Fa {cos cos r cos( r ) sin sin } r (4) If any three sets of reference inuts are excited as, ),, ) and, ) sequen- ( r r ( r r ( r r tially, the oututs, and for each of them can be obtained. The best estimates ˆ a n d ˆ c a n b e o b t a i n e d b y tan ˆ A B A B (6) A B A B where A sin B B sin r A cos r A cos r r B cos r cos r sin cos sin cos sin r r sin r r r cos r cos r cos cos r r r cos sin sin r r cos r r (7) When ˆ is estimated, then ˆ can easily be obtained by. (5) tan ˆ cos cos( ) cos cos( ) r r r r. (8) sin sin.4. Characteristic Values of GFMS r We have constitutionally two oscillatory modes, called recession and nutation modes, resectively. ro Two natural frequencies and daming coefficients are given by ki AB,, k kd k i A B (9) For conventional gyroscoic sensors, it is usu- ally ermissible to neglect assuming that >>. In designing our GFMS, cannot be neglected because increasing will decrease the sensitivity, as exressed in Equation. So, there exists some limitations on to measure a small force. The natural frequencies and daming coefficients for servomechanisms are given by s3 s4 K C P3 K E P4,, KD3 s3 CK P3 KD4 s4 EK P4 () To attain the gain attenuation of turntables to disturbance torques less than -4 [db], we have K () P 3. Design and Construction of GFMS 3.. Design Secifications The foregoing consideration allows us to 65

5 Proceedings of the 7 th International Conference on Force, Mass, Torque and Pressure Measurements, IMEKO TC3, 7- Set., Istanbul, Turkey design the GFMS. The design secifications Table 3. Characteristics of turntable are determined as, Measurable force range ~.3 [N] Measurable outut range ~4[rad/s] Measuring error (within [s]) -6 [N] Gain attenuation of turntables (to reaction torques) -4 [db] The constants of the gyro are summarized in Table. The characteristics and the feedback loo gains of the GFMS and the turntables are given in Table and 3. Table. Characteristics of gyro-rotor mass of gyro-rotor M.4 - kg moment of inertia I z.45-6 kgm revolution n sin angular moment length of lever system a 4 rm.37 - kgm /s m natural frequencies s3 3 rad/s daming coefficients s4 rad/s ζ s3. ζ s4. feedback loo gains K P3 sensitivity for disturbance Figure. K D3.883 K P4 8 K D4.3 G 3 G 4 4 db 4 db Frictional torques in two swivels will cause instrumental errors. In the figure, to transmit the force acting on the detector to the gyro-rotor without loss of accuracy, flexure ivots and steel belts are arranged in lace of ordinary in-joints as essential rotating arts of a lever. Table. Characteristics of gyro-sensor natural frequencies 5. rad/s Force detector Inner turntable G 3 Torque motor 4.8 rad/s Lever Steel belt Gyro rotor daming coefficients ζ. ζ. feedback loo gains k k i k d.68-3 Flexure ivot orizontal gimbals G Vertical gimbals G 3.. Construction of GFMS Figure. Construction model of GFMS An examle of rototye GFMS is shown in 66

6 Proceedings of the 7 th International Conference on Force, Mass, Torque and Pressure Measurements, IMEKO TC3, 7- Set., Istanbul, Turkey 4. Characteristics of Resonse 4.. Resonses of GFMS Now we consider resonses of the GFMS with numerical simulations using these values and the following requirements: Force alied : F=.3 [N] (Initial load : F =.3 [N]) GFMS [5]. 4.. Error Analysis () Definition of Errors We will now some systematic errors caused by various influences introduced in the GFMS. Errors of the magnitudes of the force F and the angles of incidence are defined by Angles of incidence : 45, 6 [ d e g ] Angles for ste-changes in reference inuts: F ˆ ˆ Fˆ F () r, r = [deg] Time interval for ste-changes of r and r : t = [s] Figure 3 and 4 show resonses of the measured where Fˆ, ˆ and ˆ are the oututs to the inuts F, and, resectively. () and From Equations and 3, we get the following equations: oututs Fˆ, and in the GFMS. It is confirmed that three oututs aroach resective inut values within about [s]. Facos kd cos sin ( r ) K K P4 P3 P4 Fa sin ( r ) K (3) The effect of nutation aears as transient, but it quickly dams out. The error F, which is in the steady-state of Fˆ, is also shown in Figure 5. that It is obvious F after [s] reresents sustained oscillation, which indicates the measured errors caused by small vibrations of the turntables due to the reaction torque of the The first terms of and involve the os- cillation comonent and the second terms of them do the dc-comonents. It should be noted that the dc-comonents are due to the estimation errors, namely tracking errors in servomechanisms. On the other hand, the amlitudes of the oscillation comonents are roortional to F and inversely rortional to K P3 and K P4. 67

7 Estimation error of desired value [deg] Difference [deg] Tilt of rotor axis [deg] Measured error and [deg] Measured error F [N] Measured force F[deg] Rotation angle [deg] Measured error F [N] Proceedings of the 7 th International Conference on Force, Mass, Torque and Pressure Measurements, IMEKO TC3, 7- Set., Istanbul, Turkey (3) F For the secial case when outut Fˆ is given by = =, the cos( ) cos cos( ) cos, (8) we can finally get the following equation: ˆ F. (4) a Fa F ( cos cos ). (9).3 6 (G 4 ) [ -6 ]. 4 (G 3 ). Fig. 4 Resonse of measured force F Fig. 5 Resonse of rotation angle of turntables - Fig. 6 Resonse of measured error F. From this it follows that the GFMS will now maintain its virtually =.5 erfect linearity of - recession -4 rate against F. To find steady- -6 state value =. we ut and get 3 4 Fig. 7 Tilt of rotor axis in initial load Tdue to daming coefficient x (5). Substituting Equation and 5 into 4,we get the following equation in steady state F ˆ F{cos cos cos( ) Inut sinforce sin [N] } (6) Thus, the error Equation. =. =. =.5 r + r Fig. Estimation errors of desired value due to inut force F F can be obtained from F F[{cos cos cos( ) sin} ] (7) It is obvious that - the estimation errors Inut force F [N] Fig. 8 Resonses of measured error and due to inut force F is strongly affected by F and [6] Relevance between kd and Finally, we will examine that there exists [ the -8 ] uer and lower bounds on the derivative [ -7 ] gain k d in Equation In the derivation rocess of errors, it is assumed that the -3-4 deflection in steady state. owever, will not be Inut vanished foece [F] for a while just after F and rotation angles excites on the GFMS. Since the measured error increases as increases, it is crucial to settle to zero within the rescribed time interval for ste-changes of - -3 F F Inut force F [N] Fig. 9 Resonses of measured error F due to inut force F r + r Fig. Differences between desired values r and r. ere, using the small aroximations for and by The k d has been introduced for the daming of nutation mode, as shown in Figure 6. Taking the Lalace transform of the first and 68

8 Proceedings of the 7 th International Conference on Force, Mass, Torque and Pressure Measurements, IMEKO TC3, 7- Set., Istanbul, Turkey Equation - and -, the transfer function k d cause instability. As shown in Figure 6, the between the deflection (s) and the inut F(s)a can be obtained by where Bs c kd s ( s) F( s) a () ( s) ( s) ( As cs)( Bs cs kds) ( s k s ki ) () which denotes the characteristic equation of the gyroscoe itself. From Equation, it fol- lows that is directly roortional to k d. Choice of the k d is imortant for obtaining meaningful sensors. A sketch of resonses of for various is given in Figure 6. The larger to vanish leads to unaccetable resonse. The condition for stability follows from Equation as smaller becomes, the shorter the settling time for zeroing can be achieved. For less than.5, the resonse shows oscillatory attern due to nutation mode. Thus, it is emhasized that for the choice of the k d we have an aroriate range between the uer and lower bounds. 4.4 Effect of F on Errors Figure 7 and 8 show the effect of the inut force F on errors these Figures, the symbols, and F. In and denote the amlitude and the dc-comonent of oscillatory errors. It can be seen that F has been gradually increased as F gets larger. This fact verifies that affected by, and F. F can be heavily kd Bk, () and k k i B A (3) kd k From the oint of stability, small values of the 69

9 Estimation error of desired value [deg] Difference [deg] Tilt of rotor axis [deg] Measured error and [deg] Measured error F [N] Measured force F[deg] Rotation angle [deg] Measured error F [N] Proceedings of the 7 th International Conference on Force, Mass, Torque and Pressure Measurements, IMEKO TC3, 7- Set., Istanbul, Turkey Figure 9 shows the estimation errors + r 6 and. + in servomechanisms. These 4 curves r.3. exhibit the exactly same atterns as Fig. 4 Resonse of measured force F and, as shown in Figure 7. It can be errors. There errors indicate theoretical limits of (G the 4 ) accuracy of the GFMS. (G 3 ) 6.Conclusions Fig. 5 Resonse of rotation angle of turntables [ -6 ] - A new gyroscoic force measuring system Fig. 6 Resonse of measured error F (simly called GFMS) was investigated as a =. =.5 =. =.5 =. 3 4 Fig. 7 Tilt of rotor axis in initial load due to daming coefficient Inut force F [N] Fig. 8 Resonses of measured error and due to inut force F [ -7 ] F F Inut force F [N] Fig. 9 Resonses of measured error F due to inut force F. r + r + Fig Inut force [N] Fig. Estimation errors of desired value due to inut force F [ -8 ] r + r Inut foece [F] Fig. Differences between desired values and rotation angles concluded that and can be mainly determined by the estimation errors. To enable us to areciate very high accuracies of the GFMS, the estimation errors should be reduced as small as ossible. Figure shows the tracking errors of turntables for the reference inuts r and r. The maximum error is little excess of -7 [deg]. For ractical alication, we can see that these errors do not influence the total amount of the estimation errors ( less than - [deg] ). It is also concluded that and vectorial force sensor to measure the magnitude and the direction of the force alied externally.the control structures and the estimator lay imortant roles in determining the erformance of the GFMS. In order to illustrate the effectiveness, some numerical analysis and simulations were conducted. The characteristics of both transient resonses and errors were discussed and the limitations of the GMFS was clarified. At resent, the GFMS has been in the works, and the ractical alicability and the erformance of it will be exlored in industrial environment. are heavily varied with the estimation 7

10 Proceedings of the 7 th International Conference on Force, Mass, Torque and Pressure Measurements, IMEKO TC3, 7- Set., Istanbul, Turkey 7. References []. Kurosu S., Kurosu M. Adachi, and Kamimura K., Dynamical characteristics of gyroscoic weight measuring device, ASME journal of Dynamic System, Measurement and Control, 997, Vol.9, []. Waagenbau W., Wöhl J., Gmb & Co., Postfach 7, D-74 Pfedelbach. [3]. Stevens C., & Son Ltd. Stevens-Wöhwa gyroscoic force measuring system Contact Person for Paer: Dr. Shigeru Kurosu Address : Oyama National College of Technology 77 Nakakuki, Oyama-city, Jaan Tel : Fax: kurosu@oyama-ct.at.ac.j [4]. Kasahara M., Kurosu, Adachi S. M., and Kamimura K., Analysis of gyroscoic sensor alied to force measurement vectorical measurement of horizontal force, IMEKO TC3/APMF 98,. 8-36, 998 [5]. Kurosu S., Kasahara M., and Kodama K., Gyroscoic Force Measuring System, Proceedings of IMEKO XVI World Congress ( Vienna, Austria ), () [6]. Kasahara M., Kurosu S., Adachi M., and Kamimura K., Analysis of a Gyroscoic Force Measuring System in Three-Dimensional Sace, Measurement 8, () 7