CHAPTER 3: TORQUE SESOR 13 Chapter 3 Torque Senor Thi chapter characterize the iue urrounding the development of the torque enor, pecifically addreing meaurement method, tranducer technology and converter technology. In addition, the ytem equation of the enor will be formulated for ue in control deign. 3.1 Meaurement Method In torque enor deign, reearcher have ued a variety of method to infer the torque applied to a link. Specifically, two baic approache are employed. 3.1.1 Open Loop or Indirect Method Open loop method often infer the etimated value of the torque applied to the link by relying on a priori knowledge of joint parameter uch a the link, motor, and tranmiion term and ening the applied voltage and current. Since mot model uffer from parameter error or implified aumption, the joint torque etimate i often erroneou. For example, the relationhip of current to the applied torque i often determined by the following linear equation:
CHAPTER 3: TORQUE SESOR 14 applied K T I (3.1) where K T i the torque contant of the motor and I i the current applied to the motor. Many application ue Equation (3.1) and a meaurement of current to approximate the torque applied to a link. Thi method i imprecie ince it doe not take into account the complexitie of the gearing mechanim, non-linear effect of friction, back EMF and backlah - all of which contribute to lo of torque from the motor to the link. In model that do include thee effect, the complexity and computational power required often reduce enor performance to bandwidth on the order of 10 to 100 Hz. 3.1. Feedback or Direct Method Feedback method require ome form of tranducer attached to the motor haft to ene the torque applied. Thee method typically involve ening ome form of diplacement or train applied by the motor to ome tructure, uch a a haft or a flexible member. In thee method, the connecting member i deigned in way that when torque i applied, an angular diplacement or train reult on the member. Tranducer uch a train gage, piezoceramic, or optical fiber are ued to meaure train on a member attached to the haft, wherea inductive tranducer, Hall Effect enor or encoder are ued for diplacement meaurement. Since thee method are baed on more direct reading of the torque applied to the member, very little computation power required to determine the torque. Therefore, the meaurement of the applied torque i often at a higher bandwidth and with higher performance characteritic and greater accuracy than open loop method. 3. Mechanical Senor de link C G G G G S G E ened comp Motor and Amplifier Gearing Senor Electronic Figure 3-1: Generalized Joint Torque Loop
CHAPTER 3: TORQUE SESOR 15 Baed on previou deign effort [Vicher90], the ARTISA writ i equipped with a torion pring between the tranmiion ytem and the link itelf. The ix-beam tructure hown in Figure 3- allow flexibility about the z-axi, normal to the face of the enor while providing greater tiffne along the x and y-axe. Thi flexibility in the axi of interet allow for a mall, meaurable diplacement that can be tracked in order to meaure the torque applied to the link. High tiffne along the other inplane direction reduce the diturbance effect caued by radial force, which improve enor accuracy. Through thi diplacement, the torque being applied to the link can be ened directly. θ r y θ 1 z x Figure 3-: Six Spoke Senor The relationhip between the ened torque,, and the angular deflection θ i defined a: where k θ (3.) θ θ θ1 (3.3) and k i the torional tiffne of the poked tructure. The angular deflection θ of the poke i ued to calculate by uing the mall angle approximation of δθ ª δ x, where x i the linear diplacement of the LVDT core along the x-axi, at a ditancer from the center of the enor a hown in Figure 3-. For mall deflection δθ, the equation become
CHAPTER 3: TORQUE SESOR 16 k δ x (3.4) r The enor torional tiffne, k, i determined by Equation (3.5) L M O QP 1 3r 3r k 4EI + + l 3 (3.5) l l where r i the inner radiu of the enor, i the number of poke, I i the modulu of the poke ection, l i the poke length, and E i Young' modulu. 3..1 Mechanical Senor Sytem Equation Current Link Motor k Ν m J m J l Tranmiion Torque Senor Figure 3-3: Sytem Diagram of Mechanical Senor Since the torque enor i a deigned flexibility in the tranmiion of torque from the motor to the link, it can be repreented a a linear pring a hown in Figure 3-3. For the tranmiion ytem, the motor deflection θ m i defined a: θ θ θ (3.6) m l where i the gear ratio, θ i the angular deflection of the enor and θ l i the angular deflection of the link. By umming the force on the link, the torque applied to the link, which i equivalent to the torque applied to the enor,, i defined a:.. J θ l l (3.7) where J l i the rotational ma moment of inertia and θ & l i the angular acceleration of the link. Auming zero initial condition, the following tranfer function reult:
CHAPTER 3: TORQUE SESOR 17 where all of the term have been defined previouly. θ l 1 (3.8) Jl Summing the force on the rotor provide an expreion for the torque applied by the motor, m, in term of angular deflection..... J θ + d θ d θ m m m m m where J m i the rotational ma moment of inertia of the motor, d m i the damping (3.9) coefficient of the motor and d i the damping coefficient of the enor. Auming zero initial condition, Equation (3.9) lead to the following reult: J θ + d θ d θ (3.10) m m m m m From Equation (3.), (3.6), (3.8) and (3.10), the tranfer function between the applied motor torque m and the ened torque can be derived: where m b0 3 a + a + a + a 0 1 3 (3.11) 3.. Free veru Fixed Configuration b 0 k (3.1) a 0 J m (3.13) a d d l 1 m (3.14) Jmk k a + Jl (3.15) dmk a3 J (3.16) l For any configuration of the link, Equation (3.11) provide an effective expreion for the tranfer function. In the free configuration, the link inertia i determined by the inertia and configuration of the component found after the joint. In the fixed configuration, where the link i locked in a fixed poition, the link inertia tend to infinity (J l ). Therefore, the tranfer function of the enor in the fixed configuration become:
CHAPTER 3: TORQUE SESOR 18 m b0 a + a + c 0 1 (3.17) where k c (3.18) and the other term are defined in Equation (3.1) through (3.14). Through imple manipulation, Equation (3.17) can be rewritten in the tandard econd-order tranfer function form. m ω + ζ ω + ω (3.19) where k ω Jm 1 d m 1 d l ζ + Jmk m J k (3.0) (3.1) Given the value for the term lited in Equation (3.0) and Table 3-1, the damped reonant frequency of the mechanical enor in the fixed configuration i approximately 130 Hz. Thi calculated frequency i a minimum bound on the reonant mode ince the free configuration doe not approximate the link inertia at infinity. Thi characteritic reult in an additive component to the reonant frequency and an additional pole/zero pair cloe to zero on the -plane. Term k Value 9751 -m/rad J m 80x10-6 kg-m 13.5 Table 3-1: Mechanical Senor Value
CHAPTER 3: TORQUE SESOR 19 3.3 Senor Electronic de link C G G G G S G E ened comp Motor and Amplifier Gearing Senor Electronic Figure 3-4: Generalized Joint Torque Loop Once the mechanical enor convert the applied torque into a mechanical quantity, the next tep i to elect an appropriate tranducer to convert the mechanical quantity into an electronic quantity. 3.3.1 Requirement for Ideal Tranducer A lit of the characteritic for an ideal enor provide an excellent roadmap for undertanding the engineering tradeoff involved in tranducer and decoder election for the mechanical torque enor. The following characteritic are required for an ideal direct tranducer: Infinite throw and infiniteimal reolution The enor would be able to ene any amount of torque applied Infinite bandwidth The enor would be intantaneouly capable of reporting the amount of torque applied Infinite ignal-to-noie ratio The enor would communicate the value of torque applied without diturbance Simple intallation and alignment The tranducer would allow for eay intallation and alignment in normal operation
CHAPTER 3: TORQUE SESOR 0 In addition, the pecification for the ARTISA manipulator require the decoder to include a digital format to interact with the overall control ytem. Therefore, additional characteritic include: Infinite length digital word The digital format provide infinite reolution on the digital repreentation of the information. Thi requirement i often analogou to the infinite reolution characteritic decribed earlier. Intantaneou converion The analog-to-digital converion take no time and incur no latency. Thi characteritic i analogou to the infinite bandwidth decribed earlier. In practice, thee ideal characteritic are generally impoible to attain. Hence, tradeoff occur between the different characteritic, uch a higher reolution for lower bandwidth. 3.3. Tranducer Selection Table 3- lit the characteritic for a variety of diplacement and train tranducer option. Technology Reolution Throw SR Format Strain Gauge High Short Low Analog Inductive Tranducer High Short Medium Analog Piezoceramic Medium Short Low Analog Fiber Optic High Short High Analog Magnetic Hall Effect Senor Linearly Varying Differential Tranformer Medium Medium Medium Binary High Long High Analog Table 3-: Tranducer Comparion From thi liting of tranducer, the Linearly Variable Diplacement Tranducer (LVDT) i choen. provide infinite reolution on diplacement being limited only by the converter technology and ignal-to-noie ratio
CHAPTER 3: TORQUE SESOR 1 are hielded to electromagnetic noie are conitant, tunable and not prone to breakage come in variable throw length and maintain a large linear region of ening in comparion to the diplacement ditance. Further dicuion regarding the operation of LVDT can be found in Appendix A. Once the tranducer i elected, the next tep i to chooe the proper converter for the tranducer. 3.3.3 Converter Selection LVDT generate an analog voltage ignal to determine the poition of the core. Thi ignal can be approximated a a ingle ide-band amplitude modulated (AM) ignal. The poition information i encoded in the amplitude and phae of the carrier wave with repect to the reference ignal. A number of technologie were examined and teted and are lited in Table 3-3. Technology Reolution Bandwidth 3 db point SR Format Signetic E551 10 bit Static 15 Hz Medium Analog Analog Device AD598 10 bit Static 100 Hz Medium Analog Analog Device ADS54 1 bit Static 15Hz High Digital Analog Device ADS93 14 bit Dynamic 150Hz High Digital Table 3-3: Comparion of Converter Technologie In order to decode thi ignal a well a provide an interface between the enor and the computer a pecified in Section 3.1.4, the Analog Device S93 LVDT-to- Digital Converter wa elected. 3.3.4 Converter Sytem Equation The Analog Device ADS93 LVDT-to-Digital Converter incorporate an analog tracking loop with a digital output format and combine it into a ingle package.
CHAPTER 3: TORQUE SESOR AC Ratio Bridge Bandpa Filter Phae-Senitive Demodulator Loop Compenator x v ω REF x d XXXX-1 XXXX XXXX+1 Up Down C VCO RVCO Digital Counter Reet Integrator and Digital Logic Figure 3-5: Functional Diagram of ADS93 Figure 3-5 illutrate the ub-component of the ADS93. The LVDT' econdarie are fed into the input of the ADS93 and paed into the AC Ratio Bridge. There, the ratio of the difference of the two ignal over their um i created and i compared to the etimated digital value of the ignal. The reulting error i tracked via the internal loop of the chip which demodulate the poition information from the error ignal and generate a digital repreentation through the voltage-controlled ocillator (Reet Integrator) and Digital Counter. The reulting digital word i then available for retrieval from any digital interface through the buffer onboard the ADS93. A more in-depth dicuion of the ub-component of the ADS93 i found in Appendix C. 3.3.4.1 Simplified Sytem Equation The ytem dynamic of the ADS93 are programmable through election of the reitor and capacitor that make up the uggeted Loop Compenator ubcomponent. If the ubytem are modeled in the demodulated domain (low frequency range), then the functional block diagram of the ADS93 can be implified into the form found in Figure 3-6.
CHAPTER 3: TORQUE SESOR 3 x v K T1 + 1 1 a T + 1 b g x d Figure 3-6: Simplified Block Diagram for ADS93 Uing thi implified tructure, the tranfer function between the input diplacement voltage x v and the digital word x d of the ADS93 can be approximated by Equation (3.) x x d v T1 + 1 T K 3 1 + K + T 1 + 1 a a (3.) where the value of T 1, T and K a are determined by the dicrete component of the Loop Compenator pecified by Analog Device. Further dicuion of the implified ytem model and expreion for the dicrete component in term of performance can be found in Appendix B. 3.3.4. Enhanced Sytem Equation Cloer examination of the ytem' component and a modification of the Loop Compenator provide a implified expreion for the ADS93' ytem equation. A dicued in Appendix C, the enhanced ytem equation for the ADS93 i expreed in Equation (3.3). x x d v K AC KBP KPSDKIT (3.3) + ω + p + K K K K b gc h BP AC BP PSD IT where the variable are defined in Appendix C. Uing Equation (3.3), the component of the ADS93 are choen to provide performance characteritic that benefit the overall torque loop deign. Equation (3.3)i ued extenively in determining the performance of the enor electronic.
CHAPTER 3: TORQUE SESOR 4 3.4 Summary of Sytem Equation All of the ytem component have been analyzed and Table 3-4 ummarize the ytem equation for each of the component. Subytem Sytem Equation Bandwidth Current Amplifier im R43C17 + 1 1Khz 10 v R C L + R R R + R C + Motor Senor and Gearing b c 1 17 m 1 43 m 14 43 17 m KT im ω m + ζ ω + ω g 10 14 Infinite ω (approx 130 Hz) Tranducer xv k 116. 9 r Infinite Converter xd K AC KBP KPSDKIT x + ω + p + K K K K Up to 1.5 khz v b gc h BP AC BP PSD IT Table 3-4: Summary of Sytem Equation By comparing the bandwidth of the component lited in Table 3-4, the mechanical enor i the limiting component of the open-loop ytem, a long a the ADS93 ytem i programmed to track at a very high bandwidth (on the order of 1 khz). If the converter i not capable of thi performance, the dynamic of the converter will have to be included in any control law developed. Chapter 4 develop the contraint for the joint torque loop and the control law neceary for both the ADS93 and the overall joint torque loop.