Ž .Sensors and Actuators 78 1999 143–148www.elsevier.nlrlocatersna

Compensation of nonlinear thermal bias drift of Resonant Rate Sensorusing fuzzy logic

Sung K. Hong )

( )Flight Dynamics and Control Team 3-3-5 , Agency for Defense DeÕelopment, Taejon, South Korea, 305-600

Received 7 April 1998; received in revised form 3 February 1999; accepted 8 May 1999

Abstract

Ž . ŽIn this paper, our attention is focused on the compensation of the nonlinear thermal bias zero-rate-output drift of RRS Resonant Rate.Sensor , which originates from a number of sources, including manufacturing tolerances, material inhomogeneity and inevitable

mechanical characteristic variation of the cylinder with temperature. Motivated by the capability of fuzzy logic in managing nonlinearity,Ž .the nonlinearity of bias was represented by Takagi–Sugeno TS fuzzy model over the entire range of the operating temperature. Then,

the fuzzy model was directly used for compensation of nonlinear bias drift by subtracting the estimated output from the raw data of RRS.Ž .By doing this, we can guarantee robust against temperature variations sensor performance throughout the entire operating temperature

range. q 1999 Elsevier Science S.A. All rights reserved.

Keywords: Vibratory cylinder rate gyro; Nonlinear thermal bias drift; Fuzzy logic

1. Introduction

Ž . 1A Resonant Rate Sensor RRS , an alternative to theconventional rate gyroscope based on a high speed rotorsupported in gimbals, is being considered for use in anumber of application areas such as stabilization, generalrate control, and autopilot systems in aircraft and missiles.The benefits of RRS compared to conventional gyroscopes

w xare well known 1,2 and include robustness, low powerconsumption, potential for miniaturization and hence, forlow cost. The RRS that has been developed in Inertial

Ž .Science ISI consists of a thin cylinder vibrating element,an electromagnetic drive and a pick-off element, and vibra-tion control electronics. These elements are shown in Fig.1.

Despite the potential of the RRS over conventional rategyroscope, its performance is degraded due to the manu-facturing tolerances, material inhomogeneity and inevitablemechanical characteristic variation of the cylinder with

) Tel.: q82-805-499-3191; fax: q82-805-3989; E-mail:hong1964@netsgo.com

1 Inertial Science, names it for vibratory cylinder rate gyro.

w xtemperature 1,3 . It is therefore vital to control or compen-sate these adverse sources to improve the performance ofthe gyroscope to an appropriate level. The alternativeapproach of selecting special material to minimize theeffect of temperature change on cylinder increases cost andstill results in a compromise.

In this paper, among many serious design problems inRRS, our attention is focused on the compensation of the

Ž .thermal bias zero-rate-output drift, which shows normallynonlinear characteristics. The approach adopted to mini-mize the nonlinear effect of temperature change on thecylinder is to measure the cylinder temperature and usethis value to compensate the sensor output.

Motivated by the capability of fuzzy logic in managingw xnonlinearity 6–8 , the nonlinear characteristics of the ther-

mal bias drift of the RRS is represented by the Takagi–Ž .Sugeno TS fuzzy model via experimental data and is

online compensated by TS fuzzy rules. The main idea ofthis approach is to derive several piecewise linear modelsfor some intervals of temperature by using a linear leastsquare algorithm, and fuzzy logic is used to smoothlyblend each linear model over the entire operating tempera-ture. This fuzzy model is saved into memory and used tocompensate for the temperature dependent nonlinear char-acteristics of bias drift.

0924-4247r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved.Ž .PII: S0924-4247 99 00241-1

( )S.K. HongrSensors and Actuators 78 1999 143–148144

Fig. 1. RRS, cylinder and electronic module.

By applying this approach to a RRS showing a worstŽ .performance 0.18rs in our products, we could reduce the

bias drift performance down to 0.058rs.

2. Operating principles of RRS

The operating principle of the RRS, or vibrating cylin-w xder rate gyroscope, is well-understood 2,3 , but a brief

summary is given in this section. As shown in Fig. 2, for aperfect cylinder, there are two modes of equal frequencyand identical mode shape, circumferentially spaced relativeto each other 458, but their absolute position is arbitrary.The cylinder is excited at its resonant frequency to vibratewith prescribed constant amplitude along the driÕe modeŽ .q axes by means of a phase-locked control loop. Thed

diametrically opposed driving forces represent the resultanteffect of electromagnetic drive mechanisms. In the absence

Ž .of externally imposed motion V , the drive mode will be

Fig. 2. Mode shape of the perfect cylinder.

oriented such that points of maximum radial displacementare aligned with the drive forces. When the RRS rotates,the cylinder experiences a Coriolis inertia force, which

Žcouples the second fundamental mode the response mode,.q to the drive mode. This force has a magnitude propor-r

tional to the applied rate of turn and thus provides anestimate of the applied rate. However, to achieve betterlinearity and increase sensor’s bandwidth, an alternativemethod of estimating the applied rate is used instead ofmeasuring the magnitude of vibration of the responsemode. That is, by using closed-loop control over theresponse mode, the locations of the node points are heldstationary during rotation and the control action used tonull the motion of the response mode is taken as a measureof the applied turn.

In an ideal instrument, the cylinder would be perfect sothat the natural frequencies of the q and q modes wouldd r

be equal. Also, the two drive forces would be equal inmagnitude, diametrically opposed and their frequency, p,would be equal to v. The simplified motion of the perfectcylinder gyro is described by the equations:

q q2z v q qv 2q y2GV q s2 F sin pt ,Ž .¨ ˙ ˙d d d d r

q q2z v q qv 2q q2GV q s0, 2.1Ž .¨ ˙ ˙r r r r d

where z and z are the damping ratio of the two modesd r

q and q , G is the effective mass producing the Coriolisd r

coupling between the modes, V is the applied rate, and Fis the driving force. In practice, however, the effects ofmanufacturing tolerances and material inhomogeneity meanthat perfect axially symmetrical cylinder is not achievablew x1,3 .

A simple and useful insight into the effects of imperfec-tion on the performance of the gyro can be gained bymodeling the imperfection as a small point mass attachedto the mid-surface of an otherwise perfectly axially sym-

w xmetric shell 4,5 . The added mass can be taken as arepresentation of non-uniform characteristic of the cylindermaterial. It should be noted here that two important as-sumptions have been made. The first assumption is thattwo modes will be oriented in the cylinder in such a waythat the mass will be located at a point of maximum

Ž .displacement in the drive mode q and consequently, at adŽ .point of minimum movement in the response mode q asr

shown in Fig. 3. This assumption is justified because the

Fig. 3. Modal position of the imperfect cylinder.

( )S.K. HongrSensors and Actuators 78 1999 143–148 145

Fig. 4. TS fuzzy model reasoning.

natural frequencies of the two modes are then stationarywith respect to the circumferential position of the mass,

w xwhich is in accordance with Rayleigh’s principle 4 . Ac-Ž .cordingly, such imperfection causes offset d drive forces

from the required position. The second assumption is thatŽ .the offset position d varies randomly with the operating

temperature. This assumption can be understood intuitivelyby the fact that the mechanical characteristics of thecylinder vary with temperature and thus are reflected insome of the parameters of the gyro. With these assump-

Ž .tions, the general equations of motion 2.1 can be modi-fied as:

q q2z v q y2 q GVqv 2q¨ ˙ ˙d d d d r d d

2s2 F 1yd T sin pt ,Ž . Ž .Ž .q q2z v q q2 q GVqv 2q s2d T F sin pt ,Ž . Ž .¨ ˙ ˙r r r r d r r

2.2Ž .

where d is the offset angle and T is the operating tempera-ture. If q is held at constant amplitude, as before, thed

Ž .RHS term 2d T F will give rise to a response of theresponse mode at frequency psv which will, in effect,d

be a bias. Therefore, such imperfection causes principally,a nonlinear thermal bias drift and it can be validatedthrough experiments. The way to minimize the temperaturecoefficients is to manufacture cylinders super-accuratelyand to select special materials of cylinder. However, theyincrease cost. An alternative approach adopted to minimizethe nonlinear effect of temperature change on the cylinderis to measure the cylinder temperature and use this valueto modify the electronic circuit parameters to compensatethe sensor output. For this objective, the nonlinear thermalbias drift for the entire operating temperature ranges shouldbe properly modeled.

3. TS fuzzy model of thermal bias drift and its compen-sation

It is well known that nonlinear global model can beapproximated by a set of piecewise linear local models via

Ž . w xTakagi–Sugeno TS approach 6–8 . The linear localmodels, which are to be used in the consequent part offuzzy rule, are described by:

ysaqbx , 3.1Ž .

Ž .where y is zero-rate-output bias , x is temperature, a andb are coefficients for the local model to best fit theexperimental data. The following matrix equation is theresult of the least square analysis and can be used to solvefor a and b using Kramer’s Rule:

N Ý x Ý ya s , 3.2Ž .½ 52 ½ 5b Ý xyÝ x Ý x

where N is the number of data points.The fuzzy model is described by IF–THEN rules, which

represent local linear input–output relations of nonlinearsystems. The ith rule of this fuzzy model of temperaturedependent bias is the following form:

ith Rule: IF x is L , THEN y sa qb x , 3.3Ž .i i i i

where is1,2, . . . ,r and r is the number of rules, and Li

is the fuzzy set centered at the ith operating point. Theinference performed by TS fuzzy model is an equivalent ofinterpolation of all the relevant linear local models. Thesedegrees of relevance become the weight in the interpola-tion process. Fig. 4 illustrates graphically the fuzzy reason-ing mechanism to derive an output y from a given input

Fig. 5. Membership function.

( )S.K. HongrSensors and Actuators 78 1999 143–148146

Fig. 6. Fuzzy modeling of thermal bias drift.

x. For the temperature input x, the final output of thefuzzy system can be explicitly expressed as:

r

l a qb xŽ .Ý i i iis1ys 3.4Ž .r

lÝ iis1

where l is the grade of membership of x in L . It shouldi i i

be noted that even if the rules in a TS fuzzy model involveonly linear combinations of the model inputs, the entiremodel is truly nonlinear.

The above fuzzy model of temperature dependent biascan be directly used for compensated output. That is, thebias can be compensated by:

ysyy aqbx 3.5Ž . Ž .˜where y denotes compensated zero-rate-output and y is˜raw gyro output. Hence, a set of r compensation rulestakes the following form:

ith Rule: IF x is L , THEN y sy y a qb x . 3.6Ž . Ž .˜i i i i i

Although each of the rules can be viewed as describinga ‘local’ compensator associated with ‘local’ bias, the

resulting fuzzy compensation is quite nonlinear, which canbe rewritten as:

r

l y y a qb x� 4Ž .Ý i i i iis1ys 3.7Ž .˜ r

lÝ iis1

Therefore, for the compensation of undesirable nonlin-ear characteristics of the system, this fuzzy scheme canoutperform any of its constituent linear counterparts sincenonlinear aspects of the model have been taken into con-sideration.

4. Experimental results

As mentioned earlier, the continuous model of thethermal bias drift is described by a set of fuzzy IF–THENrules with properly defined membership function as Fig. 5.With this definition, we have a total of six fuzzy rules.

Denoting x and y as operating temperature and zero-Ž .rate-output bias respectively, the piecewise linear TS

fuzzy model can be written as:

ith Rule: IF x is L , THEN y sa qb x , is1,2, PPP ,6i i i i

4.1Ž .

where a and b, the coefficients for local model to best fitthe data via linear least square analysis, are obtained as:

a s 1.2758 0.3930 y0.0874 y1.0716i

y0.6382 y1.0423

wb s y0.0054 y0.0018 0.0001 0.0036i

= x0.0021 0.0033

Fig. 6 provides a satisfactory result of fuzzy modelingwith experimental data that show nonlinear variations of

Ž .bias for the operating temperature ranges 220 K–340 K .

Ž . Ž .Fig. 7. Nonlinear thermal bias drift of RRS a profile of operating temperature, b raw data of RRS.

( )S.K. HongrSensors and Actuators 78 1999 143–148 147

Ž . Ž .Fig. 8. Compensated thermal bias drift of RRS a compensated by fuzzy logic, b compensated by polynomial curve fitting.

Based on the above fuzzy model, the nonlinear thermalbias can be compensated on-line by a set of six fuzzycompensation rules that take the following form:

ith Rule: IF x is L , THEN y sy y a qb x ,Ž .˜i i i i i

is1,2, PPP ,6 4.2Ž .where these fuzzy compensator shares the same fuzzy setŽ . Ž .L and same coefficients a and b with the fuzzy model.i

First, the experiment without implementation of fuzzycompensation was performed. To show the temperaturedependent nonlinear variation of bias, the RRS was oper-ated under the temperature profile of Fig. 7a in a tempera-ture chamber. As seen in Fig. 7b, the bias of the RRSvaries nonlinearly according to the operating temperature.

Second, to compensate the nonlinear thermal bias drift,the proposed fuzzy compensator was implemented and theresult is shown in Fig. 8a. In addition, for comparisonanother compensation technique that used 10th order con-ventional polynomial curve fitting in a least squares sensewas employed and its result is shown in Fig. 8b. Theperformance of these two compensation techniques weremeasured by the following quadratic error index:

t 2I s e t d t 4.3Ž . Ž .Hqe0

and the CPU elapsed time performed by Pentium 180 Mhzprocessor for processing one cycle of compensation loopfor one data. They are summarized in Table 1.

From these results, we can see that the proposed fuzzycompensation scheme is able to compensate the nonlinearbias drift of the RRS throughout the entire operating

Table 1The comparison of two compensation techniques

Fuzzy Polynomialcompensation curve fitting

2wŽ . xQuadratic error 8rs 18.4070 20.3133w xCPU elapsed time s 0.06 0.22

temperature range and to outperform other conventionalcompensation approaches in both performance and CPUefficiency.

5. Conclusions

In this paper, we proposed a nonlinear fuzzy compensa-tion scheme to overcome the temperature dependent non-

Ž .linear characteristics of zero-rate-output bias drift of theRRS. Motivated by the capability of fuzzy logic in manag-ing nonlinearity, the nonlinearity of bias was representedby TS fuzzy model over the entire range of operatingtemperature, where the nonlinear global model is approxi-mated by a set of linear local models that were obtained byleast square analysis piecewisely. Then the fuzzy modelwas directly used for compensation of nonlinear bias driftby subtracting the estimated output from the raw data of

ŽRRS. By doing this, we can guarantee a robust against.temperature variations sensor performance throughout the

entire operating temperature range. Finally, the experimen-tal results showed the effectiveness of the proposed scheme.

Acknowledgements

I would like to thank Dr. Don G. Kim, who is thepresident of Inertia Science, USA, for his excellent instruc-tion and support for the experiments.

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Sung K. Hong received the BS and MS degrees in 1987 and 1989,respectively, from Yonsei University, Seoul, South Korea. He worked for

Ž .the Agency for Defense Development ADD , Korea, as a ResearchEngineer from 1989 to 1994, where he participated in the design of

Ž .autopilot controller for an Unmanned Aerial Vehicle UAV . In 1998, hereceived the PhD degree in mechanical engineering from Texas A&MUniversity, College Station. He has come back to ADD in 1998, and he iscurrently a senior Research Engineer. There, he is working on the design

Ž .of the Fly-By-Wire FBW system for a fighter aircraft. His currentresearch interests include fuzzy logic controls, magnetic bearings, ratesensors, and flight control system.