A proposal for a selft-contained instrumentation system for flight research on stability and control

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DELFT

CRANFIELD

INSTITUTE OF TECHNOLOGY

A PROPOSAL FOR A SELF-CONTAINED

INSTRUMENTATION SYSTEM FOR FLIGHT

RESEARCH ON STABILITY AND CONTROL

by

March, 1974.

CRANFIELD INSTITUTE OF TECHNOLOGY

A PROPOSAL FOR A SELF-CONTAINED INSTRUMENTATION SYSTEM FOR FLIGHT RESEARCH ON STABILITY AND CONTROL

by

V. Klein and R. Gregory

SUMMARY

The concept and possible realization of a self-contained instrumentation system for the measurement of stability and control characteristics of an aircraft are described. The recommended accurary of the system is based on its relation to the accuracy of aerodynamic derivatives of an aircraft evaluated from flight data. Therefore several sets of simulated and measured data were analysed. A survey of techniques used for the evaluation of these derivatives is also presented.

CONTENTS

Page

INTRODUCTION 7 STABILITY AND CONTROL CHARACTERISTICS AND 7

THEIR MEASUREMENT

THE RELATIONSHIP BETWEEN THE ACCURACY OF AN 9 INSTRUMENTATION SYSTEM AND THE ACCURACY OF

EVALUATED AERODYNAMIC DERIVATIVES

3.1 M.S.760 PARIS Aircraft 10

3.,2 X.B.70 Aircraft 11 3.3 X-15 Aircraft 12 3.4 CONCORDE Aircraft 12 REQUIRED PERFORMANCE CHARACTERISTICS OF THE 12

INSTRUMENTATION SYSTEM

4.1 Accelerometers 13 4.2 Rate Gyros 13 4.3 Roll attitude - Vertical Gyro 14

4.4 Incidence and sideslip angle Sensors 14

4.5 Control position Transducers 14 4.6 Control force Transducers 1^ 4.7 Airspeed measurement using Transducers 15

4.8 Summary 15 CONCEPT OF THE SELF-CONTAINED SYSTEM 16

POSSIBLE REALIZATION OF THE SYSTEM 17

6.1 Accelerometers 17 6.2 Rate Gyros 18 6.3 Roll attitude - Vertical Gyro 19

6.4 Temperature measurement 20 6.5 External Transducers 20 6.6 Calibration and scaling 20 6.7 Signal conditioning 21

6.8 Filters 22

6.9 Speech 22

6.10 Multiplexing and digitising 22

6.11 Clock and formatting 23

6.12 Tape recorder 23

6.13 Power supplies 24

6.14 Construction 24

7. CONCLUSION 25

8. RECOMMENDATION 26

APPENDIX A A SURVEY OF TECHNIQUES USED FOR THE 27 EVALUATION OF AERODYNAMIC DERIVATIVES

FROM FLIGHT DATA

Al Equation of Motion methods 27

A2 Response Curve Fitting methods 31

A3 Estimates of Coefficients in 33 Equations of Motion of an aircraft

A4 Equation of Motion (EM) method 34

A5 Transformed Equation of Motion 36 (TEM) method

A6 Response Curve Fitting (RCF) 36 method

A7 Frequency Response Curve Fitting 37 (FRCP) method

LIST OF SYMBOLS 39

ACKNOWLEDGEMENTS 43

LIST OF TABLES

Table Page 1 Basic Stability and Control Characteristics 46-47

2 Aerodynamic Characteristics of an Aircraft 48 3 The influence of Bias Errors in Simulated Measured 49

Input and Output Data on the Evaluated Aerodynamic Derivatives. M.S.760 PARIS Aircraft

4 The influence of Bias Errors and Random Errors in 49 Simulated Measured Input and Output Data on the

Evaluated Aerodynamic Derivatives. M.S.760 PARIS Aircraft

5 The influence of Bias Errors in Simulated Measured 50 Phase Characteristics on the Evaluated Aerodynamic

Derivatives. M.S.760 PARIS Aircraft

6 The influence of Bias Errors in Simulated Measured 50 True Air Speed on the Evaluated Aerodynamic

Derivatives. M.S.760 PARIS Aircraft

7 The influence of Bias Errors in Simulated Measured 51 Input and Output Data on the Evaluated Aerodynamic

Derivatives. M.S.760 PARIS Aircraft

8 Performance Characteristics of the Instnomentation 52 System in M.S.760 PARIS Aircraft

9 Comparison of values of Aerodynamic Derivatives 53 estimated by the Response Curve Fitting Method

using different cost functions. M.S.760 PARIS Aircraft

10 Comparison of values of Aerodynamic Derivatives 53 estimated by the Response Curve Fitting Method

from repeated measurements. M.S.760 PARIS Aircraft

11 Comparison of Transfer Function Coefficients 54 calculated from Aerodynamic Derivatives and

estimated by the Response Curve Fitting Method. M.S.760 PARIS Aircraft

12 The influence of Bias Errors in Simulated Measured 55 Input and Output Data on the Evaluated Aerodynamic

Derivatives. XB-70 Aircraft

13

Table Page 14 The influence of Bias Errors in Simulated Measured 57

Input and Output Data on the Evaluated Aerodynamic Derivatives. X-15 Aircraft

15 Instrumentation pertinent to Stability and Control 58 Investigation. X-15 Aircraft (Ref. 3)

16 Instrximentation pertinent to Stability and Control 59 Investigation. CONCORDE Aircraft (Ref. 5)

17 The Relationship between the Coefficients in Short- 60 Period Equations of Motion and Aerodynamic

Derivatives

18 The Relationship between the Coefficients of 61 Lateral Equations of Motion and Aerodynamic

CONTENTS

LIST OF FIGURES

Fig.

1 The influence of Bias Errors in qg on the Evaluated Aerodynamic Derivatives. M.S.760 PARIS Aircraft

2 The influence of Bias Errors in n^ on the Evaluated Aerodynamic Derivatives. M.S.760 PARIS Aircraft

3 Comparison of Simulated and Calculated Responses in Pitch Rate. M.S.760 PARIS Aircraft

4 Comparison of Simulated and Calculated Responses in Normal Acceleration. M.S.760 PARIS Aircraft

5 Comparison of Simulated and Calculated Responses in Roll Rate. M.S.760 PARIS Aircraft

6 Comparison of Simulated and Calculated Responses in Yaw Rate. M.S.760 PARIS Aircraft

7 Comparison of Simulated and Calculated Responses in Lateral Acceleration. M.S.760 PARIS Aircraft

8 Comparison of Measured and Calculated Responses in Pitch Rate. M.S.760 PARIS Aircraft

9 Comparison of Measured and Calculated Responses in Filtered Normal Acceleration. M.S.760 PARIS Aircraft

10 Comparison of Measured and Calculated Responses in Incidence Angle. M.S.760 PARIS Aircraft

11 Comparison of Evaluated Aerodynamic Derivatives obtained by the Response Curve Fitting Method using different cost function and the Equation of Motion Method. M.S.760 PARIS Aircraft

12 Comparison of Time Histories measured in Flight with Analog Match. XB-70 Aircraft (Ref. 2)

13 Comparison of Time Histories measured in Flight with Analog Match. XB-70 Aircraft (Ref. 2)

14 Comparison of Time Histories measured in Flight and computed by the Equation of Motion Method. X-15 Aircraft (Ref. 4)

Fig.

15 Comparison of Time Histories measured in Flight and computed by the Equation of Motion Method -Method Function. X-15 Aircraft (Ref. 4)

16 Comparison of Time Histories measured in Flight and computed by the Response Curve Fitting Method. X-15 Aircraft (Ref. 4)

17 Comparison of Evaluated Aerodynamic Derivatives obtained by various methods. X-15 Aircraft

(Ref. 4)

18 Comparison of Time Histories measured in Flight, expected and computed by the Response Curve Fitting Method. CONCORDE Aircraft (Ref. 5) 19 Examples of Evaluated Aerodynamic Derivatives

with different sensitivity obtained by the

Response Curve Fitting Method. CONCORDE Aircraft (Ref. 5)

20 Comparison of Predicted Values with Values

obtained by the Response Curve Fitting Method and the Time Vector Method. CONCORDE Aircraft (Ref. 5) 21 Block Diagram of Instrument Pack

1 INTRODUCTION

The aircraft characteristics that are relevant to its stability and control are generally called flying qualities. These characteristics are carefully studied and analysed during design and development of an aircraft, Flight testing is then required to verify that the flying qualities actually achieved are satisfactory, or to find and eliminate imperfections.

For the judgment of stability and control characteristics of an

aircraft flying quality requirements were defined. They have been developed from extensive and continuing flight research and they are under continuous study and modification in order to keep them at the level which corresponds to the latest research and design knowledge.

For flight measurements either specially modified aircraft are used (variable-stability aircraft) or the information is gathered from the measurements on the various aircraft which represent different categories

including service type aircraft.

Very often the limiting factor for the latter measurements is the availability of the aircraft. This factor demands that the installation of measurement instruments within an aircraft, their checking and calibration must be achieved in the shortest possible time to allow maximum time for the measurement itself.

In an attempt to solve this problem some proposals on data measurement equipment have been made. The concept of an instrumentation pack had always been attractive in respect of providing a system capable of being installed quickly and allowing measurement of a sufficient number of parameters with acceptable accuracy.

The main objective of this report is to present a concept of a self-contained instrumentation pack for the measurement of stability and control characteristics of an aircraft. To create this pack the demands on basic performance characteristics of the system will be specified using the results from measured and calculated data.

2 STABILITY AND CONTROL CHARACTERISTICS AND THEIR MEASUREMENT

The stability characteristics of an aircraft provide information about its static stability and about the frequency and damping or time constant of different modes included in aircraft's motion which follows after a

disturbance from initial steady-state flight. The control characteristics predominantly define control power and control forces and also the behaviour of an aircraft during stalling and spinning.

Schemes of the basic control characteristics are arranged in several ways. First, the characteristics for the description of the controllability of an aircraft, during its manoeuvres following small changes to its initial flight regime, are given as the ratio of a quantity which describes a pilot's action to the change in variable which represents the response of an aircraft.

The second group of control characteristics are intended to provide the information about the controllability of an aircraft after the change of its configuration. These characteristics are defined by means of the necessary values of quantities describing a pilot's action to maintain the initial airspeed.

The third group of characteristics are used for the judgment of the controllability of an aircraft during manoeuvres connected with reaching some marginal value in different response variables. The characteristics are then represented by the necessary values of control deflections and/or control forces to reach

e.g.:-C, n p V or V .

Lmax , max , ^raax , max m m .

In recent years two new techniques of measurement have been introduced in connection with gathering information about stability and control

characteristics of an aircraft. The first is connected with the measurement of responses of an aircraft to atmospheric turbulence. This investigated how the turbulence influences the acceptability of the aircraft as a passenger transport, how it limits aircrafts accuracy as a gun or bombing platform but mainly how it degrades the flying and riding qualities of an aircraft.

The second technique is concerned with the evaluation of aerodynamic derivatives of an aircraft from flight data. These results can provide better understanding about flying qualities and can help to explain their possible imperfections. It is also possible to combine the evaluation of aerodynamic derivatives with the determination of basic stability and control characteristics into a common approach using transient-response flight data. These two techniques have introduced new sophisticated methods for flight data analysis and have introduced more stringent requirements on performance characteristics of an instrumentation system, mainly its accuracy.

A brief survey of basic stability and control characteristics is given in Table 1 together with the techniques for their measurement, with measured quantities and with relationships or characteristics obtained from flight data. Similar arrangements for the aerodynamic characteristics of an

aircraft evaluated from flight data are presented in Table 2. Two groups of methods for the determination of aerodynamic derivatives from measured data which are based on the least squares procedure are presented in Appendix A. The measured quantities which are used for the evaluation of stability and control characteristics or aerodynamic derivatives can be divided into three groups. In the first, there are quantities which define the response of an aircraft. These

are:-linear accelerations angular velocities

pitch and bank attitudes incidence and sideslip angles airspeed. Mach number and height load on the tail (in special occasions only) n , n , x' y' P. q. r t? ,'p a, 6 V. M, H

^t

The second group of quantities describes pilots actions and

includes:-control deflections

x], ^, c,

or control stick and pedal deflections n , 5 , C

control forces F , F , F

e' a' r

These quantities are input variables introduced by a pilot.

For the measurement of response of an aircraft to turbulent air the

random velocities of the atmosphere are regarded as the inputs. They are not

measured directly during the flight because they are included in measured

outputs of an aircraft.

The third group of measured quantities is formed by those which define

flight conditions. Very often it is sufficient to know the initial true or

equivalent airspeed, height and instantaneous weight of an aircraft.

Sometimes the pitch and bank attitudes are demanded as well.

3 THE RELATIONSHIP BETWEEN THE ACCURACY OF AN INSTRUMENTATION SYSTEM AND

THE ACCURACY OF EVALUATED AERODYNAMIC DERIVATIVES

For the measurement of stability and control characteristics of an

aircraft various types of instrumentation systems are employed. They differ

in the following ways; used transducers, the arrangement of the system and in

its performance characteristics. Often a problem arises over the required

accuracy and that which is obtained from the measured quantities and this has

a bearing on the choice.

The accuracy of an instrumentation system as its basic performance

characteristic is discussed in Ref. 1. The approach for its evaluation is

also presented and covered by an example. The accuracy of an instrumentation

system is a measure of correctness of results produced by such a system.

Therefore in the statement regarding the accuracy it is generally necessary

to indicate two values. One value means the credible bounds on systematic

errors, the other is the precision of a system.

In practise, however, the accuracy of an instrumentation system is

usually given by a single numerical value, and often it is not made clear

what the precise meaning of this number is. Because modern instrumentation

systems possess very high precision the influence of this number on the

accuracy of these systems can be neglected. In such cases the single

numerical value indicating the accuracy of the system means its bias error.

To evaluate the accuracy of the instrumentation system used for the

measurement of reliable input and transient-response data, an attempt can

be made to find the relation between this accuracy and the accuracy of

evaluated aerodynamic derivatives. Generally this is a very complicated

task because the accuracy of final results depends on the

following:-Types of errors in measured quantities and their magnitudes, design

of experiment and characteristics of an aircraft and the methods used for

the analysis of measured data. Therefore a practical solution is possible

only for a given case with some accepted simplifications.

The following investigation is based on the analysis of simulated data formed by computed responses to which the known bias errors were added. Mainly the static bias error due to change in the slope of a calibration line is considered. For these cases the 'experimental' value is obtained from

relation:-XE = y CI ± 6)

where 1006 is the bias error in % of reading value.

The dynamic bias error in an instrumentation system is considered as the change in the phase characteristic of a transducer due to change of its dynamic characteristics. It is expressed in degrees at the frequency equal to 3Hz.

In the first set of examples the simulated and measured data of M.S.760 PARIS small jet trainer aircraft are analysed. The results of measurements on the XB-70 supersonic bomber and the X-15 rocket research aircraft are taken from published reports and papers. They are complemented by two exampl using simulated data. The last example gives some information about the measured lateral stability derivatives on the CONCORDE supersonic transport aircraft.

In the analysis the response curve fitting method for the evaluation of aerodynamic derivatives was mainly used. However, some examples include the results of the equation of motion method and analog matching.

3.1 M.S.760 PARIS Aircraft

The influence of systematic errors due to a change in the slope of calibration lines of the measuring instruments used is studied on simulated data. First the effect of bias reading errors in 'measured' output quantities, defining the short period longitudinal motion was considered separately. This was then combined with the same kind of error in the input. The resulting errors in aerodynamic derivatives are plotted as the function of the reading errors in output quantities in Fig. 1 and 2. The results involving different input shapes and different computing techniques are also included.

The evaluated aerodynamic derivatives from data including errors are shown in Tables 3 and 4. The computed time histories of the output variables corresponding to the case in the last column of Table 4 are plotted in Fig. 3 and 4 together with simulated data used in the

analysis. This particular example includes the combined effect of bias and random errors.

As already stated before the bias errors in dynamic

characteristics of an instrumentation system are interpreted as the change in the 'measured' phase characteristic relating the output and the input variables of an aircraft. The results of three numerical examples are in Table 5 where dynamic errors in normal acceleration and pitch rate data are treated separately.

Finally the influence of error in measured true airspeed on the evaluated aerodynamic derivatives was studied in numerical examples the results of which are shown in Table 6. This kind of error affects the determined coefficients in the equation of motion if the measured

normal acceleration - in the longitudinal motion - or lateral

acceleration - in the lateral motion - enter the computing process. But the accuracy of non-dimensional aerodynamic derivatives is

dependent on the error in true airspeed in all cases as follows from expressions in Tables 17 and 18.

The influence of bias errors in both input and output quantities were also studied in simulated lateral motion. 4% error in reading values was introduced by multiplying the ordinates by constants as

follows:-Case 1: 0.96C, 1.04;, 1.04p, 0.96r, 0.96n

Case 2: 1.045, 0.96;, 0.96p, 1.04r, 1.04n

The errors in the evaluated aerodynamic derivatives using the response curve fitting method are presented in Table 7. Simulated data and computed results for Case 1 are plotted in Fig. 5 to 7.

The flight measurements were limited on the short period

longitudinal motion only. The basic performance characteristics of the used portion of the instrumentation system in the aircraft are given in Table 8. Details of their determination and the description of the

system are in Ref. 1. The pulse-shape elevator deflection was used as the input during the measurement. For the repeated measurements the same flight conditions were maintained within the best possible limits.

First of all the aerodynamic derivatives were evaluated from one measurement using different combinations of measured output variables ag, qg and nn. The obtained results are summarised in Table 9. The numerical values of aerodynamic derivatives include also the estimates of their standard errors. The 'closeness' of measured and fitted curves is defined by the mean squared error calculated from (A.25). Measured data and computed results from one example involving the fitting of all three outputs are plotted in Figs. 8-10 inclusive.

The results from repeated measurements are shown in Table 10. In this example the output variables qg and npg from four runs were fitted. All results from single and repeated measurements are then compared in Fig. 11.

The measured responses were also fitted separately to provide the estimates of corresponding transfer function coefficients. In these cases the original equations of motion were transformed into the canonical forms satisfying the condition for the identifiability of the system. The evaluated transfer function coefficients including their standard errors are in the first part of Table 11. The weighted mean squared errors enable the mutual comparison of the fitness in all three cases. The second part of Table 11 contains the transfer

function coefficients calculated from the previously evaluated aerodynamic derivatives.

3.2 XB-70 Aircraft

The short period longitudinal responses were calculated using the numerical data from Ref. 2. The obtained ordinates and the ordinates

of pulse input were changed multiplying them by factors which simulated different bias errors in reading values. The resulting errors in

evaluated aerodynamic derivatives are in Table 12. In Ref. 2 the

results of the analysis of the longitudinal and lateral motion applying the analog matching techniques are published. Two of those results are in Fig. 12 and 13. The accuracy of used instrumentation system and the instrumentation ranges are apparent from Table 13 which is also ta:ken from the mentioned reference.

3.3 X-15 Aircraft

The first example with calculated short period responses

demonstrates the effect of bias errors in all input-output data on the evaluated aerodynamic derivatives. Results of the response curve fitting method are in Table 14.

In Ref. 3 the measured longitudinal and lateral responses are again analysed using the analog matching technique. The characteristics of the instrumentation system in the aircraft are also contained therein and are reproduced in Table 15.

The lateral data is also analysed by several computational methods in Ref. 4. The measured and computed time histories following the

application of the equation of motion and response curve fitting method are in Figs. 14-16 inclusive. The values of obtained aerodynamic

derivatives are compared in Fig. 17.

3.4 CONCORDE Aircraft

In Table 16 the date relating to the accuracy of the

instrumentation system used during the measurement of stability and control characteristics is given. This data was extracted from Ref. 5 which also includes the results of the lateral motion analysis. Some of these results are reproduced in Figs. 18-20 inclusive. In Fig. 18 the measured time histories are compared with predicted and computed ones. The relationships of four aerodynamic derviatives obtained by the response curve fitting method as a function of Mach number are given in Fig. 19. These examples demonstrate two different

sensitivities of the solution.

In Fig. 20 the predicted values of the yawing moment coefficient due to sideslip are compared with its values evaluated by the response curve fitting and time vector method.

4 REQUIRED PERFORMANCE CHARACTERISTICS OF THE INSTRUMENTATION SYSTEM

The examples in the proceeding chapter demonstrate the relationship between the accuracy of an instrumentation system and the accuracy of evaluated aerodynamic derivatives from measured data. The results from simulated and measured data indicate that for determination of longitudinal derivatives the inaccuracy of ±4% in reading values in all input-output variables could be accepted. The same degree of accuracy will also be

acceptable for lateral motion, but to achieve reliable values in the case of less significant derivatives which are very sensitive to the errors in

measured data the demand on the accuracy will be more stringent and will require reading errors of less than ±2%.

If the output variables were measured with the greater accuracy, e.g. less than ±2% of reading values, then the inaccuracy in the input data will result in bias error in control derivatives only. Therefore it is possible to use the accurate instrumentation system for the measurement of transient-response data and less accurate transducers for the measurement of input data. From their estimated inaccuracies the credible bounds on

systematic errors in control derivatives are then known. The stated combination of two parts of an instrumentation system with different accuracies can be met in the use of an instrumentation pack together with external transducers.

Because of low frequency range of measured stability and control data and high natural frequencies of modern instruments the influence of

systematic errors due to changes in dynamic characteristics of an

instrumentation system will be less significant than the influence of static errors. It is believed that the bias error of ±4 degrees in phase

characteristics (at f = 3Hz) of transducers represented by the second-order system will be acceptable.

Finally the inaccuracy in evaluated true airspeed from measured

quantities or from reading of cockpit instruments should be less than ±4% of the corresponding value.

The required accuracies represent the overall acceptable bounds on bias errors in individual channels of the system which include transducers,

electronics and a recorder. No mention has yet been made about the random errors of an instrumentation system. But in modern systems these errors are small compared with bias errors and very often their effect is decreased by some kind of averaging during the processing of measured data.

Considering the examples in Chapter 3, their analysis and the concept of an instrumentation pack, the basic performance characteristics of

individual channels in the instrumentation system should be as follows:

4.1 Accelerometers

The system should include accelerometers with ranges of ±0.5g (two transducers), -2 to Og and -5 to 3g. The acceptable inaccuracy should be ±0.005g or ±2.0% of reading, whichever is greater.

Recommended dynamic characteristics are:- fg > 20Hz and t, = 0 . 7 with the acceptable bias error 6 = ±0.1.

If the accelerometer with a range of ±0.5g is used for the measurement of the attitude or bank angles it could measure these angles within an accuracy of ±0.3 degrees of ±2% of reading value whichever is greater.

Using the approach in Ref. 1 for the estimate of the effect of dynamic errors in an instrumentation system on measured time histories, and considering the simulated and measured data in Chapter 3 the

maximum bias error in measured acceleration due to 6 = ±0.1 will be

less than ±0.003g. ^

4.2 Rate Gyros

The system should include four rate gyros with the ranges ±6 deg/s, ±10 deg/s, ±20 deg/s and ±40 deg/s. These ranges do not cover the measurement of stalling and spinning characteristics of an aircraft.

The acceptable inaccuracy should be ±0.2 deg/s or ±2.0% of

reading, whichever is greater. Recommended dynamic characteristics are: fp > lOHz and ; = 0.7 with the acceptable bias error 6^ ± 0.05.

Because of the small value of naturÉ.1 frequency and sometimes high

values of angular acceleration during the transient-response measurement the effect of dynamic error 6^ on measured data should be carefully examined.

4.3 Roll attitude - Vertical Gyro

If the attitude and/or bank angle is to be measured and used in the process for the evaluation of aerodynamic derivatives then the instrument should have the linear operating range ±30 degrees and maximum overall inaccuracy ±0.2 degrees or ±2.0% of reading, whichever is greater.

4.4 Incidence and sideslip angle Sensors

Incidence and sideslip angle Sensors can provide the additional information for the evaluation of aerodynamic derivatives of an

aircraft. Their knowledge enables the introduction of more constraint on fitted data and, therefore, the ability to obtain more accurate results. Having measured time histories of incidence and sideslip angles the relationship between the output variables can be checked by means of the following expressions, for the short period

motion:-t

J

('IE

- V" V ^ *

"

°E

0

and for the lateral motion:

t

tf- "yE * % P E - ^E * f- *^°^ ''O ''E^^^ = ^E

•* 0 • ^ 0

0

If the response of an aircraft to turbulence is to be measured then the incidence and sideslip Sensors are vital instrioments. In addition the values of sideslip angle are also demanded during the measurement of lateral-directional statics in steady-state nonsymmetric

flights, see Table 1.

For the transient-response measurements the range of both

transducers should be ±10 degrees and for the measurement of responses to turbulence ±5 degrees. The acceptable inaccuracy should be ±0.2 degrees or ±2.0%, whichever is greater.

If a flow-direction vane is used as a sensor then its dynamics characteristics should be: fQ > 0.4,Jq~(q measured in N/m^) and c = 0.7 at sea level with the acceptable error 6 = ±0.1.

4.5 Control position Transducers

The range of these instrioments will mainly be influenced by the type of transducer, its installation and by the kind of experiment.

For the transient-response measurement the working range of transducers after their installation could be as follows: for the elevator angle ±10 degrees, for the aileron angle ±6 degrees and for the rudder angle ±6 degrees.

When the instrumentation pack is used then the transducers will be external ones and may possess low accuracy. But as stated previously the measurement of useful data for the evaluation of aerodynamic

derivatives will be still possible provided that the resulting

inaccuracy in control derivatives will be determined and its magnitude accepted.

4.6 Control force Transducers

The control force transducers will be used mainly for the

measurement of longitudinal and lateral-directional statics and for the measurement of longitudinal and lateral-directional manoeuvreability. Their ranges will be dependent on the type of aircraft used and the type of experiment therein. For the aforementioned measurements the accuracy of measured stick forces and rudder-pedal forces should be ±5N(1 Ibf) and ±50N(10 Ibf), respectively. If the measured data is to be used for the evaluation of the derivatives of control-surface-hinge-moment coefficients then the accuracy must be increased.

4.7 Airspeed measurement using Transducers

So far little is known about the required accuracy of the airspeed measurement during unsteady manoeuvres. If the maximum inaccuracy is to be less than 2% of reading value then the increments of airspeed in transient-responses should be measured within the accuracy ±0.1m/s.

On the other hand the accepted inaccuracy in measured airspeed in steady-state flights (definition of flight conditions) would be ±2kn for the speed range 50-300kn and ±4kn for the speed range 300-800kn.

4.8 Summary

It should be emphasised again that the stated requirements on the basic performance characteristics of an instrumentation system are mainly based on a small number of examples. The examples with

simulated data represent the idealised cases and do not cover the whole spectrum of tested aircraft or all possibilities in a design of

experiments e.g.: different flight conditions, input functions, etc. But the majority of these examples include the influence of bias errors in their worse combinations.

The examples from flight measurements include only those where the data regarding the accuracy of the instrumentation system used was available. All examples quoted confirm that the instrumentation system with an accuracy to within ±2% of full scale range of the transducers used can provide very reliable data for the evaluation of aerodynamic derivatives. But the success of these experiments is also conditioned by careful planning, their thorough realisation and by correct

5 CONCEPT OF THE SELF-CONTAINED SYSTEM

The instrumentation pack would be considered as an on-board digital magnetic tape recording system with a low frequency range. It would be capable of

recording twelve measured quantities at a rate of 100 points second, plus voice for identification purposes and additional points for reference time signals, calibration and correction signals.

The construction of the pack would consist of three basic units:

(a) Transducer pack.

(b) Electronic pack including power supplies fed from the aircraft 28 volt supply.

(c) Recorder pack.

which could be mounted either separately or together as one unit.

The transducer pack would contain seven transducers:

(i) three linear accelerometers with the alignment of their sensitive axes parallel to the body axes.

(ii) three rate gyros for measurement of angular velocities around the three body axes.

(iii) one vertical gyro for measurement of pitch and roll attitude.

The electronic pack shown schematically in Fig. 21 comprises basically signal conditioning units including amplifiers, filters, multiplexer circuits, an A.D.C. and logic control circuits. The electronic pack will have

facilities for ground and in-flight calibration as well as provision for the connection of a program matrix for selection of the required combinations of internal and external transducers.

There will also be an arrangement to connect a synchronised camera enabling pictures to be taken of the cockpit instrument panel. This

arrangement should enable information about airspeed. Mach number, height and full consumption to be recorded.

The power supply unit forms part of the electronic pack and is fed from the aircraft 28 volt supply.

The recorder pack will be formed by a magnetic tape recorder capable of recording the digital data plus speech.

A severe environment has not been considered and although vibration environments should not be too detrimental, it must be emphasised that sensitive instruments such as the force balance accelerometers recommended, could well saturate at frequencies above the required measurement frequency. In this particular field of work, the higher frequency response of this type of transducer can be a disadvantage. A temperature range of nominally 0°C to + 30°C has been assumed, should lower temperatures be required some form of heating could be arranged.and nominally higher temperatures should mean only a slight degradation of accuracy. This is a typical case where a preliminary environmental search would be valuable, using the Environmental Pack.

6 POSSIBLE REALIZATION OF TOE SYSTEM

The main factors taken into consideration when proposing this system were as follows:

(a) The requirements of the previous chapters.

(b) Experience of previous users. (Ref. 10-13 inclusive). (c) The availability of equipment, economy and a preference for

equipment manufactured in the United Kingdom.

(d) Maintainability and flexibility with a view to the ready

acceptance of any type of transducer either as part of the pack or external to the pack.

(e) Compatibility with known analysis systems, in particular the Cranfield Institute of Technology (CIT) facility.

Only the relevant parts of the component specifications have been reproduced in this proposal, the full specifications are available either from the manufacturers or from CIT^

6.1 Accelerometers

It is considered that the only low frequency accelerometers which meet the requirement are the force balance types, and the only units made in the U.K. are the American Schaevitz types manufactured by Electro Mechanisms Ltd. Another manufacturer Smiths Industries Ltd., have published details of their own developments but their unit is not yet in production.

The specification of the Schaevitz accelerometers appears to be comparable with units made in the U.S.A. by other manufacturers and they are the cheapest available. Use at CIT has proved them to be successful, although one disadvantage is that the maximum offset available is 1 g and this option adds £35 to the cost, making it less competitive. Another disadvantage, common to most makes, is that the standard units require centre-tapped supplies and the signal output is electrically connected to this centre tap, but this should not effect the system under consideration.

Electro Mechanisms Ltd., quote their calibration accuracy as 0.1% up to the 1 g range and 0.2% from the 2 g to 50 g range with a Best Fit Straight Line (BFSL) of ±0.05% Full Scale Range (FSR) where FSR is the total excursion from positive to negative. These errors may be

accounted for during analysis. Probably the most serious inaccuracy is the temperature coefficient of sensitivity, this is given as 0.01% per °F thus for a temperature range of 30°C this could mean a change of 0.54%. The brochure does not give a figure for zero drift with temperature, but the manufacturers state that drift is mainly due to mechanical hysteresis effects and should maintain the published figure for hysteresis of 0.02 FSR throughout the -40°C to +70°C range.

The brochure does not specify particularly well the damping ratio and no temperature coefficient is published, but the manufacturers state that damping is set by an integral capacitor and may be easily set to within the required ±0.1, also, that this should not vary more than 15% over the temperature range -40°C to +70°C. Thus, over the temperature range required, and with a natural frequency for the most sensitive unit of 50 Hz i.e. ±0.5g increasing approximately as the root of the range, this should not create a serious problem.

The only signal conditioning normally required with this type of unit is that of signal filtering as covered later in this chapter. This circuitry should not increase the error significantly and thus the acceleration system should meet the requirement under the worst case conditions. There would be a worthwhile improvement in accuracy if the temperature of the transducers were recorded, this measurement could then be used for error correction during analysis.

6.2 Rate Gyros

There are two main sources in the U.K., namely Elliott Bros. Ltd., and Smiths Industries Ltd. Because of first hand experience, the main Gyro of interest from Elliott Bros, is the American Nortronics unit which is two to three times the price of the standard Smiths Miniature Gyro, but experience and published data does indicate that there may be marginal advantages on Specification. Elliott Bros, also market the

'Supergyro', manufactured by Varo Inertial Products, a company within the Group. The price is about £400 and delivery is available within weeks. Also, the published specification of the 'Supergyro' indicates improved performance in some areas, compared with the Smiths Unit. Another unit available from Elliott Bros, is the Dual Axis Rate

Transducer (D.A.R.T.), this is marketed in the U.S.A. by the Northrop Corporation and works upon the principle of a sphere of spinning fluid namely mercury and a two crystal sensor which provides a sinusoidal output of amplitude, phase, and frequency dependent upon rate, direction and fluid velocity respectively. The claim is also that the units will sense rate in two axes and with slight modification, will also measure acceleration. The price is quoted at about £700 for the two rate version, with a delivery of about 4 months but as with the 'Supergyro', lack of first hand experience precludes recommendation.

The Smiths Miniature Rate Gyro is the cheapest and most well known although delivery can be a year or more, unless there happens to be one available 'off the shelf'. This reason alone may lead to considering the Elliott Bros, units. Taking the specification of the Smiths Miniature Gyro as being fairly typical, the major inaccuracies are quoted as non-linearity ±0.5% up to 1/2 full scale and ±2% up to from 1/2 full scale to full scale, Gyros may be purchased with inbuilt compensation for temperature coefficient of sensitivity but despite this there is still a quoted maximum change of ±2% of scale factor over the temperature range of -40°C to +70°C and from curves published Ref. 14 the zero drift is approximately -1/4% to +1/4% of full scale for the respective temperatures and centred around +20°C. Over the +30°C range considered the errors should be proportionally less.

The natural frequency of the 6 deg/s gyro is quoted as 11 Hz and has the similar relationship to scale as the accelerometers. Damping

fluid orifice, but even so, the change of damping ratio from O^C to +30°C is typically 0.7 to 0.5 respectively. In practise these units self heat by about +20°C so that after a certain period the effective working temperature range would become +20°C to +50°C, and at 50°C the internal compensation is such that the damping ratio is beginning to increase with temperature thus the effective damping ratio should only change by 0.1 or so. The initial damping ratio should be well within the required limits.

As with the accelerometers, the accuracies could be considerably improved by recording temperature.

Some aircraft 400 Hz supplies have a frequency tolerance of about ±2% and since sensitivity is directly proportional to this frequency, it is thought that the 400 Hz should be re-generated as part of the basic power supply system. Smiths Industries Ltd., manufacture a unit

for supplying three gyros, the price is quoted as £150, and it would be difficult to construct one more economically.

Signal conditioning consists mainly of reactively matching the pick-off output, scaling, synchronous de-modulating, and filtering. Again Smiths market a vinit for this purpose and the price is quoted as about £50, but considerable experience at CIT with these gyros, coupled with the necessity for a specifically designed filter, may well make it more economic to manufacture the signal conditioning circuitry. The 400 Hz polarization for the pick-off would be provided by the power supply unit above.

6.3 Roll attitude - vertical Gyro

The main contenders for supplying this instrument would be Sperry Gyroscope, Smiths Industries Ltd., or Ferranti Ltd. This is a most expensive item and it may be worth considering alternative signal sources or the use of existing units. Generally the visual indicator type of instrument would not give the accuracy required and many of them do not have any electrical signal outputs and therefore a high quality vertical gyro must be considered. Discussion with Ferranti indicated that a rebuilt unit may be available, but even so, the cost would approach £1000.

Apart from the usual errors due to the earth's rotation and

movement of the aircraft over the earth's surface, the main inaccuracies will be from the free wander typically less than 1/2 degree of arc per minute, plus errors from the pick-off device. This type of instrument usually has an a.c. pick-off similar to the rate gyros above or a synchro type of pick-off. In either case the errors are similar to those given for the rate gyros and may be compensated accordingly.

The free wander error appears as a zero shift and is the most difficult problem to overcome. Gyros of this type usually have built in erection mechanisms usually controlled by a vertical sensing mercury switch. Thus the assembly is being constantly processed to 'vertical', although this may in fact be a slight incline to help compensate for the effects of horizontal accelerations during rates of turn.

Additionally, the erection system is often switched away from the mercury sensor during roll manoeuvres in order to prevent eroneous precession. In a good gyro, the accuracy of the erection mechanism can

be better than 10 minutes of arc and usually precesses at the rate of a few degrees of arc/minute. Also, a 'fast erect' facility is usually provided which speeds the process by an order or more, thus by operating this facility for a few seconds of straight and level flight, an

accuracy within the required 0.2 degree of arc should be obtainable.

To improve upon these accuracies, it would be necessary to consider stabilised platform techniques.

The frequency of the supply to the wheel is not particularly critical and either the aircraft supply may be used or supplies from the inbuilt power supply unit.

The pick-off or synchro would be polarised from the inbuilt 400 Hz unit and the demodulation process would take a similar form to the rate gyro system above.

6.4 Temperature measurement

Although this does not appear as a requirement of the pack, from what is stated above, measurement of transducer temperature would

enhance the accuracy of the system. This would probably be most easily achieved using thermistors.

6.5 External Transducers

On the basis of a 12 channel facility, and a 7 channel transducer pack, 5 channels would remain for external inputs. As shown in Fig. 21 all channels would contain the same matched filters, thus the input connection to these would be the logical place to connect all inputs whether part of the pack or external. Transducer temperature

measurement could be fed into the matched filters, but this would be wasteful since this measurement is quasi-static and does not require this standard of filtering or the sampling rate applicable to the signal channels.

The most important external transducers will be those for

measurement of control surface positions. The simplest approach would probably be the attachment of cords to the stick and pedal which are in turn coupled to precision potentiometers. The outputs of these

potentiometers would conveniently match to the concept of the equipment outlined. The accuracy of such a system could be of the order of 1% of stick position with additional inaccuracies.contributed by the control linkages.

The alternative possibility would be to derive this information from existing autopilot or automatic control servos. But usually it is not possible to obtain direct electrical data from such servos because of safety regulations. Therefore a separate transducer would be fitted and the expected accuracy would be of order 0.5% of reading value.

6.6 Calibration and scaling

Calibrations take three forms:

(a) The initial laboratory system calibration - Where the input to output relationships are determined and scaled where

possible into convenient engineering units. From these

calibrations the compensations necessary for non-linearities, temperature compensation etc., may be determined. These calibrations would adopt normal procedures and in the case of this particular pack, its portability should minimise error due to changing configurations.

(b) Pre-flight system checkout calibrations - These sub-divide into two headings:

(i) Check calibrations - For this equipment it is thought satisfactory to simply physically move the transducer pack to make sure the equipment functions.

(ii) Test set facilities - Providing access to the logic circuitry for either pre-setting to required static situations for calibration purposes or as a facility to check the logic.sequences, plus access to the analogue signals and to the signals driving the record heads.

The test set would be connected by single plug and would not change the operation of the pack, unless by specific demand from this box.

(c) In-flight calibration injection - This should be a cockpit operated function, where the signal conditioning circuitry is broken at some convenient point and the transducer

signal is replaced by an injection of polarising voltage of known percentage amplitude, e.g. zero, ± full scale. Thus during the setting up of the analysis procedure, the

computer may be scaled to these injections giving a ratio directly related to signal but not necessarily dependent upon the absolute value of the polarising voltage. It is not thought worthwhile to apply these calibrations to the thermistor signals, since these are measurements of

secondary importance. Since the transducers within the pack have relatively high level output, these calibrations may be injected at the point of connection between

transducer and signal conditioning. An exception may be the accelerometers, which have an in-built facility for injecting calibration currents into the force feedback loop. A further alternative is that it may be more convenient to

inject at the filter inputs, this would provide an easier interface and a better way of dealing with any low level signals external to the pack.

6.7 Signal conditioning

In a system with a small number of channels, there is no point in considering shared signal conditioning and therefore, each channel would be individually conditioned, complete with filtering and presented to the multiplex circuitry at a level suitable for the multiplexer. Signal conditioning for the in-built transducers would be within the electronic pack, but it is thought that the filter input connection is the best

'interface' point and therefore any external signals would be conditioned to suit this input level.

6.8 Filters

Filters are particularly critical when data is to be digitised in order to overcome problems of aliasing and to achieve economy of data collection, the latter being particularly important when considering record duration. Sharp cut-off filters should be used for maximum effectiveness and the stability must be compatible with the accuracy of the system. Active filters built at CIT having cut-off rates of 36dB/ octave, i.e. 6th order, have proved to have stabilities of 0.1% over a temperature range of 0°C to +30°C. In terms of time, this means a 0.1% change in the 270 degree phase shift at natural frequency, i.e. 0.27 degree drift at natural frequency. For the case of this equipment 3 Hz

is the maximum frequency required and the data is to be sampled at 100 points/second, i.e. approximately 12 points/second/channel so that a cut-off frequency would be chosen at near 3 Hz.

6.9 Speech

Basically there is no problem with processing speech signals, a problem does arise, however, when considering digital formatting, Chapter 6.11.

6.10 Multiplexing and digitising

There are a number of ways in which the digitising process may be performed but two main requirements must be considered in this

particular case. Firstly, time correlation between channels is

important and secondly redundant data must be kept to a minimum in order to obtain the required one hour duration of recording time using a small tape recording machine.

From this the best approach appears to be to use Sample and Hold circuits, simultaneously triggered, thus, coupled with well defined filters, the data is stored with good time correlation, it then may be commutated sequentially by the Multiplexer to the single Analogue to Digital Converter A.D.C.

The Sample and Hold circuits may be made either with descrete components or with proprietary modular units, for example, the Analog Device modules. The sample pulse would be derived from the timing circuitry and would probably have a width of less than 10 p/s. this being short compared with the rate-of-change of the signals.

It is proposed that the data be multiplexed at relatively high level and that the Multiplexer would almost certainly be a proprietary equipment and most likely be part of the A.D.C. thus further

amplification is not anticipated. The A.D.C. would be chosen to have a data resolution of 10 data bits. The maximum possible speed of

sampling and digitising is likely to be far in excess of the requirement but the timing circuitry would control the rate and be compatible with the formatting and recording requirement. It is possible that the

Multiplexer may also contain the Sample and Hold circuitry, because some more recent techniques use individual monolithic circuit comparators for each channel, these being addressed directly, by logic circuitry, to the A.D.C. This technique is used by Micro Consultants Ltd., who claim advantages of less cross talk and reduced error due to settling times.

The errors due to these circuits should be less than 0.05% ± 1 bit.

6.11 Clock and formatting

It is impossible to propose a definite format without knowing the requirement of the analysis facility. It has been suggested than an I.B.M. 9 track on 1/2 inch tape standard be adopted, but this is an all digital standard with no facility for speech. One solution could be to deviate slightly from the standard and use an edge track for speech. Standard I.B.M. Format has the advantage of compatibility with many existing systems, but it is worth noting other possibilities, in particular serial recording since this method gives at least two advantages.

(a) Highest packing density without the complication of skew correction, etc.

(b) The possibility of using the extremely cheap domestic type of cassette tape recorder.

Against these advantages there is the electronic complication of data storage and serialising and there is little possibility of getting all the data on a simple cassette recorder.

The basic clock would take the form of a crystal oscillator

typically running at about IMHz or higher and with an accuracy of about 0.01%. Logic divide circuits would provide pulses at appropriate times as shown in Fig. 21. A matrix facility would provide access to the logic and provide a means of setting the system to select the required channels and sampling rates and provide a framing and address code. It is thought that 1 second frames would be convenient, thus giving an elapsed time signal in seconds. Also a synchronising pulse for a camera could be provided at this same interval.

6.12 Tape recorder

It is anticipated that a 1/2 inch magnetic tape recorder would be used, possibly similar to the Fell Recorder being used in the

Environmental Pack. The operation would be continuous mode, rather than incremental, this would allow almost any analogue or digital recorder to be used with a less limitation on data rate, - incremental machines seem to have an upper limit of about 1000 bytes/s and this specification requires about 2800 bytes/s - and no digital control would be required. Also, this mode is essential if speech is to be recorded. If the I.B.M. 9 track standard is adopted, the data would most likely record at 2 bytes/word and taking 556 bits/in as the maximum safe packing density for airborne use, the requirement for 12 channels at 100 samples/s gives a minimum speed of approximately 4.35 in/s and allowing for

address and timing data a speed of 5 in/s would seem appropriate. This appears to be one of the standard digital recording speeds. From this, the required tape length for 1 hour durations is 1500 ft.

The precise method of recording the digital data cannot be decided without some knowledge of the Tape Recorder and its heads, but one of the Non Return to Zero (NRZ) methods would be used and although skew correction should not be necessary at this packing density and using a

good digital recorder, it may be worth considering to allow greater freedom in recorder choice. If skew correction is required, a bi-phase recording method is necessary.

Head drive amplifiers and a speech channel have been considered as part of the electronic pack, thus the tape recorder would require no signal electronics.

6.13 Power supplies

For the equipment discussed above, the following types of supply are required:

(a) A stable supply of about ±14 volts d.c. for the accelerometers, signal conditioning, etc.

(b) A supply of 5 volts d.c. for the logic circuits. This should contain 'crowbar' over-voltage protection.

(c) Stable 400 Hz supplies for the rate gyros and a.c. pick-offs.

(d) A 400 Hz supply of greater power than (c), but not necessarily so stable, this being appropriate to the requirement of the vertical gyro.

(e) Tape recorder supplies.

(f) Any specific requirements external to this pack.

Since the pack would be self contained there should be minimal problems with electrical interference and earth loops. Therefore the Common Mode Rejection (CMR) of the signal conditioning circuitry should be sufficient to allow commoned supplies to be used for all channels. It is assumed that the 28 volt aircraft supply would be used and it is preferable to isolate the instrumentation from this because in most cases the tape recorder would be connected directly to this supply and thus the instrumentation would be grounded at this point, another ground would create the possibility of an earth loop.

A good method would be to generate the ±14 volts supplies with one of the proprietary d.c. to d.c. converters, e.g. the Burr-Brown modules. The 5 volt supply could either be generated from this or from another d.c. to d.c. converter. The stable 400 Hz supply could be generated from the standard Smiths units and the 400 Hz higher power supply could be a square wave d.c. to a.c. inverter, either made or purchased. The Tape Recorder is assumed to be self contained and any other requirements would be considered as required.

6.14 Construction

As with the Environmental Pack, a three unit construction is

envisaged, namely, transducer and electronic packs and the Tape Recorder. The electronic pack would contain all the signal conditioning and logic, plus a programming matrix and facilities for connection of external transducers and the Test Set.

Most of the electronics would be hard wired, including the matrix, the only plug connections being for inputs and outputs, power supplies, control, and the Test Set connection. In chapter 6.5, it has been assumed that external inputs would be conditioned to the filter input Signal level.

A simple control box is envisaged, enabling the tape recorder to be switched on or off during flight, together with the insertion of in-flight calibration signals and control of the vertical gyro fast erect system.

7 CONCLUSION

In recent years two new techniques of measurement were introduced in connection with gathering infoimation about stability and control

characteristics of an aircraft. The first is connected with the measurement of responses of an aircraft to atmospheric turbulence, the second with the evaluation of aerodynamic derivatives of an aircraft from flight data. The second technique has introduced new sophisticated methods for flight data analysis and has brought more stringent requirements on performance

characteristics of an instrumentation system, mainly its accuracy. To enable the installation of measurement instruments on board an aircraft, their checking and calibration within the shortest possible time the idea of an instrumentation pack was developed. The proposed pack is

considered as an on-board digital recording system with a low frequency range. It will be capable of recording twelve measured quantities, voice and

reference time signal.

The pack ideally should consist of three basic units, transducer pack, electronic pack including power supply and recorder pack. These units may be mounted separately or together in one unit.

The transducer pack would contain three linear accelerometers, three rate gyros and eventually a vertical gyro. The electronic pack would mainly be composed of signal conditioning units. It also should enable the

connection to external transducers and a synchronised camera. The electronic pack would have the special arrangement for the ground and in-flight

calibration as well as for the connection of a programme facility. The recorder pack would be formed by digital tape recorder.

The basic performance characteristics of individual channels in the proposed system were selected according to examples of the measurements for the determination of aerodynamic derivatives of an aircraft. But the use of the pack for the measurement of basic stability and control characteristics and responses to atmospheric turbulence was also considered. The proposed maximiun acceptable inaccuracies for the pack itself are slightly lower -about ±1% of full range - than actually required for transient-response data from which the accurate determination of aerodynamic derivatives is possible. The reason for this is the possibility of combining the accurate internal transducers with less accurate external ones used for the measurement of input variables and quantities defining flight conditions. With this combination of more and less accurate transducers the measurement of useful data for the evaluation of aerodynamic derivatives will be still possible provided that the resulting inaccuracy in control derivatives will be determined and its

8 RECON^IENDATION

That a transducer pack and an electronic pack be designed, built and flight-tested in a similar manner to the Environmental Pack and with similar construction, possibly to the extent of using the Fell Tape Recorder. The transducer pack should incorporate three accelerometers and three rate gyros, possibly with a facility for including a vertical gyro, but this should be considered as a separate component. The electronic pack, although having the facility of being joined to the transducer pack, should be self-contained and thus be a potential universal data conditioning and digitising unit capable of use with virtually any signal input and driving virtually any tape recorder.

It is intended to use the proposed self-contained instrumentation pack mainly for the measurement of transient-response manoeuvres. This approach in flight testing demands, however, the further development of procedures now used for the evaluation of aerodynamic derivatives of an aircraft from

flight data. The extension should be done in two directions, the evaluation of all present basic stability and control characteristics or their equivalents from dynamic measurements only and the application of the identification

methods for the evaluation of aerodynamic characteristics in those cases where the mathematical models are nonlinear and/or with time varying

coefficients. These new techniques might allow a considerable reduction in time necessary for testing and would provide more valuable information than the techniques used at present.

APPENDIX A

A SURVEY OF TECHNIQUES USED FOR THE EVALUATION OF AERODYNAMIC DERIVATIVES

FROM FLIGHT DATA

The evaluation of aerodynamic derivatives of an aircraft from measured data are considered from the aspect of parameter identification. This approach enables the general analysis of the given problem and the use of methods of statistical inference. Two groups of numerical methods which can be used for the parameter identification of a system are described. The first group is usually called the equation-of-motion or the equation error method and represents the application of multiple regression analysis to the

calculation of the coefficients in the differential equations governing the analysed system. The second group is called the response-curve-fitting method or the output error method. These are based on the fitting of measured output data or frequency response curves.

Explanation of the background of the methods in both groups will be demonstrated on a linear deterministic time-invariant system with multiple input and output. The methods will be then applied on the estimation of coefficients in equations of the longitudinal short-period motion of an aircraft. Extension on the lateral or complete longitudinal motion should be obvious.

Al Equation of Motion methods

Parameter identification will be based on the linear matrix differential equation of a given system

X = Ax + Bu , (A.l) where X is (nxl) state vector u is (mxl) input vector , A is (nxn) state matrix B is (nxm) input matrix

Initial conditions are equal to zero. Some or all of the elements of A and B are unknown.

From the experiment the measured values of input variables and measured values of the state and its derivatives were obtained. For analysis it will be assumed that only components of state vector derivatives are corrupted by noise.

Therefore:

Xg = X, Ug = u,

x„ = x + n (A. 2) E 2

where: E is measured quantity

The solution of a problem is as follows: Given Xp, Xg and u determine the unknown elements in the matrices A and B. The solution will use the minimalisation of the cost function which involves only the r-th component of the noise n„. 2 T Thus: Jj - I n2 (t)dt (A.3) where: n = x „ - A x - , - B u 2r Er r E r and = x_ - B (A.4) Er

(?)

6 represents a vector of unknown coefficients

A ,B are the corresponding rows in matrix A and B respectively.

Substituting (A.4) into (A.3) and setting the gradient of (A.3) with respect to 6 equal to zero, the solution for 6 will be:

[1 (?) ö V " K C^)

-T 0

Using the cost function J^ means the minimalisation of the equation errors. The basis for the equation-of-motion method is now established. This method represents the task of multivariable linear regression. The analytical forms of the regression relationships are given by the equations of motion in the original form (A.l) or in transformed form, for example in the frequency domain with the independent variable cu.

Where for practical computation, the integral representation of the cost function J, is replaced by its discrete version

k J. = S n2

j = l

1 ^ __. 2r

^'j^ = j!i [% ^'j^" ^ ^E (^j5 - \ " f^j^]' ^^-'^

and where k is the number of samples taken from measured time histories, then the basic assumptions for the least squares estimates of unknown coefficients can be formulated as:

E(y) = x'^B

y

(A. 7)

In these equations:

y = x„ is (k X 1) vector of measured values x (t.) X is (p X k) matrix of other known quantities

form:

g is (p X 1) vector of unknowns

I is (k X k) identity matrix

Z is (k X k) covariance matrix y

0^ is the variance of random variables

The normal equations for the estimates of unknown coefficients have the T"

XX 3 = Xy (A.8)

and the covariance matrix of B is found from the expression:

Eg = a2 (XX^)"^ (A.9)

Finally the estimate of variance is given by: E.V.2

s^ = - ^ (A. 10)

where: v. is the residuals calculated from:

V = y - x'^e (A. 11)

Experiments show there is a noise in the measured state variables as well, this causes the bias in the parameter estimates. This bias can be

reduced by using the iterative algorithm developed in Ref. 9 but not completely removed even though the mean value of the random noise is zero.

So far it has been considered that all the state variables and their derivatives are known from measurements taken. However, the measurement of the derivatives of the state variables is a difficult problem. Numerical differentiation of the state variables is straightforward but tends to decrease the signal to noise ratio. Several ways have been introduced to overcome these difficulties, of these, three have become the most common, as follows:

(a) Integration of matrix equation (A.l) governing the system allows the use of directly measured state variables and integrated state and control variables. Many calculations prove that numerical integration of experimental curves gives sufficiently accurate data for further processing is the corresponding time histories are not considered in excessively long time intervals.

(b) Shinbrot in Ref. 6 and Loeb and Cahen in Ref. 7 recommend the multiplication of the equations of motion by the method function

fy and the integration of equations between two limits, zero and T. Errors defined by (A.6) are then changed to:

n^ (v) =

2r X „ (t) f, (t)dt - a^ x (t) f (t)dt rE V ^ i j IE V 0

I