* 

NASA Technical Memorandum 89059 

ESTIMATION OF BIAS ERRORS IN MEASURED 

AIRPLANE RESPONSES USING MAXIMUM 

LIKELIHOOD METHOD 

(NASA-TM-89059) ESTIBATICN OE LIAS EliROHS 

Ibl HEBSURXD AIR€LANE RESPChrSES CSING BAXIMUM 

IZKELIHCOD EETHCC [NASA) 34 F CSCL Olc 

N87- 15977 

Uuclas 

G3/08 43367 

VLADISLAV KLEIN 

DAN R. MORGAN 

JANUARY 1987 

National Aeronautics and 

Space Administration 

Langley Rsrssrdr center 

Hampton. Virginia 23665 
SUMMARY 

A procedure for compatibility check of measured airplane responses is 

presented. This procedure includes estimation of bias errors in the measured 

data in terms of constant measurement biases and scale factors, and a comparison of reconstructed responses with those measured. The model relating airplane states and outputs is based on six-degree-of-freedom kinematic 

equations. In these equations the input variables are replaced by their 

measured values which are assumed to be without random errors. 

A maximum likelihood method is used as the estimation technique. The 

resulting algorithm is verified with simulated data and data from flight 

testing. The results from simulated data show that the increased number of 

unknown parameters and the correlation among them can degrade the accuracy of 

the estimates; however, moderate measurement noise level in the input variables has only a small effect on the estimates. The maximum likelihood estimates from flight data were compared with those obtained by using an extended 

Kalman filter and a nonlinear fixed-interval smoother. This comparison showed 

no major differences in results of all three techniques. 

i 
SYMBOLS AND ABBREVIATIONS 

A sensitivity matrix 

ax, ayt a, 

E 

9 

h 

h( 1 

J 

M 

M~~~ 

mij 

N 

n 

"m 

n 

PI q, r 

R 

P 

s( 1 

t 

2 longitudinal, lateral, and vertical accelerations, m/sec 

constant bias error in variable y 

expected value 

acceleration due to gravity, m/sec 2 

altitude, m 

nonlinear output vector used to represent measurement 

sys tem 

cost function 

information matrix 

modified information matrix, see Eq. (17) 

element of inverse information matrix 

number of data points 

measurement - noise vector 

number of measured output variables 

number of unknown parameters 

roll, pitch, and yaw velocities, rad/sec or deg/sec 

measurement - noise covariance matrix 

standard error estimate 

time 

longi tudina 1 , latera 1 , and ver tica 1 airspeed components , m/sec 

true airspeed, m/sec 

state vector 

linear position coordinates of aircraft, m 

ii 
2 

a' 'ar a 

X 

Y 

W 

2 

a 

B 

&i j 

€1' 2 E 

rl 

8 

0 

xY 

V 

5 

2 

0 

+ 

Subscripts: 

b 

E 

N 

0 

R 

V 

position coordinates of a wind vane with respect to 

aircraft center of gravity, m 

V 

position coordinates of f3 wind vane with respect to 

aircraft center of gravity, m 

V 

output vector 

transformation matrix 

measurement vector 

angle of attack, rad or deg 

sideslip angle, rad or deg 

Knonecker delta 

convergence criteria, see Eq. (16) 

input vector 

vector of unknown parameters 

pitch angle, rad or deg 

scale factor error of variable y 

residual vector 

process - noise vector 

variance 

roll angle, rad or deg 

body axes 

measured quantity 

nominal value 

initial value 

uncorrected for bias error 

wind vane 

ii i 
Matrix Exponents : 

T transpose matrix 

-1 inverse matrix 

Mathematical Notation: 

0 over symbols denotes derivative with respect to time 

A over symbol denotes estimated value 

A incremental value 

Abbreviations: 

EKF extended Kalman filter 

ML maximum likelihood 

NFIS nonlinear fixed -interva 1 smoother 

iv 
INTRODUCTION 

For more than fifteen years there have been numerous attempts to estimate 

airplane states from measured flight data. This estimation is possible 

because of well known kinematic equations relating the airplane states and 

output variables. In many cases the estimated states have been used for data 

compatibility checks, that is, for comparison of measured and predicted 

response variables of an airplane. Because the measured data are corrupted by 

random and systematic errors, it was recognized that the state estimates 

should be combined with the estimation of unknown biases (parameters) to 

obtain satisfactory results. Various methods for state and parameter estimation were applied. They can be divided into two groups: 

1. Methods for estimating separately unknown parameters and states 

(refs. 1 to 6). For parameter estimation the maximum likelihood or nonlinear 

least squares techniques are used. For state estimation the system is assumed 

to be deterministic. That is, the state variables are simply obtained by the 

integration of model equations. 

2. Methods for estimating states and parameters simultaneously using an 

extended Kalman filter (refs. 7 and 101, or nonlinear smoother (ref. 11). A 

review of various approaches to the problem of airplane state estimation is 

presented in reference 12. 

The purpose of this report is (1 1 to develop a maximum likelihood algorithm applicable to general motion of an airplane; (2) to compile an efficient 

computer program based on this algorithm; and (3) to verify both the algorithm 

and program on simulated and real flight data. The report starts with the 

formulation of model equations and estimation techniques. Then several 

examples are presented. When the real flight data are analyzed the maximum 

likelihood results are compared with those obtained by using an extended 

Kalman filter and nonlinear smoother. 

MODEL EQUATIONS 

The mathematical model used for the data compatibility check is described 

by three sets of kinematic equations with the state variables consisting of 

three linear velocities u, v, and w; three Euler angles +, 0, and J, ; and 

three linear positions xb, yb, and zb. 

are the linear accelerations ax, a 

and r. The form of the kinematic (state) equations considered can be found in 

various references (see, e.g. reference 9). 

The input variables in these equations 

and a, and angular velocities p, q, Y’ 

The following variables are measured: 

1. 

2. The airspeed VI two incidence angles fi and av , three Euler 

The inputs to the system a,, ay, az, PI q, and r. 

V angles $,e , and J, and altitude h= ‘zb. 
These variables represent the output of the system. The measured variables z are corrupted by systematic and random errors. It is assumed that 

each of #em can be expressed as 

- 

U 

V 

W 

h 

- 

where y is the true value of the output, X is the unknown scale factor error, 

b is the constant bias error, and n is the measurement noise. It is further 

assumed #at the scale factor error is equal to zero for all the input variables. This assumption will simplify an estimation procedure for remaining 

scale factor and bias errors. 

Y 

Y Y 

- . 

U 

V 

W 

; 

- 

The system of state equations is simplified by deleting #e equations for 

Xb and yb. 

tions by their measured values results in the following set of state equations 

Then, replacing the input variables in the remaining state equa7- 

- g sin0 0 w-vo 

0 u0 

v -u 0 0 

0 0000 

+ + 

d -- 

- 

P n 

9 n 

nr 

0 d 

r- . 

Q 

e= 

. 

iJ 

t- 

-sin+ 

1 sin+ tan8 

0 cos+ 

sin (p 

O cos0 

(2b) 

2 
The output equations take the form 

222 = (1 + Xv) JU + v + w 

vR + bv 

- 'R,EXa+ 'R,EYa 

vR- - rR,EYa+ 'R,EZa 

a - (1 + xa) ta;' [, 1 + ba 

xh' h = (1 + 

I$ = (1 + X $ + b 

f3=(1+X)0+be 

R 

R 0 4 

R 0 

R J, 

$= (1 + X J, 

In Eq. (3) Bv and av are the incidence angles measured by the vane. 

sideslip angle $* is measured in the Oxbyb plane, whereas f3 is measured 

between the wind vector and its projecton on the Oxbzb plane. 

angles are therefore defined as 

The 

These two 

-1 v fi* = tan - 

-1 v fi = sin - v 

U 

Sometimes the measured sideslip angle has been corrected for the c.q. offset 

prior to the compatibility check. In such case fi can be computed from the 

simplified equation as indicated above. In the output equations for hR 

and 11, the constant bias terms are omitted because the reference values 

for J, and h can be selected arbitrarily. Further, in as. (2) and (31, R 

indicates the variable uncorrected for bias errors and index E measured variable. Finally, x ya ,z and x 

of a and f3 vane with respect to airplane center of gravity. 

R 

are the position coordinates a' a B' yB' zB 

3 
PROBLEM STATEMENT 

The general form of state equations for a given system can be written as 

(5) 0 %(t) = 5 [x(t), rl(t), o,] + g [x(t,] E(t), x(0) = x 

and the discrete form of the measured equations as 

where x, rl, and z are the state, input and measurement vector 

re spec tively , 0, 

is the vector of unknown initial conditions, 5 and n are the process and 

measurement noise vectors respectively, and N is the number of data points. 

The compatibility check can be now formulated as an identification problem 

which involves the estimation of state and output variables, unknown parameters 0 and xo, and covariance matrices of 5 and n, from measured data. 

is the vector of unknown biases and scale factor errors, xo 

1 

The postulated model equations represent a nonlinear stochastic system 

with state-dependent process noise and with nonlinear output equations with an 

additive measurement noise. The state estimation in this case would be an 

extremely difficult problem. The separate estimation of unknown parameters 

would be equally complicated because of the resultinq form of the sensitivity 

equations. For these reasons, possible simplification of the problem will be 

considered. 

ESTIMATION METHOD 

The state and parameter estimation problem outlined above can be reduced 

to parameter estimation only by neglecting the process noise altogether. The 

state estimation is thus replaced by the integration of the state equations 

with the estimated values for initial conditions and bias errors. For the 

parameter estimation the maximum likelihood (ML) method is applied. The 

measurement noise in equation (6) is assumed to be zero-mean, uncorrelated, 

and gaussian, i.e., 

where the symbol 6 is the Kronecker delta. 

i,j 

The ML method finds a set of parameters by minimizing the log - 

likelihood function (see, e.g. reference 14). 

4 
where 

0 = to1 x0lT 

a* is the variance of the measurement noise in output variables and N is 

Y 

number of data points. Minimizing (7) for parameters in R gives the estimate 

of measurement-noise covariance matrix as 

T N 

i=l 

A1 

R=- C u(i) v (i) N 

The estimates of the remaining unknown parameters are given by the root of the 

equation 

A 

for R replaced by R. This root can be found by modified Newton-Raphson iteration technique (ref. 14). 

It is well known that under the above mentioned assumptions the final 

estimates of unknown parameters are consistent, asymptotically unbiased, and 

asymptotically efficient with 

In this expression AT is the transpose of the sensitivity matrix with the 

elements 

avk k = i12,....n 

ml kfi aoQ, I, = 1,2,....n 

AT =- 

P 

where nm and n P 

parameters in 0 respectively. 

are the numbers of measured output variables and unknown 

COMPUTING ALGORITHM 

The block diagram of computing procedure for the ML estimation of unknown 

parameters is presented in Figure 1. The measured data are considered in the 

form of digitized time histories of input and output variables with sampling 

interval At. The state and output equations are given by Eqs. (2) and (3) 

respectively with the unknown biases and scale factors 

5 
T 

x0= [~~t~~t~~th~~ eo# $0] 

For the integration of state equations and computing output time histories the 

starting values of unknown parameters must be specified. The biases and scale 

factors are usually set equal to zero. The starting value of initial conditions are obtained from measured variables at t=O as 

(0) cos& E(o) %,E 

u = v (0) cos 0 R,E I 

v = v (0) sinBRtE(0) 0 R,E 

ho= h (0) RIE 

+o= $p, I E( 0 

Using the modified Newton-Raphson iterative technique the unknown parameters obtained from 

A A o=o+Ao 

N (13) 

where 

and 

0 are the starting N values 

A T ^-1 -' T 

A0 = [ C Ai R Ai] C Ai R vi 

i=l i=l 

The transpose of the sensitivity matrix A is given in Table I, where the 

crosses indicate sensitivities ay /a0 = -av /a0 computed by a numerical 

method of reference 14. The remaining sensitivities in Table I are equal to a 

kR kR 

6 
known constant or zero, or are formed by the computed output variqbles. The 

estimates of elements in the measurement noise covariance matrix R are 

obtained from Eq. (9) and the residuals from Eq. (8). 

For t=O the output equations for 6 and 0 give R R 

6' 

e 

It means that for X 

and eo and b 

being either estimated or computed from IQ. (15). 

and X a linear relationship exists between +o and b (4 0 

The algorithm therefore provides two options for b and b e* 6 

The iteration process is completed when certain StOQping criterion is 

met. In this report two criteria proposed in reference 13 are adopted, i.e., 

where izl and c2 are specified, for example E,= c2 = .001. 

should be met simultaneously. When the iteration is completed the correlation 

matrix of unknown parameters is computed from 

These two criteria 

where 

-1 -1 

MMoD= 

WM W (16) 

The residuals can be examined using their time histories , autocorrelation 

functions and power spectral densities (ref. 9). 

EXAMPLES USING SIMULATED DATA 

The method developed in this study was applied to simulated data to check 

the accuracy of +&e computing algorithm and estimated parameters. !lbo sets of 

simulated data, one representing the longitudinal and the other representing 

the lateral motion of an airplane, were used. The time histories of the input 

and output variables are shown in Figures 2 and 3. The sampling interval for 

these time histories was At= 0.05 sec. The measurement noise with standard 

errors given in Table I1 was added to the simulated data. 

7 
The results of three cases using the longitudinal data with measurement 

noise only in the output variables are presented in Table 111. In Case 1 the 

initial conditions were fixed on true values, in Case 2 on values corrupted by 

measurement noise. In Case 3 the initial conditions were estimated. The 

degradation in the accuracy of the estimated parameters from Case 1 to Cases 2 

and 3 is apparent. It is probably caused by incorrect values of initial 

conditions (Case 21, by an increased number of unknown parameters (Case 3) and 

high correlation (greater than 0.9) between some parameters. The expected 

high correlation between parameters b and eo (see Eq. 15) did not materialize. e 

A similar approach was adopted in the next three cases, where both the 

input and output variables were corrupted by measurement noise and the number 

of unknown parameters was increased. The results in Table IV do not show any 

substantial deterioration in the accuracy of the estimated parameters when 

compared with the previous results in Case 2 and 3. 

In Table V two sets of results from the analysis of lateral data are 

given. 

variables 0, p, r, and 41 . The different values of bias errors did not 

change the differences between the true and mean values of estimated parameters but changed the standard errors of the parameters and the correlation 

between them. The accuracy was worse than that obtained from the longitudinal 

data. The estimation was then repeated with the initial conditions as the 

unknown parameters. This attempt, however , failed because of the estimation 

procedure divergence. The analysis of lateral data indicated a possibility of 

identifiability problems with this type OP data. To avoid that, the design of 

an optimal maneuver for more accurate estimates and with less sensitivity to 

the number of unknown parameters should be investiqated. 

These sets differ in the two values of simulated bias errors in the 

ANALYSIS OF FLIGHT DATA 

Three sets of measured flight data were analyzed. Two of them represented a longitudinal motion of an airplane, the last set was obtained from a 

combined maneuver with predominantly lateral motion. The sampling interval 

for all data was 

and e in the first run are presented in Fiqure 4. The resultinq es imates 

which include the parameter mean values , their standard errors (Cramer-Rao 

lower bound) and standard errors of the measurement noise in the output variables are summarized in Tahle VI. In Case 1 the vector of unknown parameters 

was postulated as 

2, At= 0.05 sec. The measured output variables VR, 

R 

As can be seen from the results, three pairs of estimated parameters are 

highly correlated, and the parameters ba, and eo, have large standard errors. 

As the next step, therefore, the parameters bax, wo and X were fixed on their 

estimated values. The estimation of the remaining parameters was then repeated V 

8 
in Case 2. The new results indicate no change in the mean values but lower 

standard errors of the estimates. The predicted time histories of the output 

variables are compared with those measured in Figure 4. The agreement is very 

good in all variables plotted. 

The time histories in the second run had similar form as those in the 

first one. The estimated parameters are contained in the last two columns of 

Table VI. All parameters from both runs agree well with the exception of the 

parameter %. This disagreement could be caused by the elimination of \ in 

the vector of unknown parameters in the second run. As in Run 1 the results 

also exhibit high correlation between some parameters. 

In order to further validate the results, the ML estimates from Run 1 

were compared in Table VI1 with those obtained by a nonlinear-fixed-internal 

smoothing (NFIS) technique (ref. 12) and by an extended Kalman filter (EKF) 

(ref. 9). For the ML and NFIS estimation the initial values of unknown bias 

and scale factor errors were set equal to zero whereas for the EKF these 

values were made equal to the ML estimates because of the slow convergence of 

the filter. The estimates from all the three techniques aqree well. The main 

differences are seen only in parameters 

estimated with poor accuracy in all cases. The reason for the disagreement in 

bV could be due to insufficient excitation of the airspeed during the airplane 

motion. 

and eo. The parameter eo was 

Data from the third maneuver were analyzed assuminq only the. bias errors 

in variables ay, aZ, p, q, r, 

of the less important terms were eliminated. As a result of that, the vector 

of unknown parameters was postulated as 

B, I$ and 8. After reviewing the estimates, some 

with b and b computed from Eq. (15). The measured and predicted time 

histories of output variables are plotted in Figure 5. Then, as in the previous case, the ML estimates were compared with those using NFIS and EKF 

techniques. The results are given in Table VIII. The aqreement between the 

ML and EKF estimates is good. This could be, however, due to the use of ML 

estimates as starting values for the MF technique. Therefore, more thorough 

checks should be made for better assessment of results from bath methods. The 

main differences between ML and NFIS techniques are in the parameters b and 

initial conditions V~,I$~ and eo. 

accuracy of these estimates and different values of the remaining parameters. 

+ 8 

6' 

These differences can be caused by poor 

9 
CONCLUDING REMARKS 

A maximum likelihood method was developed for the estimation of initial 

conditions and bias errors in measured airplane responses. In the development 

it was assumed that the input variables to the system represented by linear 

accelerations and angular velocities are measured without random errors. The 

model relating airplane state, input and output variables is based on sixdegree-of -freedom kinematic equations and on output equations specifying the 

measured variables. 

The resulting technique was first applied to a limited number of simulated data runs to check the accuracy of the computing algorithm and estimated 

parameters. It was demonstrated that the increased number of unknown parameters and the correlation among them can deqrade the accuracy of the estimates. At the same time it was observed that a moderate noise in measured 

inputs has only a small effect on the accuracy of the results. The lateral 

maneuver analyzed provided less accurate parameter estimates than the longitudinal one. The increased number of unknowns in the lateral case resulted in 

a divergence of the estimation procedure. The results from simulated data 

indicate a need for a design of an optimal maneuver for more accurate estimates and with less sensitivity to the number of unknown parameters. 

The maximum likelihood method developed for this report was also applied 

to the analysis of real fliqht data. Two longitudinal maneuvers similar in 

form were analyzed first. The resulting parameter estimates from both maneuvers were in good aqreement. Then the maximum likelihood estimates were 

compared with those obtained by a nonlinear-fixed-interval smoother and an 

extended Kalman filter. The comparison of the three techniques showed no main 

differences in the results. Similar conclusions were obtained from the analysis of a lateral maneuver with a strong lonqitudinal couplinq. All the 

comparisons served as a verification of the maximum likelihood technique 

presented in the report. 

10 
REFERENCES 

1. Wingrove, R. C., "Quasi-Linearization Technique for Estimating 

Aircraft States from Flight Data," AIAA Paper 72-965, Palo Alto, CA, 

1972. (Also Journal of Aircraft, May 1973, pp. 303-307.) 

2. Mulder, J. A., "Estimation of the Aircraft State in Nonsteady 

Flight," AGARD CP-172, Chap. 19, November 1974. 

3. Keskar, D. A. and Klein, V., "Determination of Instrumentation 

Errors from Measured Data Using Maximum Likelihood Method," AIAA 

Paper 80-1602, Danvers, MA, 1980. 

4. Feik, R. A., "A Maximum Likelihood Program for Non-Linear System 

Identification With Application to Aircraft Fliqht Data 

Compatibility Checking," ARL Aero Note 411, July 1982. 

5. Taylor, L. W., Jr., "Applications of Parameter Estimation in the 

Study of Spinning Airplanes," AIAA Paper 82-1309, San Diego, CAI 

1982. 

6. Reid, G. E. A., "Validation of Kinematic Compatibility of Flight 

Data Usinq Parameter Estimation Methodology, RAE Technical Report 

81020, 1981. 

7. Eulrich, B. J. and Weingarten, N. D., "Identification and Correlatio 

of the F-4E Stall/Poststall Aerodynamic Stability and Control 

Characteristics from Existing Test Data," Reprot AK-51 26-F-1 , Calspan Corp., Buffalo, NY, November 1973. (Also AFFDL-TR-73-125.) 

8. Jonkers, H. L., "Application of the Kalman Filter to Flisht Path 

Reconstruction from Flight Test Data Including Estimation of 

Instrumental Bias Error Corrections, 'I Ph.D. Thesis , Delft 

Technological U. , the Netherlands, 1976. 

9. Klein, V. and Schiess, J. R., "Compatibility Check of Measured 

Aircraft Responses Using Kinematic Equation and Extended Kalman 

Filter," NASA TN D-8514, 1977. 

10. Stalford, H. L. and Ramachandran, S., "Application of the 

Estimation-Before-Modeling (EBM) System Identification Method to the 

Yigh Angle of Attack/Sideslip Flight of T-2C Jest Trainer Aircraft, I' 

Vol. 11, NADC-76097-30, June 1978. 

11. Evans , R. J.; Goodwin, G. C.; Feik, R. A.; Martin, C. and LozanoLeal, R., "Aircraft Fliqht Data Ccmpatihility Checking Usinq Maximum 

Likelihood and Extended Kalman Filter Estimation, 'I 7th IFAC/IFORS 

Symposium on Identification and System Parameter Estimation, July 3- 

7, 1985, York, UK. 

12. Bach, R. E., ,Jr., "A Variational Technique for Smoothing Flight-Test 

and Accident Data, I' AIAA Paper 80-1601 , Danvers, Mass. , 1982. (Also 

Journal of Aircraft, July 1982. 

11 
13. Bach, R. E., Jr. and Wingrove, R. C., "Application of State 

Estimation in Aircraft Flight Data Analysis, 'I AIAA Paper 83-2087, 

Gatlinburg, TN, 1983. 

14. Murphy, P. C., "An Algorithm €or Efficient Maximum Likelihood 

Estimation and Confidence Interval Determination in NonLinear 

Estimation Problems , 'I Dr . Sci. Thesis , The George Washing ton 

University, Washington, DC, 1985. 
- 

bax 

baz 

bP 

bs 

br 

bV 

B 

a 

4 

b 

b 

b 

be 

aV 

'a 

+-I 

ae 

3r a 

uO 

"0 

wO 

h0 

$0 

60 

TABLE I. - TRANSPOSE OF SENSITIVITY MATRIX 

VR %R a vR hR +R OR *R 

X 

X 

X 

X 

X 

X 

1 

0 

0 

0 

0 

V 

0 

0 

0 

0 

0 

0 

X 

X 

X 

0 

X 

X 

0 

X 

X 

X 

X 

X 

X 

0 

1 

0 

0 

0 

0 

BV 

0 

0 

0 

0 

0 

X 

X 

X 

0 

X 

X 

0 

X 

X 

X 

X 

X 

X 

0 

0 

1 

0 

0 

0 

0 

aV 

0 

0 

0 

0 

X 

X 

X 

0 

X 

X 

0 

X 

X 

X 

X 

X 

X 

0 

0 

0 

0 

0 

0 

0 

0 

h 

0 

0 

0 

X 

X 

X 

++-I 

X 

X 

0 

0 

0 

0 

X 

X 

X 

0 

0 

0 

1 

0 

0 

0 

0 

0 

4 

0 

0 

0 

0 

0 

0 

1+X 4 X 

0 

0 

0 

0 

X 

X 

X 

0 

0 

0 

0 

1 

0 

0 

0 

0 

0 

e 

0 

0 

0 

0 

0 

X 

l+he 

0 

0 

0 

0 

X 

X 

X 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

3r 

0 

0 

0 

0 

X 

X 

3r 1+X 

x indicate sensitivities computed by a numerical method 

13 
TABLE 11. - STANDARD ERRORS OF 

MEASUREMENT NOISE 

Va r i ab 1 e 

ax, m/sec 2 

ay, m/secz 

a,, m/sec 

p, rad/sec 

q, rad/sec 

r, rad/sec 

V, m/sec 

2 

13, rad 

a, rad 

$, rad 

8, rad 

q, rad 

~~ 

Standard Error 

of Measurement 

Noise of Variable 

0.1 

0.1 

0.1 

0.002 

0.002 

0.002 

1 .o 

0.002 

0.002 

0.002 

0.002 

0.002 

14 
6g 

OVI 

8% 

EW 

EX 

EH 

WE 

cn 

IW 

*E 

HW 

HW 

nw 

X w4 

"2 5L 

E 

0 m m m .. - mmm0 

v) 

00hlou)cuo \ooooocDcuo 

rJ0000000 v 

. .. 

aa a a 

m 

a 

rJ mw 

m 

a Q ascp 

ala* 

m 0 

-3 

om 

m 

c.? 

0 

-0 

-0 

0 

49 

.$ ill 

E 

a LtLt 

ala 

ua 

al 

EU 

aal 

Ltx 

Q -4 PIE 

49 

15 
In 

03 

0 m 

0 om cnt-0 w03mrn 

COY0 00N00P0 

ooo~oooov~o oooNoooqooo . . .. .. . . 

5 a (d 

-9 5 

03 

m m v Qrn 

omv0 

t-mm ooooNvo *rno ~oo~oo~-~N

. . .. . rnQ 

In 

m 

03 

00 

m 

0 t-Q rnQrnQ 

00-0 NmO 

Q-0 

oooa ,00800 009. .. ..* 

bl 

8 

16 
m 

cu 

0 

. 

m 

0 

0 

c) 

0 

0 

r 

0 

0 

0 

0 

0 

. 

P 

10 

0 

0 

. 

IN 

0 

0 

0 

. 

m 

0 

0 

c 

. 

N 

c 

IO 

0 

0 

N 

0 

. 

a 

0 

m 

cn 

0 

0 

- 

. 0 

0 

e 

. u) 

0 

? 

. cu 

0 

c 

. 

m 

mm 

0- m cnoo-wm voofioc 

000000 

000000 ~ 

. .. 

m a a 

r. 

om - 

om u) 

0cucCOrcu 

N00000 

c0000~ 

. . .. . . II 

cumcum 

0000 

rOOOOc . .e . . 000000 

E 

0 

..-I 

JJ 

a 

I4 

al 

k 

k 

0 

u 

,c 

tJ 

c 

-4 

5 

..-I 

3 

k 

0) 

JJ 

al 

E 

a 

k 

a 

PI 

a 

17 
1 

Parameter 

TABLE VI. - ESTIMATES OF PARAMETERS 

FLIGHT DATA OF TWO DIFFERENT RUNS 

FROM LONGITUDINAZl 

RUN 1 

1 

Case 1 

.004 

- 1240 

1.792 

-1.2 

2.6 

-12.7 

.018 

-e073 

35.5 

6.0 

-2. 

.260 

.280 

.748 

a 

a 

a 

b 

a 

t 

t 

.0065 

.00040 

.0019 

.30 

.19 

.0077 

.0027 

.10 

.13 

.37 

Case 2 

A 

8 

.004 

- 1 240 

1.792 

-1 -24 

2.631 

-12.72 

.018 

- -073 

35.5 

6.0 

-.19 

.260 

.280 

.74a 

.00021 

.0018 

.098 

.0036 

.0026 

.10 

.025 

A 

8 

RUN 2 

- g 1257 

1.827 

-.5 

1 e5 

-12.87 

-e081 

37 -2 

6.5 

7.60 

.338 

.331 

1.19 

9 00022 

.00060 

.16 

.21 

.0034 

.16 

.12 

.025 

a Parameters with high correlation 

Computed from eo and 8 

Fixed parameter 

(0) RI E 

18 
Parameter 

‘a 

uoI m/sec 

wo, m/sec 

eo, deg 

s(V) I m/sec 

s(a), deg 

s(e), deg 

TABLE VII. - ESTIMATES OF PARAMETERS FROM LONGITUDINAL 

FLIGHT DATA USING THREE DIFFERENT METHODS 

1240 

1.792 

1.24 

2.631 

-12.72 

-e073 

35.5 

6.0 

-.19 

260 

280 

748 

ML 

.00021 

,001 8 

.098 

0036 

.0026 

.10 

025 

NFI S 

A 

0 

-. 122 

1.8 

*02 

2.34 

-1 3 e03 

-e050 

37.8 

6.1 

.7 

.177 

158 

1.08 

.0035 

e14 

.40 

.34 

.10 

e015 

.40 

.27 

.30 

A 

0 

-e 123 

1.74 

-. 28 

2.6 

-12.64 

-.09 

35.5 

6.1 

.4 

MF 

.0035 

. 042 

075 

.10 

. 033 

*012 

.82 

.39 

72 

a Parameters with high correlation 

Computed from eo and 8 

Fixed parameter 

(0) R, E 

19 
Parameter 

TABLE VIII. - ESTIMATES OF PARAMETERS FROM 

FLIGHT DATA USING THREE DIFFERENT METHODS 

ML 

A 

8 

-e003 

- 1801 

3.152 

1.414 

.7 

-3.5 

35 20 

-.47 

7.63 

.2 

70 

.300 

.435 

.854 

2.078 

1 .lo5 

.728 

i .0045 

.00041 

.0065 

.0038 

.10 

.037 

.084 

.029 

24 

-044 

NFIS 

I 

0 

-e018 

-a172 

3.07 

-1.9 

1.2 

-3.9 

36 a91 

.1 

7.14 

-1 -8 

.2 

.511 

.206 

.182 

1.053 

.961 

-0063 

.0024 

.050 

.13 

.18 

.43 

.038 

-17 

.092 

.58 

.17 

EKF 

8 - 

- 003 

-a181 

3.14 

-1.43 

-83 

-3.94 

35.3 

-.4 

7.4 

.2 

.7 

.0035 

.0035 

.042 

.056 

.094 

.031 

.16 

.25 

.39 

1 e6 

60 

a Parameter with high correlation 

Computed from +o, eo and 4 R, E (01, eRIE(O) 

20 
c 

Z~, E 

A 

State and output 

equation s 8 

i 

I 

Maxim urn likelihood 

algorit h m 

V 

Figure 1. - Block diagram of computing procedure for 

ML estimation of unknown parameters. 

21 
a 

14 F 

m/ sec2 

4 

I 

-+*-j 

I 

*-* 

+ ’r 

+ r ++ +b++ ++ +$ 

I I 

+- 

++ 

, 

Figure 2. - Simulated 

t, sec 

longitudinal motion of an airplane. 

22 
90 

v, 

ml sec 

70 

50 

t, sec 

- *;*** ++ 

- ++% + +’+++*%--** + 

#+ +++v tf+ 

+++ 

++ 

.H 

++%%+ *+A + m + &A++$+ 

&+w+ %-+$ %+,”.? 

I I I I 

Figure 2. - Concluded 

23 
I I I I I 

1 I L 

e, 

rad 

-1 1 

1 c 

Figure 3. - 

t, sec 

Simulated lateral motion of 

24 

an airplane 
rad/ sec 

p' 

I 

rad/ sec 

t, sec 

Figure 3. - Concluded 

25 
+ Measured 

- Computed 

I 1 

'F 

I -1 I I 

Figure 4. - Comparison of measured time histories in 

longitudinal maneuver with those computed. 

26 
r + Measured 

- Computed 42 

"R, 

m/ sec 

38 

34 

r 

.3 

aR, 

rad 

.2 

.1 

t, sec 

Figure 5. - Comparison of measured time histories in 

combined maneuver with those computed. 

27 
r + Measured 

- Computed 

- 

.4 - 

aR, 

rad 

2. 

I I 

0 

Figure 5. 

t, sec 

- Concluded 

28 
Standard Bibliographic Page 

. Report No. 2. Government Accession No. 

NASA 'I'M-89059 

. Title and Subtitle 

Estimation of Bias Errors in Measured Airplane 

Responses Using Maximum Likelihood Method 

'. Author(s) 

*Vladislav Klein and *Dan R. Morgan 

1. Performing Organization Name and Address 

NASA Langley Research Center 

Hampton, VA 23665-5225 - 

.2. Sponsoring Agency Name and Address 

3. Recipient's Catalog No. 

5. Report Date 

January 1987 

6. Performing Organization Code 

505-66-01-02 

8. Performing Organization Report NO. 

10. Work Unit No. 

11. Contract or Grant No. 

13. Type of Report and Period Covered 

TECHNICAL MEMORANDUM 

National Aeronautics and Space Administration 

Washington, DC 20546 

- 

19. Security Classif.(of this report) 20FSecurity Classif.(of this page) 21. No. of Pages 

Unclassified Unclassified 33 

14. Sponsoring Agency Code 

22. Price 

A0 3 

I 

15. Supplementary Notes 

*The George Washington University, Joint Institute for Advancement of Flight 

Sciences, Hampton, Virginia. 

16. Abstract 

A maximum likelihood method is used for estimation of unknown bias errors in 

measured airplane responses. The mathematical model of an airplane is 

represented by six-degrees-of-freedom kinematic equations. In these 

equations the input variables are replaced by their measured values which 

are assumed to be without random errors. The resulting algorithm is verified 

with a simulation and flight test data. The maximum likelihood estimates 

from in-flight measured data are compared with those obtained by using a 

nonlinear-fixed-interval-smoother and an extended Kalman filter. 

For sale by the National Technical Information Service, Springfield, Virginia 22161 

NASA Langley Form 63 (June 1985)