Document
NASA Technical Memorandum 4099
Estimation of Longitudinal
Stability and Control Derivatives
for an Icing Research Aircraft
From Flight Data
James G. Batterson and Thomas M. O’Mara
MARCH 1989
NASA
- ~~~
NASA Technical Memorandum 4099
Estimation of Longitudinal
Stability and Control Derivatives
for an Icing Research Aircraft
From Flight Data
James G. Batterson
LangZey Research Center
H a mpton, Virginia
Thomas M. O’Mara
The George Washington University
Joint Institate for Advancement of FZight Sciences
LmzgZey Research Center
Hampton, Virginia
National Aeronautics and Space Administration Office of Management Scientific and Technical Information Division Summary put in analytical predictions and wind-tunnel results as they relate to an aircraft in flight. Second, flight- The National Aeronautics and Space Adminis- derived derivatives can be used judiciously along with tration, at its Lewis Research Center, develops and those provided by analytical predictions and wind- coordinates plans and results related to icing tunnel tests to upgrade simulator math models to computer-code predictions, icing research tunnel provide a realistic set of aerodynamics for pilot-in- (IRT) measurements, and icing research flights. To the-loop simulations of icing scenarios. Such simula- allow for the comparison of research flight results tions can provide definitive information on how any with icing code predictions and IRT measurements, measured degradation of the aircraft flying qualities Lewis has collaborated with the NASA Langley actually affects its handling qualities.
Research Center for the planning and analysis of a Previous work on the estimation of stability and series of icing research flights related to the deter- control derivatives from flight tests for an iced air- mination of stability and control derivatives from re- craft has been performed by LeRC and Kohlman search flight data.
Systems Research, Inc. (KSR), as reported in refer- This paper presents the results of applying a mod- ences 2 and 3. That earlier work employed a modified ified stepwise regression algorithm and a maximum maximum likelihood technique to estimate the stabil- likelihood algorithm to flight data from a twin-engine ity and control derivative values. The accuracies of commuter-class icing research aircraft. The results these stability and control derivatives were estimated are in the form of body-axis stability and control by an approximation to the Cramer-Rao bound given derivatives related to the short-period, longitudinal by the maximum likelihood algorithm. Since it is motion of the aircraft. Data were analyzed for the known that the Cramer-Rao bound estimates indi- baseline aircraft (%niced”) and for the aircraft with cate better accuracy than is actually achievable by an artificial glaze-ice shape attached to the leading repeated experiments (ref. 4), and since the earlier edge of the horizontal tail. The results are discussed work showed small, but apparently discernible, dif- as to the accuracy of the derivative estimates and ferences in the derivative values for the “iced” versus the difference between the derivative values found for “uniced” aircraft, the first phase of the LaRC plan the baseline and the “iced” aircraft. Additional com- was to determine the expected accuracy of the esti- parisons are made between the maximum likelihood mated derivative values. To this end an ensemble of results and the modified stepwise regression results 45 maneuvers was flown at the same flight condition.
with causes for any discrepancies postulated.
These maneuvers were then individually analyzed as to longitudinal stability and control derivatives us- Introduction ing both a maximum likelihood (ML) algorithm and a stepwise regression (SR) algorithm. The ensem- The known dangers to safe flight that are caused ble standard deviation for the parameter estimates by the accumulation of ice on aircraft components have given rise to a multifaceted approach to un- was compared with the average of the Cramer-Rao bound estimates (maximum likelihood) for the en- derstanding the causes and predicting the effects of aircraft icing (ref. 1). The National Aeronautics semble and the average estimate of standard error Maneuvers were then flown and Space Administration (NASA), at its Lewis Re- (stepwise regression).
search Center (LeRC), supports an Icing Technology from constant-power, l g flight at several trim air- speeds for both the uniced and artificially iced air- Project Office in which plans and results related to craft, and the recorded data from those maneuvers icing computer-code predictions, icing research tun- were analyzed for longitudinal stability and control nel (IRT) measurements, and icing research flights derivatives using both a stepwise regression and a are coordinated. To allow for the comparison of re- I maximum likelihood algorithm. The flight test pro- search flight results with icing code predictions and IRT measurements, LeRC has collaborated with the gram also included several data compatibility ma- NASA Langley Research Center (LaRC) for the plan- neuvers for the assessment of data quality and several ning and analysis of a series of icing research flights deceleration/acceleration maneuvers for direct deter- related to the determination of stability and control mination of lift and pitching-moment curves.
derivatives from research flight data.
The purpose of this paper is to present the re- The successful estimation of stability and control sults of applying a modified stepwise regression algo- derivatives from flight data has two important ram- rithm (ref. 5) and a maximum likelihood algorithm ifications. First, the values for the derivatives can (ref. 6) to flight data from a twin-engine, commuter- be compared with values derived from the analytical class icing research aircraft. The results are in the icing codes and from the IRT. These comparisons al- form of body-axis stability and control derivatives low for an assessment of the confidence that should be related to the short-period, longitudinal motion of cost function for maximum the aircraft. Data were analyzed for the baseline likelihood algorithm aircraft (“uniced”) and for the aircraft with an ar- tificial glaze-ice shape attached to the leading edge cost function for stepwise of the horizontal tail. The results are discussed as regression algorithm to the accuracy of the derivative estimates and the aircraft mass, kg difference between the derivative values found for the baseline “uniced” and the “iced” aircraft. Additional number of data points comparisons are made between the maximum like- roll, pitch, and yaw rates, re- lihood results and the modified stepwise regression spectively, rad/sec or deg/sec results with causes for any discrepancies postulated.
The paper is organized as follows: After this intro- wing area, m duction, the aircraft, its instrumentation, and the total sum of squares (corrected flight test maneuvers will be briefly discussed fol- for mean) a section on the analysis techniques. The lowed by main part of the paper will then present the results of standard error the data analysis followed by a section summarizing average of standard errors the conclusions to be drawn from these results.
given by program ensemble-average standard vertical acceleration, g units error for a group of values wing span, m student t-distribution calculated constant bias in longitudinal, lateral, and longitudinal- accelerat ion vertical velocity components, measurement, m/sec2 respectively, m/sec calculated constant bias in total airspeed, ( u2 +v2 +w2) 1/2 , vertical-acceleration measure- m/sec ment, m/sec2 measured aircraft response or calculated constant bias in control surface input pitch-rate measurement, computed aerodynamic rad/sec coefficients calculated constant bias in linear approximation of Cmae angle-of-attack measurement, corresponding to some xo rad value calculated constant bias in measured output pitch-angle measurement, rad computed output rolling-, pitching-, and yawing-moment coefficients, corrected (free-stream) angle respectively of attack, rad or deg longitudinal-, lateral-, and experimentally measured angle vertical-force coefficients, of attack, rad or deg respectively criteria for achieving a desired mean aerodynamic chord, m prediction interval acceleration due to gravity angle of sideslip, rad or deg ( l g M 9.81 m/sec2) elevator, aileron, and rudder moments of inertia about deflection, respectively, rad or roll, pitch, and yaw axes, deg respectively, kg-m2 aerodynamic stability or control parameters product of inertia, kg-m2 the 24-sec slot while retrimming and/or housekeeping constant offset term 0 0 could be done during the 60- to 90-sec write-off 0 2 derivative with respect to ith period.
response or control surface Research personnel onboard the aircraft were the input research pilot in the cockpit left seat, an observer in the cockpit right seat, and the flight test engineer pitch and roll angles, respec- in the cabin for operating the DAS and coordinating tively, rad or deg available 24-sec test slots with the pilot.
upwash (scale-factor) correc- After each flight, the recorded data cassettes were tion for angle of attack sent to KSR where the data were reduced to engineer- ing units. The engineering-units data were written by air density, kg/m3 P KSR to a nine-track magnetic tape which was sent -2 U .
variance estimate of measure- to LaRC for stability and control analysis.
ment noise (eq. (2))
Research Flights
ac
c m , =
The four airplane configurations flown are given 2 v in the following table: Flap Flight Aircraft condition deflection designation Configuration A dot over a symbol represents the derivative with 1 Baseline (uniced) 0" 88-5 respect to time.
2 Baseline (uniced) 10" 88-6 3 Artificial ice on tail 0" 88-7
Aircraft and Instrumentation
4 Artificial ice on tail 10" 88-8 The icing research aircraft is a modified deHavil- land DHC-6 Twin Otter. It is powered by two Pratt & Whitney PT6A-20A gas turbine engines. Physical These flights were designated as 88-5, 88-6, 88-7, characteristics for the aircraft are found in table I. A and 88-8, respectively, by LeRC. Within each of these three-view drawing is presented in figure 1. Longi- flights, each 24-sec slice of individual maneuvers was tudinal control is effected by elevator and hydrauli- assigned a run number. The aircraft is limited to 10' cally actuated flaps; lateral and directional control, flaps in an icing environment as a safety precaution.
through ailerons and rudder, respectively. The ele- The artificial ice was a Styrofoam form modeled vator, ailerons, and rudder all have trim tabs.
after moderate glaze ice that had previously been The instrumentation, which was provided and in- seen and carefully photographed during natural icing KSR, included three-axis orthogonal lin- stalled by encounters and in the IRT. This shape was glued ear accelerometers, three-axis angular rate gyros, to an aluminum form that was contoured to the and three-axis attitude gyros. Angles of attack and leading edge of the horizontal tail. The aluminum sideslip were measured by balsa wood vanes installed was secured to the leading edge by attaching straps on a nose boom. Elevator, rudder, and aileron posi- to the stabilizer hinge line as shown in figure 2. Four tions were measured by potentiometers placed at the types of maneuvers were flown in each configuration: 1 control surface.
1. Data compatibility maneuvers Since flaps were set and remained constant during 2. Small-amplitude longitudinal perturbations an entire series of maneuvers, flap position was given 3. Large-amplitude longitudinal perturbations by a readout in the cockpit and was recorded by the 4. Deceleration/acceleration maneuvers flight engineer for future reference.
Data were recorded onboard at 20 samples per The data compatibility maneuvers involved excit- second; all channels were sampled within 1 msec by ing large angular rates about all axes, thus leading the KSR data acquisition system (DAS). Because of to large excursions in all Euler angles and to large buffer limitations of the KSR system, only 24 sec variations in linear accelerations, airspeed, angle of attack, and angle of sideslip. These data were ex- of the 20-samples-per-second data could be obtained before the buffer contents had to be written onto a amined using the algorithm described in reference 7 cassette tape. The write-off period was nominally 60 to assess the quality of the DAS measurements. In 90 sec. Hence, maneuvers had to be flown within these maneuvers, in which no attempt was made to to excite any of the aircraft modes, only the air data comparisons could be made between the clean and sensors, rate sensors, and linear accelerometers were “iced” derivatives at each trim airspeed.
exercised.
Data Analysis Methods The small-amplitude maneuvers were initiated by elevator “2-1” doublets designed to excite the short- The raw engineering-units data are first corrected period longitudinal motion of the aircraft. An ex- for all known instrumentation offsets from the air- ample of the control surface movement and aircraft craft center of gravity and for flow effects. Next, longitudinal responses is given in figure 3. These several maneuvers are analyzed with the maximum maneuvers were planned to excite f0.3g of verti- likelihood algorithm reported in reference 7 for the cal acceleration. However, because of an erroneous estimation of instrument biases and possible scale- accelerometer readout in the cockpit, these maneu- factor errors. This process is known as data compat- vers were sometimes of amplitude fl.Og of vertical ibility or data consistency checking. Once a data set acceleration.
is available, the identification process still has two The large-amplitude perturbations were initiated remaining stages. The first stage is the determina- by random elevator doublets superimposed on an tion of the parametric model structure. Then, the otherwise monotonically increasing angle of attack.
second stage is the estimation of the corresponding For these maneuvers, the pilot pulls steadily back on parameter values. The model structure was deter- the stick over the 24-sec test window, interrupting mined using a modified stepwise regression. This the steady pull at random intervals to apply small technique can determine significant terms among elevator doublets. In this way a large angle-of- the candidate variables and estimate corresponding attack range can be probed in a single maneuver at parameters. Each new variable chosen for entry into a constant power setting.
the regression equation is the one that has the largest The deceleration/acceleration maneuvers were correlation with the measured dependent variable performed by setting power for l.0g-level flight at (y) after adjusting for the effect on y of the vari- 120 knots and then slowly pulling back on the stick, ables already selected. This correlation is reflected decelerating at about 2 knots/sec. Then, if time by the calculated F-statistic of the variable. The se- was available in the 24-sec test window, the pilot lected parameters are estimated by minimizing the would push the stick forward again to accelerate at cost function 2 knots/sec, eventually returning to the initial flight
condition. ~r e l 2
The small-amplitude maneuvers fell into two groups. In the first group, to ascertain the accu- racy of the derivative estimates, a set of maneuvers where y(i) is the aerodynamic coefficient computed was performed in which the aircraft was trimmed at of approximately 120 knots, and a series from measured data at time i, N is the number of an airspeed
data points, and 4? + 1 is the number of parameters
of 45 small-amplitude longitudinal perturbations was executed. These 45 sets of small-amplitude maneu- (or independent variables) in the regression equation.
vers provided a statistically significant ensemble of At every step of the regression, the variables incorpo- rated into the model in previous stages and the new maneuvers, each of which should produce the same derivative estimates. Analyzing each maneuver of variable entering the model are reexamined. Any this ensemble and calculating the mean and standard variable that provides a nonsignificant contribution (due to correlation with more recently added terms) deviation for the 45 results for each of the longitudi- is removed from the model. The process of select- nal stability and control derivatives gives a measure ing and checking variables continues until no more of how much a parameter estimate might vary sta- are admitted to the equation and no more are re- tistically between two maneuvers at the same flight jected. Experience shows, however, that the model condition.
based only on the significance of individual parame- In the second group, the rest of the small- ters in the model equation can still include too many amplitude maneuvers were performed at different air- terms and, therefore, may have poor prediction ca- speeds with one or two repeats at each airspeed.
8). Therefore, several criteria for the pabilities (ref.
These maneuvers were analyzed for trends in the derivatives relative to flight condition. Because the selection of an adequate model were introduced in reference 8. Details of this model-structure determi- effect of artificial ice on lift and drag of the whole aircraft should be small and because the aircraft was nation procedure are also explained in that reference.
at different airspeeds using only the eleva- Finally, each model and each maneuver were an- trimmed alyzed by the maximum likelihood (ML) method tor tab (constant power), it was assumed that direct described in reference 6 assuming no external noise. account for an uncorrected lag present in the data The corresponding cost function has the form recording technique or in effects of the dynamics of the vane. Although increasing time shifts did “whiten” residuals, there was no physical justifica- tion for corrections of the magnitude of shift required.
Because of these problems involving angle-of- attack values and an overall inconsistency in com- where k is the number of output variables, is puted bias values for the three compatibility runs, the 45 repeated longitudinal maneuvers and 3 additional the variance estimate of measurement noise, z j and h large-amplitude maneuvers were analyzed using the z j are the measured and computed outputs, respec- algorithm mentioned above. From the analysis of
tively, and 6 is a vector of estimates of unknown
these maneuvers, mean values of upwash correction parameters.
( X , ) and all bias terms were compiled (table 111).
In addition to the parameter estimates, the ML The magnitude and consistency of bias values ob- method provides an estimate of the accuracy of each tained from analyzing all runs gave sufficient confi- parameter through an approximation to the Cramer- dence in the data acquisition system for the small Rao bound for that parameter. Theoretically, the perturbation maneuvers exciting the short-period Cramer-Rao bound presents a lower bound for the modes of the aircraft. They were not used as cor- variance of the estimates of a parameter. The ap- rections to the data because of their very low values proximation to the Cramer-Rao bound is realized by and relatively high standard errors, a result which the diagonal elements of the inverse of the Fisher in- indicates that no real biases exist.
formation matrix. Since with flight data there is no The average scale-factor estimate for upwash cor- “true” value with which to compare parameter esti- rection from the 45 repeat maneuvers was mates, agreement between the estimates of a given parameter from the ML and SR algorithms increases
X , = 0.10605
the overall confidence in that estimated value.
with the Cramer-Rao bound estimate of
Results and Discussion
s(A,) = 0.00660 Data Compatibility Three maneuvers that were designed specifically and an ensemble standard deviation of for a longitudinal data-compatibility check were analyzed using the algorithm developed in refer- SE(A,) = 0.00741 ence 7. This compares measured and predicted air- craft responses using the kinematic equations as a The upwash correction was made to all measured a model. It employs the maximum likelihood tech- in the following manner: nique for finding uncorrected systematic errors in measured responses in terms of constant bias errors
a M = (1 + A,)a
and scale-factor errors. For the symmetric longi- tudinal maneuvers, possible biases in angle of or attack (b,), pitch rate ( b q ) , pitch angle ( b e ) , lon-
( buz), and vertical accelera- a = (1 + A,)-laM
gitudinal acceleration tion ( b u z ) were estimated. The resulting values where a~ is the experimentally measured angle of are found in table 11. In addition to the six bias attack and Q is the corrected free-stream angle of values estimated, a scale factor in angle of at- attack.
(A,) was determined as a correction for up- tack wash at the tip of the nose boom. Although Accuracy of Derivative Estimates computed angle-of-attack time histories appeared to correspond well with those measured in flight, the The purpose of executing the 45 maneuvers under differences between measured and computed data identical flight conditions was to acquire a series of (residuals) still exhibited a definite structure (were flight data sets that would yield a statistically sig- not randomly distributed). (See fig. 4.) As a possi- nificant base of parameter and error estimates. The ble remedy for this situation, several candidate time benefits of such an ensemble are twofold. First, the shifts were introduced in the angle-of-attack time standard error calculated for the ensemble of param- of the histories. It was postulated that this action might eter estimates themselves gives us a measure effectiveness of the maneuvers through the repeata- 120 knots). These data were analyzed with the lin- bility of the predicted parameters. Second, the com- ear regression and maximum likelihood algorithms to parison of the average standard errors obtained from study the variations of parameters with initial flight the computer programs with the standard error for condition and between different configurations. The the ensemble gives us a sense of how realistic the results obtained from the stepwise regression routine program values are. are presented graphically in figures 5 through 8. In no case did the stepwise regression select a nonlinear The 45 repeat maneuvers were analyzed using term as a component of the aerodynamic model.
both the linear regression and the maximum like- lihood methods, and the resulting mean values of Each parameter value is plotted with f 2 a error coefficients are compiled in table IV. Accompany- bars, based on the standard error as given by the ing each parameter estimate is the average standard computer program. For the two configurations in error associated with it as given by the particular which the flaps were held at O", there are statisti- estimation algorithm, the standard error for the en- cally significant differences between iced and uniced semble of values, and the ratio of the two. The ratio flights. These differences generally decrease with in- given for each parameter is the factor by which the creasing velocity. When the flaps are extended and program-estimated standard error for that parame- held at lo", however, the distinction between iced ter should be multiplied to produce a realistic, ex- and uniced becomes less dramatic. Disturbances in pected standard error. The ratios from the maximum the flow coming off the wings caused by the extended likelihood results are seen to be greater than those flaps appear to negate a large percentage of the ef- from the linear regression. Two reasons are postu- fects of artificial ice attached to the leading edge of lated for this difference: First, the standard error for the tail section. The pitching-moment derivatives in- each ML estimate is given by a numerical approxi- dicate that the artificial ice shape induces a loss of mation to the Cramer-Rao bound. Theoretically, the elevator effectiveness (Cm6e ), a decrease in static sta- Cramer-Rao bound for an estimated parameter rep- bility (Cm, ), and lower pitch damping (Cmq ).
resents the best achievable accuracy for that param- Derivative values obtained from selected maneu- eter. The approximation to the Cramer-Rao bound vers using the maximum likelihood routine are pre- used in the algorithm that produced the ML param- sented in figure 9. Because of the convergence eter estimates presented in this paper is given by the problems mentioned earlier, the maximum likelihood diagonal elements of the inverse of an approximation algorithm was run only on several selected maneu- to the Fisher information matrix. In general, for ac- vers. The resulting values were usually in agreement tual flight data, these theoretical values are smaller with the regression-determined parameters with re- than the estimated standard error from an SR al- spect to trends, but they differed significantly in ac- gorithm. This gives, in general, smaller numbers in tual values estimated. No reason was found for those the denominators for the maximum likelihood ratios differences.
as compared with the linear regression ratios. Sec- ond, the dispersion of values for the maximum like- lihood estimates from these data was large for some Error Analysis parameters (CZq and C,, , in particular). This type of dispersion is usually due to multiple correlations A fundamental issue throughout this research is among variables in the data and unmodeled dynam- that of distinguishing between the effects of ice and ics (such as those giving rise to the structure of the angle-of-attack residuals discussed in the section on the inherent variability of predicted parameters. In addition to the error analysis performed on the val- data compatibility).
ues obtained from the 45 repeat maneuvers, the Cm, All further analysis of flight data was principally carried out using the linear regression method be- estimates for the 0" flap condition were considered cause of the consistency of results obtained through (fig. 10) for an additional analysis. This set of points it, whereas some comparison estimates were made was chosen for this portion of the analysis because each of the two groups of values (iced and uniced) using the maximum likelihood method.
lend themselves to linear approximations. Also, de- termining a prediction interval (denoted by the terms Effects of Artificial Ice Shape in brackets) in this particular case serves to quantita- Following the 45 repeat maneuvers completed tively enforce what is qualitatively a very noticeable with the aircraft in each of the 4 configurations, the difference in parameters for the iced/uniced cases.
pilot executed several additional longitudinal dou- After approximating the two groups of points by blets at varying initial airspeeds (ranging from 62 to straight lines, prediction intervals about these lines I I were calculated as follows: prediction intervals and gives an illustration of both the areas in which icing effects are discernible as well as the speed range in which we cannot attribute any change in Cm,e to the artificial ice shape. A similar analysis can be done on other parameters by fitting I where I them in a piecewise linear fashion.
linear approximation of YO(X0) b Conclusions Cmbe corresponding to some zo value Based on the analysis and results presented, the , a* following four basic conclusions are presented: 0.05 to achieve a 95-percent prediction interval 1. Any future research of this type will require determination of the cause of error found in angle-of- n number of data points attack measurements.
student t-distribution to* j 2 p - 2 2. A multiplicative relationship between ensem- ble standard error and estimated standard error from S standard error (can be one the stepwise regression routine was identified with of three substitutions listed numerical values representing this ratio assigned to below): each stability and control derivative.
average of standard errors 3. A multiplicative relationship between ensem- obtained from computer ble standard errors and those obtained as an approx- program imation to the Cramer-Rao bound as given by the maximum likelihood program was determined as a ensemble-average standard quantitative ratio for each parameter estimated.
error for values of Cmbe, 4. The effects of the artificial ice shape attached to the tail section have been shown to be measurable as changes in the longitudinal stability and control S P A ( Y ) multiplied by ratio of derivatives in all the flight test conditions addressed ensemble to program average here with the largest effects being seen in those obtained from table IV derivatives most directly determined by the flow on the tail, i.e., static stability (Cm,), pitch damping sum of squares of residuals,
s x x
N (Cm,), and elevator effectiveness (Cm,= ).
(zz - 2 ) 2 i=l The resulting interval is one within which we NASA Langley Research Center can expect 95 percent of all predicted elevator- Hampton, VA 23665-5225 January 25, 1989 effectiveness values to fall. Figure 10 presents these Here, equations (4), (6), (8), and (10) describe the
Appendix
longitudinal motion and equations ( 5 ) , (7), (9), and (11) describe the lateral motion of the aircraft. The
Postulated Models
angles of attack and sideslip are, respectively, given The six-degrees-of-freedom equations of motion b Y for the aircraft are
p v 2 s
u = - q w + r v - g sin 8+- C X 2m
pv2 s
The aerodynamic forces and moments are repre- i = - T U + p W + g cos 8 sin $+- C Y 2m sented by the coefficients Ca (where a = X , Y, Z,1, m, or n). It is postulated that these coefficients can
pvz s
2i, = -pv + qu + g cos 8 cos 4 + -
c z be written as a Taylor-series polynomial expansion 2m about an equilibrium trim condition as pV2 Sb
+ - P = qr ( I , ) I Y - I z + - ( p q + i . ) I x z
c, I X 2 I X where Oa,o and Oa,j are the unknown parameters.
For a = X , Z , or m, the independent variables xj are a, q?/2V, 6 , , and their combinations; and for along with the kinematic relations a = Y, I , or n, they are p, pb/2V, rb/2V, bar f i r , and their combinations. Furthermore,
O = q cos 4 - r sin 4
and 4 = p + ( q sin $ + r cos $)tan 0
AX^ = x j ( t ) - ~j
( t = 0 )
References 4. Maine, Richard E.; and Iliff, Kenneth W.: Application of
Parameter Estimation to Aircraft Stability and Control- 1. Federal Coordinator for Meteorological Services and Sup- The Output-Error Approach. NASA RP-1168, 1986.
porting Research: National Aircraft Icing Technology 5. Batterson, James G.: STEP and STEPSPL-Computer Plan. FCM-P20-1986, Dep. of Commerce, Apr. 1986.
Programs for Aerodynamic Model Structure Determina- 2. Ranaudo, Richard J.; Mikkelsen, Kevin L.; McKnight, tion and Parameter Estimation. NASA TM-86410, 1986.
Robert C.; Ide, Robert F.; Reehorst, Andrew L.; Jordan, 6. Murphy, Patrick C.: An Algorithm for Maximum Likeli- Jerry L.; Schinstock, William C.; and Platz, Stewart J.: hood Estimation Using an Eficient Method for Approxi- I The Measurement of Aircraft Performance and Stability mating Sensitivities. NASA TP-2311, 1984.
and Control After Flight Through Natural Icing Condi- 7. Klein, Vladislav; and Morgan, Dan R.: Estimation of Bias tions. AIAA-86-9758, Apr. 1986. Errors in Measured Airplane Responses Using Maximum I 3. Jordan, Jerry L.; Platz, Stewart J.; and Schinstock, Likelihood Method. NASA TM-89059, 1987.
William C.: Flight Test Report of the NASA Icing Re- 8. Klein, Vladislav; Batterson, James G.; and Murphy, search Airplane-Performance, Stability, and Control Af- Patrick C.: Determination of Airplane Model Structure ter Flight Through Natural Icing Conditions. NASA From Flight Data b y Using Modified Stepwise Regression.
CR-179515, 1986. NASA TP-1916, 1981.
Table I. Physical Characteristics of Research Aircraft Low High Mass, kg . . . . . . . . . . . . . . . . . . . . . . 4150 4600 Inertia: I x , kg-m2 . . . . . . . . . . . . . . . . . . . . 21300 21 790 I y , kg-m2 . . . . . . . . . . . . . . . . . . . . 30000 31 030 I z , kg-m2 . . . . . . . . . . . . . . . . . . . . 44990 48 640 I x z , kg-m2 . . . . . . . . . . . . . . . . . . . 1430 Wing: Area, m 2 . . . . . . . . . . . . . . . . . . . . .
39.02 Aspect ratio . . . . . . . . . . . . . . . . . . . 10.06 Span, m . . . . . . . . . . . . . . . . . . . . . 19.81 Mean geometric chord, m . . . . . . . . . . . . . 1.98 Airfoil section (17-percent thickness) . . . . . . . . “deHavilland High Lift” Horizonta12t ail: Area, m . . . . . . . . . . . . . . . . . . . . . 9.1 Airfoil section (inverted) . . . . . . . . . . . . . . NACA 63A213 Table 11. Estimated Bias and Scale-Factor Errors Present in Longitudinal-Data Compatibility Maneuvers for Flight 88-5 Longitudinal compatibility maneuvers for- Estimated parameter Run 33 Run 34 Run 35 b,, rad . . . . . . 0.0038 f 0.0006 0.0092 f 0.0007 -0.0047 f 0.0007 b,, rad/sec . . . . . 0.00057 f 0.00003 0.000996 f 0.2E-09 0.000645 f 0.3E-07 be, rad . . . . . . .
-0.0175 f 0.0005 -0.0173 f 0.0003 -0.0207 f 0.0004 b a s , g units . . . . .
0.123 f 0.003 0.089 f 0.030 0.002 f 0.002 b,,, g units . . . . .
0.014 f 0.001 0.004 f 0.002 -0.056 f 0.001 . . . . . . . .
0.022 f 0.005 -0.054 f 0.008 0.128 + 0.010
Aa Table 111. Estimated Bias and Scale-Factor Errors Present in Longitudinal Doublets and Large-Amplitude Maneuvers for Flight 88-5 Large-amplitude maneuvers for Average of Estimated 45 repeat parameter maneuvers Run 36 Run 37 Run 38 b,, rad . . . . . -0.00324 f 0.00092 0.000598 f 0.0019 0.00176 f 0.0023 -0.00381 f 0.00232 b,, rad/sec . . . . 0.00044 f 0.000488 0.00181 f 0.00016 0.000829 f 0.00013 0.00129 f 0.000151 be, rad . . . . . . 0.00040 f 0.00156 -0.00618 f 0.00261 -0.0189 f 0.0023 -0.0131 f 0.00253 bas, g units . . . .
0.0353 f 0.01606 -0.00288 f 0.0164 -0.0629 f 0.0150 -0.0144 f 0.0163 baz, g units . . . .
0.00339 f 0.0285 -0.0837 f 0.00561 -0.0528 f 0.00682 -0.0875 f 0.00717 A , . . . . . . .
0.10605 f 0.00660 0.0819 f 0.0142 0.1049 f 0.0255 0.1132 f 0.0227 Table IV. Mean Values of Stability and Control Derivatives Calculated From the 45 Repeat Maneuvers (a) Linear regression
I Standard errors I
( a )
( b ) (c) SE (@)/4@)
-5.66 0.0493 0.0169 2.9 -19.97 1.047 .386 2.7 -.608 .0281 .0122 2.3 -1.31 .0147 .0191 .8 -34.2 .6444 .4314 1.5 -1.74 .0248 .0131 1.9 (b) Maximum likelihood Standard errors -
0 S E W 40)
Parameter ( a ) ( b )
( c ) SE (0)/4@)
-5.28 0.3763 0.1933 1.9
cza
-2.00 17.4 .444 39.3
czq
-.039 .8457 .1264 6.7 "6 e - 1.42 .116 ,0272 4.3 Cma -41.3 10.33 .4416 23.4 c m , -1.97 .180 .0386 4.7 Cm 6 ,
5.66
Figure 1. Three-view drawing of icing research aircraft.
ORIGINAL PAGE I S
OF POOR QUALITY
d .e Elevator deflection, deg
l 2 b
-8 0 2 4 6 8 1 0 Time, sec
2 g 7pp7 z g l ; w l
-3 0 2 4 6 8 1 0 0 2 4 6 8 1 0 Time, sec Time, sec deglsec q, ‘ 1 w-, g ? ; i t s _1 Fz/\/r, -20 0 2 4 6 8 1 0 0 2 4 6 8 1 0 Time, sec Time, sec Figure 3. Control surface input and aircraft response for typical 2-1 elevator doublet.
Angle of
attack,
rad
+ Measured value
- Calculated response
.01
Angle-of -
attack
residuals,
rad
-.01
0 5 10 15 20 25 30
Time, sec
Figure 4. Results from compatibility check on angle of attack.
I r
-4.8 r
0 Uniced
-5.0 1
- 6
+ Iced
-5.2 *
C
za -5.4 .
!
-5m6 t
I I I I I
-5.8 I
I I
- I 0 [I
I
-20
* *
I I ,
czq -30
t
t d
-40
i
I t
-.4
i
# t r +** + + *
-.6
o * o o *
. * * rT
U
-.8
cZ
lie -1.0
-1.2
I I I I I
-1.4
70 80 90 100 110 120
Initial velocity, knots
Figure 5. Vertical force derivatives and f2a error bounds from modified stepwise regression for flaps at 0 ' .
-
-1.1
-
-1.2
-
-1.3
-
Cma -1.4
-
-1.5
Uniced
-
-1.6 + Iced
-1.7 I 1 I I I I
-
-50 I I I I I I
C -1.8
m5e -1.9
-2.0
- 2 . 1
~
-2.2
70 80 90 100 110 120
Initial velocity, knots
Figure 6. Pitching-moment derivatives and &2a error bounds from modified stepwise regression for flaps at 0 ' .
0 Uniced
-5.0 -4.8 F e
+ Iced
C
Za
-6.0 1
I I 1 I
-6.2 '
-1 0
r
-20
'B
czq
-30
-.4 r
-.5 -
' t
-.7
-.8 - I
e
-.9 -
-1.0
-1 .I - $ I I I 1
60 70 80 90 100
Initial velocity, knots
Figure 7. Vertical force derivatives and f2a error bounds from modified stepwise regression for flaps at 10'
-.8 [I
Uniced
+ Iced
-1.6 4
-28 r T f
-38 I I I I I
C
I I
60 70 80 90 100
Initial velocity, knots
Figure 8. Pitching-moment derivatives and f 2 a error bounds from modified stepwise regression for flaps at loo.
- 5 m 0 r +
0 Step
-5.2 1
+ Max
C -5.4 1
za
-
+
-5.6 1
+
I I I
-5.8 I I
+
+
+
+
+
-20 '
czq
-30
+
+
C
26 e
-1.0 t
I I I I
-1.1 1
70 80 90 100 110
Initial velocity, knots
Figure 9. Comparison of modified stepwise regression results with maximum likelihood results. Clean aircraft; flaps at 0 ' .
0 0
-1.3 1
-
-1.4
c m a 4
0 Step
+
+ + + Max -
-1.5
+
I I I I
-38
-39
C
+
+
-43
-42 -44
-1.9
-Im8 ~
C
-2.0 1
mtj
e 0 A
+ -2.1 1
o +
+
-2.2 I I I I 1
70 80 90 100 110
Initial velocity, knots
Figure 9. Concluded.
-1.4 r
-1.6
-1.8
C
m6
e
-2.0
Uniced
-2.2
-2.4 I I I I I I I
70 80 90 100 110 120 130
L Initial velocity, knots
I I Figure 10. Confidence intervals of 95 percent for elevator-effectiveness derivatives for iced and uniced aircraft.
I
nJnsA Report Documentation Page
Nal~onal A ~ ~ ~ U I I C S ana Space Adminisiration 1. Report No. 12. Government Accession No. 3. Recipient’s Catalog No.
NASA TM-4099 I
5. Report Date 1. Title and Subtitle Estimation of Longitudinal Stability and Control Derivatives for March 1989 an Icing Research Aircraft From Flight Data 6 . Performing Organization Code 7. Author(s) 8. Perforniiiig Organization Report No.
James G. Batterson and Thomas M. O’Mara L-16-178 10. Work Unit No.
3. Performing Organization Name and Address 505-66-01-02 NASA Langley Research Center 11. Contract or Grant No.
Hampton, VA 23665-5225 ~~ 13. Type of Report and Period Covered 12. Sponsoring Agency Name and Address Technical Memorandum National Aeronautics and Space Administration 14. Sponsoring Agency Code Washington, DC 20546-0001 15. Supplementary Notes James G. Batterson: Langley Research Center, Hampton, Virginia.
Thomas M. O’Mara: The George Washington University, Joint Institute for Advancement of Flight Sciences, Langley Research Center, Hampton, Virginia.
16. Abstract The National Aeronautics and Space Administration, at its Lewis Research Center, develops and coordinates plans and results related to icing computer-code predictions, icing research tunnel (IRT) measurements, and icing research flights. To allow for the comparison of research flight results with icing code predictions and IRT measurements, Lewis has collaborated with the NASA Langley Research Center for the planning and analysis of a series of icing research flights related to the determination of stability and control derivatives from research flight data. This paper presents the results of applying a modified stepwise regression algorithm and a maximum likelihood algorithm to flight data from a twin-engine commuter-class icing research aircraft. The results are in the form of body-axis stability and control derivatives related to the short-period, longitudinal motion of the aircraft. Data were analyzed for the baseline aircraft (“uniced”) and for the aircraft with an artificial glaze-ice shape attached to the leading edge of the horizontal tail. The results are discussed as to the accuracy of the derivative estimates and the difference between the derivative values found for the baseline and the “iced” aircraft. Additional comparisons are made between the maximum likelihood results and the modified stepwise regression results with causes for any discrepancies postulated.
18. Distribution Statement 17. Key Words (Suggested by Authors(s)) Icing Unclassified-Unlimi ted System identification Stepwise regression Cramer-Rao bounds Aircraft stability and control 19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22. Price Unclassified Unclassified 22 A03 NASA-Langley, 1989 JASA FORM 1626 0 ~ ~ 8 6 For sale by the National Technical Information Service, Springfield, Virginia 22161-2171