1、NASA , Technical Paper 1306 -Control Parameters -of ,a .Light x Airplane From “Flight .Data -. , ,. , I -, Using Two estimation Methods . . Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NASA Technical Paper 1306 Determination of Stability and Contr
2、ol Parameters of a Light Airplane From Flight Data Using Two Estimation Methods Vladislav Klein The George Wasbitzgton University Joint Institute for Advancement of Flight Scietzces Langley Research Center Hampton, Virginia National Aeronautics and Space Administration Scientific and Technical Infor
3、mation Office 1979 “ . Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-SUMMARY Values for stability and control parameters have been determined by use of the equation error method and the maximum likelihood method from maneuvering flight data for a l
4、ow-wing, single-engine, general aviation airplane. The air- plane responses were excited from steady flights at different airspeeds using the stabilator, aileron, and rudder deflections. The model of the air.plane is . based on the equations of motion with the linear aerodynamics. From the repeated
5、measurements, the two standard-deviation confidence intervals for the estimated parameters were established. These bounds are used for the comparison of param- eters determined by both methods and also for the assessment of an effect of dif- ferent input forms and power settings. The static paramete
6、rs are also compared with results from steady flights. Using these comparisons, the best values of estimated parameters were determined and their accuracies specified. INTRODUCTION The National Aeronautics and Space Administration is currently involved in extensive general aviation stall-spin studie
7、s. During the research program, several airplanes have been tested in the wind tunnel and in flight, and more tests with other airplanes are anticipated. In undertaking the stall-spin research, the airplane dynamics in prestall regimes must be understood. For that reason part of the overall program
8、includes the measurement of airplane transient maneuvers for the extraction of a complete set of stability and con- trol parameters. These parameters include aerodynamic derivatives and the Val- ues of aerodynamic coefficients corresponding to steady flight conditions. There have been several previo
9、us attempts using systems identification to determine parameters of general aviation airplanes from unsteady measurements. These attempts differ in the amount of data available, estimation techniques, and Verification of results obtained. In reference 1 the equation error method (regression analysis
10、) is applied to measured longitudinal data corresponding to good excitation of the long- and short-period modes. The same technique is used in reference 2 for the determination of the lateral derivatives from flights with different values of thrust coefficients. The equation error method, based on a
11、 least-squares technique, is very attractive because of its simplicity. It can be easily applied to each of the equations of motion separately and pro- vides direct estimates of the unknown parameters. The resulting estimates are, however, biased as a consequence of the measurement errors in the inp
12、ut and out- put variables. A second procedure used in airplane parameter estimation is the output error method. Because it usually uses the maximum likelihood estimation, it is often called the maximum likelihood method. The airplane longitudinal and lateral aero- dynamic parameters obtained by this
13、 method are presented in references 3 and 4 and are compared with aerodynamic derivatives obtained from wind-tunnel tests and theoretical predictions. The maximum likelihood estimates are theoretically Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
14、superior to those obtained from the equation error method. These estimates are asymptotically unbiased, consistent, and efficient, provided that the model of an airplane is correct and the input variables are measured without errors. However, the maximum likelihood method applied to the problem ment
15、ioned is time consuming because of its iterative nature and because all equations of motion considered enter the estimation algorithm. In sane experiments, small variances of the measurement noise, unknown modeling errors, and a limited number of data points could substantially reduce the superiorit
16、y of the maximum likelihood method to the equation error method. Under these conditions, both methods might provide identical values for the estimated parameters. Detailed description and comparison of both methods can be found in references 5 and 6. The purpose of this report is to document estimat
17、es of the stability and control parameters for one of the general aviation airplanes involved in the stall-spin program. The parameters are extracted from longitudinal and lateral maneuvers initiated from steady flights at different airspeeds. The airspeed range extends from the minimum airspeed at
18、which the airplane can still be maneu- vered to the maximum airspeed in horizontal flight. The two methods already men- tioned were applied to measured flight data in an attempt to obtain more accurate values of the stability and control parameters for the test airplane. This report first describes
19、the test airplane, instrumentation, flight tests, and data reduction. Then the mathematical model of the airplane is introduced, and the estimation methods are outlined. The results from both methods are then compared. The static parameters are also compared with the results obtained from steady fli
20、ghts. Last, the effect of input form and power setting in the estimated parameter values is demonstrated, the best values of parameters are determined, and their accuracies are specified. SYMBOLS A wing aspect ratio = ac, aX,ay,aZ reading of longitudinal, lateral, and vertical accelerometer, respec-
21、 tively, g units b wing span, m constant bias error in variable y CD drag coefficient , D/qS CL CL, t lift coefficient of tail, Lt/+ Cl rolling-moment coefficient , MX/GSb %I pitching-manent coefficient , My/iSc 2 “ Provided by IHSNot for ResaleNo reproduction or networking permitted without license
22、 from IHS-,-,-Cn yawing-moment coefficient, MZ/GSb CT thrust coefficient , T/qS CX longitudinal-force coefficient, Fx/GS CY lateral-force coefficient, F/;s cz vertical-force coefficient, FZ/+ C wing mean aerodynamic chord, m D drag, N F = chno/ln - Fx,Fy,FZ forces along X, Y, and Z body axes, respec
23、tively, N F1 rF2 terms in equations of motion defined by equations (A19) and (A20) f( 1 function which represents state-equation model 9 acceleration due to gravity, m/sec2 g( 1 function which represents output-equation model H sensitivity matrix *n stick-fixed center-of-gravity margin h distance of
24、 center of gravity aft of leading edge of wing mean chord expressed in percent of c - hno distance of aerodynamic center aft of leading edge of wing mean chord expressed in percent of c - Ix,Iy,Iz moment of inertia about X, Y, and Z body axes, respectively, kg-m2 Ixz product of inertia, kg-m2 J cost
25、 function j =fi KO term defined by equation (B16) kga,kgr,klnB,knlB terms defined by equations (B17) to (B19) L lift, N distance of aerodynamic center of tail aft of aerodynamic center of airplane without tail, m 3 Provided by IHSNot for ResaleNo reproduction or networking permitted without license
26、from IHS-,-,-It distance of aerodynamic center of tail aft center of gravity, m M Fisher information matrix Mx,My,Mz rolling, pitching, and yawing moments, respectively, N-m m mass, kg mj j main diagonal element of the M matrix N number of data points n measurement noise vector -b P roll rate, rad/s
27、ec or deg/sec p(S/lae defined in appendix A (eqs. (A8) to (11). Subscripts: E measured 0 tr immed condition t tail Superscripts: T transpose matrix -1 inverse matrix A estimated values - mean derivative with respect to time vector -+ Abbreviations: c.g. center of gravity EE equation error ML maximum
28、 likelihood rms root mean square TEST AIRCRAFT AND INSTRUMENTATION SYSTEM For this study, a four-place, low-wing, single-engine airplane was used. The control surfaces included conventional ailerons, rudder, and all-movable tail (stabilator). The basic geometric, mass, and inertia characteristics ar
29、e 7 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-I summarized in table I. The moments of inertia were measured for the airplane in its early test configuration. The airplane was later modified by the instal- lation of an onboard rocket system whic
30、h is used primarily for spin recovery. The resulting changes in the airplane configuration affected only its mass and inertia characteristics. New moments of inertia were calculated from those pre- viously measured. An analog measurement system was installed in the airplane for recording control sur
31、face deflections, stick and rudder forces, airplane response vari- ables, and other quantities defining flight and engine conditions. Control posi- tion motions (input variables) were measured by rotary potentiometers directly attached to the control surfaces. An orthogonal triad of linear accelerom
32、eters was rigidly mounted on the center line of the cockpit floor at a location close to the allowable center-of-gravity range of the airplane (fig. 1). The sensitive axes of all accelerometers were alined to the reference axes of the airplane. Incidence angles were measured by a swiveling vane moun
33、ted on booms ahead of each wing tip (fig. 1). Because the corrected readings of both vanes gave identical results, only the angle-of-attack and angle-of-sideslip data from the right vane were used for the analysis. The indicated airspeed was obtained from the airplanes air data system which consiste
34、d of a simple total pressure orifice located on each side of the fuselage. Total temperature was measured by a sensor located on the top of the fuselage. The remainder of the instrumentation system included three rate gyros, attitude gyros, signal conditioning, power supplies, and tape recorder. The
35、se components were mounted on a rack behind the front seats as shown in figure 1. A summary of measured quantities used in this study, transducers, and static characteristics of corresponding channels is presented in table 11. The root-mean-square (rms) errors were estimated from recorded signals du
36、ring the preflight and postflight ground operation of the instrumenta- tion system with the airplane engine running. Both the resolution and the rms errors are referred to the digitized data. Table I11 presents dynamic characteristics of transducers used for the measurement of airplane response. The
37、se characteristics were obtained from dynamic calibration. The equivalent time constants given in the last column of table I11 represent the approximation of the transducer dynamics by a first- order system. FLIGHT TEST AND DATA REDUCTION Airplane responses were measured in six flights. Table IV sum
38、marizes per- tinent flight test conditions and the average mass and inertia characteristics of the airplane in these flights. Mass and inertia characteristics for each run analyzed were determined from the airplane take-off weight and estimated fuel consumption during the flight. The longitudinal an
39、d lateral modes were excited separately, primarily from the trimmed level flights at the airspeeds listed in table IV. For the investi- gation of power effect, perturbations were initiated from a steady climb with full power and from a steady descent with idle power. 8 Provided by IHSNot for ResaleN
40、o reproduction or networking permitted without license from IHS-,-,-In longitudinal flights, the inputs used were stabilator deflections having the form of a pulse, a doublet, or a combination of both. In the lateral case, both the rudder and aileron were applied simultaneously. Various forms of the
41、se inputs are shown later. In all cases, the a- and B-traces were examined to determine that the atmospheric turbulence was negligible. The measured flight data were filtered with a 6-Hz low-pass filter and sam- pled at the rate of 20 samples per second. The sampled data were used to pro- duce autom
42、atic data tabulations, time history plots, and final tape for airplane parameter estimation. This tape included the following variables: time, true airspeed, incidence angles (right vane), angular velocities, attitude angles, linear accelerations, control surface deflections, and incidence angles (l
43、eft vane) . True airspeed was obtained from the indicated airspeed by applying correc- tions for measured position error of the static pressure system and by using the air density values computed from the measured air temperature and static pressure. The angle-of-attack vane readings were corrected
44、for air upwash by a multiplication constant. This constant was estimated from steady horizontal flights by comparing longitudinal accelerometer and wind vane readings. The recorded linear accelerations were converted into the acceleration of the air- planes center of gravity. The effective aileron d
45、eflection was computed as a mean value of the sum of the right and left aileron deflections. The next step preliminary to airplane parameter estimation included a compatibility check of measured response variables in steady and maneuvering flights. The relationship between variables olv, 8, aZ, and
46、ax, and , Bv, and ay was examined from the initial steady parts of various test runs. These data showed very small scatter in values of the longitudinal and lateral accelerations and sideslip. It was, therefore, assumed that the measurements of ax, ay, and Bv were corrupted only by zero-mean random
47、noise. Then the bias errors in aV, 8, aZ, and in the form of constant offsets were deter- mined. Similar bias errors in p, q, and r were found by assuming steady flight conditions. All these estimates were verified by the analysis of tran- sient maneuvers. The compatibility check of aircraft respons
48、e variables in maneuvering flights included the prediction of V, Bv, aV, 4, and the esti- mation of constant bias errors in measured data. The technique used is based on airplane kinematic equations and an extended Kalman filter and is described in reference 7. Typical results from the compatibility
49、 checks are given in figures 2 to 5 and in tables V and VI. In figure 2 the measured and predicted responses in V, aV, and 8 are compared. A similar cozparison for the variables V, Bv, av, , and 6 taken from one of the lateral maneuvers is presented in figure 4. The resulting residuals and the standard errors of the measured responses estimated from these residuals are included in
