This paper describes the application of the maximum likelihood method to estimate the aerodynamic derivatives of (i) the AGARD standard ballistic model HB-2 from supersonic free13; flight data available in literature, and (ii) ballistic mode simulated data of a surface-to-air missile.13; For the AGARD dam, two types of dynamic models have been used: (i) body axis coordinate13; system model, and (ii) hybrid type of formulation. wherein the aerodynamic forces are in wind axis13; arid the aerodynamic moments are represented in body axis coordinate system. For surface-to-air13; missile data the equations of motion are solved in fin-body axis system. The results of this analysis13; denonstrate the suitability and function adequacy of the mathematical models used, the13; consistency of data arid power of the parameter estimation methodology in generating13; aerodynamic derivatives from realistic free flight trajectories

Gopalratnam, G

Raol, JR

NAL-IR

Estimation of aerodynamic derivatives of projectiles from aeroballistic range data using maximum likelihood method

National Aerospace Laboratories Institutional Repository

UDC 629.762 Estimation of Aerodynamic Derivatives of Projectiles from Aeroballistic Range Data using Maximum Likelihood Method Dr J R Raol, Nonmember (Ms) G Gopalratnam, Non-member niis paper describes tlie application of the maxiniuni likelihood method to estimate the aerodynamic derivatives of ( i )  the AGARD sfandard ballistic niodel HB-2 from supersonic free flight data available in liferature, and (ii) ballistic mode simulated data of a surface-to-air missile. For the AGARD dam, two types of dynamic models have been used: (i) body axis coordinate system model, and (ii) hybrid type offorniulafioii. wherein flie aerodynamic forces are irr wind axis arid the aerodynamic monients are represenfed in body axis coordinale system. For surface-to-air rnissile data tlie equations offnotion are solved in fin-body axis systern. The results ofthis analysis denionsfrate the suitabiliry arid fiincliortol adequacy of the mathernutical models used, the consistoicy of data arid power of tlie parameter estimation r?iefhodology in generating aerodytiarnic derivatives froin realistic free flighf trajectories. reference area axial force coefficients in the body- coefficients of drag, lift and normal coefficients of rolling, pitching and inertia mass angular velocity components about the body-fvred axes ox, oy, 02, respec- tively fixed coordinate system force yawing moments dynamic pressure . flight path velocity axes in the body-fvred coordinate velocities in ox, oy, oz axes system angles of attack and side slip Euler angles vector of unknown initial conditions vectors of unknown parameters to be gust components in fin, body axes transformation matrices from tin to body and body to inertial axes suffuces for fin, body and gust system estimated progress for well over three decades. But the methods used for analysis place several restrictions on the equations of motion; for example, assumptions such as h e a r  aerodynamics, constant roll rate, small velocity variations, etc. In order to eliminate such restrictions and pave the way for complex and contemporary model configuration testing in ballistic ranges, advanced statistical methods of parameter estimation such as maximum likelihood (ML) and extended Kalman filter methods which have been suc- cessfully validated for aircraft flight data’s2 over the past decade have been recently extended to free flight analysis problems of missile configurations a l s ~ ~ . ~ .  This paper describes the application of the maximum likelihood method to estimate the aerodynamic derivatives of (i) AGARD standard ballistic model HB-2 from super- sonic free flight data available in literature, and (ii) ballistic mode generated simulated data of a surface-to-air missile (SAM). For AGARD data, axes systems used for repre- senting the mathematicalmodels: body axis and hybrid type formulation wherein the aerodynamic forces are in wind axis and the moments are in body axis system. For SAM data the equations of motion are solved in tin-body axis systems with appropriate transformations between fin and body axis systems. In order to be consistent with the two original data sets, the F’F’S and S1 systems of units have been used. MAXIMUM LIKELIHOOD METHOD The ML method as applied to free flight trajectory of ballistic range usually begins with the mathematical model INTRODUCTION of the flightvehicle configuration whose equations of no- tion are formulated in general terms as Free flight testing in aerohallistic ranges with aview to ~ extracting the aerodynamic parameters from the measured state quations :; (0 = J (x (1), P ) , ~  (0 =x+.~ Observation equations : motion patterns of missile configurations have been in (1) y ( t )  = h (x (t), P )  (2) Dr J R Raol and (Ms) G Gopalrainam are with National Aeronautical Laboratory. Bangalore. Ranchi, January 28, 1989. Revkd manuscript was received on where x(1) is II X 1 state vector,y(t) ism x 1 observation October 22,1990. vector, a n d  z(k) is t h e  ( n i x l )  measurement Measurement equations : L (k) =y (k) +V (k); This p p r  w w  presented and discussed a1 B e  Annual Papr Mrrling k =1, .,...__._.... $Y (3) Vol71, September 1990 l3 Body velocity to inertial velocity transformation is ( i , j , i  ) = T2 { ub vb wb 1, where T2 is a direction cosine matrix. Deterministic (gust) in ut excitation transformations are and .r (u@, vpb. ~ g b ] ~ T z  9 ‘  ( wndu - 0 ,  wind Y,  wind w } (u&v&f.wd} = TlTIUgb.V@,W@I. Angle of attack and side slip are given as a = t a d  { (wr- wd)  /(ur-ud) }; /3 = tan-’ ( ( vf- vd) / ( u f - u d )  &th and tanaT = J@., - v$2 + (w, - Wgf)Z I ( uf- U ~ )  v = (tl,-u& + ( Y p g Y  + (wrwgf)  ’ .......................... - 1 . .~ ~, {., , -., e I . .  .I.,~(* ....... e.1.. , ( I -  I , , ~ ,,/---- ..__. .... o. I.. . .  ..-- d 0 I” .................... . . . . . . . . . .  , .”. 1 . . . . . . . . . . . .  . .* --L.-,.-... . .  L_._L__I .... I. -.--l ........ .- .. ..... ._ .... .... .. ........ -0.4.. . . . . .  J 0.a70 ........ 1 u  I Fig 4 Time I)iaay march m These transformation equations are given for the sake of completion. For clarity the various transformations and operations involved in modelling are shown in Fig 3. The SAMtrajeUorywasusedasdatainputtotheMLEsoftware for estimation of aerodynamic derivatives as incorporated in this tin-body model. The estimation was started with initial values of some of the derivatives as 20% off from the original derivative values. The estimates were almost close to the original reference values. The results of time history match between S A M  trajectoryand ML predicted data are shown iu Fi4. The purpose of this exercise was to evaluate feasibility of &-body type of modelling in analysis of bal- listic type data. The various results from these three types of models seem consistent with the data sets used except for some differences. CONCLUSION A preliminary investigation of extracting non-dimen- sional stability derivatives from supersonic free flight data measurements in an aeroballiticrange using the maximum likelihood method has been successfully demonstrated. Analysis and estimation using various trajectories (AGARD, SAM) have validated the utility of various types of mathematical models including tin-body transforma- tions. Thus the parameter estimation methodology has been established for the kind of data that could arise out of aeroballistic range experiments being presently conducted in the country and similar or related realistic data of a missile or a projectile. ACKNOWLEDGMENTS The authors acknowledge the support of Defence Re- search and Development Laboratory in carrying out this work and their permission to publish this paper. The authors are thankful to Director, NAL and Head, FMCD/NAL,fortheu moralsupport. Also reviewers’ com- ments and suggestions to improve the presentation of the paper are highly appreciated. REFERENCES 1. R V JalegaonLar and S Balakrishna. ‘Effects of Flap Position on Longitudinal Parameters of HFB-320’. NAL TM SE 8602, March 1987. 2. S Srinathkumar, V Parameswaran and G Girija. %slimation of the Longitudinal Stability Characteristics of a Fighter Aircraft fmm Right Data’. Rocecdinp of rhr Nafionol Seminor on Fli& Rescorch, Tutond Development', Bangalore, September 2-3, 1987. 3. V S Mohan Ram and C V Srinath Sartry. ‘Nonlinear Maximum Likelihoad Estimation o[ Stability Derivativea from Range Data’. NALTM SE 8608. December 1986. 4. G T Chapman and D 6 Kirk. ‘A Method of Ertracting Aerodynamic Coefficients from Free Flight Data’. A I M  Journal, vol8, April 1970, p 753. 5. R V Jatcgaonkar and E Plxtschfe. ’Maximum Likelihood Estimation of Paramners in Nonlinear Flight Mechanics Systems’. Smndt IFAC Symposium on Idenfification and S y s m  PormevI  Esrimorion, UK, July 1985. 6. G N Malmlm and G T Chapman. ‘A Computer Program for Splernstically Analyzing Free Flight Data to Detsmine the Aerodynamia d An-symmetric Bodies’. NASA TN D 4766, September 1968. ._ .... Mass IL.*ih, R l  vector sampled at N discrete points. The m x 1 sequence of measurement noise V (k) is assumed to be gaussian with zero mean and covariance matrix of R.  Based on the ballistic output time history data a t  N discrete points, the parameter I ',y I in 10.a- 1 r.t15 I ' Ko,d I CF l. . in i. I L u g . , l l  , . i s  2 . 0 .  ] b - E 4 5  ' J E 4 '  nof uniformly sampled in time,acubicsplineinterpolation3 I." - - M N . Y l d  t*-t..d rmtine was utilizeu to curve tit the experimental motions and generate adequate number of uniform time-spaced data samoles to be used in the ML estimation Droeramme. J -0 2% 0 ,% 1 9  . -  The a, p trajectories were reconstructed using the for- mulaea = fl + tan-'(dz/dx)and,¶ =-p + tad(dy1dx). < I  L Fig 2 illustrates the matching between the AGARD ex- perimental trajectory and the ML predicted trajectory : * .D 1-e " {-'---I-"---- based on the models described elsewhere. The close match is obvious. Some of the important aerodynamic derivative estimates along with the $rcentagc standard deviations are summarised in Tablel. For the sake of comparison, the NASA estimates6 of the longitudinal derivatives obtained for the same set of flipht data are also shown in the table. The results indicate that the MLE method, in addition to providingvery satisfactory and acceptable estimates for the longitudinal derivatives, a h  simultaneourly determines thelateralderivativcs, whichwasnot thecase fortheNASA technique. The NASA technique was based on least squares approach to estimate CD from x ,  t data in iterative manner and gaussian least squares differential technique to further extract other derivatives from the range data. :'. -I\-" ,". I --*._..-. 1 ., . ._., ----------/- A- - m' T[~ ," _... ;".- , , . , , , I.",(. .... l l f l i , .  I Erlirnstn V-a+ Model Body Axis Model 1.739 1.207 (0.12) 1.190 (0.15) Fig 2 Free flight rime histories cD 3.590 3.155 (0.59) 4.353 (0.38) tif = rfvr-q, w f - g  sins - Ckl c., =,"a 24.400 49.524 (13.6) 48.065 (12.13) -1.259 -1.277 (0.21) -1.277 (0.20) - -90 190 (2 78) ' -92 780 (2 56) C", ( % standard dewation ) The absence of roll angle, +, trajectory information is probably the cause for fairly high uncertainty noticed in the estimation of the C derivative.The axi-symmetricity of the AGARD model configuration is established by almost identical values (with proper signs) obtained in both pitch and yaw planes, demonstrating the functional adequacy of the mathematical models used. The differences in es- timates across the models may be attributed to different degree of information processingin two non-linear models. Since the data set and the estimation procedures are the same for these two types of mathematical models, the differences in some of the estimated parameters could arise from the degree of parameter identifiability governed by these models and their associated transformations. ANALYSIS OF S A M  BALLISTIC DATA The 6-DOF trajectory of a model configuration of a surface-to-air missile (SAM) in a simulated ballistic mode was taken for analysis. The trajectory contained 'P time histories of ufi v j  wfipj qj rfi p, 6,$ and r, y, z data. The model fitted to this data is given below: and for the present case Q = 45O. V d  71, Sepember 1990 i4p 

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Estimation of aerodynamic derivatives of projectiles from aeroballistic range data using maximum likelihood method

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