2009 IEEE International Advance Computing Conference (IACC 2009)Patiala, India, 6-7 March 2009

ACCURATE COMPUTATION OF BALLISTIC TRAJECTORY USINGLEAST SQUARE APPROXIMATION WITH CROUT'S METHOD

Vinay Verma and Bhavika Somani

Defence Avionics Research Establishment (DARE), DRDO

C. V Raman Nagar, Bangalore - 560093.

vinay. drdo@ggmail. com, bhavikal 06@yahoo. com

Abstract Mach number is defined as a speed ratio referenced tothe speed of sound, i.e.

Ballistic Trajectory Computation Program is a part ofweapon delivery system and it is responsible for M V (1)accurate delivery of the weapon. Mach number (M) Aand Coefficient of drag (Cd) are critical parameters inexternal Ballistic computation and are used in Where M is Mach number, V is velocity of interest,computation of Impact Point. Cd Vs M relation is not and A is velocity of sound (at the given atmosphericavailable in a functional form. These data are available conditions) [3].in discrete form from Wind Tunnel. To accomplish thispolynomial curve fitting is done. Different numerical The temperature and density of air decrease withmethods have been tried and compared to find best altitude and so does the speed of sound. Hence a truepolynomial fit. Polynomial relation is found betweenCd and M for many cases using Least Square velocity results in a higher Mach number at higherApproximation with Crout's method. It is observed, altitudes. As the aircraft speed increases andthat the results obtained by this approach are of very approaches the speed of sound or exceeds it, there is ahigh accuracy and have improved the computation of large increase in drag. The lift characteristics changeBallistic path. A comparative result analysis between and the pitching moment characteristics change due tothe Wind Tunnel data and the estimated data is .presented. Results obtained were field tested and found compressibility efects.T eperformance tanthat the performance was within the expected accuracy crability of istis ishdependentgonethin determining various parameters such as Forward aircraft's Mach number in this high speed regimen.Throw and Impact Point.

Drag is the aerodynamic force that opposes an aircraft'sKeywords - Ballistic, Wind Tunnel, Drag, Mach motion through the air. Drag is generated by every partnumber, Weapon, Impact Point and Forward Throw. of the airplane (even the engines). Drag is a mechanical

force and is generated by the difference in velocity1. Introduction between the solid object and the fluid. It makes no

difference whether the object moves through a staticCoefficient of drag and Mach number influence various fluid or whether the fluid moves past a static solidfactors in Ballistic computation and their accuracy is object. Drag acts in a direction that is opposite to the

. . . ~~~~~~~~~~~motionof the aircraft.very critical. Weapon system aerodynamicrequirements are discussed in terms of Mach number, Drag coefficient is a non dimensional coefficient andAngle Of Attack (AOA), control deflection, altitude, determining its value is more difficult because of theroll orientation, and configuration geometry. multiple sources of drag. Drag coefficients are almost

always determined experimentally using a WindAerodynamic data is obtained by Wind Tunnel Tunnel.experiments, Free flight data and Ballistic range data.This aerodynamic data is used for range computation The Ballistic Trajectory Computation Programand also for any trajectory analysis. (BTCP) implements 3D trajectory algorithm for the

delivery of weapon from aerial platform like aircraft.Several aerodynamic parameters like aircraftkinematics and altitude, environment effects such as

978-1-4244-2928-8/09/$25.OO ( 2009 IEEE 305

gravity, pressure and temperature, physical feature of T q

weapon, height of the aircraft, effects of earth rotation P =p (4)(coriolis effect), earths geometry on the falling weapon. °Numerical integration technique like Runge-Kutta A = * R * T (5)method is used to solve the trajectory computation onthe mission computer [4]. Where 2 is temperature lapse rate, T0 is temperature

at mean sea level, q = , P0 is normal pressureThe program requires the following as initial data: R A

in milli bar, G 0 is gravity at sea level, R is universal* Initial position of aircraft. gas constant and 7 is ratio of specific heats [2].* Ejection position and speed of weapon.* Delay time of weapon delivery. Mach number M and Drag D is given by:

Mach number (M) = velocity of weapon (V) / A

..~~~~~~~~~~~~~~~V-o Aircr-aft r A *6)

.... ....8LD (6).. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ .

4Weapon Weear 20b p ~jWhere air density p is given by= _ TO

Po TCd is coefficient of drag andp0 is air density at sea

level [2].

-S j:In equation (6) we have Cd as unknown and has to bedetermined using M, but Cd for a given M is notknown and has to be interpolated using a table of

RangeTarget values for Cd Vs M from experiments / flight trials. InRange F Target other words, there is no direct functional relation

Figure 1: Ballistic Trajectory between M and Cd. The relation has to be obtainedusing numerical techniques. In this paper we have used

Once weapon is dropped from the aircraft, computation Least Squares Approximation with Crout's method to

of the trajectory for the Impact Point (target) starts. The obtain functional relation between M and Cd, whichpath of delivered weapon will be calculated at certain was put to use in field trials.time interval defined by the weapon delivery 2 The Method of Least Squares for a Discretealgorithm. The trajectory of a weapon dropped from awarplane is depicted in Figure 1. Data Set

Computing the correct path of the weapon requires the 2.1 Weierstrass Approximation Theoremvelocity and position at each defined time interval andaltitude. New velocity of the weapon, at defined time If f(x) is a continuous function in [a, b] then thereand altitude, depends on estimation of gravity at given exists a polynomial (x) of degree n suchaltitude, acceleration by considering drag and thrust existsoanpolynomialupeffect on weapon, and gravity correction to the that p. (x) - f(x)l<E, > 0 [4].acceleration components.

2.2 Least Square EstimateAcceleration at given altitude is given by:

Acceleration =THRUST - DRAG (2) Let N be the number of tabulated values of f(x.) in an

MASS interval a ,Cx <b and polynomial fitting the dataWhere MASS is the mass of the weapon [2]. l

bep (x)= YC,xi. We choose c,such that theThe parameters such as temperature T, pressure Pb,~ (x i= W hoe suhta h

and velocity of sound A at an altitude z are computed reida funtinin the troposphere region using: Z (P (xV is miimm which

T = To -.Z (3) gives the Least Squares Criterion. We form the normal

306 2009 IEEE Internlationlal Advance Computing Conference (IACC 2009)

equation by equating the first order partial derivatives corresponding Coefficient of Drag from the Windto zero and solve forci . The normal equations can be Tunnel data. This data is used for interpolation.written asXXTC=Xf, whereC=[c0 c . c ]T, LetC=[c0 cl c.1]T,f=[Cdo Cdl ..... Cd.]T

f = [f(x0) f(x,) ..... f(xn)]T and X is the Vandermonde and X be the corresponding Vandermonde matrix. Thematrix. The matrix XXTis positive definite and non- normal equation in the matrix form is given by:singular.

AC=B (7)The best least squares polynomial will always passthrough all the data points, when the order of the Where A= X XT, B = Xf. Solve for C for varioussystem is one less than the number of measurements. If values of n using LU decomposition method. LUwe try to increase the order of the fit for the same method is used to solve the equations as it gives greaternumber of measurements, number of equations will be accuracy and high precision [4]. In this paper, weless than the number of unknowns, which cannot be present LSE for obtaining relation between Cd and Msolved uniquely. for two cases:

It is also of interest to compare the residuals for various 3.1 Polynomial Fit for Cd Vs M for 3 Kg Weaponorders. As the order of polynomial increases themeasure of accuracy R-Square increases and eventually Table 1 shows M and corresponding Cd taken fromgoes to one. R-Square is the square of the correlation empirical data for 3 kg weapon [11:between the response values and the predicted responsevalues. It may be incorrect to conclude from the table M Cd (M)that it is best to pick as high an order polynomial as 0.5 0.25possible. If we do this, we only ensure that the 0.6 0.253polynomial passes through each measurement. The 0.7 0.267actual order of the polynomial has to be based on 0.75 0.278physical consideration and good engineering 0.8 0.29judgement. 0.85 0.304

Table 1: Cd Vs M for 3 kg Weapon3. Polynomial Relation Between Cd and M

Various order polynomials were considered and the

Let the polynomial fitting the data be best polynomial with maximum R is chosen. Let usn We have Mach number and construct the normal equations (7) in the matrix form

Cd(M) i.W av ahnme nCd'M-Li= using Least Squares Method, where

6 4.2 3.025 2.232 1.680212 1.28577 1.642000

A = 4.2 3.025 2.232 1.680213 1.28577 0.997202 and B 1.1626003.025 2.232 1.680212 1.28577 0.997202 0.781937 0.8460252.232 1.680213 1.28577 0.997202 0.781937 0.618726 0.629934

1.1680213 1.28577 0.997202 0.781937 0.618726 0.193304 0.4779551.28577 0.997202 0.781937 0.618756 0.493304 0.395833 0.368245

Coefficients c0 cl C2 C3 C4 c5 R-squareOrder 1 0.164961 0.155294 0 0 0 0 0.916217Order 2 0.367739 -0.467241 0.462145 0 0 0 0.999296Order 3 0.512929 -1.135336 1.465904 -0.493515 0 0 0.999937

| Order 4 |0.432482 |-0.666633 |0.459201 |0.452668 |-0.328809 0 |0.999947|[ Order 5 |0.292915 |-0.027501 |-0.162446 |-0.588578 |1.898089 |-1.076454 |0.999912|

Table 2: Polynomial Coefficients obtained for 3 kg weapon

2009 IEEE Internlationlal Advance Computing Conference (IACC 2009) 307

Now we can solve the equation AC = B using the Table 4 shows the accuracy of the program inCrout's method 14] for different orders. The equation obtaining Cd.becomes co + c1 M + c2 M2 + c3 M3 + C4 M4

+.,and the coefficients obtained using Analysis of the results found in case of 3 kg weaponCrout's method are as given in table 2. Table 3 gives is graphically represented in figure 2, and Cd errorthe comparison of the coefficients obtained from the analysis is represented graphically in figure 3.BTCP and the best fit coefficients obtained from the

Cd Vs Mach Number for 3kg WeaponMATLAB using the curve fitting tool. 0.31 CdIGsTRAL r W

FLIGHT TRIAL----MATLAB

Coefficients Polynomial Polynomialcoefficients coefficients 0.29 -_

(MATLAB-'th (BTCP - 4thE2 .2

order) order)c5 49.52_LC4 -171.2 -0.328809 o0 -,/

C3 234.4 0.452668C2 --158.3 0.459201cl 52.67 -0.666633 0.24 ------ __------ 7,

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

co -6.666 0.432482 Mach number

Table 3: Polynomial Coefficients obtained from Figure 2: Cd Vs M for 3 kg weaponMATLAB and BTCP

X 10-3 Error in Cd for 3kg Weapon

4th order polynomial is considered as it is giving 20 BTCPABmaximum 12 value. Under practical considerations

15- ---

and flight trials, it was observed that 4th order -coefficients gave the higher accuracy than the 5th 10order polynomial obtained from MATLAB, and thusgained computational efficiency. 2

L 5-

3.1.1 Accuracy of BTCP over MATLAB 0

The coefficients obtained from the BTCP 4th orderthand MATLAB 5t order is used to compute the Cd at 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

the given Mach number. Mach number

Figure 3: Error in Cd for 3 kg weaponM Cd Cd Cd

(Wind (MATLAB (BTCP - 3.2 Polynomial Fit for Cd Vs M for 25 lb WeaponL 5Tunnel 5th order) 4th ord Table 5 shows M and corresponding Cd, taken from

0.5 0.25 0.2415 0.2500 empirical data for 25 lb weapon 11]:0.55 - 0.2414 0.25000.6 0.253 0.2416 0.2530 M Cd (M)0.65 - 0.2453 0.2588 0.81 0.011170.7 0.267 0.2529 0.2672 0.84 0.011780.73 - 0.2587 0.2733 0.89 0.016490.75 0.278 0.2628 0.2777 0.91 0.020070.8 0.29 0.2740 0.2902 0.93 0.022860.82 - 0.2789 0.2955 0.94 0.024780.85 0.304 0.2875 0.3040 Table 5: Cd Vs Mach number for 25 lb weapon

Table 4: Cd obtained from MATLAB and BTCPfor 3 kg weapon The coefficients obtained using Grout's reduction are

given in table 6, and table 7 gives the comparison of

308 2009 IEEE Internlationlal Advance Computing Conference (IACC 2009)

the coefficients obtained from the BTCP and the 3.2.1 Accuracy of BTCP over MATLABcoefficients obtained from the MATLAB(considering best fit) using the curve fitting tool. 2nd The coefficients obtained eom the BTCP 2th orderorder polynomial is considered as it is giving andMATLAB 4th order is usedto compute the Cd atmaximum R2 value. the given Mach number. Result shown in table 8 is of

very high accuracy.

Coefficients c0 cl C2 C3 R-squareOrder 1 -0.077196 0.107204 - - 0.949568Order2 0.501131 -1.218480 0.757374 - 0.997141Order 3 0.252576 -0.392910 -0.154099 0.334531 0.996934

Table 6: Polynomial Coefficients obtained for 25 lb Weapon

Coefficients Polynomial coefficients ( MATLAB - 4th order) Polynomial coefficients (BTCP - 2nd order)C4 -51.96C3 178.2C2 -228 0.757374cl 129.1 -1.218480CO -27.3 0.501131

Table 7: Polynomial Coefficients obtained from MATLAB and BTCP

M Cd (Wind Tunnel) Cd (MATLAB - 4th order) Cd (BTCP - 2nd order)0.81 0.01117 0. 0159 0.01110.83 - 0.0169 0.01150.84 0.01178 0.0177 0.01200.86 - 0.0198 0.01340.88 - 0.0229 0.01540.89 0.01649 0.0248 0.01660.91 0.02007 0.0290 0.01950.92 0.0312 0.02120.93 0.02286 0.0334 0.02300.94 0.02478 0.0356 0.0250

Table 8: Cd obtained from MATLAB and BTCP for 25 lb weapon

Cd vs M for 251b Weapon x 10-3 Error in Cd for 251b Weapon0.04 2

FLIGHTTRIAL ----- MATLABMATLAB .......... -TC.

0.035..-- BTCP O 0 _. /Bm ,g'S.....,....

-2<) 0.03 T- --

4-4 - _

5°0 .0aw ,'W< L1U -6 - L _

Cs 0.02-8 --

0.015 - 1 0--_

0.01 -12 l--0.80.82 0.84 0.86 0.88 0.9 0.92 0.94 0.8 0.82 0.84 0.86 0.88 0.9 0.920.94

Mach number Mach number

Figure 4: Cd Vs M for 25 lb weapon Figure 5: Error in Cd for 25 lb weapon

2009 IEEE Internlationlal Advance Computing Conference (IACC 2009) 309

Graph shown in figure 4 represents accuracy of theresults obtained using BTCP over MATLABprovided results, and graph shown in figure 5represents error analysis for the 251b weapon.

4. ConclusionDirect relation of Cd & M has to be established caseby case depending on the data availability based onthe weapon type and hence the degree of thepolynomial depends on the data. A generalizedprogram was written so that the polynomial fitting isdone dynamically suiting the data. The program alsohighlights the error level and takes as close anaccuracy as possible. In the example taken forillustration, 3kg weapon requires a 4th degreepolynomial where as 251b weapon requires only 2nddegree. The program was successfully implementedand put to field trials by including the program in themain module of the weapon delivery system. Theresults in determining various parameters like ImpactPoint, Forward Throw etc were within the expectedaccuracy.

Acknowledgement

The authors would like to convey their gratefulthanks to Dr. U K. Revankar director DARE,Mrs.Mini Cherian, Sc 'F', Mrs.R.Pitchammal Sc 'E'for their encouragement and permission to publishthis work. Thanks are also due to Mr.J.Paramashivanand Ms.Rashmi S., consultants, DARE, for all thecomputational help and preparing the manuscript.

References

[1] Flight Trial / Wind Tunnel Data.[2] R.P.G. Collinson, Introduction to Avionics (1996).[3] Frank G. Moore, Approximate Methods for Weapon

Aerodynamics (Vol 186).[4] Curtis F. Gerald, Applied Numerical Analysis (1984).

310 2009 IEEE Internlationlal Advance Computing Conference (IACC 2009)