Flight Path Prediction of an Artillery Shell Using Feed Forward Neural Networks

A.K. Ghosh*, Ankur Singhal†, Ayush Jha‡

Department of Aerospace Engineering, Indian Institute of Technology, Kanpur-208016, INDIA

Abstract

An attempt has been made to predict flight path (trajectory) of an artillery shell

using feed forward neural networks. The proposed neural models are trained with

simulated flight data of a routinely used artillery shell. These trained neural models

have been used to predict trajectories under various atmospheric and launch

conditions. The robustness of these models with respect to measurement noise has

also been tested and found satisfactory. Because the trajectory data have a number

of noisy parameters that interact in a non-linear manner, the neural modeling is an

attractive alternative to the traditional mathematical modeling and regression

techniques. The neural modeling has the ability to accommodate non-linearity and

to generalize from the data shown during training sessions. These models can be

trained using radar tracked data of an artillery shell. Such models can

advantageously be used to predict flight path of an artillery shell fired at different

atmospheric and launch conditions.

Nomenclature

CDo = Non-dimensional drag coefficients.

CLα = Non-dimensional lift force derivative

Clp = Non-dimensional spin damping derivative

Cmα = Non-dimensional pitching moment derivative

____________________________________________________ * Associate Professor, Department of Aerospace Engg., Member AIAA † Research Associate, Department of Aerospace Engg. ‡ Graduate Student, Department of Aerospace Engg.

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AIAA Atmospheric Flight Mechanics Conference and Exhibit15 - 18 August 2005, San Francisco, California

AIAA 2005-5820

Copyright © 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

d = Diameter of the artillery shell

go = Acceleration due to gravity, m/s2

Ix = Moment of inertia about x axis, kgm2

m = Mass of the shell, kg

p = Spin, rad/s

R = Radius of earth, m

r = Distance from earth center to center of gravity of the shell, m

t = time of flight, sec.

ur = Velocity vector

V = Total velocity, m/s

, ,x yW W Wz = Head/tailwind, crosswind and vertical wind components, m/s

, ,x y z = Spatial coordinates, m

rα = Yaw of repose

ω = Rotation vectors

Ω = Earth rotation angular velocity, rad/s

ρ = Density of air, kg/m3

θ = Launch angle, rad.

6400 mils = 360o

I. Introduction

Artillery comprises an important wing of army in providing fire power during both war and cross-

border skirmishes with the enemy. Artillery shells are a class of projectiles around which much of aero-

ballistic theory was originally developed and it continues to form significant part of aero-ballistician’s

interest. The effectiveness of artillery shell is largely judged by the accuracy in hitting the targets. Various

error sources inherent in the artillery systems, together with external conditions such as wind, temperature

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variations, etc. cause dispersion of payload from it’s intended path. The actual path traversed by the shell is

compared with the predicted trajectory in order to calculate it’s accuracy.

The requirement for a trajectory program, needed for predicting flight variables are (i) trajectory to

be three dimensional, (ii) provision be made for arbitrary wind velocity, azimuth and other meteorological

conditions and (iii) non-linear aerodynamics with respect to flow incidence angle be included. Trajectory

simulation for such vehicle is generally done by solving six-degrees-of freedom equation of motion.1

However, the in determinability of many initial conditions and aerodynamic coefficients, which are

required as input, results in model not giving desired result. Even if accurate aerodynamic coefficients are

available, there are several other uncertainties, especially near the launcher exit, that would effect trajectory

predictions.

The recent interest in the evolving application of artificial neural networks (ANNs) to diverse

fields such as signal processing, pattern recognition, robotics, medical diagnosis, system identification and

control have led many researchers to explore their capability for aerospace engineering problems. The

neural modeling has been employed in solving aerospace problems such as aerodynamic modeling,2

buffet,3 fatigue crack growth,4 design of civil aircraft,5 aircraft parameter estimation from flight data6,7 etc.

Recently, Ghosh, et al8 presented an alternative approach to mathematical modeling, used hitherto for

predicting shell performance in terms of the launch angle (elevation) needed for required range. The neural

model is shown to be a viable way of modeling many input variables that affect the relationship between

the range and the launch angle. It has been envisaged that the measured range data for shells could be used

for developing neural models that would be useful in field applications, including finding the launch angles,

the time of flight and the drift angle for the specified range. However, for different range of launch angles,

all the neural models proposed8 were capable of predicting a single value for range and drift. These models

do not have the capability to predict flight path (trajectory) of the traversing shell when fired at different

launch angles. In the present work, an attempt has been made to predict flight path of an artillery shell

using neural models. The proposed neural models are trained with the simulated flight data and once

satisfactory training is achieved, these have been used to predict trajectories under varied atmospheric and

launch conditions. The robustness of these models with respect to measurement noise has also been tested

and found satisfactory. Because the trajectory data have a number of noisy parameters that interact in a

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non-linear manner, the neural modeling is an attractive alternative to the traditional mathematical modeling

and regression techniques. The neural modeling has the ability to accommodate non-linearity and to

generalize from the data shown during training sessions.

II. Generation of Simulated Radar Tracked Trajectory Data and Neural Modeling

Due to non availability of flight data (x,y,z) of a traversing shell in varying wind and

meteorological conditions, simulated data were generated using modified point mass model1 (MPM). The

linear and angular motions are modeled using following equations of motion.

( ) ααπρ

ααπρ

rVC LmdvvrC DC Dom

ddtud 2

8

2228

2++−=

rr

2

3 2( )o

R rrg ω− +

r u× r (1)

plpvCxI16

4ddtdp πρ

= (2)

( )[ ]r

43r αuv8

αv

/dtdCd

pI

m

xrr

×−=

απρ (3)

The quantities in the above equation are defined as follows:

dtxdur

= , , W-uvrrr

= R-xrrr

= , [ denotes the wind vector] ( 0,R- 0,R =r

) Wr

[ ] [ ,azimuthcoslatitudecos-(ω Ω=r ]

ΩSin[latitude],

ΩCos[latitude] Sin[azimuth])

where Ω = 7.29×10-5 rad/s (rotation of the earth),

R = 6370320 m (radius of the earth),

gο = 9.80665[1-0.0026373 Cos(2×latitude) +0.0000059 [Cos(2× latitude)]2].

The geometric, mass and moment characteristics, launch conditions and aerodynamic coefficients

of a routinely used artillery shell (supplied by the manufacturer) were fed to this model as input. These

equations were solved using fourth order Runge-Kutta method for solving simultaneous differential

equations. Equivalent constant wind (ECW), along with equivalent ballistic air temperature (BAT) and

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ballistic air density (BAD) representing non-standard atmosphere were used in simulation. Various sets of

trajectories were generated covering launch angles from 0.7mils to 802.3mils. The launch velocity was

fixed at 818m/s. During the generation of the trajectories, corresponding to different launch angles, ballistic

air temperature and ballistic air densities were varied within a range of ±10 % over the corresponding

values in standard atmosphere. The maximum variation in head/tail Wx, crosswind Wy were restricted to

10m/s. The computed flight path (x,y,z) of the projectile was obtained using MPM model. This data will be

referred to as radar tracked trajectory data or simply as measured trajectory data for future reference. For

neural modeling, data set consisting of trajectory (x,y,z) data along with the wind and meteorological

conditions were used to select appropriate input/output (I/O) pairs for the neural model. It may be

mentioned that MPM model was used since real radar tracked trajectory data were not available. In actual

practice, to generate radar-tracked data, a series of artillery shells would be fired at different launch angles

under various atmospheric conditions. The information about its location in flight (x,y,z) would be acquired

by the tracking radar. The information about wind conditions (head/tail Wx, cross wind Wy), BAT, BAD

etc. would be obtained following routine artillery procedures.1

The neural model does not require either the postulation of a mathematical problem, nor an

estimate of initial conditions at the time of the shell leaving the launcher. Whereas, functional mapping of

the I/O pairs create a black box type of neural model, the initial conditions are taken care implicitly by the

mapping. The input vector required for neural training (Fig. 1) consists of longitudinal location (X), launch

angle (θ), ballistic air temperature (BAT) and ballistic air density (BAD) as its elements and height (Z),

lateral drift (Y) consist the output vector. The feed forward neural networks for the present study were

simulated by using the neural network tool-box of MATLAB 5.3. The activation function used was the

sigmoidal function and a back propagation algorithm was used for training the network. A set of I/O pairs

selected randomly were used to arrive at a minimum number of I/O pairs required for adequate training of

the network. In real life, the number of measured I/O pairs available may be limited due to cost involved in

collecting such radar tracked data, hence the search for minimum number of data samples to achieve an

acceptable neural model.

To start with, around fifty complete simulated trajectories (corresponding to different elevations,

atmospheric conditions) were used for training session. Systematically, this number was brought down to

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seventeen and was found that further reduction in the number of the trajectories, deteriorated the validation

and prediction estimates. It may be noted that, for validation and prediction phase, different sets of

complete trajectories were used. The rule of thumb used is that, for the validation data, the mean square

error (MSE) is only of the order of two to three times or less than the MSE prescribed for the training

phase. For prediction, a set of randomly selected input data is taken and presented to the validation

network.

III. Results and Discussion

The trained neural models were applied to predict trajectories corresponding to different

atmospheric and launch conditions. The input vector required for neural training (Fig. 1) consists of

longitudinal location (X), launch angle (θ), ballistic air temperature (BAT) and ballistic air density (BAD)

as its elements and height (Z), lateral drift (Y) consist the output vector. After completion of training and

validation phase, the neural models were used for the trajectory predictions under various atmospheric and

launch conditions. A typical trajectory is presented in Fig. 2. Figure 2 shows that the predicted height (Z)

compares well with the measured data. Table 1 presents the comparison between predicted and measured

height for various atmospheric and launch conditions at a particular longitudinal location (X). It is observed

that the difference between the predicted and measured height is consistently low. A typical comparison, at

a specified height of the trajectory between predicted and measured range (longitudinal location of the fired

shell) corresponding to different elevations is presented in Table 2. Referring Col. (3) and Col. (4) of Table

2 it can be seen that error in the prediction is fairly low. Further, the same model was used to predict the

flight path of the same artillery shell, when fired under standard atmospheric conditions. To predict the

performance under standard atmospheric conditions, the input vector Wx, Wy, BAT and BAD were set to

zero. The predicted trajectory matched very closely with the measured trajectory.

In actual application, the measured trajectory will have measurement noise contained in it. In order

to investigate the robustness of this proposed method with respect to such noisy data, the neural model was

trained with noisy trajectory data. The study was conducted with 5% measurement noise. Figure 3 presents

a comparison between measured and predicted trajectory, for such case. Here again, excellent matching

between predicted and measured flight path was obtained. Further a comparison between values of

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measured and predicted drift for noisy data (5%) is also presented in Fig. 4. It could be seen that the

matching between predicted and measured drift is in good agreement. Based on these studies, it can be

concluded the proposed neural models could advantageously be applied to predict trajectory using neural

network for artillery shell.

IV. Conclusion

This study presents an alternative approach to mathematical modeling used hitherto for predicting

trajectory of an artillery shell for a given launch angle and varied meteorological conditions. The neural

model is shown to be a viable way of modeling many input variables that affect the relationship between

trajectory coordinates and launch conditions.

References

1“External Ballistics”, Text-Book of Ballistics and Gunnery, Vol.1, 1st ed., Her Majesty’s Stationary Office,

London, 1987

2Hess, R. A., “Use of Back Propagation with Feed Forward Neural Networks for the Aerodynamic

Estimation Problem,” AIAA paper 93-3638, Aug. 1993

3Jacobs, J,H., Hedgecock, C.E., Lichtenwalner, P.F., Pado, L.E., and Washburn, A.E., “Use of Artificial

Neural Networks for Buffet Environments,” Journal of Aircraft, Vol. 31, No. 4, 1994, pp. 831-836

4Pidaparti, R.M.V., and Palakal, M.J., “Neural Network Approach to Fatigue-Crack-Growth Predictions

under Aircraft Spectrum Loadings”, Journal of Aircraft, Vol. 32, No.4, 1995, pp. 825-831

5Patnaik, S.N., Guptill, J.D., Hopkins, D.A., and Lavelle, T.M., “Neural Network and Regression

Approximations in High-Speed Civil Aircraft Design Optimization”, Journal of Aircraft, Vol. 35, No. 6, 1998, pp. 839-

850

6Ghosh, A.K., Raisighani, S.C. and Khubchandani, S., “Estimation of Aircraft Lateral-Directional

Parameters Using Neural Networks”, Journal of Aircraft, Vol. 35, No.6, 1998, pp. 876-881

7Ghosh, A.K., and Raisighani, S.C., “Frequency –Domain Estimation of Parameters from Flight

Data Using Neural Networks”, Journal of Guidance, Control and Dynamics, Vol. 24, 2001, pp.525-530

8Ghosh, A.K., Raisighani, S.C. and Dehury, S.K., “Modeling of Performance Of an Artillery Shell

Using Neural Networks”, Journal of Spacecraft And Rockets, Vol. 39, No. 3, 2002, pp. 470-471

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Table 1 Comparison of Measured and Predicted Trajectory for Non-Standard

Atmospheric Conditions

Elevation, mils Range, X,

m %BAT % BAD Wx , m/s Measured Height, m

Predicted Height, m

Difference, m

60.4 323.2 10 -2 -2 18.4 32.2 -13.8017 60.4 1551.9 10 -2 -2 73.0 73.4 -0.39996 60.4 3738.9 10 -2 -2 99.1 92.9 6.230915 60.4 4418.1 10 -2 -2 84.9 79.9 5.032427 60.4 5121.5 10 -2 -2 56.8 54.1 2.757765

386.1 2321.2 -8 -1 4 873.1 869.2 3.855486 386.1 3575.0 -8 -1 4 1294.7 1297.2 -2.46594 386.1 4166.0 -8 -1 4 1479.0 1483.7 -4.66857 386.1 11775.1 -8 -1 4 2575.1 2585.7 -10.5766 386.1 12454.9 -8 -1 4 2502.7 2514.5 -11.8495 386.1 15746.3 -8 -1 4 1495.2 1520.4 -25.2589 386.1 16344.9 -8 -1 4 1174.0 1199.0 -25.0186 738.6 2584.1 6 -2 -5 2190.6 2194.8 -4.18856 738.6 4136.2 6 -2 -5 3387.3 3398.3 -11.0331 738.6 5937.9 6 -2 -5 4626.8 4639.2 -12.3282 738.6 6134.4 6 -2 -5 4750.6 4762.4 -11.7322 738.6 6672.2 6 -2 -5 5076.7 5086.1 -9.38713 738.6 10882.2 6 -2 -5 6837.4 6819.9 17.39744 738.6 13419.5 6 -2 -5 7028.9 7025.4 3.550655 738.6 13873.0 6 -2 -5 6983.2 6982.9 0.204872 738.6 18542.8 6 -2 -5 4952.1 4926.2 25.96127

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Table 2 Comparison of Measured and Predicted Range for Standard Atmospheric

Condition

Elevation, mils

Measured Range (X), m

Measured Range at Height of Comparison (X), m

Predicted Range at Height of Comparison, m

Height at Which Comparison is made, m

738.6 22800 22705 22698 8.0 657.1 22100 21955 21995 82.0 575.2 21100 20925 20985 108.0 521.5 20300 20110 20172 108.0 451.8 19100 18957 19005 67.0 386.1 17800 17654 17695 54.0 140.8 10700 10587 10603 6.0 60.4 6100 6005 6066 3.0

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∑ ∑ ff Height/Drift Coordinate

InputLayer Output

Layer

λ −+ /x 1 ∑ ∑ f f

∑ ∑ f f ∑∑ ff

∑ ∑∑ ff

∑ ∑ f f ∑ ∑ ff

f =

f

1 e

X X θ Wx/ Wz

% BAT % BAT % BAD % BAD

Fig. 1 Schematic for the Neural Model

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Fig. 2 Comparison of measured and predicted trajectory for non-standard atmospheric condition

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Fig. 3 Comparison of measured and predicted trajectory for non-standard atmospheric condition with 5% measurement noise

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Fig.4 Comparison of measured and predicted drift for non-standard atmospheric condition

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