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Three-Dimensional MidcourseGuidance Using NeuralNetworks for Interception of

Ballistic Targets

EUN-JUNG SONG

MIN-JEA TAHKKorea Advanced Institute of Science and Technology

A suboptimal midcourse guidance law is obtained for

interception of free-fall targets in the three-dimensional (3D)

space. Neural networks are used to approximate the optimal

feedback strategy suitable for real-time implementation. The fact

that the optimal trajectory in the 3D space does not deviate much

from a vertical plane justifies the use of the two-dimensional

(2D) neural network method previously studied. To regulate the

lateral errors in the missile motion produced by the prediction

error of the intercept point, the method of feedback linearization

is employed. Computer simulations confirm the superiority of

the proposed scheme over linear quadratic regulator (LQR)

guidance and proportional navigation (PN) guidance as well as

its approximating capability of the optimal trajectory in the 3D

space.

Manuscript received May 5, 1999; revised May 9, 2001; releasedfor publication December 11, 2001.

IEEE Log No. T-AES/38/2/11431.

Refereeing of this contribution was handled by T. F. Roome.

Authors current addresses: E-J. Song, School of Mechanicaland Aerospace Engineering, Seoul National University, Seoul,Korea; M. J. Tahk, Division of Aerospace Engineering, Dept. of Mechanical Engineering, Korea Advanced Institute of Science andTechnology, 373-1, Kusong Yusong Taejon 305-701, South Korea,E-mail: ([email protected]).

0018-9251/02/$17.00 c 2002 IEEE

I. INTRODUCTION

For most ground-based defense systems beingconsidered today, an essential element of midcourseguidance is to guide the missile with the maximumprobability of collision. Midcourse guidance is oftenformulated as an optimal control problem to shapethe trajectory to maximize the terminal energy or tominimize the flight time. The implementation of the

optimal midcourse guidance law is not easy sincea nonlinear two-point boundary value problem isto be solved to obtain the optimal trajectory. Directnumerical solutions to this problem introducesa heavy in-flight computational burden and theconvergence characteristics may not be acceptable.Solving the problem in real time is often not feasible.Furthermore, a feedback guidance law is not readilyprovided from the numerical solution.

Several techniques such as a perturbationprocedure [13], linear quadratic regulator (LQR)with a database of the optimal trajectories [4, 5],and modified proportional guidance [6] have been

previously proposed for the real-time implementationof optimal midcourse guidance. Recently, theapproximation ability of the artificial neural network [7] is used to derive an on-board guidance algorithmsuitable for real-time implementation [8]. The keyidea of this approach is to train a neural network toextract the functional relationship between optimalcommands and missile states from the set of theoptimal trajectories computed in advance. Then, thetrained network constitutes a feedback guidance law,which allows the missile to adapt perturbations in thetarget states and its own trajectory as well as to followthe optimal trajectory. For implementation, onlythe weights of the network are needed to be stored.The neural-network approach has been extended fortwo-dimensional (2D) midcourse guidance againstmoving targets [9]. For intercept point prediction, atime-to-go estimator using neural networks has beendevised to consider the time-varying characteristicsof the missile velocity. Also, a more elaborateneural-network scheme has been derived toimprove the robustness to variations in the launchcondition [10]. For this purpose, a new input patternthat is robust to the launch condition has beenemployed.

In this article, the neural-network approach isextended for three-dimensional (3D) midcourseguidance of a missile system to interceptnonmaneuvering targets decelerated by atmosphericdrag. Although many applications of neural networkson missile guidance and control have appeared withgrowing interest [1113], no application to the 3Dmidcourse guidance problem has been attempted.Direct extension of the previous neural-network method to the 3D midcourse guidance problem isnot plausible since the number of data to be learned

404 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 38, NO. 2 APRIL 2002

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by the neural network is tremendously larger thanthe 2D case. The neural network should also have avery complicated structure. To avoid this difficulty, wepropose a guidance law for which the neural-network approach is used for the guidance in the verticalplane but the feedback linearization technique [14]for lateral control. This approach is motivated by thefact that the optimal trajectory is confined to a verticalplane if the correct intercept point is known and the

missile is launched vertically or toward the interceptpoint. Hence, if the error in prediction of the interceptpoint is corrected during the initial phase of the flight,then the proposed 3D guidance method is expected towork well.

To predict the intercept point accurately, we needto precisely compute the time to go of the missile. Forthe prediction of the missiles flight time, we use anadditional neural network that learns the time-to-gocharacteristics from the optimal trajectory data. Sincethe target is supposed to be intercepted at a highaltitude, it is reasonable to assume that the targetmotion is affected only by the gravity forces. Hence,the target trajectory is a Keplerian orbit and the futureposition can be computed without direct integration of the equation of motion.

In this study, a trade-off study with other existentmethods such as LQR with a database of optimaltrajectories and proportional navigation (PN) guidanceis performed to demonstrate the effectiveness of theproposed neural-network approach. The performanceof each method is evaluated in terms of memoryrequirements, optimality, and intercept performance.

This paper is organized as follows. Themathematical model of a typical medium-rangesurface-to-air missile is introduced first. Theformulation and development of the midcourseguidance law using neural networks are thendescribed, and the prediction algorithm with thetime-to-go estimator is explained. Next, simulationresults are presented to investigate the performance of the proposed scheme. Finally, the conclusions of thiswork are summarized.

II. MATHEMATICAL MODEL

For problem formulation, the Earth-centeredEarth-fixed (ECEF) frame and the navigationnorth, east, down (NED) frame are used [15]. Theinterrelationship between the coordinate systems andthe state variables are illustrated in Fig. 1, where denotes the Earth rotational speed. The Earth isassumed to be a spherically symmetric body. The statevariables are the missile position in the ECEF frame(r , , ), the missile velocity relative to the ECEFframe v, and the flight-path angle and heading angle, respectively. The control variables are the angle of attack and the angle of total lift direction , calledthe bank angle (the missile need not be a bank-to-turn

Fig. 1. Geometry of coordinate frames ( x, y, z: inertial frame, xe, ye, ze: ECEF).

type). The equations of motion are given by

_r = vsin (1)

_ =vcos sin

r cos (2)

_ =vcos cos

r(3)

_v =(T cos D)

m g sin

+ r 2(cos 2 sin cos sin cos cos ) (4)

_ =(T sin + L)sin

mvcos +

vsin cos sin r cos

+r 2 sin cos sin

vcos

2 cos sin cos cos

+ 2 sin (5)

_ =(T sin + L)cos

mv

g cos v

+vcos

r

+r 2

v (cos 2 cos +sin cos sin cos )

+ 2 sin cos (6)

where

L = 12 v2SC L, C L = C L( 0)

D = 12 v2SCD , CD = CD0 + kC

2 L:

The aerodynamic derivatives C L, CD0 , and k are givenas functions of Mach number M , which is a functionof v and the altitude h:

C L = C L( M ), CD0 = CD0 ( M ), k = k ( M ):

For the spherical Earth model, the expression for g isas follows:

g = g0ReRe + h2

where g0 is the gravity magnitude at the surface of the Earth and Re is an effective radius of the Earth.The missile mass and thrust are given as functions of time t:

m = m(t), T = T(t):

SONG & TAHK: THREE-DIMENSIONAL MIDCOURSE GUIDANCE USING NEURAL NETWORKS 405

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The position and velocity of the target are providedby a ground support system to the missile forcomputation of the intercept point. This studyconsiders a ballistic target that is only affected by thecentral gravitational field.

III. REAL-TIME SUBOPTIMAL MIDCOURSEGUIDANCE

For long-range and medium-range missiles, theoptimal trajectory shaping ensures an extended rangeand more favorable endgame conditions. However,its direct formulation based on optimal control theoryresults in a two-point boundary value problem, forwhich the computational time is too long for real-timeimplementation. To solve this problem, a suboptimalguidance method using neural-network approximationhas been suggested in [8, 11]. A neural network istrained to learn the closed-loop optimal commandin terms of the current missile states and terminalconditions,

u(t) = g(x(t), x f ) (7)

using the optimal trajectory data generated off-line.In [11], a terminal guidance law is synthesized tominimize the kill vehicles divert propellent. Theyshow that the terminal guidance law implemented bya neural network can increase the engagement range.The feasibility of the neural-network approximationhas also been investigated for a midcourse guidanceproblem constrained in the vertical plane [9].Modifications of this work have also been proposedin [10] to provide robustness against variations in themissile launch condition. One of the most importantaspects of neural-network design is how to constructthe training data [16]. Reference [10] shows thatreorganization of the input patterns reduces thesensitivity of the neural-network guidance schemeto the trajectory perturbations produced by thelaunch error. In this study, we use the _-feedback guidance law proposed in [10], for which the optimalangle-of-attack command is assumed to bedependent on the line-of-sight (LOS) rate _ ratherthan the flight-path angle ;

= (v, _, x x f ,h h f ) (8)where

_= (h h f )vcos + ( x x f )vsin

( x x f )2 + ( h h f )2: (9)

A straightforward way to extend the 2Dneural-network guidance method to the 3D guidanceproblem is to train a neural network to learn the 3Doptimal trajectory data. However, this requires a largeamount of training data as well as a complicatedstructure of the neural network:

= g(v, _pitch , _yaw , x x f , y y f , h h f ):(10)

Fig. 2. Definition of guidance plane.

On the other hand, if the effect of Earth rotationis neglected and the missile is launched vertically,the optimal missile motion is confined within avertical plane determined by the missiles initialposition and intercept point, denoted as the guidanceplane in Fig. 2. Hence, if prediction of the interceptpoint is accurate, the optimal 2D missile motionin the guidance plane can approximate the optimal3D motion. The 3D guidance command is thendecomposed into two commands; one to track theoptimal trajectory in the guidance plane and another tohandle the lateral motion of the missile not to deviatefrom this plane. The neural-network guidance law of (8) is used to implement the former while feedback linearization is employed for the latter.

A. 3D Guidance Law

The 3D guidance law generates commands forthe angle of attack and the bank angle . Using theneural-network guidance law of (8), is commandedas

(v, _,q ( xN I xN M )2 + ( xE I xE M )2 , xD I xD M )(11)

where ( xN I , xE I , x

D I ) and ( x

N M , x

E M , x

D M ) are the predicted

intercept point and current missile position in theNED frame defined at the launch position of themissile, respectively.

The role of the bank-angle command is to steerthe missile to the direction of the predicted interceptpoint given by

= tan 1

xE I x

E M

xN I xN M : (12)The dynamics in (5) is rewritten as_ =

(T sin + L)sin mvcos

+ (13)

where represents the last four terms of theright-hand side (RHS) of (5). These terms areproduced by the rotation and roundedness of theEarth. Inspection of (13) shows that we can choose


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Fig. 3. Guidance law implementation for intercept in 3D space.(a) Neural-network guidance. (b) LQR with database of optimal

trajectories.

as

= sin 1

mvcos (ulateral )

T sin + L , jj 2

(14)

to cancel the nonlinear terms, and to linearize the mapfrom ulateral to , which is described by

_ = ulateral : (15)

Thus, the tracking control problem can be handled byusing linear control theory. If ulateral is chosen as

ulateral = k c( ) (16)

then the proper choice of the parameter k c enables the

missile to maintain . Here, k c is chosen as inverse of the time constant of the closed-loop dynamics:

_ = k c( ): (17)

In this study, the command of (13) is implementedwithout . Note that can be neglected in themedium-range missile case since it is sufficientlysmall when compared with the first term in (13). Theproposed guidance law shown in Fig. 3(a) consistsof neural-network guidance for the vertical missilemotion and a controller for lateral control with anadditional block for prediction of the intercept point.

Conventional guidance laws can also be employedfor the lateral command. For pure PN guidance [17],the bank-angle command is designed as follows

ayaw c = Nv _yaw (18)

where N is the effective navigation constant and _yawis the yaw component of the LOS rate of the predictedintercept point with respect to the missile body axes.Then, from

ayaw c =1m

(T sin + L)sin (19)

Fig. 4. Intercept geometry.

should be commanded as

= sin 1(mNv _yawT sin + L): (20)B. Intercept Point Prediction

Since the target is assumed a ballistic objectfalling in the central gravitational field, the targettrajectory is Keplerian; the initial condition and the

gravitational field determine the trajectory. Abovean altitude of approximately 60 km, aerodynamicloads are much smaller than the gravitational forces[18]. Consequently, a fairly adequate representation of the target trajectory can be obtained by ignoring theaerodynamic forces entirely.

A missile-target intercept geometry in the 3Dspace is depicted in Fig. 4, where A denotes thecurrent target position, B the current missile position,and I the predicted intercept point. The targettrajectory from A to I is a Keplerian orbit satisfying

rT

r I

()=

1 cos

cos2

T+

cos( T + )

cos T,

=v2T

=rT

(21)

where ( )T denotes the target states, ( ) I the interceptpoint, and the central angle, respectively. I and I are computed by rotating ~rT around ~rT ~vT by. Therefore, given ~rT and ~vT, there is a uniquerelationship between ~r I and . Also, the time from Ato I is given by [18]

tTgo =rTf tan T(1 cos )+ (1 )sin g

vT cos Tf (2 )(1 cos )=( cos2 T)+cos( T + )=cos Tg

+2rT

vT (2= 1)3=2 tan

1 (2= 1)1=2

cos T cot( =2) sin T: (22)For the missile, the rough approximation of t M go (timeto go from B to I ) by Range =v is not appropriatesince there are significant changes in the missilevelocity during the midcourse guidance phase. Here,an additional neural network is employed to estimatet M go as proposed in [9]. This neural network is trainedto learn the t M go -function of the optimal trajectory,


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which is assumed to be

t M go = t M go (v, , x x f , h h f ): (23)

Assume that the deviation of from is sufficientlysmall, then t M go in the 3D space can be approximatedby the t M go estimator valid for 2D vertical-planeguidance

t M go = t M go (v, ,, x

N I x

N M , x

E I x

E M , x

D I x

D M )

t M go (v, ,q ( xN I xN M )2 + ( xE I xE M )2 , xD I x

D M ):

(24)

Both tTgo and t M go are given as functions of and I () isgiven by the root of the equation

t M go () = tTgo (): (25)

The Earth rotation neglected in the above formulationproduces some errors. But the accuracy in predictionof the intercept point improves gradually as themissile approaches the target.

C. Trade-Off StudyNeural-network guidance is compared with two

existing approaches. One is LQR based on a databaseof the optimal trajectories (LQR guidance) [5], andthe other is proportional navigation guidance (PNguidance) which is a conventional guidance law usedfrequently. The implementation issue of each methodis briefly discussed in the following.

1) Neural-Network Guidance : The angle-of-attack command in (11) is implemented by the multilayerfeedforward networks:

= glinear Xk wk gsigmoid X j wkj gsigmoid Xi w ji zi!!!(26)

where g() is an activation function, wkj is thesynaptic weight from the neuron j to the neuronk , and zi is the ith element of the input vector

[v _q ( xN I xN M )2 + ( xE I xE M )2 xD I xD M 1]. Wheneverthe input vector is applied to the sensory nodes of thenetwork, the command is obtained by propagating itthrough the network, layer by layer.

2) LQR Guidance : The command of this

guidance law is a combination of a nominal commandu obtained from a database of optimal trajectoriesand a real-time correction command u to correctperturbations from the reference trajectory;

u = u + u : (27)

u is calculated by solving the linear quadraticproblem defined as [19]

J = xTf S xf + Z t f

t0( xTQ x + u TR u) dt (28)

with the dynamic constraints

_x = A x + B u

=@f @x (t,x,u ) x + @f @u (t,x,u ) u (29)

where _x = f (t, x,u ) is the nonlinear missile dynamics.Then, the guidance law is given as

u = K (t) x = R 1BTP x (30)

where P satisfies the time-varying matrix Ricattiequation

_P = PA ATP Q + PBR 1BTP ,

P (t f ) = S:(31)

Since the in-flight computation of (31) is difficult,the precomputed gain K (t) and the correspondingreference trajectory ( t, x,u ) are stored prior to launch.Therefore, its implementation requires a largermemory size than that of neural-network guidancewhich requires only the weights of neural networks.Fig. 3(b) shows the block diagram of the LQRguidance implementation. The efficient constructionof the database is another problem [20].

3) PN Guidance : The pitch and yaw-axesacceleration commands are calculated by

apitch c = N pitch v _pitch + g cos (32)

ayaw c = N yaw v _yaw : (33)

It is simpler than neural-network guidance but thenonoptimality during the midcourse guidance phasedegrades its intercept performance as shown in thenumerical results.

IV. NUMERICAL RESULTS

A. Neural-Network Design

For the missile model illustrated in Fig. 5, theneural-network guidance law and the tgo -estimator aredesigned. The optimal control problem is defined tominimize the performance index

J = t f (34)

subject to inequality constraints

j(t)j 5, 0 t 57 s

((t) = 0, t > 57 s) :(35)

Then, it is converted to a parameter optimizationproblem [21], and solved by the sequential quadraticprogramming (SQP) method [22]. Our conversionmethod is direct shooting. Once the initial guessesof the discretized control history (t) and the finaltime t f are given, the state differential equations areintegrated explicitly using the 4th-order RungeKutta


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Fig. 5. Missile data. (a) Mass and thrust profile. (b) Aerodynamic derivatives.

approach. Terminal conditions are chosen as the set of 9 points

( x f , h f ) = f (40,40),(40,60),(40,80),(60,40),

(60,60),(60,80),(80,40),(80,60),(80,80) g (km)

in the vertical plane. The same vertical launchcondition

0 = 90, v0 = 27 m/s, ( x0 ,h0) = (0,0) km

(36)

is applied to all cases. The rotation of the Earth isneglected here so that the problem is a 2D one.

Fig. 6 shows the 9 optimal flight trajectories usedfor the training of the neural networks, for whichthe procedure is identical to that described in [8].The dotted lines represent the expected region of target intercept. The structure of the neural network for vertical-plane guidance is 2 hidden layers, with7 and 6 neurons in each layer, repectively, and thetgo -estimator is composed of the same number of hidden layers with 5 and 4 units in each layer.

B. Performance Evaluation

The proposed neural-network guidance lawcombined with the tgo -estimator, shown in Fig. 3(a),


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Fig. 6. Optimal trajectory data used for neural-network training.

TABLE ISimulation Results Without Prediction Errors

Optimal NN Guidance (3D)Target t f (sec) t f (sec) (error %) MD (m) (tgo ) (sec)

Case 1 58.70 58.73 (0.05) 270.34 0.15Case 2 59.58 59.59 (0.02) 330.51 0.13Case 3 66.46 66.55 (0.14) 433.03 0.21

Note : MD : miss distance. (tgo ) = 1 =t f R t f

0 jtgotrue tgo

estimated jdt :average estimation error of tgo .

is tested by computer simulation. Depending onthe initial position and velocity of the target, threedifferent scenarios illustrated in Fig. 7 are consideredin the 3D space. The 3D midcourse guidance law isapplied until the time of intercept without a terminalhoming phase. The feedback gain for correction ischosen as k c = 0 :4 and the predicted intercept pointis updated at every 5 s. For comparision, the optimaltrajectory to the scenario is also calculated by usingthe SQP method.

1) Neural-Network Guidance : Table I showssimulation results for the case of no initial predictionerrors. The launch direction of the missile isdetermined by the predicted intercept point. It showsthat the performance of the 3D guidance law iscomparable to that of the optimal trajectory. Themiss distances obtained here can be easily reduced if homing guidance starts several kilometers away fromthe intercept point.

In the second performance test described inTable II(A), the effect of initial prediction errors isconsidered; the initial heading angle of the missileis commanded toward the target, not the predicted

Fig. 7. Target initial conditions.

intercept point. The 2D guidance law is again appliedto intercept a fictitious target fixed at the terminaltarget position obtained by the simulation resultsof the 3D guidance law, as shown in Fig. 6. In thiscase, the rotation of the Earth is ignored. In spite of the prediction error and the effect of Earth rotation,the guidance performance is not much degraded;specifically, the increase in flight time does not exceed0.14%. This table also shows that the performance of the 3D guidance law is not much different from theideal 2D guidance case. For medium range missiles,the rotation of the Earth does not produce a significanteffect on the missile trajectory.

Although the choice of k c = 0 :4 satisfies thesuboptimal performance, a higher value ( k c = 5 :0)during the initial flight can reduce the flight time


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Fig. 8. Simulation results of case 1. (a) Flight trajectory. (b) Time to go (sec). (c) Azimuth angle.

TABLE IISimulation Results With Prediction Errors

A. k c = 0 :4

NN Guidance (3D) NN Guidance (2D)target t f (sec) (error %) MD (m) (tgo ) (sec) t f (sec) MD (m) (tgo ) (sec)

Case 1 58.74 (0.07) 403.05 0.16 58.73 193.84 0.15Case 2 59.59 (0.02) 336.94 0.13 59.60 193.69 0.15Case 3 66.55 (0.14) 447.31 0.20 66.58 81.43 0.23

B. k c = 5 :0

NN Guidance (3D)target t f (sec) (error %) MD (m) (tgo ) (sec)

Case 1 58.73 (0.05) 282.22 0.14Case 2 59.59 (0.02) 330.09 0.13Case 3 66.55 (0.14) 434.57 0.21

further by a small amount, as can be seen in TablesII(A) and II(B). Note that these results are close to theresult obtained for the case of no prediction errors.However, prediction errors can induce saturationof the bank-angle command so that k c should becarefully chosen by considering various interceptscenarios.


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Fig. 8. (Continued.) (d) Bank angle. (e) Angle of attack. (f) Velocity. (g) Flight-path angle. (h) Flight trajectory in east-height plane.(i) Flight trajectory in east-north plane.

Fig. 8 shows the simulation results of Case 1for k c = 0 :4. In Fig. 8(a), the discrepancy from theoptimal flight trajectory is too small to be observed.

Fig. 8(b) shows that the predicted time to go of themissile coincides well with the true time to go. Thedirection of the predicted intercept point is also


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TABLE IIISimulation Results of Other Guidance Laws

LQR Guidance PN GuidanceTarget t f (sec) (error %) MD (m) (tgo ) (sec) t f (sec) (error %) MD (m)

Case 1 59.00 (0.51) 209.57 0.12 59.64 (1.60) 1338.27Case 2 59.73 (0.25) 475.51 0.02 60.58 (1.68) 2386.79Case 3 67.94 (2.23) 3218.82 1.44 67.59 (1.70) 19931.55

close to the heading angle of the optimal trajectory asshown in Fig. 8(c). It takes about 10 s for the missileto achieve its heading in the direction of . On theother hand, the choice of k c = 5 :0 makes the missileachieve the commanded heading more quickly. Itaccompanies larger bank-angle commands than thecase of k c = 0 :4 during the initial period as shown inFig. 8(d). The angle of attack, velocity, and flight-pathangle are shown in Figs. 8(e)8(g), respectively. It isobserved that the 3D guidance case is quite close tothe optimal trajectory as well as the ideal 2D guidanceso that the proposed guidance law can be effectivelyused for the 3D midcourse guidance problems.

2) LQR and PN Guidance : A massive trade-off study between neural-network guidance and twoexistent approaches described in Section III isconducted. The optimal trajectories used to obtainneural-network guidance are chosen as nominaltrajectories for LQR guidance. Although nominaltrajectories should be selected carefully to coverthe entire missile flight envelope, for the purposeof comparison it is reasonable to assume commontrajectories are available for both guidance laws.For each trajectory, the optimal gain matrix iscomputed by backward integration of the matrixRiccati equations. The time to go of the missile is alsocalculated by interpolating the flight-time data of theoptimal trajectories as briefly illustrated in Fig. 3(b).The simulation results are presented in Table III. Case1 and Case 2 show performance comparable to theoptimal but their flight times are slightly longer thanthose of neural-network guidance. It results from thefact that the LQR correction command to compensatethe off-nominal condition is different from the realoptimal command for the same condition. The timehistories of the angle of attack, velocity, flight-pathangle, and vertical flight trajectory in Figs. 8(e)8(h)show some differences between LQR guidance andthe optimal solution. As the flight condition deviatesmore from the nominal value, nonoptimality increases,resulting in degraded performance (see Case 3 inFig. 6). Therefore, apart from the advantage of simpleimplementation, the neural-network approach issuperior to LQR guidance in terms of ability to dealwith off-nominal (not-trained) flight conditions.

In a similar manner, PN guidance is appliedagainst the terminal target position of the 3Dneural-network guidance law. The performanceevaluated against the fixed target is expected to be

similar to that against the predicted intercept pointwhose variation is so small, as shown in Fig. 8(a).Both N in (32) and (33) are set to 5. As shownin Table III, PN guidance produces larger missdistances and longer flight times in all cases thanboth of LQR guidance and neural-network guidance.From Figs. 8(e) and 8(h), we see that the angle of attack during the initial period of the flight is toosmall to reach the altitude of the target. On the otherhand, Fig. 8(i) shows that the lateral channel has noproblem. It again confirms our claim that the synthesisof the optimal command is not necessary in the lateralchannel while its employment is important in the

vertical channel.

V. CONCLUSION

The neural-network guidance method, whichhas been previously studied for real-time midcourseguidance in the 2D space, is extended to the case of 3D flight for interception of nonmaneuvering ballistictargets. By decomposing the 3D guidance probleminto the 2D guidance problem in the vertical planeand the lateral control problem, a computationallyefficient guidance law is obtained. For the verticalmotion, a neural network is trained to produce the

optimal angle-of-attack command for tracking of theprojection of the optimal trajectory onto the verticalguidance plane. For lateral control, the method of feedback linearization is used to steer the missiletoward the direction of the predicted intercept point.An additional neural network is employed as atgo -estimator to predict the intercept point. Computersimulations show that the proposed guidance lawgenerates a good approximation of the 3D optimaltrajectory while providing better performance thanthe existing methods such as LQR guidance and PNguidance.

REFERENCES[1] Cheng, V. H. L., and Gupta, N. K. (1986)

Advanced midcourse guidance for air-to-air missiles. Journal of Guidance, Control, and Dynamics , 9, 2 (1986),135142.

[2] Menon, P. K. A., and Briggs, M. M. (1990)Near-optimal midcourse guidance for air-to-air missiles. Journal of Guidance, Control, and Dynamics , 13 , 4 (1990),596602.

[3] Dougherty, J. J., and Speyer, J. L. (1997)Near-optimal guidance law for ballistic missileinterception. Journal of Guidance, Control, and Dynamics , 20 , 2 (1997),355362.


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[4] Imado, F., Kuroda, T., and Miwa, S. (1990)Optimal midcourse guidance for medium-range air-to-airmissiles. Journal of Guidance, Control, and Dynamics , 13 , 4 (1990),603608.

[5] Imado, F., and Kuroda, T. (1992)Optimal missile guidance system against a hypersonictarget.AIAA Paper 92-4531.

[6] Newman, B. (1996)Strategic intercept midcourse guidance using modifiedzero effort miss steering. Journal of Guidance, Control, and Dynamics , 19 , 1 (1996),107112.[7] Hornik, K., Stinchcombe, M., and White, H. (1990)Multilayer feedforward networks are universalapproximators. Neural Networks , 2 (1990), 359366.

[8] Song, E. J., Lee, H., and Tahk, M. J. (1996)On-line suboptimal midcourse guidance using neuralnetworks.In Proceedings of the 35th SICE Annual Conference ,Tottori University, Japan, 1996, 13131318.

[9] Song, E. J., and Tahk, M. J. (1998)Real-time midcourse guidance with intercept pointprediction.Control Engineering Practice , 6, 8 (1998), 957967.

[10] Song, E. J., and Tahk, M. J. (1999)Real-time midcourse missile guidance robust to launchconditions.Control Engineering Practice , 7, 4 (1999), 507515.

[11] Cottrell, R. G., Vincent, T. L., and Sadati, S. H. (1996)Minimizing interceptor size using neural networks forterminal guidance law synthesis. Journal of Guidance, Control, and Dynamics , 19 , 3 (1996),557562.

Eun-Jung Song received the M.S. degree in 1997 and the Ph.D. degree in 2000,both in aerospace engineering, from Korea Advanced Institute of Science andTechnology.

She visited University of California at Los Angeles as a postdoctoralresearcher from 2000 to 2001. Currently, she is a postdoctoral researcher of theSchool of Mechanical and Aerospace Engineering, Seoul National University,Korea. Her major research interests include neural networks, missile guidance,and aircraft navigation.

Min-Jea Tahk received the B.S. degree from Seoul National University, Seoul,Korea, and the M.S. and Ph.D. degrees from the University of Texas at Austin, in1983 and 1986, all in aerospace engineering.

From 1976 to 1981 he was a research engineer at the Agency for DefenseDevelopment, and from 1986 to 1989 he was employed by Integrated Systems,Inc., Santa Clara, CA. He is presently Associate Professor of AerospaceEngineering at Korea Advanced Institute of Science and Technology, Taejon,Korea. His research interests include target tracking, missile guidance, flightcontrol, and evolutionary computation.

[12] Fu, L. C., Chang, W. D., Yang, J. H., and Kuo, T. S. (1997)Adaptive robust bank-to-turn missile autopilot designusing neural networks. Journal of Guidance, Control, and Dynamics , 20 , 2 (1997),346354.

[13] Geng, Z. J., and McCullough, C. L. (1997)Missile control using fuzzy cerebellar model arithmeticcomputer neural networks. Journal of Guidance, Control, and Dynamics , 20 , 3 (1997),557565.

[14] Khalil, H. K. (1996) Nonlinear Systems .Englewood Cliffs, NJ: Prentice-Hall, 1996, 8185.

[15] Siouris, G. M. (1993) Aerospace Avionics Systems: A Modern Synthesis .San Diego, CA: Academic Press, 1993.

[16] Zurada, J. M. (1992) Introduction to Artificial Neural Systems .St. Paul, MN: West Publishing, 1992, 214216.

[17] Ghose, D. (1996) Lecture Notes on Guidance Laws and Their Applications .Automatic Control Research Center, Seoul, Korea, 1996,90115.

[18] Regan, F. J., and Anandarkrishnan, S. M. (1993)Dynamics of atmospheric re-entry.AIAA, Washington, DC, 1993.

[19] Bryson, A. E., Jr., and Ho, Y. C. (1975) Applied Optimal Control .Halsted Press, 1975, 146174.

[20] Hardtla, J. W., and Milligan, K. H. (1987)Design and implementation of a missile guidance law

derived from modern control theory.AIAA paper 87-2447.[21] Hull, D. G. (1997)

Conversion of optimal control problems into parameteroptimization problems. Journal of Guidance, Control, and Dynamics , 20 , 1 (1997),5760.

[22] Lawrence, C., Zhou, J. L., and Tits, A. L. (1996)Users Guide for CFSQP Version 2.4: A C Code forSolving (Large Scale) Constrained Nonlinear (Minimax)Optimization Problems, Generating Iterates Satisfying AllInequality Constraints.TR-94-16rl, Institute for Systems Research, University of Maryland, College Park, MD, 1996.


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