I. INTRODUCTION

CIANN-DONG YANG FEI-BIN IISIAO National Cheng Kung University Etiwan

FANG-BO YEII nnghai University 'hiwan

Generalized Guidance Law for Homing Missiles

Most modern air-to-air and surface-to-air missile systems use a form of proportional navigation guidance (PNG) in the homing phase of flight, owing to its ease of implementation and to its efficiency. However, there exist many situations where simple PNG shows up less favorably; this fact gives rise to the discovery of various guidance laws discussed in the literature, with each being superior to the conventional PNG in some aspects of considerations. These include augmenting PNG considered by Arbenz [l] and Siouris [2], quasioptimal PNG in a series of papers by Axelband and Hardy [3-51, predicted guidance law proposed by Kim e t al. [6], and more recently, generalized PNG by Yang and Yeh [7,8]. Though these refincd PNG have proved to increase the effectiveness and validity of PNG, it is evident that their implementation also results in much complexity in measurement and The utmost purpose. of using a guidance law for a honling

~~

missile is to turn the heading of the nussile toward some desired

direction as rapidly as possible by conunanding the acceleration

of the missile proportional to the angular rate of such direclion.

Such a direction may be simply along the line of sight, the missile collision course, o r any other appropriate direction. The

generalized guidance law presented here covers a wide variety

of guidance laws; some have appeared in the literature, while most of them are totally new. Based on the exact nonlinear

equations of motion, general analytical expressions for capture area, missile commanded acceleration, and homing time duralion are derived in closed forns. An optinial nonlinear guidance law originating from this generalized guidance law, which minimizes a weighted linear conlbination of the time of capture and energy

expenditure, is also derived. The efficiency and mechanization of the generalized guidance law and the comparisons with various

existing guidance laws are demonstrated in a simulation sludp

The miss distance due to seeker/aulopilol dynanlics, maneuvering

targets, acceleration saturation, and measurement bias is also investigated.

Manuscript received November 10, 1987.

IEEE Log No. 26879.

analysis. The purpose in the present study is to seek

a versatile guidance law which may possess the advantages of the various refined OPG, while sharing the benefit of ease of implementation with PNG. The work of [SI throws a new light on this idea and indeed, as we see later, it is the PNG itself that gives the clue to the problem. Before starting the attack on the problem, it is advisable to inspect the operation of the PNG for a moment. A PNG homing system measures the rate of rotation of light of sight (LOS) from the missile to the target. The guidance input is assigned proportional to the LOS rate, and requires the missile to turn in the corrective direction, i.e., in the direction to reduce the LOS rate to zero. A question arising from this observation is what will happen if the direction that the missile is pursuing is a particular one other than LOS, and whether the interception strategy using this new heading direction results in a better performance. The choice of such a direction may depend on the mission requirements, missile performance, and the acccssibility of target information, etc.

In a point of view of kinematics, any particular direction in space can be described uniquely by three orthogonal unit vectors and the associated spatial coordinates. Since the general three-dimensional attack situation can be dealt with by resolving the components of the rate vector of this chosen direction into the two lateral missile coordinate axes and employing the same structure of mechanization, only - the two-dimensional attack is considered here. We propose the concept of generalized guidance law as follows. The general idea of PNG for a homing missile

Authors' addresses: C.-D. Yang and E-B. Hsiao, Institute of Aeronautics and Astronautics, National Cheng Kung University, Etinan, Biwan, Republic of China; E-B. Yeh, Department of Mathematics, n n g h a i University, Taichung, 'hiwan, Republic of China.

is to turn the heading of the missile toward some desired direction as rapidly as possible by commanding

0018-9251/89/0300-0197 $1.00 @ 1989 ICEE the missile's accelerationswhich are proportional to the angular rate of such direction. This direction is

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-25, NO. 2 MARCH 1989 197

Fig. 1. Planar pursuit geometry.

expressed by

L = f ( r , e ) e , + &(‘,+e. (1)

The expression for L is referred to a moving coordinate system ( r ,8) with origin fixed at the missile and with two orthogonal unit vectors e, and eo being along and perpendicular to the LOS, respectively. In a pursuit geometry, r is regarded as the relative distance between missile and target, and 8 as the angle of LOS with respect to some inertial reference (see Fig. 1). Both f and g are continuous functions of r and 8, and a proportionality constant k is considered to be embedded in the expression of f and g. The generalized guidance law requires that the missile acceleration be commanded to be equal to the time rate of L, i.e.,

a,,, = L

= Cf( r , e ) - g(r,@)e, + ( g ( r , @ ) + f ( r , e ) Q e ,

= Cf’(r,e) -g(r,B))ber + ( f ( r , e ) +g’(r,e))beo

where dots denote differentiation with respect to time t , and primes denote differentiation with respective to e.

In the case when f ( r , e ) = k = constant and g(r , e ) = 0, the generalized guidance law reduces to a PNG. The following paragraph deals with the general behaviors off and g. Specific forms of f and g are discussed in Section I11 where some special cases of generalized guidance law, appearing in the literature, are considered. In Section 11, the nonlinear equations of motion a re derived for a n ideal missile pursuing a maneuvering target under the generalized guidance law. By solving these nonlinear equations, the analytical expressions for capture area, missile commanded acceleration, and homing time duration are obtained in Section 111. Section IV is dedicated to the determination of the optimal functions off and g which minimize a weighted linear combination of time of capture and energy expenditure. Finally, a numerical simulation is employed in Section V to demonstrate the superiority of the optimal guidance law. The effect of seeker/autopilot dynamics, measurement bias, saturation of commanded acceleration, and target acceleration on the miss distance is also investigated in Section V.

II. MISSILE KINEMATICS

Assume a missile M is attempting to capture a target T as shown in Fig. (1). Let V denote the target velocity relative to the missile, and V, and V, denote the velocity vectors for missile and target, respectively. In the polar coordinate system ( r ,6) mentioned previously, V, V,, and V, have the expressions

V = V,e, + Voe,

= re, + d e , (3 )

V m = G e m = Vm(coS(~m)er + sin(ym)e~) (4)

and

VI = T/e,

= V,(cos(yt)er + s i n ( y t ) ~ ) (5a)

= VTrer + VTOQ (5b)

where VT, and VTO are the components of the target velocity and are assumed to be known a priori. The path angles of missile and target are denoted by ynI and 7,, respectively.

target; then we have Let r be the position vector from the missile to the

i = V = V,e, + ?‘,ee = V,e, - Vme,,. (6)

Multiplying both sides of (6) by e, and eo, respcctively, yields the scalar equations

V, = V,e, . e, - Vme,,, . e,

= V, cos(yt) - V m COS(?,) (7)

and

VO = V,e, . e, - Vme, . eo

= V, sin(?,) - V, sin(?,,,). (8)

Differentiating (4) with respect to time,

a, = Vm = Vmem + ~ , e , , ,

= (ri, cos(?,) - V m ~m sin(?,) - V m b sin(yr,i ) > e r

+ ( V m sin(?,> + VmTmCoS(ym) + I/nlecos(r,,l>>e~.

(9) Equating the expressions for missile accelcration in (9) and (2), yields

V,COS(~,,~)- V,Tmsin(ym) = f - g b + V,,18sin(y,,l)

(10) and

I‘, sin(?,) + ~ m ~ m cos(ym) = f b + B - ~mecos(yn,).

(11) Again, differentiating (5a) and (5b) with respcct to time and equating the results, we have

Ccos(y,) - 6 3 , sin(?,) = VT, - ?‘Toe + V,esin(yt)

(12)

198 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. A E S - 2 5 . NO. 2 MARC11 1989

and

V, ~in(7,) + xjt COS(^/^) = VT, + v T , ~ - v , ~ c o s ( ~ , ) .

(13) Differentiating V, in (7) with respect to time, and applying the results of (10) and (12), yields

Vr = V,cos(y,) - ~ j , sin(y,)

- Vm cos(7m) + v m j m sin(ym)

= VT, - VTee + Vee - (f’ - g)B. (143)

Similarly, differentiating Ve in (8) with respect to time, and applying the results of (11) and (13), yields

Ve = I‘, sin(?,) + xjt COS(^,)

- v m sin(?,) - v m 7 m cos(ym)

= VT, + V T , ~ - V,e - cf + g’)B. (14b)

Equations (14) are the governing equations for the missile kinematics using the generalized proportional navigation law. They can be rewritten as

i: - re2 = F(r(e) ,e)e (15a) re + 214 = ~ ( r ( 8 ) , 8 ) 8

where

F ( r ( Q 8 ) = -(f’(r(Q@) - g ( r ( Q 8 ) ) + %,(e) - vTe(8) and

G ( r ( 0 8 ) = - c f ( r ( W ) + g ’ ( W J 9

+ J‘;,(e) + v T r ( 8 ) .

As it appears, (15) are a set of nonlinear differential equations. In a conventional approach, the analysis for PNG had been based on linearizing the pursuit geometry about a nominal collision course [9-111. Such analysis has demonstrated the optimality of the PNG against a nonmaneuvering target. The main drawback of the linearized approach is that the results only apply in a relatively small region of the state space, where the trajectory linearization is valid. On the other hand, the analytical solutions of (15) are never discussed in the literature excepting a very special solution corresponding to F ( 8 ) = 0 and G(8) = const. # 0 (true PNG (TPNG)) which is derived by Guelman [12] for a missile pursuing a nonmaneuvering target (VT, (~) = 0, VTe(8) = 0). Even in this simple case, the resolution by Guelman’s method seems tedious. It requires a sequence of variable transformations to reduce the nonlinear equations to a set of ordinary equations and thus greatly restricting its application in more general cases. Nevertheless, the analytical study of TPNG has revealed an important fact that the capture is restricted for the cases where the initial conditions belong to a predetermined set of circles, the so-called capture area. For a guidance law other than TPNG,

~

YANG, HSIAO, L YEH: GENERALIZED GUIDANCE LAW FOR HOMING MISSILES

the capture area is no longer a circle, and different f and g result in a different form of capture area. Thus, a systematic approach for describing the general behavior of the capture area becomes indispensable in the guidance law design for homing missiles. A recent work by Yang and Yeh [SI has discussed thc solution of (15) for the special case when f and g are independent of the relative distance r , and the target is nonmaneuvering. The purpose of Section 111 is to introduce a new solution technique which provides us not only a closed-form solution of (15) but also some distinguished properties of the generalized guidance law.

Ill. CLOSED-FORM SOLUTION

To obtain the nonlinear equations in (15), the independent variable t is changed to aspect angle 8. This concept of a changing independent variablc is a key issue in what follows. Equation (14) reduces to

Elimination of Ve between (16a) and (16b), yiclds

v:‘ + v, = -(f” + f ) + VTr + v;,. The solution of the above equation can be obtained immediately as

Vr(8) = Acos(8) + Bsin(8) - f ( r ( 8 ) , 8 ) + V T , ( ~ ) .

Substituting Vr(8) into (16a), we have an exprcssion for Ve(8). That is,

By using the initial conditions

Vr(8) and Ve(8) become

Ve(8) = Rsin(8 - 80 + a ) + Fe(r(8),8) (17b)

where

and

199

Without loss of generality, we choose 80 = cr in the following discussion. The hodograph is determined by eliminating 8 between (17a) and (17b). In order €or the missile to intercept the target with a finite acceleration and with a finite time of duration, we must have the constraints

a t the end of pursuit

i.e., Ve(8,) = 0 and Vr(8r) < 0. By means of (17), these conditions a re reduced to the forms

Rsin(8f) + Fe(0,Of) = 0

Rcos(8,) + F,(O,Of) < 0 (1%)

(18b)

where 8, is the aspect angle at the end of pursuit. By substituting for 8, determined by substituting (1%) into (18b), we determine the range of initial conditions for Vr0 and Veo, in which the interception can be completed. The inequality thus obtained from (18b) defines the generalized capture area. For a powerful guidance law, the capture area must be large and this poses restrictions on the choices of f ( r , 8 ) and g(r,8).

Having derived the velocity components V, and Ve in terms of aspect angle 8, we are now in a situation to establish the time histories of these quantities. The relation between r and 8 must be established first. Using (17) and the equalities V, = r, V, = re, yields

(Fe(r,8) + Rsin(8))dr + r(Fr(r,8) + Rcos(8))de = 0.

This differential equation can be solved by introducing the integrating factor X(r,8), which is determined by the following equations

(19)

X(r,8) -r(F,(r,8) + Rcos(8))

a a -(rF,(r$)) - z F e ( " 8 )

do. (20) - ar - Fe(r,8) + Rsin(8)

Then the relation 9 ( r , 8 ) = C can be obtained by solving the following exact differential equation

A( r , @)(Fe( r ,8) + R sin(8))dr

+ X(r,8)r(Fr(r,8) + Rcos(8))dO = 0.

The determination of X needs a solution of the associated ordinary differential equation in (20), which in general, is not trivial; however, for the existing guidance laws today, the determination of X is surprisingly simple. We assume, in the following derivation, that the variable r is eliminated by the use of the relation 9 ( r , 8 ) = C, and the index r in the expressions involving functions of r is neglecting hereafter.

Define the angular momentum per unit mass for the relative coordinate system to be

h = rV0 = r2e. (21) With this new variable, (16b) can be rewritten as

it = +g'(e))ve + v;e(e)ve - v,,(e)v,. Recalling = 8dh/d8 and the expression for Ve(8) , we have the following important relation between h and 8.

(22) - - d h - -(f(O> +g'(O)) + V;,(e> - V m d 8 h Rsin(8) + Fe(8)

The integration of this equation gives

h = hoexp(-@(8)) (23a)

where @(e) is defined by

Note if f(8) and g(8) are trigonometric functions of 8, then @(e) can always be expressed by elementary functions or elliptical functions. Once h has been obtained, one can determine the missile acceleration. From (21), we get the LOS rate as

1 = -(Rsin(8) h0 + Fe(8))2exp(@(8)). (24)

Substituting this into (2), we have an expression for missile acceleration as a function of 8, i.e.,

x [(f'(e) - g(o>>er + + g'(Q>>eeI. (25)

From (24), it can b e seen that 8 is a monotonically increasing o r decreasing (depending on the sign of ho) function o f t . This also justifies the use of 8 as an independent variable.

The relative distance r is found to be

h Ve

Rsin(8) + Fe (8)

r = -

(26) - - ho exp(- @ (6))

and the duration of the pursuit rf can be obtaincd from the integration of (26). This leads to

and rf is determined by setting 8 to O f , wherc 8f is obtained from the solution of (18a). These explicit expressions for capture area, missile acceleration, and homing time duration are quite useful in the selection of guidance laws. Whatever f(8) and g(8)

WO IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. A E S - 2 5 , NO. 2 MARCII 1989

YT

y M I

t 1

+ d

Fig. 2. Mechanization of GPN.

Tracker

Dynamics

are chosen to be, those relations allow us to evaluate the performance of the missile immediately. The implementation of generalized guidance law is shown in Fig. 2.

When compared with PNG, the only additional variable to be measured in the mechanization of generalized guidance law is the relative distance r . The guidance law gives the commanded missile acceleration a,,, according to (25). This mechanization is very similar to that of conventional PNG and shares the benefits of easy implementation and efficiency. The most remarkable property of generalized proportional navigation is its versatility. Any type of guidance laws with any special functions can be synthesized by a proper choice of f ( r , 0 ) and g(r,8). Several guidance laws have stemmed from the concept of generalized guidance law and are discussed successively in the following paragraph. These include true proportional navigation (TPN), generalized true proportional navigation (GTF”), predicted guidance law, and optimal guidance law.

,- Guidance Missile

8 Laws + 4 Autopilot + Kinematics

g ( e . 9 ) Dynamics f ( r , e )

found by the use of (18) as

Rsin(0,) = 0 (29a)

(29b) Rcos(0,) + K < 0.

If we assume a positive value of VOO, i.e., the LOS moves in the counterclockwise direction, then thc impact is achieved at 8, = ?r and the capture area is characterized by the inequality

R - < I K

or equivalently,

(V,o + K)* + V;o < K2.

This is a circle centered at (-K,O) with radius K . The same result is obtained in Guelman’s paper [12] by a more complicated approach. A further detailed discussion about TPN is included in [12, 131 and we do not pay further attention to it here.

A. True Proportional Navigation B. Generalized True Proportional Navigation

This type of guidance law is the most popular one used today and is also the simplest one in its mathematical form. For this case, the corresponding f(0) and g(0) is given by

f ( r , 0 ) = K ; g(r ,O) = 0

and hence the function @(e) dcfined in (23) reduces to

where, to get a clear understanding and to avoid

Generalized true proportional navigation (GTPN) is defined similar to TPN but with the propcrty that the missile acceleration is not necessarily applicd normal to LOS (see Fig. 1). In this case, the function f(0) and g(8) are given by

f ( r , O ) = Kcos($); g ( r , 0 ) = Ksin($)

where $ is a constant angle between the dircction of the missile acceleration and the direction normal t o the LOS. With this kind of f(0) and g(O), the function @(e) is obtained, for a nonmaneuvering targct, as

4 - A 1 1: Rsin(@) + A 2 @(e) =

mathematical complexity, a nonmaneuvering target is assumed in (28). The capture area for TPN is then

where A1 = -Kcos($) and A2 = -Ksin($). According to the value of q, the integration can be dividcd into

YANG, HSIAO, & YEH: GENERALIZED GUIDANCE LAW Fori HOMING MISSILES 20 1

four cases, i.e.,

The first case corresponds to TPN and the result is already given in A. For cases 2, 3, and 4, the integration can be found from any standard mathematical table as

X ( R t a n ( O / 2 ) + A 2 - , / P )

R tan(0/2) + A2 + JG- '

if R<X2 I

I if R = X2

To make the derivations independent on initial conditions, we introduce the following dimensionless variables

b = d C 2 ( 1 - + (1 - A z ) ~ ; 0 = b/C;

c = IV,o/Veol - -

K = -V,& XI = Xcos(@); A2 = Asin($)

r = t / t f .

For case 2), the integration of (22) gives

X2 tan(O/2) + b - d b 2 - A:

X2 tan(O/2) + b + ,/- (31) nl --

Ah =

where A is the initial value of the right-hand side term of (31) and

Fig. 3. Capture area of GPN for various values of Q.

From (27) and (31), h is related to r as

-1-m h -2X2h 1 - m

+ A(l + m)

where C, is determined from the initial condition - h = l at r = 0 .

For the missile to intercept the target in a finite time, we must have from (18) or from (32) directly that

m < 1

or equivalently,

-(1- 2Asin(@)) c2 < 1 - 2Xcos($) . (33)

This capture area is depicted in Fig. 3. The performance of the missile is restricted to the right-hand side region of each curve representing the different value of q. The total capture area according to the generalized proportional navigation (GPN) is characterized by the right side region of the envelope described by thc equation

c2 + 1 2 m -

A = (34)

It is obvious from Fig. 3 that GPN has a largcr capturc area than that obtained by TPN ($ = 0) in the region where C is small.

(32) with the terminal condition The final time of pursuit end can be found from

- h = O at r = l .

The result is depicted in Fig. 4 for C = 0.5. A remarkable property of GPN is that the final time is

202 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-25 , NO. 2 MARCH 1989

zn 40 fin an

t degree 1

Fig. 4. Homing time duration for various values of y5. (y5 = 0 comsponds to ’IT”.)

very sensitive to the variation of 1c, when t,h is small. From this observation, it is noted that the interception time can be drastically reduced if I) is increased from zero to a small positive value. A further study shows that a large value of 1c, results in an unbounded missile acceleration.

From the several results mentioned above, it is suggested that GPN be used under the condition where C is small. A small value of C represents a small closing rate or a large target tangential velocity relative to the missile. Under these conditions, the GPN has larger capture area and shorter interception time than those of TI” and the increment of the missile acceleration from that of TI” is small when 1c, is kept small.

C. Prediction Guidance Law

The main idea of the P R G is to predict a straight-line collision course and to turn the heading of the missile toward this collision course as rapidly as possible. The interception geometry is shown in Fig. 5. The predicted collision points 0; can be made at any stage of engagement by projecting current relative interceptor-target velocities forward into the point of closest approach. Referring to Fig. 5, we have

MjO; V, Tjo; v, - 1 7 -

where 11 represents the instantaneous speed ratio. Let [ = m, then from the vector relation

I L L

M;O; = Mi7;: + T;O;

Squaring both sides of the above equation and solving for e, we have

cos(8) + Jq2 - sin2(8) r e = q 2 - 1

where 8 is the angle between e, and e,. Using this expression for E , e,, becomes

e,,, = l e , 7 7 + 1 ( J z j - cos(01) e,.

The missile is then commanded to turn the heading toward this direction as rapidly as possible. The first term in the expression of e, requires the information of target motion, however, under the case when the missile has a good speed superiority over the target, this term can be neglected safely. Therefore, our choice of f ( r , 8 ) and g(r ,8) for P R G becomes

(35) where the direction of target velocity is used as the refcrence line, and 7 is the missile-to-target speed ratio. If we let 7 approach infinity, PRG reduces to TPN. A numerical simulation was given for PRG in [6], where the advantages of P R G were indicated. Based on the present closed-form study, further remarkable properties of P R G are revealed. Substituting (35) into (17), yields

-v,(q = Rcos(e) + (JZ- cos(^)) 17

(36a)

-V,(8) = Rsin(8). (36b)

The hodograph is determined by eliminating 8 between (36a) and (36b) as shown in Fig. 6. A n important observation is that the intersection of LOS and the hodograph determines uniquely the relative velocity components V, and V,. The terminal aspect angle is found from (36b) as T and the capture arca is

YANG, IISIAO, & YEII: GENERALIZED GUIDANCE LAW FOR HOMING MISSILES 203

Fig. 6. The hodograph for predicted guidance law.

characterized by (36a) as

or equivalently

(Xf(O0) - 1)2 + (l/C)? < X*(1 + 1/?))2. (37)

This is depicted in Fig. 7. The capture area of the missile is restricted to the right-hand side region of each curve representing different values of q. It can be seen from this figure that the larger q is, the smaller the capture area will be, and that TPN (q + CO) has the smallest capture area. Notice that in proportional navigation r] is indeed not a constant value but here we may treat it as the averaged missile-to-target speed ratio, or merely as a design parameter. The corresponding function @(e) for PRG can be found from (34b) as

where f ( O ) is defined in (35). By substituting @(e) into (25) and (27), the

missile acceleration and the homing time duration can be found immediately. For commanded missile acceleration, we prefer the PRG. The reader can sce from Fig. 8 that the missile acceleration becomes larger as r] increases. The homing time duration obtained from (27) is shown in Fig. 9, where it is observed that is approaches a constant as 11 increases without bound. We then conclude that for a proper choice of rj (not too small or too large), the P R G is superior to TPN at least with respect to the factors considered above.

The guidance laws presented in A, B, and C are only three special examples of the generalized

guidance law. From the above discussion, it is revealed that the closed-form solution of the exact nonlinear equations and the analytical expressions for the capture area, homing time duration, and missile acceleration commanded allow the guidance designer to get immediately a preliminary understanding of the specific guidance law to choose without resorting to a heavy numerical simulation and hence, to reduce the numbers of trial-and-error procedures in the preliminary design phase. O n the other hand, the present approach provides a systematic approach to the determination of an optimal guidance law, which is a central topic considered in Section IV.

IV. OPTIMAL GUIDANCE LAW

The missile commanded acceleration undcr generalized guidance law is given, according to (2), as

a,,, = (ule, + u2eO)B (38)

where u1 = f ' ( r (O) ,O) - g(r(O),O) and u 2 = f ( r ( Q 8 ) + g'(r(O),O). Different forms of f ( r , O ) and g(r ,8 ) result in different performances, and it is intcresting to know which of the f ( r , O ) and g(r ,8 ) give the optimal missile performance, especially when there is shorter time of duration and smaller expenditurc of maneuvering energy. It has been shown earlier that 8 is a monotonically increasing function o f t (assumc a positive value of ho). Thus the guidance problem for achieving shorter time of capture and smaller expenditure of maneuvering energy is to find a control T such that the point capture r(O,) = 0 is assured while the performance index

J = Of + pJb T 2 d 8 (39) 00

204 IEEE TRANSACTIONS ON AEROSPACE ANI) ELEC?'IIONIC SYSTEMS VOL. AES-2.5, NO. 2 MARC1 I lOS9

I 2 : 4 s G 7 A Fig. 7. Capture area for predicted guidance law.

is minimized. The magnitude of the commanded acceleration is T , i.e.,

u1 = Tsin(4); u2 = Tcos(4).

The weighting factor p is selected in such a way that the required acceleration is realistic. The range of parameter p and initial conditions (V,o,Voo), which ensure the point of capture condition, are characterized in the following derivation.

(16), are The equations of motion, according to (38) and

V,! = VO + V;, - VTO - u1, V,(O) = V,O (40a)

vi = -vr + Vi0 + v'r - v#(&J) = v#O (40b)

where Vi, and Vio are given and are continuous functions of 8, and u1 and u2 are determined by optimization theory.

This optimization problem is solved by applying the maximum principle [14] by which a special case corresponding to u1 = 0 has been treated in [15]. Define an additional state variable E such that

t = 1 + p T 2 (41)

with <(eo) = 80. Then the Hamiltonian for the system of (39), (40), and (41) becomes

H = Xv,(Vo + V;, - VTB - Tsin(4))

+ .\v@(-V, vi0 vj", - TCOS($))

+ X((1+ pT2). (42)

Ab, = AV# (43a)

A b 0 = -AV, (43b)

x; = o . (43c)

The adjoint variables are defined by the equations

l*l 20 .0 -

15 .0 .

10.0

5.0

I I I I I I 40 60 80 100 1Zu 140 160 180

1 I *

e (degree)

Fig. 8. Commanded missile acceleration for predicted guidance law.

, 20 '5 (velocity ratio) 10

Fig. 9. Homing time duration for prcdicted guidance law.

With the boundary conditions given by

YANG, HSIAO, & YEH: GENERALIZED GUIDANCE LAW FOR HOMING MI9911 FS

~

206 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-7.5, NO. 2 MAllCII 1969

The adjoint system (43) can be solved immediately as

A1.e = Acos(8 - 8,) (45a) AV, = Asin(8 - Of). (45b)

The optimal control law aH/aT = 0 gives

and aH/&# = 0 gives

A v Ave

tan(@) = 2.

Combining the above equation with (44), we have

and A T = - - 2P'

Then the optimal controls become

(47b) A

U; = - - c q o - e,). 2 P

The resulting missile acceleration is

A B M e = --8(sin((e - @,)e, + cos(8 - 8 , ) ~ ) . (48)

2P Instead of a constant, as in the case of TPN, the

optimal controls U; and U; are now trigonometric functions of the aspect angle 8; however, in the neighborhood of the pursuit end, we have sin(8 - 8,) z 0 and cos(8 - 0,) z 1 and the optimal control U; and U; can be approximated by a constant; this, in turn, implies that the optimality of the conventional proportional navigation is only valid in the vicinity of the pursuit end.

The optimal f * ( r , 8 ) and g*(r ,8) arc related to U; and U; as

4 * __- df* g" = U ; - + f * = U ; . d8 d6'

Solving for f * and g', yields

f * ( r (8 ) , (e) = C1 cos(8) + Cz sin(@

A - -((e - 0f)sin(8 - 8,) (49a)

2 P

and

g"(r (0 ) ,8 ) = -C1 sin((e) + C~cos(8)

where C, and C2 are two constants determincd from the initial conditions for f ( r (6 ) ,0 ) and g ( r ( 8 ) , 0 ) at

0 = 80 and A is determined from the initial conditions for V, and Ve as follows.

Substituting the optimal controls U ; and U; into (40), we have following expressions for the state variables V, and Ve.

V, = Bcos(8 - 8,) + Dsin(8 - 8,)

(50b) A + -((e - ef)cos((e - e,) + vTe 2P

where the four parameters A , B, D , and Of can be determined in terms of V,O, Ve and p by thc terminal conditions

H(8,) = 0; Ve((e,) = 0. (51) This leads to

D = - V T @ ( O f )

By joining the initial conditions

v r ( O 0 ) = v r o ; Ve(6o) = veo (52) we have

A + -(& - ~ , ) c o s ( ~ o - (e,) + Vre(80). (53b) 2P

Note that V, = r e and the condition Ve(B,-) = 0 guarantees that the point of capture is rcachcd with a finite LOS rate, i.e., with a finite expenditure of maneuvering energy. To ensure a finite timc of capture one additional condition is needed, i.e.,

Vr(0,) < 0.

This inequality characterizes the capture arca for Vro, Veo, and p which are compatible with the point of capture condition.

To obtain the time history of 8, we introduce the specific angular momentum h as before in (40b) and use the optimal controls U; and U;. This lcads to

or

After integration, this yields

With this equation and the relation 8 = V:/h, we have

As mentioned earlier, t is a monotonically increasing function of 8 and vice versa provided 80 is positive. The time of capture is obtained by letting 8 = 8, in (55).

V. NUMERICAL EXAMPLES

Simulation has been employed to demonstrate the superiority of the optimal guidance law (OPG). A comparison is made with TPN where the commanded missile acceleration is given by

aMc = - N V,&Q (56)

and N is a navigation Constant. All the results given below have been nondimensionalized with respect to the initial values of the corresponding variables.

It has been shown in the previous sections that when a missile obeys the law of generalized guidance law, the miss distance is always zero if the initial values of some kinematic variables lie within the capture area. However, experience has shown that the reduction of miss distance to zero is very difficult. The reason for this apparent paradox is due to the practical limitations imposed by the factors such as finite control stiffness, system and measurement noises, varying stability characteristics with respect to speed and altitude, and bias caused by the imperfect components within the guidance system, etc. In the following simulation study, we consider some of these effects for missiles using the OPG. A maneuvering target is considered here. We assume that the target tries to escape according to TPN guidance law, i.e., the commanded target acceleration is in the form

a, = Re

where R is a proportionality constant indicating the relative maneuverability of the target. The mechanization of the missile guidance loop using the OPG is like the one described in Fig. 1 with f ( r ,O) and g ( r , 8 ) given in (49).

1. TmclierlAutopilot Dynamics. Preliminary design work frequently demands the approximate determination of the required dynamic response of

, I ( / 5s distance

Fig. 10. Miss distance due to variation of time constant of missile dynamic system.

a missile and its control system. A rule of thumb which has been successfully employed for terminal phase guidance, particularly in the case of the proportional navigation, is that the overall equivalent time constant of the system should be less than one-tenth of the flight time of the terminal phase so that the control system may completely "solve out" one command before the next command is transmittcd. To understand the effect of the missile dynamic on the effectiveness of the OPG, we simply assume that tracker/autopilot dynamics can be modeled as a first-order lag.

The influence of the equivalcnt time constant on the miss distance is described in Fig. 10, where the miss distance is nondimensionalized with respect to the initial range ro, and the time constant T is nondimensionalized with respect to the homing time duration ff. It shows that the miss distance increases in a very steep way as the equivalent time constant increases to a certain value. The trajectories of the missile pursuing a maneuvering target by using OPG under a first-order lag are depicted in Fig. 11. The results for TPN are also shown in Fig. 11 for comparison. The miss distance is rcpresentcd by the number labclled beside a capture circle. For example, the number 35 labelled for OPG means the miss distance is equal to 35 x 10-4r0. It can be seen that O P G has a smaller miss distance than TPN. Also, OPG requires a commanded acceleration smaller than that of TPN at the beginning stage of the interception. This fact is also observed in Fig. 12 where the curves of commanded missile acceleration versus the dimensionless time T have been plotted.

2. Samrntion of Coninlanded Missile Accelerntion. !t is not always possible to design the missile with

YANG, IISIAO, & YEH: GENERALIZED GUIDANCE LAW FOR HOMING MISSILES 207

11.25 11.5 11.75 I .I1 s/,:

Trajectories of missile with dynamic lag for pursuing maneuvering target. Fig. 11.

Fig. 12. Time histories of commanded missile acceleration for OPG and TI".

sufficient acceleration capacity to allow it to obey the law of OPG in all situations. Therefore, whenever the interception problem is such that the control system calls for more acceleration than the missile can provide, the missile tends to deviate from OPG, which results in a miss distance. The effect of saturation of commanded acceleration is shown in Fig. 12 where it is assumed that the maximum acceleration the missile can provide is 1.2VooVro/ro. It can be seen that in order to obey the OPG, the missile does not leave the saturated state until the vicinity of impact point is achieved. Averagely speaking, since TPN necessitates a larger acceleration command at the beginning stage of interception, it is more susceptible to acceleration saturation. The actual missile trajectories under the effect of acceleration saturation a re demonstrated in Fig. 13 where the equivalent missile time constant is assumed to be 0.005 t f and the target is escaping

via TF" law with proportionality constant 0.1. As expected, the miss distance is enlarged due to saturation effect.

the target maneuverability is the magnitude of the proportionality constant R for the target to escape from missile via TPN law. For a high maneuverability target, the missile requires a large expenditure of maneuvering energy to finish the interception and is frequently accompanied by a long homing time duration. In some cases, the missile performance is dominated by the saturation of the commanded acceleration and then the expenditure of maneuvering energy must be kept as low as possible at the expense of homing time duration, while, in some other cascs, the situation may be reversed. The expenditure of the maneuvering energy and homing time duration has been compromised via the weighting coefficient p introduced in (39). The influence of p and the target maneuverability R on homing time duration t f can be observed in Fig. 14 and Fig. 15. The time of capture required by TPN is obtained via the formula

3. Target Maneuverability. A n indication of

t f rf =-ro/l/,o

1 = I - 1 + (Vro/Veo)*(1 - 2 N )

which is derived from (32) by letting /\2 = 0. The rf for OPG can be found by the use of (55). The rcsults are depicted in Fig. 14. It can be seen that over the most range of p and N which are compatible with the point capture condition, the time of capture for OPG is lower than that of TPN. Especially, when (Vro/Voo)*(l - 2 ~ ) is close to -1, TPN requires a time which is several times longer than that requircd by OPG. Moreover, the lowest possible Tf which can be attained by TPN is equal to 1, while the rf can be reduced arbitrarily to 0 by OPG. For a mancuvering target, it can be seen from Fig. 14 that the homing time duration ts increases drastically as R exceeds

m IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-25 , NO. 2 MARC1 i 1989

fig. 13. Missile trajectories for OPG when commanded acceleration is saturated.

0.5. If a small value of p is employed, i.e., most of the weighting has been put on the requirement of interception time, it can delay the deterioration of the required homing time duration, when the missile is pursuing a target with high maneuverability (see Fig. 15). However, the commanded missile acceleration could be large at the same time.

If the control system contains a n inadvertent bias, which demands a fixed acceleration in addition to that called for by the tracking signals, then the commanded missile acceleration may not decrease to zero at the end of pursuit as can be seen from Fig. 16. Under the bias effect of commanded acceleration, the required missile acceleration becomes larger at the pursuit end, if a smaller miss distance is desired.

Fig. 17 shows the effect of the measurement bias of LOS angle on the missile commanded acceleration. The corresponding missile trajectories are described in Fig. 18 with T = 0.005 t f and R = 0.1. It is observed that due to this LOS bias the missile is moving oppositely to the direction of the target flight path during the beginning stage of intersection. Hence, it calls for a n additional acceleration to bring the missile velocity vector into the right direction. Of course, the larger the bias is, the larger the recovering energy will be.

derived in the previous section is in the form of

4. Systent and Measurentent Bias.

5. A Sintplifed OPG. The optimal guidance law

A aMc = --(sin(@ - @,)e, + cos(@ - @,)e,>e

2P

where A and 0, are the solutions of the transcendental equations (53a) and (53b). If the target information is not available, nor the digital processor used to provide such an on-line calculation, then having a further simplification of this guidance law is warranted. If the terms relating to the target motion have been neglected in (53), and assumed to have a small value of

Fig. 14. Homing time duration versus weighting coefficients for various curvatures of target flight path.

80 - Of, then (53a) and (53b) reduce to

= I A 1 - _ - 4P A

where C = Il/ro/Veol. Solving for A and 80 - O f , we have

A = 2 p + JG

By using this linearized approximation, the corresponding commanded missile acceleration is depicted in Fig. 12. It can be seen that this approximate optimal solution requires an intermediate value of the missile acceleration, when compared with those required by OPG and TPN. This approximation approaches the actual optimal solution when the value of C becomes larger and larger as can be seen from (Sib), since in such a case, the assumption of a small value of 00 - 8, is justified.

YANG, HSIAO, & YEH: GENERALIZED GUIDANCE LAW FOR HOMING MISSILES 209

1 .5 - f

1.0

0.5

I 0 . 5

GO

U

I 1.0 11.5 *

Vr, Fig. 16. Time history of missile acceleration in presence of acceleration bias of actuator output.

a 1 2.0-

1.5-

2.0-

1.0

I I

0 .5 1.0 0.5 R (dimensionless Curvature1

Fig. 15. Homing time duration versus target curvature for various weighting coefficients.

VI. CONCLUDING REMARKS

In this paper the concept of a generalized guidance law is presented and the closed-form solution of a homing missile pursuing a maneuvering target according to gencralized guidance laws is given. It has been shown that the existing guidance laws, appearing in the literature, a re merely special cases of the one proposed here. The derived gencralized forms of capture area, missile acceleration, and homing time duration allow us to get a quick insight into the performance of the guidance laws being considered,

210 IEEE TRANSACTIONS ON AEROSPACE Al

Fig. 17. Time history of missile acceleration i n prescncc of nieasurenient bias in LOS.

and pave the way to the discovery of new guidance laws.

The problem of finding a nonlinear optimal guidance law for a homing missile with commandcd acceleration applied to the most desirable direction so as to capture a maneuvering target with a predetcrmincd trajectory while minimizing a weighted

\ID ELECTRONIC SYSI'EMS VOL. AES-2.5. NO. 2 MARC11 1989

T-O.flfl5 t f R.0.1

0 .25 1 I UIS Bias fradl

// I ’/ 0.05

Xlr.

Fig. 18. Trajectories of missile in presence of LOS bias.

linear combination of time of capture and energy expenditure, has been solved in a closed form. The derived optimal control law is equal to the LOS rate multiplied by a trigonometric function of aspect angle, which can be implemented easily. From the numerical simulation, the resulting guidance law appears to yield a significant advantage over the TPN. Simulations also have shown the performance of the missile according to this optimal guidance law under the imperfectness such as the saturation of the commanded acceleration, delay due to missile dynamic, target acceleration, and bias of the missile commanded acceleration.

REFERENCES

Arbenz, K. (1970) Proportional navigation of nonstationary targets. IEEE Transactions on Aerospace and Electronic S)stettu (July 1970), 455457.

Comparison between proportional and augmented proportional navigation. Nachrichtentechnische ZeitschriJi (July 1974), 278-280.

Axelband, E. Z, and Hardy, E W. (1970) Quasi optimal proportional navigation. IEEE Transactions on Automatic Control, AI-15 (Dec. 1970), 6204%.

Axelband, E. Z. , and Hardy, I;: W. (1969) Quasi optimal proportional navigatiob In Proceedings of tlw Second Hawaii Itnerriatiottal Cotifcrcnce on System Sciences, 1969, pp. 417421.

Axelband, E. Z., and IIardy,.E W. (1970) Optimal feedback missile guidance. I n Proceedings of the Third Ilawaii Inrernational Con fcrcnce on Systctn Sciences, 1970, pp. 874-877.

A new guidance law for homing missiles. Joirrrwl of Guidance, Control and @narnics, 8 (May-June 1985), 402404.

Siouris, G. M. (1974)

Kim, Y. S., Cho, H. S., and Bien, Z (1985)

Yang, C. D., Yeh, E B., and Chen, J. H. (1987) The closed-form solution of generalized proportional navigation. Jocunal of Guidance and Control, 10 (Mar.-Apr. 1987), 216218.

The closed-form solution for a class of guidance laws. Journal of Guidance, Control and @namics, 10 (July-Aug. 1987), 412415.

Effects of performance index/constraint combination on optimal guidance laws for air-to-air missiles. In Proceedings of NAECON, May 1979, pp. 765-771.

Comparison of optimal control and differential game intercept missile guidance laws. Journal of Giiidarice and Control, 4 (Mar.-Apr. 1981), 109-115.

Optimal intercept guidance for short-range tactical missiles. AIAA Journal, 9 (July 1971), 1414-1415.

The closed-form solution of tNe proportional navigation. IEEE Transactions on Aerospace and Electronic S~srettu,

Yang, C. D., and Yeh, E B. (1987)

Anderson, G. M. (1979)

Anderson, G. M. (1981)

Cottrel, R. G. (1971)

Guelman, M. (1976)

AES-12 (July 1976), 472-482. Guelman, M. (1972)

A qualitative study of proportional navigation. IEEE Transactions on Aerospace and Electronic S)Bstctns, AES-7 (July 1972), 637-643.

Bryson, A. E., and Ho, Y. C. (1969) Applicd Optirnal Control. Waltham, Mass.: Blaisdell, 1969.

Optimal proportional navigation. Journal of Guidance, Control and @riatnics (July-August l988), 375-377.

Yang, C. D., and Yeh, E B. (1988)

YANG, IISIAO, & YEH: GENERALIZED GUIDANCE LAW FOR HOMING MISSILES 21 1

Ciann-Dong Yang received the B.S., M.S. and Ph.D. degrees in aeronautics and astronautics from National Cheng Kung University, Taiwan, in 1983, 1985 and 1987. He is currently an instructor with the Department of aeronautics and astronautics, National Cheng Kung University. His interests are mainly in missle guidance and control design, and system theory.

Fei-Bin Hsiao received the B.S. and M.S. degrees in power mechanical engineering from National Tsing Hua University, Taiwan, in 1976 and 1979, and the Ph.D. degree of Aerospace Engineering from the University of Southern California, Los Angeles, in 1985.

From 1976 to 1979 he was a Teaching Assistant at Tsing Hua University and served as a Research Assistant at USC during 1979-1985. He has been an assistant professor with the Institute of Aeronautics and Astronautics, National Cheng Kung University, ever since. His interests a re mainly in the fields of experimental aerodynamics, aircraft performance testing, and flight dynamics control. He is now a member of A I M , AASROC, STAMROC, and CSME.

Fang-Bo Yeh received the Ph.D. degree in mathematics from Glasgow University, Glasgow, UK, in 1983. From 19S3 to 1986, he was an associate professor in the department of Tunghai University and the Institute of Aeronautics and Astronautics Cheng Kung University, Taiwan, where he is currently profcssor of Mathematics. His research interests are in multivariable control, system theory and flight control.

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