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JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010)42

Modern Homing Missile Guidance Theoryand Techniques

Neil F. Palumbo, Ross A. Blauwkamp, and Justin M. Lloyd

INTRODUCTIONClassical guidance laws, with proportional navigation

(PN) being the most prominent example, had proven to

be effective homing guidance strategies up through the1960s and early 1970s. By the mid-1970s, however, thepredicted capability of future airborne threats (highly

maneuverable aircraft, supersonic cruise missiles, tacti-cal and strategic ballistic missile reentry vehicles, etc.)indicated that PN-guided weapons might be ineffective

against them. However, by that time, the application of

optimal control theory to missile guidance problems had

sufciently matured, offering new and potentially prom-

ising alternative guidance law designs.13 Around this

time, the computer power required to mechanize suchadvanced algorithms also was sufcient to make their

application practical.

Most modern guidance laws are derived using linear-

quadratic (LQ) optimal control theory to obtain

analytic feedback solutions.1, 4, 5 Many of the modern

lassically derived homing guidance laws, such as proportional

navigation, can be highly eective when the homing mis-

sile has signifcantly more maneuver capability than the threat.

As threats become more capable, however, higher perormance is required rom the

missile guidance law to achieve intercept. To address this challenge, most modern guid-

ance laws are derived using linear-quadratic optimal control theory to obtain analytic

eedback solutions. Generally, optimal control strategies use a cost unction to explicitly

optimize the missile perormance criteria. In addition, it is typical or these guidancelaws to employ more sophisticated models o the target and missile maneuver capabil-

ity in an eort to improve overall perormance. In this article, we will present a review

o optimal control theory and derive a number o optimal guidance laws o increasing

complexity. We also will explore the advantages that such guidance laws have over

proportional navigation when engaging more stressing threats


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JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) 43-

formulations take target maneuver into account to dealwith highly maneuvering target scenarios (particularlytrue for terminal homing guidance). The availabilityof target acceleration information for the guidance lawvaries, depending on targeting sensor capability andtype and the specic guidance law formulation. Typi-cally, there also is an explicit assumption made aboutthe missile airframe/autopilot dynamic response charac-

teristics in the modern formulations. We will show laterthat PN is an optimal guidance law in the absence ofairframe/autopilot lag (and under certain other assumedconditions). To some extent, the feedback nature of thehoming guidance law allows the missile to correct forinaccurate predictions of target maneuver and otherunmodeled dynamics (see Fig. 1). However, the require-ment for better performance continues to push optimalguidance law development, in part by forcing the con-sideration (inclusion) of more detailed dynamics of theinterceptor and its target. It is interesting that most ifnot all modern guidance laws derived using optimal con-trol theory can be shown to be supersets of PN.

In the following sections, we rst provide a cursoryreview of dynamic optimization techniques with a focuson LQ optimal control theory. Using this as background,we then develop a number of terminal homing guid-ance laws based on various assumptions, in the order ofassumption complexity. In our companion article in thisissue, Guidance Filter Fundamentals, we introduceguidance ltering, with a focus on Kalman guidancelter techniques.

REVIEW OF LQ OPTIMAL CONTROLHere, we start by considering a general nonlinear

dynamics model of the system to be controlled (i.e., theplant). This dynamics model can be expressed as

( ) ( ( ), ( ), ) .t t t tx f x u=$ (1)

In Eq. 1, x is the n-dimensional state vector of realelements ( ),x uR Rn m! ! is the control vector, and

t represents time (later we will provide more detail asto the structure of the state and control vectors for thehoming guidance problem). With this general system, we

associate the following scalar performance index:

( ( ), ( ), ) ( ( ), )

( ( ), ( ), ) .

J t t t t t

L t t t dt

x u x

x u

f f

t

t

0 0 0

f

0

=

+ # (2)

In this equation, [t0, tf] is the time interval of inter-est. The performance index comprises two parts: (i)a scalar algebraic function of the nal state and time

(nal state penalty), ( ( ), )t tx f f , and (ii) a scalar inte-gral function of the state and control (Lagrangian),

( ( ), ( ), ) .L t t t dtx uf0

t

t# The choice of ( ( ), )t tx f f and ( ( ), ( ), )L t t t dtx uf

0

t

t#(this choice is a signicant part

of the design problem) will dictate the nature of theoptimizing solution. Thus, the performance index

is selected to make the plant in Eq. 1 exhibit desiredcharacteristics and behavior (transient response, stabil-

ity, etc.).For our purposes, the optimal control problem is

to nd a control, ( )tu* , on the time interval [t0, tf] thatdrives the plant in Eq. 1 along a trajectory, ( )tx* , suchthat the scalar performance index in Eq. 2 is minimized.

It is difcult to nd analytic guidance law expressionsfor such general nonlinear systems. Therefore, we will

turn our attention to a subset of optimal control that

can yield tractable analytic solutions, known as the LQoptimal control problem.

Targetsensors

Guidancefilter

Guidancelaw

Airframe/propulsion

Inertialnavigation

Autopilot

Targetmotion

Targetdirection

Target stateestimates

Accelerationcommands

Actuationcommands

Vehiclemotion

Kalman FilterTheory(1960s)

Optimal ControlTheory(1960s)

ServomechanismTheory

(1930s1950s)

Figure 1. The traditional guidance, navigation, and control topology or a guided missile comprises guidance flter, guidance law,

autopilot, and inertial navigation components. Each component may be synthesized by using a variety o techniques, the most

popular o which are indicated here in blue text.


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N. F. PALUMBO, R. A. BLAUWKAMP, and J. M. LLOYD


The LQ Optimal Control Problem

Here, we assume that the nonlinear model in Eq. 1can be linearized about an equilibrium point ( , )x u0 0 and represented by the time-varying linear dynamicsystem described in Eq. 31, 4:

( ) ( ) ( ) ( ) ( ) .t t t t tx A x B u= +$ (3)

In this model (as in the nonlinear case), x Rn! is thestate vector, u Rm! is the control vector, and t repre-sents time. Here, however, A(t) and B(t) are the time-varying Jacobian matrices

=

( ( , , )/tf x u x x xu u

0

0

2 2 = and=

( , , )/tf x u u x xu u

0

0

2 2 = , respectively)

of appropriate dimension.Using the shorthand notation

S( ) ( ) ( )t t tz z SzT

2_ ,

we next dene the quadratic performance index (aspecial case of the performance index shown in Eq. 2)shown in Eq. 4:

dt .

( ( ), ( ), ) ( )

( ) ( )

J t t t t

t t

x u x

x u( ) ( )

f

t tt

t

Q

Q R

0 0 0 21 2

21 2 2

f

f

0

=

+ +8 B#(4)

In Eq. 4, the following assumptions are made: the ter-minal penalty weighting matrix, Qf, is positive semi-

denite (the eigenvalues of Qf are 0, expressed asQf 0); the state penalty weighting matrix, Q(t), ispositive semi-denite (Q(t) 0); and the control pen-alty matrix, R(t), is positive denite (R(t) > 0). Thus, theLQ optimal control problem is to nd a control, ( )tu* ,

such that the quadratic cost in Eq. 4 is minimized sub-ject to the constraint imposed by the linear dynamicsystem in Eq. 3.

To solve this continuous-time optimal control prob-lem, one can use Lagrange multipliers, ( ) t , to adjointhe (dynamic) constraints (Eq. 3) to the performanceindex (Eq. 4).1, 5 Consequently, an augmented cost func-tion can be written as

( )

( ( ) ( ) )

( ) ( ( ) ( )

( ) ( ) ( ))

.

J t

t t

t t t

t t t

dt

x

x u

A x

B u x

( ) ( )

f

t t

Tt

t

Q

Q R

21 2

21 2 2

f

f

0

=

+

+

+

+

l

o

R

T

SSSSS

V

X

WWWWW#

(5)

Referring to Eq. 5, we dene the Hamiltonian function

H as shown:

( ( ), ( ), ) ( ) ( )

( )( ( ) ( ) ( ) ( )) .

H t t t t t

t t t t t

x u x u

A x B u

( ) ( )t t

T

Q R21 2

21 2= +

+ +

(6)

Then, from the calculus of variations, four necessaryconditions for optimality must be satised5 to solve ourproblem: state, costate, boundary, and stationarity con-ditions must all hold. These four conditions are listed inTable 1.

From Table 1 (and based on our previous descriptionof the dimension of the plant state vector), it can be seenthat there is a system of 2n dynamic equations that must

be solved; n equations must be solved forward in timeover [t0, tf] (state equations), and n equations must besolved backward in time over [tf, t0] (costate equations).We further note that the equations are coupled. Apply-ing the stationarity condition to Eqs. 36 yields the fol-lowing result for the control:

( ) ( ) ( ) ( ) .t t t tu R B T1= (7)

Using Eq. 7 in the state and costate necessary conditions,and taking into account the boundary condition, leadsto the following two-point boundary value problem:

( )( )

( ) ( )

( ) ( )

( )( )

,

( )

( ) ( ).

tt

tt

tt

tt

t

t t

x AQ

B R BA

x

x

Q x

given

TT

f f f

1

0

=

=

$

$= = ;G G E) 3

(8)

Here, we are concerned with solution methods thatcan yield analytic (closed-form) solutions rather thaniterative numerical or gain scheduling techniques. Wenote, however, that the sparseness/structure of the con-stituent plant and cost matricese.g., A(t), B(t), Q(t),and R(t)will dictate the ease by which this can beaccomplished. Qualitatively speaking, the level of dif-

culty involved in obtaining analytic solutions is relatedprimarily to the complexity of state equation couplinginto the costate equations, most notably by the structureofQ(t) and, to a lesser extent, by the structure ofA(t). Aswe will see later, however, this fact does not negate ourability to derive effective guidance laws.

Given a suitable system structure in Eqs. 7 and 8 (asdiscussed above), one conceptually straightforward wayto solve this problem is to directly integrate the costateequations backward in time from tf to tt0 using theterminal costate conditions and then integrate the state

Table 1. Necessary conditions or optimality.

Condition Expression

State

Equation( ( ), ( ), )/ ( ) ( ),H t t t t t t tx u x for 02 2 $= o

Costate

Equation( ( ), ( ), )/ ( ) ( ),H t t t t t t tx u x for f2 2 #=

o

Stationarity ( ( ), ( ), )/ ( ) ,H t t t t t tx u u 0 for 02 2 $=

Boundary ( ( ), )/ ( ) ( ), ( )t t t ttx x x givenf f f f 02 2 =


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downrange, for example, while the y/z axis can repre-sent either crossrange or altitude, respectively (we will

use y below). For simplicity, we assume a at-Earthmodel with an inertial coordinate system that is xedto the surface of the Earth. Furthermore, we will assumethat the missile and target speeds, vM and vT ,respectively, are constant.

In Fig. 2, the positions of the missile (pursuer) M andtarget (evader) T are shown with respect to the origin (O)of the coordinate system, rM and rT , respectively. Thus,the relative position vector [or line-of-sight (LOS) vector,as it was previously dened] is given by r r r

T M= , and

the relative velocity vector is given by r v v vT M

_ =o .From Fig. 2, note that an intercept condition is satisedif yt = 0 and V

c

> 0 (i.e., a collision triangle condition).As illustrated, for this condition to be satised, the mis-sile velocity vector must lead the LOS by the lead angleL. Determining the lead angle necessary to establish acollision triangle is, implicitly, a key purpose of the guid-ance law. How this is done is a factor of many things,including what measurements are available to the guid-ance system and what assumptions were made duringformulation of the guidance law (e.g., a non-maneuver-ing target was assumed).

The homing kinematics can be expressed in vari-ous ways. Here, we concentrate on Cartesian formsof expression. From a previous section, the target-to-missile range was dened as rR = , and targetmis-sile closing velocity was dened as V R v 1

c r:_ /o ,

where the LOS unit vector is /R1 rr= . Thus, refer-

ring to Fig. 2, expressions for targetmissile relativeposition, relative velocity, and relative acceleration

are given below, where we have dened the quantities, , ,v a v av a v aandM M M M T T T T

_ _ _ _ .

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

cos sin

cos cos

sin sin

sin sin

cos cos

r r R R

v v v v L

v v L

a a a a

a a

r 1 1 1 1

v 1 1 1

1

a 1 1 1

1

x x y y x y

x x y y T T M x

T T M y

x x y y T T M x

T T M y

= + = +

= + = +

+ +

= + = +

+

6 6

6

6

6

6

@ @

@

@

@

@

"

"

"

"

,

,

,

,

(17)

Referring again to Fig. 2, we note that if the clos-ing velocity is positive (Vc > 0), then we only need to

actively control the kinematics in the y/z coordinate toachieve an intercept. That is, if Vc > 0 and the missileactively reduces and holds ry to zero by appropriatelyaccelerating normal to the LOS, then rx will continueto decrease until collision occurs. We will assume thiscondition holds and disregard the x components in thefollowing analysis.

The homing kinematics shown in Eqs. 17 are clearlynonlinear. In order to develop guidance laws using linearoptimal control theory, the equations of motion mustbe linear. Referring to the expression for relative posi-tion in Eqs. 17, note that for l very small, the y-axiscomponent of relative position is approximately given

by ryRl. Moreover, for very small T and l (e.g.,near-collision course conditions), the y-axis compo-nent of relative acceleration is approximately given byayaTaM. Hence, given the near-collision courseconditions, we can draw the linearized block diagramof the engagement kinematics as shown in Fig. 3. Cor-respondingly, we will express the kinematic equations

M= Missile flight path angle

rM= Missile inertial position vector

T = Target flight path angle

= LOS angle

rT = Target inertial position vector

vM= Missile velocity vector

vT= Target velocity vector

L = Lead angle

R = Range to target

aM= Missile acceleration,normal to LOS

aT= Target acceleration,normal to VT

rx = Relative position x(rTx rMx)

ry = Relative position y(rTy rMy)

rx

rM

xI

M

T

yI/z

I

T

vMaM

aTvT

rTry

Target(T)

Missile(M)

LR

Origin(O)

Figure 2. Planar engagement geometry. The planar intercept

problem is illustrated along with most o the angular and Carte-

sian quantities necessary to derive modern guidance laws. The

x axis represents downrange while the y/z axis can represent

either crossrange or altitude. A at-Earth model is assumed with

an inertial coordinate system that is fxed to the surace o the

Earth. The positions o the missile (M) and target (T) are shown

with respect to the origin (O) o the coordinate system. Dieren-

tiation o the targetmissile relative position vector yields relative

velocity; double dierentiation yields relative acceleration.

Missile acceleration

+

Target

acceleration

R

1aT

vy ry

aM

Relativevelocity

Relativeposition

Figure 3. Linear engagement kinematics. Planar linear homing

loop kinematics are illustrated here. The integral o targetmissile

relative acceleration yields relative velocity; double integration

yields relative position. The LOS angle, , is obtained by dividing

relative position by targetmissile range.


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MODERN HOMING MISSILE GUIDANCE THEORY AND TECHNIQUES


of motion in state-space form. To this end, we dene

the state vector x xx T1 2

_ 6 @ , the control u, the plantdisturbance w, and the pseudo-measurement y where

, , ,x r x v u a w ay y M T1 2_ _ _ _ (any residual target

acceleration is treated as a disturbance), and y R_ l

(the linearized Cartesian pseudo-measurement is com-

posed of range times the LOS angle). Given these deni-

tions, the kinematic equations of motion in state-space

form are written as

( ) ( ) ( ) ( ), (0)

( ) ( )

,

, , .

t t u t w t

y t t

x Ax B D x x

Cx

A B C D0

0

1

0

0

11 0

0

1

0= + + =

=

= = = =

o

; ; 6 ;E E @ E(18)

These equations will form the basis for developing a

number of terminal homing guidance laws in the fol-

lowing subsections. In some cases, the equations are

modied or expanded as needed to reect additional

assumptions or special conditions.

DERIVATION OF PN GUIDANCE LAW VIA

LQ OPTIMIZATIONIn the previous article in this issue, Basic Principles

of Homing Guidance, a classical development of PN

guidance was given based on that found in Ref. 12. In

the present article, we will leverage off the discussion of

the previous subsection and develop a planar version of

the PN guidance law using LQ optimization techniques.

To start, we will state the key assumptions used to

develop the guidance law:

We use the linear engagement kinematics modeldiscussed previously and, hence, the state vector is

( ) ( ) ( )t x t x tx T1 2= 6 @ (the state vector comprisesthe components of relative position and relativevelocity perpendicular to the reference x axis shownin Fig. 2).

All the states are available for feedback.

The missile and target speeds, v vM M

= andv vT T

= , respectively, are constant.

The missile control variable is commanded acceler-ation normal to the reference x axis (u = ac), which,

for very small LOS angles (l), is approximately per-pendicular to the instantaneous LOS.

The target is assumed to be nonmaneuvering

(aT= 0) and, therefore, the linear plant disturbanceis given to be w = 0.

The missile responds perfectly (instantaneously) to

an acceleration command from the guidance law(aM ac).The system pseudo-measurement is rela-tive position y = x1.

With these assumptions in mind, and referring to Eq. 18,the LQ optimization problem is stated as

( ) ( )

( ) ( )

( ),

minJ t u t dt

t t u t

x u x

x x

x

0

0

1

0

0

1

1 0

Subject to:

( )u tf

t

t

Q0 0 0 2

1 221 2

f

f

0

= +

= +

( ), ( ),t t t

( ) ( ) .y t t=

o

^ h

; ;

6

E E@

#

(19)

In words, nd a minimum-energy control u(t) on thetime interval [t0, tf] that minimizes a quadratic func-tion of the nal (terminal) relative position and relativevelocity and subject to the specied dynamic con-straints. In Eq. 19, the terminal performance weightingmatrix, Qf, is yet to be specied. We dene the scalarb > 0 as the penalty on relative position at time tf (i.e.,nal miss distance) and scalar c 0 as the penalty onrelative velocity at tf (c specied as a positive nonzerovalue reects some desire to control or minimize theterminal lateral relative velocity as is the case for a ren-dezvous maneuver). Given the penalty variables, Q

fis

the diagonal matrix given in Eq. 20:

.b

cQ

0

0

f= ; E (20)

The choice ofb and c is problem-specic, but for inter-cept problems we typically let b, c = 0, and for ren-dezvous problems we have b, c.

General Solution or Nonmaneuvering Target

We can solve the design problem posed in Eq. 19 byusing the LQ solution method discussed previously. Wedene t t tgo f_ and initially assume the scalar vari-ables b and c are nite and nonzero. Using Maple, wesymbolically solve Eqs. 15 and 16 to obtain an analyticexpression for Pc(t). We then use Pc(t) in Eq. 10 to obtainthe following general guidance law solution:

( )

( ) ( )

.

u t

t

ct x t x t t3

1

1

gobt go

ct

go go bt

cgo

1

23

4

21

1 31

2

go

go

go

3

2

=

+ +

ct1+ + + +

1 ct+^

` ch

j mR

T

SSSSS

V

X

WWWWW

(21)

We emphasize the fact that this general solution isrelevant given the assumptions enumerated above,particularly the fact that a nonmaneuvering target andperfect interceptor response to commanded accelerationare assumed.

The structure of the guidance law u1(t), shown above,can be expressed as

( ) ( , )/ ( , , ) ,u t N t t z tQ x Qf go go f go12= u8 B


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20.0

Targetdirectionof travel

20

15

10

5

000.5 0.5 1.51.0

19.8

19.60.2 0 0.2 0.4 0.6

Downrange(km)

Crossrange (km)

b, c0.1, 100100, 100100, 1100, 0.1100, 0

Terminal penalties

Start (missile trajectories)

Figure 4. Rendezvous trajectory. A plot o intercept trajectories

(in downrange and crossrange) are shown with dierent terminal

penalties or relative position and relative velocity, respectively.

The target trajectory is traveling rom right to let (shown as the

dashed black line) at the top. The target velocity is a constant

500 m/s. The initial missile velocity is 1000 m/s with no heading

error. The fnal time is 20 s. The inset highlights the endgame

geometries in each case. Notice that as the terminal penalty

on relative velocity is increased, the missile trajectory tends to

bow out such that the fnal missile velocity can be aligned with

the target velocity. Similarly, as the terminal penalty on relative

position increases, the fnal miss distance is reduced.

where the quantity

( , ) /N tQ 3 1f go bt goct

34

go

go

3_ + +ct1 +u ^c h m

is the effective navigation ratio and , ,z tx Qf go^ h com-prises the remainder of the numerator in Eq. 21. Thus,

for this general case, the effective navigation ratio is not

a constant value.

It is clear that mechanization of this guidance lawrequires feedback of the relative position and relative

velocity states to compute the control solution. Typically,

only the relative position pseudo-measurement is avail-

able, formed using measurements of LOS angle and

range (assuming a range measurement is available). In

our companion article in this issue, Guidance Filter

Fundamentals, we discuss ltering techniques that

enable us to estimate any unmeasured but observ-

250

200

150

100

50

050 10 2015

Acce

leration(m/s2)

tgo(s)

b, c

0.1, 100

100, 100

100, 1

100, 0.1

100, 0

Terminal penalties

Figure 5. Rendezvous acceleration. A plot o called-or accel-

erations or intercept trajectories with diering terminal relative

position and relative velocity penalties is shown here. The engage-

ment is the same as in Fig 4. As can be seen, substantially more

acceleration is required as the penalty on the terminal relative

velocity is increased.

able states for feedback. From above, we also see thattime-to-go (tgo) is needed in the guidance law. Later, wealso discuss how one can estimate time-to-go for use inthe control solution.

Employing Eq. 21 in a simple planar engagementsimulation, we can show what effect a terminal rela-tive velocity penalty will have (in combination with aterminal relative position penalty) on the shape of themissile trajectory. In Fig. 4, a missile is on course to inter-cept a constant-velocity target; it can be seen that the

missile trajectory shape changes for different terminalrelative position and velocity penalties (i.e., variationsin the penalty weights b and c). Note that, in this simpleexample, there is no assumption that controlled missileacceleration is restricted to the lateral direction. Figure 5shows the resulting acceleration history for each case.Note that with a nonzero terminal penalty on relativeposition in combination with no penalty on relativevelocity (the blue curve in Fig. 5) the missile does notmaneuver at all; this is because the target maneuverassumption implicit in the guidance law derivation (i.e.,the target will not maneuver) happens to be correct inthis case, and the missile is already on a collision course

at the start of the engagement.

Special Case 1: PN Guidance Law

If we assume that c = 0 and we evaluateb"3

( )lim u t1 =uPN(t), then Eq. 21 collapses to the well- known Carte-sian form of the PN guidance law:

( ) ( ) ( ) .u t x t x t tPN t go3

1 2go2= +6 @ (22)


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Leveraging the discussion above for Eq. 21, we can seethat the effective navigation ratio in Eq. 21 has now col-lapsed to become

PNN 3=u .

In Eq. 22, the quantity in square brackets now repre-sents the miss distance that would result if the missileand target did not maneuver over the time period [t, tf],which often is referred to as the zero-effort-miss(ZEM). We want to emphasize that the guidance law

ZEM is an estimate of future miss that is parameterizedon the assumptions upon which the guidance law wasderived (linear engagement, constant velocity, non-maneuvering target, etc.). Hence, the PN ZEM (in they axis, for example) is given by ZEMPN = ry(t) + vy(t)tgo.Note that the accuracy of the guidance law ZEM esti-mate is directly related to how well the guidance law willperform in any particular engagement (actual nal missdistance, maximum commanded acceleration, amountof fuel used, etc.).

Under the current stated assumptions, we can showthat Eq. 22 is equivalent to the traditional LOS rateexpression for PN, which was shown in the previous

article in this issue, Basic Principles of Homing Guid-ance, to be a NV

Mc cl= o . We rst differentiate ry Rl

to obtain v RRy . l l+o o . Next, recalling that R =o

Vv 1cr

: = and noting that we can express range-to-go as R = Vctgo, we have the following relationship forLOS rate:

.V t

r v t

c go

y y go

2l =

+o (23)

Examining the traditional LOS rate expression for PN,as well as Eqs. 22 and 23, it can be seen that if we setN = 3, then the traditional LOS rate expression forPN and Eq. 22 are equivalent expressions. Hence, theoptimal PN navigation gain isN = 3.

Special Case 2: Rendezvous Guidance Law

For the classic rendezvous problem, we desire to con-verge on and match the velocity vector of the target atthe nal time tf. If we evaluate ,b c"3

( ) ( )lim u t u t1 REN= ,then Eq. 21 collapses to the Cartesian form of the ren-dezvous (REN) guidance law

( ) ( ) ( ) .u tt

x t x t t6

go

go2 1 32

2REN = +8 B (24)

Example Results

The example in this section is illustrated by Fig. 6,where we demonstrate how PN performance degrades if(i) the interceptor response is not ideal (i.e., the actualinterceptor response deviates from that assumed in thederivation of PN) and (ii) the target performs an unan-ticipated maneuver some time during the terminal

homing portion of the engagement. A planar (down-range and crossrange) Simulink terminal homing sim-ulation was used to examine the PN performance. Toavoid layering of complexity in the results, all simula-tion noise sources (seeker noise, gyro noise, etc.) wereturned off for this study. However, a simplied guidancelter still was used in the loop to provide estimates ofrelative position and velocity to the PN guidance law,

thereby having some effect on overall guidance per-formance. Figure 6a illustrates nominal missile andtarget trajectories in a plot of downrange versus cross-range. The total terminal homing time is about 3 s. Asis evident, at the start of terminal homing, the missileis traveling in from the left (increasing downrange)and the target is traveling in from the right (decreas-ing downrange). The missile, under PN guidance, isinitially on a collision course with the target, and thetarget is, initially, non-maneuvering. At 2 s time-to-go,the target pulls a hard turn in the increasing crossrangedirection.

Recall that PN assumes that the missile acceleration

response to guidance commands is perfect or instan-taneous (i.e., no-lag), and that the target does notmaneuver. Thus, we will examine the sensitivity of PNto these assumptions by simulating a non-ideal mis-sile acceleration response in addition to target maneu-ver. We will parametrically scale the nominal missileautopilot time constant (100 ms for our nominal case),starting with a scale factor of 1 (the nominal or near-perfect response) up to 5 (a very sluggish responseof approximately 500 ms) and examine the effect onguidance performance. Figure 6b illustrates the accel-eration step response of the interceptor for 100-, 300-,and 500-ms time constants. Figures 6c and 6d pres-ent the simulation results for levels of target hard-turnacceleration from 0 to 5 g. Figure 6c shows nal missdistance versus target hard-turn acceleration level forthe three different interceptor time constants. As canbe seen, as the missile time response deviates fromthe ideal case, guidance performance (miss distance)degrades as target maneuver levels increase. Figure 6dillustrates the magnitude of missile total achieved accel-eration for variation of target maneuver glevels shownand for the three missile time constants considered.As can be seen, when the missile autopilot responsetime deviates further from that which PN assumes,

respectively higher acceleration is required from themissile; this is further exacerbated by higher targetglevels.

EXTENSIONS TO PN: OTHER OPTIMAL HOMING

GUIDANCE LAWSIn Eq. 21, a general guidance law that is based on

some specic assumptions regarding the (linearized)


9/18



Figure 6. PN perormance versus time constant. (a) A planar missiletarget engagement with a

plot o downrange versus crossrange. The total terminal homing time is approximately 3 s. At the

start o terminal homing, the missile is traveling in rom the let (increasing downrange) and thetarget is traveling in rom the right (decreasing downrange). (b) The acceleration step response o

the missile interceptor or 100-, 300-, and 500-ms time constants. (c and d) Simulation results or

a PN-guided missile versus varying levels o target hard-turn acceleration rom 0 to 5 g. (c) Final

miss distance versus target hard-turn acceleration level or the three dierent interceptor time

constants. The graph illustrates the act that, or non-ideal interceptor response, PN-homing guid-

ance perormance degrades with increasing target maneuver levels. This degradation worsens as

the autopilot response becomes more sluggish. Notice that as the autopilot response approaches

the ideal case (blue curve), the miss distance becomes nearly insensitive to target maneuver. For

an ideal autopilot response, PN-homing would result in acceleration requirements o three times

the target maneuver. (d) The magnitude o total achieved missile acceleration or the same varia-

tion o target maneuver g levels and or the three missile time constants considered. As the missile

autopilot response time deviates urther rom that which PN assumes, increasingly higher accel-

eration levels are required rom the missile.

engagement kinematics,target maneuver (actually,a lack of it), and intercep-tor response characteristics(the perfect response to anacceleration command) isshown. We also showed thatthe general optimal guid-

ance law derived there col-lapses to the well-known PNguidance law if we assigncost function components asc = 0 and b (see Eqs. 19and 20). For completenesssake, here we will derive anumber of related optimalguidance laws under differ-ing assumptions regardingtarget maneuver and inter-ceptor response models.For certain scenarios, these

assumptions can have asignicant impact on guid-ance law performance. Forexample, in Example Results,it was observed that, duringthe endgame, PN (which isderived assuming no targetmaneuver) can call for sig-nicant acceleration fromthe interceptor when pittedagainst a target pulling ahard-turn maneuver. Hence,one can develop a guidance

law that specically takestarget acceleration intoaccount. Of course, mecha-nization of such a guidancelaw will be more complexand will require the feed-back of additional informa-tion (e.g., target maneuver).In addition, if the derivationassumptions become too spe-cic, the resulting guidancelaw may work well if those

assumptions actually hold,but performance might rap-idly degrade as reality devi-ates from the assumptions.

Constant Target

Acceleration Assumption

Here, all of the previousassumptions hold with the

0 1 2 3 4 50

10

20

30

40

Target maneuver (g)

Acceleration

(g)

0 1 2 3 4 50

0.5

1.0

1.5

2.0

2.5

3.0

Target maneuver (g)

Finalmissdista

nce(m)

0 500 1000 1500 2000 2500 3000 3500 40000

1

2(a)

(b)

(c) (d)

Downrange (m)

Crossrange(km)

0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

0.5

1.0

1.5

Time (s)

Stepresponse

(g)

Missile Target

Step input command

0.1 s 0.3 s 0.5 sAutopilot time constant

exception of target maneuver; we will assume that the target is pulling a hard-turn

maneuver (i.e., constant acceleration in a particular direction). Therefore, we aug-

ment the previous (PN) state vector (where ,x r x vy y1 2_ _ ) to include a target

acceleration state x aTy3 _ , leading to x x xxT

1 2 3_ 6 @ . As before, the control u is

missile acceleration u aM

_^ h; the plant (process) disturbance is not considered whenderiving the guidance law (w = 0), but it will come into play when developing a target


10/18



acceleration estimator; and the pseudo-measurement, again, is relative positiony x1_^ h. With these modeling assumptions in mind (particularly that x aTy3 _

constant), the LQ optimization problem is stated as

( ) ( )

( ) ( ) ( ) ,

.

minJ t u t dt

t t u t

x u x

x x

x

0

0

0

1

0

0

0

1

0

0

1

01 0 0

Subject to:

( )u tf

t

t

Q0 0 0 2

1 221 2

f

f

0

= +

= +

( ), ( ),t t t

( ) ( )y t t=

o

^ h

> >6 H H@

#

(25)

As before (Eq. 20), the terminal penalty matrix is dened as Qf = diag{b, c, 0}.Following a solution procedure identical to that outlined previously, we obtain thefollowing general solution:

( )

( ) ( ) ( )

. ( )u tt

ct x t x t t x t t3

1

1

26go

bt go

ct

go go bt

cgo go bt

cgo

2 23

4

21

1 31

2 21

61 2

32

go

go

go go

3

2 2

=

+ +

+ + +

1 ct+

ct ct1 1+ + + +

^

` c c

h

j m mR

T

SSSSS

V

X

WWWWW

If we compare the guidance law in Eq. 26 to our previous result (Eq. 21), it is

clear that the only difference is in the numerator; Eq. 26 includes the addition of a

(time-varying) gain multiplying the target acceleration state. Therefore, in addition to

requiring estimates of relative position, relative velocity, and time-to-go, the new law

also requires an estimate of target acceleration. This requirement has implications on

the guidance lter structure, as we will see later. More important, estimating target

acceleration given a relative position (pseudo-)measurement can be very noisy unless

the measurement quality is very good. Hence, if sensor quality is not sufcient, lead-

ing to a target acceleration estimate that is too noisy, then guidance law performance

could actually be worse than if we simply used PN to begin with. These factors must

be carefully considered during the missile design phase.

Special Case 1: Augmented PN

If we assume that c = 0 and we evaluateb"3

( ) ( )lim u t u t2 APN= , then Eq. 26 col-

lapses to the well-known augmented PN (APN) guidance law:

( ) ( ) ( ) ( ) .u tt

x t x t t x t t3

APNgo

go go2 1 2 21

32= + +8 B (27)

Leveraging the discussion above, if we compare Eq. 27 to the expression for PN given

in Eq. 22, the only difference is the addition of the a tT go21 2

term in the numeratorof the APN guidance law. Thus, for APN, the effective navigation ratio is the same

as in PN guidance (APN

N 3=u ), but the ZEM estimate is now given by ZEMAPN

=

( ) ( ) ( )r t v t t a t ty y go T go21+ + 2 .

Special Case 2: Augmented REN Guidance Law

If we evaluate,b c"3

( ) ( )lim u t u tAREN2 = , then Eq. 26 collapses to the augmented

REN (AREN) guidance law:


11/18



( ) ( ) ( ) ( ) .u tt

x t x t t x t6

ARENgo

go2 1 32

2 3= + +8 B (28)

Notice that, unlike the APN law given in Eq. 27, theAREN guidance law calls for a direct cancellation of thetarget maneuver term in the acceleration command.

Example Results: Maneuver Requirements for

PN Versus APN

Here, in a similar approach to that found in Ref. 13,we compare the maneuver requirements for PN andAPN guidance laws versus a target that pulls a hard-turnmaneuver under ideal conditions (e.g., no sensor mea-surement noise or interceptor lag). Recall that the PNguidance law is derived assuming that the target doesnot maneuver, whereas the key APN assumption is thatthe target pulls a hard turn.

Assuming that the interceptor responds perfectly toPN acceleration commands, using Eq. 22 we can writethe following second-order dif ferential equation:

( ) ( ) ( )

[ ( ) ( ) ( )] .dt

d r t a tt t

N r t t t tvy Tf

y y f2

2

2= + (29)

Note that we have left the navigation gain as thevariable N. Next, we assume zero initial conditionsry(0) = 0, vy(0) = 0and use Maple to solve Eq. 29, thusgiving analytic expressions for ry(t) and vy(t). We thentake ry(t) and vy(t) and reinsert them into Eq. 22 toobtain the following expression for missile accelerationcaused by a hard-turn target maneuver:

( )

.a tN

Ntt a

21 1

M

f

NT

2

PN= c m= G (30)

For the APN case, we employ an analogous proce-dure, using Eq. 27 as the starting point. For this case, weobtain the following expression for missile acceleration,given that the target pulls a hard-turn maneuver:

( ) .a t Ntt

a2

1

Mf

N

T

2

APN= ; E (31)

Figure 7 illustrates a comparison of PN and APNacceleration requirements for various guidance gainvalues versus a target pulling a hard turn, via Eqs. 30and 31, respectively. Referring to Fig. 7, we see thatPN (solid lines) demands an increasing level of mis-sile acceleration as ight time increases. In fact, for aguidance gain of 3, PN requires three times the accel-eration of the target to effect an intercept; hence, thewell-known 3-to-1 ratio rule of thumb. If we increasethe guidance gain, theory says that the theoretical

3-to-1 rule of thumb can be relaxed. In practice, how-ever, higher gains may lead to excessive noise through-put, perhaps negating any theoretically perceivedbenets.

Unlike PN, APN (dashed lines in Fig. 7) is moreanticipatory in that it demands maximum accelerationat the beginning of the engagement and less accelera-tion as engagement time increases. Moreover, note thatfor a guidance gain of 3, APN (theoretically) requireshalf the acceleration that PN does when engaginga target that is pulling a hard turn. Note that, for aguidance gain of 4, the theoretical acceleration require-ments for PN and APN are the same and, for gains

above 4, APN demands more acceleration than PN(although saturating the acceleration command early isnot nearly the problem that saturating late is).

Constant Target Jerk Assumption

The assumptions stated during the deriva-tion of APN are still valid with the exceptionof target maneuver; here, we will assume thatthe target acceleration is linearly increasing (i.e.,

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0

0.5

1.0

1.5

2.0

2.5

3.0

Normalized time (t/tf)

Normalizedmissileacceleration(aC

/aT

)

Guidance gain

N= 3

N= 4

N= 5

N= 6

Guidance law

PN

APN

Figure 7. A comparison is made o PN and APN accelera-

tion requirements or various guidance gain values versus a

target pulling a hard turn. PN results (solid lines) indicate that

PN demands an increasing level o missile acceleration as ight

time increases. Notice that or a guidance gain o 3, PN requires

three times the acceleration o the target to eect an intercept;hence, the well-known 3-to-1 ratio rule o thumb. Increasing the

guidance gain can theoretically relax the 3-to-1 rule o thumb,

but higher gains may lead to excessive noise throughput. The

graph illustrates that APN (dashed lines) is more anticipatory in

that it demands maximum acceleration at the beginning o the

engagement and less as engagement time increases. Note that or

a guidance gain o 3, APN (theoretically) requires hal the accel-

eration that PN does when engaging a target that is pulling a

hard turn.


12/18



constant jerk in a particular direction). This may be a reasonable assumption,for example, in order to develop a terminal homing guidance strategy for useduring boost-phase ballistic missile defense, where it is necessary to engage anddestroy the enemy missile while it is still boosting. In such a context, it is pos-sible that a linearly increasing acceleration model (assumption) better reects actualtarget maneuver (acceleration) as compared to the APN assumption. Therefore,we augment the previous (PN) state vector (where ,x r x v

y y1 2_ _ ) to include atarget acceleration state x aTy3 _ and a target jerk state x x jTy4 3_ =o , leading to

x x x xxT

1 2 3 4_ 6 @ . As before, the control u is missile acceleration (u a M_ ), theplant (process) disturbance is not considered (w= 0), but it will come into play whendeveloping a target state estimator, and the pseudo-measurement, again, is rela-tive position ( xy 1_ ). With these modeling assumptions in mind (particularly thatx jTy4 _ / constant), the LQ optimization problem is stated as

( ) ( )

( ) ( )

( ),

] ( ) .

minJ t u t dt

t t u t

t

x u x

x x

x

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1

0

0

1

0

0

1 0 0 0

Subject to:

( )u tf

t

t

Q0 0 0 2

1 221 2

f

f

0

= +

= +

( ), ( ),t t t

( ) [y t =

o

^ h

R

T

SSSSS

R

T

SSSSS

V

X

WWWWW

V

X

WWWWW

#

(32)

Dening the terminal penalty matrix to be Qf = diag{b, c, 0, 0}, and followinga solution procedure identical to that outlined previously, we obtain the followinggeneral solution:

( )

( ) ( ) ( ) ( )

. ( )u tt ct

x t x t t x t t x t t3

1 1

1 1 1

33go

bt go

ct

ct ct

bt

cgo

ct

bt

cgo bt

cgo

3 23

4

2 1 3 2 21

62

32

61 3

43

go

go

go go

go

go

go go

3

2 2 2

=

+ + +

+ + + + + + + + 1 +

^

` c c c

h

j m m mR

T

SSSSS

V

X

WWWWW

If we compare the guidance law in Eq. 33 to our previous result (Eq. 26), we see,again, that the only difference is in the numerator; Eq. 33 includes the addition of a(time-varying) gain multiplying the target jerk state. Analogous to the previous case,this has (additional) implications on the guidance lter structure. More important,estimating target jerk given a relative position (pseudo-)measurement can be verynoisy unless the measurement quality is excellent. If sensor quality is not sufcient,then guidance law performance could be signicantly worse than if we simply usedPN or APN to begin with.

Special Case 1: E xtended PN

If we assume that c= 0 and we evaluateb"3

( ) ( )lim u t u tEPN3 = , then Eq. 33 collapses

to the extended PN (EPN) guidance law

( ) ( ) ( ) ( ) ( ) .u tt

x t x t t x t t x t t3

EPNgo

go go go2 1 2 21

32

61

43= + + +8 B (34)

By this time, a pattern should be emerging regarding the current line of guid-ance laws. For example, if we compare PN (Eq. 22), APN (Eq. 27), and EPN (Eq. 34),we see that the effective navigation ratios for these three cases are all the sameconstant:


13/18



Figure 8. Comparison o ZEM

removal against a boosting

threat. Plots o the predicted

ZEM or PN, APN, and EPN guid-

ance laws are illustrated. Here,

truth is computed by integrat-

ing orward a duplicate o the

threat truth model and propa-

gating the kill vehicle ballisti-cally. The model curve reects

the assumed ZEM calculation

or the particular guidance

law, using truth data. The esti-

mate curve is this same calcu-

lation with estimated data (via

noisy measurements), as must

be used in practice. Intercept

is achieved at tgo = 0 in each

case, since the ZEM goes nearly to zero. The magnitude o the ZEM is plotted; in each plot, the portion o the model-based ZEM and

estimated ZEM to the right o 12 s is in the opposite direction rom the true ZEM.

PN APN EPN

N N N 3= = =u u u .

It is the ZEM estimates that evolve from

( ) ( )ZEM r t v t tPN y y go= + , to

( )ZEM ZEM a t tAPN PN T go21= + 2 , and now to

( )ZEM ZEM j t tEPN APN T go61 3= + .

With regard to EPN, the addition of target accelera-tion and target jerk states will dictate a more complex

guidance lter structure, and it may be very sensitiveto sensor noise and actual target maneuver modali-ties as compared with PN or APN. We also note that,

if we evaluate,b c"3

( )lim u t3 , then Eq. 33 collapses to

the AREN guidance law previously given in Eq. 28.A simulation study of PN, APN, and EPN guid-

ance laws against a boosting threat illustrates thebenets of better matching the target assumptions tothe intended target. In this engagement, an exoatmo-spheric kill vehicle intercepts a threat that accelerates(boosts) according to the ideal rocket equation, with amaximum 8.5-gthreat acceleration occurring at inter-cept. Figure 8 shows how the ZEM prediction for eachguidance law compares with the true ZEM as eachevolves over time and under the control of the relevantguidance law. Each of the models of ZEM has signi-cant error initially (the direction is wrong, causing thetruth to increase while the assumed ZEM decreases),but for an assumed target maneuver model that moreclosely matches the actual target maneuver (i.e., EPN),this error is much less and the sign of the ZEM is cor-rect sooner. The curves also show some trade-off in

estimate quality (noise) as more derivatives are usedin the calculation. Figure 9 demonstrates the resultingacceleration commands and fuel usage for the differentguidance laws via logging of the resultant DV, which isdened as

( ) ,daMt

tf

0

#

where aM is the achieved acceleration vector. Therequired DV (translating to fuel usage) for PN, at1356 m/s, is substantially more than APN, at 309 m/s,or EPN, at 236 m/s. The required acceleration capabil-ity of the kill vehicle also is substantially different, with

PN requiring 27 gcapability, APN requiring 3.7 g, andEPN requiring 3.2g.

0

2

4

0

2

ZEM(

km)

0 5 10 15

0

2

tgo

(s)

PN

APN

EPN

Truth Model Estimate

0

100

200

PN

tgo (s)

APN

|am

|(m/s)

EPN

0 5 10 15

0

500

1000

Vused

(m/s)

Figure 9. Acceleration command history and cumulative V

used or PN, APN, and EPN guidance laws versus a boosting

threat. The progression is rom right to let as time-to-go (tgo)

decreases toward intercept in this single-simulation run.


14/18



Non-Ideal Missile Response Assumption

Here, the assumptions stated during the derivation of APN are still valid, save forthat pertaining to a perfect missile response. Instead, we will add the more realisticassumption that the missile responds to an acceleration command via the rst-orderlag transfer function

( ) / ( ) ,a s a sTs 1

1M c

=+

hence the reference to a non-ideal missile response. The time constant, T, is a compos-ite (roll-up) function of the missile response at a specic ight condition and dependslargely on the missile aerodynamic characteristics and ight control system design.

We augment the previous (APN) state vector (where , ,x r x v x aandy y Ty1 2 3_ _ _

) to include a missile acceleration state x aMy4 _ , leading to x x x xxT

1 2 3 4_ 6 @ .(Note that the fourth state here is missile acceleration, not target jerk as was the case

when deriving EPN guidance law.) As before, the control u is missile acceleration(u a

M_ ), the plant (process) disturbance is not considered (w = 0), but it will come into

play when developing a target acceleration estimator, and the pseudo-measurement,

again, is relative position (y x1_ ). In addition, we add a missile accelerometer mea-

surement. With these modeling assumptions in mind, the LQ optimization problem is

stated as

( ) ( )

( )

( ) ( ),

( ) ( ) .

minJ t u t dt

t t u t

y t t

x u x

x x

x

0

0

0

0

1

0

0

0

0

1

0

0

0

1

0

0

0

0

1 0 0 0

0 0 0 1

Subject to:

( )u t f t

t

T T

Q0 0 0 2

1 221 2

1 1

f

f

0= +

= +

=

( ), ( ),t t t

o r

r

^ hR

T

SSSSS

R

T

SSSSS

;

V

X

WWWWW

V

X

WWWWW

E

#

(35)

Here, we do not consider a terminal velocity penalty in order to reduce over-

all guidance law complexity, which leads to the terminal penalty matrix given by

Qf= diag{b, 0, 0, 0}. Thus, following an identical solution procedure to that outlined

previously, we obtain the following general guidance law solution:

/T

3 6 6 2 12 3

( ) ( ) ( ) ( ). ( )u

te e

e x t x t t t x t T e x t6 1 136

/ /

/

T

t

bT T

t

T

t

T

t

T

tt T t T

T

tt T

go T

tt

go

go4 2

2

6 2

1 2 21 2

32

4

go go go gogo go

gogo

go gogo

3 2

2

3

3=

+ + +

+ + + +`

c` `j

mj j8 B

Upon examination of Eq. 36, it becomes clear that the non-ideal missile response

assumption adds additional complexity to the guidance law structure (remember that

we have not considered a terminal penalty on relative velocity, i.e., c = 0). To bettervisualize this complexity, consider the constant target acceleration guidance law

given in Eq. 26. If we takec 0"

( )lim u t2 , we obtain the following result:

c 0=

( )( ) ( ) ( )

.u tt

x t x t t x t t3

1go bt

go go2 2 3

1 2 21

32

go3

=+

+ +> H (37)The structure of Eq. 37 is signicantly less complex than that given in Eq. 36 despite

the fact that the cost function for both is identical (i.e., b is nite and c = 0).If we take

b"3( )lim u t4 , we obtain the well-known optimal guidance law (OGL)

referred to in many texts (see Refs. 7, 13, and 14 for examples):


15/18



Missile acceleration

Target acceleration

Guidance law

Missileresponse

Relativeposition

Relative velocity

Kr

KaM

KaT

Kv

aTry

vy

+++ +

Guidance law

Gain

0

0 0

Kr

KaM

KaT

Kv

PN APN OGL

t

3

go2

t3

go t3

go

t

3

go2

23

t

N

go2

u

tN

go

u

N2

u

OGL

1N

T

t

T

t

T

t

T

te e

T

t

T

te

3 6 6 2 12 3

6

/ /

/

go go go go t T t T

go go t T

2

2

3

3

2

2

go go

go

_

+ +

+u

ec c

om m

NTT

te

t1

2 /go t T

go2

go+u c m

Figure 10. The eedback structure o the PN, APN, and OGL guidance laws is depicted here. The relative complexity o the dierent

guidance laws is established as we add additional assumptions regarding the engagement and missile response characteristics. The

diagram emphasizes the act that a substantial increase in complexity arises when the assumptions move rom an ideal to non-ideal

interceptor response assumption.

( ) ( ) ( ) ( ). ( )u

te e

e x t x t t t x t T e x t6

3 6 6 2 12 3

1 138

/ /

/ /

OGLgo

T

t

T

t

T

t

T

t

T

tt T t T

T

tt T

go go T

tt T

2

2

2

1 2 21 2

32

4go

go go go gogo go

gogo

gogo

2

2

3

3=

+ +

+ + + +`

c` `j

mj j8 B

Referring back to the APN law presented in Eq. 27, a couple of important points arenoted. First, we compare the APN ZEM estimate given by

go go( ) ( ) ( )ZEM x t x t t t x tAPN 1 2 21 2

3= + +

with that in Eq. 38 and see that the OGL ZEM estimate is

( )ZEM ZEM T e x t1 /

OGL APN T

tt T2

4

go

go= +` j .In addition, the effective navigation ratio for APN is given by

APNN 3=u . In contrast,

from Eq. 38,OGL

Nu is time-varying and can be expressed as shown below:

OGL

6

.N

e e

e

3 6 6 2 12 3

1

/ /

/

T

t

T

t

T

t

T

tt T t T

T

t

T

tt T

2

2

go go go gogo go

go gogo

2

2

3

3_

+ +

+u

c` `

mj j

(39)

For illustrative purposes, Fig. 10 depicts the feedback structure of the PN, APN,and OGL guidance laws discussed this far and thus helps to establish the relative

complexity of the different guidance laws as we add additional assumptions regardingthe engagement and missile response. From Fig. 10, it is obvious that a substantialincrease in complexity arises when the assumptions move from an ideal to non-idealinterceptor response assumption.

Example Comparisons o PN, APN, and OGL

In this example, Fig. 11 illustrates the miss distance and called-for accelerationstatistics for PN, APN, and OGL guidance laws versus a target pulling a 5-ghard turn.The Monte Carlo data are displayed in cumulative probability form. From Fig. 11, we


16/18



see a distinct trend considering PN versus APN versus OGL guidance laws. Giventhat PN is derived assuming the target is not maneuvering, we expect the perfor-mance to degrade if the target does maneuver. It also is not surprising to see that,statistically, more acceleration is required versus APN or OGL. The improvementfrom APN to OGL is explained by the fact that APN assumes a perfect missileresponse to acceleration commands and OGL assumes that the missile responds asa rst-order lag (refer back to Eq. 27 compared with Eq. 38).

TIMETOGO ESTIMATIONAs shown in the previous section, Extensions to PN: Other Optimal Homing

Guidance Laws, many modern guidance laws require an estimate of time-to-go (tgo),which is the time it will take the missile to intercept the target or to arrive at theclosest point of approach (CPA). The tgo estimate also is a critical quantity for mis-siles that carry a warhead that must detonate when the missile is close to the target.For example, recall the general optimal guidance law shown in Eq. 36. This guid-ance law can be expressed as3ZEM, where

( ) ( ) ( ) ( )ZEM x t x t t t x t T e x t1 /go go T

tt T

1 2 21 2

32

4go

go= + + +` j8 B

and the effective navigation ratio,, is shown in Eq. 40, where b and T representthe terminal miss penalty and assumed missile time constant, respectively (see thesectionNon-Ideal Missile Response Assumption, wherein Eq. 36 was derived, for afurther description):

.

N

e e

e

3 6 6 2 12 3

6 1

/ /

/

bT T

t

T

t

T

t

T

tt T t T

T

t

T

tt T

6 2

2

go go go gogo go

go gogo

3 2

2

3

3=

+ + +

+u

c` `

mj j

(40)

It is clear from Eq. 40 that is a function of tgo. Figure 12 illustrates the tgodependence of the general optimal guidance law effective navigation ratio for three

0 0.33 0.66 0.98 1.31 1.64 1.97 2.30 2.62 2.950

20

40

60

80

100

0 5 10 15 20 25 30 35 40 450

20

40

60

80

100

Maximum interceptor acceleration (g)

Closest point of approach (ft)

Probability(%)

Probability(%)

APN OGL PN Figure 11. The cumulative prob-

ability perormance o PN, APN,

and OGL guidance laws versus a 5-g

hard-turn target is shown. Both the

cumulative probability o the maxi-

mum interceptor acceleration and

CPA or 100-run Monte Carlo sets are

plotted. With these graphs, it is easy

to quickly ascertain the probability

o achieving the x-axis parameter

value (e.g., maximum acceleration).

In both graphs, lines that are more

vertical and arther to the let are

considered more desirable. All noise

and error sources are turned o, but

the target maneuver start time is ran-

domized over the last second o ter-

minal homing. Note that the actual

missile response model is non-ideal,

and it varies with the ight condi-

tions o the missile trajectory.

values of terminal miss pen-alty. The curves are normal-ized with respect to the missiletime constant T. Referring toFig. 12, consider the b = 1000curve. As /t T 2go " , achievesits maximum value and thenreduces as /t T 0go " . Clearly,this guidance law gain curveevolves in a way that placesmuch greater emphasis on ZEM

at certain times near intercept.Imagine that the actual (true)tgo is 2 s but that the estimateof tgo is in error and biasedtoward the positive directionby four missile time constants(4T). Then, from Fig. 12 wecan see that the guidance gainwould be about one-seventh ofwhat it should be at /t T 2go = ,thereby not placing optimalemphasis on ZEM at that time,and degrading overall guidanceperformance.

The simplest time-to-goestimation scheme uses mea-surements (or estimates) ofrange and range rate. Considerthe engagement geometry ofFig. 13, where vM = missilevelocity, vT = target veloc-ity, r = targetmissile relative


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0 1 2 3 4 5 6 7 8 9 10

0

70

80

Guidancegain60

50

30

40

20

10

Terminal penalty

tgo/T(s)

b=10 b= 100 b= 1000

3 6 6

2 12 3

N

bT T

t

T

t

T

t

T

te e

T T

te

6

61

/ /

/

go go go go t T t T

go t T

3 2

2

2

2

3

3

go go

go

_

+ + +

+

u

ec

om

Figure 12. The normalized tgo dependence o the eective

navigation ratio or the OGL is shown here or three terminalpenalty values. Normalization is with respect to the missile time

constant T. We note that, as tgo approaches infnity, the eective

navigation ratio always approaches 3.

position, and v = targetmissile relative velocity. Refer-ring to Fig. 13, we assume that the missile can measureor estimate relative range (R r= ) to the target andrange rate (Ro ) along the LOS to the target. If we assumethat the missile and target speeds are constant, then onecan estimate time-to-go as

.tRRgo =t o (41)

Another common approach to estimating time-to-goalso assumes that the missile and target speeds are con-stant. Dene t t t_D ) , where t is the current time andt) is a future time. Thus, given estimates of targetmissilerelative position and relative velocity at the current timet, the future targetmissile relative position at time t) isgiven as

( ) ( ) ( ) .t t t tr r v D= +) (42)

At the CPA, the following condition holds:

( ) ( ) .t tr v 0: =) ) (43)

This condition is illustrated in Fig. 13 by the perpendic-ular line from the target to the relative velocity. Basedon our assumptions, using Eq. 42 in Eq. 43, and recog-nizing the constant velocity assumption, we obtain thefollowing expression for t t

go/D :

( ) ( )

( ) ( ).t

t t

t t

v v

r v

go:

:= (44)

Using this expression for tgo in Eq. 42, we obtain thetargetmissile relative separation at the CPA:

( ) ( )

( ) [ ( ) ( )] ( ) [ ( )( ) ( )]

( ) ( )

[ ( ) ( )] ( )

t t

t t t t t t t

t t

t t t

rv v

r v v v r v

v v

v r v

CPA :

: :

:

# #

=

.=

(45)

Conceptually, the differences between time-to-go estimation using Eq. 41 rather than Eq. 44 can beexplained by using Fig. 13. For this discussion, and with-

out loss of generality, we assume that the missile velocityis constant and that the target is stationary. Referring toFig. 13, we see that Eq. 41 estimates the ight time forthe missile to reach point P. However, Eq. 44 estimatesthe time it takes for the missile to reach the CPA. Ifthe missile and target have no acceleration (the up-frontassumption during the derivation), then Eq. 44 is exact.

CLOSING REMARKSIn this article, we have focused on developing

homing guidance laws by using optimal control tech-

niques. To this end, a number of modern guidance laws(PN and beyond) were derived using LQ optimal con-trol methods. We note that, regardless of the specicstructure of the guidance law (e.g., PN versus OGL), wedeveloped the relevant guidance law assuming that allof the states necessary to mechanize the implementa-tion were (directly) available for feedback and uncor-rupted by noise (recall we referred to this as the perfectstate information problem). In a companion article in

TargetMissile

R= ||r ||

CPA

LOS

vM

vT

v

P

Figure 13. The missiletarget (planar) engagement geometry

is shown here. This depiction places the LOS to the target along

the x axis o the coordinate system. Missile and target velocity

vectors are indicated asvM and

vT, respectively. The relative veloc-

ity,v, makes an angle with the LOS and passes through the

points indicated by CPA and P.


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JOHNS HOPKINS APL TECHNICAL DIGEST VOLUME 29 NUMBER 1 (2010)

this issue, Guidance Filter Fundamentals, we pointto the separation theorem, which states that the opti-mal solution to this problem separates into the optimaldeterministic controller (i.e., the perfect state informa-tion solution) driven by the output of an optimal stateestimator. Thus, in that article, we discuss guidanceltering, which is the process of taking raw (targeting,inertial, and possibly other) sensor data as inputs and

estimating the relevant signals (estimates of relativeposition, relative velocity, target acceleration, etc.) uponwhich the guidance law operates.

REFERENCES

1Bryson, A. E., and Ho, Y.-C., Applied Optimal Control, HemispherePublishing Corp., Washington, DC (1975).

2Cottrell, R. G., Optimal Intercept Guidance for Short-Range Tacti-cal Missiles,AIAAJ. 9(7), 14141415 (1971).

3Ho, Y. C., Bryson, A. E., and Baron, S., Differential Games andOptimal Pursuit-Evasion Strategies, IEEETrans. AutomaticControlAC-10(4), 385389 (1965).

4Athans, M., and Falb, P. L., Optimal Control: An Introduction to theTheory and Its Applications, McGraw-Hill, New York (1966).

5Jackson, P. B., TOMAHAWK 1090 Autopilot Improvements: Pitch-Yaw-Roll Autopilot Design, Technical Memorandum F1E(90)U-1-305,

JHU/APL, Laurel, MD (1 Aug 1990).6 Basar, T., and Bernhard, P., H-Innity Optimal Control and RelatedMinimax Design Problems, Birkhuser, Boston (1995).

7Ben-Asher, J. Z., and Yaesh, I.,Advances in Missile Guidance Theory,American Institute of Aeronautics and Astronautics, Reston, VA(1998).

8Grewal, M. S., and Andrews, A. P., Kalman Filtering Theory and Prac-tice, Prentice Hall, Englewood Cliffs, NJ (1993).

9Pearson, J. D., Approximation Methods in Optimal Control,J. Elec-tron. Control13, 453469 (1962).

10Vaughan, D. R., A Negative Exponential Solution for the MatrixRiccati Equation, IEEE Trans. Autom. Control 14(2), 7275 (Feb1969).

11Vaughan, D. R., A Nonrecursive Algebraic Solution for the DiscreteRiccati Equation, IEEETrans.Autom. Control15(10), 597599 (Oct1970).

12Pue, A. J., Proportional Navigation and an Optimal-Aim Guidance Tech-nique, Technical Memorandum F1C (2)-80-U-024, JHU/APL, Laurel,MD (7 May 1980).

13Zames, G., Feedback and Optimal Sensitivity: Model ReferenceTransformations, Multiplicative Seminorms, and ApproximateInverses, IEEETrans.Autom. Control26, 301320 (1981).

14Shneydor, N. A., MissileGuidanceandPursuit:Kinematics, DynamicsandControl, Horwood Publishing, Chichester, England (1998).

Neil F. Palumbo is a member of APLs Principal Professional Staff and is the Group Supervisor of the Guidance,

Navigation, and Control Group within the Air and Missile Defense Department (AMDD). He joined APL in 1993

after having received a Ph.D. in electrical engineering from Temple University that same year. His interests include

control and estimation theory, fault-tolerant restructurable control systems, and neuro-fuzzy inference systems.

Dr. Palumbo also is a lecturer for the JHU Whiting Schools Engineering for Professionals program. He is a member

of the Institute of Electrical and Electronics Engineers and the American Institute of Aeronautics and Astronautics.

Ross A. Blauwkamp received a B.S.E. degree from Calvin College in 1991 and an M.S.E. degree from the University

of Illinois in 1996; both degrees are in electrical engineering. He is pursuing a Ph.D. from the University of Illinois.

Mr. Blauwkamp joined APL in May 2000 and currently is the supervisor of the Advanced Concepts and Simulation

Techniques Section in the Guidance, Navigation, and Control Group of AMDD. His interests include dynamic games,

nonlinear control, and numerical methods for control. He is a member of the Institute of Electrical and Electronics

Engineers and the American Institute of Aeronautics and Astronautics.Justin M. Lloyd is a member of the APL Senior

Professional Staff in the Guidance, Navigation, and Control Group of AMDD. He holds a B.S. in mechanical engineer-

ing from North Carolina State University and an M.S. in mechanical engineering from Virginia Polytechnic Insti-

tute and State University. Currently, Mr. Lloyd

is pursuing his Ph.D. in electrical engineering

at The Johns Hopkins University. He joinedAPL in 2004 and conducts work in optimization;

advanced missile guidance, navigation, and con-

trol; and integrated controller design. For further

information on the work reported here, contact

Neil Palumbo. His email address is neil.palumbo@

jhuapl.edu.

The Authors

Ross A. BlauwkampNeil F. Palumbo Justin M. Lloyd

TheJohns Hopkins APL Technical Digest can be accessed electronically at www.jhuapl.edu/techdigest.

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