Modern Homing Missile Guidance Theory and Techniques

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

lassically derived homing guidance laws, such as proportional navigation, can be highly effective when the homing missile has significantly more maneuver capability than the threat. As threats become more capable, however, higher performance is required from the missile guidance law to achieve intercept. To address this challenge, most modern guidance laws are derived using linear-quadratic optimal control theory to obtain analytic feedback solutions. Generally, optimal control strategies use a cost function to explicitly optimize the missile performance criteria. In addition, it is typical for these guidance laws to employ more sophisticated models of the target and missile maneuver capability in an effort to improve overall performance. In this article, we will present a review of optimal control theory and derive a number of optimal guidance laws of increasing complexity. We also will explore the advantages that such guidance laws have over proportional navigation when engaging more stressing threats

INTRODUCTION Classical guidance laws, with proportional navigation optimal control theory to missile guidance problems had (PN) being the most prominent example, had proven to sufficiently matured, offering new and potentially prom- be effective homing guidance strategies up through the ising alternative guidance law designs.1–3 Around this 1960s and early 1970s. By the mid-1970s, however, the time, the computer power required to mechanize such predicted capability of future airborne threats (highly advanced algorithms also was sufficient to make their maneuverable aircraft, supersonic cruise missiles, tacti- application practical. cal and strategic ballistic missile reentry vehicles, etc.) Most modern guidance laws are derived using linear- indicated that PN-guided weapons might be ineffective quadratic (LQ) optimal control theory to obtain against them. However, by that time, the application of analytic feedback solutions.1, 4, 5 Many of the modern

42 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) $ formulations take target maneuver into account to deal x()t= f((x t), u()t,) t . (1) with highly maneuvering target scenarios (particularly true for terminal homing guidance). The availability In Eq. 1, x is the n-dimensional state vector of real of target acceleration information for the guidance law elements ()x!!RRn, u m is the control vector, and varies, depending on targeting sensor capability and t represents time (later we will provide more detail as type and the specific guidance law formulation. Typi- to the structure of the state and control vectors for the cally, there also is an explicit assumption made about homing guidance problem). With this general system, we the missile airframe/autopilot dynamic response charac- associate the following scalar performance index: teristics in the modern formulations. We will show later that PN is an optimal guidance law in the absence of J((x t0), u()t 0 ,) t0 =  ((x tf), t f) airframe/autopilot lag (and under certain other assumed t f (2) conditions). To some extent, the feedback nature of the + L((x t), u()t,) t dt . #t homing guidance law allows the missile to correct for 0 inaccurate predictions of target maneuver and other unmodeled dynamics (see Fig. 1). However, the require- In this equation, [t0, tf] is the time interval of inter- ment for better performance continues to push optimal est. The performance index comprises two parts: (i) guidance law development, in part by forcing the con- a scalar algebraic function of the final state and time sideration (inclusion) of more detailed dynamics of the (final state penalty),  ((x tf), t f), and (ii) a scalar inte- interceptor and its target. It is interesting that most if gral function of the state and control (Lagrangian), t not all modern guidance laws derived using optimal con- f # L((x t), u()t,) t dt . The choice of  ((x tf), t f) t0 t trol theory can be shown to be supersets of PN. and # f L((x t), u()t,) t dt (this choice is a significant part In the following sections, we first provide a cursory t0 review of dynamic optimization techniques with a focus of the design problem) will dictate the nature of the on LQ optimal control theory. Using this as background, optimizing solution. Thus, the performance index we then develop a number of terminal homing guid- is selected to make the plant in Eq. 1 exhibit desired ance laws based on various assumptions, in the order of characteristics and behavior (transient response, stabil- assumption complexity. In our companion article in this ity, etc.). issue, “Guidance Filter Fundamentals,” we introduce For our purposes, the optimal control problem is * guidance filtering, with a focus on Kalman guidance to find a control, u ()t , on the time interval [t0, tf] that filter techniques. drives the plant in Eq. 1 along a trajectory, x*()t , such that the scalar performance index in Eq. 2 is minimized. It is difficult to find analytic guidance law expressions REVIEW OF LQ OPTIMAL CONTROL for such general nonlinear systems. Therefore, we will Here, we start by considering a general nonlinear turn our attention to a subset of optimal control that dynamics model of the system to be controlled (i.e., the can yield tractable analytic solutions, known as the LQ “plant”). This dynamics model can be expressed as optimal control problem.

Kalman Filter Optimal Control Servomechanism Theory Theory Theory (1960s) (1960s) (1930s–1950s)

Target Target state Acceleration Actuation direction estimates commands commands Target motion Target Guidance Guidance Autopilot Airframe/ sensors filter law propulsion

Inertial navigation

Vehicle motion

Figure 1. The traditional guidance, navigation, and control topology for a guided missile comprises guidance filter, guidance law, autopilot, and inertial navigation components. Each component may be synthesized by using a variety of techniques, the most popular of which are indicated here in blue text.

JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) 43 N. F. PALUMBO, R. A. BLAUWKAMP, and J. M. LLOYD The LQ Optimal Control Problem Then, from the calculus of variations, four necessary 5 Here, we assume that the nonlinear model in Eq. 1 conditions for optimality must be satisfied to solve our problem: state, costate, boundary, and stationarity con- can be linearized about an equilibrium point (,x0 u0) and represented by the time-varying linear dynamic ditions must all hold. These four conditions are listed in Table 1. system described in Eq. 31, 4: From Table 1 (and based on our previous description $ of the dimension of the plant state vector), it can be seen x()t= A() t x()t + B() t u()t . (3) that there is a system of 2n dynamic equations that must be solved; n equations must be solved forward in time In this model (as in the nonlinear case), x ! Rn is the over [t , t ] (state equations), and n equations must be state vector, u ! Rmis the control vector, and t repre- 0 f solved backward in time over [t , t ] (costate equations). sents time. Here, however, A(t) and B(t) are the time- f 0 We further note that the equations are coupled. Apply- varying Jacobian matrices ing the stationarity condition to Eqs. 3–6 yields the following result for the control: 2 2 x= x and 2 2 x= x , respectively) ((f x,,ut)/ x 0 f(, x u,)t / u 0 u= u u= u 0 0 u()t= –( R–1 t)(BT t)( t). (7) of appropriate dimension. Using the “shorthand” notation z()t2 _ zT () t Sz()t , S Using Eq. 7 in the state and costate necessary conditions, we next define the quadratic performance index (a and taking into account the boundary condition, leads special case of the performance index shown in Eq. 2) to the following two-point boundary value problem: shown in Eq. 4: $ –1 T x()t 1 2 x()t A()t –(BRt) B J((x t ), u()t,) t = x()t $ = T , 0 0 0 2 f Q =G =–(Q t) –(A t) G;()t E f ()t (8) tf (4) x()t given + 1 x()t2 + u() t 2 dt . 0 . 2 #t Q()t R()t ()t= Q x() t 0 8 B ) f f f 3

In Eq. 4, the following assumptions are made: the ter- Here, we are concerned with solution methods that minal penalty weighting matrix, Qf, is positive semi- can yield analytic (closed-form) solutions rather than definite (the eigenvalues of Qf are 0, expressed as iterative numerical or gain scheduling techniques. We Qf  0); the state penalty weighting matrix, Q(t), is note, however, that the sparseness/structure of the con- positive semi-definite Q( (t)  0); and the control pen- stituent plant and cost matrices—e.g., A(t), B(t), Q(t), alty matrix, R(t), is positive definite R( (t) > 0). Thus, the and R(t)—will dictate the ease by which this can be LQ optimal control problem is to find a control, u*()t , accomplished. Qualitatively speaking, the level of dif- such that the quadratic cost in Eq. 4 is minimized sub- ficulty involved in obtaining analytic solutions is related ject to the constraint imposed by the linear dynamic primarily to the complexity of state equation coupling system in Eq. 3. into the costate equations, most notably by the structure To solve this continuous-time optimal control prob- of Q(t) and, to a lesser extent, by the structure of A(t). As lem, one can use Lagrange multipliers, ()t , to adjoin we will see later, however, this fact does not negate our the (dynamic) constraints (Eq. 3) to the performance ability to derive effective guidance laws. index (Eq. 4).1, 5 Consequently, an augmented cost func- Given a suitable system structure in Eqs. 7 and 8 (as tion can be written as discussed above), one conceptually straightforward way to solve this problem is to directly integrate the costate 1 2 Jl = x() t equations backward in time from tf to t  t0 using the 2 f Q f terminal costate conditions and then integrate the state R 1 2 2 V S 2 ((xt)(+ u t))W (5) t Q()t R()t fS T W + # S +  ()t((A t)( x t) W dt . Table 1. Necessary conditions for optimality. t0 S + B()t u() t –(xo t)) W Condition Expression T X State 2H((x t), u()t,) t /(2 t)(= xo t), for t$ t Referring to Eq. 5, we define the Hamiltonian function Equation 0 H as shown: Costate 2H((x t), u()t,) t /(2x t)–= o () t , for t# t Equation f 1 2 1 2 H((x t), u()t,) t = x()t + u()t Stationarity 2H((x t), u()t,) t /(2u t),= 0 for t $ t0 2 Q()t 2 R()t (6) T +  ()t((A t)( x t)(+ B t)( u t)) . Boundary 2((xtf), t f)/ 2 x() tf= ()t f ,(x t0) given

44 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) MODERN HOMING MISSILE GUIDANCE THEORY AND TECHNIQUES equations forward in time using the costate solutions From linear system theory, we know that the solution and initial conditions on the states. This process can be to Eq. 13 can be expressed by the following matrix done by hand or by using any available symbolic solu- exponential: tion software (Maple, Mathematica, etc.). Another way W()tgo W()tgo =0 to solve the two-point boundary value problem speci- = exp()tgo , 1 Y()t Y()t =0 fied in Eq. 8 employs the sweep method. This technique = go G = go G (14) assumes that the state x()t and costate ()t satisfy the –ABRB–1 T linear relation shown below over the interval t ∈ [t , t ]  _ . 0 f = QAT G and given an (as yet) unknown matrix function Pc(t): ()t= P() t x()t . (9) In Eq. 14,  is the Hamiltonian matrix. With regard to c ease of determining analytic solutions, in an analogous Using the assumed relation in Eq. 9, the control in Eq. 7 way to our previous discussion, the complexity of the can be written as system structure (and the system order) will dictate how difficult it may be to obtain an analytic solution to the –1 T matrix exponential exp(t ). Once an analytic solution u()t= –( R t)(BPt)(c t)(x t). (10) go is found, the exponential matrix solution is partitioned as shown: To find Pc(t) such that the control (Eq. 10) is com- pletely defined, we differentiate Eq. 9 and make use of ␺()t ␺ () t the dynamic equations in Eq. 8. Doing so leads to a 11 go 12 go exp()⌿tgo / . (15) requirement to solve the following matrix Riccati dif- =␺21 ()tgo ␺22 () tgo G ferential equation: Using Eq. 15, the relation P (t ) = Y(t )W–1(t ), –(PPo t)(= t)(AAt)( + T t)(P t) c go go go c c c Y(0)  Qf, W(0) = I, and the initial condition from –1 T Eq. 12 (P (t = 0) = Q ), the solution to the matrix Ric- –(PBc t)( t)(RBt)( t)(Pc t) (11) c go f cati differential equation Pc(tgo) can be expressed as + QP()t,(c t f).= Qf

The optimal control is determined by first solving the Pc()t go =21 () tgo + 22 ()t goQ f 6 @–1 (16) matrix Riccati differential equation backward in time 11 ()tgo + 12 () tgo Qf . 6 @ from tf to t and then using this solution in Eq. 10. There are many ways to solve this equation. For completeness, From Eq. 16, it becomes clear that the existence of P (t ) we present a matrix exponential method of solution. c go is equivalent to having a nonsingular 11(tgo) + 12(tgo)Qf. (We do not explore this important issue here, but the Solving the Matrix Riccati Differential Equation via interested reader is referred to Refs. 1 and 4–11 for fur- Matrix Exponential Method ther treatment of the subject.) Using Eq. 16 in Eq. 10, the optimal control then is fully specified. Hand calcu- We first want to rewrite Eq. 11 in terms of the time- to-go variable defined as t_ t – t. We use the fact that lations (for systems of low order) or symbolic solution go f software can be employed to mechanize this technique. dtgo/dt = –1 (for fixed tf) to express Eq. 11 in terms of tgo as shown (note that plant matrices must be time- independent for this technique): THE PLANAR INTERCEPT PROBLEM o T PPc()t go= c() tgo AA + Pc()t go In general, the guidance process takes place in three- dimensional space. However, such analysis can be com- –(PBt )(RB–1 T PQt),+ (12) c go c plex and is beyond the scope of this article. Thus, here PQc(t go = 0). = f we will consider the formulation of the planar intercept (pursuit-evasion) problem that we will use, subsequently, Clearly, this matrix differential equation is quadratic in in the derivation of a number of different modern guid- Pc(tgo) and is of dimension n. Note, however, that if we ance laws. This approach is not overly restrictive or –1 assume that Pc(tgo) takes the form Pc(tgo) = Y(tgo)W (tgo), unrealistic in that many (if not most) guidance law then its solution may (instead) be found by solving the implementations, including modern ones, use the same homogeneous linear matrix differential equation of approach, i.e., planar guidance laws are devised and dimension 2n shown in Eq. 13: implemented in each of the maneuver planes. Figure 2 –1 T illustrates the planar (two-dimensional) engagement d W()tgo –ABRB W()tgo (13) geometry and defines the angular and Cartesian quan- = T . dtgo =Y()tgo G = QA G= Y()tgo G tities depicted therein. In Fig. 2, the x axis represents

JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) 45 N. F. PALUMBO, R. A. BLAUWKAMP, and J. M. LLOYD

␥M = Missile flight path angle aM = Missile acceleration, The homing kinematics can be expressed in vari- ␥ = Target flight path angle normal to LOS T a = Target acceleration, ous ways. Here, we concentrate on Cartesian forms ␭ = LOS angle T normal to VT of expression. From a previous section, the target-to- r = Missile inertial position vector L = Lead angle M missile range was defined as R = r , and target–mis- rT = Target inertial position vector rx = Relative position x (rTx – rMx) _/o : v = Missile velocity vector r = Relative position y (r – r ) sile closing velocity was defined as VRc ––v 1r, M y Ty My where the LOS unit vector is 1= r/R. Thus, refer- vT = Target velocity vector R = Range to target r ring to Fig. 2, expressions for target–missile relative y /z I I position, relative velocity, and relative acceleration ␥ T aT are given below, where we have defined the quantities vT v__v,, a a v _ v , and a _ a . ␥ M M M M T T T T vM T aM Target (T) R r r=rx 1 x + ry1 y = Rcos()␭ 1x + R sin()␭ 1y ␭ L T 6 @ 6 @ ␥ ry M v=vx 1 x + vy1 y = –(vTTcos␥ )– vM cos()L+␭ 1x ␭ 6" ,@ Missile + vTTsin()␥ –( vM sin L+␭) 1y (17) (M) r 6" ,@ M a=ax 1 x + ay1 y = aTTsin()␥+ aM sin() ␭ 1x 6" ,@ xI + aTTcos()␥–( aM cos ␭) 1y Origin rx 6" ,@ (O) Referring again to Fig. 2, we note that if the closing velocity is positive (V > 0), then we only need to Figure 2. Planar engagement geometry. The planar intercept c actively control the kinematics in the y/z coordinate to problem is illustrated along with most of the angular and Carte- achieve an intercept. That is, if V > 0 and the missile sian quantities necessary to derive modern guidance laws. The c actively reduces and holds r to zero by appropriately x axis represents downrange while the y/z axis can represent y accelerating normal to the LOS, then r will continue either crossrange or altitude. A flat-Earth model is assumed with x to decrease until collision occurs. We will assume this an inertial coordinate system that is fixed to the surface of the condition holds and disregard the x components in the Earth. The positions of the missile (M) and target (T) are shown following analysis. with respect to the origin (O) of the coordinate system. Differen- The homing kinematics shown in Eqs. 17 are clearly tiation of the target–missile relative position vector yields relative nonlinear. In order to develop guidance laws using linear velocity; double differentiation yields relative acceleration. optimal control theory, the equations of motion must be linear. Referring to the expression for relative posi- downrange, for example, while the y/z axis can repre- tion in Eqs. 17, note that for l very small, the y-axis sent either crossrange or altitude, respectively (we will component of relative position is approximately given use y below). For simplicity, we assume a flat-Earth by ry  Rl. Moreover, for very small T and l (e.g., model with an inertial coordinate system that is fixed near-collision course conditions), the y-axis compo- to the surface of the Earth. Furthermore, we will assume nent of relative acceleration is approximately given by that the missile and target speeds, vM and vT , ay  aT – aM. Hence, given the near-collision course respectively, are constant. conditions, we can draw the linearized block diagram In Fig. 2, the positions of the missile (pursuer) M and of the engagement kinematics as shown in Fig. 3. Cor- target (evader) T are shown with respect to the origin (O) respondingly, we will express the kinematic equations of the coordinate system, rM and rT , respectively. Thus, the relative position vector [or line-of-sight (LOS) vector, as it was previously defined] is given by r= rTM– r , and Relative Relative o velocity position the relative velocity vector is given by r_ v = vTM– v . + vy ry 1 From Fig. 2, note that an intercept condition is satisfied aT ∫ ∫ ␭ t Target – R if y = 0 and Vc > 0 (i.e., a collision triangle condition). As illustrated, for this condition to be satisfied, the mis- acceleration sile velocity vector must lead the LOS by the lead angle aM L. Determining the lead angle necessary to establish a Missile acceleration collision triangle is, implicitly, a key purpose of the guidance law. How this is done is a factor of many things, Figure 3. Linear engagement kinematics. Planar linear homing including what measurements are available to the guid- loop kinematics are illustrated here. The integral of target–missile ance system and what assumptions were made during relative acceleration yields relative velocity; double integration formulation of the guidance law (e.g., a non-maneuver- yields relative position. The LOS angle, λ, is obtained by dividing ing target was assumed). relative position by target–missile range.

46 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) MODERN HOMING MISSILE GUIDANCE THEORY AND TECHNIQUES of motion in state-space form. To this end, we define With these assumptions in mind, and referring to Eq. 18, T the state vector x _ x1 x 2 , the control u, the plant the LQ optimization problem is stated as disturbance w, and 6 the pseudo-measurement@ y where t x__ r ,,x v u__ a , w a (any residual target 1 2 1 f 2 1y 2 y MT min J x()t0,( u t 0), t0 =2 x()tf + 2 u() t dt u() t Q #t acceleration is treated as a disturbance), and y_ Rl ^ h f 0 (the linearized Cartesian pseudo-measurement is com- 0 1 0 Subject to: xo ()t= x() t + u() t , (19) posed of range times the LOS angle). Given these defini- ;;0 0EE–1 tions, the kinematic equations of motion in state-space y() t = 1 0 x()t . form are written as 6 @ o In words, find a minimum-energy control u(t) on the x()t= Ax() t + Bu() t +Dw() t ,( x0 = x 0) time interval [t0, tf] that minimizes a quadratic func- y() t = Cx()t (18) tion of the final (terminal) relative position and relative 0 1 0 0 AB=, = ,,CD=1 0 = . velocity and subject to the specified dynamic con- ;;0 0EE–1 6 @ ;1E straints. In Eq. 19, the terminal performance weighting matrix, Qf, is yet to be specified. We define the scalar These equations will form the basis for developing a b > 0 as the penalty on relative position at time tf (i.e., number of terminal homing guidance laws in the fol- final miss distance) and scalar c  0 as the penalty on lowing subsections. In some cases, the equations are relative velocity at tf (c specified as a positive nonzero modified or expanded as needed to reflect additional value reflects some desire to control or minimize the assumptions or special conditions. terminal lateral relative velocity as is the case for a rendezvous maneuver). Given the penalty variables, Qf is the diagonal matrix given in Eq. 20: DERIVATION OF PN GUIDANCE LAW VIA b 0 LQ OPTIMIZATION Q = . (20) f ;0 cE In the previous article in this issue, “Basic Principles of Homing Guidance,” a classical development of PN The choice of b and c is problem-specific, but for inter- guidance was given based on that found in Ref. 12. In cept problems we typically let b → , c = 0, and for ren- the present article, we will leverage off the discussion of dezvous problems we have b → , c → . the previous subsection and develop a planar version of the PN guidance law using LQ optimization techniques. General Solution for Nonmaneuvering Target To start, we will state the key assumptions used to develop the guidance law: We can solve the design problem posed in Eq. 19 by using the LQ solution method discussed previously. We • We use the linear engagement kinematics model define tgo _ tf – t and initially assume the scalar vari- discussed previously and, hence, the state vector is ables b and c are finite and nonzero. Using Maple, we T x()t= x1()t x 2 ()t (the state vector comprises symbolically solve Eqs. 15 and 16 to obtain an analytic the components6 of@ relative position and relative expression for Pc(t). We then use Pc(t) in Eq. 10 to obtain velocity perpendicular to the reference x axis shown the following general guidance law solution: in Fig. 2). • All the states are available for feedback. u1 () t = R V 1 +1 ct x() t + 1 +1 ct + c x() t t • The missile and target speeds, vM = vM and S 2 go 1 3 go bt2 2 go W 3 ` j go v = v , respectively, are constant. S c m W.(21) T T 2 ctgo tgo S 3 W • The missile control variable is commanded acceler- S 1 + 3 1 + ctgo + 4 W S btgo ^ h W T X ation normal to the reference x axis (u = ac), which, for very small LOS angles (l), is approximately perpendicular to the instantaneous LOS. We emphasize the fact that this general solution is relevant given the assumptions enumerated above, • The target is assumed to be nonmaneuvering particularly the fact that a nonmaneuvering target and (aT = 0) and, therefore, the linear plant disturbance perfect interceptor response to commanded acceleration is given to be w = 0. are assumed. • The missile responds perfectly (instantaneously) to The structure of the guidance law u1(t), shown above, an acceleration command from the guidance law can be expressed as (a  a ).The system pseudo-measurement is rela- M c u 2 tive position y = x . u1() t = N(,Qf t go)/t go z(, x Q f,)tgo , 1 8 B

JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) 47 N. F. PALUMBO, R. A. BLAUWKAMP, and J. M. LLOYD

Target 250 direction 20.0 of travel Terminal penalties b, c Terminal penalties 200 0.1, 100 19.8 b, c

) 100, 100 0.1, 100 2 100, 100 100, 1 19.6 100, 1 150 100, 0.1 –0.2 0 0.2 0.4 0.6 100, 0.1 100, 0 100, 0 20 100 Acceleration (m/s

50 15

0 0 5 10 15 20 10 tgo (s)

Figure 5. Rendezvous acceleration. A plot of called-for accel- Downrange (km) Downrange erations for intercept trajectories with differing terminal relative 5 position and relative velocity penalties is shown here. The engage- Start (missile trajectories) ment is the same as in Fig 4. As can be seen, substantially more acceleration is required as the penalty on the terminal relative 0 –0.5 0 0.5 1.0 1.5 velocity is increased. Crossrange (km) able states for feedback. From above, we also see that Figure 4. Rendezvous trajectory. A plot of intercept trajectories time-to-go (tgo) is needed in the guidance law. Later, we (in downrange and crossrange) are shown with different terminal also discuss how one can estimate time-to-go for use in penalties for relative position and relative velocity, respectively. the control solution. The target trajectory is traveling from right to left (shown as the Employing Eq. 21 in a simple planar engagement dashed black line) at the top. The target velocity is a constant simulation, we can show what effect a terminal rela- 500 m/s. The initial missile velocity is 1000 m/s with no heading tive velocity penalty will have (in combination with a error. The final time is 20 s. The inset highlights the endgame terminal relative position penalty) on the shape of the geometries in each case. Notice that as the terminal penalty missile trajectory. In Fig. 4, a missile is on course to inter- on relative velocity is increased, the missile trajectory tends to cept a constant-velocity target; it can be seen that the “bow out” such that the final missile velocity can be aligned with missile trajectory shape changes for different terminal the target velocity. Similarly, as the terminal penalty on relative relative position and velocity penalties (i.e., variations position increases, the final miss distance is reduced. in the penalty weights b and c). Note that, in this simple example, there is no assumption that controlled missile where the quantity acceleration is restricted to the lateral direction. Figure 5 shows the resulting acceleration history for each case. ct u 3 go Note that with a nonzero terminal penalty on relative N(,Qf t go)/_ 3 1 + 3 1 + ctgo + 4 c btgo ^ h m position in combination with no penalty on relative velocity (the blue curve in Fig. 5) the missile does not is the effective navigation ratio and zx,, Qf t go com- maneuver at all; this is because the target maneuver prises the remainder of the numerator ^in Eq. 21.h Thus, assumption implicit in the guidance law derivation (i.e., for this general case, the effective navigation ratio is not the target will not maneuver) happens to be correct in a constant value. this case, and the missile is already on a collision course It is clear that mechanization of this guidance law at the start of the engagement. requires feedback of the relative position and relative velocity states to compute the control solution. Typically, Special Case 1: PN Guidance Law only the relative position pseudo-measurement is avail- If we assume that c = 0 and we evaluate lim u() t = able, formed using measurements of LOS angle and b " 3 1 uPN(t), then Eq. 21 collapses to the well- known Carte- range (assuming a range measurement is available). In sian form of the PN guidance law: our companion article in this issue, “Guidance Filter Fundamentals,” we discuss filtering techniques that u() t =3 x() t + x() t t . (22) PN t2 1 2 go enable us to estimate any unmeasured but observ- go 6 @

48 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) MODERN HOMING MISSILE GUIDANCE THEORY AND TECHNIQUES

Leveraging the discussion above for Eq. 21, we can see homing portion of the engagement. A planar (down- that the effective navigation ratio in Eq. 21 has now col- range and crossrange) Simulink terminal homing sim- lapsed to become Nu = 3. ulation was used to examine the PN performance. To PN In Eq. 22, the quantity in square brackets now repre- avoid layering of complexity in the results, all simula- sents the miss distance that would result if the missile tion noise sources (seeker noise, gyro noise, etc.) were and target did not maneuver over the time period [t, tf], turned off for this study. However, a simplified guidance which often is referred to as the zero-effort-miss filter still was used in the loop to provide estimates of (ZEM). We want to emphasize that the guidance law relative position and velocity to the PN guidance law, ZEM is an estimate of future miss that is parameterized thereby having some effect on overall guidance per- on the assumptions upon which the guidance law was formance. Figure 6a illustrates nominal missile and derived (linear engagement, constant velocity, non- target trajectories in a plot of downrange versus cross- maneuvering target, etc.). Hence, the PN ZEM (in the range. The total terminal homing time is about 3 s. As y axis, for example) is given by ZEMPN = ry(t) + vy(t)tgo. is evident, at the start of terminal homing, the missile Note that the accuracy of the guidance law ZEM esti- is traveling in from the left (increasing downrange) mate is directly related to how well the guidance law will and the target is traveling in from the right (decreas- perform in any particular engagement (actual final miss ing downrange). The missile, under PN guidance, is distance, maximum commanded acceleration, amount initially on a collision course with the target, and the of fuel used, etc.). target is, initially, non-maneuvering. At 2 s time-to-go, Under the current stated assumptions, we can show the target pulls a hard turn in the increasing crossrange that Eq. 22 is equivalent to the traditional LOS rate direction. expression for PN, which was shown in the previous Recall that PN assumes that the missile acceleration article in this issue, “Basic Principles of Homing Guid- response to guidance commands is perfect or instan- o ance,” to be aMc = NVcl. We first differentiate r  Rl taneous (i.e., no-lag), and that the target does not o o y o to obtain vy . Rl+ R l. Next, recalling that R = maneuver. Thus, we will examine the sensitivity of PN v: 1r = –Vc and noting that we can express range-to- to these assumptions by simulating a non-ideal mis- go as R = Vctgo, we have the following relationship for sile acceleration response in addition to target maneu- LOS rate: ver. We will parametrically scale the nominal missile r+ v t autopilot time constant (100 ms for our nominal case), o y y go l = 2 . (23) starting with a scale factor of 1 (the nominal or near- Vc t go perfect response) up to 5 (a very “sluggish” response of approximately 500 ms) and examine the effect on Examining the traditional LOS rate expression for PN, guidance performance. Figure 6b illustrates the accel- as well as Eqs. 22 and 23, it can be seen that if we set eration step response of the interceptor for 100-, 300-, N = 3, then the traditional LOS rate expression for and 500-ms time constants. Figures 6c and 6d pres- PN and Eq. 22 are equivalent expressions. Hence, the ent the simulation results for levels of target hard-turn “optimal” PN navigation gain is N = 3. acceleration from 0 to 5 g. Figure 6c shows final miss distance versus target hard-turn acceleration level for Special Case 2: Rendezvous Guidance Law the three different interceptor time constants. As can be seen, as the missile time response deviates from For the classic rendezvous problem, we desire to con- the ideal case, guidance performance (miss distance) verge on and match the velocity vector of the target at degrades as target maneuver levels increase. Figure 6d the final time t . If we evaluate lim u() t = u() t , f b, c " 3 1 REN illustrates the magnitude of missile total achieved accel- then Eq. 21 collapses to the Cartesian form of the ren- eration for variation of target maneuver g levels shown dezvous (REN) guidance law and for the three missile time constants considered. As can be seen, when the missile autopilot response 6 2 u() t =x() t + x() t t . (24) time deviates further from that which PN assumes, REN t2 1 3 2 go go 8 B respectively higher acceleration is required from the missile; this is further exacerbated by higher target Example Results g levels. The example in this section is illustrated by Fig. 6, where we demonstrate how PN performance degrades if EXTENSIONS TO PN: OTHER OPTIMAL HOMING (i) the interceptor response is not ideal (i.e., the actual interceptor response deviates from that assumed in the GUIDANCE LAWS derivation of PN) and (ii) the target performs an unan- In Eq. 21, a general guidance law that is based on ticipated maneuver some time during the terminal some specific assumptions regarding the (linearized)

JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) 49 N. F. PALUMBO, R. A. BLAUWKAMP, and J. M. LLOYD engagement kinematics, 2 (a) target maneuver (actually, a lack of it), and intercep- 1 tor response characteristics Missile Target (the perfect response to an acceleration command) is Crossrange (km) 0 0 500 1000 1500 2000 2500 3000 3500 4000 shown. We also showed that Downrange (m) the general optimal guid- ) g 1.5 ance law derived there col- (b) Step input command lapses to the well-known PN 1.0 guidance law if we assign cost function components as 0.5

c = 0 and b →  (see Eqs. 19 ( response Step 0 and 20). For completeness’ –0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 sake, here we will derive a Time (s) number of related optimal Autopilot time constant guidance laws under differ- 0.1 s 0.3 s 0.5 s 3.0 40 ing assumptions regarding (c) (d) target maneuver and inter- ) 2.5 ceptor response models. m ( 30 ) e g For certain scenarios, these c

2.0 ( a n t

assumptions can have a o n s i a ti d significant impact on guid- 1.5 r 20 e l ss i ance law performance. For e cc example, in Example Results, m 1.0 A a l n it was observed that, during i 10 F the endgame, PN (which is 0.5 derived assuming no target maneuver) can call for sig- 0 0 0 1 2 3 4 5 0 1 2 3 4 5 nificant acceleration from Target maneuver (g) Target maneuver (g) the interceptor when pitted against a target pulling a Figure 6. PN performance versus time constant. (a) A planar missile–target engagement with a hard-turn maneuver. Hence, plot of downrange versus crossrange. The total terminal homing time is approximately 3 s. At the one can develop a guidance start of terminal homing, the missile is traveling in from the left (increasing downrange) and the law that specifically takes target is traveling in from the right (decreasing downrange). (b) The acceleration step response of target acceleration into the missile interceptor for 100-, 300-, and 500-ms time constants. (c and d) Simulation results for account. Of course, mecha- a PN-guided missile versus varying levels of target hard-turn acceleration from 0 to 5 g. (c) Final nization of such a guidance miss distance versus target hard-turn acceleration level for the three different interceptor time law will be more complex constants. The graph illustrates the fact that, for non-ideal interceptor response, PN-homing guid- and will require the feed- ance performance degrades with increasing target maneuver levels. This degradation worsens as back of additional informa- the autopilot response becomes more sluggish. Notice that as the autopilot response approaches tion (e.g., target maneuver). the ideal case (blue curve), the miss distance becomes nearly insensitive to target maneuver. For In addition, if the derivation an ideal autopilot response, PN-homing would result in acceleration requirements of three times assumptions become too spe- the target maneuver. (d) The magnitude of total achieved missile acceleration for the same varia- cific, the resulting guidance tion of target maneuver g levels and for the three missile time constants considered. As the missile law may work well if those autopilot response time deviates further from that which PN assumes, increasingly higher accel- assumptions actually hold, eration levels are required from the missile. but performance might rap- idly degrade as reality deviates from the assumptions. 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- Constant Target ment the previous (PN) state vector (where x1__ ry, x 2 v y) to include a target T Acceleration Assumption acceleration state x3 _ aTy, leading to x _ x1 x 2 x 3 . As before, the control u is 6 @ Here, all of the previous missile acceleration u_ aM ; the plant (process) disturbance is not considered when assumptions hold with the deriving the guidance^ law (wh = 0), but it will come into play when developing a target

50 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) MODERN HOMING MISSILE GUIDANCE THEORY AND TECHNIQUES acceleration estimator; and the pseudo-measurement, again, is relative position y_ x1 . With these modeling assumptions in mind (particularly that x3 _ aTy ^constant),h the LQ optimization problem is stated as

t 1 2 1 f 2 min J x()t0,( u t 0), t0 =x()tf + u() t dt u() t 2 2 #t ^ h Qf 0 0 1 0 0 Subject to: xo ()t = 0 0 1 x()t + –(1 u t), (25) >0 0 0HH> 0 y() t = 1 0 0 x()t . 6 @

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

R V 1 +1 ct x() t + 1 +1 ct + c x() t t + 1 1 +1ct + 2c x ()t t2 S 2 go 1 3 go bt2 2 go 2 6 go bt2 3 go W 3 S` j c go m c go m W u2() t = .(26) 2 ctgo tgo S 3 W S 1 + 3 1 + ctgo + 4 W S btgo ^ h W T X

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 filter 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 sufficient, leading 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 evaluate lim u() t = u() t , then Eq. 26 col- b " 3 2 APN lapses to the well-known augmented PN (APN) guidance law:

3 1 2 uAPN () t =2 x1() t + x 2() t tgo + 2 x3()t tgo . (27) tgo 8 B

Leveraging the discussion above, if we compare Eq. 27 to the expression for PN given 1 2 in Eq. 22, the only difference is the addition of the 2 aT t go term in the numerator of the APN guidance law. Thus, for APN, the effective navigation ratio is the same as in PN guidance (Nu = 3), but the ZEM estimate is now given by ZEM = APN APN 1 2 ry() t +v y() t tgo + 2 aT()t t go .

Special Case 2: Augmented REN Guidance Law If we evaluate lim u() t = u() t , then Eq. 26 collapses to the augmented b, c " 3 2 AREN REN (AREN) guidance law:

JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) 51 N. F. PALUMBO, R. A. BLAUWKAMP, and J. M. LLOYD

u() t =6 x() t + 2 x() t t+ x ()t . (28) Guidance gain Guidance law AREN 2 1 3 2 go 3 N = 3 N = 5 PN t )

8 B T go N = 4 N = 6 APN /a

C 3.0 Notice that, unlike the APN law given in Eq. 27, the AREN guidance law calls for a direct cancellation of the 2.5 target maneuver term in the acceleration command. 2.0

Example Results: Maneuver Requirements for 1.5

PN Versus APN 1.0 Here, in a similar approach to that found in Ref. 13, we compare the maneuver requirements for PN and 0.5 APN guidance laws versus a target that pulls a hard-turn 0 maneuver under ideal conditions (e.g., no sensor mea- Normalized missile acceleration ( a 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 surement noise or interceptor lag). Recall that the PN Normalized time (t/tf) guidance law is derived assuming that the target does not maneuver, whereas the key APN assumption is that Figure 7. A comparison is made of PN and APN accelera- the target pulls a hard turn. tion requirements for various guidance gain values versus a Assuming that the interceptor responds perfectly to target pulling a hard turn. PN results (solid lines) indicate that PN acceleration commands, using Eq. 22 we can write PN demands an increasing level of missile acceleration as flight the following second-order differential equation: time increases. Notice that for a guidance gain of 3, PN requires three times the acceleration of the target to effect an intercept; 2 hence, the well-known 3-to-1 ratio rule of thumb. Increasing the d r() t = a() t – N [(r t)(+ v t)( t –)t ]. (29) dt2 y T (–t t)2 y y f guidance gain can theoretically relax the 3-to-1 rule of thumb, f but higher gains may lead to excessive noise throughput. The graph illustrates that APN (dashed lines) is more anticipatory in Note that we have left the navigation gain as the that it demands maximum acceleration at the beginning of the variable N. Next, we assume zero initial conditions— engagement and less as engagement time increases. Note that for ry(0) = 0, vy(0) = 0—and use Maple to solve Eq. 29, thus a guidance gain of 3, APN (theoretically) requires half the accel- giving analytic expressions for ry(t) and vy(t). We then eration that PN does when engaging a target that is pulling a take ry(t) and vy(t) and reinsert them into Eq. 22 to hard turn. obtain the following expression for missile acceleration caused by a hard-turn target maneuver: 3-to-1 rule of thumb can be relaxed. In practice, how- N t N –2 aM () t = 1–– 1 aT . (30) ever, higher gains may lead to excessive noise through- PN N – 2 = c tf m G put, perhaps negating any theoretically perceived benefits. For the APN case, we employ an analogous proce- Unlike PN, APN (dashed lines in Fig. 7) is more dure, using Eq. 27 as the starting point. For this case, we anticipatory in that it demands maximum acceleration obtain the following expression for missile acceleration, at the beginning of the engagement and less accelera- given that the target pulls a hard-turn maneuver: tion as engagement time increases. Moreover, note that for a guidance gain of 3, APN (theoretically) requires N t N –2 half the acceleration that PN does when engaging aM () t = 1 –.aT (31) a target that is pulling a hard turn. Note that, for a APN 2 ; tf E guidance gain of 4, the theoretical acceleration requirements for PN and APN are the same and, for gains Figure 7 illustrates a comparison of PN and APN above 4, APN demands more acceleration than PN acceleration requirements for various guidance gain (although saturating the acceleration command early is values versus a target pulling a hard turn, via Eqs. 30 not nearly the problem that saturating late is). and 31, respectively. Referring to Fig. 7, we see that PN (solid lines) demands an increasing level of missile acceleration as flight time increases. In fact, for a Constant Target Jerk Assumption guidance gain of 3, PN requires three times the accel- The assumptions stated during the deriva- eration of the target to effect an intercept; hence, the tion of APN are still valid with the exception well-known 3-to-1 ratio rule of thumb. If we increase of target maneuver; here, we will assume that the guidance gain, theory says that the theoretical the target acceleration is linearly increasing (i.e.,

52 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) MODERN HOMING MISSILE GUIDANCE THEORY AND TECHNIQUES

constant jerk in a particular direction). This may be a reasonable assumption, for example, in order to develop a terminal homing guidance strategy for use during boost-phase ballistic missile defense, where it is necessary to engage and destroy the enemy missile while it is still boosting. In such a context, it is pos- sible that a linearly increasing acceleration model (assumption) better reflects actual target maneuver (acceleration) as compared to the APN assumption. Therefore, we augment the previous (PN) state vector (where x1__ ry, x 2 v y) to include a target acceleration state x3 _ aTy and a target jerk state x4_ xo 3 = jTy, leading to T x _ x1 x 2 x 3 x 4 . As before, the control u is missile acceleration (u_ aM), the plant6 (process) disturbance@ is not considered (w = 0), but it will come into play when developing a target state estimator, and the pseudo-measurement, again, is relative position (y _ x1). With these modeling assumptions in mind (particularly that x4 _/ jTy constant), the LQ optimization problem is stated as

t 1 2 1 f 2 min J x()t0,( u t 0), t0 =x()tf + u() t dt u() t 2 2 #t ^ h Qf 0 R0 1 0 0V R 0 V S W S W 0 0 1 0 –1 Subject to: xo ()t = S Wx()t + S Wu() t , (32) S0 0 0 1W S 0 W S0 0 0 0W S 0 W T X T X y() t = [1 0 0 0](x t).

Defining the terminal penalty matrix to be Qf = diag{b, c, 0, 0}, and following a solution procedure identical to that outlined previously, we obtain the following general solution:

R ct ct ct V S 1 +go x() t + 1 +go + c x() t t +1 1 + go + 2c x ()t t2 + 11 + 3c x() t t3 W 2 1 3 bt2 2 go 2 6 bt2 3 go 6 bt2 4 go 3 S` j c go m c go m c go m W u3 () t = S W.(33) 2 ctgo tgo S 1+3 1 + ct + W bt3 go 4 S go ^ h W T X

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 filter structure. More important, estimating target jerk given a relative position (pseudo-)measurement can be very noisy unless the measurement quality is excellent. If sensor quality is not sufficient, then guidance law performance could be significantly worse than if we simply used PN or APN to begin with.

Special Case 1: Extended PN If we assume that c = 0 and we evaluate lim u() t = u() t , then Eq. 33 collapses b"3 3 EPN to the extended PN (EPN) guidance law

3 1 2 1 3 uEPN() t =2 x1() t + x 2() t tgo +2 x3()t tgo + 6 x4() t tgo . (34) tgo 8 B

By this time, a pattern should be emerging regarding the current line of guidance 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 same constant:

JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) 53 N. F. PALUMBO, R. A. BLAUWKAMP, and J. M. LLOYD

Truth Model Estimate Figure 8. Comparison of ZEM 4 removal against a boosting PN threat. Plots of the predicted 2 ZEM for PN, APN, and EPN guidance laws are illustrated. Here, “truth” is computed by integrat- 0 ) ing forward a duplicate of the m

k threat truth model and propa- ( 2

M gating the kill vehicle ballisti-

ZE APN cally. The “model” curve reflects the assumed ZEM calculation 0 for the particular guidance 2 law, using truth data. The “esti- EPN mate” curve is this same calculation with estimated data (via 0 0 5 10 15 noisy measurements), as must t (s) go be used in practice. Intercept

is achieved at tgo = 0 in each case, since the ZEM goes nearly to zero. The magnitude of the ZEM is plotted; in each plot, the portion of the model-based ZEM and estimated ZEM to the right of 12 s is in the opposite direction from the true ZEM.

NNu= u = Nu = 3. estimate quality (noise) as more derivatives are used PN APNEPN in the calculation. Figure 9 demonstrates the resulting It is the ZEM estimates that evolve from acceleration commands and fuel usage for the different guidance laws via logging of the resultant DV, which is ZEMPN = ry()t + v y()t tgo , to defined as tf 1 2 a ()d  , ZEMZAPNP=EM NT + 2 a() t tgo , and now to # M t0 1 3 where a is the achieved acceleration vector. The ZEMZEPNA=EM PN + jT() t t go . M 6 required DV (translating to fuel usage) for PN, at 1356 m/s, is substantially more than APN, at 309 m/s, With regard to EPN, the addition of target accelera- or EPN, at 236 m/s. The required acceleration capabil- tion and target jerk states will dictate a more complex ity of the kill vehicle also is substantially different, with guidance filter structure, and it may be very sensitive PN requiring 27 g capability, APN requiring 3.7 g, and to sensor noise and actual target maneuver modali- EPN requiring 3.2 g. ties as compared with PN or APN. We also note that, if we evaluate lim u() t , then Eq. 33 collapses to b, c"3 3 the AREN guidance law previously given in Eq. 28. PN APN EPN A simulation study of PN, APN, and EPN guid- 200 ance laws against a boosting threat illustrates the

benefits of better matching the target assumptions to (m/s) 100 | the intended target. In this engagement, an exoatmo- m a | spheric kill vehicle intercepts a threat that accelerates 0 (boosts) according to the ideal rocket equation, with a 1000 maximum 8.5-g threat acceleration occurring at intercept. Figure 8 shows how the ZEM prediction for each guidance law compares with the true ZEM as each 500 evolves over time and under the control of the relevant V used (m/s) guidance law. Each of the models of ZEM has signifi- ⌬ 0 0 5 10 15 cant error initially (the direction is wrong, causing the t go (s) truth to increase while the assumed ZEM decreases), but for an assumed target maneuver model that more Figure 9. Acceleration command history and cumulative ∆V closely matches the actual target maneuver (i.e., EPN), used for PN, APN, and EPN guidance laws versus a boosting this error is much less and the sign of the ZEM is cor- threat. The progression is from right to left as time-to-go (tgo) rect sooner. The curves also show some trade-off in decreases toward intercept in this single-simulation run.

54 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) MODERN HOMING MISSILE GUIDANCE THEORY AND TECHNIQUES Non-Ideal Missile Response Assumption Here, the assumptions stated during the derivation of APN are still valid, save for that pertaining to a perfect missile response. Instead, we will add the more realistic assumption that the missile responds to an acceleration command via the first-order lag transfer function a() s /(a s),= 1 M c Ts + 1 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 specific flight condition and depends largely on the missile aerodynamic characteristics and flight control system design. We augment the previous (APN) state vector (where x1__ ry,,x 2 v y and x3 _ aTy T ) to include a missile acceleration state x4 _ aMy , leading to x _ x1 x 2 x 3 x 4 . (Note that the fourth state here is missile acceleration, not target jerk6 as was the case@ when deriving EPN guidance law.) As before, the control u is missile acceleration (u_ aM), 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 measurement. With these modeling assumptions in mind, the LQ optimization problem is stated as t 1 2 1 f 2 min J x()t0,( u t 0), t0 =x()tf + u() t dt u() t 2 2 #t ^ h Qf 0 R0 1 0 0 V R 0 V S W S W S0 0 1 –1 W S 0 W Subject to: xo ()t = xr ()t + u() t , (35) S0 0 0 0 W S 0 W S 1 W S 1 W S0 0 0 – TTW S W T X T X 1 0 0 0 y() t = xr ()t . ;0 0 0 1E

Here, we do not consider a terminal velocity penalty in order to reduce overall 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 t tgo 2 go go –/tgo T 1 2 2 –tgo/T 6 + e–(1 x1t)(+ x 2 t)( tgo + tgo x3 t)– T + e–(1 x4 t) u = ` T j ` T j8 2 ` T j B .(36) 4 2 t t2 t3 t t 6 go go go go –/t T –/2t T go +3 + 6 – 6 + 2 – 12 ego – 3 e go bT3 c T T2 T3 T m

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 better visualize this complexity, consider the constant target acceleration guidance law given in Eq. 26. If we take lim u() t , we obtain the following result: c"0 2 1 2 x1() t +x 2() t tgo + x3()t tgo u() t = 3 2 . (37) 2 c= 0 2 3 tgo 1 + 3 > btgo H

The structure of Eq. 37 is significantly less complex than that given in Eq. 36 despite the fact that the cost function for both is identical (i.e., b is finite and c = 0). If we take lim u() t , we obtain the well-known “optimal” guidance law (OGL) b " 3 4 referred to in many texts (see Refs. 7, 13, and 14 for examples):

JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) 55 N. F. PALUMBO, R. A. BLAUWKAMP, and J. M. LLOYD

Guidance law Gain Guidance law PN APN OGL Target acceleration u KaT 3 3 N Kr 2 2 2 tgo tgo tgo Relative velocity K 3 3 Nu v Kv t t vy go go tgo r + + y + Missile 3 Nu aT ⌺ K ⌺ KaT 0 ∫ ∫ r + response 2 2 – Relative – position 2 t –NTu go –/t T K 0 0 + e go – 1 KaM aM 2 c T m tgo 2 tgo tgo –/t T 6 + e go – 1 u _ cT m c T m N OGL 2 3 Missile acceleration tgo tgo tgo tgo –/t T –/2t T 3+ 6 – 6+ 2 – 12 ego – 3 e go e T T2 T3 T o

Figure 10. The feedback structure of the PN, APN, and OGL guidance laws is depicted here. The relative complexity of the different guidance laws is established as we add additional assumptions regarding the engagement and missile response characteristics. The diagram emphasizes the fact that a substantial increase in complexity arises when the assumptions move from an ideal to non-ideal interceptor response assumption.

t t tgo 2 go go –/tgo T 1 2 2 –/tgo T 6 + e–(1 x1t)(+ x 2 t)( tgo + tgo x3 t)– T + e–(1 x4 t) u = ` T j ` T j8 2 ` T j B .(38) OGL 2 t t2 t3 t t go go go go –/t T –/2t T go 3+ 6 ––6 + 2 12 ego – 3 e go c T T2 T3 T m

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

1 2 ZEMAPN = x1()t + x 2()t tgo + 2 tgo x3()t with that in Eq. 38 and see that the OGL ZEM estimate is t 2 go –/t T ZEMZ= EM – T+ e go – 1 x() t . OGL APN ` T j 4 In addition, the effective navigation ratio for APN is given by Nu = 3. In contrast, APN from Eq. 38, Nu is time-varying and can be expressed as shown below: OGL

t 2 t go go –/t T 6 + e go – 1 Nu _ `T j ` T j . (39) OGL t t2 t3 t go go go go –/t T –/2t T 3+ 6 – 6 + 2 – 12 ego – 3 e go c T T2 T3 T m

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 regarding the engagement and missile response. From Fig. 10, it is obvious that a substantial increase in complexity arises when the assumptions move from an ideal to non-ideal interceptor response assumption.

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

56 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) MODERN HOMING MISSILE GUIDANCE THEORY AND TECHNIQUES

APN OGL PN Figure 11. The cumulative prob- 100 ability performance of PN, APN,

) 80 and OGL guidance laws versus a 5-g %

( hard-turn target is shown. Both the y t 60 cumulative probability of the maxi- bili a 40 mum interceptor acceleration and ob

r CPA for 100-run Monte Carlo sets are P 20 plotted. With these graphs, it is easy 0 to quickly ascertain the probability 0 0.33 0.66 0.98 1.31 1.64 1.97 2.30 2.62 2.95 of achieving the x-axis parameter Closest point of approach (ft) value (e.g., maximum acceleration). 100 In both graphs, lines that are more vertical and farther to the left are 80 considered more desirable. All noise (%) y 60 and error sources are turned off, but the target maneuver start time is ran- 40 domized over the last second of ter-

Probabili t 20 minal homing. Note that the actual missile response model is non-ideal, 0 0 5 10 15 20 25 30 35 40 45 and it varies with the flight condi- Maximum interceptor acceleration (g) tions of the missile trajectory. see a distinct trend considering PN versus APN versus OGL guidance laws. Given values of terminal miss pen- that PN is derived assuming the target is not maneuvering, we expect the perfor- alty. The curves are normal- mance to degrade if the target does maneuver. It also is not surprising to see that, ized with respect to the missile statistically, more acceleration is required versus APN or OGL. The improvement time constant T. Referring to from APN to OGL is explained by the fact that APN assumes a perfect missile Fig. 12, consider the b = 1000 response to acceleration commands and OGL assumes that the missile responds as curve. As tgo / T " 2, Ñ achieves a first-order lag (refer back to Eq. 27 compared with Eq. 38). its maximum value and then reduces as tgo / T " 0. Clearly, this guidance law gain curve TIME-TO-GO ESTIMATION evolves in a way that places As shown in the previous section, Extensions to PN: Other Optimal Homing much greater emphasis on ZEM Guidance Laws, many modern guidance laws require an estimate of time-to-go (tgo), at certain times near intercept. which is the time it will take the missile to intercept the target or to arrive at the Imagine that the actual (true) closest point of approach (CPA). The tgo estimate also is a critical quantity for mis- tgo is 2 s but that the estimate siles that carry a warhead that must detonate when the missile is close to the target. of tgo is in error and biased For example, recall the general optimal guidance law shown in Eq. 36. This guid- toward the positive direction ance law can be expressed as Ñ 3 ZEM, where by four missile time constants (4T). Then, from Fig. 12 we t 1 2 2 go –/t T can see that the guidance gain ZEM= x ()t + x ()t t + t x ()t– T + ego – 1 x ()t 8 1 2 go 2 go 3 ` T j 4 B would be about one-seventh of what it should be at tgo / T = 2, and the effective navigation ratio, Ñ, is shown in Eq. 40, where b and T represent thereby not placing optimal the terminal miss penalty and assumed missile time constant, respectively (see the emphasis on ZEM at that time, section Non-Ideal Missile Response Assumption, wherein Eq. 36 was derived, for a and degrading overall guidance further description): performance. The simplest time-to-go tgo 2 tgo –/tgo T estimation scheme uses mea- u 6 T T + e – 1 N = `2j ` 3 j . (40) surements (or estimates) of tgo tgo tgo tgo 6 –/tgo T –/2tgo T range and range rate. Consider 3 +3 + 6 ––62 + 2 3 12 e– 3 e bT c T T T T m the engagement geometry of Fig. 13, where vM = missile It is clear from Eq. 40 that Ñ is a function of tgo. Figure 12 illustrates the tgo velocity, vT = target veloc- dependence of the general optimal guidance law effective navigation ratio for three ity, r = target–missile relative

JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) 57 N. F. PALUMBO, R. A. BLAUWKAMP, and J. M. LLOYD

t 6 go –/ P + e tgo T – 1 T2 c T m Nu _ CPA t t2 t3 t 6 go go go go –/t T –/2t T +3 + 6 – 6 + 2– 12ego – 3 e go bT3 e T T2 T3 T o v M v

Terminal penalty 80 b =10 b = 100 b = 1000 vT 70 ␬ R = ||r||

n 60 Missile LOS Target

gai 50 e e c n

a 40 Figure 13. The missile–target (planar) engagement geometry d i

u 30 is shown here. This depiction places the LOS to the target along G the x axis of the coordinate system. Missile and target velocity 20 – – vectors are indicated as vM and vT, respectively. The relative veloc- 10 ity, v–, makes an angle  with the LOS and passes through the 0 points indicated by CPA and P. 0 1 2 3 4 5 6 7 8 9 10 t go/T (s) r()t: v() t t = – . (44) go v()t: v() t Figure 12. The normalized tgo dependence of the effective navigation ratio Ñ for the OGL is shown here for three terminal penalty values. Normalization is with respect to the missile time Using this expression for tgo in Eq. 42, we obtain the target–missile relative separation at the CPA: constant T. We note that, as tgo approaches infinity, the effective navigation ratio always approaches 3. r()t[( v t)(: vt)] –( v t)[r()t() t : v()t ] r = CPA v()t: v() t position, and v = target–missile relative velocity. Refer- (45) [(vt)(## r t)] v()t ring to Fig. 13, we assume that the missile can measure = . or estimate relative range (R = r ) to the target and v()t: v() t range rate (Ro ) along the LOS to the target. If we assume that the missile and target speeds are constant, then one Conceptually, the differences between time-to- can estimate time-to-go as go estimation using Eq. 41 rather than Eq. 44 can be explained by using Fig. 13. For this discussion, and with- t R tgo = – o . (41) out loss of generality, we assume that the missile velocity R is constant and that the target is stationary. Referring to Another common approach to estimating time-to-go Fig. 13, we see that Eq. 41 estimates the flight time for also assumes that the missile and target speeds are con- the missile to reach point P. However, Eq. 44 estimates stant. Define Dt_ t) – t, where t is the current time and the time it takes for the missile to reach the CPA. If t) is a future time. Thus, given estimates of target–missile the missile and target have no acceleration (the up-front relative position and relative velocity at the current time assumption during the derivation), then Eq. 44 is exact. t, the future target–missile relative position at time t) is given as CLOSING REMARKS ) r()t= r() t + v()tD t . (42) In this article, we have focused on developing homing guidance laws by using optimal control tech- At the CPA, the following condition holds: niques. To this end, a number of modern guidance laws (PN and beyond) were derived using LQ optimal con- r()t)): v() t = 0 . (43) trol methods. We note that, regardless of the specific structure of the guidance law (e.g., PN versus OGL), we This condition is illustrated in Fig. 13 by the perpendic- developed the relevant guidance law assuming that all ular line from the target to the relative velocity. Based of the states necessary to mechanize the implementa- on our assumptions, using Eq. 42 in Eq. 43, and recog- tion were (directly) available for feedback and uncor- nizing the constant velocity assumption, we obtain the rupted by noise (recall we referred to this as the “perfect following expression for Dt/ tgo : state information problem”). In a companion article in

58 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) MODERN HOMING MISSILE GUIDANCE THEORY AND TECHNIQUES

this issue, “Guidance Filter Fundamentals,” we point 5Jackson, P. B., TOMAHAWK 1090 Autopilot Improvements: Pitch- Yaw-Roll Autopilot Design, Technical Memorandum F1E(90)U-1-305, to the separation theorem, which states that the opti- JHU/APL, Laurel, MD (1 Aug 1990). mal solution to this problem separates into the optimal 6 Basar, T., and Bernhard, P., H-Infinity Optimal Control and Related deterministic controller (i.e., the “perfect state informa- Minimax Design Problems, Birkhäuser, Boston (1995). tion solution”) driven by the output of an optimal state 7Ben-Asher, J. Z., and Yaesh, I., Advances in Missile Guidance Theory, estimator. Thus, in that article, we discuss guidance American Institute of Aeronautics and Astronautics, Reston, VA (1998). filtering, which is the process of taking raw (targeting, 8Grewal, M. S., and Andrews, A. P., Kalman Filtering Theory and Prac- inertial, and possibly other) sensor data as inputs and tice, Prentice Hall, Englewood Cliffs, NJ (1993). estimating the relevant signals (estimates of relative 9Pearson, J. D., “Approximation Methods in Optimal Control,” J. Elec- position, relative velocity, target acceleration, etc.) upon tron. Control 13, 453–469 (1962). which the guidance law operates. 10Vaughan, D. R., “A Negative Exponential Solution for the Matrix Riccati Equation,” IEEE Trans. Autom. Control 14(2), 72–75 (Feb 1969). 11 REFERENCES Vaughan, D. R., “A Nonrecursive Algebraic Solution for the Discrete Riccati Equation,” IEEE Trans. Autom. Control 15(10), 597–599 (Oct 1Bryson, A. E., and Ho, Y.-C., Applied Optimal Control, Hemisphere 1970). Publishing Corp., Washington, DC (1975). 12Pue, A. J., Proportional Navigation and an Optimal-Aim Guidance Tech- 2Cottrell, R. G., “Optimal Intercept Guidance for Short-Range Tacti- nique, Technical Memorandum F1C (2)-80-U-024, JHU/APL, Laurel, cal Missiles,” AIAA J. 9(7), 1414–1415 (1971). MD (7 May 1980). 3Ho, Y. C., Bryson, A. E., and Baron, S., “Differential Games and 13Zames, G., “Feedback and Optimal Sensitivity: Model Reference Optimal Pursuit-Evasion Strategies,” IEEE Trans. Automatic Control Transformations, Multiplicative Seminorms, and Approximate AC-10(4), 385–389 (1965). Inverses,” IEEE Trans. Autom. Control 26, 301–320 (1981). 4Athans, M., and Falb, P. L., Optimal Control: An Introduction to the 14Shneydor, N. A., Missile Guidance and Pursuit: Kinematics, Dynamics Theory and Its Applications, McGraw-Hill, New York (1966). and Control, Horwood Publishing, Chichester, England (1998).

Neil F. Palumbo is a member of APL’s 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 School’s Engineering for Professionals program. He is a member Theof the Institute of ElectricalAuthors 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 engineering 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 joined APL in 2004 and conducts work in optimization; advanced missile guidance, navigation, and control; and integrated controller design. For further information on the work reported here, contact Neil Palumbo. His email address is neil.palumbo@ Neil F. Palumbo Ross A. Blauwkamp Justin M. Lloyd jhuapl.edu.

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

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