I#

30 Jan 90 Interim

j A Passive Homing Missile Guidance Law Based on New Target

.J Maneuver Models C: F08635-87-K-0417

* - Jason L. Speyer; Minjea Tahk; Kevin D. Kim

* :! Debra Harto, Program Manager, AFATL/FXG

University of Texas at AustinDepartment of Aerospace Engineeringand Engineering Mechanics

Austin TX 78712-1085

o Air Force Armament LaboratoryN AFATL/FXG

Eglin Air Force Base, Florida 32542-5434 AFATL-TP-90-05

Technical Paper D T ICSELEC. ...-- APR 12 1990

Approved for public release; distribution unlimited. S " T f

A new stochastic dynamic target model is proposed on the assumption thatcertain targets execute evasive maneuvers orthogonal to their velocity vector.Along with this new acceleration dynamic model, the orthogonality is alsoenforced by the addition of a fictitious auxiliary measurement. The targetstates are estimated by the modified gain extended Kalman filter, and theangular target maneuver rate is constructed on-line. A guidance law thatminimizes a quadratic performance index subject to the assumed stochasticengagement dynamics that includes state dependent noise is derived. Thisguidance law is determined in closed form where the gains are an explicitfunction of the estimated target maneuver rate as well as time to go. Thenumerical simulation for the two-dimensional angle-only measurement caseindicates that the proposed target model leads to significant improvementin the estimation of the target states. Furthermore, the effect onterminal miss distance using this new guidance scheme is given and comparedto the Gauss-Markov model. / .- /

Kalman filter, target estimation; linear quadratic guidance law; 12guidance law-

UNCLASSIFIED .._UNCLAJYII4D . SAR

IQ (~412 11

A Passive Homing Missile Guidance Law Based on New Target Maneuver Models*

Jason L. Speyert and Kevin D. KimlThe University of Texas at Austin

Austin, Texas

Minjea Tahk

Korea Advanced Institute of Science and TechnologySeoul, Korea

Abstract ing angle-only measurements is relatively poor. Usually,the target tracking problem is approached by model-

A new stochastic dynamic target model is proposed ing target acceleration with a first-order Gauss-Markovon the assumption that certain targets execute evasive model and applying the extended Kalman filter(EKF).maneuvers orthogonal to their velocity vector. Along One difficulty with the Gauss-Markov model is thatwith this new acceleration dynamic model, the orthog- the assumed large process noise spectral density inducesonality is also enforced by the addition of a fictitious Kalman filter divergence even when the target maneu-auxiliary measurement. The target states are estimated ver is not present and can lead to a target accelerationby the modified gain exto-.ed Kalman filter, and the magnitude estimate which exceeds the actual maximum.angular target maneuver rate is constructed on-line. A Another problem is that the target motion is not wellguidance law that minimizes a quadratic performance represented by Gauss-Markov diffusion process.index subject to the assumed stochastic engagement dy- In an effort to alleviate these problems, the circu-namics that includes state dependent noise is derived. lar target model has been proposed as a target motionThis guidance law is determined in closed form where model, where the phase angle is a Brownian motion pro-the gains are an explicit function of the estimated target cess and the acceleration magnitude can be either a ran-maneuver rate as well as time to go. The numerical @im- dom variable or a bounded stochastic process. This tar-ulation for the two-dimensional angle-only measurement get model was suggested in [4] using concepts extractedcase indicates that the proposed target model leads to from [9,10].significant improvement in the estimation of the target By including a priori knowledge of the target motion,states. Furthermore, the effect on terminal miss distance improved estimates of the target states can be obtained.using this new guidance scheme is given and compared This idea is included in (6,7] by using a target accelera-to the Gauss-Markov model. tion model which employs a target mean jerk term. For

conventional targets such as winged aircraft, the longi-

1. Introduction tudinal acceleration component is often negligible com-pared to the lateral component in evasive maneuvers.

The target tracking problem for homing missile guid- This notion fits the circular target model where the an-ance involves the problem of estimating large and rapidly gular rate term is estimated to account for the actualchanging target accelerations. The time history of target dynamics of the coordinated turn. This model is pre-motion is inherently a jump process where the acceler- sented in Section 2. However, an approximate state ex-ation levels and switching times are unknown a priori. pansion is required to handle the unknown angular rateDue to this arbitrary and unpredictable nature of tar- in the target model. This approximate dynamical sys-get maneuverability, target acceleration cannot easily be tem used for estimation is presented in Sections 3.1 andmodeled. 3.2. Furthermore, in Section 3.3 the orthogonality be-

A considerable number of tracking methods for ma- tween target velocity and acceleration can be viewed as

neuvering targets have been proposed and developed a kinematic constraint where compliance is enforced byfor both new target models and filtering techniques[1]- including this constraint as a pseudo-measurenient[?,bj.[8]. In spite of the numerous modeling and filtering The approximate target dynamics and pseudoineasure-techniques available, target acceleration estimation us- ment are included in the modified gain extended Kalman

filter(MGEKF)[11] and is presented in Section 4. The*This work was supported by the Air Force Arraament Labo- MGEKF is selected because of its superior performance

ratory, Eglin AFB, under contract FOS635-87-K-0417. over the EKF especially for bearing-only problems. Int Currently, Professor, Dept. of Mechanical and Aerospace En- Section 5, a linear quadratic guidance law is derived forgineering, UCLA, Fellow AIAA.

lGraduate Re.earch Assistant, Member AIAA. this circular target model. This guidance law remains aSAssistant Professor, Dept. of Mechanical Engineering, Mer- linear function of the estimated states, but the guidance

ber A' '.',. gains obtained in closed form are a nonlinear function

1 lA o

of the estimated rotation rate and time to go. Finally, a In order to see how the current model approximatesnumerical simulation is performed for a two-dimensional the assumed nonlinear target dynamics(l), consider ahoming missile intercept problem. Both the estimation deterministically equivalent case. Integration of (2) withprocess and the terminal miss are enhanced by the new 0 = 0 and JIw > 0 yieldsmodels and the associated estimator and guidance law aT TaT

in comparison with the Gauss-Markov model , = -sinwt, V. = -- Tcosw - VT + !. (4)

2. Target Acceleration Model where the initial conditions are 14 = 0 and V. = -VT.Adding the square of each component and moving all

In this section the circular target acceleration model terms to the left hand side gives

is presented, and the dynamic consistency between this a) 2 aTtarget model and an assumed nonlinear target model is 2(1 - o [ - -- t] 0. (5)discussed.

2.1. Circular Target Motion For this equation to hold for all t > 0The two-dimensional homing missile guidance sce- aT

nario is described by two sets of nonlinear dynamic equa- T (6)tions of motion for the missile and target

which is equivalent to the differential equation for angu-iM = VMfCOSOM ,T = VTCOSOT lar rate in (1).IM = VMsinfM ,YT = VTsinOT Furthermore, by taking the dot product of the velocityVf = am, , T = 0 vector with thc acccleration vector, we obtain6m = aM. /VM , =T - aT/VT

VT-5T "- aT[-VT + - T sinuwt = 0 (7)where (xm , XT) and (YM , yT) are inertial coordinates, WVM and VT are the velocities, am, aM. and aT are the using (6) for all t > 0. This orthogonality between tar-accelerations, and OM and OT are the flight path angles[Fig. 2]. The subscripts M and T denote the missile consistency of the proposed target acceleration modeland target, respectively, and aM, and aM. are tangential for the filter with the nonlinear target dynamics.and normal accelerations, respectively. Only the normalcomponent of the acceleration contributes to changingangular orientations of each vehicle, and the target is 3. New Dynamic and Measurement Modelsassumed to fly at constant speed. for Estimation

The following target model is assumed to be used inthe filter. The objective is to choose a model that is The previous section dealt with a new circular modellinear in order to reduce the numerical computation of for filter implementation in order to exploit an assumedthe filter, but reasonably consistent with the nonlinear characterization of the motion of a typical target. Inmodel so that the estimates are of good quality. The this section, the stochastic dynamic equations for thetarget model for the filter in two dimensions is new target r, eI -,.re derived. Furthermore, the kine-

matic fictitio.,- surement suggested in [7,8] is alsoaT(t)= aTcos(wt + ), aT,() = arsin(wt + 6) (2) discussed.

3.1. Formulation and Approximationwhere aT is a constant which is unknown a priori, w is It6 stochastic calculus[12] applied to the Eq.( 2 ) resultsthe angular velocity to be estimated in a right-handed in a stochastic differential equation with white state-coordinate system, and 0 is a Brownian motion pro- dependent noise[9].cess to represent random target maenuver phasing with

statistics daT, -j -I a,I d -dl rt [ dt 4-

E[dO] = O, E[d02] = ®dt, E = 1i/re (3) daT, W - aT, de 0 aT,(8) 1

Here, E is the power spectral density of the process and where the elements -t in the drift coefficient are the i2re is the coherence time, the time for the standard devi- It6 correctit, terws. Note that the problem is nonlinear

ation of 0 to reach one radian. While in the previous cir- due to the unknown w. To avoid solving the nonlinearcular target model[4] the acceleration components were problem, w is approximately removed by an expansionjust a diffusion proces along a circ!e, those in the new nCfstate v.rlablIc as in [1,,)]. Def,,c ncw s.ates a.model are related to the actual target motion through aterm of physical meaning, w. a' = wa. ., = wa

S1 . ... (9)2.2. Dynamic Consistency a~ was , a~ wag

2

with the assumption Fictitious measurement

a+1 In Ref.[7,8] the filter performance is improved byH = " < 1. (10) introducing a kinematic constraint based on a priori

knowledge, which is implemented in the form of an aug-By augmenting the dynamics of these new states to (2), mented fictitious measurement. In particular, the accel-an approximate dynamical model which includes this eration vector is assumed to be related to the velocitynew target model is truncated as as

da. 0 0 -I 0 0 ax -a, VT-5T=0 (18)

da, -_ 1 0 0 0 all a. under the assumption that the target accelerates pre-da o 0 0 -1 a -a. dominantly orthogonally to its velocity vector. Whenda zero1 0 a + dO this condition is not met, the acceleration has a compo-

elements 2 nent in the direction of the target velocity asno,_a 5;T. = (19da, e i

-2 a1, a"

(11) where 9T and aT are assumed to be random vectors rep-Note that w does not appear explicitly in (11), and (11) resenting target velocity and acceleration, and 17 is theis a linear stochastic differential equation. An idea of uncertainty in the orthogonality. This idea can be im-this sort, given in (10) for a scalar problem, led to sig- plemented in the form of a discrete pseudo-measurementnificantly improved filter performance. as

3.2. Two-dimensional InterceptThe two dimensional intercept problem is developed Z2(tk) = h 2(z(tk)) + 77k

in a relative inertial coordinate system. The system dy- = VT.,aT, + VT., aT, + (20)namics are expressed in the following set of equations:

where 'Ak is a white random sequence with assumedZr = u., li = Vr, statistics= aT,, -aM., Vt = aT , , - M, ,, E[_k] = 0, E[rr/] = V241. (21)

aT. - aTcos(wt + 0), aT, = aTsin(wt +; 0).(12) Note that the variance of the measurement noise corre-

With the assumption of w < 1 and truncation of the sponds to the tightness of the constraint. In other words,target dynamics up to the second order, the ten-element the larger the variance, the more relaxed are the require-state vector is defined as ments on longitudinal acceleration. This fictitious mea-

l 1 2 2T surement is used along with the angle measurement inz [ x, yr u, v,. a2 a a a a Y a~] the modified gain extended Kalman filter described in= [X1 X2 X3 z 4 X5 z6 X7 X8 9 z 1 0 }T. the next section.

(13)Thus, in terms of the expanded state space, the linear,stochastic state differential equation is described by 4. Estimation of Target States

dx = (Fx + Bu)dt + GdO (14) In this section, the modified gain extended Kalman

where F is given by the relations in (12) and by the filter(MGEKF)[11] is derived for the circular targetcoeffient mtrix ven bxthrela in (1) Bis a model and for the fictitious and angle measurement de-coefficient matrix of x in (11) where i = 2, B fined in the previous section. Also, considered is the10 x 2 matrix of zero except for B3 1 = B42 = 1, method to reconstruct the maneuver rate using the esti-u = (aM. , aMf)T, and mated states. Given the continuous-time dynamics and

S= (0 0 0 0 - z 6 x -s XZ7 -z0 z ]T. (15) discrete-time measurements, as in the previous section,construction of the filter is completed by specifying the

3.3. Measurement time propagation and measurement up late procedures.Angle measurement 4.1 Time PropagationAngle information in discrete-time is assumed. Then, The state estimate i(t/t.i.)is propagated from the

the measurement at time tk is current time ti1 I to the next sample time t3 by integrat-ing

:l(tk) = hl((tki)+Vk = tan-'(y,/,)+vk !- Z*(tk)+V/k(16) 1(/ : - (/,t--#~)

where vk is a white random sequence with statistics P(t/ti-1 ) = FP(t/i-l)+ P(ti/t-)FT+ E(GG(GT],X(t) = FX(t) + X(t)FT + E[GE)GT].

= 0,)] = (17)

3

given the posteriori estimate i(f, i/t1_i) = i(ti 1 ) and measurement function z. Unfortunately, this type ofposteriori pseudo-error variance P(t4, 1/t- 1) = P(ti-). linearization with respect to the measurements occursThe notation R(t/ti-1 ) denotes the value of some quan- for only a few functions. It is applicable to angle mea-tity R at time I given the measurement sequence up to surements[11], but not for our new pseudo-measurement.time ti-1. The integration of the covariance of the state Therefore, we must for the pseudo-measurement revertX(t) begins with X(0) at time I = 0. Upon integrat- back to the extended Kalman filter form and defineing the equations above to the next sample time, thepropagated estimates are obtained as follows: g2(z(fi), .(t))(z(tj) - i(tI)) = h2 (z(ti)) - h2 (i(ti))

Oh2(ti) (titi-1) P~t = P~t/t -l (23)= a ,t zt) (t)

It should be noted that since the process noise is state- (28)

dependent , the integration to propagate P(t) also re- Oh2quires the integration of X(t), where E[GOGT] matrix where the expression for = is found in the sec-turns out to have nonzero elements for its lower-right ond row of H in (25)

6 x 6 matrix. Note that the approximation technique For angle measurements[l1]reduces the originally nonlinear dynamics to linear dy-namics. This allows for the closed form solutions of the hl(x(ti)) - hl(f(ti)) = gi(z(t),i(tI))(x(ti) - i(ti))propagation of the estimates rather than performing on- EI.jflz*i))(xI.'I -

line integration. =I(2)4.2 Measurement Update[ll] (29)The states are updated as follows: where

. = t(t) + K(i)[z -

K(ti) = P(4I)H(ti)T[H(tj)P(tj)HT(t) + V]- 1 E(ti) = D(ti)an-a(ti)(24) a(ti)where =[V 0

where V [ V2 I and D(ti) = .t,(ti)2 +Yr(i) 2 (30)(Xr(ti)-,(ti) + yr(tj09(t)))

A ((ti) D(ti)f-(z*(ti))i(t,)(25 H,(z*(ti)) = [sin z*(t), -cos z (4) , 0 , 0 , 0 ,

0 , 0iH 0, 0, , . .,*, 011, 0 , 0, 04, n , ... 0 As discussed in [11], g(z(ti),2(tj)) is only used in the

with update of P(ti) but not in the gain, since it was em-pirically shown that this procedure leads to an unbiased

1112 - _ 12 , " estimate of the state.+'' 4.3. Estimation of w and T.o

H3 = H124 =e, H2S = i + ', HU = .f + I Since the target angular velocity term is embedded

where the missile acceleration in the z and y directions, in the states, w should be reconstructed using the esti-

a and am,, are assumed to be measured very accu- mated states. A simple way to determine the value of

rately with on-board sensors. w is to divide the states as , = L- or -. However,The measurement update of the pseudo-error variance ax aX

is performed by since the expanded state space is originally an approxi-mated state space, this might lead to numerical errors,

P(t,) = [I - K(t,)g(z(i), (t,)P(t(t1 )[1 -K(i)g(z(t 1 ), (i))]T especially when the higher approximated terms are used.+K(tj)R(tj)K (,)T (26) By relying on the definition of the vector relation be-

tween and velocity and acceleration, the target angularwhere g(z(tj),E(ti)) is used in the update of P rather velocity can be obtained without using the augmentedthan H of Eq. (25) and is given as states for approximation. From the assumed dynamics

-Y'(ti) ! - ( ] the target angular velocity during its evasive maneuver

h(z(ti)) - h(i(t,)) = Zr(ti) ) an VT

ish 2 (z(ti)) - h 2(i(ti)) VT T

=g(z(t ), i(tj))(x(f)- i(tj)) (27)

Note that g is a 2 x 10 matrix of function explicit only Thus, by using the state estimates, w is constructed as

in the known quantities z and 2. In this sense, the func- = (32)tion h has a universal linearization with respect to the w- VTT,)I1-- (32)

4

To be used later in the controller, an estimate of time-to-go, TSo, is required, and approximated here as IX =X Y.iy, iv, a T, ,aT,]

Tgo - R _ (33) u [am., aM.] (36)

IRI IX . VI/

Then, the stochastic control problem is to find u whichwhere 1? and R are the estimates of relative range and minimizesrange rate, respectively, and the vectors ;X and V are theestimates of relative position and velocity, respectively. = E{j uTRudr + xfS)z! (37)

5. Linear Quadratic Guidance Law subject to the stochastic differential equation with statedependent noise

Based on the estimated states and the estimate of therotation rate constructed from the estimated states, a dx = [Az + Bu]dt + D(z)dO (38)

guidance law can be mechanized. In the following, astochastic guidance law is determined which minimizes where

a quadratic performance index subject to the stochasticengagement dynamics including the stochastic circular

target model(8) under the assumption that states in- 0 0 1 0 0 0 0 0

cluding the target states and the target rotation rate are 0 0 0 1 1 0 1 0

known perfectly. This assumption simplifies the deriva- A 0 0 0 0 0 1 , B = 0 1

tion of the guidance law enormously, and for this homing 0 0 0 0 -a -w 0 02 _e

problem it is shown that the solution to the stochastic 0 0 0 0 w -- 0 0 - (39)

control problem with state-dependent noise can be ob- 0 1 0 0 0 0 01tained in closed form. The solution obtained does not 0 1 0 00 0aproduce a certainty equivalence controller since the guid- 0 1 0= 0 0 0 0ance law explicitely depends upon the system statistics. D(Z) = 0 0 1 0 0 0

Note that since the noise in each cartesian directidn is -X6 0 0

correlated in the stochastic circular target model(8), and 2 01that with aT. and aT, dynamically coupled through wterm, the guidance commands in the x and y direction where c > 0. Note thatcannot be achieved independently. Thus, the optimal 6

stochastic controller for circular target model is based D(z) = D R6 X(on the minimization of the performance index D =

;i=1

1 2 + j +a2( where zj is the ih element of z and whereJ = Ej [xs + yj] + 2 o [ +a2 ,d (342 0 0 0

subject to the following stochastic system of linear dy- 0 0 0nainic equations DID,DqD - 0 0D 0 0

dx udt DDDD 5 0 D

dy = vdt 0 0 -1

du = (.- am.)dt L35 0(4L1 L0 )dv = (aT, - aM,)dt (35) (41)

dvaT -a, )d To obtain an optimal control for this class of prob-daT, = (--aT, - waT,)dt - aTdO lem, dynamic programming[13,14] is employed where thedaT,, = (- .aT, + WaT.)dt + aT~dO Hamilton-Jacobi-Bellman equation becomes

where 0 is a Brownian motion defined earlier and E[-]stands for an expectation operator. In the construction Iof the filter, the inherent nonlinearity of the target model 0 = J'(RUt)+Min{J)(Ax+Bu)+2 Ru]

was removed by an expansion of state variables. How- (42)

ever, for the guidance law formulation, the rotation rate, where J is the optimal return function and the sub-w, is assumed known, although it must be constructed scripts denote partial derivatives. The elements of theon-line from the state estimator(32). matrix A for any symmetric matrix W is defined as

For brevity of notation, define the state and controlvectors as follows Aij(W, I) = tr[Di )TWDj (t)] (43)

5

The minimization operator in (25) produces integrating the Riccati equation backwards without re-quiring the explicit evaluation of the A term. In partic-

u = R-iBTJ (44) ular, the stochastic optimal control problem essentiallydegenerates to a deterministic optimal control problem

By substituting (44) into (42), the dynamic program- although the It6 terms are retained. The solution pro-ming equation becomes cess for this deterministic control problem, explained in

1 1 Adetail in the Appendix A, produces a guidance law in0 J: +J]Az JBR -'BJ zclosed form. Note that the deterministic coefficient A22

(45) includes the statistic (). Therefore, the resulting con-The optimization problem is solved by explicitely troller is not a certainty equivalence controller. The new

showing that the equation above has a solution . Asume controller becomesJ(x,t) = xT2 s(t)x, then

S T =( t)Jt -" X ; X;,J - s ~ (46) y(f)

With this assumption, the dynamic programming equa- [aM = 0 C1 0 C2 -c4 C3 v(t)

tion is satisfied for al! x E R" if aT (t)

S+SA+ATS+A -SBR-lBTS =0, S(ij) = S1 (47) aT,(t)(53)The desired optimal controller becomes where the gains cl to c4 are an explicit function of Tgo,

w, and E asu = -R- 1 Bs .S (48)

where S is the solution of the Riccati equation and the To-(S, t) i = ir[DTSDj] leads to C= -- =T , c+

e z.ro 1,C .C we e,~zero C3 = e"t---en- -sinTh. + (coswT,0. - eT "* )

A = eees'(49) wO .. (t. _,2),

S66 -S56 C4 = C {--wT,e T + (eT"- coswT.) - - ,inwT,. 4

-S56 S 7ZT7 T T 2)

The fact that A has only nonzero elements for its . tbet = *T..(c+ +,, 2

lower-right 2 x 2 matrix allows a tractable closed-formsolution. To see the characteristics of the solution in a Fig. 1 is a block diagram for an adaptive guidancesimple manner, matrices are partitioned such that their scheme for a homing missile. Note that guidance gainslower-right block partitioned is a 2 x 2 matrix. Then are functions of t., estimated time to go, the statistic

E) and the estimated maneuver rate c'. Therefore, for

A A,, A12 1 [ Bi S =S11 S2 ] A 0 0] the bearing-only measurement system although the re-0 An ,B = 0 [S S22 , 0 J suiting stochastic guidance law is sub-optimal since the

(50) measurements are nonlinear functions of the states, thewhere S is the nonzero element block in (49). This leads explicit dependence on the estimate of the target ma-to the controller neuver rate is a new feature which should help reduce

terminal miss distance.

U -a mc B2 S1I J (51)6. Numerical Simulation

where the block matrices satisfies the decomposed Ric-cati equation For a particular engagement scenario, te perfor-

mance of the estimator using the new target nodels and-A= S,,A,, + ASS,, - S 1B, R-'BS 1 that of the guidance law are evaluated.• = rr.4j2 +S 2A22 +ATSj 2 - Sj 1BjB 1 (52) 6.1. Missile and Target model

-Sn 2 Sj2Aj2+S22A22+ATSn+AT~ -IBR tl+ 2 2+ 2ST2 +A S 1,B, R- 1 BT 12 Both target and missile are treated as point masses

and are considered in two-dimensional reference framesSince the S22 block does not affect the block matrices as shown in Fig. 2. The missile represents a highlyS11 and S12 , the optimal control law is not dependent maneuverable, short ra.ige air-to-air missile with a max-on S22 . Therefore, the closed form optimal guidance imum normal acceleration of 100g's. It is launched withlaw for this special class of problem can be obtained by a velocity Af = 0.9 at a 10,000ft altitude with zero

6

normal acceleration. After a 0.4 sec delay to clear the 0 is a Brownian motion process beginning at 0(0) = 9,launch rail, it flies by the guidance command provided the expected angle the target acceleration vector makesby the linear quadratic guidance law of Section 5 Also, with respect to the z axis at the time of launch, andto compensate for the aerodynamic drag and propul- aT. is the expected maximum acceleration of the tar-sion, the missile is modeled to have a known longitu- get. For the simulation with a coordinate system havingdinal acceleration profile : aM = 25g's for t < 2.6sec. one axis perpendicular to the initial V1, direction, 0 isaM = - 15g's for t > 2.6sec. The target model flies zero. Then, the possible nonzero elemeats of the up-at a constant speed of M = 0.9, and at an altitude of per triaigular part of the initial covariance matrix are10, 00ft. It accelerates at 9 g's either at the beginning P55 (0), P57 (0), P 59 (0), P7 7(0), P7 9(0), and P99 (0). Fur-or in the middle of the engagement. Thus, the rotation thermore, no information is available about the direc-rate of the target is 0.3 during its maneuver. tion of maneuver, and the possible maximum rotation

Two engagements. considered in the following section, rate can be either positive or negative. Thus, the oddare shown in Fig. 3. With Ri and RM denoting initial powers of D are taken as zero. This leaves only P5 ,(0),range and maneuver onset range, respectively, engage- P 59(0), P77 (0), P99 (0) as the nonzero elements. How-ment 1 is the situation where the target maneuver starts ever, a valie of ten is assigned to P66 , Ps, and Pl0 loat the beginning, and for engagement 2 the maneuver to ensure positive definiteness of the covariance matrixstarts in the middle. at the initial time._? Filter Parameters and Initial conditions 6.4. Filter ResultsIntegration of actual trajectories is performed by a The results in this section are the product of a Monte

fourth-order Runge-Kutta integrator with step size 0.02 Carlo analysis consisting of 50 filter runs. Along with theseconds. The variance for the angle measurement is cho- miss distance calculations, the plots of the estimationsen, as given in [4], to be error and the w estimates versus time are mainly con-

025 sidered. The errors are calculated as [E[e_] 2 + E[ey]2]'/ 2

V1 = a VR , Vo = (-- + 5.625 * 10-)/At r'd 2 where E[e., and E[e] are the averaged values of errors(55) over 50 simulation runs. In evaluating the actual miss

where R is range, At is filter sample time, and a is pa- distances, the filter state estimates are used in the guid-rameter which is useG in the simulation indicating dif- ance law. Moreover, the the tracking errors to be pre-ferent levels of sensor accuracy. sented below are based on the guidance law in terms of

As mentioned earlier, the variance for the pseudo- the estimated states, since the tracking errors were ob-measuremer,t can be interpreted to show how strictly served to be quite similar to the case where the actualthe orthogonality assumption between the target veloc- states are used for the guidance law.ity and acceleration is to be kept. By allowing some Fig. 4 represents the results for the engagement 1acceleration in the longitudinal direction, a reasonable where the target maneuver is initiated at t = 0 and theestimate of the variance to be used can be given. Sup- pseudo-measurement is not used. It is shown that thepose that the acceleration component in the velocity di- estimation improves with better angle measurements.rection has a normal distribution with zero mean. Then When the auxiliary pseudo-measurement is also im-with probability 0.95, a 1 g acceleration while flying with plemented in the filter, estimation performance im-VT = 970ft/sec leads to 2 a = 3.12 * 104 [ft 2/see3], proves over the case when only an angle measurementwhere a is the st,-.ndard deviation, which results in a is used. This is shown in Fig. 5 where again the targetvariance V2 = 2A4 * 10" [fj2/sec3 ]2 . starts its acceleration maneuver at the beginning of the

-Unless otherwise stated the filter is initialized at engagement(R = RM). At first, the filter with the fic-launch with the true relative position and relative veloc- titious measurement seems to work a little worse thanity component values assumed obtained from the launch the filter with angle-only measurement. Then, the fic-aircraft. Hence, the initial values for the diagonal ele- titious measurement promptly works as if it suppressedments of the covariance matrix associated with position or delayed the filter divergence. Note that the effect ofand velocity, P1 1, P22, P33 , and P44 are set to ten and it two values of pseudo-noise variance are shown.ensures positive definiteness. On the other hand, little The role of the fictious measurement is more observ-knowledge about target acceleration is assumed to be able for engagement 2 where the target maneuver be-provided at the beginning. Therefore, the initial values gins in the middle of the engagement(Rf = 4000ft).for the target acceleration and expanded states are zero. As plotted in Fig. 6, the filter equipped with only theInitial values of the covariance matrix associated with angle measurement diverges as soon as the target ma-target acceleration is calculated by resorting to the def- neuver occurs. On the other hand, when the filter is aug-inition of the target acceleration at t = 0 given in (2). mented with the fictitious measurement, it works veryThose covariances are produced in the Appendix B. The effectively. The divergence of position and velocity is no-target is expected to execute a maximum acceleration ticeably suppressed, and the acceleration estimate tta.dturn in its evasive motion, and the missile has no knowl- to return to its actual value from an instantaneous largeedge about the direction of target rotation. Note that acceleration error. With the accuracy of the angle mea-

surement increased, the target acceleration estimate af- ment of a new stochastic target acceleration model forter the maneuver onset improves faster than the filter the homing missile problem. In addition, this character-that uses poor angle measurements. This is shown in istic is also implemented in the form of an augLne:itedFig. 7. pseudo-measurement. A guidance l,-%% that minimizes

Performance of the current target models is also corn- a quadratic performance index subject to the stochas-pared with the Gauss-Markov target model. The two tic engagement dynamics is determined in closed fornimodels assume the same magnitude of target accelera- where the gains are an explicit function of thu esti ictltion. Note that in a Gauss-Markov model[3][4], A, the target maneuver rate and time to go. Preliminary resultstarget maneuver time constant and IV, the strength of for the two-dimensional case indicates that the circularthe dynamic driving noise in the model, are two param- target model is able to produce a reliable estimate in theeters but are varied relative to one another and they arc homing missile engagement. When it is augmented byessestially tuning parameters. Htowever, the tuning pa- the fictitious measurement, ti e modified gain extendedramt-ter is E in the new target model. Along with the Kalman filter using the proposed target model resultskinematic constraint incorporated as a pseudomeasure- in the significant enhancement of target state estima-ment, the modified gain extended Kalman filter is built tion. The kinematic constraint also leads to the signifi-to estimate the target states, and the guidance law[4] cant improvement in miss distance performarce for theis based on the Gauss-Markov target model. Figs. 8- Gauss-Markov target model. Comparisons of the cur-9 show a measure of how well the filter estimate the rent target models over the Gauss-Markov target modeltarget states. It is noted that the circular target model show that a significant improvement is gained in targetestimates the target state better than the Gauss-Markov state estimation and miss distance.model. It was also observed during the simulation tha,the estimation performance of the Gauss-Markov target IRA.erencesmodel has been improved with the pseudomeasurment, 1. Chang, C.B. and Tabsczynsky, . A.,"Application ofand this is reflected in the miss-distance caluclations to " Estimation fbe presented. This is because the kinematic constraintincreases the fidelity of the Gauss-Markov target model. Auomat. Conr., Vol. AC-29, 1984, pp. 98-109

Miss distances have been calculated on the basis of 2. Lin, C. F. and Shafroth, M. W., "A ComparativeEvaluation of some Maneuvering Target Tracking

50 runs of Monte Carlo simulations with an approxi- Algorithms," Proceedings of AIAA Guidance andmate error '0.02 ft due to subdiscretization near the Control Conference, 1983final time. In Table 1, the actual states are fed to the 3. Vergez, P. L. and Liefer, R. K., "Target Accel-guidance law in the Case I, and the estimated states and eration Modeling for Tactical Missile Guidance,"maneuver rate estimate are fed to the guidance law in AIAA J. of Guidance and Control, Vol.7, No.3,the Case II and III. The estimates are obtained from 1984, pp. 315-321angle-only measurements in the Case II, and from bothi- m4. Hull, D. G., Kite, P. C. and Speyer, J. L., "New Tar-angle and pseudo-measurement in the Case III. Missdistance performance is tested as more noise is intro- i dngs of AIAA Guidance and Contro Confernce,duced into the measurement and then into the dynam- 1983ics. Table 1 indicates that much of improvement comes .

rom the circular target model with additional improve- 5. Berg, R. F., "Estimation and Prediction for Maneu-ments a-hieved from the kinematic constraint. In addi vering Target Trajectores, IEEE Trans. Aukoma,..ion, miss distance has been improved by using the an- Conir., Vol. AC-28, 1983, pp. 294-304g. as6. Song, T. L., Ahn, J. Y., and Park, C., "Suboptimal,'.- and pseudo-measurement, especially as the processnoise power spectral density e in the state dependent Filter Design with Pseudomeasurements for Targetno;se term decreases. Calculation of miss distance with Tracking," IEEE Trans. Ae-rospace and Electronics,,he Gauss-Markov target model also indicates that sig- Vol. 24, 1988, pp.28-39nificant improvement is obtained in the Guass-Markov 7. Tahk, M. 3. and Speyer, J. L., "Target Trackingmodel using the kinematic constraint. For the particular Problems subject to Kinematic Constraints," Pro-scenario chosen here. circular target model augmented ceedings of the 27th IEEE Conf. on Decision andby pseudo-measurement out performs the Gauss-Markov Control, Dec. 1988tarcet model. 8. Kim, K. D.. Speyer, J. L. and Tahk, M., "Target

Maneuver Models for Tracking Estimators," Pro-ceedings of the IEEE International Conference on

7. Conclusions Control and Applications. April, 19599. Gustafson. D. E. and Speyer, 3. L.."Linear Mini-

Ti.e orthogonaiity between the target acceleration and mum Variance Filters Applied to Carrier Tracking."lociy vectors is a typical characteristic of the target IEEE Trans. Autormiat. Contr., Vol. .. C-21, 197C.

Z an, ar--to-a;, missile and it is utilized in the develop- pp.65-73

8

10. Speyer, J. L. and Gustafson, D. E.,"An Approxi- where A ,i = 1...,6 is a Lagrange multiplier. Themation Method for Estimation in Linear Systems Euler-Lagrange equations for A, arewith Parameter Uncertainty,"IEEE 7ans. Au-tonat. Contr., Vol. AC-20, 1975, pp. 354-359 i = 9, A2 = 0, - = A , -A 4 = A2

11. Song, T. L. and Speyer, J. L., " A Stochastic Analy- where the optimal control satisfies the optimality condi-sis of a Modified Gain Extended Kalman Filter with tionApplications to Estimation with Bearing only Mea- A3 A4surements," IEEE Trans. Automat. Contr., Vol. aM c = - -, CAC-30, 1985, pp. 94 0 -94 9 I-inally, the Euler-Lagrange equations with the natural

12. Jazwinski, A. 1H., "Stochastic Process and Filtering boundary conditions yieldTheory," Academic Press, 1970

13. Wonham, W. M., "Random Differential Equations Al = Xf, A2 = yf, A3 = XfTyo, A4 = Yf T goIn Control Theory," Probabilistic Methods in Ap- which gives the controlj ,zed Mathematics, Vol. 2, Academic Press, 1970

14. Bryson, A. E. and tlo, Y. C.,"Applied Optimal Con- aM = x(t 1 )T9,/c, aMf, = y(tf)T,0 /ctro! Theory," John Wiley L Sons, 1975

Appendix A where T., is the time-to-go of missile to intercept thetarget and c is the guidance law design parameter. In

Linear quadratic guidance law for order to get the guidance law in terms of the currentdeterministic circular target -nodel states, the underlining dynamics is integrated backward

In the following, the optimal deterministic guidance from tl to t. Succesive integrations of state differentiallaw for linear quadratic problem is sought for the current equations yieldcircular target model filter. The det-rmi ,istic optimalsolution can be obtained by solving the Riccati equationwithout tile A term via transition matrix approach, but aT. cos7T'aro(t) + sinT.e*"a;,(t)the use of Euler-Lagrange equation seems simpler for a, = - + cos..T,efr4ar,(t,)

this case. u = !T",xto) + u(t,)

The problem is to minimize the performance index 2c1 e+ ="-7 Jr2 l1a -+ 1 2

xj + 3 [a2 "- .- (1 co w2T, ']t)+ - 3inl'T,--)JT,(t )

l O=4Jd + V(f)

subject to the following linear dynamic system T = T,.t)+ [w(1 - co~r,,.er,.) + iinT,.eTT ]ar. (if)

1 '3.-

S+ -- o cswTe wsi=-wT,.e

U = aT - a)f.T, " )

V = 7; - -am, 6ccozT,.i )

t3 1 [r, (.W2)+ -~i----('-

aT, = -" aT, -waT, 2we "

2 +'aT,

= - aT, +waT + 4T 0 e) "" "

+ W2 'W[ T,. +-.¢sr. r)

This linear system of dynamics stems from taking Ito "w 1 W) + 2)+w (1 Cos:Te)

derivative of the corresponding nonlinear stochastic tar- + +W sin,T T)(get model (6). The w is the angular rate of target maneu-ver which is handled as a known constant in the deriva- Y = -" -M)Y(t,) - T,ov(t,)

tions. In the actual mechanization of guidance command I -

the value of i' constructed from the estimated states are , -, (-- + .T "u se d . W2-f

The variational Hamiltonian and the augmented end- (M.+w) "0point function are given by I r,. E 2 -

+ T 2Q - cos..79.e.021 2 7-7H - ca M + ca ,1 , + A 1 u "+ A2 v + A3 (ar - aM ) + W E. T e 7,

E)0 0 +Ti nT,,'f.ja°(l

+ A4 (aT, - aM,) + e(-7 -- CAT,) + A6(-7 + waT,)

1 (Z 2 2 The final states being expressed in terms of the current

2G= . +y) states via 6x6 mAtrix inversionthe optimal guidance law

9

is obtained as equation (51). As expected from dynamiccoupling in the target acceleration moael, guidance com-niands in each channel are the function of acceleration,omponents in both z and y directions.

Appendix B

State and Error varia.ce associated with target acceleration

Since the initial values for the state estimates associ-ated with target acceleration are set to zero, the stateand error variances are computed with the aid of ex-pected values of trigonometric functions such as Table 1. Comparison of miss distance

E[cos2 0] = +jcos 2 Oj1(O)dO with p(O) = 1 e ('W ,)

By using standard manipulation, the expected value of Statistic Case Range, (ft) Range 1 (ft) Miss Distance(ft)

the cos 2 0 is A = 0.1 1 6000 6000 0.57V1 = V 6000 4000 0.53

1r[1 0] = 101 I 6000 6000 10.13

1cos] + cos2 6000 4000 73.OSI F 6000 6000 9.42

and in the same manner _ 6000 4000 5.0sII 6000 6000 8.14

-[- V = 1 6000 4000 60.49E[sin2-0] = 1 cos296 2e1], v=I11 6000 5000 7.246000 4000 2.30

1 sin 2=e- 20

. E 0.01 1 6000 6000 0.35

E[cos 0 sin 0] = s2 = x 10-1 6000 4000 0.32

V2 = 106 11 6000 6000 1.30This yeilds the initial conditions for the state and error 6000 4000 0.74variances associated with target acceleration as follows. 6000 6000 0.82

6000 4000 0.54e = 0.001 Il 6000 6000 2.65II = V 600V 4000 1.71

Pss(0) = Xss(0) = . PM(0) = Xs.(0) = a- *f2, V = 106 11 6000 6000 2.13... 6000 4000 1.36

P1 7(0) =A .(0) Ta_ 2*-1, P.s(0)=Xs,(0)=,.r. * f2

, e= 0.1 II 6000 6000 4.59Pso(O) = X 59(0) = a]'._ - fl, P510(0) = X510(0) -4 a 2 sf2, 1 = V 0O 6000 4000 1.97P.(0) = XF(0) = a.. * f3, P,(0)= X(0) = a-4., * f2, V - 1TI 6000 6000 4.29

P10)(0) 0 X(0) = a2 f, * f3, Pr7(0) = X7(0) = a2 * f2, 6000 4000 1.82P 610(0) = X,, 0(0) = 7 '22 .:; " 3, P,,(0) = X,'(0) = a2" ,¢" sf1, e=011 00 60 .PT(0) = X8(0) 4 - 2 sf2, Pm(0) = X0.(0) - ,' fi, = 01 6000 6000 4.97

P720(0) = X3 0(0) =- r_2 *f2, P71(0) = X7(0) = a-. *f3, 000 =

_ 600 4000 2.07P7-0(0) 0) =-_ , V2 = 106 III 6000 j 6000 4.86PV(0) = X(0) -- 4,,3 * f2, P810 (0) X,o10(0) = 4- "

* f3, 6000_ 4000 2.01P.s(0) X.(0) a... .2' . f1, p010 (0) = Xc10(0) = 4.a * f2, = 0.001 11 6000 6000 2.65p2 010(0) )XICIO(O) =02 1:4 sf3 ________ ______________- = V0 60G0 4000 1.71

(1in202~V2 (1 -a 26) . 10, 111 6000 6000 2.34where fl = 2 f2 = --2, f3 = 2 6000 4000 1 1.49

10

MisiieMeaureent 150 POSMTON ERROR 400 TARGET ACC ERROR

Dvi~o 2300

100 .*).V000

LS 0 6 00

_________________________________TliAE(SEC) TUA(SEC

150 VELOCITY ERROR 0.4 TGT OMEGA ESTIMATE

Figure 1: Block diagram of horning missile guidance 10023

~'50 0

Y VT 0 .0.1

a a 0 1 2 3 0 1 2 3Mt Tn TINESEC) TiE(SEC)

aA -O Figure 4: Circular target model with different

-06 TARGLET Vl's(Engagement 1)

MISSILE

x

Figure 2: Inertial reference frame for missile and target

MISS ILE-TARGET TRAJECTORY 10 POSITON ERROR 400 TARGET .ACC ERRO .R

3000 Ri=6000ft, RmX)O0ft 75- e=6 2 930

!C2000 cc 50 0VU-10 8 -200x NSSI1LE 0 oV2-1()6

1000 25 100

0 0TARGETf 0 1 2 3 0 1 2 3

00 2000 40D 6000 900TIMXEPC TIME(SEC)X AIS 00 VELOCrTY ERROR TOT OMEGA ESTIMATE

ENGAGEMENT 0.

MISSILE-TARGET TRAJECTORY g5g.v 80.2

1 0.1

2000 0IL o0

- ' 0 -.0.1100-0 1 2 3 0 1 2 3

100T~dSECQ TmeSmC

K TARGET0e 0 00 400Figure 5: Circular target model with pseudo-

X AXIS measurement (Engagement 1)ENGAGEMENT 2

Figure 3: Typical missile target trajectories

,o sPOSON ERROR 400 TARGET ACC ERROR 500 POSITION ERROR 400 TARGET . ACCERROR

100 - 0=10 -2300 400 VVOV.oV2.10

6 300

6400 1V2 300OV2O 0

200 2009=10

~400. 0V2.10(6 'El /1 1 0

2 00 - *)-1200 - 100 K1

0 - -- 0 00 1 2 3 4 0 1 2 3 4 0 0 1 2 3

T2h4ESEC) TVAE(SEQ "navEs.SE TD-(SEC)

1000 VELOCrIY ERROR 04 TGT OMEGA ESTIMATE VELOCr ERROR 0TOT OMEGA ESTIMATE

Soo - 0.3 -1so 0.3

200 0 s0 ,oo--

0 1 0 -0.1 0 -0.10 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4

TrIM(SEC) TIME(SEQ TD SEQ T-M(SEc)

Figure 6: Circular target model with pseudo- Figure 8: Circular target model with pseudo-measurement(Engagement 2) measurement(Engagement 1)

100 POSMON ERROR 400 TARGET ACC ERROR 100 .POSMIOIOR 400 TARGET ACC ERROR

so =i2 300'o v.oi

6o .ovi=vo .o r 60O~6 t6 ~ ~0 V-0 0

20 100

13 00 _ (-,

0 1 2 3 o 1 2 3 4 1 2 3 4 0 1 2 3 4

2oo VELOCITY ERROR .0.4 TGT OMEGA ESTIMATE 2_O VELOC"ITY ERROR 0.4 TrOT OMEGA ESTIMATEo ~.100 -9100

so .i°so- -o. 00 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4

"reTDO SEC- TIME(S=~ "D'EE(SEC IMUtSEQ'

Figue 7: Circ ltarget model with pseudo- Figure 9: Circular target model with pseudo-measremnt( ngaemen 2)measurement(Engagement 2)

12