Content uploaded by Robert Fonod

Author content

All content in this area was uploaded by Robert Fonod on Oct 18, 2017

Content may be subject to copyright.

Target Evasion Strategy Against a Finite Set of Missile Guidance Laws

Robert Fonod and Tal Shima

Abstract— In this paper, a multiple model adaptive evasion

strategy for a target aircraft from a homing missile employing

a linear guidance law is proposed. We assume arbitrary-order

linear missile and target dynamics, bounded target control, non-

linear kinematics, and the missile employing one of a ﬁnite set of

possible guidance laws. Speciﬁc cases are numerically analyzed

in which the attacking missile uses proportional navigation,

augmented proportional navigation, or optimal guidance.

I. INTRODUCTION

Guidance laws for intercepting a moving target, such as

aircraft in this study, have traditionally been developed for

one-on-one engagements, assuming perfect information and

linearized kinematics [1]. In order to respond to a threat

from a homing missile employing such guidance laws, the

target can perform an evasive maneuver, which can be either

arbitrary or optimally adjusted against the incoming missile.

It is possible to develop an optimal evasion strategy using

optimal control theory tools, however it requires information

on the missile’s future behavior, i.e., its guidance law and pa-

rameters. A case study where such a problem was formulated

as a one-sided optimal control problem against a PN-guided

missile were presented in [2]–[4].

Although there has been substantial work in the literature

on target evasion, most of the research concentrated on PN-

guided missiles, which leaves the strategies against other

guidance laws lacking. In a recent work [5], optimal evasion

strategy for a target aircraft from a homing missile employing

a linear guidance law has been derived. The problem was

analyzed for arbitrary-order adversaries dynamics, bounded

target controls, and assuming perfect information. The un-

derlying assumption in this solution is that the missile’s

guidance strategy is exactly known to the target.

In this paper, an optimal multiple model adaptive evasion

strategy (MMAES) is proposed which greatly relaxes the

assumptions made in [5]. We assume that the missile is

chasing the target using one of a closed set of possible

linear guidance laws and guidance parameters. An EKF-

based static multiple model online identiﬁcation scheme is

used to identify the active missile guidance strategy. For each

such a guidance strategy, an optimal target evasion law is

paired. This law is derived based on a linearized model, but

implemented within the nonlinear setting. The ﬁnal target

This effort was sponsored by U.S. Air Force Ofﬁce of Scientiﬁc Research,

Air Force Material Command, under grant number FA9550-15-1-0429. The

U.S. Government is authorized to reproduce and distribute reprints for

Governmental purpose notwithstanding any copyright notation thereon.

Robert Fonod and Tal Shima are with Department of Aerospace Engi-

neering, Technion - Israel Institute of Technology, Haifa, 3200003, Israel.

Email: {robert.fonod; tal.shima}@technion.ac.il

evasive maneuver command is computed using a multiple

model adaptive control (MMAC) framework.

From implementation point of view, we discuss the ob-

servability issues if only the line of sight (LOS) angle is

measured. Considering different sets of noisy measurements

and two fusing criteria for the MMAC approach, the per-

formance of the proposed MMAES is compared through

extensive simulations to the scenario when the target has

perfect information about the attacking missile.

II. MATHEMATICAL MODE LS

In this section we present the full nonlinear kinematics and

dynamics equations of the missile-target evasion problem,

serving for analysis. Then, linearized equations, used for the

derivation of the optimal MMAES, are presented.

A. Nonlinear Kinematics and Dynamics

In Figure 1, the planar point mass missile-target engage-

ment geometry is shown. The speed, acceleration, and ﬂight-

path angles are denoted by V,a, and γ, respectively; the

range between the missile and target is r, and λis the angle

between the LOS and XIaxis. The acceleration vector of the

missile aMand the target aTare assumed to be perpendicular

to their own velocity vectors VMand VT, respectively.

OI

YI

XI

Missile

(M)

Target

(T)

λ

γM

γT

VT

VM

aT

aM

Y

X

λ0

y

a⊥

M

a⊥

T

λ0

LOS0

r

Fig. 1. Planar missile-target engagement geometry.

The engagement kinematics, expressed in a polar coordi-

nate system (r, λ)attached to the missile, is

˙r=−(VMcos(γM−λ) + VTcos(γT+λ)) ,Vr(1)

˙

λ=−(VMsin(γM−λ)−VTsin(γT+λ)) /r ,Vλ/r (2)

We denote the running time as t. The endgame initiates at

t= 0 with ˙r(t= 0) <0and terminates at t=tf, where

tf= arg

t

inf{r(t) ˙r(t)=0}, t > 0,(3)

allows to deﬁne the time-to-go, tgo, by tgo ,tf−t.

2016 European Control Conference (ECC)

June 29 - July 1, 2016. Aalborg, Denmark

978-1-5090-2591-6 ©2016 EUCA 655

The missile-target separation at t=tfis called the miss

distance, i.e., Miss =r(tf). During the endgame, the missile

and the target are assumed to move at a constant speeds. We

assume arbitrary-order linear dynamics for both the missile

and the target, deﬁned as follows

˙

xi=Aixi+Biui

ai=Cixi+diui

˙γi=ai/Vi

, i ={M , T },(4)

where xi∈Rniis the state vector of an entity’s internal state

variables, aiand uiare entity’s acceleration and acceleration

command, respectively. We also assume that |uT| ≤ amax

T.

B. Linearized Equations of Motion

If during the endgame the missile and target deviations

from the collision triangle are small, then the linearization

around the initial LOS is justiﬁed [1]. In Figure 1, the X-

axis, aligned with the LOS used for linearization, is denoted

as LOS0. The relative displacement between the target and

missile normal to this direction is y. Under linearization

assumption, the missile and target accelerations normal to

LOS0, i.e., a⊥

Mand a⊥

T, can be approximated by

a⊥

M≈kMaM, kM= cos(γM0−λ0),(5a)

a⊥

T≈kTaT, kT= cos(γT0+λ0),(5b)

where the subscript ”0” denotes the initial value around

which linearization has been performed. It is assumed that

|γM0−λ0|< π/2and |γT0+λ0|< π/2.

As soon as the collision triangle is reached and maintained,

the speed Vris constant and the interception time tfcan be

assumed ﬁxed and approximated by

˜

tf≈ −r0/Vr.(6)

Let’s deﬁne the state vector of the linearized problem as

xl=x1x2xT

MxT

TT,y˙yxT

MxT

TT.

Then, the missile-target equations of relative motion normal

to LOS0can be expressed as

˙x1=x2

˙x2=kTaT−kMaM

˙

xM=AMxM+BMuM

˙

xT=ATxT+BTuT

(7)

Using (4) and (5), the above equations can be rewritten as

˙

xl=Axl+Bu⊥

T+Cu⊥

M,(8)

A=

0 1 0 0

0 0 −˜

CM˜

CT

0 0 AM0

0 0 0 AT

,B=

0

dT

0

˜

BT

,C=

0

−dM

˜

BM

0

,

where ˜

BM=k−1

iBi,˜

Ci=kMCi,∀i∈ {M, T },u⊥

Mand

u⊥

Tbeing, respectively, the missile and target acceleration

commands normal to LOS0.

III. OPTIMAL TARGET EVA SIO N STRATE GY

In this section, a nonlinear implementation of the optimal

(linear) target evasion strategy [5] from a missile employing

a linear guidance law is presented.

A. Missile Guidance

In this paper, we consider a large family of linear missile

guidance laws, which all have the same linear form

u⊥

M=K(tgo)xl+KuT(tg o)u⊥

T,(9)

where

K(tgo) = K1K2KMKT.

In this study, we assume that the missile has perfect infor-

mation about its own and target’s states.

B. Optimal Evasion Problem - Linear Setting

Using the equations of motion (EOM) of the linearized

missile-target engagement (8) together with (9), we obtain

the EOM of the one-sided evasion problem

˙

xl=AE(tgo)xl+BE(tg o)u⊥

T,(10)

AE(tgo) =

0 1 0 0

−dMK1−dMK2−˜

CM−dMKM˜

CT−dMKT

˜

BMK1˜

BMK2AM+˜

BMKM˜

BMKT

0 0 0 AT

,

BE(tgo) = h0dT−dMKuT˜

BT

MKuT˜

BT

TiT

.

Theorem 1 ([5]): The optimal target evasion strategy

from a homing missile employing a linear guidance law of

the form (9), maximizing the following cost function

JE=y2(tf)/2(11)

subject to the EOM of (10) and under the constraint that the

target’s control |uT| ≤ amax

Tis bounded, is given by

u⊥∗

T=a⊥max

Tsign(sMT )sign(ZM T ); ZMT (0) 6= 0,(12)

where sMT is the switching function and ZMT is the well-

known zero-effort-miss (ZEM) distance, i.e.,

sMT =DEΦE(tf, t)BE(tg o),(13)

ZMT =DEΦE(tf, t)xl,(14)

where DE= [1 0 0 0].The transition matrix ΦE(tf, t),

associated with the solution of (10), satisﬁes

˙

ΦE(tf, t) = −ΦE(tf, t)AE(tgo),ΦE(tf, tf) = I.(15)

In (12), a⊥max

Tis a projection of amax

Tin the direction

perpendicular to LOS0, and is given by: a⊥max

T=kTamax

T.

If the engagement is initialized in the singular region, i.e.,

ZMT (0) = 0, or if ZM T (t) = 0 , then it was suggested in

[5] that for ZMT (t) = 0 the optimal evasion strategy u⊥∗

T

should be chosen as either u⊥max

Tor −u⊥max

T. Moreover,

it was also shown that if u⊥∗

Tis employed within the linear

setting (10), the optimal ZEM dynamics ˙

Z∗

MT is given by

˙

Z∗

MT =sign(ZM T )|sMT |a⊥max

T.(16)

656

Consequently, |ZMT |is a monotonically increasing function

of time, satisfying ∀t∈[0, tf],

sign(ZMT (t)) = sign(ZMT (tf)), ZM T (0) 6= 0.(17)

Deﬁning m=Rtf

0|sMT |a⊥max

Tdt, we obtain the expected

miss distance (under the assumption of linearity)

Missexp =|y(tf)|=|ZMT (tf)|=|ZM T (0)|+m. (18)

Note that sMT converges asymptotically to zero as tgo → ∞.

C. Nonlinear Implementation

To implement the optimal evasion strategy within the

nonlinear setting, u⊥∗

Tmust be projected in the direction

normal to the target’s velocity vector, denoted as u∗

T. As-

suming small deviations from the collision triangle yields to

kT/cos(γT+λ)≈1, hence u∗

Tcan be approximated by

u∗

T=u⊥∗

T

cos(γT+λ)≈amax

Tsign(sMT )sign(ZM T )(19)

for ZMT (0) 6= 0. Once a collision triangle is reached and

maintained, the speed Vris constant and the approximation

of the interception time ˜

tf, given by (6), can be assumed

ﬁxed throughout the engagement. Thus, the relevant compo-

nents of ΦE(˜

tf, t)can be precomputed off-line.

The displacement ynormal to LOS0can be expressed at

any time by

y=rsin(λ−λ0),(20)

Differentiating (20) with respect to time, yields

˙y=Vrsin(λ−λ0) + Vλcos(λ−λ0).(21)

Using the above expressions for the computation of ZMT re-

places the dependency on yand ˙yby the kinematics variables

Vr,Vλ,r, and λ. Moreover, assuming small deviations from

the collision triangle, thus λ−λ0≈0, the displacement y

can be assumed zero and ˙ycan be approximated as ˙y=Vλ.

IV. EST I MATO R IN TH E LOOP

The assumption of Theorem 1 is that the missile’s guid-

ance is known to the target. In this section, this assumption is

greatly relaxed by employing a multiple model-based online

identiﬁcation scheme together with the MMAC approach.

A. Static Multiple Model Estimation

In the multiple model estimation approach [6], the system

operates in one of a ﬁnite number of models. The operating

model is often called as “mode” or “regime” of the system.

Let’s assume that the guidance law of the missile uM, being

ﬁxed throughout the engagement, is one of ppossible ones

uM∈ {u1

M, . . . , up

M}.(22)

Each uj

Mis characterized by a set of ﬁve parameters, i.e.,

{Kj

1, Kj

2,Kj

M,Kj

T, Kj

uT}. These parameters exclusively de-

ﬁne the jth regime and might be functions of tgo. Each such

regime will generate different missile acceleration commands

uMand will also result in different missile trajectories.

As the engagement model is nonlinear, a mode-matched

extended Kalman ﬁlter (EKF) is used in the next [7]. The

main idea is to design and run in parallel a bank of pEKFs,

each matching a different possible regime j.

1) System Model for Estimation: It is assumed that target-

related parameters xT,γT, and VTare known to a very

high accuracy (via some navigation system). Then, based

on the constant missile speed assumption, the jth regime’s

dynamics is governed by the following set of equations

˙r=Vr

˙

λ=Vλ/r

˙

xM=AMxM+BMuj

M

˙γM= (CMxM+dMuj

M)/VM

˙

VM= 0

,(23)

where Vrand Vλare given by (1) and (2), respectively, and

uj

Mis the missile’s acceleration command obeying the jth

regime and being deﬁned as

uj

M=Kj

1y+Kj

2˙y+Kj

MxM+Kj

TxT+Kj

uTu⊥

T

cos(γM−λ).(24)

The variable yand ˙yis given by (20) and (21), respectively.

The discrete-time version of (23) for uj

Mcan be rewritten as

xk=fj

k−1(xk−1, u⊥

T),(25)

where xk,[r, λ, xT

M, γM, VM]Tis the state vector at time

tk=kTswith dim(xk) = nx,fj

k−1is a function derived by

integrating (23) from tk−1to tk, and jis the speciﬁc regime.

2) Measurement Model: The target is assumed to be

equipped with an electro-optic seeker and/or a radar. Thus,

one may acquire: a)both rand λmeasurements, or b)only

λ, i.e., bearing-only measurements. The measurement vector

zk∈Rnzis assumed to be acquired at a sampling time Tm

s

and being corrupted by a zero-mean mutually independent

white Gaussian noise sequence vk∈Rnz. The measurement

model, when all possible measurements are available, is

zk=Hxk+vk=rkλkT+vk,(26)

where vk∼ N (0,R)with R=diag(σ2

r, σ2

λ), and H∈

Rnz×nxis an appropriate measurement matrix.

3) Mode-Matched Filtering: For the given regime j, the

time propagation step from tk−1to tkis computed using

(25) and the appropriate uj

M, i.e.,

ˆ

xj

k|k−1=fj

k−1(ˆ

xk−1|k−1, u⊥

T).(27)

The jth mode state transition matrix Φj

kassociated with the

system dynamics (23) is

Φj

k= exp(Fj

xTs),(28)

where Ts,tk−tk−1is the sampling time used for time

propagation and Fj

xis the associated Jacobian matrix,

Fj

x=∂fj/∂x

x=ˆ

xj

k−1|k−1

,(29)

assumed to be ﬁxed for t∈(tk−1, tk]. The prediction error

covariance matrix of the ﬁlter matched to the jth regime is

Pj

k|k−1=Φj

kPj

k−1|k−1ΦjT

k+Qj

k,(30)

where Qj

kis an artiﬁcial process noise covariance matrix

Qj

k=ZTs

0

Φj

k(η)ΨjΦj

k(η)Tdη, (31)

657

used as a tuning matrix [6]. In (31), Ψjis a matrix which

has only one nonzero element, Ψj(4,4) = ψj, and is used

as the tuning parameter of the jth mode-matched EKF.

Using a new measurement zk, the state estimate ˆ

xj

k|kand

covariance Pj

k|kof the ﬁlter matched to the jth mode are

updated using the standard EKF equations

νj

k=zk−Hˆ

xj

k|k−1,(32)

ˆ

xj

k|k=ˆ

xj

k|k−1+Kj

kνj

k,(33)

where νj

kis the innovation of the jth mode-matched ﬁlter

and Kj

kis the Kalman gain, computed as

Kj

k=Pj

k|kHT(Sj

k)−1,(34)

with Sj

kbeing the covariance of the jth innovation process

Sj

k=HP j

k|k−1HT+R.(35)

The updated error covariance matrix is calculated using

Pj

k|k=Pj

k|k−1−Kj

kHP j

k|k−1(36)

If the measurement zkis not available (e.g., due to sensor

error, blind range of the sensors, etc.), equations (32)-(36) are

omitted and only time propagation (27)-(30) is performed.

4) Mode Probability Update: The prior probability that

uj

Mis correct (the system is in regime j) is

Prob{uj

M|Z0}=µj

0, j = 1, . . . , p (37)

where Z0is some a priori known information and

Pp

j=1 µj

0= 1, since the correct law is uM∈ {u1

M, . . . , up

M}.

Using Bayes’ rule, given the measurement data z1:k,

{zi;i= 1, . . . , k}up to time k, the posterior probability µj

k

of the mode jbeing correct is given by [6]

µj

k=p(zk|z1:k−1, uj

M)µj

k−1/p(zk|z1:k−1).(38)

Using (37) and applying the total probability, (38) results in

µj

k= Λj

kµj

k−1.Xnr

i=1 Λi

kµi

k−1, j = 1, . . . , p, (39)

where Λj

k,p(zk|z1:k−1, uj

M)is the jth regime-conditioned

likelihood function. given by

Λj

k=p(νj

k) = N(νj

k;0,Sj

k),(40)

where νj

kand Sk

kare the innovation and its covariance from

the jth mode-matched ﬁlter. It is obvious that µj

k≥0,∀j∈

{1,. . ., p}and Pp

i=1 µi

k= 1,∀k≥0. Note that in a nonlinear

and/or non-Gaussian setting, Gaussian likelihood functions

are used, although they are clearly approximations [6].

B. MMAC Conﬁguration for Target Evasion

In the MMAC approach [8], the estimation of each el-

ementary ﬁlter is fed into a “controller” (optimal evasion

strategy in our case) and the total control command can be

determined by one of the following approaches:

a)MMSE - minimum mean square error, where the target

control command uTis a weighted average of controls from

each ﬁlter-matched controller in the bank, i.e.,

uMM S E

T=Pp

j=1µj

ku∗j

T,(41)

where u∗j

T=amax

Tsign(sj

MT )sign(Zj

MT ),sj

MT and Zj

MT

represent, respectively, the switching function (13) and the

ZEM distance (14), both computed using the jth regime

mode-matched estimate ˆ

xj

k|kat time step k.

b)MAP - maximum a posteriori probability. In the MAP

sense, uTis determined as the control associated with the

maximum a posteriori probability, i.e.,

uMAP

T=u∗i

T, i = argmax

j

(µj

k).(42)

Using this scheme, as the identiﬁcation process of the

missile’s unknown parameters improves, the target’s evasion

will adapt to ﬁt the identiﬁed missiles guidance strategy.

C. Implementation Issues

If collision course conditions hold, the range cannot be

estimated from bearing-only measurements and thus the

proposed MMAES cannot be properly implemented as an

accurate estimate of tgo is required. For practical implemen-

tation, the time-to-go tgo is commonly approximated as

˜

tgo ≈ −r/Vr, Vr<0.(43)

A poor range estimation might result in inappropriate timing

of the optimal switches dictated by sMT and thus in a poor

evasion performance. One way to improve the accuracy of

˜

tgo is to improve range observability. Maneuvering away

from the collision triangle, i.e., forcing the collision triangle

to rotate, can improve the performance of the estimation pro-

cess because, by altering the LOS, the bearing measurement

will return some insights on the relative range.

It is obvious that ZMT (t)might cause chattering of

u∗

T. This chattering might happen when ZMT (0) '0and

uncertain states are used to compute ZMT . Such unwanted

chattering of u∗

Tmight lead to a non-rotating collision geom-

etry. A workaround is to apply a dead-zone-like function on

the sign of ZMT (t). By this, if |ZM T (t)|< , i.e., the ZEM

distance is smaller than some prescribed value , the target

will not change the direction of its maneuver command,

unless the sign of sMT (tg o)is changed or |ZM T (t)| ≥ .

Note that in the liner setting, |ZMT (t)|is a monotonically

increasing function of time t, see (17). By letting → ∞, we

actually assume that the monotonically increasing property

also holds for the nonlinear engagement. In that case, the

optimal nonlinear evasion strategy (19) reduces to

u∗

T≈amax

Tsign(sMT )sign(ZM T (0)), ZMT (0) 6= 0.(44)

If ZMT (0) = 0,u∗

T(0) can be chosen arbitrarily as either

umax

Tor −umax

T, and sign(ZMT (0)) ,sign(u∗

T(0)) should

be assumed constant in (44) throughout the engagement.

Let us now consider the above discussions within the

MMAC setting. Each “controller” in the bank has its own

switching function sj

MT entirely deﬁned by Φj

E. Each sj

MT

is evaluated using the tgo approximation, calculated for tk

658

using (43) with ˆ

xj

k|k. If Zj

MT (0) ≈0for all j∈ {1, . . . , p},

then u∗j

Tshould be initialized with the same value (±umax

T)

for all j. By this, uT, computed using MMSE or MAP, will

not be affected by the initial transitions (convergence) of the

probabilities µj

k, until the ﬁrst switch occurs in any sj

MT .

V. P E RF ORM ANC E ANA LYS IS

The performance of the proposed MMAES is evaluated

here using numerical simulations and nonlinear kinematics.

A. Simulation Environment and Scenario

All engagements are initiated at a horizontal separation of

5000 m between the missile and the target, thus r0= 5000 m

and λ0= 0 rad. Both missile and target have constant speed.

The target’s speed is VT= 300 m/s and the missile’s speed is

VM= 600 m/s. For the analysis we assume that the missile

and the target have a ﬁrst-order strictly proper dynamics with

time constants τM= 0.2s and τT= 0.5s. In this case,

matrices (4) degenerate to Ai=−1/τi,Bi= 1/τi,Ci= 1

and di= 0,∀i∈ {M, T }. The target’s maneuver capability

is limited to 10 g (i.e., umax

T= 10 g). No saturation is applied

on the missile acceleration command uM. Given the target’s

initial ﬂight path angle γT0, the missile’s initial ﬂight path

angle γM0is selected such that the initial collision conditions

hold, i.e., VMsin(γM0−λM0)−VTsin(γT0+λT0)=0.

We consider that the missile is guided to the target using

perfect information and one of the following “classical”

guidance laws: PN [9], APN [10], and OGL [11] with

navigation gain N0

i∈ {3,3.5,4,4.5,5}for i={PN,APN}

and weight α∈{0.002,0.004,0.006,0.008,0.01}for OGL1.

The MMAC regimes are matched exactly to the above

guidance laws and parameters, requiring in total 15 EKFs

to be run in parallel (i.e., p= 15). The prior probability of

each regime is µj

0= 1/15, j = 1, . . . , p.

The time propagation of the EKFs is performed at a

frequency of 200Hz (i.e., Ts= 1/200 s). Measurements are

acquired at a sampling frequency of 50 Hz (i.e., Tm

s= 1/50

s). The simulated measurement noises are with σr= 10

m and σλ= 1 mrad. The EKF tune parameters ψj, j =

1, . . . , p have been chosen by numerical simulations. The

integral in (31) is computed numerically using one-step

forward integration, whereas the nonlinear EOM (27) are

propagated using the RK4 algorithm. We assume a blind

range of 50 m. If r < 50, no new measurements are acquired.

B. Sample Run

A sample run planar trajectories of a target employing

MMAC with MMSE blending and bearing-only measure-

ments evading from an APN-guided missile with N0

AP N = 4

is presented are Fig. 2. The initial ﬂight path angle of the

target is γT0=π/12 rad. The resulting miss distance in this

sample case is approximately 0.38 m. It can be seen that the

applied evasion strategy forces the collision triangle to rotate

with time, thus allowing to gain some insight on the range.

1The weight αrepresents the ratio between the control effort and the

miss distance in the quadratic cost function used in the OGL derivation.

X (m)

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

-200

0

200

400

600

800

1000

1200

1400

1600

1800

2000

1s

1s

2s

2s

3s

3s

4s

4s

5s

5s

6s

6s

7s

7s

8s

8s

Y (m)

Target

Missile

Miss Distance = 0.37995 (m)

Fig. 2. Sample run planar engagement trajectories.

Figure 3 compares the timing of the optimal target maneu-

ver switches with the switches obtained using the proposed

MMAES. In this ﬁgure, the acceleration aT(red line) is

driven by the target’s acceleration command uT(black

dashed line). The upper frame shows the optimal switches

obtained when the target assumes perfect information, i.e.,

uses true states and is exactly matched to the active missile

guidance strategy. The other four frames below represent the

MMAES implemented with different MMAC/measurement

combinations. It can be seen that while the MMAC with two

measurements (i.e., rand λ) exhibits almost identically to

the perfect information case, the MMAC with bearing-only

measurements (i.e., λ) executes the optimal switches with a

slight time delay of about 0.3 s. This is due to the poorer

range estimation, thus poorer time-to-go approximation.

Fig. 3. Target acceleration proﬁles for different scenarios.

Figures 4-5 present the posterior probabilities of each

guidance law being true and the probability of each guidance

law parameter being correct as a function of time. In the

two measurements case (single measurement case), the APN

guidance law has been identiﬁed as the missile’s guidance

law after approximately 1 s (1.5 s) and the navigation gain

has also been identiﬁed approximately around 2 s (5 s). Note

that the obtained results for MMAC with MAP are very

similar to those given in Fig. 2 and Figs. 4-5, thus ommited.

659

Time (sec) Time (sec)

Fig. 4. Regime probabilities for two measurements.

Time (sec) Time (sec)

Fig. 5. Regime probabilities for bearings-only measurements.

C. Monte Carlo Study

The effect of the “estimator in the loop” on the evasion

accuracy, for four different MMAC/measurement combina-

tions, is evaluated here. The analysis is based on a set of

500 MC runs. The results are compared in terms of miss

distances. The employed missile guidance law and its naviga-

tion gain is one of those presented in Sec. V-A. For each run,

the active missile guidance was selected by using a uniform

random number generator. The studied engagement is sym-

metric with respect to X-axis, thus only positive γT0are con-

sidered being drawn from a uniform distribution on [0, π/6]

rad. Each MC run differs from the other run by different γT0,

by different noise seeds, and by different initial guesses, ˆ

x0|0,

assigned to the estimators. These guesses are sampled from

a Gaussian distribution, i.e., ˆ

x0|0∼ N(x0,P0|0), where x0

is the true relative missile-target initial state and P0|0=

diag{502,(3π/180)2,(2.5g)2,(3π/180)2,602}is the initial

estimation error covariance matrix of the ﬁlters.

Figure 6 shows the obtained miss distances by means of

the cumulative distribution function (CDF). The expected

linear miss, given by (18), is also considered for illustration.

It can be seen that the cases with two measurements achieve

almost identical evasion performances as the case when the

target has perfect information and that The MMSE and MAP

criterion for the MMAC achieve very similar evasion perfor-

mances. It is also evident that the performance deteriorates

when bearing-only measurements are considered.

Fig. 6. Cumulative distribution function of the estimated miss distance.

VI. CONCLUSION

A multiple model adaptive evasion strategy for a target

aircraft from a homing missile employing a linear guidance

strategy has been presented. Considering different sets of

noisy measurements, the evasion performance of the pro-

posed approach has been numerically studied, through Monte

Carlo simulations, for missile employing classical guidance

laws such as PN, APN, and OGL, using nonlinear kinematics

and missile provided perfect information. For the missile

and target having ﬁrst-order dynamics, it was shown that the

degradation in avoidance performance from a homing missile

may not be as serious as it could have been anticipated, even

when the target has limited maneuver capability and carries

sensors that provide bearing-only measurements.

REFERENCES

[1] P. Zarchan, Tactical and Strategic Missile Guidance, ser. Progress in

Astronautics and Aeronautics, Washington, DC, 2007, vol. 219.

[2] I. Forte, A. Steinberg, and J. Shinar, “The effects of non-linear

kinematics in optimal evasion,” Optimal Control Applications and

Methods, vol. 4, no. 2, pp. 139–152, 1983.

[3] D. Borg and P. Julich, “Proportional navigation vs an optimally evad-

ing, constant-speed target in two dimensions,” Journal of Spacecraft

and Rockets, vol. 7, no. 12, pp. 1454–1457, 1970.

[4] J. Shinar and D. Steinberg, “Analysis of optimal evasive maneuvers

based on a linearized two-dimensional kinematic model,” AIAA Jour-

nal of Aircraft, vol. 14, no. 8, pp. 795–802, 1977.

[5] T. Shima, “Optimal cooperative pursuit and evasion strategies against a

homing missile,” Journal of Guidance, Control, and Dynamics, vol. 34,

no. 2, pp. 414–425, 2011.

[6] Y. Bar-Shalom, X. R. Li, and T. Kirubarajan, Estimation with appli-

cations to tracking and navigation: theory algorithms and software.

John Wiley & Sons, 2004.

[7] V. Shaferman and T. Shima, “Cooperative multiple-model adaptive

guidance for an aircraft defending missile,” Journal of guidance,

control, and dynamics, vol. 33, no. 6, pp. 1801–1813, 2010.

[8] D. G. Lainiotis, “Partitioning: A unifying framework for adaptive

systems, ii: Control,” Proceedings of the IEEE, vol. 64, no. 8, pp.

1182–1198, 1976.

[9] L. C. Yuan, “Homing and navigational courses of automatic target

seeking devices,” Journal of Applied Physics, vol. 19, no. 12, pp.

1122–1128, 1948.

[10] V. Garber, “Optimum intercept laws for accelerating targets,” AIAA

Journal, vol. 6, no. 11, pp. 2196–2198, 1968.

[11] R. G. Cottrell, “Optimal intercept guidance for short-range tactical

missiles,” AIAA journal, vol. 9, no. 7, pp. 1414–1415, 1971.

660