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Target evasion strategy against a finite set of missile guidance laws

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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, nonlinear 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.
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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(γMλ) + VTcos(γT+λ)) ,Vr(1)
˙
λ=(VMsin(γMλ)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 ,tft.
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 xiRniis 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
MkMaM, kM= cos(γM0λ0),(5a)
a
TkTaT, kT= cos(γT0+λ0),(5b)
where the subscript ”0” denotes the initial value around
which linearization has been performed. It is assumed that
|γM0λ0|< 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=kTaTkMaM
˙
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=k1
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
dMK1dMK2˜
CMdMKM˜
CTdMKT
˜
BMK1˜
BMK2AM+˜
BMKM˜
BMKT
0 0 0 AT
,
BE(tgo) = h0dTdMKuT˜
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=amax
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), amax
Tis a projection of amax
Tin the direction
perpendicular to LOS0, and is given by: amax
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 umax
Tor umax
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 |amax
T.(16)
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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 |amax
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(λλ0),(20)
Differentiating (20) with respect to time, yields
˙y=Vrsin(λλ0) + Vλcos(λλ0).(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 λλ00, 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(γMλ).(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
k1(xk1, u
T),(25)
where xk,[r, λ, xT
M, γM, VM]Tis the state vector at time
tk=kTswith dim(xk) = nx,fj
k1is a function derived by
integrating (23) from tk1to 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
zkRnzis assumed to be acquired at a sampling time Tm
s
and being corrupted by a zero-mean mutually independent
white Gaussian noise sequence vkRnz. 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 tk1to tkis computed using
(25) and the appropriate uj
M, i.e.,
ˆ
xj
k|k1=fj
k1(ˆ
xk1|k1, 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,tktk1is the sampling time used for time
propagation and Fj
xis the associated Jacobian matrix,
Fj
x=fj/∂x
x=ˆ
xj
k1|k1
,(29)
assumed to be fixed for t(tk1, tk]. The prediction error
covariance matrix of the filter matched to the jth regime is
Pj
k|k1=Φj
kPj
k1|k1ΦjT
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)
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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=zkHˆ
xj
k|k1,(32)
ˆ
xj
k|k=ˆ
xj
k|k1+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|k1HT+R.(35)
The updated error covariance matrix is calculated using
Pj
k|k=Pj
k|k1Kj
kHP j
k|k1(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:k1, uj
M)µj
k1/p(zk|z1:k1).(38)
Using (37) and applying the total probability, (38) results in
µj
k= Λj
kµj
k1.Xnr
i=1 Λi
kµi
k1, j = 1, . . . , p, (39)
where Λj
k,p(zk|z1:k1, 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
k0,j
{1,. . ., p}and Pp
i=1 µi
k= 1,k0. 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
kuj
T,(41)
where uj
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=ui
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
Tamax
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
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using (43) with ˆ
xj
k|k. If Zj
MT (0) 0for all j∈ {1, . . . , p},
then uj
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=1i,Bi= 1i,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(γM0λM0)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.
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660
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