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Estimation Enhancement by Imposing a Relative Intercept Angle for Defending Missiles

Authors:

Abstract

A multiple-body interception scenario is considered, where an evading target aircraft launches two defending missiles as a countermeasure against an incoming homing missile. The defenders share their respective line of sight angle measurements, are aware of the target’s evasion strategy, and exploit a cooperate guidance law which can impose a relative intercept angle between them. The ability of the proposed cooperative guidance-estimation scheme to protect the targeted aircraft is analyzed. Especially, the effect of different values of the commanded relative intercept angle on the intertwined guidance-estimation performance is studied.
Estimation Enhancement by Imposing a Relative
Intercept Angle for Defending Missiles
Robert Fonodand Tal Shima
Technion - Israel Institute of Technology, 32000 Haifa, Israel
A multiple-body interception scenario is considered, where an evading target aircraft
launches two defending missiles as a countermeasure against an incoming homing missile.
The defenders share their respective line of sight angle measurements, are aware of the
target’s evasion strategy, and exploit a cooperate guidance law which can impose a relative
intercept angle between them. The ability of the proposed cooperative guidance-estimation
scheme to protect the targeted aircraft is analyzed. Especially, the effect of different
values of the commanded relative intercept angle on the intertwined guidance-estimation
performance is studied.
I. Introduction
A scenario where an attacking missile, homing onto an evading target aircraft, encounters two defending
missiles launched by the aircraft is considered. The incoming missile employs a linear guidance law against
the evading target. From the target’s perspective, possible solutions to deal with such a challenge are to
develop advanced sensors to accurately track the missile, to use more agile defenders, and/or to install a more
lethal warhead for the defenders. These options might be very often too complex, heavy, and expensive. An
alternative is to design more sophisticated guidance-estimation algorithms to improve the guidance system
of inexpensive defenders so that they will be able to achieve the required interception accuracy without
changing existing hardware. Therefore, we assume that the defenders are equipped with sensors which allow
to measure only the line of sight (LOS) angle and that they have limited maneuver capabilities.
In scenarios where multiple vehicles can share their respective LOS angle measurements, the estimation
performance can be improved by exploiting the triangulation method [1–3]. The estimation quality, however,
strongly depends upon the vehicles’ trajectories and hence on the implemented guidance strategy. In [1], two
distinct estimator design methods for cooperative target tracking are presented. It was assumed that the
vehicles (missiles) are guided to the target via non-cooperative guidance laws and that only the estimation
is performed cooperatively. As guidance and estimation are mutually intertwined, neglecting the effect of
the one onto the other and vice versa may have severe consequences. For example, if all vehicles employ the
same one-on-one guidance law (as considered in [1]) and are all launched with the same initial conditions,
then the resulting trajectories coincide (up to some unmodeled disturbances). As a consequence, all sensors
will measure the same quantity (LOS angle), causing the triangulation technique to fail. This, in turn, might
result in poor interception performance.
The work [1] was recently extended in [3] by the same authors, where the above issue was tackled by
introducing the concept of staggered launch of the missiles. The observability issue in a double-LOS relative
navigation setup was analyzed in [4, 5]. It was concluded that if the separation angle of the LOS vectors is
too small, the relative navigation system may become weakly observable or even unobservable. This problem
was addressed for the two missiles case in [2] by modulating the LOS angle through a performance index.
The missile with large initial LOS angle maximizes this index while the other one minimizes it. By this, the
separation angle of both LOS vectors during the engagement is increased and the estimation is improved.
In this paper, an intuitive approach for improving the estimation performance and thus the defending
capabilities of the two defending missiles is to impose a nonzero relative intercept angle constraints for the
Postdoctoral Fellow, Department of Aerospace Engineering, robert.fonod@technion.ac.il.
Associate Professor, Department of Aerospace Engineering, tal.shima@technion.ac.il. Associate Fellow AIAA.
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defenders. This can be easily achieved for each defender separately using one of the available one-on-one
terminal intercept angle guidance laws, see e.g., [6, 7], or, better yet and as done in this paper, one can achieve
the same goal using the optimal cooperative guidance law presented in [8] which can impose a predetermined
relative intercept angle between consecutive vehicles.
As larger intercept angles require more maneuverability and the fact that control saturation goes hand
in hand with degradation in the guidance performance, it is important therefore to study the effect of
estimation on to the guidance problem. In this paper, we rigorously analyze the effect of different values of
the commanded relative intercept angle on the intertwined guidance-estimation performance.
The remainder of this paper is organized as follows. The next section presents the mathematical models
of the target-defenders-missile engagement. The guidance laws are presented in Sec. III, followed by the
derivation of the estimator in Sec. IV. A comprehensive performance analysis is done in Sec. V, followed by
concluding remarks.
II. Multiple-Body Engagement Description
The problem consists of four entities (also referred to as bodies or vehicles): an attacking missile, an
evading target aircraft, and two defending missiles. For brevity, the target aircraft is referred as target, the
attacking missile as missile, and the defending missiles as defenders. The defenders are launched simulta-
neously by the evading target to intercept the incoming threat. The attacking missile is unaware of the
defenders and employs a known linear one-on-one guidance strategy to intercept the evading target.
II.A. Nonlinear Kinematics and Dynamics
We consider a skid-to-turn and roll-stabilized vehicles. The motion of the four bodies is assumed to transpire
in the same plane. In Figure 1 a schematic view of the planar point mass target-defenders-missile engagement
geometry is shown, where XI-OI-YIrepresents a Cartesian inertial reference frame. The missile, defender,
and target related variables are denoted by the subscripts m,d, and t, respectively. The speed, normal
acceleration, and flight-path angle are denoted by V,a, and γ, respectively. The range between the target
and missile and between the i-th defender and missile is denoted as ρtm and ρdim, respectively. The angle
between the target’s line-of-sight (LOS) to the missile and the XIaxis is denoted as λtm, while that between
the i-th defender’s LOS to the missile and the XIaxis is denoted as λdim.
OI
YI
XI
λd1m
γd1
γm
Vm
Vd1
am
ad1
ρd1m
λd2m
γd2
Vd2
ρd2m
γt
Vt
at
(xd1, yd1)(xd2, yd2)
(xt, yt)
λtm
ρtm (xm, ym)
ad2
Figure 1: Planar target-defenders-missile engagement geometry.
The missile and the defenders are considered to be from a similar class of vehicles, with their speeds
higher than of the target, that is, Vi> Vt,i∈ {d1, d2, m}. We assume that the target’s and defenders’ own
inertial state vector
xI
i=hxiyiaiγiiT
, i ∈ {t, d1, d2}(1)
is known to a very high accuracy (e.g., using inertial navigation system and/or GPS sensors), and that the
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target and each defender can transmit its own state vector to both defenders and to the other defender
without any delay, respectively. The target’s speed is assumed to be known is transmitted to the defenders.
Neglecting the gravity, the engagement kinematics, expressed in a polar coordinate system (ρim, λim ), is:
(˙ρim =Vρim,
˙
λim =Vλimim ,i∈ {t, d1, d2},(2)
where the respective relative velocities along and perpendicular to the LOS are
Vρim =Vicos(γiλim)Vmcos(γm+λim ),(3a)
Vλim =Visin(γiλim) + Vmsin(γm+λim ).(3b)
During the endgame, all four vehicles are assumed to move at a constant speed and to perform lateral
maneuvers only. Arbitrary-order linear dynamics is assumed for all four vehicles
˙xa
i=Aixa
i+Biui
ai=Cixa
i+Diui
˙γi=ai/Vi
, i ={t, d1, d2, m},(4)
where xa
iRniis the internal state vector of the i-th vehicle’s dynamics, aiand uiare the i-th entity’s
acceleration and acceleration command, respectively. The term Cixa
iis denoted as ais and represents, if it
exists, the part of the acceleration with dynamics.
We assume that the target’s and defenders’ maneuverabilities are limited to
|ui| ≤ amax
i, i ∈ {t, d1, d2},(5)
where amax
i>0is the maximal acceleration of the i-th vehicle. No saturation is considered for the missile,
thus umax
m=.
II.B. Timeline and Time-to-go
The running time is denoted as t. The endgame initiates at t= 0 with ˙ρim(t= 0) <0,i∈ {t, d1, d2}
and the particular engagement terminates at t=tf
im, where tf
im is the target-missile or defender-missile
interception time, formally defined as
tf
im = arg
t>0
inf{ρim(t)Vρim (t)=0}, i ∈ {t, d1, d2},(6)
and allowing to define the nonnegative time-to-go by
tgo
im =(tf
im t, t tf
im
0, t > tf
im
, i ∈ {t, d1, d2}.(7)
At t=tf
im, the separation ρim (tf
im)is minimal and is referred to as “miss distance” or compactly as “miss”.
We require that the target-missile engagement terminates after that of the defenders-missile, therefore
tf
im < tf
tm, i ∈ {d1, d2}.(8)
II.C. Physical Measurement Model
As the bearing measurement is a predominant one in missile guidance applications, it is assumed that each
defender is only equipped with an electro-optical seeker that measures the LOS angle λim, i ∈ {d1, d2}.
Both measurements are assumed to be acquired at the same discrete-time t=tk,k·T, where Tis the
measurement sampling period. The measurements are corrupted by a zero-mean white Gaussian noise with
standard deviation σλi, i ∈ {d1, d2}.
The physical measurement equation of the i-th defender is
zi;k=hi(xk) + vi;k=λim;k+vi;k, i ∈ {d1, d2},(9)
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where the state vector xk, used for estimation, will be defined later, and
vi;k N (0, σ2
λi), i ∈ {d1, d2}.
In Eq. (9) and in the rest of the paper, the discrete time step is indicated by a subscript k, separated by a
semicolon. The measurement noise sequences vi;k, i ∈ {d1, d2}are assumed to be mutually independent. It
is assumed that the first defender can transmit its measurements to the second defender without any delay
and vice versa. The advantage and utilization of such measurements sharing will be discussed in Sec. IV.B.
III. Guidance Laws
III.A. Perfect Information Guidance Law of the Missile
For simplicity, we will limit our derivation to three representative missile guidance laws of PN [9], APN [10],
and OGL [11]. The derivation can be extended to other missile guidance laws using the same formulation
and similar derivation steps.
The guidance law of PN, APN, and OGL is widely known in the following form
um=N0
j
Zj
(tgo
tm)2cos(γm+λtm ), j ∈ {PN,APN,OGL},(10)
where N0is the effective navigation gain, Zis the missile’s zero-effort-miss (ZEM) distance, and tgo
tm is given
by
tgo
tm =ρtm/Vρtm , Vρtm <0.(11)
The expression for the ZEM distance is different for each guidance law, as it is dependent on the model used
and assumptions made regarding the future target maneuvers.
Under the assumptions of ideal missile dynamics and no target maneuver (i.e., ut= 0), the obtained
missile guidance law guaranteeing zero miss distance is PN with
ZP N =Vρtm ˙
λtm(tgo
tm)2,(12)
where ˙
λtm and Vρtm are given by Eqs. (2) and (3a), respectively. If N0
P N = 3, this guidance law minimizes
the missile’s control effort. Extending the results to the case in which the target is assumed to perform a
constant maneuver (i.e., ut=const.), APN was obtained with
ZAP N =ZP N +(tgo
tm)2
2atcos(γtλtm).(13)
Additionally assuming that the missile’s closed-loop acceleration dynamics can be approximated by a first-
order strictly proper transfer function with a time constant τm, OGL was obtained with
ZOGL =ZAP N τ2
mψ(θtm)ams cos(γm+λtm ),(14)
where ψ(θtm)is an exponential-like function of the normalized target-missile time-to-go θtm given by
ψ(θtm) = exp(θtm ) + θtm 1, θtm =tgo
tmm.(15)
The navigation gains of PN and APN are constant, whereas that of OGL is time-varying and given by
N0
OGL(θtm ) = 6θ2
tmψ(θtm )
3+6θtm 6θ2
tm + 2θ3
tm 3e2θtm 12θtmeθtm + 6α/τ 3
m
,(16)
where αrepresents the ratio between the weights on the control effort and the miss distance in the quadratic
cost function used in the OGL formulation.
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III.B. Evasive Strategy of the Target
The optimal one-on-one perfect information evasion strategy against a homing missile employing a linear
guidance law has a bang-bang structure for bounded target acceleration [12]. The practical implementation
of such evasion strategy requires the estimate of the relative target-missile state and perfect knowledge about
the active missile guidance law and guidance parameters [13].
Note that the optimal one-on-one evasion strategy is not necessary the “best” from the defenders point of
view [14, 15]. The target’s guidance directly influences the homing missile’s trajectory, and hence, indirectly,
also the trajectories of the defenders. Thus, the target may lure the missile such that the defenders can
intercept it even when the defenders’ maneuver capability is much smaller compared to that of the missile
or the target can help to improve the estimation performance of the defenders. In this paper, we assume
that the evasion strategy of the target is arbitrary and that it is perfectly known to both defenders.
III.C. Cooperative Guidance Law of the Defenders
Here we present the guidance strategy for the defender team. The cornerstone of this strategy is based in
the recently developed cooperative optimal guidance law for imposing a relative intercept angle between
consecutive vehicles, see [8]. This law ensures cooperation between the two defenders (explicit cooperation)
and the protected target aircraft (implicit cooperation).
Explicit Cooperation to Impose a Relative Intercept Angle
To help the estimation process, we wish to impose a relative intercept angle between the defenders. Denote
the angle between the i-th defender and the missile as γdim=γdi+γm. The difference between the intercept
angles γd1mand γd2mis the relative intercept angle from the missile’s perspective. This is the angle that will
be enforced by using the guidance law derived in [8]. This cooperative guidance law minimizes the following
cost function
J=α1
2ξ2
d1m(tf
d1m) + α2
2ξ2
d2m(tf
d2m) + β
2hγd1m(tf
d1m)γd2m(tf
d2m)ci2(17)
+1
2Ztf
d1
m
0
u2
d1dt +1
2Ztf
d2m
0
η2u2
d2dt, (18)
where ξdimis the relative displacement between the missile and the i-th defender normal to the LOS used
for linearization, cis the required relative intercept angle, and αi,η, and βare nonnegative weighs.
For ideal defenders’ dynamics (i.e., zero lag), the optimal closed-form solution that minimizes the cost
function Jwas found to have the following form [8]
u
d1(t) = Nu1
Z1
(tgo
d1m)2Z1(t) + Nu1
Z2
(tgo
d1m)2Z2(t) + Nu1
Z3
Vd1
tgo
d1m
(Z3(t)Z4(t)c), t [0, tf
d1m],(19a)
u
d2(t) = Nu2
Z1
(tgo
d2m)2Z1(t) + Nu2
Z2
(tgo
d2m)2Z2(t) + Nu2
Z3
Vd2
tgo
d2m
(Z3(t)Z4(t)c), t [0, tf
d2m],(19b)
where the navigation gains are given in [8].
The ZEM distances of the first and the second defender, Z1and Z2, and the zero-effort flight-path angles
(ZEA-s) of the missile plus that of the first and second defender, Z3and Z4, are give by
Z1(t) = ξd1m+˙
ξd1mtgo
d1m+kd1mam(tgo
d1m)2/2,(20a)
Z2(t) = ξd2m+˙
ξd2mtgo
d2m+kd2mam(tgo
d2m)2/2,(20b)
Z3(t) = γd1+γm+tgo
d1mam/Vm,(20c)
Z4(t) = γd2+γm+tgo
d2mam/Vm.(20d)
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where the linearization parameter kim satisfies
kim = cos(γm;0 +λim;0), i ∈ {d1, d2},(21)
and relates the missile acceleration amto the direction normal to LOS used for the linearization.
Note that the ZEM distances and the ZEA-s of Eq. (20) are valid only under the assumption that the
missile is maintaining a known constant maneuver throughout the engagement, i.e., um=const. Addition-
ally, the guidance law of Eq. (19) was derived under the assumption that the future missile maneuver as well
as the relative states are known or are accurately measured, see [8] for further details.
Remark 1.The cost function Jenforces an explicit cooperation between the defenders, as their trajectories
are mutually dependent on each other. Letting αi→ ∞ yields a perfect intercept between the i-th defender
and the missile. Similarly, letting β→ ∞ enforces a perfect relative intercept angle c. The parameter η
controls the ratio between the weight on the first and the second defender’s control effort.
Target’s Implicit Cooperation and Prediction of the Missile’s Acceleration Profile
As the missile is homing onto the target that performs an evasive maneuver, it is apparent that the constant
missile acceleration assumption is invalidated in the scenario studied in this paper. Moreover, we consider
a highly uncertain environment where the only interference with the external environment is thanks to the
two noisy LOS measurements. All these emerged problems are to be discussed in the sequel.
The implicit cooperation of the defenders’ guidance law stems from the fact that the defenders are aware
of the evasive maneuver of the target and thus can anticipate the maneuvers it will induce on the incoming
homing missile. Thus, for a given missile guidance strategy, this information can be used to obtain amas
a function of time via numerical integration of the appropriate engagement equations. By doing so, the
constant missile acceleration assumption in the defenders’ guidance law can be relaxed. Additionally, the
potential intercept points of the i-th defender-missile engagement can be predicted, which in turn can help
to reduce the defenders’ acceleration demand, hence reduce the likelihood of control saturation.
When the future missile maneuver is known but is not constant, i.e., um6=const., then based on the
terminal projection transformation, the zero-effort variables of Eq. (20) are generally given by [7, 16]
Z(t) = DΦ(i)(tgo
im)x(i)(t) + D
tf
im
Z
t
Φ(i)(tf
im, τ )C(i)umdτ, i ∈ {d1, d2},(22)
where Dis a constant row vector that pulls out the appropriate element of the zero-effort variable, Φ(i)(tf
im, τ )
is the transition matrix associated with the homogenous solution of the linearized i-th defender-missile
engagement, and C(i)is a vector associated with the linearized one-sided problem. Based on Eq. (22), the
computation of the ZEM distances and ZEA-s of Eq. (20) are replaced by the following equations (assuming
constant missile speed Vm)
Z1(t) = ξd1m+˙
ξd1mtgo
d1m+Ztf
d1m
t
(tf
d1mτ)am(τ) cos (γm(τ) + λd1m(τ)) dτ, (23a)
Z2(t) = ξd2m+˙
ξd2mtgo
d2m+Ztf
d2m
t
(tf
d2mτ)am(τ) cos (γm(τ) + λd2m(τ)) dτ, (23b)
Z3(t) = γd1+γm+Ztf
d1m
t
˙γm(τ)=γd1+γm+1
VmZtf
d1m
t
am(τ)dτ, (23c)
Z4(t) = γd2+γm+Ztf
d2m
t
˙γm(τ)=γd2+γm+1
VmZtf
d2m
t
am(τ)dτ, (23d)
where the integral components in Eq. (23) are computed by numerical integration and time propagation of
the relevant parts of Eqs. (2) and (4), assuming that no further acceleration commands are issued by the
defenders and that the missile and the target follow the presumed maneuvering model. If amis constant
throughout the engagement, then Eq. (23) degenerates to Eq. (20).
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Implementation Issues
The variables ξdimand ˙
ξdim, which appear in Z1and Z2of Eq. (23), relate to the linearized model. To
implement the defenders’ guidance law in a nonlinear setting, we need to replace these variables by more
meaningful kinematic variables. Assuming small deviations from collision triangle, the displacement ξdim
can be reasonably well approximated by
ξdimρdim(λdimλdim;0),(24)
where λdim;0 is the LOS angle used for linearization. Differentiating Eq. (24) with respect to time yields
ξdim+˙
ξdimtgo
dim=Vρdim˙
λdim(tgo
dim)2.(25)
The left hand side of Eq. (25) is identical to the first two terms of Eqs. (23a) and (23b), respectively. By
this, ξdimand ˙
ξdimare replaced by ˙
λdimand Vρdim, defined in Eqs. (2) and (3a), respectively.
Due to the same assumption, the speed Vρdimcan be assumed constant, and the tgo
dim, defined in (7), can
be approximated by
tgo
im (ρim/Vρim , Vρim <0
0, Vρim 0, i ∈ {d1, d2}.(26)
IV. Joint Estimator Design
To properly implement the defenders’ cooperative guidance law presented in Sec. III.C,we need the zero-
effort miss/angle variables, time-to-go, and the relative geometry associated with the defenders. In real-world
scenarios, these variables cannot be measured and therefore need to be estimated. Note that such variables
may be estimated independently by each defender or cooperatively, as done for attacking missiles in [1–3].
In these works, it was assumed that each missile has its own estimator and the computed state estimates
are shared within the team. Such approach requires extra computational effort because the parameters
directly related to the opponent are redundantly estimated by each entity in the team. In this work, these
redundantly estimated parameters correspond to the missile’s acceleration, flight path angle, speed, and
pertinent guidance parameters, respectively.
IV.A. Estimation Model and Assumptions
We assume that the missile has no information about the defenders, it is not trying to evade the defenders,
and it is guided towards the target via one of the classical guidance laws of PN, APN, or OGL with fixed
guidance parameter N0
P N ,N0
AP N , or α, see Sec. III.A for more details. If the missile’s active guidance
strategy is fixed throughout the engagement, a static multiple-model approach can be used to identify the
guidance law and guidance parameters [13, 14]. If the missile switches between guidance strategies, a dynamic
multiple-model approach can be derived based on the interactive multiple-model (IMM) approach [1, 3].
In this paper, we assume that, prior to launching the defending missiles, the target successfully identified
the active guidance law of the missile and passed this information to the defenders. However, we assume
that the corresponding parameters of the missile guidance law are still unknown. These uncertain guidance
parameters must be estimated together with other (uncertain) variables, which are all used to properly
implement the defenders’ guidance strategy.
The i-th defender’s state vector of the missile in polar coordinates is
xR
dim=hρdimλdimγmxa
mVmδmiT
,(27)
where δmrepresents the unknown guidance parameter(s). In our case, δmmay stand for N0or α, depending
on the considered guidance law of the missile.
In this paper, instead of designing two estimators, one for xR
d1mand the other for xR
d2m, we design a single
estimator for the joint defenders’ state, defined as
xR
dm =hρd1mρd2mλd1mλd2mxa
mγmVmδmiT
.(28)
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It is obvious that dim(xR
dm)<dim(xR
d1m) + dim(xR
d2m). In the rest of the paper, to avoid excessive indexing,
we will represent xR
dm as x.
Assume that the parameters of the missile dynamics are known, then the model used for estimation is
given by
˙ρd1m=Vρd1m
˙ρd2m=Vρd2m
˙
λd1m=Vλd1md1m
˙
λd2m=Vλd2md2m
˙xa
m=Amxa
m+Bmum(xR
tm, δm)
˙γm=Cmxa
m+Dmum(xR
tm, δm)/Vm
˙
Vm= 0
˙
δm= 0
,(29)
where Vρdimand Vλdimare given in Eqs. (3a) and (3b), respectively, and um(xR
tm, δm)is the missile’s
acceleration command given by (10). This command depends on the relative state between the target and
the missile
xR
tm =hρtm λtm atγtiT
(30)
and the active guidance strategy of the missile.
For simplicity, we assume that the missile has perfect information about the target, but not vice versa.
Therefore, to use xR
tm in Eq. (29), we need to compute xR
tm using information that is available to the defender
team. As xI
i, i ∈ {t, d1, d2}are assumed to be known, we can therefore express ρtm and λtm as a function
of the most recent estimates of ρdimand λdim, i.e.,
ρtm =q(∆Xd1
tm)2+ (∆Yd1
tm)2+q(∆Xd2
tm)2+ (∆Yd2
tm)2
2,(31a)
λtm =
atan2 Yd1
tm,Xd1
tm+atan2 Yd2
tm,Xd2
tm
2,(31b)
where Xdi
tm is the horizontal and Ydi
tm is the vertical separation between the target and the missile from
the i-th defender’s perspective, respectively, given by
Xdi
tm = ∆xtdi+ρdimcos(λdim),xtdi
,xdixt,(32a)
Ydi
tm = ∆ytdi+ρdimsin(λdim),ytdi
,ydiyt.(32b)
It is evident that both ρdm and λdm in Eq. (31) are computed as arithmetic averages of the two perspectives.
By this approach, the robustness of the proposed estimation scheme is increased, because the effect of
deteriorating estimation accuracy from one “perspective” can be averaged out by the possible accurate
estimate from the other perspective. As atand γtare assumed to be known, xR
tm is now fully defined.
Let us denote the vector that contains all the target-defenders relative positions at time tkas
xR
td;k= [∆xtd1ytd1xtd2ytd2]T.(33)
By this, the discrete-time version of Eq. (29), used for the estimator design, can be compactly rewritten as
xk=fk1(xk1, xR
td;k),(34)
where xkis the defenders’ joint state vector xR
dm at time tk, and fk1is a vector function derived by
integrating of Eq. (29) from tk1to tk.
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IV.B. Combined Measurement Model and Information Sharing
The defenders form a measuring baseline relative to the missile in space. Different look angles of the defenders
on the missile can improve the observability of the information-sharing based estimation scheme.
By exploiting the triangulation technique from the measurements perspective, we can express the model
of the i-th physical measurement, given in Eq. (9), as a function of the other, j-th defender, variables and
the known relative position of the two defenders
zi;k=h
j(xk, xR
dd;k) + vi=atan2(∆Yji ,Xji ) + vi, i, j ∈ {d1, d2} ∧ i6=j(35)
where xR
dd;k= [∆xd1d2yd1d2xd2d1yd2d1]T, and
Xji = ∆xji +ρjm cos(λj m),xj i ,xixj,(36a)
Yji = ∆yji +ρjm sin(λj m),yj i ,yiyj.(36b)
Combining the physical measurement model of Eq. (9) with the indirect measurement model of Eq. (35)
yields to the combined measurement model
zk=
zd1;k
zd2;k
zd1;k
zd2;k
=h(xk, xR
dd;k) + vk=
h1(xk)
h2(xk)
h
2(xk, xR
dd;k)
h
1(xk, xR
dd;k)
+vk(37)
where vk= [vd1;kvd2;kvd1;kvd2;k]T,zdi;kis the physical LOS angle measurement of the i-th defender, and
the functions hiand h
iare defined in Eqs. (9) and (35), respectively.
When the i-th defender passes the missile, i.e., tgo
dim= 0, this defender does not transmit any measure-
ments to the other defender and hence only a single physical model is considered for measurement update.
In the next section, we will use the combined measurement model of Eq. (37) to design the joint estimator.
IV.C. Extended Kalman Filter
As the estimation model in Eq. (29) is nonlinear, an extended Kalman filter (EKF) will be used to estimate
the state vector defined in Eq. (28). Note, however, that other estimation methods such as various variants
of the Kalman filters, divided difference filters, particle filters, to name just a few, can be also appropriate.
The state estimate of the filter at time tkusing measurements up to time tk1,ˆxx|k1, is propagated in
time using Eq. (34) and the most up-to-dated xR
td;k. The state transition matrix Φk|k1associated with the
system dynamics of Eq. (29) can be approximated by
Φk|k1= exp(Fk1|k1T)I+Fk1|k1T, (38)
where T=tktk1is the sampling time used for time propagation, Iis the identity matrix of appropriate
dimension, and Fk1|k1is the Jacobian matrix associated with the dynamics of Eq. (34), i.e.,
Fk1|k1=∂fk1(x, xR
td)
∂x xxk1|k1
,(39)
is assumed to be fixed during the time interval (tk1, tk]. The prediction error covariance matrix is
Pk|k1= Φk|k1Pk1|k1ΦT
k|k1+Qd,(40)
where Qdis an artificial covariance matrix of the corresponding discrete process noise used as a tuning
parameter of the filter [17].
If the physical measurement from the i-th defender, zi;k, is not available (e.g., because the i-th defender
ceased to exist, or due to sensor error, blind range of the sensor, etc.), then the time propagated state
estimate ˆxk|k1is updated by
ˆxk|k= ˆxk|k1+Kkzkh(ˆxk|k1),(41)
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where Kkis the Kalman gain computed as
Kk=Pk|k1HT
kHkPk|k1HT
k+R1.(42)
with j6=i
Hk=h0. . . 1(2+j). . . 0i, R =hσ2
λdji(43)
where the index (2 + j)indicates the location of the only nonzero element of the vector Hk.
If the estimation is performed using measurements from both defenders, the measurement Jacobian
matrix Hkand the measurement noise covariance matrix Rbecomes
Hk=
0 0 1 0
0 0 0 1
0Hρ
d2d10Hλ
d2d1
Hρ
d1d20Hλ
d1d20
[0]4×4
x= ˆxk|k1
, R =diag σ2
λd1, σ2
λd2, σ2
λd1, σ2
λd2,(44)
where
Hρ
ji =xj i sin(λjm )yji cos(λj m)
Λji
, Hλ
ji =(∆xj i cos(λjm )+∆yj i sin(λjm ) + ρjm )ρjm
Λji
,(45)
and the common denominator is
Λji = ∆x2
ji + ∆y2
ji +ρ2
jm + 2ρj m [∆xji cos(λj m)+∆yji sin(λjm)].
Finally, the covariance matrix is updated using
Pk|k=Pk|k1KkHkPk|k1,(46)
V. Numerical Analysis
In this section, we analyze the ability of the cooperative target-defenders team to protect the targeted
aircraft from the attacking missile. We also study the effect of different values of the commanded relative
intercept angle, c, on the estimation as well as on the intertwined guidance-estimation problem.
V.A. Engagement Scenario and Simulation Environment
We consider a similar engagement scenario as presented in [14]. All engagements are initiated at a horizontal
separation of 5 km between the target and the missile. To model the separation effect, the defenders are
initiated at a vertical separation of ytd1= ∆ytd2=1m below the target. The defenders are launched
from the aircraft at the beginning of the engagement (t= 0 s). The target’s speed is Vt= 300 m/s and the
speed of the two defenders and the missile is equal and is Vd1=Vd2=Vm= 500 m/s. For the analysis,
it is assumed that the missile and the target have first-order strictly proper dynamics with time constants
τm= 0.2s and τt= 0.5s. Thus, matrices in Eq. (4) degenerate to Ai=1i,Bi= 1i,Ci= 1, and
di= 0,i∈ {m, t}. We consider ideal dynamics for both defenders. The target’s maneuver capability is
limited to amax
t= 5 g. No saturation is applied on the missile’s acceleration command, i.e., amax
m=
g. The defenders’ maneuverability belongs to the closed set amax
d∈ Umax
d={10,20,30,40,∞} g, where
g is the standard acceleration due to the gravity (g = 9.80665 m/s2). Note, the omitted subscript "i" in
amax
dindicates that both defenders are equally concerned, i.e., amax
d=amax
d1=amax
d2. We will use the
same notation simplification for other variables in the next. The missile’s initial flight path angle is chosen
such that the missile’s velocity vector points towards the initial target location, i.e., γm;0 = 0 deg. As the
defenders are launched from the aircraft’s platform, therefore the initial flight path angles of the defenders
are considered to be identical to the initial flight path angle of the target, i.e., γd;0 =γt;0. For the closed
loop MC analysis, these angles are drawn from uniform distribution of the closed interval [30,30] deg.
The missile is guided towards the target using PN guidance with N0= 4 and using perfect information.
We assumed that throughout the engagement the target applies maximum acceleration to one side. The
maneuver direction is determined based on the initial geometry as
ut=(+amax
tif γt;0 0,
amax
tif γt;0 <0,
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where γt;0 is the initial flight path angle of the target defined above. The defenders use the cooperative
guidance law of Eq. (19) with implicit target cooperation, see Eq. (23). The numerical values of the guidance
parameters are α1=α2= 105,β= 108, and η= 1. The states needed for the defenders’ guidance law
employment are estimated at each time step, using the estimator developed in Sec. IV, at a sampling rate
of 50 Hz (T= 1/50 s). The simulated measurement noises are with σλdm = 1 mrad. The filter’s tuning
parameter Qdhas been chosen by numerical simulations. The initial state of the filter is sampled from a
Gaussian distribution
ˆx0|0 N (x0, P0|0),
where x0is the true state vector and P0|0is the initial covariance matrix of the error given by
P0|0=diag n10021002(5π/180)2(5π/180)2(2.5g)2(5π/180)250222o.(47)
V.B. Sample Run Example
Two different commanded relative intercept angles c= 20 deg and c= 120 deg are considered for
sample run demonstration. The initial flight path angle of the target-defender team is γi;0 = 10 deg. The
defenders are guided towards the missile using perfect information (true state vector) and no maneuverability
limitation. The same assumptions hold for the missile.
Figure 2 and 3 present the planar trajectories and the acceleration profiles of the target, missile, and
the two defenders in the simulated sample runs, respectively. It can be seen that, although the there is a
requirement on a specific intercept angle of c= 20 deg, the maximal acceleration requirement from the
defenders is quite small, approx. 6 g, compared to the missile’s maximal acceleration being above 7 g. On
the other hand, as seen in Fig. 3b, significantly larger relative intercept angle requirement naturally leads to
much higher maneuverability requirements from the defenders.
X [m]
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Y [m]
-400
-200
0
200
400
600
800
1000
Missile
Defender 1
Defender 2
Target
Defender 1: miss=0.032 [m], intercept angle=60.84 [deg]
Defender 2: miss=0.011 [m], intercept angle=40.85 [deg]
Relative intercept angle=19.99 [deg]
(a) c= 20 [deg]
X [m]
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Y [m]
-400
-200
0
200
400
600
800
1000
Missile
Defender 1
Defender 2
Target
Defender 1: miss=0.036 [m], intercept angle=122.59 [deg]
Defender 2: miss=0.011 [m], intercept angle=1.27 [deg]
Relative intercept angle=121.32 [deg]
(b) c= 120 [deg]
Figure 2: Sample trajectories for different relative intercept angles.
V.C. Guidance Performance Evaluation in Closed Loop
The effect of different values of c∈ Dcon the intertwined guidance-estimation problem is analyzed here.
The analysis is done for various considerations of the defenders maneuverability limit amax
d∈ Umax
d. The
estimated state from the estimator is used to guide the defenders towards the missile. For comparison, full
information and saturation-free case is also examined. This case is referred in the next figures as “umax
d=
g & perfect state”. For each value of cand umax
d, a set of 500 MC simulations was run. The guidance
performance is evaluated in terms of the achieved miss distance and acceleration requirement.
In case of the miss distance, we first compute the “two defender” cumulative distribution function (CDF),
which is defined on the minimum miss of both defenders. Then, using this CDF, we compute the value of
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Time [s]
0123456
Acceleration [m/s2]
-60
-40
-20
0
20
40
60
80
Missile
Defender 1
Defender 2
Target
(a) c= 20 [deg]
Time [s]
0123456
Acceleration [m/s2]
-400
-300
-200
-100
0
100
200
300
400
Missile
Defender 1
Defender 2
Target
(b) c= 120 [deg]
Figure 3: Sample acceleration profiles for different relative intercept angles.
the miss which corresponds to the 95% of cases, i.e., Prob(miss)0.95. This quantity is denoted as miss95%
and is also known as warhead lethality range which ensures a 95% kill probability for the defender team. To
evaluate the maneuverability requirements, we consider the value of the two defender maximal acceleration
in 95% of the simulation campaign cases. We denote this value as amax
d(95%). This value is computed
analogously as miss95% is computed. Additionally to amax
d(95%), we also consider a running cost Jacc on
the acceleration profiles defined as Jacc =Rtf
d1m
0|ad1(τ)|+Rtf
d2m
0|ad2(τ)|.
Figure 4 presents the obtained results of miss95% for different intercept angles c∈ Dcand acceleration
limits umax
d∈ Umax
d. Figure 5a shows the maneuverability requirements of the defenders in terms of the
amax
d(95%) measure and Fig. 5b in terms of the running cost Jacc measure. The results of Figs. 4-5 suggest
that small values of cyield to larger miss distances as the estimation performance for these angles is
poor. Due to the same reason, such small angles also cause increase of “momentary” maximal acceleration
requirements. On the other hand, large values of crequire substantially more “overall” maneuverability
yielding to control saturation. Obviously, long-term saturation has large effects on the achievable miss, even
when accurate estimates are used. It is interesting to note that for all finite maneuverability limit cases,
i.e., for all umax
d<, there exist a plateau effect, i.e., a region of intercept angles where the obtained
miss is minimal. As expected, the performance of any perfect information guidance law is better than the
performance of the same guidance law using estimated states, see the results for umax
d={∞,?}in Fig. 4.
Commanded relative intercept angle c [deg]
0 50 100 150
Miss distance of 95 % of the runs [m]
10-2
10-1
100
101
102
103
ud
max = 10 [g]
ud
max = 20 [g]
ud
max = 30 [g]
ud
max = 40 [g]
ud
max = [g]
ud
max = [g] & perfect state
Figure 4: Values of miss95% for umax
d∈ Umax
das a function of c∈ Dc.
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Commanded relative intercept angle [deg]
0 50 100 150
Max. acceleration of 95 % of the runs [m/s2]
0
100
200
300
400
500
600
ud
max = [g]
ud
max = [g] & perfect state
(a) Values of amax
d(95%) as a function of c∈ Dc.
Commanded relative intercept angle [deg]
0 50 100 150
Mean of integral cost on acceleration [m/s]
0
500
1000
1500
2000
2500
ud
max = [g]
ud
max = [g] & perfect state
(b) Values of Jacc as a function of c∈ Dc.
Figure 5: Guidance performance - acceleration requirements for unsaturated case.
VI. Conclusions
A cooperative estimation-guidance algorithm has been presented for a team of two aircraft’s defending
missiles to intercept an attacking missile homing on to the evading target aircraft. The algorithm exploits
an explicit team cooperation in the defenders’ guidance to impose a relative intercept angle, an implicit
cooperation of the target, and a cooperative estimation scheme based on shared information. The cooperation
from the target’s point of view results from the fact that the defenders are aware of the evasion maneuvers
of the target. Thus, the defenders can predict the target-induced maneuvers on the homing missile.
The proposed joint estimation scheme is strongly linked to the defenders’ guidance when considering LOS
angle measurements only. Nonlinear simulations revealed that various relative intercept angle constraints
for the defenders have strong influence on the intertwined guidance-estimation performance of the defenders.
Small angles yield to poor estimation performance which consequently lead to control saturation and intercept
performance degradation. Angles ranging from approx. 20 deg to approx. 60 deg exhibit very good guidance
performance while maintaining modest maneuverability requirements. Larger intercept angles lead only to
negligible improvements in the interception performance but require far more agility from the defenders.
The effectiveness demonstrated by the proposed cooperative algorithm to protect the evading aircraft
from a highly maneuverable homing missile can, for a carefully selected relative intercept angle, considerably
improve the aircraft’s survivability, making it possible to design relatively inexpensive defending missiles,
without having superior maneuverability requirements, advanced sensor systems, and large lethal warhead.
Acknowledgments
This effort was sponsored by the U.S. Air Force Office of Scientific Research, Air Force Materiel 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.
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... This two-to-one interception engagement was conducted by imposing a defender intercept angle without the cooperation of the evader in a lure role. Recently, Fonod and Shima [13,14] proposed a cooperative guidance law via sharing the LOS angle between a team of aircraft. This method could enhance the intercept estimation performance by imposing an intercept angle, which relaxed the assumptions of constant acceleration and perfect information of the pursuer's actions considered in [1,2,12]. ...
... Research into cooperative strategies between an evader and multiple defenders under intercept angles constraints is relatively lacking. An intercept scenario different than scenarios addressed in [12][13][14] is considered in this study based on the finite time theory: a pursuer, an evader, and multiple defenders, focusing on cooperative defense with different intercept angle constraints and evader lure role cooperation. The three-aircraft cooperative defense scenario (Evader, Defender 1, and Defender 2) is expected to achieve superior intercept performance. ...
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Two estimators are presented, that enable cooperative target tracking of several missiles intercepting a single maneuvering target. The first estimator is a nonlinear adaptation of an interacting multiple model filter, whereas the second estimator is a multiple model particle filter. The paper develops the filters for the cooperative and non-cooperative estimation modes, and investigates their individual estimation performance, using a nonlinear two-dimensional simulation. An extensive Monte Carlo study is used to demonstrate the viability of the cooperative estimation concept, for both estimators. It is shown that the closed loop interception performance of two cooperating missiles, guided by an optimal guidance law, improves, when compared to that of non-cooperating missiles. The particle filter based estimator demonstrates hit-to-kill closed loop interception performance in the cooperative mode, but requires higher computational load than the extended Kalman filter based estimator, making the choice of estimator a tradeoff between performance and computational power.
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Aircraft protection from a missile employing a known linear guidance strategy is considered. The aircraft protects itself by using a cooperating defensive missile. Depending on the available cooperation scheme, three different cooperative linear quadratic guidance strategies are derived and investigated for arbitrary-order linear-Adversaries dynamics: 1) two-way cooperation, where the target-defender team employs its optimal cooperative strategy, 2) oneway cooperation, realized by the defender employing a classical one-on-one guidance law while the target helps by luring in the missile, and 3) one-way cooperation, realized by the target employing an arbitrary evasive strategy while the defender attempts to reach the predicted interception point. The performance of the three different cooperative strategies is compared and analyzed via simulation using the notion of Pareto fronts. It is shown that the performance of the target and the defender is highly dependent on the cooperation scheme. As expected, the two-way cooperation scheme provides the best performance. Yet, one-way cooperation schemes attain considerably better results than independent actions. Copyright © 2012 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
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Optimal-control-based cooperative evasion and pursuit strategies are derived for an aircraft and its defending missile. The aircraft-defending missile team cooperates in actively protecting the aircraft from a homing missile. The cooperative strategies are derived assuming that the incoming homing missile is using a known linear guidance law. Linearized kinematics, arbitrary-order linear adversaries' dynamics, and perfect information are also assumed. Specific limiting cases are analyzed in which the attacking missile uses proportional navigation, augmented proportional navigation, or optimal guidance. The optimal one-on-one, noncooperative, aircraft evasion strategies from a missile using such guidance laws are also derived. For adversaries with first-order dynamics it is shown that depending on the initial conditions, and in contrast to the optimal one-on-one evasion strategy, the optimal cooperative target maneuver is either constant or arbitrary. These types of maneuvers are also the optimal ones for the defender missile. Simulation results confirm the usefulness and advantages of cooperation. Specifically, it is shown how the target can lure in the attacker, allowing its defender to intercept the attacking missile even in scenarios in which the defender's maneuverability is at a disadvantage compared with the attacking missile.
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Estimation and guidance scheme is presented, which enables two missiles tracking single target cooperatively. Each missile is equipped with an infra-red sensor and measures the LOS angle only. The two missiles form a measuring baseline relative to the target in space and share measured information with each other, enabling that each identifies the target state based triangulation method. However, the observability of this double line-of-sight measuring relative navigation method is weak if the separation angle between both line-of-sight vectors is small. This paper proposes a guidance law that modulates the line-of-sight angle through a performance index. Using this guidance law, the missile with large initial ling-of-sight angle maximizes this index. Whereas the other one minimizes it. Simulation results show that the proposed estimation and guidance scheme improves the homing accuracy and can yield hit-to-kill performance. © 2013 by Yanfang Liu. Published by the American Institute of Aeronautics and Astronautics, Inc.
Book
The purpose of this chapter is to present state estimation techniques than can “adapt” themselves to certain types of uncertainties beyond those treated in earlier chapters—adaptive estimation algorithms. One type of uncertainty to be considered is the case of unknown inputs into the system, which typifies maneuvering targets. The other type will be a combination of system parameter uncertainties with unknown inputs where the system parameters (are assumed to) take values in a discrete set. The input estimation with state estimate correction technique is presented. The technique of estimating the input and, when “statistically significant,” augmenting the state with it (which leads to variable state dimension), is detailed. These two algorithms and the noise level switching technique are later compared. The design of an IMM estimator for air traffic control (ATC) is discussed in detail. Guidelines are also developed for when an adaptive estimator is really needed, i.e., when a (single model based) Kalman filter is not adequate. The chapter concludes with a brief presentation of the use of the extended Kalman filter for state and system parameter estimation. A problem solving section appears at the end of the chapter.