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Estimation Enhancement by Cooperatively Imposing
Relative Intercept Angles
Robert Fonod∗and Tal Shima†
Technion - Israel Institute of Technology, Haifa 3200003, Israel
Cooperative estimation/guidance for a team of missiles is the topic of this paper. An
example scenario is considered, where an aircraft simultaneously launches several cooper-
ative defending missiles as a countermeasure against an attacking homing missile. A new
reduced-order estimation scheme based on information sharing to cooperatively estimate
the relative states and the unknown parameters of the attacking missile is proposed. Each
defending missile shares its own noise-corrupted line of sight angle measurement with the
rest of the team. The observability of this multi-line-of-sight measuring environment is
enhanced by cooperatively imposing nonnegative relative intercept angles between con-
secutive defenders. The ability of the proposed strategy to protect the targeted aircraft
is studied for a two-defender case via extensive Monte Carlo simulations. The effect of
different values of commanded relative intercept angle on the pure estimation as well as on
the intertwined guidance-estimation performance is carefully analyzed for various consid-
erations of defenders’ maneuverability.
Nomenclature
a= normal acceleration
A,B,C,D= state space representation of the closed-loop dynamics
Dc= set of considered relative intercept angle commands
f= nonlinear equations of motion
F= state transition Jacobian matrix
h= measurement function
H= measurement Jacobian matrix
J= cost function
k= instantaneous projection (linearization) coeficient
K= Kalman filter gain matrix
N0= navigation gain of the attacking missile
Nui
Zj,Nui
∆Zj=i-th defender’s navigation gains associated with the j-th defender’s
zero-effort miss and zero effort angle error, respectively
N= Gaussian distribution
n= total number of considered defenders
na= number of active defenders
P= state error covariance matrix
Q= covariance matrix of the equivalent discrete process noise
R= measurement covariance matrix
t,tgo,tf= time, time-to-go, and interception time, respectively
∗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.
1
T= sampling period
V= speed
v= measurement noise
w= nonnegative weight
u= acceleration command
U= set of considered maneuverability limitations
x= state vector
xa= state vector of the vehicle’s dynamics
xi,yi= inertial coordinates of the i-th entity position
∆xij ,∆yij = relative displacements of the i-th entity from the j-th entity
z= measurement
Z= zero-effort miss/flight-path angle
α= weight on the defender’s miss
η= weight on the defender’s control effort
β= weight on the defenders’ relative intercept angle
ξ= defender-missile relative displacement normal to the initial LOS
γ= flight-path angle
γdim= angle between the i-th defender and the missile
ς= boolean variable indicating the status of the defender
∆c= relative intercept angle requirement
θ= normalized time-to-go
δm= unknown missile guidance parameter(s)
ε= ratio between the weight on the missile’s control effort and the miss
λ= angle between the line of sight and the XIaxis
ρ= range
σ2
λ= LOS angle measurement noise variance
τ= time constant
Φ= transition matrix
ψ= time-varying function
Ψ= tuning parameter of the Kalman filter
[0] = matrix of zeros with indicated dimension
(ˆ
·)= estimated value
(¯
·)= approximated value
Subscripts
di=i-th defending missile (defender)
m= attacking missile (missile)
t= target aircraft (target)
k= step of the discrete time tk
ρ,λ= along and normal to the line of sight
Superscripts
err = error
max = maximal value
?= perfect information assumption
†= indirect measurement model
I= inertial coordinate frame
R= relative (polar) coordinate frame
2
I. Introduction
Modern highly sophisticated aerial threats such as anti-aircraft missile, tactical ballistic missile, and
unmanned aerial vehicle (UAV) are able to engage and destroy a large class of urban and aerial targets [1].
Such threats are usually characterized by low observability and high maneuverability. Advanced air-defense
missile systems include radar-guided surface-to-air missiles and modern fighter aircraft armed with various
sensor-guided defending missiles. The major requirement for such defense systems, designed to negate these
threats, is improved interception performance, i.e., attaining small miss distance to ensure destruction of the
adversary. However, the performance of any guidance system, aimed to achieve small miss distances against
maneuvering threats, is highly reliant on the knowledge of the opponent’s acceleration, relative state, and
other uncertain parameters related to the opponent. For instance, the opponent’s acceleration cannot be
directly measured and therefore has to be estimated based on available noise-corrupted measurements. The
unavoidable estimation errors may degrade the interception performance. Development of advanced sensor
systems, use of more agile missiles, and/or deployment of more lethal warhead are some of the few possible
options to deal with the problem of uncertainty. However, these options might be very often too complex,
heavy, and expensive. An alternative is to design more sophisticated guidance and estimation algorithms
to improve the guidance system of inexpensive missiles, without deteriorating the required interception
accuracy.
Gimballed electro-optical seekers are usually positioned at the front tip of the missile. Often, the size of
the seeker and its supporting systems dictate the missile’s front tip shape, which in turn affects maneuver-
ability, volume, and aerodynamic constraints. Most tactical missiles are equipped with affordable infra-red
(IR) sensors, which allow to measure the line-of-sight (LOS) angle between the pursuer and the opponent.
Fixed opponent (target) localization using bearings-only measurements is an observable process even with-
out an observer (e.g., missile) maneuver. Oshman and Davidson [2] used the determinant of the Fisher
information matrix to optimize the observer’s trajectories for bearings-only stationary opponent localization
problem. However, the estimation performance for a target tracking problem, in the presence of maneuver
uncertainty and noise-corrupted bearings-only measurements, is limited [3–5]. Nardone and Aidala [3] and
Hepner and Geering [4] showed that certain types of maneuvers do not necessarily guarantee observability in
target tracking problems where only a single LOS angle measurement is available. For example, employing
proportional navigation (PN) guidance law attempts to null the LOS rate. As a consequence, range and
range-rate are not observable [5]. A solution to improve range observability is to maneuver away from the
collision course. This causes the LOS to rotate which in turn gives some insights on the relative range. Based
on this idea, Battistini and Shima [6] proposed a guidance logic which exploits the information content of
the error covariance matrix’s eigenvalues.
In recent years, multi-missile counterattack against aerial threats has been conceived as a very effective
way to survive, as it may significantly increase the success rate of such countermeasure [7]. In scenarios where
multiple missiles can share their respective LOS angle measurements, the estimation performance can be
improved by exploiting the triangulation structure [8–12]. The estimation quality, however, strongly depends
upon the missiles’ trajectories and hence on the implemented guidance law. Shaferman and Oshman [11]
proposed two estimation methods for cooperative target tracking based on information sharing of multiple
missiles. It was assumed that the missiles are guided towards the target via a given one-on-one guidance law,
and that only the estimation is performed cooperatively. However, guidance and estimation are mutually
intertwined. Neglecting the effect of the one onto the other and vice versa may have severe consequences.
For example, when all missiles employ the same one-on-one guidance law (as considered in [11]) and are all
fired with the same initial conditions, then the resulting missile trajectories coincide (up to some unmodeled
disturbances). As a consequence, all sensors will measure the same quantity, causing the triangulation
technique to fail and thus making the range unobservable. This, in turn, might result in poor interception
performance. The work of [11] was recently extended in [12] by the same authors, where the above issue
was resolved by introducing the concept of staggered launch of the missiles. In this work, the optimal
staggering was derived based on a linear model and a deterministic approximation of the stochastic estimation
process. Chen and Xu [8, 9] analyzed the observability issue in a double-LOS relative navigation setup. They
concluded that if the separation angle between 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
by Liu et al. [10] 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.
3
The work presented in this paper sets the above concepts into a specific problem of aircraft protection
from an attacking homing missile. In this paper, the aforementioned concepts of multimissile attack and
information sharing are related to a specific problem of an active aircraft defense from an an incoming
homing missile. The aircraft in this study may be a manned or an unmanned aerial vehicle. Among systems
developed in the past decades to increase aircraft’s protection capabilities are electronic countermeasures
(jammers) and various kinds of decoys (e.g., chaff or flares). The aircraft may also perform evasive maneuvers,
which can be either arbitrary [13–15] or optimally adjusted against the incoming missile [16–18]. In case of
arbitrary maneuvers, Zarchan [13] suggested a random telegraph approach. A periodic sine wave maneuver
with a random phase but a frequency that is matched to the missile’s navigation gain and time constant were
discussed by Zarchan [14] and Ohlmeyer [15]. Shima [16] derived optimal evasive maneuvers against a homing
missile employing a known linear guidance law. This work was recently extended by Turetsky and Shima [17]
for a case where the missile performs multiple switches between known linear guidance laws and by Fonod
and Shima [18] for a case where the guidance law of the missile is unknown. All these countermeasures may
not provide sufficient protection against agile and advanced adversarial systems. An alternative solution is
to launch one or several defending missiles (defenders) to intercept the incoming threat before it reaches
the aircraft. As the defenders and the targeted aircraft (target) can cooperate with each other to intercept
the attacking missile, such an active self-defense system results in a special multiple-body pursuit-evasion
guidance scenario which is different from the typical one-on-one engagements.
Most of the works in the literature on active aircraft protection deal with guidance strategies for the
aircraft defender team as a three-body problem (target, missile, and a single defender). The common
assumptions are that the attacking missile is homing onto the target using perfect information, is unaware
of the defender(s), and employs a linear one-on-one guidance law. Asher and Matuszewski [19] discussed the
possible application of an optimal control-based guidance law for such a scenario. Boyell [20] presented the
first results on kinematics of the three-body problem, where closed form relations were derived for constant-
bearing collision courses. Shneydor [21], in his comments on Boyell’s work [20], simplified one of the collision
conditions. In a later work under the same assumptions, Boyell [22] obtained a closed-form expression for
the intercept point in target-centered coordinates. Shinar and Silberman [23] investigated the three-body
problem as a three-player two-team game. Rusnak et al. [24] studied optimal strategies for the three-body
problem (including the missile) in a linear quadratic (LQ) game setting. It was shown in [24] that as the
weight on the defender command tends to zero, the optimal strategies of the target and the missile are
identical in form to those obtained in the one-on-one game without the defender. Ratnoo and Shima [25]
proposed an interception method where the defending missile is commanded to be on the LOS between
the attacking missile and the aircraft for all time, while the aircraft follows some predetermined trajectory.
This command to LOS (CLOS) defender guidance resulted in improved performance against missiles using
proportional navigation guidance law. However, this approach requires the defender to have at least the
same speed as the attacker. Yamasaki et al. [26] presented a modified CLOS guidance law which further
improves the defender CLOS performance. Shima [16] derived an optimal control based cooperative evasion
and pursuit strategy for an aircraft and its defending missile for the case where the attacking missile uses
a known linear guidance strategy. The sole objective was to minimize the defender-missile miss distance.
Prokopov and Shima [27] analyzed different types of cooperation, assuming the attacking missile is unaware
of the presence of the defender and its guidance law is known. In contrast to [16], the methods proposed in
[27] additionally minimize the target’s acceleration requirements. Perelman et al. [28] obtained an analytical
solution to the LQ differential game and studied the conditions for the existence of a saddle point solution.
Garcia et al. [29] extended these results by allowing the defender’s turning rate to be constrained.
All the previous cooperative guidance laws on the three-body problem assumed in their derivation that
perfect information is available both for the homing missile and for the target-defender team. Shaferman and
Shima [30] presented a multiple model adaptive guidance/estimation approach to identify the active guidance
strategy of the incoming missile. In this work, the defender’s guidance law is matched to the identified
guidance strategy of the missile and the target’s maneuver minimizes the control effort requirements of the
defender by performing an appropriately timed single direction maneuver switch. Furthermore, the method
presented in [30] assumes that the target and the defender are equipped with either radar or electro-optic
sensors.
While all the works presented earlier emphasize the optimization and/or cooperation of the aircraft with a
single defender, this paper considers a case when multiple defenders are fired simultaneously to cooperatively
intercept the attacking missile. We assume that the defenders have limited maneuver capabilities and are
4
equipped with affordable IR sensors only, which provide noisy LOS angle measurements. We also assume
that each defender can share without any delay its own-ship measurement with the other defenders and
that the target-defenders team has imperfect a priori information about the relative state and guidance
parameters of the missile. The missile is assumed to be guided towards the target using one of the classical
missile guidance laws of PN, augmented PN (APN), and optimal guidance law (OGL). However, unlike
in [30], we consider a “fire and forget” policy which does not require the target to be equipped with any
sensor system that measures the bearing and/or the range to the missile. The only assumptions made with
respect to the target are that, prior to launching the defenders, its future maneuver strategy was successfully
communicated to all defenders and that its location is known to the defenders to high accuracy. For sake of
generality, we assume that the guidance strategy of the target is arbitrary.
In this paper, we significantly extend the cooperative estimation-enhancing guidance concept of [31] by
re-deriving the solution for a team of n > 1aircraft defending missiles and by demonstrating the viability of
the proposed concept by additional simulation results. We propose a new reduced-order estimation scheme
based on information sharing to cooperatively estimate all the defenders-missile relative states and missile’s
related parameters such as speed, acceleration, flight-path angle, and its guidance-related parameters. This
scheme enables the entire estimation algorithm to be implemented in a centralized manner on a sole defender.
Therefore, the computational requirements on the on-board computer of the remaining defenders can be
significantly reduced. Considering the observability issues of multi LOS measuring environment discussed
earlier, and motivated by the recent developments in terminal intercept angle missile guidance theory [32–34],
the idea in this paper is to enhance/ensure observability by imposing nonnegative intercept angle constraints
between the defenders. This can be achieved for each defender separately using one of the available one-
on-one terminal intercept angle guidance laws, e.g., Ryoo et al. [32] and Shaferman and Shima [33]. Or,
better yet and as done in this paper, one can achieve the same goal using the optimal guidance law of
Shaferman and Shima [34] which can impose a predetermined relative intercept angle between consecutive
defenders based on explicit cooperation of all defenders. By imposing nonnegative relative intercept angle
constraints, the terminal trajectory separation between the defenders can be ensured and the observability
improved. Additionally, by exploiting the knowledge of the known future target maneuver and the fact that
the attacking missile is homing onto the target, we relax the known constant acceleration assumption in the
derivation of the guidance law presented in [34]. This strategy also helps to minimize the defenders’ control
effort as the intercept points of the defenders are implicitly taken into account in the missile’s acceleration
profile prediction phase, which is performed by numerical integration of the relevant engagement’s equations
and the use of the most recent estimates. In this paper, we extend the analysis presented in [34], where
perfect information and saturation-free scenario was considered. Through an intensive numerical study,
we rigorously analyze the effect of different values of the commanded relative intercept angle on the pure
estimation as well as on the intertwined guidance-estimation performance, while also evaluating different
acceleration requirements for the defenders.
The remainder of this paper is organized as follows. The next section presents the mathematical models
of the target-defenders-missile engagement. The joint estimator is derived in Sec. III, followed by the presen-
tation of the cooperative guidance law for the defenders in Sec. IV. A comprehensive performance analysis
is done in Sec. V, followed by concluding remarks.
II. Multiple-Body Engagement Description
The considered scenario consists of several entities (also referred to as bodies or vehicles): an attacking
missile, an evading target aircraft, and a group of n > 1aircraft defending missiles. For brevity, the target
aircraft is referred to as target, the attacking missile as missile, and the defending missiles as defenders. The
target and the ndefenders form a team against the missile.
Next, the full kinematics and dynamics equations of the target-defenders-missile engagement are pre-
sented. Then, we introduce the timeline and the considered physical measurement model.
II.A. Kinematics and Dynamics
We consider skid-to-turn and roll-stabilized vehicles. The motion of the target, all the ndefenders, and the
missile is assumed to transpire in the same plane. In Figure 1 a schematic view of the considered planar point
mass engagement geometry is shown, where XI−OI−YIrepresents a Cartesian inertial reference frame. The
5
missile, i-th defender, and target related variables are denoted by the subscripts m,di, and t, respectively.
The speed, normal acceleration, and flight-path angles are denoted by V,a, and γ, respectively. The range
between the target and the missile and between the i-th defender and the 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
λdim
γdi
γm
Vm
Vdi
am
adi
ρdim
λdi+1m
γdi+1
Vdi+1
ρdi+1m
γt
Vt
at
(xdi, ydi)(xdi+1 , ydi+1 )
(xt, yt)
λtm
ρtm (xm, ym)
adi+1
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 that of the target, i.e., Vi> Vt,∀i∈ {m, d1, . . . , dn}.
We assume that the target’s and defenders’ own inertial state vector
xI
i=hxiyiaiγiiT
, i ∈ {t, d1, . . . , dn},(1)
is known to a very high accuracy (e.g., using inertial navigation system and/or GPS sensors), and that the
target and each defender can transmit its own inertial state vector to other defenders in the team without
any delay. We also assume that the target’s speed is known or is transmitted to the defenders.
Remark 1. Only the relative positions between the cooperative vehicles (target and defenders) will be
important in the rest of the derivation. As long as all vehicles in the team can track each other accurately,
the above assumption about xI
iis not invalidated.
Neglecting the gravitational force, the engagement kinematics, expressed in a polar coordinate system
(ρim, λim )attached to the i-th entity, is:
(˙ρim =Vρim
˙
λim =Vλim/ρim
, i ∈ {t, d1, . . . , dn},(2)
where the respective relative velocities along and perpendicular to the LOS are
Vρim =−Vicos(γi−λim)−Vmcos(γm+λim ),(3a)
Vλim =−Visin(γi−λim) + Vmsin(γm+λim ).(3b)
During the endgame, all the vehicles are assumed to move at a constant speed and to perform lateral
maneuvers only. Arbitrary-order linear dynamics is assumed for all vehicles
˙xa
i=Aixa
i+Biui
ai=Cixa
i+Diui
˙γi=ai/Vi
, i ∈ {t, m, d1, . . . , dn},(4)
6
where xa
i∈Rniis the internal state vector of the i-th vehicle’s dynamics, aiand uiare the i-th entity’s
normal acceleration and acceleration command, respectively. The term Cixa
iis denoted as ais and represents,
if it exists, the part of the acceleration with dynamics (for example, an angle of attack generating lift). The
second part of the acceleration, i.e. Diui, represents the direct lift, which can be obtained immediately from
deflection of the steering mechanism such as the canard or tail (neglecting servo dynamics).
We assume that the target’s and defenders’ maneuverabilities are limited to
|ui| ≤ umax
i, i ∈ {t, d1, . . . , dn},(5)
where umax
i>0is the i-th vehicle’s maximal acceleration. No saturation is considered for um.
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, . . . , dn}.
The particular engagement terminates at t=tf
im, where tf
im is the target-missile or defender-missile inter-
ception time, formally defined as
tf
im = arg
t>0
inf{ρim(t)Vρim(t)=0}, i ∈ {t, d1, . . . , dn}.(6)
The interception time tf
im allows to define the true nonnegative time-to-go as
tgo
im =(tf
im −t, t ≤tf
im
0, t > tf
im
, i ∈ {t, d1, . . . , dn}.(7)
At t=tf
im, the separation ρim (tf
im)is minimal and is referred to as “miss distance” or compactly as “miss”.
Without loss of generality, we assume that the target-missile engagement terminates after that of the
defenders-missile, i.e., tf
dim< tf
tm,∀i∈ {1, . . . , n}, and that the defenders are numbered based on their
interception times, satisfying
tf
d1m≤tf
d2m≤. . . ≤tf
dnm.(8)
II.C. Physical Measurement Model
The bearing measurement is a pre-dominant one in missile guidance applications and it requires a relatively
inexpensive seeker. Therefore, we assume that each defender is only equipped with an IR sensor that
measures the LOS angle, i.e., the i-th defender measures only λdim. The measurements are assumed to be
contaminated by a zero-mean white Gaussian noise with standard deviation σλdimand all being acquired at
the same discrete-time t=tk,k·T, where T > 0is the measurement sampling period.
Based on the above assumptions, the physical measurement equation of the i-th defender is
zdi;k=hdi(xk) + vdi;k=λdim;k+vdi;k, i ∈ {1, . . . , n},(9)
where the state vector xk, used for estimation, will be defined later, and
vdi;k∼ N (0, σ2
λdim).
In Eq. (9), and in the rest of the paper, the discrete time step is indicated (if unavoidable) by a subscript k,
separated by a semicolon. The noise sequences vdi;k,i∈1, . . . , n, are assumed to be mutually independent.
We assume that each defender can transmit its own-ship measurement to the rest of the team without any
delay. Additionally, we assume that from the time when the i-th defender passes the missile, i.e., tgo
dim= 0,
this defender does not transmit its measurement nor its relative position to the rest of the team. The status
of the i-th measurement is characterized by the boolean variable ςidefined as
ςi=(1if measurement available,
0otherwise.(10)
The advantage and utilization of the above measurements sharing concept will be discussed in more
details in the next section.
7
III. Reduced-order Estimator Design
In practical interceptions with noisy measurements and incomplete information about the opponent,
an estimator becomes an inevitable part of any advanced guidance system. The homing accuracy of such
guidance system is then restricted by the performance of the estimator. Based on the separation theorem
[35] and the associated certainty equivalence principle [36], the common practice is to design the guidance
laws and the estimators separately.
The estimator can be implemented using a variety of techniques, e.g., various variants of the (extended)
Kalman filter [37], divided difference filter [38], sliding mode observer/differentiator [39], particle filter [11,
12], etc. If the measurements can be shared between the defenders, then the estimation can be performed
cooperatively and in a decentralized manner as done in [10–12]. In these references, it was assumed that
each teammate has its own estimator and the computed state estimates are shared within the team. In this
section, we propose a single, reduced-order, estimator design method to jointly estimate the defenders’ state
vector using shared bearings-only measurements.
III.A. Assumptions on the Missile Guidance
We assume that prior to firing the defending missiles, the target has acquired some intelligence about the
missile’s active guidance law and that this information is passed to the defenders. On the other hand, we
assume that the corresponding parameters of this guidance law are unknown, and thus need to be estimated.
For simplicity of exposition, we will focus on the three most representative missile guidance laws of PN
[40], APN [41], and OGL [42]. However, the derivation can be extended to other missile guidance laws using
the same formulation and similar derivation steps.
The guidance laws of PN, APN, and OGL have the following form
um=N0
j
Zj
(tgo
tm)2cos(γm+λtm ), j ∈ {PN,APN,OGL},(11)
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.(12)
The expression for the ZEM distance is different for each guidance law, i.e.,
ZP N =Vρtm ˙
λtm(tgo
tm)2,(13a)
ZAP N =ZP N + (tgo
tm)2atcos(γt−λtm )/2,(13b)
ZOGL =ZAP N −τ2
mψ(θtm)ams cos(γm+λtm ),(13c)
where ψ(θtm)is an exponential-like function of the normalized target-missile time-to-go θtm, given by
ψ(θtm) = e−θtm +θtm −1, θtm =tgo
tm/τm.(14)
and ˙
λtm and Vρtm are given by Eqs. (2) and (3a), respectively.
The navigation gains N0
P N and N0
AP N are constant, whereas N0
OGL is defined as a function of θtm as
follows
N0
OGL =6θ2
tmψ(θtm )
3−6θtm (ψ(θtm) + e−θtm )+2θ3
tm −3e−2θtm + 6ετ−3
m
,(15)
where εcontrols the ratio between the weights on the missile’s control effort and miss in the LQ cost used
in the OGL derivation [43].
We assume that the missile is not aware about the presence of the defenders, it is not trying to evade
from them, and 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 ε. These assumptions are realistic as most missiles can only
track a sole target and are designed to intercept the tracked target [30].
8
Remark 2. If intelligence about the incoming missile is missing or is only partial, then one can resort
to online identification techniques. For instance, a multiple-model adaptive estimation (MMAE) based
approach can be used to identify the missile’s fixedaguidance strategy [18, 30], or, in cases when the missile
switches between different guidance strategies, an interactive multiple-model (IMM) based approach can be
considered [11, 12, 44]. Both of these approaches require the target to be equipped with sensors that can
measure bearing and/or range to the missile. Consequently, if these measurements can be shared with the
defenders without any delay, then they can be used to improve the estimation accuracy of the defenders, see
for instance [30] for discussion about advantages of such target-defender sharing concept.
III.B. Estimation Model
The i-th defender’s state vector of the missile in polar coordinates is
xR
dim=hρdimλdimγmxa
mVmδmiT
,(16)
where δmrepresents the unknown guidance parameter of the missile. In our case, δmmay stand for N0
P N ,
N0
AP N , or ε, depending on the identified guidance law.
In this paper, instead of designing nestimators, i.e., for each xR
dima separate one, we design a single
estimator for the joint defenders’ state, defined as
xR
dm =hρd1m. . . ρdnmλd1m. . . λdnmxa
mγmVmδmiT
.(17)
It is obvious that dim(xR
dm)<dim(xR
d1m) + . . . + dim(xR
dnm). Thus, estimating xR
dm instead of xR
dim, i =
1, . . . , n will require less computational effort because the parameters directly related to the missile (i.e., xa
m,
γm,Vmand δm) are not redundantly estimated by each defender in the team. In the rest of the paper, to
avoid excessive indexing, we will denote xR
dm as x.
Based on the constant missile speed and missile guidance parameters assumption, the model used for
estimation is given by the following set of equations
˙ρd1m=Vρd1m
.
.
.
˙ρdnm=Vρdnm
˙
λd1m=Vλd1m/ρd1m
.
.
.
˙
λdnm=Vλdnm/ρdnm
˙xa
m=Amxa
m+Bmum(xR
tm, δm)
˙γm=Cmxa
m+Dmum(xR
tm, δm)/Vm
˙
Vm= 0
˙
δm= 0
,(18)
where Vρdimand Vλdimare given in Eqs. (3a) and (3b), respectively, and um(xR
tm, δm)is the missile’s
acceleration command given by (11). This command is a function of the target-missile relative state xR
tm
and the guidance parameter δm. The missile’s relative state vector of the target in polar coordinates is
xR
tm =hρtm λtm atγtiT
.(19)
For simplicity, we assume that the missile has perfect information about the target, but not vice versa. To
use xR
tm in Eq. (18), we need to compute xR
tm using information that is available. As xI
i, i ∈ {t, d1, . . . , dn}are
assumed to be known accurately, thus, based on the triangulation technique, ρtm and λtm can be expressed
using the most recent estimates of ρdimand λdimand the known relative positions as follows
ρ(i)
tm =q(∆Xi
tm)2+ (∆Yi
tm)2, λ(i)
tm =atan2 ∆Yi
tm,∆Xi
tm,(20)
aThe missile is not switching between different guidance laws throughout the engagement.
9
where ∆Xi
tm is the horizontal and ∆Yi
tm is the vertical separation between the target and the missile from
the i-th defender’s perspective, respectively, given by
∆Xi
tm = ∆xtdi+ρdimcos(λdim),∆xtdi
,xdi−xt,(21a)
∆Yi
tm = ∆ytdi+ρdimsin(λdim),∆ytdi
,ydi−yt.(21b)
It is obvious that any i-th perspective pair of ρ(i)
tm and λ(i)
tm can be used to compute ρtm and λtm. However,
in order to increase the robustness of the algorithm, we propose the following weighted computation
ρtm =
n
X
i=1
wiρ(i)
tm, λtm =
n
X
i=1
wiλ(i)
tm,(22)
where wi, i = 1,2, . . . , n are nonnegative weights satisfying Pn
i=1 wi= 1. These weights shall be defined
based on the designer needs. For example, when all the nsensors are deemed to be fault-free and equally
accurate, then one can define wi=ςi/na, where na=Pn
i=1 ςiis the number of active defenders. By doing
so, Eq. (22) yields to a simple arithmetic average from the naactive perspectives. Another way of defining
the weights in Eq. (22) is to consider the actual estimation accuracy of ρdimand/or λdim, e.g., by defining
the weights as: wi=ςiσ−1
i/Pn
j=1(ςjσ−1
j), where σiis the standard deviation of the estimation error of ρdim
or λdim, obtained from the filter. Finally, as atand γtare assumed to be known, the vector xR
tm is fully
defined by the available data.
Let us denote the vector that contains all the target-defenders relative positions at time tkas
xR
td;k=h∆xtd1. . . ∆xtdn∆ytd1. . . ∆ytdniT
.(23)
Using this notation, the discrete-time version of Eq. (18), used for the estimator design, can be compactly
rewritten as
xk=fk−1(xk−1, xR
td;k),(24)
where xkis the defenders’ joint state vector xR
dm at time tk, and fk−1is a vector function derived by
integrating of Eq. (18) from tk−1to tk.
Remark 3. In Eq. (18), we assumed that the parameters of the missile dynamics (Am,Bm,Cm, and Dm)
are known. If this is not the case, the missile dynamics can be approximated by a first-order dynamics, i.e.,
Am=−1/τm,Bm= 1/τm,Cm= 1, and Dm= 0, where the parameter τmcan be treated similarly as Vm
or δm, i.e., by adding τmas an additional constant state in Eq. (18).
III.C. Combined Measurement Model
If the defenders’ measurements and their relative positions are shared, the defenders form a measuring
baseline relative to the missile in space (see Fig. 1). Such information sharing might have very important
benefits, such as improved observability due to different look angels at the missile and more measurements
that are used to improve the estimation performance.
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 j-th (j6=i) defender’s relative state
and the known relative position between these two defenders, i.e., for i, j ∈ {1, . . . , n},i6=j, we have
zdi;k=h†
dj(xk, xR
ji;k) + vdi=atan2 (∆Yj i,∆Xji ) + vdi,(25)
where
∆Xji = ∆xdjdi+ρdjmcos(λdjm),∆xdjdi
,xdi−xdj,(26a)
∆Yji = ∆ydjdi+ρdjmsin(λdjm),∆ydjdi
,ydi−ydj,(26b)
and xR
ji;k=∆xdjdi∆ydjdiT. By combining the physical measurement model of Eq. (9), its health status
defined in Eq. (10), and the indirect measurement model of Eq. (25), we can write the combined measurement
10
model as
zk,
zd1;k
.
.
.
zdn;k
=h(xk, xR
dd;k) + vk=n−1
a
×
hd1(xk)h†
d2(xR
21;k). . . h†
dn(xR
n1;k)
h†
d1(xR
12;k)hd2(xk). . . h†
dn(xR
n2;k)
.
.
.....
.
.
h†
d1(xR
1n;k). . . h†
dn−1(xR
n−1n;k)hdn(xk)
ς1
ς2
.
.
.
ςn
+vk
(27)
where xR
dd;kis a vector containing the relative positions between all active defender pairs at time tk, i.e., it
contains ∆xdjdiand ∆ydjdi, for all i, j ∈ {1, . . . , n},i6=j,ςi6= 0, and ςj6= 0. The functions hdiand h†
diare
defined in Eqs. (9) and (25), respectively, zdi;kis the physical LOS angle measurement of the i-th defender,
and vk,[vd1;k. . . vdn;k]T. Note, the argument xkin h†
diwas omitted for notation simplicity. In the next
subsection, the combined measurement model of Eq. (27) will be used to design the reduced-order estimator.
III.D. Extended Kalman Filter Design
As the estimation model of Eq. (18) is nonlinear, an extended Kalman filter (EKF) is used to estimate the
state vector defined in Eq. (17). The state estimate of the filter at time tkusing measurements up to time
tk−1,ˆxx|k−1, is propagated in time using Eq. (24) and the most up-to-dated xR
td;k. The state transition
matrix Φk|k−1associated with the system dynamics of Eq. (18) can be approximated by
Φk|k−1= exp(Fk−1|k−1T)≈I+Fk−1|k−1T, (28)
where T=tk−tk−1is the sampling time used for time propagation, Iis the identity matrix of appropriate
dimension, and Fk−1|k−1is the Jacobian matrix associated with the dynamics of Eq. (24), i.e.,
Fk−1|k−1=∂fk−1(x, xR
td)
∂x x= ˆxk−1|k−1
,(29)
is assumed to be fixed during the time interval (tk−1, tk]. The prediction error covariance matrix is
Pk|k−1= Φk|k−1Pk−1|k−1ΦT
k|k−1+Qk,(30)
where Qkis a covariance matrix of the equivalent discrete process noise, i.e.,
Qk=ZT
0
eFk−1|k−1ηΨeFT
k−1|k−1ηdη, (31)
where Ψis a matrix whose only nonzero elements are Ψ(i, i)>0,i= 2n+ 1,...,2n+nm. This matrix is
used as a tuning matrix, see [37, 45] for more details.
The measurement update stage depends on whether the measurements and the relative geometries have
been shared successfully or not. Therefore, we will first present the general equations when all the measure-
ments are available, i.e., na=n, and then we will discuss the case when na< n.
The state estimate ˆxk|k−1is updated by
ˆxk|k= ˆxk|k−1+Kkzk−h(ˆxk|k−1, xR
dd;k),(32)
where h(·)is given in Eq. (27) and Kkis the Kalman gain computed as
Kk=Pk|k−1HT
kHkPk|k−1HT
k+R−1,(33)
where Hkis the measurement Jacobian matrix and Ris the measurement noise covariance matrix
Hk=∂h(x, xR
dd;k)
∂x x=ˆxk|k−1
, R =diag nσ2
λd1m. . . σ2
λdnmo.(34)
11
Finally, the covariance matrix is updated using
Pk|k=Pk|k−1−KkHkPk|k−1.(35)
and the measurement Jacobian matrix is given by
Hk=
Hρ
d1d1. . . H ρ
dnd1Hλ
d1d1. . . H λ
dnd1
.
.
.....
.
..
.
.....
.
.
Hρ
d1dn. . . H ρ
dndnHλ
d1dn. . . H λ
dndn
[0]
x= ˆxk|k−1
,(36)
where [0] is a matrix of zeros with dimension n×(3 + nm), and
Hρ
djdi=
1
na
∆xdjdisin(λdjm)−∆ydjdicos(λdjm)
Λji
ςj, i 6=j
0, i =j
,(37a)
Hλ
djdi=
1
naΩji +ρdjmρdjm
Λji
ςj, i 6=j
1, i =j
,(37b)
with Ωji = ∆xdjdicos(λdjm)+∆ydjdisin(λdjm)and the common denominator Λji is
Λji = ∆x2
djdi+ ∆y2
djdi+ρ2
djm+ 2ρdjmΩji .
In Eq. (37), ρdjmand λdjmare replaced with the appropriate values from ˆxk|k−1. The relative positions,
∆xdjdiand ∆ydjdi, defined in Eq. (26), are considered from the k-th time frame.
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 i-th row from Hk,h(·), and
zk, respectively, and the i-th row and the i-th column from Rkare eliminated.
III.E. Comments on Implementation and Observability Issues
The reduced-order estimation scheme presented earlier allows to implement the algorithm on a single defender
only (centralized approach). In such a case, at each computation cycle (time step k), the designated defender
collects all the measurements from other missiles, acquires the relative states xR
dd;kand xR
td;k, and subsequently
computes the state estimate ˆxk|kand shares it with the other defenders.
The centralized approach allows to reduce the requirements on the on-board computers for the other n−1
defenders, because all the estimation-related computations are performed on a single on-board computer of
the designated defender. Note, however, that in this formulation, the designated defender has to solve
a higher dimension estimation problem, which might lead to higher computational burden than multiple
lower-order estimation problems run by each defender separately (decentralized approach). Moreover, the
centralized approach is more prone to failures as the defenders that do not run the estimator might lack the
updated state estimates in case of communication problems.
If the reliability and robustness of the estimation algorithm is of the utmost importance, then each
defender can compute its own joint state estimate ˆxk|k. It can also share it with the rest of the team, and
perform a cross-check (consistency check) with the other n−1estimates. Such concept will lead to increased
communication overhead and to higher computation effort of all ndefenders.
From observability point of view, as only LOS angle measurements are available, the quality of the
estimation depends upon the defenders’ trajectories and hence on the implemented defenders’ guidance law.
Based on similar analysis as in [10], the i-th defender-missile range, ρdim, can be calculated using the noisy
measurements of the i-th defender (zdi) and the j-th defender (zdj) as follows
¯ρdim=ρd
ji
sin(λd
ji −zdj)
sin(zdi−zdj), i, j ∈ {1, . . . , n}, i 6=j, (38)
12
where
ρd
ji =q∆x2
djdi+ ∆y2
djdi, λd
ji =atan2 ∆ydjdi,∆xdjdi,(39)
and ∆xdjdiand ∆ydjdiare given in Eq. (26). The range ¯ρdimcan be viewed as a pseudomeasurement at
time step k. It can be shown that ¯ρdimhas a non-stationary normal distribution, i.e.,
¯ρdim∼ N ρdim, σ2
¯ρdim,(40)
where ρdimis the true range and σ¯ρdimis the standard deviation defined as
σ¯ρdim=ρd
ji
×qsin2(λd
ji −λdim)σ2
λdjm+ sin2(λd
ji −λdjm) cos2(∆λij )σ2
λdim
sin2(∆λij ),
(41)
with ∆λij ,λdim−λdjm. From Eq. (41), it can be concluded that if the difference between the LOS angle
of the i-th defender and the j-th defender, i.e., |∆λij |, becomes small (close to zero), the variance of ¯ρdim
increases, which in turn may cause significant deterioration in the estimation accuracy, especially in the
range, missile’s speed, and time-to-go estimates. Consequently, poor estimation performance might lead to
poor guidance performance.
When using multi LOS angle measurements only, the above analysis suggests that in order to achieve good
overall state estimation performance, |∆λij|needs to be kept far from zero throughout the engagement for
all i6=j. In other words, one must ensure trajectory separation between the defenders. In the next section,
we present a guidance strategy which can enforce a specified relative intercept angle between two successive
defenders. As will be shown in Sec. V, carefully selected nonzero angle constraints for the consecutive pairs
of defenders, can naturally lead to trajectory separation between them.
Remark 4. As for any information sharing estimation concept, a possible implementation challenge might
be the delayed arrival of data (measurements and/or relative positions). Yet, as long as the internal clocks
of the defenders are accurately synchronized and the data are sent with an accurate time stamp, then the
delayed information can be easily incorporated into the estimator using estimation techniques developed for
state estimation with delayed measurements [46, 47]. However, this issue has typically a smaller effect on
the performance than communication problems or jamming [34].
IV. Cooperative Guidance Law for the Defenders
In this section, we will discuss a cooperative guidance law for the team of defenders, which can ensure
trajectory separation between the defenders. This guidance law exploits an explicit cooperation of the
defenders to impose an angular geometry at the point of intercept and an implicit cooperation between the
target and the defenders. The implicit cooperation of the target stems from the fact that the defenders
are aware of the future maneuver of the target and thus can anticipate the maneuvers it will induce on the
incoming homing missile.
IV.A. Relative Intercept Angle Guidance
Here, following the exposition in [34], we briefly recall the recently developed cooperative optimal guidance
law for imposing a relative geometry in between a group of missiles and a single moving target at intercept,
while minimizing the expected miss distance and the control effort of the missiles. The problem was posed
in the LQ optimal control framework, and solutions were obtained for any team size with any linear missile
dynamics. The guidance law was derived under the assumption of linear kinematics, perfect information,
and unbounded controls. For our target-defenders-missile scenario, we will use a slightly modified version of
this guidance law to enhance estimation by imposing nonzero relative intercept angles between consecutive
defenders. Some assumptions made in [34] are also relaxed.
Let us denote the angle between the i-th defender and the missile as γdim=γdi+γm. The difference
between the intercept angles γdimand γdi+1mis the relative intercept angle from the missile’s perspective.
This is the angle that will be enforced by the presented guidance law, see Fig. 2 which depicts the relevant
13
OI
YI
XI
Vm
Vdi
Vdi+1
ξdim
LOSdi;0
λdim;0
λdi+1
m;0
LOSdi+1;0
γdi+1
−γdi
γdi+1+γm
γdi+γm
(xdi, ydi)
(xm, ym)
(xdi+1, ydi+1)
γdi
γdi+1
γm
.
Figure 2: Relative angular geometry and linearization parameters.
relative angles and linearization parameters used within the guidance law. The optimal closed-loop guidance
law of the i-th defender was found to have the following form
udi(t) =
n
X
j=1
Nui
Zj
(tgo
dim)2Zj(t)
+
n−1
X
j=1
Nui
∆Zn+j
Vdi
tgo
dimZn+j(t)−Zn+j+1(t)−∆cj, t ∈[0, tf
dim],
(42)
where the navigation gains Nui
Zjand Nui
∆Zn+jare functions of tgo
dlm, l ∈ {1, . . . , n}and are given in [34] for the
general nmissile case. For ideal defenders’ dynamics (i.e., zero lag), the ZEM distances, Zi, i = 1, . . . , n, and
the zero-effort flight-path angles (ZEA-s) of the missile plus that of the particular defender, Zn+i, i = 1, . . . , n,
are give by [34]
Zi(t) = ξdim+˙
ξdimtgo
dim+kdimam(tgo
dim)2/2,(43a)
Zn+i(t) = γdi+γm+tgo
dimam/Vm,(43b)
where ξdimis the relative displacement between the missile and the i-th defender normal to the LOS used for
linearization, denoted as LOSi;0. In Eq. (42), ∆cirepresents the required relative intercept angle between
the i-th and the (i+1)-th defender. The linearization parameter kdimsatisfies
kdim= cos(γm;0 +λdim;0), i ∈ {1, . . . , n},(44)
and relates the missile acceleration amto the direction normal to LOSi;0. In Eq. (44), γm;0 is the initial
flight path angle of the missile and λdim;0 is the i-th defender LOS angle used for linearization.
The cooperative guidance law of Eq. (42) minimizes the following cost function
J=α1
2ξ2
d1m(tf
d1m) + . . . +αn
2ξ2
dnm(tf
dnm)
+β1
2γd1m(tf
d1m)−γd2m(tf
d2m)−∆c12+. . .
+βn−1
2γdn−1m(tf
dn−1m)−γdnm(tf
dnm)−∆cn−12
+1
2Ztf
d1
m
0
η2
1u2
d1dt+. . . +1
2Ztf
dnm
0
η2
nu2
dndt,
(45)
14
where αi,ηi, and βiare nonnegative weights. This cost function enforces 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 βi→ ∞ enforces perfect intercept
angle ∆cibetween successive defenders. The parameter ηiweights the i-th defender’s control effort.
The guidance law of Eq. (42) was derived under the assumption that the future missile maneuver as
well as the relative states are known or are accurately measured. This assumption can be directly relaxed
by making use of the proposed estimation concept presented in Sec. III. However, the ZEM distances and
the ZEA-s of Eq. (43) are valid only under the assumption that the missile maintains a known constant
maneuver throughout the engagement, i.e., um(t) = const., ∀t≥0. It is apparent that for our case this
assumption is likely to be violated as the missile is homing onto the target using one of the guidance laws
presented in Sec. III.A and the target is performing some sort of (evasive) maneuvers. Next, we will discuss
the possible guidance strategies of the target and we will present the necessary modifications of the above
guidance law to cope with the maneuvering missile problem.
IV.B. Target’s Guidance and its Implicit Cooperation
For sake of generality, in this paper, we assume that the guidance strategy of the target is arbitrary. The
only assumption that we impose is that the defenders are fully aware of the future maneuvers of the target.
Examples of possible design formulations of the target’s guidance strategy are:
1. The target may perform a constant maximum acceleration maneuver to one side or an optimally ad-
justed evasion maneuver from the homing missile. For the missile guidance laws presented in Sec. III.A
and for bounded target acceleration, the resulting optimal maneuvers have a bang-bang structure [16].
2. Obviously, the optimal evasion strategy suggested in the previous point is not necessarily the “best”
from the defenders’ perspective. Exploiting the fact that the missile is homing onto the target and
that the target’s guidance strategy directly shapes the missile’s trajectory, which in turn indirectly
influences the trajectories of the defenders, the target’s guidance law can be design such that the
defenders’ control effort is minimized. By doing so, the target can lure in the attacking missile in
regions where significantly less maneuverability is required from the defenders to hit the missile under
a predefined relative intercept angle. This problem was addressed in [30] for a single defender scenario.
3. Last but not least, the target’s guidance law can be design by combining the previous two approaches
and/or by shaping the missile’s and defenders’ trajectories into regions with added information content,
see for instance [5, 48, 49].
Remark 5. The assumption on the known target’s future maneuvers implies that the target is expected to
pursue the evasive strategy that was communicated to the defenders. In case of a manned aircraft this means
that the pilot will generate evasion flight trajectories corresponding to the communicated evasion strategy.
For a UAV, full autonomy is often required, thus an autoevasive autopilot is a natural choice.
For a given missile guidance strategy, the information about the target’s future maneuvers can be very
helpful to obtain the missile acceleration profile as a function of time, i.e., am(t). This can be achieved
via numerical integration of the appropriate engagement equations. By doing so, we can relax the constant
missile acceleration assumption in Eq. (43). Additionally, the potential intercept points of the defenders
can be predicted, which in turn can help to reduce the defenders’ acceleration demand, hence reduce the
likelihood of control saturation.
Let us assume for a while that the future missile maneuver is known and it is not constant, then based
on the terminal projection transformation, the zero-effort variables of Eq. (43) are generally given by [33, 50]
Z(t) = DΦ(di)(tgo
dim)x(di)(t) + D
tf
dim
Z
t
Φ(di)(tf
dim, τ )C(di)umdτ, (46)
where i∈ {1, . . . , n},Dis a constant row vector that pulls out the appropriate element of the zero-effort
variable, Φ(di)is the transition matrix associated with the homogenous solution of the linearized i-th defender-
missile engagement, and C(di)is a vector associated with the linearized one-sided problem, see [34] for more
details. Using Eq. (46) and replacing the LOS used for linearization in Eq. (44) with the instantaneous LOS,
15
i.e., kdi= cos(γdi−λdim), the computation of the ZEM distances and ZEA-s of Eq. (43) can be replaced by
the following equations (assuming constant missile speed Vm)
Zi(t) = ξdim+˙
ξdimtgo
dim
+Ztf
dim
t
(tf
dim−τ)am(τ) cos γm(τ) + λdim(τ)dτ,
(47a)
Zn+i(t) = γdi+γm+1
VmZtf
dim
t
am(τ)dτ, (47b)
where the integral components in Eq. (47) 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. The time propagation
can be performed e.g., by using a fourth-order Runge-Kutta (4RK) algorithm and the numerical integration
can be computed by any suitable numerical integration technique, e.g., trapezoidal rule. The integration
is performed using a constant number of integration steps from tto tf
dim. Consequently, the computation
complexity at each time step is equal and the integration resolution improves as the defenders and the missile
approach each other [30]. Note, if amis constant throughout the engagement, then Eq. (47) degenerates to
Eq. (43).
Remark 6. To avoid the constant acceleration assumption and to improve accuracy, an alternative approach
known as “predictive guidance” can be considered [51]. The idea is to compute Zi, i = 1,...,2nby integrat-
ing the appropriate engagement’s equations from tto tf
dimat each time step with nulled defenders’ controls
and presumed target’s and missile’s maneuvers. Then, by definition, Zi, i = 1, . . . , n are the miss distances
obtained by this simulation. Similarly, Zn+i, i = 1, . . . , n are the intercept angles between the particular
defender and the missile obtained from the simulation. This approach tends to yield a more accurate evalu-
ation of the ZEM and the ZEA, which can be directly used in the defenders’ guidance law implementation.
Nevertheless, this approach requires considerably more computational effort as computation of Eq. (47).
IV.C. Implementation Issues
The variables ξdimand ˙
ξdim, which appear in Zi, i = 1, . . . , n of Eq. (47a), 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
ξdim≈ρdim(λdim−λdim;0).(48)
Differentiating Eq. (48) with respect to time yields
ξdim+˙
ξdimtgo
dim=−Vρdim˙
λdim(tgo
dim)2.(49)
The left hand side of Eq. (49) is identical to the first two terms of Eq. (47a), respectively. Therefore, using
Eq. (49), ξdimand ˙
ξdimcan be 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
dim≈(−ρdim/Vρdim, Vρdim<0
0, Vρdim≥0, i ∈ {1, . . . , n}.(50)
Note that in some cases, a more accurate tgo
dimestimate might be needed as the one given in Eq. (50), see
for instance the one reported in [32, 52]. Now, all variables needed for the proper implementation of the
defenders’ guidance law given by Eq. (42) are either part of the joint defenders’ state estimate ˆxk(see the
definition in Eq. (17)) or they are assumed to be known to high accuracy (Viand γifor i∈ {t, d1, . . . , dn}).
As outlined in Section III.C, the defenders’ measurements zdi;kare acquired and shared (without any
delay) at discrete time instances tk=k·T. Similarly, execution of any computer-based algorithm is performed
in discrete time intervals. Therefore, the defenders’ control commands udi(t),i= 1, . . . , n are assumed to
be computed and executed at discrete time t=tk,k·Tc, where the flight computer’s computational cycle
Tc>0is assumed to be constant and equal to the sampling rate of the measurements, i.e., Tc=T.
16
V. Simulation Study
In this section, we analyze the proposed cooperative estimation/guidance strategy using numerical simu-
lations. Our paramount interest is to study the effect of different values of the commanded relative intercept
angle: a) on the pure estimation performance, and b) on the intertwined guidance-estimation performance
while also considering different maneuverability requirements for the defenders.
First, we present the considered engagement scenario and simulation parameters, followed by a sample
run demonstration. Finally, we present two Monte Carlo (MC) studies, one evaluating the pure estimation
performance in open loop, and the second evaluating in closed loop the intertwined guidance-estimation
performance in terms of the achieved miss distance, acceleration requirement, and intercept angle precision.
V.A. Considered Engagement and Simulation Parameters
For simulation purposes, we consider two defending missiles (n= 2). Both defenders are launched from the
targeted aircraft at the beginning of the engagement. The horizontal separation between the target and the
missile is 5 [km]. The defenders are initiated at a vertical separation of ∆ytd1= ∆ytd2=−1[m] below the
target. The target’s speed is Vt= 300 [m/s] and the speed of the two defenders and the missile is equal,
i.e., Vd1=Vd2=Vm= 500 [m/s]. We consider the missile and the target having first-order strictly proper
dynamics with time constants τm= 0.2[s] and τt= 0.5[s], respectively. Thus, matrices in Eq. (4) degenerate
to Ai=−1/τi,Bi= 1/τi,Ci= 1, and Di= 0,i∈ {m, t}. We consider ideal dynamics for the defenders. 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 a uniform distribution
on the interval [−30,30] [deg]. The target’s maneuverability is limited to umax
t= 5 [g], where g = 9.80665
[m/s2] is the standard acceleration due to the gravity. The commanded relative intercept angle ∆c,∆c1
and the defenders’ maneuverability limits umax
dbelong to closed sets Dcand Ud, defined as
Dc,{5,10,...,150}[deg],Ud,{10,20,30,40,∞,∞?}[g],
where ∞?represents a special case when, in addition to the lacking maneuverability limitation, the defenders
are also assumed to have perfect information about the missile and engagement parameters, i.e., noise-free
and without an estimator in the guidance loop. In the other cases (i.e., cases without the star), the defenders
are guided using estimated states. Note that the omitted subscript "i" for umax
dindicates that both defenders
are equally concerned, i.e., umax
d=umax
d1=umax
d2. In the rest of the paper, we will use this type of notation
for other variables too.
In all considered simulations, the missile employs PN guidance law with N0= 4. This guidance law
is implemented without an estimator in the loop (perfect information) and without any maneuverability
limits, i.e., umax
m=∞?[g]. Similarly, in all simulations, the target applies a constant maximum acceleration
maneuver to one side. The maneuver direction is chosen based on the engagement’s initial geometry as
ut=(+umax
tif γt;0 ≥0,
−umax
tif γt;0 <0,
where γt;0 is the initial flight path angle of the target. The defenders employ the cooperative guidance law
of Eq. (42) with implicit target cooperation. Based on the discussions presented in Sec. IV.B, the guidance
law of Eq. (42) for n= 2 degenerates to
ud1=Nu1
Z1
(¯
tgo
d1m)2Z1+Nu1
Z2
(¯
tgo
d1m)2Z2+Nu1
∆Z3
Vd1
¯
tgo
d1mZ3−Z4−∆c,
ud2=Nu2
Z1
(¯
tgo
d2m)2Z1+Nu2
Z2
(¯
tgo
d2m)2Z2+Nu2
∆Z3
Vd2
¯
tgo
d2mZ3−Z4−∆c,
where the navigation gains and the zero-effort variables are given in the Appendix. The numerical values of
the defenders’ guidance parameters are chosen as: α1=α2= 105,β1= 108, and η1=η2= 1.
17
The estimator developed in Sec. III is implemented 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 Ψhas been chosen
by numerical simulations. The initial state of the filter is sampled from a Gaussian distribution, i.e.,
ˆx0|0∼ N (x0, P0|0),
where x0is the true state vector and P0|0is the initial covariance matrix of the error given by
qP0|0=diag n100 100 5π/180 5π/180 2.5g5π/180 50 2o.
The nonlinear equations of motion of the target-defenders-missile engagement are solved using the 4RK
algorithm. To ensure precise evaluation of the terminal guidance performance, high resolution integration
is performed when the defenders are close to the missile. After the leading defender has passed the missile,
the simulation continues to run in order to evaluate the performance of the second defender.
V.B. Sample Run Example
Before turning to a statistical MC evaluation, first we demonstrate two sample runs for two different com-
manded relative intercept angles, namely for ∆c= 20 [deg] and for ∆c= 120 [deg]. The initial flight path
angle of the target-defender team is γt;0 =γd;0 = 10 [deg] in both examples. The defenders are guided
towards the missile using perfect information and with no maneuverability limitations, i.e., umax
d=∞?.
Figure 3 and 4 present the planar trajectories and the acceleration profiles of the target, missile, and the
two defenders in the simulated sample runs, respectively. Figure 3 also contains the achieved miss distances
and relative intercept angles for the considered runs. It can be seen from Fig. 4a that, although there is
a requirement on a specific intercept angle of ∆c= 20 [deg] (achieved with 0.01 [deg] error), 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. 4b, 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 3: Sample trajectories for two different relative intercept angles.
V.C. Pure Estimation Performance Evaluation
Here, we evaluate the estimation performance of the proposed reduced-order estimation scheme for different
relative intercept angles ∆c∈ Dc. To evaluate the estimation performance in open loop, we consider again
perfect states and no saturation for the defenders, i.e., umax
d=∞?. For each value of ∆c, a set of 500 MC
runs was performed.
Note that if ∆c= 0 [deg], the resulting trajectories of the two defenders are similar. In such a case both
defenders measure the same quantity (same look angle) and, as discussed in Sec. III.E, the triangulation
18
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 4: Sample acceleration profiles for two different relative intercept angles.
technique fails, the range, missile’s speed, and time-to-go become weekly observable, and the filter diverges.
For clarity of presentation, we have omitted the case of ∆c= 0 [deg] from our analysis.
Figure 5 shows the estimation performance for a particular intercept angle of ∆c= 20 [deg]. This angle
corresponds to the one selected in the sample run analysis in the previous subsection. In addition to the
estimated states, we also depict the error between ¯
tgo
dimcomputed using true states and ¯
tgo
dimcomputed
estimated states. We denote this error as ˜
tgo
dim. It can be observed Fig. 5, that despite the relatively small
separation between the two defenders (see Fig. 3a), the estimator performs reasonably well and the standard
deviations of the errors (actual σ) are rather consistent with those predicted by the filter.
Figure 6 shows the estimation performance for various values of ∆c. The scalar measure used to compare
the estimation performances is the actual standard deviation of the estimation errors (in Fig. 5 the line
denoted as “actual σ”), evaluated at two different time instances. One being two seconds and the other
being one second prior to the termination of the leading defender. It can be seen from Fig. 6 that for small
∆cthe estimation performance is very poor. Especially notice the case of ∆c={5,10}[deg] for γerr
mwhen
the actual σof the error is higher at tgo
dm = 1 [s] than at tgo
dm = 2 [s]. This suggest that the estimate of
γmdiverges for ∆c={5,10}[deg]. In general, as the value of ∆cincreases, the estimation performance
improves. Note the fluctuation in σ(λer r
dim). This phenomenon can be explained by the fact that the variables
λd1mand λd2mare directly measured and the dynamics of the estimator does not have a significant effect
on them. On the other hand, the magnitude of σ(λerr
dim)is actually smaller than the standard deviation of
the measurement noise.
Time [s]
0246
ρerr
d1m(m)
-200
0
200
Time [s]
0246
ρerr
d2m(m)
-200
0
200 Time [s]
0 2 4 6
λerr
d1m[deg]
-0.2
0
0.2
Time [s]
0 2 4 6
λerr
d2m[deg]
-0.2
0
0.2 Time [s]
0 2 4 6
aerr
m[m/s2]
-50
0
50
Time [s]
0 2 4 6
γerr
m[deg]
-20
0
20
Time [s]
0246
Verr
m[m/s]
-100
0
100
Time [s]
0 2 4 6
N′err
-5
0
5
Time [s]
0 2 4 6
˜
tgo
d1m[s]
-0.5
0
0.5
Time [s]
0 2 4 6
˜
tgo
d2m[s]
-0.5
0
0.5
sample error
±σfilter
mean error
actual σof the error
∆c = 20 [deg]
Figure 5: Estimation performance for a particular ∆c= 20 [deg].
19
∆c[deg]
0 50 100 150
σ(ρerr
d1m) [m]
0
20
40
∆c[deg]
0 50 100 150
σ(ρerr
d2m) [m]
0
20
40
∆c[deg]
0 50 100 150
σ(λerr
d1m) [deg]
0.02
0.04
0.06
∆c[deg]
0 50 100 150
σ(λerr
d2m) [deg]
0.02
0.04
0.06
∆c[deg]
0 50 100 150
σ(aerr
m) [m/s2]
0
20
40
∆c[deg]
0 50 100 150
σ(γerr
m) [deg]
0
5
10
actual σat t= min(tf
d1m, tf
d2m) - 2 [s]
actual σat t= min(tf
d1m, tf
d2m) - 1 [s]
∆c[deg]
0 50 100 150
σ(Verr
m) [m/s]
0
20
40
∆c[deg]
0 50 100 150
σ(Nerr)
0
2
4
∆c[deg]
0 50 100 150
σ(˜
tgo
d1m) [s]
0
0.05
0.1
∆c[deg]
0 50 100 150
σ(˜
tgo
d2m) [s]
0
0.05
0.1
Figure 6: Estimation performance as a function of ∆c∈ Dc.
V.D. Intertwined Guidance-Estimation Performance Evaluation
The effect of different values of ∆c∈ Dcon the intertwined guidance-estimation problem is analyzed here
in closed loop. The analysis is done for various considerations of the defenders’ maneuverability limit, i.e.,
umax
d∈ Ud. For each value of ∆cand umax
d, a set of nmc = 500 MC simulations was run, i.e., in total
dim(Dc)×dim(Ud)×nmc runs. The guidance performance for each MC campaign is evaluated in terms of
the achieved miss distances, defenders’ acceleration requirements, and relative intercept angle errors.
For the miss distance evaluation, we first compute the “two defender” cumulative distribution function
(CDF) which is defined on the minimum miss of both defenders. Then, using the obtained CDF, we compute
the value of the miss which corresponds to the 95% of cases. This value is denoted as miss95% and is
mathematically given by
Probmin
i∈{1,...,nmc}nρ(i)
d1m(tf
d1m), ρ(i)
d2m(tf
d2m)o≤miss95%= 0.95.
where the superscript (i)denotes the i-th MC realization. The quantity miss95% is also known as “warhead
lethality range” ensuring a 95% kill probability for the defenders 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 =Ztf
d1m
0
|ad1(τ)|dτ+Ztf
d2m
0
|ad2(τ)|dτ.
Figure 7 presents the obtained CDFs of the miss for umax
d∈ Udand a particular relative intercept angle
of ∆c= 20 [deg]. Note that the x-axis in Fig. 7 uses a logarithmic scale. The results show that with
decreasing maneuverability the guidance performance deteriorates. 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={∞,∞?}[g].
In Fig. 7, using different markers, we also depicted the values of miss95% on the respective CDFs. These
markers serve as building blocks in Fig. 8 which depicts the obtained results for all the considered intercept
angles ∆c∈ Dcand acceleration limits umax
d∈ Ud. Note that the y-axis in Fig. 8 uses a logarithmic scale.
Before commenting on Fig. 8, the results of Fig. 9 need to be introduced. Fig. 9a shows the control effort of
the defenders in terms of the amax
d(95%) measure while Fig. 9b in terms of the running cost Jacc measure.
Results presented in Figs. 6, 8, and 9 suggest, except for the case of perfect information and unbounded
control, that smaller values of ∆cyield to large miss distances due to the defenders’ control saturation
and poor estimation performance. From Fig. 10 it can be observed that the “overall” maneuverability
(represented by Jacc) increases linearly with ∆cwhile the “momentary” maneuverability (represented by
amax
d) reassembles a convex function for umax
d6=∞?. The later can be explained by bad estimation accuracy
for small values of ∆c(see Fig. 6) and by the fact that large values of ∆crequire substantially more agility
20
Miss distance [m]
10-4 10-3 10-2 10-1 100101102103104
Empirical CDF of the miss distance
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
umax
d= 10 [g]
umax
d= 20 [g]
umax
d= 30 [g]
umax
d= 40 [g]
umax
d=∞[g]
umax
d=∞⋆[g]
∆c = 20 [deg]
Figure 7: Empirical CDFs of the miss for umax
d∈ Udand a particular ∆c= 20 [deg].
from the defenders. Consequently, bad estimation accuracy (resulting from small values of ∆c) may cause
control saturation, which in turn leads to larger misses. Note, however, that the control saturation is not
the only reason for large misses for small ∆c. The estimation accuracy also plays an important role, see the
behaviour of umax
d=∞[g] for ∆c∈ {5,...,40}[deg] in Fig. 8. On the other hand, despite good estimation
accuracy, limiting the defenders’ maneuverability (umax
d)worsens the target’s protection capabilities if large
intercept angles ∆care prescribed, see Fig. 8 for ∆c∈ {70,...,150}and umax
d6=∞?. Notice that for finite
maneuverability limits (umax
d<∞), there exist a plateau effect, i.e., a region of intercept angles ∆c, where
the obtained miss is minimal. It is important to note that the defenders’ control effort could be further
reduced by an appropriate design of the target’s guidance, see the discussion in Sec. IV.B.
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
umax
d= 10 [g]
umax
d= 20 [g]
umax
d= 30 [g]
umax
d= 40 [g]
umax
d=∞[g]
umax
d=∞⋆[g]
Figure 8: Values of miss95% for umax
d∈ Udas a function of ∆c∈ Dc.
Finally, Fig. 10 depicts the mean and the corresponding 1-sigma envelope of the relative intercept angle
error as a function of ∆c∈ Dc. For perfect information and no acceleration bound case (umax
d=∞?), it
can be seen from Fig. 10a that as ∆cbecomes larger, the error biases towards negative values (meaning that
the achieved relative intercept angle is smaller than the prescribed one) and the error variance increases.
Such behavior results from a trade-off in the defenders’ cost function formulation, see Eq. (45), which at
the same time penalizes the control effort, miss, and the relative intercept angle. Furthermore, the linearity
assumptions for larger ∆cis presumably also less valid. Similar conclusions can be drawn for the imperfect
21
information case (umax
d=∞)from Fig. 10b. The only difference is that the error variance also increases
for smaller values of ∆c. This is understandable as the estimation performance significantly deteriorates for
∆c→0.
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
umax
d=∞[g]
umax
d=∞⋆[g]
(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
umax
d=∞[g]
umax
d=∞⋆[g]
(b) Values of Jacc as a function of ∆c∈ Dc.
Figure 9: Guidance performance - acceleration requirements for unsaturated case.
Commanded relative intercept angle [deg]
0 50 100 150
Relative intercept angle error [deg]
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
mean
1σenvelope
(a) umax
d=∞?[g] (perfect states)
Commanded relative intercept angle [deg]
0 50 100 150
Relative intercept angle error [deg]
-15
-10
-5
0
5
10
mean
1σenvelope
(b) umax
d=∞[g] (estimated states)
Figure 10: Guidance performance - relative intercept angle error as a function of ∆c∈ Dc.
VI. Conclusions
A cooperative estimation/guidance strategy has been proposed for a team of missiles. An example
scenario is considered where these missiles are in a role of defending missiles used to cooperatively intercept an
attacking missile homing on to a target aircraft. A new reduced-order estimation scheme based on information
sharing to cooperatively estimate the relative state and the unknown parameters of the attacking missile
has been proposed under the assumption that only shared LOS angle measurements from the defenders are
available. The defenders’ guidance exploits an explicit team cooperation to impose relative intercept angle
constraints between consecutive defenders and an implicit cooperation of the target aircraft. The cooperation
from the target’s point of view stems from the fact that the defenders are aware of the evasion strategy of
the target and thus can predict the maneuvers it will induce on the homing missile.
Extensive nonlinear simulations revealed that there is a strong influence of the defenders’ cooperative
guidance on to the estimation performance and vice versa. For a team of two defenders and the target
performing a constant turn at 5 [g], it was found that imposing different relative intercept angles lead
to distinct effects on the pure estimation and on the intertwined guidance-estimation performance. Small
relative intercept angles yield to observability issues. This consequently results in control saturation and
severe degradation in the intercept performance. Relative angles ranging from approx. 30 [deg] to approx. 65
22
[deg] exhibit good estimation as well as guidance performance while maintaining modest maneuverability
requirements. Larger intercept angles lead only to negligible improvements in the estimation accuracy, while
limiting the defenders’ maneuverability worsens the target’s protection capabilities for too large intercept
angle commands.
The demonstrated capability of the proposed cooperative algorithm can, for carefully selected relative
intercept angles, considerably improve the aircraft’s survivability from a homing missile, making it possible to
design relatively inexpensive defending missiles without advanced sensor systems and large lethal warheads.
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.
Appendix: Defenders’ Guidance Parameters
The navigation gains of the defenders’ guidance law for a team size of two and η1,1are [34]
Nu1
Z1= 3kd1(tgo
d1m)3α1[tgo
d2mV2
d1C22 + 2V2
d2η2
2C21(2V2
d1+β1tgo
d1m)]/∆z
Nu1
Z2= 3kd2(tgo
d2m)2(tgo
d1m)2Vd1Vd2β1α2η2
2(6 −k2
d1(tgo
d1m)3α1)/∆z
Nu1
∆Z3= 2V2
d2β1η2
2(tgo
d1m)C21(k2
d1(tgo
d1m)3α1−6)/∆z
Nu2
Z1= 3kd1(tgo
d1m)2(tgo
d2m)2Vd1Vd2β1αa(6η2
2−k2
d2(tgo
d2m)3α2)/∆z
Nu2
Z2= 3kd2(tgo
d2m)3α2[tgo
d1mV2
d2C12 + 2V2
d1C11(2V2
d2η2
2+β1tgo
d2m)]/∆z
Nu2
∆Z3= 2V2
d1β1(tgo
d2m)C11(6η2
2−k2
d2(tgo
d2m)3α2)/∆z
where
∆z=V2
d2tgo
d1mC12 +V2
d1tgo
d2mC11C22 + 4V2
d1V2
d2C11C21 η2
2
C11 = 3 + k2
d1(tgo
d1m)3α1, C12 =β1η2
212 + k2
d1(tgo
d1m)3α1,
C21 = 3η2
2+k2
d2(tgo
d2m)3α2, C22 =β112η2
2+k2
d2(tgo
d2m)3α2,
kd1= cos(γd1−λd1m), kd2= cos(γd2−λd2m).
The zero-effort-miss distances, Z1and Z2, are
Z1(t) = −Vρd1m˙
λd1m(tgo
d1m)2
+Ztf
d1m
t
(tf
d1m−τ)am(τ) cos γm(τ) + λd1m(τ)dτ,
Z2(t) = −Vρd2m˙
λd2m(tgo
d2m)2
+Ztf
d2m
t
(tf
d2m−τ)am(τ) cos γm(τ) + λd2m(τ)dτ,
whereas the zero-effort flight-path angles, Z3and Z4, are as given in Eq.(47b).
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