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Estimation Enhancement by Imposing a Relative

Intercept Angle for Defending Missiles

Robert Fonod∗and 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 =Vλim/ρim ,i∈ {t, d1, d2},(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 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

i∈Rniis 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(γt−λtm).(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

tm/τm.(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 −3e−2θtm −12θtme−θtm + 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

d1m−τ)am(τ) cos (γm(τ) + λd1m(τ)) dτ, (23a)

Z2(t) = ξd2m+˙

ξd2mtgo

d2m+Ztf

d2m

t

(tf

d2m−τ)am(τ) cos (γm(τ) + λd2m(τ)) dτ, (23b)

Z3(t) = γd1+γm+Ztf

d1m

t

˙γm(τ)dτ =γd1+γm+1

VmZtf

d1m

t

am(τ)dτ, (23c)

Z4(t) = γd2+γm+Ztf

d2m

t

˙γm(τ)dτ =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

ξdim≈ρdim(λdim−λdim;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=Vλd1m/ρd1m

˙

λd2m=Vλd2m/ρd2m

˙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

,xdi−xt,(32a)

∆Ydi

tm = ∆ytdi+ρdimsin(λdim),∆ytdi

,ydi−yt.(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= [∆xtd1∆ytd1∆xtd2∆ytd2]T.(33)

By this, the discrete-time version of Eq. (29), used for the estimator design, can be compactly rewritten as

xk=fk−1(xk−1, xR

td;k),(34)

where xkis the defenders’ joint state vector xR

dm at time tk, and fk−1is a vector function derived by

integrating of Eq. (29) from tk−1to 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= [∆xd1d2∆yd1d2∆xd2d1∆yd2d1]T, and

∆Xji = ∆xji +ρjm cos(λj m),∆xj i ,xi−xj,(36a)

∆Yji = ∆yji +ρjm sin(λj m),∆yj i ,yi−yj.(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 tk−1,ˆxx|k−1, is propagated in

time using Eq. (34) and the most up-to-dated xR

td;k. The state transition matrix Φk|k−1associated with the

system dynamics of Eq. (29) can be approximated by

Φk|k−1= exp(Fk−1|k−1T)≈I+Fk−1|k−1T, (38)

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. (34), i.e.,

Fk−1|k−1=∂fk−1(x, xR

td)

∂x x=ˆxk−1|k−1

,(39)

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+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|k−1is updated by

ˆxk|k= ˆxk|k−1+Kkzk−h(ˆxk|k−1),(41)

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where Kkis the Kalman gain computed as

Kk=Pk|k−1HT

kHkPk|k−1HT

k+R−1.(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|k−1

, 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|k−1−KkHkPk|k−1,(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=−1/τi,Bi= 1/τi,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)|dτ +Rtf

d2m

0|ad2(τ)|dτ.

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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