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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.

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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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