Development of 3-D Modified Proportional Navigation Guidance Law against High-Speed Targets

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Development of 3-D Modified ProportionalNavigation Guidance Law against High-SpeedTargets

The work presented here proposes a terminal guidance law

against very high-speed targets. Proportional navigation guidance

(PNG) is mostly effective for interceptors and targets with

ideal dynamics and constant speeds during the collision course,

however, it is nonrobust for the objects with nonconstant speeds,

such as the interceptor equipped with thrust-vector-control

(TVC). To resolve the problem the interceptor considered

here is guided by a 3-dimensional (3-D) modified proportional

navigation guidance (MPNG) law, which is developed to account

for the interceptor’s axial acceleration. The guidance law is

potentially applicable to interceptors with TVC to provide lateral

accelerations.

NOMENCLATURE

LOS Line of sight

(XI ,YI ,ZI) Reference coordinate system

(XL,YL,ZL) LOS coordinate system

(XM ,YM ,ZM) Interceptor body coordinate system

(XT,YT,ZT) Target body coordinate system

(iI,jI,kI) Unit vectors corresponding,

respectively, to (XI ,YI ,ZI) axes

(iL, jL,kL) Unit vectors corresponding,

respectively, to (XL,YL,ZL) axes

(iM, jM,kM) Unit vectors corresponding,

respectively, to (XM ,YM ,ZM) axes

(iT, jT,kT) Unit vectors corresponding,

respectively, to (XT,YT,ZT) axes

r Relative distance between

interceptor and target

μL,ÃL Euler angles from reference

coordinate system to LOS

coordinate system

μM ,ÃM Euler angles from LOS coordinate

system to interceptor’s body

coordinate system

μT,ÃT Euler angles from LOS coordinate

system to target’s body

coordinate system

Manuscript received September 9, 2010; revised April 15 and

October 6, 2011; released for publication July 23, 2012.

IEEE Log No. T-AES/49/1/944383.

Refereeing of this contribution was handled by M-J. Tahk.

This research was sponsored by Chung Shan Institute of Science

and Technology, Taiwan, ROC under Grant XB99086.

0018-9251/13/$26.00 c° 2013 IEEE

CORRESPONDENCE 677

Ám,°m,Ãm Euler angles (flight path angles)from reference coordinate systemto interceptor body coordinatesystem

°t,Ãt Euler angles (reentry angles) frominterceptor body coordinatesystem to reference coordinatesystem

°c,Ãc Flight path control commands

(xpip,ypip,hpip) Position of predict interception point(PIP) in reference coordinatesystem

(xm,ym,hm) Position of interceptor in referencecoordinate system

Smh(Sth) Distance from missile (target) to thecollision point C in LOSH plane

VM(VT) Interceptor (target) velocity vector

AM(AT) Interceptor (target) accelerationvector

ACM Acceleration command vector ofinterceptor

Acy(Acp) Yaw (pitch) acceleration commandof interceptor

tf Interceptor flight time

tgo Time-to-go

C The collision point of interceptorand target

D Drag force of the interceptorg Gravity accelerationL Lift force of the interceptorN Navigation gainT Thrust force of the interceptorW Target weightmfuel Propellant mass of the interceptor

tb Fuel burn time of the interceptor

® Angle-of-attack of the interceptor¯ Ballistic coefficient of the target½ Air densityCmD Drag coefficient of the interceptor

CmD0(CtD0) Zero-lift drag coefficient of theinterceptor (target)

CmL Lift coefficient of the interceptor

CmL®

= @CmL=@®

st Reference area of the target

( _̧ x,_̧y,_̧z) XL,YL,ZL components of the LOS

angular velocity vector(Vmx

L,Vmy

L,Vmz

L) XL,YL,ZL components of the

interceptor velocity (Vm)

(Vtx,Vty,Vth) XI ,YI ,HI components of the targetvelocity (Vt)

(Atx,Aty,Ath) XI ,YI ,HI components of the targetacceleration (At)

I. INTRODUCTION

The proportional navigation guidance (PNG)

law [1—3] has long been adopted in air-to-air

or surface-to-air homing missiles because of its

simplicity and effectiveness in defeating targets with

low maneuverability. However, facing very high-speed

ballistic targets, the traditional PNG law may not be

competent anymore. During the past few decades, a

variety of guidance design techniques, such as linear

quadratic regulator (LQR) [4], modified proportional

guidance [5], extended proportional navigation [6],

neural-fuzzy-based guidance [7—8], and geometric

control [9], have been widely attempted to deal with

the guidance design problem for high-speed target

engagement. Traditionally, mid-course guidance is

formulated mostly as an optimal control problem

to shape the flight trajectory and to maximize the

terminal speed or minimize the flight time. However,

solving LQR problems or training neural networks in

real time is often practically infeasible, especially in

the actual defense industry. Neural network guidance

might be unreliable in practice if neural networks are

not trained adequately.

The analysis of pure PNG law in 3-dimension

(3-D) was first studied in [10]. The performance

of pure PNG law is altered while the missile and

target motions deviate from the intercept course.

In [11]—[12] the demonstration of this pure PNG

was based on a missile with constant velocity. A

3-D PNG law according to the radial basis function

(RBF) neural network was proposed in [13], which

was applied to decrease miss distance (MD) between

aircraft and missile that travel in constant speed.

The issue of 3-D PNG law was discussed in [14] by

nonlinear differential equations and vector analysis.

The results of these references have characterized

the properties of the pure PNG law based on the

nonlinear dynamics of interception situations.

However, the issue of the interceptor traveling

with axial acceleration in 3-D space is much closer

to the real-world situations of interceptor-target

engagement.

Literature detailing the issues of medium or

terminal phase guidance law design subject to

high-speed targets is very rare. Here, a 3-D modified

proportional navigation guidance (MPNG) law is

derived, which accounts for the interceptor’s axial

acceleration and variation of the target acceleration.

It is, thus, especially applicable to interceptors

configured with a thrust-vector-control (TVC)

system for attitude control. In the medium guidance

phase, the interceptor trajectory is shaped by directly

controlling its flight path toward the incoming target

for energy expenditure savings. For the homing

phase the interceptor is guided by the MPNG law for

enhancement of agility and endgame performance.

The effect of the proposed design is verified via

numerical simulation to examine its engagement

performance. MD analysis is conducted for endgame

performance verification.

678 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 49, NO. 1 JANUARY 2013

Fig. 1. 3-D interception geometry.

II. DERIVATION OF 3-D MPNG LAW

The 3-D pursuing scenario is illustrated in Fig. 1.

The line-of-sight (LOS) equations are given as

follows:

_r = Vt cosμT cosÃT¡Vm cosμM cosÃMr _μL = Vt sinμT¡Vm sinμM

rcosμL_ÃL = Vt cosμT sinÃT¡Vm cosμM sinÃM

(1)

where (μM ,ÃM) and (μT,ÃT) are Euler angles from the

LOS coordinates to the interceptor and target body

coordinate systems, respectively. The angles μL and ÃLare Euler angles from the reference coordinate to the

LOS coordinate system. The unit vectors (iL, jL,kL) in

the LOS coordinate can be expressed as

iL = cosμL cosÃLiI+cosμL sinÃLjI¡ sinμLkIjL =¡sinÃLiI+cosÃLjIkL = sinμL cosÃLiI+sinμL sinÃLjI+cosμLkI

(2)

where (iI,jI,kI) are the unit vectors corresponding,

respectively, to (XI ,YI ,ZI) axes in the reference

coordinate system.

By the definitions of Euler angles μM and ÃM ,

the unit vectors (iM,jM,kM) in the interceptor body

coordinate system are given by

iM = cosμM cosÃM iL+cosμM sinÃMjL¡ sinμMkLjM =¡sinÃMiL+cosÃMjL (3)

kM = sinμM cosÃM iL+sinμM sinÃMjL+cosμMkL:

The missile velocity vector VM in the interceptor

body coordinate is VM = VmiM. Using the results

of (3), the missile velocity vector VM in the LOS

coordinate can be divided into three components as

VM = VmiM = VmxLiL+Vmy

LjL+Vmz

LkL (4)

Fig. 2. Interception geometry of LOSV plane.

Fig. 3. Interception geometry of LOSH plane.

where

VmxL= Vm cosμM cosÃM (5)

VmyL= Vm cosμM sinÃM (6)

VmzL=¡Vm sinμM: (7)

For convenience the two-dimensional (2-D)

subspace (XL,ZL) of the LOS coordinate system

(XL,YL,ZL) is denoted as the LOSV plane, illustrated as

in Fig. 2. On the other hand the 2-D subspace (XL,YL)

is perpendicular to the LOSV plane, denoted as the

LOSH plane, and is shown as in Fig. 3.

A. Interceptor with Constant Axial Acceleration

Consider the LOSH plane and, using the

relationship of the triangle geometry, one can obtain

SthsinÁMh

=Smh(tgo)

sinÁTh=

r

sin(ÁTh¡ÁMh): (8)

Using Sth = VthLtgo yields

tgo =r sinÁMh

VthLsin(ÁTh¡ÁMh)

(9)

r sinÁTh¡ Smh(tgo)sin(ÁTh¡ÁMh) = 0 (10)

where Smh(tgo) is the distance that the interceptor has

traveled in time tgo in the LOSH plane.

CORRESPONDENCE 679

By using the LOS equations:

_r = VthLcosÁTh¡Vmh

LcosÁMh (11)

r _¾h = VthLsinÁTh¡Vmh

LsinÁMh (12)

and substituting (11) and (12) into (9), one can have

tgo =r sinÁMh

r _¾h cosÁMh¡ _r sinÁMh=

r

r _¾h cotÁMh¡ _r: (13)

From (9), (10), and (12), one gets

_¾h+1

r[Vmh

L¡ V̄mh

L] sinÁMh = 0 (14)

where the average speed V̄mhL= Smh(tgo)=tgo.

For a variable-speed interceptor heading to a

constant-speed target, Gutman’s result [15] was

_°Mh =N

·_¾h+

1

r(Vmh

L¡ V̄mh

L) sinÁMh

¸(15)

where N can be treated as the navigation guidance

gain. In particular the interceptor flies along the

interception course, with _°Mh = 0 in the LOSH plane

yielding

_¾h =¡1

r(Vmh

L¡ V̄mh

L)sinÁMh 6= 0: (16)

While _¾h = 0, by differentiating (12) and °Mh =

ÁMh+¾h, one obtains

_°Mh =¡_Vmh

L

VmhL

tanÁMh 6= 0: (17)

For practical purposes elaborating (15) in terms of

time-to-go gives

VmhL= Vmhf

L¡Amhx

Ltgo

Smh = VmhfLtgo¡

1

2Amhx

Lt2go

V̄mhL=Smh(tgo)

tgo= Vmhf

L¡ 12Amhx

Ltgo

(18)

where VmhfLis the final missile speed in the LOSH

plane and AmhxLis the missile longitudinal acceleration

in the LOS coordinates. Substituting (18) into (14)

givesr

tgo_¾h¡

1

2Amhx

LsinÁMh = 0: (19)

From the time-to-go equation (13), r=tgo =

r _¾h cotÁMh¡ _r, and substituting this into (19) gives(r _¾h cotÁMh¡ _r) _¾h¡ 1

2Amhx

LsinÁMh = 0: (20)

Recall (13) again

tgo =r

r _¾h cotÁMh¡ _r=

2r _¾hAmhx

LsinÁMh

: (21)

Using Vc =¡_r for the closing speed, the suggestedlateral command for collision in the LOSH plane is

AmhL=N 0 cosμM((Vc+ r _¾h cotÃM) _¾h¡ 1

2Amx cosμM sinÃM)

(22)where N 0 =NVm=Vc.The lateral acceleration command in the LOSH

plane can be translated in the interceptor body

coordinates by

AMHL=N 0 cosμM((Vc+ r

_̧z cotÃM)

_̧z ¡ 1

2Amx cosμM sinÃM )jM

(23)

where _̧ z is the LOS angular velocity vector in ZLcomponents in LOS coordinates. Using the same

procedures in LOSH before, the lateral acceleration

command in the LOSV plane is

AMVL=N 0[(Vc+ r

_̧y cotμM cosÃM)

_̧y ¡ 1

2Amx sinμM]

£ (sinμM sinÃM jM¡ cosÃMkM) (24)

where _̧ y is the LOS angular velocity vector in YLcomponents in LOS coordinate system.

The MPNG law generating the lateral

acceleration commands in the 3-D case can finally be

represented by

ACM =N0[(Vc+ r

_̧y cotμM cosÃM)

_̧y sinμM sinÃM

+(Vc+ r_̧z cotÃM)

_̧z cosμM ¡ 1

2Amx sinÃM]jM

¡N 0 cosÃM[(Vc+ r _̧ y cotμM cosÃM) _̧ y¡ 1

2Amx sinμM]kM: (25)

B. Interceptor with Changing Axial Acceleration

Assume that the target speed is constant and that

the interceptor has changing axial acceleration. First,

we focus on the LOSH plane. Let

AmhxL(tgo) = Amhxf

L¡ _Amhx

Ltgo (26)

where AmhxfLis the final acceleration at the predict

interception point (PIP), which is projected to the

LOSH plane, and assume that_Amhxf

Lis constant.

Then, the interceptor speed and traveling distance can

be expressed by integrating (26) and is obtained as

follows

¢VmhL= Amhxf

Ltgo¡ 1

2_Amhx

Lt2go

VmhL(tgo) = Vmhf

L¡Amhxf

Ltgo +

12_Amhx

Lt2go

Smh(tgo) = VmhfLtgo¡ 1

2Amhxf

Lt2go +

16_Amhx

Lt3go

V̄mhL= Vmhf

L¡ 1

2Amhxf

Ltgo +

16_Amhx

Lt2go:

(27)

Substituting (26) and (27) into (14) yields

r

tgo_¾h¡

1

2

μAmhxf

L¡ 23_Amhx

Ltgo

¶sinÁMh = 0: (28)


The lateral acceleration command in the LOSHplane can be translated into the interceptor body

coordinates as

AMHL=

N 0 cosμM((Vc+ r_̧z cotÃM)

_̧z ¡ 1

2(Amx+

13_Amxtgo)

£ cosμM sinÃM)jM: (29)

Similarly, the lateral acceleration command in the

LOSV plane can be obtained as

AMVL=N 0[(Vc+ r

_̧y cotμM cosÃM)

_̧y

¡ 12(Amx+

13_Amxtgo)sinμM]

£ (sinμM sinÃMjM¡ cosÃMkM): (30)

Thus, the MPNG law in the 3-D case can finally

be summarized as follows

ACM =N0[(Vc+ r

_̧y cotμM cosÃM)

_̧y sinμM sinÃM

+(Vc+ r_̧z cotÃM)

_̧z cosμM

¡ 12(Amx+

13_Amxtgo)sinÃM]jM

¡N 0 cosÃM"(Vc+ r

_̧y cotμM cosÃM)

_̧y

¡ 12(Amx+

13_Amxtgo)sinμM

#kM:

(31)

III. ENGAGEMENT STRATEGY

The guidance scheme processes three phases:

vertical launch, shaping and mid-course, and

terminal guidance phase, illustrated as in Fig. 4. In

practice target position and velocity information

in the Cartesian inertial coordinate are uplinked to

the interceptor from the ground-based radar, fire

control site, or satellite. For the estimation of target

trajectories using filters, one can refer, for example,

to [16]. The interceptor’s position, velocity, and

acceleration information in the Cartesian inertial frame

are obtained from an inertial reference unit.

A. Vertical Launch Phase

The vertical guidance follows the one developed in

[17]. During this phase the interceptor body is forced

to incline and commence the course of engagement

while it reaches a favorable altitude.

B. Shaping and Mid-Course Guidance Phase

During this phase the interceptor trajectory

is shaped by directly controlling its flight path

toward the incoming target for energy expenditure

savings. The interceptor is configured with the TVC

mechanism at the tail to provide maneuverability. The

attitude and flight path are controlled by a movable

thrust nozzle instead of the aerodynamically controlled

fins. We simply utilize flight path control to allow for

Fig. 4. Engagement scenario for vertical launch interceptor.

quick attitude changes. The key point in the design

is to predict the PIP with its coordinates denoted

by (xpip,ypip,hpip), which can be determined through

integration of the dynamic equation of the target at the

ground control center.

Based on the predicted PIP, one can get the

interceptor’s flight path control commands °c and Ãcas follows

°c = tan¡1

0@ hpip¡ hm0q(xpip¡ xm0)2 + (ypip¡ xm0)2

1A (32)

Ãc = tan¡1Ãypip¡ ym0xpip¡ xm0

!(33)

where (xm0,ym0,hm0) are the initial coordinates of the

interceptor.

The interceptor, target models, relative velocity,

and distance are applied to the PIP estimation to

evaluate the desired flight path angles °c and Ãc. The

actual flight path angles °m and Ãm are calculated

and input into the interceptor’s flight control system

for trajectory shaping. Because our interceptor is a

two-stage rocket, there still are 3 subphases which

CORRESPONDENCE 681

should be taken into consideration, i.e., the 1st stage

mid-course, coast, and 2nd stage mid-course. In our

engagement strategy the 1st stage mid-course phase

uses the flight path control. When the interceptor is in

coast, there is no guidance applied, and the interceptor

conducts free flight toward the PIP. After coast the 1st

stage is separated after the 2nd stage glint. The 2nd

stage mid-course phase is designed to adopt the same

MPNG guidance law as that of the terminal phase to

avoid causing an unsmoothed transient while changing

guidance phases.

C. Terminal Guidance Phase

For the terminal phase, the interceptor is guided

by the MPNG law for enhancement of interception

accuracy which was given by (31).

IV. SIMULATION RESULTS

By simulation we verify the guidance performance

of the MPNG law. The simulation is divided into two

scenarios, interceptor with constant and varying axial

acceleration, which is shown below, respectively. All

the simulation results of MPNG are compared with

traditional PNG.

Consider the 3-D translational equations of motion

of the interceptor’s motion:

_vm = (Tcos®¡D)=m¡ g sin°m, vm(0) = 0

(34)

_°m = (L+T sin®)cosÁm=(mvm)¡ g cos°m=Vm,°m(0) = °m0 (35)

_Ãm = (L+T sin®)sinÁm=(mvm cos°m),

Ãm(0) = Ãm0 (36)

_xm = Vm cos°m cosÃm, xm(0) = xm0 (37)

_ym = Vm cos°m sinÃm, ym(0) = ym0 (38)

_hm = Vm sin°m, hm(0) = hm0 (39)

where m is the missile mass, T is the thrust, ® is the

angle-of-attack, °m is the flight path angle, Ãm is

the azimuth angle, the lift force L= (½v2msmCmL)=2,

and the drag force D = (½v2msmCmD)=2, with

CmL = CmL®(®¡®0) and CmD = CmD

0+¹C2mL. The

aerodynamic derivatives CmL®, CmD

0, and ¹ are given

as functions of the Mach number, that is, the function

of the velocity vm and the altitude hm. sm is the

reference area of the missile, and ½ is the air density

which is measured in kg/m3 and shown to be an

accurately exponential approximation as

½=

(1:22557e¡h=9144, h < 9,144 m

1:7523e¡h=6705:6, h¸ 9,144 m:

Fig. 5. Definitions of interceptor’s flight path angle °m and Ãm.

Fig. 6. Geometry of reentry target and ground-based radar.

Referring to Fig. 5 the flight path angles °m and

Ãm can be treated as the control variables on the

vertical and horizontal plane guidance laws during the

1st stage mid-course guidance phase, respectively.

In modeling the ballistic target motion for

simulation study, we consider the target vehicle in the

reentry phase to be located over a flat, nonrotating

Earth with gravity. The geometry illustration of the

reentry target and radar system is shown in Fig. 6.

The ballistic target model in the radar coordinates

centered at the radar site can be described as

_Vtx =¡½V2t2¯

g cos°t cosÃt+Atx, Vtx(0) = Vtx0

(40)

_Vty =½V2t2¯

g cos°t sinÃt+Aty, Vty(0) = Vty0 (41)

_Vth =¡½V2t2¯

g sin°t¡g+Ath, Vth(0) = Vth0 (42)

where Vtx, Vty, and Vth are velocity components of Vtin the (XI ,YI ,HI) axes, respectively; Atx(t), Aty(t), and

Ath(t) are uncertain accelerations due to target model

uncertainties and maneuverability.


TABLE I

Parameter Settings for Generic Short/Medium Range Ballistic

Target

Item Characteristic

Length 10.0 m

Diameter 1.0 m

Reentry Velocity 1800 m/s

Reentry Weight 1200 kg

Ballistic Coefficient 2440 kg/m2

TABLE II

General Description of the Interceptor

Item Characteristic

Length 7.0 m

Diameter 0.8 m

Payload 500 kg

Range 100 km

Zero drag coefficient CmD0 = 0:45¡ (0:04=3) MachLift coefficient CmL

®= 2:93+0:34 Mach+

0:262 Mach2 +0:011 Mach3

Induced drag coefficient ¹= 0:053

Launch Weight 2540 kg

Stages Stage 1 Stage 2

Diameter 0.8 m 0.35 m

Length 3.4 m 3.6 m

Launch Weight 2330 kg 210 kg

Propellant Mass 1320 kg 65 kg

Burn-out Time 22 s 18 s

TABLE III

Performance Analysis of the Specific Guidance Gain

Item N = 3 N = 5 N = 7

MaxjAcpj (G) 3.6 3.6 28.3

MaxjAcy j (G) 4.1 1.6 9.4

MaxjAcj (G) 5.5 3.93 29.8

MD (m) 2.1 3.8 29

The incoming flight path angles °t, Ãt, and the

ballistic coefficient ¯ are given as

°t(t) = tan¡1

0@ ¡VthqV2tx +V

2ty

1A (43)

Ãt(t) = tan¡1μVty

Vtx

¶(44)

¯ =W

stCtD0(45)

where st, W, and CtD0 represent the reference area,

weight, and zero-lift drag coefficient of the ballistic

target, respectively.

Parameter settings for target (see Table I) and

interceptor (see Table II) are adopted for numerical

simulation [18]. The mass variation effect of the

interceptor is taken into account. The fuel burn-out

time is tbM1 (1st stage) and tbM2 (2nd stage),

Fig. 7. Demonstrative engagement trajectory, where interceptor

has constant axial acceleration.

respectively. We take the fuel burning rate _mfuel by

_mfuel =¡mfueltb

(46)

where mfuel is the propellant mass, which can be

found in Table II. The mass variation has to consider

the fuel burning rate and the interceptor’s stage

separation. For the ballistic target, because the

propellant mass has already burned out during the

reentry phase, the target’s mass can be treated as a

constant. A first-order transfer function (1=(0:3s+1))

is considered for attitude control of the interceptor to

reflect the dynamic delay of the TVC.

A. Interceptor with Constant Axial AccelerationScenario

A Poisson evasive model is used to emulate the

random target maneuver, with the magnitudes of

aty(t) and ath(t) being 1 g. This scenario simulates

the interceptor’s constant axial acceleration against

the target with high terminal speed and compares

the performance of PNG and MPNG. One of the

representative results for the target with the reentry

coordinates (31000, 30000, 135000) (m), °t = 70±,

and Ãt = 140± is demonstrated. After the interceptor is

launched, the engagement trajectory is recorded and is

illustrated in Fig. 7.

The different guidance gain, N = 3, 5, and 7 have

been checked in the simulation study to determine

the optimal one in MPNG. The simulation results

are summarized in Table III. When the interceptor

engages a ballistic target at medium- or higher tier in

the space judgment for a successful engagement, the

acceptable MD is considered subject to the tolerable

lateral acceleration, i.e.,

J =

Ã1

n

nXi=1

jMDij!·MDmax (47)

CORRESPONDENCE 683

Fig. 8. Time history of lateral acceleration command Acy in missile with constant axial acceleration by using: (a) PNG and (b) MPNG.

TABLE IV

Comparison of PNG and MPNG for a Constant Axial

Acceleration Interceptor

Item PNG MPNG

MaxjAcpj (G) 0.5 0.64

MaxjAcy j (G) 8.8 3.9

MaxjAcj (G) 8.81 3.95

MD (m) 1.3 0.7

TABLE V

Comparison of PNG and MPNG for a Varying Axial Acceleration

Interceptor

Item PNG MPNG

MaxjAcpj (G) 1.24 1.5

MaxjAcy j (G) 13.8 4

MaxjAcj (G) 13.9 4.3

MD (m) 8.9 1.3

Fig. 10. Time history of lateral acceleration command Acy in missile with varying axial acceleration by using: (a) PNG and (b) MPNG.

Fig. 9. Demonstrative engagement trajectory, where interceptor

has varying axial acceleration.


Fig. 11. Comparison of defensible volumes between PNG (dash line) and MPNG (solid line) based designs for interceptor with

constant axial acceleration variety against target reentry angle of (a) 45±, (b) 60±, and (c) 75±.

s.t. qA2cp(t)+A

2cy(t)· Acmax, 0· t· tf (48)

where n is the number of total Monte Carlo runs (in

this research n= 10), MDmax(= 4 m) is the allowable

MD, tf is the total fight time, and Acmax(= 4:5 g) is

the allowable lateral acceleration.

For N = 3 the MD observed is under the

requirement, but the lateral acceleration has already

exceeded the lateral acceleration limit. When N = 7

both the MD and lateral acceleration exceed the

requirements. When N = 5 the simulation result

gets less lateral acceleration with an acceptable MD.

Therefore, N is assigned as 5 in the subsequent

simulation in the study. To compare the interception

performance in PNG and MPNG (both with N = 5),

the results of the lateral acceleration commands in

the vertical (Acp) and horizontal (Acy) planes and MD

CORRESPONDENCE 685

are listed in Table IV. The lateral accelerationcommand Acy for the interceptor guided by PNG andMPNG is displayed in Fig. 8. The simulation resultsshow that the interceptor with MPNG demonstratesmore superior performance than the others with PNGin less lateral acceleration command and smaller MD.

B. Interceptor with Varying Axial Acceleration Scenario

Consider the scenario in which the interceptortravels with varying axial acceleration. The target’sinitial location is also at (31000,30000,135000) (m),and °t = 70

± and Ãt = 140± are considered. The

engagement trajectory is recorded and illustratedin Fig. 9. The results of the lateral accelerationcommands in the vertical (Acp) and horizontal (Acy)planes and MD are listed in Table V. The lateralacceleration command Acy for the interceptor guidedby PNG and MPNG is displayed in Fig. 10. Thesimulation results also show that the interceptor withMPNG demonstrates a more superior performancethan the other with PNG in less lateral accelerationcommand and smaller MD.

C. Defensible Volume

The allowable upper limit of the final MD issupposed to be 4 m so as to meet the missionrequirement. The resulting defensible volume witha comparison between PNG and MPNG laws isillustrated in Fig. 11, which shrinks with the differentreentry angles in 45, 60, and 75 deg of the target.It reveals that sensitivity of the defensible volumewill be increased with the increasing reentry angles.This is quite reasonable because the interceptor’sinitial heading error will increase as the reentry angleincreases. This leads to the fact that the PNG lawcannot adequately intercept because of the extremelylarge lateral acceleration required at the terminalinterception.For the MPNG law the defensible volume under

the target reentry angle of 75 deg is larger than thatof 45 deg. The statistics of our simulation revealsthat the defensible volume is the largest when thereentry angle is 75 deg; if the reentry angle is largerthan 75 deg, the horizontal displacement of the targetwill be small, which increases the difficulty of theinterception and leads to a smaller defensible volume.

V. CONCLUSION

A 3-D MPNG law has been developed to guidethe interceptor against high-speed targets. The MPNGlaw is sophistically designed to account for theinterceptor’s axial acceleration and the variations oftarget acceleration. The effect of the proposed designhas been verified via numerical simulation to examinethe interception performance. 3-D defensible volumeshave also been characterized to evaluate the defenseperformance of our proposed design. Comparison ofthe MPNG and PNG has been conducted to show thesuperiority of the proposed design.

YU-PING LIN

CHUN-LIANG LIN

YUN-HAO LI

Department of Electrical Engineering

National Chung Hsing University

250 Kuo-Kuang Rd.

Taichung, 402

Taiwan

E-mail:

([email protected])

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Errata: Digital GNSS PLL Design Conditioned onThermal and Oscillator Phase Noise

The above paper1 was published with an incorrect

figure. The correct Figure 6 appears below.

Fig. 6. Wiener filter block diagram. Signal estimate μ̂(n) is made

by passing the signal-plus-noise μ(n)+nμ(n) through filter G(z).

JAMES T. CURRAN

GÉRARD LACHAPELLE

University of Calgary

500 University Drive NW

Calgary, AB T2N 1N4, Canada,

E-mail: ([email protected])

COLIN C. MURPHY

University College Cork

College Rd.

Cork, Munster, Ireland.

1Curran, J. T., Lachapelle, G., and Murphy, C. C., Digital GNSS

PLL Design Conditioned on Thermal and Oscillator Phase Noise,

IEEE Transactions on Aerospace and Electronic Systems, 48, 1 (Jan.

2012), 180—196.

Manuscript received October 27, 2012.

IEEE Log No. T-AES/49/1/944384.

0018-9251/13/$10.00 c° 2013 IEEE

CORRESPONDENCE 687

Documents

Development of 3-D Modified Proportional Navigation Guidance Law against High-Speed Targets