Survey of Maneuvering Target Tracking. Part II Motion Models of Ballistic and Space Targets-KvL.pdf

I. INTRODUCTION

Survey of Maneuvering

Target Tracking.

Part II: Motion Models of

Ballistic and Space Targets

A survey of dynamic models used in maneuvering

target tracking has been reported in [64]. It, however,

does not cover motion models used for tracking

ballistic and space targets (BT), namely, ballistic

missiles, decoys, debris, satellites, projectiles, etc.

The primary reason for this omission is that these

models possess many distinctive features that differ

vastly from those covered in [64]. To supplement

[64], a survey of these models is presented here. To

our knowledge, such a survey is not available in the

literature.

Overall, a BT has a less uncertain motion than

many other types of powered vehicles, such as

maneuvering aircraft or agile missiles: Most BTs

follow a flight path that is largely predetermined

by the performance characteristics specific for a

target type, hence the name ballistic targets, although

some more advanced ballistic missiles can undergo

small maneuvers usually for retargeting. However,

this does not mean that the motion of a foreign BT

can be determined accurately. In fact, tracking a

foreignBTislikelytobeplaguedwithavarietyof

uncertainties, including those concerning trajectory

loft or depression, thrust profile management, target

weight, propellant specific impulse, sensor bias, and

atmospheric parameters. Many of these uncertainties

stem from the uncertainty in the target type, the

principal uncertainty in modeling the motion of a

foreign BT for the tracking purpose.

The entire trajectory of a BT is commonly

divided into three basic phases: boost (and possibly

a post-boost phase), ballistic (also known as coast),

and reentry. The boost phase is the powered,

endo-atmospheric flight, which lasts from launch to

thrust cutoff or burnout. It is followed by the ballistic

phase, which is an exo-atmospheric, free-flight motion

for most BTs, continuing until the atmosphere is

reached again. The atmospheric reentry begins when

the atmospheric drag becomes considerable and

endures until impact.

For the tracking purpose, only the most substantial

forces that may act on a BT are considered: thrust,

aerodynamic forces (most notably, atmospheric

drag and possibly lift), and gravity. Not all of these

forces are present at a level that significantly affects

the motion of a BT in all regimes of the trajectory.

The boost phase is characterized by a large thrust,

which in the case of rocket staging is subject to

abrupt, jump-wise changes. The effects of the drag

and gravity are also essential in this phase. After the

boost, drag is no longer present and thrust vanishes or

drops to a very low level. During the exo-atmospheric

ballistic phase the motion is governed essentially by

the gravity only. Still, small retargeting maneuvers

are possible. The reentry phase features a rapid

drag-induced deceleration with possible lateral

X. RONG LI, Fellow, IEEE

VESSELIN P. JILKOV, Member, IEEE

University of New Orleans

This paper is the second part in a series that provides a

comprehensive survey of maneuvering target tracking without

addressing the so-called measurement-origin uncertainty. It

surveys motion models of ballistic targets used for target tracking.

Models for all three phases (i.e., boost, coast, and reentry) of

motion are covered.

CONTENTS

I. Introduction

II. Ballistic (Coast) Flight

A. Gravity

B. Coordinate Accelerations

C. Coast Motion Models

III. Reentry

A. Aerodynamic Forces

B. Ballistic (Nonmaneuvering) RV Motion

C. Maneuvering RV Motion

D. Motion of Endo-Atmospheric Ballistic Targets

IV. Boost Phase

A. Accelerations

B. Kinematic Models

C. Dynamic Models

D. Example of A Profile-Based Model

V. Integration of BT Motion Models for Entire Trajectory

VI. Concluding Remarks

References

Appendix

Part I: Dynamic Models. IEEE Transactions on Aerospace and Electronic

Systems , 39 , 4 (Oct. 2003), 1333—1364.

Part II: Originally published as Ballistic Target Models. In Proceedings of

the 2001 SPIE Conference on Signal and Data processing of Small Targets ,

vol. 4473, 559—581.

Part III: Measurement Models. In Proceedings of the 2001 SPIE Conference

on Signal and Data Processing of Small Targets , vol. 4473, 423—446.

Part IV: Decision-Based Methods. In Proceedings of the 2002 SPIE

Conference on Signal and Data Processing of Small Targets , vol. 4728,

511—534.

Part V: Multiple-Model Methods. IEEE Transactions on Aerospace and

Electronic Systems , 41 , 4 (Oct. 2005), 1255—1321.

Manuscript received August 4, 2007; revised April 8, 2008; released

for publication August 25, 2008.

IEEE Log No. T-AES/46/1/935930.

Refereeing of this contribution was handled by W. Koch.

This research was supported in part by ARO Grant

W911NF-08-1-0409, ONR-DEPSCoR Grant N00014-09-1-1169,

Project 863 through Grant 2006AA01Z126, and Louisiana BoR

Grant LEQSF (2009-12)-RD-A-25.

Authors’ address: Dept. of Electrical Engineering, University of

New Orleans, New Orleans, LA 70148, E-mail: (xli@uno.edu).

0018-9251/10/$26.00 ° 2010 IEEE

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accelerations. So, the significant forces in the different

phases are boost (thrust, gravity, and aerodynamic);

coast (gravity); reentry (aerodynamic and gravity).

In general, the total acceleration vector of a BT, in

an inertial coordinate system (CS) (e.g., the ECI-CS,

see Fig. 2 in the Appendix) can be decomposed as a

vector sum:

a = a T + a A + a G = a T + a D + a L + a G

(1)

same disclaimers as made in the Introduction of

[64] apply to this survey, including those on the

restriction of point targets for temporal behaviors and

the interrelationship between dynamic models and

tracking algorithms. The reader should keep in mind

that results of many extensive studies of BT tracking

have not been published in the open literature. In

other words, much of the BT information, particularly

target-type specific information, is classified and not

open to the general public. As a result, this survey

covers only those dynamic models for BT tracking

in the open literature available to us. The emphasis

of the survey is on missile tracking, although some

models covered are applicable to tracking other

targets.

This paper is a heavily revised and updated version

of [61]. It is organized as follows: Motion models for

the simplest phase, the ballistic flight, are covered in

Section II. This is followed in Section III by a survey

of the models for reentry vehicles. Section IV then

describes models for the most sophisticated, the boost

phase. Integration of motion models for different

phases is the topic of Section V. Concluding remarks

are made in Section VI. The Appendix provides

background information for the most common CSs

used in BT tracking.

where a T , a A ,and a G denote the accelerations

induced by thrust, aerodynamic forces, and gravity,

respectively, and a A consists of accelerations due to

drag, a D ,andlift, a L . In a frame fixed to the Earth, the

relative total acceleration also includes those induced

by the Earth’s rotation.

The entire end-to-end motion of a BT can be

modeled by a “wide-band” dynamic model (e.g.,

nearly constant velocity, acceleration, jerk, or Singer

model) capable of covering all the phases. Most

models developed for the boost phase, the most

sophisticated of all three phases, can serve this

purpose. This is, however, rather crude and not in

common use. Target dynamics during the different

phases are substantially different. A succinct outline

of the main dynamic phases is given in [28]. What is

more natural and rational, as well as common practice,

is to develop different models specific for each phase

that more fully exploit the inherent characteristics of

the portion.

Throughout this paper, let the target position

and velocity vectors be p =[ X , Y , Z ] 0 and v = p =

[ _ X , _ Y , _ Z ] 0 , respectively, in whatever coordinate

system is considered. For example, p =

II. BALLISTIC (COAST) FLIGHT

A. Gravity

¡!

OP in the

(Earth-centered-inertial) ECI-CS (see Fig. 2). While

models intended for distinct phases differ significantly

in general, they have a common structure. The

kinematic part, with x =[ p 0 , v 0 ] 0 , of the state-space

models of a BT has the form

¡¡!

O S P in

While gravity is present in the entire flight of

a BT, it is the dominating, if not sole, force acting

on a coast BT. The following three gravity models,

denoted by a (0)

the (East-North-Up) ENU-CS and p =

G , a (1)

G ,and a (2)

G , are commonly used for

BT tracking.

1) Flat Earth Model : The simplest possible

model of gravity assumes a flat, nonrotating Earth.

In this model, the gravity acting on the target in the

ENU-CS is a constant:

a (0)

_ x =

(2)

G =[0,0, ¡g ]

(3)

Various models differ from one another in 1) the

choice of models of acceleration a by accounting

for different acceleration components at varying

accuracy levels, and 2) the additional models needed

for describing the evolution of unknowns in the

acceleration components.

We survey dynamic models proposed/used for

the distinct regimes of a BT. The motion phases are

presented in an order according to their sophistication

levels, rather than their chronology in the trajectory.

For each phase, we describe physics (forces or

accelerations) first and then motion models in

different CS.

Compared with [64], this survey is slightly more

tutorial in nature for the benefit of the readers less

familiar with the ballistics or aerodynamics. The

where g is the constant gravitational acceleration of

the Earth.

2) Spherical Earth Model : Assume that the Earth

and the BTs can be represented as point masses at

their centers 1 and that the gravitational forces of the

moon (and stars) can be neglected. Since the target

has a negligible mass relative to the Earth’s mass,

the gravitational acceleration a G is the solution of a

so-called restricted two-body problem, obtained by

Newton’s inverse-square gravity law [7] as

G ( ½ )= ¡ ¹

k½k 2 u ½ = ¡ ¹

k½k 3 ½ (4)

1 This holds if the Earth and the targets are spherically symmetric

with an even distribution of their masses.

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a (1)

where ½ is the vector from the Earth center to the

target, k½k its length, u ½ = ½=k½k its unit vector, and

¹ is the Earth’s gravitational constant.

3) Ellipsoidal Earth Model : More accurate

expressions for the gravitational acceleration can

be obtained by replacing the above spherical Earth

model with an ellipsoidal (or more precisely,

spheroidal) Earth model. Such a more precise

approximation–accounting for the Earth oblateness

by including the second-order gravitational harmonic

term J 2 of the Earth’s gravitational field model–is

[7, 23]

induced by the Earth’s rotation [72, 7], which consist

of the Coriolis 2 a (1)

C + a (2)

C and centrifugal a (2)

C terms:

C = −£ ( − £½ )= M − ½

(6)

where − =[ − X , − Y , − Z ] 0 is the Earth’s angular velocity

vector, v is the target’s linear velocity vector in the

noninertial Earth-fixed CS [75, 7] (Fig. 2), and the

antisymmetric matrix M − is given by

(

¡− Z

− Y

G ( ½ )= ¡ ¹

u ½ + 3

r e

k½k

a (2)

k½k 2

2 J 2

M − =

− Z

¡− X

(7)

¡− Y

− X

)

£ [(1 ¡ 5( u 0

½ u Z ) 2 ) u ½ +2( u 0

½ u Z ) u Z ]

For the boost and reentry portions of a BT in a

close vicinity of the Earth and over a short period,

the effect of the Coriolis and centrifugal forces may

be neglected. During the ballistic flight, however, this

effect is usually significant and should be accounted

for, particularly in the case of a long-range BT

(relative to the Earth’s radius).

For simplicity of presentation, let

(5)

where r e is the Earth’s equatorial radius, J 2 is a

correction constant, and u Z is the unit vector along

¡!

O Z I (see Fig. 2). The best-known Jeffery constant

J 2 represents the difference between the polar

and equatorial moment of inertia. It quantifies the

oblateness of the Earth and is approximately equal

to one third of the ellipticity of the Earth. Even more

accurate models are available. For example, the

specifications of the World Geodetic System 1984

(WGS-84) ellipsoid and the Earth Gravitational Model

(EGM 96) are given in great detail in [77].

The boost and reentry phases are relatively

short in range compared with the Earth radius and

take place in a close vicinity of the Earth. Thus

a flat, nonrotating Earth model may be adequate,

particularly in the presence of other more dominating

uncertainties. On the contrary, the coast flight of a

long-range BT comprises a much greater range, and

thus accounting for the Earth sphericity (and even

ellipticity) and rotation is essential. The spherical

model (4) is classical and has been most commonly

used in a variety of BT tracking applications

[90, 51, 55, 85, 29, 99, 56, 5, 9]. It has been used

for BT tracking over a short range and/or period, for

example, as a gravity model as an integral part of

the total acceleration within a boost (boost-to-coast)

motion model [85, 29, 5, 9] or for track initiation

purposes [99]. However, it may be inadequate

for more demanding BT tracking applications, in

particular, precision tracking of coast targets over

a long time period or at a low data rate. For a

long-range coast BT, gravity is the sole or dominating

acceleration and thus needs more accurate models

such as the ellipsoidal Earth or WGS-84 model.

a ( i G , I

g i , noninertial CS , i =0,1,2 (8)

be the total acceleration induced by the Earth, where

a ( i G was given by (3)—(5) and g i = a ( i G

a ( i )

E =

expression in an Earth-fixed frame.

¡a C is its

C. Coast Motion Models

In the exo-atmospheric coast phase, no thrust

is applied and no drag is experienced. The motion

may be considered governed by the acceleration a E

induced by the Earth alone, with other factors (e.g.,

perturbations) neglected. Thus, the total acceleration is

a = a E . It follows from (2) that the state-space models

of a coast target with the state vector x =[ p 0 , v 0 ] 0 have

the form

a ( i )

_ x =

(9)

E was given by (8). So, coast modeling of

the BT amounts to selection of an appropriate model

for a E .

1) Inverse-Square Model in ECI-CS :Inths

case, the acceleration part of the state-space model

(9) is described by the inverse-square model (4)

in the E CI-CS as a ( i )

E = a (1)

G ( p )= ¡¹p=kpk 3 ,where

X 2 + Y 2 + Z 2 .

A trajectory described by (9) with (4) is confined

to the so-called orbital plane and is a part of a conic

B. Coordinate Accelerations

If the target motion is considered in a noninertial

frame fixed to the Earth (e.g., the ENU-CS, ECF-CS,

and body frames), its relative total acceleration

2 The Coriolis acceleration is induced by the rotation of the Earth

that causes the Coriolis effect–the apparent deflection of a body

in motion with respect to the Earth, as seen by an observer on the

Earth. This effect appears in a noninertial CS [7].

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includes the coordinate accelerations a C = a (1)

a (1)

C =2 − £v =2 M − v , a (2)

where a ( i )

kpk =

orbit, governed by the Keplerian motion equation [7].

A dynamic model is used in target tracking mainly

for propagating the target state (i.e., state prediction),

known as the Kepler problem in astrodynamics,

and for covariance prediction. Although the model

given by (9) with (4) is highly nonlinear, the state

propagation with it can be done in an efficient

manner, through the algorithm outlined below.

Given x ( t 0 )=[ p 0

G ( p )ofthe

state-space model (9) has the following form [7, 29]:

E = a (2)

1+ c e

kpk 2

kpk

1 ¡ 5

G ( p )= ¡ ¹

1+ c e

kpk 2

kpk

a (2)

kpk 3

1 ¡ 5

0 ] 0 at time t 0 the predicted target

state x ( t )=[ p 0 , v 0 ] 0 at t = t 0 + T (with T> 0) is given

by [7]

0 , v 0

1+ c e

kpk 2

kpk

3 ¡ 5

1 ¡ ¸ 2

kp 0

T¡ ¸ p

p =

k C

p 0 +

¹ S

v 0

(10)

¹¸ ( ³S¡ 1)

kp 0

1 ¡ ¸ 2 C

kp 0

where c e =(3 = 2) J 2 r e .

This model was chosen in [29], [28] as the coast

model for a 6-state EKF-based ballistic filter in the

ECI-CS implemented in a multiple-model tracking

system. However, as pointed out in [29], a model

mismatch caused by moderate trajectory perturbations

may lead to a track divergence. This effect can be

alleviated by introducing small fictitious zero-mean

process noise. This technique was used in [29] to

adapt the model for the post-boost phase of the

trajectory, which may involve small maneuvers.

In the early work on BT tracking (e.g., satellite

orbit determination [51]), the target dynamics was

usually considered deterministic (i.e., without process

noise). This often led to divergence of the EKF.

Introducing fictitious process noise is an effective

means to account for such factors as model errors,

neglected perturbations, nonlinearities, and computer

roundoff errors. While the state prediction remains

unaffected if additive zero-mean process noise input is

present, the prediction error covariance is increased

by the process noise covariance Q , and thus the

possibility for the EKF to diverge is reduced. The

price paid is a possible accuracy degradation of

the filter when the deterministic model is indeed

adequate. This is closely related to the problem of

noise identification and adaptive filtering. See, e.g.,

[51], [71], [60]. More recent work along these lines

for tracking an orbital target can be found in [44],

where Q was tuned adaptively using the most recent

state estimate and covariance in a gravity-gradient

model with the inverse-square law. Although a

number of ways to choose Q have been proposed

in the literature, in practice Q still remains a design

parameter, which is adjusted/tuned based mostly on

engineering experience and intuition [23, 12]. More

comprehensive discussions of this issue can be found

in [63] and [64].

3) Model with J 2 Correction in ENU-CS :ts

straightforward to obtain the state-space form of the

acceleration model in the ENU-CS associated with

this gravity model as

v =

kkpk p 0 +

v 0

¹=kpk and can be computed accurately and

efficiently via the Newton iteration scheme for solving

a so-called time-of-flight equation, as given below

for the case of an elliptical orbit (i.e., a =2 =kp 0

k¡

kv 0

k 2 =¹> 0) [7]:

1. Initialize

® := 1 ¡ a k p 0

, ¯ := p 0

0 v 0

¸ := a p

¹ , ° :=

kp 0

¹ :

2. Repeat the following until jT¡¿j<² :

³ := a¸ 2 , C := 1 ¡ cos

³¡ s in

S :=

, ¿ := ®¸ 3 S + ¯¸ 2 C + °¸

d¿

d¸ := ®¸ 2 C + ¯¸ (1 ¡³S )+ °

d¿

d¸

¡ 1

¸ := =¸ +

( T¡¿ ) :

A useful program-like pseudocode (with a few

small typographical errors) of the above algorithm

for all conic (i.e., elliptical, parabolic, and hyperbolic)

trajectories can be found in [99] (see also [89], [19]),

along with the computation of the Jacobian F ( t , t 0 )=

@x ( t ) =@x ( t 0 ) necessary for an extended Kalman filter

(EKF) [51, 4]. Explicit evaluation of the Jacobian by

direct differentiation of (4) can be found in [55]. For a

comprehensive treatment of the theoretical background

and solutions to the Kepler problem, the reader is

referred to [7].

2) Model with J 2 Correction in ECI-CS :A

refined model for the gravitational acceleration is

based on the ellipsoidal Earth model (5). Note that

u p =[ X , Y , Z ] 0 =kpk and u Z =[0,0,1] 0 in the ECI-CS.

a E = g 2 = a (2)

G ( ½ ) ¡ 2 M − v¡M − ½ (11)

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Similar to (4), the acceleration part a ( i )

where C , S ,and ³ are known functions of ¸ .The

variab le ¸ is defined through its time derivative by

¸ =

where ½ = p + r =[ X , Y , Z + krk ] 0 ,with r being the

vector from the Earth center to the origin of the

ENU-CS, and krk = r e + Z S is the sum of the Earth

radius and the altitude of the origin.

Conversion to the radar face CS and the

corresponding RUV-CS (see the Appendix) and of its

application in the EKF can be found in [23]. A similar

model is given in [70] with inverse-square gravity,

an ellipsoidal Earth, and Coriolis and centrifugal

acceleration terms in the radar face CS, along with

the respective EKF with RUV measurements.

4) Inverse-Square Model in ENU-CS :W ththe

spherical Earth’s gravity, the model (11) is simplified

jt k ). For the propagation of the

associated error covariance needed in, e.g.,

Kalman filtering, it is proposed in [56] to add, in

effect, component-uncoupled small noise w ( t k )=

[ w X , w Y , w Z ] 0 ( t k )to[ ˆ X , ˆ Y , ˆ Z ] 0 ( t k

jt k ) in the dynamics

equation, where cov( w )=diag( q X , q Y , q Z )isa

design parameter. As a result, what is proposed

is a coordinate-“uncoupled,” state-dependent,

piecewise-constant, non-zero-mean white-noise

acceleration model, using the terminology of [4],

[64]. There seems room for improvement here by a

better way of adding noise or by adding temporally or

spatially correlated noise, but at a price of increased

complexity.

Note that the motions along the X , Y , Z directions

described by this model are not actually uncoupled

over multiple time steps. The coupling arises from the

dependence of the acceleration on the common target

state in (14). This model has been reported in [56] to

provide performance-competitive with that of the EKF

based on the fully coupled nonlinear inverse-square

model-for several BT tracking scenarios with a high

sampling rate (of 4 Hz). It seems worthwhile to try

this model in conjunction with some more precise

acceleration models, rather than (12), as originally

proposed in [56].

The underlying idea of this “acceleration

compensation” [35] approach is not restricted to

coast models or the specific form of the acceleration

model used: The piecewise-constant approximation

canbedirectlyappliedinanysituationwherethe

total acceleration is available as a function of the

position and velocity. Indeed, the same approach

was used in [52] to account for gravity in a simple

kinematic model of gravity turn motion during

boost (see Section IVB2); similarly, in [35] a

thrust acceleration term derived from a (nearly)

constant-acceleration (CA) filter estimates was

included in the total acceleration to predict the

boost motion of a BT; an additional drag-induced

acceleration term was included in [31], along with

the gravity and thrust terms. In general, this approach

is simple for implementation, but its accuracy

may be inadequate in the case of a low sampling

rate.

6) Models in Other Coordinates : Models in

other CS (e.g., radar face CS and spherical CS) can

be obtained by conversion from the ENU-CS. For

example, an explicit expression of the inverse-square

model (neglecting the Coriolis and centrifugal forces)

can be found in [15].

where ! is the Earth rotation rate, and

0 ¡ sin Á cos Á

sin Á 0 0

¡ cos Á 0 0

(13)

where Á is the latitude of the origin of the ENU-CS.

in this case (i.e., − X =0).

Like the models considered before, the standard

EKF technique is directly applicable to this model

in the ENU-CS. The linearization needed for error

covariance propagation may not be sufficiently

accurate for relatively long propagation time periods.

For cases with high sampling rates, however, a simple

yet accurate piecewise-constant acceleration model

[56] could be used, as discussed next.

5) Piecewise-Constant Acceleration Model :The

idea of this approach, proposed in [56], is simple

and natural. At each propagation time step t k

jt k )=[ X , ˆ _ X , Y , ˆ _ Y , Z , ˆ _ Z ] 0 ,thatis, a ( t k

jt k )= a ( x ( t k

jt k )).

For example, if (12) is used to model the total

acceleration, then

a ( t k

jt k )= g 1 ( x ( t k

jt k )) :

(14)

jt k )is

constant over the time period [ t k , t k +1 ), the following

discrete-time dynamics model along the X direction is

clearly obtained

X ( t k +1 )

_ X ( t k +1 )

1 t k +1

¡t k

¸·

X ( t k )

_ X ( t k )

0 1

¡t k ) 2 = 2

t k +1

ˆ X ( t k

jt k ) :

¡t k

III. REENTRY

Likewise for Y and Z directions.

This state-dependent piecewise-constant

acceleration model is simple and straightforward

During reentry, two significant forces are

always present: the gravity and atmospheric drag. If

maneuvers are possible, aerodynamic lift must also

100

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for application. It is used for state prediction

to get x ( t k +1

!t k +1 ;

first sample (i.e., compute) the continuous-time

total acceleration process at the estimated point

x ( t k

As such, under the assumption that the target motion

is uncoupled along X , Y ,and Z directions, and the

acceleration a ( t )=[ ¨ X ( t ), ¨ Y ( t ), ¨ Z ( t )] 0 = a ( t k

( t k +1

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