A Comparison of Different 

Guidance Schemes for a Direct 

Fire Rocket With a Pulse Jet 

Control Mechanism

by Thanat Jitpraphai, Bradley Burchett, 

and Mark Costello

ARL-CR-493 April 2002

prepared by

Thanat Jitpraphai, Bradley Burchett, and Mark Costello

Department of Mechanical Engineering

Oregon State University

Corvallis, OR 97331

under contract

DAAD17–01–P–1038

Approved for public release; distribution is unlimited.
Army Research Laboratory

Aberdeen Proving Ground, MD 21005-5066

ARL-CR-493 April 2002

A Comparison of Different 

Guidance Schemes for a Direct 

Fire Rocket With a Pulse Jet 

Control Mechanism

Thanat Jitpraphai, Bradley Burchett, and Mark 

Costello

Oregon State University

prepared by

Thanat Jitpraphai, Bradley Burchett, and Mark Costello

Department of Mechanical Engineering

Oregon State University

Corvallis, OR 97331

under contract

DAAD17–01–P–1038

Approved for public release; distribution is unlimited.
ii

Abstract

Compared to gun launch ammunition, uncontrolled direct fire atmospheric

rockets are terribly inaccurate, to the point where they are used most effectively 

on the battlefield as area weapons. Dispersion characteristics can be dramatically 

improved by outfitting the rocket with a suitable control mechanism and sensor 

suite. In the work reported here, a lateral pulse jet control mechanism is

considered. The lateral pulse jet mechanism consists of a finite number of small 

thrusters spaced equally around the circumference of the rocket. Using a

simulation model that includes projectile, flight control system, and inertial

measurement unit dynamics, three different control laws are contrasted, namely, 

proportional navigation guidance, parabolic and proportional navigation

guidance, and trajectory tracking control laws. When the number of individual 

pulse jets is small, a trajectory tracking control law provides superior dispersion 

reduction. However, as the number of pulse jets is increased, the relative

performance of the parabolic and proportional navigation guidance control law 

is slightly better than the trajectory tracking control law. When the number of 

pulse jets is small, the proportional navigation guidance, as well as the parabolic 

and proportional navigation guidance control laws, exhibits large mean miss

distance. All control laws appear to be equally susceptible to accelerometer and 

gyroscope errors that corrupt inertial measurement unit rocket state feedback.
iii

Contents

List of Figures v

1. Introduction 1

2. Simulation Model 2

2.1 Flight Control Laws.................................................................................2

2.1.1 PNG.............................................................................................3

2.1.2 PAPNG........................................................................................5

2.1.3 TT.................................................................................................6

2.2 Lateral Pulse Jet Firing Logic ..................................................................6

3. Results 7

4. Conclusions 9

5. References 21

Distribution List 23

Report Documentation Page 29
iv

INTENTIONALLY LEFT BLANK.
v

List of Figures

Figure 1. Schematic of a direct fire rocket with a lateral pulse jet........................11

Figure 2. LOS reference frame schematic.............................................................11

Figure 3. PNG control law in the horizontal plane...............................................12

Figure 4. Parabolic trajectory in the LOS reference frame....................................12

Figure 5. Parabolic navigation guidance control law in the vertical plane..........13

Figure 6. TT control law........................................................................................13

Figure 7. The i

th individual lateral pulse jet firing logic........................................14

Figure 8. Altitude vs. range. .................................................................................15

Figure 9. Cross range vs. range.............................................................................15

Figure 10. Comparison of dispersion radius for pulse jet impulse = 1 N-s..........16

Figure 11. Comparison of mean miss distance for pulse jet impulse = 1 N-s. .....16

Figure 12. Comparison of dispersion radius for pulse jet impulse = 5 N-s..........17

Figure 13. Comparison of mean miss distance for pulse jet impulse = 5 N-s. .....17

Figure 14. Comparison of dispersion radius for pulse jet impulse = 10 N-s........18

Figure 15. Comparison of mean miss distance for pulse jet impulse = 10 N-s.....18

Figure 16. Effect of accelerometer bias on dispersion radius. ..............................19

Figure 17. Effect of accelerometer bias on mean miss distance............................19

Figure 18. Effect of gyroscope bias on dispersion radius.....................................20

Figure 19. Effect of gyroscope bias on mean miss distance..................................20
vi

INTENTIONALLY LEFT BLANK.
1

1. Introduction

Direct fire atmospheric rockets are commonly used munitions on the battlefield 

and are launched from both land and air delivery platforms. The popularity of 

these weapons stems from the low system weight and relatively low unit cost. 

Compared to gun-launched projectiles, uncontrolled direct fire rockets are very 

inaccurate and are viewed as area weapons on the battlefield. Poor accuracy 

characteristics of direct fire rockets stems from the fact that the rocket enters the 

atmosphere with low speed. Any atmospheric or motion disturbances

experienced by the rocket at launch map into comparatively large aerodynamic 

angles of attack causing aerodynamic jump that leads to high impact point

dispersion.

In order to improve impact point accuracy performance, weapon system

developers are now considering the use of active flight control. A possible 

control mechanism to suit this purpose is a lateral pulse jet control mechanism. 

The lateral pulse jet mechanism consists of a ring of small thrusters mounted 

near the nose of the rocket. Each thruster on the ring imparts a single, short 

duration, large lateral force to the rocket. Design of a lateral pulse jet flight 

control system simplifies determining the firing time for each thruster. The 

lateral pulse jet control mechanism falls into the category of an impulse control 

mechanism. This is in contrast to the vast majority of flight control mechanisms, 

such as deflecting fins that continuously generate a control force to track a

specific trajectory. While highly developed techniques such as proportional

navigation guidance (PNG) are well known in missile guidance where

continuous control is available, few have focused attention on the use of impulse 

control mechanisms, particularly for direct fire rockets. Harkins and Brown [1] 

developed a simple control scheme to reduce dispersion of a direct fire rocket 

which only required angular rate feedback. For the example configuration

considered, impact point dispersion was reduced by a factor of 4. Jitpraphai and 

Costello [2] studied the same problem using a trajectory tracking flight control 

law which required an inertial measurement unit (IMU). Rocket dispersion was 

reduced by as much as a factor of 100, depending on the lateral pulse jet

configuration. Calise and El-Shirbiny [3] developed a lateral pulse jet control law 

for a spin-stabilized gun-launched projectile based on proportional navigation 

guidance, but modified to account for trajectory bending due to gravity.

The work reported here uses dynamic simulation to compare the merits of

different flight control laws specifically applied to a direct fire atmospheric

rocket equipped with a lateral pulse jet control mechanism. Candidate flight 

control laws include: PNG, parabolic and proportional navigation guidance

(PAPNG), and trajectory tracking (TT). Parametric studies considering the effect 
2

of the number of individual pulse jets, the impulse of each thruster, and sensor 

errors on dispersion, and mean miss distance are reported.

2. Simulation Model

The system under consideration is an atmospheric rocket equipped with a flight 

control system. The control mechanism system is a ring of lateral pulse jets, an 

IMU, and associated electronics. The coupled dynamics of the rocket, IMU, and 

flight control law are included in the simulation model deployed in this study. A 

schematic of the overall system is shown in Figure 1.

As is typical in flight dynamic modeling of projectiles, the rocket is modeled as a 

rigid body described by six degrees of freedom (6-DOF). The 6-DOF are the 

three position components of the mass center and the three Euler orientation 

angles. Forces and moments acting on the body include weight, air loads, main 

rocket motor thrust, and lateral pulse jet thrust. Details on the dynamic model of 

the rocket are available elsewhere and are omitted here for brevity [2, 4]. Sensor 

feedback used by the flight control law is obtained with a conventional IMU. 

The IMU contains three translational accelerometers and three rate gyroscopes 

that provide the acceleration of the IMU location and the angular velocity of the 

rocket with respect to the ground expressed in the rocket reference frame. The 

IMU estimates the position, orientation, and velocity of the rocket by numerically 

integrating kinematic and dynamic equations. Merhav [5] provides details on 

conventional IMU dynamic equations. Accuracy of the IMU is largely dependent 

on two factors, namely, accuracy of the initial position, orientation, and velocity 

used by the IMU as it begins to integrate the dynamics equations and errors in 

the accelerometers and gyroscopes. In order to model the effect of sensor errors 

in the dynamics of the IMU, filtered Gaussian random noise and bias is added to 

each of the sensors.

Simulation of the controlled rocket system entails numerically integrating 21 

nonlinear ordinary differential equations, 12 equations for the rocket, and 9

equations for the IMU. These equations are integrated forward in time using an 

adaptive time step Runge-Kutta method [6].

2.1 Flight Control Laws

Three different control laws are considered for a direct fire atmospheric rocket 

with a lateral pulse jet control mechanism. The control laws consist of control 

logic to generate an appropriate error signal and control logic to fire the lateral 

pulse jets. The lateral pulse jet firing logic is the same for all three flight control 

laws and is described at the end of this section.
3

2.1.1 PNG

Developed during World War II, PNG is a well-known guidance technique for 

tactical and strategic missiles and will act as a benchmark control law. PNG 

issues command accelerations calculated during flight that are proportional to 

the line-of-sight (LOS) angular rate, l&, and the rocket closing velocity, VC. 

Mathematically, the guidance law can be stated as shown in equation (1):

AC C = NV l

&

(1)

In equation (1), N is the proportional navigation constant that is typically set 

between 3 and 5 [7].

The horizontal plane PNG control law is derived by considering Figure 2. Point 

P represents a moving projectile that is fired from a launcher towards a target at 

point T. The LOS frame ( L L KLI J

v v v

, , ) is defined with its origin at point P. The LI

v

unit vector lies on the horizontal plane projection of the line between the

projectile and the target. The LJ

v

 unit vector lies on the horizontal plane and the 

KL

v

 points down. The relationship between the inertial reference frame and the 

LOS frame is given by a rotation matrix defined by equation (2):

[ ]

0

0

0 0 1

H H

H H

I L L

I L LI L

I L L

I c s I I

J s c J T J

K K K

l l

l l

    −               = =                      

r r r

r r r

r r r

where lH and lH

&

 are determined by equations (3) and (4):

1

tan T

H

T

y y

x x

l

−

  −

=     −

and

2 2

( ) ( )

( ) ( )

T T

H

T T

y x x x y y

x x y y

l

− − + −

=

−+−

& &

&

Equation (4) has been simplified with the assumption that the target is stationary. 

The horizontal acceleration command is given by equation (5):

ANu YC = HHH l

% &

H H H

u cxsy = + l l & &

.

,

.

.

.

(1)

(2)

(3)

(4)

(5)

(6)
4

Figure 3 shows the flight control system block diagram of the horizontal

guidance law.

The vertical plane PNG control law can be derived in the same manner. The final 

vertical plane guidance commands are given by equations (7)–(10):

1

tan T

V

T

z z

x x

l

−

  −

=     −

2 2

( ) ( )

( ) ( )

T T

V

T T

z x x x z z

x x z z

l

− − + −

=

−+−

& &

&

AZC = N u VVV l

% &

V V V

u cxsz = + l l & &

Equation (8) has been simplified with the assumption that the target is stationary. 

The vertical acceleration command magnitude is given by equation (9).

The total acceleration command in the LOS frame is given by equation (11):

AC AYC L ZC L XC B YC B ZC B = J +=++ A K A I A J A K

r rrrrr % %

The command acceleration components are transformed to the projectile body 

frame by first transforming the command acceleration components from the LOS 

frame to the inertial frame and then to the body frame. Equation (12) provides 

the transfer formula:

[ ][ ]

XC 0

YC IB LI YC

ZC ZC

A

A T T A

A A

          =  

       

%

%

Note that the tilde symbol (~) indicates that a quantity is described in the LOS 

reference frame. In equation (12), TIB is the inertial to body reference frame 

transformation:

IB

c c c s s

T s s c c s s s s c c s c

c s c s s c s s s c c c

q y q y q

f q y f y f q y f y f q

f q y f y f q y f y f q

  −

 

= − +

+ −  

The magnitude of the off-axis command acceleration is used as the error signal 

that is input to the lateral pulse jet firing logic. Equation (15) provides the error 

signal phase angle:

2 2 Γ = + A A YC ZC

,

,

,

.

.

.

.

,

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)
5

1

tan ZC

YC

A

A

g

−  

=    

2.1.2 PAPNG

The PAPNG control law uses the same horizontal plane acceleration command 

as PNG. Thus, AYC

%

 is given by equation (5). In the vertical plane, a parabolic 

guidance law is employed to determine command acceleration in the vertical

LOS plane to compensate for trajectory bending due to rocket weight. A desired 

parabolic trajectory is continuously updated during flight. Figure 4 shows the 

desired parabolic trajectory described by equation (16) [3].

2

1 2 ˆ ˆ ˆ ˆ P TPP z z = + + K x K x

where ˆPx and ˆPz are components of the projectile position in the target reference 

frame, and ˆTx and ˆTz are components of the target position in the target 

reference frame. The target reference frame unit vectors, ,

T T I J

v v

 and KT

v

 are

shown in Figure 4. The KT

v

 axis points toward the target point. The vectors TI

v

and KT

v

are parallel but in opposite directions to LI

v

 and KL

v

. The parameters K1

and K2 are constants determined such that a parabolic trajectory that emanates 

from the projectile passes through the target and has a specific trajectory angle, 

b F, at the target point.

tan( ) K1 = bF (17)

2 2

2 2

1

2

( ) ( )

( ( ) ( ) ˆ ˆ )

x x y y

K x x y y z z

K

T T

T T T P

− + −

− − + − + −

=

(18)

The command acceleration component in the vertical LOS plane is determined 

by equation (19):

t

~ (b − b)

=

L D

ZC

V

A

Figure 5 shows the diagram of the vertical guidance law. In equation (19), VL is 

the magnitude of projectile velocity in the LOS frame, b is the flight path angle 

of the projectile in the LOS frame, and t is the acceleration command time 

constant. b D is the desired flight path angle of the projectile, which is computed 

using the parabolic trajectory as shown in Figure 5. The acceleration command 

in the LOS frame is given in equation (20). It is transformed to the body

reference frame using sequences of rotations shown in equation (22).

.

,

.

.

.

(15)

(16)

(17)

(18)

(19)
6

C ZC L YC L ZC KL A A s I A J A c

v v v v

b b

~ ~ ~

= − + −

(20)

C XC B YC B AZC KB A A I A J

v v v v

= + +

(21)

[ ] [ ]





















−

−

=





















β

β

ZC

YC

ZC

IB LI

ZC

YC

XC

A

~

c

A

~

A

~

s

T T

A

A

A

Using equation (22), the error magnitude and phase of the off-axis command 

acceleration is computed using equations (14) and (15).

2.1.3 TT

A predetermined command trajectory is assumed to be known prior the launch 

of the rocket. This command trajectory is a desired path of the rocket from the 

launcher to the target. For direct fire rockets, a command ballistic trajectory is 

available from the fire control system and can be downloaded to the rocket just 

prior to launch. A schematic of the flight control system block diagram is shown 

in Figure 6. 

The TT flight control system first compares the measured position of the

projectile (x, y, z) to the commanded trajectory ( C C Cx , y ,z ) to form a position 

error vector in the inertial frame. The trajectory error is converted to the rocket 

body frame using equation (23):

[ ]

X C

Y IB C

Z C

e x x

e Tyy

e z z

    −

      = −  

        −

The magnitude and phase angle of the error in the off-axis plane of the rocket are 

denoted Γ and g , and are defined by equations (24) and (25), respectively. The 

error magnitude is used in the firing decision and the error phase angle is used in 

the selection of pulse jet to be fired:

2 2

Y Z Γ = + e e (24)

( )

1

tan / Z Y g e e

−

=

(25)

2.2 Lateral Pulse Jet Firing Logic

A schematic diagram of the lateral pulse jet firing logic is given in Figure 7. The 

same logic is used for all the control laws previously described. At each

computation cycle in the flight control system, a sequence of checks are

.

.

.

.

,

.

(20)

(21)

(22)

(23)

(24)

(25)
7

conducted that govern firing of individual lateral pulse jets. The conditions that 

must be satisfied for an individual lateral pulse jet to fire are as follows:

• The error magnitude must be greater than a specified tolerance, namely

THRES e :

THRES Γ > e

• The time elapsed since the last lateral pulse jet firing must be greater than a 

specified duration, THRES Δt . This condition implies a specified delay time 

between two consecutive firings,

THRES t −t > Δt

*

where *t is the time of the most recent pulse jet firing.

• The difference between the error phase angle and the individual pulse jet 

force must be less than a specified angle, THRES d :

2

PJ fJ i p g g d THRES

Δ

− − − < &

(28)

• The individual pulse jet under consideration has not been fired.

The first two checks are valid for all lateral pulse jets while the last two checks 

are specific to a given lateral pulse jet.

3. Results

The rocket configuration used in the simulation study is a representative direct 

fire rocket that is a 1.4-m-long, fin-stabilized rocket with three pop-out fins on 

the rear of the projectile. The lateral pulse jet ring is located 1.2 m from the base 

of the rocket. The main rocket motor burns for 1.1 s and imparts an impulse to 

the rocket of 6200 N-s. During the main rocket motor burn, the forward velocity 

of the rocket is increased from 44 m/s to 770 m/s. The rocket weight, mass 

center location from the base of the rocket, roll inertia, and pitch inertia before 

and after burn is 10.4/7.2 kg, 0.85/0.86 m, 0.0077 kg/0.0058 m2, and 1.83 kg/

1.61 m2, respectively. Nominally, the rocket exits the launcher with the following 

initial conditions: x = 0.0 m, y = 0.0 m, z =-30.5 m, f = 0.0°, q = 4.14°, y = 0.0°, 

u = 43.7 m/s, v = 0.0 m/s, w = 0.11 m/s, p = 51.5 rad/s, q = !0.18 rad/s, and 

r = 0.0 rad/s.

Using the control laws previously described, uncontrolled and controlled

trajectories are compared for the example rocket configuration against a nominal 

command trajectory. The rocket is launched at an altitude of 30.5 m toward a 

.

,

.

(26)

(27)

(28)
8

target on the ground with altitude and cross range equal zero at a range of 

3000 m. The ring contains 32 individual lateral pulse jets where each individual 

pulse jet imparts an impulse of 5 N-s on the projectile body over a time duration 

of 0.01 s. The pulse jet elapsed time threshold is set to 0.2 s. The pulse jet angle 

threshold is set to 0.2°. For the controlled rocket with TT flight control system 

the window size is 1.5 m. The controlled rocket with PNG or PAPNG flight 

control system has a window size setting of 6 m/s2. These window sizes were 

optimized through parametric studies in which dispersion was minimized.

Figures 8 and 9 compare the trajectory response obtained using the different 

guidance schemes. These two figures show a desired controlled trajectory, an 

uncontrolled trajectory, a PNG controlled trajectory, a PAPNG controlled

trajectory, and a TT controlled trajectory. Without the use of pulse jet control, the 

rocket falls short of the target with a miss distance of 52.5 m. The PAPNG and 

TT control laws significantly reduce miss distance from 52.5 m in the

uncontrolled case to 1.6 and 0.7 m in the controlled case, respectively. Note that 

the PNG control law actually increases miss distance relative to the uncontrolled 

case. This is due to the fact that the PNG control law initially commands a 

downward vehicle acceleration when in fact an upward command acceleration is 

required to reduce miss distance. Toward the end of the trajectory, the lateral 

pulse jets do not process sufficient control power to maneuver to the target.

If several projectiles are fired from the same weapon with the same firing setting, 

their points of impact will be different due to system uncertainty. For direct fire 

rockets, impact points are defined as the altitude and cross range of the projectile 

at a specified down range distance. The mean impact point of a group of fired 

rockets with respect to the target center is a measure of the accuracy of the 

weapon. The degree of scatter of the impact points about the mean impact point 

defines dispersion. Both accuracy and dispersion determine whether a particular 

weapon can hit an intended target.

Scatter in projectile impact points is caused by a myriad of weapon system 

inaccuracies including launcher manufacturing tolerances, launcher vibration, 

propellant ignition variation, temperature variations of the launcher or projectile, 

atmospheric turbulence and gusts, atmospheric pressure variation, etc. For

weapon system variations connected with the launch dynamics, the end result is 

a variation in the initial launch conditions for each projectile fired. Notable 

launch conditions that create significant impact point scatter are the initial

angular velocity of the projectile. To simulate launch conditions that cause

dispersion, the initial pitch and yaw rate are modeled as independent Gaussian 

random variables with a mean and standard deviation of –0.18/0.3 and

0/0.3 rad/s, respectively. The dispersion radius is defined as the radius of a 

circle that emanates from the mean impact point and contains 67% of the impact 

points.
9

Figures 10–15 compare the dispersion radius and mean impact point at a range 

of 3000 m for PNG, PAPNG, and TT control laws using different combinations of 

pulse jets. The dispersion radius and mean impact point statistics are computed 

with a sample size of 200.

Figures 10 and 11 consider 1 N-s pulses, while Figures 12 and 13 consider 5 N-s 

pulses, and Figures 14 and 15 consider 10 N-s pulses. In all cases, dispersion 

radius is steadily reduced as the number of pulses is increased. For a relatively 

small number of pulses, the TT control law provides the lowest dispersion. 

However, as the number of pulses is increased, the PAPNG control law improves 

relative to the TT control law and obtains slightly lowest dispersion for a large 

number of pulse jets. For the PNG and PAPNG control laws, the mean miss 

distance initially increases as the number of pulse jets is increased and then is 

reduced as the number of pulse jets continues to increase. For a small number of 

pulse jets, the mean miss distance is greater that the uncontrolled case. During 

the thrusting phase of flight, the PNG and PAPNG control laws tend to over 

compensate by pulsing too often and are unable to close on the target. The TT 

control law steadily reduces the mean miss distance as the number of pulse jets is 

increased.

Figures 16 and 17 examine the sensitivity of the different pulse jet control laws to 

accelerometer bias errors for the case of 24 individual pulse jets, each with an 

impulse of 10 Ns. The bias errors for each accelerometer in the IMU are

independent Gaussian random variables with zero mean. A sample size of 200 is 

used to compute the statistics shown in these figures. As the standard deviation 

of accelerometer bias is increased, dispersion is steadily increased for all control 

laws. The slope of dispersion radius vs. standard deviation of accelerometer bias 

error is slightly larger for the TT control law, indicating a larger sensitivity to 

accelerometer bias errors. The mean miss distance increases as the standard 

deviation of accelerometer bias error increases for the PNG control law and is flat 

for the PNG and TT control laws. Figures 18 and 19 are similar to Figures 16 and 

17 except gyroscope bias errors in the IMU are evaluated. The slope of

dispersion radius vs. gyroscope bias error is lowest for the TT control low

indicating the TT control law is comparatively less sensitive to gyroscope bias 

error. The mean miss distance steadily increases as a function of gyroscope bias 

error for all control laws.

4. Conclusions

With proper selection of flight control system parameters and adequately sized 

lateral pulse jets, dispersion and accuracy characteristics of a direct fire rocket 

equipped with a ring of lateral pulse jets can be drastically improved with a 
10

proportional navigation guidance, proportional and parabolic navigation

guidance, or trajectory tracking control law. The proportional navigation

guidance control law provided the least dispersion reduction and also generates 

the largest mean miss distance. This is mainly from target patterns below the 

target due to gravity effects not accounted for the control law. The proportional 

and parabolic guidance control law obtained low dispersion and mean miss 

distance performance when the number of pulse jets is large. The proportional 

and parabolic guidance control law is comparatively more sensitive to gyroscope 

bias errors than accelerometer bias errors. The TT control law generated low 

dispersion and miss distance characteristics, particularly when the number of 

pulse jets is relatively small. While the proportional navigation guidance control 

law and the proportional and parabolic navigation guidance control law

generated impact patterns off target when the number of pulse jets was low, the 

TT control law consistently generated impact patterns nearly centered on the 

target. This is due to the fact that the TT control law forms an error signal from 

position, whereas the other two control laws use acceleration as the basis for an 

error signal.
11

Pop Out 

Fins

B

I

r

KB

r

B

J

r

Main Rocket 

Motor

Lateral Pulse

 Jet Ring

KI

r

I

I

r

I

J

r

CG

1

2

3

4

5

6

Pulse Jet

B

J

r

KB

r

i

th

Ji

f

IMU

Figure 1. Schematic of a direct fire rocket with a lateral pulse jet.

TARGET

T

PROJECTILE

P

lH

I

J

v

I

I

v

L

J

v

L

I

v

Figure 2. LOS reference frame schematic.

Body 

Frame

Inertial

Frame Pulse 

Geometry 

Schematic
12

x +

y

T

x

T

y

LOS Angular Rate

Inertial to LOS

Transformation

x

y

z

f

q

y

&

&

&

P

xˆ

*

N H

L

u

L

v

wL

lH

& AYC

2

∗

2

∗

+

-

-

+

Figure 3. PNG control law in the horizontal plane.

L

I

v

KL

v

KT

v

T

I

v

VP/ I

v

TARGET

( T

z

)

0,0, )

PROJECTILE

( P P

x z

) )

,0, )

b

bD

− bF

Figure 4. Parabolic trajectory in the LOS reference frame.
13

Parabolic Trajectory 

Constant

z

x

z

x

T

T

2

∗

P

xˆ

*

D

L

L

L

w

v

u

P

T

z

z

ˆ

ˆ

VL

AZC

Inertial to Target

Transformation

Tan-1

F

b

*

b

2

K2

K1

Tan-1

Tan-1

-

 

+

+

2

∗

2

∗

+

Figure 5. Parabolic navigation guidance control law in the vertical plane.

γ

Γ

Inertial to Body Transformation

Down Range

t Table Look-Up

Cross Range

t Table Look-Up

Altitude

t Table Look-Up

Magnitude 

and Phase

Ts+1

s

xˆ

yˆ

zˆ

f

ˆ

yˆ

yC

xC

zC

ex

ey

ez

q

ˆ

+

+

+

Figure 6. TT control law.

-

-

-

x

y

z

f

?

y
14

γ

2

ΔPJ g&

Unit Fire

Function

*

Individual

Thrust

Table

Look-up

t

t

π

i

fJ

Firing 

Decision

Γ

Window Size, eTHRES

TJi

-

Signal 

Hold 

(Time)

+

+

+

+

-

-

Figure 7. The i

th individual lateral pulse jet firing logic.
15

0 500 1000 1500 2000 2500 3000

0

10

20

30

40

50

60

70

80

90

100

Range (m)

Altitude (m)

Uncontrolled

Desired 

PNG 

PAPNG 

TT 

Figure 8. Altitude vs. range.

0 500 1000 1500 2000 2500 3000

-20

-15

-10

-5

0

5

Range (m)

Cross Range (m)

Uncontrolled

Desired 

PNG 

PAPNG 

TT 

Figure 9. Cross range vs. range.
16

0 5 10 15 20 25 30

0

20

40

60

80

100

120

140

160

Dispersion Radius (m)

Number of Pulse Jets

PNG 

PAPNG

TT 

Figure 10. Comparison of dispersion radius for pulse jet impulse = 1 N-s.

0 5 10 15 20 25 30

0

5

10

15

20

25

30

Mean Miss Distance (m)

Number of Pulse Jets

PNG 

PAPNG

TT 

Figure 11. Comparison of mean miss distance for pulse jet impulse = 1 N-s.
17

0 5 10 15 20 25 30

0

20

40

60

80

100

120

140

160

Dispersion Radius (m)

Number of Pulse Jets

PNG 

PAPNG

TT 

Figure 12. Comparison of dispersion radius for pulse jet impulse = 5 N-s.

0 5 10 15 20 25 30

0

10

20

30

40

50

60

70

80

Mean Miss Distance (m)

Number of Pulse Jets

PNG 

PAPNG

TT 

Figure 13. Comparison of mean miss distance for pulse jet impulse = 5 N-s.
18

0 5 10 15 20 25 30

0

20

40

60

80

100

120

140

160

Dispersion Radius (m)

Number of Pulse Jets

PNG 

PAPNG

TT 

Figure 14. Comparison of dispersion radius for pulse jet impulse = 10 N-s.

0 5 10 15 20 25 30

0

10

20

30

40

50

60

70

80

90

100

Mean Miss Distance (m)

Number of Pulse Jets

PNG 

PAPNG

TT 

Figure 15. Comparison of mean miss distance for pulse jet impulse = 10 N-s.
19

0 2 4 6 8 10 12 14

0

2

4

6

8

10

12

14

16

18

Dispersion Radius (m)

Standard Deviation of Accelerometer Bias (ft/s2)

PNG 

PAPNG

TT 

Figure 16. Effect of accelerometer bias on dispersion radius.

0 2 4 6 8 10 12 14

0

1

2

3

4

5

6

7

8

9

10

Mean Miss Distance (m)

Standard Deviation of Accelerometer Bias (ft/s2)

PNG 

PAPNG

TT 

Figure 17. Effect of accelerometer bias on mean miss distance.
20

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

0

2

4

6

8

10

12

14

16

18

Dispersion Radius (m)

Standard Deviation of Gyroscope Bias (rad/s)

PNG 

PAPNG

TT 

Figure 18. Effect of gyroscope bias on dispersion radius.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

0

10

20

30

40

50

60

Mean Miss Distance (m)

Standard Deviation of Gyroscope Bias (rad/s)

PNG 

PAPNG

TT 

Figure 19. Effect of gyroscope bias on mean miss distance.
21

5. References

1. Harkins, T., and T. Brown. “Using Active Damping as a Precision-Enhancing 

Technology for 2.75-inch Rockets.” ARL-TR-1772, U.S. Army Research

Laboratory, Aberdeen Proving Ground, MD, 1999.

2. Jitpraphai, T., and M. Costello. “Dispersion Reduction of a Direct Fire Rocket 

Using Lateral Pulse Jets.” Journal of Spacecraft and Rocket, vol. 38, no. 5, 2001.

3. Calise, A., and H. El-Shirbiny. “An Analysis of Aerodynamic Control for 

Direct Fire Spinning Projectiles.” Proceedings of the 2001 American Institute of 

Aeronautics and Astronautics Guidance, Navigation, and Control Conference,

AIAA 2001-4217, Montreal, Canada, 2001.

4. Gast, R., S. Morris, and M. Costello. “Simulation of Shot Impacts for the 

M1A1 Tank Gun.” Journal of Guidance, Control, and Dynamics, vol. 23, no. 1, 

pp. 53–59, 2000.

5. Merhav, S. Aerospace Sensor Systems and Applications. New York: SpringerVerlag, 1996.

6. Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical 

Recipes in Fortran 77: The Art of Scientific Computing, Vol. 1. Cambridge: Press 

Syndicate of the University of Cambridge, 1992.

7. Zarchan, P. “Tactical and Strategic Missile Guidance.” Progress in Astronautics 

and Aeronautics, vol. 124, pp. 139–180, American Institute of Aeronautics and 

Astronautics, Inc., Washington, DC, 1990.
22

INTENTIONALLY LEFT BLANK.
NO. OF NO. OF

COPIES ORGANIZATION COPIES ORGANIZATION

23

2 DEFENSE TECHNICAL

INFORMATION CENTER

DTIC OCA

8725 JOHN J KINGMAN RD

STE 0944

FT BELVOIR VA 22060-6218

1 HQDA

DAMO FDT

400 ARMY PENTAGON

WASHINGTON DC 20310-0460

1 OSD

OUSD(A&T)/ODDR&E(R)

DR R J TREW

3800 DEFENSE PENTAGON

WASHINGTON DC 20301-3800

1 COMMANDING GENERAL

US ARMY MATERIEL CMD

AMCRDA TF

5001 EISENHOWER AVE

ALEXANDRIA VA 22333-0001

1 INST FOR ADVNCD TCHNLGY

THE UNIV OF TEXAS AT AUSTIN

3925 W BRAKER LN STE 400

AUSTIN TX 78759-5316

1 US MILITARY ACADEMY

MATH SCI CTR EXCELLENCE

MADN MATH

THAYER HALL

WEST POINT NY 10996-1786

1 DIRECTOR

US ARMY RESEARCH LAB

AMSRL D

DR D SMITH

2800 POWDER MILL RD

ADELPHI MD 20783-1197

1 DIRECTOR

US ARMY RESEARCH LAB

AMSRL CI AI R

2800 POWDER MILL RD

ADELPHI MD 20783-1197

3 DIRECTOR

US ARMY RESEARCH LAB

AMSRL CI LL

2800 POWDER MILL RD

ADELPHI MD 20783-1197

3 DIRECTOR

US ARMY RESEARCH LAB

AMSRL CI IS T

2800 POWDER MILL RD

ADELPHI MD 20783-1197

ABERDEEN PROVING GROUND

2 DIR USARL

AMSRL CI LP (BLDG 305)
NO. OF NO. OF

COPIES ORGANIZATION COPIES ORGANIZATION

24

3 AIR FORCE RSRCH LAB

MUNITIONS DIR

AFRL/MNAV

G ABATE

101 W EGLIN BLVD

STE 219

EGLIN AFB FL 32542

1 CDR WL/MNMF

D MABRY

101 W EGLIN BLVD STE 219

EGLIN AFB FL 32542-6810

20 OREGON STATE UNIVERSITY

DEPT OF MECHL ENGRG

M COSTELLO

CORVALLIS OR 97331

4 CDR

US ARMY ARDEC

AMSTA AR CCH

J DELORENZO

S MUSALI

R SAYER

P DONADIO

PICATINNY ARESENAL NJ

07806-5000

7 CDR

US ARMY TANK MAIN

ARMAMENT SYSTEM

AMCPM TMA

D GUZIEWICZ

R DARCEY

C KIMKER

R JOINSON

E KOPOAC

T LOUZIERIO

C LEVECHIA

PICATINNY ARESENAL NJ

07806-5000

1 CDR

USA YUMA PROV GRND

STEYT MTW

YUMA AZ 85365-9103

1 DIR

BENET LABORATORIES

SMCWV QAR

T MCCLOSKEY

WATERVLIET NY 12189-5000

10 CDR

US ARMY TACOM

AMCPEO HFM

AMCPEO HFM F

AMCPEO HFM C

AMCPM ABMS

AMCPM BLOCKIII

AMSTA CF

AMSTA Z

AMSTA ZD

AMCPM ABMS S W

DR PATTISON

A HAVERILLA

WARREN MI 48397-5000

1 CDR

USAOTEA

CSTE CCA 

DR RUSSELL

ALEXANDRIA VA 22302-1458

2 DIR

US ARMY ARMOR CTR & SCHL

ATSB WP ORSA 

A POMEY

ATSB CDC

FT KNOX KY 40121

1 CDR

US ARMY AMCCOM

AMSMC ASR A

MR CRAWFORD

ROCK ISLAND IL 61299-6000

2 PROGRAM MANAGER

GROUND WEAPONS MCRDAC

LTC VARELA

CBGT

QUANTICO VA 22134-5000

4 COMMANDER

US ARMY TRADOC

ATCD T

ATCD TT

ATTE ZC

ATTG Y

FT MONROE VA 23651-5000
NO. OF NO. OF

COPIES ORGANIZATION COPIES ORGANIZATION

25

1 NAWC

F PICKETT

CODE C2774 CLPL

BLDG 1031

CHINA LAKE CA 93555

1 NAVAL ORDNANCE STATION

ADVNCD SYS TCHNLGY BRNCH

D HOLMES

CODE 2011

LOUISVILLE KY 40214-5001

1 NAVAL SURFACE WARFARE CTR

F G MOORE

DAHLGREN DIVISION

CODE G04

DAHLGREN VA 22448-5000

1 US MILITARY ACADEMY

MATH SCI CTR OF EXCELLENCE

DEPT OF MATHEMATICAL SCI

MDN A MAJ DON ENGEN

THAYER HALL

WEST POINT NY 10996-1786

3 DIR

SNL

A HODAPP

W OBERKAMPF

F BLOTTNER

DIVISION 1631

ALBUQUERQUE NM 87185

3 ALLIANT TECH SYSTEMS

C CANDLAND

R BURETTA

R BECKER

7225 NORTHLAND DR

BROOKLYN PARK MN 55428

3 DIR USARL

AMSRL SE RM

H WALLACE

AMSRL SS SM

J EIKE

A LADAS

2800 POWDER MILL RD

ADELPHI MD 20783-1145

1 OFC OF ASST SECY OF ARMY

FOR R&D

SARD TR

W MORRISON

2115 JEFFERSON DAVIS HWY

ARLINGTON VA 22202-3911

2 CDR USARDEC

AMSTA FSP A

S DEFEO

R SICIGNANO

PICATINNY ARESENAL NJ

07806-5000

2 CDR USARDEC

AMSTA AR CCH A

M PALATHINGAL

R CARR

PICATINNY ARESENAL NJ

07806-5000

5 TACOM ARDEC

AMSTA AR FSA

K CHIEFA

AMSTA AR FS 

A WARNASCH

AMSTA AR FSF

W RYBA

AMSTA AR FSP

S PEARCY

J HEDDERICH

PICATINNY ARESENAL NJ

07806-5000

5 CDR US ARMY MICOM

AMSMI RD

P JACOBS

P RUFFIN

AMSMI RD MG GA

C LEWIS

AMSMI RD MG NC

C ROBERTS

AMSMI RD ST GD

D DAVIS

RSA AL 35898-5247

3 CDR US ARMY AVN TRP CMD

DIRECTORATE FOR ENGRNG

AMSATR ESW

M MAMOUD

M JOHNSON

J OBERMARK

RSA AL 35898-5247
NO. OF NO. OF

COPIES ORGANIZATION COPIES ORGANIZATION

26

1 DIR US ARMY RTTC

STERT TE F TD

R EPPS

BLDG 7855

REDSTONE ARSENAL AL

38598-8052

2 STRICOM

AMFTI EL

D SCHNEIDER

R COLANGELO

12350 RESEARCH PKWY

ORLANDO FL 32826-3276

1 CDR OFFICE OF NAVAL RES

CODE 333

J GOLDWASSER

800 N QUINCY ST RM 507

ARLINGTON VA 22217-5660

1 CDR US ARMY RES OFFICE

AMXRO RT IP TECH LIB

PO BOX 12211

RESEARCH TRIANGLE PARK NJ

27709-2211

4 CDR US ARMY AVN TRP CMD

AVIATION APPLIED TECH DIR

AMSATR TI 

R BARLOW

E BERCHER

T CONDON

B TENNEY

FT EUSTIS VA 23604-5577

3 CDR NAWC

WEAPONS DIV

CODE 543400D

S MEYERS

CODE C2744

T MUNSINGER

CODE C3904

D SCOFIELD

CHINA LAKE CA 93555-6100

1 CDR NSWC

CRANE DIVISION

CODE 4024

J SKOMP

300 HIGHWAY 361

CRANE IN 47522-5000

1 CDR NSWC

DAHLGREN DIV

CODE 40D

J BLANKENSHIP

6703 WEST HWY 98

PANAMA CITY FL 32407-7001

1 CDR NSWC

J FRAYSEE

D HAGEN

17320 DAHLGREN RD

DAHLGREN VA 22448-5000

5 CDR NSWC

INDIAN HEAD DIV

CODE 40D D GARVICK

CODE 4110C L FAN

CODE 4120 V CARLSON

CODE 4140E H LAST

CODE 450D T GRIFFIN

101 STRAUSS AVE

INDIAN HEAD MD 20640-5000

1 CDR NSWC

INDIAN HEAD DIV

LIBRARY CODE 8530

BLDG 299

101 STRAUSS AVE

INDIAN HEAD MD 20640

2 US MILITARY ACADEMY

MATH SCI CTR OF EXCELLENCE

DEPT OF MATHEMATICAL SCI

MDN A 

MAJ D ENGEN

R MARCHAND

THAYER HALL

WEST POINT NY 10996-1786

3 CDR US ARMY YUMA PG

STEYP MT AT A

A HOOPER

STEYP MT EA

YUMA AZ 85365-9110
NO. OF NO. OF

COPIES ORGANIZATION COPIES ORGANIZATION

27

6 CDR NSWC

INDIAN HEAD DIV

CODE 570D J BOKSER

CODE 5710 L EAGLES

J FERSUSON

CODE 57 C PARIS

CODE 5710G S KIM

CODE 5710E S JAGO

101 STRAUSS AVE ELY BLDG

INDIAN HEAD MD 20640-5035

1 B KIM

MICHIGAN STATE UNIVERSITY

2120 ENGINEERING BLDG

EAST LANSING MI 48824-1226

2 INDUSTRIAL OPERATION CMD

AMFIO PM RO

W MCKELVIN

MAJ BATEMAN

ROCK ISLAND IL 61299-6000

3 PROGRAM EXECUTIVE OFFICER

TACTICAL AIRCRAFT PROGRAMS

PMA 242 1

MAJ KIRBY R242

PMA 242 33

R KEISER (2 CPS)

1421 JEFFERSON DAVIS HWY

ARLINGTON VA 22243-1276

1 CDR NAVAL AIR SYSTEMS CMD

CODE AIR 471

A NAKAS

1421 JEFFERSON DAVIS HWY

ARLINGTON VA 22243-1276

3 ARROW TECH ASSOCIATES INC

R WHYTE

A HATHAWAY

H STEINHOFF

1233 SHELBOURNE RD SUITE D8

SOUTH BURLINGTON VT 05403

3 US ARMY AVIATION CTR

DIR OF COMBAT DEVELOPMENT

ATZQ CDM C

B NELSON

ATZQ CDC C 

T HUNDLEY

ATZQ CD

G HARRISON

FORT RUCKER AL 36362

ABERDEEN PROVING GROUND

3 CDR

USA ARDEC

AMSTA AR FSF T

R LIESKE

J WHITESIDE

J MATTS

BLDG 120

1 CDR

USA TECOM

AMSTE CT

T J SCHNELL

RYAN BLDG

3 CDR

USA AMSAA

AMXSY EV

G CASTLEBURY

R MIRABELLE

AMXSY EF

S MCKEY

47 DIR USARL

AMSRL CI HC

S WILKERSON

AMSRL SL BE

A MIKHAIL

AMSRL WM B

W CIEPIELA

A W HORST JR

AMSRL WM BA

F BRANDON

T G BROWN (5 CPS)

L BURKE

J CONDON

B DAVIS

T HARKINS (5 CPS)

D HEPNER

M HOLLIS

V LEITZKE

D LYON 

A THOMPSON

AMSRL WM BC

M BUNDY

G COOPER

J GARNER

B GUIDOS

J NEWILL

V OSKAY

P PLOSTINS (4 CPS)
NO. OF

COPIES ORGANIZATION

28

ABERDEEN PROVING GROUND (CONTD)

J SAHU

K SOENCKSEN

AMSRL WM BD

B FORCH

AMSRL WM BE

E M SCHMIDT

AMSRL WM BF

B HAUG

P HILL

J LACETERA

J WALL

AMSRL WM MB

J BENDER

W DRYSDALE

AMSRL WM RP

J BORNSTEIN

C SHOEMAKER

AMSRL WM SG

T ROSENBERGER

AMSRL WM TC

R COATES
29
30

INTENTIONALLY LEFT BLANK.