Prediction of Swerving Motion of a Dual-Spin Projectile 

With Lateral Pulsejets in Atmospheric Flight 

by Bradley Burchett, Allen Peterson, and Mark Costello 

ARL-CR-570 November 2005 

prepared by 

Oregon State University 

Corvallis, OR 97331-6001 

under contract 

W911QX-04-P0270 

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

Aberdeen Proving Ground, MD 21005-5066 

ARL-CR-570 November 2005 

Prediction of Swerving Motion of a Dual-Spin Projectile 

With Lateral Pulsejets in Atmospheric Flight 

Bradley Burchett, Allen Peterson, and Mark Costello

Oregon State University 

prepared by 

Oregon State University 

Corvallis, OR 97331-6001 

under contract 

W911QX-04-P0270 

Approved for public release; distribution is unlimited. 
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1. REPORT DATE (DD-MM-YYYY)

November 2005 

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3. DATES COVERED (From - To) 

October 2004–October 2005 

5a. CONTRACT NUMBER 

W911QX-04-P0270 

5b. GRANT NUMBER 

4. TITLE AND SUBTITLE 

Prediction of Swerving Motion of a Dual-Spin Projectile With Lateral Pulsejets in 

Atmospheric Flight 

5c. PROGRAM ELEMENT NUMBER 

5d. PROJECT NUMBER 

622618.H80 

5e. TASK NUMBER 

6. AUTHOR(S) 

Bradley Burchett, Allen Peterson, and Mark Costello 

5f. WORK UNIT NUMBER 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 

Oregon State University 

Corvallis, OR 97331-6001 

8. PERFORMING ORGANIZATION 

 REPORT NUMBER 

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S) 

U.S. Army Research Laboratory 

ATTN: AMSRD-ARL-WM-BC 

Aberdeen Proving Ground, MD 21005-5066 

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 NUMBER(S) 

ARL-CR-570 

12. DISTRIBUTION/AVAILABILITY STATEMENT 

Approved for public release; distribution is unlimited. 

13. SUPPLEMENTARY NOTES 

Point of contact for this report is Peter Plostins, U.S. Army Research Laboratory, ATTN: AMSRD-ARL-WM-BC, Aberdeen 

Proving Ground, MD 21005-5066 

14. ABSTRACT 

Using the linear theory for a dual-spin projectile in atmospheric flight, closed form expressions were obtained for swerving 

motion under the action of lateral pulsejets. Trajectory results generated by the linear theory equations and a fully nonlinear 

seven degree-of-freedom dual-spin projectile model agreed favorably. The analytic solution provided a relatively straight 

forward and computationally efficient means of trajectory estimation which is useful within smart weapon flight control systems. 

In order to accurately predict the impact point using the analytic solution, the dual-spin projectile linear model must be updated 

periodically. Terminal impact point prediction degrades rapidly as the linear model update interval is increased beyond a critical 

value. Control authority, as defined by the change in impact location due to a pulsejet firing, steadily decreases as a function of 

projectile down range position. 

15. SUBJECT TERMS

dual-spin projectile, pulsejet control, linear theory, lateral pulse jets, guidance navigation and control 

16. SECURITY CLASSIFICATION OF: 

19a. NAME OF RESPONSIBLE PERSON

Peter Plostins 

a. REPORT 

UNCLASSIFIED 

b. ABSTRACT 

UNCLASSIFIED

c. THIS PAGE 

UNCLASSIFIED

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OF ABSTRACT 

UL 

18. NUMBER 

OF PAGES 

30 19b. TELEPHONE NUMBER (Include area code) 

410-278-8878 

Standard Form 298 (Rev. 8/98) 

Prescribed by ANSI Std. Z39.18
iii 

Contents 

List of Figures iv

List of Tables iv

1. Introduction 1

2. Dual-Spin Projectile Dynamic Model 2

3. Analytic Solution 5

4. Pulsejet Force and Moment Conditions 10

5. Results 10

6. Conclusions 16

7. References 17

Appendix. Dual-Spin Linear Theory Coefficients 19

List of Symbols, Abbreviations, and Acronyms 21

Distribution List 23 
iv 

List of Figures 

Figure 1. Projectile position coordinate definitions........................................................................2 

Figure 2. Projectile orientation definitions. ....................................................................................3 

Figure 3. Cross range vs. range.....................................................................................................12 

Figure 4. Altitude vs. range...........................................................................................................12 

Figure 5. Pitch angle vs. range......................................................................................................13 

Figure 6. Yaw angle vs. range. .....................................................................................................14 

Figure 7. Roll rate vs. range..........................................................................................................14 

Figure 8. Target prediction error vs. model update interval. ........................................................15 

Figure 9. Induced pulsejet swerve vs. range to target...................................................................15 

List of Tables 

Table 1. Projectile physical properties..........................................................................................11 

Table 2. Projectile state conditions. ..............................................................................................11 
1 

1. Introduction 

With clever integration of microelectromechanical sensors and actuators into new projectile 

configurations, future weapon systems hope to achieve a notable leap in accuracy and lethality. 

Because some sensors and control mechanisms require different levels of spin to operate 

properly, new multicomponent projectile configurations are being investigated. The dual-spin 

projectile configuration is one such concept that consists of forward and aft components, 

connected through a bearing, which roll at different rates. 

A potential control mechanism for smart weapons trajectory control is a cluster of lateral 

pulsejets. Each pulsejet imparts a relatively short duration, yet large lateral body force on the 

projectile. By mounting a set of pulsejets on either the forward or aft section of a dual-spin 

projectile, a specific trajectory can be tracked. When designing a projectile pulsejet flight 

control system to minimize terminal miss distance, pulsejet firing logic relies on real-time 

estimation of the effect of firing a particular jet on the miss distance at an arbitrary time and 

system state. In theory, this can be accomplished by numerically integrating the nonlinear 

equations of motion from a given point to the target. However, from a practical point of view, 

real-time numerical integration of the nonlinear equations of motion from a given state to the 

target at each flight control system computation cycle represents too large a computational 

burden on the on-board microprocessor. Hence, computationally-efficient trajectory prediction 

of smart weapon projectile configurations is needed for insertion into control laws. Along these 

lines, Guidos and Cooper (1) developed closed form expressions for swerve of a rigid nonrolling 

projectile with a lateral pulsejet applied. Their analysis allows for finite duration impulse loads 

and showed that under mild conditions on spin rate, singular and finite duration impulses yield 

the same effect on the trajectory. 

Although the dynamics of dual-spin spacecraft have been extensively studied (2–12), work 

on the atmospheric flight mechanics of dual-spin projectiles has been comparatively sparse. 

Smith et al. (13) developed a dynamic model of a dual-spin projectile and employed this model 

to simulate a smart artillery shell. More recently, Costello and Peterson (14) reported a more 

general dual-spin projectile dynamic model which permits asymmetric mass properties and 

generalized bearing friction. This model was subsequently used to develop a linear theory and 

stability boundaries for this configuration. The work reported here extends the dual-spin 

projectile linear equations to account for the effect of lateral pulsejets in the prediction of angular 

and swerving motion. The swerving dynamics are solved in closed form, resulting in 

computationally-simple algebraic expressions for the projectile trajectory under the influence of 

lateral pulsejets. 
2

2. Dual-Spin Projectile Dynamic Model 

The dynamic model for a dual-spin projectile admits a total of seven degrees-of-freedom, 

including the components of the position vector of the mass center and the Euler yaw, pitch, aft 

body roll, and forward body roll angles. Figures 1 and 2 provide indicate coordinates used to 

specify position and orientation for the dual-spin projectile dynamic model. Linear projectile 

theory is used to simplify the highly nonlinear dynamic equations to a set of linear ordinary 

differential equations. More details on projectile linear theory can be found in McCoy (15). The 

linear equations developed by Costello and Peterson (14) neglected gravity, since their primary 

interest was dynamic stability, and also did not include lateral pulsejet forces in the formulation. 

II

G

JI

JG

x

y 

-z

KI

JJG

Figure 1. Projectile position coordinate definitions. 
3

ψ

θ I I

G

J

G

K I

JJG

Figure 2. Projectile orientation definitions. 

Equations 1–10 remove these two limitations and represent a modified set of dynamic equations 

describing the motion of a dual-spin projectile in atmospheric flight. 

V m

SDC V XO

2

ρ ′ = − . (1) 

F p F V

D φ ′ = . (2) 

A p A V

D φ ′ = . (3) 

⎭

⎬

⎫

⎩

⎨

⎧

+

⎭

⎬

⎫

⎩

⎨

⎧

⎥

⎦

⎤ ⎢

⎣

⎡ =

⎭

⎬

⎫

⎩

⎨

⎧

′

′

N

M

p

p

T L

H J

p

p

A

F

A

F

. (4) 

0 0 /( )

0 0 / /( )

/ / /( )

/ / /( )

I

I

IX I YY

IX I YY

v A Dv Y D mV

w AD w Dg V Z D mV

q BD CD E F q R Z D VI

r CD BD F E r R Y D VI

⎧ ⎫ ⎡ ⎤⎧ ⎫ ′ − − ⎧ ⎫ ⎪ ⎪ ⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ′ − +

= + ⎢ ⎥ ⎨ ⎬ ⎨⎬ ⎨ ⎬ ′ − − ⎪ ⎪ ⎪⎪ ⎪ ⎪

⎩ ⎭ ⎣ ⎦⎩ ⎭ ′ − ⎩ ⎭

. (5) 

q V

D θ ′ = . (6) 

r V

D

ψ ′ = . (7) 
4

x′= D . (8) 

D

y = v+ ψD

V ′ . (9) 

D z = w- θD

V ′ . (10) 

In equations 1–10, the ‘ symbol denotes differentiation with respect to nondimensional arc 

length,s . The constants used in the previous equations are defined as follows:

( ) 2 N A

ρ S D A= C m

⎡ ⎤

⎢ ⎥ ⎣ ⎦

. (11) 

( ) ( ) 2 22

F A

NPA NPA

T fx fx F ax ax A

YY

ρSD mD D C C B= R +r p + R +r p mI V

⎡ ⎤ ⎛⎞ ⎡ ⎤ ⎛ ⎞

⎢ ⎥ ⎢ ⎥⎜ ⎟⎜ ⎟ ⎣ ⎦ ⎝⎠ ⎣ ⎦ ⎝ ⎠

  . (12) 

 2 T MA

YY

ρSD mD C= C

m I

⎡ ⎤ ⎡ ⎤

⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦

. (13) 

 2

2 2

MQ

T

YY

ρSD mD C

E =

m I

⎡ ⎤ ⎡ ⎤

⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦

. (14) 

F A

XX F XX A

T

YY

D (I p + I p ) F =

V I

. (15) 

3

4

F

LP V

F F

XX XX

ρSD C DC H= –

I VI

. (16) 

DCV J = F VIXX

. (17) 

DCV T = A VIXX

. (18)

3

4

A ρSD C DC LP V L= – A A I VI XX XX

. (19)

2

2

F

ρSD VC

M = DD

FIxx

. (20)

2

2

A

ρSD VCDD N = AIxx

. (21)

( ) ( ) F A C = R +r C + R +r C MA fx fx NA ax ax NA . (22) 

TF A 2 2

YY YY f fx YY a ax I = I +m r +I +m r . (23) 
5

The impulse force components YI and ZI represent impulse forces that are applied to the 

projectile in the nonrolling reference frame. These loads and the associated moment arm, RIX , 

are effective forces and moments. Due to aerodynamic interaction, the force and moment 

generated by the lateral pulsejets is in general different than what would be measured on a test 

stand. Also, for this report, the impulse forces are mathematically treated as singular impulses. 

The bearing coupling rolling resistance is assumed to be of the hydrodynamic type. 

3. Analytic Solution 

Pulsejet forces and moments that appear in equation 5 are assumed to occur over a relatively 

short duration of time compared to the motion of the projectile; hence, their effect is impulsive in 

nature. An impulsive force acting on a rigid body creates a discontinuity in the translational and 

rotational velocity of the body at the time of the impulse (16). To simulate motion of the 

projectile with pulsejets acting on the system, a piecewise solution is constructed that consists of 

free-flight and pulsejet discontinuity segments. Expressions for free-flight solution segments are 

given in this section, while the pulsejet state discontinuity conditions are provided in the next 

section. 

As seen in equation 1, the total velocity is dynamically uncoupled from the system, but appears 

in a nonlinear fashion in the remaining state dynamic equations. As is customary in projectile 

linear theory (15), it is assumed that the total velocity changes slowly with respect to the other 

states of the system so it can be treated as a constant in the other dynamic equations. When 

viewed in this light, the equations of motion become linear and the equations of motion for angle 

of attack and roll motion become uncoupled. 

The total velocity of the projectile exhibits exponential decay as the round flies down range. 

Equation 24 provides the total velocity solution: 

-ρSDC s/( m) 2 X0 V =V e 0 . (24) 

The solution to the roll rate equations is aided by first defining the decay rates, as shown in 

equation 25. 

2 2 4

1,2 2

H + L ± H - 2HL+ L + KJ

σ = . (25)

With the roll rate decay rate given by equation 25, the roll rate solution can be written as 
6

( )

( )

1

2

1 11 12 12

12 1 1

2 21 22 22

2 2

1 11

1 1 ,

s

F

s

p Le

L e

σ

σ

σ

σσ σ σ

σ

σ σ

⎧⎪ ⎡⎛ ⎞ ⎤

= − Γ+ Γ − Γ ⎨ ⎢⎜ ⎟ ⎥ − ⎪⎩ ⎣⎝ ⎠ ⎦

⎡ ⎤ ⎛ ⎞ ⎫⎪ + − Γ+ Γ − Γ ⎢ ⎥ ⎜ ⎟ ⎬ ⎣ ⎦ ⎝ ⎠ ⎪⎭ (26) 

and

1

2

11 12 12

12 1 1

21 22 22

2 2

1 1

1 1 ,

s

A

s

T

p e

e

σ

σ

σσ σ σ

σ σ

⎧⎪⎛ ⎞ = Γ+ Γ − Γ ⎨⎜ ⎟ − ⎪⎩⎝ ⎠

⎛ ⎞ ⎫⎪ +Γ + Γ − Γ ⎜ ⎟ ⎬ ⎝ ⎠ ⎪⎭ (27)

where 

( ) , 11 F0 2 A0 Γ = Tp + L −σ p (28)

( ) , Γ12 = TM + L −σ 2 N (29)

( ) , 21 F0 1 A0 Γ = Tp + σ − L p (30)

and 

( ) . Γ22 = TM + σ 1 − L N (31)

As given in equation 5, the equations of motion for epicyclic pitching and yawing behavior 

consists of coupled motion of v , w , q , and r . The slow and fast modes of the epicyclic 

dynamic equations are given by equations 32 and 33: 

λ = σ + iφ FFF (32) 

and

λ = σ + iφ SSS , (33) 

where 

( ) ( )

( ) ⎥

⎦

⎤ ⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

+ −

−

+ − − = F A E

AF B

F C

A E F F

2 2 1

4

1

2 2 σ , (34) 

 ( ) ( )

( ) ⎥

⎦

⎤ ⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

+ −

−

− − − = F A E

AF B

F C

A E F S

2 2 1

4

1

2 2 σ , (35) 

F [F F 4C ] 2

1 2 φ = + − , (36) 
7

and 

S [F F 4C ] 2

1 2 φ = − − . (37) 

Corresponding to the modes given in equations 32 and 33, the right and left Eigenvector matrices 

are given by equations 38 and 39, respectively. 

( ) ( ) ( ) ( )⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

− + − − + −

+ − + −

− −

=

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎣

⎡

D

i R S

D

i R S

D

i K Q

D

i K Q

D

R S

D

R S

D

K Q

D

K Q

i i i i

T

T

T

T

2 2 2 2

2 2 2 2

1 1 1 1

*

2

*

1

2

1

ξ

ξ

ξ

ξ

. (38)

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

−

−

− −

−

−

−

− −

−

− −

− +

−

− +

−

− +

−

− +

−

− + − + − +

−

− +

=

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎣

⎡

∗ ∗ ∗ ∗ ∗ ∗

∗

∗ ∗

∗

∗ ∗ ∗ ∗ ∗ ∗

∗

∗ ∗

∗

υ μ υ μ υ μ

μ

υ μ

μ

υ μ υ μ υ μ

υ

υ μ

υ

υ μ υ μ υ μ

μ

υ μ

μ

υ μ υ μ υ μ

υ

υ μ

υ

η

η

η

η

i i

i i

i i

i i

T

T

T

T

1

1

1

1

2

1

*

2

*

1

2

1

. (39)

The constants utilized in equations 38 and 39 are defined as follows. Note that the * symbol 

denotes complex conjugate. 

K = (E − A) + 2A + iF, (40)

( ) ( )( ) ( ) 2 2 Q E A AE C F i F E A AF B =− + + −+ −+ + 44 2 2 , (41)

R = (E − A)+ 2A − iF, (42)

( ) ( )( ) ( ) 2 2 S E A AE C F i F E A AF B =− + + −− −+ + 44 2 2 , (43)

( )

, 2D

K + Q

υ = (44)

and 

( )

. 2D

K − Q

μ = (45)
8

The vector χ , which represents the initial conditions with the steady state portion removed, is 

used to define the homogeneous response of the epicyclic equations. 

1

0

0

0

0

00 0

0 0/

// 0

// 0

v A D

w A D Dg V

q BD CD E F

r CD BD F E

χ

−

⎧ ⎫ ⎡ ⎤⎧ ⎫ − −

⎪ ⎪ ⎢ ⎥⎪ ⎪ ⎪⎪ ⎪ ⎪ − = − ⎢ ⎥ ⎨⎬ ⎨ ⎬ − ⎪⎪ ⎪ ⎪

⎩ ⎭ ⎣ ⎦⎩ ⎭ −

. (46)

Using the definitions provided by equations 38, 39, and 46, the homogeneous solution for each 

epicyclic state variable is expressed as a combination of the fast and slow modes. 

sin sin ( ) ( ) F S s s

H vF F vF vS S vS ve s e s σ σ =Ω + +Ω + φ θ φθ , (47)

sin sin ( ) ( ) F S s s we s e s H wF F wF wS S wS

σ σ =Ω + +Ω + φ θ φθ , (48)

sin sin ( ) ( ) F S s s

H qF F qF qS S qS qe s e s σ σ =Ω + +Ω + φθ φθ , (49)

and 

sin sin ( ) ( ) F S s s

H rF F rF rS S rS re s e s σ σ =Ω + +Ω + φ θ φθ , (50)

while the particular solution is given by equation 51. 

1

00 0

0 0/

// 0

// 0

P

P

P

P

v A D

w A D Dg V

q BD CD E F

r CD BD F E

−

⎧ ⎫ ⎡ ⎤⎧ ⎫ − −

⎪ ⎪ ⎢ ⎥⎪ ⎪ ⎪⎪ ⎪ ⎪ − = − ⎢ ⎥ ⎨⎬ ⎨ ⎬ − ⎪⎪ ⎪ ⎪

⎩ ⎭ ⎣ ⎦⎩ ⎭ −

. (51)

Expressions for the modal amplitudes and phase angles used in equations 47–50 are provided in 

the appendix. The total epicyclic solution is the summation of the homogeneous and particular 

solutions. 

⎪

⎪

⎭

⎪

⎪

⎬

⎫

⎪

⎪

⎩

⎪

⎪

⎨

⎧

+

⎪

⎪

⎭

⎪

⎪

⎬

⎫

⎪

⎪

⎩

⎪

⎪

⎨

⎧

=

⎪

⎪

⎭

⎪

⎪

⎬

⎫

⎪

⎪

⎩

⎪

⎪

⎨

⎧

P

P

P

P

H

H

H

H

r

q

w

v

r

q

w

v

r

q

w

v

. (52)

The solutions for the Euler pitch and yaw angles are obtained by integrating the solutions for the 

pitch rate and yaw rate, respectively. 
9

( )

( ) ( )

( ) ( )

0

cos cos sin sin

cos cos sin sin ,

F F

S S

P

F F s s

F rF rF F rF rF rF

F F

S S s s

S rS rS S rS rS rS

S S

D

r s

V

D

e s es

V

D

e s es

V

σ σ

σ σ

ψ ψ

φ σ φθ θ φθ θ β β

φ σ φθ θ φθ θ β β

= +

⎛ ⎞ − + +− + +− Ω ⎡ ⎤⎡ ⎤ ⎜ ⎟ ⎣ ⎦⎣ ⎦ ⎝ ⎠

⎛ ⎞ − + +− + +− Ω ⎡ ⎤⎡ ⎤ ⎜ ⎟ ⎣ ⎦⎣ ⎦ ⎝ ⎠ (53)

and 

( )

( ) ( )

( ) ( )

0

cos cos sin sin

cos cos sin sin ,

F F

S S

p

F F s s

F qF qF F qF qF qF

F F

S S s s

S qS qS S qS qS qS

S S

D

q s V

D

e s es

V

D

e s es

V

σ σ

σ σ

θ θ

φ σ φθ θ φθ θ β β

φ σ φθ θ φθ θ β β

= +

⎛ ⎞ − + +− + +− Ω ⎡ ⎤⎡ ⎤ ⎜ ⎟ ⎣ ⎦⎣ ⎦ ⎝ ⎠

⎛ ⎞ − + +− + +− Ω ⎡ ⎤⎡ ⎤ ⎜ ⎟ ⎣ ⎦⎣ ⎦ ⎝ ⎠ (54)

where 

2 2 β F = σ F +φ F (55)

and

2 2. β S SS = + σ φ (56)

The trajectory or swerve solution is then obtained by substituting equations 47, 48, and 49 into 

equations 9 and 10 and subsequently integrating. The final results are given by equations 57 and 58. 

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

*

*

0 0

0

0

0

2

0

0

2

0

sin sin cos

2

sin sin cos .

2

F

S

X NA vF s

F vF vF F vF

F

X NA vS s

S vS vS S vS

S

y y v s

DD V

SD C C

es s

m V

SD C C

es s

m V

σ

σ

ψ

ρ φ θ π θ πφ θ π φ

ρ φ θ π θ πφ θ π φ

⎛ ⎞

= ++ ⎜ ⎟ ⎝ ⎠

⎡ ⎤ − ⎛ ⎞ Ω

+ + −− −− − ⎢ ⎥ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠

⎡ ⎤ − ⎛ ⎞ Ω

+ + −− −− − ⎜ ⎟ ⎢ ⎥ ⎣ ⎦ ⎝ ⎠ (57)

( ) ( ) ( )( ) ( )

( ) ( )( ) ( )

0

*

*

2

0 0

0 0

0 0

0

2

0

0

2

0

1

sin sin cos

2

sin sin cos

2

X

F

S

SDC s m X

X

X NA vF s

F wF wF F wF

F

X NA vS s

S wS wS S wS

S

z mD SD z w sg e Cs D D V SDC m

SD C C

es s

m V

SD C C

e s

m V

ρ

σ

σ

ρ θ

ρ

ρ φ θ π θ πφ θ π φ

ρ φ θ π θ πφ θ π φ

⎛ ⎞⎛ ⎞ ⎛ ⎞

= +− + + − − ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠

⎡ ⎤ − ⎛ ⎞ Ω

+ + −− −− − ⎢ ⎥ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠

⎡ ⎤ − Ω

+ + −− −− − ⎢ ⎥ ⎣ ⎦ ( )s . ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ (58)

Equations 57 and 58 provide relatively simple expressions for the altitude and cross range of a 

dual-spin projectile as a function of range. Both the cross range and altitude solutions contain 
10

constant, linear, and oscillatory terms. The altitude formula has an additional parabolic term to 

account for trajectory bending due to gravity. 

4. Pulsejet Force and Moment Conditions 

The net effect of a pulsejet force is a discontinuity in the translational and rotational velocity of 

the body at the time of the impulse. The discontinuity conditions are found by transforming 

equation 5 to the Laplace domain and noting the additional impulse terms adding with the preimpulse initial conditions. The results are given by equations 59–62. 

0 0

Y DI v v

mV

+ − = + . (59)

0 0

ZI D

w w

mV

+ − = + . (60)

0 0

IX I

YY

R Z D

q q I V

+ − = − . (61)

0 0

IX I

YY

R Y D

r r

I V

+ − = + . (62)

In equations 52–55, the superscript “+” denotes after impulse and the superscript “–” denotes 

before impulse. Since the epicyclic solution amplitude and phase angles (ΩvF , ΩvS , ΩvF , ΩwF , 

ΩwS , ΩqF , ΩqS , ΩrF , ΩrS , θ vF , θ vS , θ vF , θ wF , θ wS , θ qF , θ qS , θ rF , and θ rS ) all depend on 0 v , 

w0 , q0 , and 0r – χ , the dynamic model must be updated at s

+ to accurately account for the 

discontinuity in the epicyclic solution formulas. 

5. Results 

In order to investigate the utility of the simple trajectory formulas derived previously, the results 

that follow compare the trajectory of a dual-spin projectile computed by numerically integrating 

the fully nonlinear equations of motion (13) with the linear theory analytical solution. The 

numerical solution was generated using a fixed step fourth-order Runge Kutta algorithm with a 

time step of 0.00001 s. Tables 1 and 2 list the physical parameters and state conditions for the 

projectile considered in trajectory comparison. The linear model is updated for each change in 

range of one caliber. In all cases, the numerical nonlinear simulation and the analytic closed 

form solution are in agreement. Figures 3 and 4 show the swerve of the projectile. A difference 

of approximately 10 ft between the numerical nonlinear solution and the analytic solution at a 
11

Table 1. Projectile physical properties. 

Parameter Value 

Projectile length 4.5 ft 

Projectile reference diameter 0.23 ft 

Projectile weight 15.9 lb 

Composite body mass center location 2.81 ft (from base) 

Forward body mass center location 3.85 ft (from base) 

Aft body mass center location 1.35 ft (from base) 

Forward body roll inertia 0.0022 slug ft2

Forward body pitch inertia 0.019 slug ft2

Aft body roll inertia 0.002 slug ft2

Aft body pitch inertia 0.42 slug ft2

Bearing location 3.18 ft (from base) 

Bearing viscous damping coefficient 0.0001 

Lateral pulsejet stationline location 3.8 ft (from base) 

Lateral pulsejet buttline location 0.12 ft 

Lateral pulsejet waterline location 0.0 ft 

Table 2. Projectile state conditions. 

State Variable Initial Before Pulse After Pulse 

x 0.0 ft 419.0 ft 419.0 ft 

y –0.76 ft -1.0 ft –1.0 ft 

z –287.0 ft –375.0 ft –375.0 ft 

φ F 4.6° 5.8° 5.8° 

φ A 168.1° 156.5° 156.5° 

θ 0.2° 0.2° 0.2° 

ψ –0.0007° 0.0006° 0.0006° 

V 2497.0 ft/s 2420.0 ft/s 2420.0 ft/s 

v 0.17 ft/s –0.1 ft/s 1.8 ft/s 

w –0.64 ft/s 0.4 ft/s –0.9 ft/s 

F p 7.3 rad/s 6.7 rad/s 6.7 rad/s 

A p –10.1 rad/s –83.8 rad/s –83.8 rad/s 

q –0.006 rad/s –0.01 rad/s 2.14 rad/s 

r 0.001 rad/s 0.0004 rad/s 3.04 rad/s 
12

Figure 3. Cross range vs. range. 

Figure 4. Altitude vs. range. 
13

range of 3750 ft is apparent in the altitude trace. When viewed from the rear, the projectile drifts 

slightly to the left prior to the pulsejet firing. After the pulsejet is fired, the pulsejet turns and 

drifts to the right. Toward the end of the trajectory, the lateral pulsejet has changed the cross 

range impact point by approximately 17 ft. Figures 5 and 6 show that the action of the pulsejet 

induces Euler angle changes exceeding 5°. Because of the roll orientation when the pulsejet was 

fired, the altitude response does not show an effect from the pulsejet. The fins mounted on the 

aft body along with inherent roll damping combine to create the aft body roll rate history shown 

in figure 7. The forward body roll rate history is driven by the viscous friction in the bearing and 

the roll rate of the aft body. Figure 8 displays the difference in the predicted impact point of the 

numerical nonlinear simulation and the analytic solution as a function of the linear model update 

interval. At the range of 3750 ft, the impact error prediction is below 10 ft. Updating the linear 

model used for the analytical solution beyond 4000 cal of travel results in significant increase in 

impact point error. Figure 9 examines the potential change in impact point location as a function 

of down range position. After the epicyclic dynamics have settled, the swerve solution is 

dominated by the linear term. Firing a pulsejet changes the slope of the linear term. As the 

projectile closes on the target, the potential change in position of the round at the target due to 

firing a lateral pulsejet decreases, since a reduced distance to the target reduces the linear swerve 

changes at the target. Defining control authority as the potential change in swerve at the target, 

the control authority of a lateral pulsejet decreases as the projectile travels down range. 

Figure 5. Pitch angle vs. range. 
14

Figure 6. Yaw angle vs. range. 

Figure 7. Roll rate vs. range. 
15

Figure 8. Target prediction error vs. model update interval. 

Figure 9. Induced pulsejet swerve vs. range to target. 
16

6. Conclusions 

A simple set of formulas has been developed for the swerve motion of a dual-spin projectile 

under the action of lateral pulsejets. The solutions provide a computationally low cost and 

reasonably accurate means of predicting impact points of a projectile that can be employed in 

future smart weapon flight control systems. For the best impact point prediction performance, 

the linear model that underlies the analytic solution must be periodically updated. For the 

example configuration examined here, the linear model should be updated no less than once per 

4000 cal of range. The ability of lateral pulsejets to change the impact point steadily decreases 

with range to the target.
17 

7. References 

1. Guidos, B.; Cooper, G. Closed Form Solution of Finned Projectile Motion Subjected to a 

Simple In-Flight Lateral Impulse; Paper No. AIAA-2000-0767; 38th AIAA Aerospace 

Sciences Meeting and Exhibit, Reno, NV, 2000. 

2. Likins, P. W. Attitude Stability Criteria for Dual-Spin Spacecraft. Journal of Spacecraft and 

Rockets 1967, 4, pp 1638–1643. 

3. Cloutier, G. J. Stable Rotation States of Dual-Spin Spacecraft. Journal of Spacecraft and 

Rockets 1968, 5, pp 490–492. 

4. Mingori, D. L. Effects of Energy Dissipation on the Attitude Stability of Dual Spin 

Satellites. AIAA Journal 1969, 7, pp 20–27. 

5. Fang, B. T. Energy Considerations for Attitude Stability of Dual-Spin Spacecraft. Journal 

of Spacecraft and Rockets 1968, 5, pp 1241–1243. 

6. Hall, C. D.; Rand, R. H. Spinup Dynamics of Axial Dual-Spin Spacecraft. Journal of 

Guidance, Control, and Dynamics 1994, 17, pp 30–37. 

7. Or, A. C. Resonances in the Despin Dynamics of Dual-Spin Spacecraft. Journal of 

Guidance, Control, and Dynamics 1991, 14, pp 321–329. 

8. Cochran, J. E.; Shu, P. H.; Rew, S. D. Attitude Motion of Asymmetric Dual-Spin Spacecraft. 

Journal of Guidance, Control, and Dynamics 1982, 5, pp 37–42. 

9. Tsuchiya, K. Attitude Behavior of a Dual-Spin Spacecraft Composed of Asymmetric 

Bodies. Journal of Guidance, Control, and Dynamics 1979, 2, pp 328–333. 

10. Yang, H. X. Method for Stability Analysis of Asymmetric Dual-Spin Spacecraft. Journal of 

Guidance, Control, and Dynamics 1989, 12, pp 123–125. 

11. Viderman, Z.; Rimrott, F. P. J.; Cleghorn, W. L. Stability of an Asymmetric Dual-Spin 

Spacecraft with Flexible Platform. Journal of Guidance, Control, and Dynamics 1991, 14, 

pp 751–760. 

12. Stabb, M. C.; Schlack, A. L. Pointing Accuracy of a Dual-Spin Satellite due to Torsional 

Appendage Vibrations. Journal of Guidance, Control, and Dynamics 1991, 16, pp 630–635. 

13. Smith, J. A.; Smith, K. A.; Topliffe, R. Feasibility Study for Application of Modular 

Guidance and Control Units to Existing ICM Projectiles. ARLCD-CR-79001; U.S. Army 

Armament Research and Development Command: Picatinny Arsenal, Dover, NJ, 1978. 
18

14. Costello, M.; Peterson, A. Linear Theory of a Dual-Spin Projectile in Atmospheric Flight. 

Journal of Guidance, Control, and Dynamics 2000, 23. 

15. McCoy, R. L. Modern Exterior Ballistics; Schiffer Publishing Ltd.: Atglen, PA, 1999. 

16. Ginsberg, J. H. Advanced Engineering Dynamics; Cambridge University Press: New York, 

NY, 1995. 
19 

Appendix. Dual-Spin Linear Theory Coefficients
20 

( ( ) ) ( ( ) )

2

1 1 *1

2

1 1 *1 2Re ξ η χ 2Im ξ η χ T T ΩvF = + . (A-1)

( ( ) ) ( ( ) )

2

2 2 *1

2

2 2 *1 2Re ξ η χ 2Im ξ η χ T T ΩvS = + . (A-2)

( ( ) ) ( ( ) )

2

1 1 *2

2

1 1 *2 2Re ξ η χ 2Im ξ η χ T T ΩwF = + . (A-3)

( ( ) ) ( ( ) )

2

2 2 *2

2

2 2 *2 2Re ξ η χ 2Im ξ η χ T T ΩwS = + . (A-4)

( ( ) ) ( ( ) )

2

1 1 *3

2

1 1 *3 2Re ξ η χ 2Im ξ η χ T T ΩqF = + . (A-5)

( ( ) ) ( ( ) )

2

2 2 *3

2

2 2 *3 2Re ξ η χ 2Im ξ η χ T T ΩqS = + . (A-6)

( ( ) ) ( ( ) )

2

1 1 *4

2

1 1 *4 2Re ξ η χ 2Im ξ η χ T T ΩrF = + . (A-7)

( ( ) ) ( ( ) )

2

2 2 *4

2

2 2 *4 2Re ξ η χ 2Im ξ η χ T T ΩrS = + . (A-8)

( )

( ) . 2Im

2Re

tan

1 1 *1

1 1 1 *1

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

− = −

ξ η χ

ξ η χ θ T

T

vF (A-9)

( )

( ) . 2Im

2Re

tan

2 2 *1

1 2 2 *1

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

− = −

ξ η χ

ξ η χ θ T

T

vS (A-10)

( )

( ) . 2Im

2Re

tan

1 1 *2

1 1 1 *2

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

− = −

ξ η χ

ξ η χ θ T

T

wF (A-11)

( )

( ) . 2Im

2Re

tan

2 2 *2

1 2 2 *2

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

− = −

ξ η χ

ξ η χ θ T

T

wS (A-12)

( )

( ) . 2Im

2Re

tan

1 1 *3

1 1 1 *3

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

− = −

ξ η χ

ξ η χ θ T

T

qF (A-13)

( )

( ) . 2Im

2Re

tan

2 2 *3

1 2 2 *3

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

− = −

ξ η χ

ξ η χ θ T

T

qS (A-14)

( )

( ) . 2 Im

2Re

tan

1 1 *4

1 1 1 *4

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

− = −

ξ η χ

ξ η χ θ T

T

rF (A-15)

( )

( ) . 2Im

2Re

tan

2 2 *4

1 2 2 *4

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

− = −

ξ η χ

ξ η χ θ T

T

rS (A-16)

In equations A1–A16, the notation * j τ denotes the jth row of the matrix τ .
21 

List of Symbols, Abbreviations, and Acronyms 

A

X

F CX 0 C 0 , Zero yaw axial force aerodynamic coefficient for the forward and aft bodies. 

A

X

F CX 2 C 2 , Yaw angle squared axial force aerodynamic coefficient for the forward and aft 

 bodies. 

A

NA

F CNA ,C Normal force aerodynamic coefficient for the forward and aft bodies. 

A

NPA

F CNPA ,C Magnus force aerodynamic coefficient for the forward and aft bodies. 

A

DD

F CDD ,C Roll moment aerodynamic coefficient due to fin cant for the forward and aft 

 bodies. 

A

LP

F CLP ,C Roll damping moment aerodynamic coefficient for the forward and aft bodies. 

A

MQ

F CMQ ,C Pitch rate damping moment aerodynamic coefficient for the forward and aft bodies. 

Vc Viscous damping coefficient for hydrodynamic bearing. 

D Projectile characteristic length. 

g Gravity constant. 

F

YY

F

XX I , Roll and pitch inertia of the I forward body projectile section. 

A

YY

A

XX I , Roll and pitch inertia of th I e aft body projectile section. 

m Total projectile mass. 

mA Aft body mass. 

mF Forward body mass. 

pF Roll axis component of the angular velocity vector of the forward body expressed 

in the fixed plane reference frame. 

pA Roll axis component of the angular velocity vector of the aft body expressed in the 

fixed plane reference frame. 

q,r Components of the angular velocity vector of both the forward and aft bodies 

expressed in the fixed plane reference frame. 

fx fy fz r , Fixed plane components of vector from r ,r composite center of mass to forward body 

mass center. 
22

ax ay az r , Fixed plane components of vector from r ,r composite center of mass to aft body mass 

center. 

Rfx Rfy Rfz , , Fixed plane components of vector from forward body mass center to forward body 

center of pressure. 

Rax Ray Raz , Fixed plane components of vector from , aft body mass center to aft body center of 

pressure. 

, , R

ax ay az R R Fixed plane components of vector from aft body mass center to aft body Magnus 

center of pressure. 

, , R

fx fy fz R R Fixed plane components of vector from forward body mass center to forward body 

Magnus center of pressure. 

RIX RIY RIZ , , Fixed plane components of vector from composite body mass center to the 

impulse application point. 

s Nondimensional projectile arc length. 

x, Position vector components of the com y,z posite center of mass expressed in the 

inertial reference frame. 

, Y Z I I Pulsejet force components in projectile body axes. 

u, Translation velocity com v,w ponents of the composite center of mass resolved in the 

fixed plane reference frame. 

V Magnitude of mass center velocity. 

θ ,ψ Euler pitch, and yaw angles. 

φ F

 Euler roll angle of the forward body. 

φ A

 Euler roll angle of the aft body. 
NO. OF 

COPIES ORGANIZATION 

23 

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