On the Swerve Response of Projectiles to Control Input 

by Douglas Ollerenshaw and Mark Costello 

ARL-CR-0604 April 2008 

prepared by 

Oregon State University 

Corvallis, Oregon 

and 

Georgia Institute of Technology 

Atlanta, Georgia 

under contract 

W911QX-06-C-0113

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

Aberdeen Proving Ground, MD 21005-5069 

ARL-CR-0604 April 2008 

On the Swerve Response of Projectiles to Control Input 

Douglas Ollerenshaw 

Department of Mechanical Engineering, Oregon State University 

Mark Costello 

School of Aerospace Engineering, Georgia Institute of Technology 

prepared by 

Oregon State University 

Corvallis, Oregon 

and 

Georgia Institute of Technology 

Atlanta, Georgia 

under contract 

W911QX-06-C-0113

Approved for public release; distribution is unlimited.
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 On the Swerve Response of Projectiles to Control Input 

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6. AUTHOR(S) 

 Douglas Ollerenshaw (OSU); Mark Costello (GIT) 

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 Oregon State University Georgia Institute of Technology 

 Corvallis, Oregon Atlanta, Georgia

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14. ABSTRACT 

The swerve response of fin- and spin-stabilized projectiles to control mechanism input is sometimes not intuitive and often 

surprises smart weapon designers. This report seeks to explain the basic parameters that govern swerve of projectiles excited 

by control input. By modeling the overall effect of any control mechanism as a non-rolling reference frame force or moment 

applied to the projectile, we obtain general expressions for swerve in terms of basic vehicle parameters. These compact 

expressions are used to show that maximum swerve response for a fin-stabilized projectile is achieved when the force is 

applied near the nose of the projectile, while maximum swerve response for a spin-stabilized projectile is achieved when the 

force is applied near the base of the projectile. 

15. SUBJECT TERMS

 guidance navigation and control; swerve response 

16. SECURITY CLASSIFICATION OF: 

19a. NAME OF RESPONSIBLE PERSON

 Peter Plostins 

a. REPORT 

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UNCLASSIFIED

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OF ABSTRACT 

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Standard Form 298 (Rev. 8/98) 

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iii 

Contents 

List of Figures iv 

List of Tables iv 

1. Introduction 1 

2. Simplified Analytical Swerve Solution 1 

3. Correlation to Full Six-Degree-of-Freedom Model 9 

4. Effects of Individual Parameters 15 

5. Conclusions 19 

6. References 20 

Distribution List 22 
iv 

List of Figures 

Figure 1. Vertical plane swerve response of the M829A2 fin-stabilized projectile to 1-lbf 

control input applied in the +y, +z, -y, and -z directions, with 6-DOF correlation data...... 11 

Figure 2. Magnitude of the swerve response of the M829A2 fin-stabilized projectile to a 

1-lbf control input as a function of the distance from the projectile c.g. to the point of 

application of the force, with 6-DOF correlation data......................................................... 12 

Figure 3. Phase shift of the swerve response of the M829A2 fin-stabilized projectile to a 

1-lbf control input as a function of the distance from the projectile c.g. to the point of 

application of the force, with 6-DOF correlation data.......................................................... 12 

Figure 4. Vertical plane swerve response of the M549 spin-stabilized projectile to 1-lbf 

control input applied in the +y, +z, -y, and -z directions, with 6-DOF correlation data....... 13 

Figure 5. Magnitude of the swerve response of the M549 spin-stabilized projectile to a 

1-lbf control input as a function of the distance from the projectile c.g. to the point of 

application of the force, with 6-DOF correlation data......................................................... 14 

Figure 6. Phase shift of the swerve response of the M549 spin-stabilized projectile to a 

1-lbf control input as a function of the distance from the projectile c.g. to the point of 

application of the force, with 6-DOF correlation data......................................................... 14 

Figure 7. Summary of projectile response to a control input in the positive y-direction in 

both a fin-stabilized and a spin-stabilized projectile............................................................. 17 

Figure 8. Magnitude of the swerve response of the M549 spin-stabilized projectile to a 

1-lbf control input as a function of the distance from the projectile c.g. to the point of 

application of the control force, Magnus force coefficient varied from its value to zero..... 18 

Figure 9. Phase shift of the swerve response of the M549 spin-stabilized projectile to a 

1-lbf control input as a function of the distance from the projectile c.g. to the point of 

application of the control force, Magnus force coefficient varied from its value to zero..... 18 

List of Tables 

Table 1. Summary of projectile initial conditions, physical parameters, and aerodynamic 

coefficients............................................................................................................................ 10 
1 

1. Introduction 

The continuing development of micro-electromechanical systems (MEMS) is pointing to the 

possibility of mounting complete sensor systems on medium and small caliber projectiles as part 

of an actively controlled smart munition. Two important technical challenges in achieving this 

goal are the development of small, rugged sensor suites and control mechanisms. There is 

currently a flurry of activity to create innovative physical control mechanisms. Concepts include 

pulse jets, squibs, synthetic jets (1, 2, 3), drag brakes (4, 5), deployable pins (6, 7), movable nose 

(8), movable canards (9), dual-spin projectiles (10, 11), ram air deflection (12), and internal 

translating mass (13), to name a few. 

Although the physical control mechanisms mentioned are very diverse, there is a common theme 

among all these physical control mechanisms. All concepts exert a force and/or moment on the 

projectile. Moreover, since trajectories are shaped relative to ground coordinates, the forces and 

moments are effectively applied in a non-rolling reference frame. Although the uncontrolled 

dynamics of projectiles (both fin stabilized and spin stabilized) have been extensively studied in 

the ballistics community (14), issues with regard to control response have received considerably 

less attention because of the lack of practical application of control technology to spinning 

projectiles. With the use of projectile linear theory, this report analytically investigates several 

aspects of the response of a spinning projectile to a control force and/or moment in the non-rolling 

reference frame. Simple expressions result for the swerve response magnitude and phase angle in 

terms of basic physical mass properties, aerodynamic characteristics, and the state of the air 

vehicle. These expressions provide a means toward deeper understanding of the underlying 

factors driving control response of projectiles, helping smart weapon designers to create more 

capable weapon systems. 

2. Simplified Analytical Swerve Solution 

The six-degree-of-freedom (6-DOF) rigid body dynamic model used to simulate the trajectory of 

a projectile in atmospheric flight has been well developed. The model consists of the three 

inertial components of the position vector from an inertial frame to the projectile mass center and 

the three standard Euler orientation angles. The resulting equations of motion are shown as 

equations 1 through 4 (15). 

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−

+ −

− +

=

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

w

v

u

s s c c c

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

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

z

y

x

θ φ θ φ θ

θ ψ φ θ ψ φ ψ φ θ ψ φ ψ

θ ψ φ θ ψ φ ψ φ θ ψ φ ψ

&

&

&

 (1) 
2 

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

= −

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

r

q

p

s c c c

c s

s t c t

φ θ φ θ

φ φ

φ θ φ θ

ψ

θ

φ

0 / /

0

1

&

&

&

 (2) 

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−

−

−

−

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

=

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

w

v

u

q p

r p

r q

Z m

Y m

X m

w

v

u

0

0

0

/

/

/

&

&

&

 (3) 

[] []

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−

−

−

−

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

=

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧ −

r

q

p

I

q p

r p

r q

N

M

L

I

r

q

p

0

0

0

1

&

&

&

 (4) 

In equations 1 and 2, the standard shorthand notation for trigonometric functions is used: sin (α) ≡

sα, cos (α) ≡ cα, and tan (α) ≡ tα. The forces and moments appearing in equations 3 and 4 contain 

contributions from weight, body aerodynamics, and control forces and are discussed in detail in the 

literature (16). 

Since no analytical solution can be found for the differential equations shown, they must be solved 

with numerical integration techniques. Over time, however, a series of simplifications of the 

dynamic equations has been identified, which results in an analytically solvable set of quasi-linear 

differential equations and reasonably accurate trajectories. These equations are referred to collectively as projectile linear theory (17). The dynamic equations that constitute projectile linear 

theory are given as equations 5 through 13. 

0 0

0 0 F

F

F

F

A D

v v V A D

w w B C W

H F

q q D D Q

r CB r R F H

D D

⎡ ⎤ − −

⎧ ⎫ ⎧⎫ ′ ⎢ ⎥ − ⎧ ⎫ ⎪ ⎪ ⎪⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ′ ⎢ ⎥ ⎨ ⎬ ⎨⎬ ⎨ ⎬ = + − ′ ⎪ ⎪ ⎪⎪ ⎪ ⎪

′ ⎩ ⎭ ⎩⎭ ⎩ ⎭ −

⎣ ⎦

% %

% %

% %

% %

 (5) 

3

0

8

D Cx gD V V

m V

πρ θ ′ = − (6) 

4 5

8 16

LDD LP

X X

DC DC

p V p I I

πρ πρ ′ = + (7) 

φ′ = Dp (8) 

D

q V

θ′ = % (9) 
3 

D

r

V ψ′ = % (10) 

x′ = D (11) 

D y vD V ′ = +% ψ (12) 

D z wD

V ′ = −% θ (13) 

The projectile linear theory dynamic equations use dimensionless arc length, s, as the independent 

variable. Arc length is related to time as shown in equation 14: 

0

1 t

s Vdt

D = ∫ (14) 

Additionally, the linear theory equations employ a reference frame that is aligned with the projectile axis of symmetry but does not roll. Variables in this reference frame, referred to as the noroll frame or the fixed plane frame, are denoted with a ~ superscript. The no-roll frame is related 

to the body fixed frame used in the traditional 6-DOF equations by a single axis rotation about the 

projectile axis of symmetry. For example, the no-roll frame velocity components are related to the 

body frame velocity components as shown in equation 15: 

10 0

0

0

u u

v c sv

w sc w

φ φ

φ φ

⎧⎫ ⎧⎫ ⎡ ⎤ ⎪⎪ ⎪⎪ ⎢ ⎥ ⎨⎬ ⎨⎬ = − ⎢ ⎥ ⎪⎪ ⎪⎪ ⎢ ⎥ ⎩⎭ ⎩⎭ ⎣ ⎦

%

%

%

 (15) 

For the purpose of examining basic swerve response attributable to control input, gravity and 

atmospheric winds are neglected. The constant terms in the set of four coupled equations shown 

as equation 5, referred to as the epicyclic equations, can then be described as 

3

8

D CNA A

m

πρ

= (16) 

5

16

YPA SLM

P

DC p B

I V

πρ Δ = (17) 

4

8

NA SLP

P

D C C

I

πρ Δ = (18) 

R

P

I Dp F

I V = (19) 

5

16

MQ

P

D C

H

I

πρ

= (20) 
4 

C

F

DY V

mV = (21) 

C

F

DZ W

mV = (22) 

SLC C

F

P

D Z Q I VΔ = − (21) 

SLC C

F

P

D Y R

I V

Δ = (22) 

Both the velocity (V) and the roll rate (p) are considered to be constant wherever they appear in 

equations 9 through 13 and equations 16 through 22. The terms D and m refer to the projectile 

reference diameter and the total projectile mass, respectively. The projectile inertia terms, PI and 

RI , are respectively the roll and pitch inertias. The term ρ is the atmospheric air density. The 

term CNA is the normal force coefficient. The normal force acts in a direction perpendicular to the 

projectile axis of symmetry and results from non-axial wind forces caused by yawing and pitching 

of the projectile. The normal force acts not at the projectile center of gravity (c.g.) but at a point 

called the normal force center of pressure (COP). The ΔSLP term represents the distance between 

the c.g. and the center of pressure as follows: 

SLP COP CG Δ = − SL SL (23) 

in which both the c.g. and the center of pressure are measured from the projectile base along the 

projectile stationline. CYPA is the Magnus force coefficient. The Magnus force is caused by 

unequal pressures on opposite sides of a spinning body resulting from the viscous interaction 

between the spinning surface and the surrounding atmosphere. The Magnus force itself is 

generally considered to be small enough to be neglected. However, the resulting moment must 

be considered. ΔSLM represents the distance between the c.g. and the point of application of the 

Magnus force (MAG): 

SLM MAG CG Δ = − SL SL (24) 

in which both the c.g. and the MAG application point are measured from the projectile base along 

the projectile stationline. The MAG is proportional to both spin rate and transverse angular 

velocity. Therefore, in projectiles with very low spin rates, the Magnus moment approaches zero. 

The term CMQ represents the pitch-damping moment coefficient. The pitch-damping moment is 

proportional to the transverse angular velocity of the projectile. CMQ will always be negative for a 

stable projectile. Thus, it has the stabilizing effect of decreasing the total transverse angular 

velocity of the projectile. 
5 

Finally, the terms YC and ZC represent applied control forces applied along the y and z axes of the 

projectile no-roll frame. Both forces are assumed to act at the same point (CF) on the projectile 

with a moment arm, ΔSLC , defined as 

SLC CF CG Δ = − SL SL (25) 

in which both the center of mass and the point of application of the control forces are measured 

from the rear of the projectile along the projectile axis of symmetry. 

Determining the solution of the coupled epicyclic equations is necessary before solving both the 

attitude (θ,ψ ) and the swerve ( y,z ) equations. To obtain the solution, the epicyclic equations 

are first transformed into the Laplace domain. 

( )

( )

( )

( )

( )

( )

( )

( )

1 0 0 0

0 0 0

0

0

F

F

F

F

V

v sA D s

v s sA D W

w

w s B C s

sH F

q s D D Q

q

r s C B s

F sH

D D R

r

s

− ⎧ ⎫ +

+ ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ ⎧ ⎫ ⎢ ⎥ + − ⎪ ⎪ ⎪ ⎪ + ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎨⎬ ⎨ ⎬ = −− − ⎪⎪ ⎪ ⎪ +

⎩ ⎭ − − ⎪ ⎪ ⎣ ⎦ ⎪ + ⎪ ⎩ ⎭

%

% % %

% % %

%

(26) 

It is important to note that the variable s as expressed in equation 24 is the Laplace operator, not 

arc length as described in equation 14. Equation 26 can be expanded so that each of the four 

expressions, represented by the dummy variable ζ, is in the following form 

( )

432

4 3 2 10

5432

4 3 2 10

ssss

s

sssss

ζ ζ ζ ζζ

ζ ζ ζ ζζ

η η η ηη ζ δ δ δ δδ+ + ++ = + + + ++

(27) 

in which 4321043210 ,,,,,,,,, ηζ ζ ζ ζζ ζ ζ ζ ζζ ηηηη δ δ δ δδ are all known constants expressed in terms of 

the epicyclic initial conditions and the constants A, B, C, F, H, VF, WF, QF, RF. The non-zero 

eigenvalues of the system are broken into two pairs of complex conjugates known as the fast and 

slow modes of the system. They are described as 

( ) 1 1 2 22 4 42 2 2

2 2

λF = −+ ± + ± + ± − + ± + A H iF A iB C iAF F AH iFH H (28) 

( ) 1 1 2 22 4 42 2 2

2 2

λS = −+ ± − ± + ± − + ± + A H iF A iB C iAF F AH iFH H (29) 

Thus, each term can also be written in partial fractions form as 

( ) 0 10 10

2 22 2 2 2 FF F SS S

C Fs F Ss S

s

ss s s s

ζ ζζ ζζ ζ ζ ω ω ζω ω

+ +

=+ +

+ + ++

(30) 
6 

in which the fast and slow mode natural frequencies and damping rates are described as 

ωF FF = λ λ1 2 (31) 

( 1 2 )

1 2 2

F F

F

F F

λ λ

ζ λ λ

+ = − (32) 

ωS SS = λ λ1 2 (33) 

( 1 2 )

1 2 2

S S

F

S S

λ λ

ζ λ λ

+ = − (34) 

Equations 27 and 30 can be equated to one another, with the resulting expression used to solve 

for the unknown coefficients in the numerator of equation 30. The generalized Laplace domain 

solution for each of the four epicyclic differential equations can also be expressed as shown in 

equation 35. 

( ) ( ) ( )

( ) ( )

0

0 1

1 1 2 2 2 2

0

1

1 1 2 2 2 2

FR

FR

FR FI FR FI

SR

SR

SR SI SR SI

F

C F s sF F

s s s

S

s S

S S

s s

ζ

ζ ζ

ζ ζ

ζ

ζ

ζ ζ

λ

λ ζ λλ λλ

λ

λ

λλ λλ

⎛ ⎞ ⎜ ⎟ + ⎛ ⎞ − ⎜ ⎟ =+ + ⎜ ⎟

−+ −+ ⎜ ⎟ ⎝ ⎠ ⎜ ⎟ ⎝ ⎠

⎛ ⎞ ⎜ ⎟ + ⎛ ⎞ − ⎜ ⎟ + + ⎜ ⎟

−+ −+ ⎜ ⎟ ⎝ ⎠ ⎜ ⎟ ⎝ ⎠

(35) 

in which λFR and λFI are the real and imaginary parts of the fast mode eigenvalues, and λSR and λSI

are the real and imaginary parts of the slow mode eigenvalues. Note that equation 35 is arranged 

to match standard forms found in inverse Laplace transform tables, thus allowing easy transformation of the epicyclic solutions back to the arc length domain. 

With the epicyclic solutions thus obtained, the attitude solutions can be expressed in the Laplace 

domain as 

( ) ( ) ( ) 1 0 D

s rs

Vs s ψ = + % ψ (36) 

( ) ( ) ( ) 1 0 D

s qs Vs s

θ θ = + % (37) 

Similarly, the swerve solutions can be expressed in the Laplace domain as

( ) ( ) ( ) ( ) 1 0 D D

ys vs s y Vs s s

=++ % ψ (38) 
7 

( ) ( ) ( ) ( ) 1 0 D D

zs ws s z

Vs s s

= −+ % θ (39) 

The attitude expressions shown in equations 36 and 37, along with the expressions for v s %( ), 

w s % ( ) , q s %( ), and r s %( ), as expressed in equation 35, can be substituted into the swerve expressions 

in equations 38 and 39. Taking the inverse Laplace transform of the resulting expressions yields 

arc length domain swerve solutions of the form 

2

01 2 ( ) ( cos( ) sin( ))

( cos( ) sin( ))

FR

SR

s

y y y yfc FI yfs FI

s

ysc SI yss SI

y s C Cs Cs e C s C s

e C sC s

λ

λ

λ λ

λ λ

=+ + + +

+ +

(40) 

2

01 2 ( ) ( cos( ) sin( ))

( cos( ) sin( ))

FR

SR

s

z z z zfc FI zfs FI

s

zsc SI zss SI

zs C C s C s e C s C s

e C sC s

λ

λ

λ λ

λ λ

=+ + + +

+ +

(41) 

Additionally, the down-range position of the projectile, x, expressed as a function of arc length is 

simply 

0 x( )s x Ds = + (42) 

One must be careful to note that the s term appearing in equations 40 through 41 represents arc 

length, as defined in equation 14. 

To obtain a sense of the generalized swerve response of a projectile because of applied control 

forces, we examine the case in which the projectile is fired down range with no initial pitch or yaw 

angle and with no initial perturbations of the transverse lateral and angular velocities. Assuming 

that the firing position is at the origin of the inertial reference frame, this allows us to set the initial 

conditions of all terms to zero, except for velocity and roll rate. The velocity and roll rate initial 

conditions are denoted V0 and p0. Additionally, as stated earlier, the effects of gravity and atmospheric winds are neglected here. These assumptions provide a case in which a projectile with no 

control forces applied displays no swerving motion. Subsequently, the swerve response created by 

the application of control forces will be clear. 

When we examine the swerve expressions in equations 40 and 41, a few simplifications can be 

made. First of all, note that in a stable projectile, the real parts of the fast and slow mode eigenvalues, λFR and λSR , are always negative. Therefore, the oscillatory terms in the swerve response 

decays as the projectile flies down range and can be neglected for long-term swerve response. The 

pitch-damping moment is primarily associated with the oscillatory epicyclic terms and can also be 

neglected, allowing CMQ in equation 20 to be set to zero. Additionally, as the arc length value 

becomes large, the terms containing the square of the arc length begin to dominate the swerve 

response expressions and the terms Cy0 , Cy1 , Cz0 , and Cz1 can be neglected. Finally, equation 42 

can be solved for arc length, s, and substituted into equations 40 and 41. 

The resulting simplified swerve expressions can then be expressed as a function of range solely 

in terms of projectile parameters, initial velocity, and spin rate, and control force input applied in 
8 

the no-roll frame. They can be further simplified if we express the swerve response in terms of 

its magnitude relative to the magnitude of the control input, and the phase shift relative to the 

angle of the applied control vector. A compact and informative expression for the response 

magnitude, R, results: 

( )

( )

2 2 222 2 22

0 0

2 2 2 222 2 0 0 0

4

2 2 4

C YPA SLM NA SLC SLP

R NA YPA SLM NA SLP

F x D pC V C R

V p I C mDC m V C

Δ + Δ −Δ = + Δ+ Δ

(43) 

in which FC is the magnitude of the control force, defined as 

2 2 F FF C yz = + (44) 

The expression for the phase shift of the swerve response, ΦR , is 

( ( ) )

( ) ()

1 0 0

2 22

0 0

2 2

tan

2 4

NA R NA SLC SLP YPA SLM SLC

R

YPA SLM R NA YPA SLM NA SLP SLC SLP

V p C I C mDC

Dp C I C mDC mV C

− ⎛ ⎞ Δ −Δ + Δ Δ

Φ = ⎜ ⎟ Δ + Δ − Δ Δ −Δ ⎝ ⎠

(45) 

The phase angle of the swerve response, ΦT , in the y-z plane can be found if we add the phase 

angle of the control input to the phase shift of the swerve response, ΦR , as follows: 

1 tan z

T R

y

F

F

− ⎛ ⎞

Φ = +Φ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠(46) 

Since the inverse tangent function will only result in values between -π/2 and π/2, careful 

consideration must be paid to the signs of the numerator and denominator of the arguments in 

equations 46 and 47 to ensure that the result is in the proper quadrant of the y-z plane. It is 

advisable to use the atan2 function instead to avoid confusion. 

Equations 43 and 45 can be further simplified if the control mechanism being investigated imparts 

only a pure moment upon the projectile body. The expression for the swerve response magnitude, 

R, becomes 

( )

2

2 2 222 2 0 0 0

1

2 4

C NA

R NA YPA SLM NA SLP

M xC R

V p I C mDC m V C = + Δ+ Δ

(47) 

in which MC is the magnitude of the applied control moment, defined as 

2 2 MC yz = + M M (48) 

The phase shift of the swerve response, ΦR , becomes 

( )

1 0

0

2

tan

2

NA SLP

R

R NA YPA SLM

mV C

p I C mDC

− ⎛ ⎞ Δ Φ = ⎜ ⎟ + Δ ⎝ ⎠

(49) 
9 

As before, the phase angle of the swerve response, ΦT , in the y-z plane can be found if we add 

the phase angle of the control input to the phase shift of the swerve response, ΦR . 

1 tan z

T R

y

M

M

− ⎛ ⎞

Φ = +Φ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠(50) 

These equations provide relatively compact expressions for the swerve magnitude (equations 43 

and 47) and swerve phase angle (equations 45 and 49) because of a control force applied to a point 

on the projectile (equations 43 and 45) and a pure moment applied to the projectile (equations 47 

and 49). These expressions highlight the key parameters that drive control response of projectiles 

excited by control force and moment input. However, when one is applying these formulas, it is 

important to recall that stability of the projectile is inherent in the assumptions used to achieve 

these expressions. When parameters are varied in these expressions to investigate the effects on 

swerve response, care must be taken to ensure that the stability assumption is not violated. It also 

needs to be emphasized that these equations calculate the magnitude and phase shift of the swerve 

response under the assumption that the velocity and spin rate remain constant at their initial values, 

and in turn, all Mach number dependent quantities remain constant as well. Of course, this 

assumption becomes increasingly inaccurate as the projectile proceeds down range and must be 

periodically updated for long-range trajectories. 

3. Correlation to Full Six-Degree-of-Freedom Model 

To demonstrate the accuracy of the simplified swerve equations obtained previously, two exemplar 

projectiles were chosen for which complete body aerodynamic properties are already known. The 

two projectiles chosen were the M829A2 fin-stabilized projectile and M549 spin-stabilized projectile. For both projectiles, the results obtained with equations 43 and 45 were compared to results 

from a fixed step, fourth order Runge-Kutta numerical integration of the full 6-DOF equations of 

motion given as equations 1 through 4. For both projectiles, the swerve response was evaluated at 

a range of 5,280 feet in the absence of gravity and atmospheric winds and with no initial yaw or 

pitch angle. In both cases, four input force scenarios were examined: Fy = 1 lbf & Fz = 0 lbf, 

Fy = 0 lbf & Fz = 1 lbf, Fy = -1 lbf & Fz = 0 lbf, and Fy = 0 lbf & Fz = -1 lbf. The control force 

moment arm, ΔSLC , in equations 43 and 45 was varied from 1 foot behind the projectile center of 

mass to 1 foot in front of the projectile center of mass. The 6-DOF swerve response was calculated for each input force scenario with control force moment arms of ΔSLC = -1.0 ft, ΔSLC = -0.5 ft, 

ΔSLC = 0.0 ft, ΔSLC = 0.5 ft, and ΔSLC = 1.0 ft. The 6-DOF expressions calculate all forces and 

moments in the projectile body frame, as opposed to the projectile no-roll frame used to achieve 

the swerve expressions presented here. Therefore, the control forces and moments must be 

transformed from the no-roll frame to the body frame for use in the 6-DOF expressions. 
10 

Table 1 summarizes the initial conditions and the resultant aerodynamic coefficients, along with 

the relevant physical parameters, for both projectiles as used in equations 43 and 45. 

Table 1. Summary of projectile initial conditions, 

physical parameters, and aerodynamic 

coefficients. 

 M829A2 M549 

V0 (ft/s) 5479.0 2710.0 

p 0 (rad/s) 8.7000 1674.1 

ρ (slug/ft3) 2.3785x10-3 2.3785x10-3

l R (slug*ft2) 2.3870x10-4 0.10857 

l P (slug*ft2) 0.17718 1.3964 

m (slug) 0.34461 2.9465 

D (ft) 0.08790 0.50853 

CNA 13.350 2.6314 

CYPA 0.0000 -0.9600 

CMQ -5215.8 -27.700 

ΔSLM (ft) -1.3833 -0.52920 

ΔSLP (ft) -0.50079 -0.71373 

Figure 1 shows the swerve response of the M829A2 fin-stabilized projectile in the vertical target 

plane at a down-range location of x = 5280 ft, with five 6-DOF data points included to demonstrate 

correlation. Note that the positive z-direction points downward in the negative altitude direction. 

Figures 2 and 3 show the magnitude and phase shift of the response, along with 6-DOF correlation 

data for the M829A2. Note that the magnitude of the response depends only upon the magnitude 

of the control input, not its direction. The phase shift of the response, however, does not vary with 

the magnitude of the input and also depends on the direction of the input force. As positive y is 

rightward and positive z is downward in the vertical target plane plots, a positive phase shift is in 

the clockwise direction. 

Figure 4 shows the swerve response of the M529 spin-stabilized projectile in the vertical target 

plane at a down-range location of x = 5280 ft, with five 6-DOF data points included to demonstrate 

correlation. Figures 5 and 6 show the magnitude and phase shift of the response, along with 6-DOF 

correlation data for the M549. 

For both the fin-stabilized and spin-stabilized projectiles studied here, the response as predicted 

by the simplified swerve equations is shown to correlate very well with that predicted by the full 

6-DOF simulation at this relatively short range. Though only a very small fraction of the total 

terms comprising the full linear theory swerve expressions are preserved in the simplified 

version presented, it is obvious that those terms providing the dominant effect on the swerve 

response have been retained. 
11 

Figure 1. Vertical plane swerve response of the M829A2 fin-stabilized projectile to 1-lbf control input applied 

in the +y, +z, -y, and -z directions, with 6-DOF correlation data. 
12 

Figure 2. Magnitude of the swerve response of the M829A2 fin-stabilized 

projectile to a 1-lbf control input as a function of the distance from 

the projectile c.g. to the point of application of the force, with 6-DOF 

correlation data. 

Figure 3. Phase shift of the swerve response of the M829A2 fin-stabilized 

projectile to a 1-lbf control input as a function of the distance from 

the projectile c.g. to the point of application of the force, with 6-DOF 

correlation data. 
13 

Figure 4. Vertical plane swerve response of the M549 spin-stabilized projectile to 1-lbf control input applied in the 

+y, +z, -y, and -z directions, with 6-DOF correlation data. 
14 

Figure 5. Magnitude of the swerve response of the M549 spin-stabilized 

projectile to a 1-lbf control input as a function of the distance from 

the projectile c.g. to the point of application of the force, with 

6-DOF correlation data. 

Figure 6. Phase shift of the swerve response of the M549 spin-stabilized 

projectile to a 1-lbf control input as a function of the distance from 

the projectile c.g. to the point of application of the force, with 

6-DOF correlation data. 
15 

4. Effects of Individual Parameters 

Figures 1 through 6 clearly show that the point of application of the control force has a very large 

effect on both the direction and the magnitude of the swerve response. Further, they demonstrate 

that the effects are drastically different for fin- and spin-stabilized projectiles. The simplified 

swerve expressions provide insight into the physical reasons for this behavior. In the swerve 

response magnitude and phase expressions given as equations 43 and 45, the term expressing the 

distance from the center of pressure location to the point of application of the control force, 

(ΔSLC - ΔSLP), appears repeatedly. The practical result is that if control force is applied at the center 

of pressure, (ΔSLC - ΔSLP) becomes zero and the response magnitude is at a minimum. In the case 

of a fin-stabilized projectile, which has very low spin rates and negligibly small Magnus effects as 

a result, the magnitude of the response goes to zero when the control force is applied at the center 

of pressure. 

The direction of the response is also driven by the center of pressure. A typical fin-stabilized 

projectile will have a center of pressure behind the projectile center of mass, resulting in a negative 

ΔSLP. A control force applied in front of the center of pressure leads to (ΔSLC - ΔSLP) becoming 

positive. This will result in a positive numerator and a much larger positive denominator, indicating a very small positive phase shift. Conversely, when the control force is applied behind the 

center pressure, (ΔSLC - ΔSLP) becomes negative and so do the numerator and denominator of 

equation 45. This indicates a phase shift of nearly -180 degrees. 

A spin-stabilized projectile displays the opposite behavior, resulting from the center of pressure 

being typically located ahead of the center of mass. With a positive ΔSLP, a control force applied in 

front of the center of pressure will result in a positive numerator and a negative denominator in 

equation 45, indicating a phase shift approaching 180 degrees out of phase with the direction of the 

applied force. When the control force is applied behind the center of mass, the response will be 

approximately in phase with the direction of the applied force. The Magnus moment causes a 

comparably smaller response which is 90 degrees out of phase with the direction of the applied 

control force. 

Equations 47 and 49, the expressions for swerve response magnitude and phase shift in terms of 

pure applied moments, can be analyzed similarly. For a fin-stabilized projectile, a positive moment 

will lead to a response -90 degrees out of phase with the direction of the applied moment. For 

instance, a positive moment in projectile no-roll z-direction will lead to a response in the positive ydirection. A spin-stabilized projectile will respond in the opposite direction to the same applied 

moment. Again, this effect can be tied directly to the sign of ΔSLP in equation 49. 

The physical explanation for this behavior is relatively simple. Application of a control force 

away from the c.g., or application of a pure control moment, creates a non-zero angle of attack in 
16 

the projectile. The normal force results directly from the angle of attack and, in a stable projectile, 

will create a moment equal and opposite to the moment caused by the control input. The direction 

of the response will be driven primarily by the sum of these two forces. If the sum of the control 

force and the normal force are positive, the response will be in the positive direction. When the 

control force is applied at the center of pressure, the normal force will be equal and opposite to the 

control force and the response will be driven solely by the Magnus effect, which only persists in 

spin-stabilized projectiles. If the sum of the two forces is negative, the response will be in the 

negative direction. When a pure moment is applied to a projectile without an applied control 

force, the normal force alone determines the direction of the response. In the case of a fin-stabilized projectile, with a center of pressure behind the center of mass, a positive normal force is 

necessary to counteract a positive applied control moment, leading to a response in the positive 

direction. The opposite occurs in a spin-stabilized projectile. Figure 7 graphically summarizes the 

effects of a control force in the positive y-direction applied at varying points on the projectile body. 

To demonstrate the relatively small contribution of the Magnus moment in a spin-stabilized 

projectile, figures 8 and 9 show the predicted swerve magnitude and phase shift, respectively, of 

the M549 projectile when the Magnus force coefficient is equal to -0.96, -0.48, and 0. 

The magnitude of the response is largely unaffected if the Magnus moment reduced or removed, 

except when the control force is applied near the projectile center of pressure. When that is the 

case, the normal force is near zero and the Magnus moment becomes the dominant factor in the 

response magnitude. When the Magnus moment is neglected entirely, the magnitude of response 

of the M549 becomes zero when the control force is applied at the center of pressure, as is the 

case in a fin-stabilized projectile. 

The effect of the Magnus moment becomes more apparent when we examine the phase response 

of the M549 with varied Magnus force coefficients. The Magnus moment acts 90 degrees out of 

phase with the angle of attack of the projectile. As the Magnus force coefficient is reduced, the 

portion of the response that is orthogonal to the control input diminishes. With no Magnus moment 

present, the response is almost completely in phase with a force applied behind the center of pressure and is nearly 180 degrees out of phase for a force applied in front of the center of pressure. 
17 

Figure 7. Summary of projectile response to a control input in the positive y-direction in both a fin-stabilized 

and a spin-stabilized projectile. (Magnus moments, which act 90 degrees out of phase with the angle 

of attack in spin-stabilized projectiles, are not shown.) 
18 

Figure 8. Magnitude of the swerve response of the M549 spin-stabilized 

projectile to a 1-lbf control input as a function of the distance 

from the projectile c.g. to the point of application of the control 

force, Magnus force coefficient varied from -0.96 to 0. 

Figure 9. Phase shift of the swerve response of the M549 spin-stabilized 

projectile to a 1-lbf control input as a function of the distance 

from the projectile c.g. to the point of application of the control 

force, Magnus force coefficient varied from -0.96 to 0. 
19 

5. Conclusions 

Relatively simple, closed form formulas for the magnitude and phase angle of a projectile excited 

by control forces or moments in terms of fundamental projectile flight mechanic parameters have 

been created. The swerve response formulas are remarkably accurate, given the litany of simplifications and the resulting compact form of the results. These formulas explain in a clear manner 

the control response differences between fin- and spin-stabilized projectiles, including the key role 

that the center of pressure plays in control force response. It is shown that fin-stabilized projectiles, respond in phase to control force input forward of the center of pressure and control moments, 

while spin-stabilized projectiles respond out of phase to control force input forward of the center of 

pressure and control moments. The simple formulas reported here are expected to be useful to 

smart weapon designers in bringing to light basic parameters that drive swerve response from 

different control mechanisms. 
20 

6. References 

1. Harkins, T.; Brown, T. 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.; Costello, M. Dispersion Reduction of a Direct Fire Rocket Using Lateral 

Pulse Jets. Journal of Spacecraft and Rockets 2001, 38 (6), 929-936. 

3. Amitay, M.; Smith, D.; Kibens, V.; Parekh, D.; Glezer, A. Aerodynamic Flow Control Over 

an Unconventional Airfoil Using Synthetic Jet Actuators. AIAA Journal 2001, 39 (3), 361-

370. 

4. Harkins, T.; Davis, B. Drag-Brake Deployment Method and Apparatus for Range Error 

Correction of Spinning, Gun-Launched Artillery Projectiles, United States Patent 6345785, 

2002. 

5. Hillstrom, T.; Osborne, P. United Defense Course Correcting Fuze for the Precision 

Guidance Kit Program. 49th NDIA Annual Fuze Conference, April 5-7, 2005. 

6. Massey, K.; Flick, A. Mechanical and Jet Actuators for Guiding a Small Caliber Subsonic 

Projectile, AIAA Paper AIAA-2007-3813, 25th AIAA Applied Aerodynamics Conference, 

Miami, FL, June 25-28, 2007. 

7. Massey, K.; McMichael, J.; Warnock, T.; Hay, F. Mechanical Actuators for Guidance of a 

Supersonic Projectile,” AIAA Paper AIAA-2005-4970, 23rd AIAA Applied Aerodynamics 

Conference, Toronto, Ontario, June 6-9, 2005. 

8. Costello, M.; Agarwalla, R. Improved Dispersion of a Fin Stabilized Projectile using a 

Passive Moveable Nose. Journal of Guidance, Control, and Dynamics 2002, 23 (5), 900-

903, 2000. Errata: Vol 25, No 2, pp 414. 

9. Costello, M. Extended Range of a Gun Launched Smart Projectile Using Controllable 

Canards. Shock and Vibration 2001, 8 (3-4), 203-213. 

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

Journal of Guidance, Control, and Dynamics 2000, 23 (4), 789-797. 

11. Burchett, B.; Peterson, A.; Costello, M. Prediction of Swerving Motion of a Dual-Spin 

Projectile with Lateral Pulse Jets in Atmospheric Flight. Mathematical and Computer 

Modeling 2002, 35 (7-8), 821-834. 
21 

12. Chandgadkar, S.; Costello, M.; Dano, B.; Liburdy, J.; Pence, D. Performance of a Smart 

Direct Fire Projectile Using a Ram Air Control Mechanism. Journal of Dynamic Systems, 

Measurement, and Control 2002, 164, 606-612. 

13. Rogers, J.; Costello, M. Control Authority of a Projectile Equipped with an Internal 

Translating Mass. Paper Number AIAA 2007-6492, Proceedings of the 2007 AIAA 

Atmospheric Flight Mechanics Conference, Hilton Head, SC, 2007. 

14. Murphy, C. H. Free Flight Motion of Symmetric Missiles; BRL Report No. 1216; U.S. 

Army Ballistic Research Laboratories, Aberdeen Proving Ground, MD, 1963. 

15. Costello, M.; Anderson, D. Effect of Internal Mass Unbalance on the Terminal Accuracy 

and Stability of a Projectile. AIAA Paper Number 96-3447, Proceedings of the 1996 AIAA 

Flight Mechanics Conference, San Diego, CA, 1996. 

16. McCoy, R. L. Modern Exterior Ballistics: the Launch and Flight Dynamics of Symmetric 

Projectiles; Schiffer Publishing Ltd., Atglen, PA, 1999. 

17. Hainz, L.; Costello, M. Modified Projectile Linear Theory for Rapid Trajectory Prediction. 

Journal of Guidance, Control, and Dynamics 2005, 28 (5), 1006-1014. 
22 

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