Model of Predictive Control of a Direct-Fire Projectile 

Equipped With Canards 

by Douglas Ollerenshaw and Mark Costello 

ARL-CR-558 March 2005 

prepared by 

The Department of Mechanical Engineering 

Oregon State University 

Corvallis, OR 97331 

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-558 March 2005 

Model of Predictive Control of a Direct-Fire Projectile 

Equipped With Canards 

Douglas Ollerenshaw and Mark Costello 

Weapons and Materials Research Directorate, ARL 

prepared by 

The Department of Mechanical Engineering 

Oregon State University 

Corvallis, OR 97331 

under contract 

W911QX-04-P0270 

Approved for public release; distribution is unlimited. 
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Model of Predictive Control of a Direct-Fire Projectile Equipped With Canards 

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6. AUTHOR(S) 

Douglas Ollerenshaw∗ and Mark Costello†

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Department of Mechanical Engineering 

Oregon State University 

Corvallis, OR 97331 

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∗

 Graduate research assistant 

†

Associate professor, member ASME 

14. ABSTRACT 

Launch uncertainties in uncontrolled direct-fire projectiles can lead to significant impact point dispersion, even at relatively 

short range. A model predictive control scheme for direct-fire projectiles is investigated to reduce impact point dispersion. 

The control law depends on projectile linear theory to create an approximate linear model of the projectile and quickly predict

states into the future. Control inputs are based on minimization of the error between predicted projectile states and a desired 

trajectory leading to the target. Through simulation, the control law is shown to work well in reducing projectile impact point 

dispersion. Parametric trade studies on an example projectile configuration are reported that detail the effect of prediction 

horizon length, gain settings, model update interval, and model step size. 

15. SUBJECT TERMS

guided munition, direct fire, canard control 

16. SECURITY CLASSIFICATION OF: 

19a. NAME OF RESPONSIBLE PERSON

Peter Plostins 

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iii 

Contents 

List of Figures iv

List of Tables iv

1. Introduction 1

2. Projectile Dynamic Model 2

3. Projectile Linear Theory Trajectory Solution 6

4. Model-Predictive Flight Control System 9

5. Results 12

6. Conclusions 23

7. References 24

List of Symbols, Abbreviations, and Acronyms 26

Distribution List 28
iv 

List of Figures 

Figure 1. Schematic of the position coordinates of a direct-fire projectile.....................................3 

Figure 2. Schematic of the attitude coordinates of a direct-fire projectile......................................4 

Figure 3. Uncontrolled dispersion (CEP = 113.5 ft centered at mean impact point)....................14 

Figure 4. Controlled dispersion (CEP = 0.02 ft centered at mean impact point)..........................15 

Figure 5. Typical altitude vs. range. .............................................................................................15 

Figure 6. Typical cross range vs. range. .......................................................................................16 

Figure 7. Error between linear and nonlinear trajectory solutions................................................17 

Figure 8. Required control inputs for a typical trajectory.............................................................18 

Figure 9. Controlled dispersion with sensor bias and noise applied (CEP = 2.1 ft centered at 

mean impact point). .................................................................................................................20 

Figure 10. Controlled dispersion results as the control gain, r, is varied......................................20 

Figure 11. Controlled dispersion results as the linear model update interval is varied. ...............21 

Figure 12. Controlled dispersion results as the arc length step size is varied...............................22 

List of Tables 

Table 1. Initial condition uncertainty parameters for dispersion analysis. ...................................13 

Table 2. Sensor noise and bias values...........................................................................................19 
1 

1. Introduction 

Direct-fire projectiles are fired by line-of-sight aiming and are fired from ground-based 

platforms, helicopters, and fixed wing aircraft. A number of conditions can cause rounds to miss 

an intended target. These conditions include manufacturing inaccuracies of the gun tube, 

propellant, or projectile, along with variable atmospheric conditions, firing platform motion, and 

aiming errors. With the advent of low-cost, small, rugged, microelectromechanical systems, 

dramatic reduction of dispersion for direct-fire projectiles equipped with a relatively inexpensive 

flight control system is possible. One design concept consists of a set of controllable canards 

located near the nose of the projectile. This report develops a unique flight control law tailored 

to the control of smart projectiles through the application of model predictive control and 

projectile linear theory. 

In model predictive control, a dynamic model of the plant is used to project the state into the 

future and subsequently use the estimated future state to determine control action. It has been 

found to be a practical and increasingly employed control technique (1). Currently, model 

predictive control is being applied to a wide variety of problems, spanning many different 

industries. Mei et al. (2) studied vibration reduction of a tall building experiencing wind 

excitation using model predictive control and linear quadratic guassian control strategies. They 

found that the model predictive control scheme performed well and was robust to uncertainty in 

building stiffness. Tsai and Huang (3) used a model reference adaptive predictive controller for a 

variable-frequency oil-cooling machine used in concert with dynamically complex machine 

tools. Kvaternik et al. (4) developed a generalized predictive controller for tilt rotor aeroelastic 

stability augmentation in airplane mode of flight. Using the model predictive control strategy, 

significant increases in damping of aircraft body vibration modes were achieved in a wind tunnel 

test. Slegers and Costello (5) applied model predictive control to a parafoil for autonomous 

delivery of a payload in battlefield conditions. Burchett and Costello (6) used a simplified form 

of model predictive control applied to a projectile with lateral pulsejets. Their strategy was to 

use projectile linear theory to map the projected impact point in the vertical target plane and base 

control action on projected miss distance and direction. The key difference between their 

strategy and that detailed here is that the control strategy employed by Burchett and Costello 

calculated errors only in the target plane, while the control strategy used here considers error 

along the length of the trajectory. In addition, the pulsejet control scheme used by Burchett and 

Costello is inherently discontinuous and can only be applied at a discrete number of points. 

Canard control, as applied in this report, is continuous and is applied for the full flight duration. 

Any model predictive control scheme is dependent upon the accuracy of the underlying dynamic 

model representing the plant. Under most flight conditions, the equations of motion for a 

projectile in atmospheric flight can be adequately represented by a six-degree-of-freedom rigid 
2 

body model with externally-applied aerodynamic forces and moments. The resulting differential 

equations have been shown to provide an accurate representation of projectile flight 

characteristics (7), though their inherent nonlinearity prevents direct use in model-predictive 

control applications. However, a series of manipulations and simplifications of the equations of 

motion allow closed form solution of the projectile trajectory under restricted flight conditions. 

The simplified dynamic equations and their resulting solutions have become known as projectile 

linear theory. Projectile linear theory has been extended by various authors to handle more 

sophisticated aerodynamic models (8), asymmetric mass properties (9), fluid payloads (10, 11), 

moving internal parts (12, 13), dual spin projectiles (14, 15), ascending flight (16), and lateral 

force impulses (17–20). Aerodynamic range reduction software used in spark range facilities 

utilizes projectile linear theory in estimation of aerodynamic coefficients. 

The reported work employs model predictive control and projectile linear theory for control of a 

direct-fire projectile. The basic projectile configuration under consideration is fin stabilized, and 

the fins are slightly canted to provide moderate roll rates during flight. A set of controllable 

canards located near the nose of the projectile are used as the control mechanism. The canards 

can be directed to provide swerve forces and pitch and yaw moments to the projectile. The 

control law uses an approximate closed form solution of projectile motion to predict the states of 

the projectile over a set distance known as the prediction horizon. Current and future control 

actions are determined based on minimizing the estimated error of future states. It is assumed 

that sensor feedback is provided by an onboard inertial measurement unit (IMU). Simulation 

results to establish the utility of the new model predictive flight control system design 

methodology are generated for an exemplar projectile. Parametric trade studies are conducted 

that consider the effect of the cost function weighting matrices, prediction horizon length, state 

estimation step size, and the model update interval on impact point dispersion. 

2. Projectile Dynamic Model 

The nonlinear trajectory simulation used in this study is a standard six-degree-of-freedom model 

typically used in flight dynamic modeling of projectiles. A schematic of the projectile 

configuration is shown in figures 1 and 2. The six degrees of freedom are 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 equations of motion are provided in equations 1–4 (21–23). 

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−

+ −

− +

=

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

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) 
3 

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

= −

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

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( ) s α ≡ α , cos( ) c α ≡ α , and tan( ) t α ≡ α . The forces in equation 3 contain contributions from 

weight (W), body aerodynamics (A), and the control canards (C). 

Figure 1. Schematic of the position coordinates of a direct-fire projectile. 

I I

G

KI

G

I J

G

KB

G

B J

G

BI

G

x y

w

V

u

v

− z

Control Canards 
4 

Figure 2. Schematic of the attitude coordinates of a direct-fire projectile. 

W AC

W AC

W AC

X X XX

YY Y Y

Z Z ZZ

⎧⎫⎧ ⎫⎧ ⎫⎧ ⎫ ⎪⎪⎪ ⎪⎪ ⎪⎪ ⎪ ⎨⎬⎨ ⎬⎨ ⎬⎨ ⎬ = ++

⎪⎪⎪ ⎪⎪ ⎪⎪ ⎪ ⎩⎭⎩ ⎭⎩ ⎭⎩ ⎭

. (5) 

The dynamic equations are expressed in a body-fixed reference frame, thus all forces acting on 

the body are expressed in the projectile reference frame. The weight force is shown in equation 6: 

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧−

=

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

φ θ

φ θ

θ

c c

s c

s

mg

Z

Y

X

W

W

W

, (6) 

while the aerodynamic force acting at the center of pressure of the projectile is given by equation 7: 

θ

ψ

φ

I I

G

KI

G I J

G

Control Canards 
5 

222

0 2

2 2

( )/

/

8

/

A XX

A NA

A NA

X C CvwV

Y VD C v V

Z CwV

π

ρ

⎧ ⎫ ⎧ + + ⎫ ⎪ ⎪ ⎪ ⎪ ⎨⎬ ⎨ ⎬ = −

⎪⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭

.

 (7) 

The control forces are the aerodynamic drag forces created by the control canards in the 

directions perpendicular to the axis of symmetry of the projectile: 

2 2

0

0

0

8

C

C Y

C Z

X

Y VD C

Z C

π

ρ

⎧⎫ ⎧ ⎫ ⎪⎪ ⎪ ⎪ ⎨⎬ ⎨ ⎬ = −

⎪⎪ ⎪ ⎪ ⎩⎭ ⎩ ⎭

. (8) 

The applied moments about the projectile mass center contain contributions from steady 

aerodynamics (SA), unsteady aerodynamics (UA), and the control canards (C): 

SA UA C

SA UA C

SA UA C

LL L L

MM M M

NN N N

⎧ ⎫⎧ ⎫⎧ ⎫⎧ ⎫ ⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪ ⎨ ⎬⎨ ⎬⎨ ⎬⎨ ⎬ =++

⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪ ⎩ ⎭⎩ ⎭⎩ ⎭⎩ ⎭

. (9) 

The moment components due to steady aerodynamic forces and control canard forces are 

computed with a cross product between the distance vector from the mass center to the location 

of the specific force and the force itself. The unsteady body aerodynamic moment provides a 

damping source for projectile angular motion and is given by equation 10: 

2 3

2

8 2

2

LP

DD

UA

MQ

UA

UA

MQ

pDC C

V

L

qDC M VD

V

N

rDC

V

π

ρ

⎧ ⎫ ⎪ + ⎪ ⎪ ⎪ ⎧ ⎫ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎨⎬ ⎨ ⎬ =

⎪⎪ ⎪ ⎪ ⎩ ⎭ ⎪ ⎪ ⎪ ⎪ ⎪⎩ ⎭⎪

. (10) 

The mass, mass center location, and inertial properties of the projectile are all assumed to be 

constant throughout the duration of the flight. The center of pressure location and all 

aerodynamic coefficients ( 0 , ,, ,, CC CC CC X YPA NA DD LP MQ ) depend on local Mach number and are 

computed during simulation using linear interpolation. 

The dynamic equations given by equations 1–4 are numerically integrated forward in time using 

a 4th order, fixed step Runge-Kutta algorithm. Costello and Anderson (7) present correlation of 

this dynamic model against range data for a fin-stabilized projectile. 
6 

3. Projectile Linear Theory Trajectory Solution 

The six-degree-of-freedom rigid body projectile model shown earlier consists of 12 highly 

nonlinear differential equations for which a closed form solution has not been directly found. 

Significant work has been performed to simplify the equations of motion such that an accurate 

analytical solution can be determined. In order to arrive at a set of analytically solvable ordinary 

linear differential equations, the following assumptions and simplifications are made: 

1. Rather than employing a reference frame fixed to the projectile body, projectile linear 

theory uses an intermediate reference frame which is aligned with the projectile axis of 

symmetry but does not roll. Lateral translational and rotational velocity components 

described in this frame, known as the no-roll frame or the fixed plane frame, are denoted 

with a ~ superscript. Components of the linear and angular body velocities in the fixed 

plane frame are computed from body frame components of the same vector through a 

single axis rotation transformation. For example, the body frame components of the 

projectile mass center velocity are transformed to the fixed plane by 

10 0

0

0

u u

v c sv

w sc w

φ φ

φ φ

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







. (11) 

2. A change of variables is made from the velocity along the projectile axis of symmetry, u, to 

the total velocity, V. Equations 12 and 13 relate V and u and their derivatives: 

22 2 22 2 V uvw uvw = ++ = ++   ; (12) 

uu vv ww uu vv ww V

V V

++ ++ = =           . (13) 

3. Dimensionless arc length, s, is used as the independent variable instead of time, t. 

Equation 14 defines dimensionless arc length: 

0

1 t

s Vdt

D = ∫ . (14) 

Equations 15 and 16 relate time and arc length derivatives of a dummy variable ζ. Dotted terms 

refer to time derivatives and primed terms denote arc length derivatives: 

V

D ζ ζ ⎛ ⎞ = ′ ⎜ ⎟ ⎝ ⎠

 ; (15) 
7 

2

V V

D V

ζ ζ ζ ⎛ ⎞⎛ ⎞ ′ ′ = +′′ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

 . (16) 

4. Euler pitch and yaw angles are assumed to be small so that 

sin( ) cos( ) 1

sin( ) cos( ) 1

θ θ θ

ψψ ψ

≈ ≈

≈ ≈ .

 (17) 

5. Aerodynamic angles of attack are small so that 

w v

V V α β ≈ ≈ . (18) 

6. The projectile is mass-balanced such that the center of gravity lies in the rotational axis of 

symmetry: 

0 xy xz yz

R xx

P yy zz

III

I I

I I I

= = =

=

= = .

 (19) 

7. Quantities V andφ are large compared to θ, ,, ,, ψ vwq and r such that products of small 

quantities and their derivatives are negligible. 

A more detailed discussion of the development of projectile linear theory is provided by McCoy 

(21). Application of the above stated assumptions leads to a set of coupled linear differential 

equations, with the exception that the total velocity, V , the roll rate, p , and the pitch angle, θ , 

appear in nonlinear fashion in many of the equations. To remedy this, the assumption is made 

that V changes slowly with respect to the other variables and is thus considered to be constant, 

V V= 0 , when it appears as a coefficient in all dynamic equations except its own. In addition, the 

roll rate and pitch angle are held constant, p = p0 and θ = θ0 , only when they appear in 

nonlinear fashion. The equation for the total velocity is shown as equation 20: 

0

3

0

8

X D C DgS V V

m V

−πρ θ ′ = − . (20) 

The remaining 11 equations can be written as 
8 

0

0

0

0

0

0 0 0000 0 0 0

0 0 0000 0 0 0

/ / 0000 0 0 0

/ / 0000 0 0 0

0 0 0 0 0000 0 0 0

/ 0 0 0 0000 0 0

0 / 0 0 0000 0 0

0 0 0 0 0000 0 0 /

0 0 / 0 0000 0 0 0

0 0 0 / 000

v A D

w A D

q BD CD H F

r CD BD F H

x

y DV D

z DV D

D V

D V

D V

p

φ

θ

ψ

⎧ ⎫′ − −

⎪ ⎪′ − ⎪ ⎪

⎪ ⎪′ − ⎪ ⎪′ − ⎪ ⎪

⎪ ⎪′ ⎪ ⎪ ⎨ ⎬′ =

⎪ ⎪′ − ⎪ ⎪

⎪ ⎪′

⎪ ⎪′ ⎪ ⎪

⎪ ⎪′ ⎪ ⎪ ⎪ ⎪′ ⎩ ⎭









0

0

0

0

00 0 0 0

0 0 0 0 0000 0 0

F

F

F

F

c F

v V

w W

q Q

r R

x D

y

z

K p P

φ

θ

ψ

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

⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎣ ⎦ ⎪ ⎪ ⎩ ⎭ ⎪ ⎪ ⎩ ⎭









, (21) 

where 

3

8

D CNA A

m

πρ

= , (22) 

5

0

16 0

YPA m

P

D C p SL B

I V

πρ ∆ = , (23) 

4

8

NA

P

D C SL C

I

πρ ∆ = , (24) 

0

0

R

P

I Dp F

I V = , (25) 

5

16

MQ

P

D C

H

I

πρ

= , (26) 

5

16

LP

C

R

D C K

I

πρ

= , (27) 

( )

3

* 00 8 F NA Y

D V C v VC

m

πρ

= −  , (28) 

( ) 0

3

* 00

0 8 F NA Z

gDC D W C w VC

V m

θ πρ =+ −  , (29) 

3

0 *

* 00

0 8 2

YPA m

F NA Z C

P

D DC p SL v Q w C SL V C SL I Vπρ ⎛ ⎞ ∆ = − ∆− + ∆ ⎜ ⎟ ⎝ ⎠

  , (30) 
9 

3

0 *

* 00

0 8 2

YPA m

F NA Y C

P

D DC p SL w R v C SL V C SL

I V

πρ ⎛ ⎞ ∆ = − ∆− − ∆ ⎜ ⎟ ⎝ ⎠

  , and (31) 

4

0

8

DD

F

R

VDC P

I

πρ

= . (32) 

Closed form solutions to the above equations can be found, though the results are omitted here. 

A more detailed treatment of the solutions to the projectile linear theory equations can be found 

in reference (23). 

The variablesCY 0 and CZ 0 are aerodynamic trim forces perpendicular to the projectile axis of 

symmetry, which are created by movement of the control canards and are treated directly as 

control inputs. 

In reality, the total velocity, V , does not remain constant for the duration of the flight; therefore, 

the total velocity must be periodically measured throughout the trajectory and updated in the 

remaining equations. The center of pressure location and the aerodynamic coefficients 

( 0 , ,, ,, CC CC CC X YPA NA DD LP MQ ), which all depend on local Mach number, must also be 

recomputed each time V is updated. The effect of the length of the update interval on the 

accuracy of the model was studied by Burchett et al. (19). 

4. Model-Predictive Flight Control System 

The model-predictive controller uses the linearized model of the system to propagate the states 

forward in time over an interval known as the prediction horizon(H p ) (24). Control action is 

based on comparison of the predicted states and a predetermined desired trajectory over the 

prediction horizon. As the prediction step is marched forward, so too is the prediction horizon—

a process referred to as the “receding horizon principle.” The control action at each step is 

determined by minimizing a quadratic cost function, defined as 

( ) ( ) T T J W Y Q W Y U RU = − −+   . (33) 

The matrix W contains the desired system outputs,w , over the length of the prediction horizon. 

The desired system outputs,w , at each prediction step consists of the desired x , y , and z

coordinates at that time instant. These values need to be loaded into the onboard computer prior 

to projectile launch. 
10 

1

2

k

k i

k

ki ki

k i

k Hp

w

x

w

W wy

z

w

+

+

+

+ +

+

+

⎧ ⎫

⎪ ⎪ ⎧ ⎫ ⎪ ⎪ ⎪⎪ = = ⎨ ⎬ ⎨⎬ ⎪ ⎪ ⎪⎪ ⎩ ⎭ ⎪ ⎪ ⎩ ⎭

# . (34) 

The matrix Y contains the predicted system outputs, y , and the matrix U contains the calculated 

system inputs, u , as follows: 

1

2 1

1

k k

k k

k Hp k Hp

y u

y u Y U

y u

+

+ +

+ +−

⎧ ⎫⎧⎫ ⎪ ⎪⎪⎪ ⎪ = = ⎨ ⎬⎨⎬ ⎪ ⎪⎪⎪ ⎪ ⎪⎪⎪ ⎩⎭ ⎩ ⎭



  # #



. (35) 

Q and R are diagonal, positive semidefinite weighting matrices. 

In order to develop an expression for the predicted system outputs over the prediction horizon, 

the system is first cast in standard discrete state-space form: 

 xk+1 = A( ∆ s)xk + B( ∆ s)uk + F( ∆ s) (36) 

 yk = Cxk , 

where the values within the matrices A , B , and F depend on the arc length step size( ∆ s). The 

projectile linear theory expressions shown in the previous section are used to form the state space 

matrices through a 14-step loop in the control algorithm. In the first step, all the states and 

controls are set to zero and the solutions are evaluated over one arc length step to determine the 

values within the constant vector F . In the next step, the first state, u , is set equal to one, with 

the remaining states and controls still equal to zero, and the expressions are reevaluated. By 

subtracting the values of the constants, F , the coefficients making up the first column of A can 

be found. This process, consisting of setting a state variable equal to one, evaluating the linear 

theory solutions, then subtracting the constant values, is repeated for each of the remaining 10 

states to fully populate the state matrix A one column at a time. The control matrix, B , is 

formed in exactly the same manner with all 11 states equal to zero and the controls, CY 0 and CZ 0

alternately set equal to one. 

The desired outputs of the system are its center of mass position states (x, y, and z). The matrix 

C is then simply 

00001000000

00000100000

00000010000

C

⎡ ⎤

⎢ ⎥ =

⎣ ⎦

. (37) 
11 

A recursive formula can be found for ,1 kj p y jH + ≤ ≤ by substituting the expression for k j x + into 

the expression for k j y + . The result is 

1

1 1

j j

j ji ji

k j k ki

i i

y CA x CA Bu CA F − −

+ +−

= =

=+ + ∑ ∑ , (38) 

or, in matrix form, 

Y Kx K U K =+ + CA k CAB CAF  , (39) 

where 

2

3

CA

HP

CA

CA

K CA

CA

⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

#

, (40) 

2

1 2

00 0

0

0

0

CAB

HP

CB

CAB CB

K CA B CAB CB

CA B CA B CAB CB −

⎡ ⎤

⎢ ⎥

=

⎣ ⎦

...

" #

"

# # #%

"

, and (41) 

2

1

CAF

HP

CF

CAF CF

K CA F CAF CF

CA F CAF CF −

⎡ ⎤ ⎢ ⎥ + ⎢ ⎥ = ⎢ + + ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ++ + ⎥ ⎣ ⎦

#

"

. (42) 

The cost function, J , then becomes 

 ( )( ) T T

CA k CAB CAF CA k CAB CAF J W K x K U K Q W K x K U K U RU =− − − − − − + . (43) 

The minimum of the cost function is determined by selecting the control input vector that forces 

the gradient of the cost function to zero: 

U KW K x K =− − ( CA k CAF ), (44) 

where 

( ) 1 T T K K QK R K Q CAB CAB CAB

−

= + . (45) 
12 

It should be noted that U contains the optimal control inputs over the entire prediction horizon. 

At each arc length step, k , only uk is used, which is the first element of U . The first element of 

U is 

u KW Kx K k CA k CAF =−− 1 ( ), (46) 

where K1 consists only of the first M rows of K. Note that M is defined as the number of 

control inputs which, in this application, is two (CY 0 and CZ 0 ). 

It is assumed that full-state feedback is available for use in the control law; that is, 

 

x , y, z,ψ ,θ ,φ,u, v,w, p,q, and r are sensed or estimated by the IMU. Furthermore, the weapon 

that fires the projectile provides a desired trajectory leading to the target. At time = 0, the 

controller is provided with the full state of the projectile. The total velocity, V , is calculated 

from the projectile mass center velocity states and set to V0 in the linear model. The linear 

model is then used to propagate the remaining 11 states forward by ∆ s. These values are used to 

populate the A , B , and F matrices, which are sent to the MPC routine. The MPC routine 

calculates the optimal control sequence over the length of the update interval. When the 

projectile has covered the length of the first update interval, as well as every subsequent update 

interval, the controller is provided with full state feedback and the process is repeated. The 

control sequence calculated by the model predictive controller contains control inputs at 

increments of ∆ s. Linear interpolation is applied to determine control inputs between 

increments of ∆ s. 

It is important to note that the controls resulting from the earlier calculations are expressed in the 

fixed plane frame, as per assumption number 1 in the linear theory section of this report. To be 

applied to the canards, the control inputs must be converted to the conventional body fixed 

reference frame. 

Each time the Mach number is updated in the linear model, the matrices A, B, and F are updated 

as well. This in turn requires updating of the gain matrices. The size of each of these matrices, 

and hence the computational time required to calculate them, is governed by the length of the 

prediction horizon. Obviously, frequent updates to the linear model and a long prediction 

horizon provide greater accuracy in the predictor and more efficient control. These observations 

are tempered with the need to limit the computational demand placed on the onboard processor. 

5. Results 

To establish the utility of the model-predictive controller in a projectile application, a 4.5-ft long 

fin-stabilized projectile is considered. The projectile has a total weight of 22.9748 lb, a center of 

gravity location of 2.5 ft from the base, and four tail-mounted stabilization fins. The roll and 

pitch inertia of the body are 0.0057 and 1.35 slug-ft2, respectively. A set of controllable canards, 
13 

which alter the aerodynamic forces and moments, are located 4.25 ft from the base of the 

projectile. 

To model uncertainty in launch conditions, which is a primary cause of dispersion, the initial 

pitch and yaw rates, pitch and yaw angles, and body velocities are all considered to be normallydistributed random numbers with means and standard deviations that are representative of actual 

launch uncertainties. The values chosen are shown in table 1. 

Table 1. Initial condition uncertainty parameters for dispersion analysis. 

Initial Condition Mean Standard Deviation 

Pitch angle (θ) 0.1 rad 0.01 rad 

Yaw angle (ψ) 0.0 rad 0.01 rad 

Pitch rate (q) –0.18 rad/s 2.0 rad/s 

Yaw rate (r) 0.0 rad/s 2.0 rad/s 

x body velocity (u) 1143.3797 ft/s 15 ft/s 

y body velocity (v) –0.00002502 ft/s 3 ft/s 

z body velocity (w) 0.375346 ft/s 3 ft/s 

The desired trajectory is chosen as that which the projectile would follow in the absence of 

uncertainty with initial conditions of x 0 , y0 , z0 , ψ 0 ,φ 0 , and r0 = 0, θ 0 = 0.1 rad, u0 = 1143.38 

ft/s, 

v0 = –2.502x10-5 ft/s, w0= 0.375 ft/s, p0 = 51.5 rad/s, and q0 = –0.18 rad/s. The target 

location is chosen as x = 6216.613 ft, y = 0.261 ft, and z = 0.0 ft. Figure 3 shows typical 

dispersion results for 50 sample trajectories with no control applied and initial condition 

perturbations as described earlier. The circular error probable (CEP) shown in the figure is based 

on a 50% hit criterion; that is, the CEP is defined as the minimum radius of a circle centered at 

the mean impact point and containing at least 50% of the shot impact points. With no control 

applied, the CEP is 106 ft. For reference, a second CEP is shown, which, instead of being 

centered at the mean impact point, is instead centered at the target location. A 50% hit criterion 

is still used. The second CEP has a radius of 113.5 ft. 

Figure 4 shows the dispersion results with model-predictive control applied. The prediction 

horizon, H p , is chosen as 50. The error weighting matrix, Q , is chosen to be a function of range 

as follows: 

2

2

1

2

1

00 0

1000

0 00

1000

0 0

0 0

1000

p

k

k

k H

s

s

Q q

s

+

+ −

⎡ ⎤ ⎛ ⎞ ⎢ ⎥ ⎜ ⎟ ⎝ ⎠

⎛ ⎞ ⎜ ⎟ = ⎝ ⎠

⎛ ⎞ ⎜ ⎟

⎣ ⎦ ⎝ ⎠

% #

"

, (47) 
14 

Figure 3. Uncontrolled dispersion (CEP = 113.5 ft centered at mean impact point). 

in which q is chosen, in this case, to be 0.5. By defining Q in this manner, the error weighting 

is increased quadratically as the projectile flies downrange. This prevents the tendency for the 

controller to attempt to force the projectile onto the desired trajectory immediately after launch, 

leading to a large initial control input followed by subsequent control inputs of nearly zero. In 

addition, the control weighting matrix, R , is defined to be 

0 0 00

0 10 00

0 0

0 0 20

0 0 01

p

p

H

H

R r

⎡ ⎤ ⎢ ⎥ − ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

%# #

...

...

, (48) 

where r is chosen as 2.0 in this case. By defining R in this manner, the current control value is 

weighted times heavier than the control value at the end of the prediction horizon. This 

prevents large controls from being chosen at the beginning of an update interval, even if 

significant error is present. The model update interval for the case shown in figure 2 is 1000 arc 

lengths, and the arc length step size, ∆ s, is 20. These parameters provide a good baseline from 

which to begin examining the performance of the model-predictive controller. Figures 5 and 6 

show a typical controlled and uncontrolled trajectory with the model predictive control 

parameters set listed earlier. 

H p
15 

Figure 4. Controlled dispersion (CEP = 0.02 ft centered at mean impact point). 

Figure 5. Typical altitude vs. range. 
16 

Figure 6. Typical cross range vs. range. 

The model-predictive controller provides a very significant reduction in the CEP—from 106 ft, 

in the uncontrolled case, to 0.02 ft (or less than 1/4 in) with control applied. It should also be 

noted that the mean impact point is almost 0.06 ft, or ~0.75 in, above the target location. In the 

cross-range direction, however, the mean impact point is only 0.0085 ft away from the target. 

This bias error in the z-direction can be attributed directly to errors in the linear model used in 

the predictor. One of the primary assumptions upon which projectile linear theory is based is 

that the projectile maintains a small angle of attack. As the target is approached, the angle of 

attack of the projectile is forced to a small, nonzero number. Though it isn’t necessarily in 

violation of the small angle of attack assumption, it is enough to cause a small deviation between 

the trajectory predicted with the linear model and that which is arrived at by integrating the full 

six-degree-of-freedom, nonlinear equations. This error is demonstrated by plotting the error 

between the validated, full six-degree-of-freedom, nonlinear trajectory, which is solved using a 

fixed step, 4th order Runge-Kutta method, and the linear theory trajectory solution. The linear 

solution is corrected to match the nonlinear solution every 1000 arc lengths to mimic flight 

control system feedback. The control input is set equal to zero in both cases and the initial 

conditions are set to match those used in creating the desired trajectory. Figure 7 shows the 

linear theory error as a function of arc length in the x, y, and z directions. 

Note that the error is of the same order of magnitude near the end of the trajectory as the 

variation of the mean impact point in the CEP plot. It should also be noted that the error is 
17 

Figure 7. Error between linear and nonlinear trajectory solutions. 

greatest at the beginning and end of the flight, where the trajectory is furthest from horizontal. 

At the midpoint of the trajectory, where the path of flight is nearly flat and the projectile angle of 

attack is nearly zero, the error in all three spatial directions also becomes very close to zero. 

Figure 8 shows the required control inputs for the trajectory shown in figures 3 and 4. The 

magnitudes of the control inputs required to achieve the shown degree of tracking are attainable 

for a set of nose-mounted canards. 

Great care must be taken when choosing the gain values, q and r , such that the control 

values,CY 0 and CZ 0 , never exceed approximately one. Such large control inputs violate the 

small angle of attack assumption upon which the linearized model is based. As a result, the 

linear model no longer accurately approximates the true, nonlinear system, and the controller 

loses its ability to accurately predict future states. When the system is provided with state 

feedback from the IMU under these circumstances, there are very large errors and the controller 

subsequently attempts to choose a large control value to compensate. Within one to two update 

intervals, the error becomes large enough that control saturates. 

Other applications of model predictive control, such as that discussed by Mei et al. (2), use an 

iterative scheme to set a maximum control input value. In the application being discussed, where 

speed of control computation is extremely critical, iteration is not practical. If the processor is 

occupied by an iterative routine while the projectile continues to fly downrange, control is lost. 
18 

Figure 8. Required control inputs for a typical trajectory. 

A second option is to simply clip the control values at the maximum allowable value. This, too, 

presents problems as future controls are calculated under the assumption that all previous 

controls were applied exactly as calculated. When this clipping scheme is attempted, the control 

begins oscillating rapidly between both allowable extremes, quickly resulting in instability. 

The results discussed earlier assume perfect sensor feedback, which in reality can never be 

achieved. In practice, sensors possess error created by both bias and noise. These are modeled 

in the simulation by choosing normally-distributed random numbers with means of zero and 

standard deviations that are representative of commonly used IMU sensors. A bias value is 

randomly chosen for each sensor at the start of every flight simulation and retained throughout 

that particular simulation. In addition, a noise value is randomly chosen for each sensor every 

time feedback is implemented. Both the bias and noise value are added to the sensor readings at 

each update interval. The sensor bias and noise standard deviations used are summarized in 
19 

table 2. All subsequent results displayed in this report employ sensor bias and noise applied in 

this way. 

Table 2. Sensor noise and bias values. 

Sensor Function Bias Standard Deviation Noise Standard Deviation 

x and y position (ft) 0.52 0.52 

z position (ft) 1.18 1.18 

x and y velocity (ft/s) 0.10 0.08 

z velocity (ft/s) 0.16 0.13 

roll, pitch, and yaw angles (deg) 0.3 0.3 

roll, pitch, and yaw rates (deg/s) 0.05 0.01 

Sensor noise and bias become the dominant sources of error when they are applied in this 

application. As the standard deviation of the sensor noise remains constant throughout the 

projectile flight, it no longer makes sense to define the error weighting matrix as a function of 

projectile range. Doing so, while keeping the value of q low enough to avoid violating the small 

angle of attack assumption near the end of the trajectory, unnecessarily limits the control action 

near the beginning of the trajectory. The error weighting matrix, Q , is instead defined simply as 

the identity matrix multiplied by the constant gain value, q . 

Figure 9 shows dispersion results with sensor noise and bias applied. The prediction horizon, Hp, 

is again 50. The error gain, q, is set to 1. The control weighting matrix, R, is defined as shown 

in equation 48 with the gain value, r , equal to 0.2. The update interval is 1000 arc lengths and 

the arc length step size, ∆ s, is 20. As a direct result of the sensor uncertainty, the CEP radius is 

increased to 2.1 ft. 

In figure 10, the effects of changing both the control gain, r , and the prediction horizon, Hp, are 

shown. The model update interval is held constant at 1000 arc lengths and the arc length step 

size is held constant at 20 arc lengths throughout all of the simulations shown in figure 10. In 

addition, the error weighting matrix, Q, is defined as the identity matrix with the error gain, q, 

equal to 1. The control weighting matrix, R, is again defined as shown in equation 48. The 

control gain, r, is varied from 0.025 to 10 with the prediction horizon held constant. The process 

is repeated four times with values of = 25 HP , = 50 HP , = 75 HP , and = 100 HP . 

As the value of r is increased, additional weight is given to the value of the control in the cost 

function (equation 43). This in turn forces the magnitude of the chosen control values to be 

smaller, which provides less control authority. As would be expected, figure 10 shows that 

larger values of r lead to increased dispersion. However, there is a value of r below which the 

control values are allowed to be too large, leading to violation of the small angle of attack 

assumption and loss of control. This minimum value of r varies depending on the length of the 

prediction horizon. In figure 10, the lowest attempted values of r which resulted in a controllable 

trajectory are shown as the first data point for each series. 
20 

Figure 9. Controlled dispersion with sensor bias and noise applied (CEP = 2.1 ft 

centered at mean impact point). 

Figure 10. Controlled dispersion results as the control gain, r, is varied. 
21 

It is also apparent from figure 10 that, for a given value of r, there is a direct relationship 

between the length of the prediction horizon and the amount of impact point dispersion. 

Allowing the controller to take into account an increased number of the predicted states, as a 

longer prediction horizon does, leads to more intelligent control choices. It also significantly 

increases the amount of computation required at each update interval, necessitating a more 

expensive onboard processor. 

A similar study was performed to investigate the effect of the length of the linear model update 

interval on the dispersion radius. The error weighting matrix, Q, is again the identity matrix with 

the error gain, q, set to one. The control weighting matrix, R, is defined as shown in equation 48 

with the gain value, r, equal to 0.5. The arc length step size is again 20 arc lengths. The linear 

model update interval is varied from 100 to 2000 while holding the value of the prediction 

horizon constant. As before, the process is repeated four times with prediction horizon values of 

= 25 HP , = 50 HP , = 75 HP , and = 100 HP . The results are shown in figure 11. 

Figure 11. Controlled dispersion results as the linear model update 

interval is varied. 

Longer linear model update intervals lead to an increase in dispersion. This increase becomes 

more apparent for update intervals greater than 1000 arc lengths. For prediction horizon lengths 

of 25 and 50 arc length steps, update intervals greater than 1600 arc lengths led to a loss of 

control. This results from the linearized model deviating too far from the true, nonlinear system. 

Upon update, the error becomes very large and a large control is chosen to compensate, which in 

turn violates the small angle of attack assumption and causes further error in the linear model. 

Reducing the error gain, q, or increasing the control gain, r, would prevent this scenario from 
22 

occurring. However, the tradeoff would be a reduction in control authority and an increase in 

dispersion. 

The final study investigates the effects of the length of the step size, ∆ s, used by the controller to 

propagate the linear model forward. The prediction horizon is held constant at 50 steps and the 

error weighting matrix is the identity matrix with the gain, q , equal to one. The control 

weighting matrix is defined as shown in equation 48. Four values for the arc length step size 

( ∆ s = 5, ∆ s = 10, ∆ s = 20 , and ∆ s = 40) are used while the control gain, r , is varied over the 

range which provided suitable control inputs for each step size. The results can be seen in 

figure 12. 

Figure 12. Controlled dispersion results as the arc length step size is 

varied. 

As with any discrete, linear model, the length of the step size has no effect on the accuracy of the 

model itself. However, as evidenced by figure 12, the length of the arc length step size does 

have an effect on the overall accuracy of the controller. This results from an interplay between 

two competing effects. The prediction horizon length is measured in the number of steps into the 

future that are used in the calculation of the optimal control. Therefore, for a given prediction 

horizon length, increasing the arc length step size allows the predictor to take into account state 

values farther into the future. However, control values are only calculated at each step 

increment, with control values between calculation steps derived from linear interpolation. A 

large step size can therefore lead to a decrease in resolution of the controller. For arc length step 

sizes of 5, 10, and 20, these effects do little more than change the acceptable range of gains, 
23 

shifting the lines to the right on figure 12 for increasing values of ∆ s. However, at ∆ s = 40, the 

controller becomes unable to provide the necessary amount of oscillation and dispersion is 

increased dramatically. 

6. Conclusions 

This report develops a method for applying model predictive control, a proven and effective 

control technique, to a smart projectile application. The control law is shown to dramatically 

reduce the impact point dispersion caused by launch disturbances. The method uses full state 

feedback to create a linearized model of the projectile and quickly predict the future states of the 

system. These calculations can be performed by a relatively inexpensive onboard processor. As 

the predicted states depend on the states provided by the feedback loop, sensor accuracy is very 

important to the performance of the system and was shown here to be the limiting factor in 

dispersion reduction. 

Considerable opportunities exist for the control system designer to tune the model-predictive 

controller based on the desired application. It was shown that the length of the prediction 

horizon has a considerable effect on the dispersion radius, with a longer prediction horizon 

leading to a decrease in dispersion. However, a longer prediction horizon increases the size of 

the matrices used in the control calculation, which subsequently necessitates an increase in the 

processing power required to perform control calculations in a sufficiently short period of time. 

Shorter linear model update intervals lead to a decrease in dispersion as well, but with a similar 

increase in the amount of onboard computation required. The length of the arc length step size 

was shown to have little effect on dispersion as long as it remained below 20 arc lengths. 

Control and error gains should be adjusted to allow sufficient control authority without violating 

any of the assumptions upon which linear theory is based. No iterative scheme is built into the 

controller to limit the size of the control inputs, so the control system designer should run a series 

of simulations prior to launching a projectile to ensure that the control and error gains are 

properly adjusted. 
24 

7. References 

1. Camacho, E.; Bordons, C. Model Predictive Control; Springer-Verlag: London, 1999. 

2. Mei, G.; Kareem, J.; Kantor, J. Model Predictive Control of Wind-Excited Building: 

Benchmark Study. Journal of Engineering Mechanics 2004, 130 (4), 459–465. 

3. Tsai, C.; Huang, C. Model Reference Adaptive Predictive Control for a Variable-Frequency 

Oil Cooling Machine. IEEE Transactions on Industrial Electronics 2004, 51 (2), 2004. 

4. Kvaternik, R.; Piatak, D.; Nixon, M.; Langston, C.; Singleton, J.; Bennett, R.; Brown, R. An 

Experimental Evaluation of Generalized Predictive Control for Tiltrotor Aeroelastic Stability 

Augmentation in Airplane Mode of Flight. Journal of the American Helicopter Society 2002, 

47 (3), 198–208. 

5. Slegers, N.; Costello, M. Model Predictive Control of a Parafoil and Payload System. 

Proceedings of the 2004 AIAA Atmospheric Flight Mechanics Conference, Providence, RI, 

2004; paper no. 2004-4822. 

6. Burchett, B.; Costello, M. Model Predictive Lateral Pulsejet Control of an Atmospheric 

Rocket. Journal of Guidance, Control, and Dynamics 2002, 25 (5), 860–867. 

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

and Stability of a Projectile. Proceedings of the 1996 AIAA Flight Mechanics Conference,

San Diego, CA, 1996.

8. Murphy, C. H. Symmetric Missile Dynamic Instabilities—A Survey. 18th AIAA Aerospace 

Sciences Meeting, Pasadena, CA, 14–16 January 1980. 

9. Hodapp, A. E. Effect of Mass Asymmetry on Ballistic Match of Projectiles. Journal of 

Spacecraft and Rockets 1976, 13 (12), 757–760. 

10. Weber, D. J. Simplified Method for Evaluating the Flight Stability of Liquid-Filled 

Projectiles. Journal of Spacecraft and Rockets 1994, 31 (1), 130–134. 

11. Murphy, C. H. Angular Motion of a Spinning Projectile With a Viscous Liquid Payload. 

Journal of Guidance, Control, and Dynamics 1983, 6 (4), 280–286. 

12. Cobb, K. K.; Whyte, R. H.; Laird, P. K. Effects of a Moving Components on the Motion of a 

20-mm Projectile. 11th AIAA Aerodynamics Testing Conference, Colorado Springs, CO, 

18–20 March 1983; pp 94–103. 

13. Hodapp, A. E. Passive Means for Stabilizing Projectiles With Partially Restrained Internal 

Members. Journal of Guidance, Control, and Dynamics 1989, 12 (2), 135–139. 
25 

14. Soper, W. G. Projectile Instability Produced by Internal Friction. AIAA Journal 1978, 16

(1), 8–11. 

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

Journal of Guidance, Control, and Dynamics 2000, 23 (5), 789–797. 

16. Murphy, C. H. Instability of Controlled Projectiles in Ascending or Descending Flight. 

Journal of Guidance and Control 1981, 4 (1), 66–69. 

17. Cooper, G. Influence of Yaw Cards on the Yaw Growth of Spin Stabilized Projectiles. 

Journal of Aircraft 2001, 38 (2), 266–270. 

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

Simple In-Flight Lateral Impulse. 38th AIAA Aerospace Sciences Meeting and Exhibit, 

Reno, NV, 2000; paper no. AIAA-2000-0767. 

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

Projectile With Lateral Pulsejets in Atmospheric Flight. Mathematical and Computer 

Modeling 2002, 35 (1–2), 1–14. 

20. Etkin, B. Dynamics of Atmospheric Flight. Wiley: New York, 1972. 

21. McCoy, R. L. Modern Exterior Ballistics. Schiffer Military History: Atglen, PA, 1999. 

22. Murphy, C. H. Free Flight Motion of Symmetric Missiles; BRL report no. 1216; U.S. Army 

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

23. Cooper, G.; Costello M. On the Response of Projectiles to Lateral Impulsive Loads. Journal 

of Dynamic Systems, Measurement, and Control, submitted for publication, 2004. 

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New York. 
26 

List of Symbols, Abbreviations, and Acronyms 

ρ : Air density 

CNA : Normal force aerodynamic coefficient 

CMQ : Pitch rate damping moment aerodynamic coefficient 

CX 0 : Aerodynamic drag coefficient in direction parallel to projectile motion 

0 0 , C C Y Z : Aerodynamic trim coefficients perpendicular to projectile axis of symmetry 

CDD : Roll moment from fin cant 

CYPA : Magnus force 

CLP : Roll damping moment 

D : Projectile characteristic diameter 

, P R I I : Projectile precesional and rotational inertia 

m : Projectile mass 

p, q, r : Angular velocity vector components expressed in the body fixed reference 

frame 

ψ , , θ φ : Euler yaw, pitch, and roll angles 

∆SL : Stationline distance from the projectile center of pressure location to the CG 

∆SLm : Stationline distance from the projectile Magnus force location to the CG 

∆SLC : Stationline distance from the control canard location to the CG 

u, v, w : Translation velocity components of the projectile center of mass resolved in the 

body fixed reference frame 
27 

x, y, z : Position vector components of the projectile mass center expressed in the 

inertial reference frame 

V : Magnitude of the mass center velocity 

LM N , , : Total external applied moment on the rocket about the mass center expressed in 

the rocket reference frame 

X, , Y Z : Total external applied force on the rocket expressed in the rocket reference 

frame 

H p : Prediction horizon used in model predictive controller 

s : Dimensionless arc length 
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 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 

 IMNE ALC IMS 

 2800 POWDER MILL RD 

 ADELPHI MD 20783-1197 

 3 DIRECTOR 

 US ARMY RESEARCH LAB 

 AMSRD ARL CI OK TL 

 2800 POWDER MILL RD 

 ADELPHI MD 20783-1197 

 3 DIRECTOR 

 US ARMY RESEARCH LAB 

 AMSRD ARL CS IS T 

 2800 POWDER MILL RD 

 ADELPHI MD 20783-1197 

ABERDEEN PROVING GROUND

 1 DIR USARL 

 AMSRD ARL CI OK TP (BLDG 4600) 
NO. OF NO. OF 

COPIES ORGANIZATION COPIES ORGANIZATION

29

 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 UNIV 

 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 

 1 NAWC 

 F PICKETT 

 CODE C2774 CLPL 

 BLDG 1031 

 CHINA LAKE CA 93555 
NO. OF NO. OF 

COPIES ORGANIZATION COPIES ORGANIZATION

30

 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 

 1 ALLIANT TECH SYS 

 TCAAP 

 R BECKER 

 BLDG 104 

 ARDEN HILLS MN 55112 

 1 ALLIANT TECH SYS 

 C CANDLAND 

 4700 NATHAN LN 

 PLYMOUTH MN 55442 

 1 UNITED DEFENSE LP 

 R BURETTA 

 CODE M170 

 4800 E RIVER RD 

 MINNEAPOLIS MN 55421 

 3 DIR USARL 

 AMSRL SE RM 

 H WALLACE 

 AMSRL SS SM 

 J EIKE 

 A LADAS 

 2800 POWDER MILL RD 

 ADELPHI MD 20783-1145 

 

 2 CDR USARDEC 

 AMSTA FSP A 

 S DEFEO 

 R SICIGNANO 

 PICATINNY ARSENAL NJ 

 07806-5000 

 2 CDR USARDEC 

 AMSTA AR CCH A 

 M PALATHINGAL 

 R CARR 

 PICATINNY ARSENAL 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 ARSENAL 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 ENGINEERING 

 AMSATR ESW 

 M MAMOUD 

 M JOHNSON 

 J OBERMARK 

 RSA AL 35898-5247 

 

 1 DIR US ARMY RTTC 

 STERT TE F TD 

 R EPPS 

 BLDG 7855 

 REDSTONE ARSENAL AL 

 38598-8052 
NO. OF NO. OF 

COPIES ORGANIZATION COPIES ORGANIZATION

31

 2 STRICOM 

 AMFTI EL 

 D SCHNEIDER 

 R COLANGELO 

 12350 RESEARCH PKWY 

 ORLANDO FL 32826-3276 

 1 CDR OFFICE OF NAVAL RES 

 J GOLDWASSER CODE 333 

 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 

 2 CDR NSWC 

 J FRAYSEE 

 D HAGEN 

 17320 DAHLGREN RD 

 DAHLGREN VA 22448-5000 

 5 CDR NSWC 

 INDIAN HEAD DIV 

 D GARVICK CODE 40D 

 L FAN CODE 4110C 

 V CARLSON CODE 4120 

 H LAST CODE 4140E 

 T GRIFFIN CODE 450D 

 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 

 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 MICHIGAN STATE UNIV 

 B KIM 

 2120 ENGINEERING BLDG 

 EAST LANSING MI 48824-1226 
NO. OF NO. OF 

COPIES ORGANIZATION COPIES ORGANIZATION

32

 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 

 3 ARROW TECH ASSOCIATES INC 

 R WHYTE 

 A HATHAWAY 

 H STEINHOFF 

 1233 SHELBOURNE RD STE D8 

 SOUTH BURLINGTON VT 05403 

 3 US ARMY AVIATION CTR 

 DIR OF COMBAT DEV 

 ATZQ CDM C 

 B NELSON 

 ATZQ CDC C 

 T HUNDLEY 

 ATZQ CD 

 G HARRISON 

 FORT RUCKER AL 36362 

 1 US ARMY CORPS OF ENGINEERS 

 CEERD ZA T 

 W MORRISON 

 7701 TELEGRAPH RD 

 KINGMAN BLDG RM 322 

 ALEXANDRIA VA 22315 

 1 DIRCTRT OF TRAINING, DOCTRINE, 

 AND COMBAT DEV 

 ATZK TDD ORSA 

 A POMEY 

 RM 202 HARRIS HALL 

 BLDG 2368 OLD IRONSIDES AVE 

 FORT KNOX KY 40121-5000 

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 

 43 DIR USARL 

 AMSRD ARL WM 

 T ROSENBERGER 

 AMSRD ARL WM BA 

 W HORST JR 

 W CIEPELLA 

 F BRANDON 

 T BROWN (5 CPS) 

 L BURKE 

 J CONDON 

 B DAVIS 

 T HARKINS (5 CPS) 

 D HEPNER 

 V LEITZKE 

 M HOLLIS 

 A THOMPSON 

 G BROWN 

 AMSRD ARL WM BB 

 B HAUG 

 AMSRD ARL WM BC 

 P PLOSTINS (4 CPS) 

 G COOPER 

 B GUIDOS 

 J SAHU 

 M BUNDY 

 D LYON 

 AMSRD ARL WM BC 

 J BENDER 

 J GARNER 

 S WILKERSON
NO. OF 

COPIES ORGANIZATION 

33

 W DRYSDALE 

 R COATES 

 A MIKHAL 

 J WALL 

 AMSRD ARL WM BD 

 B FORCH 

 AMSRD ARL WM BE 

 M SCHMIDT 

 AMSRD ARL WM BF 

 AMSRD ARL WM BR 

 C SHOEMAKER 

 J BORNSTEIN 
34

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