A Nonlinear Model Predictive Observer for 

Smart Projectile Applications 

by Mark Costello and Ryan Letniak 

ARL-CR-0603 March 2008 

prepared by 

Georgia Institute of Technology 

Atlanta, Georgia 

and 

American Institute of Aeronautics and Astronautics 

Reston, Virginia 

under contract 

W911QX-06-C-0113

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

Aberdeen Proving Ground, MD 21005-5069 

ARL-CR-0603 March 2008 

A Nonlinear Model Predictive Observer for 

Smart Projectile Applications 

Mark Costello 

School of Aerospace Engineering, Georgia Institute of Technology 

Ryan Letniak 

American Institute of Aeronautics and Astronautics 

prepared by 

Georgia Institute of Technology 

Atlanta, Georgia 

and 

American Institute of Aeronautics and Astronautics 

Reston, Virginia 

under contract 

W911QX-06-C-0113

Approved for public release; distribution is unlimited.
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4. TITLE AND SUBTITLE 

 A Nonlinear Model Predictive Observer for Smart Projectile Applications 

5c. PROGRAM ELEMENT NUMBER

 

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 AH80 

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6. AUTHOR(S) 

 Mark Costello (GIT); Ryan Letniak (AIAA) 

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Georgia Institute of Technology American Institute of Aeronautics and Astronautics

Atlanta, Georgia Reston, Virginia

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 ARL-CR-0603 

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 U.S. Army Research Laboratory 

 Weapons and Materials Research Directorate 

 Aberdeen Proving Ground, MD 21005-5069

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14. ABSTRACT 

The work reported here presents a nonlinear model predictive observer specialized to smart projectiles. The observer is based 

on a two-step process consisting of an initial state predictor followed by a state estimate corrector. We generate the predictor 

state by simply integrating the projectile’s fixed plane equations of motion forward in time while the corrector step updates 

the state estimate by the minimization of the difference between measurements and model-predicted measurements over a 

finite duration of time in the past. Since the proposed method permits a general nonlinear representation of the projectile, the 

corrector step requires the solution of a nonlinear minimization problem which is subsequently solved with the use of a 

damped Newton procedure. The nonlinear model predictive observer is exercised on a smart projectile that employs a typical 

array of sensors including global positioning system, a three-axis gyroscope, three-axis accelerometers, and a three-axis 

magnetometer. Results with this smart weapon observer are promising since the observer is capable of estimating the full 

state of the projectile, including orientation and translational velocity, when realistic noise and bias errors are included in the 

sensor measurements. Relative to other existing observers, the proposed observer is computationally intensive because of the 

need to solve a nonlinear minimization problem at each computational cycle of the observer. 

15. SUBJECT TERMS

 nonlinear model; smart projectiles 

16. SECURITY CLASSIFICATION OF: 

19a. NAME OF RESPONSIBLE PERSON

 Peter Plostins 

a. REPORT 

 UNCLASSIFIED 

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UNCLASSIFIED

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OF ABSTRACT 

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OF PAGES 

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 410-306-0800 

Standard Form 298 (Rev. 8/98) 

Prescribed by ANSI Std. Z39.18
iii 

Contents 

List of Figures iv 

List of Tables iv 

1. Introduction 1 

2. Projectile Flight Dynamic Model 2 

3. Sensor Model 5 

4. Nonlinear Model Predictive Observer 6 

5. Sample Results 9 

6. Conclusions 11 

7. References 21 

Nomenclature 23 

Distribution List 24 
iv 

List of Figures 

Figure 1. Position definitions......................................................................................................... 2 

Figure 2. Orientation definitions.................................................................................................... 3 

Figure 3. Measurement discretization............................................................................................ 7 

Figure 4. Single time step iterations. ........................................................................................... 10 

Figure 5. Trade study results........................................................................................................ 11 

Figure 6. Noise filtering............................................................................................................... 12 

Figure 7. Projectile range............................................................................................................. 13 

Figure 8. Projectile cross range.................................................................................................... 13 

Figure 9. Projectile altitude.......................................................................................................... 14 

Figure 10. Projectile roll angle..................................................................................................... 14 

Figure 11. Projectile Euler angle θ. ............................................................................................. 15 

Figure 12. Projectile Euler angle ψ.............................................................................................. 15 

Figure 13. Projectile velocity u.................................................................................................... 16 

Figure 14. Projectile velocity v.................................................................................................... 16 

Figure 15. Projectile velocity w................................................................................................... 17 

Figure 16. Projectile rate p........................................................................................................... 17 

Figure 17. Projectile rate q........................................................................................................... 18 

Figure 18. Projectile rate r. .......................................................................................................... 18 

Figure 19. Estimate error in range. .............................................................................................. 19 

Figure 20. Estimate error in φ. ..................................................................................................... 19 

Figure 21. Estimate error in u. ..................................................................................................... 20 

Figure 22. Estimate error in p. ..................................................................................................... 20 

List of Tables 

Table 1. Sensor noise standard deviation..................................................................................... 10 
1 

1. Introduction 

Smart projectiles promise to provide drastic improvements in impact point performance of artillery 

shells, mortars, grenades, and direct fire tank rounds. Although the potential benefit of these new 

weapons is quite substantial, the technological challenges are also difficult. In contrast to unguided projectiles, smart projectiles must include hardware in the form of a sensor suite, a 

microprocessor, and a physical control mechanism along with associated software. These added 

components must be small to fit within geometric constraints and rugged to withstand the very 

harsh launch environment. To constrain cost, the number of on-board sensors must also be 

minimized. A key aspect of the flight control system software is the logic that blends available 

sensor data together to form a usable feedback signal for active control. This is the classic state 

estimator or observer problem. The state estimator is often a critical element within a smart 

weapon flight control system and generally affects impact performance of the weapon in a major 

way. The use of on-board sensors for projectile state observation is plagued by the noise inherent 

in physical sensors, while the use of on-board state propagators is plagued by inaccuracies inherent 

in simple dynamic models (1). 

For almost 50 years, the work horse of state estimation has been the celebrated Kalman filter (2 

through 9), which allows for a combination of imperfect sensor information and a linear model of 

the overall system to be blended in a manner that provides an optimal state estimate for linear 

systems. The extended Kalman filter (10, 11) is the typical way that estimators are designed for 

nonlinear physical systems. In the extended Kalman filter, the nonlinear system is effectively 

approximated by a linear system, and the standard Kalman filter is used on the linear system. 

Although Kalman filters and extended Kalman filters are used extensively in practice, they are not 

without problems. System errors and measurement errors in a Kalman filter are usually assumed 

to be zero-mean Gaussian process with a known covariance. Modeling errors seldom assume this 

form, and measurement error statistics are often not well known, leaving users to select error 

covariance in a trial-and-error fashion. This filter-tuning process can be time consuming. In 

some applications, it can be difficult to get the estimates to converge. 

An alternative to a Kalman filter is a nonlinear observer that works directly with the nonlinear 

dynamic system and not a locally linearized approximation of the system. Because of a large 

number of applications, a significant literature base has amassed for nonlinear observers (12, 13, 

14). Algebraic methods have been employed to transform certain classes of nonlinear systems into 

observer normal form, where observers can be constructed with guaranteed convergence 

properties. Another approach to construct nonlinear observers lies in variable structure systems 

theory. With the Lyapunov theory, extensions of standard linear observer theory can be made to 

prove asymptotic stability of nonlinear observers. It is also possible to employ model predictive 

control techniques to form nonlinear model predictive observers. The focus of the current report is 
2 

the use of a nonlinear model predictive observer to estimate the state of a smart projectile. The 

observer is based on a two-step process consisting of an initial state predictor followed by a state 

estimate corrector. We generate the predictor state by simply integrating the projectile’s fixed 

plane equations of motion forward in time while the corrector step updates the state estimate by the 

minimization of the difference between measurements and model predicted measurements over a 

finite duration of time in the past. Since the proposed method permits a general nonlinear representation of the projectile, the corrector step requires the solution of a nonlinear minimization 

problem, which is subsequently solved with a damped Newton procedure. The nonlinear model 

predictive observer is exercised on a smart projectile that employs a typical array of sensors 

including global positioning system (GPS), a three-axis gyroscope, three-axis accelerometers, 

and a three-axis magnetometer. Trade-off studies are shown that vary the estimation horizon. 

2. Projectile Flight Dynamic Model 

A six-degree-of-freedom (6-DOF) rigid projectile model has been proved to accurately simulate 

the dynamics of a projectile in atmospheric flight (15). The 6 DOF are comprised of the three 

translational components describing the position of the projectile’s center of mass and the three 

Euler angles describing the orientation of the projectile with respect to a fixed inertial axis. 

Figures 1 and 2 provide a schematic of the degrees of freedom. 

Figure 1. Position definitions. 
3 

Figure 2. Orientation definitions. 

Although commonly expressed and computed in the projectile body frame, the equations of 

motion for the 6-DOF model are expressed in the no-roll frame to reduce computational 

requirements within the observer. 

0

x cc s sc u

y cs c ss v

z s cw

θψ ψ θψ

θψ ψ θψ

θ θ

⎧ ⎫ ⎡ ⎤⎧ ⎫ − ⎪⎪ ⎪⎪ ⎢ ⎥ ⎨⎬ ⎨⎬ =

⎪⎪ ⎪⎪ ⎩ ⎭ ⎣ ⎦⎩ ⎭ −

& %

& %

& %

 (1) 

1 0

01 0

001

t p

q

c r

θ

θ

φ

θ

ψ

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

& %

& %

& %

 (2) 

/

/

/

u X m rv qw

v Y m t rw ru

w Z m qu t rv

θ

θ

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

⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ + ⎩⎭ ⎩ ⎭

& % % %% % %

& % % %% %%

& % % % % %%

 (3) 

[] [] 1

0

0

0

p L rq p

q I M r rt I q

r N q rt r

θ

θ

−

⎧⎫ ⎧ ⎫ ⎛ ⎞ ⎡ − ⎤ ⎧⎫ ⎪⎪ ⎪ ⎪ ⎜ ⎟ ⎢ ⎥ ⎪ ⎪ ⎨⎬ ⎨ ⎬ ⎨⎬ = + ⎜ ⎟ ⎢ ⎥ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎩⎭ ⎩ ⎭ ⎝ ⎠ ⎣

⎢ ⎥ − − ⎦ ⎩⎭

& % % %% %

& % % % %%

& % % %% %

 (4) 
4 

The force acting on the projectile in equation 3 is comprised of the weight force (W) and the 

aerodynamic force. The aerodynamic force is split into a standard (A) and Magnus (M) 

aerodynamic force. The combination of forces is expressed in equation 5. 

wAM

wAM

wAM

XX X X

YY Y Y

ZZ Z Z

⎧⎫⎧ ⎫⎧ ⎫⎧ ⎫

⎪⎪⎪ ⎪⎪ ⎪⎪ ⎪ ⎨⎬⎨ ⎬⎨ ⎬⎨ ⎬ =++

⎪⎪⎪ ⎪⎪ ⎪⎪ ⎪ ⎩⎭⎩ ⎭⎩ ⎭⎩ ⎭

%% % %

%% % %

%% % %

 (5) 

Equation 6 provides the expression for the weight force in the no-roll coordinate system. 

 0

w

w

w

X s

Y mg

Z c

θ

θ

⎧ ⎫ ⎧− ⎫ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎨⎬ =

⎪ ⎪ ⎪⎪ ⎩ ⎭ ⎩ ⎭

%

%

%

 (6) 

Equation 7 provides the expression for the aerodynamic force in the no-roll coordinate system. 

This force acts upon the projectile at the aerodynamic center of pressure. 

2 2

02 2

2 2

0 1

0 1

8

A XX X

A A Y YB

A Z ZA

X CC C

Y VD C C

Z CC

α β

π ρ β

α

⎧ ⎫ ⎧ + + ⎫ ⎪ ⎪ ⎪ ⎪ ⎨⎬ ⎨ ⎬ =− +

⎪⎪ ⎪ ⎪ + ⎩ ⎭ ⎩ ⎭

%

%

%

 (7) 

Equation 8 provides the expression for the Magnus force in the no-roll coordinate system. The 

Magnus force acts upon the projectile at the Magnus force center of pressure. 

2 2

0

8 2

2

M

NPA

M A

A

M

NPA

A

X

pDC Y VD

V

Z

pDC

V

π α

ρ

β

⎧ ⎫ ⎪ ⎪

⎧ ⎫ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ⎨⎬ ⎨ ⎬ =

⎪⎪ ⎪ ⎪ ⎩ ⎭ ⎪− ⎪ ⎪ ⎪ ⎩ ⎭

%

% %

%

%

 (8) 

Equations 7 and 8 are based on Mach number-dependent coefficients, the aerodynamic angles of 

attack, and the total aerodynamic velocity. 

The moment acting on the projectile in equation 4 is comprised of the moment attributable to the 

standard aerodynamic force (A), the moment attributable to the Magnus aerodynamic force (M) 

and the unsteady aerodynamic moments (UA) as shown in equation 9. 

A M UA

A M UA

A M UA

LL L L

MM M M

NN N N

⎧ ⎫⎧ ⎫⎧ ⎫⎧ ⎫

⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪ ⎨ ⎬⎨ ⎬⎨ ⎬⎨ ⎬ =+ +

⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪ ⎩ ⎭⎩ ⎭⎩ ⎭⎩ ⎭

%% % %

%% % %

%% % %

 (9) 
5 

The moment attributable to the aerodynamic force is expressed in equation 10. 

0

0

0

A CAZ CAY A

A CAZ CAX A

A CAY CAX A

L RR X

M R RY

N RR Z

⊕ ⊕

⊕ ⊕

⊕ ⊕

⎧ ⎫ ⎧⎫ ⎡ ⎤ − ⎪ ⎪ ⎪⎪ ⎢ ⎥ ⎨ ⎬ ⎨⎬ = −

⎪ ⎪ ⎪⎪ ⎣ ⎦ − ⎩ ⎭ ⎩⎭

% %

% %

% %

 (10) 

The moment attributable to the Magnus force is expressed in equation 11. 

0

0

0

M CMZ CMY M

M CMZ CMX M

M CMY CMX M

L RR X

MR RY

N RR Z

⊕ ⊕

⊕ ⊕

⊕ ⊕

⎧ ⎫ ⎧⎫ ⎡ ⎤ − ⎪ ⎪ ⎪⎪ ⎢ ⎥ ⎨ ⎬ ⎨⎬ = −

⎪ ⎪ ⎪⎪ ⎣ ⎦ − ⎩ ⎭ ⎩⎭

% %

% %

% %

 (11) 

The unsteady aerodynamic moments acting on the projectile are expressed in equation 12. 

2 3

2

8 2

2

LP

DD

A

UA

MQ

UA A

A

UA

NR

A

pDC C

V

L

qDC M VD

V

N

rDC

V

π

ρ

⎧ ⎫ + ⎪ ⎪

⎧ ⎫ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ⎨⎬ ⎨ ⎬ =

⎪⎪ ⎪ ⎪ ⎩ ⎭ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭

%

% % %

%

%

 (12) 

These coefficients are projectile specific functions of the Mach number of the projectile. 

3. Sensor Model 

Within the corrector part of the observer, estimated motion of the projectile is contrasted with a 

given sensor suite to iteratively determine state estimates. Sensor suites on a smart weapon 

typically include gyroscope, accelerometer, magnetometer, and GPS measurements. To properly 

blend these data together in the observer, an analytical sensor model is required for each sensor 

type, which relates sensor readings to projectile motion via rigid body kinematics. Major error 

sources such as noise error, bias error, scale factor error, cross-axis sensitivity/misalignment, and 

sensor misposition can significantly affect sensor readings and must therefore be included in the 

sensor model. The acceleration experienced by the location of an accelerometer is 

[ ] [] ( )

x

Si

T y

i Ni Bi Ai S B P Si Si

z

Si

s

a a a s R S C r g sc

c c

θ

φ θ

φ θ

δ

δ

δ

→

⎛ ⎞ ⎛ ⎞ ⎧ ⎫ ⎧− ⎫ ⎜ ⎟ ⎜ ⎟ ⎪ ⎪ ⎪ ⎪ =++ + − ⎜ ⎟ ⎜ ⎟ ⎨ ⎬ ⎨⎬ ⎪ ⎪ ⎪⎪ ⎝ ⎠ ⎝ ⎠ ⎩ ⎭ ⎩ ⎭

r r (13) 
6 

where 

T yz

Ax Ax Ax Ax

Tx z

Ay Ay Ay Ay

T xy

Ax Az Az Az

s scc

s csc

s ccs

⎧ ⎫⎡ ⎤

⎪ ⎪⎢ ⎥ ⎨ ⎬ = ⎢ ⎥ ⎪ ⎪ ⎢ ⎥ ⎩ ⎭⎣ ⎦

r

r

r and [SS S S ] = + B BI B BI B BI (α ωω / // ) ( )( ) r r r

Also, the angular velocity of the gyroscopes is 

[ ]

y z

x Nx Bx Gx Gx Gx

x z

y Ny By Gy Gy Gy S

x y

z Nz Bz Gz Gz Gz

scc p

c s c Rq

ccs r

ωω ω

ωω ω

ωω ω

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

 (14) 

Magnetometers measure the dot product between the earth’s magnetic field vector and the sensitive axis of the magnetometer. Although they do not directly measure any rigid projectile model 

states, three magnetometers can be arranged orthogonally so that they project the earth’s magnetic 

field vector onto the projectile body frame and aid in estimating projectile orientation. 

[ ]

T

X NX X

Y NY S Y

Z NZ Z

M M cc ssc cs csc ss E

M M R cs sss cc css sc E

M M s sc cc E

θψ φθψ φψ φθψ φψ

θψ φθψ φψ φθψ φψ

θ φθ φθ

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

 (15) 

GPS measures the position and velocity of the receiver relative to the earth, which is given by the 

following: 

x Nx Bx

y Ny By

z Nz Bz

x Nx Bx

y Ny By

z Nz Bz

GG G x

GG G y

GG G z

GG G x

GG G y

GG G z

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

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ ⎩⎭

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

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ ⎩⎭

& &&

& &&

& &&

&

&

&

 (16) 

4. Nonlinear Model Predictive Observer 

Consider the projectile dynamic system just described. Using the state vector defined by x = {x,y, 

zφ,θ,ψ,uvwpqr %% % % %% ,, , ,, }T

, we can describe the dynamic system by the following: 

x& = f x( ) (17) 

Assuming that several quantities are measured that are related to the state so that the output is 

described by the following: 

y hx = ( ) (18) 
7 

It is desired to estimate the state x at any time with the use of the math model and measurements. 

The state estimate will be called x% and the measurement reconstruction uses the estimates 

y hx % % = ( ) . In order to make a state estimate, a quadratic cost function is minimized over a discrete 

horizon (see figure 3). 

Figure 3. Measurement discretization. 

() () () ()

1

1

1

2

k T

ii ii

i

J yx yx W yx yx

+

=

=− − ⎡ ⎤⎡ ⎤ ∑⎣ ⎦⎣ ⎦ %% %% (19) 

where W is positive definite and k is the selected measurement horizon. 

In solving the minimization problem, it is desirable to express the cost function as a function of 

the state estimate at j. Equation 20 shows the cost function as it relates to the current state 

estimate. 

( ) ( ) ( ) ( )

1

1

1

2

k T

ij i ij i

i

J g x yx W g x yx

+

=

=− − ⎡ ⎤⎡ ⎤ ∑⎣ ⎦⎣ ⎦ % % (20) 

g x i j ( ) % is a function that relates the state estimate at the current time to past sensor measurements. 

Its structure is discussed later in this report. 

For the state estimate to be an optimal solution, the cost function must satisfy the gradient condition. 

0 J

x

∂ = ∂% (21) 

In order to satisfy equation 21, a numerical technique is used because J is highly nonlinear. A 

Newton type method is used to solve for x% that will drive ∂J x∂ =% 0 . To achieve this condition 
8 

numerically, J is approximated as a local quadratic function (equation 22), and a perturbation 

on the state is solved for so that the gradient of the local model is zero (equation 23). 

( ) ( ) ( ) 0 0

2

0 00 0 2

1

2

T

NN N

x x

J J JJ x x x x x x

x x

∂ ∂

≈+ − + − −

∂ ∂ % %

%% %% %% % % (22) 

( ) 0 0

2

2 0 0 N

N x x

JJ J

x x

xx x

∂∂ ∂ = + −=

∂∂ ∂ % %

% % %% % (23) 

The line search parameter α is implemented in order to accelerate convergence (equation 24): 

0 0

1

2

N 0 2

x x

J J

x x

x x

α

−

⎡ ⎤ ∂ ∂ = − ⎢ ⎥

⎢ ⎥ ∂ ∂ ⎣ ⎦ % %

% % % % (24) 

When a set of nonlinear equations is solved iteratively, a starting guess is required. This initial 

guess is generated by integrating the equations of motion (1 through 4) forward one time step. 

This can be viewed as the predictor step, and solving the nonlinear equations can be viewed as 

the corrector step. 

A key aspect of this technique is linking the output at one point in time to the state at the time of 

interest. To create this link, the dynamic equations of motion can be used and applied with a 

suitable discretization. For this problem, Euler’s formula was used for its simplicity and ease of 

computation. 

x%% % j−1 = −Δ x tf x j j ( ) (25) 

This link can be used recursively to link the state at any two times. 

( ) ( ) ( ( ))

() () ( ) ( ) () () ( )

21 1

3

jj j j j j j

jj j j j j j j j

x x tf x x tf x tf x tf x

x x tf x tf x tf x tf x tf x tf x tf x

−− −

−

= −Δ = −Δ −Δ −Δ

= −Δ −Δ −Δ −Δ −Δ −Δ −Δ

%% % % % % %

%% % % % % % % %

 (26) 

Equation 24 requires the partial derivatives of the measurements. These derivatives can be computed numerically, partially numerically, and partially analytically, or completely analytically. 

Using the analytical approach, we get the following equations: 

( ) ( ) () () ( ) () () ( )

( ) ( ) () () ( ) ( ) ( ) () () ( )

1

1

2 2 2 1

2 2 2

1

2

T

k T i i

ii ii

i j j j

T T

k T i ii i

ii ii

i j j jj j

J yx yx W yx yx yx yx W

xx x

J y x yx yx yx W yx yx W yx yx W

x x xx x

+

=

+

=

∂ ⎡ ⎤ ∂ ∂

= −+− ⎢ ⎥ ∂∂ ∂ ⎢ ⎥ ⎣ ⎦

∂ ⎡ ⎤ ⎡⎤ ∂ ∂∂ ∂

= −+ +− ⎢ ⎥ ⎢⎥ ∂ ∂ ∂∂ ∂ ⎣ ⎦ ⎣⎦

∑

∑

%% %% %% %% %% %

%% %% %% %% %% %% % % %% %

(27) 

The second partial derivatives used in equation 27 can typically be neglected because they are very 

small when compared with the first partial derivative term (16). 
9 

Equation 27 requires more derivatives to be determined. 

( ) ( )

( ) ( ) 1

1

1 1 1

1

j

j j

j j j

j jj

j j

x j

x x

yx yx x

x xx

h

x tf x

x x

h f I t

x x

−

−

− − −

−

∂ ∂ ∂ = ∂ ∂∂

∂ ∂ = −Δ

∂ ∂

⎛ ⎞ ∂ ∂ = −Δ ⎜ ⎟

∂ ∂ ⎝ ⎠

%

% %

%% %% %

% %%

% % % %

% %

 (28) 

These derivatives can also be recursive. 

( ) 2 2 ( ) 2 1

2 1

j j j j

j j jj

yx yx x x

x x xx

− − − −

− −

∂ ∂ ∂ ∂ = ∂ ∂ ∂∂

%% %% % %

% % %% (29) 

( )

1 jN jN p

N

j N

j x x p

y x h f I t

xx x − −+

−

=

∂ ⎛ ⎞ ∂ ∂ = −Δ ⎜ ⎟

∂∂ ∂ ⎝ ⎠

∏% %

%

% % (30) 

5. Sample Results 

In order to exercise the methodology developed here, simulation results are generated for a sample 

system. The projectile considered is a typical 155-mm spin-stabilized projectile. The projectile is 

assumed to have a sensor suite on board which includes GPS, a three-axis accelerometer, a threeaxis magnetometer, and a three-axis gyroscope. 

Convergence of the corrector portion of the observer is shown in figure 4 for a single instant in 

time for a single magnetometer axis. In figure 4, a total of 15 points is used in the estimation 

horizon, and the numerical solution exhibits rapid convergence. The cost J with a guess at initial 

conditions was 7.1735e+004 and was reduced to 1.9027977e-001 and 1.9027976e-001 after 

estimates 4 and 5, respectively. The cost had not only converged, but the estimated output was 

very acceptable for the entire horizon. These results are typical for all measurements. 

A trade study was conducted to determine an appropriate number of points in the measurement 

horizon k. A number of various horizons were selected, ranging from three data points to 42 data 

points. The measurement data were populated with noise with standard deviations shown in table 1 

and remained the same for each case in the study. Also, the initial conditions at j = 1 were the same 

for each case of the study. The results of this trade study are shown in the figure 5. As the number 

of points in the prediction increases, the sensor noise becomes increasingly filtered. From this trade 
10 

study, it was determined that a horizon of 21 data points gave satisfactory estimates without 

requiring intense computations. 

0 0.05 0.1 0.15

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (sec)

Magnetometer - Y axis

Est 1 Est 2 Est 3 Est 4 Est 5 Actual

Figure 4. Single time step iterations. 

Table 1. Sensor noise standard deviation. 

Sensor Noise Standard 

Deviation 

Sensor Noise Standard 

Deviation 

GPS Accelerometer 

x 4.8 ft x -axis 0.03 ft/sec2

y 4.8 ft y -axis 0.03 ft/sec2

z 10.8 ft z -axis 0.03 ft/sec2

x& 0.6 ft/sec Magnetometer 

y& 0.6 ft/sec x -axis 0.001 

z& 0.6 ft/sec y -axis 0.001 

Gyroscope z -axis 0.001 

p 0.1 rad/sec 

q 0.1 rad/sec 

r 0.1 rad/sec 
11 

0 0.5 1 1.5 2 2.5

0

1

2

3

4

5

6

7

8

Time (sec)

State Estimate Error - y (ft)

3 6 9 12 15 18 21 30 42

Figure 5. Trade study results. 

To further illustrate how greatly this estimator can reduce to amplitude and frequency of noise, 

figure 6 compares the sensor output, estimated state, and actual state for a single GPS signal. 

With a measurement horizon of 21 points, state estimates were found over a full projectile 

trajectory. The comparisons of each of the states with the state estimates are plotted in figures 7 

through 22. Also plotted are the some of the errors between the actual state and the state estimate 

along the trajectory. 

6. Conclusions 

A nonlinear model predictive observer specialized to smart projectiles reported here can successfully predict the full state vector of a projectile trajectory. The observer is based on a two-step 

process consisting of an initial state predictor followed by a state estimate corrector. We generate 

the predictor state by simply integrating the projectile’s fixed plane equations of motion forward in 

time while the corrector step updates the state estimate by the minimization of the difference 
12 

between measurements and model predicted measurements over a finite duration of time in the past. 

Since the method permits a general nonlinear representation of the projectile, the corrector step 

requires the solution of a nonlinear minimization problem which is subsequently solved with a 

damped Newton procedure. The nonlinear model predictive observer is exercised on a smart 

projectile that employs a typical array of sensors including GPS, a three-axis gyroscope, three-axis 

accelerometers, and a three-axis magnetometer. Results with this smart weapon observer are 

promising since the observer is capable of estimating the full state of the projectile, including 

orientation and translational velocity, when realistic noise and bias errors are included in the sensor 

measurements. Relative to other existing observers, however, the proposed observer is computationally intensive because of the need to solve a nonlinear minimization problem at each compu--

tation cycle of the observer. 

0 0.5 1 1.5 2 2.5

-15

-10

-5

0

5

10

15

20

Time (sec)

Cross Range (ft)

State

Output

Estimate

Figure 6. Noise filtering. 
13 

0 1 2 3 4 5 6 7 8

0

0.5

1

1.5

2

x 104

Time (sec)

Range (ft)

State

Estimate

Figure 7. Projectile range. 

0 1 2 3 4 5 6 7 8

-5

0

5

10

15

20

25

30

Time (sec)

Cross Range (ft)

State

Estimate

Figure 8. Projectile cross range. 
14 

0 1 2 3 4 5 6 7 8

0

500

1000

1500

2000

2500

3000

3500

Time (sec)

Altitude (ft)

State

Estimate

Figure 9. Projectile altitude. 

0 1 2 3 4 5 6 7 8

-4

-3

-2

-1

0

1

2

3

4

Time (sec)

φ (rad)

State

Estimate

Figure 10. Projectile roll angle. 
15 

0 1 2 3 4 5 6 7 8

0.1

0.15

0.2

0.25

Time (sec)

θ (rad)

State

Estimate

Figure 11. Projectile Euler angle θ. 

0 1 2 3 4 5 6 7 8

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

Time (sec)

ψ (rad)

State

Estimate

Figure 12. Projectile Euler angle ψ. 
16 

0 1 2 3 4 5 6 7 8

1800

2000

2200

2400

2600

2800

Time (sec)

u (ft/s)

State

Estimate

Figure 13. Projectile velocity u. 

0 1 2 3 4 5 6 7 8

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Time (sec)

v (ft/s)

State

Estimate

Figure 14. Projectile velocity v. 
17 

0 1 2 3 4 5 6 7 8

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time (sec)

w (ft/s)

State

Estimate

Figure 15. Projectile velocity w. 

0 1 2 3 4 5 6 7 8

1450

1500

1550

1600

1650

1700

Time (sec)

p (rad/s)

State

Estimate

Figure 16. Projectile rate p. 
18 

0 1 2 3 4 5 6 7 8

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Time (sec)

q (rad/s)

State

Estimate

Figure 17. Projectile rate q. 

0 1 2 3 4 5 6 7 8

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Time (sec)

r (rad/s)

State

Estimate

Figure 18. Projectile rate r. 
19 

0 1 2 3 4 5 6 7 8

-3

-2

-1

0

1

2

3

Time (sec)

Range Error(ft)

Figure 19. Estimate error in range. 

0 1 2 3 4 5 6 7 8

-1.5

-1

-0.5

0

0.5

1

1.5

Time (sec)

φ Error(rad)

Figure 20. Estimate error in φ. 
20 

0 1 2 3 4 5 6 7 8

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time (sec)

u Error(ft/s)

Figure 21. Estimate error in u. 

0 1 2 3 4 5 6 7 8

-1

-0.5

0

0.5

1

Time (sec)

p Error(rad/s)

Figure 22. Estimate error in p. 
21 

7. References 

1. Merhav, S. Aerospace Sensor Systems and Applications; Springer-Verlag, Inc.: New York, 

1996. 

2. Aldrich, G.; Krabill, W. A Versatile Kalman Technique for Aircraft or Missile State 

Estimation and Error Analysis using Radar Tracking Data. AIAA-72-838, AIAA Guidance 

and Control Conference, Stanford, CA, 1972. 

3. Widall, W. Enlarging the Region of Convergence of Kalman Filters that Encounter 

Nonlinear Elongation of Measured Range. AIAA-72-879, AIAA Guidance and Control 

Conference, Stanford, CA, 1972. 

4. Speyer, J.; Hull, D. Comparison of Several Extended Kalman Filter Formulations for 

Homing Missile Guidance. AIAA-80-1786, AIAA Guidance and Control Conference, 

Danvers, MA, 1980. 

5. Chin, H. Distributed Kalman Filter in an Integrated SAHRS/GPS Navigation System. 

AIAA-85-1878, AIAA Guidance, Navigation, and Control Conference, Snowmass, CO, 

1985. 

6. Lam, Q.; Pal, P.; Welch, R.; Grossman, W. Magnetometer Based Attitude Determination 

System using a Reduced Kalman Filter. AIAA-96-3629, AIAA/AAS Astrodynamics 

Conference, San Diego, CA, 1996. 

7. Burchett, B.; Costello, M. Specialized Kalman Filtering for Guided Projectiles. AIAA2001-1120, AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, 2001. 

8. Shkolnikov, I.; Shtessel, Y.; Zarchan, P.; Lianos, D. Simulation Study of the Homing 

Interceptor Guidance Loop with Sliding Mode Observers verses Kalman Filter. AIAA2001-4216, AIAA Guidance, Navigation, and Control Conference and Exhibit, Montreal, 

Canada, 2001. 

9. Kalman, R. E. A New Approach to Linear Filtering and Prediction Problems. Transactions 

of the ASME-Journal of Basic Engineering 1960, 82, 34-45. 

10. Stengel, R. Optimal Control and Estimations; Dover Publications, Inc.: New York, 1994. 

11. Coleman, N.; Yip, P.; May, R.; Lin, C.; Feng, X.; Yu, T. Estimation and Prediction of 

Projectile Trajectory, MET Data and Impact Point. AIAA-1998-5538, AIAA and SAE, 1998 

````````````````````````````````````````````````````````````````````````````````````````````````````````````````

``````````````````````World Aviation Conference, Anaheim, CA, Sept. 28-30, 1998. 
22 

12. Crassidis, J.; Markley, F. Predictive Filtering for Nonlinear Systems. AIAA-1996-3775, 

AIAA Guidance, Navigation, and Control Conference, San Diego, CA, July 29-21, 1996. 

13. Moraal, P. E.; Grizzle, J. W. Observer Design for Nonlinear Systems with Discrete-Time 

Measurements. IEEE Transactions on Automatic Control March 1995, 40 (3), 395-404. 

14. Michalska, H.; Mayne, D. Q. Moving Horizon Observers and Observer-Based Control. 

IEEE Transactions on Automatic Control June 1995, 40 (6), 995-1006. 

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

and Stability of a Projectile. AIAA-1996-3447, AIAA Atmospheric Flight Mechanics 

Conference, San Diego, CA, July 29-21, 1996. 

16. Press, W.; Teukolsky, S.; Vetterling, W.; Flannery, B. Numerical Recipes in C; Cambridge 

University Press, New York, pp. 681-685, 1992. 
23 

Nomenclature 

x,y,z = Inertial positions of the system mass center. 

φ, , θ ψ = Euler roll, pitch and yaw angles. 

uvw %% % , , = Velocity components of mass center in a no-roll reference frame. 

p% %% , , q r = Angular velocity components in a no-roll reference frame. 

I = Inertia matrix of system. 

aBi = Bias of ith accelerometer. 

Ni a = Noise of ith accelerometer. 

ai = Acceleration reading of ith accelerometer. 

j

Ai c = Cross axis sensitivity of ith accelerometer to the jth direction. 

j

Gi c = Cross axis sensitivity of ith gyroscope to the jth direction. 

Ei = ith component of the Earth’s magnetic field. 

GBi = Bias of ith GPS. 

GNi = Noise of ith GPS. 

Gi = Reading of ith GPS. 

MNi = Noise of ith magnetometer. 

Mi = Reading of ith magnetometer. 

RS = Rotation matrix from body frame to sensor frame. 

Si r⊕→

r = Distance vector from projectile mass center to ith accelerometer. 

Ai s = Scale factor of ith accelerometer. 

Gi s = Scale factor of ith gyroscope. 

α B / I

r = Angular acceleration of projectile with respect to the ground. 

j δ Si = Misposition of ith accelerometer in the jth direction. 

ωB / I

r = Angular velocity of projectile with respect to the ground. 

ωBi = Bias of ith gyroscope. 

ωNi = Noise of ith gyroscope. 

ωi = Angular velocity reading of ith gyroscope. 

CB = Vector component extraction operator for frame B . 

B S = Skew symmetric cross product operator for frame B . 
24 

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