ARMY RESEARCH LABORATORY

ARL-TR-3029 September 2003

Understanding Body-Fixed Sensor Output

From Projectile Flight Experiments

Thomas E. Harkins

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Army Research Laboratory

Aberdeen Proving Ground, MD 21005-5066

____________________________________________________________________________

ARL-TR-3029 September 2003

____________________________________________________________________________

Understanding Body-Fixed Sensor Output From

Projectile Flight Experiments

Thomas E. Harkins

Weapons and Materials Research Directorate

____________________________________________________________________________

Approved for public release; distribution is unlimited

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Understanding Body-Fixed Sensor Output From Projectile Flight Experiments

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Accurate measurement of in-flight kinematics significantly contributes to the development of experimental projectiles and rockets 

and to diagnostics for existing munitions and weapons systems. Ground-based instruments such as radar and cameras provide 

useful measurements but are often limited to portions of a trajectory and/or have limited resolution. On-board sensor systems 

fixed to a projectile body combined with a telemetry system can provide high resolution continuous data throughout a projectile’s 

entire trajectory.

However, there is a twofold difficulty in correctly interpreting and employing data from body-fixed sensors. First, sensor 

responses many times are affected by stimuli other than those which a sensor is intended to quantify, e.g., an angular rate sensor 

may be affected by any g forces to which it is subjected. Second, sensor systems often, of necessity, make measurements in a 

body-fixed coordinate system, and the quantities whose values are desired are best described in another coordinate system. This 

report treats issues affecting the output of the body-fixed sensors used by the U.S. Army Research Laboratory’s Weapons and 

Materials Research Directorate (Advanced Munitions Concepts Branch) in flight tests of military ordnance and provides the 

mathematics necessary to transform body-fixed measurements to earth-fixed parameters.

body-fixed sensors diagnostic fuze projectile kinetics
iii

Contents

List of Figures iv

1. Introduction 1

2. Coordinate Systems 2

3. Transformations Using Euler Angles 4

4. Angular Rates in Projectile and Earth Systems 9

5. Body-Fixed Sensor Locations and Orientations 10

6. Sensor Locations in the Aeroballistic Diagnostic Fuze 14

7. Estimating Projectile Spin Rate 17

7.1 Accel_Ring Measurements .................................. 17

7.2 Solarsonde Measurements .................................. 18

7.3 Magnetometer Measurements ................................ 19

8. Summary 20

9. References 22

Appendix A. Plane-Fixed Coordinate System 23

Distribution List 25
iv

List of Figures

Figure 1. Earth-fixed coordinate system ................................ 3

Figure 2. Projectile-fixed coordinate system.............................. 3

Figure 3. Position of a ball as measured in two coordinate systems .............. 4

Figure 4. First Euler rotation........................................ 5

Figure 5. Second Euler rotation ...................................... 6

Figure 6. Third Euler rotation ....................................... 7

Figure 7. Axial accelerometer (u) for modeled M483 trajectory................. 12

Figure 8. Sensed axial acceleration at the c.g. for modeled M483 trajectory ......... 12

Figure 9. Sensed axial acceleration 0.45 m forward of the c.g. for modeled M483

 trajectory .............................................. 13

Figure 10. Sensed axial acceleration 0.45 m forward of the c.g. and 2 mm off the spin

 axis for modeled M483 trajectory .............................. 13

Figure 11. DF2K1 Dfuze configuration ................................. 15

Figure 12. DF2K1 sensor board viewed from forward looking aft ................ 15

Figure 13. DF2K1 sensor board viewed from rear looking forward ............... 16

Figure 14. Sample of solarsonde data ................................... 18

Figure 15. Sample of radial magnetometer data............................. 19
1. Introduction 

Accurate measurement of in-flight kinematics significantly contributes to the development of 

experimental projectiles and rockets and to diagnostics for existing munitions and weapons 

systems. Such measurements can in some cases be made with high speed photography, but this 

technique is generally used for only limited portions of a projectile flight for reasons of expense 

and practicability. Also, the precision of angular measurements is limited in this methodology. 

Another measurement technique used for obtaining angle of attack data is yaw cards, but this 

technique also is low resolution and provides only a small number of discrete data points along a 

trajectory. Radar can provide position, velocity, and spin measurements but does not give 

projectile orientation information. Global positioning systems provide position and velocity 

measurements but cannot provide orientation information. On-board sensor systems fixed to a 

projectile body combined with a telemetry system can provide continuous measurements 

throughout a projectile’s entire trajectory. 

The Advanced Munitions Concepts Branch of the U.S. Army Research Laboratory’s (ARL’s) 

Weapons and Materials Research Directorate has for many years designed, built, and employed 

body-fixed sensor systems in support of military ordnance development and testing. Sometimes, 

a quantity of interest to a tester is directly measurable, and given an accurate sensor, only the 

observation or recording of sensor data is required to meet test objectives. Temperature is one 

example of such a quantity. In other cases, body-fixed sensors’ output needs to be combined 

and/or otherwise processed in order to obtain measures of desired quantities. For example, a 

post-launch, time history of both speed and heading is required to locate a flying body. 

Combining/processing is also often required for correct interpretation of output from body-fixed 

sensors responding to stimuli external to the flight body. Vector magnetometers measuring 

components of the earth’s magnetic field are an example of such sensors. 

In essence, there is a twofold difficulty in correctly interpreting and employing data from bodyfixed sensors. First, sensor responses many times are affected by stimuli other than those that a 

sensor is intended to quantify, e.g., an angular rate sensor may be affected by any g forces to 

which it is subjected. Second, sensor systems often, of necessity, make measurements in a bodyfixed coordinate system, and the quantities whose values are desired are best described in 

another coordinate system. Although coordinate transforms are a staple of high school 

trigonometry classes, it nevertheless is useful to clearly define the coordinate system conventions 

and rigorously develop the transformations employed at ARL when we provide instrumentation 

and data collection/processing support for projectile flight experiments. It is hoped that this 

elementary and somewhat tedious exposition will benefit interpreters of body-fixed sensor data 

and other readers by familiarizing them with the issues involved and thereby avoid the confusion 

that we sometimes have observed with regard to such data. 

1
To this end, three Cartesian systems commonly employed in projectile test and evaluation are 

defined and the mathematics necessary to transform among them are given in this report. 

Examples of idealized projectile-fixed sensor output from projectile trajectory simulations and 

sensor output recorded during flight experiments will illustrate this process. An additional 

coordinate system extensively employed in computer modeling and simulation of projectile 

flight is treated in Appendix A. 

2. Coordinate Systems 

The first coordinate system is convenient for describing the motion of a gun- or tube-launched 

projectile along its trajectory. This system is right-handed Cartesian with its origin at the launch 

site. This is referred to as the “earth-fixed” system herein, and parameters in this system are 

subscripted with an “E”. The axes are defined as follows (see Figure 1): 

• The XE and YE axes define a plane tangent to the earth’s surface at the origin. 

• The ZE axis is perpendicular to the earth’s surface with positive downward, i.e., in the 

direction of gravity. 

• The XE axis is chosen so that the centerline of the launcher is in the XE-ZE plane. 

Figure 1. Earth-fixed coordinate system. 

2
Down-range travel is then measured along the XE axis, deflection is measured along the YE axis 

(positive to the right when one is looking down range), and altitude is measured along the ZE axis 

(positive downward). Designation of height above the ground as a negative ZE component of 

projectile location may seem unnatural to some, but this coordinate system was employed by a 

former division chief at ARL, Dr. Charles Murphy, in his seminal work on free-flight motion of 

symmetric projectiles (Murphy, 1963), and this became the norm within ARL. 

The second system is convenient for aeroballistic computations of rigid projectiles’ flights and 

for describing the locations and orientations of such projectiles’ components. This system is 

right-handed Cartesian with its origin at the center of gravity (c.g.) of the flight body (see 

Figure 2). This is referred to as the “projectile-fixed” system herein, and parameters in this 

system are subscripted with a “P”. 

Figure 2. Projectile-fixed coordinate system. 

Most of the projectiles instrumented by ARL are either spin-stabilized or are designed to have 

some fin cant-induced roll rate to mitigate the effects of asymmetries. For such flight bodies, the 

XP axis of the projectile-fixed coordinate system usually lies along the projectile axis of 

symmetry, i.e., the spin axis (with positive in the direction of travel at launch). The YP and ZP 

axes are then oriented so as to complete the right-handed orthogonal system. Because the YP and 

ZP axes are collinear with projectile radii in the cross section through the c.g., sensors whose axes 

are parallel with this cross section are commonly termed “radial” sensors. Similarly, sensors 

whose axes are parallel with the XP axis are commonly termed “axial” sensors. 

The third coordinate system is commonly employed to specify locations on or near the earth’s 

surface (i.e., north, east, and down). This is referred to as the “navigation” system herein, and 

parameters in this system are subscripted with an “N”. Thus, north = XN, east = YN, and 

down = ZN. 

3
3. Transformations With Euler Angles 

The specification of a vector in any Cartesian system can be transformed into its corresponding 

specification in any other Cartesian system in two steps. First, the origins must be made 

coincident through a translation. The second step is to perform three successive rotations in a 

specified sequence. The three successive rotation angles are called Euler angles. This 

relationship is illustrated for the earth-fixed and projectile-fixed systems in Figure 3 where the 

gray ball’s position in the earth-fixed system is given by PE

r and its position in the projectilefixed system by PP

r . If the location as measured in the first system of second system’s origin at 

the projectile c.g. is OE

r , then the vector from that origin to the ball, as measured in the first 

system, is LE

r , which can be obtained from the translation equation PE OE LEr r r = + . We can then 

derive PP

r from LEr by performing the Euler rotations necessary to obtain the orientations of the 

 axes. P Y P X , P , and Z

Figure 3. Position of a ball as measured in two coordinate systems. 

After the translational offset to collocate the origins of the earth-fixed and projectile-fixed 

systems has been subtracted, transformations of vectors from the earth-fixed system to the 

projectile-fixed system via Euler rotations proceed as follows: 

 1. Rotate the X , system about the axis through an angle where is 

measured in the direction of rotation of the head of a right-hand screw advancing along the 

axis (see Figure 4). Call the resulting axes in which . is the angle 

E E ZE Y , ZE

Z1

ε1 ε1

ZE

1 1 X ,Y , Z =1 ZE ε1

4
between the axis and the projection of the axis into the X plane. Thus, the 

projectile’s spin axis is in the plane with its nose in the half-plane. 

XE XP E YE |

1 Z1 + X | 1 Z1 X |









1 c

b

a

1 Z1 ,Y ,

( )

( ) 1

1

cos

sin

ε

ε

E

E

E

c

-a

a

(











1

0

0

0

sin

cos ε

Figure 4. First Euler rotation. 

If the coordinates of a vector are denoted by the column matrix 

 in the earth-fixed system and denoted by  in the system, then the 

coordinates are related by the equations 

V

r





















E

E

E

c

b

a











1

1

X1

( )

( )

1

1 1 E

1 1 E

c

b sin b

a cos b

=

= +

= +

ε

ε

 (1) 

or, in matrix form, 

V1 E1VE

r r = (2) 

in which 

) ( )

( ) ( ) 









=

0

- cos

sin

E 1 1

1 1

1 ε ε

ε

 (3) 

 2. Rotate the X system about the axis through an angle measured in the 

direction of rotation of the head of a right-hand screw advancing along the Y axis (see 

Figure 5). Call the resulting axes in which . is the angle between the 

1 1 Z1 ,Y , Y1 ε 2

1

2 2 Z2 X ,Y , Y =2 Y1 ε 2

5
projectile spin axis and the axis so that the projectile spin axis lies along the axis. In 

other words, X

X1 X2

X . 2 = P

r

2 Z

3

Figure 5. Second Euler rotation. 

If the coordinates of V are denoted by in the system, then the coordinates are 

related by the equations 





















2

2

2

c

b

a

2 2 Z2 X ,Y ,

( ) ( )

( ) ( ) 2 1 2 1 2

2 1

2 1 2 1 2

c a sin c cos

b b

a a cos - c sin

ε ε

ε ε

= +

=

=

 (4) 

or, in matrix form, 

V2 E2V1 E2 E1VE

r r r = = (5) 

in which 

( ) ( )

( ) ( ) 

















 −

=

2 2

2 2

2

sin 0 cos

0 1 0

cos 0 sin

E

ε ε

ε ε (6) 

 3. Rotate the X Y , system about the axis through an angle measured in the 

direction of rotation of the head of a right-hand screw advancing along the X axis (see 

Figure 6). Call the resulting axes in which . is chosen so that 

 and consequently, .

2 2 ,

Z =

X2 ε 3

2

3 3 Z3 X ,Y ,

ZP

X = 3 = X2 XP ε 3

Y3 = YP

6
Figure 6. Third Euler rotation. 

If the coordinates of V are denoted by in the system, then the coordinates are 

related by the equations 

r





















3

3

3

c

b

a

3 3 Z3 X ,Y ,

( ) ( )

( ) ( ) 3 2 3 2 3

3 2 3 2 3

3 2

c -b sin c cos

b b cos c sin

a a

ε ε

ε ε

= +

= +

=

 (7) 

or, in matrix form, 

V3 E3V2 E3E2 E1VE

r r r = = (8) 

in which 

( ) ( )

( ) ( )



















−

=

ε ε

ε ε

3 3

3 3 3

0 sin cos

0 cos sin

0 0 1

E (9) 

Recognizing that the system is the same as the projectile-fixed system

and forming the product matrix , Equation (8) becomes 

3 3 Z3 X ,Y , P P ZP X ,Y ,

3 2 1

P TE = E E E

E

P VP TE V r r = (10)

Further, the choices of Euler angles were made in order to correspond with angles commonly 

used to describe projectile orientation throughout a trajectory. The first Euler angle, , is the 

azimuth component of the projectile heading relative to the initial shot line. In ballistic 

terminology, this is commonly called the yaw angle and is denoted by 

ε1

ψ . The second Euler 

angle,ε , is the elevation component of the projectile heading or, in ballisticians’ parlance, the 2

7
pitch angle, θ . The third Euler angle,ε , is the rotation component of the projectile orientation 

or the roll angle, φ .

1

3











7

4

1

T

T

T

cos(

sin(

sin(

cos(

sin(

sin(

cos(

cos(

−

V

r

When we perform the matrix multiplication and make the substitutions for the Euler angles, 

is seen to be

P T E











=

8 9

5 6

2 3

P

E

T T

T T

T T

T (11)

in which 

T ) cos( )

T ) cos( )sin( ) sin( ) cos( )

T ) cos( ) cos( ) cos( ) cos( )

T )sin( )

T )sin( )sin( ) cos( ) cos( )

T )sin( ) cos( ) cos( )sin( )

T sin( )

T )sin( )

T ) cos( )

9

8

7

6

5

4

3

2

1

= θ φ

= θ φ ψ − φ ψ

= θ φ ψ + φ ψ

= θ φ

= θ φ ψ + φ ψ

= θ φ ψ − φ ψ

= θ

= θ ψ

= θ ψ

 (12)

The inverse transformation of a vector from projectile-fixed coordinates to earth-fixed

coordinates is given by 

( ) P

T P

E TE V

r = (13)

Equations (11) and (13) can be used to transform any vector between the earth-fixed and 

projectile-fixed systems. Acceleration, velocity, thrust, and line of sight are examples of vectors 

whose components typically are needed in both systems. Before proceeding to examples of 

transformations of this type of sensor data, another critical characteristic of the relationship 

between the earth-fixed and projectile-fixed systems needs to be considered, i.e., the projectilefixed system is moving with respect to the earth-fixed system in both translation and orientation. 

The angular rates describing these orientation changes need to be formulated in each system as 

well. 

 1It is important to note that even though the use of ψ, θ, and φ to denote yaw, pitch, and roll is common in the 

literature, not every author defines each of these angles as is done here. Since there is no great agreement, care must

be taken always by authors to rigorously define the systems they employ and by readers to understand the authors’

systems. 

8
4. Angular Rates in Projectile and Earth Systems 

The angular velocity of the projectile-fixed system with respect to the earth-fixed system is 

described in the projectile-fixed system by a vector ( )

T

P XP , YP , ZP Ω = ω ω ω r in which ω is the 

angular velocity of the axes about the axis; 

X P

P ZP Y | XP YP ω is the angular velocity of the 

ZP | XP axes about the YP axis; and Z P ω is the angular velocity of the axes about the 

 axis. Each of these velocities is defined herein to be right-hand positive. In ballisticians’ 

terminology, this vector is denoted by 

P YP X |

ZP

= (p )

T ΩP ,q,r r . 

The angular velocity vector of the projectile-fixed system with respect to the earth-fixed system

is also given by the sum of the derivatives of the Euler angles ε1 , ε 2 , and . However, 

recalling that each of these Euler angles was defined in a different coordinate system, their 

derivatives need to be transformed into the projectile-fixed system before corresponding 

components can be added. With the ballistic terminology for the Euler angles adopted in the last 

section, ε

ε3

( T

1 = 0,0,ψ& & ) r in the coordinate system, 1 1 Z1 X ,Y , ( )

T

ε 0 &

2 = 0,θ,& r in the 

coordinate system, and 

2 2 Z2 X ,Y ,

( )

T

φ,0,0 & ε3 = &

r in the coordinate system. Thus, Ω can also be 

written in terms of the Euler angles as

3 3 Z3 X ,Y , P

r















φ

+















+ θ 















ψ

Ω = 00

0

0

3 0 E

0

2 E3 P E

& &

&

r (14)

Making the substitutions for and from Equations (6) and (9) and doing the algebra yields E2 E3

























− θ φ + ψ θ φ

θ φ + ψ θ φ

φ − ψ θ

Ω =

sin( ) cos( ) cos( )

cos( ) cos( )sin( )

sin( )

P

& &

& &

& &

r . (15) 

The relationships between the body-fixed components of projectile angular motion and the roll, 

pitch, and yaw rates are thus given by 

r sin( ) cos( ) cos( )

q cos( ) cos( )sin( )

p sin( )

= −θ φ + ψ θ φ

= θ φ + ψ θ φ

= φ − ψ θ

& &

& &

& &

 (16)

Solving for the Euler angle derivatives in terms of the projectile-fixed rates gives 

9
[ ]

( ) q sin( ) r cos( ) cos( )

q cos( ) rsin( )

p q sin( ) r cos( ) tan( )

ψ = φ + φ θ

θ = φ − φ

φ = + φ + φ θ

&

&

&

 (17)

As seen later in this report, there are commonly used, body-fixed sensor systems designed to 

measure p, q, and r and other systems that respond to . Failure to appreciate the 

differences among these quantities can lead to erroneous interpretations of sensor data. 

Conversely, appreciation of these differences sometimes can provide additional insights 

concerning projectile dynamics. 

φ θ ψ& & & , , and

5. Body-Fixed Sensor Locations and Orientations 

The location of a sensor on a flight body can greatly affect its output. In some cases, the quantity 

being measured is location dependent, e.g., acceleration at a point on the body. In other cases, 

the sensor response is affected by something besides the primary stimulus that it is designed to 

measure, e.g., the gain of an angular rate sensor might be g-sensitive. In both instances, the 

locations and the orientations of the sensors must be known in order to correctly interpret their 

output in the projectile-fixed coordinate system even before a transformation into the earth-fixed 

navigation system. Seldom, if ever, will it be possible to install sensors at arbitrary locations on 

projectiles. Although greater possibilities might exist in a developmental or test configuration, 

tactical configurations usually have only limited space at predetermined locations available for 

on-board sensors and electronics. Linear accelerometers provide an illustrative example of the 

effects of sensor location and orientation. 

The basic equation relating position (P), velocity (V), and acceleration (A) is 

. For projectile locations along a trajectory as measured in the 

earth-fixed system, the vector version of this relationship is 

P(t) P(0) V(0)t A t

t

0

t

0

= + + ∂ ∫∫

t

a

a

a

z(0) t

y(0) t

x(0) t

z(0)

y(0)

x(0)

z(t)

y(t)

x(t) t

0

t

0 Z

Y

X

E

E

E

E

E

E

E

E

E

E

E

E

∂





















+





















+





















=





















∫∫ &

&

&

 (18) 

Beginning with acceleration and velocity components in the projectile-fixed coordinate system, 

this relationship becomes 

10
( ) ( ) t

a

a

a

T

z(0) t

y(0) t

x(0) t

T

z(0)

y(0)

x(0)

z(t)

y(t)

x(t) t

0

t

0 Z

Y

X

T P

E

P

P

P

T P

E

E

E

E

E

E

E

E

E

E

∂





















+





















+





















=





















∫∫ &

&

&

 (19) 

In ballisticians’ terminology the velocity vector in the projectile-fixed coordinate system is 

 and the acceleration vector is . Thus, ballisticians rewrite Equation (19) as 





















=





















P

P

P

z

y

x

w

v

u

&

&

&





















w

v

u

&

&

&

( ) ( ) t

w

v

u

T

w(0)t

v(0)t

u(0)t

T

z(0)

y(0)

x(0)

z(t)

y(t)

x(t) t

0

t

0

T P

E

T P

E

E

E

E

E

E

E

∂





















+





















+





















=





















∫∫ &

&

&

 (20) 

Considering only the axial acceleration component, , the process required to obtain this 

quantity from projectile-fixed sensor data can be complicated. Designating a location on board a 

rigid projectile with unchanging mass properties as 

u&

( )

T

XP YP ZP L ,L ,L where each component is 

measured from the projectile’s c.g., the ballisticians’ equation for the axial acceleration at that 

point is given by 

( ) ( ) ( ) P XP YP ZP

2 2 AX = u& + qw − rv − q + r L + pq − r& L + pr + q& L (21) 

Because gravity cannot be sensed by an accelerometer on board a projectile in free flight, the 

idealized output ( ) XP S of an axial accelerometer at this location is thus 

S u qw rv ( q r )L (pq r)L (pr q)L g sin( ) P XP YP ZP

2 2

X = & + − + − − + − & + + & + θ (22) 

In order to simplify notation, sensed acceleration components parallel to the projectile-fixed axes 

are usually referred to as Acc_I (+XP direction), Acc_J (+YP direction), and Acc_K (+ZP

direction). Similarly, sensor offsets are usually given as ∆i, ∆j, and ∆k. Thus, Equation (22) is 

usually written as 

Acc _ I u qw - rv ( q r ) i (pq - r) j (pr q) k gsin( ) 2 2 = & + + − − ∆ + & ∆ + + & ∆ + θ (23) 

The difference between axial acceleration ( ) and an idealized axial accelerometer’s output 

(Acc_I) can be better appreciated graphically. With a six-degree-of-freedom computer model, 

the trajectory of an M483 artillery projectile was simulated for a launch at 25

u&

o

 quadrant elevation 

and 340-m/s muzzle velocity. Figure 7 shows the axial acceleration throughout the trajectory 

and Figure 8 shows the output of an ideal axial accelerometer at the projectile c.g., L=( . 

The difference between the two curves is primarily attributable to the absence of the gravity 

component in the sensed acceleration in Figure 8 because 

T 0,0,0)

qw − rv < 0.0003g . 

11
-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

Axial Acceleration (g's)

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Time Since Muzzle Exit (s)

Figure 7. Axial acceleration ( u& ) for modeled M483 trajectory.

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

Sensed Acceleration (g's)

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Time Since Muzzle Exit (s)

Figure 8. Sensed axial acceleration at the c.g. for modeled M483 

trajectory.

When the axial accelerometer is installed in the fuze, as is typical in flight experiments, the 

location would be approximately 0.45 m forward of the c.g. The sensed axial acceleration at this 

location, , is shown in Figure 9. For an accelerometer 2 mm off the projectile 

spin axis, e.g., , the sensed acceleration is shown in Figure 10.

( T L = 0.45,0,0

L = 0.45,

)

( )

T 0.0014,0.0014

12
-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

Sensed Acceleration (g's)

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Time Since Muzzle Exit (s)

Figure 9. Sensed axial acceleration 0.45 m forward of the c.g. 

for modeled M483 trajectory.

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

Sensed Acceleration (g's)

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Time Since Muzzle Exit (s)

Figure 10. Sensed axial acceleration 0.45 m forward of the c.g. 

and 2 mm off the spin axis for modeled M483 

trajectory.

Despite the lack of resemblance of the curve in Figure 10 to the curve in Figure 7, axial 

acceleration ( ) can be derived from accelerometer output, given initial conditions, knowledge 

of the location of an axial accelerometer, and knowledge of the angular velocity, , and 

acceleration, , of the projectile. These rates are estimated by other on-board sensors. 

u&

p&,

( T

p,q,r)

( )

T

q&,r&

In the preceding examples, the accelerometers were assumed to be exactly situated at known 

points and to have their sensitive axes oriented exactly parallel to the projectile spin axis. 

Neither assumption is realizable in an actual system because of location and alignment 

13
uncertainties arising from tolerances in the manufacturing and assembly processes. Additionally, 

linear accelerometers typically have some level of sensitivity to accelerations in directions other 

than the direction of the measurement axis. These uncertainties must be resolved in order to 

correctly interpret the sensor output and to derive desired quantities from the output. We can 

accomplish this resolution by calibrating the individual sensors either in the laboratory or in flight. 

6. Sensor Locations in the Aeroballistic Diagnostic Fuze

The aeroballistic diagnostic fuze (DFuze) (U.S. Patent 6349652) was developed at ARL as a 

high-g, projectile-borne system for non-intrusive measurements of projectile diagnostics and 

aerodynamic performance. The current nominal DFuze instrumentation configuration 

(designated as the DF2K1 series) combines patented optical sensors, a unique constellation of 

accelerometers and magnetic sensors, analog and digital electronics, and telemetry components. 

DFuzes can and have been built to conform to individual projectile form factor requirements. 

The DF2K1 DFuze maintains the interface characteristics of a standard North Atlantic Treaty 

Organization-compatible 155-mm artillery fuze. An example is shown in the upper half of 

Figure 11. Two of the four solar light-indicating transducers (SLITs) (U.S. Patent 5909275) 

used in the solarsonde system for measuring projectile heading and projectile crossing rate with 

respect to the sun can be seen in this view. As the schematic in the lower half of Figure 11 

shows, the radial axes of the projectile-fixed coordinate system are defined so that SLIT No. 1 is 

in the +XP|ZP plane. Locations within the DFuze body of all the components are specified, 

and laboratory calibrations are used to better measure installed locations and orientations 

of the sensing elements. The locations are then transformed to the projectile-fixed system by 

the addition of the components of the offset of the DFuze from the projectile c.g. For rigid 

projectiles with unchanging mass properties, this is a fixed offset. For flexible projectiles, 

projectiles with components in relative motion (e.g., de-spun nose), and projectiles with 

changing mass properties (e.g., through fuel consumption or cargo ejection), c.g. can change in 

flight, and location-sensitive, projectile-fixed sensors’ output will be affected. 

Besides the four solarsonde SLITs, a DF2K1 DFuze contains six accelerometers, a temperature 

sensor, and a magnetometer mounted on an ARL-designed board. This board is indexed to the 

DFuze housing so that the sensor locations are duplicated in each DFuze within manufacturing 

and installation tolerances. Figure 12 shows the forward side of the board as installed within the 

DFuze with the projectile-fixed axes and identifying letters superimposed on the picture. The 

component labeled A is a 3-axis magnetometer with its axes nominally oriented parallel to the 

projectile axes and with the same parity as the projectile axes. The output of some magnetometers (and other non-inertial sensors) is affected by the inertial forces to which the 

magnetometers are subjected while they measure the primary stimulus. Because of this, the 

location parameters as well as the orientation parameters of these sensors must be known in 

14
order to correctly interpret their output. In order to simplify notation, each magnetometer’s three 

axes are usually referred to as Mag_I (+XP direction), Mag_J (+YP direction), and Mag_K (+ZP

direction). The component labeled “B” is a single-axis accelerometer used to measure thrust and 

drag. This sensor’s axis is ideally situated on the projectile’s XP axis and oriented in the +XP

direction and is commonly called Acc_I. The axial component of this sensor’s location is 

its offset from the c.g. of the assembled projectile with fuze. Component “C” is a temperature 

sensor, which can be used to compensate for the variations in bias and/or scale factor with 

temperature, which are characteristic of many sensors. The remaining unlabeled components are 

various supporting electronics. 

(∆i)

Figure 11. DF2K1 DFuze configuration.

Figure 12. DF2K1 sensor board viewed 

from forward looking aft. 

15
On the aft side of the board (see Figure 13), there are five accelerometers, labeled D, E, F, G, and 

H. Accelerometer D is a bi-axial device whose sensitive axes are ideally situated on the 

projectile spin axis and oriented parallel to the YP and ZP axes. The axial location of 

accelerometer D is measured from the c.g. of the projectile with fuze assembly. These sensor 

axes are usually called Acc_J (-YP direction) and Acc_K (+ZP direction). The orientation of the 

Acc_J axis in the -YP direction is an artifact of accelerometer D being mounted on the aft side of 

the sensor board. A parity correction of the Acc_J output is made during data reduction to 

provide acceleration measurements in the projectile-fixed coordinate system. Accelerometers E, 

F, G, and H are single-axis devices whose output is combined on board the projectile to provide 

an estimate of the projectile spin rate. This collection of sensors is called the accelerometer spin 

ring or the accel_ring for short. In ballisticians’ notation, the equations for sensed acceleration at 

a point in the +YP and the +ZP directions are 

S v ru pw ( ) pq r L ( p r )L (qr p)L - g cos( )sin( ) P P YP ZP

2 2

Y = & + − + + & X + − − + − & θ φ (24) 

S w pv qu ( ) pr q L (qr p)L ( p q )L g cos( ) cos( ) P P P ZP

2 2

Z = & + − + − & X + + & Y + − − − θ φ (25) 

With the simplified notation of Equation (22), these equations become

Acc _ J v ru pw ( ) pq r i ( p r ) j (qr p) k - g cos( )sin( ) 2 2 = & + − + + & ∆ + − − ∆ + − & ∆ θ φ (26) 

Acc _ K w pv qu ( ) pr q i (qr p) j ( p q ) k g cos( ) cos( ) 2 2 = & + − + − & ∆ + + & ∆ + − − ∆ − θ φ (27) 

Figure 13. DF2K1 sensor board viewed from rear looking 

forward. 

16
Before any flight experiment, Equations (22), (24), and (25) or alternatively, (23), (26), and (27) 

need to be evaluated for anticipated projectile dynamics in order to determine required 

accelerometer characteristics such as dynamic range and frequency response. The natures of the 

accelerations to which the projectile-fixed sensors will be subjected and the natures of the 

acceleration components in an earth-fixed or navigation coordinate system can be very different. 

The sensor characteristics necessary to obtain projectile-fixed measurements of sufficient 

accuracies to meet required earth-fixed measurement accuracies are the design performance 

requirements. These requirements cannot be understood without an appreciation of the 

relationship between a sensor’s location and orientation and that sensor’s stimuli. 

7. Estimating Projectile Spin Rate 

Spin rate is an important characteristic of any projectile’s flight. This rate (p in ballisticians’ 

notation) is defined in the projectile-fixed system as the angular velocity of the YP- ZP axes 

about the XP axis. The DFuze does not include any sensor that directly measures spin rate. 

Because of size, high-g survivability, and cost requirements and the necessity for sometimes 

estimating spin rates of 300 Hz and more, traditional angular rate measurement devices such as 

gyroscopes have not been included in the DFuze. Perhaps there will someday be micro-electromechanical devices suitable for inclusion in future DFuzes, but at present, spin rate estimates are 

made from accel_ring, solarsonde, and magnetometer measurements. 

7.1 Accel_Ring Measurements 

The four accelerometers in the accel_ring (see Figure 13) are installed in opposed pairs about the 

projectile spin axis with the sensor axes of each pair co-linear with the diameter upon which they 

are situated and co-directional in orientation. That is, the sensitive axes of accelerometers E, F, 

G, and H are oriented in the +YP, +YP, +ZP, and +ZP directions, respectively. The differences 

between the sensed accelerations from two such pairs on the YP axis (F,E) and the ZP axis (H,G) 

are 

Acc _ K Acc _ K ( k k )( p q )

Acc _ J Acc _ J ( j j )( p r )

2 2

H G H G

2 2

F E F E

− = ∆ − ∆ − −

− = ∆ − ∆ − − (28) 

If j j k k R, then ∆ E = ∆ F = ∆ G = ∆ H =















 + − + − = +

2

r q Acc _ J Acc _ J Acc _ K Acc _ K 4R p

2 2

2

F E H G (29) 















 + = + − + −

2

r q

p 4R

Acc _ J Acc _ J Acc _ K Acc _ K 2 2

F E H G 2

 (30) 

17
This arithmetic is done on board the DFuze, and the telemetered measurement is the quantity 

given by Equation (30). When 

2

r q p

2 2

2 +

>> , as is often the case, a good estimate of the 

projectile spin rate is obtained from the accel_ring output by 















 +

≈ +

2

r q p p

2 2

2

. (31) 

7.2 Solarsonde Measurements 

The four SLIT optical sensors installed in the DF2K1 DFuze are designed to produce a 

significant output when aligned with the sun and almost no output when not aligned. On board a 

spinning projectile, these sensors provide a pulse train, which when combined with calibration 

data, yields a measurement of the angle between the spin axis and the vector to the sun (σs, 

called the solar aspect angle) and a solar roll history. Figure 14 shows a sample of telemetered 

solarsonde data from a DFuze mounted in a stationary spin fixture. The signals from all four 

SLITs are combined on board the projectile into a single output stream. The spikes on the graph 

of this combined waveform occur when the respective SLITs are aligned with the sun during a 

projectile rotation. The combination of gains and parities of the four SLITs are varied so that the 

output from each sensor can be identified (as is indicated by the numerals on the plot). Also 

shown are two simple period measurements that can provide a solar roll rate estimate. 

Figure 14. Sample of Solarsonde data 

18
If a projectile’s spin rate (p) or solar aspect angle (σS) changes within a roll cycle, error is 

introduced into the period-based solar roll rate estimate. In this instance, a more accurate roll 

rate estimate is obtained by the correlation of the solar aspect angle at each SLIT alignment with 

the roll angle at that alignment measured during laboratory calibration. Summation of the 

amount of roll required for each succeeding SLIT alignment yields an accumulated roll angle 

versus time history. First and second derivatives give estimates of solar roll rate and roll 

acceleration, respectively. These estimates are more accurate than those obtained from period 

measurements because of the greater frequency of roll orientation measurements. This 

methodology is typically employed in the processing of flight data. 

7.3 Magnetometer Measurements 

Figure 15 shows the data from one of the radial magnetometer axes (Mag_J) during the same

time period. Period measurements between signal extrema are used to estimate projectile roll 

rate with respect to the magnetic field. In contrast to the SLITs in the solarsonde system, which 

are designed to be nearly impulsive in their responses, vector magnetometers provide continuous 

response where the amplitude of the output signal varies directly with the projection of the local 

magnetic field onto the sensor axis. With knowledge of the local magnetic field and a sensor that 

is accurately located, aligned, and calibrated, we can directly derive projectile heading with 

respect to the magnetic field from an amplitude measurement. When, as is usually the case with 

actual implementations, any of these assumptions is violated, advanced processing techniques 

are employed to remove the effects of these errors. 

Figure 15. Sample of radial magnetometer data. 

19
For these solar and magnetic sensor systems, what is being measured is not the projectile spin 

rate but the rate at which the optical and magnetic sensors are swept past the sun and the earth’s 

magnetic field, respectively. The equations for these rates are the same as the equation for the 

derivative of the Eulerian roll angle (Equation [17]) with appropriate change of variables. With 

S and M subscripts to indicate solar and magnetic, respectively, these equations are, with the 

necessary changes, 

p [ qsin( ) r cos( ) ]tan( ) φS = + φS + φS θS & (32) 

φM = p + [ qsin(φM ) + r cos(φM ) ]tan(θM ) & (33) 

Since solar roll rate and magnetic roll rate are entirely analogous, further discussion is restricted 

to solar roll rate. With some manipulation, Equation (32) becomes 

[ ]

p [ ] cos( ) tan( )

cos( ) tan( ) cos( )

qsin( ) r cos( ) p

tan( ) cos( )

cos( ) p qsin( ) r cos( )

S S S

S S

S

S S

S

S

S

S S S

= + ψ θ θ

θ θ 



 





θ

φ + φ = +

θ 



 





θ

θ φ = + φ + φ

&

&

 (34) 

With the recognition that the term [ cos( ) S S ψ& θ ] is the component of projectile yawing rate 

perpendicular to the plane defined by the projectile spin axis and the solar vector, that 

θs = 90 - σs, and the realization that Equation (34) holds for any earth-fixed field, the relationship 

between projectile roll rate with respect to any such field and projectile spin rate can be stated 

thus. Roll rate is the sum of spin rate and the product of the out-of-plane yawing rate and the 

tangent of the complement of the angle between the spin axis and the field vector. Roll rate and 

spin rate are equal when there is no yawing motion (ψ& = 0)

0

 or the spin axis is perpendicular to 

the field vector (σ = 90,∴θ = 0) and unequal when ψ& ≠ and σ ≠ 90 . This relationship and the 

relationship between spin rate and accel_ring data have sometimes provided crucial insight into 

projectile dynamics not attainable from other measurements. 

8. Summary 

Body-fixed sensors combined with a telemetry system provide measurements of projectile inflight dynamics not achievable with stationary or remote sensing systems such as radar or 

cameras. Body-fixed sensor systems on board the projectiles provide continuous data throughout 

a flight, but many of the desired performance metrics are defined in an earth-fixed coordinate 

system. It is therefore crucial that the relationship between these two systems be understood for 

20
the proper design of projectile-fixed sensor systems and for the correct interpretation of sensor 

data obtained during flight experiments. 

The mathematics describing this relationship have been presented herein and examples have 

been provided for the sensor systems included within the DFuze measurement system, i.e., 

accelerometers, vector magnetometers, and solarsondes. Because these mathematics hold for 

any vector definable in these coordinate systems, this work is applicable to any sensor system

measuring vector quantities, e.g., angular rate sensors, velocimeters, gimbaled seekers, etc. 

21
9. Reference 

1. Murphy, C.H. Free Flight Motion of Symmetric Missiles; BRL-1216; Ballistics Research 

Laboratory: Aberdeen Proving Ground, MD, July 1963. 

22
Appendix A. Plane-Fixed Coordinate System 

The basic force and moment equations from which the free flight equations of motion of a rigid 

projectile are derived are simplified when formulated in a non-rolling coordinate system with the 

same X axis as the projectile-fixed system. The plane-fixed coordinate system is such a nonrolling system with the added stipulation that the plane-fixed Y axis is in the horizontal plane. 

Thus, the plane-fixed Z axis is in the vertical plane containing the spin (X) axis. Designating the 

axes with an FP subscript, the plane-fixed system is shown in Figure A-1. 

Figure A-1. Plane-fixed coordinate system. 

The plane-fixed system is used extensively in computer trajectory simulations since the 

differential equations of motion can be evaluated at larger integration time steps in this 

coordinate system than in the projectile-fixed system. The plane-fixed system already was seen 

in Section 3 of this report although it was not so named at that time. The system that 

resulted from the second Euler rotation is the plane-fixed system. The transformation matrix 

from earth- to plane-fixed vectors, given by substituting the Euler angles and evaluating the 

matrix multiplication in Equation (5), is 

2 2 Z2 X ,Y ,





















θ ψ θ ψ θ

− ψ ψ

θ ψ θ ψ − θ

=

sin( ) cos( ) sin( )sin( ) cos( )

sin( ) cos( ) 0

cos( ) cos( ) cos( )sin( ) sin( )

TFP

E (A-1) 

The transformation matrix from the plane-fixed to the projectile-fixed system has already been 

given in Equation (9). Substituting the Euler roll angle, this becomes 

23




















− φ φ

= φ φ

0 sin( ) cos( )

0 cos( ) sin( )

0 0 1

TP

FP (A-2) 

24
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ISSAQUAH WA 98027

 1 RAYTHEON 

ATTN RON KREBS 

BLDG M11 M/S 10 

PO BOX 11337

TUCSON AZ 85734-1337

ABERDEEN PROVING GROUND

 2 DIRECTOR 

US ARMY RSCH LABORATORY 

ATTN AMSRL CI OK (TECH LIB) 

BLDG 305 APG AA

 2 DIR USARL 

ATTN AMSRL WM T ROSENBERGER 

AMSRL WM B A HORST 

BLDG 4600 

26
NO. OF 

COPIES ORGANIZATION 

16 DIR USARL 

ATTN AMSRL WM BA D LYON

J CONDON B DAVIS 

T HARKINS (5) D HEPNER 

 G KATULKA V LEITZKE 

P MULLER P PEREGINO 

A THOMPSON T BROWN 

 M WILSON

BLDG 4600 

 2 DIR USARL 

ATTN AMSRL WM BC P PLOSTINS

 B GUIDOS 

 BLDG 390

 1 DIR USARL 

AMSRL WM BF S WILKERSON 

 BLDG 390

 2 DIR USARL 

AMSRL WM RP C SHOEMAKER 

 J BORNSTEIN 

 BLDG 1121 

 1 DIR USARL 

ATTN AMSRL WM MB C HOPPEL 

 BLDG 4600

 2 DIR USARL 

ATTN AMSRL WM T B BURNS 

 AMSRL WM TC R COATES 

 BLDG 309

2 CDR US ARMAMENT RD&E CTR 

ATTN AMSTA AR FST T R LIESKE 

 F MIRABELLE 

 BLDG 120

 1 CMDR USA DTC 

ATTN CSTE DTC TTM J SCHNELL 

 RYAN BLDG

 2 CDR ATC 

 ATTN K MCMULLEN 

 G MATRICCIANI 

 BLDG 359

27