ARMY RESEARCH LABORATORY

A Point-wise Solution for the

Magnetic Field Vector

Andrew A. Thompson

ARL-TR-2633 JANUARY 2002

Approved for public release; distribution is unlimited.
MATLAB® is a registered trademark of The MathWorks.

The findings in this report are not to be construed as an official Department of the Army position

unless so designated by other authorized documents.

Citation of manufacturer’s or trade names does not constitute an official endorsement or approval

of the use thereof.

Destroy this report when it is no longer needed. Do not return it to the originator.
i

Army Research Laboratory

Aberdeen Proving Ground, MD 21005-5066

_____________________________________________________________________

ARL-TR-2633 January 2002

_____________________________________________________________________

A Point-wise Solution for the Magnetic

Field Vector

Andrew A. Thompson

Weapons & Materials Research Directorate

_____________________________________________________________________

Approved for public release; distribution is unlimited.

_____________________________________________________________________
ii

_____________________________________________________________________

Abstract

_____________________________________________________________________

Magnetometers offer measurements that define the angle to the

magnetic field. Knowledge of this restricts the axis of interest to lie on

a cone centered on the magnetic field direction with an angle

proportional to the measurement of magnetic field strength. With this

restriction, formulae are derived for finding the relationship between

the magnetic field pointing vector, the spin axis, and the magnitude of

the magnetic field. Methods based on both calibrated and

uncalibrated magnetometers are discussed. These formulae are also

useful for the calibration of magnetometers.
iii

ACKNOWLEDGMENTS

The author would like to thank Bernard Guidos of the U.S. Army Research

Laboratory for his suggestions that improved this report.
iv

INTENTIONALLY LEFT BLANK
v

Contents

1. Introduction ...................................... 1

2. Background....................................... 1

3. Point-wise Solutions ................................ 2

4. Complete Solutions................................. 8

5. Solutions That Use Assumptions ....................... 13

5.1 Software....................................... 13

6. Conclusions....................................... 14

References ........................................... 15

Distribution List ........................................ 17

Report Documentation Page................................ 21

Figures

 1. Estimation Variance as a Function of Noise and Angle........... 4

 2. Sensitivity of Equation 7 ............................... 6

 3. Sensitivity of Equation 10 .............................. 7

 4. Error Magnitudes of Complete Solution ..................... 11

 5. Relationship Between Inner Product and Error ................ 12

 6. Relationship Between Determinant Measure and Error........... 12
vi

INTENTIONALLY LEFT BLANK
1

A POINT-WISE SOLUTION FOR THE MAGNETIC FIELD VECTOR

1. Introduction

Inertial navigation systems (INS) are being developed for projectiles and rockets.

These systems have higher spin rates than the traditional INS. In these

applications, the largest errors are associated with the orientation or attitude of

the body. Attitude estimation is vital to inertial navigation. Errors in attitude

result in accelerometers being misaligned with respect to the INS reference

frame. Within an INS, this misalignment will lead to accelerometer integration in

the wrong directions and in time, will lead to large errors in position. Angular

rate sensors can be integrated to estimate a body’s attitude. Errors attributable to

bias and random walk cause these estimates to drift away from the actual values.

Also, angular rate sensors are sensitive to vibration. A method to obtain an

independent measurement of an aspect of attitude could aid this process through

the formulation of a Kalman filter or allow the engineer to bypass angular rate

sensors and obtain a direct estimate of attitude. Knowledge of the attitude of a

body’s spin axis will improve the performance of an INS.

Magnetometers offer measurements that define the angle to the magnetic field.

Knowledge of this angle restricts the axis of interest to lie on a cone centered on

the magnetic field direction with an angle proportional to the measurement of

magnetic field strength. With this restriction, formulae are derived for finding

the relationship between the magnetic field pointing vector, the spin axis, and

the magnitude of the magnetic field. Methods based on calibrated and

uncalibrated magnetometers are discussed. These formulae are also useful for

the calibration of magnetometers.

2. Background

Conley and Patton [1] discuss a method to find the spin axis of a sounding rocket

via a solar sensor and one magnetometer. Their method uses iterative nonlinear

least squares, Euler angles, and inner products to find the orientation of the spin

axis. The inner product of two normalized vectors is the cosine of the angle

between them. This method requires at least one spin cycle of data to implement.

Harkins and Hepner [2] discuss a method to find the orientation of the spin axis

by the use of solar and magnetometer information. We can find the angle

between the spin axis and the magnetic field by using the zero crossings of two

magnetometers. This method requires at least one spin cycle to implement.
2

Harkins and Hepner named their magnetometer configuration a MAGSONDE

(MAGnetic SONDE). A MAGSONDE is a device that uses two magnetometers to

obtain estimates of the angle to the magnetic field. One magnetometer is in a

plane that is orthogonal to the spin axis; the other is in the plane formed by the

spin axis and the magnetometer orthogonally to the spin axis.

The magnetic coordinate system is formed by the spin axis of the projectile, a

vector orthogonal to it in the same plane as the spin axis and the vector defining

the direction to the magnetic field, and a third vector orthogonal to the previous

two. If the spin axis and the magnetic field vectors are known, then the

coordinate system can be defined via cross products. The first dimension is the

spin axis; the second is the cross product of the third dimension and the spin

axis; the third is the normalized cross product of the spin axis and magnetic field.

A magnetometer in this frame aligned orthogonally to the spin axis will read

zero when it is aligned with the third dimension and will give a maximum

reading when aligned with the second dimension.

Since the magnetic field direction is assumed to be known, the accuracy to which

the spin axis can be estimated will determine system performance. Finding the

angle to the magnetic field gives one restriction on the spin axis; this angle

defines a cone about the magnetic field vector upon which the spin axis must lie.

The use of this with a second constraint will define the spin axis. For a projectile,

the magnetic coordinate system will change during the flight because of the

change in the orientation of the spin axis. The magnetic roll angle is the rotation

about the spin axis, measured so that when the magnetometer output is zero and

the output is increasing, the measure of the angle is zero.

3. Point-wise Solutions

Formulae are developed to find the angle between the magnetic field and the

spin axis at any point of time. The accuracy of these equations can be assessed

through simulations. Software was developed to implement the simulations and

to find the theoretical performance of the formulae during noise-induced

perturbations. These performance levels yield an ideal performance level that can

only be approached in practice.

The idea is to form a ratio of the signals and eliminate the magnitude factor or to

form a linear combination that will allow extraction of desired signal

information. The background information for this report is contained in Harkins

and Hepner [2], who use zero crossings of the magnetometers to find the

direction to the magnetic field.
3

Sensor 1 is aligned orthogonally to the spin axis. The measurement from Sensor 1

is

s1 m s M = M sin sin , (1)

in which m is the angle between the spin axis and the magnetic field, s is the

roll angle (about the spin axis), and M is the magnitude of the magnetic field. If

Sensor 2 is at an angle of to the spin axis in the same plane as Sensor 1, then the

measurement from Sensor 2 is

s2 m m s M =cos M cos +sin M sin sin . (2)

If the magnitude of the first term is greater than the largest magnitude of the

second term, this signal will not cross zero. If angles are chosen randomly, then

50% of the time, Measurement 2 will not contain zero crossings. For each of these

measurements, if the magnitude of the magnetic field is known, then each

measurement can be thought of as the inner product of the normalized

magnetometer axis and the magnetic field direction. Forming the ratio of these

two measurements makes the solution amplitude independent. After

simplification, the following result is obtained. = +sin

sin

cos cot( )

M

M

s ms1

s2 (3)

The two measurements and the angle of Sensor 2 ( ) are known, so the known

quantities are moved to the right-hand side. = −sin

M

M

cos

1

sin

cot( )

s1

s2

sm (4)

The sine of the magnetic roll angle can be found by reference to the zero

crossings of Sensor 1. It will be the time since the last zero crossing divided by

the time between zero crossings multiplied by 2 . This isolates the cotangent

function; thus, we can find the angle to the magnetic field by using the

arctangent function. For obvious reasons, this solution will not be stable as the

value of Sensor 1 approaches zero. Away from that region, it should be possible

to calculate the angle to the magnetic pointing vector with Equation (4). In the

region where Sensor 1 is near zero, the inverse ratio could be used to find the

angle to the magnetic field. −

= M M sin

M

sin

cos

tan

s2 s1

s1

s m

(5)

However, the factor involving the sine of the magnetic roll angle will not permit

a solution as it approaches zero as the output from Sensor 1 does. An error
4

analysis of Equation (4) will provide regions of stable solutions for certain noise

levels.

Software was designed to assess the sensitivity of the solution. At the indicated

roll angles, the variance of the solution when each signal had Gaussian noise

with standard deviations of 0.01 to 0.07 radian is depicted in Figure 1.

Figure 1. Estimation Variance as a Function of Noise and Angle.

The solution starts with a roll angle of 180 degrees and ends with a roll angle of

172 degrees. The solution is symmetrical about 90 degrees. As expected, the

solution should be most stable for roll angles near 90 degrees (about 1.5 radians).

At a noise level of 0.01 radian (0.57 degree), the solution’s standard deviation for

roll angles of 3 radians (172 degrees) is on the order of 5 degrees. This decreases

to about 1.5 degrees when the roll angle is near 90 degrees.

A linear model will not fit well for magnetic roll angles within 0.5 radian

(29 degrees) of 0 and 180 degrees. For roll angles not in this range, when the

noise level is increased, a linear relationship between the standard deviation of
5

the input noise in radians and the variance of the solution is observed for the

noise levels of 0.01 to 0.07 radian. Figure 1 clearly shows that the curvature

increases with noise level and proximity to 180- or 0-degree roll angles.

Another method to find the pointing angle to the magnetic field can be devised

by the use of the measurement from Sensor 2. When Sensor 1 has zero output,

recall that Equation (4) is not appropriate for this situation. This is at a roll angle

of 0 or 180 degrees in the magnetic coordinate system. Using Equation (2), we

can find

(6)

In this situation, the strength of the magnetic field needs to be known. When

Sensor 1 measures zero, the magnitude of Sensor 2 defines the direction to the

magnetic field. For example, if the output from sensor 2 were 0, this would

indicate that the vector to the magnetic field is 90 degrees from the spin axis. The

value of Sensor 2 indicates how far from orthogonal the magnetic field vector is

pointing. Note that this formula would allow the calculation of field strength if

the direction to the magnetic field were known or could be estimated; thus, it

could be used for calibration.

The information available if Sensor 2 is zero can also be used to find the direction

to the magnetic field. The assumption that the output is zero allows Equation (2)

to be solved for the angle to the magnetic field. −

=

sin( )

cot( ) a tan

s M

. (7)

This solution requires the roll angle in the magnetic coordinate system to be

found. Note: The solution is determined by the magnetic roll angle and this is

equivalent to a timing parameter. Based on the assumption that the roll rate is

constant, the amount of time between zeros corresponds to a portion of the

rotational cycle. For Sensor 1, the time between zeros corresponds to 180 degrees.

If the Sensor 2 measurement is greater than the Sensor 1 measurement, then the

zero crossings will occur between 180 and 360 degrees. The ratio of the zero

crossings will give the portion of 180 degrees where Sensor 2 was less than zero.

Subtracting this ratio from 1 will give the portion of 180 degrees before the first

zero and after the second. To find the magnetic roll angle at the first zero of

Sensor 2, one can use the following equation.

(1 ratio)/ 2 s = + − . (8)

=

cos( )

cos 2

M

M

a

s m
6

This formula can be used to find the angle to the magnetic field, based on the

ratio of each sensor’s time durations between zeros. The magnetic roll angle at

the first zero of the cycle can be found, based on the ratio and then inserted into

the formula to obtain the angle to the magnetic field. This result in the

Equation (7) yields the angle to the magnetic field.

Figure 2 shows the performance of this method of solution for 10,000

replications. Note that in this diagram, both the noise level and the standard

deviation are in radians; the conversion factor to degrees is 57.296. Since the

value of the ratio is equivalent to knowledge of the magnetic roll angle, the

performance of Equation (7) depends on both the accuracy of the measurement

of the roll angle and the actual value of the magnetic roll angle. Harkins and

Hepner [2] discuss another method of using zero crossings to solve for the spin

axis angle to the magnetic field.

Figure 2. Sensitivity of Equation 7.

A variation of Equation (7) can be found by the use of the output of Sensor 1 as

input to Equation (2) when the reading is zero. Solving for the desired angle

yields

[ ] −= tan

M

M

a cos s1 m

. (9)
7

Use of this requires the knowledge of the perceived magnetic field strength.

The separation between the measurements and the magnetic roll angle contains

the necessary information to determine the angle between the magnetic field and

the spin axis. A modification of the difference between the two measurements

can be used. One can find the measure of the angle between the spin axis and

magnetic field by multiplying the value of Sensor 1 by the sine of the angle of

sensor offset and then subtracting this value from the output of Sensor 2. The

result is −

=

cos( ) M

M sin( )M

a cos s2 s1 m

. (10)

This result does not require knowledge of the magnetic roll angle; however, the

strength of the magnetic field needs to be ascertained. Note that Equation (10) is

the same as Equation (9) if Measurement 2 is 0. The fidelity of the formula will be

determined by the accuracy of the measurements and the accuracy in the

estimate of magnetic field strength. As the argument of the arc cosine function

approaches 1, errors in the measurement and magnitude terms can drive the

argument to be greater than 1. With an input noise of 0.57 degree for both

measurements and the magnitude, the standard deviation of the solution was on

the order of 1 degree for magnetic vectors between 50 and 130 degrees. Figure 3

shows the relationship based on a simulation; 10,000 replications were run for

each magnetic field angle.

Figure 3. Sensitivity of Equation 10.
8

As with the first function for magnetic pointing angle, the errors are greatest at 0

and 180 degrees where the sine of the magnetic roll angle is zero. This simply

indicates that as the spin axis and the direction to the magnetic field become

collinear, Equation 10 loses its fidelity or usefulness.

4. Complete Solutions

In this section, a method to find the orientation of the spin axis is discussed. The

attitude of a vector can be found, given the angles between the vector and two

known directions. Of interest is the situation when the vector to be found is the

spin axis and the known directions are the magnetic field and the direction to the

sun or the solar vector. Each known direction and angle will yield a cone of

possibilities in three-dimensional space; the intersection of the two cones will

yield two possible directions or points. If a magnetometer is being used, it is

possible to select one point, based on the time history of the magnetometer

output. For one point, the output should be increasing; for the other, the output

should be decreasing.

An Euler sequence is used to represent the attitude of one coordinate system

(e.g., a body-fixed system) in terms of another (such as the INS reference system).

The Euler aerospace sequence is one of the 12 possible Euler sequences for

relating coordinate systems. The first rotation is about the Z-axis ( ), followed by

a rotation about the new Y-axis ( ), and the final rotation takes place about the

newest X-axis ( ). When the aerospace sequence is used, the final X-axis is

typically the spin axis of the body. For spinning projectiles, the angular rate

about the spin axis is typically constant over a spin cycle. The validity of this

assumption is critical to the method discussed. In addition, it is assumed that the

magnetic field vector is known. Conley and Patton [1] use these two assumptions

to develop their attitude estimator. The magnetometer output is proportional to

the projection of the magnetic field onto the magnetometer. By describing the

magnetometer in terms of Euler angles, we can develop a set of equations by

considering the output to be the inner product of the magnetic field vector with

the magnetometer orientation vector. With these equations, a method can be

developed to find the desired Euler angles. Although Conley and Patton used the

satellite ephemeris Euler sequence, their method can be recast via the aerospace

sequence.

Next, assume that a magnetometer is placed orthogonally to the spin axis. Let the

body-fixed coordinate system be defined so that the spin axis is the X-axis and

the magnetometer is along the Y-axis. Starting with a reference coordinate

system, the transition matrix to the body-fixed coordinate system (based on the

aerospace sequence) is where the periods indicate the end of the term.
9

+ − +

− +

−

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

cos sin cos . sin sin

cos sin . sin sin sin . cos cos

sin cos . cos sin sin

cos cos . sin cos . sin .

For a rotation matrix, the inverse and transposed matrices are equivalent. The Xaxis in body-fixed coordinates can be represented by the vector (1,0,0).

Transforming this to the reference coordinate system yields the first row as the

vector result. The spin axis can be represented in terms of the Euler angles. The

inner product of the spin axis (from the first row) and the direction of the

magnetic field will yield an equation containing Euler angles that could be

useful. The magnetometer is along the Y-axis, which corresponds to the second

row of the matrix. The inner product of the Y-axis with the normalized direction

to the magnetic field will equal the cosine of the angle between the two. From

this matrix, the spin axis and two orthogonal vectors perpendicular to the spin

axis can be found in terms of Euler angles.

Returning to Equation (1), we find that the cosine of the angle between the

magnetometer and the magnetic field is equivalent to multiplying the sine of the

angle between the magnetic field and the spin axis by the sine of the magnetic

roll angle. By dividing the magnetometer output by the maximum possible

output (calibrated strength of the field), we can find these cosines. By taking

many measurements over a spin cycle, we can develop a system of equations

that contains more equations than unknowns. The next step in the process

requires knowledge of the roll angle about the spin axis. Any of the possible spin

axes could produce the measurement sequence. Note that the zero value of the

magnetic roll angle is unique for each candidate spin axis.

If the magnetometer value is known and associated with a magnetic roll angle, a

given spin axis will be defined. Conley and Patton [1] suggest using a

configuration that allows a measurement induced by the sun’s energy. Using the

solar pointing angle, they show that it is possible to solve for the Euler angle

about the spin axis in terms of the other two Euler rotations. This measurement

fixes the ray on which the spin axis must lie. The magnetometer measurements

then allow the spin axis to be adjusted along the ray defined by the first

measurement. This is a recursive process in which subsequent adjustments are

almost orthogonal (one along a radius and the other along the circumference).

Conley and Patton do not discuss the issue associated with errors in the

orientation of the magnetometer leading to errors associated with the spin axis

and vice versa. With their procedure, knowledge of one is assumed to find the

other in a recursive algorithm. They demonstrated that this procedure converged

for the problem they investigated.
10

A closed form solution can be devised, given the existence of two known

directions (solar and magnetic fields) and the projection of a third desired

direction (spin axis) onto the two known directions. The problem described in

Conley and Patton fits this description. A verbal description of the

implementation of a closed form solution follows. First, assume that the

directions are normalized so that each is represented by a vector to the unit

sphere. For each measurement, the desired direction will be on a cone

determined by the known direction and the projection (or measurement). The

circle formed by the cone and the unit sphere is in a plane that is normal to the

known direction and contains the point along the known direction that is in

proportion to the projection or inner product. It is possible to describe this plane

with an equation that uses the known direction and interprets the measurement

as an inner product.

A second plane, based on the other direction, can also be formulated as an

equation. The intersection of these two planes forms a line, and finally, the

intersection of this line with the unit sphere yields two candidate directions that

fulfill the conditions. One candidate solution will be associated with increasing

measurements and one with decreasing measurements. The solution based on

this requires multiplication, addition, and only one square root operation. Note

that if the two known directions are collinear, the planes will be parallel and no

solution will exist. As the known directions approach collinearity, the solution

becomes unstable.

A simulation to observe the behavior of the solution was devised. Randomly

selected directions are chosen for the two known directions, and a third

randomly chosen direction was chosen as the axis of interest. Noise was added to

the projection of the axis of interest on each of the two known axes, and this

result was sent to a routine that is based on the previously described solution. An

error in the projection value affects the equation of the plane by moving it up or

down along the direction to which it is orthogonal. For planes with almost

collinear normals, a small change in the separation between the planes can have

a large influence on their intersection. The difference between the solution and

the desired direction in degrees is reported as the system error. Normally,

distributed input errors with standard deviations of 0.005 radian or 0.2865 degree

were used. System errors are displayed in Figure 4; the percentage of

observations below a system error value is displayed.

This type of plot is referred to as a quantile plot. A quantile plot is the graph of

the percentage of the data that is less than a given value; the median will occur at

the abscissa value of 50. It is apparent that in 90% of the cases, the error is less

than 2.5 degrees. In addition to the system error (the dependent variable), the

inner product of the known axes and the determinant of all three vectors were

reported as independent variables. Also, an indicator variable was used to

indicate that the real part of a complex solution was used. Complex solutions

occur as the closest point of approach of an external line to the unit sphere. If two
11

vectors are nearly collinear, their inner product will be close to 1. The

determinant of three vectors indicates the volume they form; if this volume is

close to 0, the vectors are nearly collinear. When the norms of each vector are 1,

the volume can never exceed 1.

Figure 4. Error Magnitude of Complete Solution.

To make the two measures of collinearity commensurate, the determinant was

subtracted from 1 so that values of 1 would represent collinear input sets, and

values close to 1 would represent directions that were nearly collinear. Figure 5

shows the error plotted against the inner product with complex solutions in

green. The error increases as the inner product approaches 1; also, the complex

solutions are distributed throughout the distribution. Figure 6 shows the

determinant measure with complex solutions in green. As 1-d increases, the

maximum error slowly increases until the independent variable passes 0.8, after

which, the increase accelerates. The determinant-based measure is seen to be a

better indicator of the overall system error than the inner product; however, in a

typical situation, only two axes are known. The determinant measure shows that

all complex solutions occur when the collinearity of the three directions is high.

The linear lower boundary for the complex solutions can be understood as the

error level needed to force the solution to become complex for the collinearity

indicator.
12

Figure 5. Relationship Between Inner Product and Error.

Figure 6. Relationship Between Determinant Measure and Error.
13

A model of system error could be made with the degree of collinearity as the

predictive variable. If the system becomes collinear, then the system error would

be infinite. From Figure 6, it can be seen that the reciprocal of the determinant (d,

not [1-d]) could be used to predict an upper boundary on system error. The

numerator could be adjusted to fit different levels of input noise. A model of this

type would allow the system engineer or experiment coordinator to predict

system performance for known geometries. Desirable geometries occur when the

determinant is close to 1. Geometries where the determinant is close to 0 should

be avoided, since large system errors will dominate system performance.

5. Solutions That Use Assumptions

In some situations, it is possible to use domain knowledge about the situation to

find the attitude of the system. Specific knowledge about system behavior can be

used to develop algorithms. For example, if the change in the spin axis is known

to have a restriction, this knowledge can be used to orient the system. For an

artillery projectile, we know that the change in the spin axis takes place mainly in

the plane defined by the spin axis and gravity. This could be used in conjunction

with magnetic field information to orient the system. If the magnetic field vector

is parallel to the surface of the earth, then when the magnetometer reaches its

maximum peak-to-peak reading, the spin axis is also parallel to the surface of the

earth. This would yield two candidate systems, one of which could be

eliminated. If the magnetometer is properly calibrated, the angle to the magnetic

field and the attitude of the spin axis would also be known. Assuming that

changes in the spin axis occur in the plane formed by the spin axis and gravity

will enable computations based on the differential peak-to-peak voltages to

orient the spin axis.

For an individual magnetometer spinning about an axis, the maximum and

minimum readings will occur when the device is in the plane of the magnetic

field and the spin axis (equal readings indicate that the spin axis is aligned with

the magnetic field). These readings, coupled with knowledge of the magnetic

field and an estimate of the spin axis, can be used to orient the system.

5.1 Software

The software used to evaluate formula performance is available as a MATLAB

toolbox. Eighteen routines were used in the analysis and solution of formulae

presented in this report. This package can be used to evaluate other questions

related to MAGSONDE data processing.
14

6. Conclusions

The methods discussed in this report can be used to increase the uses of

magnetometers and to calibrate magnetometers. As aids to attitude estimation,

magnetometers offer a way to determine a restriction of the body-fixed

coordinate system. Knowledge of the cone on which the spin axis must lie,

coupled with a second restriction, can lead to a direct estimate of a system’s

attitude.

Error analyses that were completed for the individual procedures indicate the

quality of the estimate for various noise levels and parts of the magnetic roll

cycle. More comprehensive models could be developed for the noise level/roll

angle parameter space to provide the designer with a tool that indicates expected

performance of a magnetometer processing method. The software developed for

this study can be used to investigate individual cases.

These new formulae could be used in conjunction with angular rate sensors to

improve their performance. Providing an independent observation of attitude to

a Kalman filter would make it possible to estimate bias and scale factor errors for

angular rate sensors.

The ability to estimate the angle between the spin axis and the magnetic field at

any point in time will lead to real-time estimation of attitude. This increases the

possibility of on-board navigation for spinning bodies.
15

References

1. Conley, J.M., and R.B. Patton, ”An Attitude Determining System For

Spinning Rockets,” BRL R-1291, U.S. Army Ballisitic Research Laboratory,

Aberdeen Proving Ground, MD, December 1965.

2. Harkins, T.E., and D.J. Hepner, ”MAGSONDE: A Device for Making

Angular Measurements on Spinning Projectiles With Magnetic Sensors,”

ARL-TR-2310, U.S. Army Research Laboratory, Aberdeen Proving Ground,

MD, December 2000.
16

INTENTIONALLY LEFT BLANK
17

NO. OF

COPIES ORGANIZATION

1 ADMINISTRATOR

DEFENSE TECHNICAL INFO CTR

ATTN DTIC OCA

8725 JOHN J KINGMAN RD STE 0944

FT BELVOIR VA 22060-6218

1 DIRECTOR

US ARMY RSCH LABORATORY

ATTN AMSRL CI AI R REC MGMT

2800 POWDER MILL RD

ADELPHI MD 20783-1197

1 DIRECTOR

US ARMY RSCH LABORATORY

ATTN AMSRL CI LL TECH LIB

2800 POWDER MILL RD

ADELPHI MD 20783-1197

1 DIRECTOR

US ARMY RSCH LABORATORY

ATTN AMSRL D D SMITH

2800 POWDER MILL RD

ADELPHI MD 20783-1197

4 DIR ARL

ATTN A LADAS B PIEKARSKI

J PRICE J EICKE

2800 POWDER MILL RD

ADELPHI MD 20783-1197

1 TACOM/ARDEC

ATTN AMSTA AR CCF A

W KONICK

2800 POWDER MILL RD

ADELPHI MD 20783-1197

1 CDR WL/MNMF

ATTN R MABRY

101 W EGLIN BLVD STE 219

EGLIN AFB FL 32542

2 CDR WL/MNAV

ATTN G WINCHENBACH

G ABATE

EGLIN AFB FL 32542

3 CDR PM SADARM

ATTN SSAE GCSS SD A BAHIA

W VOGT D CARLUCCI

PICATINNY ARSENAL NJ

07806-5000

NO. OF

COPIES ORGANIZATION

8 CDR US ARMY

ARMAMENT RD&E CTR

ATTN AMSTA AR AET A S CHUNG

 M AMORUSO

AMSTA AR FSP S PEARCY

AMSTA AR FSP I D COLLETT

AMSTA FSP A V ILARDI

 N GRAY R SICIGNANO

AMSTA AR AEC T S LONGO

PICATINNY ARSENAL NJ

07806-5000

2 ALLIANT TECHSYSTEMS

ATTN C CANDLAND R DOHRN

600 SECOND ST NE

HOPKINS MN 55343-8384

1 LOCKHEED-MARTIN - SANDERS

ATTN MARK CARLSON

NCA1-2078 95 CANAL ST

NASHUA NH 03061

2 LOCKHEED-MARTIN - SANDERS

ATTN P ZEMANY J DEMPSEY

PO BOX 868

NASHUA NH 03061-0868

8 JHU APL

ATTN R BENSON M BOEHME

H CHARLES B D’AMICO

R DENNISON W DEVEREUX

A DRIESMAN D WICKENDEN

1110 JOHNS HOPKINS RD

LAUREL MD 20723-6099

1 CDR NSWC

ATTN CODE 40D J BLANKENSHIP

6703 WEST HIGHWAY 98

PANAMA CITY FL 32407

1 BULL SYSTEMS ASSOC

ATTN R BODENSCHATZ

96 HIGH ST

WARRENTON VA 20186-2901

1 CDR NAWC/WPNS

ATTN CODE 543200 G BORGEN

BLDG 311

PT MUGU CA 93042-5001
18

NO. OF

COPIES ORGANIZATION

1 NOESS INC

ATTN A BOUTZ

4200 WILSON BLVD STE 900

ARLINGTON VA 22203

1 BOEING NORTH AMERICAN

AUTONETICS ELECTR SYS DIV

ATTN G BRAND

3370 MIRALOMA AVE

ANAHEIM CA 92803-3105

1 AUBURN UNIVERSITY

ATTN R BARRETT

211 AEROSPACE ENG BLDG

AUBURN UNIV AL 36849-5338

7 CDR NSWC/IHDIV

ATTN CODE 4120 V CARLSON

CODE 5710 E EAGLES

CODE 4110C L FAN

CODE 40D D GARVICK

CODE 450D T GRIFFIN

CODE 440C4 E LITCHER

CODE 57 C PARAS

101 STRAUSS AVE

INDIAN HEAD MD 20640-5000

1 NATIONS INC

ATTN R CARPENTER

12249 SCIENCE DR STE 150

ORLANDO FL 32826

3 STRICOM PM ITTS

ATTN R LONGENBACH

 D SCHNEIDER

AMCPM ITTS I

 R COLANGELO

12350 RESEARCH PKWY

ORLANDO FL 32826-3276

3 CHLS STARK DRAPER LAB INC

ATTN J CONNELLY J ELWELL

J SITOMER

555 TECHNOLOGY SQ

CAMBRIDGE MA 02139-3563

5 NAWC WEAPONS DIV

ATTN D CRABTREE

CODE C3923 S GATTIS

CODE C3904 D SCOFIELD

S MEYER L ROLLINGSON

CHINA LAKE CA 93555-6100

NO. OF

COPIES ORGANIZATION

1 CDR MICOM

ATTN AMSMI RD ST GD D DAVIS

REDSTONE ARSENAL AL 35898

1 UNIV OF CA AT BERKELEY

ATTN G DELORY

SPACE SCIENCE LAB MS 7450

BERKELEY CA 94720-1740

1 SAIC

ATTN J DISHON III

16701 W BERNARDO DR

SAN DIEGO CA 92127

 2 ONR

ATTN J GOLDWASSER

P MORRISON

800 N QUINCY ST RM 507

ARLINGTON VA 22217-5660

2 DIR US ARMY RTTC

ATTN STERT TE F TD R EPPS

S HAATAJA

REDSTONE ARSENAL AL 35898

1 PATUXENT RIVER NAS

ATTN R FAULSTITCH

UNIT 1 47765 RANCH RD

PAX RIVER NAS MD 20670-1469

4 NASA/GSFC

ATTN B FLOWERS E RANSONE

N SCHULTZ A TORRES

SOUNDING ROCKET PROGRAM

WALLOPS ISLAND VA 23337

1 ALLIANT DEFENSE

ATTN A GAUZENS

PO BOX 4648

CLEARWATER FL 33758-4648

1 SAIC

ATTN J GLISH

3800 W 80TH ST STE 1090

BLOOMINGTON MN 55431

1 THE AEROSPACE CORP

ATTN H HELVAJIAN

2350 E EL SEGUNDO BLVD M5/753

EL SEGUNDO CA 90245-4691
19

NO. OF

COPIES ORGANIZATION

6 NSWC/DD

ATTN G33 FRAYSSE

G32 ELLIS

G61 LARACH

G34 WENDT A EVANS

 D HAGAN

17320 DAHLGREN RD

DAHLGREN VA 22448-5100

4 ARROW TECH ASSOC

ATTN W HATHAWAY M STEINHOFF

J WHYTE R WHYTE

PO BOX 4218

SOUTH BURLINGTON VT 05403

1 TI

ATTN T HICKS

8816 CROSSWIND DR

FORT WORTH TX 76719

2 CDR US ARMY YPG

ATTN STEYP TD ATO A HOOPER

 C RAMSDELL

YPG AZ 85365-9106

2 ROCKWELL-COLLINS

ATTN M JOHNSON R MINOR

350 COLLINS RD NE

CEDAR RAPIDS IA 52498

1 UCLA

ATTN JACK JUDY

68-121 ENGINEERING IV

BOX 951594

LOS ANGELES CA 90095-1594

1 IMI INC

ATTN T JUNEAU

2140 SHATTUCK AVE STE 205

BERKELEY CA 94703

1 SWALES AEROSPACE

ATTN QUANG LAM

5050 POWDER MILL RD

BELTSVILLE MD 20705

1 LUTRONIX CORP

ATTN GARY LEE

13627 PORTOFINO DR

DEL MAR CA 92014

NO. OF

COPIES ORGANIZATION

1 SUNY

DEPT OF MATHMATICS &

 COMPUTER SCIENCE

ATTN R MARCHAND

FREDONIA NY 14063

1 ARDEC

ATTN M MATTICE

FIRE CONTROL LDIV

PICATINNY ARSENAL NJ

 07806-5000

1 GEORGIA TECH RSCH INST

ATTN J MCMICHAEL

AEROSPACE TRANSPORTATION

 & ADVANCED SYSTEMS LAB

7220 RICHARDSON RD

SMYRNA GA 30080

1 USAF

ATTN JOHN MERTS

303 N SEVENTH ST STE 109

EGLIN AFG FL 32542

1 AEROVIRONMENT INC

ATTN C MIRALLES

4685-3H INDUSTRIAL ST

SIMI VALLEY CA 93063

1 SAIC

ATTN M PALMER

1410 SPRING HILL RD STE 400

MCLEAN VA 22102

1 GEORGIA TECH

ATTN D PAREKH

7220 RICHARDSON RD

SMYRNA GA 30080

1 UNIV OF CA AT BERKELEY

ATTN AL PIASANO

DEPT OF MECH & ELEC ENG &

 COMPUTER SCIENCES

5101-B ETCHEVERRY HALL

BERKELEY CA 94720-1740

1 OPTEC

ATTN R REDMOND

4501 FORD AVE PARK CTR IV

ALEXANDRIA VA 22302-1435
20

NO. OF

COPIES ORGANIZATION

1 MRDEC

ATTN P RUFFIN

REDSTONE ARSENAL AL 35898-5000

1 IDA

ATTN R SINGER

1801 N BEAUREGARD ST

ALEXANDRIA VA 22311-1772

1 ANALOG DEVICES

ATTN R SULOUFF

21 OSBORNE ST

CAMBRIDGE MA 02139

1 US ARMY CECOM

ATTN J VIG

FORT MONMOUTH NJ 07703-5203

1 TECOM-RTTC

ATTN K WHIGHAM

BLDG 7855

REDSTONE ARSENAL AL

35898-8052

1 MARK CLYMER

BOX 325

MYSTIC CT 06355

ABERDEEN PROVING GROUND

2 DIRECTOR

US ARMY RSCH LABORATORY

ATTN AMSRL CI LP (TECH LIB)

BLDG 305 APG AA

1 CDR ATC

ATTN A ELLIS

BLDG 400

1 CDR ATC

ATTN K MCMULLEN

BLDG 359

8 DIRECTOR

US ARMY RSCH LABORATORY

ATTN AMSRL WM BA T BROWN

B DAVIS J CONDON

T HARKINS D HEPNER

M HOLLIS V LEITZKE

D LYONS A THOMPSON (4)

BLDG 4600

NO. OF

COPIES ORGANIZATION

2 DIRECTOR

US ARMY RSCH LABORATORY

ATTN AMSRL WT TD F GREGORY

N GNIAZDOWSKI

BLDG 309

DIRECTOR

US ARMY RSCH LABORATORY

ATTN AMSRL WT TB R LOTTERO

BLDG 309

2 DIRECTOR

US ARMY RSCH LABORATORY

ATTN AMSRL WM BC B GUIDOS

D WEBB

BLDG 390

2 DIRECTOR

US ARMY RSCH LABORATORY

ATTN AMSRL SL BD J COLLINS

L MOSS

BLDG 1068

2 DIRECTOR

US ARMY RSCH LABORATORY

ATTN AMSRL IS CD B BODT

P KASTE

BLDG 321

1 DIR ECBC

ATTN AMSS RRT PT D WEBER

BLDG E3516

5183 BLACKHAWK RD

APG-EA
REPORT DOCUMENTATION PAGE

1. AGENCY USE ONLY (Leave blank)

4. TITLE AND SUBTITLE

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

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

11. SUPPLEMENTARY NOTES

12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

14. SUBJECT TERMS

13. ABSTRACT (Maximum 200 words)

5. FUNDING NUMBERS

20. LIMITATION OF ABSTRACT

NSN 7540-01-280-5500

Form Approved

OMB No. 0704-0188

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, 

gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this 

collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson 

Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503.

2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

6. AUTHOR(S)

8. PERFORMING ORGANIZATION

 REPORT NUMBER

10. SPONSORING/MONITORING 

 AGENCY REPORT NUMBER

15. NUMBER OF PAGES

16. PRICE CODE

17. SECURITY CLASSIFICATION

 OF REPORT

18. SECURITY CLASSIFICATION

 OF THIS PAGE

19. SECURITY CLASSIFICATION

 OF ABSTRACT

Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. Z39-18

298-102

ARL-TR-2633

January 2002

Unclassified Unclassified Unclassified

Final

U.S. Army Research Laboratory

Weapons & Materials Research Directorate 

Aberdeen Proving Ground, MD 21005-5066

Thompson, A.A. (ARL)

 

U.S. Army Research Laboratory

Weapons & Materials Research Directorate 

Aberdeen Proving Ground, MD 21005-5066

21

25

PR: AH80 A Point-wise Solution for the Magnetic Field Vector

Approved for public release; distribution is unlimited.

Magnetometers offer measurements that define the angle to the magnetic field. Knowledge of this restricts the axis of interest to

lie on a cone centered on the magnetic field direction with an angle proportional to the measurement of magnetic field strength.

With this restriction, formulae are derived for finding the relationship between the magnetic field pointing vector, the spin axis,

and the magnitude of the magnetic field. Methods based on both calibrated and uncalibrated magnetometers are discussed. These

formulae are also useful for the calibration of magnetometers.

magnetometer

navigation