ARMY RESEARCH LABORATORY

A Point-wise Solution for theMagnetic Field Vector

Andrew A. Thompson

ARL-TR-2633 JANUARY 2002

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Army Research LaboratoryAberdeen Proving Ground, MD 21005-5066_____________________________________________________________________

ARL-TR-2633 January 2002_____________________________________________________________________

A Point-wise Solution for the MagneticField Vector

Andrew A. ThompsonWeapons & Materials Research Directorate

_____________________________________________________________________

Approved for public release; distribution is unlimited._____________________________________________________________________

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_____________________________________________________________________

Abstract_____________________________________________________________________

Magnetometers offer measurements that define the angle to themagnetic field. Knowledge of this restricts the axis of interest to lie ona cone centered on the magnetic field direction with an angleproportional to the measurement of magnetic field strength. With thisrestriction, formulae are derived for finding the relationship betweenthe magnetic field pointing vector, the spin axis, and the magnitude ofthe magnetic field. Methods based on both calibrated anduncalibrated magnetometers are discussed. These formulae are alsouseful for the calibration of magnetometers.

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ACKNOWLEDGMENTS

The author would like to thank Bernard Guidos of the U.S. Army ResearchLaboratory for his suggestions that improved this report.

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Contents

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

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

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

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

5. Solutions That Use Assumptions ....................... 135.1 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

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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 theseapplications, the largest errors are associated with the orientation or attitude ofthe body. Attitude estimation is vital to inertial navigation. Errors in attituderesult in accelerometers being misaligned with respect to the INS referenceframe. Within an INS, this misalignment will lead to accelerometer integration inthe wrong directions and in time, will lead to large errors in position. Angularrate sensors can be integrated to estimate a body’s attitude. Errors attributable tobias 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 anindependent measurement of an aspect of attitude could aid this process throughthe formulation of a Kalman filter or allow the engineer to bypass angular ratesensors and obtain a direct estimate of attitude. Knowledge of the attitude of abody’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 onthe magnetic field direction with an angle proportional to the measurement ofmagnetic field strength. With this restriction, formulae are derived for findingthe relationship between the magnetic field pointing vector, the spin axis, andthe magnitude of the magnetic field. Methods based on calibrated anduncalibrated magnetometers are discussed. These formulae are also useful forthe calibration of magnetometers.

2. Background

Conley and Patton [1] discuss a method to find the spin axis of a sounding rocketvia a solar sensor and one magnetometer. Their method uses iterative nonlinearleast squares, Euler angles, and inner products to find the orientation of the spinaxis. The inner product of two normalized vectors is the cosine of the anglebetween 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 axisby the use of solar and magnetometer information. We can find the anglebetween the spin axis and the magnetic field by using the zero crossings of twomagnetometers. This method requires at least one spin cycle to implement.

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Harkins and Hepner named their magnetometer configuration a MAGSONDE(MAGnetic SONDE). A MAGSONDE is a device that uses two magnetometers toobtain estimates of the angle to the magnetic field. One magnetometer is in aplane that is orthogonal to the spin axis; the other is in the plane formed by thespin axis and the magnetometer orthogonally to the spin axis.

The magnetic coordinate system is formed by the spin axis of the projectile, avector orthogonal to it in the same plane as the spin axis and the vector definingthe direction to the magnetic field, and a third vector orthogonal to the previoustwo. If the spin axis and the magnetic field vectors are known, then thecoordinate system can be defined via cross products. The first dimension is thespin axis; the second is the cross product of the third dimension and the spinaxis; 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 readzero when it is aligned with the third dimension and will give a maximumreading when aligned with the second dimension.

Since the magnetic field direction is assumed to be known, the accuracy to whichthe spin axis can be estimated will determine system performance. Finding theangle to the magnetic field gives one restriction on the spin axis; this angledefines 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 thechange in the orientation of the spin axis. The magnetic roll angle is the rotationabout the spin axis, measured so that when the magnetometer output is zero andthe 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 thespin axis at any point of time. The accuracy of these equations can be assessedthrough simulations. Software was developed to implement the simulations andto find the theoretical performance of the formulae during noise-inducedperturbations. These performance levels yield an ideal performance level that canonly be approached in practice.

The idea is to form a ratio of the signals and eliminate the magnitude factor or toform a linear combination that will allow extraction of desired signalinformation. The background information for this report is contained in Harkinsand Hepner [2], who use zero crossings of the magnetometers to find thedirection to the magnetic field.

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

sm1s sinsinMM φσ= , (1)

in which σm is the angle between the spin axis and the magnetic field, φs is theroll angle (about the spin axis), and M is the magnitude of the magnetic field. IfSensor 2 is at an angle of λ to the spin axis in the same plane as Sensor 1, then themeasurement from Sensor 2 is

smm2s sinsinMsincosMcosM φσλ+σλ= . (2)

If the magnitude of the first term is greater than the largest magnitude of thesecond term, this signal will not cross zero. If angles are chosen randomly, then50% of the time, Measurement 2 will not contain zero crossings. For each of thesemeasurements, if the magnitude of the magnetic field is known, then eachmeasurement can be thought of as the inner product of the normalizedmagnetometer axis and the magnetic field direction. Forming the ratio of thesetwo measurements makes the solution amplitude independent. Aftersimplification, the following result is obtained.

λ+φ

σλ= sinsin

)cot(cos

M

M

s

m

1s

2s (3)

The two measurements and the angle of Sensor 2 (λ) are known, so the knownquantities are moved to the right-hand side.

λ−

λ=

φσ

sinM

M

cos

1

sin

)cot(

1s

2s

s

m (4)

The sine of the magnetic roll angle can be found by reference to the zerocrossings of Sensor 1. It will be the time since the last zero crossing divided bythe time between zero crossings multiplied by 2π. This isolates the cotangentfunction; thus, we can find the angle to the magnetic field by using thearctangent function. For obvious reasons, this solution will not be stable as thevalue of Sensor 1 approaches zero. Away from that region, it should be possibleto calculate the angle to the magnetic pointing vector with Equation (4). In theregion where Sensor 1 is near zero, the inverse ratio could be used to find theangle to the magnetic field.

λ−φ

λ=σsinMM

M

sin

costan

1s2s

1s

s

m

(5)

However, the factor involving the sine of the magnetic roll angle will not permita solution as it approaches zero as the output from Sensor 1 does. An error

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

Software was designed to assess the sensitivity of the solution. At the indicatedroll angles, the variance of the solution when each signal had Gaussian noisewith 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 of172 degrees. The solution is symmetrical about 90 degrees. As expected, thesolution 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 forroll angles of 3 radians (172 degrees) is on the order of 5 degrees. This decreasesto 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 thenoise level is increased, a linear relationship between the standard deviation of

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the input noise in radians and the variance of the solution is observed for thenoise levels of 0.01 to 0.07 radian. Figure 1 clearly shows that the curvatureincreases 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 devisedby 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 angleof 0 or 180 degrees in the magnetic coordinate system. Using Equation (2), wecan find

(6)

In this situation, the strength of the magnetic field needs to be known. WhenSensor 1 measures zero, the magnitude of Sensor 2 defines the direction to themagnetic field. For example, if the output from sensor 2 were 0, this wouldindicate that the vector to the magnetic field is 90 degrees from the spin axis. Thevalue of Sensor 2 indicates how far from orthogonal the magnetic field vector ispointing. Note that this formula would allow the calculation of field strength ifthe direction to the magnetic field were known or could be estimated; thus, itcould be used for calibration.

The information available if Sensor 2 is zero can also be used to find the directionto the magnetic field. The assumption that the output is zero allows Equation (2)to be solved for the angle to the magnetic field.

φ

λ−=σ)sin()cot(

tanas

M

. (7)

This solution requires the roll angle in the magnetic coordinate system to befound. Note: The solution is determined by the magnetic roll angle and this isequivalent to a timing parameter. Based on the assumption that the roll rate isconstant, the amount of time between zeros corresponds to a portion of therotational 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 thezero crossings will occur between 180 and 360 degrees. The ratio of the zerocrossings 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 firstzero and after the second. To find the magnetic roll angle at the first zero ofSensor 2, one can use the following equation.

2/)ratio1(s −π+π=φ . (8)

=

)cos(cos 2

λσ

M

Ma s

m

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This formula can be used to find the angle to the magnetic field, based on theratio of each sensor’s time durations between zeros. The magnetic roll angle atthe first zero of the cycle can be found, based on the ratio and then inserted intothe formula to obtain the angle to the magnetic field. This result in theEquation (7) yields the angle to the magnetic field.

Figure 2 shows the performance of this method of solution for 10,000replications. Note that in this diagram, both the noise level and the standarddeviation are in radians; the conversion factor to degrees is 57.296. Since thevalue of the ratio is equivalent to knowledge of the magnetic roll angle, theperformance of Equation (7) depends on both the accuracy of the measurementof the roll angle and the actual value of the magnetic roll angle. Harkins andHepner [2] discuss another method of using zero crossings to solve for the spinaxis 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 asinput to Equation (2) when the reading is zero. Solving for the desired angleyields

[ ]

−= λσ tanM

Mcos a s1

m . (9)

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

The separation between the measurements and the magnetic roll angle containsthe necessary information to determine the angle between the magnetic field andthe spin axis. A modification of the difference between the two measurementscan be used. One can find the measure of the angle between the spin axis andmagnetic field by multiplying the value of Sensor 1 by the sine of the angle ofsensor offset and then subtracting this value from the output of Sensor 2. Theresult is

λ

λ−=σM)cos(

M)sin(Mcosa 1s2s

m

. (10)

This result does not require knowledge of the magnetic roll angle; however, thestrength of the magnetic field needs to be ascertained. Note that Equation (10) isthe same as Equation (9) if Measurement 2 is 0. The fidelity of the formula will bedetermined by the accuracy of the measurements and the accuracy in theestimate of magnetic field strength. As the argument of the arc cosine functionapproaches 1, errors in the measurement and magnitude terms can drive theargument to be greater than 1. With an input noise of 0.57 degree for bothmeasurements and the magnitude, the standard deviation of the solution was onthe order of 1 degree for magnetic vectors between 50 and 130 degrees. Figure 3shows the relationship based on a simulation; 10,000 replications were run foreach magnetic field angle.

Figure 3. Sensitivity of Equation 10.

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As with the first function for magnetic pointing angle, the errors are greatest at 0and 180 degrees where the sine of the magnetic roll angle is zero. This simplyindicates that as the spin axis and the direction to the magnetic field becomecollinear, 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. Theattitude of a vector can be found, given the angles between the vector and twoknown directions. Of interest is the situation when the vector to be found is thespin axis and the known directions are the magnetic field and the direction to thesun or the solar vector. Each known direction and angle will yield a cone ofpossibilities in three-dimensional space; the intersection of the two cones willyield two possible directions or points. If a magnetometer is being used, it ispossible to select one point, based on the time history of the magnetometeroutput. For one point, the output should be increasing; for the other, the outputshould 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 forrelating coordinate systems. The first rotation is about the Z-axis (ψ), followed bya rotation about the new Y-axis (θ), and the final rotation takes place about thenewest X-axis (φ). When the aerospace sequence is used, the final X-axis istypically the spin axis of the body. For spinning projectiles, the angular rateabout the spin axis is typically constant over a spin cycle. The validity of thisassumption is critical to the method discussed. In addition, it is assumed that themagnetic field vector is known. Conley and Patton [1] use these two assumptionsto develop their attitude estimator. The magnetometer output is proportional tothe projection of the magnetic field onto the magnetometer. By describing themagnetometer in terms of Euler angles, we can develop a set of equations byconsidering the output to be the inner product of the magnetic field vector withthe magnetometer orientation vector. With these equations, a method can bedeveloped to find the desired Euler angles. Although Conley and Patton used thesatellite ephemeris Euler sequence, their method can be recast via the aerospacesequence.

Next, assume that a magnetometer is placed orthogonally to the spin axis. Let thebody-fixed coordinate system be defined so that the spin axis is the X-axis andthe magnetometer is along the Y-axis. Starting with a reference coordinatesystem, the transition matrix to the body-fixed coordinate system (based on theaerospace sequence) is where the periods indicate the end of the term.

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√√√√√√√

↵

�

�������

�

+−+

+−−

).cos()cos(.cossinsin

sincos

.cossincos

sinsin

.sincos.sinsinsin

coscos

.cossin

sinsincos.sin.cossin.coscos

φθφθψ

φψφθψ

φψ

φθφθψ

φψφψ

φθψθθψθψ

For a rotation matrix, the inverse and transposed matrices are equivalent. The X-axis 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 thevector result. The spin axis can be represented in terms of the Euler angles. Theinner product of the spin axis (from the first row) and the direction of themagnetic field will yield an equation containing Euler angles that could beuseful. The magnetometer is along the Y-axis, which corresponds to the secondrow of the matrix. The inner product of the Y-axis with the normalized directionto the magnetic field will equal the cosine of the angle between the two. Fromthis matrix, the spin axis and two orthogonal vectors perpendicular to the spinaxis can be found in terms of Euler angles.

Returning to Equation (1), we find that the cosine of the angle between themagnetometer and the magnetic field is equivalent to multiplying the sine of theangle between the magnetic field and the spin axis by the sine of the magneticroll angle. By dividing the magnetometer output by the maximum possibleoutput (calibrated strength of the field), we can find these cosines. By takingmany measurements over a spin cycle, we can develop a system of equationsthat contains more equations than unknowns. The next step in the processrequires knowledge of the roll angle about the spin axis. Any of the possible spinaxes could produce the measurement sequence. Note that the zero value of themagnetic roll angle is unique for each candidate spin axis.

If the magnetometer value is known and associated with a magnetic roll angle, agiven spin axis will be defined. Conley and Patton [1] suggest using aconfiguration that allows a measurement induced by the sun’s energy. Using thesolar pointing angle, they show that it is possible to solve for the Euler angleabout the spin axis in terms of the other two Euler rotations. This measurementfixes the ray on which the spin axis must lie. The magnetometer measurementsthen allow the spin axis to be adjusted along the ray defined by the firstmeasurement. This is a recursive process in which subsequent adjustments arealmost orthogonal (one along a radius and the other along the circumference).Conley and Patton do not discuss the issue associated with errors in theorientation of the magnetometer leading to errors associated with the spin axisand vice versa. With their procedure, knowledge of one is assumed to find theother in a recursive algorithm. They demonstrated that this procedure convergedfor the problem they investigated.

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A closed form solution can be devised, given the existence of two knowndirections (solar and magnetic fields) and the projection of a third desireddirection (spin axis) onto the two known directions. The problem described inConley and Patton fits this description. A verbal description of theimplementation of a closed form solution follows. First, assume that thedirections are normalized so that each is represented by a vector to the unitsphere. For each measurement, the desired direction will be on a conedetermined by the known direction and the projection (or measurement). Thecircle formed by the cone and the unit sphere is in a plane that is normal to theknown direction and contains the point along the known direction that is inproportion to the projection or inner product. It is possible to describe this planewith an equation that uses the known direction and interprets the measurementas an inner product.

A second plane, based on the other direction, can also be formulated as anequation. The intersection of these two planes forms a line, and finally, theintersection of this line with the unit sphere yields two candidate directions thatfulfill the conditions. One candidate solution will be associated with increasingmeasurements and one with decreasing measurements. The solution based onthis requires multiplication, addition, and only one square root operation. Notethat if the two known directions are collinear, the planes will be parallel and nosolution will exist. As the known directions approach collinearity, the solutionbecomes unstable.

A simulation to observe the behavior of the solution was devised. Randomlyselected directions are chosen for the two known directions, and a thirdrandomly chosen direction was chosen as the axis of interest. Noise was added tothe projection of the axis of interest on each of the two known axes, and thisresult was sent to a routine that is based on the previously described solution. Anerror in the projection value affects the equation of the plane by moving it up ordown along the direction to which it is orthogonal. For planes with almostcollinear normals, a small change in the separation between the planes can havea large influence on their intersection. The difference between the solution andthe desired direction in degrees is reported as the system error. Normally,distributed input errors with standard deviations of 0.005 radian or 0.2865 degreewere used. System errors are displayed in Figure 4; the percentage ofobservations below a system error value is displayed.

This type of plot is referred to as a quantile plot. A quantile plot is the graph ofthe percentage of the data that is less than a given value; the median will occur atthe abscissa value of 50. It is apparent that in 90% of the cases, the error is lessthan 2.5 degrees. In addition to the system error (the dependent variable), theinner product of the known axes and the determinant of all three vectors werereported as independent variables. Also, an indicator variable was used toindicate that the real part of a complex solution was used. Complex solutionsoccur as the closest point of approach of an external line to the unit sphere. If two

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vectors are nearly collinear, their inner product will be close to 1. Thedeterminant of three vectors indicates the volume they form; if this volume isclose 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 wassubtracted from 1 so that values of 1 would represent collinear input sets, andvalues close to 1 would represent directions that were nearly collinear. Figure 5shows the error plotted against the inner product with complex solutions ingreen. The error increases as the inner product approaches 1; also, the complexsolutions are distributed throughout the distribution. Figure 6 shows thedeterminant measure with complex solutions in green. As 1-d increases, themaximum error slowly increases until the independent variable passes 0.8, afterwhich, the increase accelerates. The determinant-based measure is seen to be abetter indicator of the overall system error than the inner product; however, in atypical situation, only two axes are known. The determinant measure shows thatall complex solutions occur when the collinearity of the three directions is high.The linear lower boundary for the complex solutions can be understood as theerror level needed to force the solution to become complex for the collinearityindicator.

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Figure 5. Relationship Between Inner Product and Error.

Figure 6. Relationship Between Determinant Measure and Error.

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A model of system error could be made with the degree of collinearity as thepredictive variable. If the system becomes collinear, then the system error wouldbe 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. Thenumerator could be adjusted to fit different levels of input noise. A model of thistype would allow the system engineer or experiment coordinator to predictsystem performance for known geometries. Desirable geometries occur when thedeterminant is close to 1. Geometries where the determinant is close to 0 shouldbe 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 tofind the attitude of the system. Specific knowledge about system behavior can beused to develop algorithms. For example, if the change in the spin axis is knownto have a restriction, this knowledge can be used to orient the system. For anartillery projectile, we know that the change in the spin axis takes place mainly inthe plane defined by the spin axis and gravity. This could be used in conjunctionwith magnetic field information to orient the system. If the magnetic field vectoris parallel to the surface of the earth, then when the magnetometer reaches itsmaximum peak-to-peak reading, the spin axis is also parallel to the surface of theearth. This would yield two candidate systems, one of which could beeliminated. If the magnetometer is properly calibrated, the angle to the magneticfield and the attitude of the spin axis would also be known. Assuming thatchanges in the spin axis occur in the plane formed by the spin axis and gravitywill enable computations based on the differential peak-to-peak voltages toorient the spin axis.

For an individual magnetometer spinning about an axis, the maximum andminimum readings will occur when the device is in the plane of the magneticfield and the spin axis (equal readings indicate that the spin axis is aligned withthe magnetic field). These readings, coupled with knowledge of the magneticfield 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 formulaepresented in this report. This package can be used to evaluate other questionsrelated to MAGSONDE data processing.

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6. Conclusions

The methods discussed in this report can be used to increase the uses ofmagnetometers and to calibrate magnetometers. As aids to attitude estimation,magnetometers offer a way to determine a restriction of the body-fixedcoordinate 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’sattitude.

Error analyses that were completed for the individual procedures indicate thequality of the estimate for various noise levels and parts of the magnetic rollcycle. More comprehensive models could be developed for the noise level/rollangle parameter space to provide the designer with a tool that indicates expectedperformance of a magnetometer processing method. The software developed forthis study can be used to investigate individual cases.

These new formulae could be used in conjunction with angular rate sensors toimprove their performance. Providing an independent observation of attitude toa Kalman filter would make it possible to estimate bias and scale factor errors forangular rate sensors.

The ability to estimate the angle between the spin axis and the magnetic field atany point in time will lead to real-time estimation of attitude. This increases thepossibility of on-board navigation for spinning bodies.

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References

1. Conley, J.M., and R.B. Patton, ”An Attitude Determining System ForSpinning 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 MakingAngular Measurements on Spinning Projectiles With Magnetic Sensors,”ARL-TR-2310, U.S. Army Research Laboratory, Aberdeen Proving Ground,MD, December 2000.

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Final

U.S. Army Research LaboratoryWeapons & Materials Research Directorate Aberdeen Proving Ground, MD 21005-5066

Thompson, A.A. (ARL)

U.S. Army Research LaboratoryWeapons & Materials Research Directorate Aberdeen Proving Ground, MD 21005-5066

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PR: AH80A 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 tolie 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. Theseformulae are also useful for the calibration of magnetometers.

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