- Email: [email protected]
Influence of sensor imperfections to electronic compass attitude accuracy
Sensors and Actuators A 155 (2009) 233–240
Contents lists available at ScienceDirect
Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna
Influence of sensor imperfections to electronic compass attitude accuracy J. Vˇcelák ∗ , J. Kubík Tyndall National Institute, Lee Maltings, Prospect Row, Cork, Ireland
a r t i c l e
i n f o
Article history: Received 25 February 2009 Received in revised form 20 July 2009 Accepted 30 August 2009 Available online 8 September 2009 Keywords: Compass Magnetic sensor Accelerometer Attitude
a b s t r a c t Analysis of various errors of orientation and motion sensors and their influence on resulting compass attitude information are investigated and quantified in this paper. A triple-axis acceleration sensor and a triple-axis magnetic sensor are considered to form a compass module giving full orientation information (attitude). 3D-compass module with PCB-fluxgate sensors and off-shelf acceleration sensor is introduced. Three main sensor error sources are discussed separately and their contribution to final compass heading accuracy is investigated. These are sensor orthogonality error, linearity error and ADC quantization noise. The influence of these errors is simulated on artificial group of test cases which evenly covers various orientations of the compass module. It was found that the errors in the accelerometric system have major influence on the heading accuracy. It was concluded that even low-resolution AD converters may have only a minor influence on the system accuracy, while the dominant error sources are sensor triplet nonorthogonality and sensor non-linearity. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Low cost, miniature electronic compass integrated circuits (ICs), multi-axis motion and orientation sensors became marketavailable during the last years. Their major application areas are handheld devices such as GPS receivers, mobile phones and PDAs, arm watches and small Attitude and Heading Reference Systems (AHRS). While in some of the applications compass accuracy is not important (arm watch and GPS receiver) in other applications the accuracy plays a major role and sensor imperfections have to be considered. There are several ICs which are dedicated to work as a part of or as a complete electronic compass on the market today. These components are designed to achieve low manufacturing costs, small footprint and low power consumption. An electronic compass module consists of a dual-axis or a triple-axis magnetometer and a tilt sensor (e.g. triple-axis accelerometer) [1]. Considering the complexity of a compass module we distinguish basic compasses without tilt compensation and fully electronically compensated compasses that can provide Pitch and Roll information in addition to Heading data completing the object attitude information. The market-available electronic integrated compass modules are e.g. Honeywell’s HMC6052, HMC6343 and HMC6352 [2] and Aichi Steel’s AMI 601 and AMI501 [3]. There are wide range of magnetic and acceleration sensors available on the mar-
∗ Corresponding author at: Tyndall National Institute, MAI, Lee Maltings, Prospect Row, Cork, Ireland. Tel.: +353 214904298. E-mail addresses: [email protected] (J. Vˇcelák), [email protected] (J. Kubík). 0924-4247/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2009.08.024
ket besides the integrated compass ICs: Honeywell’s HMC1001-2, HMC1021-2, HMC1043 and HMC5843 [2], Philips’ KMZ51-2 [4], STMicroelectronics’ LIS3L02 [5] and Analog Devices’ ADXL345 [6]. In the case of more than one sensor die in the IC package, the sensitivity axes are often assumed as ideally perpendicular one to another and any imperfections are neglected. These imperfections include: • Individual sensor linearity error. • Sensor noise (including analogue-to-digital converter (ADC) quantization noise if digital conversion is in place). • Dissimilar sensitivity of each sensor. • Dissimilar offset of each sensor. • Angular deviations of sensitivity axes of individual sensors. • Sensor hysteresis. All of the imperfections above contribute to the final accuracy of a compass system utilizing such sensors. Although there are existing methods to suppress some of these errors [7–10], they are rarely used in consumer applications due to the substantial computational power required being in direct contradiction to the low-cost device requirement. This paper will provide the reader a brief survey of final compass heading errors caused by majority of the typical sensor imperfections. 3D-compass module used for some of the presented experiments will be introduced and some of the errors and their influence on the compass readings will be presented and discussed. A designer or a user of a compass module may then decide which error is negligible and which one is worth suppressing by software algorithms.
234
J. Vˇcelák, J. Kubík / Sensors and Actuators A 155 (2009) 233–240
Fig. 1. 3D-compass magnetometer and accelerometer. Fig. 3. 3D plot of test case group.
2. 3D-compass module An electronic compass module with fluxgate sensors was developed. The sensors were manufactured by the printed circuit board technology with square or oval shape patterned soft-magnetic material (Vitrovac 6025X) core embedded between PCB laminate layers [11]. Excitation and pick-up coil patterns and vias were manufactured by the standard printed circuit board technology. Sensitivity axes are in the plane of the board, and in case of dualaxis sensor, perpendicular to each other. Sensors are excited by the pulse current (10 kHz), 100 mA peak-to-peak. The compass module uses one dual-axis and one single-axis fluxgate sensor to form a full triple-axis magnetometer (Fig. 1). The triple-axis accelerometer LIS3L02AS5 [5] was used as a tilt sensor. Six sensors’ outputs are digitized by the 24-bit analogue-to-digital converters and processed by 8-bit PIC18F4550 microprocessor. The data are sent to the PC or PDA via wireless Bluetooth interface or via USB interface. The data are then used to show attitude in the form of an artificial horizon with heading indication (Fig. 2). A dedicated graphical user interface was also developed for PDA and PC in NI LabVIEW. 3. Basic algorithm to calculate compass attitude values Simple electronic compasses equipped with only dual-axis magnetic sensor without any tilt compensation use Eq. (1) to calculate Heading . Hx , Hy , Hz , are the components of measured Earth’s
Fig. 2. 3D-compass module.
magnetic field vector and the subscript b denotes vector coordinate measured with respect to body reference frame (sensors are fixed to the object and their sensitivity axes are parallel to the longitudinal, lateral and vertical object axes): heading :
Hyb
= arctg
(1)
Hxb
More versatile compass modules with a tilt compensation using a triple-axis accelerometer calculate attitude values according to Eqs. (2)–(4). These equations are implemented in embedded software in 3D-compass module. It should be noted that they are valid only in a steady position of the compass or during a uniform rectilinear motion: roll :
˚ = arctg
pitch :
Gyb
(2)
Gzb Gxb
= arctg
2 Gyb
heading :
= arctg
(3)
2 + Gzb
Hyb · cos ˚ − Hzb · sin ˚ Hxb · cos + Hyb · sin · sin ˚ + Hzb · sin · cos ˚
(4)
4. Test cases The electronic compass is a device for determining the orientation with respect to a navigation frame. The orientation values are usually given in terms of Euler’s angles called Heading (yaw, ), Pitch (elevation, ) and Roll (bank, ˚). All three Euler’s angles are called attitude and express three degrees of freedom of compass orientation in a certain location. The simulation algorithm used for further error estimation artificially sets the orientation of a compass module. The set of attitude triads is called a group of test cases and should evenly cover the complete attitude range of a compass module. There is a set of ideal gravitational and geomagnetic field vector components associated with each of the test cases. These vector components are used in a mathematical model of a non-ideal compass to calculate the attitude information affected by the simulated sensor imperfection. Knowing the correct attitude value, the attitude errors are calculated for each of attitude triads in the test case group. The graphical representation of the test case group in 3D space is shown in Fig. 3. The AB vector is aligned with compass longitudinal axis x. The attitude of the compass is changed in the way that A point remains in the same position while B point is moving on the surface of a sphere on the line indicated in blue color. Simultaneously the roll of the device is changing as well. The trajectory of the point B in space could be described by parametric equations (5) and
J. Vˇcelák, J. Kubík / Sensors and Actuators A 155 (2009) 233–240
235
shown in Fig. 3 for N = 200, r = 1000, ˚range = ±180◦ , range = ±80◦ , = 360◦ and Nturns = 10. Such a test case group ensures that all three attitude variables of heading, pitch and roll will be equally distributed among test case group triads.
t·
x = r · cos
t·
range N
t·
range N
y = r · cos
z = r · sin pitch :
range N
=t·
· cos
t · Nturns
range 2N
t · Nturns
range 2N
· sin
(5)
range N
range 2N ˚range · 2N ˚=t· Nturns + 1 = t · Nturns
heading : roll :
Fig. 4. Simplification of sensor triplet orthogonality error.
and ϕy are known: 1 + 2N – total number of test cases, r – sphere radius, range – pitch range (≈), range – heading range (2), ˚range – roll range (), Nturns – number of spiral turns, t = −N, N – parameter.
A(x =0◦ ,z =0◦ ,z =90◦ ) =
5. Influence of sensor’s orthogonality error Angular deviation of the sensor sensitivity axis from the ideal sensitivity axis of orthogonal sensor triplet is called orthogonality error. The user usually assumes that the triple-axis magnetometer and accelerometer form ideal sensor triplets with sensor sensitivity axes perpendicular to each other. Nevertheless, the final accuracy of the sensor sensitivity axes placement is often limited by the production process and technology used. The methods of precise determination of the real sensor sensitivities, offsets and angular deviation are presented in Refs. [7,8,12]. While sensor offsets and sensitivities could be determined easily, determination of the true angles between the sensors is quite complicated. If we consider an ideal 3D orthogonal reference system xyz and a real sensor sensitivity axis S placed within this ideal system, the angles and ϕ could be defined as the angle between S and its projection ps to x–y plane and the angle between ps and ideal x axis respectively. Angles and ϕ could be defined for all three sensors’ sensitivity axes. The measured vector H with coordinates in ideal reference system xyz is then transformed to Hb vector measured by the real sensors using Eq. (6):
Hxb Hyb Hzb
⎡
cosx · cos x
sin x · cos x
cos z · cos z
sin y · cos y sin z · cos z
= ⎣ cos y · cos y
=A·H H b
⎤ Hx sin y ⎦ · Hy sin x sin z
Hz
In order to simplify the general matrix A it is convenient to set one of the true sensor axes identical to one of the axes of the ideal coordinate system (e.g. z axis). Then the general transformation matrix A is simplified to expression (7) and the corresponding situation is shown in Fig. 4. The matrix A is used to simulate the imperfections of the sensor triplet in this paper. The inverse matrix A−1 could be easily used to correct sensors’ imperfections when the angles x , y
cos x cos y · cos y 0
0 sin y · cos y 0
sin x sin y 1
(7)
Since three different angles x , y and ϕy define the sensor triplet orthogonality imperfection as parameters, the complexity of the simulation task and result representation can become complicated. The accuracy of the sensor placement in xy plane is usually much better than positioning of z sensor with respect to xy plane. In the case of an anisotropic magnetoresistance (AMR) integrated triaxial sensor circuit, the x and y sensors may be manufactured on a single die, while the z sensor may be on an another die. In order to simplify the situation, we will consider only two variable angles x = y = and ϕy = ϕ. Generally means deviation of the z axis sensor from the direction perpendicular to xy plane and ϕ represents angle between x and y sensors. The resulting maximal Roll, Pitch and Heading error heading = f(, ϕ) could be then conveniently shown in 3D plots. When a compass module is used in a real application, two types of attitude errors could be estimated for the test case group. RMS error (standard deviation) is calculated using Eq. (8) while maximal error is calculated using Eq. (9). xi represents the difference between known attitude angle and calculated attitude angle affected by sensor nonorthogonality for each of the test cases. RMS error does not reflect the maximal error which may become significantly larger than the RMS value for certain singular orientations of the compass. Therefore we have decided to present maximal error and an example of peak-to-peak to RMS ratio (Eq. (10) and Fig. 8). While the RMS error could be low, the peak-to-peak to RMS ratio could be up to 6, which means that absolute maximal error is 6 times higher than the RMS. Due to practical reasons, the knowledge of maximal error is more convenient for the user because he could be sure that output attitude value is always within the given tolerance interval:
(6)
RMS =
1 2 ( xi ) , N−1
xi = xi − xi
(8)
N
xi – simulated value with sensor imperfections, xi – original ideal value. MAX =
max( xi ) − min( xi ) 2
p–p 2 · MAX error ratio = RMS RMS
(9) (10)
236
J. Vˇcelák, J. Kubík / Sensors and Actuators A 155 (2009) 233–240
Fig. 5. Roll error due to accel. ortho. error.
Fig. 7. Heading error due to accel. ortho. error.
The test case group with 400 attitude triads was used to investigate error in various compass orientations. Fig. 5 shows Roll error which is influenced by sensor deviation angle only as could be expected from Eq. (2). Fig. 6 shows Pitch error and Fig. 7 shows resulting Heading error when only accelerometer orthogonality error is considered. Fig. 8 shows (p–p)/RMS error ratio for maximal Heading error presented in Fig. 7. It could be seen that (p–p)/RMS error ratio could reach up to 6. Fig. 9 represents maximal Heading error when only magnetometer orthogonality error is considered. The shape of combined Heading error when both magnetometer and accelerometer orthogonality errors are taken into account depends on the relation between them and could not be easily expressed, however the absolute values do not exceed 16◦ for selected range of the angles and ϕ angular deviations (an example is presented in Fig. 10). 6. Linearity error suppression The sensors used in electronic compasses are usually AMR, magnetoimpedance (MI) or fluxgate sensors. The feedback comFig. 8. (p–p)/RMS Heading error ratio.
Fig. 6. Pitch error due to accel. ortho. error.
Fig. 9. Heading error due to mag. ortho. error.
J. Vˇcelák, J. Kubík / Sensors and Actuators A 155 (2009) 233–240
237
Fig. 12. Attitude values and Heading error for single test case group. Fig. 10. Combined heading error.
pensation of measured magnetic field is convenient from the perspective of suppressing the sensor linearity error, but is not favorable in case of low power applications since the feedback current fed into the compensation coils increases the total power consumption. The linearity error of an AMR sensor (without feedback compensation loop) could reach up to 0.5% of the full scale (FS) [3]. The linearity error could be sufficiently suppressed by mathematical correction in microprocessor. The influence of such suppression is demonstrated on an example of PCB-fluxgate sensors used in the 3D-compass module. The linearity of each sensor was calibrated separately in three independent measurements using PC-controlled triaxial Helmholtz coil system. The range of generated magnetic field was ±60 T. The linearity error of squareshaped dual-axis sensor reached of ±0.8% FS for x axis, ±0.6% FS for y axis and the error of the racetrack-shaped single-axis sensor was ±1.9% FS for z axis. The linearity error waveform was approximated by the 9th order polynomial. The readout values Hx , Hy and Hz are corrected for linearity error using Eq. (11) with N = 9. Obviously, the polynomial coefficients pn differ for each sensor: Hcorr = Hread +
N
n pn · Hread
(11)
n=0
The measured linearity error, the polynomial approximation and the residual error are shown in Fig. 11. The correction algorithm according to Eq. (11) is a part of the embedded compass module software and is calculated in real-time. The results of mathematical linearity error correction for the PCB-fluxgate sensors used in
the 3D-compass module are summarized in Table 1. It can be seen from Fig. 11 and from Table 1 that while x and y sensor linearity error is corrected well, the correction is not that successful in case of z sensor. The possible reason is that this sensor is single axis with a different core shape and the transfer characteristic is more influenced by temperature changes. The comparison of sensor linearity is given in Table 2 for commercially produced magnetometers and compass modules. The linearity error is recalculated to FS = ±100 T for a clear comparison of the linearity errors provided by the manufacturers for various FS ranges. 7. Influence of sensor linearity error on calculated attitude The polynomial approximation of magnetic sensor linearity error was used to model the influence of linearity error on calculated compass heading in the case of uncorrected sensor linearity error. The example of heading error waveform evaluated for one test case group (magnetometer linearity error ±1.5% FS considered) is presented in Fig. 12 including the waveform of modeled attitude values in test cases triads. It could be seen that the maximal final heading error is ±4◦ and the corresponding RMS error is 1.7◦ . The influence of accelerometer and magnetometer linearity error has been examined separately and then the influence of both sensor errors is presented in Fig. 13 (linearity error expressed in % of FS where for magnetometers FS = ±100 T and for accelerometers FS = ±2000 mg; 1 mg = 0.00981 m s−2 ). This figure could be used for estimation of the final accuracy of the combined system as well as for commercially produced magnetometer and compass mod-
Fig. 11. PCB-fluxgate sensor linearity error axes x, y and z (before and after math correction).
238
J. Vˇcelák, J. Kubík / Sensors and Actuators A 155 (2009) 233–240
Table 1 3D-compass PCB-fluxgate linearity error. Sensor axis
Linearity error original ±1450 nT ±725 nT ±2400 nT
x y z
Linearity error after compensation ±119 nT ±99 nT ±450 nT
Lin error in %FS (FS = ±100 T)
Lin error after compensation RMS
0.12% 0.10% 0.45%
44 nT 32 nT 145 nT
Table 2 Compass modules and sensors parameters. Sensor type
FS ±400 T ±200 T ±600 T ±600 T ±300 T ±60 T
HMC5843 HMC1001 HMC1021 HMC1043 AMI601 3D compass
Lin error (%FS)
Lin error recalculated to FS = ±100 T
Sensor description
±0.2% ±0.05% ±0.1% ±0.1% ±1.3% ±1%
±0.8% ±0.1% ±0.6% ±0.6% ±3.9% ±0.6%
AMR, 3-axis, digital output, 12 bit AMR, 1- and 2-axis, analog out AMR, 1- and 2-axis, analog out AMR, 3-axis, analog out MI, 3-axis, digital out, 12 bit PCB-fluxgate, 3-axis, analog out
Table 3 Contribution of the ADC quantization noise to final compass heading error. ADC
Magnet sensor
Accel. sensor
Heading error, magnetometer considered only
Heading error, accel. sensors considered only
Pitch error, accel. sensors considered only
Roll error, accel. sensor considered only
Heading error, both sensors considered
Nom. NOB
ENOB
LSB (nT)
LSB (mg)
Max ± (◦ )
RMS (◦ )
Max ± (◦ )
RMS (◦ )
Max ± (◦ )
RMS (◦ )
Max ± (◦ )
RMS (◦ )
Max ± (◦ )
RMS (◦ )
8 10 12 14 16
7 9 10 12 14
1562.5 390.6 195.3 48.8 12.2
31.25 7.81 3.91 0.98 0.24
2.5 0.7 0.3 0.1 0.03
1.3 0.3 0.17 0.04 0.01
4 1.2 0.46 0.13 0.03
1.5 0.4 0.19 0.05 0.01
1.2 0.4 0.2 0.04 0.01
0.5 0.13 0.07 0.02 0.003
3 1.2 0.4 0.1 0.02
1 0.3 0.13 0.04 0.007
5 1.5 0.6 0.2 0.04
1.9 0.5 0.2 0.06 0.01
ules when the sensor parameters are known. It may be seen that the influence of accelerometer linearity error to the final heading accuracy is approximately two times higher than the contribution of magnetometer linearity error. 8. Influence of sensor noise and ADC quantization noise Although the sensor noise could be an issue in case of accuracy-demanding applications, it is usually neglected in lowcost applications. While peak-to-peak noise of typical AMR sensor is in the range of units to tens of nT [13] the quantization noise of ADCs (8–16 bit) is usually much higher or at least comparable. Therefore the following chapter will be focused mainly on ADC quantization noise which is the driving factor for ADC selection. Sensor output values are always digitized by ADC for further mathematical processing. The ADC is very often integrated in the magnetometer/accelerometer module or in the P used. These
ADCs are usually successive approximation or low-resolution –˙ type converters. The Nominal Number of Bits (NNOB), the estimated achievable Effective Number of Bits (ENOB), magnetic field and field of gravity corresponding to 1 Least Significant Bit (LSB) of ENOB and the attitude error resulting from the ADC quantization error based on uniformly distributed quantization noise within the range of ±1/2 LSB of ENOB are expressed in Table 3. The full scale of magnetometer and accelerometer ADCs is set to ±100 T and ±2000 mg respectively. Apart from the ADC NNOB, the user should also evaluate sensor FS versus ADC NNOB. The example could be given for magnetometer module HMC5843 (Nominal 12 bit, FS = ±400 T) where nominal LSB corresponds to 195 nT and AMI601 (Nominal 12 bit, FS = ±300 T) where nominal LSB corresponds to a lower value of 146 nT providing higher resolution despite the same NNOB of the ADC. The influence of magnetometer and accelerometer quantization noise is considered separately and then the final accuracy influenced by both sensors’ quantiza-
Fig. 13. Combined Heading error due to accel. and mag. linear. error (detail on right).
J. Vˇcelák, J. Kubík / Sensors and Actuators A 155 (2009) 233–240
Fig. 14. Max. heading error due to ADC quantization noise.
Fig. 15. Contribution of accel. and mag. ortho. error to total compass heading error.
tion noise is presented in Fig. 14. Note that the modeled calculation of compass heading was performed with floating point numerical representation. Rounding errors will emerge in case of limited fixed point calculations, but these are not subject of this paper. 9. Conclusion Heading accuracy of an electronic compass was simulated with respect to several possible sensor error sources. A group of test cases was introduced. The test group for each simulated error value consisted of 400 different orientations of the compass module evenly covering the full attitude value range. Potential sources of compass errors were listed and three of these error sources were modeled and investigated separately: sensor axial orthogonality, sensor linearity error and ADC quantization noise. Final compass heading error was estimated for each case in the group and statistic parameters of the group were determined and presented as maximal peak-to-peak (p–p) error, standard deviation and (p–p)/RMS ratio in the 3D plots. It was found that orthogonality error has a very substantial contribution to the final heading accuracy. Accelerometer orthogonality error has a major influence on the resulting Heading accuracy since the error of Roll and Pitch calculated from accelerometers outputs further propagates to the calculation of Heading (Eqs. (2)–(4)). In case of limited accelerometer placement precision, the orthogonality error should be compensated numerically. Generally, when the worst cases are taken into account it could be said that the contribution to the final heading error is given by the lines presented in Fig. 15. Nevertheless, when the orthogonality errors are comparable for magnetometer and accelerometer, the combined error is driven almost exclusively by accelerometer error contribution. It has to be mentioned that a simplification of the general axial devi-
239
ation error has been used in order to present final errors easily. The shape of the resulting heading errors presented in Figs. 9 and 10 may vary according to the mutual relation of both errors. Sensor linearity error plays also substantial role in the final compass heading accuracy. It could be seen from Fig. 13 that again the accelerometer imperfection is the major source of the compass heading error. The 3D-compass module with PCB-fluxgate sensors and accelerometer was introduced and linearity error suppression was demonstrated on this device system. Surprisingly it has been demonstrated that low-resolution ADCs with 10–12 bit could be used also when low total heading error (less than 2◦ ) is required. Nevertheless the reader has to consider that all quantization noise was recalculated to ADC FS = ±2000 mg for accelerometer and FS = ±100 T for magnetometer. When other types of errors are taken into account, further increase of the ADC resolution does not bring reasonable increase of compass heading accuracy. The demonstrated methods are applicable to estimate the heading accuracy of any compass module including commercially produced compass IC modules from the sensor parameters given by the manufacturer. The results presented above could be conveniently used for components selection in case of a new compass design. Corresponding magnetic sensor, accelerometer and matching ADC could be easily chosen to meet the final heading error requirement. The combination of the following components could be used as an example of well-matched design: • magnetometer (linearity error 0.8% FS, max orthogonal deviation ±0.4◦ ), • accelerometer (linearity 0.5% FS, max orthogonal deviation ±0.2◦ ), • ADC 12 bit for both magnetometer and accelerometer. When these components are combined to form an electronic compass, the final maximal heading error should not be more than of ±1.5◦ (0.5◦ RMS). Acknowledgments This research was funded by Enterprise Ireland (EI) Commercialization Fund Technology Development grant no. CFTD/05/IT/316. References [1] M.J. Caruso, Application of magnetoresistive sensors in navigation systems, Sens. Actuators (February) (1997) 15–21 (SAE SP 1220). [2] http://www.ssec.honeywell.com. [3] http://www.aichi-mi.com. [4] http://www.nxp.com. [5] http://www.st.com/stonline/products/families/sensors/motion sensors.htm. [6] http://www.analog.com. [7] J. Merayo, P. Brauer, F. Primdahl, J.R. Petersen, O.V. Nielsen, Scalar calibration of vector magnetometers, Meas. Sci. Technol. 11 (2) (2000) 120–132. [8] J. Vˇcelák, Application of Magnetic Sensors for Navigation Systems, vol. 2, Shaker-Verlag, 2009, ISBN 978-3-8322-7738-3. [9] J. Vˇcelák, P. Ripka, J. Kubík, A. Platil, P. Kaˇspar, AMR navigation systems and methods of their calibration, Sens. Actuators A 123–124 (2005) 122–128. [10] J. Vˇcelák, P. Ripka, A. Platil, J. Kubík, P. Kaˇspar, Errors of AMR compass and methods of their compensation, Sens. Actuators A 129 (2006) 53–57. [11] J. Kubik, L. Pavel, P. Ripka, P. Kaspar, Low-power printed circuit board fluxgate sensor, IEEE Sens. J. 7 (1–2) (2007) 179–183. [12] J. Vcelak, Calibration of triaxial fluxgate gradiometer, J. Appl. Phys. 99 (08) (2006) D913. [13] H. Hauser, P.L. Fulmek, b P. Haumer, M. Vopalensky, P. Ripka, Flipping field and stability in anisotropic magnetoresistive sensors, Sens. Actuators A: Phys. 106 (September (1–3)) (2003) 121–125.
Biographies Jan Vˇcelák received his Ing. degree (M.Sc. equivalent) in March 2003. He successfully defended his Ph.D. thesis “Application of magnetic sensors for nav-
240
J. Vˇcelák, J. Kubík / Sensors and Actuators A 155 (2009) 233–240
igation systems” under supervising of Dr. Petr Kaˇspar in November 2007 at the Faculty of Electrical Engineering of Czech Technical University in Prague. In 2006/2007 he worked in Ricardo Prague on development of SW&HW for hybrid vehicles. He has joined Tyndall NI in Cork in December 2007. His research interests include sensors (magnetometers, accelerometers and gyroscopes), multisensor systems calibration and simulation, algorithms for compass, AHRS and IMUs. Jan Kubík received his Ing. degree (M.Eng. equivalent) in the field of computercontrolled measurement systems in March 2003 at the Faculty of Electrical
Engineering of Czech Technical University in Prague. He successfully defended his Ph.D. thesis “PCB-Fluxgate Sensors” under supervision of Prof. Pavel Ripka in December 2006 at the same university. He also worked at the National University of Ireland in Galway (2001 and 2002) and at the Lappeenranta University of Technology in Lappeenranta, Finland (2003). He has joined the Tyndall National Institute in Cork in September 2006 as a postdoctoral researcher. His research interests range from magnetic field sensors design and characterisation to computer-controlled measurement systems development, FEM modelling of magnetic systems and microcontroller applications.
Contents lists available at ScienceDirect
Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna
Influence of sensor imperfections to electronic compass attitude accuracy J. Vˇcelák ∗ , J. Kubík Tyndall National Institute, Lee Maltings, Prospect Row, Cork, Ireland
a r t i c l e
i n f o
Article history: Received 25 February 2009 Received in revised form 20 July 2009 Accepted 30 August 2009 Available online 8 September 2009 Keywords: Compass Magnetic sensor Accelerometer Attitude
a b s t r a c t Analysis of various errors of orientation and motion sensors and their influence on resulting compass attitude information are investigated and quantified in this paper. A triple-axis acceleration sensor and a triple-axis magnetic sensor are considered to form a compass module giving full orientation information (attitude). 3D-compass module with PCB-fluxgate sensors and off-shelf acceleration sensor is introduced. Three main sensor error sources are discussed separately and their contribution to final compass heading accuracy is investigated. These are sensor orthogonality error, linearity error and ADC quantization noise. The influence of these errors is simulated on artificial group of test cases which evenly covers various orientations of the compass module. It was found that the errors in the accelerometric system have major influence on the heading accuracy. It was concluded that even low-resolution AD converters may have only a minor influence on the system accuracy, while the dominant error sources are sensor triplet nonorthogonality and sensor non-linearity. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Low cost, miniature electronic compass integrated circuits (ICs), multi-axis motion and orientation sensors became marketavailable during the last years. Their major application areas are handheld devices such as GPS receivers, mobile phones and PDAs, arm watches and small Attitude and Heading Reference Systems (AHRS). While in some of the applications compass accuracy is not important (arm watch and GPS receiver) in other applications the accuracy plays a major role and sensor imperfections have to be considered. There are several ICs which are dedicated to work as a part of or as a complete electronic compass on the market today. These components are designed to achieve low manufacturing costs, small footprint and low power consumption. An electronic compass module consists of a dual-axis or a triple-axis magnetometer and a tilt sensor (e.g. triple-axis accelerometer) [1]. Considering the complexity of a compass module we distinguish basic compasses without tilt compensation and fully electronically compensated compasses that can provide Pitch and Roll information in addition to Heading data completing the object attitude information. The market-available electronic integrated compass modules are e.g. Honeywell’s HMC6052, HMC6343 and HMC6352 [2] and Aichi Steel’s AMI 601 and AMI501 [3]. There are wide range of magnetic and acceleration sensors available on the mar-
∗ Corresponding author at: Tyndall National Institute, MAI, Lee Maltings, Prospect Row, Cork, Ireland. Tel.: +353 214904298. E-mail addresses: [email protected] (J. Vˇcelák), [email protected] (J. Kubík). 0924-4247/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2009.08.024
ket besides the integrated compass ICs: Honeywell’s HMC1001-2, HMC1021-2, HMC1043 and HMC5843 [2], Philips’ KMZ51-2 [4], STMicroelectronics’ LIS3L02 [5] and Analog Devices’ ADXL345 [6]. In the case of more than one sensor die in the IC package, the sensitivity axes are often assumed as ideally perpendicular one to another and any imperfections are neglected. These imperfections include: • Individual sensor linearity error. • Sensor noise (including analogue-to-digital converter (ADC) quantization noise if digital conversion is in place). • Dissimilar sensitivity of each sensor. • Dissimilar offset of each sensor. • Angular deviations of sensitivity axes of individual sensors. • Sensor hysteresis. All of the imperfections above contribute to the final accuracy of a compass system utilizing such sensors. Although there are existing methods to suppress some of these errors [7–10], they are rarely used in consumer applications due to the substantial computational power required being in direct contradiction to the low-cost device requirement. This paper will provide the reader a brief survey of final compass heading errors caused by majority of the typical sensor imperfections. 3D-compass module used for some of the presented experiments will be introduced and some of the errors and their influence on the compass readings will be presented and discussed. A designer or a user of a compass module may then decide which error is negligible and which one is worth suppressing by software algorithms.
234
J. Vˇcelák, J. Kubík / Sensors and Actuators A 155 (2009) 233–240
Fig. 1. 3D-compass magnetometer and accelerometer. Fig. 3. 3D plot of test case group.
2. 3D-compass module An electronic compass module with fluxgate sensors was developed. The sensors were manufactured by the printed circuit board technology with square or oval shape patterned soft-magnetic material (Vitrovac 6025X) core embedded between PCB laminate layers [11]. Excitation and pick-up coil patterns and vias were manufactured by the standard printed circuit board technology. Sensitivity axes are in the plane of the board, and in case of dualaxis sensor, perpendicular to each other. Sensors are excited by the pulse current (10 kHz), 100 mA peak-to-peak. The compass module uses one dual-axis and one single-axis fluxgate sensor to form a full triple-axis magnetometer (Fig. 1). The triple-axis accelerometer LIS3L02AS5 [5] was used as a tilt sensor. Six sensors’ outputs are digitized by the 24-bit analogue-to-digital converters and processed by 8-bit PIC18F4550 microprocessor. The data are sent to the PC or PDA via wireless Bluetooth interface or via USB interface. The data are then used to show attitude in the form of an artificial horizon with heading indication (Fig. 2). A dedicated graphical user interface was also developed for PDA and PC in NI LabVIEW. 3. Basic algorithm to calculate compass attitude values Simple electronic compasses equipped with only dual-axis magnetic sensor without any tilt compensation use Eq. (1) to calculate Heading . Hx , Hy , Hz , are the components of measured Earth’s
Fig. 2. 3D-compass module.
magnetic field vector and the subscript b denotes vector coordinate measured with respect to body reference frame (sensors are fixed to the object and their sensitivity axes are parallel to the longitudinal, lateral and vertical object axes): heading :
Hyb
= arctg
(1)
Hxb
More versatile compass modules with a tilt compensation using a triple-axis accelerometer calculate attitude values according to Eqs. (2)–(4). These equations are implemented in embedded software in 3D-compass module. It should be noted that they are valid only in a steady position of the compass or during a uniform rectilinear motion: roll :
˚ = arctg
pitch :
Gyb
(2)
Gzb Gxb
= arctg
2 Gyb
heading :
= arctg
(3)
2 + Gzb
Hyb · cos ˚ − Hzb · sin ˚ Hxb · cos + Hyb · sin · sin ˚ + Hzb · sin · cos ˚
(4)
4. Test cases The electronic compass is a device for determining the orientation with respect to a navigation frame. The orientation values are usually given in terms of Euler’s angles called Heading (yaw, ), Pitch (elevation, ) and Roll (bank, ˚). All three Euler’s angles are called attitude and express three degrees of freedom of compass orientation in a certain location. The simulation algorithm used for further error estimation artificially sets the orientation of a compass module. The set of attitude triads is called a group of test cases and should evenly cover the complete attitude range of a compass module. There is a set of ideal gravitational and geomagnetic field vector components associated with each of the test cases. These vector components are used in a mathematical model of a non-ideal compass to calculate the attitude information affected by the simulated sensor imperfection. Knowing the correct attitude value, the attitude errors are calculated for each of attitude triads in the test case group. The graphical representation of the test case group in 3D space is shown in Fig. 3. The AB vector is aligned with compass longitudinal axis x. The attitude of the compass is changed in the way that A point remains in the same position while B point is moving on the surface of a sphere on the line indicated in blue color. Simultaneously the roll of the device is changing as well. The trajectory of the point B in space could be described by parametric equations (5) and
J. Vˇcelák, J. Kubík / Sensors and Actuators A 155 (2009) 233–240
235
shown in Fig. 3 for N = 200, r = 1000, ˚range = ±180◦ , range = ±80◦ , = 360◦ and Nturns = 10. Such a test case group ensures that all three attitude variables of heading, pitch and roll will be equally distributed among test case group triads.
t·
x = r · cos
t·
range N
t·
range N
y = r · cos
z = r · sin pitch :
range N
=t·
· cos
t · Nturns
range 2N
t · Nturns
range 2N
· sin
(5)
range N
range 2N ˚range · 2N ˚=t· Nturns + 1 = t · Nturns
heading : roll :
Fig. 4. Simplification of sensor triplet orthogonality error.
and ϕy are known: 1 + 2N – total number of test cases, r – sphere radius, range – pitch range (≈), range – heading range (2), ˚range – roll range (), Nturns – number of spiral turns, t = −N, N – parameter.
A(x =0◦ ,z =0◦ ,z =90◦ ) =
5. Influence of sensor’s orthogonality error Angular deviation of the sensor sensitivity axis from the ideal sensitivity axis of orthogonal sensor triplet is called orthogonality error. The user usually assumes that the triple-axis magnetometer and accelerometer form ideal sensor triplets with sensor sensitivity axes perpendicular to each other. Nevertheless, the final accuracy of the sensor sensitivity axes placement is often limited by the production process and technology used. The methods of precise determination of the real sensor sensitivities, offsets and angular deviation are presented in Refs. [7,8,12]. While sensor offsets and sensitivities could be determined easily, determination of the true angles between the sensors is quite complicated. If we consider an ideal 3D orthogonal reference system xyz and a real sensor sensitivity axis S placed within this ideal system, the angles and ϕ could be defined as the angle between S and its projection ps to x–y plane and the angle between ps and ideal x axis respectively. Angles and ϕ could be defined for all three sensors’ sensitivity axes. The measured vector H with coordinates in ideal reference system xyz is then transformed to Hb vector measured by the real sensors using Eq. (6):
Hxb Hyb Hzb
⎡
cosx · cos x
sin x · cos x
cos z · cos z
sin y · cos y sin z · cos z
= ⎣ cos y · cos y
=A·H H b
⎤ Hx sin y ⎦ · Hy sin x sin z
Hz
In order to simplify the general matrix A it is convenient to set one of the true sensor axes identical to one of the axes of the ideal coordinate system (e.g. z axis). Then the general transformation matrix A is simplified to expression (7) and the corresponding situation is shown in Fig. 4. The matrix A is used to simulate the imperfections of the sensor triplet in this paper. The inverse matrix A−1 could be easily used to correct sensors’ imperfections when the angles x , y
cos x cos y · cos y 0
0 sin y · cos y 0
sin x sin y 1
(7)
Since three different angles x , y and ϕy define the sensor triplet orthogonality imperfection as parameters, the complexity of the simulation task and result representation can become complicated. The accuracy of the sensor placement in xy plane is usually much better than positioning of z sensor with respect to xy plane. In the case of an anisotropic magnetoresistance (AMR) integrated triaxial sensor circuit, the x and y sensors may be manufactured on a single die, while the z sensor may be on an another die. In order to simplify the situation, we will consider only two variable angles x = y = and ϕy = ϕ. Generally means deviation of the z axis sensor from the direction perpendicular to xy plane and ϕ represents angle between x and y sensors. The resulting maximal Roll, Pitch and Heading error heading = f(, ϕ) could be then conveniently shown in 3D plots. When a compass module is used in a real application, two types of attitude errors could be estimated for the test case group. RMS error (standard deviation) is calculated using Eq. (8) while maximal error is calculated using Eq. (9). xi represents the difference between known attitude angle and calculated attitude angle affected by sensor nonorthogonality for each of the test cases. RMS error does not reflect the maximal error which may become significantly larger than the RMS value for certain singular orientations of the compass. Therefore we have decided to present maximal error and an example of peak-to-peak to RMS ratio (Eq. (10) and Fig. 8). While the RMS error could be low, the peak-to-peak to RMS ratio could be up to 6, which means that absolute maximal error is 6 times higher than the RMS. Due to practical reasons, the knowledge of maximal error is more convenient for the user because he could be sure that output attitude value is always within the given tolerance interval:
(6)
RMS =
1 2 ( xi ) , N−1
xi = xi − xi
(8)
N
xi – simulated value with sensor imperfections, xi – original ideal value. MAX =
max( xi ) − min( xi ) 2
p–p 2 · MAX error ratio = RMS RMS
(9) (10)
236
J. Vˇcelák, J. Kubík / Sensors and Actuators A 155 (2009) 233–240
Fig. 5. Roll error due to accel. ortho. error.
Fig. 7. Heading error due to accel. ortho. error.
The test case group with 400 attitude triads was used to investigate error in various compass orientations. Fig. 5 shows Roll error which is influenced by sensor deviation angle only as could be expected from Eq. (2). Fig. 6 shows Pitch error and Fig. 7 shows resulting Heading error when only accelerometer orthogonality error is considered. Fig. 8 shows (p–p)/RMS error ratio for maximal Heading error presented in Fig. 7. It could be seen that (p–p)/RMS error ratio could reach up to 6. Fig. 9 represents maximal Heading error when only magnetometer orthogonality error is considered. The shape of combined Heading error when both magnetometer and accelerometer orthogonality errors are taken into account depends on the relation between them and could not be easily expressed, however the absolute values do not exceed 16◦ for selected range of the angles and ϕ angular deviations (an example is presented in Fig. 10). 6. Linearity error suppression The sensors used in electronic compasses are usually AMR, magnetoimpedance (MI) or fluxgate sensors. The feedback comFig. 8. (p–p)/RMS Heading error ratio.
Fig. 6. Pitch error due to accel. ortho. error.
Fig. 9. Heading error due to mag. ortho. error.
J. Vˇcelák, J. Kubík / Sensors and Actuators A 155 (2009) 233–240
237
Fig. 12. Attitude values and Heading error for single test case group. Fig. 10. Combined heading error.
pensation of measured magnetic field is convenient from the perspective of suppressing the sensor linearity error, but is not favorable in case of low power applications since the feedback current fed into the compensation coils increases the total power consumption. The linearity error of an AMR sensor (without feedback compensation loop) could reach up to 0.5% of the full scale (FS) [3]. The linearity error could be sufficiently suppressed by mathematical correction in microprocessor. The influence of such suppression is demonstrated on an example of PCB-fluxgate sensors used in the 3D-compass module. The linearity of each sensor was calibrated separately in three independent measurements using PC-controlled triaxial Helmholtz coil system. The range of generated magnetic field was ±60 T. The linearity error of squareshaped dual-axis sensor reached of ±0.8% FS for x axis, ±0.6% FS for y axis and the error of the racetrack-shaped single-axis sensor was ±1.9% FS for z axis. The linearity error waveform was approximated by the 9th order polynomial. The readout values Hx , Hy and Hz are corrected for linearity error using Eq. (11) with N = 9. Obviously, the polynomial coefficients pn differ for each sensor: Hcorr = Hread +
N
n pn · Hread
(11)
n=0
The measured linearity error, the polynomial approximation and the residual error are shown in Fig. 11. The correction algorithm according to Eq. (11) is a part of the embedded compass module software and is calculated in real-time. The results of mathematical linearity error correction for the PCB-fluxgate sensors used in
the 3D-compass module are summarized in Table 1. It can be seen from Fig. 11 and from Table 1 that while x and y sensor linearity error is corrected well, the correction is not that successful in case of z sensor. The possible reason is that this sensor is single axis with a different core shape and the transfer characteristic is more influenced by temperature changes. The comparison of sensor linearity is given in Table 2 for commercially produced magnetometers and compass modules. The linearity error is recalculated to FS = ±100 T for a clear comparison of the linearity errors provided by the manufacturers for various FS ranges. 7. Influence of sensor linearity error on calculated attitude The polynomial approximation of magnetic sensor linearity error was used to model the influence of linearity error on calculated compass heading in the case of uncorrected sensor linearity error. The example of heading error waveform evaluated for one test case group (magnetometer linearity error ±1.5% FS considered) is presented in Fig. 12 including the waveform of modeled attitude values in test cases triads. It could be seen that the maximal final heading error is ±4◦ and the corresponding RMS error is 1.7◦ . The influence of accelerometer and magnetometer linearity error has been examined separately and then the influence of both sensor errors is presented in Fig. 13 (linearity error expressed in % of FS where for magnetometers FS = ±100 T and for accelerometers FS = ±2000 mg; 1 mg = 0.00981 m s−2 ). This figure could be used for estimation of the final accuracy of the combined system as well as for commercially produced magnetometer and compass mod-
Fig. 11. PCB-fluxgate sensor linearity error axes x, y and z (before and after math correction).
238
J. Vˇcelák, J. Kubík / Sensors and Actuators A 155 (2009) 233–240
Table 1 3D-compass PCB-fluxgate linearity error. Sensor axis
Linearity error original ±1450 nT ±725 nT ±2400 nT
x y z
Linearity error after compensation ±119 nT ±99 nT ±450 nT
Lin error in %FS (FS = ±100 T)
Lin error after compensation RMS
0.12% 0.10% 0.45%
44 nT 32 nT 145 nT
Table 2 Compass modules and sensors parameters. Sensor type
FS ±400 T ±200 T ±600 T ±600 T ±300 T ±60 T
HMC5843 HMC1001 HMC1021 HMC1043 AMI601 3D compass
Lin error (%FS)
Lin error recalculated to FS = ±100 T
Sensor description
±0.2% ±0.05% ±0.1% ±0.1% ±1.3% ±1%
±0.8% ±0.1% ±0.6% ±0.6% ±3.9% ±0.6%
AMR, 3-axis, digital output, 12 bit AMR, 1- and 2-axis, analog out AMR, 1- and 2-axis, analog out AMR, 3-axis, analog out MI, 3-axis, digital out, 12 bit PCB-fluxgate, 3-axis, analog out
Table 3 Contribution of the ADC quantization noise to final compass heading error. ADC
Magnet sensor
Accel. sensor
Heading error, magnetometer considered only
Heading error, accel. sensors considered only
Pitch error, accel. sensors considered only
Roll error, accel. sensor considered only
Heading error, both sensors considered
Nom. NOB
ENOB
LSB (nT)
LSB (mg)
Max ± (◦ )
RMS (◦ )
Max ± (◦ )
RMS (◦ )
Max ± (◦ )
RMS (◦ )
Max ± (◦ )
RMS (◦ )
Max ± (◦ )
RMS (◦ )
8 10 12 14 16
7 9 10 12 14
1562.5 390.6 195.3 48.8 12.2
31.25 7.81 3.91 0.98 0.24
2.5 0.7 0.3 0.1 0.03
1.3 0.3 0.17 0.04 0.01
4 1.2 0.46 0.13 0.03
1.5 0.4 0.19 0.05 0.01
1.2 0.4 0.2 0.04 0.01
0.5 0.13 0.07 0.02 0.003
3 1.2 0.4 0.1 0.02
1 0.3 0.13 0.04 0.007
5 1.5 0.6 0.2 0.04
1.9 0.5 0.2 0.06 0.01
ules when the sensor parameters are known. It may be seen that the influence of accelerometer linearity error to the final heading accuracy is approximately two times higher than the contribution of magnetometer linearity error. 8. Influence of sensor noise and ADC quantization noise Although the sensor noise could be an issue in case of accuracy-demanding applications, it is usually neglected in lowcost applications. While peak-to-peak noise of typical AMR sensor is in the range of units to tens of nT [13] the quantization noise of ADCs (8–16 bit) is usually much higher or at least comparable. Therefore the following chapter will be focused mainly on ADC quantization noise which is the driving factor for ADC selection. Sensor output values are always digitized by ADC for further mathematical processing. The ADC is very often integrated in the magnetometer/accelerometer module or in the P used. These
ADCs are usually successive approximation or low-resolution –˙ type converters. The Nominal Number of Bits (NNOB), the estimated achievable Effective Number of Bits (ENOB), magnetic field and field of gravity corresponding to 1 Least Significant Bit (LSB) of ENOB and the attitude error resulting from the ADC quantization error based on uniformly distributed quantization noise within the range of ±1/2 LSB of ENOB are expressed in Table 3. The full scale of magnetometer and accelerometer ADCs is set to ±100 T and ±2000 mg respectively. Apart from the ADC NNOB, the user should also evaluate sensor FS versus ADC NNOB. The example could be given for magnetometer module HMC5843 (Nominal 12 bit, FS = ±400 T) where nominal LSB corresponds to 195 nT and AMI601 (Nominal 12 bit, FS = ±300 T) where nominal LSB corresponds to a lower value of 146 nT providing higher resolution despite the same NNOB of the ADC. The influence of magnetometer and accelerometer quantization noise is considered separately and then the final accuracy influenced by both sensors’ quantiza-
Fig. 13. Combined Heading error due to accel. and mag. linear. error (detail on right).
J. Vˇcelák, J. Kubík / Sensors and Actuators A 155 (2009) 233–240
Fig. 14. Max. heading error due to ADC quantization noise.
Fig. 15. Contribution of accel. and mag. ortho. error to total compass heading error.
tion noise is presented in Fig. 14. Note that the modeled calculation of compass heading was performed with floating point numerical representation. Rounding errors will emerge in case of limited fixed point calculations, but these are not subject of this paper. 9. Conclusion Heading accuracy of an electronic compass was simulated with respect to several possible sensor error sources. A group of test cases was introduced. The test group for each simulated error value consisted of 400 different orientations of the compass module evenly covering the full attitude value range. Potential sources of compass errors were listed and three of these error sources were modeled and investigated separately: sensor axial orthogonality, sensor linearity error and ADC quantization noise. Final compass heading error was estimated for each case in the group and statistic parameters of the group were determined and presented as maximal peak-to-peak (p–p) error, standard deviation and (p–p)/RMS ratio in the 3D plots. It was found that orthogonality error has a very substantial contribution to the final heading accuracy. Accelerometer orthogonality error has a major influence on the resulting Heading accuracy since the error of Roll and Pitch calculated from accelerometers outputs further propagates to the calculation of Heading (Eqs. (2)–(4)). In case of limited accelerometer placement precision, the orthogonality error should be compensated numerically. Generally, when the worst cases are taken into account it could be said that the contribution to the final heading error is given by the lines presented in Fig. 15. Nevertheless, when the orthogonality errors are comparable for magnetometer and accelerometer, the combined error is driven almost exclusively by accelerometer error contribution. It has to be mentioned that a simplification of the general axial devi-
239
ation error has been used in order to present final errors easily. The shape of the resulting heading errors presented in Figs. 9 and 10 may vary according to the mutual relation of both errors. Sensor linearity error plays also substantial role in the final compass heading accuracy. It could be seen from Fig. 13 that again the accelerometer imperfection is the major source of the compass heading error. The 3D-compass module with PCB-fluxgate sensors and accelerometer was introduced and linearity error suppression was demonstrated on this device system. Surprisingly it has been demonstrated that low-resolution ADCs with 10–12 bit could be used also when low total heading error (less than 2◦ ) is required. Nevertheless the reader has to consider that all quantization noise was recalculated to ADC FS = ±2000 mg for accelerometer and FS = ±100 T for magnetometer. When other types of errors are taken into account, further increase of the ADC resolution does not bring reasonable increase of compass heading accuracy. The demonstrated methods are applicable to estimate the heading accuracy of any compass module including commercially produced compass IC modules from the sensor parameters given by the manufacturer. The results presented above could be conveniently used for components selection in case of a new compass design. Corresponding magnetic sensor, accelerometer and matching ADC could be easily chosen to meet the final heading error requirement. The combination of the following components could be used as an example of well-matched design: • magnetometer (linearity error 0.8% FS, max orthogonal deviation ±0.4◦ ), • accelerometer (linearity 0.5% FS, max orthogonal deviation ±0.2◦ ), • ADC 12 bit for both magnetometer and accelerometer. When these components are combined to form an electronic compass, the final maximal heading error should not be more than of ±1.5◦ (0.5◦ RMS). Acknowledgments This research was funded by Enterprise Ireland (EI) Commercialization Fund Technology Development grant no. CFTD/05/IT/316. References [1] M.J. Caruso, Application of magnetoresistive sensors in navigation systems, Sens. Actuators (February) (1997) 15–21 (SAE SP 1220). [2] http://www.ssec.honeywell.com. [3] http://www.aichi-mi.com. [4] http://www.nxp.com. [5] http://www.st.com/stonline/products/families/sensors/motion sensors.htm. [6] http://www.analog.com. [7] J. Merayo, P. Brauer, F. Primdahl, J.R. Petersen, O.V. Nielsen, Scalar calibration of vector magnetometers, Meas. Sci. Technol. 11 (2) (2000) 120–132. [8] J. Vˇcelák, Application of Magnetic Sensors for Navigation Systems, vol. 2, Shaker-Verlag, 2009, ISBN 978-3-8322-7738-3. [9] J. Vˇcelák, P. Ripka, J. Kubík, A. Platil, P. Kaˇspar, AMR navigation systems and methods of their calibration, Sens. Actuators A 123–124 (2005) 122–128. [10] J. Vˇcelák, P. Ripka, A. Platil, J. Kubík, P. Kaˇspar, Errors of AMR compass and methods of their compensation, Sens. Actuators A 129 (2006) 53–57. [11] J. Kubik, L. Pavel, P. Ripka, P. Kaspar, Low-power printed circuit board fluxgate sensor, IEEE Sens. J. 7 (1–2) (2007) 179–183. [12] J. Vcelak, Calibration of triaxial fluxgate gradiometer, J. Appl. Phys. 99 (08) (2006) D913. [13] H. Hauser, P.L. Fulmek, b P. Haumer, M. Vopalensky, P. Ripka, Flipping field and stability in anisotropic magnetoresistive sensors, Sens. Actuators A: Phys. 106 (September (1–3)) (2003) 121–125.
Biographies Jan Vˇcelák received his Ing. degree (M.Sc. equivalent) in March 2003. He successfully defended his Ph.D. thesis “Application of magnetic sensors for nav-
240
J. Vˇcelák, J. Kubík / Sensors and Actuators A 155 (2009) 233–240
igation systems” under supervising of Dr. Petr Kaˇspar in November 2007 at the Faculty of Electrical Engineering of Czech Technical University in Prague. In 2006/2007 he worked in Ricardo Prague on development of SW&HW for hybrid vehicles. He has joined Tyndall NI in Cork in December 2007. His research interests include sensors (magnetometers, accelerometers and gyroscopes), multisensor systems calibration and simulation, algorithms for compass, AHRS and IMUs. Jan Kubík received his Ing. degree (M.Eng. equivalent) in the field of computercontrolled measurement systems in March 2003 at the Faculty of Electrical
Engineering of Czech Technical University in Prague. He successfully defended his Ph.D. thesis “PCB-Fluxgate Sensors” under supervision of Prof. Pavel Ripka in December 2006 at the same university. He also worked at the National University of Ireland in Galway (2001 and 2002) and at the Lappeenranta University of Technology in Lappeenranta, Finland (2003). He has joined the Tyndall National Institute in Cork in September 2006 as a postdoctoral researcher. His research interests range from magnetic field sensors design and characterisation to computer-controlled measurement systems development, FEM modelling of magnetic systems and microcontroller applications.