Method of identifying and eliminating magnetic compass deviation V.V.Meleshko, S.L. Lakoza, S.A.Sharov Aerospace & Navigation Instruments Dept. National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute» Kiev, Ukraine
[email protected],
[email protected] Abstract — It is considered method of identifying and eliminating deviation of any type magnetic compass. It is used Poisson model for describing object’s magnetic field. This approach allows to describe magnetic deviation more precise in compare with Archibald Smith technique that uses magnetic deviations model in the form of a Fourier series with an approximate calculation of the Fourier coefficients. In a short time the method allows to determine the parameters of the Poisson model when object moves. It doesn’t require performing special deviation. Using these parameters you can calculate the deviation and eliminate it with power or algorithmically means. The method allows to determine both the horizontal and heeling deviation. Method is suitable for large objects such as ships and aircraft, and for the small object such as unmanned maneuverable vehicle or different navigational robots. Keywords – magnetic compass; magnetic deviation; Poisson model.
I.
INTRODUCTION
Magnetic compass has been important navigational device for many mobile objects for many years. Its main advantage is autonomous work without electrical power often [1-3]. It is installed on ships, airplanes, cars. Magnetic compass can use the magnetic needle, and can use various types of magnetometers together with the pendulum or accelerometers. Now in compasses small-sized magnetoresistive magnetometers are widely used [6]. In complex with micromechanical gyros and accelerometers it allows build attitude and heading reference system (AHRS). It is significantly expanded the scope of such AHRS application due size miniaturization and power consumption. AHRS is used by unmanned aerial vehicles, various types of robots, medical equipment, etc. A common problem of magnetic compasses and magnetometers are errors so called magnetic deviation. There are caused by distortion of earth magnetic field from magnetic fields of the moving object. Methods for eliminating of magnetic deviation have been improved from ancient times to the present day [4, 5]. This paper shows novel promising method of identifying and eliminating magnetic compass deviation.
II. MAGNETIC DEVIATION CALCULATION It is known, magnetic deviation is the difference between the compass (device) heading K к and magnetic heading K м :
δ Kк K м . In the 19th century Archibald Smith introduced an approximate description of the magnetic deviation in the form of series [1, 2]
= A + B sinKK + C cosKK + D sin2KK +E cos2KK, (1) where A, B, C, D, E - deviation coefficients. According to Smith deviation coefficients are calculated after making measurements on the 8 headings. Large object alignment on to the 8 headings respectively to the ground reference or accurate gyrocompass is highly difficult process. Coefficients of semicircular deviation are [2]
B (δ90 δ270 ) / 2 ; С (δ0 δ180 ) / 2 ,
(2)
where the deviation index corresponds to the heading at which the measurement is carried out. Coefficients’ calculating using the Fourier series has the following equation:
B
1 δ90 δ270 sin(45o )(δ45 δ315 δ225 δ135 ) ; 4 (3)
C
1 δ0 δ180 cos(45o )(δ45 δ315 δ225 δ135 ) . 4
1
Numerical analysis shows that the use of the approximate eq. (2) instead of the exact eq. (3) may give a calculating error in the deviation up to 30%. The exact value of the magnetic deviation can be obtained using the magnetic compass value of and headings that can be derived from Fig. 1.
Here, apart from the previously mentioned, a, b, c, d, e, f, g, h, k - Poisson parameters describing the magnetic soft iron, P, Q, R – hard iron components from installation object. Considering magnetometers errors, errors of roll and pitch sensors the expression for magnetic device heading will be
Magnetic heading can be found [7]
( X cos γ Z sin γ) . K м arctg ( X sin γ Z cos γ)sin θ Y cosθ
(4)
According to the Fig.1 in (4) X, Y, Z - components of the Earth's magnetic field (EMF) in the projections on the body frame; γ, θ - the object’ angles of roll and pitch.
K к1 arctg
wherein
( X 'cos γ Z 'sin γ) , (7) ( X 'sin γ Z 'cos γ)sin θ Y 'cos θ
θ, γ, X ' and etc. – device’s values.
Magnetic deviation corresponds
δ1 Kк1 K м
(8)
Determination errors of the magnetic deviation, i.e. the difference of values calculated using (1) and (8), can be represented by the graph on Fig. 2. [8]
Fig.1. Coordinate system and angles Compass heading can be described by the equation
K м arctg
( X cos γ Z sin γ) . (5) ( X sin γ Z cos γ)sin θ Y cosθ
In (5) X ', Y ', Z ' - magnetic induction components at the installation place of magnetic compass. They can be described by the Poisson model X = X + aX + bY + cZ +P; Y = Y + dX + eY + fZ + Q; Z = Z + gX + hY +kZ + R.
(6)
Fig.2. Errors of deviation’s calculation
Fig. 2 shows it is necessary to use the exact eq. (8) for calculating the magnetic deviation. The inputs of (8) are components of the induction at installation place of magnetic compass measured by three-axis magnetometer, roll and trim angles measured by the gyroscopic system and EMF induction components X, Y, Z. These components can be found using the horizontal H and vertical components Zg of induction vector T. They can be calculated using the National Geophisical data Centre calculator from the information on the latitude and longitude of the object. If the object doesn’t have three axes magnetometer then you have to calculate the components X ', Y ', Z ' according to (6) using information
2
on the Poisson parameters a, b, c, d, e, f, g, h, k, and the components P, Q, R.
The vector components of vector
X H c12 Z g c13 ,
III. DETERMINATION METHOD OF POISSON MODEL PARAMETERS
Y H c22 Z g c23 ,
Let represent the model (6) in vector-matrix form introducing the following vectors and matrices.
X ' T' Y ' , Z '
a b F d e g h
c f, k
P Μ Q ; R
0 Tg H . Z g
X T Y , Z
magnetic induction vector of hard iron effects,
Tg - EMF
vector components (horizontal H and vertical -Zg) in projections on the navigational frame. The vector relation
Z H c32 Z g c33 . Magnetic heading is related to the true heading relation
ψ with
ψ м ψ D , where D - magnetic declination.
Rewrite the expression (6) in the form
X ' X a b Y ' Y d e Z ' Z g h
Here T ' - the sum of Earth magnetic field vector and the object magnetic field vector in installation place, F - soft iron parameter Poisson matrix, T - magnetic induction vector in projections on body frame Ox C yC zC (Fig.1), Μ - the
T will result in
c X P f Y Q k Z R
(9)
T ' (I F)CTg M CTg FCTg M .
(10)
or as
Magnetometers triad model can be represented as
T associated with the vector Tg using
T C bg Tg , where C bg (further C) - direction
cosines matrix between navigation and body frames, the relative position of which is set using angles of magnetic heading, heel and roll. TABLE I.
U mx m11 U m my 21 U mz m31
m12 m22 m32
m13 X U mx 0 nmx m23 Y U my 0 nmy ; m33 Z U mz 0 nmz
DIRECTION COSINES MATRIX
C bg xC
ξм
ηм
ζ
C11
C12
C13
yC
C21
C22
C23
zC
C31
C32
C33
C12 cos γsin ψ м sin γsin θ cos ψ м ,
C13 sin γ cosθ , C22 cosθ cos ψ м , C23 sin θ , C32= sin γsin ψ м cos γsin θ cos ψ м , C33 cos γ cosθ .
m11 Here K mu = m21 m 31
m12 m22 m32
U mx 0 m13 m23 , Um0 = U my 0 - scale m33 U mz 0
factors matrix and vector of zero offset, respectively. Vector measurement noise
nm [ nmx nmy nmz ]T. The output signal of the magnetometer in a voltage
Um K mu T Um0 nm ,
Um K mu (I F)T M
Um0 nm .
3
Random noise is smoothed by averaging signals.
Δi FTi M FCi Tg M ,
The magnetometer output signal in units of the magnetic induction
Tm K m (I F)T M
T0 ,
(11)
i - number of measurement. Difference vector is T Δ ΔX ΔY ΔZ .
where
Tm - the induction vector measured by magnetometer induction, K m - scale factors normalized matrix, T0 -
Represent (11) in the unfolded state
where
ΔX i a b ΔYi d e ΔZ g h i
magnetometer zero offset. Next parameters are needed for determination: Poisson matrix parameter components and hard iron components M of magnetic induction.
c Xi P f Yi Q . k Zi R
ΔX i a b ΔYi d e ΔZ g h i
Taking into account the non-linearity of dependence T CTg , determination of coefficients is difficult to realize analytically.
X c P i Y f Q i . Z k R i 1
It proposed the following method: 1. Through a predefined time intervals under arbitrary or predetermined object’s positions using gyro horizon compass are measured: - Heading angles
When the number of measurements is
ψ , roll γ and pitch ϑ;
- The output magnetometers signals
ΔX 1 ... ΔX n a b ΔY1 ... ΔYn d e ΔZ ... ΔZ g h n 1
Um .
2. Find EMF components H and Zg, declination D using coordinates from the satellite system and National Geophysical Data Centre calculator.
i = 1 ... n
X ... X n c P i Y ... Yn . (12) f Q i Z ... Z n k R i 1 ... 1
Let introduce the matrices notation
3. Perform calculations: 3.1. Calculate the direction cosines matrix C and then vector T CTg . 3.2. Introduce a correction magnetometer calibration parameters
using the K mp и T0p
passport
ΔX 1 ... ΔX n a b B ΔY1 ... ΔYn , S d e ΔZ ... ΔZ g h n 1
X i ... X n c P Y ... Yn f Q , A i . ... Z n Z i k R 1 ... 1
Then compact form of (12) is B=S A p 1 m
T1 (K ) (Tm T ) . As a result,
p 0
Using the least squares method (LSM)
S B Aт A Aт , 1
T1 (I F)T M .
3.3. For each of the defined positions in p.1 it are calculated differences
T1 CTg FT M .
Let introduce the notation
Δ T1 CTg .
where
A Aт A A т
1
is the pseudoinverse matrix.
In another form,
3.4. Measured magnetometer’s series value under object moves (circulation and pitching) can be represented as
4
Thus, we find the estimation of the Poisson parameters and magnetic induction vector of hard iron.
estimation errors, deg.
ST (NT N)1 NT BT , where N AT .
According to the obtained estimates deviation can be hold and eliminated with magnets or power current coils, or with algorithmic compensation.
0.05
0
-0.05 0
20
40
60
80
100
120
140
160
180
Deviation, deg.
10
IV. SIMULATION It was done simulation of determination the Poisson parameters and hard iron magnetic induction vector. It was calculated the magnetic deviation and eliminated it algorithmically. It were analyzed variants for problem solving on different moving objects (ship, vessel, aircraft, vehicle, small-sized object). The following characteristics of the devices were used: for gyro horizon compass (or similar system) we accept RMS measurement error in roll and pitch o o angles - 0.1 , heading - 0.3 . For magnetometers were considered next parameters: zero offset - T0 = [1.35 2 2.5], noise level - σ m 10 nT . We define a ship pitching with
5 0 after compensation
-5 -10 0
20
40
60
80
100
120
140
160
180
20
40
60
80
100
120
140
160
180
Heading, deg.
200 150 100 50 0
0
Time, s.
Fig. 4. Deviation error and its eliminating for aircraft
o
amplitude of 5 degrees for trim and yaw, 10 - roll; object heading is constant. Random component of the pitching -
σ p 1 . Time discretization is 1s. The graphs on Fig. 3 show
Deviation, deg.
estimation errors, deg.
the heading of the ship, calculated deviation value, the estimation of deviation error corresponding to the residual o deviation in the range 0…0.1 . Similar residual deviation value is obtained when the ship was pitching.
V.
Carried research leads to make conclusion that the proposed method of identifying and eliminating magnetic compass deviation allows to make compensation of magnetic deviation with high accuracy. To eliminate deviations user don’t need to perform special object maneuvers. Used equipment doesn’t lay increased requirements. Work time does not exceed several minutes. In developed method the condition for good work is presence of object spatial motion (pitching) with range up to 3°.
0.2 0.1 0 -0.1 -0.2 0
20
40
60
80
100
120
140
160
[1]
5
[2]
0 after compensation
-5 -10 0
REFERENCES
180
[3] 20
40
60
80
100
120
140
160
180
10
Heading, deg.
CONCLUSIONS
[4]
5 0 -5 -10 0
20
40
60
80
100
120
140
160
180
[5]
Time, s.
Fig. 3. Deviation error and its eliminating for a vessel [6] [7]
For aircraft on a turn with a roll in 17 degrees residual deviation does not exceed 0.05 degrees (Fig. 4)
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