Maximum likelihood calibration of a magnetometer using inertial sensors

Maximum likelihood calibration of a magnetometer using inertial sensors
 Maximum likelihood calibration of a
magnetometer using inertial sensors
Manon Kok and Thomas Schön
Linköping University Post Print
N.B.: When citing this work, cite the original article.
Original Publication:
Manon Kok and Thomas Schön, Maximum likelihood calibration of a magnetometer using
inertial sensors, 2014, In Proceedings of the 19th IFAC World Congress, 2014, 92-97.
Series: World Congress, ISSN 1474-6670; Volume# 19, Part# 1.
ISBN: 978-3-902823-62-5.
Postprint available at: Linköping University Electronic Press
Maximum likelihood calibration of a
magnetometer using inertial sensors
Manon Kok ∗ Thomas B. Schön ∗∗
Division of Automatic Control, Linköping University, SE-581 83
Linköping, Sweden (email: [email protected])
Department of Information Technology, Uppsala University,
SE-751 05 Uppsala, Sweden (e-mail: [email protected])
Abstract: Magnetometers and inertial sensors (accelerometers and gyroscopes) are widely used
to estimate 3D orientation. For the orientation estimates to be accurate, the sensor axes need to
be aligned and the magnetometer needs to be calibrated for sensor errors and for the presence
of magnetic disturbances. In this work we use a grey-box system identification approach to
compute maximum likelihood estimates of the calibration parameters. An experiment where
the magnetometer data is highly disturbed shows that the algorithm works well on real
data, providing good calibration results and improved heading estimates. We also provide
an identifiability analysis to understand how much rotation is needed to be able to solve the
calibration problem.
Keywords: Magnetometers, calibration, inertial sensors, maximum likelihood, grey-box system
identification, sensor fusion.
Inertial sensors (3D accelerometers and 3D gyroscopes)
in combination with 3D magnetometers are in many applications used to obtain orientation estimates using an
extended Kalman filter (EKF). When the sensor is subject
to low accelerations, the accelerometer measurements are
dominated by the gravity component from which it is
possible to deduce information about the inclination. The
magnetometer measures the local magnetic field vector
and its horizontal component can be used to obtain heading information. The orientation estimates are only accurate when the sensor axes of the inertial sensors and the
magnetometers are aligned and when the magnetometer is
properly calibrated. This calibration consists of two parts.
First, the magnetometer needs to be calibrated for errors
inherent in the sensor, for instance non-orthogonality of
the magnetometer sensor axes. Second, it needs to be calibrated for the presence of magnetic disturbances. These
magnetic disturbances are frequently present due to the
mounting of a magnetometer and cause a constant disturbance that needs to be calibrated for. When using a
magnetometer it is therefore always advisable to calibrate
it before use.
Our main contribution is a new practical algorithm for
calibration of a magnetometer when we also have access
to measurements from inertial sensors rigidly attached to
the magnetometer. The algorithm implements a maximum
likelihood (ML) estimator to find the calibration parameters. These parameters account for magnetometer sensors
errors, the presence of constant magnetic disturbances
caused by mounting the magnetometer close to magnetic
? This work is supported by MC Impulse, a European Commission,
FP7 research project and CADICS, a Linnaeus Center funded by the
Swedish Research Council (VR).
Fig. 1. Calibration results for experimental data where the
original magnetometer data is plotted in red and the
calibrated magnetometer data is plotted in blue.
objects and misalignments between the magnetometer and
the inertial sensors. An illustration of the results obtained
from calibrating a magnetometer using the proposed algorithm is available in Fig. 1.
In many practical applications, mounting the magnetometer onto for instance a car or a boat severely limits the
rotational freedom of the sensor. A secondary contribution
of this work is a quantification of how much rotation is
needed to be able to solve the calibration problem. This
is done via an identifiability analysis, deriving how much
rotation is needed in the case of perfect measurements.
Many recent magnetometer calibration approaches are
based on ellipse fitting. These calibration algorithms use
the fact that when rotating a magnetometer, its measurements should lie on a sphere if the magnetometer measures
a constant local magnetic field vector, i.e. the norm of the
magnetic field should be constant. The presence of magnetic disturbances or magnetometer sensor errors leads to
an ellipsoid of data instead. Many calibration algorithms
are used to obtain improved orientation estimates. Hence,
the radius of the sphere, i.e. the actual magnitude of
the local magnetic field vector, is irrelevant and we can
without loss of generality assume that its norm is equal
to 1. The calibration has therefore been successful when
the calibrated measurements can be seen to lie on a (unit)
sphere (recall Fig. 1). The problem of fitting an ellipsoid
of data to a sphere was considered for example by Gander
et al. [1994]. As shown by Markovsky et al. [2004] the
ordinary least squares estimate is inconsistent when using
measurements corrupted by noise. Different calibration
approaches have been developed to overcome this problem,
see e.g. Gebre-Egziabher et al. [2006], Renaudin et al.
An ellipse fitting method can only solve the calibration
up to an unknown rotation. Combining the magnetometer
with inertial sensors requires the sensor axes to be aligned,
i.e. the sphere needs to be oriented such that the magnetometer sensor axes are aligned with the inertial sensor
axes. A few recent approaches add a second step to the calibration procedure to estimate the misalignment between
the magnetometer and the inertial sensor axes [Vasconcelos et al., 2011, Li and Li, 2012, Salehi et al., 2012, Bonnet
et al., 2009]. They first use an ellipse fitting approach
and determine the misalignment between the magnetometer and the inertial sensor axes in a second step. These
approaches generally use the accelerometer measurements
for estimating the misalignment, but discard the gyroscope
measurements. Troni and Whitcomb [2013] focus on using
the gyroscope measurements to obtain an estimate of the
misalignment. Our work follows a similar approach, but
makes use of both the gyroscope and the accelerometer
measurements and aims at obtaining a maximum likelihood estimate by combining both steps into a nonlinear
optimization problem.
Our solution to the magnetometer calibration problem
makes use of a nonlinear state space model describing the
sensor’s orientation qt and its measurements,
qt+1 = ft (qt , ut , θ) + B(qt )vt (θ),
yt = ht (qt , θ) + et (θ).
Here, qt denotes the state variable representing the sensor
orientation encoded using a unit quaternion. Furthermore,
ut and yt denote observed input and output variables,
respectively. The dynamics is denoted by ft (·) and the
measurement model is denoted by ht (·). Finally, vt and et
represent mutually independent i.i.d. process and measurement noise, respectively and B(qt ) is a matrix describing
how the noise vt affects the state qt . The model is introduced in detail in Section 3.
The calibration problem is formulated as a maximum
likelihood grey-box system identification problem. Hence,
based on N measurements of the observed input u1:N =
{u1 , . . . , uN } and output y1:N = {y1 , . . . , yN } variables,
find the parameters θ that maximize the likelihood function,
θbML = arg max pθ (y1:N ),
where Θ ⊆ Rnθ . Using conditional probabilities and the
fact that the logarithm is a monotonic function we have
the following equivalent formulation of (2),
θbML = arg min −
log pθ (yt | y1:t−1 ),
where we have used the convention that y1:0 , ∅. The
ML estimator (3) enjoys well-understood theoretical properties including strong consistency, asymptotic normality,
and asymptotic efficiency [Ljung, 1999]. The state space
model (1) is nonlinear, implying that there is no closed
form solution available for the one step ahead predictor
pθ (yt | y1:t−1 ). This can be handled using sequential
Monte Carlo methods (e.g. particle filters and particle
smoothers), see e.g. Schön et al. [2011], Lindsten and
Schön [2013]. However, for the magnetometer calibration
problem under study it is sufficient to make use of a more
pragmatic approach; we simply approximate the one step
ahead predictor using an extended Kalman filter (EKF).
The result is
pθ (yt | y1:t−1 ) ≈ N yt | ybt|t−1 (θ), St (θ) ,
where N yt | ybt|t−1 (θ), St (θ) denotes the probability density function for the Gaussian random variable yt with
mean value ybt|t−1 (θ) and covariance St (θ). Here, St (θ) is
the residual covariance from the EKF [Gustafsson, 2012].
Inserting (4) into (3) results in the following optimization
kyt − ybt|t−1 (θ)k2S −1 (θ) + log det St (θ),
2 t=1
which we can solve for the unknown parameters θ. The
problem (5) is non-convex, implying that a good initial
value for θ is required.
3.1 Dynamic model
We model the orientation qt as the sensor’s orientation
from the body frame b to the navigation frame n at time t,
expressed as qtnb . The body frame b is the coordinate frame
of the inertial sensor with its origin in the center of the
accelerometer triad and its axes aligned with the inertial
sensor axes. The navigation frame n is aligned with the
earth gravity and the local magnetic field vector. The
dynamic model of the orientation (1a) takes the gyroscope
measurements as an input, which are modeled as
yω,t = ωt + δω + vω,t ,
where ωt denotes the angular velocity, δω denotes the gyroscope bias and vω,t ∼ N (0, Σω ). The dynamic equation for
the orientation states is then given by [Gustafsson, 2012]
= I4 + T2 L(yω,t − δω ) qtnb + T2 L̄(qtnb )vω,t .
Here, I4 denotes a 4 × 4 identity matrix, T denotes the
sampling time, the matrices L̄(qtnb ) and L(yω,t − δω ) are
given by
−q1 −q2 −q3
0 −ω1 −ω2 −ω3
0 ω3 −ω2 
 q −q3 q2 
L̄(q)= 0
, L(ω)= 1
, (8)
q3 q0 −q1 
ω2 −ω3
0 ω1 
−q2 q1 q0
ω3 ω2 −ω1
where the subscripts denote the different components of
the respective vectors.
3.2 Measurement model
The measurements in (1b) consist of the accelerometer
measurements ya,t and the magnetometer measurements
ym,t . The former are modeled as
ya,t = Rtbn (ant − g n ) + ea,t ≈ −Rtbn g n + ea,t ,
where ant denotes the sensor’s acceleration which is assumed to be approximately zero, g n denotes the gravity
vector in navigation frame and ea,t ∼ N (0, Σa ). The
matrix Rtbn is the rotation matrix representation of the
quaternion qtbn = (qtnb )c , where c denotes the quaternion
conjugate [Hol, 2011].
The magnetometer measurements are modeled as
ym,t = DRtbn mn + o + em,t ,
where the local magnetic field is denoted by m and
em,t ∼ N (0, Σm ). Note that mn is assumed to be constant.
The calibration matrix D and offset vector o contain
corrections for both the magnetometer sensor errors, the
presence of magnetic disturbances and the misalignment
between the magnetometer and inertial sensor axes. The
magnetometer and inertial sensor axes are assumed not to
be mirrored with respect to each other. Kok et al. [2012]
provide a more detailed derivation and discussion of this
3.3 Parameter vector
The parameter vector θ to be estimated consists first
of all of the calibration matrix D, the offset vector o
and the local earth magnetic field mn as discussed in
Section 3.2. Furthermore, the gyroscope bias δω introduced
in Section 3.1 and the noise covariances Σω , Σa and Σm
of the three sensors are treated as unknown parameters.
The different components of the unknown parameters θ
therefore consist of
D ∈ R3×3 ,
o∈R ,
mn ∈ {R3 : ||mn ||22 = 1, mnx > 0, mny = 0},
δ ω ∈ R3 ,
Σω ∈ {R
Σa ∈ {R
Σω 0, Σω = ΣT
ω },
Σa 0, Σa = Σa },
Σm 0, Σm = ΣT
m },
Σm ∈ {R
where mnx and mny denote the x- and y- component of mn ,
Although (11c) and (11e) - (11g) suggest that constrained
optimization is needed, it is possible to circumvent this via
a suitable reparametrization. The covariance matrices can
be parametrized in terms of their Cholesky factorization,
leading to only 6 parameters for each 3 × 3 covariance
matrix. The local magnetic field can be parametrized using
only one parameter denoted by d [Kok et al., 2012].
The resulting calibration algorithm is given in Algorithm 1. The optimization problem is solved using a
Broyden-Fletcher-Goldfarb-Shanno (BFGS) method with
damped BFGS updating [Nocedal and Wright, 2006].
Algorithm 1 Magnetometer and inertial calibration
b 0 , ob0 , m
b ω,0 , Σ
b a,0 ,
(1) Set i = 0 and initialize D
b n0 , δbω,0 , Σ
b m,0 .
(2) Repeat,
(a) Run an EKF using the current
b a,i 03×3
n b
Di , obi , m
b i , δω,i , Qi = Σω,i , Ri =
b m,i
03×3 Σ
and obtain {b
yt|t−1 }N
t=1 , S1:N .
(b) Determine θi+1 using the numerical gradient of
the objective function in (5), its approximate
Hessian and a line search algorithm.
b i+1 , obi+1 , m
b ω,i+1 , Σ
b a,i+1 ,
(c) Obtain D
b ni+1 , δbω,i+1 , Σ
b m,i+1 from θbi+1 .
(d) Set i := i + 1 and repeat from 2(a) until convergence.
Since the optimization problem (5) is a non-convex problem, it is important to start Algorithm 1 with a good
initialization of the parameters. We will do this using a
two-step approach.
In a first step the ellipsoid of magnetometer data is
mapped onto a unit sphere, using an ellipse fitting approach [Gander et al., 1994]. The second step is based
on the invariance of the inner product of two vectors
under rotation and is similar to approaches in Vasconcelos
et al. [2011], Li and Li [2012], Salehi et al. [2012], Bonnet
et al. [2009]. This requires estimates of the inclination
which can be obtained by running the EKF using only the
accelerometer and gyroscope data. These two steps result
b 0 , ob0 and m
in initial estimates D
b n0 and are described in
more detail in Kok et al. [2012]. The gyroscope bias and
the noise covariance matrices can be initialized based on
a small period of stationary data.
For magnetometer calibration, the sensor needs to be
rotated in all possible orientations so the magnetometer
measurements describe (a part of) an ellipsoid. Work on
magnetometer calibration generally assumes that the sensor can be “sufficiently” rotated for the calibration parameters to be identified. Often, however, magnetometers are
mounted in such a way that their movement is more or less
constrained to the 2D plane [Wu et al., 2013a,b]. Although
some approaches do consider the case of limited rotation,
to the best of the authors’ knowledge there is no study on
how much rotation is sufficient. We will now answer this
question via an identifiability analysis. Note that our analysis assumes noise-free measurements and it therefore gives
a lower bound on the number of measurements needed to
identify the calibration parameters. In many applications,
rotations around the z-axis are possible, while rotations
around the other axes are constrained. We therefore ask
the question of how much rotation around the x- and yaxes is needed in addition to rotation around the z-axis.
Identifiability analysis of the calibration parameters θc =
vec D oT d , where d is used to parametrize mn as
described in Section 3.3, can be performed by studying
local observability of an extended system [Walter, 1982].
This extended system consists for our problem of the state
qt (see (1a)) and the parameters θ. Assuming noise-free
and bias-free measurements, the orientation at different
time steps is known from the inertial measurements, save
for the initial heading. The analysis can therefore be
limited to the study of θc , where it needs to be kept in
mind that the initial heading, denoted by α, is unknown.
Discrete-time observability analysis considers the system
of equations
ym,1 = DR1bn (α)mn + o = hm (R1bn , α, θc ),
ym,2 = DR2bn (α)mn + o = hm (R2bn , α, θc ),
ym,K = DRK
(α)mn + o = hm (RK
, α, θc ).
We will analyze the identifiability of the parameters θc
assuming that the initial heading is known. The identifiability of the initial heading will be discussed separately
afterwards. The parameters θc are said to be locally idenbn
such that we can solve for
tifiable if there exists an R1:K
θc in (12).
Due to the nonlinear nature of (12) it is typically hard to
analyze it directly. Instead, observability is often studied
by considering a linearized version of (12), making use of
the Jacobian
∇hm (R1bn , α, θc )
∇hm (R2bn , α, θc )
J =
∇hm (RK
, α, θc )
where ∇ denotes the derivative with respect to θc . If (13)
is full rank for all θc , the parameters are said to be locally
weakly identifiable [Nijmeijer and van der Schaft, 1990].
The Jacobian J can be built by stacking different measurements on top of each other. For notational simplicity,
we consider 90◦ -rotations around the axes. Note that the
analysis is equally valid for any other amount of rotation.
For rotations of 0◦ , 90◦ and −90◦ around the z-axis, 90◦
and −90◦ around the x-axis and 90◦ and −90◦ around the
y-axis, the Jacobian becomes
∂mx 
mx I3
 03×3
 03×3
 m I
 x 3
J (θc )= mx I3
 m I
 z 3
−mz I3
m x I3
−mx I3
−mz I3
mz I3
vec D
mz I3
mz I3
mz I3
D3 ∂d + D1 ∂d
D3 ∂m
∂d + D2 ∂d 
z − D ∂mx 
D3 ∂m
∂mz 
03×3 I3 D1 ∂d − D2 ∂d 
∂mx 
03×3 I3 D2 ∂m
∂d + D1 ∂d ,
∂mx 
−mx I3 I3 D1 ∂d − D3 ∂d 
∂mz 
mx I3 I3 D3 ∂m
∂d − D1 ∂d 
} |{z} |
where Di denotes the i’th column of the matrix D. For
clarity, the contributions of the different rotation axes are
separated by dashed lines. For each column it is explicitly
indicated which part of the derivative with respect to θc
it represents.
The rank of (14) can be studied for subsets of its rows.
Considering only rotations around the z-axis, i.e. the first
J (θc ) full rank
#meas z-axis
#meas x-axis
#meas y-axis
Table 1. Summary of how many measurements (i.e. lower bound) around which axes
are needed for identifiability of the calibration
parameters θc .
three rows in (14), rank J (θc ) = 9 as long as mx 6= 0
(the calibration is not performed on the magnetic north
or south pole). The matrix will not gain any rank from
adding more measurements around the z-axis. Adding two
measurements around the x- and/or y-axis can be shown
to lead to rank J (θc ) = 13 under very mild conditions
on the calibration matrix D in case the calibration is
not performed on the equator or on a magnetic pole. A
sufficient condition for this is that det D 6= 0. On the
equator and the magnetic poles it is also possible to obtain
identifiability of all calibration parameters. In these cases,
however, stricter requirements on the measurements are
needed. The results on how many measurements around
which axes are needed are summarized in Table 1. It is
straightforward to show that also α is identifiable based
on these measurements, provided that the calibration is
not performed on one of the magnetic poles. In this case,
the initial heading is not identifiable for any number
of measurements since the horizontal component of the
magnetic field is zero.
The identifiability analysis above was done assuming
90◦ rotations between each measurement. The same result holds, however, for any other difference in rotations.
Although the amount of rotation does not influence the
identifiability of the calibration parameters, it will indeed
influence the quality of the estimates. This can be understood in terms of the condition number [Kailath et al.,
2000], i.e. the ratio between the maximum and minimum
singular value of the Jacobian (14). Any singular values
being zero implies that θc is not identifiable, but a smaller
condition number also implies that certain parameters are
more difficult to estimate. Fig. 2 shows the singular values
of the Jacobian in (14) for different amounts of rotation.
Five measurements are considered, corresponding to three
measurements around the z-axis, one around the x-axis
and one around the y-axis for D = I3 , o = (0 0 0)
and m equal to that in Linköping, Sweden. The amount
of rotation between the measurements is assumed to be
equal for all four differences and is varied between 0◦ (red)
and 90◦ (blue). As can be seen, the larger the amount of
rotation, the smaller the condition number of (14).
It can therefore be concluded that rotation around the zaxis only is never enough to identify the calibration parameters θc . The parameters will, however, become identifiably
already with very little excitation in the other directions
as summarized in Table 1. More excitation around the
different axes will always lead to better estimates as can
be concluded from Fig. 2.
6.1 Experimental setup
An experiment has been conducted with an Xsens MTi-100
inertial measurement unit (IMU), collecting inertial and
Norm magnetic field [a.u.]
Singular values
Fig. 2. Singular values of the Jacobian in (14) for small
rotation ranges (red) to large rotation ranges (blue).
Fig. 3. Experimental setup where a calibration experiment
is performed outdoors. An Xsens MTi-100 IMU (orange box) together with a magnetic disturbance is
placed in a aluminum block.
magnetometer data using a sampling frequency of 100 Hz.
The experimental setup is shown in Fig. 3. The data has
been collected outdoors to ensure that the local magnetic
field is as homogeneous as possible. A magnetic disturbance has been rigidly attached to the sensor, causing
severe magnetic disturbances. Both the block in which the
IMU is placed and the table shown in Fig. 3 are made of
non-magnetic material.
The sensor is slowly rotated around all three axes for
roughly 2 minutes. The collected magnetometer data is
shown in red in Fig. 1. Note that for plotting purposes
this data has been downsampled to 1 Hz.
To aid the convergence of the optimization problem (5), a
few iterations of step 2 in Algorithm 1 were run for parts of
the parameter vector, first estimating only D, o, mn and
δω and afterwards only estimating Σω , Σa and Σm . The
results from these steps were used as an initial estimate
to obtain the final maximum likelihood estimate of the
parameters θ.
6.2 Experimental results
The calibration resulted in the following calibration matrix
b and offset vector ob
0.74 −0.14 0.02
b = −0.12 0.68 0.01 , ob =
1.22 .
−0.04 0.43 1.00
Correcting the magnetometer measurements with the obtained calibration matrix and offset vector leads to the
corrected measurements shown in blue in Fig. 1, which
Before calibration
After calibration
Time [s]
Fig. 4. Norm of the magnetic field measurements before
(red) and after (blue) calibration.
Fig. 5. Normalized residuals St
(yt − ybt|t−1 ) from the
EKF after calibration for the original data set (left)
and the validation data set (right). A Gaussian distribution (red) is fitted to the data.
can be seen to lie on a unit sphere. To better visualize the
deviation from the unit norm, the norm of the magnetic
field before and after calibration is depicted in Fig. 4. The
norm after calibration can be seen to lie around 1.
A histogram of the normalized residuals after calibration
from the EKF St
(yt − ybt|t−1 ) is shown in Fig. 5. A
Gaussian distribution has been fitted to the normalized
residuals showing that the residuals are indeed more or
less Gaussian distributed.
6.3 Validation
The calibration results have been applied to a second data
set recorded with the same experimental setup as in Fig. 3.
The results are very similar to the results displayed in
Fig. 1 and Fig. 4, implying that the calibration parameters
can well be used to calibrate the magnetometer measurements of the other data set. As can be seen from Fig. 5, the
resulting residuals still look fairly Gaussian, save for the
presence of a small number of outliers. Closer inspection
of the residuals and the data shows that these outliers are
the result of high accelerometer measurements due to a
small impact during the collection of the validation data
6.4 Heading estimation
An important goal of magnetometer calibration is to facilitate good heading estimates. To check the quality of
Calibrated mag [a.u.]
The authors would like to thank Laurens Slot, Dr. Henk
Luinge and Dr. Jeroen Hol from Xsens Technologies for
their support in collecting the data sets and for interesting
Time [s]
Fig. 6. Calibrated magnetometer data of an experiment
rotating the sensor into 24 different sensor orientations where the blue, green and red lines represent the
data from the x-, y- and z-axis of the magnetometer,
z up
z down
x up
x down
y down
Table 2. Difference in estimated orientation
between two subsequent rotations around the
sensor axes using calibrated magnetometer
data. The values represent the deviation in
degrees from 90◦ .
the heading estimates after calibration, the block in which
the sensor is placed (shown in Fig. 3) is rotated around
all axes. This block has right angles and it can therefore be placed in 24 orientations that differ from each
other by 90◦ . The calibrated magnetometer data of this
experiment is shown in Fig. 6. Orientation estimates are
determined by taking the mean value of 500 magnetometer
and accelerometer samples in each orientation and using
the accelerometer to estimate the sensor’s inclination and
the magnetometer data to estimate its heading. After calibration, we expect the difference of the estimated heading
between each subsequent rotation to be 90◦ . Note that
when rotating around an axis, the sensor is always rotated
back to its initial position, enabling the computation of
4 orientation differences per rotation axis. Table 2 enlists
the deviation from 90◦ between two subsequent rotations.
As can be seen, the deviation is very small, indicating
that good heading estimates are obtained after calibration.
Note that the metal object causing the magnetic disturbance as shown in Fig. 3 physically prevents the setup
to properly be placed in all orientations around the yaxis. Rotation around the y-axis with the y-axis pointing
upwards has therefore not been included in Table 2.
A grey-box system identification approach has been used
to compute maximum likelihood estimates of magnetometer calibration parameters for a magnetometer in combination with inertial sensors. The results show that the
algorithm works well on real data and leads to improved
heading estimates.
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