A Novel Procedure for In-field Calibration of Sourceless
Inertial/Magnetic Orientation Tracking Wearable Devices
D. Campoloa , M. Fabrisa, G. Cavalloa, D. Accotob , F. Kellerc , E. Guglielmellia
a Biomedical
Robotics & EMC Lab
Neuroscience and
Neural Plasticity Lab
Università Campus Bio-Medico
via Longoni, 83
00155 Roma - Italy
c Developmental
Abstract— Recent research in the emerging field of Phenomics aims at developing unobtrusive and ecological technologies which allow monitoring the behavior of infants and
toddlers. Orientation tracking devices based on accelerometers
and magnetometers represent a very promising technology since
orientation in 3D space can be derived by solely relying upon
the direction of the natural geomagnetic and gravitational fields
which constitute an absolute coordinate frame of reference, i.e.
sourceless.
Many commercially available devices allow on-board calibration by means of addition of external circuitry, mainly used to
generate artificial fields which act on the sensor itself as a known
forcing input. Addition of external circuits is a major drawback
in applications such as the one of interest, where the technology
has to be worn by infants.
When external fields, (e.g. gravitational and geomagnetic
fields) are present, alternative calibration techniques are possible
which rely on predefined orientation sequences of the sensor.
In standard procedures, prior knowledge of the external field
(magnitude and direction) as well as accuracy in performing
the predefined orientation sequences contribute to determine
the calibration parameters.
In this work, a novel procedure for in-field calibration of magnetometric sensors is presented which does not rely on previous
knowledge of magnitude and direction of the geomagnetic field
and which does not require accurately predefined orientation
sequences. Such a method proves especially useful in clinical
applications since the clinician is no longer compelled to execute
accurate calibration protocols.
Index Terms— Wearable Technology, Posture Tracking, Movement Analysis, Inertial/Magnetic Sensors.
I. I NTRODUCTION
Motion tracking finds application in a broad set of fields,
ranging from virtual reality to medicine, from film industry to
biomedical research. As shown in [1] and references therein,
although no “silver bullet” exists, motion tracking can count
on a “respectable arsenal” of available technologies, each with
its own advantages and disadvantages with respect to a given
application.
At Campus Bio-Medico University, research is being carried out in the emerging field of Phenomics [2] with the
aim of developing unobtrusive and ecological technologies
corresponding author: d.campolo@unicampus.it
b CRIM
Center for Research
in Microengineering
Scuola Superiore Sant’Anna
P.zza Martiri della Libertà, 33
56127 Pisa - Italy
which allow monitoring the behavior of infants and toddlers,
especially during their first two years of life.
Behavior monitoring includes, for example, tracking a
child’s posture, tracking the head direction (which mainly
relates to the child’s attention), tracking position and/or
orientation of toys while the child plays with them, etc. . .
At this moment, the most suitable technologies are being
screened for this purpose. The selected ones will be deployed
to be embedded in a child’s everyday environment, e.g. in toys
and clothes. Special attention is paid to technologies which do
not require costly equipment (e.g. photogrammetric systems)
and/or a structured environment (e.g. motion analysis laboratories). The technology of interest should be able to work in
clinical settings as well as at home and should be operated
by minimally trained personnel, e.g. the child’s caregiver.
In this sense, orientation tracking based on inertial/magnetic sensors [3], [4], [5] represents a promising technology since, as shown in next sections, orientation of a rigid
body can be measured solely relying upon gravitational and
geomagnetic fields, which are present everywhere on earth,
without the need of other sources of fields, i.e. sourceless.
Accelerometers and magnetometers are nowadays available
in packages small enough to be worn or embedded into toys
and can be used to track position/orientation in unstructured
environments.
Section II develops the mathematics needed to relate the
inertial/magnetic sensors read-outs to the spatial orientation
of the system itself.
Emphasis is given to the geometrical description since geometrical considerations are also at the basis of the proposed
calibration procedure.
Orientation in space is described in terms of rotation
matrices, fully exploiting the geometry of the problem. An
invariant, i.e. coordinate-free, formulation has been adopted
which presents, to the authors’ knowledge, a certain degree
of novelty, at least when applied to compass-based navigation
systems [6], [7].
In Section III, the proposed calibration procedure is described.
Section IV presents the experimental verification of the
proposed calibration procedure and the discussion of the
results.
Finally, conclusive remarks are provided in Section V.
II. S OURCELESS I NERTIAL /M AGNETIC O RIENTATION
T RACKING
Accelerometers, in static conditions, directly provide a
measure of the gravity vector g which is always vertical with
respect to the earth surface.
Magnetometers measure the geomagnetic field b when no
other magnetic source is present. Geomagnetic field has an
horizontal component b which points towards North and a
vertical component b ⊥ which depends on the latitude. Define
now a third vector:
∆
(1)
h =g×b
where “×” is the vector product in 3D Euclidean space. It is
worth noting that h is never null (since g and b are never
collinear) and always points towards East. The three vectors
g, b, and h are therefore independent and can be used to
define a convenient fixed coordinate frame {x 0 , y0 , z0 }:
- z0 : unitary vector pointing Up
- x0 : unitary vector pointing East
- y0 : unitary vector pointing North
This can be expressed in invariant (i.e. valid in every coordinate system) geometrical terms:
∆
z0
= −g/g
x0
= h/h
y0
= z0 × x0 = h × g/h × g
∆
∆
where g = g ≈ 9.8 m/s2 . It is worth noting how
components simplify with respect to the fixed coordinate
frame:
g =
b =
h =
0 x0
0 x0
h x0
+
+
+
0 y0
b y 0
0 y0
−
+
+
g z0
b⊥ z0
0 z0
(2)
where h, the scalar equivalent of Eq.(1), is defined as:
∆
h = g b
(3)
Let now {x1 , y1 , z1 } be an orthonormal frame (referred
to as moving frame) defined via the sensitive axes of the
sensorized system (the tilted box in Fig. 1) whose orientation
should be determined with respect to the fixed frame.
With reference to Fig. 1, the fixed frame {x 0 , y0 , z0 }
and the moving frame {x 1 , y1 , z1 } are related by a rototranslation. The translation is ignored hereafter (the origins
of the two coordinate frames shall always be imagined as
coincident) since only relative orientation is of interest. The
rotation can be expressed by a matrix R defined as:

 0 1
x · x x0 · y1 x0 · z1
(4)
R =  y0 · x1 y0 · y1 y0 · z1 
z0 · x1 z0 · y1 z0 · z1
R
Up
North
z0
East
y0
y1
z1
x1
x0
Fig. 1. Fixed and moving coordinate frames. The rotation matrix R can be
viewed as: i) an operator which transforms each versor of the fixed frame
(x0 , y0 , z0 ) into the corresponding versor of the moving frame (x1 , y1 , z1 );
ii) a change of coordinates mapping components of a vector expressed in the
moving frame into the components of same vector expressed with respect to
the fixed frame.
where “·” represents the Euclidean scalar product. A peculiar
property of rotation matrices is that the inverse and the
transpose coincide, i.e.:

 0 1
x · x y0 · x1 z0 · x1
R−1 = RT =  x0 · y1 y0 · y1 z0 · y1 
(5)
x0 · z1 y0 · z1 z0 · z1
Accelerometers and magnetometers are displaced along
with the moving frame and provide read-outs which correspond, respectively, to the three components (g x , gy , gz ) of the
gravitational field 1 and to the three components (b x , by , bz ) of
the magnetic field with respect to moving coordinate frame.
By the invariance of Eq.(1), auxiliary read-outs (h x , hy , hz )
with respect to the moving frame can readily be defined:

 
 
 

hx
gx
bx
gy bz − gz by
 hy  =  gy  ×  by  =  gz bx − gx bz  (6)
hz
gz
bz
gx by − gy bx
Geometrically, each

 gx = g · x1
gy = g · y 1

gz = g · z1
which can

gx
 gy
gz
be expressed as:

 hx = h · x1
hy = h · y 1

hz = h · z1
be compacted into an equivalent
 
g · x1 b · x1
b x hx
b y hy  =  g · y 1 b · y 1
b z hz
g · z1 b · z1
By substituting

gx
 gy
gz
or equivalently:

R−1
set of read-outs can

 bx = b · x1
by = b · y 1

bz = b · z1
matrix notation:

h · x1
h · y1 
(7)
h · z1
Eq.(2), Eq.(7) can be rewritten


b x hx
0
0
by hy  = R−1  0
b
b z hz
−g b⊥
gx
=  gy
gz
bx
by
bz
 
hx
0
hy  ∗  0
−g
hz
0
b
b⊥
as:

h
0 
0
(8)
−1
h
0 
0
where “∗” is usual the matrix product.
1 In static conditions the gravitational field is the only source of acceleration. In general, the apparent acceleration should also be considered but in
applications such as the one of interest it can be negligible, see [3].
Eq.(8) simply states that R −1 , besides being an operator
which rotates vectors, can also be interpreted as a change
of coordinates which transforms components of a vector
expressed in the fixed frame into components of same vector
expressed with respect to the moving frame.
Recalling that R = (R−1 )T for rotation matrices and that
h was defined in Eq.(3), the following holds:


hy
hz
hx
1
R =  gx b⊥ + g bx gy b⊥ + g by gz b⊥ + g bz  (9)
h
−gx b
−gy b
−gz b
with respect to the fixed frame (i.e. [0 b b⊥ ]T ) are constant
throughout space 2 . As the device is oriented in space, the
moving reference attached to it (refer to Fig. 1) is also
subjected to the same orientation and the field components in
the moving frame (i.e. [b x by bz ]T ) change accordingly. As
pointed out in the previous section, this change of coordinates
is fully determined by the rotation matrix R.
Rotations in space are linear transformations which act on
vectors preserving their modules (a major property of rotation
matrices):
Eq.(9) allows determining the rotation matrix R once the
sensors read-outs (g x , gy , gz ) and (bx , by , bz ) are known,
remaining values (h x , hy , hz ) can be computed via Eq.(6).
b2x + b2y + b2z = constant = b2 + b2⊥
III. M AGNETOMETER C ALIBRATION P ROCEDURE
In the previous section, it was implicitly assumed that
accelerometers and magnetometers would provide direct
measurement of the components of the gravitational fields
(gx , gy , gz ) and of the geomagnetic field (b x , by , bz ). In fact
sensors are just transducers and provide an output voltage v
that, in the best scenario, is proportional to variations of the
measurand m. In practical terms: v = k m + v o , where k
and vo respectively represent the linear gain and the offset
value. A calibration procedure is needed to determine such
coefficients in order to derive the measurand m = (v −v o )/k,
i.e. the fields components to be used in formulas such as
Eq.(9).
Parameters vo and k are easily determined when situations
exist where the measurand assumes (at least) two known
values. In the case of accelerometers, the measurand can
easily assume values 0, +g and −g by simply aligning (e.g. by
means of mechanical set-ups such as a pendulum) the sensor’s
axis, respectively, orthogonally, parallel and anti-parallel with
the vertical direction. When it comes to the magnetic field,
alignment of the sensor’s axis with the field’s direction is not
straightforward. For this reason an additional field has to be
generated.
Commercially available devices such as the Honeywell
HMC105X, a family of multi-axes magneto-resistive sensors,
contain purposefully designed “offset straps”, i.e. spirals of
metallization that couple with each sensitive axis of the device
producing an additional magnetic field. Such patented feature
can be used for auto-calibration of the sensor.
Such procedure requires addition of extra circuitry used
to drive each offset strap. In applications where infants are
supposed to wear such technology, reduction of components
is highly desirable.
For this reason a procedure which would only rely on
the natural geomagnetic field distribution and that would not
require any accurate alignment was investigated, as presented
in the following.
A. Proposed Calibration Procedure
The proposed procedure solely relies upon uniformity of
the natural geomagnetic field. This means that components
This means that, as the device is being displaced and oriented
in space, the geomagnetic vector is seen by the moving frame
as a time-varying vector of constant module, i.e. the trajectory
of its end-point (b x , by , bz ) is bound to stay on a sphere
centered in the origin and of radius:
b = b2 + b2⊥ .
(10)
In terms of sensor output voltages (v x , vy , vz ), considering
different offset voltages (v ox , voy , voz ) and different linear
gains (kx , ky , kz ) for each axis, the sphere becomes now an
off-centered ellipsoid:
vx − vox
kx
2
+
vy − voy
ky
2
+
vz − voz
kz
2
= b2
or equivalently:
vx − vox
b kx
2
+
vy − voy
b ky
2
+
vz − voz
b kz
2
= 1 (11)
Such an ellipsoid is uniquely identified by the six parameters
(vox , voy , voz , b kx , b ky , b kz ), the first three identify the center of the ellipsoid, the remaining three identify the semi-axis
length of the ellipsoid.
At least six independent equations are needed in order to
determine the unknown parameters. Such equations can be
derived as part of the calibration procedure: a sequence of N
orientations in space allows measuring the sensor read-outs
(vxi , vyi , vzi ) relatively to each orientation i = 1 . . . N .
As a result of such measurements, a set of N equations
becomes available, the i-th equation being:
vxi − vox
b kx
2
+
vyi − voy
b ky
2
+
vzi − voz
b kz
2
= 1 (12)
Because of the errors (measurement noise) affecting the
sensor read-outs, least-squares fitting methods applied to a
larger number (N 6) of measurements can be deployed to
obtain an estimate of the six unknown parameters which is
less sensitive to measurement noise.
2 Such requirement is not too strict although some care should be taken,
refer to [3] for details of usage in clinical practice.
+
p
N
3 The
4 It’s
appendix at the end of this paper provides explanation for this choice.
worth recalling that translations do not influence the sensors.
z−axis
y−axis
x−axis
2.3
2.2
2.1
2
1.9
1.8
i=1
A. Experimental Setup
In order to validate the previously described calibration
procedure, an experimental setup consisting of:
- one HMC1051: a 1-axis magnetometer sensing the field
in the z−direction;
- one HMC1052: a 2-axis magnetometers sensing the field
in the xy−plane;
- three amplifying stages: one for each axis;
was developed.
Fig. 2 shows how the signal coming from each axis of the
device was amplified by means of an operational amplifier in
a differential configuration.
The device was placed in a wooden box and randomly
(manually) moved around in 3D space for a few seconds while
amplified data were being acquired and stored on a computer
for later processing. Care was taken not to simply translate 4
the box but to provide random orientation as well.
Fig. 3 shows the experimental data as acquired voltages at
the output of the three amplifying stages.
R3
2.4
e2i (p)
IV. E XPERIMENTAL V ERIFICATION
In this section, the experimental setup and how the experimental data are derived is first discussed. Then the numerical
procedure that implements the least-squares method to fit
acquired data is presented. As a result, the six parameters
defining the ellipsoid that best fits the experimental data are
determined together with the statistics of the fitting error.
R2
Rd
1.7
1.6
0
100
Fig. 3.
200
300
400
number of acquisitions
500
600
Measurement sequence: 3-axis sensor’s amplified read-outs.
2.3
2.3
2.2
2.2
2.1
2.1
x
min
output
+
2.5
c2
(13)
where p = [x0 y0 z0 a b c]T is the vector of 6 unknown
parameters which fully determine the ellipsoid.
As shown next, such vector p will be eventually determined
by numerically solving:
_
x
b2
R1
Fig. 2. Differential amplification (R1 = R2 = 2.2 kΩ and R3 = R4 =
150 kΩ) of the magneto-resistive bridge (Ra = Rb = Rc = Rd = 850 Ω,
nominal values) relative to a single axis.
output voltage [V]
+
R4
Rc
Rb
v [V]
a2
Ra
v [V]
(xi − x0 )2
(yi − y0 )2
(zi − z0 )2
+
+
−1=0
2
2
a
b
c2
where i = 1 . . . N and where obvious changes of notations
(e.g. vxi → xi , vox → x0 , b kx → a, etc. . . ) are made.
Least-squares fitting is probably one of the most widely
used approaches for estimating ellipses’ parameters [9]. In
general there exist no exact solution satisfying all of the N
equation, what can be found is an estimate which minimizes a
given error-of-fit (EOF) function. Several choices are possible
for the EOF, see [9] for details.
In this paper, the following EOF was chosen 3 :
∆
ei (p) = (xi − x0 )2 + (yi − y0 )2 + (zi − z0 )2 ·
1
1 − (xi −x0 )2 (yi −y
2
(zi −z0 )2
0)
+5V
Magnetoresistive
bridge
B. Least-Squares Fitting
For sake of clarity, rewrite the N nonlinear equations in
Eq.(12) as:
2
2
1.9
1.9
2.2
2.1
2.1
2
2
1.9
1.9
1.8
vy [V]
1.7
1.8
1.9
1.8
vy [V]
v [V]
z
1.7
1.9
1.8
v [V]
z
Fig. 4. Left: “cloud” of measurements, i.e. the trajectory of measurement
sequences in 3D space. Right: best fitting ellipsoid (thin lines) superimposed
with cloud of measurements (thick lines).
B. Numerical Procedure
Considering the 3D space of measurements (x i , yi , zi ), the
sequence of N measurements appears as a cloud of points
(see the left plot in Fig. 4) distributed along the surface of
the ellipsoid yet to be determined (presence of measurement
noise causes these points not to perfectly lie on the surface).
In order to numerically determine the ellipsoid (i.e. its
six parameters), calculations were carried out in the MATLAB environment, specifically making use of lsqnonlin
(a MATLAB function for nonlinear least-squares problems).
Such a function requires two input arguments:
- the EOF: implemented as another function returning an
N dimensional vector [e 1 (p) e2 (p) . . . eN (p)]T , here
each ei (p) is defined as in Eq.(13);
- an initial guess p0 of the unknown parameter p.
As an initial guess for first three parameters, the coordinates
of center of mass of the cloud of measurements were chosen
(every measurement is given a unitary mass). For the remaining three parameters (representing the semi-axis length of the
ellipsoid), the mean radius between the previously guessed
center of mass and the measured data points was chosen.
In particular:
∆
p0 = [x̄ ȳ z̄ r̄ r̄ r̄ ]T
where x̄, ȳ, z̄, and r̄ are defined as follows:
x̄ =
N
1 xi ;
N i=1
r̄ =
ȳ =
N
1 yi ;
N i=1
z̄ =
N
1 zi ;
N i=1
N
1 (xi − x̄)2 + (yi − ȳ)2 + (zi − z̄)2
N i=1
Remark: a “good” initial guess leads the lsqnonlin to
quickly converge towards the optimal solution. The more uniform is the distribution of the measurements on the ellipsoid,
the closer is the center of mass (x̄, ȳ, z̄) to the center of the
ellipsoid. A “good” guess of the center of mass also leads to
a “good” guess of the radius.
The proposed initial guess together with the EOF proposed
in Eq.(13) led the lsqnonlin to converge to the vector:
p = [2.07 1.94 1.87 0.27 0.27 0.07]T
(V olts)
The relative ellipsoid (centered in x 0 = 2.07 V , y0 =
1.94 V , z0 = 1.87 V and with semi-axis lengths a = b =
0.27 V and c = 0.7 V ) is shown in the right plot of Fig. 4,
where the cloud of measurements is also superimposed.
The goodness of the fitting, besides the graphical representation, can be inferred by statistics of the errors e i (p):
- average: −0.38 mV
- RMS: 0.15 V
C. Discussion of Results
The best fitting ellipsoid, besides being off-centered due to
bridge offsets, revealed different sensitivities among different
axes. In particular, the z-axis (c = 0.07 V ) is much less
sensitive than the x and y axis. This is due to the fact that
an HMC1051 was used to sense the field in the z-axis while
an HMC1052 was used for the remaining ones. In practical
situations, different devices may display different sensibilities
for a variety of reasons. The proposed calibration procedure
proved capable of overcoming such problems.
In order to guarantee convergence of the least-squares
algorithm, as previously remarked, no restriction is posed
on the calibration trajectory of measurements except for the
uniformity of its distribution. This directly translates into a
clinical protocol which requires the human operator (who,
for example, wears such devices on his/her upper limbs) to
perform a set of predetermined actions (lifting an arm, pointing right or left etc. . . ) which need to be only qualitatively
described, i.e. a procedure which is more suitable to a clinical
practice.
V. C ONCLUSION AND F UTURE W ORK
In this work a novel procedure for in-field calibration of
inertial/magnetic wearable devices for orientation tracking
was presented. Although several techniques for calibrating
such devices already exist, a novel method was investigated
with the specific aim of being deployed in clinical practice,
where existing procedures often prove impractical and lead
to disuse.
Emphasis was placed on the fact that a human operator
would perform the orientation sequence needed for calibration and that such orientation sequence should be only
qualitatively described, requiring no particular dexterity or
performance accuracy.
A geometrical description of the problem was used to
show that, although randomly generated, all the measurements
were expected to lie on the surface of an ellipsoid. The
least-squared fitting method was used to determine such
an ellipsoid described by six parameters representing three
offsets and three gains values needed for calibrating a 3-axis
magnetic device (this work focused on magnetometers since
accelerometers generally pose much fewer problems).
Accurate calibration was finally obtained by having a large
number of random orientation sequences rather than a few
accurately performed ones. To this end, the least-squares
fitting approach proved essential both to exploit the large
number of measurements and to gain robustness with respect
to measurement noise.
As future work, robustness of the proposed calibration
procedure will be statistically tested against several clinical
protocols (of increasing complexity) performed by minimally
trained personnel.
ACKNOWLEDGEMENT
The authors would like to thank Oliver Tonet for useful
discussions on least-squares methods.
A PPENDIX
E RROR - OF -F IT F UNCTION
In this section a brief explanation of the reasons behind
the choice of Eq.(13) as a candidate error-of-fit function is
provided. For sake of simplicity, the problem will be stated
in terms of ellipses (2D) since generalization to ellipsoids in
3D space is straightforward.
(xP,yP)
A
(xo,yo)
P
B
O
Fig. 5. Distance of a point P from an ellipse: although BP represents the
true geometrical distance, AP is used instead as an analytically convenient
approximation.
The purpose of an error-of-fit function is defining a sort (not
necessarily positive definite) of distance of a generic point
from an ellipse.
With reference to Fig. 5, for a point P of coordinates
(xP , yP ) the most intuitive choice would be the geometrical
distance BP . Unfortunately for a generic point P off the ellipse, the analytical derivation of B is not so straightforward.
As an analytically convenient approximation, the length of
AP can be used instead, where the point A is simply where
the ray OP intersects the ellipse.
Define the following function:
2 2
x − xO
y − yO
∆
+
F (x, y) =
a
b
and let F (x, y) = 1 be the equation of the ellipse. A generic
point P of coordinates (x P , yP ) will therefore lie on the
ellipse if F (xP , yP ) = 1, outside the ellipse if F (xP , yP ) > 1
and inside the ellipse if F (xP , yP ) < 1.
Since by definition the point A lies on the ellipse, it is
straightforward computing its coordinates (x A , yA ):
1
xA
xO
xP − xO
=
+
yA
yO
F (xP , yP ) yP − yO
The length of OA is simply:
xO )2 + (yA − yO )2
dOA = (xA − 1
= F (xP ,yP ) (xP − xO )2 + (yP − yO )2
= F (xP1,yP ) dOP
Since O, A and P are collinear, d AP = dOP − dOA , i.e.:
1
2
2
dAP = (xP − xO ) + (yP − yO ) 1 −
F (xP , yP )
Eq.(13) is simply a generalization of the previous formula
to the 3D case.
Note: dAP is not strictly speaking a distance because it is
not positive definite but it does not matter since in the leastsquares method its squared value will be used instead.
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