A method for deriving displacement data during cyclical movement using an inertial sensor
1 Structure and Motion Laboratory, The Royal Veterinary College, Hawkshead
Lane, Hatfield, Hertfordshire, AL9 7TA, UK
2 Structure and Motion Laboratory, Royal National Orthopaedic Hospital,
Brockley Hill, Stanmore, Middlesex, HA7 4LP, UK
* Author for correspondence (e-mail: tpfau{at}rvc.ac.uk)
Accepted 25 April 2005
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Summary |
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Key words: inertial sensor, accelerometer, motion analysis, gait, horse, kinematics
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Introduction |
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One important application of this approach is the quantification of
external work. External work, the work that an individual performs against the
environment that results in changes in the potential energy
(Ep) and kinetic energy (Ek) of the
centre of mass (CoM), represents a major component of the mechanical work of
locomotion (Cavagna, 1975) and
is often evaluated as a measure of locomotor efficiency on different surfaces
(Ferris et al., 1998
), in
locomotion with prosthetic limbs (Skinner
and Effeney, 1985
; Tesio et
al., 1998
) or to justify gait choice
(Minetti et al., 1999
).
Vertical displacements and velocities in three dimensions are needed to
calculate changes in Ep and Ek.
External work is traditionally calculated either from force plate data or by
summing individual body segment Ek and
Ep. Both are time-consuming approaches; the former
requires a force plate and knowledge of initial conditions, and the latter
requires a large set of marker coordinates and knowledge of segmental inertial
properties (Buchner et al.,
2000
; Minetti et al.,
1999
). These analyses are therefore rarely undertaken and are
usually confined to treadmill locomotion or low-speed locomotion over a force
platform. Since the influences on external work of factors such as incline,
surface stiffness and gait are of interest, this represents a major
limitation. Another disadvantage is the differences between treadmill and
over-ground locomotion that have been reported for both horses and humans
(Elliott and Blanksby, 1976
;
Nigg et al., 1995
). In the
horse, when compared to over-ground locomotion, treadmill locomotion displays
several key mechanical differences including reduced vertical trunk
displacement and increased stride length
(Barrey et al., 1993
;
Buchner et al., 1994b
).
A good estimate of CoM displacement and velocity can be gained from overall
trunk movement since limb movement is out of phase in the majority of bipedal
and quadrupedal gaits and changes of potential energy of the limbs are small
(data from Wilson et al.,
2003). This is especially true for cursorial animals due to their
relatively low limb mass (e.g. 5.8% and 5.5% of the total body mass for the
hind limbs and forelimbs of the horse, respectively;
Buchner et al., 1997
). Although
limb, head and visceral movements cause the CoM to change location within the
trunk, visceral movements can only be quantified by the use of force platforms
as ergometers (Cavagna, 1975
).
If head movement is considered an important factor for CoM movement, it can
easily be captured by an additional marker (in case of optical motion capture)
or an additional sensor (this approach) and included in the calculations of
the CoM. However, if the CoM can be considered to lie in a fixed position,
then the six degree of freedom (DoF) movement of that position can be
determined either from the movement of at least three markers in three
dimensions or the six DoF position of a single point that lies a known
distance and orientation from the assumed CoM. This simplifies the measurement
procedure; however, motion analysis data are still required.
Here, we are interested in examining movement outside the laboratory when
collection of motion capture or force plate data is not feasible. The approach
is to integrate acceleration data to determine position. Through the advent of
low-cost, miniature triaxial accelerometers, direct measurement of
acceleration has become feasible. However, the process of determining velocity
and position by integration from acceleration is more problematic than the
reverse. Errors rapidly accumulate during the integration process and
additional knowledge in the form of initial conditions is required for
determination of integration constants. Attempts to track motion by
integration of accelerometer signals are therefore often unsuccessful unless
low-pass filtering is permitted at each integration or very high quality,
expensive and bulky equipment is used
(Barrey et al., 2001;
Barrey and Galloux, 1997
;
Leleu et al., 2002
). The
technique is therefore usually the preserve of military engineering.
In the integration process, changes in accelerometer orientation must be
accounted for since an accelerometer measures acceleration relative to its
orientation rather than to the earth or global coordinate system. This
underlies the application of accelerometers as inclinometers in electronic
spirit levels, where they determine the component of gravity that acts
orthogonal to the level. Accelerometer orientation has previously been
determined in two ways; by optical motion capture systems
(Hedrick et al., 2004), which
encounter the inherent problems discussed above, and by angular rate
gyroscopes. The angular velocity output of a gyroscope can be integrated to
determine orientation, so three orthogonal gyroscopes can be used to sense the
orientation of a triaxial accelerometer. Again, integration errors will
accumulate over time, resulting in drift, and knowledge of initial orientation
is required. One solution is to use the known orientation of the earth's
gravitational field, but this becomes difficult during movement and does not
enable correction for the integration errors. It also does not allow for
correction of heading. Alternatively, it is possible to obtain an absolute
measure of orientation using the earth's magnetic field, as in a compass,
which uses the horizontal components for heading determination. The earth's
magnetic field also has a vertical component (hence why an un-weighted compass
needle will dip), and a triaxial magnetometer can therefore sense absolute
two-dimensional (2D) orientation, since it is not possible to sense rotations
about the axis of the earth's magnetic field. A combination of accelerometers
and magnetometers will therefore give absolute three-dimensional (3D)
orientation, except at the magnetic North and South pole (where gravity and
earth magnetic field are parallel).
Sub-miniature accelerometers, gyroscopes and magnetometers, with a mass of 1 g or less and costing as little as $10, are readily available. These MEMS (micro-electro-mechanical systems) accelerometers contain silicon beams, which deform during acceleration. Resultant changes in capacitance are processed within the chip, which outputs a voltage relative to the applied acceleration. Miniature gyroscopes that sense Coriolis forces when rotations are applied to a pair of oscillating tines have been developed for applications such as movement compensation in hand-held video cameras, and small magnetometers are widely employed in electronic compasses.
Several companies have developed inertial sensors combining all three technologies, which give orientation in a global coordinate system (relative to the earth's magnetic and gravitational fields). These are used in a wide array of applications, such as virtual reality systems, where they sense the orientation of the head mounted display of the subject. Since they give orientation information and can sense 3D acceleration, these transducers provide the basis for determining position by integration of acceleration. However, as alluded to above, the integration process from acceleration to velocity and displacement remains challenging. Here, we focus on the implementation of an appropriate integration procedure that, in combination with the available orientation estimates, allows us to track movements with six DoF. This process would allow the application of miniature inertial sensors for an enormous range of animal tracking and locomotion studies, making studies of real-life activities under non-laboratory conditions possible.
The approach proposed is based on the cyclical nature of locomotion. This means that it is possible to mean subtract the data, effectively meaning that the sensor path is constrained to return to its starting point over a cycle. The period used for mean subtraction will represent a trade-off between minimising the accumulation of integration errors and capturing features that extend over a longer period of time (for instance, tripping and jumping).
For example, during steady-state, level locomotion it is reasonable to assume that changes in the Ek and Ep of the trunk between strides are either small or zero. This means that trunk displacement data should follow a closed loop (subtracting the effect of forward motion). The average velocity over the stride (or several strides) should be zero, and the average forwardbackward and side-to-side acceleration should be zero. Furthermore, the measured average vertical acceleration should equal g (the gravitational constant). Thus, stride-by-stride mean subtraction of acceleration and of the calculated velocity vectors before integration theoretically enables determination of the integration constants. Applying mean subtraction over individual strides constrains the movement pattern to steady-state, level locomotion. However, relaxing the mean subtraction to a series of strides (a standard technique for dealing with data drift) enables evaluation of inter-stride variability with the cost of greater accumulation of integration errors.
Here, we develop an approach for deriving displacement data from a commercial orientation sensor by integration and mean subtraction. The results and errors of our approach are compared to the `gold standard' of 3D optical motion analysis.
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Materials and methods |
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Three experiments were undertaken:
Experiment 1 accuracy of optical motion capture system
The accuracy of the optical motion capture system under our experimental
conditions was determined by attaching two spherical retro-reflective markers
(diameter 30 mm) to the turntable of a record player, which was then spun at
33 and 45 r.p.m. in the field of view of the two-camera 3D motion capture
system (MCU240; ProReflex, Qualisys Ltd, Gothenburg, Sweden).
The accuracy of the motion capture system was tested based on the
circularity of the path of the spherical markers on the turntable of the
record player. A least-mean-square error-based gradient descent method was
used to find the best match of the 3D marker positions to a circle in 3D space
(Barker, 2004).
Experiment 2 accuracy of MT9 inertial sensor compared with optical motion capture determined during treadmill locomotion
Whilst not the envisaged application, treadmill locomotion is used to
enable validation of many consecutive strides in a short period of time. An
MT9 inertial sensor was mounted in a custom-made harness constructed of
malleable casting material (Dynacast, Smith and Nephew, Wound Management,
Hull, UK) over the 4th thoracic vertebra (the withers) of a Thoroughbred horse
(Fig. 1). A cable ran from the
sensor, via an overhanging beam, to a laptop computer and battery
located beside the treadmill. Serial data were collected from the sensor
via custom software written in Microsoft Visual C++ (Microsoft
Corporation, Redmond, WA, USA) using the sensor's software development kit
(SDK; Xsens). A wand with three orthogonal arms, each bearing a
retro-reflective spherical motion capture marker, was fixed to the sensor
(Fig. 2). The positions of
these markers were recorded at 240 Hz using the 3D optical motion capture
system validated in experiment 1. The cameras were positioned 3 m to the left
of the treadmill. The calibrated volume of the cameras was comparable to the
calibrated volume used in experiment 1. A treadmill-centred right-handed
Cartesian coordinate system (global system) was defined with the following
conventions: the positive x-axis was pointing towards the front of
the treadmill, the positive y-axis was pointing towards the left side
of the treadmill and the positive z-axis was pointing upwards. These
directions correspond to craniocaudal, mediolateral and dorsoventral movements
of the withers of the horse. Three DoF tracking of the three real markers was
performed using QTrac software (Qualisys Ltd) in the treadmill-centred system.
Based on the positions of these three markers, six DoF tracking of one virtual
marker located at the centre of the inertial sensor was carried out using
custom software in MATLAB (The Mathworks Inc., Natick, MA, USA).
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The horse was habituated to the high-speed treadmill (Sato, Upsala, Sweden)
before the start of the study (Buchner et
al., 1994a). After a warm-up period, the horse was exercised at
increments of speed between 1.4 m s-1 and 9.0 m s-1. At
each speed increment, the horse was allowed to settle into a regular gait
prior to simultaneous collection of 15 s of inertial sensor, optical motion
capture and accelerometer data. After discarding the first 5 s of data to
allow for initializing effects of the inertial sensing to settle, this allowed
us to capture a sequence of strides at each gait (walk, trot and canter).
Audio transcription software (Transcriber;
http://www.etca.fr/CTA/gip/Projets/Transcriber/)
was used to identify the times of foot-on from the accelerometer voltage
output. A mean absolute error of 2 ms for foot-on has been determined for this
system (Witte et al., 2004).
Inertial sensor and optical motion capture data were processed using custom
software written in MATLAB.
The position of a virtual marker created at the centre of the inertial sensor using the positions of the three optical motion capture markers was used to compare the optical motion capture data with inertial sensor data. The results were subdivided into the accuracy of orientation determination and the accuracy of measurement of linear displacements.
Orientation
First, we were interested in the accuracy of the inertial sensor for
measurement of absolute orientation. Errors in absolute orientation will
propagate into the integration process from acceleration to velocity and
displacement since sensor accelerations have to be projected into the global
coordinate system based on absolute orientation derived from the sensor.
However, small absolute errors in orientation will have a minimal effect on
the projection of the acceleration values.
The orientation algorithm of the inertial sensor provided orientation data in the earth reference frame in the form of Euler angles, quaternions or rotation matrices. Euler angles were used for the purposes of graphical display, whilst data processing was carried out using rotation matrices, which do not suffer the problem of gimbal lock. Euler angles (roll, pitch and heading) represent rotations from the sensor system into the earth reference system (horizontal and magnetic North; Fig. 3). The conventions used set the order of rotations to roll first (rotation around the sensor x-axis), then pitch (rotation around the sensor y-axis) and then heading (rotation around the sensor and then also the earth reference z-axis).
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![]() | (1) |
Subsequently, a rotation matrix, Rsg, representing a
rotation from the sensor system into the motion capture system (global system)
was derived using the following equation, with Euler angles obtained from
Eqn 1 (subscripts and time
dependencies have been omitted on the right-hand side of the equation in order
to improve readability):
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In addition, minor misalignment between the motion analysis marker wand and
the inertial sensor was corrected for. A correction vector was determined from
values of roll, pitch and heading for both methods recorded simultaneously
while the inertial sensor was mounted in the harness equipped with the optical
motion capture wand. Euler angles of the optical motion capture system were
then corrected using the following equation:
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The last two terms of Eqn 3 were used for correction of the misalignment between the optical motion capture and the inertial sensor coordinate system (compare also Eqn 1), and the last three terms eliminated the misalignment between the optical motion capture marker wand and the sensor.
High linear accelerations over a sustained period of time during trot and canter locomotion resulted in a malfunction of the sensor orientation output, which is designed to deal with comparatively low accelerations observed in human locomotion. Therefore, sensor accelerations were low-pass filtered using a 100-sample moving average (effective 3 dB cut-off: 1.1 Hz). The exact realisation of the low-pass filter in this processing step is uncritical, its only purpose being the elimination of acceleration peaks. An additional processing step was required that simulated an initial fixed orientation of the sensor, since capturing was started in full motion of the horse. Five seconds of simulated static data, consisting of sensor acceleration and angular velocity means and magnetometer first frame values, were added to the beginning of the sensor data. These modified data were fed into the sensor orientation algorithm to give the final orientation estimates.
The quality of the orientation output of the inertial sensor was evaluated by means of median and interquartile ranges of the differences between the aligned Euler angles for the two systems.
Linear displacement
The goal was to derive linear displacements in the global coordinate system
from the linear accelerations in the sensor coordinate system. This required a
projection of the sensor accelerations into the global coordinate system
followed by a double integration of the accelerations.
First, the accelerations were projected from the sensor coordinate system
into the global system based on the sensor orientation output using the
following equation.
![]() | (4) |
During the integration process, sensor drift resulted in accumulation of integration errors (Euler integration), and suitable integration constants were required. In the special case of steady-state locomotion, both of these challenges were overcome by assuming cyclical sensor movement. This meant that differences in position (and velocity) between strides or a series of strides were set to zero. Throughout all experiments, this constraint was implemented by stride-by-stride mean subtraction of x, y and z accelerations and x, y and z velocities using a context window of variable length for calculation of the mean. The results of the integration process are x, y and z displacements in the global coordinate system. The x, y and z displacements derived from the inertial sensor output were compared with those from the optical motion capture system.
The process of integration is summarised in a step-by-step fashion below:
The implemented mean subtraction using a context window of several strides represents a trade-off between reducing integration errors caused by sensor drift and allowing the integration process to reproduce non-cyclical movements for individual strides. Whereas the use of longer context windows (more strides) will increase drift-related errors, it will enable observation of changes over several strides.
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Original and filtered linear displacements derived from the inertial sensor and optical motion capture were compared, and median and interquartile ranges of the differences evaluated.
Experiment 3 trunk movement of a Thoroughbred horse during treadmill locomotion
The inertial sensor was attached to the horse with the same custom-made
harness used in experiment 2 and was used to measure the x, y and
z displacements and the roll, pitch and heading angles of the 4th
thoracic vertebra while the horse performed a range of speeds and gaits on a
high-speed treadmill. The experimental set-up and initial data processing were
as described for experiment 2. For each speed increment, inertial sensor data
were derived and, in addition to the processing described for experiment 2,
data were interpolated to percentage of stride and the mean stride was
calculated (six last strides at walk, 12 last strides at trot and 17 last
strides at canter). Displacement and orientation amplitudes were
calculated.
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Results |
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Experiment 2 accuracy of MT9 inertial sensor determined by direct comparison with optical motion capture
A total of 35 strides were analysed (6 strides at walk, 12 strides at trot
and 17 strides at canter). Displacement data from the inertial sensor
integration process (using a context window of one stride to each side of the
current stride) followed the optical motion capture data nicely over a series
of strides (Fig. 4). Mean
x, y and z displacement and roll, pitch and heading traces
obtained using the inertial sensor were virtually indistinguishable from those
obtained using optical motion capture (Fig.
5; Table 1). There
was, however, a small offset between the inertial sensor and optical motion
capture roll and pitch angles. The inertial sensor tended to underestimate the
absolute value of roll and overestimate the absolute value of pitch in all
gaits, with the magnitude of the offset increasing through the gaits from walk
to trot and canter. These errors were gait dependent
(Table 1). After high-pass
filtering, mean outputs for x, y and z displacements for
both methods were still almost identical
(Fig. 6;
Table 1) and, in addition,
decreased standard deviations confirmed the efficacy of the procedure to
remove inter-stride differences for both optical motion capture and inertial
sensor outputs. Compared with the optical motion capture system, 50% of the
values for the x, y and z displacement obtained from the
inertial sensor were found within (1.9, +1.7) mm, (0.8, +0.9)
mm, (0.6, +0.6) mm for walk, (2.8, +1.4) mm, (0.9, +0.9)
mm, (4.3, +4.9) mm for trot and (3.6, +6.2) mm, (4, +3.8)
mm, (4.5, +5.1) mm for canter (Table
1). Compared with the range of motion (true values derived from
optical motion capture data), these values correspond to a relative error of
(3.3, +3.0)%, (2.6, +2.9)%, (2.5, +2.4)% for walk,
(6.5, +3.2)%, (2.6, +2.6)%, (5.6, +6.4)% for trot and
(3.2, +5.5)%, (6.7, +6.3)%, (3.3, +3.7)% for canter
locomotion (Table 1).
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The difference (error) between the inertial sensor and the optical motion capture was calculated for each sample across all strides. Error histograms as well as median and interquartile ranges for x, y and z displacements and roll, pitch and heading angles are given to qualitatively and quantitatively demonstrate the ability of the inertial sensor to reproduce the six DoF position of the sensor. For the original data, x, y and z displacements and roll, pitch and heading angles showed error distributions with increasing spread from walk to trot and canter (Fig. 7). With the exception of roll and pitch angles, the errors were distributed around the origin. Roll error data were distributed around median values of 1.4°, 2.6° and 5.4° (with a range of motion of 8.0°, 7.7° and 12.5°), and pitch error data were distributed around median values of 1.0°, 1.7° and 2.4° (with a range of motion of 9.1°, 8.0° and 18.5°) for walk, trot and canter, respectively. After high-pass filtering, the error histograms for linear displacements showed narrower distributions, as confirmed by lower interquartile ranges (Fig. 8; Table 1). Interquartile ranges were reduced by between 14 and 89% by high-pass filtering. For each gait, the highest error reduction was found for the y direction, in which the non-cyclical components were most prominent.
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Ranges of displacement (maximum minimum value within a stride) for x, y and z displacement were calculated at each speed (Fig. 9). The range of displacement in the x (craniocaudal) direction was independent of speed for all gaits. In the y (mediolateral) direction, walk showed no speed dependence, but amplitude increased with increasing speed at both trot and canter. In the z (dorsoventral) direction, only canter showed an effect of speed on amplitude with the range of motion decreasing with increasing speed.
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Discussion |
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The proposed context-dependent stride-wise mean subtraction process applied
to acceleration and velocity represents a trade-off between minimizing
integration errors from sensor drift over short time intervals (single
strides) and capturing non-cyclical components of movement. A context window
of three strides (one to the left and one to the right of the current stride)
proved suitable and gave good results for x, y and z
displacements in all gaits. In general, smaller context windows resulted in
improved cyclical displacement estimates, whereas increasing the context
number resulted in overestimations of the non-cyclical parts especially in the
y direction. This approach to integration is limited as it only
resolves for sudden acceleration changes (constant offsets) between strides
and thus linear changes in velocity within a stride. This oversimplifies the
actual conditions found during animal locomotion. Non-linear, smooth changes
of velocity over several strides would represent a better model, but
additional knowledge about changes in position and/or velocity over a longer
time period would be required to be able to impose more-realistic constraints
under these conditions. However, this also means that stride-to-stride changes
are small and the context-dependent mean subtraction is a good approximation.
The additional information needed for changes in velocity may be derived from
further sensors on the subject, such as global positioning system devices
(Witte and Wilson, 2004), or
from further analysis of the magnetometer signals and might enable us to
further enhance the quality of the sensor displacement output. Now global
positioning system units with 4 Hz update rate are available, which are
especially promising for this. Potentially, changing the configuration of the
context-dependent mean subtraction to ignore the current stride will enable us
to capture short time deviations from the steady-state pattern.
High-pass filtering proved suitable for separating cyclical and
non-cyclical components of the movement across all gaits. At a filter cut-off
of 1 Hz, most of the stride-to-stride variations were eliminated successfully
with the biggest improvements found in the y direction. This reflects
the fact, that the y direction, which corresponds to a lateral
movement of the horse, showed the highest inter-stride variability before
filtering, which is caused by both sudden and gradual lateral drift, i.e. big
changes between two strides or a series of small changes for a number of
strides. For different applications, a different cut-off frequency might be
more suitable. The average improvement over all directions was 68% for walk,
60% for trot and 51% for canter. Although a period of 1 s represented a
different number of cycles for each gait (with stride frequencies varying from
1 Hz for walk to 2 Hz for canter), changing the cut-off frequency to
reflect this did not change the results considerably.
Accurate knowledge of the timing of the stride is crucial for the
integration process (step 4 in the integration procedure), since a cyclical
movement (over a series of strides) is assumed for the mean subtraction. In
this study, a limb-mounted accelerometer was used to directly measure the
timing of each left front leg footfall. This system is accurate to within 2 ms
(less than an inertial sensor sample) across all speeds and gaits on a hard
surface (Witte et al., 2004).
Errors introduced into the integration can therefore be regarded as
negligible. Theoretically, it would be possible to characterise features of
the inertial sensor output corresponding to foot placement. However, the
indirect and gait-dependent nature of the linkage between the foot and the
trunk will inevitably result in reduced accuracy. When possible, the direct
method used here is preferable given the importance of appropriate stride
segmentation.
The sensor is adequately small and lightweight to be used in field studies (35 g; 39x54x28 mm, width x length x height). Several sensors can easily be mounted on a single subject, allowing relative movements of body segments to be tracked. Since each sensor gives six DoF position information, there is no need to assume rigid links between the segments, and a limited set of sensors is sufficient to capture whole limb movements. However, the experimental conditions have to be evaluated carefully when interpreting the results. The current technique is designed for steady-state movements and gives good results for individual strides (see results without high-pass filtering). In addition, high-pass filtering is an appropriate method for averaging over strides. Whereas experimental set-ups with considerable variations in, for example, slope or surface are still difficult to interpret, field experiments on constant slopes, or comparing different surfaces (with constant or slowly changing surface characteristics), are well within the scope of the technique. In addition, compared with force plate analysis, this technique enables us to capture a series of strides and is cheap in comparison to a multiple-camera motion capture system under field conditions.
In the study described here, data were acquired into a laptop computer alongside the treadmill via a serial connection. However, the versatility of the sensor is enhanced because the data stream can be easily logged on the animal into a hand-held computer (depending on the size of the animal) or can be transmitted via a high-speed telemetry link to a stationary computer. Links capable of a sufficiently high bandwidth and range include Bluetooth®, DECT (digital enhanced cordless telecommunications) or wireless networking technology (802.11a-h). When logging is started during locomotion and orientation data are required, the described post-processing step has to be implemented in order to simulate an initial fixed orientation of the sensor. Here, we used a simple moving average filter (effective low pass cut-off of 1.1 Hz) for smoothing the acceleration and gyroscope data. Although the design of this filter is not critical, its only purpose being to simulate fixed orientation (around the mean acceleration), the post-processing step might have to be adapted to the experimental conditions.
Rigid mounting of the sensor on the subject was very important in order to capture accurately overall trunk movement. This was achieved here using the custom-made harness (Fig. 2). There was minimal movement of the sensor relative to the subject, although this was difficult to quantify.
Conclusion
Inertial sensors make it possible to capture cyclical movements with
comparable accuracy to optical motion capture systems. This enables the study
of (quasi) steady-state field locomotion. The extraction of non-cyclical
components of movement is a future goal.
List of symbols and abbreviations
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Acknowledgments |
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