sensors
Article
Eﬀect of IMU Design on IMU-Derived Stride Metrics
for Running
Michael V Potter *, Lauro V Ojeda
, Noel C Perkins and Stephen M Cain
Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA;
lojeda@umich.edu (L.V.O.); ncp@umich.edu (N.C.P.); smcain@umich.edu (S.M.C.)
* Correspondence: mvpotter@umich.edu
Received: 18 March 2019; Accepted: 5 June 2019; Published: 7 June 2019


Abstract: Researchers employ foot-mounted inertial measurement units (IMUs) to estimate the
three-dimensional trajectory of the feet as well as a rich array of gait parameters.
However,
the accuracy of those estimates depends critically on the limitations of the accelerometers and angular
velocity gyros embedded in the IMU design. In this study, we reveal the eﬀects of accelerometer
range, gyro range, and sampling frequency on gait parameters (e.g., distance traveled, stride length,
and stride angle) estimated using the zero-velocity update (ZUPT) method.
The novelty and
contribution of this work are that it: (1) quantiﬁes these eﬀects at mean speeds commensurate with
competitive distance running (up to 6.4 m/s); (2) identiﬁes the root causes of inaccurate foot trajectory
estimates obtained from the ZUPT method; and (3) oﬀers important engineering recommendations
for selecting accurate IMUs for studying human running. The results demonstrate that the accuracy
of the estimated gait parameters generally degrades with increased mean running speed and with
decreased accelerometer range, gyro range, and sampling frequency. In particular, the saturation
of the accelerometer and/or gyro induced during running for some IMU designs may render those
designs highly inaccurate for estimating gait parameters.
Keywords: wearable sensors; ZUPT; sensor requirements; gait
1. Introduction
Studies of running biomechanics suggest that measured kinematic parameters (e.g., joint angles,
stride frequency, stride length) may lead to the insight necessary to improve running performance and
reduce injury risk [1,2].
Miniature inertial measurement units (IMUs) are an attractive option
for analyzing human performance outside of traditional laboratory environments due to their
relatively low cost, simple setup, and portability [3]. In one application, foot-mounted IMUs provide
three-dimensional foot accelerations and angular rotational velocities from which foot trajectories
(and associated gait parameters) are derived during walking/running [4–10]. Doing so requires
minimizing the accumulated drift error in the estimated foot velocity and position using the so-called
“zero-velocity update” (ZUPT) method [4,5,11]. The ZUPT method exploits the fact that the foot is
nearly stationary at some time during the stance phase and uses that condition to estimate the foot
velocity drift error for each gait cycle.
Prior studies conﬁrm that the ZUPT method yields accurate foot trajectory estimates for walking
gait [9,12,13] and running gait with modest speeds (up to 4.36 m/s) [14]. However, little research
addresses the requirements that ensure accurate trajectory estimates, particularly at faster running
speeds, such as those observed in competitive middle- and long-distance running (up to 6.5 m/s) [15–17].
One limitation is that the sensor never achieves exactly zero-velocity, even in walking [18], and this
assumption becomes increasingly suspect for faster running speeds. Another limitation lies with the
range and sampling frequency of the inertial sensors themselves. Bailey and Harle [19] investigated
Sensors 2019, 19, 2601; doi:10.3390/s19112601
www.mdpi.com/journal/sensors
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the eﬀect of IMU sampling frequency and accelerometer range on foot trajectory estimates and found
that errors increase with increased running speed and decreased sampling frequency. While the errors
observed in [19] were relatively small, the study considered modest running speeds (2.3–3.4 m/s)
that are well below those of typical competitive middle- and long-distance running (up to 6.5 m/s).
Additionally, the experiments in [19] were conducted on a treadmill rather than running overground
which may also have inﬂuenced the conclusions. For example, signiﬁcant diﬀerences may arise in
gait kinematics when comparing walking on a treadmill versus overground [20]. Additionally, in a
pilot study [21], the authors evaluated estimates of IMU-derived running speed using a treadmill.
They observed that ﬂuctuations in the treadmill belt speed, especially at higher running speeds with
accompanying larger ground reaction forces, generated signiﬁcant discrepancies between the reported
belt speed and IMU-derived estimates of running speed. The speed ﬂuctuations (i.e., accelerations)
of the treadmill belt render it a non-inertial frame; thus, IMU-measured accelerations and angular
velocities cannot be directly integrated to yield accurate estimates of foot velocity and position
relative to the belt (as assumed using the ZUPT method). These treadmill-based limitations were
a primary factor in modifying the pilot study protocol to the overground-based protocol of the
study presented in this paper. Recently, Mitschke, et al. [22] examined the impact of accelerometer
range on IMU-derived estimates of stride length, velocity, and tibial acceleration for overground
running. They found no signiﬁcant degradation in stride parameter estimates when the accelerometer
range was ±32 g or greater, but signiﬁcant degradation with smaller accelerometer ranges. Similar
to [19], the study [22] considered only modest running speeds (up to 3.6 m/s) and did not disclose
the fundamental reasons for the inaccurate estimates within the ZUPT method. In addition to the
limitations imposed by accelerometer range and sampling frequency considered in these prior studies,
we hypothesize that gyro range may also impose limitations on achieving accurate foot trajectory
estimates for running. The eﬀect of gyro range on stride estimates has likely not been studied previously
because many commercial IMUs are unlikely to experience gyro saturation at the modest speeds
observed in past studies. However, we hypothesize that the eﬀects of gyro range on these estimates
will become increasingly important at higher running speeds.
The objective of this study is to reveal the impact of an IMU’s accelerometer range, gyro range,
and sampling frequency on estimated stride parameters (i.e., stride length, stride angle, and total
distance traveled) during overground walking and running up to competitive distance running speeds
(up to 6.4 m/s) and over a wide range of sensor ranges and sampling frequencies typically found in
commercially-available IMUs. This study also addresses the impact of gait speed on the estimated
distance traveled in the presence of no saturation of the IMU signals. The novelty and contribution
of this work are that it: (1) quantiﬁes these eﬀects at mean speeds commensurate with competitive
distance running (up to 6.4 m/s); (2) identiﬁes the root causes of inaccurate foot trajectory estimates
obtained from the ZUPT method; and (3) oﬀers important engineering recommendations for selecting
accurate IMUs for studying human running. The results of this study will aid coaches and researchers
in selecting appropriate IMUs to study stride parameters in outdoor environments and at speeds up to
6.4 m/s using the ZUPT method.
2. Materials and Methods
2.1. Subjects and Experimental Protocol
Six healthy subjects (3 female, 3 male; mean (standard deviation) age 24.2 (±6.0) years, height
1.71 (±0.12) m, mass 68.4 (±15.2) kg) were recruited for this study. All subjects veriﬁed that they felt
capable of completing the experimental protocol. Informed consent was obtained from all subjects
and the study was approved by the University of Michigan IRB. Two IMU designs were employed.
The ﬁrst (IMU 1) provides a high accelerometer range and the second (IMU 2) provides a high sampling
frequency as reported by the IMU speciﬁcations in Table 1. Each subject wore both IMUs strapped
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together and placed on the bridge of both feet as shown in Figure 1 (only right foot shown) and secured
with a strap and tape to limit their movement with respect to each other and to the foot.
Table 1. Speciﬁcations for IMU 1 and IMU 2.
IMU 1
IMU 2
Model
Opal
Custom design
Manufacturer
APDM
Insight Sports, Ltd.
Sampling Frequency (Hz)
128
1000
Accelerometer Range (g)
±200
±32
Angular Gyro Range (deg/s)
±2000
±4000
Figure 1. Two inertial measurement unit (IMU) designs and the means of attachment to a foot. (a) IMU 1
(Opal, APDM, left) and IMU 2 (custom design, Insight Sports, Ltd., right); (b and c) Side and front
views of attachment of both IMUs to the instep via a Velcro strap. Sensor axes are denoted by X,
Y, and Z (X and Z largely lie in foot sagittal plane and Y largely points to subject’s left). Note that
these axes are illustrated to aid interpretation of the raw data signals presented in Figures 4 and 8,
and that the zero-velocity update (ZUPT) method herein does not require speciﬁc sensor alignment to
anatomical axes.
We employed a method similar to [4] to assess the accuracy of ZUPT-based foot trajectory estimates
outside of laboratory environments as follows. Each subject completed ten straight 100-meter trials
on (level) asphalt. The subjects completed the ﬁrst two trials at a perceived slow walk and fast walk,
respectively. For the third through to the tenth trial, subjects ran at increasing speeds from a perceived
slow jog (third trial) up to maximal sprint (tenth trial). For each trial, the subjects started at rest
with the front of both shoes aligned with the start line. The subjects then walked/ran 100 meters and
stopped with the front of both shoes aligned with the ﬁnish line. Rest between trials was self-selected
by the subjects.
Since all trials started and ended with the subject at rest, the time to complete each trial was
readily identiﬁed from the start and end of signiﬁcant acceleration (magnitude). The known 100-meter
distance traveled was divided by the trial time to yield the mean speed for each trial. We employed the
mean speed as an independent variable in the analyses below.
2.2. Overview of the ZUPT Method
While the ZUPT method is generally known, we provide an overview for the reader’s beneﬁt
in following the discussions oﬀered later in this paper. The ZUPT method used here draws largely
from [4] and [9]. The method begins with estimating the instantaneous orientation of the IMU relative
to an inertial frame as further detailed in [23]. The orientation is described by the rotation matrix,
R, that deﬁnes the orientation of the sensor’s three orthogonal axes [ˆis, ˆjs,ˆks] (corresponding to x, y,
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and z sensor axes, respectively) relative to the orthogonal axes [ˆiw, ˆjw,ˆkw] of a world (i.e., inertial) frame
(corresponding to x, y, and z world axes, respectively, with z pointing in the direction of gravity) per:


ˆiw
ˆjw
ˆkw


= R


ˆis
ˆjs
ˆks


(1)
Following [23], the angular velocity is integrated to estimate R at each time step with a Kalman
ﬁlter used to correct drift error in the tilt angle. The three components of IMU-measured acceleration
(axs, ays, azs) yield the acceleration components in the world frame (axw, ayw, azw) through:


axw
ayw
azw


= R


axs
ays
azs


−


0
0
g


(2)
where g is the acceleration of gravity. These acceleration components in the inertial frame are used to
estimate velocity in the inertial frame using the fact that the foot-mounted IMU returns to zero-velocity
during the stance phase as described below.
First, zero-velocity times are identiﬁed as times of minimum angular velocity magnitude during
the stance phase. A stride is deﬁned by the time interval between two successive zero-velocity times
tn−1 < t < tn. Thus, the stride time is:
ts = tn − tn−1
(3)
and the number of data samples, l, in each stride is:
l = ts × Fs + 1
(4)
where Fs is the sampling frequency of the IMU. For each stride, the initial velocity is set to zero at
tn−1 and the acceleration components in the inertial frame (2) are integrated to estimate the velocity
components in the inertial frame (´vxw, ´vyw, ´vzw). Because it is assumed that the velocity at tn returns to
zero, the estimated velocity error (per sample) in the stride is:
→
Verror =


´vxw
´vyw
´vzw


(t = tn) ×
1
l − 1,
(5)
assuming linear drift. For each sample number, k ∈ (1, l), within the stride (e.g., k = 3 for third
sample in stride), a linear velocity drift error correction is applied to each of the three world frame
velocity components:


vxcorr
vycorr
vzcorr


=


´vxw
´vyw
´vzw


−
→
Verror × (k − 1).
(6)
The corrected velocity for the whole trial is integrated to estimate the foot position throughout
the trial. The position estimates are segmented by zero-velocity times and used to estimate the stride
length and stride angle as follows. Figure 2 gives a two-dimensional illustration of the stride length
and angle.
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Figure 2. Illustration of stride length , ∆, and stride angle, θ. Orange dots on footprints represent
consecutive zero-velocity times.
Identiﬁed zero-velocity times deﬁne the start position,
→
Sn, and end position,
→
En, of the nth stride
as given by:
→
Sn =


Sxw,n
Syw,n
Szw,n


,
(7)
→
En =


Exw,n
Eyw,n
Ezw,n


,
(8)
which denote the X, Y, and Z components of the start and end positions in the world frame for the
nth stride. The stride length, ∆, is calculated as the total three-dimensional displacement of the IMU
(and thus the foot) between the start and end of the stride. We note that in many biomechanical studies,
stride length often refers to only the anterior–posterior component of the foot displacement; however,
in using this technology for biomechanical analyses, there is a precedent to use alternative deﬁnitions
of stride length, such as the total horizontal displacement of the foot, such as in [14]. Other studies
make assumptions about the sensor/foot orientation, such as assuming a particular sensor axis is
perfectly aligned with the medial–lateral axis of the subject [22] or impose a level ground assumption
to constrain vertical drift [24]. While such assumptions may be useful in simplifying calculations or in
interpreting some of the other foot parameters obtainable using similar ZUPT method applications
(e.g., if interested in the foot roll and pitch angles, precise alignment of sensor and anatomical axes
may be helpful or even necessary), we do not impose such restrictions in our method (e.g., our method
can be used for non-level walking/running and our method does not require any alignment between
the sensor and anatomical axes). In particular, the stride length (as deﬁned for this study) for the nth
stride is:
∆n =
r
(Exw,n − Sxw,n)2 +

Eyw,n − Syw,n
2 + (Ezw,n − Szw,n)2.
(9)
The stride angle, θ, is calculated as the three-dimensional angle between successive strides’ vectors.
First, we deﬁne the stride vector for the nth stride as:
→
Dn =


Exw,n − Sxw,n
Eyw,n − Syw,n
Ezw,n − Szw,n


.
(10)
The nth stride angle is thus computed as:
θn = tan−1



→
Dn ×
→
Dn+1

→
Dn·
→
Dn+1


,
(11)
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with × and · being the standard vector cross and dot products, respectively. We again note that the
estimated stride parameters used in this study (stride length and stride angle) do not require any
assumptions of a particular alignment of the sensor axes on the foot.
2.3. Data Processing and Analysis
Each IMU’s acceleration and angular velocity data were used to estimate the foot’s
three-dimensional trajectory throughout each trial using the ZUPT method described above. The stride
lengths were added (for both the left and right foot and then averaged) to yield the estimated total
distance traveled (Dcalc) during each trial. The cumulative distance error:
Derr = (Dcalc/Dtruth − 1) × 100%,
(12)
is reported for each trial, where Dtruth is the known distance traveled (100 meters in this study).
To investigate the eﬀect of accelerometer range, the raw accelerometer data from IMU 1,
that possesses an acceleration range ±200 g, were numerically truncated to seven smaller ranges;
namely 100, 75, 50, 24, 16, 10, and 6 g. For example, to investigate the eﬀect of a 16 g accelerometer,
any acceleration outside the range of −16 g < a < 16 g was set to the corresponding limit (−16 g
or 16 g) to simulate sensor saturation at that limit. These seven ranges were chosen because they
are typical of commercial IMU designs. Note that the acceleration data never exceeded ±100 g in
any trial and thus we used the data with the ±100 g accelerometer range as the baseline for this
analysis. After the raw accelerometer data were modiﬁed in this manner, the ZUPT method was used
(with the modiﬁed accelerometer data and raw angular velocity data as input) to estimate the foot
trajectories. The cumulative distance error (12) was computed as a function of the mean speeds for each
accelerometer range. These data were then ﬁt to a linear mixed-eﬀects model [25] to test the statistical
signiﬁcance of the following: (1) the eﬀect of mean gait speed on the cumulative distance error with
no accelerometer saturation (i.e., using the 100 g range accelerometer), and (2) the eﬀect of smaller
accelerometer ranges on these estimates (i.e., using the 75, 50, 24, 16, 10, and 6 g range accelerometers).
The statistical model and its full results are detailed in Appendix A. Similarly, to investigate the
eﬀect of gyro range, the raw angular velocity data from IMU 1, which possesses an angular velocity
range of ±2000 deg/s, were also numerically truncated to four smaller ranges; namely 1500, 1000, 750,
and 500 deg/s (i.e., ranges common in commercially available IMU designs). For this analysis, we used
the data with the ±2000 deg/s gyro range as the baseline. After the raw angular velocity data were
modiﬁed in this manner, the ZUPT method was used (with the modiﬁed angular velocity data and raw
accelerometer data as input) to estimate the foot trajectories and cumulative distance error as described
above. An analogous statistical model to the one described above was also employed, but investigating
gyro range instead of accelerometer range.
Additionally, for both accelerometer and gyro range eﬀects, the amount of data that was lost due
to truncation was quantiﬁed. To accomplish this, we compared the integrated area under the truncated
signal and that of the non-truncated signal. In particular, to quantify the data lost due to accelerometer
saturation, we deﬁned the percent data loss as:
Ldata =

1 −
R 
→a trun
dt
R 
→a non
dt

 × 100%,
(13)
where

→a trun
 is the acceleration magnitude of the truncated signal,

→a non
 is the corresponding
acceleration magnitude of the non-truncated signal, and dt is the time per sample. The integration is over
the length of the trial. The percent data loss due to angular velocity saturation was deﬁned analogously.
Note that IMU 1 was speciﬁcally chosen for studying the eﬀects of accelerometer and gyro ranges
and we veriﬁed that all data were within the design ranges (±200 g and ±2000 deg/s) for IMU 1 in all
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trials. By contrast, accelerometer saturation arose in IMU 2 (±32 g) at higher running speeds. However,
IMU 2 was speciﬁcally chosen to study the eﬀect of sampling frequency due to the sampling frequency
limitations of IMU 1 (128 Hz). IMU 2 possesses a sampling frequency of 1000 Hz, far beyond the
minimum 250 Hz rate recommended in [19] for obtaining accurate foot position and velocity estimates
using the ZUPT method in running at speeds up to 3.4 m/s. For the purpose of studying the eﬀect of
sampling frequency, the accelerometer and gyro data from IMU 2 (1000 Hz) were also down-sampled
to four smaller sampling frequencies (500, 250, 125, and 62.5 Hz) typical of commercial IMU designs.
To that end, we employed two down-sampling methods as further described in the Results section.
For these analyses, we used the data with the 1000 Hz sampling frequency as the baseline.
In addition to studying the cumulative distance error, we also report stride-to-stride variations in
the diﬀerences of individual stride length and stride angle estimates as deﬁned above (e.g., the standard
deviation of the stride length diﬀerence over a trial) as these data also reveal important conclusions.
To this end, the stride length diﬀerence for a particular stride and trial (e.g., stride 2 for subject 1 and
trial 1) was deﬁned as:
∆di f = ∆base − ∆est,
(14)
where the ∆base is the estimated stride length using the baseline (non-saturated/non-downsampled)
IMU data and ∆est is the estimated stride length using the truncated or downsampled data generated
as discussed above.
3. Results
3.1. Eﬀect of Accelerometer Range
The statistical analysis (Table A1 of Appendix A) reveals a signiﬁcant eﬀect of speed on cumulative
distance error (p < 0.01) with the 100 g accelerometer range despite no observed accelerometer or
gyro signal saturation in any of the trials with this range. Additionally, the statistical analysis reveals
that using an accelerometer range of 24 g or below leads to signiﬁcantly greater degradation of the
estimated distance traveled with speed (p < 0.01 for 24 g, p < 0.001 for 16 g, 10 g, and 6 g) relative to
the 100 g range. The accelerometer ranges of 75 g and 50 g revealed no statistically signiﬁcant eﬀects
compared to 100 g. These ﬁndings are observable in Figure 3 which illustrates the cumulative distance
error versus mean running speed for the original (100 g) accelerometer range and for each of the six
truncated accelerometer ranges utilizing data from IMU 1. As illustrated, subjects achieved mean
speeds up to 6.4 m/s with the upper end of this range, similar to speeds observed in elite distance
running [15]. Because the peak accelerations rarely exceeded 50 g, the cumulative distance errors for
the 100, 75, and 50 g accelerometer ranges are visually indistinguishable on this scale as expected
from the statistical results. Additionally, note that the cumulative distance error results converge
across accelerometer ranges with decreased running speeds because these speeds generally yield lower
accelerations (e.g., for the slowest trial, accelerations never exceeded 6 g, yielding identical cumulative
distance errors for all accelerometer ranges considered).
For walking and low running speeds (i.e., <2.2 m/s), the illustrated results largely conﬁrm the
cumulative distance errors reported by others for walking [4,9]. Additionally, for these low speeds,
the IMU-estimated cumulative stride distances are nearly independent of accelerometer range. This is
expected since the peak accelerations rarely exceeded 6 g for walking and low running speeds. However,
for high running speeds, signiﬁcant portions of each stride cycle generated accelerations larger than 6 g,
leading to the large observable degradations in the estimated cumulative distance at the higher running
speeds with decreased accelerometer range. Importantly, Figure 3 shows that accelerometers with
ranges exceeding 50 g yield cumulative distance errors no greater than 5% for all mean speeds observed
in this study (up to 6.4 m/s). Thus, depending on the accuracy needs for a particular use of these
estimates, the ZUPT method may yield acceptable results (i.e., errors of 5% or less) even at the highest
speeds observed in this study, provided no saturation arises in the accelerometer (and gyro) signals.
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By contrast, at the opposite extreme, errors exceeding 30% are observable for the 6 g accelerometer
where signiﬁcant saturation occurs.
Figure 3. Eﬀect of accelerometer range and mean running speed on the cumulative distance error. Note
that errors arising from the 100, 75, and 50 g accelerometers are indistinguishable on this scale. The
shaded region indicates estimates within ±5% error.
The degradation of estimates of the cumulative distance traveled with increased speed and
decreased accelerometer range traces to saturation in the accelerometer signals. Figure 4 illustrates the
eﬀect of truncating the accelerometer range for sample walking (Figure 4a) and running (Figure 4b)
trials. In the sample walking trial, the three acceleration components never exceed 6 g and therefore
distance estimates based on any of the accelerometer ranges considered (6 g through 100 g) yield
essentially identical results. However, in the sample running trial, the acceleration components often
exceed 6 g and for signiﬁcant portions of the gait cycle. The data loss leads to signiﬁcant foot trajectory
errors largely due to how the ZUPT method corrects for velocity drift error as described in detail in
the Discussion.
Figure 4. Eﬀect of accelerometer range limits on acceleration data for a sample (a) walking trial (mean
speed 1.4 m/s) and a sample (b) running trial (mean speed 5.8 m/s). Acceleration data is saturated
when the slope is zero as is most apparent for the X-axis acceleration of the 6 g accelerometer for the
running trial. Shaded areas indicate stance phase.
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We note that the presence of saturation does not necessarily lead to poor estimates. In particular,
acceptable results may still be obtainable if the amount of data that is lost due to saturation remains
small. To illustrate what might be “acceptable” levels of saturation for the accelerometer range,
we further quantiﬁed the amount of data that is lost due to saturation for each trial and accelerometer
range and present the relationship of the cumulative distance error and amount of data lost due
to saturation in Figure 5. These results show that the cumulative distance errors remain below 5%
when the percentage of acceleration data lost due to saturation is below 1.5%, irrespective of the
accelerometer range.
Figure 5. Cumulative distance error versus percentage of acceleration data lost due to saturation for all
trials and for each accelerometer range. The shaded region indicates estimates within ±5% error.
Beyond studying errors in the cumulative distance traveled, we also considered diﬀerences in
stride length and stride angle estimates on a stride by stride basis. To this end, we compared the
estimated length and angle of each stride (where stride angle refers to the angle between successive
strides as deﬁned in the Methods) using truncated accelerometer data to the same quantities estimated
from untruncated accelerometer data, employing the 100 g accelerometer as the benchmark. Figure 6
illustrates the standard deviation of the resulting diﬀerences in the individual stride length (Figure 6a)
and stride angle (Figure 6b) estimates. When the variation is large, the agreement between truncated
and baseline estimates of the given parameter for individual strides is small. These variations increase
strongly with mean speed and decreased accelerometer range. As both mean speed and accelerometer
range contribute to accelerometer saturation, they also signiﬁcantly impact the estimates of these
metrics on an individual stride basis.
Figure 6. Dependence of the standard deviation of diﬀerence in (a) stride length and (b) stride angle
estimates with mean speed and accelerometer range. The diﬀerences are computed with respect to the
results of the 100 g accelerometer as the benchmark.
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3.2. Eﬀect of Gyro Range
The statistical analysis (Table A2 of Appendix A) reveals a signiﬁcant eﬀect of speed on cumulative
distance error (p < 0.01) with the 2000 deg/s gyroscope range despite no observed accelerometer or
gyro signal saturation in any of the trials with this range. Additionally, the statistical analysis reveals
that using a gyro range of 750 deg/s or below leads to signiﬁcantly greater degradation of the estimated
distance traveled with speed (p < 0.05 for 750 deg/s, p < 0.001 for 500 deg/s) versus the 2000 deg/s
range. Gyro ranges of 1500 deg/s and 1000 deg/s reveal no statistically signiﬁcant eﬀects compared to
2000 deg/s. These ﬁndings are observable in Figure 7 which illustrates the cumulative distance error
versus mean running speed for the ﬁve gyro ranges considered utilizing data from IMU 1. Because
the angular velocities rarely exceeded 1000 deg/s, the distance error is nearly the same for the 2000,
1500, and 1000 deg/s range gyros. The distance errors remain within 5% for all gyro ranges of at least
1000 deg/s for the entire range of mean speeds studied herein (up to 6.4 m/s).
Figure 7. Eﬀect of gyro range and mean running speed on the cumulative distance error. Note that
distance errors arising from the 2000 and 1500 deg/s gyros are indistinguishable on this scale. The
shaded region indicates estimates within ±5% error.
As in Figure 3, the cumulative distance traveled is underestimated at faster (running) speeds
and this underestimation increases with speed and decreased gyro range. However, the cumulative
distance traveled is often overestimated at slower (walking) speeds and this overestimation increases
with decreased gyro range; observe the slower (walking) trials with a 500 deg/s range gyro.
The underestimation versus overestimation traces to saturation in distinct portions of the stride
cycle as revealed in Figure 8 for sample walking (Figure 8a) and running (Figure 8b) trials. Observe in
Figure 8a that the Y-axis angular velocity for the 500 deg/s gyro exhibits saturation during a modest
fraction of the stance phase near toe-oﬀ (end of the stance phase) for walking. By contrast, Figure 8b
reveals that for maximal sprinting, the same angular velocity component saturates during toe-oﬀ,
heel-strike (beginning of the stance phase), and for a signiﬁcant portion of the swing phase. The
portion of the stride cycle in which data is lost leads to overestimation versus underestimation because
of how it impacts the ZUPT algorithm as described in detail in the Discussion.
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Figure 8. Eﬀect of gyro range limits on angular velocity for a sample (a) walking trial (mean speed
1.4 m/s) and a sample (b) running trial (mean speed 5.8 m/s). Angular velocity data is saturated when
the slope is zero as is most apparent for the Y-axis angular velocity of the 500 deg/s gyro for the running
trial. Shaded areas indicate stance phases.
As with the accelerometer, we note that the presence of saturation in the gyro does not necessarily
lead to poor estimates; in particular, acceptable results may still be obtainable if the amount of data that
is lost due to saturation remains small. To illustrate what might be “acceptable” levels of saturation for
gyro signals, we further quantify the amount of data that is lost due to saturation for each trial and
gyro range and present the relationship of cumulative distance error and amount of data lost due to
saturation in Figure 9. These results show that cumulative distance errors remain below 5% when the
percentage of angular velocity data lost due to saturation is below 2.6%, regardless of the gyro range.
Figure 9. Cumulative distance error versus percentage of angular velocity data lost due to saturation
for all trials and for each gyro range. The shaded region indicates estimates within ± 5% error.
As in the previous section, we also considered diﬀerences in stride length and stride angle
estimates on a stride by stride basis. To this end, we compared the estimated length and angle of each
stride (where stride angle refers to the angle between successive strides as deﬁned in the Methods)
using truncated gyro data to the same quantities estimated from untruncated gyro data, employing
the 2000 deg/s gyro as the benchmark. Figure 10 illustrates the standard deviation of the resulting
diﬀerences in the individual stride length (Figure 10a) and stride angle (Figure 10b) estimates. When the
Sensors 2019, 19, 2601
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variation is large, the agreement between the truncated and baseline estimates of the given parameter
for individual strides is small. These variations increase strongly with mean speed and decreased gyro
range. As both mean speed and gyro range contribute to gyro saturation, they also signiﬁcantly impact
the estimates of these metrics on an individual stride basis.
Figure 10. Dependence of the standard deviation of the diﬀerence in (a) stride length and (b) stride
angle estimates with mean speed and gyro range. The diﬀerences are computed with respect to the
results of the 2000 deg/s gyro as the benchmark.
3.3. Eﬀect of Sampling Frequency
Sampling methods can vary widely in commercial IMUs. In particular, one or more ﬁlters are
commonly employed within the IMU hardware and/or software before data is output at the IMUs
speciﬁed sampling frequency. Therefore, an IMU having a higher sampling frequency (speciﬁcation)
does not necessarily imply it will lead to superior estimates of stride parameters in the context of this
study. Because the ﬁlters and sampling methods are generally hidden to the user and vary between
manufacturers, we considered the eﬀect of sampling frequency by studying two simple sampling
methods, including both an extreme method (no ﬁltering before down-sampling) and a common
method (low pass ﬁlter before down-sampling). For both methods, data from IMU 2 was utilized
as that IMU design yields data at a high (1000 Hz) sampling frequency. Neither sampling method
demonstrated a statistically signiﬁcant eﬀect for the interaction of speed and sampling frequency on the
cumulative distance error except for the most extreme downsampling used in this study (Method 1 at
the lowest sampling frequency). See Appendix A for full statistical results. Because this one exception
represents a most unrealistic scenario and because no other sampling method and sampling frequency
combinations studied herein revealed statistically signiﬁcant eﬀects for this interaction, we oﬀer no
further results for this eﬀect. However, signiﬁcant diﬀerences do arise in the estimated stride lengths
and stride angles of the individual strides as reported below.
Method
1
constitutes
simple
down-sampling
performed
without
ﬁltering
(e.g.,
when down-sampling from 1000 to 500 Hz, every other sample is retained).
This overly
simplistic approach introduces aliasing eﬀects and hence sub-optimal results [26,27].
Figure 11
illustrates the standard deviation of diﬀerences in stride length (Figure 11a) and stride angle
(Figure 11b) estimates using Method 1 as functions of both mean speed and sampling frequency.
The diﬀerences are with respect to the same quantities computed using the original data (i.e., data
sampled at 1000 Hz). The standard deviation of the diﬀerence from simple down-sampling quickly
grows (i.e., increasing variation) with increasing mean speed and decreasing sampling frequency.
For example, at the lowest sampling frequency (62.5 Hz), the standard deviation in stride length
Sensors 2019, 19, 2601
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diﬀerence becomes a signiﬁcant fraction of the stride length at higher mean speeds. Thus, the reliability
of these measures is signiﬁcantly impacted by both mean speed and sampling frequency.
Figure 11. Dependence of the standard deviation of the diﬀerence in (a) stride length and (b) stride
angle estimates with mean speed and sampling frequency. Method 1: down-sampling without
ﬁltering. The diﬀerences are computed with respect to the results of the 1000 Hz sampling frequency as
the benchmark.
Method 2 follows a more common strategy known as decimation [28,29], which consists of low
pass ﬁltering prior to down-sampling. We used the decimate function in MATLABTM [30] which utilizes
a low pass Chebyshev Type I ﬁlter (inﬁnite impulse response, order 8) before down-sampling the
data. Figure 12 illustrates the results from Method 2, analogous to those of Method 1. The results still
illustrate increased diﬀerences in estimates with increased speed and decreased sampling frequency,
but signiﬁcantly less than that observed using Method 1 (Figure 11). For example, the variation of
the stride length diﬀerence is reduced by nearly a factor of ﬁve (compare scales of Figures 11a and
12a). These results suggest that sensor hardware that employs well-designed ﬁlters can signiﬁcantly
mitigate the adverse impact of limited sampling frequencies.
Figure 12. Dependence of the standard deviation of the diﬀerence in (a) stride length and (b) stride
angle estimates with mean speed and sampling frequency. Method 2: low pass ﬁltering before
down-sampling (using MATLABTM decimate function). The diﬀerences are computed with respect to
the results of the 1000 Hz sampling frequency as the benchmark. Note the diﬀerences in the y-axis
scales compared to Figure 11.
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4. Discussion
Overall, this study highlights the importance of proper sensor selection in order to estimate
accurate gait parameters from foot-mounted IMUs using the ZUPT method. Accurate estimates of the
cumulative distance traveled are possible upon limiting acceleration and angular velocity saturation.
Importantly, we observed that the cumulative distance error using the ZUPT method remains below
5% when acceleration saturation is limited to 1.5% and when angular velocity saturation is limited to
2.6%; refer to Figures 5 and 9.
We also observed that gait parameter estimates degrade with higher mean speeds even without
sensor saturation (p < 0.01); refer to Appendix A. However, the results conﬁrm that ZUPT-based
algorithms yield accurate estimates for some applications (i.e., less than 5% cumulative distance
error) over the entire range of mean speeds studied herein (up to 6.4 m/s) contingent on the IMU
design. Importantly, lower range inertial sensors yield signiﬁcant errors in gait parameter estimates
at higher mean speeds due to (increasingly larger) data saturation. Interestingly, saturation may
produce both overestimates and underestimates of the cumulative distance traveled depending on
which signal (acceleration or angular velocity) is saturated, in which part of the stride cycle most of
the saturation occurs, and the mean speed. These errors arise from error sources within the ZUPT
method (detailed in the Methods) as follows. The gyro data is employed to estimate the orientation of
the IMU (via integrating angular velocity) and this is critical to accurately resolving the acceleration
into the world frame. The orientation estimates are corrected for drift error using a Kalman ﬁlter
based on the core assumption of zero-mean Gaussian gyro noise. However, this assumption is violated
when the angular velocity saturates, leading to inaccurate estimates of orientation and thus improper
resolution of acceleration in the world frame. Subsequent integration of poorly resolved (and even
possibly saturated) acceleration yields inaccurate estimates of velocity and position. Additionally, even
with proper IMU orientation estimates, saturation of accelerometer signals will create velocity drift
errors that do not increase linearly in time between the zero-velocity update times as assumed in the
ZUPT method.
These error sources suggest an intuitive explanation for why the estimated total distance traveled
is increasingly underestimated with increased speed and decreased accelerometer range. In this
study, the majority of acceleration data lost due to saturation was acceleration directed opposite to the
direction of travel (i.e., deceleration). Consequently, the uncorrected velocity in the direction of travel
was overestimated at the end of a stride. However, when the (linear) velocity-drift correction was then
applied, it consistently lead to underestimated velocity in the direction of travel and corresponding
underestimated stride length. By contrast, there is no parallel explanation for why saturated gyro data
may lead to both over and under estimates of the stride length. In particular, note that saturation of
gyro data creates errors farther upstream in the ZUPT algorithm and speciﬁcally in the orientation
estimation. Errors in the orientation estimation may yield both over and under estimates of the stride
length. Despite these several error sources, accurate velocity and position estimates are still obtained if
either the percentage of the missing sensor data remains small (as described above) or if the saturation
occurs along a sensor axis that does not contribute signiﬁcantly to the estimate.
A naïve user may be tempted to conclude that it is always best to select an IMU with the largest
ranges for acceleration and angular velocity to always avoid saturation. However, increased range
often comes with the tandem penalty of reduced resolution (e.g., if range is increased, but bit resolution
is not) as well as increased sensor noise, which may both defeat the apparent advantage of higher range
sensors. Consequently, there could well be instances where an IMU possessing an accelerometer that
admits minor saturation yields superior stride parameter estimates relative to one possessing a higher
range accelerometer that admits no saturation. Of course, a superior concept is to employ multiple
accelerometers and/or rate gyros with increasing (and even slightly overlapping) ranges, a concept not
studied herein.
We note that in estimating the total distance traveled, symmetrically distributed stride length
errors (i.e., some overestimated and some underestimated) may cancel, leading to accurate estimates
Sensors 2019, 19, 2601
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of total distance traveled. Therefore, it is important to understand how individual stride length
and angle estimates are aﬀected by sensor parameters. We chose to study these eﬀects using the
reported standard deviation of the stride length (and stride angle) diﬀerences, where these diﬀerences
were compared to the baseline estimates (using the maximal sensor parameters). By using both the
cumulative error in the distance traveled and the standard deviations of stride length and stride angle
diﬀerences, we were able to reach sound conclusions of how IMU parameters aﬀect individual stride
length estimates as reported herein despite not explicitly having stride by stride ground truth data.
In particular, we note that standard deviations of stride length and stride angle diﬀerences appear to
converge as sensor parameters (ranges and sampling frequency) approach the nominal parameters for
that sensor (Figures 6 and 10–12). This apparent convergence suggests that (as one would expect) the
baseline estimates are likely the best available estimates and thus the diﬀerences likely correspond
to degradations in the estimates. However, we also acknowledge that independent ground truth
estimates of individual stride lengths and angles are required to conﬁrm this conclusion and that
data was not available in the present study. The convergence in standard deviations also suggests
that improvements in estimates of individual stride parameters when going beyond the ranges and
sampling frequencies utilized by the sensor in this study will be minor compared to the degradation
eﬀects due to sensor limitations demonstrated in this study.
For these experiments, sampling frequency showed no signiﬁcant impact on estimates of the total
distance traveled (except in one limited condition); however, it signiﬁcantly impacted estimates of the
individual stride parameters (stride length and stride angle). In particular, the variance of the individual
stride parameters was signiﬁcantly inﬂuenced by the ﬁltering/sampling method employed. This eﬀect
was demonstrated using two simple down-sampling methods. While this analysis demonstrated
diﬀerences in stride parameter estimates with a reduction of the sampling frequency on a single
IMU, caution must be exercised when comparing sampling frequencies between IMU designs. Many
factors of IMU design in addition to sampling frequency (e.g., sensor hardware, sensor placement)
impact stride parameter estimates. Thus, it remains possible for an IMU with a modest sampling
frequency (e.g., 128 Hz) to yield superior stride parameter estimates to another IMU design with a
higher sampling frequency (e.g., 1000 Hz).
Finally, we describe several limitations of this study which we also believe do not alter the core
conclusions. First, we acknowledge that the sensors available for this study both had limitations
(in range and sampling frequency) that may aﬀect the accuracy of the calculated stride metrics presented
herein. Despite these limitations, we demonstrated important conclusions about the eﬀects of the
sensor parameters on the selected stride parameters. We duly note additional factors not studied
herein that may impact the accuracy of stride estimates (e.g., sensor noise, sensor bandwidth, sensor
resolution, Kalman ﬁlter tuning, etc.) that motivate future studies. Second, these experiments were
conducted over a relatively short (100 m) distance. However, the results for the cumulative distance
error are expected to hold for any (i.e., longer) distance because the cumulative distance error is
equivalent to a percent error in the mean stride length estimates. Third, in the statistical analyses
detailed in Appendix A, we did not evaluate subject-speciﬁc eﬀects (i.e., we removed the eﬀect of
subjects on our results by treating subject as a random eﬀect). Future studies could investigate the eﬀect
of subject demographics (e.g., weight, height, etc.) on estimates of gait parameters using the ZUPT
method. Fourth, we note that our study is not well suited for traditional statistical power analyses.
Thus, we remind the reader that caution should be employed when interpreting the eﬀects that were
not found to be signiﬁcant in the statistical analyses as they may be subject to type-II statistical errors
(i.e., an eﬀect not found to be signiﬁcant does not guarantee that there is no eﬀect). However, we also
note that the potential presence of type-II errors (for the eﬀects not observed to be signiﬁcant) in no
way diminishes the importance of the eﬀects that were observed to be signiﬁcant and their associated
conclusions. Fifth, we suggest that future studies investigate the impact of sensor properties on other
stride parameters obtainable from the ZUPT method (e.g., foot clearance, foot roll angle, etc.).
Sensors 2019, 19, 2601
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5. Conclusions
Appropriate selection of the ranges and sampling frequencies of the inertial sensors embedded in
IMUs is crucial for accurately estimating foot trajectories (hence gait parameters) from foot-mounted
IMUs, and particularly for the speeds associated with competitive distance running. In this study,
we investigated the eﬀects of mean gait speed and sensor parameters on estimates of stride parameters.
The novelty and contribution of this work are that it: (1) quantiﬁes these eﬀects at mean speeds
commensurate with competitive distance running (up to 6.4 m/s); (2) identiﬁes the root causes
of inaccurate foot trajectory estimates obtained from the ZUPT method; and (3) oﬀers important
engineering recommendations for selecting accurate IMUs for studying human running. Estimates of
the cumulative distance traveled (from the individual stride length estimates) degrade with speed;
however, across the range of mean speeds studied here, estimates remained within 5% of ground truth
if there was no or minor saturation of the accelerometer (1.5% or less) or gyro (2.6% or less) signals as
deﬁned herein. In particular, the reported experiments required accelerometer ranges of at least 50 g
and gyro ranges of at least 1000 deg/s to avoid signiﬁcant errors in estimates of the cumulative distance
traveled for mean running speeds up to 6.4 m/s. Errors that arise due to sensor saturation trace to
core assumptions that are violated in the underlying estimation procedure based on the ZUPT method
(i.e., zero-mean Gaussian noise, zero-velocity, and linear velocity drift assumptions). For applications
similar to the ones described in this paper, accurate results remain possible even with modest sampling
frequencies (e.g., 128 Hz), provided well-designed ﬁlters are employed.
Author Contributions: Conceptualization, M.V.P. and S.M.C.; Formal analysis, M.V.P.; Funding acquisition,
N.C.P.; Investigation, M.V.P. and S.M.C.; Methodology, M.V.P. and S.M.C.; Supervision, L.V.O., N.C.P., and S.M.C.;
Writing—original draft, M.V.P.; Writing—review and editing, L.V.O., N.C.P., and S.M.C.
Funding: Parts of this research were funded by US Army Contracting Command-APG, Natick Contracting
Division, Natick, MA, contract number W911QY-15-C-0053. This material is also based upon work supported
by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE 1256260.
Any opinions, ﬁndings, and conclusions or recommendations expressed in this material are those of the authors
and do not necessarily reﬂect the views of the National Science Foundation.
Acknowledgments: The authors would like to thank Corey Powell for his help with the statistical analyses for
this study.
Conﬂicts of Interest: The authors declare no conﬂict of interest.
Appendix A —Full Statistical Results from Linear Mixed-Eﬀects Models
We report results for four statistical analyses that reveal the eﬀects of accelerometer range, gyro
range, and downsampling by two methods. For each of the four analyses, we utilized a linear
mixed-eﬀects model to test the eﬀects of speed, sensor parameter (sensor range or sampling frequency),
and the combination of speed and sensor parameter on the cumulative distance error. In each model,
we removed subject eﬀects by treating the subject as a random eﬀect. Because we are interested in
the eﬀect of mean speed on the cumulative distance error without saturation, we treated mean speed
as a continuous variable and ﬁxed eﬀect. We are also interested in the interaction of mean speed
and the sensor parameter (i.e., how speed aﬀects the cumulative distance error for a change in the
sensor parameter versus that eﬀect for the baseline sensor parameter) and included this interaction
as a ﬁxed eﬀect. The model also calculated the eﬀects of the sensor parameter alone (eﬀect of only
sensor parameter independent of speed) and a y-axis intercept (cumulative distance error with a
mean speed of 0 m/s) for the baseline sensor condition; however, we did not include those additional
ﬁndings as they did not add to the conclusions or the interpretation of the presented results. In each
analysis, we used estimates obtained from the original (non-truncated or non-downsampled) IMU
data as the baseline. All statistical analyses were run in R statistical software using lme4 and lmerTest
packages [25,31,32]. The function lmer was used to ﬁt the models and the function conﬁnt was used to
compute 95% conﬁdence intervals for all estimates.
Sensors 2019, 19, 2601
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The four tables below present results from this model for each of the four analyses. The ﬁrst
(shaded) row in each table reports the eﬀect of speed only on the cumulative distance error for the
baseline sensor condition. This includes an estimated linear slope for the cumulative distance error
versus speed (i.e., an estimated slope of 1%/(m/s) would indicate that the cumulative distance error
increases by an estimated 1% for each 1 m/s increase in mean speed) for the baseline sensor condition
where a negative value means that the error is negative as deﬁned in this paper (i.e., underprediction
of cumulative distance). This slope also represents the sensitivity of the cumulative distance error to
mean speed. In the remaining (unshaded) rows, we report the interactions of mean speed and the
sensor parameter (i.e., how a change to sensor parameters compared to the baseline sensor condition
impacts the sensitivity of the cumulative distance error to mean speed). The estimated slopes in these
rows denote the estimated diﬀerence in the slope of the cumulative distance error versus speed over
that for the baseline condition. Thus, these entries report the additional error sensitivity to speed
for the speciﬁed change in the sensor parameter relative to the baseline. For example, in Table A1,
the estimated cumulative distance speed error is negative and it grows by −0.39%/(m/s) for the baseline
sensor. The error sensitivity then grows (i.e., worsens) by an additional −0.57%/(m/s) for a sensor
employing an accelerometer with a 24 g range. The third and fourth columns report the signiﬁcance
(p-value) and the 95% conﬁdence interval for the estimated slope/error sensitivity.
Table A1. Eﬀect of speed on the cumulative distance versus accelerometer range. First (shaded) row
reports the estimated slope of the cumulative distance versus mean speed using the baseline sensor
(100 g range accelerometer). Remaining (unshaded) rows report estimated diﬀerences in this slope
for IMUs with indicated acceleration ranges compared to the baseline IMU (i.e., the increased error
sensitivity to speed with the indicated reductions in acceleration range relative to the baseline range).
Speed Eﬀect
Estimated Slope
(%/(m/s))
p-Value
95% Conf. Int.
Baseline (100g)
−0.39
<0.01
(−0.65, −0.13)
75 g vs. Baseline
0.00
0.99
(−0.37, 0.37)
50 g vs. Baseline
−0.03
0.88
(−0.39, 0.34)
24 g vs. Baseline
−0.57
<0.01
(−0.94, −0.21)
16 g vs. Baseline
−1.08
<0.001
(−1.45, −0.72)
10 g vs. Baseline
−2.24
<0.001
(−2.60, −1.87)
6 g vs. Baseline
−5.70
<0.001
(−6.06, −5.33)
Table A2. Eﬀect of speed on the cumulative distance versus gyro range. First (shaded) row reports the
estimated slope of cumulative distance versus mean speed using the baseline sensor (2000 deg/s gyro).
Remaining (unshaded) rows report estimated diﬀerences in this slope for IMUs with indicated angular
velocity ranges compared to the baseline IMU (i.e., the increased error sensitivity to speed following
the indicated reductions in angular velocity range relative to the baseline range).
Speed Eﬀect
Estimated Slope
(%/(m/s))
p-Value
95% Conf. Int.
Baseline (2000 deg/s)
−0.41
<0.01
(−0.68, −0.13)
1500 deg/s vs. Baseline
0.00
0.99
(−0.39, 0.39)
1000 deg/s vs. Baseline
0.05
0.81
(−0.34, 0.44)
750 deg/s vs. Baseline
−0.40
<0.05
(−0.80, −0.01)
500 deg/s vs. Baseline
−3.08
<0.001
(−3.47, −2.69)
Sensors 2019, 19, 2601
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Table A3.
Eﬀect of speed on the cumulative distance versus sampling frequency for Method 1
(downsampling). First (shaded) row reports the estimated slope of the cumulative distance traveled
versus mean speed using the baseline sensor (1000 Hz). Remaining (unshaded) rows report the
estimated diﬀerences in this slope for IMUs with the indicated sampling frequency compared to
the baseline IMU (i.e., the increased error sensitivity to speed following the indicated reductions in
sampling frequency relative to the baseline sampling frequency).
Speed Eﬀect
Estimated Slope
(%/(m/s))
p-Value
95% Conf. Int.
Baseline (1000 Hz)
−1.12
<0.001
(−1.48, −0.77)
500 Hz vs. Baseline
−0.09
0.73
(−0.59, 0.41)
250 Hz vs. Baseline
−0.13
0.63
(−0.63, 0.38)
125 Hz vs. Baseline
−0.26
0.32
(−0.76, 0.24)
62.5 Hz vs. Baseline
0.94
<0.001
(0.44, 1.44)
Table A4.
Eﬀect of speed on the cumulative distance versus sampling frequency for Method 2
(decimation). First (shaded) row reports the estimated slope of the cumulative distance traveled versus
mean speed using the baseline sensor (1000 Hz). Remaining (unshaded) rows report the estimated
diﬀerences in this slope for IMUs with the indicated sampling frequency compared to the baseline IMU
(i.e., the increased error sensitivity to speed following the indicated reductions in sampling frequency
relative to the baseline sampling frequency).
Speed Eﬀect
Estimated Slope
(%/(m/s))
p-Value
95% Conf. Int.
Baseline (1000 Hz)
−1.17
<0.001
(−1.50, −0.85)
500 Hz vs. Baseline
−0.01
0.96
(−0.47, 0.45)
250 Hz vs. Baseline
−0.03
0.90
(−0.49, 0.43)
125 Hz vs. Baseline
0.08
0.74
(−0.38, 0.54)
62.5 Hz vs. Baseline
0.12
0.60
(−0.33, 0.58)
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Effect of IMU Design on IMU-Derived Stride Metrics for Running.
Potter, Michael V,Ojeda, Lauro V,Perkins, Noel C,Cain, Stephen M