sensors
Article
Estimating Stair Running Performance Using
Inertial Sensors
Lauro V. Ojeda 1,*
ID , Antonia M. Zaferiou 2, Stephen M. Cain 1
ID , Rachel V. Vitali 1,
Steven P. Davidson 1, Leia A. Stirling 3
ID and Noel C. Perkins 1
1
Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA;
smcain@umich.edu (S.M.C.); vitalir@umich.edu (R.V.V.); stevepd@umich.edu (S.P.D.);
ncp@umich.edu (N.C.P.)
2
Department of Orthopedic Surgery, Rush University Medical Center, Chicago, IL 60612, USA;
antonia_zaferiou@rush.edu
3
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology,
Boston, MA 02139, USA; leia@mit.edu
*
Correspondence: lojeda@umich.edu; Tel.: +1-734-647-1803
Received: 10 October 2017; Accepted: 13 November 2017; Published: 17 November 2017
Abstract: Stair running, both ascending and descending, is a challenging aerobic exercise that many
athletes, recreational runners, and soldiers perform during training. Studying biomechanics of stair
running over multiple steps has been limited by the practical challenges presented while using
optical-based motion tracking systems. We propose using foot-mounted inertial measurement units
(IMUs) as a solution as they enable unrestricted motion capture in any environment and without
need for external references. In particular, this paper presents methods for estimating foot velocity
and trajectory during stair running using foot-mounted IMUs. Computational methods leverage
the stationary periods occurring during the stance phase and known stair geometry to estimate foot
orientation and trajectory, ultimately used to calculate stride metrics. These calculations, applied to
human participant stair running data, reveal performance trends through timing, trajectory, energy,
and force stride metrics. We present the results of our analysis of experimental data collected on
eleven subjects. Overall, we determine that for either ascending or descending, the stance time is the
strongest predictor of speed as shown by its high correlation with stride time.
Keywords: wearable sensors; inertial measurement units; motion tracking; human performance;
stair running
1. Introduction
We present a method for using inertial measurement units (IMUs) to measure the kinematics and
performance of stair running. Running on stairs is a mechanically challenging task. Stair ascent (both
walking and running) challenges the body to achieve center of mass translation forward and upward
against gravity (repeatedly generating upward ground reaction forces larger than the downward
bodyweight force). Therefore, studying stair ascent can provide insights into an individual's aerobic
conditioning [1], athletic strength and lower extremity power [2], and performance [3]. Stair descent,
in contrast, challenges the body to achieve the desired forward and downward trajectory while
controlling and leveraging the assistance of gravity. Therefore, stair descent performance is often
studied in clinical populations to assess the level of lower extremity joint stability and control [4].
Furthermore, each footfall needs to land on the relatively small surface of each step, therefore,
successful performance of both stair ascent and decent require body coordination across multiple
body segments in order to avoid trips or falls. Overground running has been studied extensively from
different points of view [5,6]; a detailed review of early research being provided by Novacheck [7].
Sensors 2017, 17, 2647; doi:10.3390/s17112647
www.mdpi.com/journal/sensors
Sensors 2017, 17, 2647
2 of 14
On the other hand, in-depth biomechanical analysis of stair running has been limited by inadequate
biomechanical tracking tools. Optical-based motion capture systems and instrumented walkways,
which are commonly used for studying gait, are limited by practical challenges in order to appropriately
position cameras for the desired motion capture volume. Consequently, past studies of stair climbing
focus on the functional walking pace [8–10] and have estimated overall energy expenditure [11], basic
timing measures [12], and joint angles [6].
In contrast, we propose using foot-mounted IMUs as a motion capture instrument. Body-worn
IMUs enable human motion analysis in outdoor and other contextually-relevant settings (e.g., training
facilities, game settings, obstacle courses) and have been used in a wide array of biomechanics
applications; see, for example, [13–20]. Our approach uses foot-mounted IMUs to measure the foot
kinematic variables (acceleration and angular velocity) during stair running. Doing so enables one to
track a large number of steps, to understand transient and steady state running on stairs, and to also
deduce performance measures.
IMUs are portable, unobtrusive, and unconstrained (e.g., they do not need external references)
motion tracking devices. However, IMU data (and quantities computed therefrom) are affected
by several sources of error (e.g., bias instability, scale factor errors, acceleration, and temperature
sensitivity) that must be accounted for during motion tracking applications [21]. In this paper, we
present specialized algorithms that address these sources of error to estimate the foot trajectory
and velocity during stair running.
In particular we extend the Zero velocity UPdaTe (ZUPT)
algorithm [22], which has been validated to provide accurate foot motions [14,23], by adding additional
drift corrections speciﬁc to the constraints of stair running (known riser dimensions). We further
employ those estimates to deduce metrics for evaluating stair running performance and explore
the metrics utilizing experimental data collected on 11 subjects. We hypothesized that the metrics
that could be deﬁned were related to the overall speed, thereby providing an ability to assess stair
running techniques.
2. Materials and Methods
We tested 11 healthy volunteer subjects (three female, eight male; age:
25.6 ± 3.7 years;
mean ± SD). The University of Michigan IRB approved the study and all subjects provided informed
consent. Subjects were instructed to run up a long staircase at maximum speed, without skipping
treads. After pausing for approximately ten seconds, the subjects ran down the same ﬂight of stairs
at maximum speed returning to the starting position, again without skipping treads. The staircase
provided 16 strides total during the steady state (eight left and eight right). Subjects were not instructed
which foot to begin stepping with for the task. The staircase rise height was 18 cm and the depth
was 30 cm.
The subjects wore two IMU data loggers (Opals, APDM, Portland OR, USA; 128 Hz sampling,
±6 g acceleration, ±2000 deg/s angular rate), one mounted on each shoe afﬁxed using athletic
tape to the top of the laces (see Figure 1). The IMUs measure and store three components of linear
acceleration (af = [ax, ay, az]) from the on-board accelerometer and three components of angular
velocity (ωf =

ωx, ωy, ωz
) from the on-board angular rate gyro, both relative to the sensor-ﬁxed
axes (x, y, z). These sensor axes deﬁne the IMU frame of reference. We also deﬁne a navigation frame
that overlaps with the IMU frame during initialization. The navigation frame remains afﬁxed to the
world during the experiment, while the IMU frame moves with the subject’s foot. Since the IMU
sensor measurements are relative, there is no need to follow a strict anatomical calibration. However,
since the IMU reference frame determines the navigation frame during initialization, it is advisable to
approximately align the IMU axes to the desired navigation frame (see Figure 1). In-depth explanations
of how (strap-down) IMUs are used, particularly for navigation applications, are provided in [24,25].
Major results from this ﬁeld that we employ are summarized below.
Sensors 2017, 17, 2647
3 of 14
Sensors 2017, 17, 2647 
3 of 14 
 
 
(a) 
(b)
(c) 
Figure 1. IMU data logger setup: (a) APDM IMU device showing the IMU sensor axes; (b) IMU 
attached to the shoe showing the IMU frame axes convention used in this paper; and (c) video stills 
showing the IMUs mounted on a subject's shoe climbing stairs ascent. 
2.1. Orientation Estimation 
Estimating the foot trajectory from IMU data begins with first estimating the orientation of the 
foot-mounted IMU. For this purpose, we choose a quaternion (ࢗ)  representation of the IMU 
orientation. Unlike the more common Euler angle representation that suffers from gimbal-lock, the 
quaternion representation readily describes any arbitrary sequence of rotations [26]. Quaternions 
represent an orientation as a rotation angle about a rotation axis. Thus, quaternions are defined using 
four parameters, one defining the angle of rotation and three defining the axis of rotation (e.g., three 
direction cosines). The four quaternion parameters satisfy the differential equation: 
ࢗሶ = ࢗ ∘ࢹ
2 (1) 
ࢹ=ൣ0,߱௫, ߱௬, ߱௭൧ 
(2) 
in which the operator ∘ denotes quaternion multiplication [25,27] and ࢹ is a four-element vector 
containing the aforementioned measured angular velocity components ൫߱௙൯. Thus, the solution of 
(1) using the measured ࢹ yields the gyro-estimated orientation of the IMU as a function of time. 
The gyro-estimated orientation will inevitably drift due to sensor errors, including bias drift, 
scale factor errors, and acceleration sensitivity. Our algorithm fuses the gyro-estimated orientation 
with accelerometer-estimated tilt angles from vertical (roll and pitch). This is achieved using a 
Kalman filter [28,29]. When the IMUs are mounted on the feet, the foot and the attached IMU are 
essentially stationary during specific time periods (ݐ௦) for the stance phase of each stride. The 
stationary periods are detected by observing the gyroscope and accelerometer measurements  
(see [14] Section 2.1 for more information about how ݐ௦ is determined). During stationary periods, 
the accelerometer measures the components of gravity (ܩ) along each sense axis. These measures 
are used to form accelerometer-estimated roll and pitch angles (ࢠ=[߶௔,ߠ௔]) per: 
߶௔ = sinିଵ ቀܽ௫
ܩ ቁ 
(3) 
ߠ௔=−sinିଵ ൬ ܽ௬ܩ
cos ߶௔
൰ 
(4) 
Next, we use the gyroscope-estimated quaternion (ࢗ) value to calculate the equivalent Euler 
angles ൫࢞ = ൣ߶௚,ߠ௚,߰௚൧൯, which also includes the estimated yaw angle ൫߰௚൯ (that is temporarily 
ignored as it cannot be detected from the accelerometers). The Kalman filter states ൫࢞ෝ = ൣ߶෠,ߠ෠൧൯ are 
estimated as a combination of the gyroscope-based and accelerometer-based tilt estimates. We 
assume that all gyroscope error contributions and accelerometer-based tilt errors can be modeled as 
zero mean Gaussian noise. Since the process and measurement covariance errors are sensor-
dependent only, once the Kalman filter is tuned the parameters are valid for all participants. The 
Figure 1. IMU data logger setup: (a) APDM IMU device showing the IMU sensor axes; (b) IMU
attached to the shoe showing the IMU frame axes convention used in this paper; and (c) video stills
showing the IMUs mounted on a subject's shoe climbing stairs ascent.
2.1. Orientation Estimation
Estimating the foot trajectory from IMU data begins with ﬁrst estimating the orientation of the
foot-mounted IMU. For this purpose, we choose a quaternion (q) representation of the IMU orientation.
Unlike the more common Euler angle representation that suffers from gimbal-lock, the quaternion
representation readily describes any arbitrary sequence of rotations [26]. Quaternions represent
an orientation as a rotation angle about a rotation axis. Thus, quaternions are deﬁned using four
parameters, one deﬁning the angle of rotation and three deﬁning the axis of rotation (e.g., three direction
cosines). The four quaternion parameters satisfy the differential equation:
.q = q ◦ Ω
2
(1)
Ω =

0, ωx, ωy, ωz

(2)
in which the operator ◦ denotes quaternion multiplication [25,27] and Ω is a four-element vector
containing the aforementioned measured angular velocity components (ω f ). Thus, the solution of (1)
using the measured Ω yields the gyro-estimated orientation of the IMU as a function of time.
The gyro-estimated orientation will inevitably drift due to sensor errors, including bias drift,
scale factor errors, and acceleration sensitivity. Our algorithm fuses the gyro-estimated orientation
with accelerometer-estimated tilt angles from vertical (roll and pitch).
This is achieved using
a Kalman ﬁlter [28,29]. When the IMUs are mounted on the feet, the foot and the attached IMU are
essentially stationary during speciﬁc time periods (ts) for the stance phase of each stride. The stationary
periods are detected by observing the gyroscope and accelerometer measurements (see [14] Section 2.1
for more information about how ts is determined). During stationary periods, the accelerometer
measures the components of gravity (G) along each sense axis. These measures are used to form
accelerometer-estimated roll and pitch angles (z = [φa, θa]) per:
φa = sin−1 ax
G

(3)
θa = − sin−1
 ayG
cos φa

(4)
Next, we use the gyroscope-estimated quaternion (q) value to calculate the equivalent Euler
angles (x = [φg, θg, ψg]), which also includes the estimated yaw angle
Sensors 2017, 17, 2647
4 of 14
once the Kalman ﬁlter is tuned the parameters are valid for all participants. The updated state is then
converted back to its corresponding quaternion value. Figure 2 illustrates a block diagram of this
orientation estimation algorithm.
Sensors 2017, 17, 2647 
4 of 14 
 
updated state is then converted back to its corresponding quaternion value. Figure 2 illustrates a 
block diagram of this orientation estimation algorithm. 
 
Figure 2. Angular velocity components measured by the gyroscope are integrated once to obtain 
orientation estimates ࢞. The accelerometer components are used to estimate tilt (roll and pitch) during 
stationary periods ݐ௦. The Kalman filter bounds the tilt errors by fusing the gyro-based orientation 
and accelerometer-based tilt to establish the “corrected orientation”  ࢞ෝ.  
2.2. Foot Trajectory Estimation 
The resulting orientation estimates are used to resolve the foot IMU frame-acceleration 
components ൫ࢇ௙൯ into the navigation frame acceleration components (ࢇ௡). The ݖ-axis component of 
the resultant world-referenced acceleration ࢇ௡ will be affected by gravity ܩ: 
ࢇ௪ = ࢘௙
௡ࢇ௙+ܩ (5) 
in which ࢘௙
௡ is the rotation matrix from the foot IMU frame to the navigation frame as computed 
from the quaternion ࢗ [24,25]. Next, integrating ࢇ௡ once and then twice yields the foot IMU velocity 
(࢜) and position (࢖): 
࢜ = ࢜௢ + නࢇ௡݀ݐ
௧
௧೚
 
(6) 
࢖ = ࢖௢ + න ࢜݀ݐ
௧
௧೚
 
(7) 
Since the experiment starts with a stationary phase, the initial velocity (࢜௢) is zero and at a 
position (࢖௢) also designated as zero. However, this can be generalized to a non-zero initial velocity 
or position for applications that require such. Examples of software implementations  
of (1)–(7) are found in [30,31]. 
The velocity estimated from (6) is often polluted by residual drift error (deriving from both the 
gyro and the accelerometer) which leads to (often slowly varying) velocity errors. The velocity drift 
error can be estimated and (approximately) eliminated using the following procedure. During the 
stationary times (ݐ௦) any remaining estimated velocity during these times can be assumed to be 
caused by drift error. These velocity errors are used to correct both the velocity (6) and position (7) 
estimates using an algorithm known as the Zero velocity UPdaTe (ZUPT). A block diagram for the 
ZUPT algorithm is illustrated in Figure 3 and further details of its implementation can be found  
in [14,22]. 
࢞
න ݂(ݐ)݀ݐ
௧
௧೚
Kalman
Filter
࢞
Corrected
Orientation
࢞ෝ
Angular
Velocity
Tilt
Estimation
ࢠ
Linear
Acceleration
Noise
+
Orientation
ݐ௦
ࢇ௙
࣓ࢌ
Figure 2. Angular velocity components measured by the gyroscope are integrated once to obtain
orientation estimates x. The accelerometer components are used to estimate tilt (roll and pitch) during
stationary periods ts. The Kalman ﬁlter bounds the tilt errors by fusing the gyro-based orientation and
accelerometer-based tilt to establish the “corrected orientation” ˆx.
2.2. Foot Trajectory Estimation
The resulting orientation estimates are used to resolve the foot IMU frame-acceleration
components (af ) into the navigation frame acceleration components (an). The z-axis component
of the resultant world-referenced acceleration an will be affected by gravity G:
aw = rn
f af + G
(5)
in which rn
f is the rotation matrix from the foot IMU frame to the navigation frame as computed from
the quaternion q [24,25]. Next, integrating an once and then twice yields the foot IMU velocity (v) and
position (p):
v = vo +
Z t
to
andt
(6)
p = po +
Z t
to
vdt
(7)
Since the experiment starts with a stationary phase, the initial velocity (vo) is zero and at a
position (po) also designated as zero. However, this can be generalized to a non-zero initial velocity or
position for applications that require such. Examples of software implementations of (1)–(7) are found
in [30,31].
The velocity estimated from (6) is often polluted by residual drift error (deriving from both
the gyro and the accelerometer) which leads to (often slowly varying) velocity errors. The velocity
drift error can be estimated and (approximately) eliminated using the following procedure. During
the stationary times (ts) any remaining estimated velocity during these times can be assumed to
be caused by drift error. These velocity errors are used to correct both the velocity (6) and position
(7) estimates using an algorithm known as the Zero velocity UPdaTe (ZUPT). A block diagram for
the ZUPT algorithm is illustrated in Figure 3 and further details of its implementation can be found
in [14,22].
Sensors 2017, 17, 2647
5 of 14
Sensors 2017, 17, 2647 
5 of 14 
 
 
Figure 3. The accelerometer measurements are resolved in the world coordinate frame using the 
corrected orientation. The resultant accelerations are integrated twice to determine velocity and 
position. During stationary periods  ݐ௦, any remaining velocity is considered an error and its value is 
used to reset the position and acceleration errors. 
2.3. Elevation Correction 
Since the riser (step height) and tread (step depth) dimensions of the stairs are known, we add 
an additional correction to the position estimate. In particular, we designed a single-state Kalman 
filter that makes corrections to the IMU-derived vertical foot position (࢞ = [݌௭]) knowing the riser 
height (ܪ) and the number of steps (݊) to yield an elevation observation per footfall (ࢠ=[ܪ݊]). The 
filter makes corrections to its state (࢞ෝ = [݌௭
ෞ]) whenever the foot reaches a new tread during the 
stationary time  (ݐ௦) . The filter assumes that the state and observation are both affected by 
uncorrelated white noise. A block diagram showing this filter is illustrated in Figure 4. Finally, we 
apply a linear interpolation in order to provide backward corrections to obtain the complete foot 
trajectory for each stride. 
 
Figure 4. A Kalman filter makes foot elevation corrections using the known step height (riser), during 
each stationary time ݐ௦. 
2.4. Gait Timing Variables 
We used a wavelet analysis to establish the beginning (foot-strike) and end (toe-off) of each 
foot/ground contact period [32]. This approach is effective at identifying gait events because when 
the foot strikes or leaves the ground, the acceleration and angular velocity signals contain 
significantly more high-frequency content than at other times of the gait cycle. The wavelet analysis 
is used to identify time points when the measured signals contain significant content above 20 Hz, 
corresponding to either foot-strikes or toe-offs. Foot-strike time (ݐ௦௧௥௜௞௘) was defined as the time 
when the foot first contacts a tread. For running on stairs, the toe is more likely to contact the tread 
first (whereas, during flat-surface walking the heel contacts the ground first). The initial contact 
ݐ௦௧௥௜௞௘ estimation does not require it to be a heel or toe specifically. Toe-off time ൫ݐ௢௙௙൯ is defined as 
the time when the foot first loses contact with the tread. The durations of the major phases of the gait 
Corrected
Orientation
Linear
Acceleration
×
ݐ௦
ࢇ௙
න ݂(ݐ)݀ݐ
௧ೞ
௧೚
න ݂(ݐ)݀ݐ
௧ೞ
௧೚
࢘௙
௡
Corrected
Position
݂݀(ݐ)
݀ݐ
න ݂(ݐ)݀ݐ
௧ೞ
௧೚
࢖
࢜ࢇ௡
Velocity
Error
Accel.
Error
Position
Error
−
−
Kalman
Filter
࢞
Corrected
Elevation
࢞ෝ
Elevation
ࢠ
Stair
Height
Noise
+
ݐ௦
ܪ݊
Noise
+
݌௭
Figure 3. The accelerometer measurements are resolved in the world coordinate frame using the
corrected orientation. The resultant accelerations are integrated twice to determine velocity and
position. During stationary periods ts, any remaining velocity is considered an error and its value is
used to reset the position and acceleration errors.
2.3. Elevation Correction
Since the riser (step height) and tread (step depth) dimensions of the stairs are known, we add
an additional correction to the position estimate. In particular, we designed a single-state Kalman
ﬁlter that makes corrections to the IMU-derived vertical foot position (x = [pz]) knowing the riser
height (H) and the number of steps (n) to yield an elevation observation per footfall (z = [Hn]).
The ﬁlter makes corrections to its state (ˆx = [ ˆpz]) whenever the foot reaches a new tread during the
stationary time (ts). The ﬁlter assumes that the state and observation are both affected by uncorrelated
white noise. A block diagram showing this ﬁlter is illustrated in Figure 4. Finally, we apply a linear
interpolation in order to provide backward corrections to obtain the complete foot trajectory for
each stride.
Sensors 2017, 17, 2647 
5 of 14 
 
 
Figure 3. The accelerometer measurements are resolved in the world coordinate frame using the 
corrected orientation. The resultant accelerations are integrated twice to determine velocity and 
position. During stationary periods  ݐ௦, any remaining velocity is considered an error and its value is 
used to reset the position and acceleration errors. 
2.3. Elevation Correction 
Since the riser (step height) and tread (step depth) dimensions of the stairs are known, we add 
an additional correction to the position estimate. In particular, we designed a single-state Kalman 
filter that makes corrections to the IMU-derived vertical foot position (࢞ = [݌௭]) knowing the riser 
height (ܪ) and the number of steps (݊) to yield an elevation observation per footfall (ࢠ=[ܪ݊]). The 
filter makes corrections to its state (࢞ෝ = [݌௭
ෞ]) whenever the foot reaches a new tread during the 
stationary time  (ݐ௦) . The filter assumes that the state and observation are both affected by 
uncorrelated white noise. A block diagram showing this filter is illustrated in Figure 4. Finally, we 
apply a linear interpolation in order to provide backward corrections to obtain the complete foot 
trajectory for each stride. 
 
Figure 4. A Kalman filter makes foot elevation corrections using the known step height (riser), during 
each stationary time ݐ௦. 
2.4. Gait Timing Variables 
We used a wavelet analysis to establish the beginning (foot-strike) and end (toe-off) of each 
foot/ground contact period [32]. This approach is effective at identifying gait events because when 
the foot strikes or leaves the ground, the acceleration and angular velocity signals contain 
significantly more high-frequency content than at other times of the gait cycle. The wavelet analysis 
is used to identify time points when the measured signals contain significant content above 20 Hz, 
corresponding to either foot-strikes or toe-offs. Foot-strike time (ݐ௦௧௥௜௞௘) was defined as the time 
when the foot first contacts a tread. For running on stairs, the toe is more likely to contact the tread 
first (whereas, during flat-surface walking the heel contacts the ground first). The initial contact 
ݐ௦௧௥௜௞௘ estimation does not require it to be a heel or toe specifically. Toe-off time ൫ݐ௢௙௙൯ is defined as 
the time when the foot first loses contact with the tread. The durations of the major phases of the gait 
Corrected
Orientation
Linear
Acceleration
×
ݐ௦
ࢇ௙
න ݂(ݐ)݀ݐ
௧ೞ
௧೚
න ݂(ݐ)݀ݐ
௧ೞ
௧೚
࢘௙
௡
Corrected
Position
݂݀(ݐ)
݀ݐ
න ݂(ݐ)݀ݐ
௧ೞ
௧೚
࢖
࢜ࢇ௡
Velocity
Error
Accel.
Error
Position
Error
−
−
Kalman
Filter
࢞
Corrected
Elevation
࢞ෝ
Elevation
ࢠ
Stair
Height
Noise
+
ݐ௦
ܪ݊
Noise
+
݌௭
Figure 4. A Kalman ﬁlter makes foot elevation corrections using the known step height (riser), during
each stationary time ts.
2.4. Gait Timing Variables
We used a wavelet analysis to establish the beginning (foot-strike) and end (toe-off) of each
foot/ground contact period [32]. This approach is effective at identifying gait events because when the
foot strikes or leaves the ground, the acceleration and angular velocity signals contain signiﬁcantly
more high-frequency content than at other times of the gait cycle. The wavelet analysis is used to
identify time points when the measured signals contain signiﬁcant content above 20 Hz, corresponding
to either foot-strikes or toe-offs. Foot-strike time (tstrike) was deﬁned as the time when the foot ﬁrst
contacts a tread. For running on stairs, the toe is more likely to contact the tread ﬁrst (whereas, during
ﬂat-surface walking the heel contacts the ground ﬁrst). The initial contact tstrike estimation does not
require it to be a heel or toe speciﬁcally. Toe-off time (to f f ) is deﬁned as the time when the foot ﬁrst
loses contact with the tread. The durations of the major phases of the gait cycle are important indicators
of stair-climbing performance. In particular, we consider the durations of: (1) the entire stride; (2) the
Sensors 2017, 17, 2647
6 of 14
stance phase; and (3) the swing phase. The stride time tstride is measured as the time it takes from one
foot-strike to the next foot-strike of the same foot during steady state. The stance time tstance is the time
difference between two consecutive foot-strike and toe-off events. The swing time tswing is the time
difference between two consecutive toe-off and foot-strike events:
tstride = ∆tstrike
(8)
tstance = to f f − tstrike
(9)
tswing = tstride − tstance
(10)
We calculate the percentage of time that the subjects remain in the stance phase:
tps = 100 × tstance
tstride
(11)
Assuming left-right gait symmetry [33], a tps value larger than 50% indicates the existence of a
double support phase (when both feet are in contact with the ground simultaneously).
2.5. Gait Kinematic and Kinetic Variables
Beyond the timing of gait events, our approach provides the full trajectory and orientation of the
feet, which are useful for understanding stair running performance. Foot clearance (c) is deﬁned as
the foot height (pz) difference between the times of the local maximum (tmax) and minimum (tmin)
around foot-strike:
c = pz(tmax) − pz(tmin)
(12)
In particular, for every stride we identify the local minimum foot height (tmin) after the tstrike
and before to f f . For stair ascending, tmax is deﬁned as the time when the local maximum foot height
occurs just prior to foot-strike (swing phase) while, for stair descending, it is identiﬁed after the foot
strike and, in most cases, before toe-off (stance phase). Examples showing the typical distribution
of local minimum and maximum times in the different gait cycles for stair running (both ascending
and descending) are shown in Figure 5. One interpretation of the clearance, c is that it indicates how
subjects minimize tripping risk as they plan for advancing to the next step (i.e., larger value of c could
imply a more careful foot trajectory planning that provides a safer margin to clear the steps).
Sensors 2017, 17, 2647 
6 of 14 
 
cycle are important indicators of stair-climbing performance. In particular, we consider the durations 
of: (1) the entire stride; (2) the stance phase; and (3) the swing phase. The stride time ݐ௦௧௥௜ௗ௘ is 
measured as the time it takes from one foot-strike to the next foot-strike of the same foot during 
steady state. The stance time ݐ௦௧௔௡௖௘ is the time difference between two consecutive foot-strike and 
toe-off events. The swing time ݐ௦௪௜௡௚ is the time difference between two consecutive toe-off and foot-
strike events: 
ݐ௦௧௥௜ௗ௘ = Δݐ௦௧௥௜௞௘ 
(8) 
ݐ௦௧௔௡௖௘ =ݐ௢௙௙−ݐ௦௧௥௜௞௘
(9) 
ݐ௦௪௜௡௚ =ݐ௦௧௥௜ௗ௘ −ݐ௦௧௔௡௖௘ 
(10) 
We calculate the percentage of time that the subjects remain in the stance phase: 
ݐ௣௦ = 100 ×ݐ௦௧௔௡௖௘
ݐ௦௧௥௜ௗ௘
 
(11) 
Assuming left-right gait symmetry [33], a ݐ௣௦ value larger than 50% indicates the existence of a 
double support phase (when both feet are in contact with the ground simultaneously). 
2.5. Gait Kinematic and Kinetic Variables 
Beyond the timing of gait events, our approach provides the full trajectory and orientation of the 
feet, which are useful for understanding stair running performance. Foot clearance (ܿ) is defined as 
the foot height (݌௭) difference between the times of the local maximum (ݐ௠௔௫) and minimum (ݐ௠௜௡) 
around foot-strike: 
ܿ = ݌௭(ݐ௠௔௫) − ݌௭(ݐ௠௜௡) (12) 
In particular, for every stride we identify the local minimum foot height (ݐ௠௜௡) after the ݐ௦௧௥௜௞௘ 
and before ݐ௢௙௙. For stair ascending, ݐ௠௔௫ is defined as the time when the local maximum foot height 
occurs just prior to foot-strike (swing phase) while, for stair descending, it is identified after the foot 
strike and, in most cases, before toe-off (stance phase). Examples showing the typical distribution of 
local minimum and maximum times in the different gait cycles for stair running (both ascending and 
descending) are shown in Figure 5. One interpretation of the clearance, ܿ is that it indicates how 
subjects minimize tripping risk as they plan for advancing to the next step (i.e., larger value of ܿ 
could imply a more careful foot trajectory planning that provides a safer margin to clear the steps).  
(a) 
(b)
Figure 5. Estimated foot trajectory and speed for running over three treads during ascending (a) and 
descending (b). Close up of a steady state running gait showing the major stride events times: toe-
off ݐ௢௙௙ (green dots), foot strike ݐ௦௧௥௜௞௘ (red dots), maximum elevation ݐ௠௔௫ (black dots), minimum 
elevation ݐ௠௜௡ (yellow dots); and gait phases: stance phase ݐݏݐܽ݊ܿ݁ (blue curves) and swing phase 
ݐݏݓ݅݊݃ (red curves). 
Figure 5. Estimated foot trajectory and speed for running over three treads during ascending (a)
and descending (b). Close up of a steady state running gait showing the major stride events times:
toe-off to f f (green dots), foot strike tstrike (red dots), maximum elevation tmax (black dots), minimum
elevation tmin (yellow dots); and gait phases: stance phase tstance (blue curves) and swing phase tswing
(red curves).
Sensors 2017, 17, 2647
7 of 14
The estimated foot IMU velocity (6) is used to compute a proxy for the foot kinetic energy per
unit of mass kem per using the following formulation:
kem = k
m = |v|
2
2
(13)
where |v| denotes the average magnitude of the foot speed calculated over the duration of every stride
tstride (8).
During stair running, the foot rotates with the majority of rotation manifesting in changes in pitch
θ. We estimate the “bounce angle” θbounce as the angular displacement in pitch from foot-strike to
toe-off as follows:
θbreak = |θ(tstrike) − θ(tmin)|
(14)
θprop = |θ(to f f ) − θ(tmin)|
(15)
θbounce = θbreak + θprop
(16)
Here, the “braking angle” θbreak is computed as the change in foot pitch from the contact time
tstrike until the foot reaches its minimum elevation during the stance phase. The “propulsion angle”
θprop is computed as the change in foot pitch from the time of minimum elevation until toe-off
to f f . The resulting bounce angle could be related to ankle stiffness used during propulsion [34],
which implicates performance outcomes [35] (i.e., stiffer ankles limit the time delay, or, “give” in the
transmission of forces up the kinetic chain) or risk for injury [36].
By estimating the impulse between foot-strike and toe-off events, we also estimate the foot vertical
ground reaction force per unit of mass g f m per:
g f m = fz
m = ∆vz
∆t
(17)
∆vz = vz(to f f ) − vz(tstrike)
(18)
where the time increment ∆t equals the tstance (9).
2.6. Statistical Analysis
In our analysis, we eliminated the ﬁrst and the last step from each stair run, as we considered
them to be transition steps that differ from the approximately steady state stepping that is the focus
of our study. We also assumed left-right foot symmetry and pooled these data within the statistical
analysis. This study does not consider or use the anthropometric characteristics of the participants.
To evaluate how the gait timing, kinematic, and kinetic parameters were related to the stride times
(speed), we performed a simple linear regression for each relationship to determine: the R-squared
value (R2) to quantify the variation explained by the relation; the slope of the relation (b) between
the metric of interest and the stride time; and the statistical signiﬁcance of the slope (pb). The simple
linear regression assumptions of normality and constant variance of the residual were assessed
using the Lilliefors test and Engle’s Auto Regressive Conditional Heteroskedasticity (ARCH) test,
respectively. When these conditions were not met, a transformation of the variables was performed
and the simple linear regression was ﬁt to the transformed variables to assess if the relationship trends
were consistent. Comparison of the variation between tswing and tstance was assessed using an F-test.
We use a two-sample t-test to compare the ascending and descending conditions for the tps, c, and
θbounce variables. Finally, we use a one-sample t-test to determine if g f m was different than zero.
3. Results and Discussion
Figure 5 shows an example of the estimated foot elevations and velocity magnitudes against time
for a subject running while ascending (Figure 5a) and descending (Figure 5b) the stairs. The trajectories
Sensors 2017, 17, 2647
8 of 14
illustrate several steady state strides with labelled times for foot-strike, toe-off, maximum elevation,
minimum elevation, and gait phases.
The above algorithm yields estimates of the full (three-dimensional) trajectories, as well as
(three-dimensional) foot orientation angles. Figure 6 presents a foot trajectory in space (elevation
plotted versus forward position) as well as the foot pitch angle and for the same sample steps
considered in Figure 5.
Using speed alone as the criterion, stair running performance can then be quantiﬁed by the
stride time (shorter average stride time predicting greater average speed since step lengths are
deﬁned/constrained by the stairs geometry). Figures 7–12 compare the individual stride times (vertical
axis) against all other metrics, including the additional gait timing, kinematic, and kinetic variables
deﬁned above (horizontal axes). In these ﬁgures, each dot represents one stride during steady state,
and each color represents one subject. We also provide the equation of the linear ﬁt, R2, and pβ for
each relation.
Sensors 2017, 17, 2647 
8 of 14 
 
The above algorithm yields estimates of the full (three-dimensional) trajectories, as well as 
(three-dimensional) foot orientation angles. Figure 6 presents a foot trajectory in space (elevation 
plotted versus forward position) as well as the foot pitch angle and for the same sample steps 
considered in Figure 5. 
Using speed alone as the criterion, stair running performance can then be quantified by the stride 
time (shorter average stride time predicting greater average speed since step lengths are 
defined/constrained by the stairs geometry). Figures 7–12 compare the individual stride times 
(vertical axis) against all other metrics, including the additional gait timing, kinematic, and kinetic 
variables defined above (horizontal axes). In these figures, each dot represents one stride during 
steady state, and each color represents one subject. We also provide the equation of the linear fit, R2, 
and ݌ఉ for each relation. 
(a) 
(b)
Figure 6. Foot trajectory (black curve) and pitch angle ߠ (colored lines) for ascending (a) and 
descending (b) stairs. The colors distinguish the distinct gait cycles across successive treads. 
3.1. Gait Timing Variables 
Our data analysis shows that in either direction (stair ascent or decent), the stride time ݐ௦௧௥௜ௗ௘ 
was mainly predicted by the stance time ݐ௦௧௔௡௖௘ as measured by high correlation (R2 value for ascent 
0.84, ݌௕ < 0.001; R2 for descent 0.92, ݌௕ < 0.001); refer to Figure 7. Thus, shorter ݐ௦௧௔௡௖௘ values are 
strong predictors of overall speed (shorter stride times) during both stair ascent and decent. 
(a) 
(b)
Figure 7. Stance ݐݏݐܽ݊ܿ݁ and stride time ݐݏݐݎ݅݀݁ relationship for ascending (a) and descending (b) 
stairs. Each dot represents one stride, and each color represents one subject. Overall speed is largely 
determined by the stance phase. 
Figure 6. Foot trajectory (black curve) and pitch angle θ (colored lines) for ascending (a) and descending
(b) stairs. The colors distinguish the distinct gait cycles across successive treads.
3.1. Gait Timing Variables
Our data analysis shows that in either direction (stair ascent or decent), the stride time tstride was
mainly predicted by the stance time tstance as measured by high correlation (R2 value for ascent 0.84,
pb < 0.001; R2 for descent 0.92, pb < 0.001); refer to Figure 7. Thus, shorter tstance values are strong
predictors of overall speed (shorter stride times) during both stair ascent and decent.
Sensors 2017, 17, 2647 
8 of 14 
 
The above algorithm yields estimates of the full (three-dimensional) trajectories, as well as 
(three-dimensional) foot orientation angles. Figure 6 presents a foot trajectory in space (elevation 
plotted versus forward position) as well as the foot pitch angle and for the same sample steps 
considered in Figure 5. 
Using speed alone as the criterion, stair running performance can then be quantified by the stride 
time (shorter average stride time predicting greater average speed since step lengths are 
defined/constrained by the stairs geometry). Figures 7–12 compare the individual stride times 
(vertical axis) against all other metrics, including the additional gait timing, kinematic, and kinetic 
variables defined above (horizontal axes). In these figures, each dot represents one stride during 
steady state, and each color represents one subject. We also provide the equation of the linear fit, R2, 
and ݌ఉ for each relation. 
(a) 
(b)
Figure 6. Foot trajectory (black curve) and pitch angle ߠ (colored lines) for ascending (a) and 
descending (b) stairs. The colors distinguish the distinct gait cycles across successive treads. 
3.1. Gait Timing Variables 
Our data analysis shows that in either direction (stair ascent or decent), the stride time ݐ௦௧௥௜ௗ௘ 
was mainly predicted by the stance time ݐ௦௧௔௡௖௘ as measured by high correlation (R2 value for ascent 
0.84, ݌௕ < 0.001; R2 for descent 0.92, ݌௕ < 0.001); refer to Figure 7. Thus, shorter ݐ௦௧௔௡௖௘ values are 
strong predictors of overall speed (shorter stride times) during both stair ascent and decent. 
(a) 
(b)
Figure 7. Stance ݐݏݐܽ݊ܿ݁ and stride time ݐݏݐݎ݅݀݁ relationship for ascending (a) and descending (b) 
stairs. Each dot represents one stride, and each color represents one subject. Overall speed is largely 
determined by the stance phase. 
Figure 7. Stance tstance and stride time tstride relationship for ascending (a) and descending (b) stairs.
Each dot represents one stride, and each color represents one subject. Overall speed is largely
determined by the stance phase.
Sensors 2017, 17, 2647
9 of 14
Due to the restrictions imposed by the stair design, subjects are relatively constrained during
the swing phase. Regardless of speed, the feet must travel approximately the same distance. Thus,
one expects less variation in tswing than in tstance. This expectation is supported by the smaller
standard deviation of tswing (SD for ascent 0.020 s, for descent 0.022 s) compared to that for the tstance
(SD for ascent 0.044 s, for descent 0.049 s) across all subjects (F(153, 153) = 4.67, p < 0.001 for ascent;
F(153, 153) = 5.06, p < 0.001 for descent). During stair ascent, subjects provide just enough speed to
reach the next tread, since otherwise they risk missing, tripping, or overshooting, making the task
either dangerous or inefﬁcient. As a result, there is a lower correlation between tstride and tswing during
ascent (R2 value 0.24) (see Figure 8a). During stair descent, however, subjects have more freedom to
choose higher speeds during the swing phase by using their muscles to break less, as shown by the
higher correlation between tswing and tstride for stair descent (R2 value 0.60) (see Figure 8b). This gain
in speed comes at the expense of having to accommodate for higher foot-strike impacts and increasing
fall risk.
Sensors 2017, 17, 2647 
9 of 14 
 
Due to the restrictions imposed by the stair design, subjects are relatively constrained during the 
swing phase. Regardless of speed, the feet must travel approximately the same distance. Thus, one 
expects less variation in ݐ௦௪௜௡௚ than in ݐ௦௧௔௡௖௘. This expectation is supported by the smaller standard 
deviation of ݐ௦௪௜௡௚ (SD for ascent 0.020 s, for descent 0.022 s) compared to that for the  ݐ௦௧௔௡௖௘ (SD 
for ascent 0.044 s, for descent 0.049 s) across all subjects (F(153, 153) = 4.67, p < 0.001 for ascent;  
F(153, 153) = 5.06, p < 0.001 for descent). During stair ascent, subjects provide just enough speed to 
reach the next tread, since otherwise they risk missing, tripping, or overshooting, making the task 
either dangerous or inefficient. As a result, there is a lower correlation between ݐ௦௧௥௜ௗ௘ and ݐ௦௪௜௡௚ 
during ascent (R2 value 0.24) (see Figure 8a). During stair descent, however, subjects have more 
freedom to choose higher speeds during the swing phase by using their muscles to break less, as 
shown by the higher correlation between ݐ௦௪௜௡௚ and ݐ௦௧௥௜ௗ௘ for stair descent (R2 value 0.60) (see 
Figure 8b). This gain in speed comes at the expense of having to accommodate for higher foot-strike 
impacts and increasing fall risk.  
 
(a) 
(b)
Figure 8. Swing ݐ௦௪௜௡௚ and stride time ݐ௦௧௥௜ௗ௘ relationship for ascending (a) and descending (b) 
stairs. The greater correlation during stair descent indicates that subjects likely generate speed gains 
during the swing phase. 
Finally, we observe that when running downstairs, subjects do so more carefully, as manifested 
in a greater (t(306) = −15.65, ݌ < 0.001) percentage of time ݐ௣௦ (11) that the subjects remain in the 
stance phase while descending (ascending: 44.3 ± 10.8%, descending: 53.4 ± 13.5%; mean/SD). We 
conclude that ݐ௦௧௥௜ௗ௘ is highly correlated with  ݐ௦௧௔௡௖௘ and therefore speed is determined largely by 
the ability of the subjects to generate enough impulse to reach the next step in the shortest period of 
time. 
Table 1 presents a summary of the gait timing variables. To summarize, both ݐ௦௧௔௡௖௘ and ݐ௦௪௜௡௚ 
have significant relationships to speed. However,  ݐ௦௧௔௡௖௘ shows the highest correlation, indicating 
the potential to be a better predictor.  
Table 1. Gait cycle timing variables for running while ascending and descending stairs. 
Direction 
࢚࢙࢚࢘࢏ࢊࢋvs  ࢚࢙࢚ࢇ࢔ࢉࢋ R2/࢈ ࢚࢙࢚࢘࢏ࢊࢋvs ࢚࢙࢝࢏࢔ࢍ R2/࢈࢚࢙࢚ࢇ࢔ࢉࢋ SD (s) 
࢚࢙࢝࢏࢔ࢍ SD (s) 
 ࢚࢖࢙ Mean ± SD (%) 
Ascent 
0.84/1.05 
0.24/1.22 † 
0.044 
0.020 
44.3 ± 10.8 
Descent 
0.92/1.25 † 
0.60/2.27 † 
0.049 
0.022 
53.4 ± 13.5 
† Does not meet constant variance assumption. 
3.2. Gait Kinematic and Kinetic Variables 
While the estimated slope between ݐ௦௧௥௜ௗ௘  (speed) and foot clearance ܿ  for ascent was 
significant, there is a negligible relationship between these variables as seen by the low R2 value (R2 
for ascent 0.03, ݌௕ = 0.05). There was a significant linear relation for descent (R2 = 0.34, ݌௕ < 0.001) 
(see Figure 9). During descent, subjects clear the steps with a smaller average clearance relative to 
ascent (ascending: 0.06 ± 0.02 m, descending: 0.02 ± 0.02 m; mean ± SD; t(306) = 17.49; ݌ < 0.001), in 
Figure 8. Swing tswing and stride time tstride relationship for ascending (a) and descending (b) stairs.
The greater correlation during stair descent indicates that subjects likely generate speed gains during
the swing phase.
Finally, we observe that when running downstairs, subjects do so more carefully, as manifested in
a greater (t(306) = −15.65, p < 0.001) percentage of time tps (11) that the subjects remain in the stance
phase while descending (ascending: 44.3 ± 10.8%, descending: 53.4 ± 13.5%; mean/SD). We conclude
that tstride is highly correlated with tstance and therefore speed is determined largely by the ability of
the subjects to generate enough impulse to reach the next step in the shortest period of time.
Table 1 presents a summary of the gait timing variables. To summarize, both tstance and tswing
have signiﬁcant relationships to speed. However, tstance shows the highest correlation, indicating the
potential to be a better predictor.
Table 1. Gait cycle timing variables for running while ascending and descending stairs.
Direction
tstride vs. tstance R2/b tstride vs. tswing R2/b
tstance SD (s)
tswing SD (s)
tps Mean ± SD (%)
Ascent
0.84/1.05
0.24/1.22 †
0.044
0.020
44.3 ± 10.8
Descent
0.92/1.25 †
0.60/2.27 †
0.049
0.022
53.4 ± 13.5
† Does not meet constant variance assumption.
3.2. Gait Kinematic and Kinetic Variables
While the estimated slope between tstride (speed) and foot clearance c for ascent was signiﬁcant,
there is a negligible relationship between these variables as seen by the low R2 value (R2 for ascent
0.03, pb = 0.05). There was a signiﬁcant linear relation for descent (R2 = 0.34, pb < 0.001) (see Figure 9).
During descent, subjects clear the steps with a smaller average clearance relative to ascent (ascending:
Sensors 2017, 17, 2647
10 of 14
0.06 ± 0.02 m, descending: 0.02 ± 0.02 m; mean ± SD; t(306) = 17.49; p < 0.001), in some cases by
rolling the foot on the nose of the tread as they transition to the next tread. Smaller average clearance
enables the foot to follow a more linear trajectory (see Figure 6), which can be more energy efﬁcient as
explained in the next section.
Sensors 2017, 17, 2647 
10 of 14 
 
some cases by rolling the foot on the nose of the tread as they transition to the next tread. Smaller 
average clearance enables the foot to follow a more linear trajectory (see Figure 6), which can be more 
energy efficient as explained in the next section. 
(a) 
(b)
Figure 9. Foot clearance ܿ and stride time ݐ௦௧௥௜ௗ௘ relationship for ascending (a) and descending (b) 
stairs. Descent is accomplished with an overall smaller clearance relative to ascent. 
A significant linear relationship between the foot kinetic energy ݇݁݉ (13) and ݐ௦௧௥௜ௗ௘ exists (R2 
for ascent: 0.53, ݌௕ < 0.001; R2 for descent: 0.8, ݌௕ < 0.001) (see Figure 10), with faster subjects 
exhibiting higher kinetic energy. During stair ascent, a fraction of the kinetic energy is consumed just 
to clear the nose of the steps safely and, as a result, the foot describes a parabolic trajectory (see  
Figure 6a) in strong contrast with the linear trajectory exhibited during descent (see Figure 6b). 
(a) 
(b)
Figure 10. Kinetic energy per unit of mass ݇݁݉ and stride time ݐݏݐݎ݅݀݁ relationship for ascending (a) 
and descending (b) stairs. In stair ascent, a fraction of the kinetic energy is consumed in order to safely 
clear the nose of the treads. 
Example variations in the pitch angle during stair running are illustrated in Figure 6. The pitch 
variations ߠ௕௢௨௡௖௘  (14)–(16) do not have a linear relationship with ݐ௦௧௥௜ௗ௘  (Figure 11) during 
ascending (R2 for ascent: 0.01, ܾ = 0, ݌௕ = 0.36) and have a moderate relationship during descending 
(R2 for descent: 0.32, ܾ = 3 × 10−3, ݌௕ < 0.001). The average bounce angle during ascent is smaller than 
the average bounce angles during descent (ascending: 45.0 ± 9.2 deg, descending: 66.2 ± 10.4 deg; 
mean ± SD; t(306) = −19.029; ݌ < 0.001). It is noteworthy that during the stair ascent smaller bounce 
angles are indicative of an increase in ankle stiffness which, in turn, increases vertical velocity [35].  
Figure 9. Foot clearance c and stride time tstride relationship for ascending (a) and descending (b) stairs.
Descent is accomplished with an overall smaller clearance relative to ascent.
A signiﬁcant linear relationship between the foot kinetic energy kem (13) and tstride exists (R2 for
ascent: 0.53, pb < 0.001; R2 for descent: 0.8, pb < 0.001) (see Figure 10), with faster subjects exhibiting
higher kinetic energy. During stair ascent, a fraction of the kinetic energy is consumed just to clear
the nose of the steps safely and, as a result, the foot describes a parabolic trajectory (see Figure 6a) in
strong contrast with the linear trajectory exhibited during descent (see Figure 6b).
Sensors 2017, 17, 2647 
10 of 14 
 
some cases by rolling the foot on the nose of the tread as they transition to the next tread. Smaller 
average clearance enables the foot to follow a more linear trajectory (see Figure 6), which can be more 
energy efficient as explained in the next section. 
(a) 
(b)
Figure 9. Foot clearance ܿ and stride time ݐ௦௧௥௜ௗ௘ relationship for ascending (a) and descending (b) 
stairs. Descent is accomplished with an overall smaller clearance relative to ascent. 
A significant linear relationship between the foot kinetic energy ݇݁݉ (13) and ݐ௦௧௥௜ௗ௘ exists (R2 
for ascent: 0.53, ݌௕ < 0.001; R2 for descent: 0.8, ݌௕ < 0.001) (see Figure 10), with faster subjects 
exhibiting higher kinetic energy. During stair ascent, a fraction of the kinetic energy is consumed just 
to clear the nose of the steps safely and, as a result, the foot describes a parabolic trajectory (see  
Figure 6a) in strong contrast with the linear trajectory exhibited during descent (see Figure 6b). 
(a) 
(b)
Figure 10. Kinetic energy per unit of mass ݇݁݉ and stride time ݐݏݐݎ݅݀݁ relationship for ascending (a) 
and descending (b) stairs. In stair ascent, a fraction of the kinetic energy is consumed in order to safely 
clear the nose of the treads. 
Example variations in the pitch angle during stair running are illustrated in Figure 6. The pitch 
variations ߠ௕௢௨௡௖௘  (14)–(16) do not have a linear relationship with ݐ௦௧௥௜ௗ௘  (Figure 11) during 
ascending (R2 for ascent: 0.01, ܾ = 0, ݌௕ = 0.36) and have a moderate relationship during descending 
(R2 for descent: 0.32, ܾ = 3 × 10−3, ݌௕ < 0.001). The average bounce angle during ascent is smaller than 
the average bounce angles during descent (ascending: 45.0 ± 9.2 deg, descending: 66.2 ± 10.4 deg; 
mean ± SD; t(306) = −19.029; ݌ < 0.001). It is noteworthy that during the stair ascent smaller bounce 
angles are indicative of an increase in ankle stiffness which, in turn, increases vertical velocity [35].  
Figure 10. Kinetic energy per unit of mass kem and stride time tstride relationship for ascending (a) and
descending (b) stairs. In stair ascent, a fraction of the kinetic energy is consumed in order to safely
clear the nose of the treads.
Example variations in the pitch angle during stair running are illustrated in Figure 6. The pitch
variations θbounce (14)–(16) do not have a linear relationship with tstride (Figure 11) during ascending
(R2 for ascent: 0.01, b = 0, pb = 0.36) and have a moderate relationship during descending (R2 for
descent: 0.32, b = 3 × 10−3, pb < 0.001). The average bounce angle during ascent is smaller than
the average bounce angles during descent (ascending: 45.0 ± 9.2 deg, descending: 66.2 ± 10.4 deg;
Sensors 2017, 17, 2647
11 of 14
mean ± SD; t(306) = −19.029; p < 0.001). It is noteworthy that during the stair ascent smaller bounce
angles are indicative of an increase in ankle stiffness which, in turn, increases vertical velocity [35].
Sensors 2017, 17, 2647 
11 of 14 
 
(a) 
(b)
Figure 11. Bounce angle ߠ௕௢௨௡௖௘ and stride time ݐݏݐݎ݅݀݁ correlation for ascending (a) and descending 
(b) stairs. Lower bounce angle during stair ascent is related to impulsive motion. 
We calculated foot vertical ground force ݂݃݉ using (17), and determined that there was 
moderate correlation between ݐ௦௧௥௜ௗ௘ and ݂݃݉ for both ascent and descent (R2 for ascent: 0.45, 
݌௕  < 0.001; R2 for descent: 0.21, ݌௕  < 0.001) (see Figure 12). The ݂݃݉ mean value shows that 
ascending stairs requires generating a non-zero reaction force (0.09 ± 0.03 N/kg; mean ± SD;  
t(153) = 40.97, p < 0.001), whereas the descending force was not statistically different from zero  
(0.0 ± 0.02 N/kg, mean ± SD; t(153) = −1.96, p = 0.052). This suggests distinct mechanisms for running 
on stairs with ascending requiring changes in momentum (impulses), while descending requires 
maintaining momentum. Ascending stairs requires generating the necessary force needed to propel 
the body upwards and forwards; conversely, during descending the muscles have less resistance (as 
supported by the increase in bounce angle) allowing gravity to do the work.  
The kinematic and kinetic variables are summarized in Table 2. In summary, we determine that 
clearance, ܿ, is only correlated to speed during stair descent. We found that some ݇݁݉ is lost during 
stair ascent because of the foot parabolic trajectory required to clear safely the steps. The foot angle 
ߠ௕௢௨௡௖௘ shows ankle stiffness during stair ascent versus compliance during stair descent. The effect 
of ߠ௕௢௨௡௖௘ is also evident in ground forces ݂݃݉ being large for stair ascent and negligible for stair 
descent. 
 
(a) 
(b)
Figure 12. Vertical ground reaction force per unit of mass ݂݃݉ and stride time ݐݏݐݎ݅݀݁ relationship 
for ascending (a) and descending (b) stairs. Stair ascent employs significantly larger impulses relative 
to descent. 
 
 
Figure 11. Bounce angle θbounce and stride time tstride correlation for ascending (a) and descending (b)
stairs. Lower bounce angle during stair ascent is related to impulsive motion.
We calculated foot vertical ground force g f m using (17), and determined that there was moderate
correlation between tstride and g f m for both ascent and descent (R2 for ascent: 0.45, pb < 0.001;
R2 for descent: 0.21, pb < 0.001) (see Figure 12). The g f m mean value shows that ascending stairs
requires generating a non-zero reaction force (0.09 ± 0.03 N/kg; mean ± SD; t(153) = 40.97, p < 0.001),
whereas the descending force was not statistically different from zero (0.0 ± 0.02 N/kg, mean ± SD;
t(153) = −1.96, p = 0.052). This suggests distinct mechanisms for running on stairs with ascending
requiring changes in momentum (impulses), while descending requires maintaining momentum.
Ascending stairs requires generating the necessary force needed to propel the body upwards and
forwards; conversely, during descending the muscles have less resistance (as supported by the increase
in bounce angle) allowing gravity to do the work.
The kinematic and kinetic variables are summarized in Table 2. In summary, we determine that
clearance, c, is only correlated to speed during stair descent. We found that some kem is lost during
stair ascent because of the foot parabolic trajectory required to clear safely the steps. The foot angle
θbounce shows ankle stiffness during stair ascent versus compliance during stair descent. The effect of
θbounce is also evident in ground forces g f m being large for stair ascent and negligible for stair descent.
Sensors 2017, 17, 2647 
11 of 14 
 
(a) 
(b)
Figure 11. Bounce angle ߠ௕௢௨௡௖௘ and stride time ݐݏݐݎ݅݀݁ correlation for ascending (a) and descending 
(b) stairs. Lower bounce angle during stair ascent is related to impulsive motion. 
We calculated foot vertical ground force ݂݃݉ using (17), and determined that there was 
moderate correlation between ݐ௦௧௥௜ௗ௘ and ݂݃݉ for both ascent and descent (R2 for ascent: 0.45, 
݌௕  < 0.001; R2 for descent: 0.21, ݌௕  < 0.001) (see Figure 12). The ݂݃݉ mean value shows that 
ascending stairs requires generating a non-zero reaction force (0.09 ± 0.03 N/kg; mean ± SD;  
t(153) = 40.97, p < 0.001), whereas the descending force was not statistically different from zero  
(0.0 ± 0.02 N/kg, mean ± SD; t(153) = −1.96, p = 0.052). This suggests distinct mechanisms for running 
on stairs with ascending requiring changes in momentum (impulses), while descending requires 
maintaining momentum. Ascending stairs requires generating the necessary force needed to propel 
the body upwards and forwards; conversely, during descending the muscles have less resistance (as 
supported by the increase in bounce angle) allowing gravity to do the work.  
The kinematic and kinetic variables are summarized in Table 2. In summary, we determine that 
clearance, ܿ, is only correlated to speed during stair descent. We found that some ݇݁݉ is lost during 
stair ascent because of the foot parabolic trajectory required to clear safely the steps. The foot angle 
ߠ௕௢௨௡௖௘ shows ankle stiffness during stair ascent versus compliance during stair descent. The effect 
of ߠ௕௢௨௡௖௘ is also evident in ground forces ݂݃݉ being large for stair ascent and negligible for stair 
descent. 
 
(a) 
(b)
Figure 12. Vertical ground reaction force per unit of mass ݂݃݉ and stride time ݐݏݐݎ݅݀݁ relationship 
for ascending (a) and descending (b) stairs. Stair ascent employs significantly larger impulses relative 
to descent. 
 
 
Figure 12. Vertical ground reaction force per unit of mass g f m and stride time tstride relationship for
ascending (a) and descending (b) stairs. Stair ascent employs signiﬁcantly larger impulses relative
to descent.
Sensors 2017, 17, 2647
12 of 14
Table 2. Kinematic and kinetic variables for running while ascending and descending stairs.
Direction
tstride vs. c
R2/b
tstride vs. kem
R2/b
tstride vs. θbounce
R2/b
tstride vs. gfm
R2/b
c
Mean ± SD
(m)
θbounce
Mean ± SD
(deg)
gfm
Mean ± SD
(N/Kg)
Ascent
0.03/−0.38 ‡,*
0.53/−0.11 ‡
0.01/0.0 †,*
0.45/−1.21
0.06 ± 0.02
45.0 ± 9.2
0.09 ± 0.03
Descent
0.34/2.44 †
0.80/−0.17 ‡
0.32/3 × 10−3
0.21/1.45
0.02 ± 0.02
66.2 ± 10.4
0.0 ± 0.02
* b Not statistically signiﬁcant. † Constant variance assumption not met. ‡ Normality assumption not met.
For every simple linear regression relation, the assumptions of normality and constant variance of
the residuals were tested (see Tables 1 and 2). For the cases that did not meet the assumptions, we used
data transformation algorithms to correct for distribution skewness as described in [37,38] and veriﬁed
the signiﬁcance of the relationships when assumptions were met. To facilitate the interpretation of the
measures, we presented the relationships for the variables prior to transformation. It is important to
note that while the slopes may differ with the transformed variables, the direction and signiﬁcance of
the relationship would not be expected change the results presented.
Finally, it is important to note that the sensors that we use have limited operational range that
may inﬂuence some of the outcomes, in particular the vertical acceleration during the foot-strike could
be underestimated. We believe that the ﬁnal effect of this limitation in our calculations is small due
to the short duration of this event, the elevation correction that we perform, and our stride-by-stride
basis analysis instead of the whole trajectory.
4. Conclusions
This paper presents a method for understanding the task of running on stairs (both ascending
and descending) from data harvested from foot-mounted IMUs. This understanding derives from an
algorithm that estimates the foot velocity and trajectory while correcting for sensor drift errors using
the ZUPT technique together with a known stair riser height. In studies of human mobility outside
of a controlled experimental setup, during which stair height may not be known to the researchers,
implementing a “standard” step height correction may still assist in calculating stride metrics. Timing,
kinematic, and kinetic variables are proposed as metrics of stair running performance. Results on
human subjects reveal that stair running speed is largely controlled by the stance phase, as opposed
to the swing phase. An approximate measure of foot kinetic energy illustrates greater foot energy
economy during descent versus ascent, which also follows from the near-linear foot trajectory during
descent versus the parabolic path during ascent. The IMU-derived estimates for foot clearance may
have future use in assessing trip/fall risks while the IMU-derived estimates of ground reaction and
bounce angle may have future use in assessing injury potential.
Acknowledgments: This material is based upon work supported by the US Army Contracting Command-APG,
Natick Contracting Division, Natick, MA, under contract W911QY-15-C-0053.
Author Contributions: L.V.O. developed and implemented the algorithms for computing foot trajectories and
performance parameters. L.V.O., A.M.Z., and N.C.P. developed the biomechanical analysis. S.M.C. developed the
gait event detection algorithms. L.V.O. and L.A.S. developed and completed the statistical analysis. L.V.O., A.M.Z,
S.M.C., R.V.V., S.P.D., L.A.S. and N.C.P. provided valuable insight and/or assistance in collecting the data, as well
as contributed to the preparation of the manuscript.
Conﬂicts of Interest: The authors declare no conﬂict of interest.
References
1.
Kerr, J.; Eves, F.; Carroll, D. Six-Month Observational Study of Prompted Stair Climbing. Prev. Med. 2001, 33,
422–427. [CrossRef] [PubMed]
2.
Huskey, T.; Mayhew, J.L.; Ball, T.E.; Arnold, M.D. Factors affecting anaerobic power output in the
Margaria-Kalamen test. Ergonomics 1989, 32, 959–965. [CrossRef] [PubMed]
Sensors 2017, 17, 2647
13 of 14
3.
Harris, G.R.; Stone, M.H.; O’Bryant, H.S.; Johnson, R.L. Short-term performance effects of high power, high
force, or combined weight-training methods. J. Strength Cond. Res. 2000, 14, 14–20.
4.
Protopapadaki, A.; Drechsler, W.; Cramp, M.C.; Coutts, F.J.; Scott, O.M. Hip, knee, ankle kinematics and
kinetics during stair ascent and descent in healthy young individuals. Clin. Biomech. 2007, 22, 203–210.
[CrossRef] [PubMed]
5.
De Wit, B.; De Clercq, D.; Aerts, P. Biomechanical analysis of the stance phase during barefoot and shod
running. J. Biomech. 2000, 33, 269–278. [CrossRef]
6.
Braunstein, B.; Arampatzis, A.; Eysel, P.; Brüggemann, G.P. Footwear affects the gearing at the ankle and
knee joints during running. J. Biomech. 2010, 43, 2120–2125. [CrossRef] [PubMed]
7.
Novacheck, T.F. The biomechanics of running. Gait Posture 1998, 7, 77–95. [CrossRef]
8.
Reid, S.M.; Graham, R.B.; Costigan, P.A. Differentiation of young and older adult stair climbing gait using
principal component analysis. Gait Posture 2010, 31, 197–203. [CrossRef] [PubMed]
9.
Novak, A.C.; Brouwer, B. Kinematic and kinetic evaluation of the stance phase of stair ambulation in persons
with stroke and healthy adults: A pilot study. J. Appl. Biomech. 2013, 29, 443–452. [CrossRef] [PubMed]
10.
Benedetti, M.G.; Agostini, V.; Knaﬂitz, M.; Bonato, P. Muscle activation patterns during level walking and
stair ambulation. In Applications of EMG in Clinical and Sports Medicine; Steele, C., Ed.; Available online:
https://cdn.intechopen.com/pdfs-wm/25822.pdf (accessed on 12 September 2017).
11.
Halsey, L.G.; David, A.R.; Watkins, D.A.; Duggan, B.M. The energy expenditure of stair climbing one step
and two steps at a time: Estimations from measures of heart rate. PLoS ONE 2012, 9, e100658. [CrossRef]
[PubMed]
12.
Johnson, A.N.; Cooper, D.F.; Edwards, R.H. Exertion of stair climbing in normal subjects and in patients
with chronic obstructive bronchitis. Thorax 1977, 32, 711–716. [CrossRef] [PubMed]
13.
Beyea, J.; McGibbon, C.A.; Sexton, A.; Noble, J.; O’Connell, C. Convergent validity of a wearable sensor
system for measuring sub-task performance during the timed up-and-go test.
Sensors 2017, 17, 934.
[CrossRef] [PubMed]
14.
Rebula, J.R.; Ojeda, L.V.; Adamczyk, P.G.; Kuo, A.D. Measurement of foot placement and its variability with
inertial sensors. Gait Posture 2013, 38, 974–980. [CrossRef] [PubMed]
15.
Peruzzi, A.; Della Croce, U.; Cereatti, A. Estimation of stride length in level walking using an inertial
measurement unit attached to the foot: A validation of the zero velocity assumption during stance. J. Biomech.
2011, 44, 1991–1994. [CrossRef] [PubMed]
16.
Seel, T.; Schauer, T.; Raisch, J. Joint axis and position estimation from inertial measurement data by exploiting
kinematic constraints. In Proceedings of the 2012 IEEE International Conference Control Application,
Dubrovnik, Croatia, 3–5 October 2012.
17.
Mariani, B.; Hoskovec, C.; Rochat, S.; Büla, C.; Penders, J.; Aminian, K. 3D gait assessment in young and
elderly subjects using foot-worn inertial sensors. J. Biomech. 2010, 43, 2999–3006. [CrossRef] [PubMed]
18.
McGinnis, R.S.; Cain, S.M.; Davidson, S.P.; Vitali, R.V.; McLean, S.G.; Perkins, N.C. Inertial sensor and cluster
analysis for discriminating agility run technique. IFAC 2017, 32, 150–156. [CrossRef]
19.
Chardonnens, J.; Favre, J.; Cuendet, F.; Gremion, G.; Aminian, K.A. System to measure the kinematics during
the entire ski jump sequence using inertial sensors. J. Biomech. 2013, 46, 56–62. [CrossRef] [PubMed]
20.
Vitali, R.V.; Cain, S.M.; McGinnis, R.S.; Zaferiou, A.M.; Ojeda, L.V.; Davidson, S.P.; Perkins, N.C. Method
for estimating three-dimensional knee rotations using two inertial measurement units: Validation with a
coordinate measurement machine. Sensors 2017, 17, 1970. [CrossRef] [PubMed]
21.
El-Sheimy, N.; Hou, H.; Niu, X. Analysis and modeling of inertial sensors using Allan variance. IEEE Trans.
Instrum. Meas. 2008, 57, 140–149. [CrossRef]
22.
Ojeda, L.; Borenstein, J. Non-GPS Navigation for Security Personnel and First Responders. J. Navig. 2007, 60,
391–407. [CrossRef]
23.
Ojeda, L.V.; Rebula, J.R.; Adamczyk, P.G.; Kuo, A.D. Mobile platform for motion capture of locomotion over
long distances. J. Biomech. 2013, 46, 2316–2319. [CrossRef] [PubMed]
24.
Titterton, D.; Weston, J. Strapdown Inertial Navigation Technology, 2nd ed.; American Institute of Aeronautics
and Astronautics: Reston, VA, USA, 2004; ISBN 978-0863413582.
25.
Rogers, R.M. Applied Mathematics in Integrated Navigation Systems, 3rd ed.; American Institute of Aeronautics
and Astronautics: Reston, VA, USA, 2007; ISBN 9781563479274.
Sensors 2017, 17, 2647
14 of 14
26.
Shoemake, K. Animating rotation with quaternion curves. In Proceedings of the 12th Annual Conference on
Computer Graphics and Interactive Techniques, New York, NY, USA, 22–26 July 1985; pp. 245–254.
27.
Farrell, J.A.; Barth, M. The Global Positioning System & Inertial Navigation; McGraw-Hill Professional:
New York, NY, USA, 1998; ISBN 0-07-022045-X.
28.
Kalman, R. A New Approach to Linear Filtering and Prediction Problems. J. Basic Eng. 1960, 82, 35–45.
[CrossRef]
29.
Grover, R.G.; Hwang, P.Y. Introduction to Random Signals and Applied Kalman Filtering with Matlab Exercises,
4th ed.; John Wiley & Sons Inc.: New York, NY, USA, 2012; ISBN 978-1-118-21485-5.
30.
Savage, P.G. Strapdown Inertial Navigation Integration Algorithm Design Part 1: Attitude Algorithms.
J. Guid. Control Dyn. 1998, 21, 19–28. [CrossRef]
31.
Savage, P.G. Strapdown Inertial Navigation Integration Algorithm Design Part 2: Velocity and Position
Algorithms. J. Guid. Control Dyn. 1998, 21, 208–221. [CrossRef]
32.
Cain, S.M.; McGinnis, R.S.; Davidson, S.P.; Vitali, R.V.; Perkins, N.C.; McLean, S.G. Quantifying performance
and effects of load carriage during a challenging balancing task using an array of wireless inertial sensors.
Gait Posture 2016, 43, 65–69. [CrossRef] [PubMed]
33.
Sadeghi, H.; Allard, P.; Prince, F.; Labelle, H. Symmetry and limb dominance in able-bodied gait: A review.
Gait Posture 2000, 12, 34–45. [CrossRef]
34.
Arampatzis, A.; Brüggemann, G.P.; Metzler, V. The effect of speed on leg stiffness and joint kinetics in human
running. J. Biomech. 1999, 32, 1349–1353. [CrossRef]
35.
Charalambous, L.; Irwin, G.; Bezodis, I.N.; Kerwin, D. Lower limb joint kinetics and ankle joint stiffness in
the sprint start push-off. J. Sports Sci. 2012, 30, 1–9. [CrossRef] [PubMed]
36.
Butler, R.J.; Crowell, H.P.; Davis, I.M. Lower extremity stiffness: Implications for performance and injury.
Clin. Biomech. 2003, 18, 511–517. [CrossRef]
37.
Howell, D.C. Statistical Methods for Psychology, 6th ed.; Thomson Wadsworth: Belmont, CA, USA, 2007;
ISBN 0495012874.
38.
Tabachnick, B.G.; Fidell, L.S. Using Multivariate Statistics, 5th ed.; Allyn and Bacon: Boston, MA, USA, 2007;
ISBN 0205459382.
© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access
article distributed under the terms and conditions of the Creative Commons Attribution
(CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Estimating Stair Running Performance Using Inertial Sensors.
Ojeda, Lauro V,Zaferiou, Antonia M,Cain, Stephen M,Vitali, Rachel V,Davidson, Steven P,Stirling, Leia A,Perkins, Noel C