Sensors 2011, 11, 10571-10585; doi:10.3390/s111110571
Kinematics of Gait: New Method for Angle Estimation Based on
Milica D. Djurić-Jovičić 1,2*, Nenad S. Jovičić 1 and Dejan B. Popović 1,3
1 School of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, Belgrade,
Serbia; E-Mails: firstname.lastname@example.org (N.S.J.); email@example.com (D.B.P.)
2 Tecnalia Serbia, Vladetina 13/6, Belgrade, Serbia
3 Center for Sensory Motor Interaction, Aalborg University, Fredrik Bajers Vej 7, Aalborg, Denmark
* Author to whom correspondence should be addressed; E-Mail: firstname.lastname@example.org;
Received: 1 October 2011; in revised form: 24 October 2011 / Accepted: 25 October 2011 /
Published: 7 November 2011
Abstract: A new method for estimation of angles of leg segments and joints, which uses
accelerometer arrays attached to body segments, is described. An array consists of two
accelerometers mounted on a rigid rod. The absolute angle of each body segment was
determined by band pass filtering of the differences between signals from parallel axes
from two accelerometers mounted on the same rod. Joint angles were evaluated by
subtracting absolute angles of the neighboring segments. This method eliminates the need
for double integration as well as the drift typical for double integration. The efficiency of
the algorithm is illustrated by experimental results involving healthy subjects who walked
on a treadmill at various speeds, ranging between 0.15 m/s and 2.0 m/s. The validation was
performed by comparing the estimated joint angles with the joint angles measured with
flexible goniometers. The discrepancies were assessed by the differences between the two
sets of data (obtained to be below 6 degrees) and by the Pearson correlation coefficient
(greater than 0.97 for the knee angle and greater than 0.85 for the ankle angle).
Keywords: accelerometers; ambulatory system; angles; gait assessment
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Gait analysis is important for objective assessment of the effects of rehabilitation interventions. The
most accurate systems for gait analysis are camera-based systems with reflective markers . These
systems acquire spatial movement (3D) of many markers positioned on the body, while a software
outputs the joint angles and/or other gait parameters. However, camera-based systems require a
dedicated laboratory and limit the length of the analyzed walking distances. Gait laboratories also use
force platforms to measure the ground reaction forces which typically record from only one or two
steps in the middle of the gait sequence. The platforms are 60 × 60 cm, so that aiming for the platforms
hinders the subjects’ natural gait patterns. The alternative to camera-based systems are ultrasound
systems  and magnetic tracking systems , which allow complete 3D kinematic analysis of human
Over the last decade, many gait analysis systems using non-traditional methods have been developed.
These systems, for example, use laser technology or measure near-body air flow [4,5] in order to
estimate kinematics and spatial gait parameters. Also, electronic carpets or wearable force sensors are
used for estimation of ground reaction forces, centre of pressure, and temporal gait parameters [6,7].
Since there is often a need for gait recording in various environments, portable body-mounted systems
are preferred [8,9].
Portable body-mounted systems allow data acquisition from many steps. The portable systems for
kinematics data acquisition directly measure joint angles, or they can record accelerations or angular
velocities of the body segments that carry the sensors. Measurement of joint angles can be done with
various electrogoniometers [9-11]. Particularly convenient are flexible goniometers, which measure
the relative angle between two small blocks that are fixed to the body segments (e.g., Biometrics
flexible Penny & Giles sensors). The advantages of flexible goniometers are: their output is directly
proportional to the angle and their mounting is simpler compared to some other measurement systems.
However, they are not sufficiently robust for daily clinical usage.
An alternative to goniometers, offered by the progress made in micro-electromechanical systems
(MEMS), is the use of accelerometers and gyroscopes. The advantages of these sensors include their
small size and robustness when compared with goniometers. However, the disadvantages of the
accelerometers (and gyroscopes) are computational problems for determining the angles [12-15].
Accelerometers are used for long-term monitoring of human movements, for assessment of energy
expenditure, physical activity, postural sway, fall detection, postural orientation, activity classification
and estimation of temporal gait parameters [16-21]. However, only a few papers report using only
accelerometers for angle estimation, or position and orientation estimation. In most papers some
additional types of sensors are included (gyroscopes, magnetometers, etc.) [22-25].
One method to calculate the angle is to double integrate the measured angular acceleration.
However, the double integration leads to a pronounced drift [13,14]. Several techniques have been
presented in literature for minimization of this drift. For example, Morris  identified the beginning
and the end of each walking cycle and made the signals at the beginning and the end of the cycle equal.
Tong et al.  applied a low cut high pass filter on the shank and thigh inclination angle signals.
However, these methods also removed the static and low-frequency information about the angles and
they cannot be applied to real-time processing.
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The other method for estimation of joint angles from the measured accelerations is the estimation of
the inclination angles between the segments (sensor) and the vertical, followed by the subtraction of
the angles for neighboring body segments. The results are acceptable only if the segment accelerations
are small compared to the gravity .
Adding Kalman filtering to the integration procedure decreases the drift and provides for
real-time applications, but it requires calibration and data from other sensors (accelerometers,
gyroscopes, and magnetic sensors in most cases) for error minimization, as well as noise statistics and
good probabilistic models [29-31]. These algorithms can be applied in real-time and seem to give
excellent accuracy for motions which exhibit lower accelerations than the leg segments, and which are
not exposed to impacts like those of heel contacts. For the lower extremities, the performance of the
Kalman filter is considerably reduced when measuring the orientation angles of segments that move
fast . Inertial sensors that consist of accelerometers, gyroscopes, and magnetometers, along with
Kalman filtering, allow a good accuracy for estimation of lower limb angles . However, good
accuracy of angle estimation can also be achieved using fewer sensors and much simpler algorithms
that are not sensitive to the presence of metals and ferromagnetic materials such as those that comprise
Willemsen et al.  developed a technique to estimate joint angles without integration. This
method is based on comparison of weighted accelerations of the joint (e.g., knee or ankle) obtained
from two accelerometer pairs mounted on two adjacent segments of the leg. The method requires
adequate low-pass filtering, which introduces a delay and to a certain extent hinders the real-time
applicability. Further, the accelerometer pairs need to be precisely oriented, so that their axes intersect
at the joint, which is very difficult to achieve considering that the human joints are polycentric. Also,
the distances between sensors and the joints are required for computation.
We have developed an accurate, yet simple method and instrumentation for estimation of absolute
segment and joint angles during the gait (assuming kinematics in the sagittal plane) which minimizes
the effects of drift. The proposed system is based only on accelerometer sensors, which is advantageous
because their calibration is static and less complex than the dynamic calibration required for
gyroscopes. Additional motivation for this paper was the “bad reputation” of accelerometers due to the
pronounced drift. We wanted to investigate if it is possible to use only accelerometers for angle
estimations and evaluate the precision of the results.
2. Experimental Section
2.1. Sensor System
The acquisition system that we developed for gait analysis is designed as a distributed wireless
sensor network. A set of battery powered sensor nodes is placed on the subject, one sensor node for
each leg segment of both legs. Sensor nodes establish communication with the coordinator node
through a low power 2.4 GHz wireless communication link. The coordinator node is connected using a
USB interface to the computer. Wireless communication is bidirectional, with a coordinator node
acting as a master, and the sensor nodes as slaves. The coordinator node manages network traffic and
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the USB connection with the computer. Data streams from the sensor nodes are synchronized and the
system operates with a 100 Hz sampling rate.
Sensor nodes are realized as a sandwich structure of processor and sensor board with a
Li-ion battery placed between the boards. The compact size design of sensor nodes, with dimensions
70 × 25 × 15 mm and 27 grams weight, enables comfortable wearing and does not hinder the subject’s
movements. Hardware design is based on the Texas Instrument’s CC2430 microcontroller, which
integrates a RF front end and a 8051 core in the same case. Standard microcontroller peripherals
enable interfacing to analog and digital sensors, and different sensor boards can be combined with the
same processor board.
In the configuration used in this research, the sensor board comprises two high performance 12-bit
digital accelerometers LIS3LV02 (SGS-Thomson Microelectronics, USA). The range of the sensors is
either ±2 g or ±6 g, which can be selected in the acquisition software. Accelerometers are aligned to
y axes with distance of 55 mm between centers. This configuration requires the clinician only to fix the
sensor array along the body segment, approximately at the mid section of lateral side of leg (Figure 1).
Figure 1. Setup of the sensor system. (a) photo of the sensors mounted on the body during
the gait analysis, (b) schematic of the system configuration with the coordinate systems.
Goniometers were attached to the leg segments by using double sided adhesive tape and secured
with elastic bands with Velcro endings, mounted over the sensors and around the leg segment. Sensor
nodes were placed in custom made tight sensor node-size elastic pockets placed on elastic bands with
Velcro at their ends.
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The custom-designed software, created in CVI (LabWindows, National Instruments, USA), is used
for online monitoring and storing of the acquired data.
The mechanics of importance for the analysis considers two sensors (denoted by S1 and S2), which
are mounted on a rigid rod (Figure 2). The distance between the sensors is l. The rod is freely moving
with respect to the fixed global coordinate system (O'x'y'z'), shown in Figures 1(b) and 2. The axis x' of
the global coordinate system is walking direction, and the axis y' is vertical. The center of the rod (O)
is determined by the position vector
To analyze the movement in the sagittal plane, we consider the case when the rod moves in the
O'x'y' plane (2D model). We define the vector l, which connects the centroids of the two sensors. The
positions of the accelerometers are
Figure 2. Rod with two accelerometers and the coordinate system for analysis of
movement in the sagittal plane.
Each accelerometer measures the two Cartesian components of the acceleration vector, with respect
to the local coordinate system Oxy attached to the rod. The equivalent accelerations measured by the
two sensors are:
where g is the gravity acceleration.
The difference of the signals from these two sensors is proportional to the amplitude of the vector
. In this way, we cancel out the influence of the movement of the rod centroid and of the
gravity, and retain information only about the changes of the vector l. The second derivative of the
vector l is:
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where φ is the angle between the axes x and x', ω and α are the absolute angular velocity and angular
acceleration of the rod, respectively, xi ,
The difference of the outputs from the accelerometers in the direction along the rod axis (
proportional to the square of the angular velocity, and the difference between the outputs from the
accelerometers in the perpendicular direction (
segment. The proportionality coefficient is equal to the distance between the centers of the
accelerometers. In this way we eliminated the gravity component from the signal, and eliminated the
need for precise positioning of the rod on the body segment and calibration of the system.
As explained in the introduction, one of the main problems with accelerometers is significant drift
after integration, whether the integration is performed numerically or by means of analog integrators.
A characteristic example of the drift, which resulted even with carefully calibrated accelerometers, is
presented in Figure 3.
yi , and zi are the unit vectors of the x, y, and z axes of the
zi is the unit vector of the axis of
) is proportional to the angular acceleration of the
Figure 3. Joint angle (bottom panel) and angular velocity (middle panel) obtained by
numerical integration of the measured acceleration of the segment (top panel). Dashed line
envelope on the bottom panel is fitted through the points where the knee should be fully
extended with zero degree joint angle. However, due to the integration drift, instead of
remaining approximately constant, this line has a parabolic shape.
We introduce a method for estimation of the joint angles based on digital filtering. In order to
explain the method, we use the frequency domain. According to the Laplace transform, the integration
in the time domain corresponds to multiplication by
integration corresponds to multiplication by
/ 1 s , where s is the complex frequency. On the frequency
axis (i.e., in the Fourier-transform domain, where
/ 1)j /(1
.Hence, we use a second-order low-pass filter, which mimics this multiplication.
Further in the paper, we shall write
/ 1 ω−
this multiplicative term is purely real (although negative).
s / 1
in the frequency domain, and the double
), this corresponds to multiplication by
/ 1 s , because we wish to emphasize the fact that
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Without the loss of generality, we can assume that the signal
has pronounced spectral components at
spectral components of
function is proportional to
/ 1 ω−
lt / )(ax
is nearly periodic. Hence, it
,... 2 , 1
, where T is the stride period. All relevant
should be in the roll-off region of the filter, where its transfer
. For example, the transfer function of the second-order Butterworth
2 f π=ω
is the cutoff angular frequency. On the imaginary axis, when
, we get:
In order to approximate the double integration, we pass the signal through the filter and divide the
The filter is, however, dispersive. Various spectral components have various delays and the filtered
signal will only barely resemble the actual function
bi-directional filtering, using the filtfilt function in Matlab. First, the signal is filtered in the forward
direction. Then, the filtered sequence is reversed and run back through the filter. This procedure results
in a real and positive transfer function (zero-phase distortion and zero group delay), whose order
corresponds to double the filter order. For example, if we use the first-order Butterworth filter, whose
transfer function is
) 1/ /(1)(
, the result of the bidirectional filtering is the transfer function
. Hence, to obtain the
first-order Butterworth filter with the filtfilt function and divide the result by
normalized spectrum of the angular acceleration, along with the
the transfer characteristic of the low-pas filter (Bode plot), low-pass filter combined with a high-pass
filter obtained by the filtfilt function.
. A zero-delay filter can be obtained by
/ 1 ω−
transfer function, we use the
. Figure 4 shows the
function, and the magnitude of
Figure 4. Normalized spectrum of the angular acceleration for knee angle (shown in
Figure 3) and magnitudes of the function
Butterworth low-pass filter (lpf), and function of this low-pass filter combined with a
high-pass filter (lpf + hpf).
, transfer function of a second-order
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The choice of
similar to the numerical integration. On the other hand, in order to keep the spectral components in the
roll-off region of the filter, the condition
spectral components are within the pass band of the filter, where the transfer function of the filter is
approximately constant and close to 1. Increasing further the filter bandwidth, i.e., increasing
components are not affected by the filter. However, their magnitudes are divided by
level of these components is reduced, and the result is distorted. On the contrary, the magnitudes of the
spectral components that are in the roll-off region of the filter are insensitive to the modifications of
Since filtering should replace the integration, all relevant spectral components of the gait should be in
the roll-off region of the filter, i.e., well above the cutoff frequency of the filter. By comparing the
filter amplitude characteristic, which is the modulus of (4), and
between these two functions is less than 1 dB for spectral components that are above two times the
cutoff frequency of the filter. Similarly, the error is less than 0.5 dB for spectral components above
three times the cutoff frequency. Hence, the cutoff frequency
lowest relevant spectral component of the signal is positioned between
0f followed the heuristics (Figure 5). If
0f is too low, joint angles exhibit drifting
T / 1
should be fulfilled. If this condition is not
is the gait cycle frequency), one or more
0f is taken to be higher than
0f , these
ω , so that the
, it can be verified that the error
0f of the filters is determined so that the
2 f and
Figure 5. Influence of
0f on angle estimation: filtering for several cutoff frequencies
compared with angle acquired from goniometers.
The drift can be additionally reduced if a high-pass filter is used in conjunction with the low-pass
filter. The cutoff angular frequency of the high-pass filter should be below
of the low-pass filter is not affected. The order of the high-pass filter can be selected as an additional
parameter to help keep the drift under control.
As an example, Figure 4 shows the magnitude of the transfer characteristic of the combined low-pas
filter and a high-pass filter of 8th order, which is an actual filter used in computations in this paper.
Since filtering distorts the DC level, we restore this information through the self-calibration in the
following way. Before gait initiation, the subject needs to remain standing still (and sensors immobile)
for at least two seconds. During this interval, the initial conditions are determined for each pair of
accelerometers by using them as inclinometers.
The procedure of approximating the double integration can be applied both for reconstruction of
absolute angles (the angles between the rod axes and the vertical axis of the fixed coordinate system)
ω , so that the major role
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and the reconstruction of the joint angles. For example, the knee angle is obtained directly from the
by performing the bi-directional filtering, and divide the result by
. An analogous procedure is
performed for the ankle angle.
The algorithm was tested on 27 healthy subjects walking on the ground at their natural pace.
In order to provide a more systematic validation, we additionally recorded 10 subjects (age: 26 ± 1.5
mean ± SD) walking at various speeds on treadmills (Life Fitness 9500HR and Panatta Advance Lux
1AD003), whose results are presented in this paper.
Four trials per subject were recorded. Besides walking, recording sequence also included standing
still for at least 2 s before and after each walking sequence, which was used for self-calibration and
checking. Subjects were walking with various velocities on a treadmill, starting from 0.15 m/s and
incremented by 0.05 m/s up to 2 m/s. As the reference system for this study, we used SG110 and
SG150 flexible goniometers with the joint angle units for signal conditioning (Biometrics, Gwent,
UK). Goniometers were mounted on the lateral side of the leg (at the ankle and knee joints) following
the instructions of the manufacturer. Simultaneously, the sessions were recorded with a video camera
for later analysis.
2.4. Processing of Measured Data
Based on the recorded accelerometer data, the joint angles were estimated by the proposed algorithm.
Joint angles recorded by goniometers were also computed. The accuracy of our algorithm was
evaluated in terms of the root-mean-square error (RMSE) as well as the Pearson’s correlation
coefficients (PCCs) between the goniometer results and angles provided by the proposed method.
RMSE is expressed in degrees. PCC values range between −1 and 1, where 1 represents the best
possible similarity between the two sets of angles (identical shapes). The first and the last stride were
excluded from each trial, and comparison between goniometer signals and angles provided by our
method was done on the remaining sequence. The data processing was done offline using Matlab 7.5
Two typical examples for the knee and ankle angles are shown in Figure 6. The error was defined as
the difference between the angles obtained by the proposed method and the angles obtained from
Based on the results obtained from treadmill recordings, the cutoff frequency for the knee angles
should be in the range [
f /2], where
f is the gait cycle frequency. The blue area in Figure 6
shows the family of curves estimated by our method when filtered with various frequencies in the
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Figure 6. Knee and ankle joint angles measured with goniometers and estimated from
accelerometers. Thick lines represent angles from goniometers (dashed line) and
accelerometers (solid line) for optimal filter frequency. Blue areas show the range of angle
values estimated from accelerometers when filtered with various frequencies in the range
Figure 7 shows the optimal filtering frequency versus gait cycle frequency. The squares represent
optimal points obtained by maximizing Pearson’s correlation coefficient and minimizing RMSE
between our results and the angles obtained by goniometers for each walking trial (different gait
velocity). The straight lines are obtained by fitting these data. It is obvious from Figure 7 that signals
for the estimation of the ankle angles should be filtered with about two times higher cutoff frequency
than the signals used for the estimation of the knee angles. These findings are in agreement with the
theoretical background from the previous section.
Figure 7. Optimal filtering frequencies for knee and ankle angles versus gait cycle frequency.
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The higher cutoff frequency of the filter for the ankle angle can be used because the spectrum of the
ankle angle has a very pronounced second harmonic. This is convenient, because the influence of the
drift is further suppressed.
Using the proposed algorithm, for each subject and each trial, we evaluated the PCC and RMSE
values between the angles estimated by our method and goniometer outputs. Figure 8 shows the PCC
and RMSE, respectively, as a function of normalized frequency, for various velocities. Although we
recorded velocities from 0.15 to 0.2 m/s in the steps of 0.05 m/s, Figure 8 shows results for a subset of
the recorded velocities.
Figure 8. Pearson’s correlation coefficient (PCC) and RMSE between goniometers
and estimated angles, for knee and ankle angles, vs. filtering frequency in the range
corresponds to different gait velocity (in m/s).
f /2]. The normalization is with respect to the optimal frequency. Each curve
As shown in Figure 8, all RMSE curves have broad minima, which are, on average about 1. These
results are in accordance with PCC curves, confirming that the optimum normalized cutoff frequency
is 1. Cumulative results for all walking trials and all subjects are presented in Figure 9.
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Figure 9. Boxplots presenting comparison between joint angles (for knee and ankle
angles) calculated from accelerometers and goniometers. Left: Pearson’s correlation
coefficient (PCC). Right: root-mean-square error (RMSE).
4. Discussion and Conclusions
The presented results are based on a model which assumes that the lower limbs move in the sagittal
plane. For healthy subjects, this 2D model has proven to be sufficient, because the sagittal plane is the
plane where the majority of movement takes place. Generally, for clinical applications, the proposed
method provides an acceptable accuracy for angles and high correlation coefficients with the
measurements obtained from goniometers.
In particular, PCCs for the knee angle are higher than 0.97 and RMSE is within 6º for the angle
values. Further, our results showed that a 5° RMSE is obtained for walking speeds in the frequency
∈ [0.35, 1.15]. This corresponds to all velocity curves in Figure 8 except for v = 0.15 , 1.8 ,
and 2.0 m/s (the slowest and fastest recorded walking). Regarding the ankle angles, PCCs are slightly
lower and in the range from 0.85 to 0.97, while RMSE is from 2° to 4.7°.
For both estimated joint angles (except for the extreme velocities for the knee angle), this error is in
the range of 5° mean error limit accepted by the American Medical Association to consider the
measurements reliable for the evaluation of movement impairments in a clinical context . The
accuracy of our simple system is comparable to the accuracy demonstrated in plots presented in ,
which were obtained by much more complex hardware and software.
Joint angles were determined by subtracting the absolute angles of the neighboring leg segments.
The error of our method was estimated based on joint angles and includes errors from the two segments.
In this way, the total error of the joint angle estimation is different than the error of the absolute angles.
Hence, comparison of the absolute angles with a camera system would be more appropriate for
validation of the proposed method. However, such a comparison was not possible in our experiments
because the treadmill would present a visual obstacle between cameras and markers. Since our main
goal was to investigate how our method performs for various gait velocities, the treadmill was the
important part of the experiments. Therefore, we selected goniometers as the reference system, which
have ±2° accuracy and 1° repeatability . Although electrogoniometers are prone to errors due to
potential misalignment with the femur and tibia segment in the sagittal plane, this does not affect the
validation of our method since we secured sensor units to be aligned with goniometer blocks.
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Skin motion artifacts cause errors to all body-fixed sensors. Sensors placed on thighs are more
susceptible to skin and soft tissue related motion, because the majority of femur is concealed by a
substantial amount of soft tissue. However, the errors that we report here are much smaller than errors
due to rod misalignment in the procedure proposed by Willemsen .
Another limit for this algorithm is the speed of the subject’s gait. As it can be seen from Figures 7
and 8, if the gait is very slow, the quality of our method decreases. This is due to very low angular
accelerations, whose major component comes from the impacts. This suggests that for a very slow
walk, e.g., for subjects with high levels of disability, the quality of the angle estimation may not be
acceptable. However, for very slow gait, accelerometers can be used as inclinometers and angles can
be successfully estimated in this way .
Our method does not need information about the distances to joint centers or distances between
sensor rods placed on different segments, which is one of the benefits of this algorithm. The only
condition for mounting the sensors is that they follow the segment line (to be aligned with a line
connecting adjacent joints, viz. hip and knee, knee and ankle, and along the foot and parallel to the
This algorithm could be used not only for level walking, but also for estimation of angles during
slope walking, stair climbing, or any other rhythmical (periodic) leg movements. It can also be used for
estimation of other segment and joint angles, as long as the movements are in 2D. However,
movements should be fast enough so that the angular acceleration signal is sufficiently above the noise
floor. The proposed method is suitable for postprocessing of raw data. However, it can also be
included into real-time algorithms to estimate the angles of the leg segments with a delay of one stride.
The proposed method is simple and computationally efficient. We have demonstrated that it yields
accurate shapes of the ankle and knee angles. The accuracy of the method is sufficient for quick
diagnostics of gait, as well as for applications of gait control.
The work on this project was partly supported by the Serbian Ministry of Education and Science,
Grant No. 175016.
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