Automatic filtering techniques for three-dimensional kinematics data using 3D motion capture system

Aissaoui R., Husse S., Mecheri H., Parent G., de Guise J.A.

Laboratoire de recherche en imagerie et orthopdie, cole de technologie suprieure e-mail : rachid.aissaoui@etsmtl.ca

Abstract-The purpose of this paper is to investigate the performance of three algorithms for automatic filtering of 3D displacement data. The first approach is based on power spectrum analysis of signal with autoregressive modeling approach to detect a signal bandwidth in the frequency domain. The second method uses the autocorrelation of the residual signal between filtered and unfiltered data to separate the signal bandwidth from noise. The third approach uses a singular spectrum analysis to detect the variance of the signal and reject the noise based on the eigenvalue decomposition of the signal. Overall, the highest RMS value of 0.480 m/s2 was measured in the X direction for PSA method, whereas the lowest RMS value of 0.162 m/s2 was recorded for cluster 2 for SSA method. This represents a gain of 3 in accuracy in estimating higher-order derivatives such as linear acceleration of rigid body motion. SSA method is robust and seems to behave well for different signal combinations.

I. INTRODUCTION

Three-dimensional (3D) motion capture systems are becoming widely used in the measurement of 3D human movement. They are generally used in the virtual-reality and biomechanics areas and can be divided into two categories. The first includes systems that measure the 3D location of a marker in space: they are called 3DOF-system. These systems are based on optoelectronic cameras that sense passive or active markers, but necessitate a calibration procedure [1]. The second category includes systems that measure simultaneously the location and orientation of a rigid-body in 3D space. They are called 6DOF-systems. They are generally based on electromagnetic AC or DC fields [2]. In general, 3DOF-systems can achieve a higher accuracy of about 1 mm when compared to 6DOF-systems, which are tenfold lower in accuracy and also prone to field distortion without a calibration procedure [3].

The estimation of derivatives of measured time series is an important issue in many biomechanical applications. In fact, the estimation of the forces and moments acting at human joint using inverse dynamic approach needs the computation of linear and angular acceleration of human-body segments [4]. Often this computation is obtained by twice differentiation of displacement data using a numerical approach. Since the differentiation process acts as a high-pass filter, this procedure will largely amplify the high frequency noise. It has been theoretically verified that the following inequality applies to all unbiased linear estimates of the kth order derivative [5].

2 2 1

2

(2 1)

k

f s

k

T

k

+

+ (1)

Where k is the standard deviation of the noise in the estimated kth derivative; f is the standard deviation of the noise in the measured displacement data (which is assumed to be white and additive to the measurement signal); T is the sampling period; s is the bandwidth of the signal (which is supposed to be band-limited, i.e. no information exists for frequencies above s in rad/sec); and k is the derivative order (k=1 for velocity, and k=2 for acceleration data). If the signal bandwidth is known, it will be easy to remove high frequency noise by low-pass filtering of displacement data. Unfortunately, this information about the power of the signal and noise is not known, since the frequency content of the displacement data varies for different human body segments [6], and different human activities [7]. In biomechanics, recursive digital filter such as the Butterworth low-pass filter has been extensively used by researchers. However, one disadvantage of this technique is that the user has to decide on the cut-off frequency of the filter, after repeatedly comparing the computed filtered second derivative with the finite-difference method of the unfiltered data. An attempt to compare the acceleration measured by an accelerometer and that obtained by twice differentiation of a filtered data has revealed the difficulty in choosing the cut-off frequency [8]. In [9], an automatic method based on regularized Fourier series was proposed to estimate higher-order derivatives of signal. Unfortunately, the accuracy was not reported in [9]. Moreover, it is known that the estimation of the power spectral density (PSD) of time-series signals using nonparametric approach such as periodogram suffers from three important aspects: leakage due to time truncation, resolution due to time-series length and sampling frequency, and finally bias in the estimation of PSD [10]. Parametric methods can yield higher resolutions than nonparametric methods in cases when the signal length is short. These methods use a different approach to spectral estimation; instead of trying to estimate the PSD directly from the data, they model the data as the output of a linear system driven by white noise, and then attempt to estimate the parameters of that linear system. The most commonly used linear system model is the all-pole model, a filter with all of its zeroes at the origin in the z-plane. The output of such a filter for white noise input is an autoregressive (AR) process. For

this reason, these methods are sometimes referred to as AR methods of spectral estimation.

Non-conventional approaches have also been proposed for data smoothing using wavelet-transform [11], time-frequency distribution [12], and singular spectrum analysis [16]. In spite of their superiority in detecting high signal impact acceleration, wavelet and time-frequency can generate spurious peaks in low impact signal. Moreover, non-conventional techniques [11-12] are difficult to automate, since they necessitate the adjustment of many parameters, whereas in [16] there are relatively few parameters to be fixed. Three approaches have been proposed to compute the cut-off frequency of the signal in automatic manner. The first technique, developed in [13], used an autoregressive modeling approach to estimate the power spectral density and to detect the bandwidth frequency of displacement data. This method will be called PSA for power spectrum analysis. The second technique uses the autocorrelation of the residual signal between the filtered and unfiltered data to detect also the signal bandwidth [14-15]. This method will be called AC, for computation of the autocorrelation function. The third approach is recent, and uses a non-parametric time series analysis called singular spectrum analysis (SSA) [16]. The original signal is transformed onto a Hankel matrix. This image matrix is transformed using a singular value decomposition method. A few singular values are then retained, and the filtered signal is back reconstructed based on a new Hankel matrix. Most of the previous studies comparing filtering and smoothing techniques were based on single marker comparison i.e. they do not take into account the rigidity constraint of rigid-body motion. The purpose of this paper is twofold: (i) to investigate the influence of three algorithms for automatic filtering of 3D displacement data on the acceleration variable as measured by accelerometer; (ii) to use the rigidity constraint imposed by solid motion to compute the linear acceleration vector using gyroscopic measurements.

II. FILTERING THEORY AND NUMERICAL DIFFERENTIATION

The following sections describe the automatic filtering algorithms, but only in their major steps respectively. Let S be a signal represented by its time serie {si}i=1..n sampled signal measured at equidistant time with a sampling frequency fs.

A. Power spectral analysis method (PSA) [13] The first step in PSA is to model the signal by an all-pole

model i.e. an autoregressive process.

1

( ) ( ) ( ) ( ) (2)p

k

S n A k S n k E n=

= + Where A(k) represents the model parameters, A(0)=1, and E(n) is an input white noise. This is obtained by the modified-covariance algorithm [17]. The second step corresponds to the estimation of power spectral density of the signal using the Fast Fourier Transform (FFT):

2( ( ))PSD E FFT A n= (3) The third step is to evaluate the signal noise bandwidth by computing the average of the PSD signal from 80% to 100% of the Nyquist frequency i.e. fs/2. The corresponding average value is pre-multiplied by a predefined signal to noise ratio (SNR) k. The last step is to detect the frequency, which is considered a cut-off frequency (fc) that corresponds to the noise spectral power density. Afterwards, the signal S is filtered using a second order zero-lag Butterworth filter at frequency fc.

B. Autocorrelation method (AC) [14] Let S be a raw signal, and let Sf be the same signal filtered

with a second order zero-lag Butterworth filter with a cut-off frequency f. Let the variable res correspond to the residual of the signal i.e.: fres S S= (4) The autocorrelation of the residual signal is then estimated for a predefined lag L. The frequency f is varied from Nyquist frequency to the direction of the zero one. It is thought that when the autocorrelation function is minimal, then the corresponding frequency is optimally dividing the signal information from its noise content [14]. The frequency that allows the autocorrelation function to be minimal is considered as a cut-off frequency. The signal is filtered with a Butterworth filter at that cut-off frequency.

C. Singular spectrum analysis method (SSA) [16] Let S be a signal of length N, and let L represent a window

length. The first step is to construct a Hankel matrix from the original signal by sliding a window with length L. A Hankel matrix is a matrix which has equal elements along the diagonals that are perpendicular to the main diagonal i.e: aij = ai+1,j-1; for all i 4 has been imposed to get an adequate rank for signal reconstruction. This procedure is similar to a smoothing process, and it is done in a recursive manner. The procedure stops when the difference between the root mean square (RMS) of two successive iterations is under a certain predefined threshold.

D. Numerical differentiation and rigidity constraint Once the signal S has been filtered, a simple way to compute

the linear acceleration is to numerically differentiate the signal. Let the time serie {si}i=1..n considered to be filtered then the corresponding acceleration time serie {ai} i=1..n is computed by the central difference formula as:

2 22

2

4i i i

i

s s sa

t+ += (6)

Where t denotes the sampling time. Another way to compute the linear acceleration is to express the theoretical relation between position and acceleration for exact data in rigid-body motion, which is given by the following [18]: 2( ) . TP W W P c h= + + (7) Where P is a 3xN array containing the acceleration components of N markers in 3D space; P is a 3xN array containing the x,y, and z position components of N markers; W is the 3x3 skew-symmetric angular velocity matrix characterizing an infinitesimal displacement; W is the 3x3 skew-symmetric angular acceleration matrix characterizing an infinitesimal displacement; c is the vector that includes the rigid body linear acceleration for infinitesimal displacements; h is nx1 vector one ones i.e. a unit tensor. Equation (7) holds only for exact data. In general, the angular velocity vector

[ ], ,x y z T = is estimated from the position matrix P . Let the matrices 'P and 'V represent the relative position and the relative linear velocity of N markers with respect to their centroid. The invariance concept states that the moment of inertia (MI) of the N markers remains invariant with respect to any rotation [19]. This gives the following relation: 12 ( ' ' )TMI vect V P = (8) ( ' ' ) ' 'T TMI trace P P I P P= (9) Where I is a 3x3 identity matrix, whereas trace and vect denote algebraic operators for diagonal and skew-symmetric matrices respectively. Equation (8) has been validated experimentally with a triaxial gyroscope and motion analysis in an earlier study [20].

III. EXPERIMENTAL SET-UP

Figure 1 shows the experimental set-up used in this study. Two rigid clusters made from plexiglass were fixed on a rigid tube simulating a knee rotational movement. Four reflective markers (8mm diameter) were placed in tetrahedron shape in each cluster. In addition, a pair of triaxial gyroscopes with a range of 400 /s, and a pair of triaxial accelerometers with a range of 5g were rigidly fixed on each cluster. Three-dimensional trajectories of eight markers were tracked using a motion analysis system (VICON M460, Vicon, Inc., USA). The VICON system consists of 6 CCD high resolution cameras (1000x1000) which collect the movement of the reflective markers (Fig.1) in real-time at a sampling frequency of 120 Hz. The VICON system was calibrated using the new wand calibration procedure in a volume of 2x2x2 m3. The accelerometer and the gyroscope signals were collected with the PHYSILOG system (BioAGM, Inc, Switzerland) at a sampling frequency of 120 Hz. The VICON and the PHYSILOG systems were synchronized using external trigger. The movement was induced by rotating manually the moving body around the x-axis (Fig. 1). Three rapid trials were executed for a total duration of 12 seconds.

Figure 1: knee simulator prototype (Right-hand side). Four markers are placed onto the moving rigid body in cluster 1 and four others on cluster 2. Circles represent reflective markers. (Top left) cluster with four markers and one pair of tri-axial gyroscope and tri-axial accelerometer (Bottom left). Data from the gyroscope and accelerometer were pre-processed in order to remove the drift and bias from a static position of the knee simulator. Since the accelerometer measures the gravitational vector in static position, the former has been removed from the signal in order to keep only the acceleration related to the kinematics of the rigid body. Two methods described in section II-D have been implemented: Method-1 estimates the second derivatives of the filtered signal after numerical differentiation techniques (6), whereas Method-2 estimates the second derivative of the signal using (7). In the latter, the angular velocity vector is also filtered by the same corresponding technique as the displacement raw data. For each method, three filtering techniques are applied (PSA, AC, SSA). For PSA algorithm, the number of poles from which the autoregressive process is estimated has been fixed to 20 as suggested in [13]. The signal to noise ratio (SNR) was fixed to 50 and the average power spectrum of the noise signal was estimated in the bandwidth that corresponds to the interval 80%-100% Nyquist frequency. For the AC method, the autocorrelation function of the residual has been computed for each frequency, from the Nyquist one (i.e. 60 Hz) to the closest one to zero with a step of 0.06 Hz. The minimal value of the autocorrelation function indicates the value of cut-off frequency. The SSA algorithm was performed with the following parameterization. First, the ratio between the size of time-serie and the window length was fixed to 60, i.e. a window length of 20 for a time-serie length of about 1180 samples. This ratio has been chosen since it corresponds to the one tested originally in [16]. To compare the acceleration of reflective markers to that measured by the accelerometer, two additional steps must be done. First, the gravitational vector must be removed from the measured signal of the accelerometer: this is done for each image frame using a local

coordinate fixed on the accelerometers. The local coordinate is built with the 3D position of four reflective markers using quaternion algebra. The second step is to transfer the local accelerometric data measured to the marker one, using a formula similar to that of (7). Since the accelerometer sensor and the markers cluster are fixed the transformation vector is fixed (Fig. 2), however as indicated in (7) the angular velocity vector and its derivatives are still needed. The latter are obtained by (8) and (9). For each method, algorithm and trial two parameters will be calculated. The first one is the RMS between the filtered data and the referenced accelerometric data for each marker that will be estimated along the time-serie curve; the second parameter is the RMS evaluated only at the peaks of the signal for each marker separately.

Figure 2: Mechanical alignment of the accelerometer with respect to reflective markers in the cluster. Side and top view of the rigid cluster.

IV. RESULTS

Figure 3 shows a typical pattern of linear acceleration measured by the accelerometer and estimated with a raw data from twice differentiation technique. Data in figure 3 show a cyclical movement induced manually by a single operator. The range of the acceleration in the X, Y, Z axis are respectively 1, 10 and 5 m/s2. Since the accelerometers were not oriented similarly to the global coordinate system, they actually measure the projection of the movement onto the X-axis. This is of course will not be interpreted as a movement in the X-axis direction since the cylinder undergoes a pure rotational movement. It should be noted that the acceleration estimated without filtering the initial position of markers lead to highly noised signal, especially in the direction of X and Z axis during the deceleration phases (Fig. 3.). Figure 4 shows the effect of filtering the initial displacement signal with the automatic SSA filter. In Fig. 4, the acceleration computed and measured fits very well on all parts of the signal. Table 1 shows the RMS value along all the entire data-curve of the differentiation using method-1. The trials represent different repetitions of the rotation movement in the knee simulator. The clusters represent the position of markers and accelerometers, with cluster 1 closer to the axis of rotation. Overall, the highest RMS value of 0.480 m/s2 was measured for cluster 2 in the X direction for PSA method,

0 200 400 600 800 1000 1200-10

-5

0

5

10

Y-axis

0 200 400 600 800 1000 1200-10

0

10

20

Z-axis

0 200 400 600 800 1000 1200-10

-5

0

5

10

X-axis

Figure 3: Typical acceleration patterns for rotational movement of knee simulator. The movement was performed around the x-axis (see Fig.1). Linear acceleration obtained by double differentiation of raw data for marker 1 in cluster 1 (dashed-line). Linear acceleration measured by the accelerometer at the location of marker 1 (solid-line). Time in sampling: 1200 samples correspond to 10 seconds.

0 200 400 600 800 1000 1200-10

-5

0

5

10X-

axis

0 200 400 600 800 1000 1200-10

-5

0

5

10

Y-ax

is

0 200 400 600 800 1000 1200-10

0

10

20

Z-ax

is

Figure 4: Linear acceleration obtained using SSA filtering method on displacement signal (same trial as in Fig. 3). Linear acceleration obtained by double differentiation of raw data for marker 1 in cluster 1 (thick-line). Linear acceleration measured by the accelerometer at the location of marker 1 (thin-line). Time in sampling: 1200 samples correspond to 10 seconds. whereas the lowest RMS value of 0.162 m/s2 was recorded for cluster 2 for SSA method. In general, RMS values for each of the three corresponding methods are lower for cluster 1 than cluster 2, except for SSA method in which RMS value are almost similar for the clusters. Also, the RMS values for all three methods are in the same order of magnitude, the standard deviations (Std) obtained by SSA method are one order lower than the PSA and AC methods. In fact, the lowest Std of 0.006 m/s2 was obtained by SSA, whereas the highest Std value of 0.355 m/s2 was obtained by PSA method. This variation represents an increase factor of 5900%, and shows the consistency of SSA method in estimating high-order derivatives. Table 2 shows the RMS value between measured and computed linear acceleration of two clusters with 4 markers attached on each one.

Table 1: Method-1. Average and standard deviations (Std) of the RMS error value along the duration of signal of four markers in cluster 1 and 2 for 3 trials. Units are in m/s2. Each trial represents a separate movement.

PSA SSA

X Y Z X Y Z X Y Z

trial 1, Cluster 1 mean 0,275 0,351 0,325 0,282 0,286 0,217 0,304 0,263 0,298Std 0,040 0,158 0,177 0,055 0,137 0,048 0,026 0,016 0,028

trial 1, Cluster 2 mean 0,326 0,356 0,317 0,284 0,220 0,246 0,263 0,263 0,226Std 0,244 0,149 0,230 0,132 0,023 0,034 0,017 0,009 0,019

trial 2, Cluster 1 mean 0,256 0,313 0,193 0,339 0,233 0,172 0,295 0,202 0,207Std 0,072 0,128 0,030 0,284 0,132 0,030 0,024 0,007 0,022

trial 2, Cluster 2 mean 0,480 0,354 0,402 0,381 0,201 0,174 0,203 0,222 0,162Std 0,355 0,159 0,252 0,312 0,016 0,035 0,017 0,006 0,013

trial 3, Cluster 1 mean 0,356 0,346 0,275 0,217 0,209 0,240 0,280 0,266 0,339Std 0,259 0,142 0,047 0,043 0,020 0,060 0,051 0,015 0,036

trial 3, Cluster 2 mean 0,447 0,399 0,436 0,203 0,234 0,208 0,223 0,279 0,236Std 0,304 0,184 0,248 0,045 0,014 0,026 0,010 0,013 0,025

AC

Overall, the RMS value varies from 0.176 m/s2 (SSA, cluster-2, Z axis) to 4.187 m/s2 (AC, cluster-1, X axis). It is interesting to note that the standard deviations found in Table 2 are lower for all of the PSA and AC methods, but similar to the SSA one. The low standard deviations indicate high repeatability of the experiment; however this repeatability comes with a significant bias for PSA and AC methods. It is also interesting to note that in Table 1, most of the highest RMS values are due to X-axis direction. This not the case in Table 2 since the Y-axis generates also a high RMS value, even if the major part of the displacement is recorded throughout this axis. Table 2: Method-2. Average and standard deviations (Std) of the RMS error value along the duration of signal of four markers in cluster 1 and 2 for 3 trials. Units are in m/s2. Each trial represents a separate movement.

PSA SSA

X Y Z X Y Z X Y Z

trial 1, Cluster 1 mean 1,322 0,710 0,483 1,873 1,698 0,945 0,599 0,397 0,308Std 0,093 0,046 0,046 0,141 0,120 0,091 0,041 0,019 0,027

trial 1, Cluster 2 mean 0,776 0,553 0,370 1,530 0,770 0,433 0,274 0,354 0,232Std 0,098 0,029 0,042 0,211 0,054 0,053 0,017 0,006 0,020

trial 2, Cluster 1 mean 1,100 0,907 0,606 4,187 1,205 0,648 1,009 0,350 0,234Std 0,076 0,066 0,058 0,327 0,091 0,091 0,076 0,016 0,021

trial 2, Cluster 2 mean 0,979 1,284 0,686 1,158 0,855 0,433 0,216 0,291 0,176Std 0,115 0,133 0,084 0,135 0,105 0,056 0,018 0,004 0,016

trial 3, Cluster 1 mean 1,374 1,037 0,750 1,188 0,724 0,479 0,663 0,422 0,356Std 0,097 0,075 0,072 0,085 0,045 0,043 0,047 0,021 0,035

trial 3, Cluster 2 mean 0,844 1,399 0,710 0,699 0,584 0,294 0,225 0,369 0,239Std 0,093 0,141 0,087 0,094 0,029 0,027 0,010 0,006 0,025

AC

Table 3 and 4 represent RMS values obtained on the first six consecutive peak values (either positive or negative, Fig. 2). In Table 3 the RMS-peak value varies from a low of 0.103 m/s2 (SSA, cluster-2, X-axis) to a high of 0.718 m/s2 (SSA, cluster-1, Z-axis). Standard deviations are lower by one order of magnitude for the SSA method in comparison with the other two. In Table 3, cluster 2 always presents the lowest RMS-peak value with respect to cluster-1. Table 4 shows the same peak value for data as in Table 3, but obtained with the use of rigidity constraints.

Table 3: Method-1. Average and standard deviations (Std) of the RMS error value for peak value of the signal of four markers in cluster 1 and 2 for 3 trials. Units are in m/s2. Each trial represents a separate movement.

PSA AC SSA

X Y Z X Y Z X Y Z

trial 1, Cluster 1 mean 0,289 0,424 0,593 0,374 0,420 0,261 0,284 0,520 0,814Std 0,059 0,199 0,218 0,188 0,185 0,192 0,033 0,076 0,079

trial 1, Cluster 2 mean 0,316 0,364 0,380 0,221 0,251 0,221 0,103 0,359 0,434Std 0,335 0,070 0,170 0,124 0,085 0,049 0,020 0,026 0,064

trial 2, Cluster 1 mean 0,284 0,312 0,238 0,406 0,339 0,202 0,411 0,459 0,371Std 0,124 0,096 0,094 0,381 0,164 0,044 0,033 0,024 0,033

trial 2, Cluster 2 mean 0,673 0,371 0,533 0,563 0,337 0,321 0,349 0,460 0,318Std 0,403 0,056 0,204 0,403 0,043 0,103 0,072 0,022 0,038

trial 3, Cluster 1 mean 0,346 0,490 0,337 0,220 0,345 0,240 0,289 0,595 0,718Std 0,267 0,297 0,200 0,042 0,073 0,127 0,063 0,070 0,087

trial 3, Cluster 2 mean 0,441 0,458 0,505 0,238 0,336 0,175 0,251 0,432 0,374Std 0,378 0,149 0,313 0,085 0,043 0,066 0,041 0,037 0,071

Table 4: Method-2. Average and standard deviations (Std) of the RMS error value for peak value of the signal of four markers in cluster 1 and 2 for 3 trials. Units are in m/s2. Each trial represents a separate movement.

PSA AC SSA

X Y Z X Y Z X Y Z

trial 1, Cluster 1 mean 0,916 1,341 0,980 2,583 3,083 2,464 0,466 0,776 0,755Std 0,111 0,130 0,130 0,213 0,271 0,259 0,016 0,079 0,076

trial 1, Cluster 2 mean 0,557 0,294 0,664 1,262 0,318 0,374 0,123 0,469 0,325Std 0,096 0,059 0,076 0,177 0,062 0,055 0,023 0,046 0,053

trial 2, Cluster 1 mean 1,272 1,959 1,220 5,674 1,931 0,802 0,923 0,629 0,423Std 0,130 0,124 0,112 0,439 0,125 0,109 0,065 0,024 0,026

trial 2, Cluster 2 mean 1,610 1,476 1,012 1,165 1,235 0,839 0,327 0,621 0,363Std 0,215 0,138 0,096 0,135 0,153 0,137 0,071 0,023 0,029

trial 3, Cluster 1 mean 1,492 1,467 1,007 1,195 0,846 0,520 0,600 0,836 0,679Std 0,148 0,130 0,129 0,104 0,073 0,066 0,041 0,085 0,077

trial 3, Cluster 2 mean 0,739 0,880 0,594 0,493 0,567 0,269 0,283 0,563 0,305Std 0,100 0,089 0,074 0,097 0,045 0,032 0,039 0,050 0,057

In Table 4, RMS-peak value reached its lowest value at 0.123 m/s2 (SSA, cluster-2, X-axis), and its highest value at 5.674 m/s2 (AC, cluster-1, X-axis). Standard deviations are also low by one order of magnitude for SSA technique in comparison with PSA and AC methods. In general RMS-peak values obtained from cluster-2 are lower than those from cluster-1.

V. DISCUSSION

The purpose of this study was to compare three methods for automatic filtering of three-dimensional trajectories as obtained by motion-capture systems. It is important here to note that, even if motion-capture systems are now very accurate with approximately ~1 mm accuracy in real world coordinate system and in a volume of 3 m3, filtering is still needed when estimating high-order derivatives of displacement signals as shown in Fig. 3. In general, our data reveals that the SSA method behave very well in different trials with different

clusters, whereas other methods such as PSA and AC sometimes degenerates due to the presence of noise. The RMS-curve values found in this study are lower than that estimated in [22], which ranged from 0.4 to 1.8 m/s2. It should be noted that in [22], the authors reported on the relative rankings among algorithms and not on actual acceleration. Moreover, in this earlier study, data was collected by fixing markers on human, which did not guarantee the rigidity constraint. Finally, the authors [22] ranked the PSA method as one of the best methods; however neither SSA nor AC methods were tested. Walker [21] also found that PSA performed relatively well in simulating swimming fish. In [14], the AC method was not compared to other methods; however, it was found in [15] that a similar AC method behaves well when compared to the generalized cross-validated quintic spline function. In [15], data was synthetic i.e. built with a polynomial functions in which white Gaussian noise was added. Finally, Alonso et al. [16] proposed their own algorithm for automatic filtering of noisy data using SSA method. In our knowledge, this is the first time that SSA method is compared to other methods using measured and computed accelerations in a three-dimension motion analysis setting. Our data reveal that SSA algorithm behaves well in all situations and clusters positions. Our data also reveal that linear acceleration estimate is accurate when the cluster is close to rotation axe. However, this information is counter-indicated by the use of the rigidity constraint for method-2. In fact, it is somehow surprising that the use of rigidity constraint did not enhance the accuracy of linear acceleration estimates. The rigid-body constraint uses a screw-theory to estimate the linear acceleration of 3D coordinate knowing the instantaneous linear displacement and the angular velocity vector. In our data, the angular velocity vector estimated by Eq. (8) was very close to the measurement of the gyroscope [20]. However, it is also important to note that Eq. (7) incorporates the angular velocity vector and its derivatives (i.e. the angular acceleration vector) that could have propagated nonlinear noisy signal. In this study, we filtered the angular velocity vector with the corresponding technique used for displacement data. Further study on specific filtering of rotational data should be done to maintain the rigidity constraint of rigid body motion. The three algorithms implemented here have internal parameterization already fixed by their author such as lag in AC method [14], or order of AR process in PSA [13], and window length in SSA [16]. One important feature in this study is the assumption made about noise. Most of algorithm treated the noise as white with and expectation value of sigma. SSA algorithm as proposed here made implicitly this assumption, but if not more complex implementation are necessary to distinguish between linear ergodic signal and nonlinear noise signal [23].

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