**An Advanced Hybrid Technique of DCS and JSRC for Telemonitoring of Multi-Sensor Gait Pattern**

#### **Jianning Wu \*, Jiajing Wang, Yun Ling and Haidong Xu**

College of Mathematics and Informatics, Fujian Normal University, Fuzhou 350117, China; jiajingwang\_1993@126.com (J.W.); yunling93@163.com (Y.L.); sihaijunmi@fjnu.edu.cn (H.X.)

**\*** Correspondence: jianningwu@fjnu.edu.cn

Received: 25 October 2017; Accepted: 28 November 2017; Published: 29 November 2017

**Abstract:** The jointly quantitative analysis of multi-sensor gait data for the best gait-classification performance has been a challenging endeavor in wireless body area networks (WBANs)-based gait telemonitoring applications. In this study, based on the joint sparsity of data, we proposed an advanced hybrid technique of distributed compressed sensing (DCS) and joint sparse representation classification (JSRC) for multi-sensor gait classification. Firstly, the DCS technique is utilized to simultaneously compress multi-sensor gait data for capturing spatio-temporal correlation information about gait while the energy efficiency of the sensors is available. Then, the jointly compressed gait data are directly used to develop a novel neighboring sample-based JSRC model by defining the sparse representation coefficients-inducing criterion (SRCC), in order to yield the best classification performance as well as a lower computational time cost. The multi-sensor gait data were selected from an open wearable action recognition database (WARD) to validate the feasibility of our proposed method. The results showed that when the comparison ratio and the number of neighboring samples are selected as 70% and 40%, respectively, the best accuracy (95%) can be reached while the lowest computational time spends only 60 ms. Moreover, the best accuracy and the computational time can increase by 5% and decrease by 40 ms, respectively, when compared with the traditional JSRC techniques. Our proposed hybrid technique can take advantage of the joint sparsity of data for jointly processing multi-sensor gait data, which greatly contributes to the best gait-classification performance. This has great potential for energy-efficient telemonitoring of multi-sensor gait.

**Keywords:** multi-sensor gait classification; distributed compressed sensing; joint sparse representation classification; telemonitoring of gait

#### **1. Introduction**

Recently, wireless body area networks (WBANs), as an emerging wireless sensor networks technology that is applied to the human body [1–3], has been more and more popular in gait telemonitoring applications [4–6]. In such WBANs, each sensor node usually takes the form of a small wearable device equipped with a microcontroller, accelerometer, gyroscope, wireless communication module, and battery. These wearable sensors are co-located on the body for acquiring gait data of people who freely walk in a home or outdoor setting. Moreover, all acquired data can be transmitted, via internet, to a remote terminal such as a hospital for gait monitoring or assessment by further data processing such as gait classification. This greatly contributes to the telemonitoring of gait pattern change, and people do not often visit a hospital for gait function assessment [4–7]. However, in clinical applications, continuous acquisition of the multi-sensor gait data over the long term is usually required for further data processing, in order to accurately evaluate the gait pattern change [8–10]. In such a case, there have been several challenging existing issues such as improving the energy efficiency of the sensor, getting the best classification performance, and getting a lower computation complexity. The above existing issues are strangling the WBANs-based gait telemonitoring application, and some advanced techniques for multi-sensor gait data analysis are urgently needed to tackle these existing issues [9–12].

As we know, due to the limited life of the battery, the wearable sensors are usually required to consume the least possible amount of energy for the continuous telemonitoring of gait [10–13]. How to improve the energy efficiency of the sensors has been one challenging endeavor in the WBANs-based telemonitoring applications [10–12]. Many previous studies found that most of the energy of the sensors is consumed by wireless communication, and their works mainly focused on developing the energy-aware MAC protocols for the low energy consumption of sensors. However, these protocols don't provide reliable data transfer because heavy collision is usually generated by some adjacent sensors [10,12]. In recent ten years, compressed sensing (CS)—an emerging data compression methodology that data compression and reconstruction can be perfectly performed based on data sparsity—has attracted wide attention in the study of energy-efficient gait telemonitoring applications [14]. The basic idea is that a larger amount of the acquired gait data is firstly compressed on sensors, in order to significantly reduce the energy consumption during data transmission. Then, all compressed data received in the remote terminal are perfectly reconstructed to perform further data processing [15,16]. For example, Wu et al. investigated the application of CS for the energy-efficient telemonitoring of gait using acceleration data [16]. Although significant research efforts are made, many existing works have only showed that the CS technique for energy-efficient gait telemonitoring is feasible only using single-sensor data. Theoretically, the CS technique has no ability to simultaneously compress multi-sensor data for the energy efficiency of sensors. This leads to the possible loss of the highly-correlated information regarding the intrinsic dynamics of human movement. It is essential to search for an advanced technique for jointly processing multi-sensor gait data.

Further data processing such as gait classification is also very important for gait monitoring. How to develop the best gait-classification model has been another challenging issue in gait telemonitoring applications [16–18]. Some studies have showed that the original gait data are directly used to develop the gait-classification models based on machine learning algorithms. In these studies, some commonly used traditional machine learning algorithms include artificial neural networks (ANN) [18], decision tree, *k*-nearest neighbor (KNN) [17], support vector machine (SVM) [16,19], and Hidden Markov Model (HMM) [20–22]. Comparing these methods, the HMM technique has shown higher performance in gait pattern classification using on-body wearable sensors. For instance, Mannini et al. investigated the feasibility of human physical activity classification by using on-body wearable sensors. Especially, Taborri et al. proposed a novel HMM-based distributed classifier for the detection of gait phases by using a wearable inertial sensor network. In their proposed algorithm, a distributed stochastic model and a hierarchical-weighted classification are implemented to detect gait phases by simultaneously processing the data of multi-sensors placed on different body segments of lower limbs. Recently, an emerging sparse representation classification (SRC) algorithm has widely attracted the focus of gait classification. Its basic idea is that all gait training samples are directly used to construct an over-complete dictionary, and a test gait can be sparsely represented as a linear combination of just those training samples with same gait class. The residual error criterion is defined by solving sparse representation coefficients, in order to exactly determine the class of the test gait [23–25]. Zhang et al. investigated the feasibility of a sparse representation-based gait-classification model using wearable sensor gait data [24]. Also, Allen et al. studied the practicality of the sparse representation-based gait-classification model using wearable multi-sensor gait data [25]. Although the SRC algorithm can produce a better accuracy, it has a higher computational time cost. More importantly, all above studies don't take into account the use of the gait data received at a remote terminal to develop a gait-classification model [26,27].

At the same time, other recent studies have focused on using the received gait data to develop gait-classification models. In view of the energy efficiency of sensors, recent relevant studies mainly concentrated on the combination of the CS technique with machine learning algorithms for gait classification. Their basic idea is that all compressed gait data received at a remote terminal are reconstructed by the CS technique, and the reconstructed gait data are then employed to develop gait-classification models based on machine learning algorithms [14,16]. In such works, one important step is to reconstruct gait data with a higher quality. However, the traditional CS technique has no ability to perfectly reconstruct the gait data; this is because gait data such as acceleration data are poorly sparse [14,16]. This possibly deteriorates the further gait-classification performance. Moreover, these hybrid techniques possibly produce a higher computational complexity. This motivates us to search for advanced approaches to the energy-efficient telemonitoring of multi-sensor gait [28,29].

It is well known that multi-sensor gait data generally interact in a complex fashion, an observation attributable to the intrinsic dynamics of human gait. Theoretically, we can assume multi-sensor gait data as a data ensemble with joint sparsity, in order to jointly process multi-sensors and thus capture the higher-correlation information associated with gait. This can potentially allow the best classification performance and energy efficiency of sensors. Recently, there have emerged some advanced techniques that have a superior ability to jointly process multi-signals based on data joint sparsity. Successful examples include distributed compressed sensing (DCS) [30,31] and the joint sparse representation classification (JSRC) model [32–34]. These two techniques have been widely applied in video coding [35–37], image fusion [38], and multichannel physiological monitoring [39]. So far, no studies have been reported where DCS and JSRC have been employed for jointly processing multi-sensor gait data.

In this study, a novel advanced hybrid technique of DCS and JSRC for the telemonitoring of multi-sensor gait is proposed based on data joint sparsity. Unlike the recent relevant studies, the advantage of our proposed technique is that we directly take advantage of the multi-sensor gait data compressed by DCS to develop the novel neighboring JSRC gait-classification models with better performance. This can not only avoid the complicated step of joint reconstruction of multi-sensor gait data, but can also produce the best classification performance as well as a lower computational time. In our proposed technique, the DCS technique is firstly utilized to simultaneously compress multi-sensor gait data, in order to potentially gain higher-correlation information as well as improved energy efficiency of the sensors. Then, all jointly compressed gait data are directly used to develop a novel neighboring sample-based JSRC model by defining the sparse representation coefficients-inducing criterion (SRCC), not to perform the reconstruction task. The test gait can be sparsely represented by constructing a new over-complete dictionary that includes a few local neighboring training samples containing more valuable information. This possibly produces the best classification performance as well as a lower computational time. Our proposed technique has great potential to enable continuous collection of people's gait information and accurate human gait measurement anywhere and at any time. This possibly provides benefits in telemedicine applications such as the early identification of at-risk gait of elderly in the community, as well as the monitoring of gait rehabilitation progress of people at home. The multi-sensor gait data are selected from an open wearable action recognition database (WARD), in order to validate the feasibility of our proposed method. The results show the superior classification performance of our proposed techniques to the traditional techniques such as SRC and JSRC. Moreover, a lower computational time cost is spent in our proposed technique.

The rest of the paper is organized as follows. In Section 2, we describe the DCS technique for multi-sensor gait data. A novel neighboring JSRC model for multi-sensor gait classification is presented in Section 3. In Section 4, we describe the materials and methods for evaluating our proposed technique. The evaluation results obtained from multi-sensor gait data are given in Section 5. Discussion and conclusions are presented in Sections 6 and 7, respectively.

#### **2. Distributed Compressed Sensing for Multi-Sensor Gait Data**

Theoretically, the DCS technique combines distributed source coding theory with compressed sensing theory [14,30,31]. It can take advantage of data joint sparsity to capture both inter- and intra-signal correlation among multi-sensor signals. In the DCS technique, three different joint sparsity models (JSM), such as JSM-1, JSM-2, and JSM-3 [30,31], are usually used to perform the simultaneous compression and the joint reconstruction task in the multi-signal case. In this study, since each sensor can acquire the same gait data, such as acceleration data, we assume that each sensor's data has the same sparsity pattern but with different coefficients. Here, JSM-2 is adopted to jointly process the multi-sensor gait data [30,31]. That is, all *J* sensors' gait data in WBANs are assumed as a data ensemble *X*= [*x*1, *x*2, ··· , *xJ*] *<sup>T</sup>* <sup>∈</sup> *<sup>R</sup>JN*×1, where *<sup>N</sup>* denotes the length of each sensor data *<sup>x</sup><sup>j</sup>* ∈ *<sup>R</sup>N*×1(*<sup>j</sup>* = 1, 2, ··· , *<sup>J</sup>*). Here, we assume that each sensor data *<sup>x</sup><sup>j</sup>* is sparsely represented as

$$\mathbf{x}\_{\rangle} = \mathbf{Y}\boldsymbol{\theta}\_{\rangle} \tag{1}$$

where *Ψ* denotes dictionary matrix vectors and *θ<sup>j</sup>* is the corresponding sparse vector that satisfies the same support set *Ω* ⊂ {1, 2, ··· , *N*}. Then, the multi-sensor gait data ensemble *X* can be represented as

$$X = \mathbf{Y}\boldsymbol{\theta}\tag{2}$$

where *θ* = *θ*1, *θ*2, ··· , *θj*, *θ<sup>J</sup> <sup>T</sup>* is the joint sparse coefficients vector and *Ψ* = ⎡ ⎢ ⎢ ⎢ ⎣ *Ψ*, 0, 0, ··· , 0 0, *Ψ*, 0, ··· , 0 · 0, 0, 0, ··· , *Ψ* ⎤ ⎥ ⎥ ⎥ ⎦ is the

corresponding dictionary matrix. Then, the multi-sensor gait data ensemble *X* can be rewritten as

$$\begin{bmatrix} \mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_{j\_1}, \dots, \mathbf{x}\_{j\_l} \end{bmatrix}^T = \begin{bmatrix} \Psi\_r \mathbf{0}, \mathbf{0}, \dots, \mathbf{0} \\ \mathbf{0}, \Psi\_r \mathbf{0}, \dots, \mathbf{0} \\ \vdots \\ \mathbf{0}, \mathbf{0}, \mathbf{0}, \dots, \mathbf{0} \end{bmatrix} \begin{bmatrix} \boldsymbol{\theta}\_1, \boldsymbol{\theta}\_2, \dots, \boldsymbol{\theta}\_{j\_l}, \dots, \boldsymbol{\theta}\_l \end{bmatrix}^T. \tag{3}$$

Next, we define a measurement matrix *<sup>Φ</sup>* <sup>∈</sup> <sup>R</sup>JMj×JN(*Φ*<sup>j</sup> <sup>∈</sup> <sup>R</sup>Mj×N, Mj <sup>N</sup>) as

$$
\Phi = \begin{pmatrix}
\Phi\_1 & 0 & 0 & 0 & \cdots & 0 \\
0 & \Phi\_2 & 0 & 0 & \cdots & 0 \\
\vdots & \vdots & \vdots & \vdots & \cdots & \vdots \\
0 & 0 & 0 & 0 & \cdots & \Phi\_I
\end{pmatrix}.
\tag{4}$$

Thus, the multi-sensor gait data ensemble *X* can be simultaneously compressed as

$$\mathbf{Y} = \boldsymbol{\Phi} \mathbf{X} = \begin{pmatrix} \Phi\_1 & 0 & 0 & 0 & \cdots & 0 \\ 0 & \Phi\_2 & 0 & 0 & \cdots & 0 \\ & \vdots & \vdots & \vdots & \vdots & \cdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & \Phi\_l \end{pmatrix} [\mathbf{x}\_1, \mathbf{x}\_2, \cdots, \mathbf{x}\_j, \cdots, \mathbf{x}\_l]^T \tag{5}$$

where *Y* = [*y*1, *y*2, ··· , *y<sup>j</sup>* , ··· , *yL*] *<sup>T</sup>* <sup>∈</sup> *<sup>R</sup>JM*×<sup>1</sup> is the jointly compressed multi-sensor data. Here, it is noted that each *y<sup>j</sup>* = *Φjxj*, *j* ∈ {1, 2, ··· , *J*} is performed independently of the others.

In the DCS technique, joint reconstruction is another important task. Its basic idea is to obtain an accurate solution of the joint sparse representation coefficients *θ* = *θ*1, *θ*2, ··· , *θj*, *θ<sup>J</sup> <sup>T</sup>* by solving the following *l*1-norm minimization:

$$\theta = \operatorname\*{argmin}\_{\theta} ||\theta||\_{l\_1 \times} \text{s.t. } \mathbf{Y} = \Phi \mathbf{X}. \tag{6}$$

Then, the joint reconstruction of multi-sensor data can be estimated as *X*ˆ = *Ψθ*ˆ. The commonly used joint reconstruction algorithms include the One-Step Greedy Algorithm (OSGA) [32] and the Simultaneous Orthogonal Matching Pursuit (SOMP) [31]. However, since multi-sensor gait data such as accelerometer and gyroscope data are poorly sparse, it is very difficult to guarantee the best solution of joint representation coefficients *θ*ˆ by Equation (6). In fact, the jointly compressed multi-sensors data *Y* by DCS possibly contain enough highly-correlated discriminative information associated with gait. Theoretically, the compressed multi-sensor data *Y* can be directly used to develop a JSRC-based classification model, but not to perform a joint reconstruction task. This helps to improve multi-sensor gait-classification performance.

#### **3. A Novel Neighboring JSRC Model for Gait Classification**

Theoretically, the JSRC model is used to generalize the traditional SRC model to the multi-signal case [33,40–43]. Its basic idea is to explore the joint sparse representation for the multiple input signals in the classification task, and the joint sparsity can be enforced by imposing a joint sparsity-inducing norm(such as *l*1,2-norm, *l*1,∞-norm) penalty on representation coefficients [23–25]. However, due to the large-scale gait training samples in this study, more computational time and higher memory cost are needed for the matrix inverse operation in the JSRC algorithm. This possibly yields a high computational complexity as well as a poor classification performance. It is essential to search for a few training samples to develop the best JSRC-based gait-classification models. In this study, a novel neighboring JSRC model is developed based on a few neighboring training samples selected by the sparse representation coefficients-inducing criterion (SRCC). As we know, the same gait class can be theoretically spanned by the training samples (i.e., over-complete dictionary atoms) of the same class in a low-dimensional subspace [33,41–48]. The spanned training sample can be sparsely represented in the high-dimension space. Those samples with high values of sparse representation coefficients are selected as the nearest neighbor samples because they possibly contain the more valuable information associated with gait pattern change. So, these selected neighboring samples are used to reconstruct a new over-complete dictionary for multi-sensor gait classification. The block diagram of the proposed novel neighboring JSRC model is shown in Figure 1. In order to clearly describe the novel neighboring JSRC model, we firstly describe the construction of an over-complete dictionary using all original training samples.

**Figure 1.** The block diagram of the proposed novel neighboring joint sparse representation classification (JSRC) model.

#### *3.1. Constructing an Over-Complete Dictionary for JSRC*

In this study, we assume that each sensor node is equipped with a triaxial accelerometer (*x*, *y*, *z*) and a biaxial gyroscope (*θ*, *ρ*) [24,25], in order to capture more valuable information associated with gait. Firstly, we define a vector *aj* to denote the gait data of each sensor *j*(*j* = *1*, *2*, ··· ,*J*) at time *t*, i.e.,

$$\mathbf{a}\_{j} = \left(\mathbf{x}\_{j}(\mathbf{t}), \mathbf{y}\_{j}(\mathbf{t}), \mathbf{z}\_{j}(\mathbf{t}), \theta\_{j}(\mathbf{t}), \rho\_{j}(\mathbf{t})\right)^{T} \in \mathbb{R}^{5}.\tag{7}$$

Then, a vector *a*(*t*) is defined to represent multi-sensor gait data of all *J* sensors at time *t*, i.e.,

$$\mathbf{a}(\mathbf{t}) = \left(\mathbf{a}\_1^T(\mathbf{t}), \mathbf{a}\_2^T(\mathbf{t}), \dots, \mathbf{a}\_I^T(\mathbf{t})\right)^T \in \mathbb{R}^{5\bar{\mathbf{J}}}.\tag{8}$$

Next, based on Equation (6), a matrix *S* is defined to represent the multi-sensor gait data ensemble during the length of time *h*:

$$\mathbf{S} = \begin{bmatrix} \mathbf{a}(1), \mathbf{a}(2), \cdots, \mathbf{a}(h) \end{bmatrix}^T = \begin{pmatrix} \mathbf{s}^1, \mathbf{s}^2, \cdots, \mathbf{s}^J \end{pmatrix}^T \in \mathbb{R}^{5l \times h} \tag{9}$$

where the sub-vector *sj* denotes the gait data of sensor*j* with duration *h*, *sj* = - *aj*(1),*aj*(2),··· ,*aj*(*h*) *<sup>T</sup>* <sup>∈</sup> R5×*h*.

Here, assuming that there are *L* different gait classes, we can define a new matrix *ST* that concatenates the training sample from all *L* gait classes:

$$\mathbf{S}\_{T} = [\mathbf{S}\_{1}, \mathbf{S}\_{2}, \cdots, \mathbf{S}\_{l}, \mathbf{S}\_{L}] (i = 1, 2, \cdots, L) \tag{10}$$

where sub-matrix *Si* = - *si*,1, *si*,2, ··· , *si*,*n<sup>i</sup>* represents all training sample data of gait class *i*, and *ni* denotes the total of training samples of class *i*. In such a case, we can employ the matrix *S<sup>T</sup>* to construct an over-complete dictionary that consists of *L* sub-dictionary *Si* with respect to all *L* classes [23,25,33,41,43]. Thus, based on the constructed over-complete dictionary *ST*, a test gait *Y* (*Y* = [**y**1, **y**2, ··· , *y<sup>j</sup>* , ··· , **y***J*] *<sup>T</sup>* <sup>∈</sup> *<sup>R</sup>JM*×1) with class *<sup>i</sup>* can be joint-sparsely represented as

$$\mathbf{Y} = \mathbf{S}\_T \mathbf{W} \tag{11}$$

where *W* = 0, 0, ··· , 0, *wi*,1, *wi*,2, ··· , *wi*,*ni* , 0, ··· , 0 denotes the joint sparse representation coefficients matrix whose entries are zero except those associated with gait class *i*, and the total number of samples *n* = ∑*<sup>L</sup> <sup>i</sup> ni*.

Usually, the joint sparse coefficients matrix *W* can be estimated by solving the following joint sparse optimization problem [33,41,43]:

$$\dot{W} = \operatorname{argmin} ||\mathcal{Y} - S\_T \mathcal{W}||\_F^2 \text{ s.t. } ||\mathcal{W}||\_{l\_0 \backslash l\_2} \le k \tag{12}$$

where || · ||*<sup>F</sup>* denotes the Frobenius norm, ||*W*||*l*0\*l*<sup>2</sup> refers to the mixed norm that can be phrased as performing the *<sup>l</sup>*2-norm cross the column and then the *<sup>l</sup>*0-norm along the row, i.e., ||*W*||*l*0\*l*<sup>2</sup> = ||[||*W*(1, : )||*l*<sup>2</sup> , ||*W*(2, :)||*l*<sup>2</sup> , ··· , ||*W*(*L*, :)||*l*<sup>2</sup> ]||*l*<sup>0</sup> . *k* denotes the maximum number of the nonzero coefficients in *W*.

In fact, because the *l*2/*l*<sup>0</sup> regularization norm needs the shared gait pattern across observation, the computation complexity reaches *O*(*w*2*n*) for the optimization solution of non-zero sparse coefficients *w*. That is, if large-scale training sample data are directly used to construct the over-complete dictionary, then a higher computational complexity is possibly yielded in estimating the optimization solution of *w*. Therefore, it is necessary to find a few training samples to reconstruct an over-complete dictionary for developing the JSRC-based multi-sensor gait-classification model with high quality.

#### *3.2. Reconstructing a New Over-Complete Dictionary for JSRC*

Here, the sparse representation coefficients-inducing criterion (SRCC) is used to find a few nearest neighboring samples containing the most valuable information, in order to reconstruct a new over-complete dictionary for JSRC. The detailed procedure of selecting the nearest neighbor samples is presented as follows.

**Step 1** The test sample *y<sup>j</sup>* from sensor *j* is linearly represented as

$$y\_j = \mathbf{w}\_{i,1}\mathbf{s}\_1^j + \mathbf{w}\_{i,2}\mathbf{s}\_2^j + \cdots + \mathbf{w}\_{i,n\_i}\mathbf{s}\_{n\_i}^j. \tag{13}$$

Then, the sparse representation coefficients *w* are obtained by solving the following *l*2-minimization problem:

$$\mathfrak{w} = \underset{\mathfrak{w}}{\text{argmin}} ||\mathfrak{y}\_{\mathfrak{j}} - \mathfrak{s}^{\bar{\mathfrak{j}}} \mathfrak{w}||\_{2} + \lambda ||\mathfrak{w}||\_{2} \tag{14}$$

where *<sup>λ</sup>* is a positive regularization, and the *<sup>l</sup>*2-norm is defined as ||*w*||<sup>2</sup> <sup>=</sup> <sup>∑</sup>*ni <sup>i</sup>*=<sup>1</sup> |*wi*| 2.

**Step 2** Based on the solution of *w*, the *m* training samples with the larger sparse coefficients values are selected as the nearest neighbor training samples. For gait class *i*, a new training sample set is defined as

$$A\_{l} = (\mathbf{s}\_{l,1}, \mathbf{s}\_{l,2}, \dots, \mathbf{s}\_{l,\text{nu}}).\tag{15}$$

**Step 3** Based on Equation (15), a new over-completed dictionary is reconstructed as

$$\mathbf{A}\_{T} = \begin{bmatrix} \mathbf{A}\_{1}, \mathbf{A}\_{2}, \cdots, \mathbf{A}\_{L} \end{bmatrix}^{T}. \tag{16}$$

Therefore, a test gait *Y* with class *i* can be joint-sparsely represented as

$$
\Upsilon = A\_T \mathbf{W}.\tag{17}
$$

Here, the joint sparse coefficients matrix *W* in Equation (17) can be estimated by solving the following *l*2/*l*0-minimization problem:

$$\dot{\mathbf{W}} = \operatorname\*{argmin} ||\mathbf{Y} - \mathbf{A}\mathbf{W}||\_F^2 \text{ s.t. } ||\mathbf{W}||\_{l\_0 l\_2} \le k. \tag{18}$$

The detailed procedure of solving joint sparse representation coefficients *W*ˆ is presented in Section 3.3.

#### *3.3. MBCS Technique for Solving Joint Sparse Representation Coefficients*

In this study, the Multitask Bayesian Compressive Sensing (MBCS) technique is applied to solve the joint sparse representation coefficients *W* in Section 2.2.2 [48], in order to capture more of the temporal–spatial correlation information regarding gait. Due to multi-sensor gait data that are contaminated by noise, the test gait *Y* can be represented as *Y* = *AW* + *E*, i.e.,

$$
\begin{bmatrix} y\_1 \\ y\_2 \\ \vdots \\ y\_I \end{bmatrix} = \begin{bmatrix} A\_1 \\ A\_2 \\ \vdots \\ A\_I \end{bmatrix} [\mathcal{W}\_1, \mathcal{W}\_2, \dots, \mathcal{W}\_I] + \begin{bmatrix} c\_1 \\ c\_2 \\ \vdots \\ c\_I \end{bmatrix} \tag{19}
$$

where *E* = [*e*1,*e*2, ··· ,*eJ*] *<sup>T</sup>* denotes the noise. Here, the noise *<sup>e</sup><sup>j</sup>* satisfies a Gaussian distribution with an unknown parameter *β*0, and the prior distribution function of *yj* is defined as

$$p(y\_j|\mathcal{W}\_{\flat}, \boldsymbol{\beta}\_0) = (\beta\_0/2\pi)^{N/2} \exp(-\frac{\beta\_0}{2}||y\_j - \mathbf{A}\_j \mathcal{W}\_{\flat}||\_2^2). \tag{20}$$

Then, the prior distribution function of the sparse coefficients *W<sup>j</sup>* is defined as

$$p(\mathcal{W}\_j|\boldsymbol{\beta}) = \prod\_{i=k}^{M} \mathcal{N}(\mathcal{W}\_{j,k}|0, \boldsymbol{\beta}\_k^{-1}) \tag{21}$$

where *Wj*,*<sup>k</sup>* denotes the *<sup>k</sup>*th sparse representation coefficient of sensor *<sup>j</sup>* and *<sup>N</sup>*(·|0, *<sup>β</sup>*−<sup>1</sup> *<sup>k</sup>* ) refers to the Gaussian distribution function whose mean is equal to zero and variance *σ*<sup>2</sup> = 1/*βk*.

After the optimal parameters *β*, *β*<sup>0</sup> are estimated by constructing a likelihood function, we can solve the joint sparse representation coefficient *W<sup>j</sup> J j*=1 . The production of the solution for *W<sup>j</sup> J <sup>j</sup>*=<sup>1</sup> is presented in the Appendix A, and the detailed procedure for the solution can be found in reference [48].

#### *3.4. The Definition of Minimal Residual Error Rule for Gait Classification*

Based on the estimated solutions of the joint sparse representation coefficients *W*ˆ , we firstly define the matrix indication function *δi*(·) whose entries are zero except for those associated with gait class *i*. i.e.,

$$\delta\_{\mathbf{i}}(\mathbf{\hat{W}}) = \begin{bmatrix} 0^T, \cdots, 0^T, \mathbf{W}\_{\mathbf{i}}^T, 0^T \cdots \mathbf{0}^T \end{bmatrix}^T. \tag{22}$$

Then, the approximation of test gait sample *Y* can be defined as

$$
\tilde{Y} = A \delta\_i(\mathbf{\hat{W}}).\tag{23}
$$

The minimal residual error between *Y* and *Y* can be defined as

$$\min\_{\hat{\mathbf{i}}} r\_{\hat{\mathbf{i}}}(\mathbf{Y}) = \|\mathbf{Y} - A\boldsymbol{\delta}\_{\hat{\mathbf{i}}}(\hat{\mathbf{W}})\|\_{2}. \tag{24}$$

Thus, the class label of test gait *Y* can be identified by the following minimal residual error rule [23,33,41]:

$$\hat{\mathfrak{I}} = \underset{i}{\text{argmin}} \|\mathfrak{I} - A\delta\_i(\hat{\mathfrak{W}})\|\_{2}^{2}, i = 1, 2, \dots, L. \tag{25}$$

#### **4. Materials and Methods**

#### *4.1. The Selection of Multi-Sensor Gait Data*

In this study, the multi-sensor gait data were selected from an open wearable action recognition database (WARD) database from University of California, in order to evaluate the feasibility of our proposed technique. In the selected WARD database, the multi-sensor gait data with different walking patterns are sampled from a total of 20 subjects (7 female subjects and 13 male subjects) with ages ranging from 19 to 75 years. All participants had no known injuries or abnormalities that could affect their walking pattern. The wearable motion sensor networks consist of five wireless motion sensors (two are located on the wrists, one is located on the waist, and two are positioned on the ankles, all using elastic belts). Each sensor node integrates a 3-axis (i.e., *x*-axis, *y*-axis and *z*-axis) accelerometer with a range of ±2 g and a 2-axis gyroscope with a range of ±500 dps, and each axis is reported as a 12 bit value to the sensor node. Each subject is asked to perform 13 activity or gait patterns such as stand, sit, lie down, walk forward, walk left-circle, walk right-circle, turn left, turn right, go upstairs, go downstairs, jog, jump, and push a wheelchair. Each subject is asked to perform each pattern 5 times. The sample ratio is set to 20 Hz. More details about the multi-sensor data collection are available at the website (http://www.eecs.berkeley.edu/~yang/software/WAR/).

#### *4.2. Data Preprocessing*

In this study, in order to effectively validate the feasibility of our proposed method, we randomly selected nine gait or activity pattern classes from all 13 gait patterns, such as stand, sit, walk forward, turn left, turn right, go upstairs, go downstairs, jog, jump. Firstly, the denoising for the recorded multi-sensor data is performed by using a fifth-order Butterworth with a low-pass filter at 30 Hz. Then, a sliding window technique is employed to segment the acquired time-sequence data into a sequence of fixed-length windows, in order to gain valuable correlation information from the collected multi-sensor gait data [16,24]. Here, the window is set to a length of *N* = 200 with 50% overlap, and 10 windows were randomly extracted from each subject's data. Here, each subject's data consist of the 9 different gait pattern classes data, each class including 10 windows' data (i.e., each subject's data includes 10 stand data, 10 sit data, 10 walk forward data, 10 turn left data, 10 turn right data, 10 go upstairs data, 10 go downstairs data, 10 jog data, 10 jump data). In this study, the 20 subjects' total data are used to train and test, in order to objectively evaluate the classification performance. The total sample data size is 1800 (i.e., 20 subjects × 9 gait pattern classes × 10 window samples), as shown in Table 1. In addition, we perform the classification task of nine gait patterns (walk forward, walk left, walk right, go upstairs, go downstairs, run forward, sitting), in order to further evaluate the feasibility of our proposed model for multi-sensor gait telemonitoring applications.

**Table 1.** The total amount of available sample data.


#### *4.3. Training and Testing the Gait-Classification Model*

In this work, the ten-fold cross validation strategy is adopted to objectively and accurately evaluate gait-classification performance [13–15]. Here, all 20 subjects' data are divided into ten subsets, each including 2 subjects' data. To effectively evaluate the generalization performance, the data that are used to test are not included in the training samples data. The detailed training and testing procedure is as follows: Firstly, the training set is constructed to develop a gait-classification model. Here, 18 subjects' data are randomly selected as the training set, containing a sample size of 1620 (i.e., 180 stand data, 180 sit data, 180 walk forward data, 180 turn left data, 180 turn right data, 180 go upstairs data, 180 go downstairs data, 180 jog data, 180 jump data). In the process of training the classification model, a nine-fold cross-validation scheme is employed due to the limited data size. That is, all selected 18 subjects' data are randomly divided into nine subsets, each subset including 2 subjects' data. Eight subsets are arbitrarily selected as the training set while the remaining one is used to test. The above steps are repeated until each subset is employed to test. After the classification model is yielded, the remaining two subjects' data that are not included in training set are used to test the generalization ability. Each subject's data must be used to test the generalization ability. Finally, the whole averaged classification result is obtained for all subjects.

#### *4.4. Evaluation Criteria for the Proposed Technique*

In addition, some common criteria are adopted to objectively assess the performance of the proposed gait-classification model using multi-sensor gait data, and they are respectively defined as follows.

#### (1) Compression ratio (CR)

The compression ratio is used to quantitatively evaluate the ability to compress the multi-sensor gait data, and is defined as follows:

$$CR = \frac{N - M}{N} \times 100\% \tag{26}$$

where *N* and *M* denote the original data length and the compressed data length, respectively.

(2) Accuracy

Accuracy is employed to objectively evaluate the classification performance, and it is defined as the ratio of the number of gait patterns accurately identified to the total of gait patterns in the tests.

$$Accuracy = \frac{NA}{NT} \times 100\% \tag{27}$$

where *NA* and *NT* denote the number of gait patterns accurately identified and the total number of gait patterns, respectively.

(3) Precision

Precision is defined as a measure of the ability of the classifier to detect gait patterns with class *i* in the confusion table.

$$Precision = \frac{NP}{NP + NF} \times 100\% \tag{28}$$

where *NP* and *NF* denote the number of gait patterns accurately identified as class *i* and class *j*, respectively.

(4) Recall

Recall is defined as a measure of the ability of the classifier to specify gait patterns with class *j* in the confusion table.

$$Recall = \frac{NF}{NP + NF} \times 100\%.\tag{29}$$

In addition, in this study, statistical analyses to determine the mean and standard deviation (SD) are performed for the outcomes obtained by the different classification algorithms. All indices used are tested by means of a statistical analysis such as ANOVAs, in order to further test the objectivity of

the outcomes obtained by the different classification algorithms. Here, all different algorithms that are considered as independent variables are performed on all indices.

In this study, all algorithm programs are developed in Matlab 2013a, and they are performed on a computer with an Intel(R) Core(TM) i5-3470 3.20GHz CPU, 4.00GB RAM, and Windows 7 operating system.

#### **5. Results**

#### *5.1. The Evaluation of the Effect of DCS on the Developed Neighboring JSRC*

Firstly, we evaluate the effect of the simultaneous compression of multi-sensor gait data by DCS on our developed neighboring JSRC(SRCC-JSRC) model. For comparison, we also perform a JSRC model and a KNN-based JSRC(KNN-JSRC) model. For DCS, the sparse binary matrix was selected as the random projection matrix where only two nonzero entries are included in each column. All 20 subjects' data are used to train and test, and the ten-fold cross validation strategy is adopted to objectively evaluate gait-classification performance. Firstly, we compared three JSRC-based classifications' performances based on multi-sensor gait data which are simultaneously compressed by DCS. The number of neighboring samples are selected as 20, 40 and 60, respectively. All comparative results are shown in Table 2. From Table 2, we can observe that when the number of neighboring samples is selected as 40 or 60, all neighboring sample-based JSRC models can produce a better accuracy of 94% as well as a lower computational time cost when compared with the JSRC model. These results suggest that the simultaneous compression of multi-sensor gait data by DCS can provide the valuable high-correlation information regarding gait for the JSRC model.

**Table 2.** Comparative results from the different JSRC-based gait-classification performance.


It is noted that KNN is the abbreviation of k-nearest neighbor and SRCC is the abbreviation of sparse representation coefficients-inducing criterion.

#### *5.2. The Assessment of Gait-Classification Performance Based on the Different Compression Ratios*

Next, we evaluate the classification performance of our proposed method based on the different compression ratios to examine the feasibility of the energy-efficient telemonitoring of gait. The number of neighboring samples are selected as 20 and 40, respectively. The comparative results from all JSRC-based models are shown in Figure 2. The errors of accuracy based on the different compression ratios are also presented in Figure 2. Table 3 shows the mean and SD values of the best accuracy from all selected algorithms. As shown in Figure 2 and Table 3, the SD of all accuracies can obtain lower error. From Figure 2, we obviously see that, based on the different compression ratios, our proposed method (SRCC-JSRC) can produce better accuracy than KNN-JSRC and JSRC, suggesting that the neighboring samples by our defined criteria contain the more valuable information about the test sample. In particular, when the compression ratio increases from 10 to 70%, SRCC-JSRC (*m* = 40) reaches the best accuracy. These results showed that our proposed method can yield the superior classification performance when the compression ratio is properly selected. That is, our proposed method can potentially produce the best classification performance as well as the best energy efficiency of sensors. In addition, from Figure 2, we also find that all JSRC-based gait classifications produce poor accuracy when the compression ratio is more than 70%. This suggests that the higher compression ratios could lose the more valuable correlation information regarding gait.

**Figure 2.** Comparative results of the different gait-classification performances based on the different compression ratios.

In addition, we also compared our proposed model to some tradition classification models such as SRC, naive Bayesian classifier (NBC), and KNN models. Here, the number of nearest neighboring samples are selected as 40 (i.e., *m* = 40). The comparative results are shown in Figure 3. As shown in Figure 3, our proposed model significantly outcompetes SRC, NBC, and KNN models. In this comparison, the NBC technique is the poorest. These results demonstrate that our proposed model can gain more distinctive temporal–spatial correlation information associated with gait pattern change.

**Table 3.** The statistical results the best accuracy from all selected algorithms.


**Figure 3.** Comparison of accuracy of our proposed technique with KNN, naive Bayesian classifier (NBC), and sparse representation classification (SRC).

#### *5.3. The Evaluation of Computation Time Cost Based on the Different Compression Ratios*

Meanwhile, we also evaluate the computational time cost of all JSRC-based models based on the different compression ratios. Here, statistical analyses to determine the mean and standard deviation (SD) are performed for the obtained computational time from the different classification algorithms. All comparative results are shown in Figure 4. The errors of the obtained computational time based on the different compression ratios are also presented in Figure 4. Table 4 shows the mean and SD values of the lowest computation time from all selected algorithms. As shown in Figure 4 and Table 4, the SD of all computation times can obtain lower error. From Figure 4, we can obviously see that the computation time cost of each JSRC-based model is reduced as the compression ratio increases. In comparison, our proposed method (SRCC-JSRC) has the lowest computational time cost. In particular, the lowest computational time cost of SRCC-JSRC (*m* = 20) remains only 67 ms while the compression ratio increases from 10 to 70%. These results demonstrate that our proposed method could possibly result in a lower computational time cost and better energy efficiency of sensors when the compression ratio is properly selected. Besides this, we also observe that, once the compression ratios are more than 70, all computation times significantly decline. This is because the higher compression ratios could yield fewer multi-sensor gait data to be processed.

**Figure 4.** Comparative results of computation time from the different JSRC-based gait-classification algorithms.

**Table 4.** The statistical results for the lowest computation time from all selected algorithms.


#### *5.4. Evaluation of Our Proposed Model for Gait Telemonitoring*

We perform the task of classification for nine gait patterns (walk forward, walk left, walk right, go upstairs, go downstairs, run forward, sitting, jog, jump), in order to further evaluate the feasibility of our proposed model for multi-sensor gait telemonitoring applications. Here, the number of nearest neighboring samples is selected as 40, and the compression ratio is set to 50%. Table 5 shows the confusion table for the classification of the nine gait patterns. In Table 4, the entry in the *i*th row and *j*th column refers to the count of gait patterns belonging to class *i* that are classified as class *j*. As shown in Table 5, the whole averaged classification accuracy across all nine gait patterns reaches 95%, suggesting that our proposed technique can accurately classify these nine gait patterns. In terms of individual patterns, the "turn left" and "jump" patterns obtain the maximal value of 100% and the "jog" pattern gains the best precision value of 100%. The "downstairs" pattern gains the same maximal precision and recall value of 97%, and the "sitting" pattern obtains poor precision and recall values. For the "jump" pattern, its precision is equal to 100%, whereas its recall is only 90%. The possible reason for this is that some samples of "jump" are easily misclassified as "walk forward" and "downstairs", but samples of "walk forward" are not misclassified as "jump". In conclusion, these results show that our proposed technique can capture the more distinctive and high-correlation information associated with gait for multi-sensor gait classification. Our proposed method has great potential for gait telemonitoring applications.


**Table 5.** The confusion table for classification of the nine different gait pattern classes.

In addition, the results of statistical tests based on ANOVAs are presented in Table 6. As shown in Table 6, all significance values are less than the significance level value of 0.05, suggesting that the significance of the difference among classification algorithms is at an acceptable level. This further validates the objectivity of the outcomes obtained by the different classification algorithms.

**Table 6.** The results of statistical tests for all indices based on ANOVAs.


#### **6. Discussion**

The results of present studies demonstrate that the hybrid technique of DCS and JSRC can take advantage of data joint sparsity to jointly process multi-sensor gait data, which can capture the more valuable high-correlation information hidden in multi-sensor gait data. This greatly contributes to the best classification performance as well as to the energy efficiency of sensors in WBANs-based gait telemonitoring applications. Currently, the discovery of powerful techniques for the quantitative analysis of multi-sensor gait data has become a research focus in the gait telemonitoring field [10–12]. One important issue is how to gain the higher-correlation information regarding gait for the best gait-classification performance as well as for improving the energy efficiency of sensors [13,26–28].

In the present studies, considering that both DCS and JSRC have the best ability to jointly process multi-signals based on data joint sparsity, we investigated the feasibility of the hybrid technique of DCS and JSRC for gait telemonitoring by jointly processing multi-sensor gait data [32–34]. In particular, we try to take advantage of the joint sparsity of multi-sensor gait data to develop a novel neighboring JSRC model for the best classification performance as well as lower computation time. In this study, we firstly evaluate the effect of the simultaneous compression of multi-sensor data by DCS on the JSRC-based classification model. As illustrated in Table 2, all JSRC-based gait-classification models

can yield better accuracy. This suggests that the DCS technique has the superior ability to capture the higher-correlation information regarding gait by jointly compressing multi-sensor gait data, and all compressed multi-sensor gait data containing the more valuable information significantly helps to produce the better classification performance of the JSRC-based model. These results further suggest that it is feasible that DCS and JSRC can jointly process multi-sensor gait data for gait classification based on data joint sparsity.

Next, we evaluate the feasibility of our proposed method for multi-sensor gait classification based on the different compression ratios. As shown in Figure 2, all selected JSRC models can yield a better gait-classification performance when compression ratios are properly chosen. In the comparison, our proposed model is best. This suggests that our proposed model potentially yields the best performance as well as the best energy efficiency of sensors. In addition, we also compared our proposed method to the traditional SRC, NBC, and KNN models. As shown in Figure 3, our proposed method significantly outcompetes the selected traditional methods. A possible reason for this is that the selected neighboring training samples can contain the more distinctive and high-correlation information associated with gait, which significantly improves multi-sensor gait-classification performance. However, the traditional SRC model does not gain the valuable high-correlation information as it does not jointly processes multi-sensors gait data [15,16,24,25]. Similar results in a study on hyperspectral image classification have been found in [45]. Besides this, from Figure 2, we also find that all JSRC-based gait-classification models result in poor accuracy when the compression ratios are more than 70%. This is because the compressed multi-sensor gait data possibly contains more redundant information, thus destroying the classification performance. Similar results in studies on gait telemonitoring have been also reported in [10,15,16,29].

In addition, we also evaluate the computational time cost corresponding to all JSRC-based models based on the different compression ratios. As illustrated in Figure 4, in comparison, our proposed method has the lowest computational cost. The main reasons for this are that the proposed novel neighboring JSRC algorithm has a maximum time complexity of only *T*<sup>1</sup> + *O*(*w*2*m*) (*T*<sup>1</sup> denotes the time for searching the *m* nearest neighbor samples) to produce the best classification performance. However, the traditional JSRC algorithm has a computational complexity of *<sup>O</sup>*(*w*2*n*) (*<sup>n</sup>* ≥ *<sup>m</sup>*, *<sup>n</sup>* is the total number of training sample data points) for multi-sensor gait performance [33,41,43]. In conclusion, all the above results show that our proposed technique has the potential ability to gain the best classification performance, lower computational time cost, and better energy efficiency of sensors in multi-sensor gait classification.

In this study, our proposed model is also employed to classify nine different multi-sensor gait patterns, in order to further examine the practicality of gait telemonitoring applications. As shown in Table 5, our proposed technique shows high-quality gait-classification performance. This suggests that our proposed technique may enforce the robustness of coefficient estimation, which helps to exactly solve the joint sparse representation coefficients. In particular, this is because our hybrid technique can take advantage of the joint sparsity of multi-sensor gait data to gain the more distinctive information associated with the spatial, temporal, and dynamic correlations of gait [13,26–28], which greatly contribute to identifying multi-sensor gait pattern change. Similar findings have been reported in [9,12,46].

#### **7. Conclusions**

In this study, an advanced hybrid technique of the DCS and JSRC models is proposed for multi-sensor gait telemonitoring. In our technique, both DCS and JSRC feasibly take advantage of data joint sparsity to jointly process multi-sensor gait data. All multi-sensor gait data jointly compressed by DCS contain the more valuable high-correlation information related to gait, and are directly used to develop the JSRC model. This can avoid the multi-sensor gait data reconstruction step and potentially increase the energy efficiency of the sensors. A novel neighboring JRSC model can be developed to gain the best gait-classification performance as well as a lower computational time. It is very hopeful that our proposed technique can serve as a powerful tool for processing multi-sensor gait dat in gait telemonitoring

applications. Future research paths are to find the effective methods that can significantly improve the classification performance of multi-sensor gait data while compression ratios are higher.

**Acknowledgments:** We are thankful for the Natural Science project funding from Fujian Province (No. JK2016006, JAT160098 and 2017Y0028) and the Humanities and Social Sciences project funding from Ministry of Education, China (No. 17YJAZH091). In addition, the authors wish to acknowledge the open Wearable Action Recognition Database (WARD) database from University of California, Berkeley.

**Author Contributions:** Jianning Wu conceived the original idea of the study, along with a corresponding author. He participated in the design of the algorithm and drafted the manuscript. Jiajing Wang, Yun Ling and Haidong Xu participated in the design of the algorithm and analyzed the data. All authors read and approved the submitted manuscript.

**Conflicts of Interest:** The authors declare that they have no competing interests.

#### **Appendix A**

In the MBCS technique, the set of unknown parameters *<sup>β</sup>* <sup>=</sup> {*βk*}*k*=1,2,··· ,*<sup>N</sup>* is the hyper-parameter for the prior distribution of *Wj*. Here, in order to obtain the best sparsity of coefficients *Wj*, the hyper-parameters *β*0, *β* are assumed to satisfy a Gamma distribution, and the prior probabilities of *β*0, *β* are defined, respectively, as

$$p(\beta\_0|a,b) = Ga(\beta\_0|a,b),\tag{A1}$$

$$p(\boldsymbol{\beta}|\mathbf{c},d) = \prod\_{k=1}^{M} \text{Ga}(\beta\_k|\mathbf{c},d),\tag{A2}$$

where *a*, *b*, *c*, and *d*, respectively, denote four different parameters in the Gamma distribution. When *a* = *b* = *c* = *d* = 0, the super prior probabilities of *β*0, *β* are available. So, the following maximum likelihood function of *β*0, *β* is defined as

$$\mathbb{E}\left\{\beta\_{\prime}\beta\_{0}\right\} = \operatorname\*{argmin}\_{j=1} \sum\_{j=1}^{J} \log \int d\mathcal{W}\_{j} p(y\_{j}|\mathcal{W}\_{j}, \beta\_{0}) p(y\_{j}|\beta). \tag{A3}$$

Once *β*0, *β* are solved, the posterior probability distribution of *Wj* is defined as

$$p(\mathcal{W}\_{\dot{\jmath}}|\mathcal{y}\_{\dot{\jmath}'}\boldsymbol{\beta}, \boldsymbol{\beta}\_0) = N(\mathcal{W}\_{\dot{\jmath}}|\boldsymbol{\mu}\_{\dot{\jmath}'}\sum\_{\dot{\jmath}}) \tag{A4}$$

where *μ<sup>j</sup>* = *β*0∑*j*(*Aj*) *<sup>T</sup>y<sup>j</sup>* is the mean, <sup>∑</sup>*<sup>j</sup>* = [*β*0(*Aj*) *<sup>T</sup>A<sup>j</sup>* <sup>+</sup> *<sup>Γ</sup>*] <sup>−</sup><sup>1</sup> is the covariance, and *Γ* = diag(*β*1, *β*2, ··· , *βN*) is the diagonal matrix.

In order to effectively estimate the parameters *β*0, *β*, the likelihood function is constructed as follows:

$$L(\boldsymbol{\beta}, \beta\_0) = \log p(\boldsymbol{y}\_j | \boldsymbol{\beta}, \beta\_0) \ = -\frac{1}{2} \sum\_{j=1}^{J} \left( N \log 2\pi + \log |\mathbf{C}\_j| + \boldsymbol{y}\_j \mathbf{C}\_j^{-1} \boldsymbol{y}\_j \right) \tag{A5}$$

where **C***<sup>j</sup>* = [*β*−<sup>1</sup> <sup>0</sup> *<sup>I</sup>* + *<sup>A</sup>jΓ*−1(*Aj*) *<sup>T</sup>*]. Usually, the derivative of *L*(*β*, *β*0) to *β*, *β*<sup>0</sup> is assumed to be zero, and the iterative rules of *β*, *β*<sup>0</sup> are respectively defined as follows:

$$\beta^{new} = \frac{J - \beta\_k \sum\_{k=1}^{J} \sum\_{j,(k,k)} }{\sum\_{j=1}^{J} \mu\_{j,k}^2},\tag{A6}$$

$$\beta\_0^{new} = \frac{\sum\_{k=1}^{l} \left(M - N + \sum\_{k=1}^{M} \beta\_j \sum\_{j,(k,k)}\right)}{\sum\_{j=1}^{l} \left| |y\_j - A\_j \mu\_j| \right|\_2^2}. \tag{A7}$$

When the optimal parameters *β*, *β*<sup>0</sup> are estimated, the solutions of the joint sparse representation coefficients *Wj J <sup>j</sup>*=<sup>1</sup> <sup>=</sup> *μj J <sup>j</sup>*=<sup>1</sup> are available.

#### **References**


© 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/).

*Article*

### **The Effect of the Accelerometer Operating Range on Biomechanical Parameters: Stride Length, Velocity, and Peak Tibial Acceleration during Running**

#### **Christian Mitschke \*, Pierre Kiesewetter and Thomas L. Milani**

Department of Human Locomotion, Chemnitz University of Technology, 09126 Chemnitz, Germany; pierre.kiesewetter@hsw.tu-chemnitz.de (P.K.); thomas.milani@hsw.tu-chemnitz.de (T.L.M.) **\*** Correspondence: christian.mitschke@hsw.tu-chemnitz.de; Tel.: +49-371-531-32196

Received: 11 November 2017; Accepted: 4 January 2018; Published: 5 January 2018

**Abstract:** Previous studies have used accelerometers with various operating ranges (ORs) when measuring biomechanical parameters. However, it is still unclear whether ORs influence the accuracy of running parameters, and whether the different stiffnesses of footwear midsoles influence this accuracy. The purpose of the present study was to systematically investigate the influence of OR on the accuracy of stride length, running velocity, and on peak tibial acceleration. Twenty-one recreational heel strike runners ran on a 15-m indoor track at self-selected running speeds in three footwear conditions (low to high midsole stiffness). Runners were equipped with an inertial measurement unit (IMU) affixed to the heel cup of the right shoe and with a uniaxial accelerometer at the right tibia. Accelerometers (at the tibia and included in the IMU) with a high OR of ±70 g were used as the reference and the data were cut at ±32, ±16, and at ±8 g in post-processing, before calculating parameters. The results show that the OR influenced the outcomes of all investigated parameters, which were not influenced by tested footwear conditions. The lower ORs were associated with an underestimation error for all biomechanical parameters, which increased noticeably with a decreasing OR. It can be concluded that accelerometers with a minimum OR of ±32 g should be used to avoid inaccurate measurements.

**Keywords:** operating range; accelerometer; stride length; peak tibial acceleration; running velocity; wearable sensors

#### **1. Introduction**

With the further advancement of sensor technologies and data analyzing techniques, micro-electro-mechanical sensors (MEMS) have become useful tools for biomechanical research and clinical practice [1]. It has been reported that these wearable motion sensors are an inexpensive alternative to optoelectronic systems and force plates, that they are simple to handle, cost-effective, and are suitable for field measurements [2–6]. When using these sensors, spatio-temporal and kinetic parameters during walking and running can be analyzed in clinical as well as in sportive applications. For kinetic measurements, for example, when investigating the impact loads on lower limbs during running under various conditions (e.g., footwear conditions or the influence of fatigue), the peak tibial acceleration (PTA) was examined by some authors using MEMS [7–20]. Thereby, unidirectional accelerations along the longitudinal axis of the tibia, as well as medio-lateral and anterior-posterior accelerations of the tibia were examined. Furthermore, the determination of stride frequency, of walking or running velocity (runVel), and of stride length (strLen) have also been the focus of research that utilizes MEMS [21–25]. To investigate these biomechanical parameters, individually configured sensors or commercially available inertial measurement units (IMUs: e.g., Shimmer, Achillex, or XSens) were used, which combine accelerometers and gyroscopes. In this

context, Provot et al. [26] compared a calibrated industrial accelerometer (considered as the gold standard) to an IMU accelerometer in two tests: (a) on a shaker, and (b) on the distal anteromedial aspect of the subject's tibia during running at 3.33 m/s. They concluded that IMUs can be used for valid measurements of tibial acceleration during running.

Besides the different sensor types, sensor locations, and the various sampling rates, sensors with considerably different accelerometer operating ranges (ORs) have also been used. When investigating walking or running, some studies used ORs between ±2 and ±70 g, with g being the acceleration of gravity [27–32]. However, high accelerations act on the sensor in vertical and anterior–posterior directions during the swing phase and in the first 50 ms after foot strike [11,32]. These accelerations can distinctly exceed the gravitational acceleration of 1 g. For instance, vertical accelerations of 24.62 ± 4.1 g were measured with a heel-mounted IMU accelerometer during heel strike, when running at 3.5 ± 0.1 m/s in a neutral running shoe (PUMA FAAS 500) [32]. If the accelerations exceeded the accelerometer OR, a lower accuracy and an underestimated variability of biomechanical parameters derived from accelerometer signals could result. In this context, Ziebart et al. [33] investigated the influence of accelerometer OR and of a sampling rate on peak acceleration during seven jumping tasks. They used a tri-axial accelerometer (Model 7267A, Endevco Corporation, San Juan Capistrano, CA, USA) with a high OR of ±260 g as the reference and compared the peak accelerations with two commercially available tri-axial accelerometers with an OR of ±6 g (device1: ActiGraph GT3X+, ActiGraph LLC, Pensacola, FL, USA and device2: X6-2mini, Gulf Coast Data Concepts, Waveland, MS, USA). They found that the peak impact acceleration was underestimated by up to 35%. Furthermore, the underestimation error was greater for tasks with a greater impact acceleration.

It is currently still unclear whether, and to what extent, high accelerations can influence the accuracy of running parameters when exceeding the accelerometers' OR. However, this information is necessary to determine whether differences in the investigated biomechanical parameters strLen, runVel, and in the PTA between conditions are caused by measurement errors due to an accelerometer OR that is too low, or by the investigated conditions themselves.

In addition, the sensor signal characteristics of the IMU accelerometer, which is affixed to the heel cup of a running shoe, can be influenced by the midsole stiffness of footwear [5]. The authors found that a decreasing midsole stiffness resulted in an increasing delay in specific signal characteristics, when determining the time of foot touchdown. At this time, it is still unclear whether the accuracies of strLen and runVel are also influenced by altering acceleration signal characteristics due to a change in midsole stiffness.

Since acceleration variables are related to running injuries, information about whether the ORs influences running parameters is highly relevant and could assist coaches, researchers, and clinicians in selecting the most appropriate accelerometer specification for their investigations.

Therefore, the aim of the present study was to investigate the influence of the accelerometer OR on running parameters when reducing the OR stepwise from ±70 g to ±8 g. We hypothesized that the rapid and short spikes at the beginning of ground contact and the high accelerations during the swing phase influenced the accuracy of strLen (H1), runVel (H2), and PTA (H3) significantly, depending on the footwear conditions. The running parameters were determined based on previously published methods, and accelerometers with a high OR of ±70 g were used as the reference.

#### **2. Materials and Methods**

#### *2.1. Participants*

Twenty-one recreational male heel strike runners (age: 24.4 ± 4.2 years; height: 178.2 ± 4.0 cm; weight: 74.1 ± 6.5 kg; running experience: 8.9 ± 3.3 years; training hours: 3.5 ± 1.7 h per week), free of injury for the last six months, participated in this study. High force rising rates can be observed after foot touchdown during rearfoot running [34], which results in a higher PTA for rearfoot runners than for forefoot runners [11]. Therefore, to determine the greatest effect of the reduced OR, only heel strike runners were investigated. This study was approved by the university's Ethics Committee (V-103-17-HS-CM-Bodenkontakt-25082015), and participants gave written informed consent to their participation in the study.

#### *2.2. Test Procedure*

During the test, participants wore three different commercially available running shoes, which were provided by our laboratory in men's UK size 8: Adidas AdiStar (ADIDAS); PUMA Speed 600 (PUMA); Asics Gel Nimbus 12 (ASICS). After an individual warm-up and familiarization with the measurement setup, five repeated trials were recorded on a 15-m indoor track in the three footwear conditions. The footwear conditions were investigated in a randomized order. The running speed was individual and self-selected (on average 3.60 ± 0.4 m/s), however it was constant (range: ±0.1 m/s) for each subject, for all of the 15 trials (three footwear conditions, each having five trials). Running speed was monitored using two light barriers placed 4 meters apart.

#### *2.3. Experimental Setup*

An individually configured IMU, combining a biaxial accelerometer (ADXL278, Analog Devices, OR ±70 g) and a biaxial gyroscope (IDG-650, InvenSense, OR ±2000 deg/s), was affixed to the heel cup of the right shoe (Figure 1). Wobble-free sensor fixation was achieved using double-sided adhesive tape and an inelastic strap. The sensitive axes of the IMU accelerometer measured the horizontal—forward direction—(acc\_x) and the vertical (acc\_z) acceleration of the shoe. The two sensitive axes of the gyroscope measured the angular velocities in the sagittal (ω1), and the frontal planes. Furthermore, to measure accelerations along the longitudinal axis of the tibia (acc\_T), double-sided adhesive tape was used to attach a uniaxial lightweight accelerometer (ADXL78, Analog Devices, OR ±70 g) to the shaved skin at the medial aspect mid-distance between the malleolus and the plateau of the right tibia [12,35]. An elastic strap was used to stabilize the accelerometer and to prevent excessive movements due to its own weight. All sensors operated synchronously, and the data were transmitted by cable to a data logger, which recorded these data. The data logger was secured in a waist belt. To avoid any effects from a sampling rate that was too low, the sensor sampling rate was set to 1000 Hz, which exceeds the minimum requirements to measure these parameters accurately [32].

**Figure 1.** A schematic representation of the sensor setup. Sensitive axes of the accelerometer in the heel cup inertial measurement unit are represented by arrows: horizontal forward direction (acc\_x) and vertical acceleration (acc\_z). The angular velocity of the shoe was measured in the sagittal (ω1) and frontal plane using a gyroscope. The accelerometer, located at the tibia, measured accelerations along the longitudinal axis of the tibia (acc\_T).

#### *2.4. Data Analyses*

Data from the sensors were analyzed in post-processing using MATLAB R2016b (MathWorksTM, Natick, MA, USA). Prior to all of the processing steps, the data were filtered using a zero-lag Butterworth low-pass filter (accelerometers: 4th order at 200 Hz; gyroscopes: 4th order at 50 Hz) to remove noise. In the next step, accelerations measured with an OR of ±70 g were cut at ±32, ±16, and at ±8 g when exceeding their respective thresholds. After the data cutting, running parameters were calculated for all four ORs (±70, ±32, ±16, and ±8 g) separately, as described below.

To separate the strides of the right foot in continuous data, the accelerometer signal of the IMU at the heel cup (vertical axis) was 80 Hz zero-lag high pass filtered and the first peak in the filtered signal was defined as the foot touchdown [5,36].

For each stride, the orientation angle of the shoe in the sagittal plane (θ) was calculated using integrated data from ω1. A nulling algorithm was then applied to the orientation angle of the shoe to eliminate errors resulting from integration offset and from drift error [32]. The time of two consecutive flat shoe phases (flat relative to the ground) was detected by finding the lowest absolute angular velocity in ω1 for each stride. The shoe orientation angles at these reference times were used in a linear drift model to eliminate the drift error and the integration offset. Based on Sabatini et al. [21] and Mitschke et al. [32], acc\_x, acc\_z, and the orientation angle θ were used to calculate the resulting horizontal acceleration (acc\_hor) of the right shoe (see Equation (1); Figure 1).

$$\mathbf{acc\\_hor} = \mathbf{acc\\_x} \cdot \cos(\theta) - \mathbf{acc\\_z} \cdot \sin(\theta) \tag{1}$$

Subsequently, the horizontal velocity of the shoe (v\_x) was calculated using a numerical integration of acc\_hor. To eliminate the drift error and the integration offset of the shoe velocity, the velocity of the shoe during the flat shoe phases was assumed to be temporarily equal to zero [37]. Therefore, velocity was reset to zero as the stride start condition and for error back-propagation with a linear drift model [32,38]. The mean running velocity (runVel) was calculated by averaging v\_x over time between two consecutive flat shoe phases. Furthermore, the stride length (strLen) between two consecutive flat shoe phases was calculated using integrated data from v\_x.

For kinetics, the positive peak tibial acceleration (PTA) was defined as the maximum acceleration value of the accelerometer at the tibia (Figure 1: acc\_T) [12,35].

#### *2.5. Footwear Conditions*

To quantify the shoe midsole stiffness, shoes were tested in a servo-hydraulic testing device (HC10; Zwick GmbH & Co. KG; Ulm, Germany) as described in Schwanitz and Odenwald [39]. Each footwear condition underwent a total of ten load tests of 103 load cycles each. Load cycles were applied to the heel of each running shoe by a spherically shaped stamp (50 mm). The load-time profile was derived from biomechanical measurements of ground reaction force while running at a velocity of 3.5 ± 0.1 m/s [40]. For each footwear condition, midsole deformation at the 101st load cycle was analyzed for each of the ten test sessions. Internal studies have shown that the cycles after the 100th cycle represent reliable results when testing shoes mechanically [41]. The sampling rate was 1000 frames per second, and the data were analyzed using MATLAB R2016b (MathWorksTM, Natick, MA, USA) in post-processing. The midsole stiffness in the rearfoot area between 1000 and 1500 N was calculated for each footwear condition using Equation (2).

$$\text{stiffness} = \frac{1500 \text{ N} - 1000 \text{ N}}{\text{deformation}\_{\text{F}=1500 \text{ N}} - \text{deformation}\_{\text{F}=1000 \text{ N}}} \tag{2}$$

#### *2.6. Statistical Analyses*

To compare footwear stiffness, the mean and standard deviations (mean ± SD) were calculated. Given that parameters were normally distributed, a one-way analysis of variance (ANOVA) followed

by Bonferroni post hoc tests were used to determine whether differences existed between mechanical footwear characteristics. The level of significance was set to an alpha of 0.05. Furthermore, the means of the five trials for each subject and for each footwear condition were used to calculate the group means and the standard deviations (mean ± SD), and the 95% confidence intervals (95% CI) for strLen, runVel, PTA, and all ORs. Due to normal distribution, paired sample t tests were used to compare the reference (±70 g) with the three lower ORs (±32, ±16, and ±8 g) for strLen and runVel for each of the three footwear conditions. In addition, Wilcoxon tests were used to compare the reference with the lower OR for PTA for each footwear condition. A Bonferroni correction was used to adjust the *p*-values (*p* = 0.05/3/3 = 0.006) in relation to the number of comparisons between the investigated OR (*n* = 3) and the investigated biomechanical parameters (*n* = 3). The effect size (Cohen's *d*) was calculated to quantify the magnitude of differences between the shoe characteristics and the magnitude of differences between the biomechanical parameters examined with different ORs. The coefficients were interpreted as a trivial effect (*d* < 0.2), a small effect (*d* < 0.5), a medium effect (*d* < 0.8), and a large effect (*d* ≥ 0.8) [42]. Additionally, mean differences (MDs), relative mean differences in percent (MD\_rel), and root mean square errors (RMSEs) were calculated between the reference OR and each of the lower ORs.

#### **3. Results**

#### *3.1. Footwear Characteristics*

All pair-wise comparisons reached statistical significance (*p* < 0.001). Cohen's *d* showed large effects (*d* ≥ 0.8) in all footwear conditions. The lowest rearfoot stiffness was found for ASICS (156.9 ± 0.1 N/mm) and the highest stiffness was found for ADIDAS (210.3 ± 0.4 N/mm) (Table 1).

**Table 1.** Group mean ± standard deviation (SD) for rearfoot stiffness of the three footwear conditions; statistical significance (*p* < 0.05) and large effects (*d* ≥ 0.8) were found in all footwear conditions.


#### *3.2. Biomechanical Parameters: Stride Length, Running Velocity, and Peak Tibial Acceleration*

For each footwear condition, results of the strLen and the runVel calculations, as well as the PTA values are represented in Figures 2–4 for the reference OR and for the lower ORs.

When comparing the OR of ±70 g and ±32 g, no significant differences were found for runVel for all three footwear conditions (Table 2). Distinctly greater differences to the reference were found using the ORs of ±16 g and ±8 g. Setting the OR to ±16 g resulted in significant differences, up to 3.48% (ADIDAS: MD: 0.13 m/s; *p* < 0.001), and large effects (*d* ≥ 0.86) for all three footwear conditions. When using an OR of ±8 g, a significantly lower runVel of up to 9.68% (ADIDAS: MD: 0.36 m/s; *p* < 0.001) was found.

When comparing strLen, calculated using ORs of ±70 g and ±32 g, no significant effects were found for ADIDAS, PUMA, or ASICS (Table 2). When using sensors with a lower OR, strLen calculations resulted in significant differences for all footwear conditions (MD > 2.62 cm; MD\_rel > 0.99%; *p* < 0.001), showing large effects for all comparisons (*d* > 0.84). The highest strLen difference between the reduced OR and the reference was found for Adidas. An OR of ±8 g resulted in a 9.68% lower stride length in comparison to the reference OR (MD: 26.15 cm; *p* < 0.001).

No significant differences were found for PTA when comparing an OR of ±70 g and ±32 g, and of ±70 g and ±16 g for all footwear conditions (Table 2). The comparison of the OR ±70 g

and ±8 g revealed significant differences for all footwear conditions of at least 9.76% on average (ASICS: MD: 0.61 g), which resulted in a medium effect (*d* = 0.60). The greatest difference between the OR of ±70 g and ±8 g was found for ADIDAS (MD: 2.65 g; MD\_rel: 28.17%; *p* = 0.002; *d* = 0.65). In general, we found that with higher midsole stiffnesses, the differences increased between the reference OR and ±16 g, as well as between the reference OR and ±8 g (Figure 4 and Table 2).

**Figure 2.** A boxplot of the three footwear conditions for running velocity (runVel), calculated with different accelerometer operating ranges of ±70, ±32, ±16, and ±8 g. Significant differences between stride lengths are marked with \*\* (*p* < 0.001).

**Figure 3.** A boxplot of the three footwear conditions for stride length (strLen), calculated with different accelerometer operating ranges of ±70, ±32, ±16, and ±8 g. Significant differences between running velocities are marked with \*\* (*p* < 0.001).

**Figure 4.** A boxplot of the three footwear conditions for the positive peak tibial acceleration (PTA), determined using different accelerometer operating ranges of ±70, ±32, ±16, and ±8 g. Significant differences between peak tibial accelerations are marked with \* (*p* < 0.006) and \*\* (*p* < 0.001).

**Table 2.** Mean differences (MDs), relative mean differences (MD\_rel), *p*-values of pair-wise comparisons, effect size (Cohen's *d*), and root mean square errors (RMSEs) between stride lengths (strLen), running velocities (runVel), and PTA, each determined using different accelerometer operating ranges of ±70, ±32, ±16, and ±8 g.


#### **4. Discussion**

The aim of the present study was to investigate the influence of the accelerometer OR on stride length, running velocity, and on peak tibial acceleration when reducing the OR stepwise from ±70 g to ±32, ±16, and ±8 g. Biomechanical parameters were determined based on previously published methods, and accelerometers (attached at the heel cup and at the tibia) with a high OR of ±70 g were used as the reference.

The results of this study revealed that OR influences the outcomes of stride length, running velocity, and peak tibial acceleration, which were not dependent on tested footwear conditions. Lower ORs were associated with an underestimation error for all biomechanical parameters, which increased with decreasing OR. These results confirm hypotheses H1 (strLen), H2 (runVel), and H3 (PTA). Our results show that a sensor OR needs to be carefully considered when interpreting biomechanical parameters of existing investigations and when planning future studies.

Using accelerometers with an OR of ±32 g resulted in small errors, on average of up to 0.14% for strLen and 0.14% for runVel (ADIDAS). However, insignificant differences and low RMSE values were found for all footwear conditions, and therefore, the observed differences can be considered irrelevant. The accelerations during running that were measured in our study did not critically exceed the ±32 g threshold. However, our results show that when the sensor OR was limited to ±16 g or ±8 g, distinctly higher differences, when compared to the reference, were found for runVel and strLen. Due to the lower vertical and horizontal forward accelerations (set to ±16 g and ±8 g when exceeding the respective thresholds), the resulting horizontal forward velocity of the shoe (v\_x) was calculated inaccurately when using the numerical integration of acc\_hor. Figure 5 shows v\_x for one representative stride. When observing the velocity curve progression, it seems there was a drift error after the numerical integration (frame 120 to 520), which would explain the significantly lower runVel for the OR of ±16 g and ±8 g when compared to the reference. However, to eliminate any drift error and integration offset, the shoe velocity during the flat shoe phases was assumed to be temporarily equal to zero [37]. Therefore, the velocity was reset to zero as the stride start condition and for error back-propagation with a linear drift model [32,38]. As shown in Figure 5, the horizontal forward velocity of the foot was reset to zero in both of the flat shoe phases and no additional offset was observed for these phases. We presume, therefore, that the lower velocities were the result of ORs which were too low and were not due to a drift error.

Additionally, the results show that the stiffnesses of the tested footwear conditions do not appear to noticeably influence the accuracy of runVel. Due to the high impact loads during rearfoot running, we expected that the footwear with the highest midsole stiffness in the rearfoot area (ADIDAS) would show distinctly greater differences between ±70 g and the lower ORs, than the footwear with the lowest stiffness (ASICS). However, whereas ADIDAS revealed the highest differences between the reference OR and the reduced ORs, the ASICS shoe showed higher differences between the reference OR and the reduced ORs than PUMA (medium stiffness) did. Furthermore, similar significant differences (*p* < 0.001) and effect sizes between the reference OR and the OR of ±16 g and ±8 g were found for all of the footwear conditions. Therefore, due to the high accelerations which occurred during the heel strike while running in footwear with differing midsole stiffnesses, sensors with an OR greater than ±32 g should be used to measure these accelerations accurately and to calculate runVel with high accuracy (errors lower than 0.14%).

Despite the different acceleration magnitudes which occurred during heel strike for the three footwear conditions, the calculated velocities (OR: ±70 g) were comparable using the measurements taken by the light barriers (3.60 ± 0.4 m/s): ADIDAS: 3.70 ± 0.3 m/s; PUMA: 3.66 ± 0.4 m/s; ASICS: 3.60 ± 0.3 m/s. Therefore, the method for calculating runVel can be used when the OR is >±32 g.

For strLen, effects due to the lower OR were also observed. StrLen was calculated between two consecutive flat shoe phases using the integrated data from v\_x. As expected, lower velocities for the reduced ORs also resulted in shorter strLen. The values of MD\_rel and the significant differences between the reference and the reduced ORs, including effect sizes, were similar for strLen and runVel. This is the result of the numerical integration of v\_x between two consecutive flat shoe phases when calculating strLen. Additionally, tested footwear conditions did not influence strLen calculations when using an OR of ±32 g. It seems that the critical accelerations in anterior–posterior and in vertical directions do not critically exceed the OR of ±32 g during heel strike, regardless of the midsole stiffness. However, using an OR of ±16 g can lead to strLen differences of 3.49% (MD: 9.43 cm) for each stride when running in the ADIDAS shoe. In contrast, when running in the PUMA shoe, differences of

only 0.99% (MD: 2.62 cm) were found. With a further decrease in the OR, the differences between the calculated strLen of PUMA and of ADIDAS increased. When investigating mechanical midsole characteristics of footwear in biomechanical tests (e.g., benchmark tests) using an OR that is too low (<±32 g), errors in the strLen calculation can lead to incorrect conclusions. Therefore, to avoid effects of an OR that is too low, an OR of at least ±32 g should be used to measure strLen as accurately as possible and independently from the midsole stiffness of footwear.

**Figure 5.** The horizontal velocity (forward direction) of the shoe for one representative stride with an accelerometer operating range of ±70 g (black line), ±16 g (grey dotted line), and of ±8 g (grey line), subdivided in swing phase and foot contact (until toe off, exceeding 2 g in the horizontal acceleration signal [43] and after the foot touchdown [5,36]).

For PTA, none of the subjects in this study generated higher peak tibial accelerations than ±32 g when running in the three provided footwear conditions (MD\_rel: 0.0%; RMSE: 0.0 g). However, faster running velocities (e.g., sprint runs), running in shoes with a higher midsole stiffness (e.g., barefoot or minimal shoes), or running barefoot may increase peak tibial acceleration [10,15,44]. In some cases, accelerations could then exceed 32 g and may lead to inaccurate results when measuring with an OR that is too low. Our results for PTA show that accuracy decreased up to 28.17% (ADIDAS: ±8 g) with lower OR. Thereby, we found that a higher midsole stiffness generally decreased the accuracy of PTA. The footwear condition with the stiffest midsole (ADIDAS) showed the greatest PTA for the reference OR (mean ± SD: 9.41 ± 5.3 g), whereas the footwear with the softest midsole configuration (ASICS) showed the lowest PTA (mean ± SD: 6.28 ± 3.0 g). When observing the mean values of PTA for each footwear condition, the minimum OR can be derived from these results.

To accurately determine PTA, a distinctly lower OR (≥±16 g) can be used in contrast to the strLen and runVel (≥±32 g). Due to various influencing factors (e.g., heel fat pad cushioning, foot pronation, shoe midsole deformation, skin displacement, and sensor wobbling), measured accelerations at the tibia were damped and were attenuated in contrast to the wobble-free sensor at the heel cup. Therefore, a distinctly lower OR is necessary for the accelerometer placed at the tibia to determine PTA without a significant loss of information. However, different locations of the accelerometers on the tibia can also influence the temporal and spectral parameters of PTA. Lucas-Cuevas et al. [45] found that a distally placed accelerometer measured a greater PTA and shock attenuation compared to a proximal accelerometer. In our study, the accelerometer was placed at the medial aspect mid-distance between the malleolus and plateau of the right tibia according to Henning et al. [35] and Milani et al. [12]. Therefore, a more distal sensor attachment may require a greater OR than was recommended in our study.

#### **5. Conclusions**

In literature, various accelerometers with ORs between ±2 g and ±70 g have been used to determine biomechanical parameters, although some studies did not state the OR used. In the present study, the influence of OR on biomechanical parameters was investigated. Our results are important for coaches, researchers, and clinicians in selecting the best accelerometer specification for their future investigations. Our study shows that an accelerometer OR that is set too low results in an inaccurate determination of strLen, runVel, and PTA. Especially in prolonged runs, when investigating a large number of continuous strides (e.g., a marathon [24]), an OR that is too low leads to an underestimation of strLen for each stride, which results in great differences in the total distance. Furthermore, when investigating strLen or PTA variability during prolonged runs in the context of fatigue, a low PTA or strLen variability can be misinterpreted due to an OR that is too low. Additionally, the midsole stiffnesses of the footwear conditions did not influence the accuracy of biomechanical parameters when the OR was ≥±32 g. Furthermore, significant differences and effect sizes between the reference OR and the reduced ORs (±16 g and ±8 g) were similar for all footwear conditions. However, our results show that when investigating mechanical footwear characteristics in biomechanical tests (e.g., benchmark tests), an OR that is too low (for strLen and runVel < ±32 g, for PTA < ±16 g) can lead to an approximation of biomechanical parameters, which leads to incorrect conclusions. Finally, when examining biomechanical parameters during running using accelerometers with an OR lower than ±32 g, these results should be considered carefully.

**Acknowledgments:** The authors would like to thank Lisa Peterson for proofreading the paper as a native speaker. This research was supported by Puma® Inc., Germany. The publication costs of this article were funded by the German Research Foundation/DFG and the Technische Universität Chemnitz, Chemnitz, Germany, through the funding program Open Access Publishing.

**Author Contributions:** Christian Mitschke, Pierre Kiesewetter, and Thomas L. Milani conceived and designed the experiments; Christian Mitschke and Pierre Kiesewetter performed the experiments; Christian Mitschke, Pierre Kiesewetter, and Thomas L. Milani analyzed the data; Thomas L. Milani contributed reagents/materials/analysis tools; Christian Mitschke, Pierre Kiesewetter, and Thomas L. Milani wrote the paper.

**Conflicts of Interest:** The authors declare no conflicts of interest. The funding sponsors had no role in the design of the study, in the collection, analyses, or in the interpretation of data, in the writing of the manuscript, or in the decision to publish the results.

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