A Novel Zero Velocity Interval Detection Algorithm

sensors Article

A Novel Zero Velocity Interval Detection Algorithm for Self-Contained Pedestrian Navigation System with Inertial Sensors Xiaochun Tian, Jiabin Chen, Yongqiang Han *, Jianyu Shang and Nan Li School of Automation, Beijing Institute of Technology, Beijing 100081, China; [email protected] (X.T.); [email protected] (J.C.); [email protected] (J.S.); [email protected] (N.L.) * Correspondence: [email protected]; Tel.: +86-10-6891-3431 Academic Editor: Jörg F. Wagner Received: 14 July 2016; Accepted: 21 September 2016; Published: 24 September 2016

Abstract: Zero velocity update (ZUPT) plays an important role in pedestrian navigation algorithms with the premise that the zero velocity interval (ZVI) should be detected accurately and effectively. A novel adaptive ZVI detection algorithm based on a smoothed pseudo Wigner–Ville distribution to remove multiple frequencies intelligently (SPWVD-RMFI) is proposed in this paper. The novel algorithm adopts the SPWVD-RMFI method to extract the pedestrian gait frequency and to calculate the optimal ZVI detection threshold in real time by establishing the function relationships between the thresholds and the gait frequency; then, the adaptive adjustment of thresholds with gait frequency is realized and improves the ZVI detection precision. To put it into practice, a ZVI detection experiment is carried out; the result shows that compared with the traditional fixed threshold ZVI detection method, the adaptive ZVI detection algorithm can effectively reduce the false and missed detection rate of ZVI; this indicates that the novel algorithm has high detection precision and good robustness. Furthermore, pedestrian trajectory positioning experiments at different walking speeds are carried out to evaluate the influence of the novel algorithm on positioning precision. The results show that the ZVI detected by the adaptive ZVI detection algorithm for pedestrian trajectory calculation can achieve better performance. Keywords: pedestrian navigation system (PNS); adaptive ZVI detection; SPWVD; gait frequency; ZUPT

1. Introduction Pedestrian navigation, having received more and more attention from researchers in recent years, is an important branch in the field of navigation. PNS can help personnel in missions to identify their positions in real time and to get in touch with the command center; thus the safety of the emergency rescue personnel can be greatly guaranteed in an unknown environment. In addition, in places such as an airport, theatre, underground parking, other large public places and modern cities with tall buildings, it is necessary for pedestrians to identify their positions and find targets. At present, technologies suitable for pedestrian navigation can be divided into positioning technology based on the Global Navigation Satellite System (GNSS), positioning technology based on the radio frequency (RF) signal, as well as positioning technology based on self-contained sensors. The first kind is relatively mature, but the positioning accuracy is affected in indoor environments or outdoor environments with tall buildings and trees. The PNS based on the RF signal (such as RFID [1], UWB [2], etc.) requires pre-installing signal transmitting equipment in the positioning area, which is costly and has limited application range. Compared with the other two positioning technologies, PNS based on self-contained sensors has the advantages of strong autonomy and independence because it mainly adopts sensors, such as accelerometers, gyroscopes and magnetometers, to calculate pedestrian

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position information. Considering the limited load-carrying ability of pedestrians, in this paper, self-contained sensors are adopted to design PNS by installing them on the pedestrian’s foot, and the pedestrian’s position is obtained by integrating the inertial sensors output. However, low-cost inertial sensors have low performance, and the acceleration error produces a position error that grows cubically over time [3]. Therefore, it is very important to eliminate the error accumulation of PNS for the improvement of positioning accuracy. During pedestrian’s walking process, there is a period (about 0.3–0.4 s [4]) when the sole fully contacts with ground and the foot velocity approximates zero. This period is usually called ZVI. According to the periodic existence of ZVI in pedestrian gait, ZUPT can be adopted for clearing the position error periodically, where the core of ZUPT is accurately detecting the ZVI in the pedestrian gait. Many scholars have carried out research on ZVI detection methods; for example Zhou et al. [5] used a shoe-mounted radar to detect the ZVI; Bebek et al. [6] measured the ZVI by adopting a high-resolution pressure sensor for assistance; Zhou et al. [7] detected the ZVI by embedding an RF sensor in the pedestrian’s shoe. Generally speaking, the above-mentioned methods can better detect the ZVI, but they all need additional sensors as assistance to realize the ZVI detection, adding the cost of a PNS. In addition, the PNS based on self-contained sensors can only use the output of accelerometers and gyroscopes to detect the ZVI without the assistance of external sensors. Commonly-used ZVI detection methods for PNS with self-contained sensors include the acceleration magnitude method [8], the angular velocity magnitude method [9–12], the moving variance method [13], a combination of the above methods [14–16], and so on; all of these methods have a common characteristic that the ZVI is detected by setting a threshold. The thresholds in the traditional ZVI detection methods take on fixed values. When a pedestrian walks at a constant gait frequency, the ZVI can be accurately detected by setting a fixed threshold; however, actually, pedestrians move randomly and cannot maintain a constant gait frequency all of the time. Previous research work has found out that different gait frequencies corresponds to different optimal thresholds. Therefore, when a pedestrian walks at a changing gait frequency, the adoption of a fixed threshold leads to missed or false detection in the ZVI detection results. In order to solve this problem, a novel adaptive ZVI detection algorithm based on SPWVD-RMFI is proposed in this paper, and the function relationships between the optimal ZVI detection thresholds and the gait frequency are established experimentally. During the walking process, the gait frequency is extracted in real time by adopting SPWVD-RMFI, and in the meantime, by using the function relationships between the thresholds and the gait frequency, the optimal thresholds corresponding to the current gait frequency can be obtained; thus, the adaptive adjustment of thresholds with gait frequency is realized, and the purpose of accurately detecting ZVI is achieved. The remainder of this paper is organized as follows: the architecture of the pedestrian navigation algorithm is described in Section 2; the characteristics of pedestrian gait are analyzed in Section 3, and an adaptive ZVI detection algorithm based on SPWVD-RMFI is designed, as well; the contrast experiments of ZVI detection, as well as pedestrian trajectory positioning experiments are carried out in Section 4. In the meantime, the performance of the proposed algorithm in this paper is assessed; in the last section the research results provided by this paper are summarized. 2. Pedestrian Navigation Algorithm Architecture In this paper, the PNS algorithm based on self-contained sensors contains the strapdown inertial navigation algorithm module, the adaptive ZVI detection module and the error estimation module. The algorithm architecture is shown in Figure 1.

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Accelerometer

a

Corrected state Strapdown Inertial

Gyroscope

Gait Frequency



Navigation Algorithm

Zero Velocity Detection

Adaptive Threshold Adaptive ZVI Detection

ZUPT

Kalman Filter Error estimation

Figure1.1.Pedestrian Pedestriannavigation navigationalgorithm algorithmarchitecture. architecture. Figure

2.1.Strapdown StrapdownInertial InertialNavigation NavigationAlgorithm AlgorithmModule Module 2.1. Thestrapdown strapdown inertial navigation algorithm of PNS is similar to the traditional strapdown The inertial navigation algorithm of PNS is similar to the traditional strapdown inertial inertial navigation algorithm. The navigation calculation process mainly includes attitude update, navigation algorithm. The navigation calculation process mainly includes attitude update, velocity velocity update and position update. The navigation coordinate system (n-frame) is defined as update and position update. The navigation coordinate system (n-frame) is defined as north-east-down, north-east-down, is local consistent with the local geographic system. Inrange general, the which is consistentwhich with the geographic coordinate system. Incoordinate general, the motion of the motion range of the pedestrian is relatively small, so the real-time position and trajectory will be pedestrian is relatively small, so the real-time position and trajectory will be shown in the tangential shown in the tangential frame in this paper. After collecting the IMU output data, the angular rate frame in this paper. After collecting the IMU output data, the angular rate can be used to update the can be used to and update the attitude matrix and calculate the attitude angle. the On specific this basis, transforming attitude matrix calculate the attitude angle. On this basis, transforming force measured the specific force measured in the body coordinate system (b-frame) to the navigation coordinate in the body coordinate system (b-frame) to the navigation coordinate frame and removing the harmful frame and removing the harmful acceleration, then the speed information can beonce, obtained by acceleration, then the speed information can be obtained by integrating the acceleration and the integrating the acceleration once, and the position information can be obtained by integrating the position information can be obtained by integrating the acceleration again. The basic equations of the acceleration again.navigation The basic system equations of the inertial navigation system (SINS) can be strapdown inertial (SINS) canstrapdown be expressed as follows: expressed as follows:   .n n ωb × Cb = n C n b  Cb b Cb nb nb   .n
n f b − 2ω n + ω n (1) V = C × V n + gn n n b n n en  ie b (1)  Cb f   2ie   en   V n  g n .n V n P =n V P  V n where the n-frame and b stands for the b-frame, Cbn is the attitude transformation matrix wheren represents n represents the n-frame and b stands for the b-frame, C bn is the attitude transformation b represents the turn rate from the b-frame to the n-frame, V is the velocity and P is the position. ωnb b V P is the velocity and is the position. the matrix from the b-frame to the n-frame, b n nb represents of the b-frame with respect to the n-frame; f is the specific force; ωie is the Earthrotation angular b n n n turnωrate of the b-frame with respect to the n-frame; f is the specific force;  ie is thegEarth rate; and is therotation Earth en is the rotation angular rate of the n-frame with respect to the Earth frame; n is the rotation angular rate of the n-frame with respect to the Earth frame; and angular rate;  e nvector, which approximates a constant within a range on Earth. gravitational field g n is the Earth gravitational field vector, which approximates a constant within a range on Earth. 2.2. Adaptive ZVI Detection Module

2.2. The Adaptive ZVI ZVI Detection Module adaptive detection module is used to detect the zero velocity information in the pedestrian gait and to provide a trigger condition for the Kalman filter. According to the characteristics that The adaptive ZVI detection module is used to detect the zero velocity information in the the pedestrian’s walking speed changes over time, an adaptive ZVI detection algorithm based on pedestrian gait and to provide a trigger condition for the Kalman filter. According to the SPWVD-RMFI is proposed in this paper. The novel algorithm firstly uses the SPWVD-RMFI method to characteristics that the pedestrian’s walking speed changes over time, an adaptive ZVI detection extract the gait frequency by processing the gyroscope output, and then, the function relationships algorithm based on SPWVD-RMFI is proposed in this paper. The novel algorithm firstly uses the between the ZVI detection thresholds and the gait frequency are established to calculate the optimal SPWVD-RMFI method to extract the gait frequency by processing the gyroscope output, and then, ZVI detection thresholds in real time. On this basis, the ZVI at different walking speeds can be detected the function relationships between the ZVI detection thresholds and the gait frequency are adaptively and accurately. Detailed research contents are described in Section 3. established to calculate the optimal ZVI detection thresholds in real time. On this basis, the ZVI at different walking speeds can be detected adaptively and accurately. Detailed research contents are described in Section 3.

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2.3. Error Estimation Module As low-cost inertial sensors have a large bias, the navigation parameters obtained by the SINS algorithm often contain error terms. The error estimation module uses the ZUPT to assist the Kalman filter to estimate the system state errors. The error state vectors are defined as follows: δx =

h

δφn

δV n

δPn

εb

i

∇b

(2)

where δφn is the attitude error vector, δV n is the velocity error vector, δPn is the position error vector, εb is the gyroscope bias error vector and ∇b is the accelerometer bias error vector. The discrete system state equation is: δxk + 1 = Φk δxk + wk

(3)

where δxk is the system state at time k, Φk is the state transition matrix and wk is the process noise. In this study, the expression of the state transition matrix is: 

I3 × 3 ∆t· f n × 03 × 3

   Φk =    03 × 3  03 × 3

03 × 3 I3 × 3 ∆t·I3 × 3

03 × 3 03 × 3 I3 × 3

03 × 3

03 × 3

03 × 3

03 × 3



∆t·Cbn 03 × 3 03 ×3

1 −

∆t τg

03 × 3 ∆t·Cbn 03 × 3

I3 × 3

03 × 3

03 × 3   1 − ∆t τa I3 × 3

       

(4)

where τg and τa are the correlation times of the gyroscopes and accelerometers, respectively, ∆t stands for the sample time interval and f n × is the skew symmetric matrix constructed by the specific force in the n-frame. When the ZVI is detected, the velocity error can be obtained by doing a subtraction operation between the velocity calculated by the SINS and the velocity in the ZVI; then feeding the velocity error into the Kalman filter for measurement update. The measurement equation of the system is: δzk = Hδxk + vk

(5)

where δzk is the error measurement at time k, H is the measurement matrix and vk is the measurement noise. For the ZUPT-based Kalman filter, the expression of the measurement matrix H is as follows: H =

h

03 × 3

I3 × 3

03 × 3

03 × 3

03 × 3

i

(6)

When the measurement update is performed, the system state error is estimated at every detected zero velocity; then, the navigation parameters and the inertial data can be corrected by feeding the estimated error back to the strapdown inertial navigation algorithm module. In practical applications, the filter stability can be improved by tuning the parameters in the Kalman filter, which mainly include the state estimation covariance matrix P, the process noise covariance matrix Q and the measurement noise covariance h matrix R. In this study, P and Q both are diagonal 15 × 15 matricesi

with elements as P = diag 01 × 3 , 01 × 3 , 01 × 3 , (1 × 10−2 ) I1 × 3 , (1 × 10−2 ) I1 × 3 h i and Q = diag (1 × 10−4 ) I1 × 3 , (1 × 10−4 ) I1 × 3 , 01 × 3 , 01 × 3 , 01 × 3 , respectively. R is a

diagonal 3 × 3 matrix with elements as R = (0.02 m/s)2 I3 × 3 . After tuning the Kalman filter parameters, stable experiment results can be obtained for the PNS.

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3. Adaptive ZVI Detection Algorithm Based on SPWVD-RMFI

The PNS based on foot-mounted foot-mounted inertial sensors employs the ZUPT method to estimate and error, but but the the premise premise of of ZUPT ZUPT is is to to analyze analyze the the pedestrian’s pedestrian’s gait gait characteristics characteristics correct the system error, effectively and to detect detect the the ZVI ZVI correctly. correctly. 3.1. Gait Characteristics Analysis The pedestrian shown inin Figure 2, where all all of pedestrian navigation navigationshoe shoebased basedon onself-contained self-containedsensors sensorsis is shown Figure 2, where the sensors areare integrated in in a structure to to constitute anan inertial measurement unit (IMU). The IMU is of the sensors integrated a structure constitute inertial measurement unit (IMU). The IMU fixed onon a mounting plate onon thethe heel part of of thethe right shoe. The sensors inin the x axis andand thethe y axis is fixed a mounting plate heel part right shoe. The sensors the y x axis measure the longitudinal direction and lateral direction motion parameters, respectively; the motion axis measure the longitudinal direction and lateral direction motion parameters, respectively; the parameters in the vertical are measured by the sensors the z axis. motion parameters in the direction vertical direction are measured by the in sensors in the z axis.

Figure 2. Pedestrian navigation shoe based on the IMU.

A complete complete gait gait cycle cycle shown shown in in Figure Figure 33 is is obtained obtained by by using using the the navigation navigation shoe shoe to to collect collect the the A z inertial parameters parameters of ofaapedestrian’s pedestrian’sright rightfoot footduring duringwalking. walking. The red solid line stands inertial The red solid line stands forfor thethe z axis y axis accelerometer output, and the blue dotted line represents the axis gyroscope output. The accelerometer output, and the blue dotted line represents the y axis gyroscope output. The complete complete cycleAbetween and B in the figure is divided into four stages, which respectively are gait cycle gait between and B inAthe figure is divided into four stages, which respectively are P1, stance, P1, stance, P2 and swing. P1 stands for the process from the heel striking the ground to the front sole P2 and swing. P1 stands for the process from the heel striking the ground to the front sole striking the striking the ground; during this period, thearound foot turns and theoutput gyroscope output ground; during this period, the foot turns thearound −y axis,the and-y theaxis, gyroscope is negative. is negative. Meanwhile, output of accelerometer reaches the maximum strikes the Meanwhile, the output ofthe accelerometer reaches the maximum when the heelwhen strikesthe theheel ground. After ground. the front contacts the ground completely, there which is a period duringoutputs which are the the front After sole contacts thesole ground completely, there is a period during the sensors’ sensors’ outputs are approximately constant gyroscope output is approximately zero,output and the approximately constant (the gyroscope output(the is approximately zero, and the accelerometer is accelerometer output is approximately the gravitational is also called ZVI approximately the gravitational acceleration). This periodacceleration). is also calledThis ZVI period (the stance phase shown (the stance3).phase in Figure After theP2, ZVI, the from lift foot P2,the starts from heeloff of the in Figure Aftershown the ZVI, the lift 3). foot stage, starts thestage, heel of right footthe lifting right foot off the ground to the moment of the toe gyroscope off, during output which the gyroscope output remains ground tolifting the moment of toe off, during which remains negative. After that, negative. Afterlifts that, foot the liftsleg offbegins the ground, the and leg begins to swing the body the right foot offthe theright ground, to swing the body movesand forward (the moves swing forward (the swing phase in Figureoutput 3). Thecorresponding gyroscope output corresponding theswing gait change phase shown in Figure 3). shown The gyroscope to the gait change intothe phase in positive. the swing phase positive. After the swing thestrikes heel ofthe the right foot strikes ground is After the is swing phase, the heel of the phase, right foot ground again, whichthe marks the again, which marks the beginning beginning of another gait cycle. of another gait cycle.

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Figure 3. Pedestrian Figure 3. Pedestrian gait gait cycle. cycle.

3.2. Adaptive ZVI Detection 3.2. Adaptive ZVI Detection The analysis of the pedestrian’s gait characteristics in Section 3.1 indicates that the output of each The analysis of the pedestrian’s gait characteristics in Section 3.1 indicates that the output of sensor in ZVI has a significant difference from other motion stages of the gait cycle, namely that the each sensor in ZVI has a significant difference from other motion stages of the gait cycle, namely that gyroscope output approximates zero and that the acceleration output in the vertical direction is the gyroscope output approximates zero and that the acceleration output in the vertical direction is approximately the gravitational acceleration. According to these obvious features, the ZVI can be approximately the gravitational acceleration. According to these obvious features, the ZVI can be extracted by adopting the zero velocity detection method. The adaptive ZVI detection algorithm extracted by adopting the zero velocity detection method. The adaptive ZVI detection algorithm based on SPWVD-RMFI in this paper mainly consists of two parts: one part is the gait frequency based on SPWVD-RMFI in this paper mainly consists of two parts: one part is the gait frequency analysis method based on SPWVD-RMFI, which can accurately extract the gait frequency during analysis method based on SPWVD-RMFI, which can accurately extract the gait frequency during pedestrian’s walking process at any time; the other part is an adaptive ZVI detection algorithm, which pedestrian’s walking process at any time; the other part is an adaptive ZVI detection algorithm, which can adaptively set the threshold for ZVI detection according to the change of pedestrian gait can adaptively set the threshold for ZVI detection according to the change of pedestrian gait frequency frequency and finally achieve the accurate detection of ZVI. Details are described as follows. and finally achieve the accurate detection of ZVI. Details are described as follows. 3.2.1. on SPWVD-RMFI SPWVD-RMFI 3.2.1. Gait Gait Frequency Frequency Extraction Extraction Based Based on The analysis in Section 3.1 3.1 shows shows that that during during the the pedestrian’s pedestrian’s walking walking process, process, the the gyroscope gyroscope The analysis in Section signal of the y axis can reflect the periodic change of the pedestrian gait completely and signal of the y axis can reflect the periodic change of the pedestrian gait completely and clearly. clearly. Therefore, thegait gaitfrequency frequency at any canobtained be obtained by adopting a time-frequency Therefore, the at any timetime can be by adopting a time-frequency analysisanalysis method. method. Commonly-used time-frequency analysis such methods, such as Fourier short time Fourier(STFT) transform Commonly-used time-frequency analysis methods, as short time transform [17] (STFT) [17] and Wigner–Ville distribution (WVD) [18], can obtain the signal’s time-frequency and Wigner–Ville distribution (WVD) [18], can obtain the signal’s time-frequency information, but both information, have the STFT has the a lower calculation methods havebut theirboth own methods deficiencies: thetheir STFTown has a deficiencies: lower calculation precision; WVD is unable to precision; the WVD is unable to suppress the cross interference items [19]. The SPWVD adopted in suppress the cross interference items [19]. The SPWVD adopted in this study is an improved WVD, this study is an improved WVD, which employs two independently controlled smooth window which employs two independently controlled smooth window functions g(t) and h(τ ) respectively (t ) and and h( )frequency functions respectively in theThe time domainisand frequency domain. Themethod SPWVDwith is a in the timegdomain domain. SPWVD a time-frequency analysis time-frequency analysis method with higher in time-frequency focus;interference it can effectively higher performance in time-frequency focus;performance it can effectively suppress the cross items suppress the the cross interference items of and improve the information extraction precession of time-frequency and improve extraction precession time-frequency simultaneously. The SPWVD information Thex(SPWVD expression of continuous signal x ( t ) is: expression ofsimultaneously. continuous signal t) is: Z ∞



Z ∞ 





τ

 τ

 tt)) ·xx((ε + ) )x· x(∗ (ε − ) d)dεe e − j2π df τ dτ  h(hτ() )−∞gg((ε − 22 22 −∞

SPWVD SPWVD (t, f()t, f=) = where,

*

Z ∞

τ τ j2πfτ x(t +  ) · x*∗ (t − )-je2− dτ 2 WVD(t, f ) = −∞ x(t  )  x (t  2)e  fτ d

WVD(t, f ) =





2

2

-j 2 fτ

(7) (7)

(8) (8)

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where t is the time of x ( t ) , f is the frequency of x ( t ) and x* (t ) is the complex conjugate where of x ( t t) is . the time of x(t), f is the frequency of x(t) and x∗ (t) is the complex conjugate of x(t). Because is discrete discrete rather rather than than continuous, continuous, the thecontinuous continuoustime timesignal signal xx((tt)) Because the the sensor sensor output output is needs needs to to be be discretized. discretized. Then, Then, the the discrete discrete SPWVD SPWVD can can be be expressed expressedas: as: (Q1 −1)/2( Q11 1) / 2

)= )e ˆf ) = ( n, f ∑ h(m)he( m−j2πkm/N SPWVD(n,SPWVD m   ( Q 1) / 2 ˆ

Q22 − 1) / 2 ((Q 2 1)/2

∑

-j 2  km / N

m   ( Q11 1) / 2

m=−(Q1 −1)/2

p   ( Q22 1) / 2

* g ( pg)(px)(·nx(n p− pm+ )  xm ()n ·  x∗ (pn −m p) − m) (9) (9)

p=−(Q2 −1)/2

where Q and Q are the window lengths of h(k ) and g (n) respectively. n , fˆ , p and m where Q1 1 and Q2 2are the window lengths of h(k) and g(n) respectively. n, ˆf , p and m are the are the discretization of the continuous discretization forms offorms the continuous variablesvariables t, f , ε andt τ., f ,  and  . Collecting aa group group of of yy axis axis gyroscope gyroscope output output data data during during the the walking walking process, process, the the time time domain domain Collecting waveform of of the the data data is is shown shown in in Figure Figure 4.4. It It can can be be seen seen that that the the pedestrian’s pedestrian’s walking walking speed speed is is waveform relatively stable, and the local enlarged drawing shows that the pedestrian gait frequency is about relatively stable, and the local enlarged drawing shows that the pedestrian gait frequency is about 0.72 Hz. Hz. After After implementing implementing Fourier Fourier transform, transform, the the amplitude-frequency amplitude-frequency characteristic characteristic of of the the yy axis axis 0.72 gyroscope output output is is shown shown in in Figure Figure 5, 5, in in which which the the frequency frequency 1.44 1.44 Hz Hz is is the the cyclic cyclic component component with with gyroscope the strongest strongest energy, energy, and and itit is is approximately approximately to to two-times two-times the the pedestrian pedestrian gait gait frequency frequency (0.7202 (0.7202 Hz). the Hz). Meanwhile,other otherfrequency frequency components with strong energy are 2.161 Hz, 2.881 Hz3.601 andHz, 3.601 Hz, Meanwhile, components with strong energy are 2.161 Hz, 2.881 Hz and which which are approximately three-times, four-times and five-times the pedestrians gait frequency, are approximately three-times, four-times and five-times the pedestrians gait frequency, respectively. respectively. This shows frequencies that multiple frequencies existgyroscope in the y axis gyroscope output. This shows that multiple exist in the y axis output. Magnitude (rad/s) (rad/s) Magnitude

10 5 0 -5 -10

0

10

20

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50 60 Time (s)

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Magnitude (rad/s) (rad/s) Magnitude

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-10 40

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42.43 42

43.80 43

44

45.16

48.0

46.57

45 46 Time (s)

47

49.38

48

49

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Figure 4. The time domain waveform of the y axis gyroscope output and its local enlarged drawing. Figure 4. The time domain waveform of the y axis gyroscope output and its local enlarged drawing. 2.5

1.44 Hz

Magnitude Magnitude

2

Human gait frequency 0.7202 Hz

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1

2.881 Hz

3.601 Hz

0.5

0

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0.5

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Figure Figure 5. 5. The The Fourier Fourier spectrum spectrum of of the the yy axis axis gyroscope gyroscope output. output.

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Furthermore, Furthermore, the the SPWVD SPWVD was was used used to to extract extract the the time-frequency time-frequency information information of of the the yy axis axis gyroscope output. Figure 6 is the time-frequency spectrum of the y axis gyroscope output. gyroscope output. Figure 6 is the time-frequency spectrum of the y axis gyroscope output.The Thecolor color enclosing enclosinglines linesrepresent representthe thesignal signalamplitude amplitudeat atthe thecorresponding correspondingtime timeand andfrequency, frequency,and andthe thesame same color enclosing lines represent the distribution of the frequency with the same signal amplitude. color enclosing lines represent the distribution of the frequency with the same signal amplitude.ItItisis very veryclear clearthat thatthe thetime-frequency time-frequencyspectrum spectrumcontains containsmulti-peak multi-peakspectral spectrallines linesand andthat thatthe themaximum maximum signal amplitude appears within the range of 1.3–1.55 Hz. Hence, if directly extracting the ridge lineline of signal amplitude appears within the range of 1.3–1.55 Hz. Hence, if directly extracting the ridge the spectrum to obtain the time-frequency information, all of the multiple frequency constituents are of the spectrum to obtain the time-frequency information, all of the multiple frequency constituents interference items,items, whichwhich will make difficult to identify the gait frequency. In addition, 5 are interference will itmake it difficult to identify the gait frequency. In Figures addition, and 6 show that the frequency doubling energy is higher than the gait frequency energy and that the Figures 5 and 6 show that the frequency doubling energy is higher than the gait frequency energy frequency doubling is a range rather a value, features of the frequency and that the frequency doubling is athan range ratherwhich than amakes value,the which makes the gait features of the less gait prominent and increases the extraction difficulty. frequency less prominent and increases the extraction difficulty. 4 3.5

Frequncy (Hz)

3 2.5 2 1.5 1 0.5 0

0

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50 60 Time (s)

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Figure6.6.The TheSPWVD SPWVDof ofthe theyyaxis axisgyroscope gyroscopeoutput. output. Figure

In order to accurately identify the pedestrian gait frequency and effectively filter the multiple In order to accurately identify the pedestrian gait frequency and effectively filter the multiple frequency constituents, this paper puts forward a time-frequency information extraction method frequency constituents, this paper puts forward a time-frequency information extraction method based on SPWVD-RMFI, whose principle diagram is shown in Figure 7. The f , 2 f , 3 f 0 , 4 f 0 based on SPWVD-RMFI, whose principle diagram is shown in Figure 7. The f0 , 02 f0 , 3 f00 , 4 f0 and 5 f0 5 f are the ranges of the gait frequency, frequency doubling, frequency tripling, frequency and 0 are the ranges of the gait frequency, frequency doubling, frequency tripling, frequency quadrupling quadrupling frequency respectively, quintupling, respectively, the y axisoutput. gyroscope output. Among those and frequencyand quintupling, of the y axis of gyroscope Among those frequency frequency ranges, thedoubling frequency hasenergy the highest energy intensity, and the has gaitthe frequency ranges, the frequency hasdoubling the highest intensity, and the gait frequency second has the second highest energy intensity. In addition, multi-peaks may appear in all of the multiple highest energy intensity. In addition, multi-peaks may appear in all of the multiple frequency ranges. f human could frequency the ranges. Therefore, pedestrian gait frequency thefrequency second or with the N-th Therefore, pedestrian gait the frequency f human could be the second or thebe N-th the frequency with the highest energy intensity. The size of N depends on the number and the energy highest energy intensity. The size of N depends on the number and the energy intensity of the intensity of in thethe multi-peaks the frequency However, of the multiple frequencies multi-peaks frequency in doubling range. doubling However,range. all of the multipleallfrequencies and the number and number ofcontained the multi-peaks contained in the multiple ranges are unknown during of thethe multi-peaks in the multiple frequency rangesfrequency are unknown during walking. Hence, f walking. Hence, under the premise of unknown frequency range to extract the under the premise of unknown gait frequency range gait f0 , how to extract the pedestrian frequency 0 , how gait fpedestrian accurately and intelligently is a daunting task. Therefore, the time-frequency extraction method f gait frequency accurately and intelligently is a daunting task. Therefore, the human human based on SPWVD-RMFI is proposed in this paper, and its principle is listed as follows: time-frequency extraction method based on SPWVD-RMFI is proposed in this paper, and its principle is listed as follows: (1) Using the SPWVD to extract the time-frequency spectral line of the y axis gyroscope output. (2) theSPWVD frequency to the largest time-frequency spectral peak1 corresponding (1) Extract Using the to fextract the time-frequency spectralpeak line in ofthe thefirst y axis gyroscope output. line, as is shown in Figure 7; f peak1 is in the range of frequency doubling (2 f0 ). (2) Extract the frequency f peak1 corresponding to the largest peak in the first time-frequency (3) Extract the frequency f peak2 corresponding to the second largest peak in the first time-frequency spectral line, as is shown in Figure 7; f peak1 is in the range of frequency doubling ( 2 f 0 ). spectral line, and judge the relationship between f peak2 and 0.75 f peak1 . If f peak2 < 0.75 f peak1 , (3) itExtract the frequency corresponding to the namely secondthat largest in frequency the first f indicates that f peak2 is in thepeakrange of one time frequency, f peak2 ispeak the gait 2 ftime-frequency to line, time tand process; otherwise, 0.75 f peakif2 fand 1 during human 1 corresponding peak2 > spectral judgethe thewalking relationship between . If, 0.75 f peakf peak1 1 it indicates that f peak2 is still in the frequency doubling range, which means that it is necessary f 2  0.75fpeak1 , it indicates that f peak 2 is in the range of one time frequency, namely that f peak 2 topeak continue to extract the frequency f peak N corresponding to the N-th largest peak in the first is the gait frequency fhuman1 corresponding to time t1 during the walking process; otherwise, if

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fpeak2  0.75fpeak1 , it indicates that f peak 2 is still in the frequency doubling range, which means that fpeak2  0.75fpeak1 , it indicates that f peak 2 is still in the frequency doubling range, which means that

it is necessary to continue to extract the frequency fpeak N corresponding to the N-th largest peak fpeak N corresponding to the N-th largest9peak of 18 in the first time-frequency spectral line; meanwhile, the relationship between fpeak N and in the first time-frequency spectral line; meanwhile, the relationship between fpeak N and 0.75 f peak 1 needs to be judged according to the above-mentioned steps until the judge condition time-frequency spectral line; meanwhile, thethe relationship betweensteps f peak until 0.75 f peak1condition needs to to be judged according to above-mentioned judge 0.75 f peak 1 needs N andthe f f 1 corresponding f N  0.75 f peak1 is fulfilled. fpeak N until At this moment, steps is the frequency bepeakjudged according to the above-mentioned thegait judge conditionfhuman < 0.75 f peak1 peak N f peak N  0.75 f peak1 is fulfilled. At this moment, fpeak N is the gait frequency human 1 corresponding N in f peak fNhuman is fulfilled. At this the moment, f peak gait frequency timeort1equal during to time t1 during walking process. is an integer greatertothan to N is theThe 1 corresponding f N t1 during to time theThe walking process. The in is an integer greater than or equal to peak N the walking process. N in f is an integer greater than or equal to one. peak N one. one. (4) Analyze other spectral lines lines in (4) Analyze other time-frequency time-frequency spectral in (1) (1) according according to to Steps Steps (2)–(3); (2)–(3); then, then, the the gait gait (4) frequency Analyze other spectral lines in (1) according to Steps (2)–(3); then, the gait frequency ffhumantime-frequency of any other time t during the walking process can be obtained, where i is i i humani of any other time ti during the walking process can be obtained, where i is an integer greater than or equal to one. frequency fhumani of any other time ti during the walking process can be obtained, where i is an integer greater than or equal to one. an integer greater than or equal to one.

is necessary Sensorsit2016, 16, 1578 to continue to extract the frequency

Amplitude Amplitude

f human f human peak 3 f peak 3

f peak1 peak21 ffpeak ff peak 2 peak 4 f peak 4

f peak 5 f peak 5

Frequency

f0 2 f0 3 f0 4 f0 5 f0 Frequency f0 2 f0 3 f0 4 f0 5 f0 0.75 f peak1 0.75 f peak1 Figure 7. Principle diagram of the time-frequency extraction method based on SPWVD-RMFI. Figure 7. Principle diagram of the time-frequency extraction method based on SPWVD-RMFI. Figure 7. Principle diagram of the time-frequency extraction method based on SPWVD-RMFI.

According to the above description, the principle of the time-frequency extraction method based According description, the of time-frequency According to to the the above above description, the principle principle of the the 8. time-frequency extraction extraction method method based based on SPWVD-RMFI can be shown by the flow chart in Figure on SPWVD-RMFI can be shown by the flow chart in Figure 8. on SPWVD-RMFI can be shown by the flow chart in Figure 8. Gyroscope data (y-axis ) Gyroscope data (y-axis )

Using SPWVD to extract the timefrequency spectral line Using SPWVD to extract the timefrequency spectral line Extract f peak1 corresponding to the largest peak in the time-frequency Extract f peak1 corresponding to the spectral line at time ti largest peak in the time-frequency spectral line at time N=N+1 ti N=N+1 Extract f peakN corresponding to the Nthlargestf (N=2,3,4...) peak in the timeExtract peakN corresponding to the Nthfrequency spectral line at time ti largest (N=2,3,4...) peak in the timefrequency spectral line at time ti

f peak N  0.75 f peak1 ?

No

f peak N  0.75 f peak1 ?

No

Yes Yes Obtain the gait frequency at time ti : f human i  f peak N Obtain the gait frequency at time ti : f human i  f peak N

Figure 8. Flow chart of the time-frequency extraction method based on SPWVD-RMFI. Figure 8. Flow chart of the time-frequency extraction method based on SPWVD-RMFI. Figure 8. Flow chart of the time-frequency extraction method based on SPWVD-RMFI.

During the walking process, the SPWVD-RMFI method proposed in this paper can extract the gait frequency correctly, which provides the foundation for the thresholds’ selection in the adaptive ZVI detection algorithm.

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3.2.2. Adaptive ZVI Detection On the basis of the gait frequency obtained by the SPWVD-RMFI method in Section 3.2.1, this section designs an adaptive ZVI detection algorithm. Through many experiments and statistical analysis, the thresholds at different walking speeds are obtained. Meanwhile, the function relationships between the optimal thresholds and the gait frequency are established. Furthermore, the algorithm can calculate the optimal ZVI detection thresholds based on the real-time gait frequency, and finally, the adaptive ZVI detection is achieved. The principle of the adaptive ZVI detection algorithm is as follows. Define |a(ti )| as the magnitude of the acceleration and σa (ti ) as the moving variance of the acceleration. During the walking process, they can be calculated by the following two equations: q

ax (ti )2 + ay (ti )2 + az (ti )2

(10)

v u 2 u 1 m+w−1  t σa (ti ) = |a(ti )| − |aw | n i∑ =m

(11)

|a(ti )| =

where ax (ti ), ay (ti ) and az (ti ) respectively represent the accelerometers output in the x axis, y axis and z axis at time ti , w is the window width and |aw | is the mean value of the acceleration in the window. According to the fact that the sensor output in the ZVI is approximate to a constant, the ZVI in the human gait can be extracted by setting thresholds to Equations (10) and (11). The detail is shown in the following equation: ( Ra1 (ti ) < |a(ti )| < Ra2 (ti ) (12) σa (ti ) < Rσ (ti ) where Ra1 (ti ) is the minimum threshold of the acceleration magnitude at time ti , Ra2 (ti ) is the maximum threshold of the acceleration magnitude at time ti and Rσ (ti ) is the threshold of the acceleration variance at time ti . When the judgment conditions in Equation (12) are fulfilled at the same time, we consider the pedestrian’s walking speed at ti to be approximately zero. For the traditional fixed threshold methods, Ra1 (ti ), Ra2 (ti ) and Rσ (ti ) are all fixed empirical values, which are only applicable to the ZVI detection at a constant walking speed rather than a changing one. In order to solve this problem and improve the robustness of the ZVI detection method, in this study, Ra1 (ti ), Ra2 (ti ) and Rσ (ti ) are all set as dynamic values, which are directly related to the gait frequency f human i at time ti . The amount of experiments on different gait frequencies have been done to obtain the relationships between the thresholds (Ra1 , Ra2 and Rσ ) and the gait frequency f human . By collecting the gait data of the pedestrian who walks at an increasing speed and analyzing the statistical feature of the thresholds at different gait frequencies, the relationships between the thresholds and the gait frequency are shown in Figure 9. The blue solid lines in Figure 9 are the relationship curves between the three thresholds and the gait frequency. The change tendency of the blue solid lines indicates that Ra1 decreases with the increase of the gait frequency; Ra2 and Rσ increase with the increase of the gait frequency. The main cause for the changes of the three thresholds are: when the pedestrian’s walking speed increases gradually, the dynamic nature of foot movement is gradually enhanced, and the change of the signal amplitude is intensified, which leads to the corresponding changes of the ZVI detection thresholds (Ra1 , Ra2 and Rσ ). After fitting the experimental thresholds with the gait frequency in Figure 9, the relationships between the ZVI detection thresholds (Ra1 , Ra2 and Rσ ) and the gait frequency f human are shown as the red dotted lines. Their function expressions are: Ra1 = λ1 ∗ f human + b1

(13)

2 Ra2 = λ2 ∗ f human + λ3 ∗ f human + b2

(14)

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Rσ = λ4 ∗ f human + b3

(15)

where λ1 , λ2 , λ3 , λ4 , b1 , b2 and b3 are the fitting coefficients of the functions; meanwhile, λ1 = −1.48, b1 = 10.29, λ2 = 4.03, λ3 = −4.0, b2 = 11.35, λ4 = 2.84, b3 = −1.12. According to the function relationships between the thresholds and the gait frequency in Equations (13)–(15), the ZVI detection Sensors 2016, 16, 1578 11 of 18 thresholds at different walking speeds can be obtained, and the ZVI can be detected adaptively. 9.6

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Figure 9. The relationships between the thresholds and the gait frequency. (a) The minimum Figure 9. The relationships between the thresholds and the gait frequency. (a) The minimum threshold threshold of the acceleration to the gait(b) frequency; (b) The maximum threshold of the of the acceleration magnitudemagnitude to the gait frequency; The maximum threshold of the acceleration acceleration to the gait(c) frequency; (c) The threshold of thevariance acceleration to the gait magnitudemagnitude to the gait frequency; The threshold of the acceleration to thevariance gait frequency. frequency.

4. Experiment Validation

The blue solid lines in Figure 9 are the relationship curves between the three thresholds and the In this section, we firstly give a brief description of the PNS hardware; then, accuracy with of thethe Ra1the gait frequency. The change tendency of the blue solid lines indicates that decreases novel adaptive ZVI detection algorithm proposed in this paper is assessed by comparing with the traditional ZVI detection algorithm; finally, pedestrian trajectory positioning experiments at different main causespeeds for theare changes thresholds are: when pedestrian’s walking speed increases walking carried of outthe to three evaluate the influence of the the novel algorithm on positioning accuracy.

increase of the gait frequency; Ra 2 and R increase with the increase of the gait frequency. The gradually, the dynamic nature of foot movement is gradually enhanced, and the change of the signal 4.1. System Description amplitude is Hardware intensified, which leads to the corresponding changes of the ZVI detection thresholds R ). After ( Ra1 , RUnlike fitting theaexperimental thresholds with the frequency Figure 9, the a 2 and vehicles and airplanes, pedestrian has a limited ability of gait carrying a load.inDue to this,

R )small and in thesize gait frequency relationships between thetoZVI detection thresholdsof( R the IMU in the PNS has match the characteristics being light weight, and being a1 , R a 2 and low power to make thered pedestrian move normally, so expressions that long time f human are shown as the dotted lines. Their function are:navigation can be realized. With all of the characteristics considered, the IMU MTi-G (as shown in Figure 10) produced by Xsens 1  the f humantesting  b1 prototype. The size of the IMU is only Technologies B.V. (Netherlands) is selectedRto build (13) a1  57 × 42 × 24 mm, and the weight is 58 g [20]. Meanwhile, the IMU integrates three MEMS gyroscopes, 2 three MEMS accelerometers and a triaxial sensor in a small structure. The full range of the Ra 2  magnetic (14) 2  f human  3  f human  b2 accelerometer is ±15 g, and the gyroscope is ±1000◦ /s. During the experimental process, the IMU is installed on the pedestrian’s right foot, and the sampling rate is set as 100 Hz.

R  4  f human  b3

(15)

where 1 , 2 , 3 , 4 , b1 , b2 and b3 are the fitting coefficients of the functions; meanwhile,

1   1.48 , b1  10.29 , 2  4.03 , 3   4.0 , b2  11.35 , 4  2.84 , b3   1.12 . According to the function relationships between the thresholds and the gait frequency in Equations (13)–(15), the ZVI detection thresholds at different walking speeds can be obtained, and the ZVI can be detected adaptively. 4. Experiment Validation In this section, we firstly give a brief description of the PNS hardware; then, the accuracy of the novel adaptive ZVI detection algorithm proposed in this paper is assessed by comparing with the traditional ZVI detection algorithm; finally, pedestrian trajectory positioning experiments at different walking speeds are carried out to evaluate the influence of the novel algorithm on positioning accuracy.

of of the the characteristics characteristics considered, considered, the the IMU IMU MTi-G MTi-G (as (as shown shown in in Figure Figure 10) 10) produced produced by by Xsens Xsens Technologies B.V. (Netherlands) is selected to build the testing prototype. The size of the IMU Technologies B.V. (Netherlands) is selected to build the testing prototype. The size of the IMU is is only only 57 57 ×× 42 42 ×× 24 24 mm, mm, and and the the weight weight is is 58 58 gg [20]. [20]. Meanwhile, Meanwhile, the the IMU IMU integrates integrates three three MEMS MEMS gyroscopes, gyroscopes, three three MEMS MEMS accelerometers accelerometers and and aa triaxial triaxial magnetic magnetic sensor sensor in in aa small small structure. structure. The The full full range of the accelerometer is ±15 g, and the gyroscope is ±1000 °/s. During the experimental process, Sensors 2016, 16, 1578 12 of 18 range of the accelerometer is ±15 g, and the gyroscope is ±1000 °/s. During the experimental process, the IMU is installed on the pedestrian’s right foot, and the sampling rate is set as 100 Hz. the IMU is installed on the pedestrian’s right foot, and the sampling rate is set as 100 Hz.

Figure 10. IMU for for the the foot-mounted foot-mounted PNS. Figure 10. 10. Xsens foot-mounted PNS. PNS. Figure Xsens IMU

4.2. 4.2. ZVI ZVI Detection Detection Experiment Experiment 4.2. The IMU output data were collected when the walked at speed The IMU IMUoutput output collected when the experimenter experimenter at aa(86slow slow speed The datadata werewere collected when the experimenter walked at awalked slow speed steps/min) (86 steps/min) and a fast speed (116 steps/min); then, with the traditional fixed threshold method, the (86 steps/min) and a fast speed (116then, steps/min); then, with thefixed traditional fixedmethod, threshold the and a fast speed (116 steps/min); with the traditional threshold themethod, ZVI in the ZVI in the pedestrian gait was detected. The upper part of Figure 11 is the ZVI detection result for ZVI in the gait pedestrian gait was detected. Theofupper ofthe Figure 11 is the ZVI detection for pedestrian was detected. The upper part Figurepart 11 is ZVI detection result for slowresult walking. slow walking. The result shows that the fixed threshold method can detect the ZVI in the human gait slow walking. The result shows that the fixed threshold method can detect the ZVI in the human gait The result shows that the fixed threshold method can detect the ZVI in the human gait accurately if the accurately the thresholds are appropriately. The of Figure 11 ZVI detection result accurately if if are set set appropriately. The bottom bottom Figure 11 is is the the ZVIfor detection result thresholds arethe setthresholds appropriately. The bottom of Figure 11 is theofZVI detection result fast walking, for fast walking, in which the selected ZVI detection thresholds are exactly the same as the fixed for fast walking, in which the selected ZVI detection thresholds are exactly the same as the in which the selected ZVI detection thresholds are exactly the same as the fixed threshold for fixed slow threshold for walking. The result shows that same thresholds will lead false thresholdThe for slow slow Thethe result shows that the thewill same thresholds willmissed lead to to detections false and and missed missed walking. resultwalking. shows that same thresholds lead to false and in the detections in ZVI results when the walking changes. For false detections detections in the the ZVI detection detection when thechanges. walking speed speed changes.false For example, example, detections ZVI detection results when theresults walking speed For example, detectionsfalse appear in the appear in the marked parts ①, ③ and ④, where the stance phase in the gait is detected as aa motion appear in the marked parts ①, ③ and ④, where the stance phase in the gait is detected as motion 1 3 and , 4 where the stance phase in the gait is detected as a motion state; in addition, marked parts , state; addition, missed appears in the marked part ②, where the length of in state; in indetection addition,appears missed detection detection appears marked ②, of where thethe length of ZVI ZVI gait in the the 2the missed in the marked partin , where thepart length ZVI in pedestrian is pedestrian gait is shortened. Therefore, the fixed threshold method has a better ZVI detection effect pedestrian Therefore, gait is shortened. Therefore, fixed threshold method has a better effect shortened. the fixed thresholdthe method has a better ZVI detection effectZVI for detection single walking for walking speed, if pedestrian walks at speeds, false or detections for single single speed, but but if the the pedestrian walksfalse at different different falsewill or missed missed speed, but walking if the pedestrian walks at different speeds, or missedspeeds, detections appear detections in the ZVI will appear in the ZVI detection result. will appear in the ZVI detection result. detection result. 20 20 10 10 0 -100 -10 -20 -20 -40 -40 -60 -60 -80 -80

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Figure Figure 11. 11. ZVI ZVI detection detection results results with with the the fixed fixed threshold threshold method method at at different different walking walking speeds. speeds. The The red red Figure 11. ZVI detection results with the fixed threshold method at different walking speeds. The red lines lines are are the the zz axis axis accelerometer accelerometer outputs. outputs. The The blue blue lines lines are are the the ZVI ZVI detection detection results; results; −10 −10 represents represents lines are the z axis accelerometer outputs. The blue lines are the ZVI detection results; −10 represents ZVI; ZVI; and and 10 10 indicates indicates non-ZVI. non-ZVI. ZVI; and 10 indicates non-ZVI.

Meanwhile, adopting the adaptive ZVI detection algorithm to detect the ZVI in the pedestrian gait, the results are shown in Figure 12. It can be seen that when the pedestrian walks at a slow speed, the extraction effect of ZVI by the adaptive ZVI detection algorithm is the same as adopting the fixed

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Meanwhile, adopting the adaptive ZVI detection algorithm to detect the ZVI in the pedestrian 13 of 18 gait, the results are shown in Figure 12. It can be seen that when the pedestrian walks at a slow speed, the extraction effect of ZVI by the adaptive ZVI detection algorithm is the same as adopting the fixed thresholdmethod methodin inFigure Figure11; 11;but butwhen whenthe thepedestrian pedestrianwalks walksat ataafast fastspeed, speed,the thenovel noveladaptive adaptiveZVI ZVI threshold detectionalgorithm algorithmcan canstill stillaccurately accuratelydetect detectthe theZVI ZVIwithout withoutfalse falseand andmissed misseddetections. detections. detection

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ZVI Detection with Adaptive Threshold (Slow Walking)

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Figure 12. 12. ZVI ZVI detection detection results results with with the the adaptive adaptive ZVI ZVI detection detection algorithm algorithm at at different different walking walking Figure speeds. The red lines are the z axis accelerometer outputs. The blue lines are the ZVI detection results; speeds. The red lines are the z axis accelerometer outputs. The blue lines are the ZVI detection results; −10 ZVI; and and 10 10 indicates indicates non-ZVI. non-ZVI. − 10 represents represents ZVI;

In order to further compare the ZVI detection performance between the adaptive ZVI detection In order to further compare the ZVI detection performance between the adaptive ZVI detection algorithm and the traditional fixed threshold method, the experimenter walked 200 steps (the actual algorithm and the traditional fixed threshold method, the experimenter walked 200 steps (the actual ZVI number is 100) at the speeds of 80 steps/min, 100 steps/min and 120 steps/min, respectively. ZVI number is 100) at the speeds of 80 steps/min, 100 steps/min and 120 steps/min, respectively. Table 1 shows the results of ZVI detected by the fixed threshold method and the adaptive ZVI Table 1 shows the results of ZVI detected by the fixed threshold method and the adaptive ZVI detection detection algorithm. While applying the fixed threshold method, the threshold of 100 steps/min algorithm. While applying the fixed threshold method, the threshold of 100 steps/min (Th100) is (Th100) is selected as the detection threshold for the three walking speeds. selected as the detection threshold for the three walking speeds. Table 1. ZVI detection results at different walking speeds. Table 1. ZVI detection results at different walking speeds. Detected ZVI Number Walking Speed Actual ZVI Fixed Threshold DetectedAdaptive ZVI Detection ZVI Number (Steps/min) Number Method Algorithm Walking Speed (Steps/min) Actual ZVI Number Fixed Threshold Adaptive ZVI Detection 80 100 102 Method 100 Algorithm 100 100 100 100 80 100 102 100 120 100 94 100 100 100 100 100 120 100 94 100

The data in Table 1 indicate that Th100 has the best ZVI detection result for a 100-steps/min walking speed, and the detected ZVI number is 100, which is the same as the actual ZVI number. For The data in Table 1 indicate that Th100 has the best ZVI detection result for a 100-steps/min the other two walking speeds, 80 steps/min and 120 steps/min, the ZVI detected by the fixed walking speed, and the detected ZVI number is 100, which is the same as the actual ZVI number. threshold method includes two false detections and six missed detections, respectively. However, the For the other two walking speeds, 80 steps/min and 120 steps/min, the ZVI detected by the fixed ZVI number detected by the adaptive ZVI detection algorithm is 100 for the three different walking threshold method includes two false detections and six missed detections, respectively. However, the speeds, the same as the actual ZVI number. Its root cause lies in the fact that the proposed method ZVI number detected by the adaptive ZVI detection algorithm is 100 for the three different walking can adaptively provide optimal thresholds for ZVI detection with the gait frequency changing, which speeds, the same as the actual ZVI number. Its root cause lies in the fact that the proposed method avoids the limited application of the fixed threshold only for a single walking speed and effectively can adaptively provide optimal thresholds for ZVI detection with the gait frequency changing, which reduces the rate of false and missed detections. avoids the limited application of the fixed threshold only for a single walking speed and effectively reduces the rate of false and missed detections.

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Therefore, the design and implementation of the adaptive ZVI detection algorithm enhances the Therefore, the design and implementation of the adaptive ZVI detection algorithm enhances the accuracy of ZVI detection at different walking speeds, which further improves the positioning effect accuracy of ZVI detection at different walking speeds, which further improves the positioning effect of of PNS and makes it have better robustness and stronger practical application. PNS and makes it have better robustness and stronger practical application. 4.3. Pedestrian Trajectory Positioning Experiment 4.3. Pedestrian Trajectory Positioning Experiment Furthermore, pedestrian trajectory positioning experiments at different walking speeds Furthermore, pedestrian trajectory positioning experiments at different walking speeds were carried were carried out to assess the influence of the novel adaptive ZVI detection algorithm on out to assess the influence of the novel adaptive ZVI detection algorithm on positioning accuracy. positioning accuracy.

Rectangular Route Experiment Rectangular Route Experiment The pedestrian trajectory positioning experiment was carried out on a 49 m-wide and 13.9 m-long The pedestrian trajectory positioning experiment was carried out on a 49 m-wide and rectangular route on campus. Facing north, the experimenter started and walked a loop along the 13.9 m-long rectangular route on campus. Facing north, the experimenter started and walked a loop route at “slow-normal-fast-slow-normal-fast-slow” speed alternately and finally returned to the start along the route at “slow-normal-fast-slow-normal-fast-slow” speed alternately and finally returned point. The total walking distance was 125.8 m, and them, total wastime 119.6 s. The data to the start point. The total walking distance was 125.8 andwalking the total time walking was 119.6raw s. The output by the accelerometers and gyroscopes during the walking process are shown in Figure raw data output by the accelerometers and gyroscopes during the walking process are shown in13, indicating that the accelerometer in the z axis has a better sensing theonfoot movement and Figure 13, indicating that the accelerometer in the z axiseffect has aon better effect sensing the foot that the amplitude ofthe theamplitude acceleration changes greatly; the gyroscope the y axisinhas obvious movement and that of the acceleration changes greatly; theingyroscope the yanaxis has perception of foot rotation, amplitude of the angularofrate In addition, can an obvious perception of and foot the rotation, and the amplitude thechanges angular greatly. rate changes greatly.itIn also be found from Figure 13 that the amplitude and gait frequency of the sensors’ output change with addition, it can also be found from Figure 13 that the amplitude and gait frequency of the sensors’ change with the walking speed. When the walking increases, the amplitude and gait theoutput walking speed. When the walking speed increases, the speed amplitude and gait frequencies increase, frequencies and vice versa.increase, and vice versa. Acceleration (m/s2 )

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Figure 13. Output of the IMU during walking. Figure 13. Output of the IMU during walking.

By adopting the adaptive ZVI detection algorithm to analyze the IMU output during the walking By adopting the adaptive algorithm to analyze the IMU output14. during the walking process, the detection results ZVI of thedetection gait frequency and ZVI are provided in Figure The upper part process, the detection of the gait frequency and that ZVI the are amplitude provided inand Figure 14. The upper part of of Figure 14 is the yresults axis gyroscope output, showing the cycle of the angular Figure is thewith y axis output, showing that the the “slow-normal-fast-slow-normal-fast-slow” amplitude and the cycle of the angular rate rate 14 change thegyroscope walking speed, which matches speedwith changing characteristic. The middle part Figure 14 shows the real-time gait frequency change the walking speed, which matches theof“slow-normal-fast-slow-normal-fast-slow” speed extracted by the adaptive ZVImiddle detection algorithm. canshows be seenthe that the time-frequency curveextracted clearly changing characteristic. The part of FigureIt 14 real-time gait frequency reflects the gaitalgorithm. frequency It change the walking process, which provides by and the accurately adaptive ZVI detection can beduring seen that the time-frequency curve clearlythe and real-time gait frequency for selecting dynamic thresholds in the process of adaptive ZVI detection. accurately reflects the gait frequency change during the walking process, which provides the real-time bottom of 14 isdynamic the ZVI detected by in thethe adaptive detection ZVI algorithm, and The the result gaitThe frequency forFigure selecting thresholds processZVI of adaptive detection. bottom shows that there is no false and missed detection during the walking process. of Figure 14 is the ZVI detected by the adaptive ZVI detection algorithm, and the result shows that

there is no false and missed detection during the walking process.

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15 of 18 15of of18 18 15 Gait Angularrate rate(rad/s) (rad/s) GaitFrequency Frequency(Hz) (Hz) Angular

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Figure 14. Gait frequency and ZVI detectionresults. results.In Inthe thebottom bottomof ofthe thefigure, figure,−10 −10 represents ZVI Figure Gait frequency and ZVI detection results. In bottom ZVI Figure 14.14. Gait frequency and ZVI detection of the figure, −10represents represents ZVI and 10 indicates non-ZVI. indicates non-ZVI. andand 10 10 indicates non-ZVI.

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Then, using using the the ZUPT-aided ZUPT-aided pedestrian pedestrian navigation navigation algorithm algorithm to to calculate calculate the the pedestrian pedestrian Then, Then, using the ZUPT-aided pedestrian navigation algorithm to calculate the pedestrian trajectory. trajectory.The Theresults resultsare areshown shownin inFigure Figure15. 15.The Thered reddashed-dotted dashed-dottedline lineand andthe theblue bluedashed dashed line trajectory. line Therepresent results are shown in Figure 15. Theby redthe dashed-dotted line and thenavigation blue dashed line represent the the trajectories calculated ZUPT-aided pedestrian algorithm with the represent the trajectories calculated by the ZUPT-aided pedestrian navigation algorithm with the trajectories calculated by the ZUPT-aided pedestrian navigation algorithm with the traditional fixed traditionalfixed fixedthreshold thresholdmethod methodand andthe theadaptive adaptiveZVI ZVIdetection detectionalgorithm, algorithm,respectively. respectively.The Thegreen green traditional threshold method and the adaptive ZVI The detection algorithm, respectively. The green solidthat lineafter is the solid line is the reference trajectory. local enlarged drawings of Figure 15 show solid line is the reference trajectory. The local enlarged drawings of Figure 15 show that after reference trajectory. The local enlarged drawings of Figure 15 show that after obtaining the ZVI obtaining the ZVI by the fixed threshold method, the calculated trajectory deviates from the reference obtaining the ZVI by the fixed threshold method, the calculated trajectory deviates from the referenceby thetrajectory, fixed threshold calculated trajectory fromdirection. the reference trajectory, and the trajectory, andthe themethod, positionthe error morethan than min indeviates theeast-west east-west direction. Themain main causeof ofthe the and position error isismore 11m the The cause position error is more 1 exist m infalse the direction. The cause offixed the position error is that position error thatthan there exist falseeast-west andmissed missed detections ofmain ZVIwith with the fixed threshold method position error isisthat there and detections of ZVI the threshold method when the experimenter walks at different speeds. However, after obtaining the ZVI by the adaptive there exist false and missed detections of ZVI with the fixed threshold method when the experimenter when the experimenter walks at different speeds. However, after obtaining the ZVI by the adaptive ZVIat detection algorithm, thecalculated calculated trajectory fits with the reference trajectory well.The Thealgorithm, position walks differentalgorithm, speeds. However, aftertrajectory obtainingfits the ZVIthe byreference the adaptive ZVI detection ZVI detection the with trajectory well. position error at the end point is 0.69 m, accounting for 0.55% of the total walking distance. Thus, the ZVI is theerror calculated trajectory fits with referencefortrajectory Thewalking positiondistance. error at Thus, the end point at the end point is 0.69 m,the accounting 0.55% of well. the total the ZVI detected by the the adaptive adaptive ZVI detection algorithm for pedestrian pedestrian trajectory calculation can achieve achieve 0.69 m, accounting for 0.55%ZVI of the total walking distance. Thus, thetrajectory ZVI detected by thecan adaptive ZVI detected by detection algorithm for calculation higher positioning accuracy. higher positioning accuracy. detection algorithm for pedestrian trajectory calculation can achieve higher positioning accuracy. 50 50

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Figure15. 15.Results Resultsof ofwalking walkingalong along arectangle rectangletrajectory. trajectory. Thered red dashed-dottedline line andthe theblue blue Figure Figure 15. Results of walking along aarectangle trajectory. The The reddashed-dotted dashed-dotted lineand and the blue dashedline linerepresent representthe thetrajectory trajectorycalculated calculatedby bythe theZUPT-aided ZUPT-aidedpedestrian pedestriannavigation navigationalgorithm algorithm dashed dashed line represent the trajectory calculated by the ZUPT-aided pedestrian navigation algorithm withthe thetraditional traditionalfixed fixedthreshold thresholdmethod methodand andthe theadaptive adaptiveZVI ZVIdetection detectionalgorithm, algorithm,respectively. respectively. with with thegreen traditional fixed threshold and the adaptive ZVI detection algorithm, respectively. The solidline line the referencemethod trajectory. The green solid isisthe reference trajectory. The green solid line is the reference trajectory.

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We onon a longer path, as as shown in We also also conducted conductedthe thepedestrian pedestriantrajectory trajectorypositioning positioningexperiment experiment a longer path, shown Figure 16. The experimenter walked a loop on the campus runway at “normal-slow-fast-slow” walking in Figure 16. The experimenter walked a loop on the campus runway at “normal-slow-fast-slow” speed. totalThe walking distance distance was about 430 m, and walking time wastime 427 s. The red walkingThe speed. total walking was about 430the m, total and the total walking was 427 s. and blue curves are the pedestrian trajectories calculated by the ZUPT-aided pedestrian navigation The red and blue curves are the pedestrian trajectories calculated by the ZUPT-aided pedestrian algorithm the fixed threshold andmethod the adaptive ZVIadaptive detection algorithm, navigationwith algorithm with the fixedmethod threshold and the ZVI detectionrespectively. algorithm, From Figure 16, we can learn that the pedestrian position error by adopting the fixed threshold method respectively. From Figure 16, we can learn that the pedestrian position error by adopting the fixed (5.1% of the travelled distance) is larger than the one with the adaptive threshold algorithm (3.5% of threshold method (5.1% of the travelled distance) is larger than the one with the adaptive threshold the travelled distance). The reason is thatThe when the experimenter walks at different speeds, false or algorithm (3.5% of the travelled distance). reason is that when the experimenter walks at different missed detections appear when applying the fixed threshold method, which further leads to or speeds, false or missed detections appear when applying the fixed threshold method, which false further missed corrections in the ZUPT process. However, the adaptive ZVI detection algorithm can adjust leads to false or missed corrections in the ZUPT process. However, the adaptive ZVI detection the thresholds in realthe time accordingintoreal thetime gait according frequency,towhich ensures the proper of algorithm can adjust thresholds the gait frequency, whichexecution ensures the ZUPT and can correct the navigation error adequately. In addition, the trajectories calculated by the proper execution of ZUPT and can correct the navigation error adequately. In addition, the two methods share thebycommon the heading drifts gradually with the increase trajectories calculated the two characteristics methods sharethat the common characteristics that the heading drifts of the walking distance, the reason for which is that ZUPT fails to correct the heading error [21,22], gradually with the increase of the walking distance, the reason for which is that ZUPT fails to correct leading to the accumulation of heading error calculated SINS.error Heading drift, another the heading error [21,22], leading to the accumulation of by heading calculated by SINS.important Heading problem in theimportant PNS, is anproblem issue that be discussed in athat future drift, another in will the PNS, is an issue willstudy. be discussed in a future study.

Figure 16. Results of walking around the playground in a circle. The red line represents the pedestrian Figure 16. Results of walking around the playground in a circle. The red line represents the pedestrian trajectory calculated by the ZUPT-aided pedestrian navigation algorithm with the fixed threshold trajectory calculated by the ZUPT-aided pedestrian navigation algorithm with the fixed threshold method; the the blue blue line line is method; is the the pedestrian pedestrian trajectory trajectory calculated calculated by by the the ZUPT-aided ZUPT-aided pedestrian pedestrian navigation navigation algorithm with the adaptive ZVI detection algorithm. algorithm with the adaptive ZVI detection algorithm.

5. Conclusions 5. Conclusions This paper designs a novel adaptive ZVI detection algorithm based on SPWVD-RMFI for the This paper designs a novel adaptive ZVI detection algorithm based on SPWVD-RMFI for the PNS PNS with self-contained sensors. The novel algorithm can extract the pedestrian’s gait frequency with self-contained sensors. The novel algorithm can extract the pedestrian’s gait frequency during during walking in real time. With the establishment of the function relationships between ZVI walking in real time. With the establishment of the function relationships between ZVI detection detection thresholds and gait frequency, the problem of setting the thresholds adaptively for ZVI thresholds and gait frequency, the problem of setting the thresholds adaptively for ZVI detection at detection at different walking speeds is solved, which realizes the accurate detection of ZVI. different walking speeds is solved, which realizes the accurate detection of ZVI. Compared with the Compared with the traditional fixed threshold method, the novel algorithm has better robustness traditional fixed threshold method, the novel algorithm has better robustness and higher detection and higher detection precision of ZVI. At the same time, the results of ZVI detection contrast precision of ZVI. At the same time, the results of ZVI detection contrast experiments and pedestrian experiments and pedestrian trajectory positioning experiments at different walking speeds show that trajectory positioning experiments at different walking speeds show that the novel algorithm can the novel algorithm can effectively suppress the false and missed detections existing in the traditional fixed threshold method. Meanwhile, the ZVI detected by the adaptive ZVI detection algorithm for pedestrian trajectory calculation has better performance. In addition, the SPWVD-RMFI algorithm

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effectively suppress the false and missed detections existing in the traditional fixed threshold method. Meanwhile, the ZVI detected by the adaptive ZVI detection algorithm for pedestrian trajectory calculation has better performance. In addition, the SPWVD-RMFI algorithm presented in this paper can also be applied to the motion frequency detection of human health monitoring and the step length estimation of PNS based on the pedestrian dead reckoning algorithm. In a future study, we will pay close attention to the problem of heading correction for PNS. By adopting the adaptive ZVI detection algorithm proposed in this paper for ZUPT and combing the heading correction algorithm, the positioning accuracy of the PNS can be further improved. Acknowledgments: This work was supported by the National Natural Science Foundation of China (Grant No. 91120010). Author Contributions: Xiaochun Tian designed the algorithm and wrote the paper. Jiabin Chen and Yongqiang Han conceived of and designed the experiments. Jianyu Shang and Nan Li performed the experiments and analyzed the data. All of the authors contributed to the paper correction and improvements. Conflicts of Interest: The authors declare no conflict of interest.

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