# A Robust Dead Reckoning Algorithm Based on Wi-Fi FTM and Multiple Sensors

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- To improve the traditional multi-sensor-based dead reckoning method, a multi-pattern-based step detection and location updating algorithm is proposed in order to adapt to complex indoor walking modes.
- (2)
- A real-time ranging model based on Wi-Fi FTM is presented which can effectively reduce the Wi-Fi ranging error caused by clock deviation, non-line-of-sight (NLOS), and multipath propagation.
- (3)
- Based on the fusion of Wi-Fi ranging model and multi-pattern-based dead reckoning method, DRWMs is proposed. The combination of the real-time Wi-Fi FTM ranging model and the multi-sensor estimation method effectively improves the accuracy and stability of final dead reckoning.

## 2. Theoretical Framework

#### 2.1. Positioning Method Based on Wi-Fi FTM

_{1}(k) is the timestamp when the FTM framework first sent by Wi-Fi AP, t

_{2}(k) is the timestamp when the FTM signal arrives to the mobile terminal, t

_{3}(k)is the timestamp when the mobile terminal returns the ACK signal to Wi-Fi AP, t

_{4}(k) is the timestamp when the ACK signal is finally received by the Wi-Fi AP, and the parameter n is the number of FTMs per burst among one ranging period. Generally, the protocol excludes the processing time on the mobile terminal by subtracting (t

_{3}(k) – t

_{2}(k)) from the total round-trip time (t

_{4}(k) – t

_{1}(k)), which represents the time from the instant the FTM message is being sent (t

_{1}(k)) to the instant the ACK is being received (t

_{4}(k)). This calculation is repeated for each FTM-ACK exchange, and the final RTT is the average over the number of FTMs per burst. In this paper, the parameter FTMs per burst n is set as 30 to minimize the measurement noise, so it will take more time to complete this procedure compared to a smaller FTMs per burst [14]. The sampling rate of the RTT depends on the hardware performance of the processor and bandwidth of Wi-Fi, and the frequency of the processor should satisfy pico-seconds fine-grained requirements according to [13].

**C**is the speed of light and D

_{RTT}is the final distance calculated by one FTM period.

**x**

_{p}is the localization result, j is the number of APs, x

_{j}and y

_{j}are the position of Wi-Fi AP, and D

_{RTT}(j) is real-time RTT data received from each Wi-Fi AP. It should be noted that the number of APs needed to solve the above equation is at least one more than the dimension of the

**x**

_{p}. In theory, we can get more accurate positioning results by increasing the number of APs [28].

#### 2.2. Multi-Pattern-Based Dead Reckoning via Multiple Sensors

#### 2.2.1. Multi-Pattern-Based Step Detection and Step-Length Estimation

_{1}is the timestamp when the peak or valley of Z-axis value is detected during one step period and A

_{y}is the acceleration data from Y-axis.

_{1}. When the conditions K

_{1}> 0 and K

_{2}> 0 are met, in the case of walking forward, m

_{1}is the timestamp when the peak of Z-axis value is detected. In case of walking backward, m

_{1}is the timestamp when the valley of the Z-axis value is detected.

_{2}is the timestamp when the peak of Z-axis value is detected during one step period and A

_{x}is the acceleration data from X-axis. Figure 4 shows the comparison of left lateral walking and right lateral walking. The tester walked left laterally for 6 m and then walked right laterally to the original point. The red dots show the peaks or valleys of the Z-axis data, as corresponding to the red stars, which represent the X-axis data at the moment m

_{2}. When the conditions K

_{3}< 0 and K

_{4}< 0 are met, in case of left lateral walking, m

_{2}is the timestamp when the peak of the Z-axis value is detected. When the conditions K

_{3}> 0 and K

_{4}> 0 are conformed, in case of right lateral walking, m

_{2}is the timestamp when the peak of the Z-axis value is detected. Only a texting pattern is considered in this work; as shown in Figure 2, subtle tilt does not affect the recognition results because the Equations (4) and (5) calculate the changing trend of the acceleration data.

_{max}and A

_{min}are the maximum and minimum values of the Z-axis acceleration during one step period and K is the ratio of the real and the estimated distance:

#### 2.2.2. Location Update

_{1}and C

_{2}. When using the traditional DR algorithm, the walking trajectory still shows from point B to point D, perpendicular to the actual trajectory of the pedestrian.

_{k}is the step-length and θ

_{k}is the difference of heading angle compared with initial heading which is provided by the calibrated magnetometer [32]. The real-time heading is calculated by an EKF filter which combines the outputs of the gyroscope and magnetometer [30]. A threshold $\Delta h$ is used to eliminate the heading jitter when walking and keep the heading unchanged when the heading increment is smaller than $\Delta h$. S

_{k}is the flag of detected walking patterns, including walking forwards and backwards. S

_{k}is set to 1 when a walking forwards pattern is detected, while S

_{k}is set to −1 when a walking backwards pattern detected. In case of other walking patterns, S

_{k}is set to 0. U

_{k}is the flag of lateral walking patterns. U

_{k}is set to −1 when a left lateral walking pattern detected, and U

_{k}is set to 1 when a right lateral walking pattern detected. In case of other walking patterns, U

_{k}is set to 0. The real-time location of pedestrians can be accurately updated using the above equation. Compared with the traditional location updating method provided in [30], when backwards or lateral walking patterns happen, the proposed location updating algorithm can effectively classify the different walking patterns and go on to decrease the positioning error caused by misjudgment.

#### 2.3. Challenges of Indoor Positioning for Pedestrians

## 3. Ranging Model of Wi-Fi FTM

#### 3.1. Model of Clock Deviation Error

_{delay}, which is inconsistent between different hardware structures and processing methods of the signal—similar as TOA and DOA technology. Δt

_{delay}in one period of RTT can be described as follows:

_{true}is the true ranging result after subtracting Δt

_{delay}from real-time measurement result RTT

_{measurement}—t is defined in Equation (1).

_{random}can be assumed as Gaussian-distributed variables with zero mean and variance after calibration, which is described in Equation (10):

_{delay}exists before ranging which can’t be easily detected directly by hardware. As such, before using the Wi-Fi ranging system, some calibration measurements have to be done to eliminate the initial clock deviation Δt

_{delay}. Δt

_{random}exists during the ranging process which can result in signal fluctuation. Both errors cannot be fully eliminated by calibration and filter, so a further processing algorithm is needed.

#### 3.2. Model of NLOS and Multipath Propagation

**P**

_{i}, and location of the mobile terminal is indicated as

**P**. Taking the effect of NLOS and multipath into consideration can result in the following model:

_{i}is the measured distance, i is used to differentiate different APs, L

_{0}is the extra ranging distance caused by multipath,

**P**= [x

_{0}y

_{0}]

^{T}indicates the location of mobile terminal,

**P**

_{i}is the location of Wi-Fi AP, ‖

**P**−

**P**

_{i}‖ is the matrix norm that indicates the Euclidean distance between mobile terminal and Wi-Fi AP, e

_{i}is the NLOS error which indicates the difference between the final propagation distance of the signal and the true straight line distance when LOS path is lacked [45], d

_{random}is the random error of measurements which confront to a zero-mean Gaussian distribution with a variance of 0.25 [14], and Δt

_{random}is defined in Equation (10). We assume that e

_{i}is much larger than d

_{random}, with a boundary of b

_{i}, which always indicates the distance between mobile terminal to the farthest AP. Since d

_{random}is always between −0.5m and 0.5m according to [14], we move e

_{i}to the left side, square both sides, and neglect the d

^{2}

_{random}:

_{random}can be obtained as:

_{i}:

_{i}< b

_{i}. Assuming that the number of available Wi-Fi APs is N:

_{i}< b

_{i}, ${\mathrm{max}}_{{e}_{i}}f({e}_{i})$ can be divided into two cases:

_{i}<= b

_{i}, then ${\mathrm{max}}_{{e}_{i}}f({e}_{i})=\mathrm{max}\{f(0),f({L}_{i}),f({b}_{i})\}$.

_{i}> b

_{i}, then ${\mathrm{max}}_{{e}_{i}}f({e}_{i})=\mathrm{max}\{f(0),f({b}_{i})\}$.

**‖P‖**

^{2}, r = L

_{0}

^{2}, k

_{i}= 2L

_{0}‖

**P**−

**P**

_{i}‖ creates the following equation:

## 4. Integrated Localization Based on Wi-Fi FTM and PDR

#### 4.1. System Model Based on Unscented Kalman filter

^{T}are the 2D coordinates of the pedestrian at time t − 1, S(t) is the real-time step-length,

**v**is the Gaussian noise with a noise matrix

**Q**,$\mathit{\psi}$ is a unit matrix, and $\mathit{B}={[{S}_{t}\mathrm{sin}{\theta}_{t}+{U}_{t}\mathrm{cos}{\theta}_{t}{S}_{t}\mathrm{cos}{\theta}_{t}-{U}_{t}\mathrm{sin}{\theta}_{t}]}^{\mathrm{T}}$, indicates the difference of heading angle compared with initial heading. S

_{k}and U

_{k}are defined in Equation (6) and are used for multi-pattern step detection. The initial location (x(0), y(0)) is provided by the LS algorithm, which is defined in Equation (3).

**U**is the random error of Wi-Fi FTM with a noise matrix

**R**, $\mathit{U}={[{d}^{1}{}_{\mathrm{random}},{d}^{2}{}_{\mathrm{random}},\cdots ,{d}^{j}{}_{\mathrm{random}}]}^{T}$, and d

_{random}is defined in Equation (12). x(t) and y(t) are estimated by the system state equation. The Euclidean distance between the predicted position and each Wi-Fi AP is calculated as observation value, j is the number of Wi-Fi AP, and x

_{j}and y

_{j}indicate the position of each AP.

#### 4.2. Data Fusion via Unscented Kalman filter

- (1)
- Getting sigma point set based on the previous location $\stackrel{\wedge}{\mathit{X}}(t|t)$ and the corresponding weight:$$(\begin{array}{l}{\mathit{X}}^{(\eta )}(t|t)=[\stackrel{\eta =0}{\stackrel{\wedge}{\mathit{X}}(t|t)},\stackrel{\eta =1~\beta}{\stackrel{\wedge}{\mathit{X}}(t|t)+\sqrt{(\beta +\lambda )\mathit{\varphi}(t|t)}},\stackrel{\eta =\beta +1~2\beta}{\stackrel{\wedge}{\mathit{X}}(t|t)-\sqrt{(\beta +\lambda )\mathit{\varphi}(t|t)}}]\\ {w}^{(0)}=\frac{\lambda}{\beta +\lambda},\eta =0\\ {w}^{(\eta )}=\frac{\lambda}{2(\beta +\lambda )},\eta =1~2\beta \end{array}$$
- (2)
- Further prediction of $2\beta +1$ sigma point sets, $\eta =0,1,2,\cdots ,\beta +1$:$${\mathit{X}}^{(\eta )}(t+1|t)=\mathit{\psi}{\mathit{X}}^{(\eta )}(t|t)+\mathit{B}S(t+1)+v$$
- (3)
- Weighting sigma point set, getting predicted value and covariance matrix.$$\stackrel{\wedge}{\mathit{X}}(t+1|t)={\displaystyle \sum _{\eta =0}^{2\beta}{w}^{(\eta )}{\mathit{X}}^{(\eta )}(t+1|t)}$$$$\mathit{\varphi}(t+1|t)={\displaystyle \sum _{\eta =0}^{2\beta}w(\eta )[\stackrel{\wedge}{\mathit{X}}(t+1|t)}-{\mathit{X}}^{(\eta )}(t+1|t)]{[\stackrel{\wedge}{\mathit{X}}(t+1|t)-{\mathit{X}}^{(\eta )}(t+1|t)]}^{T}+\mathit{Q}$$
- (4)
- Getting the sigma point set again using UT transform based on the predicted state value.$${\mathit{X}}^{(\eta )}(t+1|t)=[\stackrel{\wedge}{\mathit{X}}(t+1|t),\stackrel{\wedge}{\mathit{X}}(t+1|t)+\sqrt{(\beta +\lambda )\mathit{\varphi}(t+1|t)},\stackrel{\wedge}{\mathit{X}}(t+1|t)-\sqrt{(\beta +\lambda )\mathit{\varphi}(t+1|t)}]$$
- (5)
- Further prediction of observation based on 2n + 1 sigma point sets of prediction, $\eta =0,1,2,\cdots ,2\beta +1$.$${\mathit{Z}}^{(\eta )}(t+1|t)=\left[\begin{array}{l}\sqrt{{({x}^{(\eta )}(t+1|t)-{x}_{0})}^{2}+{({y}^{(\eta )}(t+1|t)-{y}_{0})}^{2}}\\ \sqrt{{({x}^{(\eta )}(t+1|t)-{x}_{2})}^{2}+{({y}^{(\eta )}(t+1|t)-{y}_{2})}^{2}}\\ \vdots \\ \sqrt{{({x}^{(\eta )}(t+1|t)-{x}_{j})}^{2}+{({y}^{(\eta )}(t+1|t)-{y}_{j})}^{2}}\end{array}\right]$$
- (6)
- Weighting sigma point sets, getting predicted observation value, and corresponding covariance matrix.$$\stackrel{\wedge}{\mathit{Z}}(t+1|t)={\displaystyle \sum _{\eta =0}^{2\beta}{w}^{(\eta )}{\mathit{Z}}^{(\eta )}(t+1|t)}$$$${\mathit{\varphi}}_{{z}_{t}{z}_{t}}={\displaystyle \sum _{\eta =0}^{2\beta}w(\eta )[}{\mathit{Z}}^{(\eta )}(t+1|t)-\stackrel{\wedge}{\mathit{Z}}(t+1|t)]{[{\mathit{Z}}^{(\eta )}(t+1|t)-\stackrel{\wedge}{\mathit{Z}}(t+1|t)]}^{T}+\mathit{R}$$$${\mathit{\varphi}}_{{x}_{t}{z}_{t}}={\displaystyle \sum _{\eta =0}^{2\beta}w(\eta )[}{\mathit{X}}^{(\eta )}(t+1|t)-\stackrel{\wedge}{\mathit{Z}}(t+1|t)]{[{\mathit{X}}^{(\eta )}(t+1|t)-\stackrel{\wedge}{\mathit{Z}}(t+1|t)]}^{T}+\mathit{R}$$
- (7)
- Calculating the Kalman gain.$$\mathit{K}(t+1)={\mathit{\varphi}}_{{x}_{t}{z}_{t}}{\mathit{\varphi}}_{{z}_{t}{z}_{t}}{}^{-1}$$
- (8)
- System status and covariance updating.$$\stackrel{\wedge}{\mathit{X}}(t+1|t+1)=\stackrel{\wedge}{\mathit{X}}(t+1|t)+\mathit{K}(t+1)[\mathit{Z}(t+1)-\stackrel{\wedge}{\mathit{Z}}(t+1|t)]$$$$\mathit{\varphi}(t+1|t+1)=\mathit{\varphi}(t+1|t)-\mathit{K}(t+1){\mathit{\varphi}}_{{z}_{t}{z}_{t}}{\mathit{K}}^{T}(t+1)$$

## 5. Experimental Results of DRWMs

#### 5.1. Evaluation of Multi-Pattern-Based Dead Reckoning

#### 5.2. Experiment Results of Wi-Fi FTM-Based Ranging Model

#### 5.3. Experiment Results of DRWMs Algorithm

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Walking Pattern | True Steps | Detected Steps | Misclassification Steps | Error Rate |
---|---|---|---|---|

Forward | 100 | 98 | 2 (Not detected) | 2% |

Backward | 100 | 95 | 4 (Forward), 1(Not detected) | 5% |

Left Lateral | 100 | 92 | 5 (Forward), 3(Not detected) | 8% |

Right Lateral | 100 | 93 | 4 (Forward), 3(Not detected) | 7% |

Walking Pattern | True Distance/m | Detected Distance/m | Error Rate |
---|---|---|---|

Forward | 50 | 48.62 | 2.76% |

Backward | 50 | 48.34 | 3.32% |

Left Lateral | 50 | 47.58 | 4.84% |

Right Lateral | 50 | 47.91 | 4.18% |

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## Share and Cite

**MDPI and ACS Style**

Yu, Y.; Chen, R.; Chen, L.; Guo, G.; Ye, F.; Liu, Z. A Robust Dead Reckoning Algorithm Based on Wi-Fi FTM and Multiple Sensors. *Remote Sens.* **2019**, *11*, 504.
https://doi.org/10.3390/rs11050504

**AMA Style**

Yu Y, Chen R, Chen L, Guo G, Ye F, Liu Z. A Robust Dead Reckoning Algorithm Based on Wi-Fi FTM and Multiple Sensors. *Remote Sensing*. 2019; 11(5):504.
https://doi.org/10.3390/rs11050504

**Chicago/Turabian Style**

Yu, Yue, Ruizhi Chen, Liang Chen, Guangyi Guo, Feng Ye, and Zuoya Liu. 2019. "A Robust Dead Reckoning Algorithm Based on Wi-Fi FTM and Multiple Sensors" *Remote Sensing* 11, no. 5: 504.
https://doi.org/10.3390/rs11050504