# A Doppler-Tolerant Ultrasonic Multiple Access Localization System for Human Gait Analysis

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## Abstract

**:**

## 1. Introduction

## 2. Description of Problem

## 3. Proposed Method

#### 3.1. Signal Design

#### 3.1.1. Up and Down Chirp for Multiple Access

#### 3.1.2. Orthogonal Frequency Division Multiple Access

#### 3.2. Doppler Shift Compensation

## 4. Experimental Procedure

#### 4.1. Channel Multiple Access

#### 4.2. Gait Tracking with the Proposed System

#### Unscented Kalman Filter

- (A)
- Prediction
- Generate the matrix of sigma points (${\chi}_{k-1}$) and weights (${W}_{i}^{mean},{W}_{i}^{cov}$) for unscented transformation. Here n is the number of states and $n=6$ and $\lambda =-3$.$$\begin{array}{cc}\hfill {\chi}_{k-1}=& \left[{m}_{k-1}\phantom{\rule{1.em}{0ex}}...\phantom{\rule{1.em}{0ex}}{m}_{k-1}\right]+\hfill \\ & \sqrt{n+\lambda}\times \left[0\phantom{\rule{1.em}{0ex}}\sqrt{{P}_{k-1}}\phantom{\rule{1.em}{0ex}}-\sqrt{{P}_{k-1}}\right]\hfill \\ \hfill {W}_{0}^{mean}=& \frac{\lambda}{n+\lambda},\phantom{\rule{1.em}{0ex}}{W}_{0}^{cov}={W}_{0}^{mean}\hfill \\ \hfill {W}_{i}^{mean}=& {W}_{i}^{cov}=\frac{1}{2(n+\lambda )}\phantom{\rule{1.em}{0ex}}\mathrm{for}\phantom{\rule{1.em}{0ex}}i=1,\dots ,2n\hfill \end{array}$$
- Transmit the sigma points through the dynamic model.$${\widehat{\chi}}_{k,i}=f\left({\chi}_{k-1,i}\right)\phantom{\rule{1.em}{0ex}}i=1,\dots ,2n+1\phantom{\rule{-6.0pt}{0ex}}$$
- Predict the mean state matrix ${m}_{k}^{-}$ and co-variance matrix ${P}_{k}^{-}$ using weight matrices ${W}_{i-1}$.$$\begin{array}{c}{m}_{k}^{-}=\sum _{i}{W}_{i-1}^{mean}{\widehat{\chi}}_{k,i}\hfill \\ {P}_{k}^{-}=\sum _{i}{W}_{i-1}^{cov}({\widehat{\chi}}_{k,i}-{m}_{k}^{-}){({\widehat{\chi}}_{k,i}-{m}_{k}^{-})}^{T}+{Q}_{k-1}\hfill \end{array}$$

- (B)
- Update
- Generate the matrix of sigma points.$$\begin{array}{cc}\hfill {\chi}_{k}^{-}=& \left[{m}_{k}^{-}\phantom{\rule{1.em}{0ex}}\dots \phantom{\rule{1.em}{0ex}}{m}_{k}^{-}\right]\hfill \\ & +\sqrt{n+\lambda}\times \left[0\phantom{\rule{1.em}{0ex}}\sqrt{{P}_{k}^{-}}\phantom{\rule{1.em}{0ex}}-\sqrt{{P}_{k}^{-}}\right]\hfill \end{array}$$
- Transmit the sigma points through the dynamic model.$${\widehat{Y}}_{k,i}=h\left({\chi}_{k,i}^{-}\right)$$
- Compute the mean ${\mu}_{k}$, co-variance of the measurement ${S}_{k}$ and cross co-variance of the sate and measurement ${C}_{k}$ using weight matrices ${W}_{i}$.$$\begin{array}{c}{\mu}_{k}=\sum _{i}{W}_{i}^{mean}{\widehat{Y}}_{k,i}\hfill \\ {S}_{k}=\sum _{i}{W}_{i}^{cov}({\widehat{Y}}_{k,i}-{\mu}_{k}){({\widehat{Y}}_{k,i}-{\mu}_{k})}^{T}+{R}_{k}\hfill \\ {C}_{k}=\sum _{i}{W}_{i}^{cov}({\chi}_{k,i}^{-}-{m}_{k}^{-}){({\widehat{Y}}_{k,i}-{\mu}_{k})}^{T}\hfill \end{array}$$
- Compute the Kalman gain K, state mean ${m}_{k}$ and co-variance ${P}_{k}$ after obtaining measurement update ${y}_{k}$ and update mean state and co-variance.$$\begin{array}{c}K={C}_{k}{S}_{k}^{-1}\hfill \\ {m}_{k}={m}_{k}^{-}+K({y}_{k}-{\mu}_{k})\hfill \\ {P}_{k}={P}_{k}^{-}-K{S}_{k}{K}^{T}\hfill \end{array}$$

#### 4.3. Unscented Raunch-Tung Striebel Smoother (URTS)

## 5. Results and Discussions

#### 5.1. Calibration of Reference System

#### 5.2. Performance in Environments with Multiple Access Interference

#### 5.3. Performance of Doppler Shift Compensation

#### 5.4. Performance of Gait Tracking with Proposed Method

#### Extraction of Spatial and Temporal Parameters

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

3D | Three Dimensional |

NLOS | Non-Line of Sight |

LOS | Line of Sight |

RMSE | Root Mean Square Error |

PCC | Pearson’s Correlation Coefficient |

UKF | Unscented Kalman Filter |

URTS | Unscented Raunch-Tung Striebel Smoother |

IC | Integrated Circuit |

MAE | Mean Absolute Error |

RF | Radio Frequency |

## Appendix A. Steps for URTS

- Calculate the matrix of sigma points$${\chi}_{k}=\left[{m}_{k}\phantom{\rule{1.em}{0ex}}\dots \phantom{\rule{1.em}{0ex}}{m}_{k}\right]+\sqrt{(n+\lambda )}\left[0\phantom{\rule{1.em}{0ex}}\sqrt{{P}_{k}}\phantom{\rule{1.em}{0ex}}-\sqrt{{P}_{k}}\right]$$
- Transmit the sigma points through the dynamic model$${\widehat{\chi}}_{k+1,i}=f\left({\chi}_{k,i}\right),\phantom{\rule{1.em}{0ex}}i=1,\dots ,2n+1$$
- Predict the mean ${m}_{k+1}^{-}$, covariance ${P}_{k+1}^{-}$ and cross-covariance ${C}_{k+1}$ using weight matrices ${W}_{i-1}$.$$\begin{array}{c}{m}_{k+1}^{-}=\sum _{i}{W}_{i-1}^{mean}{\widehat{\chi}}_{k+1,i}\hfill \\ {P}_{k+1}^{-}=\sum _{i}{W}_{i-1}^{cov}({\widehat{\chi}}_{k+1,i}-{m}_{k+1}^{-}){({\widehat{\chi}}_{k+1,i}-{m}_{k+1}^{-})}^{T}+{Q}_{k}\hfill \\ {C}_{k+1}=\sum _{i}{W}_{i-1}^{cov}({\chi}_{k,i}-{m}_{k}){({\widehat{\chi}}_{k+1,i}-{m}_{k+1}^{-})}^{T}\hfill \end{array}$$
- Compute the smoother gain, $S{G}_{k}$ and final smoothed mean, ${m}_{k}^{s}$ and covariance ${P}_{k}^{s}$.$$\begin{array}{c}S{G}_{k}={C}_{k+1}{\left[{P}_{k+1}^{-}\right]}^{-1}\hfill \\ {m}_{k}^{s}={m}_{k}+S{G}_{k}({m}_{k+1}^{s}-{m}_{k+1}^{-})\hfill \\ {P}_{k}^{s}={P}_{k}+S{G}_{k}({P}_{k+1}^{s}-{P}_{k+1}^{-})S{G}_{k}^{T}\hfill \end{array}$$

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**Figure 1.**Illustration of up and down chirp rates for two signals with 39–41 kHz/9 ms and 41–39 kHz/9 ms chirp signals.

**Figure 2.**Comparison of (

**a**) auto-correlation and (

**b**) cross-correlation of ${S}_{1}$ and ${S}_{2}$ for up and down chirp signals and (

**c**)auto correlation and (

**d**) cross-correlation of $Tx1$ and $Tx2$ for orthogonal chirp signals.

**Figure 8.**The comparison of the range measurements from one of the lower limbs to a fixed anchor node extracted from the proposed ultrasonic system using up-chirp, down-chirp, a mean of up and down-chirp and motion capture system for a walking speed of 0.83 m/s.

**Figure 9.**The comparison of the 3D coordinates of one marker attached to one of the lower limbs extracted from the proposed ultrasonic system and motion capture system for a walking speed of 0.83 m/s.

**Figure 10.**Bland-Altman plots comparing (

**a**) step lengths and (

**b**) stride time calculated from the proposed system with the motion capture system at 0.83 m/s walking speed for one foot of a subject. The upper and lower limits are set to mean difference ± 2× standard deviation (SD).

Transmitter No. | Range to Anchor No. | MAE Using Up-Chirp (mm) | MAE Using Down-Chirp (mm) | MAE after Doppler Correction (mm) |
---|---|---|---|---|

1 | 1 | 103.72 | 99.03 | 9.45 |

2 | 105.45 | 100.31 | 10.85 | |

3 | 94.95 | 85.88 | 11.92 | |

4 | 93.99 | 84.34 | 12.16 | |

2 | 1 | 269.90 | 109.16 | 10.88 |

2 | 270.88 | 116.09 | 10.95 | |

3 | 241.58 | 97.61 | 13.13 | |

4 | 241.93 | 95.78 | 16.43 |

Subject | Axis | 0.28 m/s | 0.56 m/s | 0.83 m/s | |||
---|---|---|---|---|---|---|---|

L | R | L | R | L | R | ||

1 | X | 11.77 | 13.42 | 10.66 | 12.28 | 14.73 | 18.77 |

Y | 9.02 | 12.92 | 12.56 | 11.30 | 16.07 | 17.59 | |

Z | 13.84 | 17.81 | 14.72 | 11.87 | 21.37 | 19.17 | |

2 | X | 8.17 | 25.50 | 12.83 | 7.99 | 15.63 | 8.60 |

Y | 11.97 | 18.26 | 13.41 | 11.57 | 20.93 | 14.24 | |

Z | 14.21 | 26.67 | 18.47 | 14.06 | 25.56 | 12.73 | |

3 | X | 19.84 | 18.75 | 22.62 | 11.51 | 21.84 | 14.81 |

Y | 9.22 | 10.08 | 15.66 | 11.49 | 17.62 | 13.29 | |

Z | 21.94 | 24.27 | 25.09 | 12.72 | 23.34 | 13.96 | |

4 | X | 16.04 | 22.23 | 22.51 | 19.56 | 25.61 | 23.83 |

Y | 11.55 | 14.40 | 16.29 | 13.65 | 21.30 | 15.93 | |

Z | 17.67 | 23.87 | 29.99 | 20.61 | 28.39 | 22.16 | |

5 | X | 19.45 | 18.62 | 18.34 | 19 | 18.72 | 9.04 |

Y | 16.38 | 16.42 | 15.24 | 18.50 | 17.15 | 16.42 | |

Z | 21.97 | 26.65 | 21.55 | 22 | 23.40 | 16.02 |

**Table 3.**Mean and Standard deviation (SD) of error between the gait parameters estimated from proposed and motion capture system at a walking speed of 0.28 m/s. E[SL] and E[ST] represents the error in step length and stride time respectively.

Subject | Foot | E[SL] (mm) | E[ST] (ms) | ||
---|---|---|---|---|---|

Mean | SD | Mean | SD | ||

1 | left | −30.95 | 11.59 | 0.36 | 10.7 |

right | 4.02 | 13.95 | 0.18 | 9 | |

2 | left | −26.42 | 8.36 | 0.23 | 20 |

right | 18.63 | 25.46 | −0.9 | 26.4 | |

3 | left | −19.95 | 16.61 | −1.1 | 26.9 |

right | 4.11 | 10.72 | 0.81 | 11 | |

4 | left | −17.74 | 11.98 | 0.00 | 16.9 |

right | 9.96 | 13.26 | 0.49 | 16.9 | |

5 | left | −16.21 | 21.39 | 1.2 | 24.2 |

right | 10.51 | 18.21 | −0.59 | 26.5 |

**Table 4.**Mean and Standard deviation(SD) of error between the gait parameters estimated from proposed and motion capture system at a walking speed of 0.83 m/s. E[SL] and E[ST] represents the error in step length and stride time respectively.

Subject | Foot | E[SL] (mm) | E[ST] (ms) | ||
---|---|---|---|---|---|

Mean | SD | Mean | SD | ||

1 | left | −60.34 | 15.30 | −0.43 | 14 |

right | 8.73 | 20.71 | 0.00 | 9.7 | |

2 | left | −62.27 | 17.76 | 0.19 | 10.7 |

right | 7.00 | 9.32 | −0.37 | 8.5 | |

3 | left | −48.17 | 33.17 | 1.9 | 41.1 |

right | −6.00 | 21.64 | −0.2 | 19.9 | |

4 | left | −49.99 | 29.53 | 0.00 | 43.4 |

right | 6.09 | 21.28 | 0.38 | 18.7 | |

5 | left | −37.27 | 21.85 | 0.21 | 30.9 |

right | 11.71 | 13.55 | −0.40 | 12.5 |

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## Share and Cite

**MDPI and ACS Style**

Ashhar, K.; Khyam, M.O.; Soh, C.B.; Kong, K.H.
A Doppler-Tolerant Ultrasonic Multiple Access Localization System for Human Gait Analysis. *Sensors* **2018**, *18*, 2447.
https://doi.org/10.3390/s18082447

**AMA Style**

Ashhar K, Khyam MO, Soh CB, Kong KH.
A Doppler-Tolerant Ultrasonic Multiple Access Localization System for Human Gait Analysis. *Sensors*. 2018; 18(8):2447.
https://doi.org/10.3390/s18082447

**Chicago/Turabian Style**

Ashhar, Karalikkadan, Mohammad Omar Khyam, Cheong Boon Soh, and Keng He Kong.
2018. "A Doppler-Tolerant Ultrasonic Multiple Access Localization System for Human Gait Analysis" *Sensors* 18, no. 8: 2447.
https://doi.org/10.3390/s18082447