# Application of Feedforward and Recurrent Neural Networks for Fusion of Data from Radar and Depth Sensors Applied for Healthcare-Oriented Characterisation of Persons’ Gait

## Abstract

**:**

## 1. Introduction

- The localisation of a person walking around the monitored area according to various predefined movement scenarios.
- The estimation of several parameters, carrying information important for medical experts, on the basis of the estimated movement trajectories.

## 2. Compared Methods of Data Fusion

#### 2.1. Method Based on Nonlinear Autoregressive Network with Exogenous Inputs

- Four input neurons—the input data represent two pairs of the x-y coordinates acquired by means of the impulse-radar sensor and depth sensor;
- Six neurons in a single hidden layer;
- Two output neurons—the output data represent one pair of fused x-y coordinates; the output values, delayed by two time instants, are fed back to the hidden layer;

#### 2.2. Method Based on Multilayer Perceptron

#### 2.3. Method Based on Kalman Filter

- ${x}_{n}={\left[{x}_{n}\text{\hspace{0.17em}}{x}_{n}^{\left(1\right)}\text{\hspace{0.17em}}{y}_{n}\text{\hspace{0.17em}}{y}_{n}^{\left(1\right)}\right]}^{\mathrm{T}}$ is a state vector representative of the two-dimensional coordinates $({x}_{n},\text{\hspace{0.17em}}{y}_{n})$, corresponding to a time instant ${t}_{n}$, and the velocities along these dimensions $\left({x}_{n}^{\left(1\right)},{y}_{n}^{\left(1\right)}\right)$;
- ${F}_{n}$ and ${\Gamma}_{n}$ are the matrices of the form:$${F}_{n}=\left[\begin{array}{cccc}1& {\Delta}_{n}& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& {\Delta}_{n}\\ 0& 0& 0& 1\end{array}\right]\text{}\mathrm{and}\text{}{\Gamma}_{n}=\left[\begin{array}{cc}{\Delta}_{n}^{2}/2& 0\\ {\Delta}_{n}& 0\\ 0& {\Delta}_{n}^{2}/2\\ 0& {\Delta}_{n}\end{array}\right]$$
- ${\alpha}_{n}={\left[{\alpha}_{x,n}\text{\hspace{0.17em}}{\alpha}_{y,n}\right]}^{\mathrm{T}}$ is a vector modelling the acceleration, whose elements are assumed to be the realisations of a zero-mean bivariate normal distribution with a known covariance matrix ${\Sigma}_{\alpha}=\mathrm{diag}\left\{{\sigma}_{\alpha ,x}^{2},{\sigma}_{\alpha ,y}^{2}\right\}$.

- $H$ is an observation matrix of the form:$$H=\left[\begin{array}{c}\begin{array}{cccc}1& 0& 0& 0\end{array}\\ \begin{array}{cccc}0& 0& 1& 0\end{array}\end{array}\right]$$
- ${\eta}_{n}={\left[{\eta}_{x,n}\text{\hspace{0.17em}}{\eta}_{y,n}\right]}^{\mathrm{T}}$ is a vector representative of the observation noise corrupting the data, whose elements are assumed to be realisations of a zero-mean bivariate normal distribution with a known covariance matrix ${\Sigma}_{\eta ,n}$.

- 1.
- Determination of the pre-estimate ${\widehat{x}}_{n}^{\mathrm{pre}}$ of the state vector and the pre-estimate ${\widehat{P}}_{k}^{\mathrm{pre}}$ of its covariance matrix:

- 2.
- Calculation of the estimate ${\widehat{z}}_{n}$ of the observation vector:

- 3.
- Calculation of the so-called innovation vector:$${\zeta}_{n}={\tilde{z}}_{n}-{\widehat{z}}_{n}$$$${\widehat{\Sigma}}_{\zeta ,n}=H{\widehat{P}}_{n}^{\mathrm{pre}}{H}^{\mathrm{T}}+{\Sigma}_{\eta ,n}$$

- 4.
- Calculation of the final estimate of the state vector:$${\widehat{x}}_{n}={\widehat{x}}_{n}^{\mathrm{pre}}+{G}_{n}{\zeta}_{n}$$

## 3. Extraction of Healthcare-Related Parameters

#### 3.1. Detection of Motion

#### 3.2. Estimation of Walking Direction and Moment of Turning

#### 3.3. Estimation of Travelled Distance

#### 3.4. Estimation of Walking Speed

## 4. Methodology of Experimentation

#### 4.1. Acquisition of Measurement Data

- Two impulse-radar sensors based on Novelda NVA6201 (Novelda, Oslo, Norway) chip working in the frequency range 6.0–8.5 GHz [46], and having the data acquisition rate of 10 Hz;
- An infrared depth sensor being a part of the Microsoft Kinect V2 device (Microsoft, Redmond, WA, United States) [47], having the data acquisition rate of 30 Hz.

- In experiment EXP#1, a person walked forth and back along a serpentine trajectory, among the obstacles occluding that person (see Figure 4b); 30 walks were performed and the walking speed was kept constant at $v=0.5$ m/s.

- In experiment EXP#2, a person walked clockwise and counter-clockwise along a rectangle-shaped trajectory (see Figure 4c), at six different values of the walking speed $v\in \left\{0.5,\text{\hspace{0.17em}}0.6,\text{\hspace{0.17em}}0.7,\text{\hspace{0.17em}}0.8,\text{\hspace{0.17em}}0.9,\text{\hspace{0.17em}}1.0\right\}$ m/s. For each value of the walking speed, 20 walks were performed.

#### 4.2. Criteria for Performance Evaluation

#### 4.2.1. Indicators of Uncertainty of Estimation of Monitored Person’s Position

- The area under the empirical cumulative distribution function $F\left(\xi \right)$ in the interval $\xi \in \left[0,1\right]$ m, denoted with ${\mathrm{A}}_{\mathrm{ECDF}}$, taking the value from the interval $\left[0,1\right]$;

- The mean position error (MEAE);
- The maximum position error (MAXE);
- The standard deviation of the position errors (STDE).

#### 4.2.2. Indicators of Uncertainty of Estimation of Healthcare-Related Parameters

- The mean error determined with respect to the corresponding reference value (ME);
- The standard deviation of that error (SE).

## 5. Results of Experimentation

#### 5.1. Uncertainty of Estimation of Monitored Person’s Position

- The fused estimates of the person’s position are characterised by lower bias and dispersion when compared with the radar-data-based and depth-data-based estimates; moreover, the fused estimates are only slightly affected by the obstacles occluding a person. It should be stressed that in the case of the methods based on the artificial neural networks, the significant increase in the accuracy of the position estimation has been achieved despite the fact that the artificial neural networks were trained only on the synthetic data.
- In experiment EXP#1, where the monitored person was occluded by the obstacles, the more accurate estimates of the trajectories have been obtained by means of the NARX method than by means of the MLP method. This result can be explained by the fact that in the case of the recurrent neural network, the prediction of the person’s position at each time instant is based on the past position of that person. When the monitored person “disappears” behind the obstacle, larger measurement errors may corrupt the radar data and the depth data, but the recurrent neural network may react to such sudden changes and mitigate their influence on the result of the data fusion.
- If the values of the uncertainty indicators, calculated on the basis of the fused data, are concerned, the best overall results have been obtained for the NARX method: it is reflected in the lowest values of the mean error and the median error, as well as the ${\mathrm{A}}_{\mathrm{ECDF}}$ indicator. Even though the values of the maximum error and the standard deviation of the errors are slightly greater when compared to the other methods, it does not affect the overall performance of the NARX method.

#### 5.2. Uncertainty of Estimation of Healthcare-Related Parameters

#### 5.2.1. Experiment EXP#1

- In the case of the radar data, the number of turns is significantly underestimated. This may be explained by the smoothing of the radar data during their preprocessing: as a result, two consecutive turns are often treated as one.
- In the case of the depth data, the travelled distance is significantly underestimated. This may be explained by the obstacles occluding the monitored person and making the tracking impossible.
- The radar-data-based and depth-data-based estimates of the mean walking speed are similarly accurate.

- The values of the uncertainty indicators are generally lower than the uncertainty indicators obtained for the radar data and for the depth data.
- The estimates of the number of turns, obtained for the MLP method and the NARX method, are considerably more accurate than those estimates obtained for the KF method—in the case of the last method, the number of turns for each realisation of the scenario has been overestimated by almost one. This may be explained by the corruption of the depth data related to “disappearing” of the monitored person behind the obstacles and the inability of the KF method to mitigate this phenomenon.

#### 5.2.2. Experiment EXP#2

- In the case of the estimates of the number of turns, the values of the ME indicator, determined on the basis of the radar data, are generally lower than zero—meaning that, regardless of the walking speed, the number of turns is underestimated; on the other hand, the values of the ME indicator, determined on the basis of the data fused by the KF method, are greater than zero, which means that the number of turns is overestimated. The best results are obtained for the depth data and for the data fused by means of the MLP method and the NARX method—the values of the ME indicator, determined on the basis of those data, are not significantly affected by the walking speed of the person.
- In the case of the estimates of the travelled distance and the estimates of the walking speed, the values of the ME indicator, determined on the basis of the data fused by means of the KF method, are slightly better than the values of this indicator, determined on the basis of the data fused by means of the other methods. Moreover, for all the methods of data fusion, the values of the ME indicator decrease with the increase in the walking speed of the monitored person. This phenomenon is caused by the deviations of the estimates of the walking trajectory around the corners of that trajectory—the greater the walking speed, the smoother the trajectory becomes and the smaller the estimates of the distance and walking speed.

## 6. Conclusions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Artificial neural networks used for fusion of data: the nonlinear autoregressive network with exogenous inputs (

**a**) and the multilayer perceptron (

**b**); ${x}_{\mathrm{r}}$ and ${y}_{\mathrm{r}}$ denote the coordinates acquired by means of the impulse-radar sensor; ${x}_{\mathrm{d}}$ and ${y}_{\mathrm{d}}$ denote the coordinates acquired by means of the depth sensor; ${x}_{\mathrm{f}}$ and ${y}_{\mathrm{f}}$ denote the fused coordinates; ${\mathrm{h}}_{1},\text{\hspace{0.17em}}\dots ,\text{\hspace{0.17em}}{\mathrm{h}}_{8}$ denote the neurons in the hidden layers; ${\mathrm{o}}_{1}$ and ${\mathrm{o}}_{2}$ denote the neurons in the output layers.

**Figure 3.**Synthetic data used for training the neural networks: the radar data (

**left column**), the depth data without occlusion (

**middle column**) and the depth data with occlusion (

**right column**). Each graph for radar data depicts 40 superimposed sequences of synthetic data, while each graph for depth data depicts 20 superimposed sequences of synthetic data.

**Figure 4.**Experimental setup (

**a**) and the movement scenarios considered in the experiments: the serpentine trajectory used in EXP#1 (

**b**), and the rectangle-shaped trajectory used in EXP#2 (

**c**). The reference points, i.e., the points where marks have been placed on the floor, are indicated with the crosses.

**Figure 5.**Estimates of the trajectories obtained in experiment EXP#1, on the basis of the radar data (

**a**), the depth data (

**b**), and the fused data (

**c**–

**e**); the black dashed lines denote the reference trajectories.

**Figure 6.**Zoom on the empirical cumulative distribution functions characterising the position errors, obtained in EXP#1.

**Figure 7.**Estimates of the trajectories obtained in experiment EXP#2, on the basis of the radar data (

**a**), the depth data (

**b**), and the fused data (

**c**–

**e**); the black dashed lines denote the reference trajectories.

**Figure 8.**Zoom on the empirical cumulative distribution functions characterising the position errors, obtained in EXP#2.

**Figure 9.**Estimates of the trajectories obtained in EXP#1, by means of the NARX method (

**a**), and the WINFNS method described in [37] (

**b**); for the NARX method, the value of the ${\mathrm{A}}_{\mathrm{ECDF}}$ indicator is 0.85, while for the WINFNS method, it is 0.83.

**Figure 10.**Uncertainty indicators characterising the estimates of the numbers of turns, obtained in EXP#2.

**Figure 11.**Uncertainty indicators characterising the estimates of the travelled distance, obtained in EXP#2.

**Figure 12.**Uncertainty indicators characterising the estimates of the walking speed, obtained in EXP#2.

**Table 1.**Values of the uncertainty indicators, calculated on the basis of the position errors, obtained in EXP#1.

Uncertainty Indicator | Radar Data | Depth Data | MLP | NARX | KF |
---|---|---|---|---|---|

MEAE [m] | 0.22 | 0.17 | 0.16 | 0.15 | 0.16 |

MEDE [m] | 0.20 | 0.18 | 0.14 | 0.12 | 0.16 |

MAXE [m] | 0.63 | 0.41 | 0.63 | 0.67 | 0.70 |

STDE [m] | 0.12 | 0.07 | 0.11 | 0.10 | 0.08 |

${\mathrm{A}}_{\mathrm{ECDF}}$ | 0.78 | 0.83 | 0.84 | 0.85 | 0.84 |

**Table 2.**Values of the uncertainty indicators, calculated on the basis of the position errors, obtained in EXP#2.

Uncertainty Indicator | Radar Data | Depth Data | MLP | NARX | KF |
---|---|---|---|---|---|

MEAE [m] | 0.14 | 0.12 | 0.07 | 0.07 | 0.09 |

MEDE [m] | 0.13 | 0.12 | 0.07 | 0.07 | 0.08 |

MAXE [m] | 0.60 | 0.33 | 0.32 | 0.34 | 0.43 |

STDE [m] | 0.08 | 0.05 | 0.04 | 0.05 | 0.06 |

${\mathrm{A}}_{\mathrm{ECDF}}$ | 0.86 | 0.88 | 0.93 | 0.93 | 0.91 |

Uncertainty Indicator | Radar Data | Depth Data | MLP | NARX | KF |
---|---|---|---|---|---|

Number of turns | |||||

ME | –6.00 | –0.73 | –0.13 | 0.07 | 0.93 |

SE | 1.08 | 0.91 | 1.36 | 1.34 | 1.20 |

Travelled distance [m] | |||||

ME | –1.43 | –6.86 | –0.74 | –0.79 | –0.53 |

SE | 0.42 | 0.13 | 0.30 | 0.33 | 0.25 |

Walking speed [m/s] | |||||

ME | –0.03 | –0.04 | –0.02 | –0.02 | –0.01 |

SE | 0.01 | 0.00 | 0.01 | 0.01 | 0.01 |

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**MDPI and ACS Style**

Mazurek, P.
Application of Feedforward and Recurrent Neural Networks for Fusion of Data from Radar and Depth Sensors Applied for Healthcare-Oriented Characterisation of Persons’ Gait. *Sensors* **2023**, *23*, 1457.
https://doi.org/10.3390/s23031457

**AMA Style**

Mazurek P.
Application of Feedforward and Recurrent Neural Networks for Fusion of Data from Radar and Depth Sensors Applied for Healthcare-Oriented Characterisation of Persons’ Gait. *Sensors*. 2023; 23(3):1457.
https://doi.org/10.3390/s23031457

**Chicago/Turabian Style**

Mazurek, Paweł.
2023. "Application of Feedforward and Recurrent Neural Networks for Fusion of Data from Radar and Depth Sensors Applied for Healthcare-Oriented Characterisation of Persons’ Gait" *Sensors* 23, no. 3: 1457.
https://doi.org/10.3390/s23031457