# TDOA-Based Target Tracking Filter While Reducing NLOS Errors in Cluttered Environments

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Preliminary Information

#### 3.1. Time Difference of Arrival (TDOA)

#### 3.2. Kalman Filter (KF)

#### 3.2.1. Prediction

#### 3.2.2. Measurement Update

## 4. Transmitter Tracking While Decreasing NLOS Error in TDOA Measurements

#### 4.1. System Models

#### 4.2. Definitions and Assumptions

- (A1) Considering unknown, cluttered environments, at least three LOS receivers exist among all receivers. However, these LOS receivers are not known in advance.
- (A2) Considering LOS receivers, the measurement noise ${n}_{i}$ in Equation (2) has a Gaussian distribution with zero mean and variance ${\sigma}^{2}$ that is not known in advance.
- (A3) The transmitter exists inside a bounded workspace, whose boundary is known in advance.

#### 4.3. Least-Squares Estimation (LSE) for Solving the TDOA Localization

#### 4.4. NLOS Error Reduction Algorithm

- 1
- Let ${I}_{0}=\{{r}_{1},{r}_{2},\dots ,{r}_{N}\}$ define the set of all receivers. Initially, we set $K=N-3$.
- 2
- From ${I}_{0}$, we compute all receiver sets, such that each receiver set has K receivers. The number of total receiver sets is ${C}_{N}^{K}$. Let ${I}_{c}$ define each receiver set. Using all receivers in each receiver set ${I}_{c}$, one calculates the transmitter estimate ${\widehat{\mathbf{Z}}}_{c}$ utilizing the LSE solution (Equation (18)) in Section 4.3. In addition, we calculate the associated $\overline{Res}$, as defined in Equation (19).
- 3
- Under Equation (20), a reliable estimate has a smaller $\overline{Res}\left({I}_{c}\right)$. Therefore, we find a receiver set with the minimum $\overline{Res}$. Let ${I}_{min}\in \{{I}_{1},{I}_{2},\dots ,{I}_{{C}_{N}^{K}}\}$ define the found receiver set.
- 4
- Let $\parallel {I}_{min}\parallel $ define the number of elements in ${I}_{min}$. From ${I}_{min}$, we compute all receiver sets, such that each receiver set has $\parallel {I}_{min}\parallel -1$ receivers. In this way, we build $\parallel {I}_{min}\parallel $ new receiver sets, $\{{I}_{c},C=1,2,\dots ,\parallel {I}_{min}\parallel \}$. For the $\parallel {I}_{min}\parallel $ new receiver sets, one computes the transmitter estimate utilizing the LSE solution (Equation (18)) in Section 4.3. In addition, one utilizes Equation (19) to derive the associated $\overline{Res}$. Among the $\parallel {I}_{min}\parallel $ new receiver sets, one searches for the set with the minimum $\overline{Res}$. Let ${I}_{min}$ define the found receiver set.
- 5
- If $\parallel {I}_{min}\parallel $ becomes 3, then jump to the next step. Else, jump to step [4].
- 6
- Derive the fused estimate $\widehat{\mathbf{E}}$ utilizing Equation (23). If $K\ne {K}_{limit}$ and $\widehat{\mathbf{E}}$ is outside the bounded workspace, then update K under the ReceiverSelectOrder. Then, jump to Step [2]; else, this algorithm is finished, and we select $\widehat{\mathbf{E}}$ in Equation (23) as the algorithm output.

#### Exception Handling

#### 4.5. IMM KF

## 5. MATLAB Simulations

#### 5.1. Monte Carlo (MC) Simulations

#### 5.2. Scenario 1

^{2}. From 150 to 180 s, the transmitter varies its orientation with a change rate of $rat{e}_{k}^{a}=-3$ degrees per second. The simulation is finished after 300 s have elapsed.

#### 5.3. Scenario 2

^{2}. From 150 to 180 s, the transmitter varies its orientation with a change rate of $rat{e}_{k}^{a}=-1$ degree per second. The simulation is finished after 300 s have elapsed.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Scenario 1. The transmitter’s location at every sample-stamp is plotted with a red cross. The start point of the transmitter is marked with a black circle, and the end point of the transmitter is marked with a black diamond. Reflected signals can be generated due to obstacles, which are plotted with rectangles in the workspace. As the transmitter moves, an LOS receiver may become an NLOS receiver, and vice versa. At the moment when the simulation ends, LOS receivers are plotted with green asterisks, and NLOS receivers are plotted with black asterisks.

**Figure 2.**$RMS{E}_{k}$ with respect to sample-stamp k (scenario 1). We set $\sigma =5/C$ s, which implies that the distance noise in LOS measurements is 5 m. The proposed filters ($\left[Pro\right]$ and $IMM\left[Pro\right]$) outperform all other location methods.

**Figure 3.**Scenario 2. The transmitter’s position at every sample-stamp is plotted with a red cross. The start point of the transmitter is marked with a black circle, and the end point of the transmitter is marked with a black diamond. Reflected signals can be generated due to obstacles, which are plotted with rectangles in the workspace. As the transmitter moves, an LOS receiver may become an NLOS receiver, and vice versa. At the moment when the simulation ends, LOS receivers are plotted with green asterisks, and NLOS receivers are plotted with black asterisks.

**Figure 4.**$RMS{E}_{k}$ with respect to sample-stamp k (scenario 2). We set $\sigma =5/C$ s, which implies that the distance noise in LOS measurements is 5 m. The proposed filters ($\left[Pro\right]$ and $IMM\left[Pro\right]$) outperform all other location methods, considering the localization accuracy.

**Figure 5.**$RMS{E}_{k}$ with respect to sample-stamp k (scenario 2). We set $\sigma =10/C$ s. The proposed filters ($\left[Pro\right]$ and $IMM\left[Pro\right]$) outperform all other location methods.

**Table 1.**Computational load analysis (simulation of Figure 2).

$\mathit{Alg}.$ | $\mathit{One}\phantom{\rule{3.33333pt}{0ex}}\mathit{MC}\phantom{\rule{3.33333pt}{0ex}}\mathit{Time}$ |
---|---|

$IMM\left[Pro\right]$ | 2 s |

$\left[Pro\right]$ | 2 s |

$\left[Apo\right]$ | 4 s |

$\left[Su\right]$ | 78 s |

$\left[Yang\right]$ | 35 s |

**Table 2.**Computational load analysis (simulation of Figure 5).

$\mathit{Alg}.$ | $\mathit{One}\phantom{\rule{3.33333pt}{0ex}}\mathit{MC}\phantom{\rule{3.33333pt}{0ex}}\mathit{Time}$ |
---|---|

$IMM\left[Pro\right]$ | 3 s. |

$\left[Pro\right]$ | 3 s. |

$\left[Apo\right]$ | 6 s. |

$\left[Su\right]$ | 74 s. |

$\left[Yang\right]$ | 48 s. |

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**MDPI and ACS Style**

Kim, J.
TDOA-Based Target Tracking Filter While Reducing NLOS Errors in Cluttered Environments. *Sensors* **2023**, *23*, 4566.
https://doi.org/10.3390/s23094566

**AMA Style**

Kim J.
TDOA-Based Target Tracking Filter While Reducing NLOS Errors in Cluttered Environments. *Sensors*. 2023; 23(9):4566.
https://doi.org/10.3390/s23094566

**Chicago/Turabian Style**

Kim, Jonghoek.
2023. "TDOA-Based Target Tracking Filter While Reducing NLOS Errors in Cluttered Environments" *Sensors* 23, no. 9: 4566.
https://doi.org/10.3390/s23094566