# Comprehensive Evaluations of NLOS and Linearization Errors on UWB Positioning

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Least Square

**P**is the priori weight matrix.

#### 2.2. Extended Kalman Filter

**F**represents the state transfer matrix;

**I**is the unit vector; ${w}_{\dot{p}}$ is the velocity noise.

**Q**is the system noise covariance matrix;

**R**is the measurement noise covariance matrix; ${H}_{k+1}$ is the Jacobian matrix.

#### 2.3. Robust Particle Filter

- (1)
- Setting the initial value ${X}_{0}$;
- (2)
- Particle initialization: Sampling particle (${x}_{i,0}$, i = 1, 2, …, m) is generated from the prior probability distribution $p({X}_{0})$;
- (3)
- Predictive step$${x}_{i,1}^{}=F{x}_{i,0}^{}+{q}_{i}$$
**N**(0,**Q**); - (4)
- Update step

**R**, and then, the corresponding weight ${w}_{i,1}^{}$ of the particle ${x}_{i,1}^{}$ is calculated [37] by

- (5)
- Normalization of weights$${w}_{i,1}^{}=\frac{{w}_{i,1}^{}}{{\displaystyle \sum {w}_{i,1}^{}}}$$

- (6)
- Particle resampling

- (7)
- Optimal state estimate

## 3. Tests and Evaluation

#### 3.1. Data Quality Analysis

#### 3.2. Evaluations Based on the Simulation Test

#### 3.3. Evaluations Based on the Field Test

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Part of the used equipment; (

**b**) the trajectory of the field test; (

**c**) UWB base station distribution; (

**d**) time-series of observed distances.

**Figure 3.**Residual histogram of UWB data in the simulation experiment (

**a**) and these in the field experiment (

**b**).

**Figure 8.**Positioning error statistics of different algorithms in static simulation experiments: (

**a**) 3D positioning error statistics of different algorithms; (

**b**) horizontal positioning error statistics of different algorithms; (

**c**) vertical positioning error statistics of different algorithms.

**Figure 12.**Positioning error statistics of different algorithms in the field experiment: (

**a**) 3D positioning error statistics of different algorithms; (

**b**) horizontal positioning error statistics of different algorithms; (

**c**) vertical positioning error statistics of different algorithms.

Algorithm | Advantages | Disadvantages |
---|---|---|

Bancroft | This algorithm does not need to iterate the direct solution method with algebraic analytic properties and the solution speed is fast [40]. | This algorithm cannot achieve the optimal solution in terms of statistical characteristics. |

LS | The optimal matching method between data is found by finding the sum of minimum error squares [41]. | It does not have error resistance, a small amount of gross error can cause unreliable parameter estimation, and the number of iterations and position accuracy are affected by the initial value. |

EKF | The EKF linearizes the nonlinear system locally and can be applied to the weak nonlinear system. | When linearizing the nonlinear equation, EKF retains only one term coefficient, and the truncation error caused by discarding the higher-order term will have a significant impact on positioning accuracy [21]. |

UKF | The UKF approximates the posterior probability density function of the nonlinear system through UT transformation, and computes the mean value and covariance of the state vector, avoiding the linearized truncation error [42]. | UKF has poor robustness in the case of system state mutation, and its accuracy is easily affected. |

CKF | CKF algorithm uses spherical radial volume criterion to approximate the state posterior distribution of optimal estimation. CKF can not only overcome the shortcomings of UKF in high and strong nonlinear state estimation. However, it also has higher filtering accuracy [25]. | The standard CKF requires Cholesky decomposition of the covariance of the posterior state when constructing volume points. This requires that the covariance of the transfer is a non-negative definite matrix, and the decomposition operation not only consumes time, but also reduces the stability of the increment algorithm [43]. |

PF | The core idea of the PF algorithm is to randomly select a group of particles to replace the posterior probability distribution of the current system state [44]. When the system is in a nonlinear environment, compared with other algorithms, this algorithm has better filtering performance and the ability to deal with the influence of non-Gaussian noise, so it is more and more widely used in the positioning system. | When the target state changes or bad measurement occurs, the tracking performance of the PF algorithm will decrease. The calculation is large and the calculation time is long. |

RPF | The influence of anomaly observation is weakened and the accuracy of parameter estimation and the reliability of filtering are improved [45]. | The calculation is large and the calculation time is long. |

**Table 2.**Specification parameters of devices (https://www.nooploop.com/en/, accessed on 9 May 2023).

Parameters | LinkTrack P |
---|---|

Size | 60.3 × 29 × 9 mm |

Weight | 33.3 g |

Maximum Communication Distance | 500 m |

Recommended Distance | 300 m |

Maximum Sampling Rate | 200 Hz |

Recommended Sampling Rate | 20 Hz |

One, two-dimensional Accuracy | 10 cm |

Three-dimensional Accuracy | 30 cm |

Frequencies | 4/4.5 GHz |

Band-wide | 499.2 MHz |

Bancroft | LS | EKF | UKF | CKF | PF | RPF | |
---|---|---|---|---|---|---|---|

Average (m) | −0.03 | −0.02 | −0.0008 | 0.0005 | 0.002 | 0.001 | 0.002 |

STD (m) | 0.09 | 0.07 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 |

Max (m) | 0.56 | 0.33 | 0.34 | 0.30 | 0.32 | 0.30 | 0.30 |

Bancroft | LS | EKF | UKF | CKF | PF | RPF | |
---|---|---|---|---|---|---|---|

Average (m) | 0.25 | −0.05 | 0.007 | 0.004 | 0.15 | 0.07 | 0.10 |

STD (m) | 1.33 | 0.76 | 0.66 | 0.66 | 0.87 | 0.68 | 0.67 |

Max (m) | 19.25 | 14.56 | 12.77 | 12.73 | 15.23 | 13.28 | 12.96 |

Bancroft | LS | EKF | UKF | CKF | PF | RPF | |
---|---|---|---|---|---|---|---|

Time (s) | 1.5 | 1.2 | 3.8 | 10.7 | 6.8 | 19.3 | 24.6 |

Bancroft | LS | EKF | UKF | CKF | PF | RPF | |
---|---|---|---|---|---|---|---|

Time (s) | 2.8 | 5.8 | 6.3 | 21.9 | 13.6 | 45.4 | 51.6 |

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## Share and Cite

**MDPI and ACS Style**

Li, Y.; Gao, Z.; Xu, Q.; Yang, C.
Comprehensive Evaluations of NLOS and Linearization Errors on UWB Positioning. *Appl. Sci.* **2023**, *13*, 6187.
https://doi.org/10.3390/app13106187

**AMA Style**

Li Y, Gao Z, Xu Q, Yang C.
Comprehensive Evaluations of NLOS and Linearization Errors on UWB Positioning. *Applied Sciences*. 2023; 13(10):6187.
https://doi.org/10.3390/app13106187

**Chicago/Turabian Style**

Li, Yan, Zhouzheng Gao, Qiaozhuang Xu, and Cheng Yang.
2023. "Comprehensive Evaluations of NLOS and Linearization Errors on UWB Positioning" *Applied Sciences* 13, no. 10: 6187.
https://doi.org/10.3390/app13106187