# A Novel Hybrid NN-ABPE-Based Calibration Method for Improving Accuracy of Lateration Positioning System

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## Abstract

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## 1. Introduction

- A new hybrid procedure based on ABPE and NNs is used to correct the positioning system measurements;
- Different neural network architectures are employed in order to find the optimally tuned parameters for the proposed calibration problem, e.g., 16 neural network architectures with 10 learning algorithms and 12 different activation functions for hidden layers are trained and validated in MATLAB environment to learn and predict measured positions;
- The performance of the novel hybrid NN-ABPE-based method in terms of both the set-up time and accuracy is compared to the state-of-the-art calibration methods, i.e., mapping with a distortion model, Bias and Scale Factor Estimation (BSFE), and Apparent Beacon Position Estimation (ABPE). Experimental results obtained in two different scenarios (environment with and without obstacles) confirmed the effectiveness of the proposed methodology to predict positioning system measurement errors in real-world situations.

## 2. Methods

#### 2.1. Position Correction in Positioning Systems

#### 2.1.1. Distortion Model

#### 2.1.2. Apparent Beacon Position Estimation

#### 2.1.3. Bias and Scale Factor Estimation

#### 2.1.4. Neural Networks

**I**is the input data,

**W**is the matrix of the weights, $\mathbf{\Theta}$ is the matrix of bias values, and f is the vector of activation functions for consecutive layers.

#### 2.1.5. Hybrid NN-ABPE Method

**Offline stage.**The calibration stage starts with the equipment setup, where beacons are placed in arbitrary positions surrounding the workspace. Next step is the data collection phase, where the pattern ${\mathbf{P}}_{\mathbf{j}}=({x}_{j},\phantom{\rule{0.166667em}{0ex}}{y}_{j}),\phantom{\rule{0.277778em}{0ex}}j=1,\cdots ,n$ of n reference points is assumed. The pattern $\mathbf{P}$ is chosen such that it uniformly covers the workspace with a desired resolution. The receiver is subsequently placed at consecutive points in the pattern. At each of these positions the beacon–receiver distances ${d}_{ij}$ are measured via UWB positioning system. The distances ${d}_{ij}$ and the pattern $\mathbf{P}$ are necessary as the input for the next stage of the algorithm.

**Online stage.**In the online stage, the distances ${d}_{ij}$ between the beacons and the receiver are measured, and the initial position estimate ${\mathbf{r}}^{\prime}=({x}^{\prime},\phantom{\rule{0.166667em}{0ex}}{y}^{\prime})$ is provided through NLS solver where the beacon positions ${\mathbf{A}}_{\mathbf{i}}=({X}_{i},\phantom{\rule{0.166667em}{0ex}}{Y}_{i}),\phantom{\rule{0.277778em}{0ex}}i=1,\cdots ,m$ are set according to the ABPE estimation obtained in the calibration stage. This position estimate ${\mathbf{r}}^{\prime}$ is further improved by setting it as the input of the neural network and acquiring the appropriate output. The NN used in this stage represents the one with the best validation performance obtained within the training process in the offline stage. The output of the network ${\mathbf{r}}^{\prime \prime}=({x}^{\prime \prime},\phantom{\rule{0.166667em}{0ex}}{y}^{\prime \prime})=net({\mathbf{r}}^{\prime},\mathbf{W},\mathbf{\Theta},\mathbf{f})$ is the corrected estimate for the position of the receiver and is the final output of the hybrid method. By achieving such output, the proposed calibration method is able to predict a more accurate estimate of the receiver position while simultaneously mitigating the systematic error.

## 3. Experimental Results

#### 3.1. Experiment 1

#### 3.2. Experiment 2

#### 3.3. Experiment 3

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Testing RMSE for 12 different activation functions in experiment 1. Red lines show the median, the blue boxes encompass the 25th and the 75th percentiles, the whiskers represent the range and the plus signs indicate outliers.

**Figure 5.**Measuring position patterns with different number of densities ${\rho}_{n}$ and number of points n used for training of the neural networks in experiment 2.

**Figure 6.**RMSE for ‘purelin’ activation function and different densities in experiment 2. Red lines show the median, the blue boxes encompass the 25th and the 75th percentiles, the whiskers represent the range and the plus signs indicate outliers.

**Figure 7.**Measuring position patterns with different number densities ${\rho}_{n}$ and number of points n used for training of the neural networks in experiment 3.

**Figure 8.**RMSE for ‘purelin’ activation function and different patterns in experiment 3. Red lines show the median, the blue boxes encompass the 25th and the 75th percentiles, the whiskers represent the range and the plus signs indicate outliers.

No. | Learning Algorithm | Acronym |
---|---|---|

1 | Levenberg–Marquardt back-propagation | LM |

2 | Bayesian regularization | BR |

3 | Resilient back-propagation | RP |

4 | Scaled conjugate gradient back-propagation | SCG |

5 | Gradient descent back-propagation | GD |

6 | Gradient descent with momentum back-propagation | GDM |

7 | Gradient descent with momentum and adaptive learning rule back-propagation | GDMA |

8 | Powell–Beale conjugate gradient back-propagation | PB |

9 | Fletcher–Powell conjugate gradient back-propagation | FP |

10 | Polak–Ribiére conjugate gradient back-propagation | PR |

No. | Architecture | No. | Architecture | |
---|---|---|---|---|

1 | 3 | 9 | 3-3-3 | |

2 | 5 | 10 | 5-5-5 | |

3 | 10 | 11 | 3-5-10 | |

4 | 15 | 12 | 5-10-15 | |

5 | 3-3 | 13 | 3-3-3-3 | |

6 | 5-5 | 14 | 5-5-5-5 | |

7 | 5-10 | 15 | 3-3-10-10 | |

8 | 3-15 | 16 | 5-5-10-15 |

**Table 3.**Best results for 12 activation functions in the experiment 1. Best six activation functions according to minimum RMSE are highlighted. Bold text indicates the best value in the column.

Activation Function | Arch | Alg | RMSE_Best [cm] | |||
---|---|---|---|---|---|---|

Max | Min | Median | Average | |||

logsig | 10 | 1 | 1.93 | 1.06 | 1.22 | 1.24 |

tansig | 6 | 1 | 1.31 | 0.82 | 1.14 | 1.13 |

softmax | 11 | 8 | 31.07 | 1.21 | 2.44 | 8.58 |

radbas | 6 | 1 | 6.43 | 0.95 | 1.35 | 1.58 |

compet | 3 | 3 | 36.77 | 15.46 | 26.42 | 26.63 |

tribas | 2 | 9 | 3.79 | 1.78 | 2.25 | 2.35 |

hardlim | 4 | 3 | 15.77 | 8.88 | 10.98 | 11.44 |

hardlims | 4 | 2 | 15.43 | 7.71 | 10.59 | 10.66 |

poslin | 15 | 7 | 47.41 | 3.08 | 25.70 | 19.11 |

purelin | 9 | 1 | 1.01 | 0.93 | 0.99 | 0.99 |

satlin | 7 | 2 | 1.43 | 0.96 | 1.14 | 1.16 |

satlins | 11 | 9 | 17.98 | 1.39 | 1.99 | 3.21 |

**Table 4.**Experiment 1 results for ‘purelin; activation function—best, average, and standard deviation for the testing set.

Arch | LM [cm] | BR [cm] | RP [cm] | SCG [cm] | GD [cm] | GDM [cm] | GDMA [cm] | PB [cm] | FP [cm] | PR [cm] | |
---|---|---|---|---|---|---|---|---|---|---|---|

3 | Best | 0.99 | 0.98 | 0.96 | 0.97 | 0.96 | 0.96 | 0.95 | 0.97 | 0.96 | 0.97 |

Ave | 1.00 | 3.31 | 1.00 | 1.00 | 4.15 | 4.16 | 1.51 | 1.00 | 1.00 | 1.00 | |

Std | 0.01 | 3.29 | 0.03 | 0.04 | 5.78 | 5.81 | 3.34 | 0.02 | 0.02 | 0.04 | |

5 | Best | 0.99 | 0.97 | 0.97 | 0.97 | 0.95 | 0.95 | 0.96 | 0.98 | 0.97 | 0.98 |

Ave | 0.99 | 2.58 | 1.02 | 1.00 | 1.15 | 1.14 | 1.03 | 1.00 | 1.00 | 1.00 | |

Std | 0.01 | 2.95 | 0.04 | 0.03 | 0.37 | 0.36 | 0.05 | 0.02 | 0.01 | 0.01 | |

10 | Best | 0.99 | 0.95 | 0.96 | 0.96 | 0.97 | 0.97 | 0.93 | 0.98 | 0.98 | 0.97 |

Ave | 0.99 | 3.38 | 1.02 | 1.00 | 0.99 | 0.99 | 1.05 | 1.00 | 1.00 | 1.00 | |

Std | 0.00 | 3.97 | 0.06 | 0.04 | 0.02 | 0.02 | 0.10 | 0.02 | 0.04 | 0.03 | |

15 | Best | 0.96 | 0.97 | 0.97 | 0.97 | 0.98 | 0.98 | 0.97 | 0.97 | 0.98 | 0.97 |

Ave | 1.00 | 2.48 | 1.02 | 1.00 | 0.99 | 0.99 | 1.05 | 1.00 | 1.00 | 1.01 | |

Std | 0.02 | 2.91 | 0.04 | 0.02 | 0.02 | 0.02 | 0.08 | 0.03 | 0.03 | 0.04 | |

3-3 | Best | 0.99 | 0.97 | 0.96 | 0.97 | 0.95 | 0.95 | 0.95 | 0.97 | 0.97 | 0.98 |

Ave | 0.99 | 0.99 | 1.00 | 1.00 | 3.96 | 5.54 | 3.97 | 1.00 | 1.00 | 1.02 | |

Std | 0.01 | 0.01 | 0.03 | 0.02 | 6.87 | 9.42 | 9.00 | 0.02 | 0.03 | 0.05 | |

5-5 | Best | 0.99 | 0.97 | 0.93 | 0.96 | 0.95 | 0.96 | 0.97 | 0.97 | 0.96 | 0.97 |

Ave | 0.99 | 0.99 | 1.00 | 1.00 | 1.00 | 7.79 | 2.68 | 1.00 | 1.00 | 1.01 | |

Std | 0.00 | 0.01 | 0.04 | 0.04 | 0.02 | 16.33 | 8.23 | 0.02 | 0.02 | 0.05 | |

5-10 | Best | 0.96 | 0.97 | 0.97 | 0.97 | 0.98 | 0.98 | 0.97 | 0.97 | 0.97 | 0.97 |

Ave | 0.99 | 1.65 | 1.01 | 1.00 | 1.47 | 9.77 | 1.54 | 1.00 | 1.00 | 1.01 | |

Std | 0.01 | 4.68 | 0.04 | 0.01 | 3.36 | 17.28 | 3.61 | 0.03 | 0.03 | 0.03 | |

3-15 | Best | 0.99 | 0.96 | 0.97 | 0.96 | 0.98 | 0.98 | 0.97 | 0.97 | 0.98 | 0.98 |

Ave | 0.99 | 0.98 | 1.01 | 1.01 | 1.00 | 12.80 | 2.53 | 1.01 | 1.00 | 1.00 | |

Std | 0.01 | 0.01 | 0.05 | 0.05 | 0.02 | 19.22 | 6.17 | 0.04 | 0.02 | 0.02 | |

3-3-3 | Best | 0.93 | 0.97 | 0.98 | 0.97 | 0.96 | 0.97 | 0.95 | 0.95 | 0.96 | 0.97 |

Ave | 0.99 | 5.76 | 1.02 | 1.01 | 6.39 | 9.68 | 6.88 | 1.27 | 1.00 | 1.01 | |

Std | 0.01 | 10.34 | 0.04 | 0.03 | 9.12 | 12.53 | 12.01 | 1.93 | 0.02 | 0.03 | |

5-5-5 | Best | 0.99 | 0.96 | 0.94 | 0.99 | 0.97 | 0.97 | 0.98 | 0.98 | 0.97 | 0.97 |

Ave | 0.99 | 4.04 | 1.03 | 1.00 | 2.61 | 17.18 | 5.89 | 1.01 | 1.01 | 1.00 | |

Std | 0.01 | 8.35 | 0.07 | 0.02 | 6.51 | 22.91 | 12.88 | 0.04 | 0.04 | 0.02 | |

3-5-10 | Best | 0.96 | 0.97 | 0.97 | 0.96 | 0.96 | 0.97 | 0.94 | 0.97 | 0.98 | 0.97 |

Ave | 0.99 | 5.14 | 1.01 | 1.01 | 1.78 | 26.39 | 2.93 | 1.01 | 1.13 | 1.00 | |

Std | 0.01 | 10.49 | 0.03 | 0.04 | 4.21 | 25.65 | 7.61 | 0.04 | 0.86 | 0.03 | |

5-10-15 | Best | 0.99 | 0.96 | 0.94 | 0.97 | 0.98 | 0.98 | 0.93 | 0.97 | 0.96 | 0.97 |

Ave | 0.99 | 2.54 | 1.01 | 1.00 | 1.00 | 30.82 | 1.02 | 1.00 | 1.00 | 1.01 | |

Std | 0.01 | 6.19 | 0.04 | 0.02 | 0.02 | 27.45 | 0.06 | 0.01 | 0.03 | 0.04 | |

3-3-3-3 | Best | 0.98 | 0.97 | 0.96 | 0.97 | 0.94 | 0.96 | 0.95 | 0.94 | 0.96 | 0.97 |

Ave | 1.00 | 18.11 | 1.02 | 1.00 | 10.02 | 15.59 | 14.10 | 1.01 | 4.70 | 1.00 | |

Std | 0.02 | 13.87 | 0.04 | 0.04 | 13.75 | 15.33 | 16.02 | 0.04 | 7.76 | 0.03 | |

5-5-5-5 | Best | 0.99 | 0.97 | 0.97 | 0.97 | 0.98 | 0.98 | 0.97 | 0.96 | 0.94 | 0.97 |

Ave | 0.99 | 17.53 | 1.01 | 1.00 | 1.52 | 22.82 | 6.67 | 1.01 | 1.16 | 1.00 | |

Std | 0.01 | 13.93 | 0.04 | 0.02 | 3.66 | 24.46 | 15.89 | 0.04 | 0.87 | 0.03 | |

3-3-10-10 | Best | 0.99 | 0.97 | 0.95 | 0.95 | 0.97 | 0.98 | 0.96 | 0.97 | 0.97 | 0.97 |

Ave | 1.00 | 20.83 | 1.00 | 1.01 | 1.93 | 35.23 | 5.39 | 1.01 | 1.65 | 1.02 | |

Std | 0.01 | 13.54 | 0.03 | 0.03 | 6.56 | 27.02 | 10.59 | 0.05 | 3.11 | 0.10 | |

5-5-10-15 | Best | 0.98 | 0.97 | 0.96 | 0.94 | 0.98 | 0.98 | 0.97 | 0.97 | 0.97 | 0.96 |

Ave | 0.99 | 19.39 | 1.01 | 1.02 | 1.00 | 55.34 | 2.65 | 1.01 | 1.01 | 1.01 | |

Std | 0.00 | 13.07 | 0.03 | 0.06 | 0.02 | 75.14 | 8.13 | 0.04 | 0.04 | 0.06 |

**Table 5.**Position RMSE for the different patterns and methods in experiment 2—without obstacles (in [cm]). Best result for each of the patterns is presented in bold.

Pattern | n | Raw [cm] | DQM [cm] | ABPE [cm] | BSFE [cm] | NN—ABPE Logsig [cm] | NN—ABPE Tansig [cm] | NN—ABPE Softmax [cm] | NN—ABPE Radbas [cm] | NN—ABPE Purelin [cm] | NN—RAW Purelin [cm] | NN—ABPE Satlin [cm] | IR [%] |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

#2.1 | 197 | 6.94 | 5.98 | 6.29 | 5.75 | 5.01 | 4.66 | 5.98 | 5.09 | 5.45 | 6.65 | 4.11 | 5.22 |

#2.2 | 50 | 6.94 | 6.18 | 6.38 | 5.74 | 5.33 | 5.29 | 5.41 | 5.16 | 4.38 | 4.10 | 4.96 | 23.69 |

#2.3 | 24 | 6.94 | 6.81 | 6.33 | 5.72 | 4.74 | 5.17 | 5.47 | 5.03 | 4.39 | 4.79 | 4.92 | 23.25 |

#2.4 | 15 | 6.94 | 6.68 | 6.72 | 6.15 | 5.49 | 5.61 | 5.38 | 5.76 | 4.37 | 5.06 | 4.95 | 28.94 |

#2.5 | 10 | 6.94 | 7.67 | 6.34 | 5.90 | 5.46 | 5.21 | 5.36 | 5.49 | 4.39 | 4.95 | 5.86 | 25.59 |

#2.6 | 6 | 6.94 | 10.91 | 9.60 | 7.89 | 7.90 | 7.33 | 7.23 | 10.42 | 4.59 | 4.73 | 8.39 | 41.83 |

#2.7 | 6 | 6.94 | 8.32 | 6.40 | 7.14 | 5.18 | 5.26 | 5.28 | 7.01 | 4.50 | 4.89 | 5.59 | 36.97 |

#2.8 | 5 | 6.94 | 8.09 | 9.39 | 6.01 | 9.02 | 9.56 | 8.45 | 38.30 | 4.66 | 4.75 | 11.93 | 22.46 |

**Table 6.**Position RMSE for the different patterns and methods in experiment 3—with obstacles (in [cm]). Best result for each of the patterns is presented in bold.

Pattern | n | Raw [cm] | DQM [cm] | ABPE [cm] | BSFE [cm] | NN—ABPE Logsig [cm] | NN—ABPE Tansig [cm] | NN—ABPE Softmax [cm] | NN—ABPE Radbas [cm] | NN—ABPE Purelin [cm] | NN—ABPE Satlin [cm] | IR [%] |
---|---|---|---|---|---|---|---|---|---|---|---|---|

#3.1 | 189 | 25.19 | 17.09 | 20.63 | 21.13 | 3.93 | 3.42 | 5.62 | 4.42 | 1.90 | 3.86 | 91.01 |

#3.2 | 48 | 25.19 | 17.84 | 21.51 | 21.26 | 15.91 | 13.76 | 15.86 | 17.84 | 9.08 | 13.21 | 57.29 |

#3.3 | 23 | 25.19 | 19.54 | 20.76 | 21.99 | 8.22 | 8.27 | 8.30 | 8.78 | 7.22 | 8.00 | 67.17 |

#3.4 | 14 | 25.19 | 20.18 | 21.40 | 22.39 | 10.74 | 10.14 | 10.56 | 10.96 | 7.67 | 9.87 | 65.74 |

#3.5 | 10 | 25.19 | 18.89 | 21.03 | 21.26 | 10.34 | 10.41 | 10.18 | 10.49 | 7.70 | 9.52 | 63.78 |

#3.6 | 6 | 25.19 | 21.29 | 26.36 | 22.32 | 30.00 | 32.66 | 28.74 | 56.05 | 9.74 | 29.20 | 56.36 |

#3.7 | 6 | 25.19 | 20.34 | 26.86 | 22.60 | 16.29 | 19.52 | 17.27 | 39.13 | 8.22 | 18.92 | 63.63 |

#3.8 | 5 | 25.19 | 23.74 | 24.32 | 24.62 | 11.32 | 28.08 | 15.14 | 62.87 | 8.05 | 14.42 | 67.30 |

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Petrović, M.; Ciężkowski, M.; Romaniuk, S.; Wolniakowski, A.; Miljković, Z.
A Novel Hybrid NN-ABPE-Based Calibration Method for Improving Accuracy of Lateration Positioning System. *Sensors* **2021**, *21*, 8204.
https://doi.org/10.3390/s21248204

**AMA Style**

Petrović M, Ciężkowski M, Romaniuk S, Wolniakowski A, Miljković Z.
A Novel Hybrid NN-ABPE-Based Calibration Method for Improving Accuracy of Lateration Positioning System. *Sensors*. 2021; 21(24):8204.
https://doi.org/10.3390/s21248204

**Chicago/Turabian Style**

Petrović, Milica, Maciej Ciężkowski, Sławomir Romaniuk, Adam Wolniakowski, and Zoran Miljković.
2021. "A Novel Hybrid NN-ABPE-Based Calibration Method for Improving Accuracy of Lateration Positioning System" *Sensors* 21, no. 24: 8204.
https://doi.org/10.3390/s21248204