# 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

- Kwon, W.; Park, J.H.; Lee, M.; Her, J.; Kim, S.H.; Seo, J.W. Robust Autonomous Navigation of Unmanned Aerial Vehicles (UAVs) for Warehouses’ Inventory Application. IEEE Robot. Autom. Lett.
**2020**, 5, 243–249. [Google Scholar] [CrossRef] - Shi, D.; Mi, H.; Collins, E.G.; Wu, J. An Indoor Low-Cost and High-Accuracy Localization Approach for AGVs. IEEE Access
**2020**, 8, 50085–50090. [Google Scholar] [CrossRef] - Krug, R.; Stoyanov, T.; Tincani, V.; Andreasson, H.; Mosberger, R.; Fantoni, G.; Lilienthal, A.J. The Next Step in Robot Commissioning: Autonomous Picking and Palletizing. IEEE Robot. Autom. Lett.
**2016**, 1, 546–553. [Google Scholar] [CrossRef][Green Version] - Kamei, K.; Ikeda, T.; Shiomi, M.; Kidokoro, H.; Utsumi, A.; Shinozawa, K.; Miyashita, T.; Hagita, N. Cooperative customer navigation between robots outside and inside a retail shop—An implementation on the ubiquitous market platform. Ann. Telecommun. Ann. Télécommun.
**2012**, 67, 329–340. [Google Scholar] [CrossRef] - Demesure, G.; Defoort, M.; Bekrar, A.; Trentesaux, D.; Djemai, M. Navigation Scheme with Priority-Based Scheduling of Mobile Agents: Application to AGV-Based Flexible Manufacturing System. J. Intell. Robot. Syst.
**2016**, 82, 495–512. [Google Scholar] [CrossRef] - Sprunk, C.; Lau, B.; Pfaff, P.; Burgard, W. An accurate and efficient navigation system for omnidirectional robots in industrial environments. Auton. Robot.
**2017**, 41, 473–493. [Google Scholar] [CrossRef] - Papapostolou, A.; Chaouchi, H. Scene analysis indoor positioning enhancements. Ann. Télécommun.
**2011**, 66, 519–533. [Google Scholar] [CrossRef] - Schindhelm, C.; Macwilliams, A. Overview of Indoor Positioning Technologies for Context Aware AAL Applications. In Ambient Assisted Living; Springer: Berlin/Heidelberg, Germany, 2011; pp. 273–291. [Google Scholar] [CrossRef]
- Alarifi, A.; Al-Salman, A.; Alsaleh, M.; Alnafessah, A.; Al-Hadhrami, S.; Al-Ammar, M.A.; Al-Khalifa, H.S. Ultra Wideband Indoor Positioning Technologies: Analysis and Recent Advances. Sensors
**2016**, 16, 707. [Google Scholar] [CrossRef] - Farid, Z.; Nordin, R.; Ismail, M. Recent Advances in Wireless Indoor Localization Techniques and System. J. Comput. Netw. Commun.
**2013**, 2013, 185138. [Google Scholar] [CrossRef] - Marano, S.; Gifford, W.; Wymeersch, H.; Win, M. NLOS Identification and Mitigation for Localization Based on UWB Experimental Data. Sel. Areas Commun. IEEE J.
**2010**, 28, 1026–1035. [Google Scholar] [CrossRef][Green Version] - Sinriech, D.; Shoval, S. Landmark configuration for absolute positioning of autonomous vehicles. IIE Trans.
**2000**, 32, 613–624. [Google Scholar] [CrossRef] - Loevsky, I.; Shimshoni, I. Reliable and efficient landmark-based localization for mobile robots. Robot. Auton. Syst.
**2010**, 58, 520–528. [Google Scholar] [CrossRef] - Aksu, A.; Kabara, J.; Spring, M.B. Reduction of location estimation error using neural networks. In Proceedings of the First ACM International Workshop on Mobile Entity Localization and Tracking in GPS-Less Environments, San Francisco, CA, USA, 19 September 2008; pp. 103–108. [Google Scholar]
- Pelka, M.; Goronzy, G.; Hellbrück, H. Iterative approach for anchor configuration of positioning systems. ICT Express
**2016**, 2, 1–4. [Google Scholar] [CrossRef][Green Version] - Pierlot, V.; Droogenbroeck, M. BeAMS: A Beacon-Based Angle Measurement Sensor for Mobile Robot Positioning. IEEE Trans. Robot.
**2014**, 30, 533–549. [Google Scholar] [CrossRef][Green Version] - Meissner, P.; Steiner, C.; Witrisal, K. UWB positioning with virtual anchors and floor plan information. In Proceedings of the 2010 7th Workshop on Positioning, Navigation and Communication, Dresden, Germany, 11–12 March 2010; pp. 150–156. [Google Scholar] [CrossRef]
- Soltani, M.; Motamedi, A.; Hammad, A. Enhancing Cluster-based RFID Tag Localization using artificial neural networks and virtual reference tags. In Proceedings of the International Conference on Indoor Positioning and Indoor Navigation, Montbeliard, France, 28–31 October 2013; Volume 54, pp. 1–10. [Google Scholar] [CrossRef]
- Motamedi, A.; Soltani, M.; Hammad, A. Localization of RFID-equipped assets during the operation phase of facilities. Adv. Eng. Inform.
**2013**, 27, 566–579. [Google Scholar] [CrossRef] - Krapež, P.; Munih, M. Anchor Calibration for Real-Time-Measurement Localization Systems. IEEE Trans. Instrum. Meas.
**2020**, 69, 9907–9917. [Google Scholar] [CrossRef] - Wolniakowski, A.; Ciężkowski, M. Improving the Measurement Accuracy of the Static IR Triangulation System Through Apparent Beacon Position Estimation. In Proceedings of the 2018 23rd International Conference on Methods & Models in Automation & Robotics (MMAR), Miedzyzdroje, Poland, 27–30 August 2018; pp. 597–602. [Google Scholar] [CrossRef]
- Ciężkowski, M.; Romaniuk, S.; Wolniakowski, A. Apparent beacon position estimation for accuracy improvement in lateration positioning system. Measurement
**2020**, 153, 107400. [Google Scholar] [CrossRef] - Zekavat, R.; Buehrer, R.M. Handbook of Position Location: Theory, Practice and Advances; John Wiley & Sons: Hoboken, NJ, USA, 2011; Volume 27. [Google Scholar]
- Dardari, D.; Closas, P.; Djuric, P. Indoor Tracking: Theory, Methods, and Technologies. Veh. Technol. IEEE Trans.
**2015**, 64, 1263–1278. [Google Scholar] [CrossRef][Green Version] - Hartley, R. Theory and Practice of Projective Rectification. Int. J. Comput. Vis.
**1999**, 35, 115–127. [Google Scholar] [CrossRef] - Ronda, J.; Valdes, A. Geometrical Analysis of Polynomial Lens Distortion Models. J. Math. Imaging Vis.
**2019**, 61, 252–268. [Google Scholar] [CrossRef][Green Version] - Nelder, J.A.; Mead, R. A simplex method for function minimization. Comput. J.
**1965**, 7, 308–313. [Google Scholar] [CrossRef] - Vuković, N.; Petrović, M.; Miljković, Z. A comprehensive experimental evaluation of orthogonal polynomial expanded random vector functional link neural networks for regression. Appl. Soft Comput.
**2018**, 70, 1083–1096. [Google Scholar] [CrossRef] - Miljković, Z.; Petrović, M. Intelligent Manufacturing Systems—With Robotics and Artificial Intelligence Backgrounds, 1st ed.; Faulty of Mechanical Engineering, University of Belgrade: Belgrade, Serbia, 2021; 409p. [Google Scholar]
- Miljković, Z.; Aleksendrić, D. Artificial Neural Networks—Solved Examples with Theoretical Background, 2nd ed.; Faculty of Mechanical Engineering, University of Belgrade: Belgrade, Serbia, 2018; 225p. [Google Scholar]
- Wang, L.; Zeng, Y.; Chen, T. Back propagation neural network with adaptive differential evolution algorithm for time series forecasting. Expert Syst. Appl.
**2015**, 42, 855–863. [Google Scholar] [CrossRef] - Petrović, M.; Miljković, Z.; Babić, B. Integration of process planning, scheduling, and mobile robot navigation based on TRIZ and multi-agent methodology. FME Trans.
**2013**, 41, 120–129. [Google Scholar] - Petrović, M.; Miljković, Z.; Babić, B.; Vuković, N.; Čović, N. Towards a conceptual design of intelligent material transport using artificial intelligence. Stroj. Časopis Za Teor. Praksu Stroj.
**2012**, 54, 205–219. [Google Scholar] - Petrović, M.; Wolniakowski, A.; Ciezkowski, M.; Romaniuk, S.; Miljković, Z. Neural Network-Based Calibration for Accuracy Improvement in Lateration Positioning System. In Proceedings of the 2020 International Conference Mechatronic Systems and Materials (MSM), Bialystok, Poland, 1–3 July 2020. [Google Scholar] [CrossRef]

**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