# Analysis and Accuracy Improvement of UWB-TDoA-Based Indoor Positioning System

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## Abstract

**:**

## 1. Introduction

- (i)
- A theoretical study of the precision of the position estimates is performed based on a CRLB analysis for round-robin scheduling and an anisotropic representation of the signal-to-noise ratio function of the 3D radiation pattern of the anchor antennas.
- (ii)
- A geometrical study of the 2D IPS domain is carried out, defining bifurcation envelopes that bound the areas where the IPS is predicted to fail. This complements the CRLB analysis, which does not predict regions of failure. Together, they define the so-called flyable area in which positioning is reliable.
- (iii)
- Experiments using an existing IPS with four anchors and a static tagged object are used to validate the precision and failure predictions and to estimate the bias (inaccuracy).
- (iv)
- A debiasing filter is developed to increase the accuracy of the static position estimates, which is then tested for both static and moving tagged objects.

## 2. Theoretical Study of Precision and Failure

#### 2.1. IPS under Study

`bitcraze`’s

`Loco Positioning System`[51] and consists of a drone to be localised and four transceiver anchors positioned at the vertices and facing the centre of a $4\times 4\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}$ domain, as shown in Figure 1. All antennas under study are at a height of 20 cm above the floor; the drone is mounted on a sliding ground-referenced measurement system parallel to the floor equipped with a laser pointer aligned with the onboard UWB antenna to achieve reference positioning with high precision (±1 mm) and accuracy (see Figure 2). This experimental setup is used in Section 4 and Section 5, where the regularly spaced markers on the floor are the sampling positions to be used.

#### 2.2. CRLB Analysis for Pseudo-Range Multilateration with Round-Robin Scheduling

`Loco Positioning System`[51,53] is as in Equation (3).

_{0}is the signal-to-noise power ratio at a threshold distance ${r}_{0}$ from the ${i}^{\mathrm{th}}$ anchor under consideration, c is the signal propagation speed, and B is the bandwidth of the received signal. The SNR

_{0}varies with the view angle $\theta $ if the antenna has some directionality. In order to evaluate the SNR($\mathbf{x}$), we use the Friis formula for noise, which provides the relation between the signal gain (over noise) and distance between the transmitter and receiver for different channel frequencies.

#### 2.2.1. Signal-to-Noise Ratio Formulation

#### 2.2.2. Radiation Pattern of the DW1000 Anchor Antenna

`BitCraze`seemingly recommending that they be placed in pairs facing each other and forming a 90-degree angle with the floor (see figure with eight anchors placed in a room in [56]). Furthermore, this study provides additional variables for the optimal design problem formulation of IPSs, namely, the antenna orientations.

#### 2.2.3. Analytical Results of CRLB Analysis

#### 2.3. Bifurcation Envelope Analysis

#### 2.3.1. Bifurcation Curve

#### 2.3.2. Bifurcation Envelope

- 1.
- The unique-solution area, defined as the intersection of all concave areas outside each green bifurcation envelope (i.e., not including anchors).
- 2.
- The region with acceptable precision returned by the CRLB analysis (the convex hull).

## 3. Filter Design

#### 3.1. Proposed Filter Design

`Crazyflie 2.0`nano-quadcopter and the IPS is

`bitcraze`’s

`Loco Positioning System`[56]. This is setup already equipped with an Extended Kalman Filter (EKF) [57,58], which transforms raw sensor measurements into better estimates of the state of the drone (i.e., higher precision). The EKF developers note that the position estimates are affected by a measurement bias which appears to be non-uniform in space. In other words, the quadcopter is estimated to be in a position that is shifted from the actual one. To address this issue, we proposed that a debiasing filter be incorporated here after the built-in EKF.

#### 3.2. Debiasing Filter

- 1.
- The bias values are available only at a limited set of points, and therefore they need to be interpolated to cover the continuous domain.
- 2.
- The bias to be subtracted from a measured position to obtain the actual one is a function of the actual position itself.

## 4. Design of Experiments

#### 4.1. IPS Bias Map Generation

#### 4.2. DF Calibration and Validation Setup

#### 4.3. DF Validation under Dynamic Setup Conditions

#### 4.4. Square Path Experiment Setup

## 5. Results and Discussion

#### 5.1. Proof of Accuracy Improvement

#### 5.2. Dynamic Validation of Debiasing

_{IPS-1}on the bottom edge is obtained using the data cloud (cyan colour in Figure 18) of ten experiments on the edge from $(0.5,0.5)$ to $(3.5,0.5)$. The same dynamic issues that were pointed out in the validation experiment in Figure 17 persist in the square-path experiment (yellow regions in Figure 18). Assuming that the DF is insufficient to address the intrinsic problems of the IPS under study, we believe it may be useful to isolate this misbehaviour and provide statistics on the data unaffected by this (hence, the raw and sel. columns in Table 2).

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

IPS | Indoor Positioning System |

ToA | Time of Arrival |

TDoA | Time Difference of Arrival |

UWB | Ultra-Wideband |

CRLB | Cramér–Rao Lower Bound |

GNSS | Global Navigation Satellite System |

GDoP | Geometric Dilution of Precision |

CRLB | Cramér–Rao Lower Bound |

EKF | Extended Kalman Filter |

DF | Debiasing filter |

## Appendix A. Radial Basis Function Network Implementation

**Figure A1.**RBFN of bias values on marker points (red stars) for estimating the position of the x (

**left**) and y (

**right**) components.

**Figure A2.**RBFN surface interpolating bias values of estimated position of the x (

**left**) and y (

**right**) components.

## Appendix B. Reference CRLB Analysis

## Appendix C. Initial Filter Design

- 1.
- Extended Kalman Filter
- 2.
- Saturation (and artificial smoothing)
- 3.
- Correction of position via fourth-order Adams–Moulton (AM4) method
- 4.
- Debiasing filter

**Figure A3.**Proposed filtering process, consisting of four steps and returning estimates identified with their respective filter symbols: $\overline{\mathbf{x}}$ stands for saturated, $\tilde{\mathbf{x}}$ for dynamically corrected, and $\widehat{\mathbf{x}}$ for debiased. Superscripts ${}^{\mathrm{b}}$ and ${}^{\mathrm{g}}$ refer to the body frame and inertial frame, respectively.

#### Appendix C.1. Saturation and Smoothing Filter

**Figure A4.**Representation of the presented saturation filter on velocity estimates in ith direction; blue shows the original EKF estimate and red shows the correction. Note that the measurements are discrete and represented by the peaks; the linear interpolation between measurements is only for visualisation purposes.

#### Appendix C.2. Adams–Moulton Filter

**Figure A5.**Weight function for averaging EKF state estimations and AM4 predictions; ${v}_{i\phantom{\rule{0.166667em}{0ex}}\mathrm{f}}$ is the flipping velocity, ${v}_{i\phantom{\rule{0.166667em}{0ex}}\mathrm{max}}$ is the maximum expected speed, and ${\alpha}_{\mathrm{min}}$ is a calibration parameter.

## References

- Elsanhoury, M.; Makela, P.; Koljonen, J.; Valisuo, P.; Shamsuzzoha, A.; Mantere, T.; Elmusrati, M.; Kuusniemi, H. Precision Positioning for Smart Logistics Using Ultra-Wideband Technology-Based Indoor Navigation: A Review. IEEE Access
**2022**, 10, 44413–44445. [Google Scholar] [CrossRef] - Roy, P.; Chowdhury, C. A Survey of Machine Learning Techniques for Indoor Localization and Navigation Systems. J. Intell. Robot. Syst.
**2021**, 101, 63. [Google Scholar] [CrossRef] - Chen, D.; Neusypin, K.; Selezneva, M.; Mu, Z. New Algorithms for Autonomous Inertial Navigation Systems Correction with Precession Angle Sensors in Aircrafts. Sensors
**2019**, 19, 5016. [Google Scholar] [CrossRef] [PubMed] - Rong, H.; Gao, Y.; Guan, L.; Zhang, Q.; Zhang, F.; Li, N. GAM-Based Mooring Alignment for SINS Based on An Improved CEEMD Denoising Method. Sensors
**2019**, 19, 3564. [Google Scholar] [CrossRef] - Widodo, S.; Shiigi, T.; Hayashi, N.; Kikuchi, H.; Yanagida, K.; Nakatsuchi, Y.; Ogawa, Y.; Kondo, N. Moving Object Localization Using Sound-Based Positioning System with Doppler Shift Compensation. Robotics
**2013**, 2, 36–53. [Google Scholar] [CrossRef] - Schott, D.J.; Saphala, A.; Fischer, G.; Xiong, W.; Gabbrielli, A.; Bordoy, J.; Höflinger, F.; Fischer, K.; Schindelhauer, C.; Rupitsch, S.J. Comparison of Direct Intersection and Sonogram Methods for Acoustic Indoor Localization of Persons. Sensors
**2021**, 21, 4465. [Google Scholar] [CrossRef] [PubMed] - Arbula, D.; Ljubic, S. Indoor Localization Based on Infrared Angle of Arrival Sensor Network. Sensors
**2020**, 20, 6278. [Google Scholar] [CrossRef] - Mahmoud, A.A.; Ahmad, Z.U.; Haas, O.C.; Rajbhandari, S. Precision indoor three-dimensional visible light positioning using receiver diversity and multi-layer perceptron neural network. IET Optoelectron.
**2020**, 14, 440–446. [Google Scholar] [CrossRef] - Alarifi, A.; Al-Salman, A.; Alsaleh, M.; Alnafessah, A.; Al-Hadhrami, S.; Al-Ammar, M.; Al-Khalifa, H. Ultra Wideband Indoor Positioning Technologies: Analysis and Recent Advances. Sensors
**2016**, 16, 707. [Google Scholar] [CrossRef] [PubMed] - Tsang, Y.P.; Wu, C.H.; Ip, W.; Ho, G.; Tse, M. A Bluetooth-based Indoor Positioning System: A Simple and Rapid Approach. Annu. J. IIE
**2015**, 35, 11–26. [Google Scholar] - Zhao, X.; Xiao, Z.; Markham, A.; Trigoni, N.; Ren, Y. Does BTLE measure up against WiFi? A comparison of indoor location performance. In Proceedings of the European Wireless 2014; 20th European Wireless Conference, Barcelona, Spain, 14–16 May 2014; pp. 1–6. [Google Scholar]
- Ezhumalai, B.; Song, M.; Park, K. An Efficient Indoor Positioning Method Based on Wi-Fi RSS Fingerprint and Classification Algorithm. Sensors
**2021**, 21, 3418. [Google Scholar] [CrossRef] [PubMed] - Abusara, A.; Hassan, M.S.; Ismail, M.H. Reduced-complexity fingerprinting in WLAN-based indoor positioning. Telecommun. Syst.
**2016**, 65, 407–417. [Google Scholar] [CrossRef] - Sahota, H.; Kumar, R. Sensor Localization Using Time of Arrival Measurements in a Multi-Media and Multi-Path Application of In-Situ Wireless Soil Sensing. Inventions
**2021**, 6, 16. [Google Scholar] [CrossRef] - Sakpere, W.; Oshin, M.A.; Mlitwa, N.B. A State-of-the-Art Survey of Indoor Positioning and Navigation Systems and Technologies. S. Afr. Comput. J.
**2017**, 29, 145–197. [Google Scholar] [CrossRef] - Hernandez, L.A.M.; Arteaga, S.P.; Perez, G.S.; Orozco, A.L.S.; Villalba, L.J.G. Outdoor Location of Mobile Devices Using Trilateration Algorithms for Emergency Services. IEEE Access
**2019**, 7, 52052–52059. [Google Scholar] [CrossRef] - Mosleh, M.F.; Zaiter, M.J.; Hashim, A.H. Position Estimation Using Trilateration based on ToA/RSS and AoA Measurement. J. Phys. Conf. Ser.
**2021**, 1773, 012002. [Google Scholar] [CrossRef] - Neirynck, D.; Luk, E.; McLaughlin, M. An alternative double-sided two-way ranging method. In Proceedings of the 2016 13th Workshop on Positioning, Navigation and Communications (WPNC), Bremen, Germany, 19–20 October 2016. [Google Scholar] [CrossRef]
- Jamil, F.; Iqbal, N.; Ahmad, S.; Kim, D.H. Toward Accurate Position Estimation Using Learning to Prediction Algorithm in Indoor Navigation. Sensors
**2020**, 20, 4410. [Google Scholar] [CrossRef] [PubMed] - Mahida, P.; Shahrestani, S.; Cheung, H. Deep Learning-Based Positioning of Visually Impaired People in Indoor Environments. Sensors
**2020**, 20, 6238. [Google Scholar] [CrossRef] [PubMed] - Alraih, S.; Alhammadi, A.; Shayea, I.; Al-Samman, A.M. Improving accuracy in indoor localization system using fingerprinting technique. In Proceedings of the 2017 International Conference on Information and Communication Technology Convergence (ICTC), Jeju, Republic of Korea, 18–20 October 2017. [Google Scholar] [CrossRef]
- Alhammadi, A.; Alraih, S.; Hashim, F.; Rasid, M.F.A. Robust 3D Indoor Positioning System Based on Radio Map Using Bayesian Network. In Proceedings of the 2019 IEEE 5th World Forum on Internet of Things (WF-IoT), Limerick, Ireland, 15–18 April 2019. [Google Scholar] [CrossRef]
- Alhammadi, A.; Hashim, F.; Rasid, M.F.A.; Alraih, S. A three-dimensional pattern recognition localization system based on a Bayesian graphical model. Int. J. Distrib. Sens. Netw.
**2020**, 16, 155014771988489. [Google Scholar] [CrossRef] - European Union Agency for the Space Programme. Galileo Initial Services. 2021. Available online: https://www.euspa.europa.eu/european-space/galileo/services/initial-services (accessed on 18 August 2021).
- Yeh, S.C.; Hsu, W.H.; Su, M.Y.; Chen, C.H.; Liu, K.H. A study on outdoor positioning technology using GPS and WiFi networks. In Proceedings of the 2009 International Conference on Networking, Sensing and Control, Okayama, Japan, 26–29 March 2009. [Google Scholar] [CrossRef]
- Ghavami, M.; Michael, L.; Kohno, R. Ultra Wideband Signals and Systems in Communication Engineering; John Wiley & Sons: Hoboken, NJ, USA, 2007. [Google Scholar]
- Innocente, M.S.; Grasso, P. Self-organising swarms of firefighting drones: Harnessing the power of collective intelligence in decentralised multi-robot systems. J. Comput. Sci.
**2019**, 34, 80–101. [Google Scholar] [CrossRef] - Bezas, K.; Tsoumanis, G.; Angelis, C.T.; Oikonomou, K. Coverage Path Planning and Point-of-Interest Detection Using Autonomous Drone Swarms. Sensors
**2022**, 22, 7551. [Google Scholar] [CrossRef] [PubMed] - Cramér, H. Mathematical Methods of Statistics (PMS-9); Princeton University Press: Princeton, NJ, USA, 1999; Volume 9. [Google Scholar]
- Chen, C.S.; Chiu, Y.J.; Lee, C.T.; Lin, J.M. Calculation of Weighted Geometric Dilution of Precision. J. Appl. Math.
**2013**, 2013, 953048. [Google Scholar] [CrossRef] - Sieskul, B.; Kaiser, T. Cramer-Rao Bound for TOA Estimations in UWB Positioning Systems. In Proceedings of the 2005 IEEE International Conference on Ultra-Wideband, Zurich, Switzerland, 5–8 September 2005. [Google Scholar] [CrossRef]
- Amigo, A.G.; Closas, P.; Mallat, A.; Vandendorpe, L. Cramér-Rao Bound analysis of UWB based Localization Approaches. In Proceedings of the 2014 IEEE International Conference on Ultra-WideBand (ICUWB), Paris, France, 1–3 September 2014. [Google Scholar] [CrossRef]
- Zhang, J.; Kennedy, R.A.; Abhayapala, T.D. Cramér-Rao Lower Bounds for the Synchronization of UWB Signals. EURASIP J. Adv. Signal Process.
**2005**, 2005, 293649. [Google Scholar] [CrossRef] - Alhakim, R.; Raoof, K.; Simeu, E.; Serrestou, Y. Cramer–Rao lower bounds and maximum likelihood timing synchronization for dirty template UWB communications. Signal Image Video Process.
**2011**, 7, 741–757. [Google Scholar] [CrossRef] - D’Amico, A.A.; Mengali, U.; Taponecco, L. Cramer-Rao Bound for Clock Drift in UWB Ranging Systems. IEEE Wirel. Commun. Lett.
**2013**, 2, 591–594. [Google Scholar] [CrossRef] - Mallat, A.; Louveaux, J.; Vandendorpe, L. UWB based positioning: Cramer Rao bound for Angle of Arrival and comparison with Time of Arrival. In Proceedings of the 2006 Symposium on Communications and Vehicular Technology, Liege, Belgium, 23 November 2006. [Google Scholar] [CrossRef]
- Silva, B.; Pang, Z.; Akerberg, J.; Neander, J.; Hancke, G. Experimental study of UWB-based high precision localization for industrial applications. In Proceedings of the 2014 IEEE International Conference on Ultra-WideBand (ICUWB), Paris, France, 1–3 September 2014. [Google Scholar] [CrossRef]
- Grasso, P.; Innocente, M.S. Theoretical study of signal and geometrical properties of Two-dimensional UWB-based Indoor Positioning Systems using TDoA. In Proceedings of the 2020 6th International Conference on Mechatronics and Robotics Engineering (ICMRE), Barcelona, Spain, 12–15 February 2020. [Google Scholar] [CrossRef]
- Compagnoni, M.; Notari, R.; Antonacci, F.; Sarti, A. A comprehensive analysis of the geometry of TDOA maps in localization problems. Inverse Probl.
**2014**, 30, 035004. [Google Scholar] [CrossRef] - Dulman, S.O.; Baggio, A.; Havinga, P.J.; Langendoen, K.G. A geometrical perspective on localization. In Proceedings of the First ACM International Workshop on Mobile Entity Localization and Tracking in GPS-Less Environments—MELT ’08, San Francisco, CA, USA, 19 September 2008. [Google Scholar] [CrossRef]
- Park, K.; Kang, J.; Arjmandi, Z.; Shahbazi, M.; Sohn, G. Multilateration under Flip Ambiguity for UAV Positioning Using Ultrawide-Band. ISPRS Ann. Photogramm. Remote. Sens. Spat. Inf. Sci.
**2020**, V-1-2020, 317–323. [Google Scholar] [CrossRef] - Li, Y.L.; Shao, W.; You, L.; Wang, B.Z. An Improved PSO Algorithm and Its Application to UWB Antenna Design. IEEE Antennas Wirel. Propag. Lett.
**2013**, 12, 1236–1239. [Google Scholar] [CrossRef] - Lim, K.S.; Nagalingam, M.; Tan, C.P. Design and construction of microstrip UWB antenna with time domain analysis. Prog. Electromagn. Res. M
**2008**, 3, 153–164. [Google Scholar] [CrossRef] - Chahat, N.; Zhadobov, M.; Sauleau, R.; Ito, K. A Compact UWB Antenna for On-Body Applications. IEEE Trans. Antennas Propag.
**2011**, 59, 1123–1131. [Google Scholar] [CrossRef] - Chen, Z.N. UWB antennas: Design and application. In Proceedings of the 2007 6th International Conference on Information, Communications & Signal Processing, Singapore, 10–13 December 2007. [Google Scholar] [CrossRef]
- Compagnoni, M.; Pini, A.; Canclini, A.; Bestagini, P.; Antonacci, F.; Tubaro, S.; Sarti, A. A Geometrical–Statistical Approach to Outlier Removal for TDOA Measurements. IEEE Trans. Signal Process.
**2017**, 65, 3960–3975. [Google Scholar] [CrossRef] - Compagnoni, M.; Notari, R. TDOA-based Localization in Two Dimensions: The Bifurcation Curve. Fundam. Inform.
**2014**, 135, 199–210. [Google Scholar] [CrossRef] - Kaune, R.; Horst, J.; Koch, W. Accuracy Analysis for TDOA Localization in Sensor Networks. In Proceedings of the 14th International Conference on Information Fusion, Chicago, IL, USA, 5–8 July 2011. [Google Scholar]
- Kaune, R. Accuracy studies for TDOA and TOA localization. In Proceedings of the 2012 15th International Conference on Information Fusion, Singapore, 9–12 July 2012; pp. 408–415. [Google Scholar]
- Grasso, P.; Innocente, M.S. Debiasing of position estimations of UWB-based TDoA indoor positioning system. In Proceedings of the UKRAS20 Conference: “Robots into the Real World” Proceedings, Lincoln, UK, 14–17 April 2020. [Google Scholar] [CrossRef]
- Bitcraze. Loco Positioning System: TDOA Principles. 2020. Available online: http://www.bitcraze.io/documentation/repository/lps-node-firmware/2020.09/functional-areas/tdoa_principles/ (accessed on 18 August 2021).
- Kay, S.M. Fundamentals of Statistical Processing; Prentice Hall: Hoboken, NJ, USA, 1993; Volume I. [Google Scholar]
- Bitcraze. Loco Positioning System: TDOA2 vs. TDOA3. 2020. Available online: http://www.bitcraze.io/documentation/repository/lps-node-firmware/2020.09/functional-areas/tdoa2-vs-tdoa3/ (accessed on 18 August 2021).
- DW1000 IEEE802.15.4-2011 UWB Transceiver—Datasheet v2.09. 2015.
- DWM1000 IEEE 802.15.4-2011 UWB Transceiver Module—Datasheet v1.3. 2015.
- Bitcraze. Getting Started with the Loco Positioning System. 2022. Available online: https://www.bitcraze.io/documentation/tutorials/getting-started-with-loco-positioning-system/ (accessed on 14 November 2022).
- Mueller, M.W.; Hamer, M.; D’Andrea, R. Fusing ultra-wideband range measurements with accelerometers and rate gyroscopes for quadrocopter state estimation. In Proceedings of the 2015 IEEE International Conference on Robotics and Automation (ICRA), Seattle, WA, USA, 26–30 May 2015. [Google Scholar] [CrossRef]
- Mueller, M.W.; Hehn, M.; D’Andrea, R. Covariance Correction Step for Kalman Filtering with an Attitude. J. Guid. Control. Dyn.
**2017**, 40, 2301–2306. [Google Scholar] [CrossRef]

**Figure 1.**Not-to-scale diagram of the $4\times 4$ m

^{2}2D IPS being studied: (a) adjustable stands for the transmitting anchor antennas (A0–A3); (b) measurement points distributed every 50 cm in each direction; and (c) mobile stand for the object to be localised.

**Figure 2.**Dynamic experiment setup: mobile stand on rail; (a,b) beginning and end of rail; (h) optical obstacles (pins); (g) optical infrared sensor; (c) pcu, batteries, and motors; (d)

`Crazyflie 2.0`; (e) direction of movement; (f) laser pointer. Optical sensor aligned with drone’s antenna.

**Figure 3.**(

**a**) Original experimental radiation pattern sections on the $\theta $, ${\varphi}_{1}$, and ${\varphi}_{2}$ planes; (

**b**) approximation procedure forcing identical values on intersections; and (

**c**) radial projection of the approximated radiation pattern sections. These are used to reconstruct the 3D radiation pattern.

**Figure 5.**Precision level sets and colourmaps for symmetric and random anchors. The magenta trapezoid is the convex hull of four anchors (modified from [38], with permission).

**Figure 6.**Bifurcation curves, bifurcation envelopes (green line), convex hull of anchors (magenta trapezoid, acceptable precision), and flyable area (yellow shade) for three and four anchors.

**Figure 7.**Expected position from the debiasing filter ($\widehat{{\mathbf{x}}}$) when applied to a measured posistion ($\overline{{\mathbf{x}}}$) in 2D. The cloud of actual positions (${{\mathbf{X}}}_{{i}{j}}$) is constrained by the boundary ${\Omega}$.

**Figure 9.**Diagram of derivation of debiasing functions in x direction ${}^{\mathrm{x}}\beta \left(\mathbf{x}\right)$ and y direction ${}^{\mathrm{y}}\beta \left(\mathbf{x}\right)$.

**Figure 10.**Precision map of x component (${\pm}^{\mathrm{x}}\sigma $) and y component (${\pm}^{\mathrm{y}}\sigma $) of position.

**Figure 11.**Accuracy map of x component (${\pm}^{\mathrm{x}}b$) and y component (${\pm}^{\mathrm{y}}b$) of position.

**Figure 12.**Debiasing function for measured x component (${}^{\mathrm{x}}\beta $) and y component (${}^{\mathrm{y}}\beta $) of position.

**Figure 13.**Visualisation of rail and IPS position measurements, where ${L}_{r}$ is the total length of the rail.

**Figure 14.**Square path experiments (

**a**) from [57] and (

**b**) carried out in this paper. (

**a**) Drone flying in auto-pilot along a desired square path (black) (from [57]). Both the EKF estimate (blue) and the actual position (red) are shifted from the desired path. (

**b**) Proposed experiment following a fixed square path (black). The EKF + DF estimation (red) is expected to be more accurate than the EKF-only estimation (blue).

**Figure 17.**Dynamic experiment at average cruise velocity of 0.33 m/s with x spanning 0 m to 4 m at constant y = 2 m, where ($x,y$): position estimate with EKF-only; ($\widehat{x},\widehat{y}$): debiased position; (${x}_{\mathrm{ref}}$): actual position on rail; and (${v}_{x}$): estimated instantaneous velocity in x-direction.

**Figure 18.**Square-path experiment results, with flying domain delimited by four anchors. The vehicle starts moving from $(0.5,0.5)$ following the positive x-axis direction. IPS-1 uses EKF only, while IPS-2 uses EKF + DF. Problematic regions of the path are highlighted in yellow. The overall experiment shows a clear improvement when incorporating the proposed DF.

**Table 1.**Representative results of dynamic validation. The RMSEs of an IPS with and another without DF (IPS-2 and IPS-1, respectively) are compared to demonstrate the accuracy improvement of the former. The average performance difference is shown in columns $\Delta x$ and $\Delta y$. The average cruise velocity is shown in the last column.

RMSE_{x,avg} [cm] | RMSE_{y,avg} [cm] | ||||||||
---|---|---|---|---|---|---|---|---|---|

dir. | $\mathit{x}$[m] | y[m] | IPS-1 | IPS-2 | $\Delta \mathit{x}$ | IPS-1 | IPS-2 | $\Delta \mathit{y}$ | ${\mathit{v}}_{\mathrm{avg}}$ |

hor. | [0, 4] | 1 | 12.7 | 6.8 | 5.9 | 10.0 | 7.9 | 2.1 | 0.58 |

hor. | [0, 4] | 2 | 12.0 | 8.1 | 3.9 | 6.7 | 4.3 | 2.4 | 0.44 |

hor. | [0, 4] | 3 | 12.6 | 8.0 | 4.6 | 9.3 | 8.0 | 1.3 | 0.43 |

ver. | 1 | [4, 0] | 15.6 | 10.3 | 5.4 | 9.4 | 6.8 | 2.7 | 0.58 |

ver. | 2 | [4, 0] | 10.3 | 8.0 | 2.3 | 15.8 | 10.1 | 5.7 | 0.51 |

ver. | 3 | [4, 0] | 11.4 | 9.4 | 2.0 | 15.3 | 12.2 | 3.1 | 0.42 |

**Table 2.**Square-path experiment results. The RMSEs of an IPS with and another without DF (IPS-2 and IPS-1, respectively) are compared to show the accuracy improvement. The ‘raw’ heading refers to full data stream and the ‘sel.’ heading refers to the undamaged data stream (i.e., no misbehaviour). Average improvement with DF is shown by $\Delta $.

RMSE_{IPS-1} [cm] | RMSE_{IPS-2} [cm] | |||||||
---|---|---|---|---|---|---|---|---|

Edge | dir. | x[m] | y[m] | Raw | sel. | Raw | sel. | $\Delta $ [cm] |

bot | hor. | [0.5, 3.5] | 0.5 | 9.2 | 9.5 | 7.5 | 4.7 | 4.8 |

right | ver. | 3.5 | [0.5, 3.5] | 12.6 | 12.5 | 9.0 | 8.3 | 4.2 |

top | hor. | [3.5, 0.5] | 3.5 | 6.0 | 5.5 | 5.7 | 4.6 | 0.9 |

left | ver. | 0.5 | [3.5, 0.5] | 15.8 | 15.2 | 8.8 | 6.7 | 8.5 |

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## Share and Cite

**MDPI and ACS Style**

Grasso, P.; Innocente, M.S.; Tai, J.J.; Haas, O.; Dizqah, A.M. Analysis and Accuracy Improvement of UWB-TDoA-Based Indoor Positioning System. *Sensors* **2022**, *22*, 9136.
https://doi.org/10.3390/s22239136

**AMA Style**

Grasso P, Innocente MS, Tai JJ, Haas O, Dizqah AM. Analysis and Accuracy Improvement of UWB-TDoA-Based Indoor Positioning System. *Sensors*. 2022; 22(23):9136.
https://doi.org/10.3390/s22239136

**Chicago/Turabian Style**

Grasso, Paolo, Mauro S. Innocente, Jun Jet Tai, Olivier Haas, and Arash M. Dizqah. 2022. "Analysis and Accuracy Improvement of UWB-TDoA-Based Indoor Positioning System" *Sensors* 22, no. 23: 9136.
https://doi.org/10.3390/s22239136