# Finite Element Modelling of a Field-Sensed Magnetic Suspended System for Accurate Proximity Measurement Based on a Sensor Fusion Algorithm with Unscented Kalman Filter

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Prototype and Modelling of the Electromagnetic Actuator with Integrated Proximity Sensor

## 3. Magnetic Force Modelling with the Finite Element Method

#### 3.1. Coupling FEMM with Matlab and Data Analysis

#### 3.2. Modelling of Electromagnetic Force ${F}_{e}$

#### 3.3. Model Structure Selection ${F}_{e}$—Stage One

#### 3.4. The ${F}_{1e}$ Coefficient Estimation and ${F}_{e}$ Model Derivation—Stage Two

## 4. Distance Measurement and Sensor Fusion Algorithm

#### 4.1. Unscented Kalman Filter and Sensor Fusion Algorithm

- Initialization of the UKF- filter:$$\begin{array}{l}{\widehat{x}}_{0}=E\left[{x}_{0}\right],\\ {P}_{0}=E\left[\left({x}_{0}-{\widehat{x}}_{0}\right){\left({x}_{0}-{\widehat{x}}_{0}\right)}^{T}\right].\end{array}$$Variable ${\widehat{x}}_{0}$ represents initial states of nonlinear system given in Equation (17) and ${P}_{0}$ is an initial covariance matrix of the state variable ${x}_{0}$.
- Prediction:UT-transformation, calculation of 2L + 1 sigma points:$$\begin{array}{ll}{\chi}_{0,k-1}^{a}={\widehat{x}}_{k-1},& \\ {\chi}_{i,k-1}^{a}={\widehat{x}}_{k-1}+\sqrt{\left(L+\lambda \right){P}_{k-1}},& i=1,\dots ,L,\\ {\chi}_{i,k-1}^{a}={\widehat{x}}_{k-1}-\sqrt{\left(L+\lambda \right){P}_{k-1}},& i=L+1,....,2L,\end{array}$$$$\begin{array}{ll}{W}_{0}^{m}=\lambda /\left(L+\lambda \right),& \\ {W}_{0}^{c}=\lambda /\left(L+\lambda \right)+\left(1-{\alpha}^{2}+\beta \right),& \\ {W}_{i}^{m}={W}_{i}^{c}=1/\text{\hspace{0.17em}}\left(2L+2\lambda \right),& i=1,\dots ,2L.\end{array}$$The variable $L$ is the number of system states and $\lambda $ is the scaling parameter; $\lambda ={\alpha}^{2}\left(L+{k}_{i}\right)-L$. Parameter $\alpha $ determinates the spreads of the sigma points, ${k}_{i}$ is a secondary scaling parameter and $\beta $ is used to incorporate prior knowledge of the distribution ${\mathit{\chi}}_{k}^{a}$. The weights ${W}^{m}$ and ${W}^{c}$ represent mean weighting factor and estimation error covariance weighting factor respectively [42].
- Time Update$${\mathsf{\chi}}_{k|k-1}^{a}=F\left[{\mathsf{\chi}}_{k-1}^{a},{u}_{k-1}\right],$$$${\widehat{x}}_{k|k-1}={\displaystyle \sum _{i=0}^{2L}{W}_{i}^{m}{\mathsf{\chi}}_{i,k|k-1}^{a}},$$$${P}_{k|k-1}={\displaystyle \sum _{i=0}^{2L}{W}_{i}^{c}\left[{\chi}_{i,k|k-1}^{a}-{\widehat{x}}_{k|k-1}\right]}{\left[{\chi}_{i,k|k-1}^{a}-{\widehat{x}}_{k|k-1}\right]}^{T}+{Q}_{k},$$$${\mathsf{\Upsilon}}_{k|k-1}=H\left[{\mathsf{\chi}}_{k|k-1}^{a}\right],$$$${\widehat{y}}_{k|k-1}={\displaystyle \sum _{i=0}^{2L}{W}_{i}^{m}{\mathsf{\Upsilon}}_{i,k|k-1}}.$$
- Measurement Update Equation$${P}_{{y}_{k}{y}_{k}}={\displaystyle \sum _{i=0}^{2L}{W}_{i}^{c}\left[{\mathsf{{\rm Y}}}_{i,k|k-1}-{\widehat{y}}_{k|k-1}\right]}{\left[{\mathsf{{\rm Y}}}_{i,k|k-1}-{\widehat{y}}_{k|k-1}\right]}^{T}+{R}_{k},$$$${P}_{{x}_{k}{y}_{k}}={\displaystyle \sum _{i=0}^{2L}{W}_{i}^{c}\left[{\chi}_{i,k|k-1}^{a}-{\widehat{x}}_{k|k-1}\right]}{\left[{\mathsf{{\rm Y}}}_{i,k|k-1}-{\widehat{y}}_{k|k-1}\right]}^{T},$$$${\mathit{\kappa}}_{k}={P}_{{x}_{k}{y}_{k}}{P}_{{y}_{k}{y}_{k}}^{-1},$$$${\widehat{x}}_{k|k}={\widehat{x}}_{k|k-1}+{\mathit{\kappa}}_{k}\left({y}_{k}-{\widehat{y}}_{k|k-1}\right),$$$${P}_{k|k}={P}_{k|k-1}-{\mathit{\kappa}}_{k}{P}_{{y}_{k}{y}_{k}}{\mathit{\kappa}}_{k}^{T}.$$

#### 4.2. Deployment of the UKF Filter

## 5. Experimental Results

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Popovic, R.S. Hall Effect Devices, 2nd ed.; CRC Press-Taylor & Francis Group: Boca Raton, FL, USA, 2003. [Google Scholar]
- Ramsden, E. Hall-Effect Sensors: Theory and Applications, 2nd ed.; Elsevier: Amsterdam, The Netherlands; Newnes: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Yan, L.; Zhu, B.; Jiao, Z.X.; Chen, C.Y.; Chen, I.M. An orientation measurement method based on Hall-effect sensors for permanent magnet spherical actuators with 3d magnet array. Sci. Rep.
**2014**. [Google Scholar] [CrossRef] [PubMed] - Yan, L.; Zhu, B.; Jiao, Z.X.; Chen, C.Y.; Chen, I.M. Hall-sensor-based orientation measurement method in three-dimensional space for electromagnetic actuators. In Proceedings of the 2014 IEEE International Conference on Automation Science and Engineering (CASE), New Taipei, Taiwan, 18–22 August 2014; pp. 182–187.
- Krause, P.C.; Wasynczuk, O.; Sudhoff, S.D. Analysis of Electric Machinery and Drive Systems; IEEE Press: Piscataway, NY, USA, 2002. [Google Scholar]
- Fontana, M.; Salsedo, F.; Bergamasco, M. Novel magnetic sensing approach with improved linearity. Sensors
**2013**, 13, 7618–7632. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Li, J.H.; Chiou, J.S. Digital control analysis and design of a field-sensed magnetic suspension system. Sensors
**2015**, 15, 6174–6195. [Google Scholar] [CrossRef] [PubMed] - Li, J.H.; Chiou, J.S. GSA-tuning IPD control of a field-sensed magnetic suspension system. Sensors
**2015**, 15, 31781–31793. [Google Scholar] [CrossRef] [PubMed] - Lin, C.M.; Lin, M.H.; Chen, C.W. SoPC-based adaptive PID control system design for magnetic levitation system. IEEE Syst. J.
**2011**, 5, 278–287. [Google Scholar] [CrossRef] - Mehrtash, M.; Khamesee, M.B. Design and implementation of LQG/LTR controller for a magnetic telemanipulation system—Performance evaluation and energy saving. J. Microsyst. Technol.
**2011**, 14, 1135–1143. [Google Scholar] [CrossRef] - Shameli, E.; Khamesee, M.B.; Huissoon, J.P. Nonlinear controller design for a magnetic levitation device. Microsyst. Technol.
**2007**, 13, 831–835. [Google Scholar] [CrossRef] - Elbuken, C.; Khamesee, M.B.; Yavuz, M. Design and implementation of a micromanipulation system using a magnetically levitated MEMS robot. IEEE/ASME Trans. Mechatron.
**2009**, 14, 434–445. [Google Scholar] [CrossRef] - An, S.; Ma, Y.; Cao, Z. Applying simple adaptive control to magnetic levitation system. In Proceedings of the Second International Conference on Intelligent Computation Technology and Automation, Changsha, China, 10–11 October 2009; pp. 746–749.
- Lin, F.J.; Chen, S.Y.; Shyu, K.K. Robust dynamic sliding-mode control using adaptive RENN for magnetic levitation system. IEEE Trans. Neural Netw.
**2009**, 20, 938–951. [Google Scholar] [PubMed] - Yang, Z.J.; Kunitoshi, K.; Kanae, S.; Wada, K. Adaptive robust output-feedback control of a magnetic levitation system by K-filter approach. IEEE Trans. Ind. Electron.
**2008**, 55, 390–399. [Google Scholar] [CrossRef] - Gentili, L.; Marconi, L. Robust nonlinear disturbance suppression of a magnetic levitation system. Automatica
**2003**, 39, 735–742. [Google Scholar] [CrossRef] - Kashif, I.; Yasir, S.; Abdullah, S.S.; Amjad, M.; Munaf, R.; Suhail, K. Modeling and control of magnetic levitation system via fuzzy logic controller. In Proceedings of the 4th International Conference on Modeling, Simulation and Applied Optimization (ICMSAO), Kuala Lumpur, Malaysia, 19–21 April 2011. [CrossRef]
- Castanedo, F. A review of data fusion techniques. Sci. World J.
**2013**, 2013, 704504. [Google Scholar] [CrossRef] [PubMed] - Kalman, R.E. A new approach to linear filtering and prediction problems. J. Basic Eng.
**1960**, 82, 35–45. [Google Scholar] [CrossRef] - Lei, H. Auto regressive moving average (ARMA) modeling method for Gyro random noise using a robust Kalman filter. Sensors
**2015**, 15, 25277–25286. [Google Scholar] - Evensen, G. Data Assimilation: The Ensemble Kalman Filter; Springer: Berlin, Germany, 2007. [Google Scholar]
- Naumović, M.B. Modeling of a didactic magnetic levitation system for control education. In Proceedings of the International Conference on Telecommunications in Modern Satellite, Cable and Broadcasting Services–TELSIKS2003, Niš, Serbia and Montenegro, 1–3 October 2003; pp. 783–786.
- Hajjaji, E.A.; Ouladsine, M. Modeling and nonlinear control of magnetic levitation system. IEEE Trans. Ind. Electron.
**2001**, 48, 831–838. [Google Scholar] [CrossRef] - Qin, Y.M.; Peng, H.; Ruan, W.J.; Wu, J.; Gao, J.C. A modeling and control approach to magnetic levitation system based on state-dependent ARX model. J. Process. Control
**2014**, 24, 93–112. [Google Scholar] [CrossRef] - Shameli, E.; Khamesee, M.B.; Huissoon, J.P. Frequency response identification and dynamic modeling of a magnetic levitation device. In Proceedings of the ASME 2007 International Mechanical Engineering Congress and Exposition American Society of Mechanical Engineers, Seattle, DC, USA, 11–15 November 2007; pp. 1635–1639.
- Shameli, E.; Khamesee, M.B.; Huissoon, J.P. Real-time control of a magnetic levitation device based on instantaneous modeling of magnetic field. Mechatronics
**2008**, 18, 536–544. [Google Scholar] [CrossRef] - Coulomb, J.; Meunier, G. Finite element implementation of virtual work principle for magnetic or electric force and torque computation. IEEE Trans. Magn.
**1984**, 20, 1894–1896. [Google Scholar] [CrossRef] - Meessen, K.J.; Paulides, J.J.H.; Lomonova, E.A. Force calculations in 3-D cylindrical structures using fourier analysis and the Maxwell stress tensor. IEEE Trans. Magn.
**2013**, 49, 536–545. [Google Scholar] [CrossRef] - Griffiths, D.J. Introduction to Electrodynamics, 3rd ed.; Prentice Hall: New Jersey, NJ, USA, 1998. [Google Scholar]
- Naumović, M.B. Nonlinear state observation in a didactic magnetic levitation system. In Proceedings of the International Scientific Conference on Information, Communication and Energy Systems and Technologies (ICEST 2004), Bitola, Macedonia, 16–19 June 2004; pp. 473–476.
- Furlani, E.P. Permanent Magnet and Electromechanical Devices: Materials, Analysis and Applications; Academic Press: New York, NY, USA, 2001. [Google Scholar]
- Robertson, W.; Cazzolato, B.; Zander, A. Axial force between a thick coil and a cylindrical permanent magnet: Otpimizing the geometry of an electromagnetic actuator. IEEE Trans. Magn.
**2012**, 48, 536–545. [Google Scholar] [CrossRef] - Reddy, J.N. An Introduction to the Finite Element Method, 3rd ed.; McGraw-Hill: Singapore, 2006. [Google Scholar]
- Jin, J. The Finite Element Method in Electromagnetics; John Wiley and Sons: New York, NY, USA, 1993. [Google Scholar]
- Meeker, D. FEMM42.exe Software. Available online: http://www.femm.info (accessed on 6 August 1998).
- Benamimour, T.; Bentounsi, A.; Djeghloud, H. CAD of electrical machines using coupled FEMM-MATLAB softwares. In Proceedings of the 3rd International Conference on Electric Power and Energy Conversion Systems (EPECS), Istanbul, Turkey, 2–4 October 2013; pp. 1–6.
- Gould, N.; Toint, P.L. Preprocessing for quadratic programming. Math. Program.
**2004**, 100, 95–132. [Google Scholar] [CrossRef] - Sabatini, A.M. Variable-state-dimension Kalman-based filter for orientation determination using inertial and magnetic sensors. Sensors
**2012**, 12, 8491–8506. [Google Scholar] [CrossRef] [PubMed] - Ligorio, S.; Sabatini, A.M. Extended Kalman filter-based methods for pose estimation using visual, inertial and magnetic sensors: Comparative analysis and performance evaluation. Sensors
**2013**, 13, 1919–1941. [Google Scholar] [CrossRef] [PubMed] - Wan, A.E.; Van der Merwe, R. The unscented Kalman filter for nonlinear estimation. In Proceedings of the Adaptive Systems for Signal Processing, Communications, and Control Symposium, Lake Louise, Canada, 1–4 October 2000; pp. 153–158.
- Julier, S.J. The scaled unscented transformation. In Proceedings of the American Control Conference, Anchorage, KY, USA, 8–10 May 2002; pp. 4555–4559.
- Julier, S.J.; Uhlmann, J.K. Uncsented filtering and nonlinear estimation. Proc. IEEE
**2004**, 92, 401–422. [Google Scholar] [CrossRef] - Särkkä, S. On Unscented Kalman filtering for state estimation of continues-time nonlinear systems. IEEE Trans. Autom. Control
**2007**, 52, 1631–1641. [Google Scholar] [CrossRef] - Kokotovic, P.V. The joy of feedback nonlinear and adaptive. Control Syst. Mag.
**1992**, 12, 7–17. [Google Scholar] [CrossRef] - Galeazzi, R. Observer backstepping control for variable speed wind turbine. In Proceedings of the American Control Conference, Washington, WA, USA, 17–19 June 2013; pp. 1036–1043.

**Figure 1.**Two electromagnetic actuators with integrated proximity HE sensor-EMAwS; (

**a**) Vertical movement of the body and (

**b**) Horizontal movement of the body.

**Figure 4.**Finite element method with FEMM software for EMAwS. (

**a**) Computational mesh; (

**b**) Calculated values of magnetic force and field density.

**Figure 8.**Residue value of model fitting ${F}_{1e}$, ${F}_{2e}$ over current characteristics from $0\text{}\mathrm{A}\text{}\mathrm{to}\text{}1.4\text{}\mathrm{A}$.

**Figure 9.**Comparison of model fitting with models ${F}_{1e}$—(

**a**) and ${F}_{2e}\u2014$—(

**b**) on current characteristics $0\text{}\mathrm{A},0.6\text{}\mathrm{A}$ and $1.2\text{}\mathrm{A}$.

**Figure 11.**Static characteristics of Hall voltage in regard to applied coil voltage—(

**a**) and the magnet proximity—(

**b**).

**Figure 12.**Sensor fusion algorithm with Unscented Kalman Filter for distance measurement of a levitating magnetic object.

**Figure 14.**(

**a**) Schematic of an algorithm structure and (

**b**) Flow chart of real time algorithm execution.

**Figure 15.**Feedback control of EMAwS with nonlinear Backstapping controller, comparison between direct proximity measurement and sensor fusion-UKF proximity algorithm.

**Figure 16.**Feedback control of EMAwS with PID controller, comparison between direct proximity measurement and sensor fusion-UKF proximity algorithm.

**Figure 17.**Frequency spectrum of the proximity measurements (direct measurement and sensor fusion) with Backstepping controller.

**Figure 18.**Frequency spectrum of the proximity measurements (direct measurement and sensor fusion) with PID controller.

**Figure 19.**Comparison of the UKF and EKF sensor fusion algorithms with nonlinear backstepping feedback controller.

Parameters | Value |
---|---|

Solenoid high (h_{s}) | 25 mm |

Solenoid flange high (h_{se}) | 5 mm |

Solenoid flange diameter (w_{s}) | 40 mm |

Centre hole diameter (p) | 5 mm |

Inside winding diameter (w_{sin}) | 20 mm |

Ferrite magnetic permeability (µ/µ_{0}) | 450 |

Permanent magnet-neodymium | N52 |

Permanent magnet diameter (w_{m}) | 11 mm |

Permanent magnet high (h_{m}) | 5 mm |

Coil wire diameter AGW30 | 0.255 mm |

Number of turns | 30 |

Weight of the magnet | 4.35 g |

$\mathit{p}$ | ${\mathit{L}}_{\mathit{d}}$ | Current Characteristic Difference | Mean Value | Standard Deviation |
---|---|---|---|---|

1 | ${L}_{1}$ | $\left[0.2\u20130\right]\text{}\mathrm{A}$ | 0.0012 | 0.6740 |

2 | ${L}_{2}$ | $\left[0.4\u20130.2\right]\text{}\mathrm{A}$ | 0.0012 | 0.6740 |

3 | ${L}_{3}$ | $\left[0.6\u20130.4\right]\text{}\mathrm{A}$ | 0.0012 | 0.6740 |

4 | ${L}_{4}$ | $\left[0.8\u20130.6\right]\text{}\mathrm{A}$ | 0.0012 | 0.6740 |

5 | ${L}_{5}$ | $\left[1\u20130.8\right]\text{}\mathrm{A}$ | 0.0012 | 0.6740 |

6 | ${L}_{6}$ | $\left[1.2\u20131\right]\text{}\mathrm{A}$ | 0.0012 | 0.6740 |

7 | ${L}_{7}$ | $\left[1.4\u20131.2\right]\text{}\mathrm{A}$ | 0.0012 | 0.6740 |

**Table 3.**Parameter estimation of the model ${F}_{1e}$ and ${F}_{2e}$ , by current characteristics $0\text{}\mathrm{A}\u20131.4\text{}\mathrm{A}$.

Current | Model ${\mathit{F}}_{1\mathit{e}}$ | Model ${\mathit{F}}_{2\mathit{e}}$ | |||
---|---|---|---|---|---|

$\mathit{a}$ | $\mathit{b}$ | Residual $({\mathit{J}}_{1})$ | $\mathit{c}$ | Residual $({\mathit{J}}_{2})$ | |

$0\text{}\mathrm{A}$ | $5.999\times {10}^{-9}$ | $7.587\times {10}^{-8}$ | $6.409\times {10}^{-5}$ | $6.031\times {10}^{-9}$ | $6.424\times {10}^{-5}$ |

$0.2\text{}\mathrm{A}$ | $5.98\times {10}^{-9}$ | $88.6\times {10}^{-8}$ | $6.421\times {10}^{-5}$ | $6.415\times {10}^{-9}$ | $1.13\times {10}^{-4}$ |

$0.4\text{}\mathrm{A}$ | $6.012\times {10}^{-9}$ | $174.4\times {10}^{-8}$ | $6.399\times {10}^{-5}$ | $6.781\times {10}^{-9}$ | $2.033\times {10}^{-4}$ |

$0.6\text{}\mathrm{A}$ | $6.005\times {10}^{-9}$ | $263.4\times {10}^{-8}$ | $6.437\times {10}^{-5}$ | $7.135\times {10}^{-9}$ | $3.615\times {10}^{-4}$ |

$0.8\text{}\mathrm{A}$ | $5.985\times {10}^{-9}$ | $357.1\times {10}^{-8}$ | $6.427\times {10}^{-5}$ | $7464\times {10}^{-9}$ | $5.512\times {10}^{-4}$ |

$1\text{}\mathrm{A}$ | $5.994\times {10}^{-9}$ | $450.6\times {10}^{-8}$ | $6.395\times {10}^{-5}$ | $7.775\times {10}^{-9}$ | $7.867\times {10}^{-4}$ |

$1.2\text{}\mathrm{A}$ | $5.989\times {10}^{-9}$ | $543.1\times {10}^{-8}$ | $6.397\times {10}^{-5}$ | $7.892\times {10}^{-9}$ | $9.183\times {10}^{-4}$ |

$1.4\text{}\mathrm{A}$ | $6.007\times {10}^{-9}$ | $637.2\times {10}^{-8}$ | $6.403\times {10}^{-5}$ | $8.134\times {10}^{-9}$ | $12.437\times {10}^{-4}$ |

Parameter $\mathit{a}$ | Parameter $\mathit{b}$–$\left(\mathit{i}\right)$ | |
---|---|---|

Mean $\overline{\mathit{a}}$ | $\mathit{m}$ | Residual $({\mathit{J}}_{3})$ |

$5.996\times {10}^{-9}$ | $4.509\times {10}^{-6}$ | $2.0486\times {10}^{-14}$ |

**Table 5.**Residual value of objective function ${J}_{1}$ with final model ${F}_{e}$ and FEMM results.

Current | Model ${\mathit{F}}_{\mathit{e}}\left(\mathit{x},\mathit{i}\right)=5.996\times {10}^{-9}{\mathit{x}}^{-4}+\text{}5.996\times {10}^{-9}\cdot \mathit{i}\cdot {\mathit{x}}^{-2}$ |
---|---|

Residual $({\mathit{J}}_{1})$ | |

$0\text{}\mathrm{A}$ | $6.413\times {10}^{-5}$ |

$0.2\text{}\mathrm{A}$ | $6.411\times {10}^{-5}$ |

$0.4\text{}\mathrm{A}$ | $6.402\times {10}^{-5}$ |

$0.6\text{}\mathrm{A}$ | $6.399\times {10}^{-5}$ |

$0.8\text{}\mathrm{A}$ | $6.417\times {10}^{-5}$ |

$1\text{}\mathrm{A}$ | $6.399\times {10}^{-5}$ |

$1.2\text{}\mathrm{A}$ | $6.401\times {10}^{-5}$ |

$1.4\text{}\mathrm{A}$ | $6.416\times {10}^{-5}$ |

RMS | Backstepping Controller | PID Controller |
---|---|---|

Direct measurement | $12.32\times {10}^{-5}$ | $15.71\times {10}^{-5}$ |

Sensor fusion with UKF | $1.52\times {10}^{-5}$ | $1.68\times {10}^{-5}$ |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Chowdhury, A.; Sarjaš, A.
Finite Element Modelling of a Field-Sensed Magnetic Suspended System for Accurate Proximity Measurement Based on a Sensor Fusion Algorithm with Unscented Kalman Filter. *Sensors* **2016**, *16*, 1504.
https://doi.org/10.3390/s16091504

**AMA Style**

Chowdhury A, Sarjaš A.
Finite Element Modelling of a Field-Sensed Magnetic Suspended System for Accurate Proximity Measurement Based on a Sensor Fusion Algorithm with Unscented Kalman Filter. *Sensors*. 2016; 16(9):1504.
https://doi.org/10.3390/s16091504

**Chicago/Turabian Style**

Chowdhury, Amor, and Andrej Sarjaš.
2016. "Finite Element Modelling of a Field-Sensed Magnetic Suspended System for Accurate Proximity Measurement Based on a Sensor Fusion Algorithm with Unscented Kalman Filter" *Sensors* 16, no. 9: 1504.
https://doi.org/10.3390/s16091504