# Development of Magnetic-Based Navigation by Constructing Maps Using Machine Learning for Autonomous Mobile Robots in Real Environments

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Magnetic Navigation Method

- Updating the location particles given the robot’s movement.
- Computing the likelihood of each particle based on the most current sensor measurement and the observation model$$\omega =\frac{1}{\sqrt{2\pi \sigma}}exp\left(\right)open="("\; close=")">-\frac{{({z}_{map}-{z}_{sensor})}^{2}}{2\sigma}$$
- Resampling particles according to $\omega $ when necessary.

## 3. Machine Learning for Generating Magnetic Map

#### 3.1. Gaussian Process Regression (GPR)

#### 3.2. Sparse Gaussian Process Regression (SGPR)

- Compute the vector ${k}_{\ast}$ and matrix $\mathrm{K}+{\sigma}_{n}^{2}{I}_{n}$ and store them in memory.
- Compute the inverse matrix ${(\mathrm{K}+{\sigma}_{n}^{2}{I}_{n})}^{-1}$ and store it in memory.
- Calculate ${k}_{\ast}^{T}{(\mathrm{K}+{\sigma}_{n}^{2}{I}_{n})}^{-1}m$ and store it in memory.

#### 3.2.1. Subset of Data Approximation (SoD)

#### 3.2.2. The Inducing Variables Method

## 4. Kernel Function for High-Accuracy Localization

#### 4.1. Kernel Function

- (1)
- Radial Basis Function (RBF) Kernel$$k({x}_{p},{x}_{q})={\sigma}_{f}^{2}exp\left(\right)open="("\; close=")">-\frac{\mathbf{r}}{2{l}^{2}}$$
- (2)
- Exponential Kernel$$k({x}_{p},{x}_{q})={\sigma}_{f}^{2}(1+\sqrt{3}\mathbf{r})exp(-\sqrt{3}\mathbf{r})$$
- (3)
- Matern 3/2 Kernel$$k({x}_{p},{x}_{q})={\sigma}_{f}^{2}\left(\right)open="("\; close=")">1+\frac{\sqrt{3}\mathbf{r}}{l}$$
- (4)
- Matern 5/2 Kernel$$k({x}_{p},{x}_{q})={\sigma}_{f}^{2}\left(\right)open="("\; close=")">1+\frac{\sqrt{5}\mathbf{r}}{l}+\frac{5{\mathbf{r}}^{2}}{3{l}^{2}}$$
- (5)
- Exponential + Cosine Kernel$$k({x}_{p},{x}_{q})={\sigma}_{f}^{2}(1+\sqrt{3}\mathbf{r})exp(-\sqrt{3}\mathbf{r})+\frac{\pi}{4}cos\left(\right)open="("\; close=")">\frac{\pi \mathbf{r}}{2}$$
- (6)
- Exponential + Matern 3/2 Kernel$$k({x}_{p},{x}_{q})={\sigma}_{f}^{2}(1+\sqrt{3}\mathbf{r})exp(-\sqrt{3}\mathbf{r})+{\sigma}_{f}^{2}\left(\right)open="("\; close=")">1+\frac{\sqrt{3}\mathbf{r}}{l}$$
- (7)
- RBF + Exponential + Matern 3/2 Kernel$$k({x}_{p},{x}_{q})={\sigma}_{f}^{2}exp\left(\right)open="("\; close=")">-\frac{{\mathbf{r}}^{2}}{2{l}^{2}}exp\left(\right)open="("\; close=")">-\frac{\sqrt{3}\mathbf{r}}{l}$$

#### 4.2. Experiments and Discussions

- REAL: 5, 10, 20, 30, 40, 50
- HICity: 10, 50, 100, 500, 1K, 2K

#### 4.2.1. Magnetic Maps

- Exponential kernelThe Exponential kernel generates magnetic maps that are considerably smoother than those generated by the RBF Kernel. Smoother maps have been shown to improve localization accuracy when combined with MCL [34]. However, the maps generated by this kernel seem to be too smooth, which could limit the ability of the localization algorithm to differentiate between nearby points (as both have similar intensities), lowering its potential localization accuracy.
- Matern 3/2 kernelContrary to the exponential kernel, the Matern 3/2 generates maps that accentuate magnetic disturbances (higher peaks and valleys), while still generating smooth maps. As it can be observed, compared to the RBF kernel, several peaks are combined into larger ones.
- Matern 5/2 kernelSimilar to the Matern 3/2 kernel, the Matern 5/2 kernel also generates maps that accentuate magnetic disturbances. However, it does not tend to combine peaks, showing the same patterns as the RBF Kernel. Compared with the RBF kernel and Matern 3/2 kernel, it is hard to assess which would yield higher localization accuracies, hence the requirement to test the actual localization accuracy that can be achieved with them.
- Exponential + Cosine kernelThe Cosine kernel has periodicity in one dimension, but when combined with other kernels, the result does not show such periodicity. The main idea when testing the Exponential + Cosine kernel was to see if the training found some periodicity in the data. As can be seen, when compared with the Exponential kernel, this kernel has no significant differences. This means that no such periodicities were dominant in the data.
- Exponential + Matern 3/2 kernelAs both the Exponential and the Matern 3/2 kernels showed similar maps, we combined them to see if their combination would increase localization accuracy. As expected, the resulting maps are smooth and somewhat in the middle between the exponential and Matern in terms of the height of its peaks and valleys.
- Exponential + Cosine + RBF kernelThe RBF, Exponential, and Cosine kernels were combined to see if the addition of several kernels would improve localization accuracy.

#### 4.2.2. Localization Accuracy Using Different Magnetic Maps

## 5. SGPR for Generating Large-Scale Magnetic Map

#### 5.1. Mapping

- GPR.
- GPR with clustered data (KM-GPR).
- SGPR.

#### 5.1.1. Memory Consumption

#### 5.1.2. Generating Time

#### 5.2. Effect to Localization

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviation

i.i.d | independent and identically distributed |

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**Figure 1.**Travel distance estimation based on a magnetic fluctuation. The robot is given a magnetic map that stores the magnetic intensity with respect to distance.

**Figure 2.**Differences in variance among the seven types of kernel functions proposed this time. $\mathbf{r}$ is the Euclidean distance of the input, $\mathbf{r}\hspace{0.17em}=\hspace{0.17em}|{x}_{p}-{x}_{q}|$. (

**a**) RBF Kernel (

**b**) Exponential Kernel (

**c**) Matern 3/2 Kernel (

**d**) Matern 5/2 Kernel (

**e**) Exponential + Cosine Kernel (

**f**) Exponential + Matern 3/2 Kernel (

**g**) RBF + Exponential + Cosine Kernel.

**Figure 3.**“MAUV”, the hardware used in the experiment. (

**a**) MAUV with the cowl. The cowl was developed considering the requirements of a service robot for transporting people while ensuring safety [51]. (

**b**) MAUV without the cowl. (

**c**) Appearance and specifications of the magnetic sensors “3DM-DH”.

**Figure 4.**“REAL” and “HICity”, environments used in this experiment. (

**a**,

**c**) show the geometric map and the trajectory of the robot; when it measures magnetic information it is superimposed with a purple line. (

**b**,

**d**) show the environment in this experiment.

**Figure 5.**Magnetic map in the experimental environment. The strength is indicated by the color bar on the RGB scale. (

**a**) RBF Kernel (

**b**) Exponential Kernel (

**c**) Matern 3/2 Kernel (

**d**) Matern 5/2 Kernel (

**e**) Exponential + Cosine Kernel (

**f**) Exponential + Matern 3/2 Kernel (

**g**) RBF + Exponential + Cosine Kernel.

**Figure 6.**Example of localization result. Compare the results of odometry-only localization and magnetic navigation method localization for the specified path indicated by the blue dots with those indicated by the yellow and green dots.

**Figure 7.**Graphs of memory consumption for each method in REAL’s magnetic mapping for GPR, KM-GPR, and SGPR with 5, 40, and 50 inducing points are shown. (

**a**) Graph showing all methods, (

**b**) enlarged graph of SGPR, and (

**c**) enlarged graph of GPR.

**Figure 8.**Graph of memory consumption for each method of magnetic mapping in HICity. GPR, KM-GPR, and SGPR with 10, 1K, and 2K inducing points are shown. (

**a**) Graph showing all the methods, (

**b**) enlarged graph of KM-GPR, and (

**c**) enlarged graph of SGPR with 10 inducing points.

**Figure 10.**Experimental results of localization. Magnetic map created by GPR and SGPR is displayed as a two-dimensional color map, and each specified route and the trajectory of localization are superimposed. (

**a**–

**c**) and (

**d**–

**f**) show the results of localization for each specified route using the maps created by GPR and SGPR, respectively.

**Figure 11.**Color map of the variance stored in REAL’s magnetic map. (

**a**) using GPR, (

**b**) using SGPR with five inducing variable points. The white lines represent the trajectories of the three magnetic sensors mounted on the robot during the acquisition of magnetic information.

Path | # Sample | Elapsed Time (mean ± 2std) [s] | Memory Consumption [MiB] | ||||||
---|---|---|---|---|---|---|---|---|---|

Full GP | SGPR | Full GP | SGPR | ||||||

Desktop PC | DGX-1 | Desktop PC | DGX-1 | Desktop PC | DGX-1 | Desktop PC | DGX-1 | ||

1 | 540 | 0.26 ($\pm 0.29$) | 0.68 ($\pm 0.78$) | 1.71 ($\pm 0.71$) | 2.11 ($\pm 0.79$) | 178.379 | 215.223 | 124.738 | 160.602 |

2 | 356 | 0.17 ($\pm 0.15$) | 0.45 ($\pm 0.45$) | 1.49 ($\pm 0.36$) | 2.21 ($\pm 1.65$) | 146.918 | 182.457 | 125.832 | 162.594 |

3 | 876 | 0.66 ($\pm 0.64$) | 1.55 ($\pm 1.66$) | 2.15 ($\pm 1.14$) | 2.52 ($\pm 0.77$) | 247.039 | 294.352 | 124.336 | 161.336 |

4 | 859 | 0.65 ($\pm 0.64$) | 1.59 ($\pm 1.56$) | 1.81 ($\pm 1.45$) | 2.36 ($\pm 1.14$) | 243.086 | 290.250 | 124.945 | 161.465 |

5 | 362 | 0.15 ($\pm 0.15$) | 0.38 ($\pm 0.39$) | 1.59 ($\pm 0.96$) | 2.02 ($\pm 0.97$) | 143.902 | 183.395 | 126.637 | 162.480 |

6 | 224 | 0.08 ($\pm 0.05$) | 0.22 ($\pm 0.14$) | 1.27 ($\pm 0.23$) | 1.88 ($\pm 1.22$) | 130.973 | 169.520 | 125.418 | 161.199 |

7 | 211 | 0.09 ($\pm 0.09$) | 0.26 ($\pm 0.26$) | 1.42 ($\pm 0.41$) | 1.69 ($\pm 0.44$) | 130.016 | 169.371 | 124.629 | 161.844 |

8 | 246 | 0.12 ($\pm 0.02$) | 0.24 ($\pm 0.07$) | 4.97 ($\pm 3.16$) | 2.55 ($\pm 1.27$) | 132.621 | 170.117 | 124.910 | 162.012 |

9 | 196 | 0.08 ($\pm 0.07$) | 0.23 ($\pm 0.15$) | 1.71 ($\pm 0.56$) | 1.94 ($\pm 0.44$) | 129.906 | 168.516 | 125.336 | 161.133 |

10 | 223 | 0.08 ($\pm 0.04$) | 0.22 ($\pm 0.12$) | 5.89 ($\pm 6.95$) | 7.81 ($\pm 4.16$) | 131.219 | 168.621 | 125.074 | 161.285 |

11 | 287 | 0.10 ($\pm 0.06$) | 0.26 ($\pm 0.17$) | 3.78 ($\pm 3.47$) | 6.08 ($\pm 5.72$) | 139.516 | 174.492 | 125.156 | 161.816 |

Desktop PC | NVIDIA DGX-1 ^{1} | |
---|---|---|

Processor | Intel Core i7-9700 [email protected] GHz × 8 ^{2} | Dual 20-Core Intel Xeon E5-2698 v4 2.2 GHz |

RAM | 32 GB (Crucial CT16G4SFD8266 × 2 ^{3}) | 512 GB 2,133 MHz DDR4 RDIMM |

Storage | Samusung SSD 860 EVO MZ-76E500 ^{4} | 4X 1.92 TB SSD RAID 0 |

OS | Ubuntu 20.04.1 LTS | Ubuntu 18.04.1 LTS |

^{1}https://www.nvidia.com/en-us/data-center/dgx-1/?ncid=van-dgx-1 (accessed on 1 March 2021);

^{2}https://www.crucial.com/memory/ddr4/ct2k16g4dfd8266 (accessed on 1 March 2021);

^{3}https://ark.intel.com/content/www/us/en/ark/products/191792/intel-core-i7-9700-processor-12m-cache-up-to-4-70-ghz.html (accessed on 1 March 2021);

^{4}https://s3.ap-northeast-2.amazonaws.com/global.semi.static/Samsung_SSD_860_EVO_Data_Sheet_Rev1.pdf (accessed on 1 March 2021).

**Table 3.**Memory and computational cost for each computational process in GPR and SGPR. N is input data points, D is input dimensions, and M is inducing points.

Calculation Process | Memory Consumption Order | Computational Order | |
---|---|---|---|

GPR | Covariance vector: ${k}_{\ast}$ | $\mathcal{O}\left(ND\right)$ | $\mathcal{O}\left(ND\right)$ |

Covariance matrix: $\mathrm{K}$ | $\mathcal{O}\left({N}^{2}D\right)$ | $\mathcal{O}\left({N}^{2}D\right)$ | |

Inverse matrix of $\mathrm{K}$: ${\mathrm{K}}^{-1}$ | $\mathcal{O}\left({N}^{2}\right)$ | $\mathcal{O}\left({N}^{3}\right)$ | |

Matrix product: ${k}_{\ast}^{T}{\mathrm{K}}^{-1}m$ | $\mathcal{O}\left(N\right)$ | $\mathcal{O}\left({N}^{2}\right)$ | |

SGPR | Covariance matrix: ${\mathrm{K}}_{MM},{\mathrm{K}}_{MN}$ | $\mathcal{O}(NM+{M}^{2})$ | $\mathcal{O}(NM+{M}^{2})$ |

Diagonal matrix: $\mathsf{\Lambda}$ | $\mathcal{O}\left(N\right)$ | $\mathcal{O}\left(N\right)$ | |

Matrix: ${\mathbf{Q}}_{MM}$ | $\mathcal{O}\left({M}^{2}\right)$ | $\mathcal{O}\left(N{M}^{2}\right)$ | |

Inverse matrix: ${\mathbf{Q}}_{u}^{-1}$ | $\mathcal{O}\left({M}^{2}\right)$ | $\mathcal{O}\left({M}^{3}\right)$ |

**Table 4.**Results of localization experiments. The results of this experiment summarize the calculation of the mean and standard deviation as well as the minimum and maximum values from 100 localization experiments.

RBF | Exponential | Matern 3/2 | Matern 5/2 | Exponential + Cosine | Exponential + Matern 3/2 | Exponential + Cosine + RBF | |
---|---|---|---|---|---|---|---|

Ave (±2std) [m] | 0.616 ($\pm 1.01$) | 0.715 ($\pm 0.869$) | 0.578 ($\pm 0.638$) | 0.495 ($\pm 0.558$) | 0.788 ($\pm 1.46$) | 0.693 ($\pm 0.667$) | 0.619 ($\pm 0.654$) |

Min [m] | 0.295 | 0.340 | 0.301 | 0.266 | 0.286 | 0.317 | 0.293 |

Max [m] | 3.07 | 3.18 | 1.96 | 2.13 | 4.09 | 1.86 | 2.32 |

**Table 5.**Result of time when generating magnetic map (data:mean(±2×std)(s)). All the results are calculated as the mean and standard deviation of the 10 experimental results.As for the magnetic mapping in HICity, the time could not be measured in the case of GPR and 2K inducing variable points due to the forced termination of the program by insufficient memory.

REAL | Hardware | 5 | 10 | 20 | 30 | 40 | 50 | GPR |

Desktop PC | 2.61 ($\pm 2.32$) | 4.55 ($\pm 1.28$) | 7.27 ($\pm 2.82$) | 10.9 ($\pm 0.72$) | 14.5 ($\pm 0.26$) | 17.7 ($\pm 0.52$) | 75.0 ($\pm 5.89$) | |

DGX-1 | 18.78 ($\pm 22.9$) | 29.30 ($\pm 17.34$) | 43.71 ($\pm 18.54$) | 58.26 ($\pm 0.98$) | 67.36 ($\pm 5.72$) | 77.77 ($\pm 1.72$) | 127.6 ($\pm 20.31$) | |

HICity | Hardware | 10 | 50 | 100 | 500 | 1K | 2K | GPR |

Desktop PC | 15.4 ($\pm 4.90$) | 78.7 ($\pm 0.66$) | 166.4 ($\pm 1.04$) | 1197.3 ($\pm 11.4$) | 2753.3 ($\pm 22.7$) | - | - | |

DGX-1 | 63.8 ($\pm 22.46$) | 228.1 ($\pm 2.65$) | 456.7 ($\pm 4.66$) | 2670.3 ($\pm 6.77$) | 5426.4 ($\pm 11.7$) | 11358.1 ($\pm 761.5$) | 2315.4 ($\pm 66.9$) |

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## Share and Cite

**MDPI and ACS Style**

Takebayashi, T.; Miyagusuku, R.; Ozaki, K.
Development of Magnetic-Based Navigation by Constructing Maps Using Machine Learning for Autonomous Mobile Robots in Real Environments. *Sensors* **2021**, *21*, 3972.
https://doi.org/10.3390/s21123972

**AMA Style**

Takebayashi T, Miyagusuku R, Ozaki K.
Development of Magnetic-Based Navigation by Constructing Maps Using Machine Learning for Autonomous Mobile Robots in Real Environments. *Sensors*. 2021; 21(12):3972.
https://doi.org/10.3390/s21123972

**Chicago/Turabian Style**

Takebayashi, Takumi, Renato Miyagusuku, and Koichi Ozaki.
2021. "Development of Magnetic-Based Navigation by Constructing Maps Using Machine Learning for Autonomous Mobile Robots in Real Environments" *Sensors* 21, no. 12: 3972.
https://doi.org/10.3390/s21123972