# High Accuracy Passive Magnetic Field-Based Localization for Feedback Control Using Principal Component Analysis

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

- A method using concurrent field measurements of a moving permanent magnet (PM) to infer its precise position in real-time is presented. This method employs a sensor array to provide bijective relationships between measurements of magnetic flux density (MFD) and position as well as spatially extending the sensing range. Instead of directly mapping sensor outputs to position, PCA is used to determine an optimized set of transformed measurements for ANN mapping.
- Through numerical simulations, the effects of geometric parameters (including the PM dimensions and sensor spacing and location) on sensing accuracy is examined. Simulated measurements are corrupted with artificial Gaussian noise to explore practical implementation issues of the system.
- Using an ironless brushless linear motor as a platform for analysis, the experimental performance of a 9-sensor field-based system using PCA optimized ANN mapping is investigated. The tracking error resulting from the closed-loop control of the system using the field-based system is compared with an optical encoder. The tracking performance using the field-based system gives a similar response of that obtained using the optical encoder with 1 µm resolution.

## 2. Materials and Methods

- A distributed spatial network of sensors to uniquely relate position of a magnetic source to its measured field from the sensors, and
- An approach using ANNs and PCA to optimally relate multiple concurrent field measurements to position coordinates and minimize time-consuming computation.

**M**= M

**e**of the PM is perpendicular to the motion path x. The aggregate MFD as measured by a network of n single-axis sensors at an arbitrary position of the magnetic source can be denoted by Equation (1):

_{z}_{i}is the individually measured MFD along the z-axis (

**e**) from the ith sensor at the absolute lateral position x of the magnetic source. The vector

_{z}**B**contains the individual f

_{i}for each sensor, which could represent analytical field models (single dipole [23], distributed multipole model [24,25] or hybrid [26]) or experimental field measurements at each spatial location d

_{i}.

**B**. Essentially, the inverse expression of

**B**

^{−1}, is necessary and sufficient for field-based sensing. However, extracting an analytical expression for f

_{i}

^{−1}is difficult due to the high degree of non-linearity and non-uniqueness of analytical field models [27,28]. A numerical method capable of handling discontinuities at the magnet surface involving manipulation of the scalar magnetic potential to compute magnetic fields in a current free space around a magnet can be found in [29].

#### 2.1. Inducing Bijectivity with a Spatial Network

_{i}± R in Equation (3), its effective positional sensing range is only R, [d

_{i}− R, d

_{i}] or [d

_{i}, d

_{i}+ R], because of its inability to distinguish between the two symmetric magnetic fields in Equation (2). The issue with symmetry is that multiple x locations result in the same f

_{i}(x) value, preventing these locations from being distinguished and hence unique from one another using only f

_{i}(x). In other words, for a given continuous interval x, f

_{i}must be a strictly increasing or decreasing monotonic function if the entire interval were to be unique. This criterion can be assessed by analyzing the spatial derivative of f

_{i}$\left(\frac{\partial f}{\partial x}\right)$.

**B**of multiple sensors do not exhibit the above symmetry because each f

_{i}has distinct axis of symmetry about d

_{i}. As best illustrated using Figure 2 where f

_{i}is approximated by a triangle function for simplicity, the composite vector

**B**(x) will be unique for $\Vert B\left(x\right)\Vert >0$ as long as d

_{i}

_{+1}– d

_{i}< R (i = 1,…,n–1). This stipulation ensures that local axes of symmetries inherent due to Equation (2) are no longer present. As illustrated in Figure 2, while x

_{1}and x

_{2}may share the same f

_{1}value, f

_{1}(x

_{1}) = f

_{1}(x

_{2}), the composite

**B**at both values are different and hence unique, ${\rm B}\left({x}_{1}\right)\ne {\rm B}\left({x}_{2}\right)$. Moreover, the effective positional sensing range of the sensor network is extended to [d

_{1}− R, d

_{n}+ R] as long as $\Vert B\left(x\right)\Vert >0$ throughout that specified range for x. In summary, the spatial network of sensors induces uniqueness for

**B**as well as increases the range where this uniqueness holds.

**B**is bijective, the inverse model

**B**

^{−1}exists and is bijective as well. This property allows measurements to be mapped uniquely to position coordinates, which is the fundamental mechanics of a sensing system. As analytical solutions to the inverse model are not available (especially if numerous sensors are involved), a function fitting approach is adopted to solve for

**B**

^{−1}for real-time feedback. With this mapping approach, the desired inverse model

**B**

^{−1}is approximated by a fitted analytical artificial function. Look-up tables (LUT) and conventional least squares (LS) using basis functions of polynomials and sinusoidals are commonly used methods in creating such mappings but ANN mapping is preferred as the latter are more adaptable and scalable when managing multiple inputs and outputs.

#### 2.2. PCA Optimized ANN Mapping

**B**), k neurons in the hidden layer and a single output neuron (x position at

**B**). In order to construct this mapping, the bijective range of motion [d

_{1}− R, d

_{n}+ R] is discretized into N data points, resulting to total of N ANN training-target sets. The positional estimate ${\widehat{x}}_{v}$ of the neural network can be mathematically represented as:

**ω**is a k × n matrix containing the weighing coefficients ω

_{ji}between the i-th input node and the j-th hidden node,

**Ω**is a 1 × k matrix containing weight coefficients

**Ω**

_{j}between the j-th hidden node and output node,

**b**is a vector containing the biases of each of the k hidden nodes, c is the bias of the solitary output node, and the subscript v ($1\le v\le N$data points) is an integer representing the training set index. For this single hidden layer network, there are a total of (n + 1)k weighing coefficients (

**Ω**,

**ω**) and (k + 1) biases (

**b**, c) which are determined offline during backpropagation training. To numerically execute Equation (4) for every estimate, it will consist of the following simple and efficient scalar arithmetic operations: (n + 1)k multiplications, (n + 1)k additions, and k Sigmoid functional evaluations. The root mean squared error (RMSE) defined in Equation (5) is used to evaluate of the performance of the neural network:

_{1}, …, S

_{i}…, S

_{n}) can be represented by the principal components (P

_{1}, …, P

_{q}, …, P

_{n}) such that the greatest variance of the measurements lie on the 1st principal component P

_{1}, the second greatest on the 2nd principal component P

_{2}and so on [30]. Rather than using correlated sensor field measurements as individual inputs to the ANN and graphically illustrated in Figure 4, an optimized subset of m principal components (from the full set of n principal components) can be applied to Equation (4) directly and this represents an advantage of requiring fewer ANN inputs while retaining the critical bijectivity of

**B**.

**A**= [

**B**

_{1}…

**B**

_{v}…

**B**

_{N}], the full suite of n principal components of

**A**can be determined by finding an orthonormal matrix

**P**such that the covariance matrix of

**G**(that contains the collated transformed field measurements of

**A**) is a diagonal matrix [30]:

**G**=

**PA**and

**P**is a square n × n matrix obtained from the offline computation of the singular value decomposition (SVD) or eigenvectors of:

**P**contain the n principal components coefficients of

**A**; and the eigenvector with the highest associated eigenvalue is the (first) principal component. The relative numerical value of the eigenvalues denotes the statistical significance of each principal component. To explicitly compute the qth principal component from n sensor measurements, the following arithmetic expression is used:

**C**

_{q}is a 1 × n vector constructed from the qth row of

**P**. Numerically computing m principal components requires mn scalar multiplications and m(n − 1) scalar additions. Note this is independent on the ANN architecture (k).

## 3. Results

#### 3.1. Numerical Simulation

_{o}M

_{o}where μ

_{o}is the magnetic permeability of free space; and M

_{o}is a specified residual magnetization (or magnetic moment per unit volume):

_{1}= 0) where it defines the PM boundary’s projection on x regardless of the actual dimensions and parameters of the system. The absolute field sensitivity (AFS) is defined by the partial spatial derivative:

#### 3.1.1. Singular Sensor Geometrical Considerations

_{1}= 0) single sensor configuration (S

_{1}, n = 1) are evaluated for a variety of β and γ values. The results are compiled in Figure 6, where the top, middle and bottom rows illustrate the numerically simulated MFD and AFS for γ = 0.25, 1 and 4 respectively. Within each plot, the effects of β (=1, 0.5 and 2) are compared. For all plots, the lateral displacement is spatially zeroed (d

_{1}= 0) such that the centerlines of the PM and sensor align vertically. In addition, the field sensitivity is presented in logarithmic scale to accentuate the differences and only the positive spatial domain ($0\le X\le 4$) is illustrated as the magnetic field is symmetric about X = 0.

#### 3.1.2. Dual-Sensor PCA ANN Mapping Analysis

_{1},

_{2}, n = 2), where the parameters (β = 0.5, γ = 0.25) are selected to yield the highest AFS per unit peak MFD.

_{21}= (d

_{2}− d

_{1})/w, on the corresponding measured MFD over the domain of $-2\le X\le 2$. Applying PCA using the process outlined in Equations (6) and (7), the two principal components for each of the three distinct normalized sensor spacing (1, 2 and 4) are compared in Figure 7b. For each sensor spacing, the PCA coefficients and associated variability for both principal components are tabulated in Table 1.

_{21}= 4.

#### 3.1.3. Multi-Sensor PCA ANN Mapping Analysis

_{1,2,3,4,5,6}; n = 6), with variable spacing. This continued analysis illustrates how effortlessly PCA scales with larger sensor networks while offering the added advantage of improved noise attenuation. The same PCA process as outlined in Equations (6) and (7), can be applied to these six inputs and the six resultant principal components are spatially plotted in Figure 9. The relative variability of each principal component expressed as a percentage and PCA coefficient matrix are consolidated in Table 2.

#### 3.2. Experimental Investigation

**B**by the Hall sensors as well as directly from the quadrature output of the optical encoder.

_{1}, h

_{2}and h

_{3}). Taking the sensor noise into consideration, a modified signal-to-noise ratio (SNR) defined by Equation (11) is used to characterize the variability in the measured signal:

#### 3.2.1. PCA-ANN Field-Based Localization

#### 3.2.2. Closed-Loop Tracking Performance

## 4. Discussion

#### 4.1. Numerical Results from Singular Sensor Geometrical Considerations

#### 4.2. Numerical Results from Dual & Multi-Sensor PCA ANN Mapping Analysis

_{1}and S

_{2}) in Figure 7a are symmetrical about X = 0, at least one of the corresponding principal components (P

_{1}and P

_{2}) are asymmetrical. In fact, the first principal component when δ

_{21}= 4 is a strictly increasing monotonic function. While the other sensor spacing configuration require both principal components to uniquely relate position, employing only the monotonic principal component allows the use of a single input ANN (as opposed to a two input ANN) for mapping as presented in Figure 8 albeit possessing a higher mapping error.

_{21}= 4 with a value of 7.53 × 10

^{−6}. The spatial plot in Figure 8 demonstrates a relatively consistent position error when using both P

_{1}and P

_{2}but noticeably higher at the ends when only P

_{1}is employed.

#### 4.3. Experimental Results from PCA ANN Field-Based Localization

_{1}, S

_{4}remains fully saturated throughout the motion range. This is reflected in the low SNR in Table 4 of 7.0. As h is increased, saturation effects and low SNR become less of an issue. At h

_{2}and h

_{3}, only S

_{9}has SNR less than 28 dB. However, increasing h further is undesirable as it will result in reduced SNR across all sensors.

_{1}is clearly bijective for all three values of h as reflected in Figure 14. Hence, for any h value, the 1st principle component is sufficient to uniquely infer position.

_{3}, it was over 90%). In addition, the SNR for the 1st principal component for all h (over 60 dB) was noticeably higher than any of the individual sensors SNR in Table 5. At 60 dB, σ/Q is 1000.

_{1}(single input ANN model), an RMSE of 10 µm or less is attained for all values of h. With the exception for h

_{1}(possibly due to the high degree of saturation across the sensors), this was achieved with only five hidden nodes. The results suggest that with PCA, the mapping performance is relatively insensitive to h, but related to the variability of the corresponding principle component. The larger the variability, the smaller the mapping errors.

_{2}, using the first two and all three principal components lowers the RMSE to 2.5 µm and 2.4 µm, respectively, at k = 35. This modest improvement reiterates the significance of the first principal component in mapping accuracy.

_{1}, h

_{2}and h

_{3}respectively) which require larger computational overheads and memory. Moreover, the RMSE of using with and without PCA filtered measurements are comparatively insignificant. Even with hidden nodes as low as k = 10, the absolute error distribution in Figure 15c demonstrates that the mapping accuracy at h

_{3}using only three principle components (P

_{1,2,3}) as ANN inputs is comparative to that using eight sensor measurements (S

_{1–8}) throughout the range of motion.

#### 4.4. Experimental Results from Closed-Loop Tracking Performance

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 3.**Two layer fully-connected feedforward ANN used for functional mapping. (Biases not shown).

**Figure 5.**Ratio of number of arithmetic operations using PCA-ANN to ANN-only mapping. Lower ratios (darker areas) denote instances of k (# of hidden nodes) and n (# of ANN inputs) where PCA-ANN is computationally more efficient. (

**a**) m = 1, (

**b**) m = 2, (

**c**) m = 3.

**Figure 6.**Normalized MFD (top) and absolute sensitivity (bottom) for various aspect and separation ratios; γ = l/(2w) and β = h/(2w). (

**a**) γ = 0.25, (

**b**) γ = 1 and (

**c**) γ = 4.

**Figure 7.**Concurrent MFD measurements (

**a**) and transformed principal components (

**b**) from three 2-sensor (S

_{1}and S

_{2}) configuration of varying normalized sensor spacing.

**Figure 8.**ANN absolute lateral position mapping error at δ

_{21}= 4 using both and single principal components.

**Figure 10.**Absolute error comparison between PCA assisted mapping and mapping without PCA at different levels of noise corruption. (

**a**) Zero Gaussian noise; (

**b**) 1% Gaussian noise.

**Figure 13.**Composite

**B**measurements from a nine sensor network at various separation distances. (

**a**) h

_{1}= 1.905 mm (3/4 rotations), (

**b**) h

_{2}= 3.81 mm (3/2 rotations) and (

**c**) h

_{3}= 5.715 mm (9/4 rotations).

**Figure 16.**Feedback response comparison between field-based sensing and optical encoder. (

**a**) Tracking performance; (

**b**) Close-up view of the response at the peak in (

**a**).

Sensor Spacing | PCA | ANN RMSE (Normalized) | |||||
---|---|---|---|---|---|---|---|

% Variability | Coefficients | ||||||

δ_{21} | P_{1} | P_{2} | C_{11} | C_{12} | C_{21} | C_{22} | |

1 | 52.2 | 47.8 | 1/√2 | −1/√2 | 1/√2 | 1/√2 | 1.60 × 10^{−5} (P_{1,}P_{2}) |

2 | 96.0 | 4.0 | 1/√2 | 1/√2 | −1/√2 | 1/√2 | 9.00 × 10^{−6} (P_{1,}P_{2}) |

4 | 78.5 | 21.5 | 1/√2 | −1/√2 | −1/√2 | −1/√2 | 7.53 × 10^{−6} (P_{1,}P_{2}) |

1.13 × 10^{−3} (P_{1} only) |

PCA % Variability | ANN Position Mapping | ||||||||
---|---|---|---|---|---|---|---|---|---|

P_{1} | P_{2} | P_{3} | P_{4} | P_{5} | P_{6} | Noise | Inputs | ANN RMSE (Normalized) | |

71.6 | 24.0 | 4.1 | 0.31 | 0.040 | 0.011 | 0% | P_{123} | 2.85 × 10^{−6} | |

PCA Coefficient Matrix | S_{1–6} | 2.40 × 10^{−6} | |||||||

C_{ij} | i = 1 | 2 | 3 | 4 | 5 | 6 | 1% | P_{123} | 1.09 × 10^{−2} |

j = 1 | 0.309 | −0.577 | 0.332 | −0.002 | 0.541 | −0.408 | S_{1–6} | 3.13 × 10^{−3} | |

2 | −0.309 | −0.577 | −0.332 | −0.002 | −0.541 | −0.408 | 10% | P_{123} | 1.05 × 10^{−1} |

3 | 0.499 | −0.160 | 0.246 | −0.649 | −0.436 | 0.230 | S_{1–6} | 3.32 × 10^{−2} | |

4 | −0.499 | −0.160 | −0.246 | −0.649 | 0.436 | 0.230 | |||

5 | 0.394 | 0.376 | −0.573 | −0.280 | 0.127 | −0.529 | |||

6 | −0.394 | 0.376 | 0.573 | −0.280 | −0.127 | −0.529 |

Field Sensor Network | PM (Grade N42) | |||||
---|---|---|---|---|---|---|

n | d_{1} (mm) | d_{i}_{+1} − d_{i} (mm) | 2w (mm) | l (mm) | c (mm) | M_{o} (A/m) |

9 | −7.27 | 4.09 | 12.7 | 6.35 | 4.76 | 4.67 × 10^{5} |

h (mm) | SNR (dB) | ||||||||
---|---|---|---|---|---|---|---|---|---|

S_{1} | S_{2} | S_{3} | S_{4} | S_{5} | S_{6} | S_{7} | S_{8} | S_{9} | |

1.905 | 54.1 | 59.1 | 56.5 | 7.0 | 53.4 | 59.4 | 57.2 | 37.3 | 32.0 |

3.81 | 52.1 | 57.6 | 55.6 | 40.3 | 53.1 | 58.0 | 55.1 | 42.2 | 21.6 |

5.715 | 49.6 | 54.0 | 53.9 | 46.4 | 52.9 | 55.9 | 51.9 | 42.6 | 27.0 |

h (mm) | % Variability, (SNR, dB) | ||||||||
---|---|---|---|---|---|---|---|---|---|

P_{1} | P_{2} | P_{3} | P_{4} | P_{5} | P_{6} | P_{7} | P_{8} | P_{9} | |

1.905 | 79.55 | 14.54 | 5.49 | 0.22 | 0.12 | 0.04 | 0.018 | ~0 | ~0 |

(64.0) | (56.6) | (52.4) | (38.4) | (35.9) | (31.8) | (27.6) | (5.9) | (4.8) | |

3.81 | 86.81 | 10.67 | 1.98 | 0.37 | 0.11 | 0.04 | 0.004 | ~0 | ~0 |

(63.0) | (53.9) | (46.6) | (39.3) | (34.1) | (29.9) | (19.1) | (10.7) | (−0.5) | |

5.715 | 91.50 | 8.20 | 0.27 | 0.03 | 0.001 | 0.0002 | 0.0001 | ~0 | ~0 |

(61.0) | (50.6) | (35.8) | (26.3) | (12.8) | (4.4) | (1.2) | (−0.2) | (−1.4) |

© 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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Foong, S.; Sun, Z. High Accuracy Passive Magnetic Field-Based Localization for Feedback Control Using Principal Component Analysis. *Sensors* **2016**, *16*, 1280.
https://doi.org/10.3390/s16081280

**AMA Style**

Foong S, Sun Z. High Accuracy Passive Magnetic Field-Based Localization for Feedback Control Using Principal Component Analysis. *Sensors*. 2016; 16(8):1280.
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**Chicago/Turabian Style**

Foong, Shaohui, and Zhenglong Sun. 2016. "High Accuracy Passive Magnetic Field-Based Localization for Feedback Control Using Principal Component Analysis" *Sensors* 16, no. 8: 1280.
https://doi.org/10.3390/s16081280