# Novel Information-Driven Smoothing Spline Linearization Method for High-Precision Displacement Sensors Based on Information Criterions

^{*}

## Abstract

**:**

^{−5}level. Validation experiments were carried out on two different types of displacement sensors to benchmark the performance of the proposed method compared to the polynomial models and the the non-smoothing cubic spline. The results show that the proposed method with the new modified Akaike Information Criterion stands out compared to the other linearization methods and can improve the residual nonlinearity by over 50% compared to the standard polynomial model. After being linearized via the proposed method, the residual nonlinearities reach as low as ±0.0311% F.S. (Full Scale of Range), for the 1.5 mm range chromatic confocal displacement sensor, and ±0.0047% F.S., for the 100 mm range laser triangulation displacement sensor.

## 1. Introduction

#### 1.1. Background and Importance of Sensor Linearization

- (1)
- Nonlinearity measurement: measure the discrete nonlinearity deviations of the sensor using a traceable instrument with higher accuracy;
- (2)
- Nonlinearity reconstruction: construct an accurate mathematical model of nonlinearity from the discrete measurement data;
- (3)
- Nonlinearity correction: implement a real-time or off-line correction of the new measurement results using the constructed linearization model.

#### 1.2. Review of Current Research on Sensor Linearization

#### 1.3. Objective and Arrangement of This Paper

## 2. Principle

#### 2.1. The Polynomial Linearization Model

#### 2.2. The Non-Smoothing Spline Linearization Model (SP-FEDF)

#### 2.3. The Proposed Information-Driven Smooth Spline Linearization Model

**Criterion strategy 1**: The smoothing spline by the Bayesian Information Criterion (SP-BIC).

**Criterion strategy 2**: The smoothing spline by the Akaike Information Criterion (SP-AIC).

**Criterion strategy 3**: The smoothing spline by the corrected Akaike Information Criterion (SP-AICc).

**Criterion strategy 4**: The smoothing spline by the new Modified Akaike Information Criterion (SP-MAIC).

^{−5}range. The measurement data are therefore fine observations of the system’s nonlinearity characteristics. In contrast to applications like in ecology, these applications have a low noise–signal ratio. In such low-noise applications, it is possible for the BIC, AIC, and AICc metric to select an over-simplified model due to excessive penalty, thus restricting the improvement of linearization accuracy. More information from the measurement data should be incorporated into the linearization model to make better predictions compared to large noise datasets. Hereby, the adjustment coefficient is introduced to alleviate the penalty term in AIC, where the new modified Akaike information criterion (MAIC) is formulated by:

^{−5}noise to range ratio.

## 3. Methodology

## 4. Experiments

#### 4.1. Theoretical Validation and Simulation

#### 4.1.1. The Influence Factors for Sensor Linearization Methods

**Factor 1. The local undulation A**: The local undulation refers to the local variability exhibited by a model function. Intuitively, the freedom of a model decides its complexity, which is positively correlated to its capability to handle local undulation. Compared to polynomial models, the information-driven smoothing splines are more effective to handle local undulation because they allow for higher freedom and can approximate the local undulations better via piecewise reconstruction.

**Factor 2. The noise level N**: The noise-resistant capabilities of a linearization method is essential for achieving good linearization results. Ideally, a linearization model should demonstrate consistent high performance across different noise levels. This is key in physical applications since noise levels vary from one condition to another. For the proposed information-driven linearization methods, their superiority compared to the non-smoothing spline lie in the ability to adapt to the noise data, and thus, can achieve better linearization accuracy.

**Factor 3. The number of sampling points P**: The number of sampling points could also influence the linearization results. As a general rule, more sampling points can furnish linearization models with more physical information to achieve better linearization results. Nevertheless, in practical conditions, the number of sampling points is commonly restricted to the hundreds, considering the time and effort required for measurement. Too many samples could also lead to overfitting. This trade-off must be balanced to avoid overfitting.

#### 4.1.2. Monte Carlo Simulation of Linearization Methods

#### 4.1.3. Comparison of the Linearization Results

#### 4.2. Verification Experiments of Linearization Methods

#### 4.2.1. Verification Experiment on the Chromatic Confocal Displacement Sensor (CCDS)

^{−5}mm and the noise–range ratio is 5.86 × 10

^{−5}, considering the 1.5 mm measurement range of the CCDS.

#### 4.2.2. Verification Experiment on the Laser Triangulation Displacement Sensor (LTDS)

^{−3}mm and the variance–range ratio was 1.243 × 10

^{−5}. The change trend of evaluation criterions for polynomials and for smoothing spline was calculated and shown in Figure 10a and Figure 10b, respectively.

## 5. Results

#### 5.1. Linearization Results on the CCDS

#### 5.2. Linearization Results on the LTDS

#### 5.3. Comparison of the Information Criterions for the Proposed Linearization Method

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**The two typical nonlinear functions for modeling physical systems. (

**a**) The cubic polynomial function. (

**b**) The sigmoid function.

**Figure 4.**Performance of the linearization methods on different local undulation. (

**a**) Linearization residuals on the CPF. (

**b**) Linearization residuals on the SF. (

**c**) Relative residuals to the SP-MAIC of the spline methods on the CPF. (

**d**) Relative residuals to the SP-MAIC of the spline methods on the SF. (

**e**) The best EDFs selected via smoothing spline methods at each local undulation for CPF. (

**f**) The best EDFs selected via smoothing spline methods at each local undulation for SF.

**Figure 5.**Performance of the linearization methods on different noise level. (

**a**) Linearization residuals on the CPF. (

**b**) Linearization residuals on the SF. (

**c**) Relative residuals to the SP-MAIC of the spline methods on the CPF. (

**d**) Relative residuals to the SP-MAIC of the spline methods on the SF. (

**e**) The best EDFs selected by smoothing spline methods at each noise level for CPF. (

**f**) The best EDFs selected by smoothing spline methods at each noise level for SF.

**Figure 6.**Performance of the linearization methods on different number of samples. (

**a**) Linearization residuals on the CPF. (

**b**) Linearization residuals on the SF. (

**c**) Relative residuals to the SP-MAIC of the spline methods on the CPF. (

**d**) Relative residuals to the SP-MAIC of the spline methods on the SF. (

**e**) The best EDFs selected by smoothing spline methods at each number of samples for CPF. (

**f**) The best EDFs selected by smoothing spline methods at each number of samples for SF.

**Figure 7.**Linearization experiments on a self-developed CCDS. (

**a**) Experimental setup. (

**b**) Working principle of the CCDS. (

**c**) The measured nonlinearity curve. (

**d**) The distribution of measurement noise compare to Gaussian distribution.

**Figure 8.**Change curve of model selection criterions on CCDS. (

**a**) Criterions on the polynomial model. (

**b**) Criterions on the smoothing spline model.

**Figure 9.**Linearization experiments on a self-developed LTDS. (

**a**) Experimental setup. (

**b**) Working principle of the LTDS. (

**c**) The measured nonlinearity curve. (

**d**) The distribution of measurement noise compare to Gaussian distribution.

**Figure 10.**Change curve of model selection criterions on LTDS. (

**a**) Criterions on the polynomial model. (

**b**) Criterions on the smoothing spline model.

**Figure 11.**Linearization model performance on CCDS. (

**a**) Validated nonlinearity residual error. (

**b**) RMSE comparison of nonlinearity residuals for different models. (

**c**) PV comparison of nonlinearity residuals for different models.

**Figure 12.**Linearization model performance on LTDS. (

**a**) Validated nonlinearity residual error. (

**b**) RMSE comparison of nonlinearity residuals for different models. (

**c**) PV comparison of nonlinearity residuals for different models.

Reference | Model | Targeted Sensor | Summary of Contents |
---|---|---|---|

[19] | PL with periodic term | Laser interferometer | Developed a system to correct sub-fringe periodic nonlinearity using a capacitive sensor, achieving picometer-level repeatability. |

[20] | Linear regression | Scale of motion axes | Utilized standard height artefacts to calibrate motion axes’ scale via linear regression, achieving ±0.043% F.S. for a coherence scanning interferometer. |

[21] | PL | Inductive displacement sensor | Developed a laser interferometer for linearizing a 13 µm range sensor via third-order PL, attaining a theoretical nonlinearity of ±0.012% F.S.. |

[23] | Not applicable | Capacitive displacement sensor | Devised a Fabry–Perot interferometer to calibrate nanometer sensors up to a 300 µm range, with picometer resolution and nanometer uncertainty. |

[28] | PL | Laser displacement sensor | Adopted a third-order PL for linearizing two non-contiguous intervals, reducing nonlinearity by up to 70% compared to linear interpolation. |

[31] | PWL | General on-chip smart sensors | Implemented PWL for hardware linearization of smart sensors, focusing on cost-efficiency and memory optimization. |

[32] | PWL | General types of sensors | Enhanced PWL models with alterable divisions for simplicity and high efficiency, achieving more than a 30% error reduction. |

[33] | PWL | J-type and K-type thermocouple | Employed the included angle method for optimized PWL model construction, obtaining a nonlinearity of ±0.12% F.S. for a J-type thermocouple and ±0.41% F.S. for a K-type thermocouple. |

[34] | PWL | General types of sensors | Constructed a PWL model that limits the maximum approximation error while retaining computational efficiency for microcontroller applications. |

[35] | ANN | General types of sensors | Applied ANN in sensor linearization. Reduced nonlinearity of resistor sensors from ±3.33% F.S. to ±0.42% F.S.. |

[29,36] | PL and ANN | K-type thermocouple and thermistor | Applied both PL and ANN in linearizing thermal sensors. Achieved ±0.001% F.S. nonlinearity for a thermocouple and ±0.065% F.S. for a thermistor. |

[37] | ANN | Thermistor | Showcased a 1-5-1 shallow ANN outperforming the hardware linearization method in linearization range and accuracy. |

[39,40] | Bezier curve | Potentiometer displacement sensor | Employed Bezier curve modeling for sensor linearization, improving the position error from 14.2 mm to 4.2 mm. |

[41] | S-spline | General on-chip smart sensors | Developed a recursive S-spline for on-chip linearization, achieving ±0.005% F.S. for an S-type thermocouple and ±0.175% F.S. for an NTC thermistor. |

[42] | S-spline | Laser displacement sensor | Employed S-spline for contour reconstruction, reducing the measurement error from 282.91 µm to 252.55 µm in measuring an API thread. |

Model | Polynomial | Proposed Information-Driven Smoothing Spline | Cubic Spline (SP-FEDF) | |||
---|---|---|---|---|---|---|

SP-BIC | SP-AIC | SP-AICc | SP-MAIC | |||

EDF | 40 | 59 | 94 | 68 | 143 | 200 |

Validation | 2.337605 | 2.755046 | 1.343160 | 2.249387 | 0.922966 | 0.967733 |

Test 1 | 2.225379 | 2.718632 | 1.321536 | 2.208768 | 1.038585 | 1.050816 |

Test 2 | 2.415095 | 2.738590 | 1.308218 | 2.204588 | 0.880576 | 0.942672 |

Test 3 | 2.373813 | 2.813443 | 1.182333 | 2.269050 | 0.875963 | 0.999185 |

Mean Nlt. (F.S.) | ±0.0779% | ±0.0919% | ±0.0424% | ±0.0742% | ±0.0311% | ±0.0333% |

Model | Polynomial | Proposed Information-Driven Smoothing Spline | Cubic Spline (SP-FEDF) | |||
---|---|---|---|---|---|---|

SP-BIC | SP-AIC | SP-AICc | SP-MAIC | |||

EDF | 38 | 53 | 75 | 59 | 138 | 200 |

Validation | 20.008912 | 30.479826 | 19.482484 | 25.349988 | 13.764306 | 14.194509 |

Test 1 | 20.393406 | 30.371380 | 17.061518 | 25.258444 | 11.234457 | 11.685411 |

Test 2 | 23.133073 | 25.753753 | 15.416428 | 20.621292 | 8.939692 | 9.006504 |

Test 3 | 23.800611 | 26.495056 | 13.468149 | 21.371787 | 8.319668 | 8.794299 |

Mean Nlt. (F.S.) | ±0.0112% | ±0.0138% | ±0.0077% | ±0.0112% | ±0.0047% | ±0.0049% |

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## Share and Cite

**MDPI and ACS Style**

Zhang, W.-H.; Dai, L.; Chen, W.; Sun, A.; Zhu, W.-L.; Ju, B.-F.
Novel Information-Driven Smoothing Spline Linearization Method for High-Precision Displacement Sensors Based on Information Criterions. *Sensors* **2023**, *23*, 9268.
https://doi.org/10.3390/s23229268

**AMA Style**

Zhang W-H, Dai L, Chen W, Sun A, Zhu W-L, Ju B-F.
Novel Information-Driven Smoothing Spline Linearization Method for High-Precision Displacement Sensors Based on Information Criterions. *Sensors*. 2023; 23(22):9268.
https://doi.org/10.3390/s23229268

**Chicago/Turabian Style**

Zhang, Wen-Hao, Lin Dai, Wang Chen, Anyu Sun, Wu-Le Zhu, and Bing-Feng Ju.
2023. "Novel Information-Driven Smoothing Spline Linearization Method for High-Precision Displacement Sensors Based on Information Criterions" *Sensors* 23, no. 22: 9268.
https://doi.org/10.3390/s23229268