# Sensitivity Analysis of RV Reducer Rotation Error Based on Deep Gaussian Processes

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Analysis of the Factors Influencing the Rotation Error of RV-40E Reducer

#### 2.1. RV Reducer Transmission Principle and Structural Composition Analysis

#### 2.2. Analysis of Factors Influencing the Rotation Error of RV Reducer

#### 2.3. RV Reducer Rotation Error Sample Data Acquisition

## 3. Construction of Prediction Model of RV Reducer Rotation Error Based on the DeepGP Model

- Determine the range of values for each rotation error-influencing factor based on the actual production research, and then use the OLHS technique to collect the sample points for the prediction model’s training set.
- Construct a rotation error prediction model using the DeepGP model and determining the structure and parameters of the model.
- Use a validation set to evaluate the prediction model’s accuracy in order to make it easier to conduct a sensitivity analysis of each individual influencing element in the following section.

#### 3.1. Sample Point Extraction Based on the OLHS Method

^{15}sub-regions, which significantly increases the computer’s workload by excessively subdividing the spatial sample area. Moreover, there may be a part of the design space missing due to too many areas, so the LHS method is obviously not suitable for the task of sample point sampling in this paper. Based on the LHS approach, the OLHS method can ensure that all design points are uniformly distributed across the design space by evenly and orthogonally distributing them in the design space region of the test factors [23].

- 1.
- Each dimensional axis is subdivided into 10 equal intervals, thereby dividing the sample space range into 10
^{15}sub-regions with dimensions based on the 15 variables that affect the rotation error of the reducer, such as the $\mathsf{\Delta}{r}_{rp}$. - 2.
- Obtaining 200 sub-regions of an optimal Latin hypercube sampling matrix with uniform distribution based on the space-filling optimality criterion, space-filling optimality criterion is shown in the following equation:

- 3.
- Following that, a sample point is selected within each of the extracted sub-regions and each parameter sampling, point is then used as the coordinate component of each dimension of the sampling point.

#### 3.2. Process of Building Rotation Error Prediction Model

#### 3.2.1. Deep Gaussian Processes Model

^{2}), the mean absolute error (MAE), and the root mean squared error (RMSE) are also used to evaluate the local and global accuracy of the developed model. The formulas for calculating the four performance indicators are provided below:

#### 3.2.2. Construction of RV Reducer Rotation Error Prediction Model

^{2}of the model’s ability to predict the validation set collected above. The wolf count (P) of the population is set to 30, and the number of iterations (N) is set to 100. It is determined through an iterative search that the prediction model with the highest R

^{2}value is constructed when the number of nodes in the first hidden layer is six and the number of nodes in the second hidden layer is three for the validation set, as well as the prediction accuracy of this model. As a result, Figure 6 depicts the structure of the RV reducer rotation error prediction model developed using the DeepGP model.

#### 3.2.3. Accuracy Check of Rotation Error Prediction Model

^{2}, MAE, and RMSE are chosen as the performance metrics to evaluate the model’s regional and overall accuracy. In order to reduce the chance factor in the test data, the validation set was evenly divided into two for the model regression testing. Table 5 shows the test results for the two validation sets. In this study, the prediction model has a prediction accuracy of more than 95% for both validation sets. The prediction model in this paper has a low prediction error and good fitting accuracy.

^{2}value of the prediction model presented in this article reaches 96%. The model presented in this paper has stable mapping and high prediction accuracy, whereas the R

^{2}value of the other two models is just over 75%. Figure 8 shows that the values predicted in this article are more in line with the true value, while the rotation error predicted by the other two models deviates from the measured values and are, in most cases, smaller than the measured values. Due to the fact that this paper is based on the actual production and processing plant data, the other two models may not account for the influence of the friction force and disregard the influence of certain components, such as the bearings, in the modeling process.

## 4. Sensitivity Analysis of Factors Influencing the Rotation Error of RV Reducers Based on Sobol Method

#### 4.1. Sobol Sensitivity Analysis Method

#### 4.2. Sensitivity Analysis Process for Rotation Error Influencing Factors Based Sobol Method

#### 4.2.1. Results of Total Effects of Factors Influencing the Rotation Error of RV Reducer

#### 4.2.2. Analysis of the Coupling Effect of Factors Influencing the Rotation Error of RV Reducer

#### 4.2.3. Comparison of Sensitivity Analysis Results

## 5. Conclusions

- Using the OLHS method, actual production facilities were sampled to ensure the accuracy of the created prediction models, as well as the dependability and applicability of the SA results. Using the Deep GP model, a high-precision prediction model for the rotation error of each RV reducer was developed, and the accuracy was compared with the equivalent method and virtual prototypes to demonstrate the validity and accuracy of the prediction model, thereby establishing the conditions for the sensitivity analysis of the RV reducer’s rotation error.
- On the basis of the prediction model, a global sensitivity analysis of the variables affecting the rotation error of the RV reducer was conducted using the Sobol method. The primary cause of the reducer’s rotation error is the second-stage cycloidal pin-wheel transmission mechanism, with the influence of the planetary gear transmission and the planetary output carrier being small and negligible.
- The second-order sensitivity index is used to evaluate how multiple influencing elements interact to affect the rotation error of the reducer. There is significantly more coupling between $\delta J$ and $\delta t$, as well as between $\delta {r}_{rp}$ and $\delta J$. The coupling effect between other variables has little effect on the rotation error. The total order sensitivity index of each factor is calculated by adding its main effects and the coupling of those effects to other influences. The results of this paper’s sensitivity analysis were contrasted with the results of Taylor’s midvalue theorem to confirm the originality and precision of the analysis results.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic diagram of RV-40E reducer structure principle. (

**a**) The structure of RV reducer; (

**b**) Description of what is contained in the second panel. 1. Input shaft with sun gear; 2 (2′). Planetary gear; 3 (3′). Crankshaft; 4 (4′). Cycloid gear; 5. Needle tooth shell; 6. Planet carrier; 7. Flange plate; 8. Needle gear.

**Figure 7.**Two model diagrams of RV-40E Reducer. (

**a**) Equivalent spring error model of RV-40E reducer. (

**b**) Virtual prototype of RV-40E reducer.

**Figure 9.**The main effects and total effects of each influencing factor of RV reducer rotation error.

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

Number of teeth of sun gear and planetary gear | 18/54 |

Number of teeth of cycloid gear (z_{c}) and needle gear | 39/40 |

Eccentricity of crankshaft (a) (mm) | 1.5 |

Circle center radius of needle tooth shell (δr_{p}) (mm) | 82 |

Radius of needle gear (δr_{rp}) (mm) | 4 |

Total transmission ratio (i) | 121 |

Short amplitude factor (k_{1}) | 0.73 |

Input power | 1.05 (kW) |

Nominal torque | 377 (N∙m) |

Sources of Influencing Factors | No. | Influencing Factor of Rotation Error | Code Name | The Range/mm |
---|---|---|---|---|

The primary involute planetary part | 1 | Center distance error of input shaft and planetary gear | $\mathsf{\Delta}{f}_{a}$ | 0.15~0.50 |

2 | Radial error between gear and ring | $\mathsf{\Delta}{F}_{r}$ | 0.20~0.30 | |

3 | Fit clearance of gear and shaft | $\mathsf{\Delta}{C}_{k}$ | 0.001~0.003 | |

The second stage cycloidal pinwheel part | 4 | Amount of equidistant modification of cycloid gear | $\mathsf{\Delta}{r}_{rp}$ | 0.016~0.048 |

5 | The amount of radial-moving modification of cycloid gear | $\mathsf{\Delta}{r}_{p}$ | 0.026~0.078 | |

6 | Clearance of inside hole of cycloid gear and slewing bearing | $\sigma $ | 0~0.01 | |

7 | Radius error of needle tooth center circle | $\delta {r}_{p}$ | 0.005~0.015 | |

8 | Needle gear radius error | $\delta {r}_{rp}$ | 0.01~0.03 | |

9 | Matching clearance between the pin gear and needle gear hole | $\delta J$ | 0.001~0.003 | |

10 | Circular position error of the needle gear hole | $\delta t$ | 0~0.01 | |

11 | Eccentricity error of crankshaft | $\delta a$ | 0.01~0.03 | |

12 | Clearance of crankshaft and slewing bearing | $\mathsf{\Delta}r$ | 0~0.002 | |

Planetary output carrier | 13 | Eccentricity error of planet carrier | $Eh$ | 0.005~0.015 |

14 | Carrier wheel crankshaft center distance error | $\mathsf{\Delta}Fa$ | 0~0.01 | |

15 | Bearing hole phase error of planet carrier | ${\mathsf{\Delta}}_{2}$ | 0.03~0.05 |

Sample Point | $(\mathsf{\Delta}{\mathit{f}}_{\mathit{a}},\mathsf{\Delta}{\mathit{F}}_{\mathit{r}},\mathsf{\Delta}{\mathit{C}}_{\mathit{k}},\mathsf{\Delta}{\mathit{r}}_{\mathit{r}\mathit{p}},\mathsf{\Delta}{\mathit{r}}_{\mathit{p}},\mathit{\sigma},{\mathit{r}}_{\mathit{p}},{\mathit{r}}_{\mathit{r}\mathit{p}},\mathit{\delta}\mathit{J},\mathit{\delta}\mathit{t},\mathit{a},\mathsf{\Delta}\mathit{r},\mathit{E}\mathit{h},\mathsf{\Delta}\mathit{F}\mathit{a},{\mathsf{\Delta}}_{2})$ | Rotation Error/’ |
---|---|---|

$1$ | $\left(0.037,0.200,0.001,0.047,0.044,0.009,82.015,4.027,0.001,0.001,1.519,0.001,0.011,0.006,0.037\right)$ | $0.059$ |

$2$ | $\left(0.164,0.297,0.002,0.038,0.072,0.003,82.011,4.016,0.002,0.005,1.525,0.001,0.011,0.001,0.041\right)$ | $1.238$ |

$3$ | $\left(0.325,0.277,0.001,0.022,0.047,0.001,82.013,4.030,0.001,0.004,1.528,0.002,0.010,0.008,0.034\right)$ | $1.538$ |

$\cdots $ | $\cdots $ | $\cdots $ |

$100$ | $\left(0.164,0.224,0.003,0.030,0.076,0.002,82.009,4.026,0.002,0.003,1.516,0.002,0.008,0.005,0.041\right)$ | $1.483$ |

$101$ | $\left(0.420,0.258,0.002,0.028,0.063,0.005,82.012,4.024,0.002,0.002,1.529,0.001,0.008,0.002,0.044\right)$ | $2.161$ |

$\cdots $ | $\cdots $ | $\cdots $ |

$199$ | $\left(0.436,0.242,0.003,0.036,0.042,0.004,82.015,4.018,0.003,0.001,1.515,0.001,0.011,0.03,0.047\right)$ | $1.217$ |

$200$ | $\left(0.286,0.233,0.002,0.044,0.068,0.002,82.008,4.026,0.001,0.003,1.510,0,0.007,0.008,0.050\right)$ | $1.040$ |

Kernel Function | Expression |
---|---|

The Linear kernel | $k\left(x,{x}^{\prime};\theta \right)=x{x}^{\prime}$ |

The absolute-exponential kernel (RBF) | $k\left(x,{x}^{\prime};\theta \right)={\sigma}^{2}\left(-\frac{1}{2}{\left[\frac{\left(x-{x}^{\prime}\right)}{l}\right]}^{2}\right)$ |

The absolute-exponential kernel | $k\left(x,{x}^{\prime};\theta \right)={\sigma}^{2}exp\left(\frac{\left|x-{x}^{\prime}\right|}{l}\right)$ |

The Periodic covariance kernel function (PER) | $k\left(r\right)={\sigma}^{2}exp\left[-\frac{2si{n}^{2}\left(\pi \left|x-{x}^{\prime}\right|/p\right)}{{l}^{2}}\right]$ |

Performance Metrics | Validation Set 1 | Validation Set 2 |
---|---|---|

MAPE | 0.05533 | 0.07584 |

R^{2} | 0.97871 | 0.95913 |

MAE | 0.06059 | 0.07085 |

RMSE | 0.07053 | 0.08143 |

Performance Metrics | The Proposed Method | Equivalent Spring Method | Virtual Prototype |
---|---|---|---|

MAPE | 0.06533 | 0.17326 | 0.20461 |

R^{2} | 0.96846 | 0.78542 | 0.76941 |

MAE | 0.08059 | 0.13212 | 0.12941 |

RMSE | 0.07053 | 0.16819 | 0.20851 |

Sensitivity Analysis Methods | Characteristics of Sensitivity Methods | |
---|---|---|

Local sensitivity analysis | FAST | Simple operation, ignores parameter interactions, low model applicability. |

Global sensitivity analysis | Morris | Compared the output results of adjacent parameters in the parameter space, inefficient. |

LH-OAT | Although the benefits of both the random one-factor-at-a-time and LHS sampling techniques are taken into account, the computer program is complex. | |

GLUE | Combining the benefits of Rivest-Shamir-Adleman techniques and fuzzy mathematics to rank sensitivities as scatter plots. | |

Sobol | The sensitivity indices are solved using the Monte Carlo sampling technique, which can distinguish independently between parameter-independent and parameter-interacting sensitivities. |

Label | Influencing Factor of Rotation Error | Total Effects |
---|---|---|

a | $\mathsf{\Delta}{f}_{a}$ | 0.012414 |

b | $\mathsf{\Delta}{F}_{r}$ | 0.025427 |

c | $\mathsf{\Delta}{C}_{k}$ | 0.007129 |

d | $\mathsf{\Delta}{r}_{rp}$ | 0.269588 |

e | $\mathsf{\Delta}{r}_{p}$ | 0.144105 |

f | $\sigma $ | 0.387645 |

g | $\delta {r}_{p}$ | 0.112615 |

h | $\delta {r}_{rp}$ | 0.531029 |

i | $\delta J$ | 0.111341 |

j | $\delta t$ | 0.568193 |

k | $\delta a$ | 0.021295 |

l | $\mathsf{\Delta}r$ | 0.228339 |

m | $Eh$ | 0.014437 |

n | $\mathsf{\Delta}Fa$ | 0.012762 |

o | ${\mathsf{\Delta}}_{2}$ | 0.009721 |

**Table 9.**Comparison of the sensitivity indexes of the factors influencing the rotation error obtained by the two methods.

Infuencing Factor of Ratation Error | Taylor Sensitivity Method | Sobol | ||
---|---|---|---|---|

Rotation Error Caused by Each Factor | Sensitivity Coefficient | Sensitivity Index | Total Effects | |

$\mathsf{\Delta}{f}_{a}$ | $\frac{\mathsf{\Delta}{f}_{a}\times 180\times 60}{\pi \times i\times {r}_{1}}$ | - | 0.025 | 0.0124 |

$\mathsf{\Delta}{F}_{r}$ | $\frac{\mathsf{\Delta}{F}_{r}\times 180\times 60}{\pi \times i\times {r}_{1}}$ | - | 0.025 | 0.0254 |

$\mathsf{\Delta}{r}_{rp}$ | $\frac{2\times \mathsf{\Delta}{r}_{rp}}{a\times {Z}_{c}}$ | 1 | 1 | 0.269588 |

$\mathsf{\Delta}{r}_{p}$ | $\frac{2\times \mathsf{\Delta}{r}_{p}\times \sqrt{1-{k}_{1}^{2}}}{a\times {z}_{c}}$ | $\sqrt{1-{k}_{1}^{2}}$ | 0.68 | 0.144105 |

$\delta {r}_{p}$ | $\frac{2\times \delta {r}_{p}\times \sqrt{1-{k}_{1}^{2}}}{a\times {z}_{c}}$ | $\sqrt{1-{k}_{1}^{2}}$ | 0.68 | 0.112615 |

$\delta {r}_{rp}$ | $\frac{-2\times \delta {r}_{rp}}{a\times {z}_{c}}$ | $1$ | 1 | 0.5312 |

$\delta J$ | $\frac{\delta J}{a\times {z}_{c}}$ | 0.5 | 0.5 | 0.1111 |

$\delta t$ | $\frac{{k}_{1}\times \delta J}{a\times {z}_{c}}$ | ${k}_{1}$ | 0.73 | 0.56819 |

$\mathsf{\Delta}r$ | $\frac{180\times 60\times \mathsf{\Delta}r}{\pi \times a}$ | $\frac{e{z}_{c}}{2a}$ | 0.679487 | 0.228339 |

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## Share and Cite

**MDPI and ACS Style**

Jin, S.; Shang, S.; Jiang, S.; Cao, M.; Wang, Y. Sensitivity Analysis of RV Reducer Rotation Error Based on Deep Gaussian Processes. *Sensors* **2023**, *23*, 3579.
https://doi.org/10.3390/s23073579

**AMA Style**

Jin S, Shang S, Jiang S, Cao M, Wang Y. Sensitivity Analysis of RV Reducer Rotation Error Based on Deep Gaussian Processes. *Sensors*. 2023; 23(7):3579.
https://doi.org/10.3390/s23073579

**Chicago/Turabian Style**

Jin, Shousong, Shulong Shang, Suqi Jiang, Mengyi Cao, and Yaliang Wang. 2023. "Sensitivity Analysis of RV Reducer Rotation Error Based on Deep Gaussian Processes" *Sensors* 23, no. 7: 3579.
https://doi.org/10.3390/s23073579