# Modelling of Material Removal in Abrasive Belt Grinding Process: A Regression Approach

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## Abstract

**:**

## 1. Introduction

_{A}(grinding process constant), K

_{A}(constant of resistance of the workpiece with grinding ability of the belt), k

_{t}(belt wear factor), V

_{b}(grinding rate), V

_{w}(feed-in rate), L

_{w}(machining width), and F

_{A}(normal force). The model states that the overall material removal rate (MRR) r is either proportional or inversely proportional to belt grinding parameters as shown in Equation (1). However, this model does not take into account the interaction between belt grinding parameters.

_{v}, and polishing pressure, P [10]. The constant, C, is established to denote other influential parameters and it is determined experimentally for each polishing system.

_{w}, to be a function of normal load, Fn, sliding distance, S, and hardness of the softest contacting surface, H, while K is a dimensionless constant [11].

_{A}, K and C) need to be determined after many exhaustive physical experiments. Developing analytical models for such nonlinear processes with a large number of parameters and assumptions may introduce biases and will not be a viable option to model the process. Though the correlation of individual parameters on material removal is understood using ANOVA and statistical techniques, their combined consequence on effective material removal is not well established [12]. The adaptive neuro-fuzzy inference system (ANFIS) model with sigmoidal membership function has been used to determine material removal in a belt grinding process [12].

## 2. Theoretical Basis

#### 2.1. Abrasive Belt Grinding

#### 2.2. Multiple Linear and Stepwise Regression

^{2}the most are considered significant whereas others are removed.

#### 2.3. Artificial Neural Networks

#### 2.4. Adaptive Neuro-Fuzzy Inference System

_{1}): Every adaptive node i in the L

_{1}has a node function.

_{i}(or B

_{i}

_{–2}) generating a linguistic label coupled with the node as given by Equation (12). The membership function for A (or B) can be any, such as a sigmoidal membership function given by Equation (13).

_{i}, a

_{i}) is the parameter set. These are called premise parameters. As the values of the parameters change, the shape of the membership function varies.

_{2}): Every node in L

_{2}is a fixed node labelled ∏. Each node calculates the firing strength of each rule, which is the output using the simple product operator. The rule premises result is evaluated as the product of all of the incoming signals and given by the Equation (14)

_{3}): The ratio of the i

^{th}rule’s firing strength to the sum of all of the rule’s firing strengths is calculated by Equation (15) in L

_{3}. The output of L

_{3}is called normalised firing strengths.

_{4}): Every node $i$ in L

_{4}is an adaptive node with a node function. The nodes compute a parameter function on the L

_{4}output. Parameters in this L

_{4}are referred to as consequent parameters.

_{i}is a normalised firing strength from L

_{3}and (p

_{i}, q

_{i}, r

_{i}) is the parameter set for the node.

_{5}): L

_{5}has a single fixed node labelled Σ, which computes the overall output as the summation of all of the incoming signals, as shown in Equation (17). The Σ gives the overall output of the constructed adaptive network, having the same functionality as the Sugeno fuzzy model.

#### 2.5. Support Vector Regression

#### 2.6. Random Forest

_{i}which basically is a regression decision tree. Unlike linear models, RF can capture the non-linear interaction between the features and the target. One of the most significant advantages of RF over DT is that the algorithm works on bootstrapping [35]. The RF creates a lot of individual DT by re-sampling the data many times with replacement and makes the final prediction at a new point by averaging the predictions from all the individual binary regression trees on this point. Averaging over all the decision trees results in a reduction of variance thereby enhancing the accuracy of the prediction. The accuracy of the RF can be estimated from observations that are not used for individual trees otherwise called “out of the bag data” (OOB) as shown in Equation (27)

## 3. Experimental Setup

#### 3.1. Methodology

#### 3.2. Taguchi Design of Experiments (DoE) and Data Collection

_{27}orthogonal array (five-factor, three-level) model. The familiarity of the probable interactions of the belt grinding parameters on material removal was not known therefore the Taguchi-based design of experiments (DoE) methodology was chosen Table 2 shows a list of belt grinding parameters (factors) and their levels used during the belt grinding trials. Material removal was quantified based on the depth of cut, which was measured using a Mitutoyo stylus profilometer with a tip radius of five microns with the accuracy of two decimal places used measure to the depth of cut. The depth of cut was calculated as the distance from the deepest point in the ground path from the unground surface of the workpiece as shown in Figure 7.

## 4. Regression Modelling for Abrasive Belt Grinding Process

#### 4.1. Multiple Linear Regression Based Modelling

^{2}) gave a value of 0.886, which implied that the fitness of the linear model was inadequate.

#### 4.2. Stepwise Regression-Based Modelling

^{2}) of value 0.975 increased when the regression took in a quadratic form which also took into account the interaction effect between the belt grinding parameters. The order in which predictor variables were removed or added could provide valuable information about the quality of the predictor variables. Referring to the t-stat value of the developed regression model, especially between the predictor interaction from Table 5, it was apparent that grit parameter played a dominant role. The estimated coefficients of the regression model were significant when the interaction happened with the grit size predictor.

^{2}statistical metric values from both approaches, it was clear that the latter performed better than the former as it incorporated a quadratic form, which addressed the influence of the interaction between the grinding parameters and the nonlinear behaviour of the belt grinding process. The proposed methodologies on the multilinear regression and the stepwise multilinear regression provided a useful tool to predict material removal depending on the grinding parameters. The multilinear model may not have been the best choice according to the prediction accuracy and R

^{2}, but still, we could use it to find the nature of the relationship between the two variables. Interpreting the performance of the model, it was apparent that the data were intrinsically nonlinear and a straight-line relationship should never be assumed in the belt grinding process. The use of a quadratic form in the stepwise regression helped to tune the model to have a better fit.

#### 4.3. Artificial Neural Networks Based Modelling

**.**

^{2}of 0.981 suggested a satisfactory fitness model. The results lead to the conclusion that the proposed models could be used to predict the depth of cut in the belt grinding process effectively. Though this network correlates the parameters on its own, as ANN essentially functions as a “black box”, it is trivial to evaluate the association between each independent variable and the dependent variable inside a neural network. Moreover, there is no specific rule for determining the structure of the ANN as the network is configured based on trial and error.

#### 4.4. Adaptive Neuro-Fuzzy Inference System Based Modelling

^{2}calculated based on the fitted regression line was 0.980 which also indicated a satisfactory fit of the model. The prediction results showed that by employing a hybrid learning algorithm such as ANFIS, the quality of generated relevant fuzzy if-then rules could be tuned to model the material removal behaviour.

#### 4.5. Support Vector Regression Based Modelling

- Define the data, predictors, and response for learning and testing.
- Decide a fitting kernel function (linear, Gaussian, RBF, polynomial, etc.).
- Select an ideal model for training on the input data with predictors and response. The training is done by using a Bayesian optimisation to model material in MATLAB.
- Validation of the testing data set.

^{2}calculated based on the fitted regression line was 0.980 which also showed the good fit of the model.

#### 4.6. Random Forest Based Modelling

- A bootstrap sample $\left({x}_{1},{x}_{2},{x}_{3},\dots .{x}_{n}\right)$ of size, $N$ is to be drawn from the training data consisting of the five predictors and the response variable.
- For each bootstrap sample ${x}_{i}$, a regression tree model is constructed by optimising the parameters, such as the number of trees ${t}_{n}$ and leaf size based on MSE error.
- Prediction at a new point Z is achieved by aggregating the predictions of the regression tree models.
- The accuracy of the RF model is calculated based on the deviation of the predicted value, x, from the ideal value in the validation data set.

^{2}calculated based on the fitted regression line was of value 0.975 which also showed that goodness of fit of the RF model was good. Although RF seems more of a “black box” approach compared to regression trees since individual trees cannot be assessed separately, it still provides means for interpretation in giving measures of variable importance.

## 5. Conclusions

- Observing the performance of the multilinear and stepwise regression models, it was seen that belt grinding parameters were intrinsically nonlinear and a straight-line relationship assumption could not satisfy the material removal.
- Although predicted values using ANN networks were close to the measured values, it functioned as a black box model correlating the parameters on its own, and the structure determination was based on trial and error.
- SVR modelling implemented using a Gaussian kernel function showed good accuracy on material removal. However, the drawback of the model was attributed to the selection of the kernel function that was based on trial and error.
- The ANFIS model developed in this research work had acceptable deviations between the predicted and the real results and was viable to predict the depth of cut in the abrasive belt grinding process compared to other regression techniques. The performance of the six algorithms in terms of RMSE is summarised in Figure 26.
- In addition, ANFIS could interpret the relationship between the input parameters towards the output behaviour which was not possible using other modelling techniques.
- The random forest model which was based on the frequency table was not able to predict at higher accuracy compared to other complex predictive models even though it performed better than the multilinear regression.
- Analysis of the sigmoidal membership function after training in ANFIS, interpretation on the variable importance results from RF, and the coefficient from stepwise regression techniques indicated that the grit size factor in the belt grinding parameters was the most significant factor on the material removal process.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**The regression line of support vector regression (SVR) is shown with the loss function and slack variables.

**Figure 6.**Compliant abrasive belt grinding experimental setup [12].

**Figure 8.**Schematic illustration of the multilinear regression model for prediction of the material removal.

**Figure 9.**(

**a**). Comparison of the observed and predicted depth of cut using multilinear regression; (

**b**). Statistical analysis fit of the multilinear regression model.

**Figure 10.**(

**a**). Comparison of observed and predicted depth of cut using stepwise regression; (

**b**) Statistical analysis fit of the stepwise regression model.

**Figure 11.**Residual plot observed between the multilinear regression model and stepwise multilinear regression model.

**Figure 12.**Schematic illustration of an artificial neural network (ANN) model for prediction of the material removal.

**Figure 14.**(

**a**). Comparison of observed and predicted depth of cut using neural network regression; (

**b**) Statistical analysis fit of the neural network regression model.

**Figure 15.**Adaptive Neuro-Fuzzy Inference System (ANFIS) model for belt grinding showing inputs and output [12].

**Figure 16.**Change in shape of the Gaussian bell membership function for each input before and after training.

**Figure 17.**(

**a**). Comparison of observed and predicted depth of cut using ANFIS with Gaussian bell membership function; (

**b**) Statistical analysis fit of the ANFIS model with Gaussian bell membership.

**Figure 20.**(

**a**) Comparison of observed and predicted depth of cut using SVR; (

**b**) Statistical analysis fit of the SVR model.

**Figure 22.**Optimisation of random forest (RF) parameters (number of grown trees) using mean square error (MSE).

**Figure 25.**(

**a**). Comparison of observed and predicted depth of cut using RF regression; (

**b**) Statistical analysis fit of the RF regression model.

Investigators | Contribution |
---|---|

Y. Wang et al. [13] | Developed a controllable material removal strategy to control the acting force and grinding dwell time by modelling the global and local material removal process of belt grinding. A finite element method (FEM) has been adopted to calculate the local force and global grinding model based on the Hertz contact theory. |

X. Ren et al. [14] | Established a simulation system using Surfel to visualise the material removal process interactively and to optimise the tool path planning. |

S. Wu et al. [15] | Presented a comprehensive platform to simulate the belt grinding system incorporating a kinematic model of the robot for tool path planning, dynamic model of the robot joint, along with the material removal model of the grinding process. |

S. Mezgahani et al. [16] | Performed a comparative study of contact pressure and abrasive grit size to material removal keeping parameters such as speed of workpiece, tool hardness, cycle time, coolant, abrasive feed, and tool wear constant. The study showed that a decrease in grain size results in more ploughing action rather than cutting action. |

J. Shibata et al. [17] | Offered a metal removal model incorporating the belt wear factor to explain the belt grinding characteristics quantitatively. |

A. Khellouki et al. [18] | Theoretically modelled contact conditions between abrasive film and the surface and investigated the effect of average contact pressure, contact duration and the number of active grains in the contact. |

V. T. Thien et al. [19] and Y. Sun et al. [20] | Demonstrated that pressure distribution obtained from pressure films can be correlated with a Hertzian model under different loads and hardness of the polymer wheel. |

H. Lv et al. [21] | Presented a material removal modelling technique for free-form surface using an echo state network. |

W. Wang et al. [22] | Proposed a grinding depth predicting frame working using a local stress model and a local material removal model taking into account the contact wheel deformation. |

Y. Sun et al. [11] and V. Pandiyan et al. [23] | Proposed a novel methodology using a dynamic pressure sensor to predict material removal considering belt grinding parameters such as force, workpiece geometry and different types of contact wheel geometry. |

Y. J. Wang et al. [24] | Demonstrated that the nonlinear material model performs better than the linear material removal model. |

Parameter | Unit | Levels | ||
---|---|---|---|---|

L1 | L2 | L3 | ||

RPM | (m/min) | 250 | 500 | 700 |

Feed | (mm/s) | 10 | 20 | 30 |

Force | (N) | 10 | 20 | 30 |

Rubber hardness | (Shore A) | 30 | 60 | 90 |

Grit Size | - | 60 | 120 | 220 |

**Table 3.**Taguchi design of experiments (DoE) using the L

_{27}orthogonal array and the corresponding depth of cut [12].

Trial No. | Factors | MRR | ||||
---|---|---|---|---|---|---|

RPM | Feed | Force | Hardness | Grit | Depth of Cut | |

(m/min) | (mm/s) | (N) | (Shore A) | (μm) | ||

1 | 250 | 10 | 10 | 30 | 60 | 65.60076 |

2 | 250 | 10 | 10 | 30 | 120 | 25.87109 |

3 | 250 | 10 | 10 | 30 | 220 | 13.34471 |

4 | 250 | 20 | 20 | 60 | 60 | 86.10453 |

5 | 250 | 20 | 20 | 60 | 120 | 44.20156 |

6 | 250 | 20 | 20 | 60 | 220 | 23.53456 |

7 | 250 | 30 | 30 | 90 | 60 | 93.8753 |

8 | 250 | 30 | 30 | 90 | 120 | 54.33391 |

9 | 250 | 30 | 30 | 90 | 220 | 23.55062 |

10 | 500 | 10 | 20 | 90 | 60 | 142.9324 |

11 | 500 | 10 | 20 | 90 | 120 | 86.37583 |

12 | 500 | 10 | 20 | 90 | 220 | 59.38035 |

13 | 500 | 20 | 30 | 30 | 60 | 120.6638 |

14 | 500 | 20 | 30 | 30 | 120 | 57.50747 |

15 | 500 | 20 | 30 | 30 | 220 | 45.55799 |

16 | 500 | 30 | 10 | 60 | 60 | 77.47286 |

17 | 500 | 30 | 10 | 60 | 120 | 26.08495 |

18 | 500 | 30 | 10 | 60 | 220 | 13.54166 |

19 | 700 | 10 | 30 | 60 | 60 | 134.8952 |

20 | 700 | 10 | 30 | 60 | 120 | 76.88529 |

21 | 700 | 10 | 30 | 60 | 220 | 58.97687 |

22 | 700 | 20 | 10 | 90 | 60 | 103.8255 |

23 | 700 | 20 | 10 | 90 | 120 | 56.9663 |

24 | 700 | 20 | 10 | 90 | 220 | 35.31606 |

25 | 700 | 30 | 20 | 30 | 60 | 114.009 |

26 | 700 | 30 | 20 | 30 | 120 | 56.65924 |

27 | 700 | 30 | 20 | 30 | 220 | 44.31528 |

Parameter | Value |
---|---|

Training set | 70% |

Testing set | 30% |

Method | Forward stepwise regression |

Starting model | Linear |

Upper limit | Quadratic |

S. No | Predictors | Estimate | Std Error | t-Stat | p-Value |
---|---|---|---|---|---|

1 | (Intercept) | 86.755 | 7.956 | 10.904 | 5.392 × 10^{–21} |

2 | RPM | 0.049856 | 0.0084825 | 5.8776 | 2.3937 × 10^{–8} |

3 | Feed | 2.0067 | 0.51282 | 3.9131 | 13.509 × 10^{–5} |

4 | Force | −2.7861 | 1.2896 | −2.1605 | 03.2243 × 10^{–4} |

5 | Rubber hardness | 1.7358 | 0.18569 | 9.3481 | 8.2594 × 10^{–17} |

6 | Grit size | −1.3237 | 0.062244 | −21.266 | 3.2042 × 10^{–48} |

7 | RPM: grit size | −0.00014814 | 4.1438 × 10^{–5} | −3.575 | 4.6498 × 10^{–4} |

8 | Feed: force | 0.062239 | 0.011008 | 5.654 | 7.1536 × 10^{–8} |

9 | Feed: rubber hardness | −0.069002 | 0.0091514 | −7.5401 | 3.4527 × 10^{–12} |

10 | Force: grit size | −0.0035275 | 0.00096854 | −3.642 | 3.6624 × 10^{–4} |

11 | Rubber hardness: grit size | −0.0010501 | 0.00031645 | −3.3184 | 1.1237 × 10^{–3} |

12 | Force^2 | 0.084463 | 0.029849 | 2.8297 | 5.265 × 10^{–3} |

13 | Grit size^2 | 0.0039407 | 0.00018283 | 21.554 | 6.5639 × 10^{–49} |

Parameter | Value |
---|---|

Maximum number of epochs to train | 200 |

Performance goal | 0 |

Backpropagation method | Bayesian regularisation |

Initial μ | 0.005 |

Hidden layers | 10, 20, 30 |

Training | 70% of the dataset |

Testing | 30% of the dataset |

Weight function | Dot product |

Activation function | tansig |

Predictors | 5 |

Response | 1 |

Hidden Layers | Epochs |
---|---|

10 | 151 |

20 | 62 |

30 | 40 |

Parameter | Value |
---|---|

andMethod | Prod |

orMethod | Max |

defuzzMethod | Wtaver (Weighted average) |

impMethod | Prod |

aggMethod | Max |

Membership function | /Sigmoidal/Gaussian bell/Gaussian/Bell-shaped |

Learning rules | Gradient descent algorithm |

Membership Function | RMSE |
---|---|

Sigmoidal membership [12] | 5.648 |

Gaussian bell membership | 6.7941 |

Gaussian membership | 7.100 |

Bell-shaped membership | 7.0341 |

Parameter | Value |
---|---|

Epsilon ($\mathsf{\epsilon}$) | 0.12457 |

Box Constraint (C) | 989.65 |

Optimisation Method | Bayesian optimisation |

Kernel | Gaussian RBF |

Kernel scale ($\gamma $) | 0.10894 |

Parameter | Value |
---|---|

In bag fraction | 0.25% of the training dataset |

Method used by trees | Regression |

Min leaf size | 5 |

Number of trees | 50 |

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**MDPI and ACS Style**

Pandiyan, V.; Caesarendra, W.; Glowacz, A.; Tjahjowidodo, T.
Modelling of Material Removal in Abrasive Belt Grinding Process: A Regression Approach. *Symmetry* **2020**, *12*, 99.
https://doi.org/10.3390/sym12010099

**AMA Style**

Pandiyan V, Caesarendra W, Glowacz A, Tjahjowidodo T.
Modelling of Material Removal in Abrasive Belt Grinding Process: A Regression Approach. *Symmetry*. 2020; 12(1):99.
https://doi.org/10.3390/sym12010099

**Chicago/Turabian Style**

Pandiyan, Vigneashwara, Wahyu Caesarendra, Adam Glowacz, and Tegoeh Tjahjowidodo.
2020. "Modelling of Material Removal in Abrasive Belt Grinding Process: A Regression Approach" *Symmetry* 12, no. 1: 99.
https://doi.org/10.3390/sym12010099