Enhance Displacement Amplification Ratio of Micro-Gripper Compliant Mechanisms Using Bridge-Type Amplifier Based on Make a Decision Criteria and Grey-Taguchi Method

3.1 Set up simulation

According to the study [1], the displacement and stress are affected by the distance between adjacent FHs, the FHs' length and the connected objects' length, the thickness of the FH, and other structural parameters have low sensitivity. This was also demonstrated by Kee-Bong Choi and colleagues from Korea Institute of Machinery and Materials, South Korea [10]. In this study, the variables were selected as depicted in Figure 2 and listed in Table 1. First, the length of the FH is the L variable along with three levels: 3 mm, 3.5 mm, and 4 mm. Next, the D variable is the distance between the centers of the two FHs; it has three levels: 0.8, 1, and 1.2; its unit is mm. The T variable is the thickness of FHs; it has three levels: 0.2, 0.3, and 0.4; its unit is mm. Finally, the R variable is the radius between the rigid link and flexure hinges; it has three levels: 0.4, 0.6, and 0.8; its unit is mm.

Twenty-seven cases were created with Minitab software, and the SolidWorks software designed twenty-seven models. Then, twenty-seven instances were imported into ANSYS' static structure module to analyze stress and displacement. The FEM was carried out in this module as presented in Section 2.2.

Table 2 depicts the simulation results from ANSYS. The 6th column shows the result of displacements, and the 8th column shows the result stress. According to this table, changing the input variables causes the output position and the stress of the gripper to change with different values in columns 6^th, 8^th. The results of twenty-seven cases are different, which proves that the input variable has an influence on the results.

Table 1. The factors and their levels

Table 2. L27 orthogonal array and simulation results

Table 3. The values were obtained by the MEREC method

3.2 Determine weight

The weight for every criterion must be determined to select the optimal case. In this work, the MEREC method was applied to determine the weight. The MEREC method, as presented in Section 2.4, and Eqs. (3)-(8) results were listed in Table 3. Column 2^nd was the result of H_ij values for output displacement when Eq. (3) was used for calculating, and column 3^rd was the result of H_ij values for output maximum principal stress when Eq. (4) was used for calculating. Column 4^th was the result of S_i values when Eq. (5) was used for calculation.

Columns 5^th and 6^th were the result of S_ij's values for output displacement and output maximum principal stress when Eq. (6) was used for calculating. Columns 7^th and 8^th resulted from e_j values for output displacement and output maximum principal stress when Eq. (7) was used for calculation. Finally, the weight results obtained for the two criteria of displacement were 0.7633 and stress were 0.2367 when Eq. (8) was used for calculation.

3.3 Grey relational analysis

Grey theory was frequently applied to systems with unclear models or incomplete knowledge [28]. Using a large number of discrete inputs, this effectively addressed the uncertainty problem [29].

GRA, a component of gray systems theory, is appropriate for solving numerous factors [30, 31], as follows:

Step 1: Determine the values of the objective

The largest value is the optimal in Eq. (9), and the smallest value is optimal in Eq. (10).

$D_i^*=\frac{D_i^{(0)}(k)-\min D_i^0(k)}{\max D_i^{(0)}(k)-\min D_i^{(0)}(k)}$                                    (9)

$D_i^*=\frac{\max D_i^{(0)}(k)-D_i^0(k)}{\max D_i^{(0)}(k)-\min D_i^{(0)}(k)}$                                    (10)

where, $D_i^{(0)}(k)$ are the values of D_i and S_t obtained from the method mentioned in Section 2.2.

Step 2: Calculate deviation

$\Delta_{0 i}=\left\|D_0^*(k)-D_i^*(k)\right\|$                                    (11)

$\Delta_{\min }=\max _{\forall j \in i} \min _{\forall k}\left\|D_0^*(k)-D_j^*(k)\right\|$                                    (12)

$\Delta_{\max }=\max _{\forall j \in i} \max _{\forall k}\left\|D_0^*(k)-D_j^*(k)\right\|$                                    (13)

Step 3: Estimate the grey relational coefficient (GRC) ($\gamma$)

$\gamma_i(k)=\frac{\Delta_{\min }+\xi \Delta_{\max }}{\Delta_{0 i}+\xi \cdot \Delta_{\max }}$                                    (14)

where, $\xi$=0.5.

Step 4: Compute GRG ($\psi_i$)

$\psi_i=\sum_{k=1}^n w_k \gamma_i(k)$                                    (15)

where, n is the total of the test and w_k is the weight of each objective obtained by the MEREC method.

Step 5: Compute the rank of the GRG value. The highest value of GRG was ranked first in the optimal case.

Table 4. Results of the grey relational analysis

The results of Eqs. (9)-(15) were presented in Table 4 and were archived by substituting the obtained D_i and S_t values from Table 2, where D_i^*(1) and D_i^*(2) are the values of the objective functions in which a larger displacement is better, and the smaller stress is better, respectively; Δ_oi(1) and Δ_oi(2) are the deviation values of the two criteria; and γ_i(1) and γ_i(2) are the grey relational coefficients of two criteria. Finally, ψ_i represents the GRG values. The optimal case was obtained by ranking the GRG values according to the criterion that the highest GRG value is the best. Therefore, the third case is the optimum case with the design variable at T₁L₁D₃R₃.

3.4 TOPSIS method

With multi-objective optimization problem was achieving high amplification while minimizing stress. The TOPSIS approach [32, 33] was used to confirm the optimal stress and displacement values of the innovative BT amplifier, as follows:

Step 1: Determine the objective's normalized values.

$n_{i j}=\frac{u_{i j}}{\sqrt{\sum_{i=1}^n u_{i j}^2}}$                                    (16) 

where, u_ij are the values of D_i and S_t obtained from the method mentioned in Section 2.2.

Step 2: Determine the objective's weighted normalized values.

$v_{i j}=\mathrm{w}_i n_{i j}$                                    (17)

where, w_i is the weight of each objective obtained by the MEREC method.

Step 3: Calculate the highest and lowest values of v_ij

$v^{+}=\left(v_1^{+}, v_2^{+}, \ldots v_n^{+}\right)$                                    (18)

$v^{-}=\left(v_1^{-}, v_2^{-}, \ldots v_n^{-}\right)$                                    (19)

Step 4: Determine values $K_i^{+}$ and $K_i^{-}$ of the optimal criteria

$K_i^{+}=\sqrt{\sum_{j=1}^n\left(v_{i j}-v_j^{+}\right)^2}$                                    (20)

$K_i^{-}=\sqrt{\sum_{j=1}^n\left(v_{i j}-v_j^{-}\right)^2}$                                    (21)

Step 5: Determine the values of CC_i

$CC_i=\frac{K_i^{-}}{K_i^{+}+K_i^{-}}$                                    (22)

Step 6: Calculate the highest value of the CC_i ranked first, which is the optimal case.

Using Eqs. (16)-(19) of the TOPSIS method to calculate the maximum and minimum values of the objective function, Table 5 shows the calculated results. The values of K_i⁺ and K_i^- were determined using Eqs. (20)-(21). The CC_i values were then calculated using Eq. (22). The largest value of CC_i was ranked as the third case, indicating that the third instance was the best. The optimal CC_i value obtained was 0.9508. Twenty-seven CC_i values were not identical. This difficulty so implies that the intended variables had a considerable effect on output displacement and stress. This result is identical to that discovered via Grey-R-A and FEA.

Table 5. Results of the TOPSIS method

3.5 SAW method

To improve the dependability of the GRA and TOPSIS approaches, the SAW (Simple Additive Weighting) method, which is extensively used in decision making for multi-objective issues [34]. In this study, the confirmation of the optimal values of the D_i and the S_t was performed using the SAW method [35] as follows:

Step 1: Determine the normalized values of each criterion

The largest value is the optimal in Eq. (23), and the smallest value is optimal in Eq. (24).

$n_{i j}=\frac{u_{i j}}{\max u_{i j}}$                                    (23)

$n_{i j}=\frac{\min u_{i j}}{u_{i j}}$                                    (24)

where, u_ij are the values of D_i and S_t obtained from the method mentioned in Section 2.2.

Step 2: Determine the sum of the weight-normalized values

$v_i=\sum_{j=1}^n w_j \cdot n_{i j}$                                    (25)

where, w_j is the weight of each objective obtained by the MEREC method.

Step 3: Calculate the highest value of the v_i ranked first, which is the optimal case.

Table 6. Results of the SAW method

Table 6 displays all of the SAW method's outcomes. The values in columns 2 and 3 were computed using Eqs. (23)-(24). The values of V_i were calculated using Eq. (25). The case with the highest V_i value was ranked third, indicating that it was the best. The ideal V_i value was 0.94498. Twenty-seven V_i values did not match. This difficulty indicates that the planned variables had a significant impact on output displacement and stress.

This result is identical to that found via gray relational analysis, finite element analysis, and the TOPSIS method.

3.6 WASPAS method

Furthermore, optimum findings were produced utilizing the WASPAS approach [36, 37]. WASPAS is also among the best methods for determining decision procedures for discovering the optimal situation that fulfills numerous criteria or objectives. It was completed as follows:

Step 1: Use Eq. (23) and Eq. (24) to determine n_ij; next determine v_ij, Q_i and P_i

$v_{i j}=w_j \cdot n_{i j}$                                    (26)

where, w_j is the weight of each objective, obtained by the MEREC method.

$Q_i=\sum_{j=1}^n v_{i j}$                                    (27)

$P_i=\prod_{j=1}^n\left(v_{i j}\right)^{w_i}$                                    (28)

Step 2: Determine A_i

$A_i=\lambda \cdot Q_i+(1-\lambda) \cdot P_i$                                    (29)

Step 3: Calculate the highest value of the A_i ranked as the first, which is the optimal case.

Table 7. Results of the WASPAS method

Table 7 displays all of the WASPAS method's outcomes. The values in columns 4 and 5 were calculated using Eq. (26). Eq. (27) were calculated Q_i's value. Eq. (28) were calculated P_i's value. The A_i values were then determined with Eq. (29). The highest value of A_i was assigned to the third scenario, indicating that it was the best. The ideal A_i value was 0.9048. Twenty-seven A_i values did not match. This difficulty indicates that the planned variables had a significant impact on output displacement and stress.

This result optimally corresponds to the grey relational analysis, the TOPSIS method, and the SAW method.

3.7 VIKOR method

The VIKOR approach [38, 39] confirmed the ideal situation for the four indicated methods. It is assumed that compromise is appropriate for conflict resolution and that the decision maker chooses the closest option to the ideal, therefore the alternatives are examined against all set criteria. This is likewise an effective and dependable multi-objective optimization method, which is carried out through the following steps:

Step 1: Calculate the maximum and minimum value of each criterion

$u_j^{+}=\max u_{i j}$                          (30)

$u_j^{-}=\min u_{i j}$                          (31)

where, u_ij the values of D_i and S_t are obtained from the method mentioned in Section 2.2.

Step 2: Determine the r_ij values

$r_{i j}=\left(\left|u_j^{+}-u_{i j}\right|\right) /\left(\left|u_j^{+}-u_j^{-}\right|\right)$                          (32)

Step 3: Determine the S_i values

$S_i=\sum_{j=1}^n w_j r_{i j}$                          (33)

where, w_j is the weight of each objective obtained by the MEREC method.

Step 4: Determine the R_i values

$R_i=\max \left(w_j \cdot r_{i j}\right)$                          (34)

Step 5: Determine the maximum and minimum values of the S_i values and the R_i values

$S^{-}=\min \left(S_i\right)$                          (35)

$S^{+}=\max \left(S_i\right)$                          (36)

$R^{-}=\min \left(R_i\right)$                          (37)

$R^{+}=\max \left(R_i\right)$                          (38)

Step 6: Determine the Q_i values

$\begin{gathered}Q_i=0.5\left(S_i-S^{-}\right) /\left(S^{+}-S^{-}\right)+0.5\left(R_i-R^{-}\right) /\left(R^{+}-R^{-}\right)\end{gathered}$                          (39)

Step 7: The best alternative, according to the compromise ranking list, is the first with the lowest with the condition that:

Condition 1:

$Q_i(2)-Q_i(1) \geq D Q$                          (40)

where,

(2) = alternative with the second order in ranking Q_i

(1) = alternative with the best order in ranking Q_i

$D Q=1 /(m-1)$                          (41)

where, m is the number of alternatives.

Condition 2:

Alternative (1) must be ranked best in S_iand R_i

Table 8 presented all of the VIKOR method's outcomes. The values in columns 4 and 5 were computed using Eqs. (33)-(38). The values of Q_i were determined by using Eq. (39). The lowest value of Q_i was ranked third case when the Q_i value was 0.00000, In this table, the values of S_i, R_i, and Q_i were used to determine the optimal ranking (7^th column).

Table 8. Results of the VIKOR method

According to Eq. (40):

Q_i(2) – Q_i(1)=0.18079-0.000=0.18079 > DQ=0.038461

Where: Q_i (1) = 0.000 is the alternative with the smallest option in Q_i ranking.

Q_i(2)=0.18079 is the alternative with the 2^nd smallest option in Q_i ranking

DQ=1/(27-1)=0.038461 (Eq. (41))

And according to Figure 5, the third case is the smallest of S_i, R_i, and Q_i. Therefore, the Q_i values were determined according to the criterion that the lower value is the best. The third case is the optimal case.

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Figure 5. Ranking of Q_i, S_i, and R_i

3.8 Results of signal to noise analysis

The impacts of the levels of the design variables were confirmed by using the Taguchi technique based on the study of S/N. The GRG values and CC_i values in Table 4 and Table 5 were used to analyze S/N. The S/N-A results of GRG and CC_i were listed in Table 9 and Table 10, respectively. The delta values were calculated as the maximum values minus the minimum values. The values in the table revealed that the maximum values are the optimal cases for the design variables. In Table 9, the first rank indicates a variable (T), which significantly affected the GRG value; next is variable (D), a variable (R), and finally, a variable (L). The data from Table 9 was also utilized to generate the graph shown in Figure 6. The levels of the design variables are presented on the horizontal axis, with the S/N values presented on the vertical axis. The variable (T) has a considerable effect on the GRG, next is a variable D, followed by a variable R, and finally by a variable L. The influence grows as the graph's slope increases. In summary, the designed dimensions have a significant influence on the stress and displacement of GMSBTCM. The high peak on the graph indicates that at the position obtained, the optimal levels of the design variables were T1, L1, D3, R3, and that is the third case.

In Table 10, the first rank indicates a variable (T), which significantly affected the CC_i value; next is a variable (D), a variable (L), and finally a variable (R). The data from Table 10 was also utilized to generate the graph shown in Figure 7. The levels of the design variables are presented on the horizontal axis, with the S/N values presented on the vertical axis. The influence grows as the graph's slope increases. The variable (T) has a considerable effect on the CC_i values; next to is variable D, followed by variable L, and finally by variable R. So, the designed dimensions have a significant influence on the stress and displacement of GMSBTCM. The high peak on the graph indicates that at the position obtained, the optimal levels of the design variables were T1, L1, D3, R3, and that is the third case.

Table 9. Response table for signal to noise ratios of GRG

Table 10. Response table for SN ratios of CC_i

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Figure 6. The plot means of S/N of GRG

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Figure 7. The plot means of S/N of CC_i

Table 11. Response table for signal to noise ratios of V_i

Table 12. Response table for signal to noise ratios of A_i

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Figure 8. The plot means of S/N of V_i

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Figure 9. The plot means of S/N of A_i

The impacts of the levels of the design variables were confirmed by using the Taguchi technique based on the study of S/N. The V_i values and A_i values in Table 6 and Table 7 were used to analyze S/N. The S/N-A results of V_i and A_i were listed in Table 11 and Table 12, respectively. The delta values were calculated as the maximum values minus the minimum values. The values in the table revealed that the maximum values are the optimal cases for the design variables. In Tables, the first rank indicates a variable (T), which significantly affected the V_i value and A_i value; next is a variable (D), a variable (R), and finally, a variable (L). The data from this tables were also utilized to generate the graph shown in Figure 8 and Figure 9. The levels of the design variables are presented on the horizontal axis, with the S/N values presented on the vertical axis. The variable T has a considerable effect on the V_i and A_i values; next is a variable D, followed by a variable R, and finally by a variable L. The influence grows as the graph's slope increases. In summary, the designed dimensions have a significant influence on the stress and displacement of GMSBTCM. The high peak on the graph indicates that at the position obtained, the optimal levels of the design variables were T1, L1, D3, R3, and that is the third case.

The results of the analysis of variance (ANOVA) of GRG were presented in Table 13. In this table, column 1 is the designed dimensions and the interaction of the designed dimensions. Column 2 the degree of freedom of the designed dimensions, the interaction of the designed dimensions, and the error. The column 3 is the sequential sum of squares. The column 4 is the contribution percentage of the designed dimensions and the interaction of the designed dimensions. The column 5 is the adjusted sum of squares. The values of column 3 and column 5 are to equal each other. The column 6 is adjusted mean square. Columns 7 and 8 are the F-value and P-value. The F-values are greater than 2 and the P-values are less than 0.05. This indicates that the designed dimensions have strongly affected the GRG, or the displacement and the stress of the GMSBTCM. And the influence of specific design dimensions is as follows: the variable (T) has a considerable effect on the GRG values or the displacement and equivalent stress; next is variable D, followed by variable R, and finally by variable L. This is clearly shown in the percentage contribution column of the design dimensions. In addition, the results of the ANOVA achieved the following R-square, R-square (adjusted), and R-square (pred): values 97.55%, 96.38%, and 95.37%, as presented in Table 14.

The results of the ANOVA analysis of CC_i were presented in Table 15. In this table, column 1 is the designed dimensions and the interaction of the designed dimensions. Column 2: the degree of freedom of the designed dimensions, the interaction of the designed dimensions, and the error. The column 3 is the sequential sum of squares. The column 4 is the contribution percentage of the designed dimensions and the interaction of the designed dimensions. The column 5 is the adjusted sum of squares. The values of the column 3 and column 5 are equal each other. The column 6 is adjusted mean square. Columns 7 and 8 are F-value, and P-value. The F-values are greater than 2 and the P-values are less than 0.05. This indicates that the designed dimensions have strongly affected the CC_i, or the displacement and the stress of the GMSBTCM. And the influence of specific design dimensions is as follows: the variable (T) has a considerable effect on the CC_i values or the displacement and equivalent stress, next is variable D, followed by variable L, and finally by variable R. This is clearly shown in the percentage contribution column of the design dimensions. In addition, the results of the ANOVA achieved the following R-square, R-square (adjusted), R-square (pred), values: 99.92%, 99.64%, and 98.34%, as presented in Table 16.

Table 13. Analysis of variance of GRG

Table 14. Model summary of GRG

Table 15. Analysis of variance of CC_i

Table 16. Model summary CC_i

Table 17. Analysis of variance of V_i

Table 18. Model summary of V_i

The results of the ANOVA analysis of V_i were presented in Table 17. In this table, column 1 is the designed dimensions and the interaction of the designed dimensions. Column 2: the degree of freedom of the designed dimensions, the interaction of the designed dimensions, and the error. The column 3 is the sequential sum of squares. The column 4 is the contribution percentage of the designed dimensions and the interaction of the designed dimensions. The column 5 is the adjusted sum of squares. The values of column 3 and column 5 are equal to each other. The column 6 is adjusted mean square. Columns 7 and 8 are F-value, and P-value. The F-values are greater than 2 and the P-values are less than 0.05. This indicates that the designed dimensions have strongly affected the V_i, or the displacement and the stress of the GMSBTCM. And the influence of specific design dimensions is as follows: the variable (T) has a considerable effect on the V_i values or the displacement and equivalent stress, next is variable D, followed by variable L, and finally by variable R. This is clearly shown in the percentage contribution column of the design dimensions. In addition, the results of the ANOVA achieved the following R-square, R-square (adjusted), R-square (pred), values: 99.89%, 99.54%, and 97.85%, as presented in Table 18.

The results of the ANOVA analysis of A_i were presented in Table 19. In this table, column 1 is the designed dimensions, and the interaction of the designed dimensions. Column 2: the degree of freedom of the designed dimensions, the interaction of the designed dimensions, and the error. The column 3 is the sequential sum of square. The column 4 is the contribution percentage of the designed dimensions and the interaction of the designed dimensions. The column 5 is the adjusted sum of squares. The values of column 3 and column 5 are equal to each other. The column 6 is adjusted mean square. The columns 7 and 8 are F-value, and P-value. The F-values are greater than 2 and the P-values are less than 0.05. This indicates that the designed dimensions have strongly affected the A_i, or the displacement and the stress of the GMSBTCM. And the influence of specific design dimensions is as follows: the variable (T) has a considerable effect on the A_i values or the displacement and the stress, next is variable D, followed by variable L, and finally by variable R. This is clearly shown in the percentage contribution column of the design dimensions. In addition, the results of the ANOVA achieved the following R-square, R-square (adjusted), and R-square (pred), values: 99.9%, 99.56%, and 97.96%, as presented in Table 20.

Table 19. Analysis of variance of A_i

Table 20. Model summary of A_i

Table 21. Analysis of variance of Q_i

Table 22. Model summary of Q_i

The results of the ANOVA analysis of Q_i were presented in Table 21. In this table, column 1 is the designed dimensions and the interaction of the designed dimensions. Column 2: the degree of freedom of the designed dimensions, the interaction of the designed dimensions, and the error. The column 3 is the sequential sum of squares. The column 4 is the contribution percentage of the designed dimensions and the interaction of the designed dimensions. The column 5 is the adjusted sum of squares. The values of column 3 and column 5 are equal to each other. The column 6 is adjusted mean square. The columns 7 and 8 are F-value, and P-value. The F-values are greater than 2 and the P-values are less than 0.05. This indicates that the designed dimensions have strongly affected the Q_i, or the displacement and equivalent stress of the GMSBTCM. And the influence of specific design dimensions is as follows: the variable (T) has a considerable effect on the Q_i values or the displacement and equivalent stress, next is variable D, followed by variable R, and finally by variable L. This is clearly shown in the percentage contribution column of the design dimensions. In addition, the results of the ANOVA achieved the following R-square, R-square (adjusted), and R-square (pred) values: 99.86%, 99.4%, and 97.19%, as presented in Table 22. Through the R-square of the ANOVA analysis results of GRG, CC_i, V_i, A_i, and Q_i, it was shown that the ANOVA analysis results showed that the best result was the TOPSIS method, next to is the WASPAS method, SAW method, VIKOR method, and finally the grey relational analysis.

3.9 Discussion of optimization methods

Table 23 provides a synthesis of the optimal rankings of the five implemented methods. The Saw and Waspas methods yield very similar ranking results in this table. There are only two cases: Case 10 is ranked 21 with the SAW method and 20 with the WASPAS method, while Case 13 is the opposite, with the Saw method ranking at 20 and Waspas at 21. These two methods share similarities in their calculations, yielding almost identical results. Generally, all 5 methods rank Case 3 as the best optimal and Case 19 as the badest optimal (rank 27). Cases 2, 12, 17, 22, 23, and 25 have 4 methods with identical rankings; the remaining cases differ between the two methods.

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Figure 10. The optimal result of ranking

The similarity in rankings is shown in Figure 10 when the ranking lines of the five methods have the same similarity and look in the same direction.

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Figure 11. The result of output displacement

Table 23. The optimal result of ranking