Optimal Design of a Six-Dimensional Force Sensor with Three-Beam Structure

Chen, Xiliang; Wang, Zhiqiang; Xie, Zhengyu; Li, Ning; Sun, Yiwen

doi:https://doi.org/10.1155/2023/4902624

Journal of Sensors

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Research Article | Open Access

Volume 2023 | Article ID 4902624 | https://doi.org/10.1155/2023/4902624

Optimal Design of a Six-Dimensional Force Sensor with Three-Beam Structure

Xiliang Chen,^1,2Zhiqiang Wang,^2,3Zhengyu Xie,²Ning Li,¹and Yiwen Sun²

Academic Editor: Roshan Prakash Joseph

Received13 Mar 2023

Revised29 Sept 2023

Accepted30 Oct 2023

Published27 Nov 2023

Abstract

A six-dimensional force sensor with a three-beam structure was designed, and its strain and stress distribution under various load conditions were analyzed based on finite element simulation technology. According to the strain distribution, the Wheatstone bridge scheme of the sensor is designed, and the output voltage and sensitivity formulas in each direction are deduced. The optimal patch location was determined according to the strain distribution on the centerline of the strain beam. Finally, the size parameters of the elastic body were optimized based on the response surface method and multiobjective genetic algorithm, which greatly improved the sensitivity of the sensor in all directions. It can provide a useful guidance for the optimal design of the six-dimensional force sensor with a three-beam structure.

1. Introduction

As an important space force sensing element, the six-dimensional force sensor has been researched and explored by experts and scholars at home and abroad. In 1975, American scholars P. C. Waston and S. H. Drake designed a six-dimensional force/moment sensor with three vertical ribs [1], which has the advantages of simple structure, good lateral effect, and large bearing capacity, but the disadvantage is that the vertical effect is poor. The interdimensional coupling is large, which means the force/torque applied to the sensor in each dimension will have a large impact on the output signals of the sensor, resulting in interdimensional coupling problems. To avoid such disadvantages, Stanford University in the US designed a six-dimensional wrist force sensor with a “Maltess Cross” structure [2], which has the advantages of a symmetrical and compact structure and small interdimensional coupling, but the overload capacity and dynamic characteristics are poor. Subsequently, Belgium, Israel, Japan, China, South Korea, and other countries successively carried out the research [3–7] and designed a variety of six-dimensional force sensors with different structural forms.

The optimization of the structural parameters of the elastic body is one of the important research in the design of six-dimensional force sensors [8–11], which directly determines the overall performance of the sensors. The optimization of the structural parameters of the elastic body is one optimization scheme that defines certain objectives, such as sensitivity, quality, etc., that are optimized under certain constraints, such as size range, stiffness, safety margin, etc., to obtain the best structure parameters. Due to the complexity of the elastic body structure, it is difficult to establish an accurate mathematical model with an analytical solution [12]. Hence, the finite element method (FEM) is commonly used to solve such problems. Kim [13] used FEM to design and simulate a six-axis wrist force/moment sensor as an example. Among the various indicators of the sensor, sensitivity is one of the most important indicators, which will greatly affect the overall performance of the sensor [14]. In view of the above problems, a six-dimensional force sensor with a three-beam structure is designed in this paper, whose structure is simulated by using the finite element software ANSYS. The Wheatstone bridge scheme of the sensor is designed according to the strain distribution, the output voltage and sensitivity formulas of the six-dimensional force sensor in each direction were derived, and the optimal patch location was determined according to the strain distribution of the centerline of the strain beam surface. Finally, the optimal structural parameters of the elastic body were determined by the response surface method (RSM) and multiobjective genetic algorithm (MOGA), and the results showed that the sensitivity of the optimized sensor was greatly improved in all directions.

2. Finite Element Analysis of the Elastic Body

The designed structure of the three-beam elastic body is shown in Figure 1, including the outer flange, central platform, flexible beam, and strain beam. The three strain beams AB, CD, and EF form an angle of 120° with each other. Among them, the outer flange diameter is D, the central platform diameter is d, the strain beam width is W, the strain beam height is H, and the flexible beam thickness is t.

The elastic body of the initial six-dimensional force sensor is analyzed by the finite element software. The strain-sensitive area and the stress distribution under the load of each working condition are understood. The measuring range and initial size parameters of the six-dimensional force sensor are shown in Table 1, and the elastomer material is 2024-T6, as shown in Table 2.

Import the 3D model into ANSYS, where fixed constraints are applied to the bottom of the outer flange, and seven load cases are applied to the upper surface of the central platform, namely (1) = 200 N; (2) = 200 N; (3) = 200 N; (4) = 8 N m (5) = 8 N m; (6) = 8 N m; and (7) fully loaded.

Under the condition of = 200 N, the strain beam CD and EF have large strain on the sides, as shown in Figure 2(a), and the maximum von-Mises stress is 49.15 MPa, which occurs at the connection between the strain beam and the flexible beam, as shown in Figure 2(b). Under the condition of = 200 N, the strain beams AB, CD, and EF have large strain on the sides, as shown in Figure 3(a), the maximum von-Mises stress is 48.48 MPa, which occurs at the connection between the strain beam and the flexible beam, as shown in Figure 3(b). Under the condition of = 200 N, the upper and lower surfaces of the strain beams AB, CD, and EF have large strain, as shown in Figure 4(a), the maximum von-Mises stress is 36.43 MPa, which occurs at the connection between the flexible beam and the outer flange, as shown in Figure 4(b). Under the condition of = 8 N m, the upper and lower surfaces of the strain beams CD and EF have large strain, as shown in Figure 5(a), and the maximum von-Mises stress is 76.03 MPa, which occurs at the connection between the strain beam and the central platform, as shown in Figure 5(b). Under the condition of = 8 N m, the upper and lower surfaces of the strain beams AB, CD, and EF have large strain, as shown in Figure 6(a), the maximum von-Mises stress is 81.65 MPa, which occurs at the connection between the strain beam and the central platform, as shown in Figure 6(b). Under the condition of = 8 N m, the strain beams AB, CD, and EF have a larger strain on the sides, as shown in Figure 7(a), the maximum von-Mises stress is 40.74 MPa, which occurs at the connection between the strain beam and the central platform, as shown in Figure 7(b). Under fully loaded conditions, the maximum von-Mises stress is 228.94 MPa, which occurs at the connection between the strain beam and the central platform, as shown in Figure 8.

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3. Strain Gauge Layout and Wheatstone Bridge Scheme

From Figures 2–7, it can be found that when the two opposite surfaces on the strain beam are subjected to the force in the sensitive direction, the strain values must be one positive and one negative. Therefore, according to this characteristic, six half-bridge circuits are assembled on the three-beam structure elastic body, and each bridge can be reused to detect forces in multiple directions and then obtain the voltage output in the corresponding direction through algebraic operations. The strain gauge layout and Wheatstone bridge scheme designed according to the above ideas are shown in Figure 9. The strain gauges S1–S12 are pasted on the four surfaces of each strain beam, and the bridge resistance R is a fixed resistor whose resistance value is the same as that of the strain gauges S1–S12.

The resistance of the strain gauge is as follows:where ρ is the resistivity of the material, L is the length of the resistance, A is the cross-sectional area of the resistance, ν is the Poisson’s ratio of the material, ε is the average strain of the strain gauge, and K is the sensitivity coefficient of the strain gauge.

When the elastic body is subjected to the force in the positive direction of , the resistance values of strain gauges S9 and S12 increase, and the resistance values of S10 and S11 decrease, the output of half-bridge circuits are as follows:where SG5–SG6 are the output voltage of the bridge, U is the supply voltage of the Wheatstone bridge, ΔR is the change value of the resistance of strain gauge, and is the average strain of strain gauge in the corresponding bridge.

When the elastic body is subjected to the force in the positive direction of , the resistance values of strain gauges S7, S10, and S12 increase, and the resistance values of S8, S9, and S11 decrease, the output of half-bridge circuits are as follows:

When the elastic body is subjected to the force in the positive direction of , the resistance values of strain gauges S1, S3, and S5 increase, and the resistance values of S2, S4, and S6 decrease, the output of half-bridge circuits are as follows:

When the elastic body is subjected to the force in the positive direction of , the resistance values of strain gauges S3 and S6 increase, and the resistance values of S4 and S5 decrease, the output of half-bridge circuits are as follows:

When the elastic body is subjected to the force in the positive direction of My, the resistance values of strain gauges S2, S3, and S5 increase, and the resistance values of S1, S4, and S6 decrease, the output of half-bridge circuits are as follows:

When the elastic body is subjected to the force in the positive direction of , the resistance values of strain gauges S7, S9, and S11 increase, and the resistance values of S8, S10, and S12 decrease, the output of half-bridge circuits are as follows:

In summary, the signal distribution of the six half-bridge circuits under uniaxial load is obtained, as shown in Table 3.

It is easy to get the output voltage and sensitivity of the six-dimensional force sensor in each direction:where U_Fx–U_Mz is the output voltage of each direction, is the average strain of the strain gauge in the relevant bridge, and S_Fx–S_Mz is the sensitivity of each direction of the sensor.

4. Optimization of the Pasting Position of the Strain Gauge

In order to obtain the best pasting position of the strain gauge, the path mapping technology provided by ANSYS postprocessing is used to analyze the strain of each node of the elastic body. The position of the centerline of the side and the upper surface of the strain beam is, respectively, selected as the defined path, and the length of the strain beam is used as the path definition interval, and the strain distribution of the path interval under each load condition is obtained, as shown in Figure 10. The abscissa indicates the distance between each node of the defined path and the edge of the central platform, and the ordinate indicates the strain value of each node along the sensitive direction.

It can be seen from Figure 10 that the nodal strain value decreases with the increase of the distance under any load condition and changes sharply due to stress concentration in the range of 0–1 mm and the change in the range of 1–5 mm is relatively gentle, and the strain value is large. The strain value in the range of 5–8 mm is small. Considering the sensitivity and stability of the six-dimensional force sensor comprehensively, the strain gauge should be pasted within the range of 1–5 mm. This paper uses Minebea YJ-1 strain gauge (length is 3.3 mm, width is 2.5 mm), and S1–S12 strain gauges are pasted in the range of 1–4.3 mm, as shown by the shadow in Figure 10. Finally, the strain values of S1–S12 strain gauges under each load condition are obtained through finite element simulation.

Summarizing the finite element results and calculating the sensor sensitivity (K = 2) according to Equation (20), the data in Table 4 can be obtained. Under the uniaxial load condition, the maximum von-Mises stress is 81.65 MPa, which is far less than the material fatigue limit of 140 MPa. Under the fully loaded condition, the maximum von-Mises stress is only 228.94 MPa, which is far less than the material tensile yield strength of 340 MPa. The sensor sensitivity is from 0.4068 to 1.0263 mV/V. The average sensitivity is only about 0.67 mV/V, and the sensitivity is relatively small. It can be seen that the structural parameters of the elastic body have a large space for optimization.

5. Parameter Optimization of Elastic Body Based on RSM

The RSM describes the relationship between the output response of the system and the input variables of the system by approximately constructing a polynomial with a clear analytical expression. The RSM can be used to predict the output response of the elastic body under different structural parameters, such as the maximum von-Mises stress, strain, quality, etc., so as to realize the optimal design of the elastic body [15]. The process is shown in Figure 11.

After sensitivity analysis and parameter screening, the variable factors that have little influence on the stress and strain of the elastic body are eliminated. At the same time, considering the limitation of the shape size of the six-dimensional force sensor, the main influencing variables are selected as central platform diameter, strain beam height, strain beam width, and flexible beam thickness. Using the response surface optimization module integrated with ANSYS software, the design of the experiments method adopts the CCD and uses the Kriging space interpolation method for modification to increase the accuracy of the response surface model [16]. The structural size of the elastic body is taken as the input variable, and the maximum von-Mises stress, and the average strain of the six bridge strain gauges under each load condition are the output variables. Among them, the value range of central platform diameter d is (25, 40 mm), the value range of strain beam height H is (4, 8 mm), the value range of strain beam width W is (4, 8 mm), and the value range of flexible beam thickness t is (0.8, 1.5 mm). The responses are (1) at load case Fx = 200 N, the maximum von-Mises stress EQS_Fx, the average strain of strain gauges S9–S12 ε_Fx; (2) at load case Fy = 200 N, the maximum von-Mises stress EQS_Fy, the average strain of strain gauges S7–S12 ε_Fy; (3) at load case Fz = 200 N, the maximum von-Mises stress EQS_Fz, the average strain of strain gauges S1–S6 ε_Fz; (4) at load case Mx = 8 N m, the maximum von-Mises stress EQS_Mx, the average strain of strain gauges S3–S6 ε_Mx; (5) at load case My = 8 N m, the maximum von-Mises stress EQS_My, the average strain of strain gauges S1–S6 ε_My; (6) at load case Mz = 8 N·m, the maximum von-Mises stress EQS_Mz, the average strain of strain gauges S7–S12 ε_Mz; (7) at fully loaded case, the maximum von-Mises stress EQS_Full.

The correlation coefficient R² of the finally established response surface model is 0.999, indicating that the fitted response surface model has a very high precision and can be used for optimal design.

In order to take into account the service life and sensitivity of the six-dimensional force sensor, the optimization constraints are set as follows: the maximum von-Mises stress under the uniaxial full load condition is not greater than the fatigue limit of the material, and the maximum von-Mises stress under the fully loaded condition is not greater than the tensile yield strength of the material. The optimization goal is to maximize sensor sensitivity; that is, the average strain ε_Fx–ε_Mz reaches the maximum value. Then, the optimization problem of the elastic body can be expressed by the following formulae [17]:

Find the design variables:

Objective:

And satisfy the constraints:where are the weight factor, in this paper, .

MOGA is a modeling method that simulates natural evolution and is widely used to solve complex optimization problems. It supports multiple objectives and constraints and aims at finding the global optimum [18]. In this paper, the algorithm is configured with an initial sample size of 3,000, generating 600 samples per iteration and finding 3 candidate samples in a maximum of 20 iterations. After rounding, the diameter of the central platform d is 25 mm, the height of the strain beam H is 4.7 mm, the width of the strain beam W is 7.7 mm, and the thickness of the flexible beam t is 1.1 mm. The optimized size parameters are substituted into the model for finite element analysis.

Under the condition of Fx = 200 N, the maximum von-Mises stress EQS_Fx is 72.17 MPa, and the average strain ε_Fx is 263.0 , as shown in Figure 12. Under the condition of Fy = 200 N, the maximum von-Mises stress EQS_Fy is 75.55 MPa, and the average strain ε_Fy is 179.3 , as shown in Figure 13. Under the condition of Fz = 200 N, the maximum von-Mises stress EQS_Fz is 58.05 MPa, and the average strain ε_Fz is 371.9 , as shown in Figure 14. Under the condition of Mx = 8 N m, the maximum von-Mises stress EQS_Mx is 122.27 MPa, and the average strain ε_Mx is 841.9 , as shown in Figure 15. Under the condition of My = 8 N m, the maximum von-Mises stress EQS_My is 129.07 MPa, and the average strain ε_My is 648.4 , as shown in Figure 16. Under the condition of Mz = 8 N m, the maximum von-Mises stress EQS_Mz is 50.27 MPa, and the average strain ε_Mz is 335.5 , as shown in Figure 17. Under the condition of fully loaded, the maximum von-Mises stress EQS_Full is 333.84 MPa, as shown in Figure 18.

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Summarizing the finite element results and calculating the sensor sensitivity (K = 2) according to Equation (20), the data in Table 5 can be obtained. Under the uniaxial full load condition, the maximum von-Mises stress is 129.07 MPa, which is close to but not exceeding the fatigue limit of the material; under the fully loaded condition, the maximum von-Mises stress of the elastic body is 333.84 MPa, which is close to but not exceeding the tensile yield strength of the material; the sensitivity in the Fx direction is 0.526 mV/V, increased by 29.3% compared with the initial structure, the sensitivity in the Fy direction is 0.5379 mV/V, increased by 26.2%, the sensitivity in the Fz direction is 1.1157 mV/V, increased by 98.7%, the sensitivity in the Mx direction is 1.6838 mV/V, increased by 89.5%, the sensitivity in the My direction is 1.9452 mV/V, increased by 89.5%, and the sensitivity in the Mz direction is 1.0065 mV/V, increased by 40.7%. In summary, under the condition that the six-dimensional force sensor meets the strength requirements, the sensitivity in all directions is greatly improved compared with the initial structure. A very good optimization effect has been achieved.

6. Conclusion

In this paper, a six-dimensional force sensor with a three-beam structure is designed. Based on the finite element simulation technology, its strain and stress distribution under various load conditions are analyzed. According to the strain distribution, the Wheatstone bridge scheme of the sensor is designed. The output voltage and sensitivity formula of the six-dimensional force sensor in each direction are obtained from the bridge circuit. According to the strain distribution of the centerline of the surface of the strain beam, the optimal patch position is determined. Finally, based on the RSM, the size parameters of the elastic body were optimized so that the sensitivity of the sensor in all directions was greatly improved, among which the sensitivity in the Fx direction increased by 29.3%, the Fy direction increased by 26.2%, the Fz direction increased by 98.7%, the Mx direction increased by 89.5%, the My direction increased by 89.5%, and the Mz direction increased by 40.7%.

Compared with the maltess cross beam, the number of strain gauges required to be pasted on three-beam is reduced from 24 to 12. The use of a three-beam structure can greatly reduce the manufacturing cost of sensors and make the sensors easy to miniaturize. This paper can provide useful guidance for the optimal design of a six-dimensional force sensor with a three-beam structure.

However, the present work in this paper only considers the optimization of sensitivity and does not involve other indicators of the sensor. In the future, both sensitivity and stiffness will be considered for optimization and conduct relevant verification through the prototype sensor.

Data Availability

The data that support the findings of this study are available from Taizhou Zhongqing Technology Co. Ltd. Restrictions apply to the availability of these data, which were used under license for this study. Data are available from the authors upon reasonable request and with the permission of Taizhou Zhongqing Technology Co. Ltd.

Conflicts of Interest

The authors have no conflicts to disclose.

Acknowledgments

The authors would like to acknowledge the funding support from the Taizhou Zhongqing Technology Co. Ltd. (HJ-2019-174), the Science Technology Department of Zhejiang Province (LQ20E050011), and the Taizhou University of China.

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Copyright © 2023 Xiliang Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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