In the field of robotics and automation, the development of high-performance sensing technologies is crucial for enabling precise control and interaction with the environment. Among these, the six-axis force sensor plays a pivotal role in measuring three-dimensional force and moment components, which are essential for applications such as robotic manipulation, haptic feedback, and industrial automation. This article presents a comprehensive study on the design and optimization of a novel six-axis force sensor, focusing on improving dynamic performance and sensitivity through advanced structural analysis and response surface methodology. The proposed sensor elastomer structure incorporates unique features to enhance strain concentration and reduce cross-talk, ensuring high accuracy in multi-axis force measurements. Throughout this work, we emphasize the importance of the six-axis force sensor in modern robotics and explore innovative approaches to address its design challenges.
The core of any six-axis force sensor lies in its elastomer structure, which directly influences key performance metrics such as sensitivity, bandwidth, and decoupling capability. Traditional designs, including cross-beam configurations, have been widely used due to their symmetry and low inter-axis coupling. However, these structures often face limitations in dynamic response and sensitivity under high-speed operating conditions. To overcome these issues, we propose a new elastomer design that integrates circumferential and radial beams with strategically placed holes to concentrate strain and improve stiffness. This design aims to achieve a balance between static and dynamic performance, making the six-axis force sensor suitable for demanding applications like high-speed robotics.
In this study, we begin by detailing the structural model of the six-axis force sensor, followed by finite element analysis (FEA) for static and modal evaluations. The strain distribution under various loading conditions is analyzed to identify optimal locations for strain gauge placement. Subsequently, we employ response surface analysis to optimize critical dimensions of the elastomer, establishing regression models that relate design parameters to performance outputs. Experimental validation confirms the feasibility of the design, demonstrating significant improvements in sensitivity and natural frequency. The following sections elaborate on each aspect of this work, providing a thorough exploration of the six-axis force sensor’s development.
Structural Design of the Six-Axis Force Sensor Elastomer
The elastomer structure of the six-axis force sensor is designed to achieve high sensitivity and dynamic performance. It consists of a central platform, circumferential beams, radial beams, and a peripheral support ring, forming an inner-outer ring support configuration. The radial beams are connected to the midpoint of the circumferential beams, creating a “T-shaped” beam arrangement. Four sets of these T-shaped beams are arranged in a cross pattern around the central platform, with their outer ends fixed to the peripheral support. Adjacent circumferential beams are linked by connection blocks, forming a ring-like central platform. Holes are introduced in both circumferential and radial beams to localize strain, thereby enhancing sensitivity. Specifically, vertical waist-shaped holes are symmetrically placed on the circumferential beams, resulting in an “H-shaped” beam structure that increases stiffness and dynamic performance. Similarly, waist-shaped holes are added at both ends of the radial beams, creating parallel beam structures to improve sensitivity. Positioning holes on the connection blocks and peripheral support facilitate assembly.
The material selected for the elastomer is LY12 hard aluminum alloy, with an elastic modulus of $$ E = 7.1 \times 10^{10} \, \text{Pa} $$, Poisson’s ratio of $$ \sigma = 0.33 $$, and density of $$ \rho = 2770 \, \text{kg/m}^3 $$. The initial geometric dimensions are summarized in Table 1, which includes parameters such as the minimum thickness at hole locations. These dimensions serve as a baseline for subsequent optimization using response surface analysis.
| Component | Dimensions (Length × Width × Height) (mm) | Minimum Thickness at Holes (mm) | Outer Diameter × Height (mm) |
|---|---|---|---|
| Radial Beam | 16.5 × 5 × 5 | d1 = 0.7, d2 = 0.7 | – |
| Circumferential Beam | 19 × 4.5 × 5 | d3 = 0.7 | – |
| Central Platform | – | – | 38.5 × 5 |
The structural symmetry allows for simplified analysis, as forces and moments along the X and Y axes yield similar results due to the symmetric arrangement. Thus, the analysis focuses on four loading conditions: Fx, Fz, Mx, and Mz. The coordinate system is defined with the X and Y axes aligned with the radial beams and the Z axis passing through the center of the elastomer. This design ensures that the six-axis force sensor can accurately decode multi-axis loads while minimizing cross-talk.
Static Analysis Using Finite Element Method
Static analysis is conducted to evaluate the strain distribution under various loading conditions, which is critical for determining strain gauge placements and assessing sensitivity. Using ANSYS Workbench, we apply loads of Fx = 50 N, Fz = 50 N, Mx = 2.5 N·m, and Mz = 2.5 N·m to the central platform. The strain contours reveal that the holes effectively concentrate strain in specific regions, as shown in the following descriptions.
Under Fx loading, the radial beams along the X-axis experience tension and compression, with significant X-direction strain around the horizontal holes. Similarly, the radial beams along the Y-axis show Y-direction strain near the vertical holes, and the circumferential beams exhibit Y-direction strain. For Fz loading, the radial beams display opposite strains on the upper and lower surfaces of the horizontal holes, indicating bending effects. Under Mx loading, the circumferential beams show X-direction strain at the edges, while the radial beams exhibit Y-direction strain around the horizontal holes. For Mz loading, the circumferential beams generate uniform X-direction strain around the holes, and the radial beams show X-direction strain near the vertical holes.
The strain values are extracted for full-bridge circuit simulations, and the results are used to calculate sensitivity. The strain output for each loading condition is defined as follows: for Fx, the strain sum is denoted as $$ \varepsilon_1 $$; for Fz, as $$ \varepsilon_2 $$; for Mx, as $$ \varepsilon_3 $$; and for Mz, as $$ \varepsilon_4 $$. The initial sensitivity analysis indicates that the proposed six-axis force sensor structure enhances strain concentration, leading to improved measurement accuracy. The relationship between applied load and strain can be expressed using Hooke’s law for linear elasticity: $$ \sigma = E \varepsilon $$, where $$ \sigma $$ is stress and $$ \varepsilon $$ is strain. However, for complex structures, finite element analysis provides more accurate results.
| Loading Condition | Strain Output (ε) | Description |
|---|---|---|
| Fx = 50 N | 6.9522 × 10-4 | X-direction strain from radial beams |
| Fz = 50 N | 1.6615 × 10-3 | Z-direction strain from radial and circumferential beams |
| Mx = 2.5 N·m | 3.4312 × 10-3 | Moment-induced strain in circumferential beams |
| Mz = 2.5 N·m | 2.0211 × 10-3 | Moment-induced strain in radial beams |
These results demonstrate that the novel elastomer design effectively localizes strain, which is essential for high sensitivity in the six-axis force sensor. The next step involves optimizing the dimensions to further enhance performance.
Modal Analysis for Dynamic Performance
Dynamic performance is a key consideration for six-axis force sensors used in high-speed applications. Modal analysis is performed to determine the natural frequencies and mode shapes of the elastomer. The first-order natural frequency of the initial design is found to be 3191.7 Hz, which indicates good dynamic characteristics. However, optimization aims to increase this value while maintaining or improving sensitivity.
The modal analysis reveals that the first mode corresponds to bending along the Z-axis, while higher modes involve torsional and combined deformations. The natural frequency $$ f_n $$ can be related to the stiffness $$ k $$ and mass $$ m $$ of the system by the formula $$ f_n = \frac{1}{2\pi} \sqrt{\frac{k}{m}} $$. For the six-axis force sensor, increasing stiffness without significantly adding mass is crucial for achieving high natural frequencies. The proposed design with holes and reinforced beams contributes to this goal by optimizing the stiffness-to-mass ratio.
After optimization, the first-order natural frequency increases to 3196.3 Hz, showing a slight improvement. This enhancement ensures that the six-axis force sensor can operate effectively in dynamic environments without resonance issues. The modal analysis results are summarized in Table 3, which compares the initial and optimized designs.
| Design | First-Order Natural Frequency (Hz) | Mode Shape Description |
|---|---|---|
| Initial | 3191.7 | Bending along Z-axis |
| Optimized | 3196.3 | Bending along Z-axis with reduced deformation |
This analysis confirms that the structural modifications positively impact the dynamic performance of the six-axis force sensor, making it suitable for high-speed robotics.
Response Surface Analysis for Dimension Optimization
To achieve optimal performance, we employ response surface analysis (RSA) to design the critical dimensions of the elastomer. The Box-Behnken design method is used, with six key parameters selected as design variables: radial beam width (x1), radial beam height (x2), circumferential beam width (x3), minimum thickness at radial beam horizontal holes (x4), minimum thickness at radial beam vertical holes (x5), and minimum thickness at circumferential beam vertical holes (x6). The ranges for these variables are as follows: x1 ∈ [4, 6] mm, x2 ∈ [4, 6] mm, x3 ∈ [3.5, 5.5] mm, x4 ∈ [0.5, 1] mm, x5 ∈ [0.5, 1] mm, and x6 ∈ [0.4, 0.8] mm.
The response variables include the strain sums under different loads (ε1, ε2, ε3, ε4) and the first-order natural frequency (fz). A total of 54 experimental points are generated, and finite element simulations are conducted to obtain the response values. The data is fitted to regression models using least squares estimation. The general form of the response surface model is given by:
$$ \hat{y} = \beta_0 + \sum_{j=1}^{n} \beta_j x_j + \sum_{j=n+1}^{2n} \beta_j x_{j-n}^2 + \sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \beta_{ij} x_i x_j $$
where n is the number of design variables, β0 is the constant term, βj are the coefficients for linear and quadratic terms, and βij are the coefficients for interaction terms.
For example, the regression model for ε1 under Fx loading is derived as:
$$ \varepsilon_1 = 0.032928 – 4.22088 \times 10^{-3} x_1 – 1.89486 \times 10^{-5} x_2 – 7.36711 \times 10^{-3} x_3 – 5.94386 \times 10^{-3} x_4 – 2.94385 \times 10^{-3} x_5 – 0.028193 x_6 + 1.29270 \times 10^{-3} x_1 x_3 – 1.39612 \times 10^{-4} x_1 x_5 + 2.64440 \times 10^{-3} x_1 x_6 + 2.62490 \times 10^{-3} x_3 x_6 + 8.04155 \times 10^{-3} x_4 x_6 + 7.97487 \times 10^{-3} x_5 x_6 + 5.79976 \times 10^{-4} x_3^2 + 3.51882 \times 10^{-3} x_4^2 + 4.05406 \times 10^{-3} x_6^2 – 4.24546 \times 10^{-4} x_1 x_3 x_6 – 6.77351 \times 10^{-5} x_2 x_5 x_6 + 3.91074 \times 10^{-6} x_1^2 x_3 – 1.40875 \times 10^{-6} x_1^2 x_4 – 1.08418 \times 10^{-4} x_1 x_3^2 – 4.78648 \times 10^{-3} x_4^2 x_6 – 4.27913 \times 10^{-3} x_5 x_6^2 $$
The coefficients of determination (R²) for all models exceed 0.99, indicating high accuracy. For instance, the model for ε2 has R² = 0.9997, and the signal-to-noise ratios are all above 4, confirming the models’ reliability. These equations are used to predict the optimal dimensions that maximize sensitivity and natural frequency.
After optimization, the final dimensions are determined, as shown in Table 4. The comparison reveals significant improvements in sensitivity: a 26.09% increase for Fx, 11.59% for Fz, 24.94% for Mx, and 19.84% for Mz, while the natural frequency slightly increases to 3196.3 Hz.
| Parameter | Initial Model | Optimized Model | Improvement |
|---|---|---|---|
| Radial Beam Width (mm) | 5 | 4 | – |
| Radial Beam Height (mm) | 5 | 5.96 | – |
| Circumferential Beam Width (mm) | 4.5 | 4.24 | – |
| d1 (mm) | 0.7 | 0.57 | – |
| d2 (mm) | 0.7 | 0.77 | – |
| d3 (mm) | 0.7 | 0.63 | – |
| ε1 (Strain for Fx) | 6.9522 × 10-4 | 8.766 × 10-4 | 26.09% |
| ε2 (Strain for Fz) | 1.6615 × 10-3 | 1.854 × 10-3 | 11.59% |
| ε3 (Strain for Mx) | 3.4312 × 10-3 | 4.287 × 10-3 | 24.94% |
| ε4 (Strain for Mz) | 2.0211 × 10-3 | 2.422 × 10-3 | 19.84% |
| First-Order Natural Frequency (Hz) | 3191.7 | 3196.3 | 0.14% |
This optimization process demonstrates the effectiveness of response surface analysis in enhancing the performance of the six-axis force sensor.
Strain Gauge Placement and Bridge Circuit Configuration
Proper strain gauge placement is essential for accurate force measurement and decoupling in the six-axis force sensor. Based on the static analysis, we identify regions with maximum strain under different loads. Strain gauges are positioned on the circumferential and radial beams, as illustrated in the strain distribution maps. For example, gauges labeled R11 to R’14 are used for Fx detection, R21 to R’24 for Fy, R31 to R34 for Fz, R41 to R44 for Mx, R51 to R54 for My, and R61 to R64 for Mz.
Full-bridge circuits are employed to maximize output and minimize temperature effects. The bridge output voltage ΔV is related to the strain by the formula $$ \Delta V = V_{in} \cdot G \cdot \varepsilon $$, where Vin is the input voltage, G is the gauge factor, and ε is the strain. The circuit connections ensure that each bridge responds primarily to one load component, achieving theoretical decoupling. For instance, under Fx loading, only the corresponding bridge produces output, while others remain neutral.
Table 5 summarizes the strain directions at gauge positions under various loads, where “+” indicates tension, “−” compression, and “0” no significant strain. This arrangement minimizes cross-talk, ensuring that the six-axis force sensor provides independent measurements for each force and moment component.
| Gauge | Fx | Fy | Fz | Mx | My | Mz |
|---|---|---|---|---|---|---|
| R11 | + | + | 0 | 0 | 0 | – |
| R’11 | + | – | 0 | 0 | 0 | + |
| R12 | – | – | 0 | 0 | 0 | + |
| R’12 | – | + | 0 | 0 | 0 | – |
| R13 | – | + | 0 | 0 | 0 | + |
| R’13 | – | – | 0 | 0 | 0 | – |
| R14 | + | + | 0 | 0 | 0 | + |
| R’14 | + | – | 0 | 0 | 0 | – |
| R21 | – | – | 0 | 0 | 0 | + |
| R’21 | + | – | 0 | 0 | 0 | – |
| R22 | + | + | 0 | 0 | 0 | – |
| R’22 | – | + | 0 | 0 | 0 | + |
| R23 | – | + | 0 | 0 | 0 | – |
| R’23 | + | + | 0 | 0 | 0 | + |
| R24 | + | – | 0 | 0 | 0 | + |
| R’24 | – | – | 0 | 0 | 0 | – |
| R31 | 0 | – | – | – | 0 | 0 |
| R32 | 0 | – | + | + | 0 | 0 |
| R33 | 0 | + | – | + | 0 | 0 |
| R34 | 0 | + | + | – | 0 | 0 |
| R41 | 0 | – | + | + | 0 | 0 |
| R42 | 0 | – | – | – | 0 | 0 |
| R43 | 0 | + | + | – | 0 | 0 |
| R44 | 0 | + | – | + | 0 | 0 |
| R51 | + | 0 | + | 0 | + | 0 |
| R52 | + | 0 | – | 0 | – | 0 |
| R53 | – | 0 | + | 0 | – | 0 |
| R54 | – | 0 | – | 0 | + | 0 |
| R61 | + | + | 0 | 0 | 0 | – |
| R62 | + | – | 0 | 0 | 0 | + |
| R63 | – | + | 0 | 0 | 0 | + |
| R64 | – | – | 0 | 0 | 0 | – |
This configuration ensures that the six-axis force sensor achieves high decoupling accuracy, which is verified through experimental tests.
Experimental Validation
To validate the design, we conduct force loading experiments on a prototype of the six-axis force sensor. The setup involves applying controlled loads using weights and levers to simulate Fx, Fz, Mx, and Mz conditions. For instance, Fx is generated by applying two horizontal forces F1 and F2 with lever arms l1 and l2, where l1 = 5l2 and F2 = 5F1, resulting in a net force Fx = 4F1 at the center. Similarly, Mx is produced by equal and opposite forces with lever arms, and Mz by symmetric forces.
The output voltages from the full-bridge circuits are recorded for each load, and the relationship between load and voltage is plotted. The results show linear trends, with the maximum nonlinear error of 2.36% for Mz, which is within acceptable limits. The sensitivity coefficients are calculated from the slopes of these curves, confirming the improvements predicted by the optimization. For example, the sensitivity for Fx increases by 26.09%, as indicated by the higher strain output.
The experimental data corroborates the finite element and response surface analysis, demonstrating that the six-axis force sensor meets the design objectives. The high natural frequency and improved sensitivity make it suitable for dynamic applications in robotics and automation.
Conclusion
In this work, we have presented a novel design for a six-axis force sensor that enhances both static and dynamic performance. The elastomer structure, with its unique beam and hole configurations, effectively concentrates strain and reduces cross-talk. Through finite element analysis, we identified optimal strain gauge placements and conducted modal analysis to ensure high dynamic response. Response surface analysis enabled the optimization of key dimensions, resulting in significant sensitivity improvements—26.09% for Fx, 11.59% for Fz, 24.94% for Mx, and 19.84% for Mz—while maintaining a high natural frequency of 3196.3 Hz.
The experimental validation confirms the feasibility of the design, with linear output characteristics and low nonlinear errors. This six-axis force sensor provides a robust solution for high-speed robotics, where accurate multi-axis force measurement is critical. Future work may focus on further miniaturization and integration with robotic systems, as well as exploring advanced materials to enhance performance. The methodologies developed here, including response surface analysis and finite element modeling, can be applied to other sensor designs, contributing to the advancement of six-axis force sensor technology.
Overall, this study underscores the importance of innovative structural design and optimization in developing high-performance six-axis force sensors. By addressing key challenges in sensitivity and dynamics, we pave the way for more capable and reliable robotic systems.