In the field of robotics and precision engineering, the demand for compact and high-performance force sensing solutions has grown significantly. As robots evolve toward lightweight and miniaturized designs, there is a pressing need for six-axis force sensors that can be seamlessly integrated into confined spaces, such as robotic joints or end-effectors, without compromising accuracy. Traditional six-axis force sensor designs, including Stewart platforms and cross-beam structures, often face limitations in miniaturization due to their complex assembly, high part counts, or reliance on numerous strain gauges. For instance, cross-beam configurations typically require up to 24 strain gauges, which increases size, cost, and complexity. To address these challenges, we propose a novel composite beam structure that combines three horizontal and three vertical beams in a symmetric arrangement. This design reduces the outer diameter of the elastomer to 40 mm, with a fully assembled sensor diameter of 45 mm, making it suitable for embedded applications in modern robotic systems.
The core innovation lies in the elastomer structure, which features an upper outer flange, a central platform, a lower outer flange, and six strain beams—three horizontal and three vertical—arranged at 120-degree intervals. This configuration leverages both bending and shear deformations to enhance sensitivity while maintaining structural integrity. Key parameters include the outer flange diameter (D), central platform diameter (d), horizontal beam width (W1), height (H1), vertical beam length (L2), width (W2), and height (H2). By optimizing these dimensions, we aim to maximize strain output under various load conditions, thereby improving the sensor’s performance metrics such as sensitivity, linearity, and minimal cross-talk.
To accurately model the behavior of the elastomer under load, we employ the Timoshenko beam theory, which accounts for shear deformation and rotational inertia—critical factors for short beams where the span-to-height ratio is less than 5. Unlike the Euler-Bernoulli beam theory, which assumes infinite shear stiffness, the Timoshenko approach provides a more precise representation of strain distribution in miniaturized components. The fundamental equations governing the Timoshenko beam are as follows:
$$M(x) = EI \frac{d\phi(x)}{dx}$$
$$F(x) = KGA \left( \frac{d\omega(x)}{dx} – \phi(x) \right)$$
Here, \( M(x) \) represents the bending moment, \( F(x) \) is the shear force, \( E \) is Young’s modulus, \( G \) is the shear modulus, \( I \) is the moment of inertia, \( A \) is the cross-sectional area, and \( K \) is the shear coefficient (typically 1.2 for rectangular sections). The variables \( \omega(x) \) and \( \phi(x) \) denote the deflection and rotation at position \( x \), respectively. The strain at any point on the beam surface can be derived as:
$$\epsilon(x) = -\frac{h}{2} \frac{d\phi(x)}{dx}$$
where \( h \) is the beam height. This equation is essential for determining the optimal locations for strain gauge placement, as it relates surface strain to the beam’s geometric and material properties.
We conducted finite element analysis (FEA) using ANSYS Workbench to evaluate the strain distribution under individual load cases: \( F_x = 50 \, \text{N} \), \( F_y = 50 \, \text{N} \), \( F_z = 50 \, \text{N} \), \( M_x = 2 \, \text{N·m} \), \( M_y = 2 \, \text{N·m} \), and \( M_z = 2 \, \text{N·m} \). The elastomer material was 2024-T6 aluminum alloy, with properties summarized in Table 1. The FEA results revealed that horizontal beams experience significant bending under \( F_z \), \( M_x \), and \( M_y \) loads, while vertical beams deform primarily under \( F_x \), \( F_y \), and \( M_z \). For example, under \( F_x \) loading, the maximum strain occurs near the lower end of the vertical beams, whereas under \( F_z \), the highest strain is at the junction of horizontal beams and the central platform. This strain localization guides the placement of strain gauges to capture the most sensitive responses.
| Parameter | Value |
|---|---|
| Density (kg/m³) | 2780 |
| Young’s Modulus (GPa) | 73.1 |
| Poisson’s Ratio | 0.33 |
| Tensile Yield Strength (MPa) | 340 |
| Fatigue Limit (MPa) | 140 |
Based on the FEA results, we designed a Wheatstone bridge circuit configuration using 12 strain gauges—a reduction from the 24 typically required in cross-beam sensors—which contributes to the miniaturization of the six-axis force sensor. The gauges are positioned on the upper and lower surfaces of horizontal beams and the left and right sides of vertical beams, as shown in the schematic. This arrangement forms six half-bridge circuits, each corresponding to a specific load component. The output voltage for each half-bridge can be expressed as:
$$U_i = \frac{\Delta R}{2R} U_0 = \frac{1}{2} K U_0 \epsilon_i$$
where \( U_0 \) is the excitation voltage, \( \Delta R \) is the change in gauge resistance, \( R \) is the nominal resistance, \( K \) is the gauge factor, and \( \epsilon_i \) is the average strain in the direction of interest. For instance, under \( F_x \) loading, gauges on vertical beams produce positive and negative strain responses, leading to differential voltages that isolate the \( F_x \) component. Table 2 summarizes the output signals for each half-bridge under single-axis loads, demonstrating how the circuit design minimizes cross-talk by leveraging symmetric strain patterns.
| Load | U1 | U2 | U3 | U4 | U5 | U6 |
|---|---|---|---|---|---|---|
| Fx | — | — | — | — | \( \frac{1}{2} K U_0 \epsilon_{Fx1} \) | \( -\frac{1}{2} K U_0 \epsilon_{Fx2} \) |
| Fy | — | — | — | \( -\frac{1}{2} K U_0 \epsilon_{Fy1} \) | \( \frac{1}{2} K U_0 \epsilon_{Fy2} \) | \( \frac{1}{2} K U_0 \epsilon_{Fy3} \) |
| Fz | \( \frac{1}{2} K U_0 \epsilon_{Fz1} \) | \( \frac{1}{2} K U_0 \epsilon_{Fz2} \) | \( \frac{1}{2} K U_0 \epsilon_{Fz3} \) | — | — | — |
| Mx | — | \( \frac{1}{2} K U_0 \epsilon_{Mx1} \) | \( -\frac{1}{2} K U_0 \epsilon_{Mx2} \) | — | — | — |
| My | \( \frac{1}{2} K U_0 \epsilon_{My1} \) | \( -\frac{1}{2} K U_0 \epsilon_{My2} \) | \( -\frac{1}{2} K U_0 \epsilon_{My3} \) | — | — | — |
| Mz | — | — | — | \( \frac{1}{2} K U_0 \epsilon_{Mz1} \) | \( \frac{1}{2} K U_0 \epsilon_{Mz2} \) | \( \frac{1}{2} K U_0 \epsilon_{Mz3} \) |
To enhance the performance of the six-axis force sensor, we employed the Box-Behnken Design (BBD) response surface methodology (RSM) to optimize the elastomer geometry. The goal was to maximize the average strain at strain gauge locations while constraining the maximum stress to avoid yielding or fatigue failure. The design variables included the central platform radius (\( x_1 \)), horizontal beam height (\( x_2 \)), vertical beam height (\( x_3 \)), and vertical beam length (\( x_4 \)), with bounds set as follows: \( 7 \, \text{mm} \leq x_1 \leq 9 \, \text{mm} \), \( 2 \, \text{mm} \leq x_2 \leq 4 \, \text{mm} \), \( 2 \, \text{mm} \leq x_3 \leq 4 \, \text{mm} \), and \( 7 \, \text{mm} \leq x_4 \leq 9 \, \text{mm} \). The optimization objectives were defined as functions \( f_1 \) to \( f_6 \), representing the average strain under \( F_x \), \( F_y \), \( F_z \), \( M_x \), \( M_y \), and \( M_z \) loads, respectively:
$$f_1 = 2(\epsilon_{Fx9} + \epsilon_{Fx10})$$
$$f_2 = \epsilon_{Fy7} + \epsilon_{Fy8} + \epsilon_{Fy9} + \epsilon_{Fy10} + \epsilon_{Fy11} + \epsilon_{Fy12}$$
$$f_3 = 3(\epsilon_{Fz1} + \epsilon_{Fz2})$$
$$f_4 = 2(\epsilon_{Mx3} + \epsilon_{Mx4})$$
$$f_5 = \epsilon_{My1} + \epsilon_{My2} + \epsilon_{My3} + \epsilon_{My4} + \epsilon_{My5} + \epsilon_{My6}$$
$$f_6 = 3(\epsilon_{Mz7} + \epsilon_{Mz8})$$
Stress constraints were imposed to ensure durability: \( f_7 = \max \sigma_{Mx} \leq 140 \, \text{MPa} \), \( f_8 = \max \sigma_{My} \leq 140 \, \text{MPa} \), and \( f_9 = \max \sigma_{\text{Full}} \leq 340 \, \text{MPa} \). Using BBD, we generated 25 experimental points and fitted second-order polynomial models via Kriging interpolation to improve accuracy. The response surface models for \( f_1 \) to \( f_9 \) are given below, with coefficients derived from regression analysis:
$$f_1 = (7.2 – 1.25x_1 – 13.19x_2 – 2.83x_3 + 11.63x_4 + 0.077x_1^2 + 1.673x_2^2 + 2.165x_3^2 + 0.066x_4^2 – 0.027x_1x_2 – 0.068x_1x_3 + 0.085x_1x_4 + 0.380x_2x_3 – 0.350x_2x_4 – 2.252x_3x_4) \times 10^{-5}$$
$$f_2 = (-10.4 + 0.41x_1 – 13.76x_2 – 2.85x_3 + 15.56x_4 – 0.007x_1^2 + 1.842x_2^2 + 2.335x_3^2 – 0.068x_4^2 – 0.112x_1x_2 – 0.035x_1x_3 + 0.075x_1x_4 + 0.302x_2x_3 – 0.390x_2x_4 – 2.505x_3x_4) \times 10^{-5}$$
$$f_3 = (529 – 45.8x_1 – 149.1x_2 – 9.8x_3 + 2.2x_4 + 0.02x_1^2 + 7.85x_2^2 + 0.79x_3^2 – 0.08x_4^2 + 10.29x_1x_2 – 0.04x_1x_3 + 0.02x_1x_4 + 1.02x_2x_3 – 0.13x_3x_4) \times 10^{-5}$$
$$f_4 = (2743 – 235x_1 – 770.7x_2 – 41.4x_3 + 10x_4 + 2.57x_1^2 + 47.94x_2^2 + 3.18x_3^2 – 0.33x_4^2 + 44.4x_1x_2 – 0.43x_1x_3 + 0.16x_1x_4 + 6.06x_2x_3 – 0.68x_2x_4 – 0.68x_3x_4) \times 10^{-5}$$
$$f_5 = (3142 – 264x_1 – 892.9x_2 – 47.2x_3 + 13x_4 + 2.35x_1^2 + 55.31x_2^2 + 3.66x_3^2 – 0.42x_4^2 + 51.62x_1x_2 – 0.47x_1x_3 + 0.23x_1x_4 + 7.09x_2x_3 – 0.77x_2x_4 – 0.86x_3x_4) \times 10^{-5}$$
$$f_6 = (-154 + 9.1x_1 – 57x_2 – 61.4x_3 + 106.7x_4 – 0.72x_1^2 + 6.35x_2^2 + 27.87x_3^2 – 0.97x_4^2 + 0.54x_1x_2 – 0.26x_1x_3 + 0.04x_1x_4 – 0.23x_2x_3 – 0.53x_2x_4 – 20.42x_3x_4) \times 10^{-5}$$
$$f_7 = 1277 – 97.2x_1 – 297.3x_2 – 22.8x_3 – 35.5x_4 + 3.03x_1^2 + 29.66x_2^2 + 1.97x_3^2 + 1.5x_4^2 + 6.41x_1x_2 + 1.53x_1x_3 + 1.28x_1x_4 + 0.53x_2x_3 + 0.37x_2x_4 – 0.32x_3x_4$$
$$f_8 = 472 – 25x_1 – 248.8x_2 + 11.5x_3 + 66.7x_4 – 1.57x_1^2 – 18.16x_2^2 – 1.89x_3^2 – 4.57x_4^2 + 8.44x_1x_2 – 0.64x_1x_3 + 1.08x_1x_4 + 3.6x_2x_3 + 0.37x_2x_4 – 1.08x_3x_4$$
$$f_9 = 1878 – 155.3x_1 – 467.8x_2 – 9.1x_3 – 17.8x_4 + 3.31x_1^2 + 37.23x_2^2 + 1.42x_3^2 + 2.08x_4^2 + 23.48x_1x_2 – 0.39x_1x_3 + 0.03x_1x_4 + 0.59x_2x_3 – 4.55x_2x_4 – 0.29x_3x_4$$
Analysis of variance (ANOVA) confirmed the statistical significance of these models, with p-values below 0.01 and R-squared values exceeding 99%, indicating high reliability. We then applied a multi-objective genetic algorithm (MOGA) to solve the optimization problem, setting an initial population of 4,000, 800 samples per iteration, and a maximum of 20 iterations. After convergence, the optimal parameters were found to be \( x_1 = 7.011 \, \text{mm} \), \( x_2 = 2.876 \, \text{mm} \), \( x_3 = 2.007 \, \text{mm} \), and \( x_4 = 8.996 \, \text{mm} \). For practicality, these were rounded to \( x_1 = 7 \, \text{mm} \), \( x_2 = 2.9 \, \text{mm} \), \( x_3 = 2 \, \text{mm} \), and \( x_4 = 9 \, \text{mm} \). Table 3 compares the performance before and after optimization, showing substantial improvements in strain sensitivity across all load directions.
| Parameter | Optimized (C-1) | Initial (C-2) | Improvement Factor |
|---|---|---|---|
| f1 (εFx) | 4.75 × 10⁻⁴ | 3.00 × 10⁻⁴ | 1.5833 |
| f2 (εFy) | 5.44 × 10⁻⁴ | 3.48 × 10⁻⁴ | 1.5632 |
| f3 (εFz) | 5.34 × 10⁻⁴ | 3.08 × 10⁻⁴ | 1.7338 |
| f4 (εMx) | 3.012 × 10⁻³ | 1.993 × 10⁻³ | 1.5113 |
| f5 (εMy) | 3.478 × 10⁻³ | 2.302 × 10⁻³ | 1.5109 |
| f6 (εMz) | 2.653 × 10⁻³ | 1.222 × 10⁻³ | 2.171 |
| f7 (σMx, MPa) | 136.96 | 106.82 | — |
| f8 (σMy, MPa) | 139.93 | 118.97 | — |
| f9 (σFull, MPa) | 255.07 | 200.98 | — |
A prototype of the six-axis force sensor was fabricated based on the optimized design, and static calibration experiments were conducted to validate its performance. The calibration setup involved applying known loads using weights and fixtures, with care taken to account for additional forces and moments due to fixture geometry. For example, fixture A was used for \( F_x \), \( F_y \), and \( M_z \) calibration, while fixture B was employed for \( F_z \), \( M_x \), and \( M_y \). Loads were applied in increments from 0 to 50 N for forces and 0 to 2 N·m for moments, with voltage outputs recorded from the six half-bridges. The relationship between the applied load matrix \( \mathbf{F} \) and the output voltage matrix \( \Delta \mathbf{U} \) is linear, as evidenced by the calibration curves, allowing us to use least squares method for decoupling.
The decoupling matrix \( \mathbf{G} \) was computed as \( \mathbf{G} = \mathbf{F} \Delta \mathbf{U}^T (\Delta \mathbf{U} \Delta \mathbf{U}^T)^{-1} \), resulting in:
$$\mathbf{G} = \begin{bmatrix}
81.703 & 1.239 & 11.031 & -0.172 & 13.672 & -9.711 \\
-8.508 & 76.562 & -6.527 & 1.637 & -15.354 & 29.801 \\
17.717 & 6.504 & 88.341 & -20.573 & -39.659 & -10.168 \\
0.374 & 0.251 & -0.056 & 2.348 & 0.015 & 0.868 \\
-0.239 & -0.186 & 1.073 & -0.103 & 2.731 & -0.602 \\
-0.929 & 0.126 & -1.063 & 0.111 & 0.506 & 3.664
\end{bmatrix}$$
To evaluate the sensor’s accuracy, we defined Type I and Type II errors. Type I error represents the deviation in the measured value of a load component when only that component is applied, while Type II error indicates cross-talk when other components are loaded. The errors are calculated as:
$$\text{Type I Error} = \frac{\text{Measured Value at Full Scale}}{\text{Full Scale Value}} \times 100\%$$
$$\text{Type II Error} = \frac{\text{Measured Value of Other Component}}{\text{Full Scale Value of That Component}} \times 100\%$$
Using the decoupling matrix, we processed the full-scale load data and obtained the error matrix \( \mathbf{E} \). The diagonal elements of \( \mathbf{E} \) correspond to Type I errors, while off-diagonal elements represent Type II errors. The results are summarized in Table 4, demonstrating that the six-axis force sensor achieves high precision with minimal cross-coupling.
| Load | Fx | Fy | Fz | Mx | My | Mz |
|---|---|---|---|---|---|---|
| Fx | 0.18% | 0.02% | 0.02% | 0.42% | 0.37% | 1.44% |
| Fy | 0.79% | 0.42% | 0.44% | 0.02% | 0.71% | 0.08% |
| Fz | 0.80% | 0.32% | 0.41% | 0.04% | 1.62% | 0.34% |
| Mx | 0.01% | 0.29% | 0.43% | 0.05% | 0.32% | 0.35% |
| My | 0.25% | 0.22% | 0.25% | 1.46% | 0.49% | 0.14% |
| Mz | 1.67% | 0.56% | 0.54% | 0.35% | 0.47% | 0.60% |
The Type I errors for \( F_x \), \( F_y \), \( F_z \), \( M_x \), \( M_y \), and \( M_z \) are 0.18%, 0.42%, 0.41%, 0.05%, 0.49%, and 0.60%, respectively, all within 1%. The maximum Type II error is 1.665%, which occurs in the \( M_z \) direction when \( F_x \) is applied. These results confirm that the composite beam structure effectively minimizes cross-talk and enhances sensitivity, making the six-axis force sensor suitable for high-precision applications in robotics.
In conclusion, the development of this small-scale six-axis force sensor with a composite beam structure addresses the challenges of miniaturization and performance optimization. Through finite element analysis, response surface methodology, and multi-objective genetic algorithm optimization, we achieved significant improvements in strain sensitivity—ranging from 1.51 to 2.17 times the initial values—while maintaining stress within safe limits. The prototype validation via static calibration and least squares decoupling demonstrated low Type I and Type II errors, underscoring the sensor’s accuracy and minimal cross-coupling. This design not only reduces the sensor’s footprint but also leverages advanced modeling techniques to ensure reliability in demanding environments. Future work may focus on dynamic calibration, temperature compensation, and integration with robotic control systems to further enhance the applicability of the six-axis force sensor in real-world scenarios.