As a mechanical engineer focused on sensor technology, I have dedicated significant effort to designing and analyzing a six-axis force sensor with a three-beam elastomer structure. The six-axis force sensor is a critical component in robotic systems, enabling precise measurement of three-dimensional force and moment information. This capability is essential for applications such as robot teaching, contour tracking, precision assembly, and grinding or polishing operations. The elastomer, as the core element of the six-axis force sensor, directly influences its performance metrics including sensitivity, accuracy, and decoupling efficiency. The three-beam configuration offers advantages like compact size, structural simplicity, and cost-effectiveness, which are pivotal for the widespread adoption and industrialization of six-axis force sensors. Despite extensive research on cross-beam and parallel elastomer structures, the three-beam design has received limited attention, necessitating in-depth investigation to unlock its potential.
In my design, the three-beam elastomer consists of an outer flange, a central platform, floating beams, and strain beams arranged at 120-degree intervals. This symmetric configuration ensures uniform strain distribution under multi-axial loading. Strain gauges S1 to S12 are strategically placed on the elastomer: S1, S3, and S5 on the upper surfaces of the strain beams; S2, S4, and S6 on the lower surfaces; and S7 to S12 on the side surfaces. The precise positioning of these strain gauges is crucial for capturing strain variations induced by different load components. The dimensions and material properties of the elastomer are summarized in the following tables to provide a clear foundation for analysis.
| Parameter | Value (mm) |
|---|---|
| Strain Beam Width (W) | 6 |
| Strain Beam Height (H) | 5.5 |
| Strain Beam Length (L) | 15 |
| Floating Beam Width (w) | 1.1 |
| Floating Beam Length (l) | 20 |
| Central Platform Diameter (D) | 25 |
| Parameter | Value |
|---|---|
| Material Code | 2024-T6 |
| Density (kg/m³) | 2780 |
| Elastic Modulus (GPa) | 73.1 |
| Poisson’s Ratio | 0.33 |
| Yield Strength σ₀.₂ (MPa) | 340 |
The strain analysis under various load components is fundamental to understanding the behavior of the six-axis force sensor. When a positive Fx force is applied, the side surfaces of strain beams CD and EF experience bending strain, resulting in tension in strain gauges S9 and S12 and compression in S10 and S11. For a positive Fy force, all three strain beams AB, CD, and EF exhibit bending on their sides, with S7, S10, and S12 in tension and S8, S9, and S11 in compression. Under Fz loading, the upper and lower surfaces of the strain beams deform, causing S1, S3, and S5 to tension and S2, S4, and S6 to compression. Moment loads induce similar strain patterns: Mx causes bending in beams CD and EF上下 surfaces, leading to tension in S3 and S6 and compression in S4 and S5; My affects all beams, with S2, S3, and S5 in tension and S1, S4, and S6 in compression; and Mz results in side surface bending, tensioning S7, S9, and S11 while compressing S8, S10, and S12. This comprehensive strain response is summarized in the table below to facilitate bridge circuit design.
| Load Component | Tension Strain Gauges | Compression Strain Gauges |
|---|---|---|
| Fx | S9, S12 | S10, S11 |
| Fy | S7, S10, S12 | S8, S9, S11 |
| Fz | S1, S3, S5 | S2, S4, S6 |
| Mx | S3, S6 | S4, S5 |
| My | S2, S3, S5 | S1, S4, S6 |
| Mz | S7, S9, S11 | S8, S10, S12 |
Based on the strain characteristics, I designed a Wheatstone half-bridge circuit to convert mechanical strain into electrical signals. Each bridge consists of strain gauges on adjacent arms paired with fixed resistors R, supplied by a voltage U. The output voltage SG for each bridge is derived from the strain-induced resistance changes. For instance, in bridge SG1 under +Fz loading, strain gauge S1 experiences tension (resistance increase ΔR) and S2 compression (resistance decrease ΔR), resulting in an output voltage calculated as:
$$ SG1 = \frac{(S1 – S2)R}{S1 \cdot S2 + (S1 + S2)R + R^2} \cdot U \approx \frac{\Delta R}{2R} \cdot U = \frac{1}{2} K U \epsilon_{Fz1} $$
where K is the gauge factor and ε_{Fz1} is the average strain. Similar calculations apply to other bridges, and the overall bridge output matrix SG is defined as:
$$ \mathbf{SG} = \begin{bmatrix} SG1 \\ SG2 \\ SG3 \\ SG4 \\ SG5 \\ SG6 \end{bmatrix} $$
The nominal output voltage matrix U for the six-axis force sensor is then constructed from linear combinations of SG components, representing each load dimension:
$$ U_{Fx} = SG5 – SG6 = \frac{K U}{2} (\epsilon_{Fx1} – \epsilon_{Fx2}) = \frac{K U}{2} \epsilon_{Fx} $$
$$ U_{Fy} = SG4 – SG5 – SG6 = \frac{K U}{2} (\epsilon_{Fy1} – \epsilon_{Fy2} – \epsilon_{Fy3}) = \frac{K U}{2} \epsilon_{Fy} $$
$$ U_{Fz} = SG1 + SG2 + SG3 = \frac{K U}{2} (\epsilon_{Fz1} + \epsilon_{Fz2} + \epsilon_{Fz3}) = \frac{K U}{2} \epsilon_{Fz} $$
$$ U_{Mx} = SG2 – SG3 = \frac{K U}{2} (\epsilon_{Mx1} – \epsilon_{Mx2}) = \frac{K U}{2} \epsilon_{Mx} $$
$$ U_{My} = SG2 + SG3 – SG1 = \frac{K U}{2} (\epsilon_{My1} + \epsilon_{My2} – \epsilon_{My3}) = \frac{K U}{2} \epsilon_{My} $$
$$ U_{Mz} = SG4 + SG5 + SG6 = \frac{K U}{2} (\epsilon_{Mz1} + \epsilon_{Mz2} + \epsilon_{Mz3}) = \frac{K U}{2} \epsilon_{Mz} $$
These equations are consolidated into a matrix form:
$$ \mathbf{U} = \begin{bmatrix} U_{Fx} \\ U_{Fy} \\ U_{Fz} \\ U_{Mx} \\ U_{My} \\ U_{Mz} \end{bmatrix} = \mathbf{K} \cdot \mathbf{SG} $$
where K is a coefficient matrix derived from the bridge combinations. The relationship between the force matrix F and the output U is expressed through a decoupling matrix B:
$$ \mathbf{F} = \mathbf{B} \cdot \mathbf{U} = \mathbf{B} \cdot (\mathbf{K} \cdot \mathbf{SG}) = (\mathbf{B} \cdot \mathbf{K}) \cdot \mathbf{SG} = \mathbf{B’} \cdot \mathbf{SG} $$
Here, B′ is the transformed decoupling matrix that simplifies calibration. Using the least squares method, B′ is computed from calibration data:
$$ \mathbf{B’} = \mathbf{F} \cdot \mathbf{SG}^T \cdot (\mathbf{SG} \cdot \mathbf{SG}^T)^{-1} $$
This approach reduces computational complexity and enhances the practicality of the six-axis force sensor.
To validate the design and decoupling method, I conducted finite element analysis (FEA) using ANSYS software. The elastomer model was constrained at the outer flange bottom, and loads were applied to the central platform across 10 load steps. The first six steps involved single-axis calibration loads, while steps 7-10 comprised composite loads to test decoupling accuracy. The applied loads and corresponding bridge outputs are detailed in the following tables.
| Load Step | Fx (N) | Fy (N) | Fz (N) | Mx (N·m) | My (N·m) | Mz (N·m) |
|---|---|---|---|---|---|---|
| 1 | 300 | 0 | 0 | 0 | 0 | 0 |
| 2 | 0 | 300 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 300 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 8 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 8 | 0 |
| 6 | 0 | 0 | 0 | 0 | 0 | 8.00 |
| 7 | 100 | 200 | 300 | 2.00 | 5.00 | 8.00 |
| 8 | 250 | 150 | 30 | 1.00 | 1.50 | 6.00 |
| 9 | 35 | 68 | 280 | 0.50 | 0.90 | 6.48 |
| 10 | 88 | 38 | 180 | 7.46 | 4.68 | 1.69 |
| Load Step | SG1 | SG2 | SG3 | SG4 | SG5 | SG6 |
|---|---|---|---|---|---|---|
| 1 | -3.01×10⁻⁴ | 1.51×10⁻⁴ | 1.51×10⁻⁴ | -1.58×10⁻⁶ | 9.54×10⁻⁴ | -9.52×10⁻⁴ |
| 2 | 1.39×10⁻⁸ | -2.61×10⁻⁴ | 2.61×10⁻⁴ | 1.10×10⁻³ | -5.51×10⁻⁴ | -5.51×10⁻⁴ |
| 3 | 1.06×10⁻³ | 1.06×10⁻³ | 1.06×10⁻³ | 5.51×10⁻⁶ | 6.45×10⁻⁷ | 3.32×10⁻⁶ |
| 4 | -1.10×10⁻⁷ | 1.65×10⁻³ | -1.65×10⁻³ | 1.30×10⁻⁵ | -7.63×10⁻⁶ | -1.26×10⁻⁶ |
| 5 | -1.90×10⁻³ | 9.51×10⁻⁴ | 9.50×10⁻⁴ | -9.77×10⁻⁶ | -1.07×10⁻⁵ | 1.42×10⁻⁵ |
| 6 | -1.64×10⁻⁸ | -3.77×10⁻⁸ | 2.09×10⁻⁸ | 9.24×10⁻⁴ | 9.23×10⁻⁴ | 9.22×10⁻⁴ |
| 7 | -2.26×10⁻⁴ | 1.94×10⁻³ | 1.47×10⁻³ | 1.66×10⁻³ | 8.66×10⁻⁴ | 2.49×10⁻⁴ |
| 8 | -5.01×10⁻⁴ | 4.85×10⁻⁴ | 3.35×10⁻⁴ | 1.24×10⁻³ | 1.21×10⁻³ | -3.75×10⁻⁴ |
| 9 | 7.43×10⁻⁴ | 1.16×10⁻³ | 1.07×10⁻³ | 1.00×10⁻³ | 7.33×10⁻⁴ | 5.15×10⁻⁴ |
| 10 | -5.63×10⁻⁴ | 2.74×10⁻³ | -2.64×10⁻⁴ | 3.44×10⁻⁴ | 3.92×10⁻⁴ | -1.45×10⁻⁴ |
Using the single-axis load data (steps 1-6), I calculated the nominal output voltage matrix U, which exhibits strong correlation with the respective load components, as shown below.
| Load Step | U_Fx | U_Fy | U_Fz | U_Mx | U_My | U_Mz |
|---|---|---|---|---|---|---|
| 1 | 1.91×10⁻³ | -3.28×10⁻⁶ | 1.47×10⁻⁷ | 1.23×10⁻⁸ | 6.03×10⁻⁴ | 1.11×10⁻⁷ |
| 2 | 4.60×10⁻⁷ | 2.20×10⁻³ | 8.20×10⁻⁸ | -5.22×10⁻⁴ | 5.42×10⁻⁸ | 8.31×10⁻⁷ |
| 3 | -2.67×10⁻⁶ | 1.54×10⁻⁶ | 3.19×10⁻³ | 3.13×10⁻⁷ | 1.06×10⁻³ | 9.47×10⁻⁶ |
| 4 | -6.37×10⁻⁶ | 2.19×10⁻⁵ | 4.02×10⁻⁷ | 3.29×10⁻³ | 6.23×10⁻⁷ | 4.12×10⁻⁶ |
| 5 | -2.49×10⁻⁵ | -1.33×10⁻⁵ | -2.26×10⁻⁷ | 1.69×10⁻⁷ | 3.80×10⁻³ | -6.25×10⁻⁶ |
| 6 | 9.77×10⁻⁷ | -9.21×10⁻⁴ | -3.32×10⁻⁸ | -5.86×10⁻⁸ | -4.41×10⁻¹⁰ | 2.77×10⁻³ |
| 7 | 6.17×10⁻⁴ | 5.46×10⁻⁴ | 3.19×10⁻³ | 4.75×10⁻⁴ | 3.64×10⁻³ | 2.78×10⁻³ |
| 8 | 1.58×10⁻³ | 4.09×10⁻⁴ | 3.19×10⁻⁴ | 1.51×10⁻⁴ | 1.32×10⁻³ | 2.08×10⁻³ |
| 9 | 2.18×10⁻⁴ | -2.46×10⁻⁴ | 2.97×10⁻³ | 8.77×10⁻⁵ | 1.49×10⁻³ | 2.25×10⁻³ |
| 10 | 5.37×10⁻⁴ | 9.72×10⁻⁵ | 1.91×10⁻³ | 3.00×10⁻³ | 3.04×10⁻³ | 5.91×10⁻⁴ |
The decoupling matrix B′ was computed from load steps 1-6 using the least squares method:
$$ \mathbf{B’} = \begin{bmatrix}
-1239.31 & 1120.90 & 514.11 & -3.29 & 156938.38 & -157101.38 \\
-932.24 & -793.54 & 1125.53 & 181062.18 & -90583.09 & -90697.00 \\
94113.78 & 94113.36 & 94136.35 & -7.16 & -3.79 & 14.35 \\
-4.04 & 2425.97 & -2425.19 & 765.77 & -383.01 & -383.53 \\
-2800.19 & 1397.80 & 1400.99 & -0.13 & -663.59 & 664.40 \\
-14.92 & -9.02 & -1.83 & 2887.00 & 2889.21 & 2892.56
\end{bmatrix} $$
To validate B′, I applied it to the bridge outputs from load steps 7-10, obtaining the force matrix F′:
$$ \mathbf{F’}_{6 \times 4} = \mathbf{B’} \cdot \mathbf{SG’}_{6 \times 4} = \begin{bmatrix}
99.999 & 250 & 35 & 87.999 \\
199.997 & 149.999 & 67.999 & 37.999 \\
300 & 30 & 280 & 180 \\
2 & 1 & 0.5 & 7.46 \\
5 & 1.5 & 0.9 & 4.68 \\
8 & 6 & 6.48 & 1.69
\end{bmatrix} $$
Comparing F′ with the applied loads in Table 4 reveals negligible errors, confirming the accuracy of the decoupling matrix and the overall design of the six-axis force sensor.
In conclusion, the three-beam elastomer structure for the six-axis force sensor demonstrates excellent performance through systematic strain analysis, bridge circuit design, and decoupling methodology. The finite element simulations validate the feasibility of the approach, providing a robust foundation for future optimizations in sensitivity, range, and miniaturization. This work underscores the potential of three-beam designs in advancing six-axis force sensor technology for industrial robotics and automation.