In the fields of automation, robotics, and aerospace, the six-axis force sensor stands as one of the most critical components for measuring three-dimensional force and torque information simultaneously. These sensors enable systems to perceive their environment through six distinct channels, facilitating high-precision control in applications such as welding, grinding, assembly, and trajectory tracking. Typically integrated at the end-effector of robotic systems, the six-axis force sensor enhances interaction quality by providing real-time feedback on external forces and moments. However, a significant challenge in their design is inter-axis coupling, where forces or torques in one direction inadvertently affect measurements in others, leading to reduced accuracy. To address this, I propose an optimized elastic body structure based on the traditional cross-beam configuration, coupled with an advanced circuit decoupling method. This design aims to achieve superior decoupling performance, higher precision, and improved structural simplicity, making it easier to manufacture while minimizing coupling errors.
The structural design of the elastic body is paramount to the performance of a six-axis force sensor. Traditional designs often employ a uniform cross-beam section, which can exacerbate coupling effects. In my approach, I have refined this by developing a variable cross-section elastic body, where the cross-sectional area gradually decreases from the central measurement platform toward the floating beams. This optimization enhances strain concentration in specific regions, improving sensitivity and reducing interference between axes. The elastic body consists of four beams arranged in a cross pattern, with fixed ends at the corners and a central platform for connecting to the measured object. Floating beams link the elastic beams to the rim, and four uniformly distributed main beams connect the central platform to the floating beams. This configuration allows for efficient load distribution and strain generation under multi-axis loading conditions.
To validate the structural improvements, I conducted finite element analysis (FEA) using ANSYS software, comparing the traditional and optimized six-axis force sensor designs. The elastic body was modeled with key dimensions as summarized in Table 1, and materials were selected for their mechanical properties—specifically, 7075 high-strength aluminum alloy with a Young’s modulus of 10^5 MPa, density of 7850 kg/m³, and Poisson’s ratio of 0.25. The model was meshed with SOLID186 elements for high accuracy, and boundary conditions were applied by fixing the screw holes and applying loads at the central platform. Static analyses under individual force and torque components revealed the strain distributions, with the optimized structure exhibiting more localized and higher strain magnitudes in targeted areas, which is crucial for precise sensing.
| Parameter | Dimension (mm) | Parameter | Dimension (mm) |
|---|---|---|---|
| B1 | 1.5 | T2 | 2 |
| B2 | 2 | R1 | 4 |
| H | 49 | R2 | 12.5 |
| L1 | 25 | R3 | 19 |
| T1 | 16 | R4 | 22.5 |
Under uniaxial loads, such as Fx = 150 N or Mz = 10 N·m, the strain profiles were mapped along paths on the beam surfaces to identify optimal locations for strain gauge placement. For instance, in the Fx case, the maximum strain occurred at approximately 1.1669 mm from the central platform, while for Fz = 150 N, it was at 2.0963 mm. Similarly, for torque components like Mx = 10 N·m and Mz = 10 N·m, peak strains were found at 2.5622 mm and 1.8892 mm, respectively. These path mapping results, illustrated through FEA deformation plots, ensure that strain gauges are positioned at points of highest sensitivity, thereby maximizing output signals and minimizing cross-talk. The strain relationships can be expressed using basic elasticity equations, such as the strain-displacement relation: $$ \epsilon = \frac{\sigma}{E} $$ where $\epsilon$ is strain, $\sigma$ is stress, and E is Young’s modulus. For bending beams, the strain can be derived from: $$ \epsilon = \frac{M \cdot y}{I \cdot E} $$ with M as the bending moment, y as the distance from the neutral axis, and I as the moment of inertia.
Building on the structural analysis, I designed a full-bridge circuit for decoupling the six-axis force sensor outputs. A total of 24 strain gauges are strategically placed on the elastic beams—12 on the left and right beams and 12 on the upper and lower beams—to form six independent Wheatstone bridges. Each bridge corresponds to a specific force or torque component: Fx, Fy, Fz, Mx, My, and Mz. The output voltage for each bridge is given by the equation: $$ \Delta U = \frac{U \cdot K}{4} \cdot (\epsilon_1 – \epsilon_2 – \epsilon_3 + \epsilon_4) $$ where U is the excitation voltage, K is the gauge factor, and $\epsilon_i$ represents the strain from the i-th gauge. For the six-axis force sensor, the overall output vector can be represented as a matrix equation: $$ \begin{bmatrix} \Delta U_{Fx} \\ \Delta U_{Fy} \\ \Delta U_{Fz} \\ \Delta U_{Mx} \\ \Delta U_{My} \\ \Delta U_{Mz} \end{bmatrix} = \frac{U \cdot K}{4} \cdot \begin{bmatrix} \epsilon_1 – \epsilon_2 – \epsilon_3 + \epsilon_4 \\ \epsilon_5 – \epsilon_6 – \epsilon_7 + \epsilon_8 \\ \epsilon_9 – \epsilon_{10} – \epsilon_{11} + \epsilon_{12} \\ \epsilon_{13} – \epsilon_{14} – \epsilon_{15} + \epsilon_{16} \\ \epsilon_{17} – \epsilon_{18} – \epsilon_{19} + \epsilon_{20} \\ \epsilon_{21} – \epsilon_{22} – \epsilon_{23} + \epsilon_{24} \end{bmatrix} $$ This configuration amplifies the strain signal by a factor of four while providing inherent temperature compensation and noise rejection through differential measurements. By placing opposing strain gauges on regions with inverse deformations, the circuit effectively decouples the outputs, reducing inter-axis interference.
Experimental validation involved comparing the outputs of the traditional and optimized six-axis force sensor designs under controlled loads. Using the path mapping technique, I determined the precise locations for strain gauge attachment to maximize bridge outputs. For example, under Fx = 150 N, bridge 1 achieved a peak output at 1.1669 mm, whereas for Fz = 150 N, bridge 3 peaked at 2.0963 mm. The outputs for other components, such as Mx and Mz, showed similar optimizations at 2.5622 mm and 1.8892 mm, respectively. To quantify decoupling performance, I analyzed the inter-axis coupling by applying uniaxial forces and torques and measuring the outputs across all bridges. The results, summarized in Table 2 for the optimized six-axis force sensor and Table 3 for the traditional design, demonstrate a significant reduction in coupling errors. For instance, in the optimized six-axis force sensor, Fx primarily affects bridge 1 with minimal cross-talk to other bridges, whereas the traditional design exhibits higher coupling, such as Fz influencing bridge 4. The coupling degree, calculated as the ratio of cross-axis output to primary output, decreased from 1.36% in the traditional six-axis force sensor to 0.23% in the optimized version, highlighting the effectiveness of the variable cross-section design.
| Bridge | Fx (100 N) | Fy (100 N) | Fz (100 N) | Mx (10 N·m) | My (10 N·m) | Mz (10 N·m) |
|---|---|---|---|---|---|---|
| Bridge 1 | -1186.92 | 0.00 | 0.54 | 0.35 | 0.00 | -0.32 |
| Bridge 2 | 0.00 | -1186.92 | 0.00 | 0.00 | 0.00 | 0.00 |
| Bridge 3 | 0.19 | 0.00 | 986.46 | -2.30 | 0.00 | -0.38 |
| Bridge 4 | -0.06 | 0.00 | -0.42 | 5811.3 | 0.00 | 0.22 |
| Bridge 5 | 0.00 | 0.00 | 0.00 | 0.00 | 5811.3 | 0.00 |
| Bridge 6 | -0.12 | 0.00 | -0.09 | 2.18 | 0.00 | 2856.08 |
| Bridge | Fx (100 N) | Fy (100 N) | Fz (100 N) | Mx (10 N·m) | My (10 N·m) | Mz (10 N·m) |
|---|---|---|---|---|---|---|
| Bridge 1 | -831.67 | 0.00 | 0.25 | -1.39 | 0.00 | -0.27 |
| Bridge 2 | 0.00 | -831.67 | 0.00 | 0.00 | 0.00 | 0.00 |
| Bridge 3 | 0.42 | 0.00 | 807.74 | 11.00 | 0.00 | 1.67 |
| Bridge 4 | -0.71 | 0.00 | 2.08 | 4675.4 | 0.00 | -1.07 |
| Bridge 5 | 0.00 | 0.00 | 0.00 | 0.00 | 4675.4 | 0.00 |
| Bridge 6 | 0.17 | 0.00 | 2.09 | 1.14 | 0.00 | 2250.13 |
Furthermore, the optimized six-axis force sensor showed enhanced strain outputs across all axes. For example, the microstrain for Fx increased by 46% to approximately 300 units, while other components like Fz and Mz also saw improvements. This boost in sensitivity, combined with the reduced coupling, underscores the advantages of the variable cross-section elastic body. The decoupling circuit further amplifies these benefits by ensuring that each bridge responds predominantly to its intended load, as evidenced by the minimal outputs in non-corresponding bridges. The overall performance can be modeled using a decoupling matrix, where the output vector $\mathbf{U}$ relates to the input force-torque vector $\mathbf{F}$ through a sensitivity matrix $\mathbf{S}$: $$ \mathbf{U} = \mathbf{S} \cdot \mathbf{F} $$ For an ideal six-axis force sensor, $\mathbf{S}$ is diagonal, but in practice, off-diagonal elements represent coupling. My optimized design minimizes these off-diagonal terms, as shown by the low values in Table 2.
In conclusion, the optimized six-axis force sensor design presented in this work demonstrates significant improvements in decoupling capability, accuracy, and sensitivity compared to traditional approaches. Through finite element analysis, I identified optimal strain gauge placements that maximize output signals while minimizing inter-axis interference. The incorporation of a full-bridge circuit design further enhances decoupling by leveraging differential measurements and strategic gauge positioning. Experimental results confirm that the variable cross-section elastic body reduces coupling degree to 0.23%, a substantial improvement over the 1.36% observed in conventional designs. This six-axis force sensor offers practical benefits such as ease of manufacturing, high precision, and robustness, making it suitable for demanding applications in robotics and automation. Future work could focus on dynamic calibration and real-world testing to further validate its performance under varying operational conditions. Overall, this six-axis force sensor represents a step forward in multi-axis force measurement technology, with potential for widespread adoption in industrial and research settings.