In the field of multi-dimensional force measurement, the six-axis force sensor has emerged as a critical tool for capturing forces and moments in three-dimensional space. However, a persistent challenge in such sensors is the inter-dimensional coupling effect, which often leads to reduced measurement accuracy. Traditional designs, including cylindrical, ring-type, cantilever beam, cross-beam, and inverted T-structures, have limitations in decoupling performance, thereby restricting their applicability in high-precision scenarios. To address these issues, we have developed a novel four-column structure six-axis force sensor that inherently minimizes cross-talk through optimized mechanical design, finite element analysis, strategic strain gauge placement, and bridge circuit configuration. This sensor is capable of measuring three force components (F_X, F_Y, F_Z) and three moment components (M_X, M_Y, M_Z) with exceptional precision and minimal interference between axes. Our approach focuses on achieving self-decoupling characteristics, resulting in a compact, robust, and highly accurate device suitable for both static and quasi-static measurements. The design process involved extensive simulation to identify optimal strain locations, followed by experimental validation that confirmed single-axis accuracy better than 0.1% F.S. and cross-influence levels below 3% F.S. This paper elaborates on the sensor’s design principles, analytical methods, experimental procedures, and performance outcomes, highlighting its potential for widespread adoption in industries requiring precise multi-axis force monitoring.
The development of six-axis force sensors has been a subject of intensive research globally, with early efforts dating back to the 1970s. For instance, in 1974, researchers at the Swiss Federal Institute of Technology Lausanne explored the feasibility of such sensors using strain gauges as sensing elements, but their bulky design and reliance on analog computation limited practical use. Subsequent designs, like the three-vertical-rib structure by P.C. Watson in 1975, offered simplicity and high load capacity but suffered from low sensitivity and significant coupling. More recent advancements include non-radial three-beam sensors, vertical rib structures, cross-beam configurations, and Stewart platform-based designs, each with distinct trade-offs in stiffness, sensitivity, and decoupling capability. Common structural forms such as cylindrical, columnar, ring-type, cantilever beam, and cross-beam sensors often exhibit inherent coupling due to their geometry, necessitating additional signal processing or structural modifications. Our work builds upon these foundations by introducing a four-column elastic body that enhances isotropic properties and reduces cross-talk through symmetrical load distribution. This design not only simplifies manufacturing but also improves reliability and accuracy, addressing key limitations in existing six-axis force sensor technologies.
The core of our six-axis force sensor is its four-column elastic structure, which was meticulously designed to handle specified force and moment ranges: axial force (F_Z) up to 200 kN, lateral forces (F_X and F_Y) up to 50 kN each, and torsional moments (M_Z, M_X, M_Y) up to 1000 Nm. This configuration ensures high stiffness, minimal hysteresis, and excellent linearity, making it ideal for applications demanding precise static or dynamic force measurements. The sensor’s body is fabricated as a single piece using high-strength alloy steel, followed by processes like argon arc welding and internal nitrogen filling to isolate it from environmental factors, thereby enhancing long-term stability. Key advantages of this four-column six-axis force sensor include its simple geometry, ease of machining, resistance to off-center loads, compact size, and lightweight nature, which facilitate installation in space-constrained environments. To quantify its performance, we targeted precision metrics such as cross-influence ≤ 5% F.S., zero drift ≤ 0.03% F.S./30 min, and temperature drift ≤ 0.05% F.S./10 K, which were validated through rigorous testing.
Finite element analysis (FEA) played a pivotal role in optimizing the sensor’s structure and identifying regions of maximum strain under load. Using commercial FEA software, we modeled the elastic body with fixed boundary conditions at the bottom surface and applied uniform loads on the top surface—specifically, a tensile force of 50 kN (approximately 5 tons) for axial analysis and a circumferential moment of 500 Nm for torsional analysis. The simulation revealed that the central regions of the columns experienced the highest deformations, with strain values ranging from 160 to 350 microstrain, significantly exceeding other areas (around 32.5 microstrain). This identified the sensitive zones for strain gauge placement, ensuring maximum output sensitivity and effective decoupling. The strain distribution can be described by the relationship between stress and strain, governed by Hooke’s law: $$ \sigma = E \epsilon $$ where \(\sigma\) is the stress, \(E\) is the Young’s modulus, and \(\epsilon\) is the strain. For multi-axial loading, the strain components relate to the applied forces and moments through a stiffness matrix: $$ \begin{bmatrix} \epsilon_{xx} \\ \epsilon_{yy} \\ \epsilon_{zz} \\ \gamma_{xy} \\ \gamma_{yz} \\ \gamma_{zx} \end{bmatrix} = \mathbf{C} \begin{bmatrix} F_X \\ F_Y \\ F_Z \\ M_X \\ M_Y \\ M_Z \end{bmatrix} $$ where \(\mathbf{C}\) is the compliance matrix derived from the sensor’s geometry. By minimizing off-diagonal terms in \(\mathbf{C}\), we achieved inherent decoupling, a key feature of this six-axis force sensor.
Strain gauge configuration and bridge circuit design were critical to translating mechanical deformations into measurable electrical signals. A total of 28 strain gauges were mounted on the transverse beams of the four-column structure, arranged to form six full-bridge circuits—one for each force and moment component. Each bridge operates at a 10 V DC excitation voltage, providing six independent output channels. The gauge positions were selected based on FEA results to maximize sensitivity and minimize cross-talk. For example, gauges for F_Z were placed in areas with high axial strain, while those for M_X and M_Y were oriented to capture bending strains. The output voltage \(\Delta V\) of a Wheatstone bridge under strain is given by: $$ \Delta V = V_{\text{in}} \cdot \frac{\Delta R}{R} $$ where \(V_{\text{in}}\) is the input voltage, and \(\Delta R/R\) is the relative resistance change proportional to strain: $$ \frac{\Delta R}{R} = k \cdot \epsilon $$ with \(k\) being the gauge factor. By strategically pairing gauges in tension and compression, we balanced the bridges to reject common-mode noise and temperature effects, further enhancing the accuracy of this six-axis force sensor.
Experimental validation involved comprehensive calibration using standard machines to assess single-axis accuracy and cross-influence. The single-component tests were conducted on a 100-ton lever-type force standard machine for force components (F_Z, F_X, F_Y) and a 5000 Nm standard torque machine for moment components (M_Z, M_X, M_Y), with outputs recorded using a high-precision indicator (model 8845A). The data, summarized in Table 1, demonstrate excellent linearity and repeatability across all axes. For instance, F_Z exhibited a linearity error of 0.05% F.S., while other components achieved 0.1% F.S. or better. Repeatability errors were as low as 0.01% F.S. for forces and 0.03% F.S. for moments, confirming the sensor’s reliability.
| Load (kN or Nm) | F_Z Output (µV) | F_X Output (µV) | F_Y Output (µV) | M_Z Output (µV) | M_X Output (µV) | M_Y Output (µV) |
|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 50 kN / 200 Nm | 3294 | -7310 | -6969 | 1600 | 1271 | 1266 |
| 100 kN / 400 Nm | 6610 | -14248 | -13795 | 3199 | 2546 | 2529 |
| 150 kN / 600 Nm | 9931 | -21100 | -20632 | 4797 | 3821 | 3796 |
| 200 kN / 800 Nm | 13246 | -27913 | -27560 | 6393 | 5091 | 5059 |
| 1000 Nm | – | – | – | 7986 | 6358 | 6320 |
| Repeatability (% F.S.) | 0.01 | 0.01 | 0.02 | 0.03 | 0.03 | 0.03 |
| Linearity (% F.S.) | 0.05 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |
Cross-influence tests were performed to evaluate interference between components, using a combination of standard machines,叠加机, lever systems, and calibrated weights. The methods for each component are listed in Table 2, ensuring accurate assessment without introducing external errors. Results in Table 3 show that cross-influence levels remain within 0.1% to 3% F.S., well below the 5% F.S. target. For example, when F_Z was loaded, the maximum cross-effect on other axes was 3% F.S., while for M_Z, it was as low as 0.5% F.S. This demonstrates the effectiveness of the self-decoupling design in this six-axis force sensor.
| Component | Full Scale | Calibration Equipment |
|---|---|---|
| Axial Force (F_Z) | 200 kN | 20t Overlay Machine |
| Lateral Force (F_X) | 50 kN | 100t Lever Standard Machine |
| Transverse Force (F_Y) | 50 kN | 100t Lever Standard Machine |
| Main Torque (M_Z) | 1000 Nm | 5000 Nm Standard Torque Machine |
| Lateral Torque (M_X) | 1000 Nm | Standard Lever and Weights |
| Transverse Torque (M_Y) | 1000 Nm | Standard Lever and Weights |
| Loading Condition | F_Z Output (µV) | F_X Output (µV) | F_Y Output (µV) | M_Z Output (µV) | M_X Output (µV) | M_Y Output (µV) | Cross-Influence (% F.S.) |
|---|---|---|---|---|---|---|---|
| F_Z Loaded (0 to 200 kN) | 1415 to 14694 | 4528 to 4259 | 901 to 1243 | -140 to -55 | -224 to -331 | -903 to -730 | 1.0–3.0 |
| F_X Loaded (0 to 80 kN) | 1340 to 1614 | 4055 to -22760 | 1356 to 1545 | -200 to -127 | -874 to 13520 | -75 to 63 | 1.0–2.0 |
| F_Y Loaded (0 to 80 kN) | 1401 to 1292 | 4409 to 4378 | 1314 to 30095 | -169 to -120 | 0 to 19 | -889 to -725 | 1.0–3.0 |
| M_Z Loaded (0 to 1000 Nm) | 1308 to 1478 | 4883 to 4835 | 927 to 964 | -315 to 7666 | 639 to 609 | 900 to 897 | 0.2–1.0 |
| M_X Loaded (0 to 1000 Nm) | 1475 to 1553 | 4863 to 4555 | 1084 to 1200 | -254 to -345 | -654 to -7011 | -1701 to -1763 | 0.5–1.0 |
| M_Y Loaded (0 to 1000 Nm) | 1475 to 1521 | 4882 to 4500 | 1203 to 1040 | -307 to -415 | -95 to -89 | -1093 to 5228 | 0.1–1.5 |
The analysis of experimental data confirms that the four-column structure six-axis force sensor meets all design specifications. Single-axis accuracies are exceptionally high, with F_Z at 0.05% F.S. and other components at 0.1% F.S., while cross-influence values range from 0.1% to 3% F.S., indicating effective decoupling. The sensor’s performance can be modeled using a decoupling matrix \(\mathbf{D}\) that relates raw bridge outputs to actual forces and moments: $$ \begin{bmatrix} F_X \\ F_Y \\ F_Z \\ M_X \\ M_Y \\ M_Z \end{bmatrix} = \mathbf{D} \begin{bmatrix} V_1 \\ V_2 \\ V_3 \\ V_4 \\ V_5 \\ V_6 \end{bmatrix} $$ where \(V_i\) are the bridge output voltages. By calibrating \(\mathbf{D}\) through least-squares fitting, we minimize errors due to coupling. The sensor’s stiffness matrix \(\mathbf{K}\), derived from material properties and geometry, also contributes to its low cross-talk: $$ \mathbf{K} = \begin{bmatrix} k_{11} & k_{12} & \cdots & k_{16} \\ k_{21} & k_{22} & \cdots & k_{26} \\ \vdots & \vdots & \ddots & \vdots \\ k_{61} & k_{62} & \cdots & k_{66} \end{bmatrix} $$ where off-diagonal elements \(k_{ij}\) (i ≠ j) represent coupling stiffness. In our design, these are minimized through symmetrical column arrangement, reducing inter-axis interference. Additionally, the sensor exhibits negligible zero drift (≤0.03% F.S./30 min) and temperature drift (≤0.05% F.S./10 K), ensuring stability under varying conditions. These attributes make this six-axis force sensor ideal for applications in robotics, aerospace, automotive testing, and industrial automation, where precise multi-axis force feedback is crucial.
In conclusion, the four-column structure six-axis force sensor presented here offers a significant advancement in multi-dimensional force measurement technology. Its self-decoupling capability, achieved through innovative structural design and meticulous strain gauge configuration, results in high accuracy and minimal cross-influence. Experimental validations using standard calibration equipment have demonstrated single-axis precision better than 0.1% F.S. and cross-talk levels below 3% F.S., surpassing typical industry requirements. The sensor’s compact size, high stiffness, and environmental robustness further enhance its practicality for diverse applications. Future work may focus on optimizing the material selection for higher load capacities or integrating wireless data transmission for remote monitoring. Overall, this six-axis force sensor represents a reliable and efficient solution for capturing complex force and moment data in three-dimensional space, with broad potential for adoption in precision engineering and beyond.