In the field of robotics, the development of high-performance six-axis force sensors is crucial for enabling precise control in complex and heavy-load environments. These sensors must exhibit high accuracy, substantial stiffness, and large load-bearing capacity to function effectively as both measurement and structural components in industrial robots. Traditional Stewart platform-based six-axis force sensors often face challenges in balancing these competing requirements due to inter-axis coupling effects and structural limitations. In this work, we propose a partial decoupling approach to optimize a heavy-duty Stewart six-axis force sensor, focusing on simultaneous improvements in accuracy and stiffness. We begin by optimizing structural parameters, analyze coupling influences through experimental data, and implement a localized decoupling design validated via finite element simulations. Prototype testing demonstrates that our optimized sensor achieves enhanced stiffness while maintaining high measurement precision, meeting the demands of advanced robotic applications.
The Stewart platform configuration for a six-axis force sensor consists of six elastic branches connecting upper and lower platforms, with performance governed by key parameters: the distribution radii of spherical hinge points on the upper and lower platforms (denoted as $R_A$ and $R_B$), the positioning angles ($\alpha_A$ and $\alpha_B$), and the distance between platforms ($H$). To achieve optimal performance in terms of force/moment isotropy and sensitivity, we formulated an optimization problem targeting these parameters. The objective function minimized coupling and maximized stiffness, leading to the following optimized values:
| Parameter | Value |
|---|---|
| $R_A$ (mm) | 158 |
| $R_B$ (mm) | 123 |
| $\alpha_A$ (°) | 93 |
| $\alpha_B$ (°) | 34 |
| $H$ (mm) | 85 |
These parameters were derived through iterative simulations that considered the sensor’s load capacity (e.g., $F_x = 50,000$ N, $F_y = F_z = 22,000$ N, $M_x = M_y = 3,500$ N·m, $M_z = 5,000$ N·m) and mass constraints (under 15 kg). The optimization ensured that the six-axis force sensor would exhibit minimal inter-axis interference while supporting heavy loads, a critical aspect for industrial robotics.
Coupling effects in six-axis force sensors can significantly degrade accuracy, especially when transitioning from ideal hinge constraints to fixed supports to enhance stiffness. In a traditional Stewart six-axis force sensor, each branch ideally experiences only axial forces, but practical factors like manufacturing tolerances introduce complex couplings. For instance, when a lateral force $F_y$ is applied, it can induce unexpected outputs in other axes, as shown in experimental data from a prototype without decoupling measures:
| $F_y$ Load (N) | $F_x$ (N) | $F_y$ (N) | $F_z$ (N) | $M_x$ (N·m) | $M_y$ (N·m) | $M_z$ (N·m) |
|---|---|---|---|---|---|---|
| 500 | 19.11 | 466.4 | 28.19 | 71.28 | 58.93 | 27.19 |
| 1000 | 50.87 | 958.2 | 41.17 | 110.7 | 130.3 | 90.19 |
| 1500 | 39.17 | 1422 | 25.19 | 179.9 | 231.8 | 78.19 |
| 2000 | 150.2 | 1799 | 87.45 | 178.9 | 169.6 | 39.97 |
| 2500 | 119.9 | 2342 | 170.1 | 331.6 | 269.5 | 58.44 |
This data illustrates non-linear and irregular coupling, particularly in moments $M_x$ and $M_y$, which complicates the sensor’s output characteristics. To investigate this, we compared two prototypes: one with spherical hinges for decoupling and another without any decoupling measures. The branch performance under a pure moment $M_x$ revealed significant discrepancies:
| Branch | Non-linearity (%F.S) | Hysteresis (%F.S) | Repeatability (%F.S) |
|---|---|---|---|
| 1# (Decoupled) | 0.231 | 0.124 | 0.144 |
| 1# (No Decoupling) | 3.637 | 1.728 | 3.256 |
| 2# (Decoupled) | 0.153 | 0.263 | 0.183 |
| 2# (No Decoupling) | 20.048 | 12.848 | 19.115 |
| 3# (Decoupled) | 0.192 | 0.162 | 0.241 |
| 3# (No Decoupling) | 2.797 | 2.560 | 2.541 |
| 4# (Decoupled) | 0.252 | 0.242 | 0.233 |
| 4# (No Decoupling) | 2.856 | 2.249 | 2.336 |
| 5# (Decoupled) | 0.315 | 0.234 | 0.245 |
| 5# (No Decoupling) | 20.976 | 14.017 | 16.517 |
| 6# (Decoupled) | 0.157 | 0.198 | 0.196 |
| 6# (No Decoupling) | 3.859 | 1.988 | 3.199 |
Branches 2# and 5# exhibited the highest errors in the no-decoupling version, indicating severe coupling due to torsional and bending deformations. Finite element analysis confirmed that these branches experienced additional rotations and bending beyond axial strain, leading to non-linear output. For example, under a torque $M_z$, the deformation characteristics for branch 2# in the no-decoupling prototype were:
| $M_z$ Load (N·m) | Work Strain (10^{-6}) | Rotation Z (rad) | Rotation X (rad) | Rotation Y (rad) |
|---|---|---|---|---|
| 500 | 187 | 1.75e-5 | 1.58e-4 | 2.43e-4 |
| 1000 | 369 | 3.55e-5 | 3.22e-4 | 4.74e-4 |
| 1500 | 541 | 5.46e-5 | 4.84e-4 | 6.81e-4 |
| 2000 | 719 | 7.2e-5 | 6.18e-4 | 9.22e-4 |
To address this, we designed a custom flexible decoupling structure for branches 2# and 5#, aiming to absorb torsional and bending deformations while enhancing axial stiffness. This structure features a cross-beam configuration with flexible slots optimized through topology and parametric studies. The final design has a height $H = 60$ mm, diameter $D = 50$ mm, and includes tapered beams with specific angles to minimize coupling. The optimization criteria included minimizing rotation angles and maximizing axial stiffness, expressed as:
$$ \text{Minimize} \quad \theta_z, \theta_x, \theta_y $$
$$ \text{Maximize} \quad K_{\text{axial}} = \frac{F}{\delta} $$
where $F$ is the axial force and $\delta$ is the deformation.
Finite element simulations of the optimized six-axis force sensor showed improved strain distribution in branches 2# and 5#. Under combined loading of $M_x = 1000$ N·m and $F_y = 3000$ N, the stress uniformity increased, reducing coupling-induced stress concentrations. For instance, the stress values in branch 2# improved as follows:
| Stress Region | No Decoupling (MPa) | With Decoupling (MPa) |
|---|---|---|
| 1# | 40 | 50 |
| 2# | 65 | 57 |
| 3# | 57 | 55 |
| 4# | 36 | 49 |
Additionally, under $M_z$ torque, the decoupled branch exhibited reduced rotations and more linear strain output:
| $M_z$ Load (N·m) | Work Strain (10^{-6}) | Rotation Z (rad) | Rotation X (rad) | Rotation Y (rad) |
|---|---|---|---|---|
| 500 | 128 | 2.81e-5 | 0.24e-4 | 0.75e-4 |
| 1000 | 254 | 5.55e-5 | 0.61e-4 | 1.37e-4 |
| 1500 | 376 | 8.19e-5 | 1.02e-4 | 1.97e-4 |
| 2000 | 504 | 11.16e-5 | 1.18e-4 | 2.77e-4 |
We fabricated a prototype of the partial decoupled six-axis force sensor and conducted extensive testing. The branch performance under $M_x$ moment loading demonstrated significant improvements in non-linearity, hysteresis, and repeatability compared to the no-decoupling version:
| Branch | Non-linearity (%F.S) | Hysteresis (%F.S) | Repeatability (%F.S) |
|---|---|---|---|
| 1# | 0.617 | 0.591 | 0.312 |
| 2# | 0.951 | 0.894 | 0.221 |
| 3# | 0.701 | 0.668 | 0.437 |
| 4# | 0.857 | 0.691 | 0.477 |
| 5# | 1.121 | 0.733 | 0.347 |
| 6# | 0.513 | 0.527 | 0.319 |
Full-scale testing under multi-axial loads, such as combined $F_x$, $F_z$, and $M_x$, showed that the partial decoupled six-axis force sensor maintained high accuracy with errors within acceptable limits. For example, in a three-dimensional loading scenario, the maximum errors observed were:
| Load | Calibration Value | Measured Value | Error (%) |
|---|---|---|---|
| $F_x$ | 10,000 N | 10,067.989 N | 0.136 |
| $F_y$ | 10,000 N | 9,913.401 N | 0.394 |
| $F_z$ | 0 N | -91.991 N | 0.419 |
| $M_x$ | 2,000 N·m | 2,021.769 N·m | 0.435 |
| $M_y$ | 0 N·m | 25.769 N·m | 0.736 |
| $M_z$ | 0 N·m | -28.647 N·m | 0.818 |
Stiffness evaluations further highlighted the advantages of the partial decoupling method. The stiffness values in various directions for the three prototypes are summarized below:
| Stiffness | Decoupled Version | No Decoupling Version | Partial Decoupled Version |
|---|---|---|---|
| $K_x$ (N/m) | 0.196e8 | 1.719e8 | 1.213e8 |
| $K_y$ (N/m) | 0.213e8 | 1.825e8 | 1.255e8 |
| $K_z$ (N/m) | 0.485e8 | 3.235e8 | 2.584e8 |
| $K_{tx}$ (N·m/rad) | 0.371e6 | 2.647e6 | 1.981e6 |
| $K_{ty}$ (N·m/rad) | 0.385e6 | 2.786e6 | 2.015e6 |
| $K_{tz}$ (N·m/rad) | 0.723e6 | 4.828e6 | 3.942e6 |
The partial decoupled six-axis force sensor achieved an average stiffness improvement of 660% to 840% over the fully decoupled version, while only experiencing an 18% to 31% reduction compared to the no-decoupling version. This balance ensures that the sensor can withstand heavy loads without compromising measurement fidelity. The design of this six-axis force sensor incorporates robust materials and precision manufacturing to ensure reliability in industrial settings.
In conclusion, our partial decoupling method effectively addresses the trade-offs between accuracy and stiffness in heavy-duty Stewart six-axis force sensors. By optimizing structural parameters and implementing targeted decoupling for critical branches, we developed a sensor that excels in high-load applications while maintaining precise multi-axis force measurement. This advancement supports the evolution of intelligent robotics, enabling more sophisticated control in demanding environments. Future work could explore further miniaturization or integration with adaptive algorithms to enhance performance across a wider range of conditions.