In this study, I focus on the structural parameter optimization and mechanical performance evaluation of a Stewart-type six-axis force sensor, which is critical for high-precision force and torque measurement in aerospace and robotic applications. The six-axis force sensor is designed to measure multi-dimensional forces and moments, and its performance heavily relies on the optimal design of structural parameters. Through theoretical analysis and numerical simulations, I aim to enhance the sensor’s isotropy, sensitivity, and overall robustness. The six-axis force sensor’s ability to handle complex loading conditions makes it indispensable in environments where accuracy and reliability are paramount. This work delves into the parameter optimization process, finite element modeling, and comprehensive mechanical analysis to ensure the six-axis force sensor meets stringent operational requirements.
The Stewart-type six-axis force sensor consists of six elastic limbs connecting upper and lower platforms, and its performance is influenced by key geometric parameters: the distribution radii of the spherical hinge points on the upper and lower platforms (denoted as \( R_2 \) and \( R_1 \)), the positioning angles of the platforms (\( \phi_2 \) and \( \phi_1 \)), and the distance between the geometric centers of the platforms (\( H_c \)). These parameters define the sensor’s configuration, as illustrated in the schematic where \( u_1 \) to \( u_6 \) represent the hinge positions on the upper platform, and \( U_1 \) to \( U_6 \) represent those on the lower platform. To optimize the six-axis force sensor, I developed an objective function that minimizes the condition number of the calibration matrix and incorporates sensitivity metrics, ensuring high isotropy and minimal noise susceptibility. The objective function is expressed as:
$$ F_{\text{min}} ( X) = k \left[ \text{cond}( G_F ) + \text{cond}( G_M ) \right] + ( k – 1 ) \left[ \sqrt{ \text{tr}( [C_F]^T [C_F] ) } + \sqrt{ \text{tr}( [C_M]^T [C_M] ) } \right] $$
where \( G \) is the calibration matrix derived from structural parameters, \( C \) is the compliance matrix, and \( F \) and \( M \) represent force and moment vectors, respectively. The condition number \( \text{cond}(G) \) serves as an isotropy indicator; values close to 1 indicate low sensitivity to system noise, which is ideal for the six-axis force sensor. The norms of \( G \) and its inverse are given by:
$$ \| G \| = \sqrt{ L^2 + \frac{R_2^2 H_c^2 + R_1^2 R_2^2 \sin^2 \phi_{12}}{L^2} } $$
and
$$ \| G^{-1} \| = \frac{ \sqrt{ 4L^2 – 3H_c^2 + 4R_2^2 H_c^2 + R_1^2 R_2^2 \sin^2 \phi_{12} } }{ 6 R_1 R_2 H_c \sin^2 \phi_{12} } \cdot L $$
where \( L \) is the length of the elastic limbs and \( \phi_{12} = \frac{|\phi_1 – \phi_2|}{2} \). Constraints were applied to ensure practical design limits, such as \( \phi_1 – \phi_2 \in [30, 100]^\circ \), \( \phi_1, \phi_2 \in [0, 120]^\circ \), \( R_1, R_2 \in [0, 200] \, \text{mm} \), \( R_2 < R_1 \), and \( H_c \in [100, 150] \, \text{mm} \). After iterative optimization, the optimal parameters for the six-axis force sensor were determined, as summarized in the table below.
| Parameter | Value |
|---|---|
| \( R_1 \) | 143 mm |
| \( R_2 \) | 118 mm |
| \( \phi_1 \) | 98° |
| \( \phi_2 \) | 31° |
| \( H_c \) | 86 mm |
Substituting these values into the condition number formula yields \( \text{cond}(G) < 8 \), which meets the technical specifications for the six-axis force sensor, indicating excellent noise immunity and isotropy. This optimization ensures that the six-axis force sensor can accurately decode external force vectors without significant error propagation.
Following parameter optimization, I proceeded to create a detailed finite element model of the six-axis force sensor to simulate its mechanical behavior under various loads. The three-dimensional model was developed using Pro/ENGINEER, with each component, including the upper and lower platforms and elastic limbs, saved in SAT format for compatibility. The materials were assigned based on actual specifications: the upper and lower loading platforms were made of super-hard aluminum alloy 7A04, and the elastic limbs were composed of TB9 titanium alloy, known for its high strength and corrosion resistance. The model was imported into ABAQUS for finite element analysis, where I performed mesh generation to ensure computational accuracy and efficiency. Given the cylindrical nature of most components, I employed quadratic tetrahedral elements (C3D10M) for meshing. For complex regions, sweeping techniques were applied after defining edges, while free meshing was used for simpler parts. The global approximate size was set to 10 mm, with local refinements ensuring 6–8 elements per circular hole. Mesh quality was verified using ABAQUS’s built-in checks, with parameters such as shape factor (< 0.0005), face corner angles (< 10° and > 160°), aspect ratio (< 5), geometric eccentricity factor (< 0.15), and element edge lengths (between 0.1 mm and 20 mm). The mesh independence study, as shown in the figure below, confirmed that an element size of ≤ 4 mm yields consistent results, leading to a model with over 1.5 million elements. The assembled finite element model of the six-axis force sensor is depicted in the following image, which illustrates the intricate mesh structure and component integration.
Interactions between components were defined using tie constraints to simulate the bonding between elastic limbs and platforms. To apply external loads, I used multi-point constraints (MPC) with beam elements, distributing forces equally across the sensor’s upper platform. The lower platform was fixed at its threaded interfaces to represent real-world mounting conditions. This setup allows for realistic simulation of the six-axis force sensor’s response to complex loading scenarios, ensuring that the model accurately captures stress, strain, and displacement distributions.
The mechanical performance of the six-axis force sensor was evaluated through modal and load analyses to assess its dynamic characteristics and structural integrity. Modal analysis, conducted using the Lanczos eigensolver in ABAQUS, determined the first six natural frequencies of the sensor, which are crucial for avoiding resonance during operation. The results, presented in the table below, show that the first natural frequency is 0.839 kHz, primarily associated with vibration of the upper platform’s dust cover edge. The frequencies of modes 1–4 are similar, as are those of modes 5–6, reflecting the sensor’s structural symmetry. Since the technical specification requires a resonance frequency above 300 Hz, this six-axis force sensor demonstrates satisfactory dynamic performance.
| Mode | Frequency (kHz) |
|---|---|
| 1 | 0.839 |
| 2 | 0.837 |
| 3 | 0.835 |
| 4 | 0.829 |
| 5 | 0.736 |
| 6 | 0.722 |
Load analysis involved applying unidirectional forces and moments within the sensor’s rated capacity, as defined in the following table, to examine stress, strain, displacement, and stiffness. The six-axis force sensor’s full-scale ranges are 1500 N for forces (\( F_x, F_y, F_z \)) and 2000 N·m for moments (\( M_x, M_y, M_z \)).
| Load Type | Value |
|---|---|
| \( F_x, F_y, F_z \) | 1500 N |
| \( M_x, M_y, M_z \) | 2000 N·m |
Under a 1500 N tensile force along the X-axis (\( F_x \)), the maximum stress in the six-axis force sensor is approximately 26.83 MPa, occurring at the edges of the elastic body’s working zone. The stress in the working zone ranges from 13 to 18 MPa, well below the yield strength of TB9 (1124 MPa), resulting in a safety factor of 61.52. The average strain is about 70 με, giving a sensitivity of 0.15 mV/V, with uniform strain distribution and good symmetry. The maximum displacement in the X-direction is \( 1.093 \times 10^{-5} \) m at the upper platform’s protective shell, and the average displacement at the mechanical interface is \( 0.987 \times 10^{-5} \) m, yielding an X-direction stiffness of \( 1.519 \times 10^8 \) N/m, which exceeds the requirement of \( \geq 1 \times 10^8 \) N/m.
For a 1500 N tensile force along the Y-axis (\( F_y \)), the maximum stress is 49.06 MPa at the elastic body’s edges, with working zone stresses between 15 and 25 MPa. The safety factor is 44.13, average strain is 100 με, sensitivity is 0.22 mV/V, and strain distribution is uniform. The Y-direction displacement peaks at \( 2.380 \times 10^{-5} \) m, with an average interface displacement of \( 1.852 \times 10^{-5} \) m, corresponding to a stiffness of \( 8.010 \times 10^7 \) N/m, meeting the \( \geq 6 \times 10^7 \) N/m specification.
Under a 1500 N tensile force along the Z-axis (\( F_z \)), the maximum stress is 34.93 MPa, with working zone stresses of 11–24 MPa, a safety factor of 46.67, average strain of 90 με, and sensitivity of 0.19 mV/V. The Z-direction displacement is \( 2.327 \times 10^{-5} \) m, average interface displacement is \( 1.792 \times 10^{-5} \) m, and stiffness is \( 8.371 \times 10^7 \) N/m, satisfying the requirement.
For moment loads, under 2000 N·m torque about the X-axis (\( M_x \)), the maximum stress is 302.93 MPa at the elastic body edges, with working zone stresses of 140–180 MPa and a safety factor of 6.22. The average strain is 500 με, sensitivity is 1 mV/V, and the average rotational displacement at the interface is \( 8.396 \times 10^{-4} \) rad, giving a torsional stiffness of \( 2.382 \times 10^6 \) N·m/rad, above the \( \geq 0.8 \times 10^6 \) N·m/rad threshold.
Under 2000 N·m torque about the Y-axis (\( M_y \)), the maximum stress is 538.7 MPa, working zone stresses are 203–245 MPa, safety factor is 4.79, average strain is 600 με, sensitivity is 1.2 mV/V, rotational displacement is \( 1.522 \times 10^{-3} \) rad, and torsional stiffness is \( 1.314 \times 10^6 \) N·m/rad.
For 2000 N·m torque about the Z-axis (\( M_z \)), the maximum stress is 313.5 MPa, working zone stresses are 160–200 MPa, safety factor is 5.62, average strain is 450 με, sensitivity is 0.9 mV/V, rotational displacement is \( 1.338 \times 10^{-3} \) rad, and torsional stiffness is \( 1.495 \times 10^6 \) N·m/rad.
To evaluate the six-axis force sensor under extreme conditions, I simulated an overload scenario with simultaneous application of all rated loads: \( F_x = F_y = F_z = 1500 \) N and \( M_x = M_y = M_z = 2000 \) N·m. The results show a maximum stress of approximately 930.4 MPa at the elastic body edges, which is below the TB9 yield strength, indicating a safety factor of 1.87. The working zone average stress is around 600 MPa, demonstrating that the six-axis force sensor maintains structural integrity even under full combined loading. This analysis confirms the robustness of the optimized six-axis force sensor design.
In conclusion, this study successfully optimized the structural parameters of a Stewart-type six-axis force sensor through theoretical modeling and numerical simulations. The optimized parameters resulted in a condition number below 8, ensuring high isotropy and low noise sensitivity. The finite element model, validated through mesh independence studies, provided reliable insights into the sensor’s mechanical behavior. Modal analysis revealed natural frequencies well above the required threshold, minimizing resonance risks. Load analyses under various unidirectional and combined loads demonstrated that the six-axis force sensor exhibits uniform stress and strain distributions, high safety factors, and stiffness values that meet or exceed specifications. The overload analysis further confirmed the sensor’s durability under extreme conditions. Overall, the six-axis force sensor design is rational and offers excellent mechanical performance, making it suitable for demanding applications in aerospace and robotics. Future work could involve experimental validation and further refinement of the six-axis force sensor for specific use cases.