In the context of space robotics, the six-axis force sensor plays a critical role as the core component of the “tactile” perception system for space manipulators. It enables real-time monitoring of environmental loads and provides essential force feedback for force and motion control, facilitating intelligent and compliant operations. However, the Stewart six-axis force sensor, a common parallel-structured sensor, faces significant challenges in achieving long service life due to stress concentration and fatigue damage under complex operational loads. This paper addresses the low service life issue by proposing a diagnostic and optimization framework based on structural parameter analysis and fatigue life theory. We derive the relationship between sensor structural parameters and life characteristics, establish evaluation indices, and employ multi-objective optimization to enhance longevity. Comprehensive simulations and experimental validations demonstrate the effectiveness of our approach in achieving stress homogenization and extending sensor lifespan.
The Stewart six-axis force sensor consists of six elastic limbs connecting upper and lower platforms via spherical joints, as illustrated in the following figure. This configuration allows each limb to primarily withstand axial tension/compression forces, enabling the measurement of spatial loads through the detection of combined deformations. The sensor’s performance is governed by five key structural parameters: the distribution radii of the upper and lower platforms (denoted as \( R_A \) and \( R_B \)), the positioning angles of the platforms (\( \alpha_A \) and \( \alpha_B \)), and the height between the platforms (\( H \)). These parameters critically influence the sensor’s isotropy, sensitivity, and fatigue life.
Based on the force screw theory and virtual work principle of Stewart parallel mechanisms, the generalized force equilibrium equation is expressed as:
$$ \sum_{i=1}^{6} f_i \phi_i = F_F + \epsilon F_M $$
where \( f_i \) represents the axial force of the \( i \)-th elastic limb, \( \phi_i \) is the unit line vector of the \( i \)-th limb axis relative to the reference frame, and \( F_F \) and \( F_M \) denote the principal vector and moment applied to the upper platform, respectively. This equation can be reformulated in matrix form as:
$$ \mathbf{F} = \mathbf{G} \mathbf{f} $$
Here, \( \mathbf{f} = [f_1, f_2, f_3, f_4, f_5, f_6]^T \) is the axial force vector of the limbs, \( \mathbf{F} = [F_F, F_M]^T \) is the external force vector, and \( \mathbf{G} \) is the first-order forward mapping influence coefficient matrix. The inverse mapping is given by the force Jacobian matrix \( \mathbf{C} = \mathbf{G}^{-1} \). The sensor’s structural performance, including force and moment isotropy, is evaluated using the condition numbers of submatrices of \( \mathbf{G} \) and \( \mathbf{C} \). Specifically, the isotropy indices are defined as:
$$ u_F = \frac{1}{\text{cond}(\mathbf{G}_F)} = \frac{\lambda_{\min}(\mathbf{G}_F \mathbf{G}_F^T)}{\lambda_{\max}(\mathbf{G}_F \mathbf{G}_F^T)} $$
$$ u_M = \frac{1}{\text{cond}(\mathbf{G}_M)} = \frac{\lambda_{\min}(\mathbf{G}_M \mathbf{G}_M^T)}{\lambda_{\max}(\mathbf{G}_M \mathbf{G}_M^T)} $$
$$ \eta_F = \frac{1}{\text{cond}(\mathbf{C}_F)} = \frac{\lambda_{\min}(\mathbf{C}_F \mathbf{C}_F^T)}{\lambda_{\max}(\mathbf{C}_F \mathbf{C}_F^T)} $$
$$ \eta_M = \frac{1}{\text{cond}(\mathbf{C}_M)} = \frac{\lambda_{\min}(\mathbf{C}_M \mathbf{C}_M^T)}{\lambda_{\max}(\mathbf{C}_M \mathbf{C}_M^T)} $$
where \( \lambda_{\min} \) and \( \lambda_{\max} \) are the minimum and maximum eigenvalues of the respective matrices, and \( \text{cond} \) denotes the condition number. Values closer to 1 indicate better isotropy. However, these indices are interdependent and cannot be optimized simultaneously.
To address the life characteristics of the six-axis force sensor, we integrate nominal stress fatigue theory. Assuming linear behavior and consistent limb properties, the axial force in each limb relates to stress \( \sigma \) by \( f = A \sigma \), where \( A \) is the equivalent cross-sectional area. Combining this with the force equilibrium equations yields:
$$ \mathbf{F} = \mathbf{G} A \sigma $$
$$ \sigma = \frac{1}{A} \mathbf{C} \mathbf{F} $$
The fatigue life is modeled using a three-parameter S-N curve:
$$ S = Q \left(1 + \frac{P}{N^\beta}\right) $$
where \( S \) is the stress amplitude, \( N \) is the number of cycles to failure, \( Q \) is the theoretical fatigue limit, and \( \beta \), \( P \) are shape parameters. For variable amplitude loading, the Miner linear cumulative damage theory is applied, and the median fatigue life under constant amplitude stress is:
$$ N_{50} = \sum_{j=1}^{n} \frac{Q P}{(\sigma_j – Q)^{1/\beta}} $$
Simplifying for normalized load spectra (\( n=1 \)), the relationship between generalized forces and life characteristics is derived as:
$$ \frac{1}{N_{50}^\beta} + \frac{1}{P} = \frac{1}{A Q P} \mathbf{C} \mathbf{F} $$
Defining \( \mathbf{Z} = \frac{1}{A Q P} \mathbf{C} \) and \( \mathbf{L} = \frac{1}{N_{50}^\beta} + \frac{1}{P} \), we obtain \( \mathbf{L} = \mathbf{Z} \mathbf{F} \). The life characteristic vector \( \mathbf{L} \) depends on the force/moment influence coefficient matrix \( \mathbf{Z} \). When the generalized force magnitude is unitary, the extremum of \( \mathbf{L} \)’s norm serves as a life performance index. Using Lagrange multipliers, the minimum norms for force and moment components are:
$$ \|\mathbf{L}_F\|_{\min} = \sqrt{\lambda_{F,\min}} $$
$$ \|\mathbf{L}_M\|_{\min} = \sqrt{\lambda_{M,\min}} $$
where \( \lambda_{F,\min} \) and \( \lambda_{M,\min} \) are the minimum eigenvalues of \( \mathbf{Z}_F^T \mathbf{Z}_F \) and \( \mathbf{Z}_M^T \mathbf{Z}_M \), respectively. Smaller values indicate better life characteristics.
For multi-objective optimization, we formulate a target function that incorporates isotropy indices and life characteristics:
$$ F_{\min}(R_A, R_B, \alpha_A, \alpha_B, H) = k_1 \frac{\lambda_{\max}(\mathbf{G}_F \mathbf{G}_F^T)}{\lambda_{\min}(\mathbf{G}_F \mathbf{G}_F^T)} + k_2 \frac{\lambda_{\max}(\mathbf{G}_M \mathbf{G}_M^T)}{\lambda_{\min}(\mathbf{G}_M \mathbf{G}_M^T)} + k_3 \frac{\lambda_{\max}(\mathbf{C}_F \mathbf{C}_F^T)}{\lambda_{\min}(\mathbf{C}_F \mathbf{C}_F^T)} + k_4 \frac{\lambda_{\max}(\mathbf{C}_M \mathbf{C}_M^T)}{\lambda_{\min}(\mathbf{C}_M \mathbf{C}_M^T)} + k_5 \sqrt{\lambda_{F,\min}} + k_6 \sqrt{\lambda_{M,\min}} $$
where \( k_1 \) to \( k_6 \) are weighting coefficients. Based on operational requirements and stress analyses, we set \( \mathbf{K} = [1, 1, 1, 1, 1, 5] \) to prioritize moment life due to higher stress impacts. The structural parameters are normalized as \( r_1 = R_A / P \), \( r_2 = R_B / P \), \( r_3 = H / P \), with \( P = R_A + R_B + H \leq 936 \, \text{mm} \). Constraints include \( r_1 + r_2 + r_3 = 1 \), \( 0 < r_i < 1 \), \( R < \min(R_A, R_B) \), and \( -30^\circ \leq \alpha_A, \alpha_B \leq 120^\circ \).
We employ a genetic algorithm to solve this optimization problem, iteratively refining parameters to minimize the objective function. The optimized structural parameters are compared with the initial design in the following table:
| Parameter | Initial Design (P=364 mm) | Optimized Design (P=380 mm) |
|---|---|---|
| Upper Platform Radius \( R_A \) (mm) | 150 | 150 |
| Lower Platform Radius \( R_B \) (mm) | 100 | 130 |
| Upper Platform Angle \( \alpha_A \) (°) | 80 | 100 |
| Lower Platform Angle \( \alpha_B \) (°) | 40 | 30 |
| Platform Height \( H \) (mm) | 114 | 100 |
The performance metrics before and after optimization are summarized below:
| Performance Index | Initial Design | Optimized Design |
|---|---|---|
| Force Isotropy \( u_F \) | 0.517 | 0.624 |
| Moment Isotropy \( u_M \) | 0.773 | 0.684 |
| Force Sensitivity Isotropy \( \eta_F \) | 0.718 | 0.609 |
| Moment Sensitivity Isotropy \( \eta_M \) | 0.484 | 0.619 |
| Force Life Index \( \|\mathbf{L}_F\|_{\min} \) | 4.702 | 5.824 |
| Moment Life Index \( \|\mathbf{L}_M\|_{\min} \) | 13.981 | 7.826 |
Post-optimization, force isotropy improves by 20.7%, while moment isotropy decreases by 11.5%. Force sensitivity isotropy drops by 15.2%, but moment sensitivity isotropy rises by 27.9%. Crucially, the force life index decreases by 23.9%, indicating slightly reduced life under pure force, but the moment life index improves by 44.0%, significantly enhancing overall longevity. This trade-off aligns with the weighting strategy emphasizing moment life.
Finite element analysis simulations validate these findings. Von Mises stress distributions under combined operational and thermal loads show that the optimized six-axis force sensor exhibits more uniform stress across the limbs, with peak stress at the strain beams reduced by over 25% compared to the initial design. Fatigue life simulations, using the Soderberg criterion with a reliability factor of 1.5, predict a 97% increase in fatigue cycles for the optimized sensor. The following table compares stress and life metrics from simulations:
| Metric | Initial Design | Optimized Design |
|---|---|---|
| Max Von Mises Stress (MPa) | 385 | 288 |
| Stress Homogeneity Index | 0.45 | 0.72 |
| Predicted Fatigue Life (Cycles) | 1.2e5 | 2.36e5 |
Experimental studies on prototype six-axis force sensors confirm the simulations. Fatigue testing under cyclic loading (R=0) reveals that the optimized sensor maintains stable zero-point output within 0.3% precision up to 240,000 cycles, whereas the initial design exceeds this threshold at 180,000 cycles. Post-fatigue, the initial sensor shows significant degradation in linearity, hysteresis, and repeatability, as detailed below:
| Limb Index | Nonlinearity (% F.S.) – Initial | Nonlinearity (% F.S.) – Optimized | Hysteresis (% F.S.) – Initial | Hysteresis (% F.S.) – Optimized |
|---|---|---|---|---|
| 1 | 0.93 | 0.23 | 1.73 | 0.15 |
| 2 | 7.11 | 2.15 | 3.85 | 0.24 |
| 3 | 1.22 | 0.22 | 1.56 | 0.13 |
| 4 | 1.36 | 0.15 | 1.25 | 0.24 |
| 5 | 5.18 | 1.82 | 2.02 | 0.23 |
| 6 | 1.06 | 0.20 | 1.99 | 0.18 |
Calibration tests under uniaxial and multidimensional loads demonstrate that the optimized six-axis force sensor meets accuracy requirements (force error <2.5%, moment error <1%), while the initial design fails due to excessive coupling and nonlinearity. Stiffness tests show a 18-24% reduction in axial and torsional stiffness for the initial sensor, indicating potential plastic deformation or crack initiation. Material analyses via ultrasonic testing and microscopy reveal internal cracks in the initial sensor’s limbs, whereas the optimized sensor retains structural integrity. Mechanical property tests further show higher yield strength (≈1050 MPa vs. ≈700 MPa) and fracture toughness (≈50 MPa·m\(^{1/2}\) vs. ≈35 MPa·m\(^{1/2}\)) for the optimized limbs.
In conclusion, our life-based optimization methodology for the Stewart six-axis force sensor effectively addresses low service life by balancing structural parameters to achieve stress homogenization. The derived life characteristics model and multi-objective optimization framework enable significant improvements in fatigue resistance, as validated through simulations and experiments. This approach provides a generalizable foundation for enhancing the reliability and longevity of six-axis force sensors in space applications, ensuring robust performance over extended missions. Future work will focus on dynamic load adaptations and real-time health monitoring for further lifecycle extensions.