In this study, we explore the fault-tolerant measurement principles and calibration testing of a pre-stressed parallel six-axis force sensor. Six-axis force sensors are critical components for detecting three force components and three moment components in three-dimensional space, widely used in aerospace, defense technology, biomedical engineering, and industrial applications. Traditional Stewart-based six-axis force sensors offer advantages such as high stiffness, structural stability, and no cumulative error. However, they are prone to branch failures due to sensitivity and accuracy trade-offs, leading to overall system failure. To address this, we investigate a pre-stressed parallel six-axis force sensor with inherent fault tolerance, enabling reliable measurement even when a branch signal fails. This work focuses on theoretical modeling, static calibration algorithms, and experimental validation under various loading conditions, including unidirectional, bidirectional combined, and partitioned loading. The results demonstrate the sensor’s capability to maintain measurement integrity in fault conditions, laying a foundation for practical applications of fault-tolerant six-axis force sensors.
The structure of the pre-stressed parallel six-axis force sensor consists of a force-measuring platform, an intermediate platform, a pre-stressed platform, sleeves, a base, adjustment shims, and seven measuring branches. These branches are divided into two groups distributed on both sides of the force-measuring platform: three branches uniformly on the upper side of the intermediate platform and four on the lower side. Pre-tightening bolts apply a pre-load force between the pre-stressed platform and the measuring branches, eliminating gaps at contact surfaces and avoiding zero-crossing issues, thereby enhancing overall stiffness. Compared to traditional Stewart structures, this six-axis force sensor offers benefits such as no stress coupling, low friction moments, minimal nonlinearity and hysteresis errors, high accuracy, and good dynamic performance. Importantly, it exhibits fault tolerance: if one branch fails (e.g., due to signal errors), the faulty branch’s output can be isolated, and the remaining six branches can reconstruct the six-dimensional external force measurement. This redundancy ensures continuous operation, which is crucial for critical tasks where system failures could lead to performance degradation or paralysis.
To analyze the fault-tolerant measurement mechanism, we first establish mathematical models for both fault-free and signal-fault conditions using screw theory and deformation coordination. For the fault-free case, the sensor has seven measuring branches. The static equilibrium equation based on screw theory is given by:
$$ \mathbf{F}_w = \mathbf{G} \mathbf{f} $$
where $\mathbf{F}_w = [F_x, F_y, F_z, M_x, M_y, M_z]^T$ is the generalized six-dimensional external load, $\mathbf{f} = [f_1, f_2, \dots, f_7]^T$ is the vector of axial forces in the measuring branches, and $\mathbf{G}$ is the force Jacobian matrix. The matrix $\mathbf{G}$ is constructed from the unit line vectors of the branches relative to the reference coordinate system:
$$ \mathbf{G} = \begin{bmatrix} \mathbf{S}_1 & \mathbf{S}_2 & \cdots & \mathbf{S}_7 \\ \mathbf{S}_{01} & \mathbf{S}_{02} & \cdots & \mathbf{S}_{07} \end{bmatrix} $$
Here, $\mathbf{S}_i$ represents the direction vector of the i-th branch, and $\mathbf{S}_{0i}$ is the moment vector. For the specific structure, the Jacobian matrix can be expressed as:
$$ \mathbf{G} = \begin{bmatrix} \frac{\mathbf{b}_1 – \mathbf{B}_1}{\|\mathbf{b}_1 – \mathbf{B}_1\|} & \frac{\mathbf{b}_2 – \mathbf{B}_2}{\|\mathbf{b}_2 – \mathbf{B}_2\|} & \cdots & \frac{\mathbf{b}_7 – \mathbf{B}_7}{\|\mathbf{b}_7 – \mathbf{B}_7\|} \\ \frac{\mathbf{B}_1 \times \mathbf{b}_1}{\|\mathbf{b}_1 – \mathbf{B}_1\|} & \frac{\mathbf{B}_2 \times \mathbf{b}_2}{\|\mathbf{b}_2 – \mathbf{B}_2\|} & \cdots & \frac{\mathbf{B}_7 \times \mathbf{b}_7}{\|\mathbf{b}_7 – \mathbf{B}_7\|} \end{bmatrix} $$
where $\mathbf{b}_i$ and $\mathbf{B}_i$ denote the position vectors of the spherical joint points on the force-measuring platform and the base platform, respectively. This model enables the mapping from branch forces to external loads in the fault-free six-axis force sensor.
When a signal fault occurs in one branch, say the i-th branch, the sensor operates with six functional branches. The measurement model is reconstructed using deformation coordination. The elastic deformation of the branches relates to the platform’s micro-displacement $\Delta \mathbf{D} = [\Delta \mathbf{d}, \Delta \boldsymbol{\theta}]^T$, where $\Delta \mathbf{d}$ is the linear displacement and $\Delta \boldsymbol{\theta}$ is the angular displacement. For the j-th branch, the deformation is:
$$ \Delta l_j = \mathbf{S}_j^T \Delta \mathbf{d} + (\mathbf{p}_j \times \mathbf{S}_j)^T \Delta \boldsymbol{\theta} = \begin{bmatrix} \mathbf{S}_j^T & (\mathbf{p}_j \times \mathbf{S}_j)^T \end{bmatrix} \Delta \mathbf{D} $$
where $\mathbf{p}_j$ is the position vector. The deformations of the static-determinate part (six branches) and the redundant branch (faulty branch) are:
$$ \Delta \mathbf{l}_s = \mathbf{G}_s^T \Delta \mathbf{D}, \quad \Delta l_i = \boldsymbol{\$}_i^T \Delta \mathbf{D} $$
Here, $\mathbf{G}_s$ is the Jacobian matrix excluding the i-th branch, and $\boldsymbol{\$}_i$ is the unit line vector of the faulty branch. The branch forces and deformations are related by:
$$ \begin{bmatrix} \mathbf{f}_s \\ f_i \end{bmatrix} = \begin{bmatrix} \mathbf{K}_s & \mathbf{0} \\ \mathbf{0} & k_i \end{bmatrix} \begin{bmatrix} \Delta \mathbf{l}_s \\ \Delta l_i \end{bmatrix} $$
where $\mathbf{K}_s$ is the stiffness matrix of the static-determinate branches, and $k_i$ is the stiffness of the faulty branch. Combining these, the external load is expressed as:
$$ \mathbf{F}_w = \left( \mathbf{G}_s \mathbf{K}_s \mathbf{G}_s^T + k_i \boldsymbol{\$}_i \boldsymbol{\$}_i^T \right) \Delta \mathbf{D} $$
Thus, the measurement model under signal fault becomes:
$$ \mathbf{F}_w = \mathbf{G}_w \mathbf{f}_s $$
where $\mathbf{G}_w = \mathbf{G}_s + k_i \boldsymbol{\$}_i \boldsymbol{\$}_i^T \mathbf{C}_s \mathbf{K}_s^{-1}$ and $\mathbf{C}_s = (\mathbf{G}_s^T)^{-1}$. This model allows the six-axis force sensor to maintain functionality by using the outputs of the six healthy branches, demonstrating its fault-tolerant capability.
For static calibration, we determine the linear mapping between the input forces and the output voltages of the measuring branches. In the fault-free case, the relationship is:
$$ [\mathbf{F}_w]_{6 \times 1} = [\mathbf{G}_c]_{6 \times 7} [\mathbf{V}]_{7 \times 1} $$
where $[\mathbf{V}]$ is the output voltage vector. With multiple loading points (k > 7), the calibration matrix $\mathbf{G}_c$ is computed using the least squares method:
$$ [\mathbf{G}_c]_{6 \times 7} = [\mathbf{F}_w]_{6 \times k} [\mathbf{V}]_{7 \times k}^T \left( [\mathbf{V}]_{7 \times k} [\mathbf{V}]_{7 \times k}^T \right)^{-1} $$
We conducted unidirectional loading tests, where forces were applied in increments of 10 N up to 100 N and moments in increments of 1 N·m up to 10 N·m. The calibration matrix and linearity errors were calculated. The results for the fault-free six-axis force sensor are summarized in the following table:
| Performance Metric | F_x | F_y | F_z | M_x | M_y | M_z |
|---|---|---|---|---|---|---|
| Linearity (%) | 0.3326 | 0.1384 | 0.4715 | 0.1156 | 0.0821 | 0.9089 |
| Precision (%) | 0.0891 | 0.3450 | 0.4881 | 0.2777 | 0.0993 | 0.0590 |
| Accuracy (%) | 0.3515 | 0.8300 | 0.4348 | 0.1797 | 0.3341 | 0.0593 |
The maximum linearity error in the main direction is 0.91% of full scale, and the maximum cross-coupling error is 1.49%. This indicates good performance of the six-axis force sensor under unidirectional loading.
We also performed bidirectional combined loading, where one direction was loaded to 70% of full scale, and the other directions were incrementally loaded to 80% of their full scale. The calibration matrix for this case is:
$$ \mathbf{G} = \begin{bmatrix}
-0.0706 & -0.0740 & 0.2305 & 0.2377 & -0.2431 & -0.2138 & -0.2314 \\
0.1790 & 0.1719 & -0.0105 & -0.0429 & 0.0222 & 0.0315 & 0.0335 \\
-0.3061 & -0.3021 & 0.2775 & 0.3595 & -0.3609 & -0.3686 & -0.3995 \\
0.0274 & 0.0163 & -0.0057 & -0.0001 & -0.0060 & -0.0063 & -0.0002 \\
-0.0085 & -0.0086 & 0.0283 & 0.0158 & 0.0162 & 0.0123 & 0.0130 \\
-0.0014 & -0.0020 & 0.0018 & 0.0030 & -0.0044 & -0.0027 & -0.0049
\end{bmatrix} $$
The linearity errors for bidirectional loading are shown below:
| Direction | F_x | F_y | F_z | M_x | M_y | M_z |
|---|---|---|---|---|---|---|
| Linearity (%) | 0.3269 | 0.1305 | 0.4528 | 0.1418 | 0.1590 | 0.9363 |
| Precision (%) | 0.1575 | 0.2244 | 0.1339 | 0.1197 | 0.1678 | 0.1353 |
| Accuracy (%) | 0.2904 | 0.5152 | 0.1334 | 0.4604 | 0.4583 | 0.2501 |
The maximum linearity error is 0.94%, and the maximum coupling error is 1.35%. The bidirectional loading results show more balanced deviations across directions compared to unidirectional loading.
For specific applications where the six-axis force sensor operates in a limited force range, we conducted partitioned loading calibration with a maximum load of 30 N for forces and 3 N·m for moments. The calibration matrix is:
$$ \mathbf{G}_H = \begin{bmatrix}
-0.0794 & -0.0808 & 0.2231 & 0.2352 & -0.2391 & -0.2189 & -0.2370 \\
0.1799 & 0.1712 & -0.0116 & -0.0411 & 0.0221 & 0.0319 & 0.0346 \\
-0.3661 & -0.3603 & 0.3457 & 0.3262 & -0.3276 & -0.3395 & -0.3670 \\
0.0238 & 0.0196 & -0.0018 & 0.0020 & -0.0040 & -0.0045 & -0.0022 \\
-0.0112 & -0.0112 & 0.0254 & 0.0145 & 0.0149 & 0.0136 & 0.0145 \\
-0.0017 & -0.0011 & 0.0016 & 0.0046 & -0.0030 & -0.0042 & -0.0033
\end{bmatrix} $$
The linearity errors for partitioned loading are lower, with a maximum of 0.65% in the main direction, indicating that task-specific calibration improves measurement accuracy for the six-axis force sensor.
Under signal fault conditions, we developed fault-tolerant calibration models for each possible branch failure. For example, if branch 1 fails, the calibration matrix $\mathbf{G}_1$ is a 6×6 matrix mapping the six healthy branch outputs to the external loads. The linearity error matrix $\xi_{L1}$ for this model is:
$$ \xi_{L1} = \begin{bmatrix}
0.7191 & 0.4297 & 0.6374 & 0.5770 & 0.5747 & 0.6769 \\
0.7414 & 1.0572 & 0.4969 & 1.9849 & 1.6889 & 2.2424 \\
2.7731 & 1.8920 & 1.4260 & 2.1997 & 2.2497 & 2.8882 \\
1.7360 & 2.0304 & 0.7355 & 2.2268 & 2.4210 & 2.3520 \\
0.4600 & 0.6971 & 0.4486 & 0.8477 & 0.8768 & 0.9994 \\
0.0177 & 0.1898 & 0.0214 & 0.0430 & 0.2576 & 0.9432
\end{bmatrix} $$
The maximum linearity error is 2.89%, and the maximum coupling error is 2.89%, which is higher than in the fault-free case but still acceptable for fault-tolerant operation. We analyzed all seven fault-tolerant models for unidirectional, bidirectional, and partitioned loading. The results show that linearity errors increase under fault conditions, but the six-axis force sensor remains functional. The error variation depends on the spatial distribution of the branches, suggesting that optimization in the design phase could improve fault-tolerant performance.
In summary, the pre-stressed parallel six-axis force sensor exhibits inherent fault tolerance, allowing continuous operation even with a branch signal failure. The mathematical models based on screw theory and deformation coordination provide a foundation for understanding its behavior. Calibration tests under various loading conditions confirm the sensor’s reliability, with linearity errors within acceptable ranges. For practical applications, task-specific calibration in smaller force intervals enhances accuracy. This study advances the development of robust six-axis force sensors for critical systems, ensuring high reliability and performance in diverse environments.