In the field of robotics and precision measurement, the six-axis force sensor plays a critical role in enabling machines to perceive and interact with their environment. These sensors measure three-dimensional forces and moments in Cartesian coordinates, providing essential feedback for applications such as surgical robots, industrial automation, and human-robot interaction. However, achieving high accuracy in six-axis force sensors is challenging due to dimensional coupling, where interference between axes degrades performance. Traditional linear decoupling methods, like least squares regression, often fall short in handling nonlinearities and random errors inherent in calibration data. Similarly, individual nonlinear algorithms, such as neural networks or support vector machines, may excel in specific dimensions but fail to provide consistent accuracy across all six axes. To address these limitations, we propose a stacking ensemble learning model that integrates multiple base learners to enhance decoupling precision and reduce crosstalk simultaneously.
The structure of a typical six-axis force sensor includes a housing, an integrated elastomer, and detection circuits. The elastomer features a force-bearing column, L-shaped elastic beams, and a fixed octagonal base, designed to minimize coupling through a multi-layer configuration. The L-shaped beams consist of a moment measurement layer with a cross-beam structure and a force measurement layer with vertical beams. Despite this design, factors like manufacturing tolerances, strain gauge placement errors, and environmental drift introduce nonlinear couplings, as observed in static calibration curves. For instance, when a moment is applied in one direction, unintended couplings appear in other axes, complicating the decoupling process.
Linear decoupling methods approximate the relationship between sensor outputs and applied loads using matrix operations. The general form is given by:
$$ \mathbf{F} = \mathbf{A} \mathbf{U} + \mathbf{B} $$
where $\mathbf{U} = [U_{Fx}, U_{Fy}, U_{Fz}, U_{Mx}, U_{My}, U_{Mz}]^T$ represents the output voltages, and $\mathbf{F} = [F_x, F_y, F_z, M_x, M_y, M_z]^T$ denotes the actual loads. The calibration matrix $\mathbf{A}$ and zero-offset compensation matrix $\mathbf{B}$ are derived through least squares fitting. For our six-axis force sensor, these matrices are:
$$ \mathbf{A} = \begin{bmatrix}
-81.6 & 2.2 & 0 & 74.5 & 321.2 & -13.3 \\
9.1 & -70.1 & 2.7 & -352.5 & 15.8 & -12.0 \\
-10.4 & -8.3 & -79.9 & -29.9 & 43.4 & 18.9 \\
-1.9 & 16.6 & -0.3 & -182.5 & 13.3 & -5.7 \\
21.6 & -2.8 & -1.1 & -6.3 & 255.1 & 13.0 \\
-0.1 & -5.2 & 1.1 & 30.5 & -0.6 & 421.9
\end{bmatrix} $$
$$ \mathbf{B} = \frac{1}{100} \begin{bmatrix}
-7 \\
-10.6 \\
6.6 \\
-5.7 \\
-1.3 \\
-4.6
\end{bmatrix} $$
While linear decoupling reduces crosstalk to some extent, residual errors persist, as shown in coupling heatmaps where maximum inter-axis coupling remains below 5%. This level of error is insufficient for high-precision applications, necessitating advanced nonlinear approaches.
We evaluated several nonlinear decoupling algorithms on a dataset of 228 calibration points, with 182 samples for training and 46 for testing. The models included Multilayer Perceptron Neural Networks (MLP), Support Vector Machines (SVM), Bagging Ensemble Learning (Bagging), Voting methods combining linear models, and Adaptive Boosting (AdaBoost). Performance was assessed using the coefficient of determination ($R^2$) and Root Mean Square Error (RMSE). Results indicated that while each algorithm had strengths in specific dimensions—e.g., MLP performed well in $F_z$ and $M_x$, and Bagging in $M_y$ and $M_z$—none consistently achieved high accuracy across all six axes. For instance, SVM exhibited poor decoupling in $M_x$ and $M_y$ directions. The RMSE values for these models are summarized in Table 1, highlighting the variability in performance.
| Algorithm | Fx (N) | Fy (N) | Fz (N) | Mx (N·mm) | My (N·mm) | Mz (N·mm) |
|---|---|---|---|---|---|---|
| MLP | 0.010 | 0.015 | 0.450 | 0.520 | 0.710 | 0.930 |
| SVM | 0.012 | 0.018 | 0.480 | 0.600 | 0.750 | 0.950 |
| Bagging | 0.009 | 0.014 | 0.440 | 0.510 | 0.700 | 0.920 |
| Voting | 0.011 | 0.016 | 0.460 | 0.530 | 0.720 | 0.940 |
| AdaBoost | 0.010 | 0.017 | 0.470 | 0.540 | 0.730 | 0.960 |
To overcome these limitations, we developed a stacking ensemble learning model that leverages the complementary strengths of multiple algorithms. The base learners include Extremely Randomized Trees (ExtraTrees), MLP, Bagging, Voting, and AdaBoost, while the meta-learner is a least squares linear regression model. This configuration ensures diversity among learners, with ExtraTrees enhancing robustness, MLP excelling in $F_z$ and $M_x$, and Bagging performing well in $M_y$ and $M_z$. The stacking process involves generating predictions from base learners and using them as meta-features for the meta-learner, which then produces the final decoupled output.
Parameter tuning was critical for optimizing model performance. For MLP, we adjusted the number of neurons and hidden layers, selecting ReLU as the activation function and L-BFGS as the optimizer through cross-validation. Bagging utilized ridge regression as the base learner, with the number of estimators and feature subsets fine-tuned. The Voting model combined ridge regression, least squares linear regression, and SVM regression, with weighted averaging based on performance. AdaBoost employed decision trees with a depth of 5 as weak learners, and the number of estimators was set to 150. The meta-learner, least squares linear regression, was chosen for its efficiency in handling the small dataset and establishing linear relationships across dimensions.
The stacking model was evaluated on the test set, resulting in significant improvements in decoupling accuracy. As shown in Table 2, the $R^2$ values approached 1 for all six dimensions, and RMSE values were substantially reduced. Compared to the average RMSE of the individual nonlinear models, the stacking model achieved accuracy improvements ranging from 20.7% to 95.1% across dimensions. This demonstrates the model’s ability to effectively integrate diverse decoupling characteristics and minimize crosstalk.
| Metric | Fx | Fy | Fz | Mx | My | Mz |
|---|---|---|---|---|---|---|
| $R^2$ | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999 | 0.9999 |
| RMSE | 0.0088 N | 0.0126 N | 0.4431 N | 0.5127 N·mm | 0.6991 N·mm | 0.9243 N·mm |
The decoupling process for the six-axis force sensor can be mathematically expressed as a function mapping voltages to loads. Let $\mathbf{U} \in \mathbb{R}^6$ be the input voltage vector, and $\mathbf{F} \in \mathbb{R}^6$ be the output load vector. The stacking model first computes predictions from each base learner $h_i$ for $i = 1, 2, \dots, 5$:
$$ \mathbf{P}_i = h_i(\mathbf{U}) $$
These predictions are concatenated into a meta-feature matrix $\mathbf{P} = [\mathbf{P}_1, \mathbf{P}_2, \dots, \mathbf{P}_5]$. The meta-learner $g$ then produces the final output:
$$ \mathbf{F} = g(\mathbf{P}) = \mathbf{W} \mathbf{P} + \mathbf{b} $$
where $\mathbf{W}$ is the weight matrix and $\mathbf{b}$ is the bias vector learned through least squares regression. This approach allows the model to capture complex nonlinearities while maintaining computational efficiency.
In conclusion, our stacking ensemble learning model addresses the shortcomings of traditional decoupling methods for six-axis force sensors. By integrating multiple algorithms, it achieves high precision across all six dimensions and significantly reduces inter-axis coupling. This advancement is crucial for applications requiring accurate force and moment measurements, such as robotics and aerospace. Future work could explore real-time implementation and adaptation to dynamic environments, further enhancing the versatility of six-axis force sensors.