In modern agricultural practices, the demand for high-precision and robust sensing technologies has grown significantly, particularly for monitoring complex loads during tillage operations. Traditional force sensors often fall short in handling the multi-directional forces and moments encountered in field conditions, leading to inaccuracies and limited applicability. To address these challenges, we developed a novel six-axis force sensor based on a radial beam structure, capable of simultaneously measuring forces and moments in three orthogonal directions. This work focuses on the structural optimization of the sensor through simulation methods and the implementation of an advanced machine learning-based decoupling algorithm to minimize cross-axis coupling errors. The integration of these approaches ensures high measurement accuracy and reliability, paving the way for enhanced agricultural machinery optimization.
The design of the six-axis force sensor builds upon the classical cross-beam structure, incorporating elements such as a loading platform, floating beams, and strain beams arranged symmetrically to distribute loads effectively. Key components include four identical strain beams and floating beams, which work in tandem to reduce mechanical coupling. The sensor’s compact design allows for a high load capacity of up to 10,000 N, making it suitable for heavy-duty agricultural applications. Structural optimization was conducted using finite element analysis (FEA) to determine the optimal dimensions of the strain beams, balancing strain sensitivity and mechanical strength. The final dimensions were set to a length of 9 mm, width of 10 mm, and height of 6 mm, as summarized in Table 1.
| Parameter | Value (mm) |
|---|---|
| Length | 9 |
| Width | 10 |
| Height | 6 |
To further enhance the sensor’s performance, we analyzed its strain response under various loading conditions using simulation software. The strain distribution across the beams was monitored to identify optimal locations for strain gauge placement. For instance, under a Z-direction moment load of 1,000 N·m, the maximum strain was observed on the side surfaces of the strain beams. The strain capacity can be modeled using the following equation for stress-strain relationship:
$$ \sigma = E \cdot \epsilon $$
where \( \sigma \) is the stress, \( E \) is the Young’s modulus, and \( \epsilon \) is the strain. The results guided the placement of strain gauges: for X-direction force measurement, gauges are positioned on the side of strain beam 4; for Y-direction force, on strain beam 5; and for Z-direction force, on the front of strain beam 4. Similarly, moment measurements involve specific beam surfaces, as detailed in Table 2.
| Measurement Type | Strain Gauge Location |
|---|---|
| X-direction force (F_X) | Side of strain beam 4 |
| Y-direction force (F_Y) | Side of strain beam 5 |
| Z-direction force (F_Z) | Front of strain beam 4 |
| X-direction moment (M_X) | Front of strain beam 6 |
| Y-direction moment (M_Y) | Front of strain beam 5 |
| Z-direction moment (M_Z) | Side of strain beam 7 |
The six-axis force sensor was subjected to static calibration tests to collect data for decoupling analysis. Loads were applied incrementally from 0 to 10,000 N for forces and 0 to 1,000 N·m for moments, with output signals recorded at each step. The calibration data revealed significant cross-axis coupling, particularly between force and moment channels, necessitating advanced decoupling techniques. Traditional methods like multiple linear regression (MLR) proved inadequate due to nonlinear coupling effects. Therefore, we employed an improved Extreme Gradient Boosting (XGBoost) machine learning model, enhanced with Bayesian optimization for hyperparameter tuning, to achieve superior decoupling accuracy.
The improved XGBoost model leverages ensemble learning by combining multiple weak predictors into a strong learner. The objective function for XGBoost is given by:
$$ \mathcal{L}(\phi) = \sum_{i=1}^{n} l(y_i, \hat{y}_i) + \sum_{k=1}^{K} \Omega(f_k) $$
where \( \mathcal{L} \) is the loss function, \( y_i \) is the true value, \( \hat{y}_i \) is the predicted value, \( l \) is a differentiable convex loss function, and \( \Omega \) is the regularization term that penalizes model complexity to prevent overfitting. The Bayesian optimization process iteratively selects hyperparameters to minimize the loss, ensuring robust performance. We compared the improved XGBoost model with MLR and Random Forest (RF) models using metrics such as the coefficient of determination (\( R^2 \)) and mean absolute error (MAE) on test datasets. The results, summarized in Table 3, demonstrate the superiority of the improved XGBoost approach.
| Loading Direction | Model | \( R^2_P \) | MAE_P |
|---|---|---|---|
| F_X | Improved XGBoost | 0.9804 | 0.026 N |
| F_Y | Improved XGBoost | 0.9418 | 0.031 N |
| F_Z | Improved XGBoost | 0.9434 | 0.039 N |
| M_X | Improved XGBoost | 0.9868 | 0.024 N·m |
| M_Y | Improved XGBoost | 0.9969 | 0.018 N·m |
| M_Z | Improved XGBoost | 0.9822 | 0.029 N·m |
| F_X | Random Forest | 0.9767 | 0.026 N |
| F_Y | Random Forest | 0.8780 | 0.046 N |
| F_Z | Random Forest | 0.9090 | 0.037 N |
| M_X | Random Forest | 0.9896 | 0.019 N·m |
| M_Y | Random Forest | 0.9989 | 0.009 N·m |
| M_Z | Random Forest | 0.9834 | 0.020 N·m |
| F_X | MLR | 0.7999 | 0.111 N |
| F_Y | MLR | 0.7784 | 0.082 N |
| F_Z | MLR | 0.7650 | 0.073 N |
| M_X | MLR | 0.7815 | 0.119 N·m |
| M_Y | MLR | 0.7909 | 0.162 N·m |
| M_Z | MLR | 0.8273 | 0.139 N·m |
The improved XGBoost model consistently outperformed other methods, with \( R^2_P \) values exceeding 0.94 for all force and moment directions. For example, in the Y-direction moment (M_Y), the \( R^2_P \) reached 0.9969, indicating nearly perfect prediction accuracy. The MAE values were also significantly lower, highlighting the model’s effectiveness in reducing coupling errors. The decoupling process involves transforming the raw output signals from the six-axis force sensor into accurate force and moment components. The general decoupling equation can be expressed as:
$$ \mathbf{F} = \mathbf{C} \cdot \mathbf{V} $$
where \( \mathbf{F} \) is the vector of forces and moments, \( \mathbf{C} \) is the decoupling matrix, and \( \mathbf{V} \) is the vector of voltage outputs from the strain gauges. In machine learning approaches, this matrix is learned through training, allowing for nonlinear mappings. The improved XGBoost model’s ability to handle complex interactions between channels makes it ideal for this six-axis force sensor application.
In addition to decoupling, we evaluated the sensor’s linearity under individual load conditions. The linearity error \( \gamma_L \) is calculated as:
$$ \gamma_L = \pm \frac{\Delta L_{\text{max}}}{Y_{\text{FS}}} \times 100\% $$
where \( \Delta L_{\text{max}} \) is the maximum nonlinear absolute error and \( Y_{\text{FS}} \) is the full-scale output. The main channels exhibited linearity errors below 0.38%, confirming the sensor’s reliability. However, cross-channel coupling introduced nonlinearities, such as a 66.56% linearity error between F_X and M_X, underscoring the need for advanced decoupling.
Overall, the integration of structural optimization and machine learning-based decoupling has resulted in a high-performance six-axis force sensor tailored for agricultural environments. The sensor’s design ensures minimal interference with existing machinery structures, preserving load transmission characteristics during operation. Future work will focus on field testing and further refinement of the decoupling algorithms to adapt to dynamic loading conditions. This research contributes to the advancement of smart agriculture by providing a reliable tool for real-time load monitoring, ultimately aiding in the optimization of agricultural machinery and improving operational efficiency.
The development of this six-axis force sensor highlights the importance of multidisciplinary approaches in solving engineering challenges. By combining mechanical design principles with computational intelligence, we have created a sensor that not only meets the demanding requirements of agricultural applications but also sets a benchmark for future innovations in force sensing technology. The continued evolution of such sensors will play a crucial role in the automation and precision of farming practices, supporting sustainable agricultural development.