In the field of robotics and precision engineering, the demand for high-performance six-axis force sensors has grown significantly due to their ability to measure forces and moments in all three spatial dimensions. These sensors are critical for applications such as robotic manipulation, aerospace testing, and industrial automation, where accurate multi-directional force feedback is essential. Among various structural designs, the Stewart platform-based six-axis force sensor stands out for its high stiffness, compact symmetry, and favorable decoupling characteristics. However, achieving both high stiffness and low inter-dimensional coupling remains a challenge, particularly when traditional hinge constraints are replaced with fixed constraints to enhance rigidity. In this article, I explore the limitations of conventional decoupling methods and propose an optimized approach that combines single-dimensional and multi-dimensional loading during calibration. By leveraging both calibration matrix solutions and BP neural networks, I demonstrate how this method effectively reduces errors in six-axis force sensor measurements, addressing issues like dimensional coupling and multi-load inaccuracies simultaneously.
The Stewart platform structure, commonly used in six-axis force sensors, consists of two parallel plates connected by six elastic branches. This configuration allows the sensor to measure spatial forces and moments by detecting deformations in the branches, which act as two-force members. The relationship between the applied forces and the branch reactions can be described using screw theory, leading to the equilibrium equation: $$ \sum_{i=1}^{6} f_i \cdot \xi_i = F + M $$ where \( f_i \) represents the axial force in the i-th branch, \( \xi_i \) is the unit line vector of the branch axis in the reference coordinate system, and \( F \) and \( M \) denote the applied force and moment vectors, respectively. This equation can be simplified into a matrix form: $$ F_W = G f $$ Here, \( F_W = [F, M]^T \) is the six-dimensional force/moment vector, \( f = [f_1, f_2, f_3, f_4, f_5, f_6]^T \) is the vector of branch forces, and \( G \) is the forward mapping matrix that relates the branch forces to the applied loads. The stiffness of the six-axis force sensor is a crucial parameter, and to meet high stiffness requirements, fixed constraints are often used instead of hinge constraints. This modification, however, introduces nonlinearities in the input-output relationship, complicating the decoupling process. The structural symmetry and minimal constraints of the Stewart platform make it ideal for maintaining geometric stability, but manufacturing imperfections can lead to significant inter-dimensional coupling, where applying a force or moment in one direction produces unwanted outputs in others.
Decoupling algorithms are essential for translating the raw sensor outputs into accurate force and moment readings. Traditional methods assume a linear relationship between inputs and outputs, which is often invalid in practical six-axis force sensors due to nonlinearities from fixed constraints and structural asymmetries. One common approach is the calibration matrix method, which involves solving for the inverse of the forward mapping matrix. Given the calibration data, the calibration matrix \( C \) is computed as: $$ C = F_S U^- $$ where \( F_S \) is the matrix of applied calibration forces and moments, \( U \) is the matrix of corresponding output voltages, and \( U^- \) is the pseudo-inverse of \( U \). This method relies on the linearity assumption, but in real-world six-axis force sensors, nonlinear effects can lead to substantial errors, especially in inter-dimensional coupling. For instance, when a moment is applied in the My or Mz direction, it may cause significant cross-talk in other axes, resulting in errors exceeding 10% in some cases. This highlights the limitation of the calibration matrix approach for high-stiffness six-axis force sensors.
To address nonlinearities, artificial neural networks, particularly BP neural networks, have been employed for decoupling in six-axis force sensors. The BP network is a multi-layer feedforward network that can model complex nonlinear mappings. In the context of a six-axis force sensor, the input layer receives the voltage signals from the sensor branches, while the output layer produces the estimated forces and moments. The network is trained using calibration data, with the goal of minimizing the mean squared error between the predicted and actual loads. The activation functions in the hidden layers, such as the Tansig function, and the linear function in the output layer, enable the network to capture nonlinear relationships. The training process, often using the Levenberg-Marquardt algorithm, adjusts the weights and biases to improve accuracy. However, when trained solely on single-dimensional loading data, the BP network may fail to generalize to multi-dimensional loading scenarios, leading to errors as high as 200% in some validations. This indicates that traditional BP neural network training, while effective for reducing inter-dimensional coupling, does not adequately handle the complexities of multi-axial loads in six-axis force sensors.
To overcome these challenges, I propose an optimized decoupling strategy that involves calibrating the six-axis force sensor with both single-dimensional and multi-dimensional loading data. This approach enriches the training dataset, allowing the decoupling algorithm to learn both isolated and combined load effects. The calibration process includes applying forces and moments in individual directions as well as in multi-axial combinations, such as four-dimensional, five-dimensional, and full six-dimensional loads. This comprehensive data collection ensures that the decoupling model accounts for the interactions between different axes. When applying the calibration matrix method to this combined dataset, the errors remain high due to persistent nonlinearities. In contrast, the BP neural network, when trained on the mixed dataset, shows significant improvement. For example, with 40 neurons in the hidden layer, the maximum error across all axes is reduced to 2.27%, and the validation on unseen multi-dimensional data yields errors below 2%, meeting the typical accuracy requirements for six-axis force sensors. This demonstrates that the combined loading approach enhances the robustness of the BP network, enabling it to handle both dimensional coupling and multi-axial loading effectively.
The effectiveness of the optimized decoupling method is validated through experimental data from a sensor prototype. The prototype has a stiffness of at least \( 1 \times 10^8 \, \text{N/m} \) and a measurement range of -1500 N to 1500 N for forces and -1500 N·m to 1500 N·m for moments. The calibration data includes multiple trials of single-dimensional and multi-dimensional loads. The error for each axis is calculated as: $$ E = \frac{\max |F – f|}{\max |F|} \times 100\% $$ where \( F \) is the applied force or moment, and \( f \) is the decoupled output. The following table compares the maximum errors for different decoupling methods, highlighting the superiority of the optimized BP neural network approach for six-axis force sensors.
| Method | Fx Error (%) | Fy Error (%) | Fz Error (%) | Mx Error (%) | My Error (%) | Mz Error (%) |
|---|---|---|---|---|---|---|
| Initial Calibration Matrix (Single-Dimensional) | 4.55 | 11.49 | 2.65 | 0.45 | 0.36 | 1.54 |
| Initial BP Neural Network (Single-Dimensional) | 0.63 | 0.48 | 1.13 | 0.22 | 0.21 | 0.32 |
| Initial Calibration Matrix (Multi-Dimensional Validation) | 8.02 | 14.00 | 15.57 | 1.37 | 2.39 | 1.57 |
| Initial BP Neural Network (Multi-Dimensional Validation) | 69.74 | 230.40 | 240.81 | 64.49 | 133.58 | 124.65 |
| Optimized Calibration Matrix (Combined Loading) | 6.12 | 13.58 | 14.24 | 1.38 | 2.13 | 1.63 |
| Optimized BP Neural Network (Combined Loading) | 2.27 | 1.85 | 1.83 | 1.14 | 0.62 | 1.73 |
As shown in the table, the optimized BP neural network method consistently achieves lower errors across all axes, with the maximum error reduced to 2.27% for Fx. This is a significant improvement over the initial methods, which struggled with inter-dimensional coupling and multi-dimensional loading. The mathematical formulation of the BP network for the six-axis force sensor can be expressed as follows: Let \( U = [U_1, U_2, U_3, U_4, U_5, U_6]^T \) be the input voltage vector, and \( F_S = [F_x, F_y, F_z, M_x, M_y, M_z]^T \) be the target output vector. The network output \( \hat{F_S} \) is given by: $$ \hat{F_S} = W_2 \cdot \sigma(W_1 \cdot U + b_1) + b_2 $$ where \( W_1 \) and \( W_2 \) are weight matrices, \( b_1 \) and \( b_2 \) are bias vectors, and \( \sigma \) is the activation function (e.g., Tansig). The training minimizes the error function: $$ E = \frac{1}{N} \sum_{i=1}^{N} ||F_S^{(i)} – \hat{F_S}^{(i)}||^2 $$ where \( N \) is the number of training samples. By incorporating multi-dimensional loading data, the network learns to decouple complex load interactions, making it highly effective for real-world applications of six-axis force sensors.
In conclusion, the optimization of decoupling algorithms for six-axis force sensors is crucial for achieving high accuracy in demanding environments. The traditional calibration matrix and BP neural network methods, when applied only to single-dimensional data, fall short in handling inter-dimensional coupling and multi-axial loads. By integrating both single-dimensional and multi-dimensional loading during calibration, and employing a BP neural network with an adequate number of hidden neurons, the decoupling performance of six-axis force sensors can be significantly enhanced. This approach reduces the maximum error to below 3%, ensuring that the sensor meets stringent accuracy requirements. Future work could explore advanced neural network architectures or hybrid models to further improve the robustness and efficiency of six-axis force sensors in dynamic applications.