In the field of robotics, the six-axis force sensor plays a critical role in enabling intelligent control by measuring forces and torques in three-dimensional space. However, dimensional coupling significantly affects the measurement accuracy of the six-axis force sensor, leading to errors in applications such as deep-sea manipulators. Traditional decoupling methods, including structural and software approaches, often struggle with nonlinearities. Structural decoupling involves complex designs and high costs, while linear software methods like least squares (LS) fail to capture nonlinear relationships. Nonlinear methods, such as BP neural networks (BPNN) and extreme learning machines (ELM), offer improvements but face issues like local optima and instability due to random parameter initialization. To address these challenges, this paper proposes a decoupling algorithm based on an improved fireworks algorithm optimized extreme learning machine (IFWA-ELM). This approach enhances the stability and accuracy of the six-axis force sensor by optimizing ELM’s initial weights and thresholds using an improved fireworks algorithm (IFWA), which incorporates adaptive explosion radii, mutation operators, and selection strategies. Experimental results on a six-axis force sensor for a 4,500 m deep-sea manipulator demonstrate that IFWA-ELM effectively reduces coupling errors, with Class I errors below 0.27% and Class II errors below 0.13%, outperforming LS, BPNN, and ELM methods.
The six-axis force sensor is essential for precise force and torque measurement in robotics, but its performance is hampered by inter-dimensional coupling. This coupling arises from structural integration and manufacturing imperfections, such as strain gauge misalignment. In this study, we focus on a resistor-based six-axis force sensor applied in deep-sea environments, where accuracy is paramount. The proposed IFWA-ELM algorithm leverages machine learning to handle nonlinear decoupling. First, we describe the fundamentals of ELM and FWA, then detail the improvements in IFWA, including adaptive radius adjustment, Gaussian mutation, and elite selection. Subsequently, we integrate IFWA with ELM to optimize network parameters, ensuring robust decoupling. Calibration experiments provide data for training and testing, and decoupling performance is evaluated using Class I and II errors. The results confirm that IFWA-ELM enhances the measurement precision of the six-axis force sensor, making it suitable for high-stakes applications.
Extreme learning machine (ELM) is a single-hidden-layer feedforward neural network known for its fast training and generalization capabilities. For a dataset with N samples (X_i, Y_i), where X_i ∈ R^n is the input vector and Y_i ∈ R^m is the output vector, the ELM model can be expressed as:
$$ \sum_{f=1}^{l} \beta_f g(W_f X_i + b_f) = T_i $$
Here, g(x) is the activation function, W_f is the input weight vector, b_f is the bias of the hidden layer neuron, β_f is the output weight vector, and T_i is the predicted output. The goal is to minimize the output error, leading to the solution for output weights β via the Moore-Penrose generalized inverse of the hidden layer output matrix H:
$$ \beta = H^+ Y $$
Despite its advantages, ELM’s random initialization of weights and biases can cause instability, and the number of hidden neurons significantly impacts performance. To mitigate this, we employ an improved fireworks algorithm (IFWA) for optimization. The fireworks algorithm is a swarm intelligence method that simulates fireworks explosions to explore the search space. In IFWA, we enhance the explosion radius, mutation operator, and selection strategy. The explosion radius for the best firework is adaptively adjusted based on the distance to a selected individual, ensuring better local search:
$$ R_{best} = \lambda \cdot d(\hat{x}, x^*) $$
where λ is a coefficient (set to 1.3), d is the distance metric, and \hat{x} is the selected individual satisfying fitness conditions. The Gaussian mutation is modified to direct mutations along the line between the current position and the best individual’s position:
$$ x_{ik} = x_{ik} + (x_{bk} – x_{ik}) \cdot e $$
where e is a Gaussian random variable. Additionally, elite selection replaces the distance-based strategy, using fitness-proportional probabilities to improve efficiency:
$$ p(x_i) = \frac{1 / f(x_i)}{\sum_{j=1}^{K} 1 / f(x_j)} $$
These improvements enhance IFWA’s global search ability and convergence speed. By integrating IFWA with ELM, we optimize the initial weights and thresholds, reducing randomness and improving decoupling accuracy for the six-axis force sensor.
Calibration experiments are crucial for establishing the relationship between input forces/torques and output voltages in the six-axis force sensor. We conducted static calibration using a standard platform with a capacity of F_x/F_y: 1,000 N, F_z: 2,000 N, and M_x/M_y/M_z: 50 Nm, and an accuracy of 0.1% F.S. The sensor was mounted on the platform, and loads were applied incrementally in each direction, from zero to full scale and back, with three repetitions to gather data. The output voltages were recorded at each loading point, and zero baseline calibration was performed initially. The results, as shown in the figure above, indicate significant coupling between dimensions, such as between M_x and F_y, due to structural and error-related factors. This data is split into 80% for training and 20% for testing, with normalization to the range [-1, 1] using the formula:
$$ \text{new_value} = 2 \times \frac{\text{old_value} – \min}{\max – \min} – 1 $$
This preprocessing ensures uniform scaling and improves network training efficiency for the six-axis force sensor decoupling.
For the decoupling experiments, we compare IFWA-ELM with LS, BPNN, and ELM. The IFWA-ELM network has an input layer with 6 nodes (voltage outputs), a hidden layer with 75 neurons (determined through analysis), and an output layer with 6 nodes (forces/torques). The activation function is sigmoid. IFWA parameters include a population size of 5, mutation sparks of 5, and maximum iterations of 500. The fitness function for optimization is the mean absolute error:
$$ \text{fitness} = \frac{1}{n} \sum_{i=1}^{n} |y_i – \hat{y}_i| $$
where y_i is the actual value and \hat{y}_i is the predicted value. To evaluate performance, we use Class I error (nonlinear error) and Class II error (coupling error), defined as:
$$ \delta_i = \frac{|F_i(P) – F_i(L)|}{F_i(FS)} $$
$$ \delta_{ij} = \frac{|F_{ij}|}{F_i(FS)} $$
where F_i(P) is the measured force/torque, F_i(L) is the applied force/torque, and F_i(FS) is the full-scale value. The hidden neuron analysis for ELM and IFWA-ELM shows that IFWA-ELM achieves lower mean squared error (MSE) with fewer neurons, as summarized in the table below. The MSE is calculated as:
$$ \text{MSE} = \frac{1}{n} \sum_{i=1}^{n} (y_i – \hat{y}_i)^2 $$
| Algorithm | Hidden Neurons | MSE |
|---|---|---|
| ELM | 30 | 99.5 |
| ELM | 80 | 30.2 |
| IFWA-ELM | 75 | 10.1 |
| PSO-ELM | 75 | 25.3 |
| FWA-ELM | 75 | 18.7 |
The decoupling results demonstrate that IFWA-ELM outperforms other methods. For instance, LS shows high errors, with maximum Class I and II errors around 2.69% and 2.655%, respectively. BPNN reduces these errors to 0.53% and 0.33%, while ELM achieves 0.312% and 0.17%. IFWA-ELM further improves this, with Class I errors below 0.27% and Class II errors below 0.13%. The following table provides a detailed comparison of average errors across dimensions for the six-axis force sensor.
| Algorithm | Direction | F_x | F_y | F_z | M_x | M_y | M_z |
|---|---|---|---|---|---|---|---|
| LS | F_x | 0.196 | 0.293 | 0.166 | 0.191 | 0.314 | 0.552 |
| F_y | 0.070 | 0.301 | 0.173 | 0.287 | 0.272 | 0.411 | |
| F_z | 2.655 | 0.527 | 2.690 | 0.105 | 0.942 | 0.355 | |
| M_x | 0.075 | 0.116 | 0.119 | 0.668 | 0.329 | 0.832 | |
| M_y | 0.711 | 0.508 | 0.250 | 0.143 | 0.586 | 0.783 | |
| M_z | 0.689 | 0.415 | 0.431 | 0.135 | 0.153 | 0.216 | |
| BPNN | F_x | 0.246 | 0.278 | 0.140 | 0.178 | 0.168 | 0.171 |
| F_y | 0.185 | 0.234 | 0.161 | 0.187 | 0.151 | 0.268 | |
| F_z | 0.138 | 0.182 | 0.291 | 0.330 | 0.282 | 0.307 | |
| M_x | 0.236 | 0.230 | 0.215 | 0.530 | 0.196 | 0.241 | |
| M_y | 0.204 | 0.190 | 0.249 | 0.153 | 0.499 | 0.147 | |
| M_z | 0.168 | 0.177 | 0.127 | 0.157 | 0.177 | 0.227 | |
| ELM | F_x | 0.126 | 0.132 | 0.059 | 0.033 | 0.055 | 0.086 |
| F_y | 0.109 | 0.102 | 0.057 | 0.035 | 0.034 | 0.132 | |
| F_z | 0.061 | 0.063 | 0.276 | 0.040 | 0.072 | 0.049 | |
| M_x | 0.085 | 0.102 | 0.123 | 0.312 | 0.083 | 0.079 | |
| M_y | 0.133 | 0.167 | 0.077 | 0.069 | 0.238 | 0.072 | |
| M_z | 0.132 | 0.147 | 0.095 | 0.062 | 0.073 | 0.164 | |
| IFWA-ELM | F_x | 0.108 | 0.095 | 0.029 | 0.012 | 0.011 | 0.027 |
| F_y | 0.073 | 0.101 | 0.036 | 0.015 | 0.016 | 0.028 | |
| F_z | 0.053 | 0.044 | 0.264 | 0.016 | 0.014 | 0.026 | |
| M_x | 0.058 | 0.116 | 0.112 | 0.205 | 0.041 | 0.046 | |
| M_y | 0.126 | 0.098 | 0.055 | 0.034 | 0.127 | 0.034 | |
| M_z | 0.099 | 0.091 | 0.045 | 0.029 | 0.032 | 0.106 |
The boxplot analysis of decoupling errors further confirms the superiority of IFWA-ELM. The errors for all six directions are concentrated below 0.1%, with fewer outliers compared to other algorithms. This consistency highlights IFWA-ELM’s robustness in handling the nonlinearities of the six-axis force sensor. The integration of IFWA ensures that ELM parameters are optimally tuned, avoiding local minima and enhancing generalization. In conclusion, the IFWA-ELM algorithm provides an effective solution for decoupling six-axis force sensors, significantly improving measurement accuracy in demanding environments like deep-sea robotics. Future work could explore real-time adaptation and integration with other sensor types for broader applications.
The six-axis force sensor is a pivotal component in modern robotics, and its decoupling is essential for high-precision tasks. Our approach combines the speed of ELM with the optimization power of IFWA, resulting in a reliable method for reducing coupling errors. The experimental validation on a deep-sea manipulator sensor underscores the practical benefits, with errors maintained within tight bounds. This advancement not only enhances the performance of six-axis force sensors but also opens avenues for their use in more complex systems. As robotics continue to evolve, such decoupling techniques will be crucial for achieving autonomous and accurate operations.