In the field of robotics and automation, the six-axis force sensor plays a crucial role in providing accurate force and torque measurements in three-dimensional space. This sensor detects forces along the x, y, and z axes (denoted as \(F_x\), \(F_y\), \(F_z\)) and moments around these axes (denoted as \(M_x\), \(M_y\), \(M_z\)). However, a significant challenge in utilizing these sensors is the inter-dimensional coupling effect, where applying a force or torque in one direction results in unintended outputs in other directions. This coupling severely degrades measurement accuracy and necessitates effective decoupling methods to isolate the true signals. Traditional approaches, such as structural decoupling through hardware improvements, often face limitations due to manufacturing complexities and cost constraints. Therefore, software-based decoupling techniques have gained prominence for their economic viability and adaptability. In this work, we propose a novel decoupling method based on an improved random forest algorithm optimized with genetic algorithms (RF-GA) to address the coupling issues in six-axis force sensors. By leveraging machine learning and evolutionary optimization, we aim to enhance the precision and efficiency of force measurement systems.
The inter-dimensional coupling in six-axis force sensors arises from factors like elastic body design imperfections, strain gauge placement errors, and material inhomogeneities. When a force is applied solely in one direction, the sensor’s output may exhibit cross-talk, leading to inaccuracies in other dimensions. For instance, applying \(F_x\) might inadvertently produce signals in \(F_y\), \(F_z\), \(M_x\), \(M_y\), or \(M_z\). This phenomenon is quantified through calibration experiments, where the relationship between input forces/torques and output voltages is characterized. Linear decoupling methods, such as least squares, have been widely used due to their simplicity. The least squares method approximates the decoupling matrix \(W\) that minimizes the error between predicted and actual forces. The matrix equation is expressed as:
$$ U \cdot W = F $$
where \(U\) is the voltage output matrix of size \(m \times 6\) (with \(m\) samples), \(W\) is the weight matrix of size \(6 \times 6\), and \(F\) is the force/torque matrix of size \(m \times 6\). The solution can be derived using the formula:
$$ W = U^T \cdot (U \cdot U^T)^{-1} \cdot F $$
Alternatively, gradient descent can be employed to iteratively minimize the loss function \(Q\):
$$ Q = \sum (U_i W – F_i)^2 $$
where the weight update is given by:
$$ W_{\text{new}} = W_{\text{old}} – \alpha \frac{\partial Q}{\partial W} $$
with \(\alpha\) as the learning rate. However, linear methods often fall short because the relationship between voltage outputs and applied forces is inherently nonlinear. Thus, nonlinear decoupling techniques, such as artificial neural networks (ANNs), have been explored. For example, backpropagation (BP) neural networks model the nonlinear mapping through multiple layers. The network structure includes an input layer (voltage signals), hidden layers with neurons, and an output layer (force predictions). The weights and biases are adjusted during training to reduce the mean squared error. Despite their accuracy, BP networks require extensive training time and are prone to overfitting.
Random forest (RF), an ensemble learning method, offers a robust alternative by combining multiple decision trees. Each tree is trained on bootstrap samples from the dataset, and the final prediction is the average of all tree outputs. This approach reduces variance and handles noise effectively. The RF algorithm involves the following steps: for a dataset of size \(N\), generate \(k\) bootstrap samples; train \(k\) regression trees; and aggregate predictions. The out-of-bag (OOB) error, calculated from samples not included in the bootstrap, provides an unbiased estimate of generalization error. The probability of a sample being omitted in bootstrap is approximately \(1/e \approx 0.368\), ensuring diversity in tree training. However, standard RF may include suboptimal trees that degrade performance. To address this, we integrate genetic algorithms (GA) to selectively retain high-quality trees, forming the RF-GA method.
The RF-GA method encodes each tree in the forest as a binary gene in a chromosome. A gene value of ‘1’ indicates the tree is selected for prediction, while ‘0’ excludes it. The fitness of a chromosome is evaluated based on decoupling accuracy, specifically the maximum Type I and Type II errors. Type I error refers to the relative error in the direction of the applied force, while Type II error represents the cross-talk error in other directions. The fitness function is defined as:
$$ \text{Fitness} = \frac{1}{(\sum \text{Max Type I}_i + \sum \text{Max Type II}_i)^3} $$
where the denominator emphasizes lower errors. GA operations—selection, crossover, and mutation—are applied over generations to evolve chromosomes toward higher fitness. Selection uses roulette wheel selection based on fitness, crossover exchanges gene segments between parents, and mutation flips bits randomly. This process identifies an optimal subset of trees that minimizes decoupling errors.
To validate the RF-GA method, we conducted experiments using a six-axis force sensor based on strain gauge detection. The sensor was mounted on a calibration platform, and forces/torques were applied using standardized weights. The output voltages were amplified, filtered, and acquired via a data acquisition card. The dataset comprised 354 samples, split into 300 for training and 54 for testing. Data normalization was performed to scale inputs to the range [0, 1] using:
$$ \text{new_value} = \frac{\text{old_value} – \min}{\max – \min} $$
where \(\min\) and \(\max\) are the feature-wise minimum and maximum values. This ensures numerical stability during model training.
We compared RF-GA against traditional methods: least squares, BP neural network, and standard random forest. For the BP network, we optimized the learning rate and hidden layer size. As shown in Table 1, the optimal configuration had 13 neurons and 60,000 training iterations, balancing accuracy and time. For random forest, we set the number of trees to 200 and maximum depth to 17 to avoid overfitting. In RF-GA, we used a population size of 30, crossover probability of 0.3, mutation probability of 0.1, and 57 generations based on convergence analysis.
| Method | Max Type I Error (%) | Max Type II Error (%) | Decoupling Time (s) |
|---|---|---|---|
| Least Squares | 80.48 | 30.27 | N/A |
| BP Neural Network | 1.21 | 0.83 | 57.24 |
| Random Forest | 1.88 | 0.51 | 0.42 |
| RF-GA | 1.05 | 1.05 | 13.23 |
The decoupling performance was evaluated using error rates defined as:
$$ \text{Error Rate} = \left| \frac{F_{\text{actual}} – F_{\text{measured}}}{F_{\text{full-scale}}} \right| \times 100\% $$
where \(F_{\text{full-scale}}\) is the maximum force in the sensor’s range. Type I errors for each direction and Type II errors for cross-coupling were computed. Results indicate that RF-GA achieves lower Type I errors compared to other methods, with most Type II errors near zero, signifying effective coupling suppression. For instance, when \(F_x\) is applied, the Type II errors in other directions are minimized. The decoupling time of RF-GA is intermediate between BP network and standard RF, making it suitable for real-time applications.
Further analysis of the six-axis force sensor data involved visualizing the relationship between applied forces and errors. Under \(F_z\) loading, coupling was particularly evident, but RF-GA reduced this effect. The algorithm’s robustness stems from GA’s ability to prune low-performing trees, enhancing the ensemble’s predictive power. The fitness evolution over generations showed consistent improvement, confirming the efficacy of the optimization. Moreover, RF-GA’s parallelism allows efficient computation, though the GA component introduces additional overhead.
In conclusion, the RF-GA method demonstrates superior decoupling performance for six-axis force sensors by combining the strengths of random forest and genetic algorithms. It addresses the limitations of linear methods and the time-consuming nature of neural networks. Future work could explore adaptive GA parameters or integration with deep learning for further enhancements. This approach contributes to the advancement of high-precision force sensing in robotics, industrial automation, and beyond.
The mathematical foundation of the RF-GA method can be extended to other multi-sensor systems. For example, the weight matrix optimization in least squares can be reformulated as a constrained problem, but nonlinear methods like RF-GA inherently capture complex interactions. The error minimization principle is universal, and the fitness function can be tailored to specific applications. Overall, the six-axis force sensor benefits greatly from machine learning-driven decoupling, ensuring accurate and reliable measurements in dynamic environments.
To elaborate on the random forest mechanism, each decision tree is built using a random subset of features and samples. The prediction for a sample \(x\) is given by:
$$ \hat{y} = \frac{1}{k} \sum_{i=1}^{k} T_i(x) $$
where \(T_i(x)\) is the prediction of the \(i\)-th tree. In RF-GA, the subset of trees is optimized via GA, so the prediction becomes:
$$ \hat{y} = \frac{1}{\sum_{j=1}^{k} g_j} \sum_{i=1}^{k} g_i T_i(x) $$
where \(g_i\) is the binary gene indicating tree selection. This selective averaging reduces noise and improves accuracy.
In calibration, the six-axis force sensor’s output voltages for each force/torque component were recorded. The coupling effects were evident, as seen in the voltage responses. For example, applying \(M_z\) resulted in outputs in \(F_x\) and \(M_y\) directions. Through decoupling, these spurious signals are suppressed. The RF-GA method achieves this by learning the inverse mapping from voltages to forces, effectively inverting the sensor’s transfer function.
The advantages of RF-GA include its resistance to overfitting, handling of high-dimensional data, and scalability. However, it requires careful parameter tuning, such as the number of trees and GA generations. In practice, for a six-axis force sensor with varying load conditions, RF-GA adapts well due to its non-parametric nature. This makes it a versatile tool for improving the performance of force measurement systems in applications like robotic manipulation, aerospace testing, and biomedical devices.
In summary, the integration of genetic algorithms with random forest presents a significant leap in decoupling technology for six-axis force sensors. By focusing on tree quality rather than quantity, RF-GA delivers high precision with reasonable computational cost. As sensors evolve towards miniaturization and higher integration, such software solutions will become increasingly vital for achieving accurate multidimensional sensing.