In modern industrial robotics, the six-axis force sensor plays a critical role in enabling high-precision tasks such as assembly, grinding, and deburring. However, the inherent coupling effects between different dimensions of the sensor often lead to inaccuracies, necessitating effective decoupling methods. Traditional approaches, including nonlinear static decoupling and basic neural network models, suffer from issues like oscillations, slow convergence, and local minima entrapment due to arbitrary initial parameter selection. To address these challenges, this paper proposes a novel decoupling algorithm based on the ant colony optimization (ACO) and backpropagation (BP) neural network, specifically designed for the six-axis force sensor. By leveraging the global optimization capabilities of ACO to determine optimal initial weights and thresholds for the BP network, the proposed method significantly enhances convergence speed and mitigates common training drawbacks. Through extensive simulations and experimental validation, we demonstrate the superiority of this approach in improving decoupling accuracy and efficiency for the six-axis force sensor.
The six-axis force sensor is essential for capturing multidimensional force and torque data in real-time applications, but its performance is often compromised by cross-coupling among axes. This coupling arises from the sensor’s mechanical structure and manufacturing imperfections, leading to erroneous readings that affect robotic reliability. Previous studies have explored various decoupling techniques, such as polynomial-based static decoupling and neural network models. For instance, early work on nonlinear static decoupling highlighted the impact of polynomial order on effectiveness, while later research employed BP neural networks to model the relationship between sensor inputs and outputs. However, the BP network’s sensitivity to initial parameters often results in suboptimal performance. In contrast, our method integrates ACO to optimize these parameters, ensuring faster and more stable convergence. This integration is particularly beneficial for the six-axis force sensor, where high-dimensional data exacerbates training complexities.
The BP neural network operates as a multilayer feedforward system that minimizes error through forward and backward propagation. Given an input vector $X = [x_1, x_2, \ldots, x_m]$ for the six-axis force sensor outputs, the hidden layer output $h_k$ is computed as:
$$ h_k = \phi_1 \left( \sum_{i=1}^m x_i w_{ik} \right), \quad i=1,2,\ldots,m; \quad k=1,2,\ldots,p $$
where $w_{ik}$ represents the weight between the input and hidden layers, $p$ is the number of hidden neurons, and $\phi_1(\cdot)$ is the activation function. The output layer then produces the predicted force and torque values $y_t$:
$$ y_t = \phi_2 \left( \sum_{k=1}^p h_k v_{kt} \right), \quad t=1,2,\ldots,n $$
Here, $v_{kt}$ denotes the weight between the hidden and output layers, $n$ is the number of output neurons (e.g., six for the six-axis force sensor), and $\phi_2(\cdot)$ is the output activation function. The overall network output for an input $X$ is:
$$ y_t = \phi_2 \left( \sum_{k=1}^p \phi_1 \left( \sum_{i=1}^m x_i w_{ik} \right) v_{kt} \right) $$
The error $e_t$ between the network output $y_t$ and the desired output $d_t$ is calculated as $e_t = d_t – y_t$. This error is propagated backward to adjust the weights $w_{ik}$ and $v_{kt}$ iteratively until the error falls below a threshold. However, the random initialization of these weights in traditional BP networks often leads to slow convergence and poor performance for the six-axis force sensor decoupling.
To overcome these limitations, we incorporate the ant colony algorithm, which mimics the foraging behavior of ants to find optimal paths. In this context, the weights and thresholds of the BP network are treated as path segments that ants explore. For a BP network with $m$ input, $p$ hidden, and $n$ output neurons, the total number of parameters $H$ is given by $H = (m + n + 1) \cdot p + n$. Each parameter corresponds to a vertex in a graph, and ants traverse these vertices to form a solution path. The probability of an ant selecting a particular parameter value $P_j(IP_i)$ from set $IP_i$ is defined as:
$$ P_c(\tau_j(IP_i)) = \frac{\tau_j(IP_i)}{\sum_{g=1}^N \tau_g(IP_i)} $$
where $\tau_j(IP_i)$ is the pheromone level associated with parameter $P_j(IP_i)$, and $N$ is the total number of elements in $IP_i$. The pheromone update rule combines local and global information:
$$ \tau(IP_i)(t + s) = \rho \tau_j(IP_i)(t) + \Delta \tau_j(IP_i) $$
$$ \Delta \tau_j(IP_i) = \sum_{k=1}^h \Delta \tau_j^k(IP_i) $$
Here, $\rho$ (0 ≤ ρ < 1) is the pheromone persistence factor, $h$ is the number of ants, and $\Delta \tau_j^k(IP_i)$ is the pheromone change contributed by ant $k$, calculated as:
$$ \Delta \tau_j^k(IP_i) = \begin{cases}
\frac{Q}{e_k}, & \text{if ant } k \text{ selects } P_j(IP_i) \\
0, & \text{otherwise}
\end{cases} $$
where $Q$ is a constant, and $e_k$ is the output error when using the parameters selected by ant $k$. This error is defined as $e_k = |O – O_q|$, with $O$ and $O_q$ being the actual and expected outputs, respectively. The ACO process iterates until all ants converge to a single path, indicating the optimal initial parameters for the BP network. This optimized initialization accelerates training and reduces the risk of local minima for the six-axis force sensor decoupling task.
In our experimental setup, we utilized a calibration testbed for the six-axis force sensor, which included loading mechanisms, the sensor本体, signal conversion circuits, and data acquisition systems. The sensor was subjected to various force and torque inputs $F = [F_x, F_y, F_z, M_x, M_y, M_z]$, and the corresponding output signals $F’ = [F’_x, F’_y, F’_z, M’_x, M’_y, M’_z]$ were recorded. The data collection process involved multiple loading scenarios to capture a comprehensive dataset for training and testing the neural network model. Below is a sample of the collected data, which illustrates the input-output relationships used in our study:
| Sample | Input Fx (N) | Input Fy (N) | Input Fz (N) | Input Mx (N·m) | Input My (N·m) | Input Mz (N·m) | Output F’x (V) | Output F’y (V) | Output F’z (V) | Output M’x (V) | Output M’y (V) | Output M’z (V) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | -5 | 0 | 0 | 0 | 0 | 0 | 726.63 | 15.60 | -11.83 | 1.77 | -16.33 | -92.87 |
| 2 | 0 | -5 | 0 | 0 | 0 | 0 | -87.93 | 768.33 | -4.87 | -6.93 | -1.57 | 33.47 |
| 3 | 0 | 0 | 4 | 0 | 0 | 0 | -1.07 | -26.63 | -658.6 | 2.03 | -15.00 | -1.57 |
This data was used to train the BP neural network with the following configuration: input layer size $m = 6$, hidden layer size $p = 11$ (determined empirically), and output layer size $n = 6$. The target mean squared relative error was set to $10^{-1}$, and ACO parameters were $\rho = 0.7$, $h = 30$, and $Q = 10$. The training process involved comparing the traditional BP network with the ACO-optimized BP network to evaluate performance metrics such as iteration count and runtime.
The simulation results demonstrated a significant improvement with the ACO-BP approach. For instance, the traditional BP network required an average of 800 iterations to achieve the target error, whereas the ACO-BP network converged in only 400 iterations. Similarly, the runtime was reduced from 5.2 seconds to 2.1 seconds, representing a 60% speedup. These findings are summarized in the table below:
| Algorithm | Average Iterations | Average Runtime (s) |
|---|---|---|
| Traditional BP | 800 | 5.2 |
| ACO-BP | 400 | 2.1 |
This enhancement is attributed to the optimal initial parameters provided by ACO, which reduce oscillations and prevent entrapment in local minima. The decoupling accuracy for the six-axis force sensor was notably higher, as the network could better approximate the complex nonlinear relationships between inputs and outputs. Additional tests with varying load conditions confirmed the robustness of the ACO-BP method, consistently yielding faster convergence and lower errors compared to traditional approaches.
In conclusion, the integration of ant colony optimization with the BP neural network offers a powerful solution for decoupling six-axis force sensors. By addressing the limitations of random parameter initialization, this method achieves superior convergence speed and stability, making it highly suitable for real-time industrial applications. The six-axis force sensor benefits from this approach through improved precision and reliability, which are crucial for advanced robotic systems. Future work could explore adaptations of this algorithm for other multidimensional sensors or dynamic decoupling scenarios, further expanding its utility in smart manufacturing environments.