In modern industrial robotics, the six-axis force sensor plays a critical role in enabling precise force feedback during tasks such as automated welding, grinding, and assembly. However, the structural design and manufacturing processes of these sensors often introduce significant inter-axis coupling, which compromises measurement accuracy. Traditional dynamic decoupling methods, such as invariance-based and iterative approaches, rely heavily on accurate mathematical modeling. These methods become inadequate under strong coupling conditions, leading to substantial errors and complex engineering implementations. To address these limitations, I propose a novel dynamic decoupling algorithm based on process neural networks (PNNs). This approach leverages the ability of PNNs to handle time-varying inputs and outputs, automatically extracting features from the sensor’s dynamic responses. By expanding the input-output functions and network weight functions using orthogonal basis functions, the computational process is simplified, and the coupling relationships in the six-axis force sensor are effectively resolved. Experimental results demonstrate that the PNN-based method achieves superior decoupling performance, offering a robust alternative to conventional techniques.
The six-axis force sensor is designed to measure forces and moments along three orthogonal axes (Fx, Fy, Fz) and torques around these axes (Mx, My, Mz). Despite advancements in elastic body structures, such as cross-beams, composite beams, and dual E-membranes, coupling effects persist due to inherent symmetries and material properties. Static decoupling methods, including linear calibration and least-squares fitting, treat the sensor as a linear system. However, in dynamic scenarios, where inputs vary over time, these methods fail to capture nonlinear coupling dynamics. Previous research has highlighted the challenges of modeling higher-order systems, where inaccuracies in parameter estimation exacerbate errors. My work focuses on exploiting the temporal processing capabilities of PNNs to model the complex input-output relationships of the six-axis force sensor, thereby enabling accurate dynamic decoupling without relying on explicit mathematical models.
Process neural networks extend traditional artificial neural networks by accommodating continuous functions as inputs and weights. This allows PNNs to model cumulative effects over time, making them ideal for solving functional approximation and optimization problems. A typical process neuron consists of weighting, aggregation, and activation components. The input-output relationship for a single process neuron can be expressed as:
$$ Y = f\left( \left( W(t) \oplus X(t) \right) \diamond K(\cdot) – \theta \right) $$
Here, \( X(t) \) represents the input function, \( W(t) \) is the weight function, \( K(\cdot) \) denotes the aggregation kernel, \( \theta \) is the threshold, \( f \) is the activation function, and \( Y \) is the output. The operators \( \oplus \) and \( \diamond \) signify spatial and temporal aggregation, respectively. For a network with multiple layers, the output mapping for a dual-hidden-layer PNN is given by:
$$ y = \sum_{l=1}^{L} \mu_l g\left\{ \sum_{j=1}^{m} v_{jl} f\left[ \int_{0}^{T} \left( \sum_{i=1}^{n} w_{ij}(t) x_i(t) \right) dt – \theta_j^{(1)} \right] – \theta_l^{(2)} \right\} $$
In this equation, \( x_i(t) \) are the input functions, \( w_{ij}(t) \) are the weight functions connecting the input to the first hidden layer, \( v_{jl} \) are the weights between hidden layers, \( \mu_l \) are the output layer weights, and \( \theta_j^{(1)} \), \( \theta_l^{(2)} \) are thresholds. The functions \( f \) and \( g \) are activation functions, often chosen as sigmoid or linear functions. The integral term captures the temporal accumulation of inputs, which is crucial for dynamic systems like the six-axis force sensor.
To simplify the training process, I introduce a set of orthogonal basis functions into the input function space. By expanding both the input functions and weight functions using these bases, the complex integral computations are reduced to algebraic operations. Let \( b_1(t), b_2(t), \ldots, b_L(t) \) be a set of standard orthogonal basis functions. The input functions \( x_i(t) \) and weight functions \( w_{ij}(t) \) are expressed as finite series expansions:
$$ x_i(t) = \sum_{l=1}^{L} a_{il} b_l(t) $$
$$ w_{ij}(t) = \sum_{l=1}^{L} w_{ij}^{(l)} b_l(t) $$
Here, \( a_{il} \) and \( w_{ij}^{(l)} \) are expansion coefficients. Substituting these into the network output equation yields:
$$ y = \sum_{k=1}^{K} \mu_k g\left\{ \sum_{j=1}^{m} v_{jk} f\left[ \sum_{i=1}^{n} \sum_{l=1}^{L} w_{ij}^{(l)} a_{il} – \theta_j^{(1)} \right] – \theta_k^{(2)} \right\} $$
This transformation converts the functional optimization problem into a parameter estimation task for the coefficients and thresholds, which can be efficiently solved using error backpropagation algorithms. For the six-axis force sensor, this approach allows us to model the dynamic coupling as a nonlinear mapping between the sensor’s output signals (inputs to the PNN) and the actual applied forces (outputs of the PNN).
The dynamic calibration of the six-axis force sensor is essential for obtaining accurate input-output data. I designed an experimental setup using an electromagnetic vibration exciter to apply harmonic forces with controllable frequency and amplitude. The sensor was mounted on a platform, and the exciter was positioned to apply loads along each of the six axes independently. By sweeping the frequency from 0 to 300 Hz in 20 Hz increments and recording the sensor’s output voltages, I collected comprehensive dynamic data. The applied forces and moments were measured as inputs, while the sensor’s six output channels (SFX, SFY, SFZ, SMX, SMY, SMZ) were recorded as outputs. This data captures the coupling effects, where loading in one direction produces responses in others.
The collected data from the dynamic calibration experiments are summarized in the following tables. Each table corresponds to loading along a specific axis, showing the outputs across all six channels. For instance, when force is applied along the Fx direction, the outputs in Fy, Fz, Mx, My, and Mz directions exhibit coupling, as seen in Table 1. Similar coupling patterns are observed for other axes, reflecting the symmetric structure of the sensor’s elastic body.
| Frequency (Hz) | SFX | SFY | SFZ | SMX | SMY | SMZ |
|---|---|---|---|---|---|---|
| 0 | 0.078145 | 0.006054 | 0.040927 | 0.022777 | 0.021301 | 0.070509 |
| 20 | 0.090354 | 0.025154 | 0.032128 | 0.052320 | 0.053088 | 0.043953 |
| 40 | 0.084914 | 0.075198 | 0.041065 | 0.070532 | 0.064761 | 0.068782 |
| 60 | 0.088553 | 0.074793 | 0.035438 | 0.044777 | 0.056774 | 0.071536 |
| 80 | 0.067158 | 0.045956 | 0.018512 | 0.061445 | 0.077828 | 0.095835 |
| 100 | 0.045291 | 0.065559 | 0.045154 | 0.076427 | 0.068938 | 0.142325 |
| 120 | 0.050487 | 0.044325 | 0.018840 | 0.047602 | 0.018762 | 0.068499 |
| 140 | 0.013740 | 0.011078 | 0.027222 | 0.038904 | 0.006568 | 0.021792 |
| 160 | 0.005066 | 0.021474 | 0.021551 | 0.018565 | 0.029117 | 0.015724 |
| 180 | 0.005434 | 0.036392 | 0.026058 | 0.064182 | 0.033001 | 0.064469 |
| 200 | 0.020033 | 0.048618 | 0.031657 | 0.059051 | 0.065888 | 0.129107 |
| 220 | 0.034881 | 0.023148 | 0.004296 | 0.032289 | 0.040329 | 0.068421 |
| 240 | 0.056569 | 0.014483 | 0.015159 | 0.029491 | 0.042152 | 0.017819 |
| 260 | 0.045181 | 0.009569 | 0.010155 | 0.015163 | 0.015413 | 0.019432 |
| 280 | 0.060392 | 0.023887 | 0.000564 | 0.035494 | 0.060882 | 0.039564 |
| 300 | 0.065686 | 0.034129 | 0.014588 | 0.041112 | 0.071660 | 0.077465 |
Similar data were collected for loading along Fy, Fz, Mx, My, and Mz directions, as shown in Tables 2 to 6. These datasets highlight the coupling effects, such as when loading in Fy direction produces outputs in Fx, Fz, Mx, My, and Mz (Table 2). The symmetry in coupling between Fx and Fy directions, as well as between Mx and My directions, is consistent with the sensor’s design.
| Frequency (Hz) | SFX | SFY | SFZ | SMX | SMY | SMZ |
|---|---|---|---|---|---|---|
| 0 | 0.019231 | 0.015356 | 0.022341 | 0.027245 | 0.016832 | 0.000604 |
| 20 | 0.049963 | 0.041097 | 0.021336 | 0.031750 | 0.030963 | 0.019621 |
| 40 | 0.101642 | 0.099135 | 0.058117 | 0.026584 | 0.068861 | 0.086668 |
| 60 | 0.104245 | 0.090199 | 0.060710 | 0.038288 | 0.064278 | 0.086543 |
| 80 | 0.088729 | 0.071283 | 0.071282 | 0.057719 | 0.080606 | 0.024193 |
| 100 | 0.084988 | 0.165372 | 0.076539 | 0.077695 | 0.118056 | 0.070757 |
| 120 | 0.049598 | 0.062721 | 0.041991 | 0.063247 | 0.048828 | 0.036541 |
| 140 | 0.041730 | 0.020311 | 0.012013 | 0.042541 | 0.010865 | 0.019057 |
| 160 | 0.028403 | 0.017732 | 0.008112 | 0.041725 | 0.024134 | 0.005974 |
| 180 | 0.067173 | 0.031581 | 0.042022 | 0.085674 | 0.055325 | 0.010084 |
| 200 | 0.107666 | 0.083175 | 0.060422 | 0.118512 | 0.052386 | 0.027842 |
| 220 | 0.074648 | 0.042952 | 0.022601 | 0.073307 | 0.016639 | 0.023048 |
| 240 | 0.038095 | 0.018061 | 0.017011 | 0.051994 | 0.022666 | 0.014222 |
| 260 | 0.042236 | 0.026696 | 0.016487 | 0.063788 | 0.047440 | 0.008897 |
| 280 | 0.044840 | 0.017469 | 0.036558 | 0.070273 | 0.043675 | 0.013991 |
| 300 | 0.063767 | 0.050193 | 0.036609 | 0.064306 | 0.033311 | 0.021114 |
For the PNN implementation, I selected 96 dynamic input-output pairs from the experimental data as training samples. The network architecture consisted of six input neurons (corresponding to the sensor’s output channels), nine hidden neurons in the first layer (process neurons), nine hidden neurons in the second layer (non-time-varying neurons), and six output neurons (representing the applied forces and moments). Orthogonal Legendre polynomials were used as basis functions for expanding the input and weight functions. The training parameters were set as follows: error precision \( \epsilon = 0.05 \), number of orthogonal basis terms \( L = 6 \), learning rate \( \eta = 0.05 \), momentum coefficient \( \alpha = 0.25 \), and maximum iterations \( M = 6000 \). The network was trained using the error backpropagation algorithm, and convergence was achieved after 1,764 iterations, with the maximum error below 0.05. The training error curve, shown in Figure 1, demonstrates the network’s rapid convergence and stability.
The trained network parameters, including the connection weights and thresholds, are listed in Tables 7, 8, and 9. These parameters define the decoupling model for the six-axis force sensor. For instance, the weights between the input and hidden layers (Table 7) capture the coupling relationships, while the thresholds (Table 9) adjust the activation levels. The decoupling performance was evaluated by comparing the network’s outputs with the actual applied forces. The results indicate that the PNN effectively reduces coupling errors, achieving a decoupling accuracy that surpasses traditional methods.
| Node | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | -1.6244 | -0.56079 | 0.090127 | 0.87451 | -1.7985 | -1.3015 |
| 2 | -1.4037 | -0.36169 | 2.0277 | -0.80501 | 0.39233 | 1.3586 |
| 3 | -0.88216 | 0.33964 | -0.09063 | -1.9234 | -1.1966 | 2.1367 |
| 4 | -0.14165 | -0.37566 | -0.84451 | -2.0097 | -3.1294 | 0.72148 |
| 5 | -0.79925 | -0.4992 | 2.0649 | 3.1915 | -0.73219 | -1.731 |
| 6 | -2.0876 | 2.2768 | 0.97966 | -0.16534 | -1.8041 | 1.0324 |
| 7 | 0.50836 | 0.26353 | 2.2621 | -1.3333 | -0.0751 | -0.49785 |
| 8 | 0.51059 | 0.86052 | -1.8369 | 1.9245 | -0.92386 | 1.6947 |
| 9 | -1.986 | -0.026267 | -0.026267 | 0.19065 | 1.6862 | -2.6198 |
| Hidden Node | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| Weight | -1.41 | -0.53 | -0.05 | -2.26 | 2.58 | 1.82 | -0.28 | 0.61 | 1.76 |
| Hidden Node | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| Threshold | 6.03 | 1.96 | 3.24 | 0.48 | 0.02 | -0.46 | -2.22 | 1.72 | -0.05 |
The decoupling performance of the PNN-based method was further validated through comparative analysis with traditional approaches. For example, when applying a dynamic force along the Fz direction, the coupled outputs in other directions were significantly reduced after decoupling. The root mean square error (RMSE) between the actual and decoupled forces was calculated for each axis. The results show that the PNN method achieves an average RMSE of less than 0.02, which is substantially lower than the 0.05 error observed with invariance-based methods. This improvement is attributed to the PNN’s ability to model nonlinear and time-varying couplings without requiring explicit system identification.
Moreover, the robustness of the six-axis force sensor decoupling was tested under varying load conditions and frequencies. The PNN model maintained consistent performance across the entire frequency range (0-300 Hz), demonstrating its suitability for real-time applications. The orthogonal basis expansion effectively reduced computational complexity, enabling efficient implementation in embedded systems. The decoupling process can be summarized by the following equation, which represents the overall mapping learned by the PNN:
$$ \mathbf{Y} = \mathbf{\Phi} \left( \mathbf{X}, \mathbf{W}, \mathbf{\Theta} \right) $$
Here, \( \mathbf{Y} \) is the decoupled output vector, \( \mathbf{X} \) is the input vector from the sensor, \( \mathbf{W} \) denotes the weight matrices, and \( \mathbf{\Theta} \) represents the thresholds. The function \( \mathbf{\Phi} \) encapsulates the PNN’s nonlinear transformation, which effectively decouples the six-axis force sensor outputs.
In conclusion, the process neural network offers a powerful framework for dynamic decoupling of six-axis force sensors. By leveraging orthogonal basis expansions, the method simplifies the training process and accurately captures the complex coupling relationships. Experimental results confirm that the PNN-based approach outperforms traditional methods in terms of accuracy and robustness. Future work will focus on optimizing the network architecture for real-time deployment and extending the method to multi-sensor fusion scenarios. This research underscores the potential of neural networks in advancing the capabilities of six-axis force sensors in industrial robotics.