The six-axis force sensor is a critical component in modern industrial automation and robotics, enabling precise measurement of forces and moments in three-dimensional space. However, a significant challenge in its application is the inherent inter-dimensional coupling, which compromises measurement accuracy. This coupling arises from the sensor’s structural design and the nonlinear interactions between different force and moment components. To address this issue, various decoupling methods have been developed, including hardware-based approaches that minimize coupling through optimized designs and software-based techniques that employ algorithms to reduce errors. Among software methods, traditional approaches like least squares fitting and basic neural networks often fall short due to their limitations in handling nonlinearities and susceptibility to local optima. In this article, I propose an enhanced decoupling strategy that combines a backpropagation (BP) neural network with an improved genetic algorithm (GA), incorporating simulated annealing (SA) to optimize the network’s parameters. This method aims to achieve higher decoupling accuracy, faster convergence, and better global search capabilities, ultimately improving the performance of six-axis force sensors in practical applications.
The core of the six-axis force sensor lies in its elastic body structure, which deforms under applied loads, generating signals that correspond to specific force and moment components. For instance, in an L-shaped fingertip six-axis force sensor, the elastic body consists of composite beams and a base, with strain gauges attached at strategic locations to measure deformations. When forces or moments are applied in different directions, the resulting voltage outputs from these strain gauges are used to calculate the loads. However, due to the sensor’s design and manufacturing tolerances, these outputs often exhibit coupling, where a load in one direction influences the readings in others. This necessitates the development of accurate decoupling models to isolate the true force and moment values from the sensor’s output signals.
To construct a decoupling model, I utilize a BP neural network due to its ability to approximate complex nonlinear relationships. The network takes the six voltage outputs from the six-axis force sensor as inputs and produces the corresponding six force and moment components as outputs. The relationship can be expressed as $U = f(F)$, where $U$ is the voltage vector and $F$ is the force vector. The goal is to train the network so that its output $N_F$ closely matches the actual applied forces $F$. However, standard BP neural networks are prone to issues like slow convergence and getting trapped in local minima. To overcome these limitations, I integrate an improved genetic algorithm that optimizes the network’s initial weights and thresholds, enhancing its performance and stability.
The performance of the six-axis force sensor is evaluated using two key error metrics: Type I error and Type II error. Type I error measures the deviation between the actual applied force and the decoupled measurement in the same direction, defined as $d_i = \frac{F(i)_R – F(i)_M}{F(i)_{max}}$, where $F(i)_R$ is the reference force, $F(i)_M$ is the measured force, and $F(i)_{max}$ is the maximum force in that direction. Type II error, or coupling error, quantifies the interference from other directions when a load is applied in one specific direction, given by $d_{iv} = \frac{F_{iv}}{F_{i_{max}}}$, where $F_{iv}$ is the force measured in direction $i$ due to a load applied in direction $v$. Minimizing these errors is crucial for achieving high accuracy in six-axis force sensor applications.
The genetic algorithm is a population-based optimization technique inspired by natural selection, which I enhance to improve its efficiency and effectiveness. In the standard GA, the fitness function, crossover probability, and mutation probability are fixed, leading to slow convergence and premature convergence to suboptimal solutions. I address this by refining the fitness function using a scaling mechanism: $f’ = f + (\bar{f} – C \cdot d)$, where $f’$ is the scaled fitness, $\bar{f}$ is the mean fitness, $d$ is the standard deviation, and $C$ is a coefficient that adapts based on population diversity. This adjustment prevents dominant individuals from overly influencing the selection process, maintaining diversity and reducing the risk of early convergence.
Furthermore, I introduce adaptive mechanisms for crossover and mutation probabilities. The crossover probability $P_c$ is defined as:
$$P_c = \begin{cases} P_{c1} – \frac{(P_{c1} – P_{c2})(f_1 – f_{avg})}{f_{max} – f_{avg}} & \text{if } f_1 \geq f_{avg} \\ P_{c1} & \text{if } f_1 < f_{avg} \end{cases}$$
where $P_{c1}$ and $P_{c2}$ are the maximum and minimum crossover probabilities, $f_{max}$ is the maximum fitness, $f_{avg}$ is the average fitness, and $f_1$ is the higher fitness of the two individuals undergoing crossover. Similarly, the mutation probability $P_m$ is given by:
$$P_m = \begin{cases} P_{m1} – \frac{(P_{m1} – P_{m2})(f_{max} – f_f)}{f_{max} – f_{avg}} & \text{if } f \geq f_{avg} \\ P_{m1} & \text{if } f < f_{avg} \end{cases}$$
where $P_{m1}$ and $P_{m2}$ are the upper and lower bounds for mutation probability, and $f_f$ is the fitness of the individual being mutated. These adaptive rates allow the algorithm to dynamically adjust its search intensity, promoting exploration in the early stages and exploitation in the later stages.
To further enhance the global search capability, I incorporate simulated annealing into the genetic algorithm. After mutation, the change in fitness $\Delta f = f’_1 – f$ is computed, where $f’_1$ is the new fitness and $f$ is the original fitness. The acceptance probability for the mutated individual is determined by the Metropolis criterion: $P = \exp(\frac{\Delta f}{T_k})$ if $\Delta f \leq 0$, and $P = 1$ if $\Delta f \geq 0$, where $T_k = T_0 \cdot \alpha^k$ is the temperature at iteration $k$, $T_0$ is the initial temperature, and $\alpha$ is a cooling rate less than 1. This approach allows the algorithm to accept worse solutions with a certain probability initially, facilitating escape from local optima, and gradually focuses on refining the best solutions as the temperature decreases.
The improved genetic algorithm optimizes the BP neural network by selecting the best initial weights and thresholds. The network architecture includes an input layer with six neurons (corresponding to the voltage outputs), a hidden layer with ten neurons, and an output layer with six neurons (representing the force and moment components). The hidden layer uses a tangent sigmoid activation function, while the output layer employs a linear activation function. The forward propagation is described by $z = IW_1 \cdot x + b_1$ and $y = LW_2 \cdot a + b_2$, where $IW_1$ is the input weight matrix, $x$ is the input vector, $b_1$ is the hidden layer bias, $LW_2$ is the output weight matrix, $a$ is the hidden layer output, and $b_2$ is the output layer bias. The optimized parameters obtained from the improved GA significantly enhance the network’s decoupling performance.
To validate the proposed method, I conduct experiments using calibration data from an L-shaped fingertip six-axis force sensor and a commercial strain-based planar cross-beam six-axis force sensor. The calibration process involves applying known forces and moments in various directions and recording the corresponding voltage outputs. The data is normalized using Min-Max normalization to eliminate dimensional effects. The improved GA-BP algorithm is compared with traditional methods, including standard BP neural networks and basic GA-BP, in terms of convergence speed, solution quality, and decoupling accuracy.
The calibration data for the six-axis force sensor is summarized in the following table, which shows the applied forces and moments along with the measured voltage outputs:
| Test No. | Input (N/N·mm) | Output (mV) | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| F_x | F_y | F_z | M_x | M_y | M_z | U_Fx | U_Fy | U_Fz | U_Mx | U_My | U_Mz | |
| 1 | 5 | 0 | 0 | 0 | 0 | 0 | -43 | -11 | -7 | 5 | 45 | 211 |
| 2 | 0 | 5 | 0 | 0 | 0 | 0 | -11 | -59 | 1 | 76 | -3 | -11 |
| 3 | 0 | 0 | 5 | 0 | 0 | 0 | 0 | -13 | -62 | 19 | 200 | 18 |
| 4 | 0 | 0 | 0 | 400 | 0 | 0 | 17 | -385 | -13 | -1 | 504 | 80 |
| 5 | 0 | 0 | 0 | 0 | 400 | 0 | -578 | -1 | 443 | -323 | -1 | 893 |
| 6 | 0 | 0 | 0 | 0 | 0 | 400 | 904 | -1 | 450 | -360 | -2 | 045 |
| 7 | 10 | 0 | 0 | 0 | 0 | 0 | -87 | -10 | 11 | -32 | 103 | -217 |
| 8 | 0 | 10 | 0 | 0 | 0 | 0 | -19 | -113 | -1 | 150 | 1 | -17 |
| 9 | 0 | 0 | 10 | 0 | 0 | 0 | 10 | 0 | -126 | 18 | 48 | -20 |
| 10 | 0 | 0 | 0 | 500 | 0 | 0 | 23 | -462 | -20 | -1 | 889 | 105 |
| 11 | 0 | 0 | 0 | 0 | 500 | 0 | -499 | -1 | 449 | -326 | -1 | 905 |
| 12 | 0 | 0 | 0 | 0 | 0 | 500 | -906 | -1 | 446 | -362 | -2 | 088 |
The improved GA-BP algorithm demonstrates superior performance in optimization, as evidenced by the fitness curve comparison. The convergence speed is faster, and the final fitness value is lower than those of traditional GA and standard BP methods. The training error of the neural network also decreases more rapidly and smoothly, indicating enhanced stability and accuracy. The decoupling results show that the improved algorithm significantly reduces both Type I and Type II errors. For the L-shaped sensor, the maximum Type I error is reduced to 0.311%, and the maximum Type II error is controlled within 0.250%. In contrast, traditional methods exhibit higher errors, with standard BP having a maximum Type I error of 1.298% and GA-BP reaching 1.371%.
The following table compares the decoupling errors across different algorithms for the six-axis force sensor:
| Error Type | Decoupling Algorithm | F_x | F_y | F_z | M_x | M_y | M_z |
|---|---|---|---|---|---|---|---|
| Type I Error (Max) | BP | 1.298% | 1.572% | 3.878% | 0.364% | 0.471% | 0.377% |
| GA-BP | 0.450% | 1.100% | 1.371% | 0.311% | 0.355% | 0.417% | |
| Improved GA-BP | 0.278% | 0.227% | 0.311% | 0.210% | 0.204% | 0.213% | |
| Type II Error (Max) | BP | 0.819% | 0.787% | 3.974% | 0.542% | 0.412% | 0.504% |
| GA-BP | 0.651% | 0.418% | 2.421% | 0.465% | 0.311% | 0.211% | |
| Improved GA-BP | 0.147% | 0.247% | 0.137% | 0.002% | 0.021% | 0.011% |
To further validate the robustness of the improved genetic algorithm, I apply it to a commercial strain-based planar cross-beam six-axis force sensor. The full-scale prediction results for forces and moments are shown in the table below, where the applied loads are compared with the decoupled outputs. The maximum errors in all directions remain within 0.250%, confirming the algorithm’s effectiveness across different sensor types.
| Direction | F_x | F_y | F_z | M_x | M_y | M_z |
|---|---|---|---|---|---|---|
| F_x | 100.140 | 0.020 | -0.090 | 0.012 | 0.010 | -0.009 |
| F_y | -0.030 | 99.960 | -0.090 | -0.020 | -0.005 | 0.003 |
| F_z | -0.080 | 0.040 | 199.550 | 0.010 | -0.010 | 0.000 |
| M_x | 0.030 | 0.020 | 0.070 | 8.010 | 0.006 | 0.006 |
| M_y | -0.070 | -0.020 | 0.050 | 0.008 | 8.020 | -0.007 |
| M_z | -0.003 | 0.090 | -0.110 | -0.010 | 0.000 | 7.980 |
The improved genetic algorithm also demonstrates advantages in computational efficiency. The decoupling time for the proposed method is approximately 3.6 seconds, which is significantly faster than other algorithms like RF-GA, which takes 13.23 seconds. This efficiency, combined with high accuracy, makes the improved GA-BP suitable for real-time applications in six-axis force sensor systems.
In comparison with existing decoupling methods, such as polynomial fitting and random forest-genetic algorithm hybrids, the improved GA-BP algorithm shows superior performance in terms of error reduction and convergence stability. For instance, polynomial methods often struggle with nonlinearities, leading to higher errors, while basic neural networks without optimization may converge slowly or inaccurately. The integration of simulated annealing with the genetic algorithm provides a balanced approach between global exploration and local exploitation, ensuring that the solution is both optimal and reliable.
In conclusion, the improved genetic algorithm optimized BP neural network offers a robust solution for decoupling six-axis force sensors. By enhancing the fitness function, adaptive crossover and mutation probabilities, and incorporating simulated annealing, the algorithm achieves faster convergence, higher accuracy, and better stability. Experimental results on multiple sensor types confirm that Type I and Type II errors are significantly reduced, with averages decreasing by 1.36% and 1.72%, respectively, compared to traditional methods. This approach effectively addresses the inter-dimensional coupling problem, improving the measurement precision of six-axis force sensors and providing a reliable foundation for their optimization in various industrial and robotic applications. Future work could focus on extending this method to dynamic load conditions and integrating it with real-time control systems for enhanced performance.