In the field of space exploration, the six-axis force sensor plays a critical role in enabling precise force and torque measurements for space manipulators. However, these sensors are often subjected to extreme temperature variations in space environments, leading to significant temperature drift errors that compromise accuracy. Traditional compensation methods, such as hardware-based approaches, face limitations in wide temperature ranges and complex debugging processes. To address this, we propose an efficient temperature compensation method based on a radial basis function (RBF) neural network optimized by an improved particle swarm optimization (IPSO) algorithm. This approach aims to minimize temperature-induced errors in the six-axis force sensor, ensuring reliable performance in space applications. In this article, we detail the principles of temperature compensation, the development of the RBF neural network model, the optimization process using IPSO, and experimental validation. Through comparative analysis with other methods, we demonstrate the superiority of our approach in reducing temperature drift errors to as low as 0.27% of the full scale, meeting the stringent requirements for space manipulators.
The six-axis force sensor is integral to space manipulators, providing multi-dimensional force and torque feedback for tasks such as docking, assembly, and maintenance. Typically constructed with strain gauges, the sensor’s output is influenced by temperature changes due to alterations in material properties, including strain gauge resistance coefficients, elastic body volume, and elastic modulus. These changes result in zero-point and sensitivity drift, which can severely impact measurement precision. For instance, in space environments where temperatures can swing between -40°C and 80°C, the six-axis force sensor may exhibit nonlinear relationships between input forces and output voltages. This nonlinearity necessitates advanced compensation techniques beyond conventional methods like least squares or basic neural networks, which often require data normalization or suffer from optimization inefficiencies. Our work focuses on developing a software-based compensation model that leverages the RBF neural network’s ability to handle nonlinear systems, enhanced by IPSO for parameter optimization, to achieve high accuracy across a broad temperature range.
The temperature compensation principle for the six-axis force sensor revolves around modeling the relationship between environmental temperature, input forces, and output voltages. The sensor system comprises the sensor body, signal conditioning circuitry, a microcontroller, and communication interfaces. When exposed to temperature variations, the output voltage $U_r$ deviates from the expected values at a reference temperature (e.g., 20°C). The input parameters include the applied force vector $F = [F_x, F_y, F_z, M_x, M_y, M_z]$, the output voltage vector $U_r = [U_{Fx}, U_{Fy}, U_{Fz}, U_{Mx}, U_{My}, U_{Mz}]$, and the temperature $T$. The goal is to map these inputs to the compensated output voltage vector $U_F = [U_{Fx\_base}, U_{Fy\_base}, U_{Fz\_base}, U_{Mx\_base}, U_{My\_base}, U_{Mz\_base}]$, which represents the sensor output at the reference temperature. This mapping is inherently nonlinear due to temperature effects, making it suitable for neural network approaches. The RBF neural network, with its single hidden layer and Gaussian activation functions, excels at transforming nonlinear problems into linearly separable ones in a higher-dimensional space. The compensation model is designed to learn this mapping, minimizing the error between the actual and desired outputs through iterative training.
To formalize the compensation model, we define the RBF neural network structure. The input layer consists of 13 neurons corresponding to the elements of $F$, $U_r$, and $T$, while the output layer has 6 neurons for the compensated voltages. The hidden layer uses Gaussian functions as activation functions, given by:
$$\phi_i = \exp\left( -\frac{\| \mathbf{x} – \mathbf{c}_i \|^2}{2\delta_i^2} \right)$$
where $i = 1, 2, \dots, I$ (with $I$ being the number of hidden neurons, set to 8 based on preliminary experiments), $\mathbf{x} = [F, U_r, T]^T$ is the input vector, $\mathbf{c}_i$ is the center vector of the $i$-th neuron, and $\delta_i$ is the width parameter. The network output $\hat{Y} = [\hat{y}_1, \hat{y}_2, \dots, \hat{y}_6]^T$ is computed as a weighted sum of the hidden layer outputs:
$$\hat{Y} = \mathbf{W} \cdot \mathbf{\Phi}$$
where $\mathbf{W}$ is the weight matrix with elements $w_{ij}$ (representing the weight from the $i$-th hidden neuron to the $j$-th output), and $\mathbf{\Phi} = [\phi_1, \phi_2, \dots, \phi_I]^T$. The performance of the network is evaluated using the mean squared error (MSE) as the loss function:
$$E = \frac{1}{2} \sum_{j=1}^{6} (y_j – \hat{y}_j)^2$$
where $y_j$ is the expected output and $\hat{y}_j$ is the network output. The parameters $\mathbf{c}_i$, $\delta_i$, and $w_{ij}$ are optimized using gradient descent, with update rules defined as:
$$\mathbf{c}_i = \mathbf{c}_i + \eta \frac{w_{ij}}{\delta_i^2} \sum_{j=1}^{6} \phi_i (y_j – \hat{y}_j) (\mathbf{x} – \mathbf{c}_i)$$
$$\delta_i = \delta_i + \eta \frac{w_{ij}}{\delta_i^3} \sum_{j=1}^{6} \phi_i (y_j – \hat{y}_j) \| \mathbf{x} – \mathbf{c}_i \|^2$$
$$w_{ij} = w_{ij} + \eta \sum_{j=1}^{6} \phi_i (y_j – \hat{y}_j)$$
where $\eta$ is the learning rate, typically set between 0 and 1. However, the standard RBF neural network may converge slowly or get trapped in local minima, necessitating further optimization.
To enhance the compensation efficiency, we employ an improved particle swarm optimization (IPSO) algorithm to optimize the RBF neural network parameters. The PSO algorithm mimics the social behavior of bird flocks, where particles represent potential solutions. Each particle has a velocity and position, updated iteratively based on personal and global best experiences. The standard PSO update equations are:
$$V_{kj}(n+1) = \omega V_{kj}(n) + c_1 r_1 [pbest_{kj}(n) – X_{kj}(n)] + c_2 r_2 [gbest_j(n) – X_{kj}(n)]$$
$$X_{kj}(n+1) = V_{kj}(n+1) + X_{kj}(n)$$
where $V_{kj}$ and $X_{kj}$ are the velocity and position of the $k$-th particle in dimension $j$, $n$ is the iteration number, $\omega$ is the inertia factor, $c_1$ and $c_2$ are learning factors, and $r_1$ and $r_2$ are random numbers in [0,1]. To overcome local optima, we introduce a nonlinear decreasing inertia factor:
$$\omega(n) = \omega_{\text{min}} + (\omega_{\text{max}} – \omega_{\text{min}}) \beta^{-(2n/n_{\text{max}})^\alpha}$$
with $\omega_{\text{max}} = 0.9$, $\omega_{\text{min}} = 0.4$, $\alpha = 3$, and $\beta = 2.8$. This ensures a gradual decrease in inertia, promoting global exploration early and local refinement later. Additionally, the learning factors are adapted as:
$$c_1 = c_{\text{max1}} – \frac{n (c_{\text{max1}} – c_{\text{min1}})}{n_{\text{max}}}$$
$$c_2 = c_{\text{min2}} – \frac{n (c_{\text{min2}} – c_{\text{max2}})}{n_{\text{max}}}$$
where $c_{\text{max1}} = 2.5$, $c_{\text{min1}} = 1.25$, $c_{\text{max2}} = 2.5$, and $c_{\text{min2}} = 1.25$. To maintain population diversity, we incorporate crossover and mutation operations from genetic algorithms. For crossover between individual best $A_o$ and global best $A_p$:
$$A_{ok} = r A_{ok} + (1 – r) A_{pk}$$
$$A_{pk} = (1 – r) A_{ok} + r A_{pk}$$
where $r$ is a random number in [0,1]. For mutation of the $s$-th individual’s $c$-th gene:
$$A_{sc} = \begin{cases} A_{sc} + (A_{sc} – A_{\text{max}}) g(n), & \text{if } q \geq q_0 \\ A_{sc} + (A_{\text{min}} – A_{sc}) g(n), & \text{if } q < q_0 \end{cases}$$
with $g(n) = 1 – r(1 – n/n_{\text{max}})^a$, where $a = 2$. The IPSO algorithm optimizes the RBF neural network parameters ($\mathbf{c}_i$, $\delta_i$, and $w_{ij}$) by treating them as particles and minimizing the error $E$. The optimization process involves initializing particles, updating velocities and positions, applying crossover and mutation, and iterating until convergence. This results in an optimal compensation model for the six-axis force sensor.
We conducted experiments to validate the proposed temperature compensation method for the six-axis force sensor. The experimental setup included a calibration frame, data acquisition system, temperature controller, and a thermal chamber capable of maintaining temperatures from -50°C to 120°C. The six-axis force sensor used in the experiments had a operational temperature range of -40°C to 80°C, with force and torque ranges as specified in Table 1.
| Parameter | Value |
|---|---|
| Fx, Fy (N) | 800 |
| Fz (N) | 1000 |
| Mx, My (N·m) | 80 |
| Mz (N·m) | 100 |
Data collection involved testing at temperature intervals of 20°C (from -40°C to 80°C), with 10 loading points in each of the six force/torque directions. The sensor was stabilized for 2 hours at each temperature before recording output voltages $U_r$. A total of 3,200 datasets were used for training, including no-load, single-direction load, and composite load points, while 800 datasets were reserved for testing. The raw output voltages exhibited significant temperature-dependent nonlinearities, as illustrated by the temperature-force output curves, where maximum sensitivity drifts reached up to 34.45 mV in the Fx direction.
The training data was fed into the initial RBF neural network model with random parameter initialization. The IPSO algorithm was configured with a population size of 100, maximum iterations of 200, velocity range of (-1, 1), and position range of (-2, 2). After optimization, the compensated output voltages showed remarkable improvement, with temperature drift errors reduced to below 0.13 mV in all directions. For example, the maximum drift in the Fx direction decreased from 34.45 mV to 0.13 mV, representing a reduction of over 99.6%. The compensated temperature-force output curves demonstrated enhanced linearity and minimal temperature influence, confirming the effectiveness of the IPSO-RBF approach.
To benchmark our method, we compared it with other compensation techniques, including piecewise-least squares, extreme learning machine (ELM), standard RBF neural network, and PSO-RBF neural network. The performance was evaluated based on the temperature drift error as a percentage of the full scale. The results, summarized in Table 2, indicate that the IPSO-RBF method achieved the lowest error of 0.27% full scale in testing, outperforming all other methods. This meets the desired threshold of 0.5% full scale for space applications, highlighting the robustness of our approach.
| Method | Training Error (%) | Testing Error (%) |
|---|---|---|
| Before Compensation | 35.44 | 35.44 |
| Piecewise-Least Squares | 3.79 | 3.48 |
| ELM | 0.54 | 0.59 |
| RBF Neural Network | 0.92 | 0.94 |
| PSO-RBF Neural Network | 0.38 | 0.42 |
| IPSO-RBF Neural Network | 0.24 | 0.27 |
The analysis of prediction error distributions further validates the superiority of the IPSO-RBF method. While other algorithms showed scattered errors, the IPSO-RBF approach consistently produced minimal deviations across all test cases. This is attributed to the enhanced global search capability of IPSO, which avoids local optima and efficiently tunes the RBF parameters. The six-axis force sensor, after compensation, maintains high accuracy even under extreme temperature conditions, making it suitable for long-duration space missions. Additionally, the software-based nature of this method allows for easy implementation and adaptation to various sensor designs, providing a versatile solution for multidimensional force sensors in harsh environments.
In conclusion, we have developed an advanced temperature compensation method for the six-axis force sensor used in space manipulators, leveraging an IPSO-optimized RBF neural network. This approach effectively addresses the challenges of temperature drift by modeling the nonlinear relationships between temperature, force inputs, and output voltages. Experimental results demonstrate that our method reduces temperature drift errors to 0.27% of the full scale, outperforming traditional and neural network-based techniques. The integration of IPSO ensures robust parameter optimization, enhancing the compensation efficiency and reliability. This work contributes to the advancement of force sensing technology in space applications, offering a practical solution for maintaining precision in extreme temperature environments. Future research could explore real-time adaptation and integration with other sensor fusion techniques to further improve performance.