As intelligent robotics rapidly integrate into industrial production and daily life, the six-axis force sensor has emerged as a critical component for enabling “touch” perception, attracting significant attention from researchers worldwide. This spatial force-sensing element is widely applied in robot force/position control, grasping and assembly, contour detection, autonomous obstacle avoidance, and human-computer interaction. Currently, improving accuracy remains a primary research focus for six-axis force sensors. However, due to factors such as inherent structural design and machining errors, these sensors often exhibit interdimensional coupling, where a force applied in one direction produces unwanted outputs in other directions, significantly compromising measurement precision. To address this issue, we investigate decoupling algorithms through a combination of error analysis, theoretical derivation, and experimental validation, aiming to reduce coupling errors without altering the sensor’s physical structure.
The six-axis force sensor typically measures forces and moments along three orthogonal axes (Fx, Fy, Fz) and torques around these axes (Mx, My, Mz). Structural configurations mainly include monolithic types (e.g., cross-beam structures) and parallel types (e.g., Stewart platform-based sensors). Monolithic sensors, fabricated from a single piece of metal, rely on strain gauges positioned at various spatial orientations to determine the applied forces. For instance, the cross-beam structure is commonly used but suffers from inherent coupling because deformation in one direction affects others. Parallel sensors, such as those with orthogonal or Stewart architectures, utilize multiple measuring branches connected by spherical joints to theoretically decouple forces structurally. However, practical limitations like machining inaccuracies, joint clearances, and frictional torques introduce coupling effects. Even in ideal parallel designs, the force distribution across branches leads to interdimensional coupling, as any single-direction force or moment results in outputs across all channels. Thus, developing effective decoupling algorithms is essential for enhancing sensor accuracy.
To mathematically model the coupling phenomenon, we analyze the input-output relationship of the six-axis force sensor. Let the generalized input force vector be denoted as $\mathbf{F_s} = [F_{sx}, F_{sy}, F_{sz}, M_{sx}, M_{sy}, M_{sz}]^T$, representing the actual forces and moments applied along and around the x, y, and z axes, respectively. The output vector, measured by the sensor, is $\mathbf{F_c} = [F_{cx}, F_{cy}, F_{cz}, M_{cx}, M_{cy}, M_{cz}]^T$. Due to coupling, the output in each direction is influenced not only by its corresponding input but also by inputs from other directions. This relationship can be expressed as a coupling matrix $\mathbf{H}$, such that:
$$\mathbf{F_c} = \mathbf{H} \mathbf{F_s}$$
Here, $\mathbf{H}$ is a 6×6 matrix where the diagonal elements represent the direct relationships (ideally 1), and the off-diagonal elements $H_{ij}$ (for $i \neq j$) quantify the coupling effects between the $i$-th input and $j$-th output. For example, $H_{12}$ describes how the x-direction force input affects the y-direction force output. In practice, these off-diagonal elements are not constants but functions of the input forces, leading to a nonlinear coupling model. Thus, for any output $F_{cj}$, we have:
$$F_{cj} = F_{si} + \sum_{i=1, i \neq j}^{6} H_{ij}(F_{si}), \quad j = 1, 2, \ldots, 6$$
where $H_{ij}(F_{si})$ is a coupling function that can be approximated using polynomial expressions. This forms the basis for our proposed decoupling algorithm, which aims to characterize these functions accurately.
Linear decoupling algorithms are commonly employed in six-axis force sensor calibration, assuming a linear relationship between inputs and outputs. The standard approach uses least squares fitting to derive a calibration matrix $\mathbf{G}$ that maps the sensor’s voltage outputs $\mathbf{U} = [U_1, U_2, \ldots, U_m]^T$ to the applied forces $\mathbf{F_s}$:
$$\mathbf{F_s} = \mathbf{G} \mathbf{U}$$
The matrix $\mathbf{G}$ is computed as $\mathbf{G} = \mathbf{F_s} \mathbf{U}^-$, where $\mathbf{U}^-$ is the pseudoinverse of $\mathbf{U}$, given by $\mathbf{U}^- = \mathbf{U}^T (\mathbf{U} \mathbf{U}^T)^{-1}$. This yields:
$$\mathbf{G} = \mathbf{F_s} \mathbf{U}^T (\mathbf{U} \mathbf{U}^T)^{-1}$$
However, this linear method often fails to account for nonlinearities caused by factors like material deformation, strain gauge misplacement, and temperature variations. As a result, residual coupling errors persist, limiting the accuracy of the six-axis force sensor. For instance, when a force is applied solely along the x-axis, outputs may appear in y and z directions due to unmodeled nonlinear couplings. To illustrate, consider a calibration experiment where forces are applied incrementally along one axis while monitoring all outputs. The data often reveal significant off-axis responses, underscoring the need for advanced decoupling techniques.
To overcome these limitations, we propose a decoupling algorithm based on polynomial fitting. This method explicitly models the coupling functions $H_{ij}(F_{si})$ as polynomials, capturing nonlinear relationships more effectively. A key advantage is its ability to handle the sensor’s zero-point drift by initializing the system such that outputs under no-load conditions are zero. This adjustment ensures that the coupling functions have no constant term, simplifying the polynomial to:
$$H_{ij}(F_{si}) = a_1 F_{si} + a_2 F_{si}^2 + \cdots + a_n F_{si}^n$$
where $a_1, a_2, \ldots, a_n$ are coefficients determined through curve fitting. The decoupling process involves applying standardized forces along each axis independently while recording the outputs in all directions. For example, when applying a force $F_{sx}$ (with other inputs zero), the outputs $F_{cy}, F_{cz}, M_{cx}, M_{cy}, M_{cz}$ are measured at multiple load points. Polynomial regression is then used to fit $H_{1j}(F_{sx})$ for $j = 2, 3, 4, 5, 6$. Similarly, forces along $F_{sy}, F_{sz}, M_{sx}, M_{sy}, M_{sz}$ are applied to derive all coupling functions. The resulting coupling matrix $\mathbf{H}$ is populated with these polynomial expressions.
Once $\mathbf{H}$ is established, decoupled outputs $\Delta \mathbf{F_c}$ are computed by subtracting the coupled components from the raw outputs:
$$\Delta F_{cj} = F_{cj} – \sum_{i=1, i \neq j}^{6} H_{ij}(F_{si}), \quad j = 1, 2, \ldots, 6$$
This approach effectively reduces interdimensional coupling, as demonstrated in our experimental validation. We conducted calibration tests on an orthogonal parallel six-axis force sensor, applying forces ranging from 10 N to 100 N in 10 N increments along each axis. The sensor’s voltage outputs were recorded, and both linear and polynomial fitting decoupling algorithms were applied. For instance, when $F_{sx}$ was applied, the outputs in other directions were fitted to polynomials. The coupling functions for $F_{cy}$ and $F_{cz}$ relative to $F_{sx}$ were found to be:
$$F_{cy} = \begin{cases} -0.00184 F_{sx} – 0.00002 F_{sx}^2, & F_{sx} \geq 0 \\ -0.00162 F_{sx} – 0.00002 F_{sx}^2, & F_{sx} < 0 \end{cases}$$
$$F_{cz} = \begin{cases} 0.00056 F_{sx} + 0.00007 F_{sx}^2, & F_{sx} \geq 0 \\ 0.00612 F_{sx} + 0.00012 F_{sx}^2, & F_{sx} < 0 \end{cases}$$
Similar procedures were followed for other inputs, such as $F_{sy}$, yielding functions like:
$$F_{cx} = \begin{cases} -0.0026900 F_{sy} – 0.0000047 F_{sy}^2, & F_{sy} \geq 0 \\ 0.002060 F_{sy} + 0.000016 F_{sy}^2, & F_{sy} < 0 \end{cases}$$
These polynomial fits accurately capture the coupling behavior, enabling precise decoupling. The effectiveness of this method is evaluated through error analysis, focusing on linearity error and maximum coupling error. Linearity error is defined as the absolute deviation between the decoupled output and the actual applied force, while maximum coupling error is the ratio of the maximum deviation in each direction to its full-scale value. For the linear decoupling algorithm, the error matrix $\mathbf{E_{L1}}$ is computed as:
$$\mathbf{E_{L1}} = \begin{bmatrix} \frac{\max(|\Delta L_{Fx1}|)}{F_{xM}} & \frac{\max(|\Delta L_{Fy1}|)}{F_{yM}} & \frac{\max(|\Delta L_{Fz1}|)}{F_{zM}} & \frac{\max(|\Delta L_{Mx1}|)}{M_{xM}} & \frac{\max(|\Delta L_{My1}|)}{M_{yM}} & \frac{\max(|\Delta L_{Mz1}|)}{M_{zM}} \end{bmatrix}^T$$
where $F_{xM}, F_{yM}, \ldots, M_{zM}$ are the full-scale values. Similarly, for the polynomial fitting decoupling, the error matrix $\mathbf{E_{L2}}$ is derived. Comparative results are summarized in the table below, highlighting the performance improvements.
| Error Type | Direction | Linear Decoupling (%) | Polynomial Fitting Decoupling (%) |
|---|---|---|---|
| Linearity Error | Fx | 0.417 | 0.403 |
| Fy | 0.925 | 0.904 | |
| Fz | 3.095 | 2.984 | |
| Mx | 0.144 | 0.108 | |
| My | 0.084 | 0.076 | |
| Mz | 0.226 | 0.189 | |
| Maximum Coupling Error | Fx | 2.509 | 1.340 |
| Fy | 1.599 | 0.472 | |
| Fz | 9.867 | 0.953 | |
| Mx | 0.546 | 0.055 | |
| My | 0.243 | 0.059 | |
| Mz | 0.313 | 0.042 |
The results demonstrate that the polynomial fitting decoupling algorithm significantly reduces both linearity and coupling errors compared to the linear method. For instance, the maximum coupling error in the Fz direction decreases from 9.867% to 0.953%, an improvement of 8.914 percentage points. Similarly, linearity errors show modest reductions, such as in Fz from 3.095% to 2.984%, a 0.111 percentage point enhancement. This confirms that the polynomial approach effectively mitigates interdimensional coupling in six-axis force sensors, leading to higher accuracy. The algorithm’s ability to model nonlinearities without structural modifications makes it a practical solution for various applications, from industrial robotics to precision measurement systems.
In conclusion, our research addresses the critical issue of coupling in six-axis force sensors through a novel decoupling algorithm based on polynomial fitting. By analyzing coupling mechanisms and deriving explicit polynomial functions, we achieve substantial error reduction. Experimental validation on an orthogonal parallel six-axis force sensor verifies the algorithm’s superiority over linear decoupling, particularly in minimizing maximum coupling errors. This method offers a software-based solution that enhances sensor precision without incurring additional hardware costs, providing valuable insights for future developments in force sensing technology. Future work could explore higher-order polynomials or machine learning techniques for further refinement, but the current approach establishes a robust foundation for improving six-axis force sensor performance.