In the field of robotics, aerospace, and advanced manufacturing, the ability to accurately measure multi-dimensional forces and torques is critical. Traditional six-axis force sensors often require complex and tedious calibration procedures after fabrication, which can be time-consuming and costly. To address this issue, I propose a novel design concept for a self-calibrating six-axis force sensor that eliminates the need for extensive external calibration. This sensor utilizes a parallel mechanism with mechanical decoupling through rolling steel balls, enabling weak coupling and full thrust force measurement. The design incorporates an orthogonal arrangement of branches to achieve high stiffness and minimal inter-dimensional coupling, facilitating self-calibration. In this article, I will detail the design principles, mathematical modeling, force analysis, simulation results, and validation of the self-calibration特性 for this innovative six-axis force sensor.
The self-calibration concept for multi-dimensional force sensors is centered on the idea that after fabrication, the sensor does not require comprehensive multi-axis loading calibration. Instead, by leveraging simple geometric relationships and internal single-axis force sensor readings, the full external force information can be derived. This approach necessitates that the sensor structure has minimal coupling between measurement branches, allowing forces to be transmitted unidirectionally. For a six-axis force sensor, this means that each component of the external force and torque can be independently measured by specific branches, reducing the complexity of calibration to that of calibrating individual single-axis sensors. The key conditions for self-calibration include the use of single-axis force sensors in each branch, a clear geometric configuration, and low interference between branches.
The structure of the proposed six-axis force sensor consists of a loading plate, a connection plate, a cover plate, a fixed base, and 16 decoupled measurement branches. The fixed base forms a frame structure that, when connected to the cover plate via bolts, creates an enclosed housing. The loading plate is attached to the connection plate using four studs, and the connection plate is linked to both the cover plate and the fixed base through the 16 branches. These branches are distributed orthogonally across the six faces of the connection plate: four on the top, four on the bottom, and two on each of the four side faces. The branches on the same face are parallel, while those on adjacent faces are perpendicular. Each branch includes a single-axis force sensor, an upper curved surface, a lower curved surface, and a steel ball. The single-axis force sensor is fixed at one end to the base or cover plate and at the other end to the planar side of the lower curved surface. The upper curved surface is attached to the connection plate, and its curved end, along with the curved end of the lower surface, encapsulates the steel ball, allowing rolling motion for mechanical decoupling. This symmetric configuration enhances structural integrity and force transmission.
The self-calibration principle of this six-axis force sensor is based on the orthogonal layout of the branches. The eight vertical branches on the top and bottom faces measure the axial force along the z-axis and the moments about the x and y axes. The eight horizontal branches on the side faces measure the axial forces along the x and y axes and the moment about the z-axis. This arrangement ensures that external loads are primarily transmitted to the corresponding branches, minimizing cross-talk. The relationship between external loads and the measurement branches is summarized in Table 1.
| External Load | Corresponding Branches | External Load | Corresponding Branches |
|---|---|---|---|
| F_x | 3, 4, 7, 8 | T_x | 9, 10, 11, 12, 13, 14, 15, 16 |
| F_y | 1, 2, 5, 6 | T_y | 9, 10, 11, 12, 13, 14, 15, 16 |
| F_z | 9, 10, 11, 12, 13, 14, 15, 16 | T_z | 1, 2, 3, 4, 5, 6, 7, 8 |
To establish the mathematical model of the six-axis force sensor, I employ screw theory, which provides a framework for analyzing the statics of parallel mechanisms. The ideal model assumes that the branches and connections are rigid, neglecting elastic deformations initially. The static equilibrium equation for the sensor is given by:
$$ \mathbf{F}_W + \sum_{i=1}^{16} f_i \mathbf{\$}_i = 0 $$
where \(\mathbf{F}_W = (\mathbf{F}, \mathbf{T})^T\) is the generalized external force applied to the connection plate, \(f_i\) is the internal force in the i-th branch, and \(\mathbf{\$}_i = (\mathbf{S}_i, \mathbf{S}_{0i})^T\) is the unit screw of the i-th branch. Here, \(\mathbf{S}_i = (\mathbf{A}_i – \mathbf{B}_i) / |\mathbf{A}_i – \mathbf{B}_i|\) and \(\mathbf{S}_{0i} = (\mathbf{B}_i \times \mathbf{A}_i) / |\mathbf{A}_i – \mathbf{B}_i|\), with \(\mathbf{A}_i\) and \(\mathbf{B}_i\) being the position vectors of the connection points on the connection plate and fixed base, respectively. This can be expressed in matrix form as:
$$ \begin{bmatrix} \mathbf{F} \\ \mathbf{T} \end{bmatrix} + \begin{bmatrix} \mathbf{S}_1 & \mathbf{S}_2 & \ldots & \mathbf{S}_{16} \\ \mathbf{S}_{01} & \mathbf{S}_{02} & \ldots & \mathbf{S}_{016} \end{bmatrix} \begin{bmatrix} f_1 \\ f_2 \\ \vdots \\ f_{16} \end{bmatrix} = 0 $$
Thus, the force mapping relationship is:
$$ \mathbf{F}_W = \mathbf{G} \mathbf{f} $$
where \(\mathbf{G}\) is the first-order static influence coefficient matrix, and \(\mathbf{f} = [f_1, f_2, \ldots, f_{16}]^T\). For the orthogonal 16-branch configuration with parameters such as connection plate width \(a = 2 \, \text{m}\), length \(b = 2 \, \text{m}\), and branch spacings \(c = 1.7 \, \text{m}\) and \(d = 1.7 \, \text{m}\), the matrix \(\mathbf{G}\) is derived as:
$$ \mathbf{G} = \begin{bmatrix}
0 & 0 & 1 & 1 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\frac{d}{2} & -\frac{d}{2} & \frac{d}{2} & \frac{d}{2} & \frac{d}{2} & \frac{d}{2} & -\frac{d}{2} & -\frac{d}{2} \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\frac{c}{2} & \frac{c}{2} & \frac{c}{2} & -\frac{c}{2} & \frac{c}{2} & -\frac{c}{2} & -\frac{c}{2} & \frac{c}{2} \\
\frac{a}{2} & -\frac{a}{2} & \frac{b}{2} & -\frac{b}{2} & \frac{a}{2} & -\frac{a}{2} & \frac{b}{2} & -\frac{b}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix} $$
This matrix illustrates the linear relationship between external loads and branch forces, highlighting the decoupling characteristics of the six-axis force sensor. However, in practical applications, elastic deformations of the branches must be considered, as the structure is highly statically indeterminate. I analyze the force distribution under typical loading conditions, such as horizontal force and moment loading, using principles of mechanics for statically indeterminate structures.
For horizontal force loading along the y-axis, the connection plate displaces by \(\Delta_1\) in the direction of the force and \(\Delta_2\) vertically due to pre-tension and symmetry. The deformation compatibility equations for the vertical branches (9 to 16) and horizontal branches (3, 4, 7, 8 and 1, 2, 5, 6) are derived based on geometric relationships. For vertical branches:
$$ (l – \Delta_2)^2 + \Delta_1^2 = \left( l – \frac{f_a l}{E A} \right)^2 $$
where \(f_a\) is the axial force in branches 9-16, \(l\) is the equivalent branch length, \(E\) is the elastic modulus, and \(A\) is the cross-sectional area. For horizontal branches perpendicular to the force:
$$ l^2 + \Delta_1^2 = \left( l + \frac{f_b l}{E A} \right)^2 $$
where \(f_b\) is the axial force in branches 3, 4, 7, 8. For branches parallel to the force, such as branches 1 and 2:
$$ (l – \Delta_1)^2 + \Delta_2^2 = \left( l – \frac{f_c l}{E A} \right)^2 $$
and for branches 5 and 6:
$$ (l + \Delta_1)^2 + \Delta_2^2 = \left( l + \frac{f_d l}{E A} \right)^2 $$
The static equilibrium equations in the z and y directions are:
$$ \sum F_z = \frac{8 f_a \Delta_1}{l + \frac{f_a l}{E A}} + \frac{4 f_b \Delta_1}{l + \frac{f_b l}{E A}} + \frac{2 f_c (l – \Delta_1)}{l + \frac{f_c l}{E A}} + \frac{2 f_d (l + \Delta_1)}{l + \frac{f_d l}{E A}} = 0 $$
$$ \sum F_y = \frac{8 f_a (l – \Delta_1)}{l + \frac{f_a l}{E A}} – \frac{4 f_b \Delta_1}{l + \frac{f_b l}{E A}} + \frac{2 f_c \Delta_1}{l + \frac{f_c l}{E A}} – \frac{2 f_d \Delta_1}{l + \frac{f_d l}{E A}} = 0 $$
These equations are solved numerically to determine the force distribution, demonstrating the decoupling behavior of the six-axis force sensor.
For moment loading about the z-axis, the connection plate undergoes rotational displacement, and the branches experience tangential deformations. The deformation compatibility for vertical branches (9-16) is:
$$ \Delta_9 = \frac{f_{ta} l}{G A} $$
where \(f_{ta}\) is the tangential force, and \(G\) is the shear modulus. For horizontal branches perpendicular to the moment axis (3, 4, 7, 8):
$$ \Delta_{10} = \frac{f_{tb} l}{G A} $$
For branches parallel to the moment axis (1, 2, 5, 6):
$$ (l + \Delta_{12})^2 + \left( \frac{f_{tc} l}{G A} \right)^2 = \left( l + \frac{f_{a5} l}{E A} \right)^2 $$
The moment equilibrium equation is:
$$ \sum T = f_{ta} c + f_{tb} a + f_{tc} b – F_z c = 0 $$
These equations are used to analyze the force distribution under moment loading, further validating the decoupling capability of the sensor.
To verify the self-calibration特性, I conduct numerical simulations and analyses. Assuming branch axial stiffness values of \(1.884 \times 10^9 \, \text{m/N}\) for horizontal branches and \(4.884 \times 10^9 \, \text{m/N}\) for vertical branches, and other parameters as mentioned, the force mapping model is applied under single-axis and composite loading conditions. The input-output relationship is given by:
$$ \mathbf{f}’ = \mathbf{\Lambda} \mathbf{F}_W $$
where \(\mathbf{f}’\) is the vector of branch sensor outputs (including preload), and \(\mathbf{\Lambda}\) is the sensitivity matrix derived from \(\mathbf{G}\). For single-axis force loading along the x-axis, the branch forces are computed, and the results show that branches parallel to the load (3, 4, 7, 8) have significant outputs, while perpendicular branches have minimal responses, indicating low coupling. Table 2 presents the branch outputs and errors for x-axis force loading up to 20 kN.
| Load (kN) | Branch 3 (kN) | Branch 4 (kN) | Branch 7 (kN) | Branch 8 (kN) | Measured Force (kN) | Error (%) |
|---|---|---|---|---|---|---|
| 2.5 | 0.67 | 0.65 | -0.60 | -0.59 | 2.48 | 0.10 |
| 5.0 | 1.31 | 1.28 | -1.22 | -1.23 | 5.04 | 0.20 |
| 7.5 | 1.95 | 1.93 | -1.83 | -1.84 | 7.55 | 0.25 |
| 10.0 | 2.58 | 2.55 | -2.48 | -2.48 | 10.08 | 0.40 |
| 12.5 | 3.18 | 3.17 | -3.12 | -3.15 | 12.62 | 0.60 |
| 15.0 | 3.81 | 3.79 | -3.76 | -3.75 | 15.11 | 0.55 |
| 17.5 | 4.85 | 4.79 | -3.91 | -3.83 | 17.38 | 0.60 |
| 20.0 | 5.88 | 5.82 | -4.68 | -3.79 | 20.17 | 0.85 |
For single-axis moment loading about the x-axis, the branch outputs for vertical branches (9-16) are analyzed, and errors are computed, as shown in Table 3. The results confirm that the sensor can accurately measure moments with minimal cross-talk.
| Load (kN·m) | Branch 9 (kN) | Branch 10 (kN) | Branch 11 (kN) | Branch 12 (kN) | Branch 13 (kN) | Branch 14 (kN) | Branch 15 (kN) | Branch 16 (kN) | Measured Moment (kN·m) | Error (%) |
|---|---|---|---|---|---|---|---|---|---|---|
| 2.5 | 0.76 | 0.76 | -0.74 | -0.73 | -0.73 | -0.73 | 0.75 | 0.76 | 2.53 | 0.15 |
| 5.0 | 1.49 | 1.48 | -1.46 | -1.47 | -1.46 | -1.47 | 1.49 | 1.48 | 5.04 | 0.20 |
| 7.5 | 2.26 | 2.24 | -2.19 | -2.20 | -2.20 | -2.18 | 2.24 | 2.28 | 7.57 | 0.35 |
| 10.0 | 2.98 | 2.98 | -2.91 | -2.93 | -2.93 | -2.92 | 2.96 | 2.97 | 10.07 | 0.35 |
| 12.5 | 3.70 | 3.68 | -3.64 | -3.65 | -3.65 | -3.64 | 3.69 | 3.71 | 12.62 | 0.60 |
| 15.0 | 4.48 | 4.46 | -4.38 | -4.38 | -4.40 | -4.39 | 4.48 | 4.47 | 15.10 | 0.50 |
| 17.5 | 5.20 | 5.19 | -5.13 | -5.13 | -5.16 | -5.14 | 5.18 | 5.18 | 17.11 | 0.55 |
| 20.0 | 6.03 | 6.10 | -5.82 | -5.88 | -6.00 | -5.80 | 6.11 | 5.88 | 20.17 | 0.85 |
For composite loading, such as simultaneous forces along the x and y axes, the branch outputs are analyzed, and the results show that forces are primarily measured by the corresponding branches, with negligible interference. This further demonstrates the decoupling performance of the six-axis force sensor, enabling self-calibration.
To validate the self-calibration principle, I perform simulations using RecurDyn software, where the sensor model is imported and flexible body analysis is conducted. The simulation model is subjected to an x-axis force of 20 kN, and the branch forces are recorded. The theoretical calculations and simulation results are compared, as shown in Figure 1 for a single branch (Branch 3) and Figure 2 for the overall sensor response. The error between theoretical and simulation results is less than 0.1%, confirming the accuracy of the model and the self-calibration capability.
In conclusion, the mechanically decoupled self-calibrating parallel six-axis force sensor presented in this article offers a innovative solution to the challenges of traditional calibration methods. The design incorporates rolling steel balls for mechanical decoupling and an orthogonal branch layout for minimal coupling. The mathematical model based on screw theory provides a clear force mapping relationship, and the analysis of elastic deformations ensures accurate force distribution. Numerical and simulation results validate the decoupling特性 and self-calibration特性, with errors within acceptable limits. This work lays the foundation for the development of advanced six-axis force sensors, particularly for heavy-load applications, and provides a reference for future research in multi-dimensional force measurement. The self-calibration approach significantly reduces the time and cost associated with sensor calibration, making it a practical choice for industrial and scientific applications.