In the field of robotics and precision engineering, the accurate measurement of multi-dimensional forces and torques is crucial for applications such as robotic manipulation, haptic feedback, and industrial automation. Traditional six-axis force sensors often face challenges related to structural complexity, sensitivity to manufacturing errors, and limited durability. To address these issues, we propose a novel shunted flexure hinge-based six-axis force sensor that leverages parallel mechanism principles and elastic elements to enhance performance. This paper presents the design, modeling, optimization, and experimental validation of the sensor, focusing on achieving high isotropy and accuracy. The term “six-axis force sensor” is central to this work, as it encapsulates the device’s capability to measure three force components and three torque components simultaneously. Throughout this study, we emphasize the importance of structural integrity and calibration for reliable six-axis force sensor applications.
The development of six-axis force sensors has been extensively researched, with common designs including Stewart platforms and elastic body-based configurations. However, existing sensors often suffer from drawbacks such as friction in joints, sensitivity to preload, and difficulties in calibration. Our approach integrates flexure hinges to eliminate backlash and wear, while a shunted structure distributes loads effectively. This design allows for the use of standard force sensors, enabling easy replacement and reducing downtime. In this work, we detail the sensor’s architecture, derive its static and stiffness models using equivalent and infinitesimal methods, and validate the models through finite element analysis (FEA). Furthermore, we employ a genetic algorithm for structural optimization to maximize isotropy, and conduct calibration experiments to verify measurement accuracy. The results demonstrate that our six-axis force sensor achieves errors below 1.05% in full-scale measurements, making it suitable for heavy-load applications. By addressing key limitations of conventional sensors, this six-axis force sensor offers a robust solution for precise force and torque sensing.
The structure of the proposed six-axis force sensor comprises a base, three elastic measuring limbs, six standard tension-compression force sensors (load cells), a loading platform, and connecting bolts. Each limb is fabricated from high-strength aluminum alloy using wire-cut electrical discharge machining, ensuring precision and durability. The limbs are symmetrically arranged around the base and loading platform, forming a parallel configuration that enhances load distribution. A key feature is the use of flexure hinges, which replace traditional spherical joints to minimize friction and hysteresis. These hinges consist of circular notches that allow elastic deformation, enabling smooth motion without lubrication. The standard force sensors, with a capacity of 50 kg each, are mounted between the limbs and the base, measuring axial forces exclusively due to the aligned revolute joints. This arrangement ensures that only tensile and compressive loads are transmitted to the sensors, improving measurement fidelity. The loading platform connects to the limbs via flexible revolute joints, facilitating the transmission of external loads. When subjected to a six-axis force or torque, the sensor’s limbs deform elastically, and the resulting forces in the load cells are used to compute the external load components through a static model. This design not only simplifies manufacturing but also allows for modular replacement of components, addressing common issues in existing six-axis force sensors.
To establish the static force model of the six-axis force sensor, we analyze the equilibrium of forces and moments acting on the structure. The sensor’s limbs are equivalent to a 3-RPS (revolute-prismatic-spherical) parallel mechanism, where the flexure hinges approximate spherical joints and the elastic segments act as prismatic joints. The external load vector is denoted as $\mathbf{F} = [F_x, F_y, F_z, T_x, T_y, T_z]^T$, where $F_x$, $F_y$, and $F_z$ are the force components, and $T_x$, $T_y$, and $T_z$ are the torque components. The forces measured by the six load cells are represented as $\mathbf{F}_l = [F_{l1}, F_{l2}, F_{l3}, F_{l4}, F_{l5}, F_{l6}]^T$. Using the principle of virtual work and the Jacobian matrix of the equivalent parallel mechanism, we derive the relationship between the external load and the measured forces. The Jacobian matrix $\mathbf{J}$ for the 3-RPS mechanism is given by:
$$ \mathbf{J} = \frac{1}{2} \begin{bmatrix}
0 & 0 & 2 & -e & -\sqrt{3}e & 0 \\
0 & 0 & 2 & 2e & 0 & 0 \\
0 & 0 & 2 & -e & \sqrt{3}e & 0 \\
\sqrt{1} & \sqrt{3} & 0 & 0 & 0 & 2e \\
-2 & 0 & 0 & 0 & 0 & 2e \\
\sqrt{1} & -\sqrt{3} & 0 & 0 & 0 & 2e
\end{bmatrix} $$
where $e$ is the distance from the center of the loading platform to the limb attachment points. The force mapping equation is derived as:
$$ \mathbf{F} = -\mathbf{J}^T \mathbf{C} \mathbf{F}_l $$
where $\mathbf{C}$ is a transformation matrix that accounts for the geometric parameters of the limbs, including the inclination angle $\beta = 20^\circ$ and a deformation coefficient $c$. For instance, $c = \frac{(L – 2l \tan \beta) \cos \beta}{2l}$, where $L$ is the length of the limb and $l$ is a specific dimension related to the flexure hinges. This model enables the computation of the six-axis force and torque components from the load cell readings, providing a foundation for sensor calibration. The accuracy of this static model is critical for the performance of the six-axis force sensor, as it directly influences measurement precision.
The stiffness analysis of the six-axis force sensor is essential for understanding its deformation under load and ensuring structural integrity. The overall stiffness matrix $\mathbf{K}$ relates the external load $\mathbf{F}$ to the deformation vector $\delta \mathbf{d} = [\delta_x, \delta_y, \delta_z, \delta_\alpha, \delta_\beta, \delta_\gamma]^T$, where $\delta_x$, $\delta_y$, $\delta_z$ are linear deformations, and $\delta_\alpha$, $\delta_\beta$, $\delta_\gamma$ are angular deformations. The stiffness model is derived by considering the compliance of the flexure hinges and the elastic limbs. The branch stiffness matrix $\mathbf{K}_p$ is diagonal, with elements corresponding to the stiffness in the actuation and constraint directions. For each limb, the stiffness in the actuation direction $k_{ai}$ and constraint direction $k_{ci}$ are calculated as:
$$ k_{ai} = \frac{2(k_e + k_s)}{\cos \beta} $$
where $k_e$ is the stiffness of the equivalent flexure hinge, and $k_s$ is the stiffness of the standard force sensor. The stiffness $k_{ci}$ is given by:
$$ k_{ci} = \frac{k_z \cos \beta}{l} $$
where $k_z$ is the rotational stiffness of the flexure hinge. For a circular flexure hinge, $k_z$ is computed using the formula:
$$ k_z = \frac{E h r^2}{12 C_r} $$
where $E$ is the elastic modulus of the material, $h$ is the thickness of the hinge, $r$ is the radius of the hinge, and $C_r$ is a coefficient derived from integration over the hinge profile. The overall stiffness matrix $\mathbf{K}$ is then:
$$ \mathbf{K} = \mathbf{J}^T \mathbf{K}_p \mathbf{J} $$
This matrix is symmetric and positive definite, reflecting the sensor’s ability to resist deformation. Using the parameters $r = 1 \, \text{mm}$, $t = 2 \, \text{mm}$, $e = 90 \, \text{mm}$, $h = 24 \, \text{mm}$, $g = 10 \, \text{mm}$, $L = 94 \, \text{mm}$, $l = 59.2 \, \text{mm}$, and $E = 71.7 \, \text{GPa}$, we compute the stiffness values as $k_{ai} = 1.028 \times 10^8 \, \text{N/m}$ and $k_{ci} = 2.355 \times 10^6 \, \text{N/m}$. The deformation under load is expressed as:
$$ \delta_x = 2.83 \times 10^{-7} F_x $$
$$ \delta_y = 2.83 \times 10^{-7} F_y $$
$$ \delta_z = 3.24 \times 10^{-9} F_z $$
$$ \delta_\alpha = 8 \times 10^{-7} T_x $$
$$ \delta_\beta = 8 \times 10^{-7} T_y $$
$$ \delta_\gamma = 1.75 \times 10^{-5} T_z $$
Finite element analysis (FEA) simulations validate these theoretical results, showing close agreement with maximum deviations of approximately 20% due to simplifications in the model. This stiffness analysis ensures that the six-axis force sensor maintains structural stability under operational loads, which is vital for accurate force measurement.
To optimize the performance of the six-axis force sensor, we focus on achieving high isotropy, which ensures uniform sensitivity and stiffness across all measurement directions. Isotropy is quantified using performance indices derived from the force and moment mapping matrices. The force isotropy $\mu_{ff}$ and moment isotropy $\mu_{fm}$ are defined as:
$$ \mu_{ff} = \frac{[\lambda_{\min}(\mathbf{G}_F^T \mathbf{G}_F)]^{1/2}}{[\lambda_{\max}(\mathbf{G}_F^T \mathbf{G}_F)]^{1/2}} $$
$$ \mu_{fm} = \frac{[\lambda_{\min}(\mathbf{G}_M^T \mathbf{G}_M)]^{1/2}}{[\lambda_{\max}(\mathbf{G}_M^T \mathbf{G}_M)]^{1/2}} $$
where $\mathbf{G}_F$ and $\mathbf{G}_M$ are submatrices of the overall mapping matrix $\mathbf{G}$ related to forces and moments, respectively. Similarly, the force sensitivity isotropy $\mu_{sf}$ and moment sensitivity isotropy $\mu_{sm}$ are:
$$ \mu_{sf} = \frac{[\lambda_{\min}(\mathbf{D}_F^T \mathbf{D}_F)]^{1/2}}{[\lambda_{\max}(\mathbf{D}_F^T \mathbf{D}_F)]^{1/2}} $$
$$ \mu_{sm} = \frac{[\lambda_{\min}(\mathbf{D}_M^T \mathbf{D}_M)]^{1/2}}{[\lambda_{\max}(\mathbf{D}_M^T \mathbf{D}_M)]^{1/2}} $$
where $\mathbf{D}$ is the inverse of $\mathbf{G}$. The overall isotropy objective function $f_h$ is a weighted sum:
$$ f_h = k_1 \mu_{ff} + k_2 \mu_{fm} + k_3 \mu_{sf} + k_4 \mu_{sm} $$
with weights $k_1 = k_2 = k_3 = k_4 = 0.25$ for balanced performance. We employ a genetic algorithm to optimize the structural parameters, including the hinge radius $r$, minimum thickness $t$, platform distance $e$, limb thickness $h$, flexure length $g$, limb length $L$, and dimension $l$. The parameter bounds are set based on design constraints: $0.5 \, \text{mm} \leq t \leq 6 \, \text{mm}$, $0.5 \, \text{mm} \leq r \leq 2 \, \text{mm}$, $9 \, \text{mm} \leq g \leq 12 \, \text{mm}$, $20 \, \text{mm} \leq h \leq 30 \, \text{mm}$, $80 \, \text{mm} \leq e \leq 100 \, \text{mm}$, $80 \, \text{mm} \leq L \leq 120 \, \text{mm}$, and $50 \, \text{mm} \leq l \leq 62 \, \text{mm}$. The optimization results show significant improvements in isotropy, as summarized in the table below.
| Parameter | Initial Value (mm) | Optimized Value (mm) |
|---|---|---|
| r | 1 | 1 |
| t | 2 | 2 |
| e | 90 | 85 |
| h | 24 | 25 |
| g | 10 | 10 |
| L | 94 | 118 |
| l | 59.2 | 53 |
| Isotropy Metric | Initial Value | Optimized Value |
|---|---|---|
| μff | 0.2857 | 0.5073 |
| μfm | 0.5714 | 0.9856 |
| μsf | 0.3226 | 0.5073 |
| μsm | 0.5987 | 0.9856 |
| fh | 0.4446 | 0.7465 |
The optimization reveals that parameters $L$ and $l$ have the most significant impact on force isotropy and sensitivity. For instance, setting $L = 200 \, \text{mm}$ and $l = 30 \, \text{mm}$ can achieve ideal isotropy of 1, though practical constraints may limit these values. This optimization enhances the overall performance of the six-axis force sensor, ensuring consistent measurement accuracy across all axes.
Experimental validation is conducted to verify the accuracy and reliability of the six-axis force sensor. A prototype is fabricated based on the optimized parameters, with a designed capacity of $F_x = F_y = \pm 1500 \, \text{N}$, $F_z = \pm 4000 \, \text{N}$, $T_x = T_y = \pm 200 \, \text{N·m}$, and $T_z = \pm 300 \, \text{N·m}$. Calibration experiments involve applying known loads in each direction and recording the output from the six load cells. For example, in the z-direction, a load of 4053 N is applied using a lever system, and the average force measured by the load cells is 260.2 N. This indicates that the elastic limbs分担 approximately 61.5% of the load, demonstrating the shunting effect that allows the use of lower-capacity sensors. The measurement errors for each direction are calculated as follows:
| Loading Direction | Measurement Error (%) |
|---|---|
| Fx | 0.58 |
| Fy | 0.61 |
| Fz | 0.38 |
| Tx | 1.05 |
| Ty | 0.98 |
| Tz | 0.71 |
Additionally, a practical test is performed by mounting the sensor on a 3-SPR parallel manipulator equipped with robotic grippers. An unknown mass (a sand bucket) is grasped, and the sensor outputs are used to compute the load components. The calculated force in the z-direction is $-65.91 \, \text{N}$, compared to the actual mass of 6.8 kg (equivalent to $66.64 \, \text{N}$), resulting in an error of 0.38%. This confirms that the sensor can be used without recalibration after component replacement, highlighting its practicality for industrial applications. The experimental results validate the static model and demonstrate that the six-axis force sensor meets the required precision for diverse loading conditions.
In conclusion, we have developed a shunted flexure hinge-based six-axis force sensor that addresses common limitations of existing designs. Through theoretical modeling, stiffness analysis, structural optimization, and experimental validation, we have demonstrated its high accuracy and robustness. The sensor achieves measurement errors below 1.05% in full-scale tests and exhibits excellent isotropy after optimization. The use of flexure hinges eliminates backlash and wear, while the shunted structure enables load distribution and modular maintenance. This six-axis force sensor is suitable for applications requiring precise force and torque measurement, such as robotic systems and industrial automation. Future work may focus on miniaturization and dynamic response analysis to further enhance its capabilities. The successful implementation of this six-axis force sensor underscores the potential of parallel mechanisms and elastic elements in advancing sensor technology.