In recent years, the demand for high-performance six-axis force sensors has grown significantly across various fields, including robotics, aerospace, and precision engineering. Traditional elastic body-based six-axis force sensors often suffer from low natural frequencies and limited dynamic ranges, typically below 2 kHz, which restricts their application in high-dynamic scenarios. To address these limitations, we have developed a self-preloaded piezoelectric six-axis force sensor that leverages multiple quartz crystal groups for force and torque measurement. This design eliminates the need for complex elastic structures, thereby enhancing dynamic performance. In this article, we present a comprehensive dynamic mathematical model of the sensor, analyze its performance using an atlas-based method, and validate the results through finite element simulations and experimental tests. The key focus is on optimizing the sensor’s sensitivity and natural frequency, with repeated emphasis on the term “six-axis force sensor” to underscore its relevance.
The self-preloaded piezoelectric six-axis force sensor utilizes a central compression structure with pre-tightening bolts, ensuring inherent preload capability without external components. This design simplifies the assembly process and improves reliability. The sensor comprises several key components: a preload nut, upper cover, outer shell, output electrodes, quartz crystal groups, preload bolts, and a base. The quartz crystals are arranged symmetrically on the base’s distribution circle, with alternating 0°Y-cut and 0°X-cut types to measure forces and moments along the X, Y, and Z axes. The working principle relies on the piezoelectric effect, where applied forces or moments cause deformation in the quartz crystals, generating electrical signals that are linearly decoupled to obtain the six-axis components. This non-elastic body approach significantly boosts the sensor’s dynamic response compared to conventional strain-based designs.
To quantitatively analyze the sensor’s performance, we established a dynamic mathematical model that accounts for the multi-degree-of-freedom vibration system. The sensor is modeled as a mass-spring system, neglecting damping effects due to the minimal energy absorption in piezoelectric materials under dynamic conditions. The general form of the vibration differential equation for the system is given by:
$$[m]\{\ddot{x}\} + [k]\{x\} = \{0\}$$
where [m] is the equivalent mass matrix, {x} is the displacement vector, [k] is the equivalent stiffness matrix, and {0} represents the zero force vector under free vibration. Assuming harmonic motion, the solution takes the form:
$$x_i = A_i \sin(\omega_{ni} t + \phi_i)$$
where A_i is the amplitude, ω_ni is the natural frequency, and φ_i is the phase angle. Substituting this into the differential equation leads to the characteristic equation:
$$|\mathbf{k} – \omega_{ni}^2 \mathbf{m}| = 0$$
Solving this eigenvalue problem yields the natural frequencies of the sensor, with the smallest value defining the overall natural frequency. The equivalent stiffness and mass for each component are derived from classical mechanics. For instance, the equivalent mass of the upper cover (m1) is calculated as:
$$m_1 = \frac{\rho_3}{6} \left( \pi r_1^2 b_1 – 4\pi r_7^2 b_7 – \pi r_6^2 b_1 \right)$$
where ρ_3 is the material density, r_1, r_6, and r_7 are radii, and b_1 and b_7 are thicknesses. The equivalent stiffness (k1) for different force directions varies; for example, under Z-axis force (F_z), it is expressed as:
$$k_1 = \frac{E_3 (\pi r_1^2 – 4\pi r_7^2 – \pi r_6^2)}{2b_1}$$
where E_3 is the elastic modulus. Similarly, for moment directions, the stiffness involves additional terms based on geometry and material properties.
The sensitivity of the six-axis force sensor is crucial for static performance. The force and moment sensitivities (S_FQ and S_MQ) are derived from the piezoelectric coefficients and structural parameters:
$$S_{FQ} = \frac{n}{N} d_{ij} \frac{k_a + k_c}{k_a + k_b + k_c}$$
$$S_{MQ} = \frac{n}{N} d_{ij} \frac{k_a + k_c}{k_a + k_b + k_c} R$$
Here, n is the number of sensitive quartz groups for a specific direction, N is the total number of groups (8 in this design), d_ij is the piezoelectric coefficient, k_a, k_b, and k_c are equivalent stiffnesses of different paths, and R is the moment radius. These formulas highlight the dependence of sensitivity on the sensor’s structural dimensions.
To analyze the impact of structural parameters on performance, we employed an atlas-based method, generating four-dimensional plots that relate sensitivity and natural frequency to parameters like upper cover thickness, base thickness, and quartz crystal radius. The following table summarizes key parameters used in the analysis:
| Component | Thickness (mm) | Radius (mm) | Ring Radius (mm) |
|---|---|---|---|
| Quartz Crystal | 0.075 | 7.5 | – |
| Upper Cover | 8.0 | – | 28 |
| Preload Bolt | 15 | 6.0 | – |
| Base | 6.0 | 29 | – |
| Outer Shell | 15.2 | – | 30.3 |
| Crystal Position | – | 20.5 | – |
From the atlas analysis, we observed that increasing the upper cover thickness enhances sensitivity but reduces natural frequency, whereas the upper cover radius has a mixed effect: it improves force sensitivity but degrades moment sensitivity. The base thickness positively influences sensitivity but negatively affects natural frequency, while the preload bolt radius increases natural frequency at the cost of reduced sensitivity. The quartz crystal radius has a more pronounced impact than electrode thickness, with larger radii favoring both sensitivity and natural frequency. These insights guide the optimization of the six-axis force sensor for specific applications.
For experimental validation, we fabricated a prototype based on the optimized parameters and conducted dynamic tests using a hammer impact method. The experimental setup involved applying impulsive forces and moments along the six axes and measuring the frequency response through Fourier analysis. The results, summarized in the table below, confirm the high natural frequencies of the sensor:
| Direction | Natural Frequency (kHz) |
|---|---|
| F_x | 10.2 |
| F_y | 9.8 |
| F_z | 22.1 |
| M_x | 21.9 |
| M_y | 22.3 |
| M_z | 9.6 |
The overall natural frequency is defined as the minimum value, 9.6 kHz, resulting in a dynamic range of approximately 3.2 kHz (one-third of the natural frequency). This exceeds the typical 2 kHz range of elastic body six-axis force sensors, demonstrating the superiority of our design. The finite element simulations, performed using ANSYS, aligned closely with the experimental results, with harmonic response analyses showing natural frequencies of 10.8 kHz for F_x, 20.4 kHz for F_z, and up to 30.8 kHz for M_z. The model incorporated SOLID98 elements for quartz crystals and SOLID186 for other components, with material properties set accordingly.
In conclusion, the self-preloaded piezoelectric six-axis force sensor offers a robust solution for high-dynamic applications. The dynamic mathematical model provides a precise tool for predicting performance, while the atlas-based method enables effective parameter optimization. Experimental tests validate the sensor’s high natural frequency and extended dynamic range, making it suitable for advanced robotics, aerospace, and other fields requiring accurate six-axis force measurements. Future work will focus on further refining the model and exploring miniaturization for broader applicability. This research underscores the importance of innovative designs in enhancing the capabilities of six-axis force sensors.
The development of this six-axis force sensor involved extensive analysis of its dynamic characteristics. The equivalent stiffness matrix [k] and mass matrix [m] were derived from the sensor’s geometry, leading to the characteristic equation for natural frequencies. For example, the stiffness k_a for the left branch (masses m2, m3, m4) is calculated as a series combination:
$$k_a = \left( \frac{1}{k_2} + \frac{1}{k_3} + \frac{1}{k_4} \right)^{-1}$$
Similarly, the sensitivity formulas were used to generate performance atlases, illustrating how changes in parameters like upper cover thickness (b1) affect F_x sensitivity and natural frequency. The mathematical model also accounts for the piezoelectric constants, with d_ij values specific to the quartz cut types. The decoupling matrix for output signals is given by:
$$
\begin{bmatrix}
F_x \\ F_y \\ F_z \\ M_x \\ M_y \\ M_z
\end{bmatrix}
\propto
\begin{bmatrix}
f_5 – f_1 \\ f_3 – f_7 \\ f_2 + f_4 + f_6 + f_8 \\ (f_2 + f_8) – (f_4 + f_6) \\ (f_6 + f_8) – (f_2 + f_4) \\ f_1 + f_3 + f_5 + f_7
\end{bmatrix}
$$
where f_i represents the output from the i-th quartz group. This linear relationship ensures minimal cross-coupling errors, a critical advantage for precise six-axis force sensor applications. The integration of these elements into a cohesive model facilitates the design of sensors with tailored performance, meeting the evolving demands of modern technology.