In the field of robotics, aerospace, and precision engineering, the ability to accurately measure multi-dimensional forces and moments is crucial for applications such as robotic tactile sensing, foot force measurement in humanoid robots, and space station docking. Among various types of multi-axis force sensors, the washer-type piezoelectric six-axis force sensor stands out due to its dynamic response characteristics and ability to overcome bottlenecks associated with elastic body-based sensors. However, achieving optimal static sensitivity in such sensors requires careful design and optimization of multiple structural parameters. In this article, I will present a comprehensive analysis of the static sensitivity of a washer-type piezoelectric six-axis force sensor, focusing on the development of an analytical mathematical model, its validation through numerical simulations, and the application of orthogonal experiments for parameter optimization. The study aims to provide a foundation for multi-parameter active design of these sensors, ensuring high performance in real-world applications.
The washer-type piezoelectric six-axis force sensor typically consists of components such as an outer shell, upper and lower covers, an inner cylinder, quartz crystal groups, and electrodes. The quartz crystal groups, arranged in an alternating pattern of X0° and Y0° cut types, serve as both sensing and transduction elements, converting external forces into electrical charge outputs. The X0° cut quartz crystals are sensitive only to axial forces, while the Y0° cut types respond to shear forces. These are distributed uniformly between the upper and lower covers, with four groups of each type placed at specific positions on a distribution circle of radius R. The distance from the force application point to the quartz crystal groups is denoted as h, which influences the sensor’s response to moments. The structural configuration ensures that the sensor can decouple and measure all six components of force and torque: Fx, Fy, Fz, Mx, My, and Mz. Understanding the internal load transfer path is essential for deriving the static sensitivity model, as the forces and moments are transmitted through various components like elastic membranes, the inner cylinder, and the housing, each contributing to the overall stiffness and sensitivity.
To derive the analytical mathematical model for the static sensitivity of the washer-type piezoelectric six-axis force sensor, I start by analyzing the load transfer mechanism within the sensor. When a six-dimensional force or moment is applied, it is distributed among the eight quartz crystal groups, resulting in partial forces such as FX1, FX5, FY3, FY7, FZ2, FZ4, FZ6, and FZ8. The sensor’s output charges, denoted as QFx, QFy, QFz, QMx, QMy, and QMz, are related to these partial forces through the piezoelectric coefficients d11 and d26 for the X0° and Y0° cut crystals, respectively. The basal area of the quartz crystal groups is S, and the electrode area is Se. The mapping between the applied forces/moments and the sensor outputs can be expressed as:
$$ \begin{bmatrix} Q_{Fx} \\ Q_{Fy} \\ Q_{Fz} \\ Q_{Mx} \\ Q_{My} \\ Q_{Mz} \end{bmatrix} = \begin{bmatrix} f_x \\ f_y \\ f_z \\ m_x \\ m_y \\ m_z \end{bmatrix} \cdot C_Q $$
where CQ is the static sensitivity matrix. Expanding this matrix yields:
$$ C_Q = \begin{bmatrix} 0.5 k_{fx} d_{26} S_e / S & 0 & 0 & 0 & b k_{my} d_{11} S_e / (r S) & 0 \\ 0 & 0.5 k_{fy} d_{26} S_e / S & 0 & -b k_{mx} d_{11} S_e / (r S) & 0 & 0 \\ 0 & 0 & k_{fz} d_{11} S_e / S & 0 & 0 & 0 \\ 0 & 0 & 0 & 4 k_{mx} d_{11} S_e / (3 r S) & 0 & 0 \\ 0 & 0 & 0 & 0 & 4 k_{my} d_{11} S_e / (3 r S) & 0 \\ 0 & 0 & 0 & 0 & 0 & k_{mz} d_{26} S_e / (R S) \end{bmatrix} $$
Here, kfx, kfy, kfz, kmx, kmy, and kmz are the load transfer coefficients, which account for the proportion of the applied force that effectively acts on the quartz crystals due to factors like manufacturing precision and assembly. The parameters b and r relate to the sensor’s geometry, such as the height and radius of components. The load transfer coefficients are derived from the stiffness of various sensor components, including the outer shell, elastic membranes, inner cylinder, and quartz crystals. For instance, when a force Fx is applied, the equivalent stiffness of different paths (e.g., through the outer shell, upper cover, and quartz groups) can be modeled using Hooke’s law. The equivalent stiffness for each component is calculated based on material properties like Young’s modulus E, Poisson’s ratio μ, thickness b, and radius r. For example, the stiffness k for a component under shear load is given by k = G S / L, where G is the shear modulus (G = E / (2 + 2μ)), S is the loaded area, and L is the thickness. For compression deformation, k = E S / L. The load transfer coefficient for a given direction, such as kfx, is then expressed as the ratio of the stiffness of the path through the quartz crystals to the total stiffness of all parallel paths.
To validate the analytical mathematical model, I compared its results with those from a numerical model based on ANSYS software. Nine different sets of structural parameters were used, as shown in Table 1, which vary dimensions like thicknesses of elastic membranes (b2, b3), quartz crystal radius (r1), and other geometric factors. For each set, the static sensitivity was computed using both the analytical model and the numerical simulation. In the numerical model, the potential difference U across the quartz crystals was converted to charge using Q = C U, where C is the capacitance, and then mapped to the six force/moment components. The results for the six-axis force sensor sensitivities are summarized in Table 2, showing close agreement between the two methods.
| Set | b2 | b3 | b5 | b6 | r1 | r3 | r6 | r7 | a1 |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 0.3 | 0.3 | 0.3 | 0.3 | 3.5 | 8 | 20 | 21 | 45 |
| 2 | 0.3 | 0.3 | 0.3 | 0.3 | 3.5 | 8 | 20 | 21.5 | 46 |
| 3 | 0.3 | 0.3 | 1.0 | 1.0 | 3.5 | 8 | 20 | 21 | 45 |
| 4 | 0.5 | 0.5 | 0.5 | 0.5 | 3.5 | 8 | 20 | 21 | 45 |
| 5 | 0.3 | 0.3 | 0.3 | 0.3 | 3.5 | 8 | 20 | 21 | 45 |
| 6 | 0.3 | 0.3 | 0.3 | 0.3 | 3.5 | 8 | 20 | 21 | 45 |
| 7 | 0.3 | 0.3 | 0.3 | 0.3 | 3.5 | 7.5 | 20 | 21 | 45 |
| 8 | 0.3 | 0.3 | 0.3 | 0.3 | 3.5 | 7.5 | 20 | 21 | 46 |
| 9 | 0.3 | 0.5 | 0.5 | 0.5 | 3.5 | 8 | 20 | 21 | 45 |
| Set | Model | Fx | Fy | Fz | Mx | My | Mz |
|---|---|---|---|---|---|---|---|
| 1 | Analytical | 0.613 | 0.613 | 0.964 | 181.707 | 181.707 | 257.948 |
| 1 | Numerical | 1.685 | 1.685 | 2.154 | 376.654 | 376.957 | 359.755 |
| 2 | Analytical | 0.701 | 0.701 | 1.000 | 188.536 | 188.536 | 317.653 |
| 2 | Numerical | 1.893 | 1.893 | 2.184 | 385.889 | 385.889 | 431.916 |
| 3 | Analytical | 0.486 | 0.486 | 0.916 | 172.788 | 172.788 | 200.744 |
| 3 | Numerical | 1.363 | 1.364 | 2.099 | 364.383 | 364.383 | 267.304 |
| 4 | Analytical | 0.506 | 0.506 | 0.907 | 171.110 | 171.110 | 222.213 |
| 4 | Numerical | 1.463 | 1.463 | 2.085 | 361.059 | 361.061 | 290.411 |
| 5 | Analytical | 0.613 | 0.613 | 0.964 | 181.707 | 181.707 | 257.948 |
| 5 | Numerical | 1.697 | 1.697 | 2.153 | 376.114 | 376.119 | 362.967 |
| 6 | Analytical | 0.605 | 0.605 | 0.954 | 180.211 | 180.282 | 268.866 |
| 6 | Numerical | 1.705 | 1.705 | 2.152 | 375.967 | 376.143 | 365.589 |
| 7 | Analytical | 0.600 | 0.600 | 0.957 | 180.368 | 180.368 | 256.124 |
| 7 | Numerical | 1.611 | 1.611 | 2.147 | 376.174 | 376.036 | 357.946 |
| 8 | Analytical | 0.590 | 0.590 | 0.952 | 179.544 | 179.544 | 255.146 |
| 8 | Numerical | 1.588 | 1.588 | 2.143 | 375.116 | 375.095 | 351.053 |
| 9 | Analytical | 0.517 | 0.517 | 0.910 | 171.505 | 171.505 | 222.713 |
| 9 | Numerical | 1.490 | 1.490 | 2.096 | 361.328 | 361.394 | 290.537 |
To quantitatively assess the correlation between the analytical and numerical models, I performed a statistical analysis using the sample correlation coefficient r and a T-test for significance. The results, shown in Table 3, indicate that for all six dimensions of the six-axis force sensor, the correlation coefficients exceed 0.98, demonstrating a strong positive linear relationship. The T-test statistics are all greater than the critical value of 2.365 at a significance level of α = 0.025, leading to the rejection of the null hypothesis that there is no linear correlation. Thus, the analytical mathematical model is valid and can be reliably used for sensitivity analysis of the washer-type piezoelectric six-axis force sensor.
| Dimension | Sample Correlation Coefficient (r) | T-test Statistic |
|---|---|---|
| Fx | 0.981 | 13.248 |
| Fy | 0.981 | 13.282 |
| Fz | 0.988 | 16.692 |
| Mx | 0.994 | 23.197 |
| My | 0.994 | 23.950 |
| Mz | 0.988 | 16.705 |
| Overall | 0.988 | 17.846 |
Given the complexity of optimizing multiple structural parameters for the six-axis force sensor, I employed an orthogonal experimental design to systematically study their effects on sensitivity. This approach allows for the consideration of multiple factors simultaneously with a reduced number of experiments, providing insights into the influence of each parameter and identifying optimal combinations. The factors selected for the orthogonal experiment include the quartz crystal radius r1, thickness b7, thicknesses of the inner and outer elastic membranes b2 and b3, and the width dimensions r5 and r6 that affect the elastic membranes. These parameters were varied at four levels, as detailed in Table 4, while keeping other geometric constraints constant, such as the sensor side length a1 = 50 mm, housing wall thickness = 2 mm, inner cylinder height b9 = 12.05 mm, inner cylinder radius r3 = 7.5 mm, and inner cylinder wall thickness = 0.5 mm.
| Level | b2 | b3 | r5 | r6 | r1 | b7 |
|---|---|---|---|---|---|---|
| 1 | 0.3 | 0.3 | 10.0 | 20.00 | 3.5 | 0.75 |
| 2 | 0.4 | 0.4 | 9.8 | 20.25 | 4.0 | 0.85 |
| 3 | 0.5 | 0.5 | 9.6 | 20.50 | 4.5 | 1.00 |
| 4 | 0.6 | 0.6 | 9.4 | 20.75 | 5.0 | 1.20 |
Using the L32(4^9) orthogonal array, I conducted 32 experimental trials, with the sensitivity index S defined as the product of the minimum force sensitivity and the minimum moment sensitivity across the three axes. The results, summarized in Table 5, were analyzed using range analysis to determine the influence of each factor. The average values K̄i for each factor at different levels and the range Ri (difference between maximum and minimum K̄i) are presented in Table 6. A larger Ri indicates a stronger influence of the factor on the sensitivity of the six-axis force sensor.
| Trial | b2 | b3 | r5 | r6 | r1 | b7 | S (pC²/N²·m) |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 232.5225 |
| 2 | 1 | 2 | 2 | 2 | 2 | 2 | 202.7267 |
| 3 | 1 | 3 | 3 | 3 | 3 | 3 | 172.7754 |
| 4 | 1 | 4 | 4 | 4 | 4 | 4 | 144.5041 |
| 5 | 2 | 1 | 1 | 2 | 2 | 3 | 217.5655 |
| 6 | 2 | 2 | 2 | 1 | 1 | 4 | 131.9425 |
| 7 | 2 | 3 | 3 | 4 | 4 | 1 | 228.8109 |
| 8 | 2 | 4 | 4 | 3 | 3 | 2 | 171.5711 |
| 9 | 3 | 1 | 2 | 3 | 4 | 1 | 321.0699 |
| 10 | 3 | 2 | 1 | 4 | 3 | 2 | 209.1862 |
| 11 | 3 | 3 | 4 | 1 | 2 | 3 | 162.8120 |
| 12 | 3 | 4 | 3 | 2 | 1 | 4 | 87.2172 |
| 13 | 4 | 1 | 2 | 4 | 3 | 3 | 224.6981 |
| 14 | 4 | 2 | 1 | 3 | 4 | 4 | 205.4642 |
| 15 | 4 | 3 | 4 | 2 | 1 | 1 | 153.5501 |
| 16 | 4 | 4 | 3 | 1 | 2 | 2 | 161.7973 |
| 17 | 1 | 1 | 4 | 1 | 4 | 2 | 331.4319 |
| 18 | 1 | 2 | 3 | 2 | 3 | 1 | 258.0675 |
| 19 | 1 | 3 | 2 | 3 | 2 | 4 | 121.6474 |
| 20 | 1 | 4 | 1 | 4 | 1 | 3 | 87.5620 |
| 21 | 2 | 1 | 4 | 2 | 3 | 4 | 226.5596 |
| 22 | 2 | 2 | 3 | 1 | 4 | 3 | 260.5570 |
| 23 | 2 | 3 | 2 | 4 | 1 | 2 | 118.0963 |
| 24 | 2 | 4 | 1 | 3 | 2 | 1 | 156.3737 |
| 25 | 3 | 1 | 3 | 3 | 1 | 2 | 187.0887 |
| 26 | 3 | 2 | 4 | 4 | 2 | 1 | 192.2524 |
| 27 | 3 | 3 | 1 | 1 | 3 | 4 | 170.9587 |
| 28 | 3 | 4 | 2 | 2 | 4 | 3 | 190.9186 |
| 29 | 4 | 1 | 3 | 4 | 2 | 2 | 163.9173 |
| 30 | 4 | 2 | 4 | 3 | 1 | 3 | 132.5496 |
| 31 | 4 | 3 | 1 | 2 | 4 | 1 | 238.4379 |
| 32 | 4 | 4 | 2 | 1 | 3 | 4 | 211.5525 |
| Factor | K̄1 | K̄2 | K̄3 | K̄4 | Range Ri |
|---|---|---|---|---|---|
| b2 | 193.9 | 188.9 | 190.2 | 186.5 | 7.4 |
| b3 | 238.1 | 199.1 | 170.9 | 151.4 | 86.7 |
| r5 | 189.8 | 190.3 | 190.0 | 189.4 | 0.9 |
| r6 | 208.0 | 196.9 | 183.6 | 171.1 | 36.8 |
| r1 | 141.3 | 172.4 | 205.7 | 240.2 | 98.8 |
| b7 | 219.3 | 202.5 | 181.2 | 156.5 | 62.8 |
The range analysis reveals that the quartz crystal radius r1 has the strongest influence on the sensitivity of the six-axis force sensor, with a range of 98.8, followed by the thickness of the outer elastic membrane b3 (range 86.7), the quartz crystal thickness b7 (range 62.8), the width dimension affecting the outer elastic membrane r6 (range 36.8), the thickness of the inner elastic membrane b2 (range 7.4), and the width dimension affecting the inner elastic membrane r5 (range 0.9). The optimal level combination, based on the highest average sensitivity index, is r1 = 5.0 mm, b7 = 0.85 mm, b2 = b3 = 0.3 mm, r5 = 9.8 mm, and r6 = 20.0 mm. This combination maximizes the sensor’s static sensitivity while adhering to geometric constraints.
To verify the practical applicability of the optimized parameters, I fabricated an experimental prototype of the washer-type piezoelectric six-axis force sensor using the preferred structural parameters, as listed in Table 7. The prototype was calibrated using a static sensitivity calibration system, which included a six-dimensional force/torque loading platform, NI data acquisition cards, charge amplifiers, and LabVIEW-based software for data processing. The calibration involved applying forces and moments at 10 loading points within the sensor’s range for each dimension and computing the sensitivity from the output charges.
| Parameter | Value | Parameter | Value | Parameter | Value |
|---|---|---|---|---|---|
| b2 | 0.3 | b7 | 1.0 | r3 | 7.5 |
| b3 | 0.3 | b8 | 0.05 | r4 | 8.0 |
| b5 | 0.3 | b9 | 12.0 | r5 | 9.8 |
| b6 | 0.3 | b10 | 12.0 | r6 | 20.75 |
| b11 | 2.5 | r1 | 5.0 | r7 | 23.0 |
| b12 | 3.5 | r2 | 5.0 | a1 | 50.0 |
The sensitivity results from the experimental prototype, along with those from the analytical and numerical models for the same parameters, are compared in Table 8. The load transfer coefficients for the experimental prototype relative to the models, denoted as EMLTC (experimental to mathematical) and ESLTC (experimental to numerical), are also calculated. The average EMLTC is 0.73, and the average ESLTC is 0.75, indicating that the experimental values are slightly lower due to factors like manufacturing tolerances and assembly variations. Additionally, the isotropy of sensitivity, which measures the uniformity across different dimensions, was evaluated for all three approaches, as shown in Table 9. The analytical and numerical models both exhibit a comprehensive isotropy of 0.71, while the experimental prototype has 0.67, demonstrating consistent performance across the models.
| Model Type | Fx | Fy | Fz | Mx | My | Mz |
|---|---|---|---|---|---|---|
| Numerical Model | 2.052 | 2.071 | 2.237 | 389.593 | 389.540 | 294.831 |
| Analytical Model | 2.016 | 2.016 | 2.306 | 362.418 | 362.418 | 280.403 |
| Experimental Prototype | 1.855 | 1.639 | 2.382 | 211.91 | 220.036 | 144.743 |
| EMLTC | 0.920 | 0.813 | 1.033 | 0.585 | 0.607 | 0.516 |
| ESLTC | 0.904 | 0.791 | 1.065 | 0.544 | 0.565 | 0.491 |
| Model Type | Force Isotropy | Moment Isotropy | Comprehensive Isotropy |
|---|---|---|---|
| Numerical Model | 0.65 | 0.76 | 0.71 |
| Analytical Model | 0.65 | 0.77 | 0.71 |
| Experimental Prototype | 0.71 | 0.62 | 0.67 |
In conclusion, the analytical mathematical model developed for the washer-type piezoelectric six-axis force sensor provides an effective tool for predicting static sensitivity and optimizing structural parameters. The strong correlation with numerical simulations and the consistent results from experimental validation underscore its reliability. The use of orthogonal experiments based on this model enables efficient multi-parameter design, highlighting the significance of factors like quartz crystal radius and elastic membrane thickness. This approach facilitates the active design of high-performance six-axis force sensors for advanced applications in robotics and precision engineering. Future work could focus on incorporating manufacturing process parameters into the models to further enhance accuracy and bridge the gap between theoretical and experimental results.