In the field of robotics and automation, the ability to precisely measure forces and moments in three-dimensional space is critical for enabling intelligent feedback and control. Six-axis force sensors, which can detect three orthogonal forces (Fx, Fy, Fz) and three orthogonal moments (Mx, My, Mz), are essential components in applications ranging from industrial robotics to aerospace systems. Among various types of force sensors, piezoelectric-based designs offer distinct advantages, including high precision, excellent dynamic response, and minimal environmental interference. This article presents the design and analysis of a novel piezoelectric six-axis force sensor utilizing a spoke-structured configuration with an eight-fulcrum support layout. The sensor employs piezoelectric quartz as the sensing element, leveraging its inherent piezoelectric effects for accurate force measurement. The design incorporates principles of parallel load distribution and multi-point measurement to achieve strong decoupling, lightweight construction, high load capacity, and elevated natural frequency. Through finite element simulations and experimental validation, the sensor’s performance is thoroughly evaluated, demonstrating its suitability for dynamic and high-load applications.
Introduction to Six-Axis Force Sensors
Six-axis force sensors are pivotal in modern automation, providing the necessary data for force control in complex environments. Traditional resistive strain-based sensors, while widely used, often suffer from limitations in dynamic response due to their lower natural frequencies. In contrast, piezoelectric sensors, which exploit the piezoelectric effect of materials like quartz, excel in dynamic measurements owing to their high stiffness and rapid response times. Piezoelectric quartz, a single-crystal material, exhibits minimal charge leakage and outstanding static and dynamic characteristics, making it ideal for both quasi-static and dynamic force sensing. The key challenge in six-axis force sensor design lies in minimizing inter-dimensional coupling, where forces in one direction inadvertently affect measurements in others. This work addresses this issue by proposing an innovative eight-fulcrum support layout that effectively decouples the six force and moment components, thereby enhancing measurement accuracy and simplifying data processing.
Measurement Principle of the Piezoelectric Six-Axis Force Sensor
The measurement principle of the proposed six-axis force sensor is grounded in the piezoelectric effects of quartz crystals. Quartz crystals can be fabricated into different types of sensing elements based on their polarization directions, allowing for the detection of both normal and shear forces. The sensor utilizes a combination of longitudinal and shear effects to measure the full six degrees of freedom. The core innovation lies in the eight-fulcrum spatial arrangement, which integrates two distinct four-point support schemes: a diamond layout for force measurement and a square layout for moment measurement. This configuration ensures that the force and moment components are independently sensed, reducing cross-talk.
The eight quartz crystal groups are uniformly distributed on a circle of radius R. Groups 1, 3, 5, and 7 (diamond layout) are responsible for measuring the three orthogonal forces Fx, Fy, and Fz. Groups 2, 4, 6, and 8 (square layout) measure the three moments Mx, My, and Mz. During assembly, the x and y crystal elements in groups 1, 3, 5, and 7 are aligned with the sensor’s X and Y axes, respectively. The x elements in groups 2, 4, 6, and 8 are oriented tangentially to the installation circle, while all z elements are aligned with the Z-axis. Based on spatial force equilibrium and lever principles, the measurement equations for the six-axis force sensor are derived as follows:
$$
\begin{cases}
F_x = f_{x1} + f_{x3} + f_{x5} + f_{x7} \\
F_y = f_{y1} + f_{y3} + f_{y5} + f_{y7} \\
F_z = f_{z1} + f_{z3} + f_{z5} + f_{z7} \\
M_x = \frac{\sqrt{2}R}{2} \times (-f_{z2} + f_{z4} + f_{z6} – f_{z8}) \\
M_y = \frac{\sqrt{2}R}{2} \times (-f_{z2′} – f_{z4′} + f_{z6′} + f_{z8′}) \\
M_z = R \times (f_{x2} – f_{x4} + f_{x6} – f_{x8})
\end{cases}
$$
Here, fxi, fyi, and fzi represent the force components measured by the i-th quartz group. This formulation ensures that the force and moment measurements are decoupled, as the diamond and square layouts operate independently. Theoretical analysis confirms that the eight-fulcrum approach effectively eliminates inter-dimensional coupling, a significant advancement in six-axis force sensor design.
Structural Design of the Spoke-Structured Sensor
The structural design of the piezoelectric six-axis force sensor is optimized for lightweight construction and high load capacity. The sensor employs a spoke-structured configuration, which includes an upper cover, a lower cover, piezoelectric quartz elements, and preload bolts. The upper and lower covers are divided into three regions: an inner load-sharing ring, a spoke mechanism, and an outer load-sharing ring. The spoke region and inner ring feature holes for preload bolt installation, while the outer ring has symmetrically arranged wiring ports. The quartz crystal groups are mounted within the spoke mechanism, and preload bolts secure the assembly, applying a preload force to eliminate gaps between components.
When an external force acts on the sensor’s upper surface, the inner and outer load-sharing rings distribute the load, ensuring that only a portion of the force is transmitted to the quartz elements via the spokes. This design protects the sensitive quartz crystals from overloads. By adjusting the areas of the inner and outer rings, the sensor’s measurement range can be tailored to specific applications. The target specifications for this six-axis force sensor include a Z-axis load capacity of 60 kN, X and Y-axis capacities of 30 kN, and moment capacities of 3 kN·m for Mx, My, and Mz, with a natural frequency exceeding 5 kHz. The spoke structure, combined with the parallel load distribution principle, enables these high performance metrics while maintaining a compact and lightweight form factor.
Simulation Analysis Using Finite Element Methods
To validate the design and measurement principles, finite element analysis (FEA) was conducted using ANSYS software. A parametric model of the six-axis force sensor was developed, incorporating material properties and boundary conditions. The model was meshed with free-grid elements, and the base of the sensor was constrained to simulate real-world mounting conditions. The materials used in the simulation are summarized in Table 1.
| Component | Density (kg/m³) | Elastic Modulus (GPa) | Poisson’s Ratio | Element Type |
|---|---|---|---|---|
| Loading Block | 7850 | 175 | 0.300 | Solid186 |
| Upper Cover | 7900 | 206 | 0.243 | Solid186 |
| Lower Cover | 7900 | 206 | 0.243 | Solid186 |
| Quartz Crystal | 2210 | 77.8 | 0.170 | Solid186 |
| Electrode | 8930 | 82.7 | 0.330 | Solid186 |
| Insulation Film | 1380 | 3.50 | 0.340 | Solid186 |
Static Simulation under Axial Force (Fz)
A normal force Fz was applied to the sensor’s surface, ranging from 0 to 60 kN in 10 kN increments. The equivalent stress distribution at 10 kN loading is shown in the simulation results. The output forces and moments were calculated using the measurement equations. The results, plotted in Figure 1, indicate that Fx, Fy, Mx, My, and Mz exhibit negligible coupling outputs, while Fz increases linearly with the applied load, confirming the sensor’s effectiveness in axial force measurement.
Static Simulation under Shear Force (Fy)
A shear force Fy was applied from 0 to 30 kN in 5 kN steps. The stress distribution at 5 kN loading is analyzed. The computed six-axis outputs demonstrate that Fx, Fz, My, and Mz have minimal coupling, whereas Fy shows a linear response. A slight coupling in Mx is observed due to the height difference between the loading point and the constraint plane, but this does not impair overall performance.
Static Simulation under Bending Moment (My)
A bending moment My was generated by applying opposing normal forces (±Fz) at a distance of 250 mm, with My ranging from 0 to 3 kN·m. The stress distribution at 1 kN·m is examined. The outputs reveal that all force components and Mx and Mz moments are nearly zero, while My increases linearly, validating the moment measurement capability.
Static Simulation under Torque (Mz)
A torque Mz was produced by applying opposing shear forces (±Fy) at 250 mm separation, with Mz varying from 0 to 3 kN·m. The stress results at 1 kN·m are used to compute the outputs. The data show that Fx, Fy, Fz, Mx, and My have minimal coupling, and Mz responds linearly, affirming the decoupling achieved by the eight-fulcrum design.
Dynamic Simulation: Modal Analysis
Modal analysis was performed to determine the natural frequencies of the six-axis force sensor. The first six mode shapes and their corresponding frequencies are listed in Table 2. The results indicate that the lowest natural frequency is above 9 kHz, exceeding the design target of 5 kHz, which ensures excellent dynamic performance for high-frequency applications.
| Mode Number | Natural Frequency (Hz) | Vibration Description |
|---|---|---|
| 1 | 9168.69 | Linear vibration along Y-axis |
| 2 | 9172.44 | Linear vibration along X-axis |
| 3 | 9308.31 | Torsional vibration around Z-axis |
| 4 | 9311.98 | Linear vibration along Z-axis |
| 5 | 9912.86 | Torsional vibration around X-axis |
| 6 | 9918.67 | Torsional vibration around Y-axis |
Calibration Experiments: Static and Dynamic Testing
A prototype of the spoke-structured piezoelectric six-axis force sensor was fabricated and subjected to comprehensive calibration experiments. The static calibration platform comprised a six-axis force calibration setup, a high-speed data acquisition card, charge amplifiers, signal conditioning circuits, and a PC for data processing. The dynamic calibration utilized an impact response method to determine the natural frequency.
Static Calibration under Three-Dimensional Forces
Forces were applied incrementally along the X, Y, and Z axes. X and Y forces were increased in 5 kN steps up to 30 kN, and Z forces in 10 kN steps up to 60 kN. Each test was repeated three times, and the average values were recorded. The calibration curves for Fx, Fy, and Fz are plotted in Figure 2, showing linear outputs with negligible coupling. The nonlinearity for each force component is calculated as follows:
$$
\text{Nonlinearity} = \frac{\text{Max Deviation}}{\text{Full Scale Output}} \times 100\%
$$
The results indicate nonlinearity values below 0.5% for all force axes, demonstrating high linearity.
Static Calibration under Three-Dimensional Moments
Moments Mx, My, and Mz were applied using calibrated levers with opposing forces, increasing from 0 to 3 kN·m in steps equivalent to 2 kN force increments. The output voltages for moments are plotted in Figure 3, revealing linear relationships with minimal inter-dimensional interference. The coupling errors are summarized in Table 3, which compares simulation and experimental values for inter-dimensional interference.
| Force/Moment | Nonlinearity (%) | Inter-Dimensional Interference (Simulation/Experiment) (%) |
|---|---|---|
| Fx | 0.12 | – |
| Fy | 0.12 | 0.31/2.50 (Fx), 0.18/3.03 (Fz) |
| Fz | 0.36 | 0.03/2.02 (Fx), 0.03/2.02 (Fy) |
| Mx | 0.31 | 0.89/1.27 (My), 0.37/2.12 (Mz) |
| My | 0.32 | 0.89/1.34 (Mx), 0.37/2.32 (Mz) |
| Mz | 0.34 | 0.14/1.25 (Mx), 0.14/1.29 (My) |
Dynamic Calibration Using Impact Response
The dynamic characteristics of the six-axis force sensor were evaluated by striking the sensor with a force hammer and analyzing the response via Fourier transform. The results, shown in Figure 4, indicate a natural frequency exceeding 6 kHz, which satisfies the requirement for dynamic measurements and aligns with the simulation predictions.
Conclusion
The design and development of a spoke-structured piezoelectric six-axis force sensor with an eight-fulcrum support layout have been presented. This sensor leverages the piezoelectric properties of quartz and innovative structural design to achieve high performance in terms of decoupling, load capacity, and dynamic response. Finite element simulations confirmed the measurement principles, showing minimal inter-dimensional coupling under various loading conditions. Experimental calibration demonstrated nonlinearity below 0.5% and inter-dimensional interference less than 4%, with a natural frequency over 6 kHz. The spoke structure effectively distributes loads and protects the sensing elements, while the eight-fulcrum layout ensures independent measurement of forces and moments. This work provides a foundational framework for the design of lightweight, high-capacity six-axis force sensors and serves as a valuable reference for future advancements in piezoelectric sensor technology. The proposed six-axis force sensor is well-suited for applications requiring precise dynamic force feedback, such as robotic manipulation and advanced manufacturing systems.