In the field of robotics and biomechanics, the accurate measurement of multi-dimensional forces and torques is crucial for applications such as human motion analysis, robotic manipulation, and aerospace engineering. Traditional six-axis force sensors often face limitations due to their large size, complex wiring, and susceptibility to environmental interference. To address these challenges, we developed a compact, wireless six-axis force sensor based on the resistive strain principle. This sensor integrates a cross-beam elastic structure, advanced signal conditioning circuits, and Bluetooth-based data transmission, enabling high-precision force and torque measurements in dynamic environments. The design focuses on miniaturization, low power consumption, and real-time decoupling to minimize cross-dimensional coupling errors. In this paper, we detail the structural optimization, hardware implementation, calibration methodologies, and decoupling algorithms that collectively enhance the sensor’s performance for motion mechanics applications.
The elastic body of the six-axis force sensor is designed using a cross-beam structure, which consists of four main beams arranged in a十字 pattern, connected to a central hub via floating beams. This configuration ensures high sensitivity and minimal radial effects. The material selected for the elastic body is LY12 hard aluminum alloy, known for its high hardness and fatigue resistance, with an elastic modulus of $E = 72 \times 10^9$ Pa and a Poisson’s ratio of $\mu = 0.33$. Finite element analysis (FEA) was conducted using ANSYS to optimize the dimensions and validate the structural integrity under load. The key parameters of the elastic body are summarized in Table 1.
| Component | Length (mm) | Width (mm) | Height (mm) | Diameter (mm) |
|---|---|---|---|---|
| Cross Beam | 27 | 3.8 | 3.8 | – |
| Floating Beam | 32 | 1 | 3.8 | – |
| Central Hub | 10 | 10 | 8 | 5 |
| Outer Rim | – | – | 8 | 74/58 |
| Mounting Hole | – | – | – | 3.2 |
The FEA model was constructed using SOLIDWORKS and analyzed in ANSYS with SOLID95 elements for high accuracy. Loads were applied separately along each axis to evaluate strain distribution. For instance, a force of $F_x = 200$ N and a moment of $M_x = 2$ N·m were simulated, with constraints applied at the mounting holes. The results indicated that the maximum strain occurred near the central hub, with a linear relationship between distance and strain along the main beams. The strain $\epsilon$ at any point can be expressed as $\epsilon = \frac{\sigma}{E}$, where $\sigma$ is the stress. Path mapping revealed that the optimal locations for strain gauge placement are 5 mm from the hub for force measurements and 11 mm for torque measurements, as these positions exhibit high linearity and sensitivity. Under full load conditions, the maximum strain was $2.682 \times 10^{-3}$, corresponding to a stress of approximately 193.10 MPa, which is well below the yield strength of LY12 (380 MPa). This ensures the sensor’s durability and safety.
The sensor’s packaging includes a base, top cover, and force application shaft, resulting in a compact design with a height of 50 mm and an outer diameter of 94 mm. The assembly uses a 7-pin self-locking connector for power and Bluetooth communication, facilitating easy integration into various systems. The structural design effectively reduces the overall size while maintaining high performance, making the six-axis force sensor suitable for confined spaces.
The hardware circuit of the six-axis force sensor comprises strain gauge bridges, amplification stages, and a signal processing unit. Each of the six force/torque components corresponds to a full-bridge circuit using BF350 strain gauges with a nominal resistance of $R_0 = 350 \Omega$ and a gauge factor of $K = 2.10$. The output voltage $U_i$ for each bridge is given by $U_i = \frac{V_{in} \cdot K \cdot \epsilon}{4}$, where $V_{in} = \pm 5$ V is the bridge excitation voltage. For example, the theoretical full-scale outputs are $F_x$: 15.57 mV, $F_z$: 21.69 mV, $M_x$: 6.55 mV, and $M_z$: 8.31 mV. The bridge configurations are designed to minimize temperature effects and cross-talk, as shown in Figure 8(b) of the original paper.
To amplify these small signals to a range of $\pm 9$ V for analog-to-digital conversion, a two-stage amplification circuit is employed. The first stage uses an AD620 instrumentation amplifier with a gain $G_1$ set by an external resistor $R_G$: $G_1 = \frac{49.4 \text{ k}\Omega}{R_G} + 1$. For $F_x$, $R_G$ is chosen to achieve $G_1 \approx 100$. The second stage utilizes a TL084 operational amplifier in a non-inverting configuration, providing a gain $G_2$ calculated as $G_2 = 1 + \frac{R_f}{R_i}$, where $R_f$ and $R_i$ are feedback and input resistors, respectively. The overall gain $G = G_1 \cdot G_2$ is tailored for each channel: 610 for $F_x$, 438 for $F_z$, 1451 for $M_x$, and 1143 for $M_z$. Power supply filtering is implemented with 100 μF and 0.1 μF capacitors to reduce noise.
| Channel | Theoretical Output (mV) | Stage 1 Gain (AD620) | Stage 2 Gain (TL084) | Total Gain |
|---|---|---|---|---|
| $F_x$, $F_y$ | 15.57 | 100 | 6.10 | 610 |
| $F_z$ | 21.69 | 100 | 4.38 | 438 |
| $M_x$, $M_y$ | 6.55 | 100 | 14.51 | 1451 |
| $M_z$ | 8.31 | 100 | 11.43 | 1143 |
Signal processing is handled by an STM32F103ZET6 ARM microcontroller, which interfaces with an AD7606 16-bit ADC for simultaneous sampling of the six voltage channels at a rate of up to 200 kSPS. The microcontroller implements a sliding average filter to reduce noise and transmits data via an HC-06 Bluetooth module. The software operates on a 2–10 ms timer interrupt, checking for serial commands and initiating data acquisition. The wireless transmission range is up to 8 m indoors, with a data rate of 100–500 Hz, sufficient for low to medium-frequency force signals. This integrated approach eliminates the need for external data acquisition cards, enhancing the six-axis force sensor’s portability and flexibility.
Calibration and decoupling are critical for achieving high accuracy in six-axis force sensors due to inherent cross-dimensional coupling. We employed a static calibration method using weights and pulleys to apply precise forces and moments. The calibration setup allows for sequential loading along each axis, with three load-unload cycles per direction to average out errors. The linearity error for each primary direction is listed in Table 3, demonstrating the sensor’s initial performance before decoupling.
| Direction | Linearity Error (%) | Direction | Linearity Error (%) |
|---|---|---|---|
| $F_x$ | 0.32 | $M_x$ | 0.32 |
| $F_y$ | 0.52 | $M_y$ | 0.22 |
| $F_z$ | 0.23 | $M_z$ | 1.22 |
To address coupling errors, we developed a decoupling algorithm based on coupling error modeling and piecewise fitting. In an ideal six-axis force sensor, the output voltage $U_i$ would directly relate to the force or moment $D_i$ as $D_i = k_i’ U_i$, where $k_i’$ is the inverse sensitivity. However, due to coupling, the actual voltage $U_i$ is influenced by all six components. The decoupling process uses a阶梯形 approach, where corrected voltages $U_i’$ are computed sequentially:
$$ U_1′ = U_1 $$
$$ U_2′ = U_2 – \sum_{i=1, i \neq 2}^{2} h_{2i}(U_i’) $$
$$ U_3′ = U_3 – \sum_{i=1, i \neq 3}^{3} h_{3i}(U_i’) $$
$$ U_4′ = U_4 – \sum_{i=1, i \neq 4}^{4} h_{4i}(U_i’) $$
$$ U_5′ = U_5 – \sum_{i=1, i \neq 5}^{5} h_{5i}(U_i’) $$
$$ U_6′ = U_6 – \sum_{i=1, i \neq 6}^{6} h_{6i}(U_i’) $$
Here, $h_{ji}(U_i)$ represents the coupling effect of the $i$-th component on the $j$-th component. To account for asymmetry in positive and negative directions, we apply piecewise quadratic fitting:
$$ h_{ji}(U_i) = \begin{cases} f_{ji}(U_i) = a_{ji} U_i^2 + b_{ji} U_i, & U_i \geq 0 \\ g_{ji}(U_i) = b_{ji}’ U_i^2 + b_{ji} U_i, & U_i < 0 \end{cases} $$
The coefficients $a_{ji}$ and $b_{ji}$ are determined from calibration data. After decoupling, the forces and moments are computed as $D_i = k_i’ U_i’$. This algorithm significantly reduces coupling errors, as shown in Table 4, which compares errors before and after decoupling. Type I error (non-linearity) remains unchanged, while Type II error (coupling) is drastically reduced, with all channels below 3%, meeting high-precision requirements for motion mechanics measurement.
| Error Type | $F_x$ (%) | $F_y$ (%) | $F_z$ (%) | $M_x$ (%) | $M_y$ (%) | $M_z$ (%) |
|---|---|---|---|---|---|---|
| Type I (Before) | 0.40 | 0.27 | 0.27 | 0.34 | 0.29 | 1.28 |
| Type I (After) | 0.40 | 0.27 | 0.27 | 0.34 | 0.29 | 1.28 |
| Type II (Before) | 4.00 | 2.09 | 2.69 | 112.47 | 111.81 | 3.85 |
| Type II (After) | 0.49 | 1.68 | 2.69 | 2.67 | 0.45 | 2.16 |
In conclusion, we have successfully designed and implemented a wireless six-axis force sensor that combines compact structure, integrated electronics, and advanced decoupling algorithms. The sensor achieves high accuracy with coupling errors below 3%, wireless transmission up to 8 m, and a sampling rate suitable for dynamic applications. Future work will focus on enhancing temperature compensation and extending the wireless range for broader applications in robotics and biomechanics. This six-axis force sensor represents a significant step forward in portable, high-precision force measurement technology.