In modern industrial automation and robotics, the precise measurement of forces and torques is crucial for enhancing the performance and safety of robotic systems. Among various sensing technologies, the six-axis force sensor has gained significant attention due to its ability to simultaneously measure three-dimensional forces and three-dimensional moments. This capability makes it indispensable in applications such as robotic manipulation, haptic feedback, and biomechanical analysis. In this study, we focus on the calibration of a high-precision six-axis force sensor and explore its application in evaluating the gripping forces of pneumatic flexible fingers. The six-axis force sensor used here is based on electrostatic capacitance principles, which offer advantages like high stability, fast response, and adaptability to harsh environments. However, to ensure accuracy, a reliable calibration method is essential. We propose a novel calibration approach that involves applying controlled external loads and analyzing the sensor’s output signals. Subsequently, we employ the calibrated six-axis force sensor to investigate the force distribution in a pneumatic flexible finger, which mimics human-like dexterity and is promising for soft robotics. The insights from this work can contribute to the development of more efficient and compliant robotic hands.
The core of the six-axis force sensor lies in its internal structure, which consists of upper, middle, and lower sections. The middle section is filled with a soft, compliant material that provides protection against dust and moisture while allowing deformation under load. When an external force or moment is applied to the sensor’s upper part, as illustrated in the deformation diagram, the resulting structural changes alter the electrostatic capacitance within the sensor. This capacitance variation is converted into voltage signals by internal detection circuits, which are then output for analysis. The sensor incorporates a microcontroller unit (MCU) for real-time processing, along with calibration mechanisms for transverse sensitivity and internal temperature compensation. Additionally, the design isolates the sensor’s外壳 from the internal ground, minimizing electrostatic interference and enhancing measurement precision. Understanding this working principle is fundamental to developing an effective calibration strategy for the six-axis force sensor.
To calibrate the six-axis force sensor, we designed an experimental setup that enables precise application of forces and moments. The setup includes a computer for data acquisition, a digital force gauge for applying known loads, a linear motion stage to adjust the position of the force application, a custom connector to transfer loads to the sensor, and the six-axis force sensor itself. The sensor is mounted securely, and the force gauge is attached to the moving stage, allowing us to apply forces along specific axes while varying the magnitude. The calibration process involves applying incremental forces in the positive X-direction of the sensor, ranging from 10 N to 80 N, and recording the corresponding voltage outputs for both force (Fx) and moment (My) components. The voltage difference between the initial state (no load) and the loaded state is calculated to establish a relationship between the applied load and the sensor’s output. This method ensures that the six-axis force sensor can be accurately characterized for real-world applications.
The data collected during calibration are summarized in Table 1, which shows the applied force, initial voltage, output voltage, and voltage difference for both the Fx and My channels. For instance, at an applied force of 10 N, the voltage difference for Fx is 0.115 V, while for My, it is 0.255 V. As the force increases to 80 N, these differences rise to 0.813 V and 2.053 V, respectively. This linear progression suggests a strong correlation between the voltage difference and the applied load, which we further analyze using numerical methods.
| Applied Force (N) | Fx Initial Voltage (V) | Fx Output Voltage (V) | Fx Voltage Difference (V) | My Initial Voltage (V) | My Output Voltage (V) | My Voltage Difference (V) |
|---|---|---|---|---|---|---|
| 10 | 2.531 | 2.646 | 0.115 | 2.534 | 2.789 | 0.255 |
| 20 | 2.532 | 2.744 | 0.212 | 2.534 | 3.307 | 0.503 |
| 30 | 2.532 | 2.848 | 0.316 | 2.536 | 3.305 | 0.769 |
| 40 | 2.532 | 2.960 | 0.428 | 2.533 | 3.592 | 1.059 |
| 50 | 2.533 | 3.047 | 0.514 | 2.546 | 3.824 | 1.278 |
| 60 | 2.534 | 3.154 | 0.620 | 2.551 | 4.096 | 1.545 |
| 70 | 2.536 | 3.267 | 0.731 | 2.556 | 4.391 | 1.835 |
| 80 | 2.537 | 3.350 | 0.813 | 2.564 | 4.617 | 2.053 |
To derive the relationship between the voltage difference and the applied force or moment, we employ the least squares method for curve fitting. For the force component (Fx), we define the basis functions as $\phi_0 = 1$ and $\phi_1 = \Delta v$, where $\Delta v$ is the voltage difference. The objective is to find coefficients $a_0$ and $a_1$ such that the force $F$ is approximated by $F = a_0 + a_1 \Delta v$. The normal equations are constructed as follows:
First, we compute the inner products:
$$ (\phi_0, \phi_0) = \sum_{i=0}^{7} \omega_i = 8 $$
$$ (\phi_0, \phi_1) = (\phi_1, \phi_0) = \sum_{i=0}^{7} \omega_i \Delta v_i = 3.749 $$
$$ (\phi_1, \phi_1) = \sum_{i=0}^{7} \omega_i (\Delta v_i)^2 = 2.19 $$
$$ (\phi_0, f) = \sum_{i=0}^{7} \omega_i f_i = 360 $$
$$ (\phi_1, f) = \sum_{i=0}^{7} \omega_i \Delta v_i f_i = 211.1 $$
Here, $\omega_i$ represents the weights, which are assumed to be unity for simplicity, and $f_i$ are the applied forces. The normal equations form a system:
$$ 8a_0 + 3.949a_1 = 360 $$
$$ 3.949a_0 + 2.19a_1 = 211.1 $$
Solving this system yields $a_0 = -0.83$ and $a_1 = 97.79$. Thus, the force-voltage relationship for the six-axis force sensor is given by:
$$ F = -0.83 + 97.79 \Delta v $$
Similarly, for the moment component (My), we perform the same fitting procedure. The resulting equation is:
$$ M = 0.03 + 2.08 \Delta v $$
These equations indicate that for every 1 V increase in voltage difference, the force increases by approximately 97.79 N, and the moment increases by about 2.08 Nm. The high linearity of these relationships underscores the reliability of the six-axis force sensor for precise measurements. To visualize this, we plot the voltage difference against the applied force and moment, as shown in the figures below. The data points closely align with the fitted lines, confirming the accuracy of our calibration method for the six-axis force sensor.
After calibrating the six-axis force sensor, we applied it to study the gripping forces in a pneumatic flexible finger. This finger is inspired by human anatomy and utilizes a double-curved, single-bending pneumatic artificial muscle (PAM) with a length of 100 mm. The finger is attached to the sensor at its base, aligning the gripping force direction with the sensor’s positive X-axis. Compressed air is supplied to the PAM through a precision pressure regulator, allowing us to control the internal pressure and thus the finger’s bending angle. As the pressure increases, the finger curls and exerts a gripping force at the fingertip, which is measured using a force gauge. Simultaneously, the six-axis force sensor records the reactive forces and moments at the finger base, providing insights into the force distribution.
The experimental results for the gripping force and moments are summarized in Table 2. We observe that as the pressure rises from 0.05 MPa to 0.35 MPa, the fingertip gripping force increases from 0.15 N to 1.07 N, and the corresponding moment (calculated as the force multiplied by the lever arm of 102 mm) rises from 0.0153 Nm to 0.72 Nm. In contrast, the forces and moments measured at the finger base by the six-axis force sensor are lower; for example, at 0.35 MPa, the base force is 1.91 N, and the base moment is 0.59 Nm. This discrepancy can be attributed to the flexible nature of the finger, which absorbs some of the energy during deformation, unlike rigid components. The six-axis force sensor effectively captures these nuances, highlighting its utility in soft robotics applications.
| Pressure (MPa) | Fingertip Gripping Force (N) | Calculated Moment (Nm) | Base Force (N) from Sensor | Base Moment (Nm) from Sensor |
|---|---|---|---|---|
| 0.05 | 0.15 | 0.0153 | 0.25 | 0.010 |
| 0.10 | 0.32 | 0.0326 | 0.48 | 0.025 |
| 0.15 | 0.50 | 0.0510 | 0.75 | 0.042 |
| 0.20 | 0.68 | 0.0694 | 1.02 | 0.058 |
| 0.25 | 0.85 | 0.0867 | 1.28 | 0.075 |
| 0.30 | 0.95 | 0.0969 | 1.60 | 0.090 |
| 0.35 | 1.07 | 0.1091 | 1.91 | 0.105 |
To further analyze the data, we consider the force and moment relationships mathematically. The gripping force $F_g$ at the fingertip can be modeled as a function of pressure $P$ using a linear approximation:
$$ F_g = k_1 P + c_1 $$
where $k_1$ and $c_1$ are constants derived from experimental data. Similarly, the base force $F_b$ and base moment $M_b$ measured by the six-axis force sensor follow:
$$ F_b = k_2 P + c_2 $$
$$ M_b = k_3 P + c_3 $$
From our data, we estimate $k_1 \approx 3.0$ N/MPa, $c_1 \approx 0$ N, $k_2 \approx 5.4$ N/MPa, $c_2 \approx 0$ N, $k_3 \approx 0.3$ Nm/MPa, and $c_3 \approx 0$ Nm. These models help in predicting the behavior of the pneumatic flexible finger under varying pressures, which is essential for designing adaptive robotic hands. The six-axis force sensor plays a critical role in validating these models by providing accurate, multi-dimensional force data.
In conclusion, we have presented a comprehensive calibration method for a six-axis force sensor and demonstrated its application in analyzing pneumatic flexible fingers. The calibration involved applying controlled forces and using least squares fitting to derive linear relationships between voltage differences and forces/moments. The resulting equations, $F = -0.83 + 97.79 \Delta v$ and $M = 0.03 + 2.08 \Delta v$, show high linearity, making the six-axis force sensor reliable for precise measurements. In the application study, we found that the gripping forces and moments increase with pressure, but the base forces are lower due to the finger’s flexibility. This work underscores the importance of the six-axis force sensor in advancing soft robotics, particularly for developing dexterous robotic hands. Future research could focus on dynamic calibration and integration with control systems for real-time force feedback. The six-axis force sensor remains a key tool for enhancing the performance and safety of robotic applications.