In the field of robotics and precision engineering, the accurate measurement of forces and torques is crucial for applications such as industrial automation, aerospace, and biomedical devices. The six-axis force sensor, capable of measuring three orthogonal forces and three moments, plays a vital role in these domains. However, the precision of a six-axis force sensor heavily depends on the calibration process, which involves applying known loads and correlating them with sensor outputs. In this article, we analyze the static calibration system for a six-axis force sensor, focusing on the various error sources that can affect measurement accuracy. We develop a comprehensive error model and propose measures to minimize these errors, thereby enhancing the overall performance of the calibration system. The analysis is based on a self-designed six-axis force sensor with specified ranges: forces along the X, Y, and Z axes (FX, FY, FZ) up to 200 N, and moments about these axes (MX, MY, MZ) up to 1 N·m. Through detailed examination, we identify that directional deviations during force application are the primary contributors to error, and we provide quantitative insights into the full-scale accuracy of each channel.
The static calibration system for a six-axis force sensor typically comprises several key components: the sensor itself, a data acquisition card, a rotary table, pulleys, and standard weights. The system is designed to apply precise forces and torques in controlled directions, allowing for the characterization of the sensor’s response. For instance, when calibrating the FX channel, the sensor is oriented such that the X-axis aligns horizontally, and a force is applied via a wire rope passing over a pulley and attached to standard weights. Similarly, other channels are calibrated by adjusting the orientation or applying force couples for moment measurements. The setup ensures that the applied loads are as close as possible to the intended directions, but inherent imperfections in the system introduce errors. Below, we describe the system in more detail and insert an illustrative image of a typical force-torque sensor setup to provide visual context.
The calibration process involves mounting the six-axis force sensor on a stable platform and connecting it to a data acquisition system that records voltage outputs corresponding to applied loads. The rotary table allows for precise angular adjustments to align the sensor’s axes with the loading directions. For force calibrations along the X, Y, and Z axes, weights are suspended via a wire rope that runs over pulleys to ensure horizontal or vertical force application. For moment calibrations, such as MX, MY, and MZ, force couples are applied at specific points on the sensor to generate pure moments. The use of standard weights ensures traceability, but factors like pulley friction, air buoyancy, and directional misalignments can lead to discrepancies between the applied and actual loads. In the following sections, we delve into a systematic error analysis, considering each potential source and its impact on the calibration accuracy of the six-axis force sensor.
Error Sources in the Calibration System
The accuracy of a six-axis force sensor calibration is influenced by multiple error sources, which can be categorized into systematic and random errors. Systematic errors, such as those from pulley friction and air buoyancy, can often be corrected, while random errors, like those from environmental fluctuations, may require statistical treatment. Below, we analyze the key error sources in detail, providing mathematical models and numerical examples based on our six-axis force sensor setup. Each subsection addresses a specific error type, including the friction moment in pulley bearings, air buoyancy effects, resolution limitations of the data acquisition card, directional deviations during force application, and inaccuracies in standard weights. By quantifying these errors, we aim to build a comprehensive error model that can be used to improve calibration procedures for six-axis force sensors.
Friction Moment in Pulley Bearings
One significant error source in the calibration system is the friction moment in the pulley bearings. When a force is applied via a wire rope over a pulley, friction in the bearing resists rotation, leading to unequal tension on either side of the pulley. This results in a reduction in the effective force applied to the six-axis force sensor. The friction moment \( T_f \) can be expressed as:
$$ T_f = \lambda F_0 \frac{D_0}{2} $$
where \( \lambda \) is the friction coefficient dependent on the bearing type and load, \( F_0 \) is the load on the bearing, and \( D_0 \) is the pitch diameter of the bearing, calculated as \( D_0 = \frac{D + d}{2} \), with \( D \) and \( d \) being the outer and inner diameters, respectively. The load on the bearing \( F_0 \) is related to the applied force \( F \) by \( F_0 = \sqrt{2} F \) due to the vector sum of tensions. The tangential force loss \( \Delta F \) due to friction can be derived from the moment balance:
$$ \Delta F \cdot R = T_f $$
where \( R \) is the pulley radius. Solving for \( \Delta F \), we get:
$$ \Delta F = \frac{\lambda F D_0}{\sqrt{2} R} $$
This force loss represents the error introduced by pulley friction. For our six-axis force sensor calibration, using typical values such as \( \lambda = 0.0015 \) (for a bearing model 6001ZZ), \( D = 28 \, \text{mm} \), \( d = 12 \, \text{mm} \), \( R = 18 \, \text{mm} \), and \( F = 200 \, \text{N} \), we calculate:
$$ \Delta F = \frac{0.0015 \times 200 \times \frac{28 + 12}{2}}{\sqrt{2} \times 18} \approx 0.2357 \, \text{N} $$
This error affects all force and moment channels to varying degrees, as it reduces the effective load applied during calibration. For moment channels, where force couples are used, the error may be compounded if multiple pulleys are involved. Therefore, regular lubrication of pulley bearings is essential to minimize this error in six-axis force sensor calibrations.
Air Buoyancy Effects
Another error source is air buoyancy, which affects the apparent weight of the standard masses used to generate forces. According to Archimedes’ principle, any object immersed in a fluid experiences an upward buoyant force equal to the weight of the displaced fluid. For a mass \( M \) with density \( \rho \), the buoyant force in air of density \( \rho_0 \) is \( \frac{\rho_0}{\rho} M g \), where \( g \) is the acceleration due to gravity. Thus, the error in the applied force due to air buoyancy is:
$$ \Delta F_b = M g \frac{\rho_0}{\rho} $$
In our calibration system, we use stainless steel weights with a density \( \rho = 7850 \, \text{kg/m}^3 \), and under standard conditions (20°C, atmospheric pressure), the air density \( \rho_0 \approx 1.2 \, \text{kg/m}^3 \). For a 20 kg mass, the error is:
$$ \Delta F_b = 20 \times 9.81 \times \frac{1.2}{7850} \approx 0.0176 \, \text{N} $$
This error is relatively small compared to other sources and can be neglected for most practical purposes in six-axis force sensor calibration. However, for high-precision applications, it should be corrected, especially when using large masses. For moment calibrations, where force couples are applied, the buoyancy effects may cancel out if symmetric loading is used, but non-symmetric setups could introduce residual errors.
Data Acquisition Card Resolution
The resolution of the data acquisition card used to read the sensor outputs also contributes to calibration errors. The card converts analog voltage signals from the six-axis force sensor into digital values, and its resolution determines the smallest detectable change in force or moment. For an n-bit card with an input range of \( \pm U_m \), the resolution \( a \) is given by:
$$ a = \frac{U_m}{2^{n-1}} $$
If the relationship between the applied force \( F_i \) and the output voltage \( U_i \) for channel i is linear, i.e., \( F_i = k_i U_i \), where \( k_i \) is the calibration coefficient, then the error due to resolution \( u_1 \) is:
$$ u_1 = a \cdot k_i $$
In our setup, we use a NET2801 data acquisition card with a 16-bit resolution and an input range of \( \pm 10 \, \text{V} \), resulting in \( a = \frac{10}{2^{15}} \approx 3 \, \text{mV} \). Using the calibration coefficients for our six-axis force sensor, we compute the errors for each channel as shown in the table below. These errors are inherent to the digital conversion process and can be reduced by using higher-resolution cards or averaging multiple readings.
| Channel | Coefficient \( k_i \) | Error \( u_1 \) |
|---|---|---|
| FX | 42.33 N/V | 0.127 N |
| FY | 41.33 N/V | 0.124 N |
| FZ | 39.33 N/V | 0.118 N |
| MX | 240 N·mm/V | 0.72 N·mm |
| MY | 413.33 N·mm/V | 1.24 N·mm |
| MZ | 340 N·mm/V | 1.02 N·mm |
Directional Deviations in Force Application
Directional deviations during force application are among the most significant error sources in six-axis force sensor calibration. Ideally, forces should be applied purely along the intended axis, but in practice, misalignments occur due to limitations in the setup, such as pulley positioning or sensor mounting. These deviations introduce cross-axis errors, where forces in one direction produce components in orthogonal directions. For example, when calibrating the FY channel, if the applied force \( F \) deviates by an angle \( \alpha \) in the YOZ plane and \( \beta \) in the XOY plane, the actual force along Y is \( F \cos \alpha \), and the errors in Y and other directions can be derived.
For force channels (FX, FY, FZ), the error due to directional deviation \( u_2 \) can be approximated as:
$$ u_2 = F \sqrt{ (1 – \cos \alpha)^2 + \sin^2 \beta } $$
For small angles, \( 1 – \cos \alpha \approx \frac{1}{2} \alpha^2 \), so:
$$ u_2 \approx F \alpha $$
assuming \( \alpha \) and \( \beta \) are of similar magnitude. In our system, the directional error is controlled within \( \pm 1^\circ \), or approximately 0.01745 radians. For a force \( F = 200 \, \text{N} \), this gives:
$$ u_2 \approx 200 \times 0.01745 = 3.49 \, \text{N} $$
For moment channels, the analysis differs. For MZ calibration, a force couple is applied with a lever arm \( l \). If the force deviates by angles \( \alpha \) and \( \beta \), the error \( u_2′ \) is:
$$ u_2′ = F l \sqrt{ (1 – \cos \alpha)^2 + (1 – \cos \beta)^2 } \approx F l \frac{\alpha^2 \sqrt{2}}{2} $$
With \( l = 0.1 \, \text{m} \) and \( F = 200 \, \text{N} \), we get:
$$ u_2′ \approx 200 \times 0.1 \times \frac{(0.01745)^2 \sqrt{2}}{2} \approx 0.211 \, \text{N·mm} $$
These errors highlight the importance of precise alignment in six-axis force sensor calibration. Using high-precision orientation devices, such as laser alignment tools, can significantly reduce directional deviations.
Inaccuracies in Standard Weights
The standard weights used to generate forces have inherent inaccuracies due to manufacturing tolerances. According to calibration standards, such as JJG 99-2006, weights have specified maximum permissible errors. For M1 grade weights, the error for a 2 kg mass is 100 mg, and for a 50 g mass, it is 3.0 mg. When multiple weights are used, the cumulative error \( u_3 \) in the applied force is:
$$ u_3 = g \sqrt{ \frac{N m_s^2}{3} } $$
where \( N \) is the number of weights, and \( m_s \) is the mass error. For force calibrations, we use ten 2 kg weights, so \( N = 10 \), \( m_s = 0.0001 \, \text{kg} \), and:
$$ u_3 = 9.81 \times \sqrt{ \frac{10 \times (0.0001)^2}{3} } \approx 0.0018 \, \text{N} $$
For moment calibrations, twenty 50 g weights are used for MZ, giving \( N = 20 \), \( m_s = 0.000003 \, \text{kg} \), and:
$$ u_3 = 9.81 \times \sqrt{ \frac{20 \times (0.000003)^2}{3} } \approx 0.013 \, \text{N·mm} $$
This error is relatively small but should be considered in high-precision applications of six-axis force sensors. Using higher-grade weights or applying corrections based on calibration certificates can mitigate this issue.
Comprehensive Error Model
Based on the analysis above, we develop a comprehensive error model for the six-axis force sensor static calibration system. The total error for each channel is a combination of the individual errors, which are assumed to be uncorrelated. Therefore, the combined standard uncertainty \( u \) can be computed as the root sum square of the contributions. The model accounts for the specific characteristics of force and moment channels, as outlined below.
For force channels (FX, FY, FZ), the total error \( u \) is given by:
$$ u = \sqrt{ \left( \frac{\lambda F D_0}{\sqrt{2} R} \right)^2 + \left( \frac{M \rho_0 g}{\rho} \right)^2 + (a k_i)^2 + (F \alpha)^2 + \left( g \sqrt{ \frac{N m_s^2}{3} } \right)^2 } $$
For moment channels (MX, MY), the air buoyancy error is zero due to force couple symmetry, so:
$$ u = \sqrt{ \left( \frac{\lambda F D_0}{\sqrt{2} R} \right)^2 + (a k_i)^2 + (F \alpha)^2 + \left( g \sqrt{ \frac{N m_s^2}{3} } \right)^2 } $$
For the MZ channel, the directional error term is modified:
$$ u = \sqrt{ \left( \frac{\lambda F D_0}{\sqrt{2} R} \right)^2 + (a k_i)^2 + \left( F l \frac{\alpha^2 \sqrt{2}}{2} \right)^2 + \left( g \sqrt{ \frac{N m_s^2}{3} } \right)^2 } $$
Using the values from our six-axis force sensor setup, we calculate the total errors and full-scale accuracies for each channel, as summarized in the table below. The full-scale error is expressed as a percentage of the full-scale range, providing a clear metric for calibration quality.
| Channel | Total Error | Full-Scale Range | Full-Scale Error (%) |
|---|---|---|---|
| FX | 3.76 N | 200 N | 1.88 |
| FY | 3.76 N | 200 N | 1.88 |
| FZ | 3.90 N | 200 N | 1.95 |
| MX | 18.82 N·mm | 1000 N·mm | 1.88 |
| MY | 18.82 N·mm | 1000 N·mm | 1.88 |
| MZ | 24.81 N·mm | 1000 N·mm | 2.48 |
The results indicate that the MZ channel has the highest full-scale error at 2.48%, primarily due to the directional deviation effects compounded by the lever arm in moment application. Other channels exhibit errors below 2%, demonstrating the overall robustness of the calibration system for the six-axis force sensor. However, further improvements can be achieved by addressing the dominant error sources.
Measures to Reduce Errors
To enhance the accuracy of six-axis force sensor calibration, several measures can be implemented based on the error analysis. First, regular lubrication of pulley bearings is essential to minimize friction-related errors. Using low-friction bearings or advanced materials can further reduce this source of error. Second, employing high-resolution data acquisition cards, such as those with 24-bit resolution, can decrease the quantization error, improving the sensitivity of force and moment measurements. Third, directional deviations should be minimized through precise alignment techniques. For instance, using laser alignment systems or digital protractors can ensure that forces are applied along the intended axes, reducing cross-talk between channels. Fourth, standard weights with higher accuracy grades, such as E1 or E2, should be used, and their masses should be regularly calibrated to account for any drifts. Additionally, environmental factors like temperature and humidity should be controlled, as they can affect air density and material properties. Finally, statistical methods, such as repeated measurements and regression analysis, can help identify and compensate for random errors, leading to more reliable calibration outcomes for six-axis force sensors.
Conclusion
In this article, we have conducted a thorough error analysis of the static calibration system for a six-axis force sensor. By examining various error sources, including pulley friction, air buoyancy, data acquisition resolution, directional deviations, and weight inaccuracies, we developed a comprehensive error model that quantifies the impact on each sensor channel. The analysis reveals that directional deviations during force application are the most significant contributor to calibration errors, particularly for moment channels like MZ. The full-scale errors for the six-axis force sensor channels are all below 2.5%, with the MZ channel being the least accurate at 2.48%. To achieve higher precision, we recommend implementing measures such as improved alignment, higher-resolution data acquisition, and the use of precision weights. This work provides a foundation for optimizing calibration procedures, ensuring that six-axis force sensors deliver reliable and accurate measurements in critical applications. Future research could explore dynamic calibration methods or advanced compensation algorithms to further enhance the performance of six-axis force sensors.