In the field of robotics and precision engineering, the accurate measurement of multi-dimensional forces and torques is crucial for applications such as robotic manipulation, wind tunnel testing, and medical devices. Among various designs, the Stewart platform-based six-axis force sensor has gained prominence due to its parallel structure, which offers high stiffness, excellent load-bearing capacity, and inherent decoupling capabilities. However, despite these advantages, the practical accuracy of such sensors is often compromised by Type I and Type II errors, which arise from structural imperfections, manufacturing tolerances, and assembly inaccuracies. Type I errors refer to the non-linearity and inconsistency in the input-output relationship for a single axis, while Type II errors, or cross-coupling errors, occur when a force or torque applied in one axis affects the outputs of other axes. These errors significantly degrade the performance of the six-axis force sensor, necessitating precise calibration to enhance measurement accuracy.
This article explores a novel calibration method for the Stewart six-axis force sensor, focusing on reducing Type I and Type II errors through a static calibration approach. I begin by explaining the fundamental mechanics of the Stewart six-axis force sensor, including the force mapping principle and the transformation matrix that relates external forces to internal axial forces in the sensor’s limbs. The transformation matrix, denoted as G, is derived solely from the sensor’s geometric parameters, but real-world deviations introduce errors. To address this, I propose a calibration scheme utilizing a calibration rod, which allows for precise application of known loads in all six degrees of freedom without the need for complex, expensive loading fixtures. The feasibility of this method is validated through theoretical analysis and finite element simulation using ANSYS software. The simulation involves modeling the sensor, performing static calibration experiments, and comparing the measurement accuracy before and after calibration. Results demonstrate a substantial reduction in both Type I and Type II errors, confirming the effectiveness of the proposed calibration method for improving the performance of the Stewart six-axis force sensor.
The Stewart six-axis force sensor operates on the principle of parallel mechanics, where an external generalized force vector applied to the top platform is transformed into axial forces along the six limbs. Mathematically, this relationship is expressed as:
$$ \mathbf{F} = \mathbf{G} \mathbf{f} $$
where $\mathbf{F} = [F_x, F_y, F_z, M_x, M_y, M_z]^T$ represents the external force and torque vector, $\mathbf{f} = [f_1, f_2, f_3, f_4, f_5, f_6]^T$ denotes the axial forces in the limbs, and $\mathbf{G}$ is the transformation matrix. The elements of $\mathbf{G}$ depend on the sensor’s structural parameters, such as the radii of the top and bottom platforms, the orientation angles, and the vertical height. Ideally, this matrix allows for direct decoupling of forces and torques. However, in practice, inaccuracies in manufacturing and assembly lead to deviations in $\mathbf{G}$, causing cross-coupling and non-linearities. For instance, a pure force applied along the z-axis might induce unexpected outputs in the moment axes, highlighting the need for calibration to recalibrate the force mapping.
The proposed calibration method aims to establish a revised mapping between the applied forces and the sensor outputs by performing static calibration experiments. In this process, known calibration loads are applied sequentially along each axis of the six-axis force sensor, and the corresponding axial force outputs are recorded. The calibration matrix $\mathbf{C}$ is then computed to minimize the discrepancies between the applied loads and the reconstructed forces. The relationship is given by:
$$ \mathbf{F}_1 = \mathbf{C} \cdot (\mathbf{f}_c – \mathbf{B}) $$
where $\mathbf{F}_1$ is the matrix of applied calibration loads, $\mathbf{f}_c$ is the matrix of measured axial forces during calibration, $\mathbf{B}$ is the bias matrix accounting for initial offsets, and $\mathbf{C}$ is the calibration matrix. Since $\mathbf{C}$ is typically non-square, the pseudo-inverse is used for computation:
$$ \mathbf{C} = \mathbf{F}_1 \cdot \mathbf{A}^- $$
with $\mathbf{A} = (\mathbf{f}_c – \mathbf{B})$ and $\mathbf{A}^- = \mathbf{A}^T (\mathbf{A} \mathbf{A}^T)^{-1}$ being the pseudo-inverse. The accuracy of $\mathbf{F}_1$ is critical, as it directly influences the precision of $\mathbf{C}$ and, consequently, the overall performance of the six-axis force sensor. Traditional calibration setups often require bulky and costly loading apparatus, but the calibration rod method offers a simplified alternative. By attaching a rod to the sensor’s top platform and applying weights at specific positions, forces and moments can be accurately generated for all six axes. For example, forces along the x and y axes are applied by hanging weights horizontally, while moments are induced by offset loading. This approach ensures high loading precision while minimizing equipment costs.
To validate the calibration method, a finite element model of a Stewart six-axis force sensor was developed in ANSYS. The sensor parameters included a top platform radius of 138.1 mm, a bottom platform radius of 138.1 mm, a vertical height of 74 mm, and specific orientation angles. The model was meshed, and material properties were assigned to simulate real-world behavior. Initially, the theoretical transformation matrix $\mathbf{G}$ was used to compute forces from axial outputs, but this led to significant errors due to assumed ideal conditions. The initial measurement accuracy was evaluated by applying single-axis loads and calculating the errors using the formula:
$$ E = \frac{\max |\mathbf{F}_s – \mathbf{F}’_s|}{\max |\mathbf{F}_s|} \times 100\% $$
where $\mathbf{F}_s$ is the applied load and $\mathbf{F}’_s$ is the computed load. The results, summarized in Table 1, show substantial Type I and Type II errors, with the maximum Type I error reaching 5.62% for the z-axis force and cross-coupling errors up to 8.99% between axes.
| Loading Direction | Computed Fx | Computed Fy | Computed Fz | Computed Mx | Computed My | Computed Mz |
|---|---|---|---|---|---|---|
| Fx | 0.288 | 0.420 | 0.140 | 0.100 | 0.070 | 0.970 |
| Fy | 0.220 | 0.150 | 1.980 | 0.100 | 0.120 | 0.120 |
| Fz | 5.620 | 5.620 | 5.620 | 5.680 | 5.670 | 5.620 |
| Mx | 0.090 | 8.950 | 0.140 | 0.260 | 0.110 | 0.130 |
| My | 8.990 | 0.000 | 0.030 | 0.070 | 0.280 | 2.320 |
| Mz | 0.460 | 0.010 | 0.030 | 0.070 | 0.510 | 0.120 |
The calibration simulation involved applying known loads in all six directions using the calibration rod method. For each axis, loads were incremented from zero to the full scale in both positive and negative directions, and the axial forces were recorded. The data was processed in MATLAB to compute the calibration matrix $\mathbf{C}$. Post-calibration, the sensor’s performance was re-evaluated by applying test loads and comparing the computed forces with the actual inputs. The results, shown in Table 2, indicate a remarkable improvement in accuracy. Type I errors were reduced to a maximum of 0.248%, and Type II errors dropped to below 2.794%, demonstrating the efficacy of the calibration method for the six-axis force sensor.
| Loading Direction | Computed Fx | Computed Fy | Computed Fz | Computed Mx | Computed My | Computed Mz |
|---|---|---|---|---|---|---|
| Fx | 0.160 | 0.001 | 0.012 | 0.011 | 0.050 | 0.016 |
| Fy | 0.001 | 0.131 | 0.004 | 0.016 | 0.004 | 0.019 |
| Fz | 0.160 | 0.018 | 0.224 | 0.020 | 0.039 | 0.023 |
| Mx | 0.008 | 2.794 | 0.008 | 0.248 | 0.038 | 0.008 |
| My | 0.800 | 0.003 | 0.001 | 0.080 | 0.072 | 0.024 |
| Mz | 0.005 | 0.001 | 0.005 | 0.002 | 0.027 | 0.009 |
In conclusion, the calibration method based on the calibration rod provides a practical and efficient solution for enhancing the accuracy of the Stewart six-axis force sensor. Through theoretical analysis and finite element simulations, I have demonstrated that this approach significantly reduces both Type I and Type II errors, leading to improved performance in real-world applications. The method’s simplicity and cost-effectiveness make it suitable for calibrating medium to small-range six-axis force sensors, offering valuable insights for future research and development in multi-dimensional force measurement. Further work could explore dynamic calibration and the integration of this method into industrial sensor systems to achieve even higher precision.
The Stewart six-axis force sensor, with its parallel architecture, remains a pivotal tool in robotics and automation. By addressing the inherent errors through systematic calibration, we can unlock its full potential for precise force and torque sensing. The proposed calibration scheme not only improves accuracy but also underscores the importance of tailored calibration techniques in the evolution of six-axis force sensor technology. As applications expand into areas like collaborative robotics and aerospace testing, such advancements will be crucial for ensuring reliable and accurate measurements.