In the field of robotics, automotive manufacturing, biomechanics, and aerospace, the six-axis force sensor plays a crucial role in measuring three-dimensional force and moment components simultaneously. Accurate calibration of the six-axis force sensor is essential to ensure high measurement precision, as it directly impacts the performance of systems relying on force feedback. Traditional calibration methods often assume ideal loading conditions, but in practice, errors in the calibration apparatus—such as misalignment in force direction or position—introduce additional forces that lead to inter-dimensional coupling. This coupling distorts the calibration matrix and degrades the accuracy of the six-axis force sensor. To address this issue, I propose a static calibration method based on Independent Component Analysis (ICA), which effectively corrects the calibration matrix by separating mixed signals and recovering the true loading forces.
The core idea of this method lies in treating the forces acting on the sensor platform as independent source signals that become mixed through the sensor’s elastic elements. By applying ICA, these mixed signals are decomposed into independent components, which are then aligned with the ideal source signals to resolve uncertainties in ordering and amplitude. This process allows for the reconstruction of the actual forces, including any additional unintentional loads, leading to a corrected calibration matrix that closely approximates the ideal one. In this article, I will detail the principles of ICA, explain the steps to eliminate indeterminacies, and present simulation results that validate the effectiveness of the approach for enhancing the performance of the six-axis force sensor.
Principles of Independent Component Analysis
Independent Component Analysis is a computational technique used to separate multivariate signals into additive, statistically independent components. The basic model of ICA, without considering noise, can be expressed as follows: Suppose we have n observed random variables x1, x2, …, xn, which are linear combinations of n unknown source signals s1, s2, …, sn. Mathematically, this is represented as:
$$x_i = a_{i1}s_1 + a_{i2}s_2 + \cdots + a_{in}s_n \quad \text{for } i = 1, 2, \ldots, n$$
where aij are the mixing coefficients. In vector-matrix form, this becomes:
$$\mathbf{X} = \mathbf{A} \mathbf{S}$$
Here, X = [x1, x2, …, xn]T is the vector of observed signals, S = [s1, s2, …, sn]T is the vector of source signals, and A is an n × n mixing matrix composed of the coefficients aij. The goal of ICA is to estimate both A and S from the observed X, by finding a separation matrix W such that:
$$\mathbf{Y} = \mathbf{W} \mathbf{X} = \mathbf{W} \mathbf{A} \mathbf{S} = \mathbf{P} \mathbf{S}$$
where Y = [y1, y2, …, yn]T is the vector of separated signals, and P is a permutation matrix with scaling. Ideally, P should be an identity matrix, meaning that the separated signals Y are identical to the source signals S. However, due to the inherent indeterminacies of ICA, Y may differ from S in terms of ordering and amplitude.
For ICA to be applicable, two key assumptions must hold. First, the source signals must be statistically independent. This independence is the foundation of the method, as it allows the separation process to maximize non-Gaussianity. Second, the independent components should have non-Gaussian distributions. According to the central limit theorem, the sum of independent random variables tends toward a Gaussian distribution. Thus, the mixed signals are more Gaussian than the source signals, and by maximizing non-Gaussianity in the separated components, ICA can achieve effective separation. In this work, I employ the FastICA algorithm, a batch-processing method known for its rapid convergence and robust performance in signal separation, which is well-suited for the static calibration of the six-axis force sensor.
Elimination of Indeterminacies in ICA
When applying ICA to the calibration of a six-axis force sensor, the separated signals often exhibit indeterminacies in ordering and amplitude. These uncertainties arise because the mixing matrix A and source signals S are not uniquely identifiable. Specifically, any permutation of the source signals or scaling of their amplitudes can yield the same observed signals. To correct the calibration matrix accurately, it is essential to resolve these indeterminacies by leveraging prior knowledge of the ideal loading conditions.
Ordering Indeterminacy Elimination
The ordering indeterminacy occurs because the separated signals Y may not correspond to the original sequence of the source signals S. In the context of the six-axis force sensor, the ideal loading forces are applied sequentially along each dimension. For instance, during calibration, forces are loaded separately along the three force axes and three moment axes, resulting in an ideal source signal matrix that is diagonal. Let S represent the ideal source signals, where each row corresponds to a dimension of the six-axis force sensor, and the columns represent time samples. For a calibration with n loading points per dimension, S can be written as:
$$\mathbf{S} = \text{diag}(s_{ii}) \quad \text{for } i = 1, 2, \ldots, 6$$
where sii = [ai1, ai2, …, ain] is the vector of ideal loads for the i-th dimension. However, due to additional forces introduced by calibration errors, the actual source signals S’ may contain non-zero elements in off-diagonal positions. After ICA separation, we obtain Y, which approximates S’ but with potential reordering.
To correct the ordering, I compute the correlation coefficient matrix between the separated signals yi and the ideal source signals sj. The correlation coefficient for two vectors m and n is given by:
$$R(\mathbf{m}, \mathbf{n}) = \frac{C(\mathbf{m}, \mathbf{n})}{\sqrt{C(\mathbf{m}, \mathbf{m}) C(\mathbf{n}, \mathbf{n})}}$$
where C(m, n) is the covariance matrix. The determinant |R| indicates the similarity between m and n; a smaller |R| implies greater similarity, with |R| = 0 for identical vectors. By calculating |R| for each pair of yi and sj, I identify the permutation that minimizes |R|, thereby aligning the separated signals with the correct dimensions of the six-axis force sensor.
Amplitude Indeterminacy Elimination
The amplitude indeterminacy stems from the fact that the source signals can be scaled arbitrarily without affecting the observed signals. In mathematical terms, if a source signal si is scaled by a factor α, and the corresponding column of A is scaled by 1/α, the observed signals remain unchanged. After ordering correction, I estimate the scaling factors for each separated signal relative to the ideal source signals. For a separated signal yi and its corresponding ideal signal si, the scaling factor pi is computed as:
$$p_i = \frac{\mathbf{y}_i}{\mathbf{s}_i}$$
This factor accounts for the amplitude difference due to mixing. However, in the presence of additional forces, the scaling must be adjusted to recover the true amplitudes. Suppose the actual source signal S’ includes an additional force vector f in some dimensions. After ICA, the separated signal yi may contain scaled versions of both the ideal force and f. By applying the scaling factor pi derived from the ideal components, I can rescale yi to obtain the corrected signal S”i = yi / pi, which includes the recovered additional forces. This step ensures that the amplitudes of all forces, including unintentional ones, are accurately restored for the six-axis force sensor calibration.
Once the ordering and amplitude indeterminacies are resolved, the corrected source signals S” are used to compute the calibration matrix C for the six-axis force sensor. The calibration matrix relates the observed signals from the elastic elements to the applied forces and moments. Specifically, it is given by:
$$\mathbf{C} = \mathbf{S”} \mathbf{X}^+$$
where X+ is the pseudoinverse of the observed signal matrix. This corrected matrix C minimizes the effects of inter-dimensional coupling and improves the overall accuracy of the six-axis force sensor.
Simulation and Validation
To validate the proposed method, I developed a simulation model of a six-axis force sensor based on the Stewart platform structure, specifically the 6/3-3 type, which is known for its force isotropy and widespread use in high-precision applications. The model consists of an upper platform, a lower platform, and six elastic bodies connecting them via spherical joints. The materials were selected to mimic real-world conditions: the elastic bodies are made of aluminum alloy H12, while the platforms are constructed from 45 steel. The coordinate system is defined with the origin at the center of the upper platform, the X and Y axes lying in the plane of the upper platform, and the Z axis pointing downward toward the lower platform.
In the finite element analysis, I meshed the platforms with a coarse grid spacing of 100 mm to reduce computational load, while the elastic bodies were finely meshed at 20 mm for accuracy. The spherical joints were modeled as frictionless contacts to simulate ideal conditions, though in practice, lubrication minimizes friction effects. This model allows for the simulation of force loading and the extraction of mixed signals from the elastic bodies, which are essential for evaluating the calibration process of the six-axis force sensor.
For the simulation, I applied ideal loading forces and moments to the upper platform of the six-axis force sensor. The forces ranged from -100 N to 100 N with a step size of Δ = 10 N, and the moments ranged from -100 N·m to 100 N·m with the same step size. This resulted in a total of n = 21 loading points per dimension, leading to an ideal source signal matrix S that is diagonal, as shown in the waveform below (represented conceptually):
| Dimension | Force Range (N or N·m) | Step Size | Waveform Description |
|---|---|---|---|
| Fx | -100 to 100 N | 10 N | Linear ramp from -100 to 100 |
| Fy | -100 to 100 N | 10 N | Linear ramp from -100 to 100 |
| Fz | -100 to 100 N | 10 N | Linear ramp from -100 to 100 |
| Mx | -100 to 100 N·m | 10 N·m | Linear ramp from -100 to 100 |
| My | -100 to 100 N·m | 10 N·m | Linear ramp from -100 to 100 |
| Mz | -100 to 100 N·m | 10 N·m | Linear ramp from -100 to 100 |
From the elastic bodies, I extracted the mixed signals X, which represent the sensor’s output under ideal loading. The standard calibration matrix C was computed using the pseudoinverse method:
$$\mathbf{C} = \mathbf{S} \mathbf{X}^+$$
This matrix C serves as the reference for evaluating the correction method. For example, a subset of the standard calibration matrix is shown below (values are rounded for clarity):
| Row | Column 1 | Column 2 | Column 3 | Column 4 | Column 5 | Column 6 |
|---|---|---|---|---|---|---|
| 1 | 0.4016 | -0.2914 | -0.2943 | -0.3458 | -0.2013 | -0.2765 |
| 2 | -0.2027 | 0.2400 | -0.2701 | -0.3816 | 0.0157 | 0.2733 |
| 3 | -0.4525 | -0.2029 | -0.2944 | 0.0115 | 0.3979 | -0.2765 |
Next, I introduced additional forces to simulate calibration errors, such as misalignment. These additional forces were applied as off-diagonal elements in the source signal matrix S’, representing unintended loads during the calibration of the six-axis force sensor. The mixed signals X’ under these conditions were recorded, and if the additional forces were ignored, the calibration matrix C’ computed as C’ = S X’+ would deviate from the standard matrix C. For instance, a subset of C’ is:
| Row | Column 1 | Column 2 | Column 3 | Column 4 | Column 5 | Column 6 |
|---|---|---|---|---|---|---|
| 1 | 0.3869 | -0.3094 | -0.2943 | -0.3534 | -0.1961 | -0.2845 |
| 2 | -0.1927 | 0.2202 | -0.2701 | -0.3807 | 0.0131 | 0.2659 |
| 3 | -0.4524 | -0.2023 | -0.2944 | 0.0053 | 0.3920 | -0.2845 |
To correct this, I applied ICA to the mixed signals X’ using the FastICA algorithm. The separated signals Y were obtained, which included both the ideal forces and the additional forces. However, these signals exhibited ordering and amplitude indeterminacies. By computing the correlation coefficients with the ideal source signals S, I realigned the separated signals. Then, I estimated the scaling factors and rescaled the amplitudes to recover the true forces, resulting in the corrected source signals S”. The waveform of S” showed the restored forces, including the additional components, as illustrated conceptually in the simulation results.
Using S”, I computed the corrected calibration matrix C” as:
$$\mathbf{C”} = \mathbf{S”} \mathbf{X’}^+$$
A subset of C” is presented below:
| Row | Column 1 | Column 2 | Column 3 | Column 4 | Column 5 | Column 6 |
|---|---|---|---|---|---|---|
| 1 | 0.4012 | -0.2918 | -0.2941 | -0.3459 | -0.2010 | -0.2766 |
| 2 | -0.2025 | 0.2396 | -0.2701 | -0.3819 | 0.0158 | 0.2733 |
| 3 | -0.4521 | -0.2029 | -0.2945 | 0.0133 | 0.3981 | -0.2765 |
To quantify the improvement, I compared the matrices C’ and C” with the standard matrix C by computing the product with the inverse of C. Let P1 = C’ C-1 and P2 = C” C-1. Ideally, these products should be identity matrices. I evaluated two metrics: the crosstalk error δct and the diagonal error Er. The crosstalk error measures how close the matrix is to a permutation matrix, defined as:
$$\delta_{ct} = \frac{1}{N} \sum_{i=1}^{N} \left( \frac{\sum_{j=1}^{N} |p_{ij}|}{\max_j |p_{ij}|} – 1 \right)$$
where pij are the elements of P, and N = 6 for the six-axis force sensor. A smaller δct indicates better performance. The diagonal error assesses the deviation from the identity matrix:
$$E_r = \frac{1}{N} \sum_{i=1}^{N} |p_{ii} – 1|$$
where pii are the diagonal elements. The results are summarized in the table below:
| Matrix | Crosstalk Error δct | Diagonal Error Er |
|---|---|---|
| P1 (Uncorrected) | 0.5986 | 0.0802 |
| P2 (Corrected) | 0.0187 | 0.0018 |
The significantly lower errors for P2 demonstrate that the corrected calibration matrix C” is much closer to the standard matrix C, confirming the effectiveness of the ICA-based method in improving the accuracy of the six-axis force sensor.
Conclusion
In this study, I have presented a novel static calibration method for the six-axis force sensor that leverages Independent Component Analysis to correct the calibration matrix. By addressing the indeterminacies in ordering and amplitude through correlation-based alignment and scaling factor estimation, the method effectively recovers the true loading forces, including additional unintentional forces caused by calibration errors. The simulation results, based on a Stewart platform model, show that the corrected matrix reduces crosstalk and diagonal errors significantly, leading to enhanced measurement precision. This approach is particularly valuable for applications requiring high accuracy, such as robotics and aerospace, where the six-axis force sensor is critical for force feedback and control. Future work could explore real-time implementation and adaptation to dynamic loading conditions, further advancing the capabilities of the six-axis force sensor in complex environments.