We first establish an appropriate coupling error model to capture the relationships between input forces and corresponding coupling errors. In the model, the input forces and output voltages of a 3-axis force sensor in X, Y, Z directions are defined as f_x, f_y, f_z and u_x, u_y, u_z, respectively.

For each dimension, output voltages are partitioned into two categories. One category includes the voltages corresponding to input forces in the same dimension, called prime voltages. The other category includes the voltages corresponding to the input forces in the other two dimensions, called coupling errors. We use u_xx, u_yy, u_zz to denote prime voltages and e_x, e_y, e_z to denote coupling errors in X, Y, Z directions, respectively. Prime voltages account for the majority of output voltages. Next, coupling errors are separated into two coupling error elements caused by input forces of different dimensions. Let (e_xy, e_xz) represent the coupling error element in X direction, where e_xy refers to the coupling error element caused by f_y, and e_xz refers to the coupling error element caused by f_z. Similarly, we split the coupling error in Y direction into e_yx and e_yz, and split the coupling error in Z direction into e_zx and e_zy. We can get: (1) { u x = u xx + e xy + e x z u y = u y y + e yx + e yz u z = u z z + e zx + e zy

Based on the observation of calibration data of multi-axis force sensors in our lab, we make the following assumptions about coupling errors.

From the above assumptions and the principle of superposition for stress, we propose a coupling error model as shown in Figure 1.

In Figure 1, there are three layers in the coupling error model: the input layer, the output layer, and the middle layer. Nine nodes in the middle layer are parallel and separated from each other. Functions ω_x(), ω_y() and ω_z(), are non-coupling functions, linearly relating prime force to prime voltage in each dimension as shown in Equation (2).

In calibration experiments, only one-dimensional force is applied to a 3-axis force sensor each time, while the output voltages of all directions are recorded simultaneously. Detailed calibration experiment process will be described in Section 3. Consequently, as for calibration data, no coupling error exists in the output voltage corresponding to the direction of the input force and no prime force exists in the output voltages of other directions. In other words, the output voltage equals the prime voltage when the direction of the output voltage is the same as the direction of the input force, and the output voltage equals the corresponding coupling error elements when the direction of the output voltage is different from the direction of the input force. For instance, during calibration experiment of X direction, a set of standard f_x(c) are applied to the force sensor while f_y(c) and f_z(c) remain zero (the subscript c represents “calibration data”). Thus we can get u_x(c) = u_xx(c), u_y(c) = e_yx(c), u_z(c) = e_zx(c). Similarly, for calibration data in Y direction, u_x(c) = e_xy(c), u_y(c) = u_yy(c), u_z(c) = e_zy(c); for calibration data in Z direction, u_z(c) = e_xz(c), u_y(c) = e_yz(c), u_z(c) = u_zz(c).Thus, coefficients k_xx, k_yy, and k_zz in Equation (2) can be calculated by linear fitting of prime forces and prime voltages of calibration data using LSM.

Functions respecting nonlinear relationships between disturbing forces and corresponding coupling error elements in every dimension are φ_yx(), φ_zx(), φ_xy(), φ_zy(), φ_xz(), φ_yz(), called coupling functions as shown in Equation (3).

Then, corrected coupling functions χ() can be obtained from Equations (2) and (3): (4) { e yx = φ yx ( f x ) = φ yx ( ω x − 1 ( u xx ) ) = χ yx ( u xx ) e zx = φ zx ( f x ) = φ zx ( ω x − 1 ( u xx ) ) = χ zx ( u xx ) e xy = φ xy ( f y ) = φ xy ( ω y − 1 ( u y y ) ) = χ xy ( u y y ) e zy = φ zy ( f y ) = φ zy ( ω y − 1 ( u y y ) ) = χ zy ( u y y ) e x z = φ x z ( f z ) = φ x z ( ω z − 1 ( u z z ) ) = χ x z ( u z z ) e yz = φ yz ( f z ) = φ yz ( ω z − 1 ( u z z ) ) = χ yz ( u z z )

During a decoupling process of an actual force perception task, one can hardly obtain the exact value of prime voltages u_xx, u_yy, u_zz for Equation (4), because the data obtained in a force perception task (called task data) are different from calibration data. The input forces in all directions are always non-zero, so the output voltages in all directions contain coupling errors and the output voltages no longer equal the prime voltages. However, a large number of experiment data show that, in most cases, coupling errors take up less than 5% of the full scale (F.S.) output voltages and the absolute values of the slopes of the corrected coupling functions are no more than 0.05. Thus, the output voltages of task data u_x(t), u_y(t), u_z(t) approximately equal the prime voltages u_xx(t), u_yy(t), u_zz(t) (the subscript t represents “task data”), and can be used as substitutions of the prime voltages u_x(t), u_y(t), u_z(t) as independent variables of Equation (4). This approximate substitution may induce second-order coupling errors. Take e_yx for example: the measured coupling error element e_yx(m) = χ_yx(u_x(t)) is calculated instead of the actual coupling error element e_yx(a) = χ_yx(u_xx(t))(the subscript m represents “measured”; the subscript a represents “actual”). The second-order coupling error E_eyx can be expressed in Equation (5), (5) E e yx = | e y x ( m ) − e y x ( a ) | u x ( F . S . ) = | χ yx ( u x ( t ) ) − χ yx ( u x x ( t ) ) | u x ( F . S . )where u_x(F.S.) represents the full scale value of output voltages in X direction.

According to the Lagrange mean value theorem, ∃ξ ∈ (u_xx(t), u_x(t)) such that (6) E e yx = | χ yx ′ ( ξ ) | | ( u x ( t ) − u x x ( t ) ) | u x ( F . S . ) ≤ 0.05 ⋅ ( 5 % u x ( F . S . ) ) u x ( F . S . ) = 0.25 %

As a result, the rate of second-order coupling errors is less than 0.25% and can be neglected in most industrial applications.

We choose ε-SVR for nonlinear approximations for the corrected coupling functions, which is much less likely to subject to overfitting problems. Also, the fitting processes of those functions are independent from one dimension to another, hence the fitting result of one dimension does not affect that of another dimensions.

Six ε-SVRs are utilized to approximate six corrected coupling functions for its generalization ability. ε-SVR learns the relationship between the input (i.e., prime voltages) and the output (i.e., corresponding coupling error elements) by adjusting the structure and parameters of a flexible model directly from training data to minimize the prediction error.

The basic idea for the case of nonlinear regression by ε-SVR is to project the input space x_i to a higher dimensional feature space by a map Φ. Then, the ε-SVR defines a linear prediction model over the mapped samples in the feature space. A nonlinear function is learned by this model while the capacity of the model is controlled by a parameter that does not depend on the dimensionality of the space [16]. As calculation with the map Φ can easily become computationally infeasible (because it is too complex), a kernel function k is introduced as the dot product of Φ, as expressed in Equation (7).

Typical kernels include the linear k ( x i , x j ) = x i T = x j, the polynomial k ( x i , x j ) = ( x i T x j + 1 ) d, and the Gaussian Function(RBF) k(x_i, x_j) = exp(−‖x_i − x_j‖²/2δ²). The RBF kernel is most frequently used and it is also the one used in our implementation.

Briefly, the ε-SVR is to solve a convex optimization problem: (8a) minimize 1 2 ‖ w ‖ 2 + C ∑ i = 1 Q ( ξ i + ξ i ∗ ) (8b) subject to { y i − w Φ ( x i ) − b ≤ ɛ + ξ i w Φ ( x i ) + b − y i ≤ ɛ + ξ i ∗ ξ i , ξ i ∗ ≥ 0where i = 1,2, … , Q are training data points, the parameter w indicates the flatness of regression function f due to the fact that kernels can be associated with flatness properties via regularization operators [19]. ξ_i, ξ i ∗ are slack variables introduced by Vapnik's ε-insensitive loss function [20], in which errors up to ε are not penalized, and all further deviations will incur a linear penalization [21]. C > 0 is the regularization parameter determining the trade-off between the flatness of regression function f ( i . e . , 1 2 ‖ w ‖ 2 ) and the total tolerance on deviations larger than ε ( i . e . , ∑ i = 1 Q ( ξ i + ξ i ∗ ) ). A graphical description of ε-SVR model is shown in Figure 2. Points on the boundaries and outside the boundaries are called Support Vectors.

To solve the optimization problem of Equation (8), a Lagrange function is constructed by introducing Lagrange multipliers a_i and a i ∗. Partial derivatives of the Lagrange function with respect to the primal variables (w, b, ξ_i, ξ i ∗) are calculated. Finally, the dual optimization problem can be derived by partial derivation of the Lagrange function and taking into account that ξ i ξ i ∗ = 0: (9a) maximize − 1 2 ∑ i , j = 1 Q ( a i − a i ∗ ) ( a j − a j ∗ ) k ( x i x j ) − ε ∑ i = 1 Q ( a i + a i ∗ ) + ∑ i = 1 Q ( a i − a i ∗ ) (9b) subject to ∑ i = 1 Q ( a i − a i ∗ ) = 0 and a i , a i ∗ ε [ 0 , C ]

Note that Support Vectors correspond to training data whose Lagrange multipliers a_i's are non-zero in Equation (9). Training data for which a_i = C are called Bounded Support Vectors, and they are located on the two boundaries in Figure 2. Training data with 0 < a_i < C are called Free Support Vectors and they are outside the boundaries [22]. Only Support Vectors contribute to the penalization in Equation (8a). Equation (9) can be solved by optimization algorithms such as Sequential Minimal Optimal, Chunking and Decomposing. The Sequential Minimal Optimal [23] is most frequently used due to its fast calculation speed. Solving the quadratic problems of Equation (9) yields the final solution of ε-SVR: (10) f ( x ) = ∑ i = 1 Q ( a i − a i ∗ ) k ( x i , x ) + b

It is shown that ε-SVR has properties of robustness to noises and gross errors even though the gross errors are part of the set of Support Vectors [24,25]. This is because the Lagrange Multipliers solved in the objective function of Equation (9) are upper bounded by the constraint of Equation (9). All Support Vectors, including the gross errors, will have Lagrange Multipliers with absolute value no more than the upper bound C. From aspect of Equation (8), as C searches for trade-off between flatness and empirical risk, if necessary, ε-SVR will sacrifice the tolerance to get a good flatness. ε-SVR also has many other advantages such as global minima and reduced likelihood of overfitting.

The whole decoupling process consists of two stages. The first stage is for establishment of the coupling error functions. The second stage is in an actual on-line force perception task. During the task, the output voltages of all directions are obtained and the coupling error model is utilized to eliminate coupling errors and calculate measured forces. Our decoupling process is described as follows:

Conduct the static calibration experiment of a 3-axis force sensor and get the calibration data, which contain sets of input forces and output voltages (f_x(c), f_y(c), f_z(c) and u_x(c), u_y(c), u_z(c)).

The above steps establish the entire coupling error model. The following steps are designed for decoupling based on the model using a 3-axis force sensor in an actual force perception task.

Figure 3 shows the flow chart of the whole decoupling process. The blue arrows indicate off-line process of the establishment of the coupling error model with calibration data. The red arrows indicate on-line process of decoupling in an actual force perception task. Note that the coupling error model and decoupling process are proposed with respect to a predefined frame based on the structural characteristics of the sensor. During an actual force perception task, if the orientation of the predefined frame is different from that of the reference frame used for force perception, the measured forces after decoupling with respect to the predefined frame should be transformed by a rotation matrix in order to align with the reference frame.

Cross validation is used to estimate the performance of decoupling by a standard MIMO RBF NN. The cross validation is repeated three times while each two sets of the zero-corrected calibration, named as “test data I” and “test data II”, are used for verification and the other four sets are used as training data. Matlab Neural Network Toolbox is used for simulations. The variance is set to 1, the maximum MSE error tolerance is set to 2.4 × 10⁻⁶.

The maximum MSE error tolerance is chosen to be small enough to ensure that all training data be selected as centers of the MIMO RBF NN.

In order to test the decoupling accuracy, interference errors γ are calculated. Taking calibration data in X direction for example, the interference error γ_x is calculated in Equation (12), (12) γ x = max | f y ( m ) | + max | f z ( m ) | f x ( F S )where f_x(FS) denotes the full scale value of loaded force in X direction. The interference error γ_x accounts for maximum coupling errors in Y direction and Z direction induced by loaded force in X direction. γ_y and γ_z are calculated in the same way. Interference errors of initial data and data after decoupling by a standard MIMO RBF NN are shown in Table 1.

Table 1 shows that the decoupling algorithm by a standard MIMO RBF is unreliable, because although some interference errors are reduced to less than 1%, some interference errors are even worse than initial data (in boldface). This is because compared with the number of training data, there are too many parameters in the MIMO RBF NN. Also, different coupling error elements are not well related. For example, there is little relationship between e_xy and e_xz. However, decoupling by a standard MIMO RBF NN means all the relationships are regarded as the same in the training process. The standard MIMO RBF NN does not have a good conformability with the calibration data shape. As a result, random noise in the training data may easily contribute to overfitting.

We also use cross validation to test the performance of our proposed decoupling algorithm. Here we take the first round of cross validation for example. To quantify the strong linearity of non-coupling functions, linear fittings are performed on the training data (plotted as blue points) to determine the slopes and strength of the correlations (plotted as black lines) as shown in Figure 6.

The linear correlation coefficients for all three directions are calculated to be more than 0.99, which correspond to assumption 1 (see Section 2). The inverse functions of non-coupling functions are obtained as: (13) { f x = ω x − 1 ( u x ) = 75.065 u x f y = ω y − 1 ( u y ) = 77.788 u y f z = ω z − 1 ( u z ) = 125.17 u z

The six corrected coupling functions are approximated by ε-SVRs. Each corrected coupling function is fitted by a separate ε-SVR, where the pre-specified parameters are set as the same value. The ε-SVR based approximations of corrected coupling functions are carried out using MATLAB and LIBSVM [31] software applications. LIBSVM [31] is an integrated software for support vector classification, distribution estimation and regression. It implements an SMO-type optimization algorithm [32] and provides an efficient interface for MATLAB. The initial parameters used to train the ε-SVR model by LIBSVM tool are listed in Table 2.

Here we also employ six separate Single Input Single Output(SISO) RBF NNs to do nonlinear approximations of the corrected coupling error functions to make a comparison with ε-SVRs. Parameters of the SISO RBF NNs are defined the same as the MIMO RBF NN, which is mentioned in Section 4.1.

Figure 7 shows the fitting results of the six corrected coupling functions.The blue points represent training data, the red points and magenta points represent two groups of test data respectively, the black curves represent fitted corrected coupling functions using ε-SVR and the red curves represent fitted corrected coupling functions using SISO RBF NN. Relationships between disturbing forces and related coupling error elements are nonlinear and distinct from one another, which correspond to assumption 2 and assumption 3 (see Section 2.1). In Figures 7(a) and 7(d), obvious overfittings happen in the fitting results of χ_xy and χ_yz using SISO RBF NNs, but the ε-SVR provides a good generalization ability to approximate different kinds of nonlinear functions without overfitting.

In the same fashion, the other two rounds of cross validation are calculated and interference errors are listed in Table 3. In Table 3, all interference errors are reduced to less than 1.6% FS. after decoupling by ε-SVR. Compared with Table 1, the decoupling accuracy is much more stable because no overfitting happens. Therefore, the proposed decoupling algorithm based on our coupling error model and ε-SVR is more robust.

Besides decoupling accuracy, another crucial criterion for a successful decoupling algorithm is the processing speed of both model establishment and decoupling an output voltage vector in an actual force perception task, especially the latter for on-line task purposes. To evaluate the processing speed, the elapsed time of the above two stages is recorded and the whole process is repeated 10 times. All calculations are done on a Windows XP Inter(R) Xeon(TM) 2 QUAD CPU, 3.0 GHz processor with 3.00 GB RAM. An overview of the averaged elapsed time is shown in Table 4. Table 4 shows that the elapsed time of the proposed algorithm in both the model establishment and the decoupling of an output voltage vector is less than decoupling by a standard RBF NN. ε-SVR proves to be fast regressions. The elapsed time of decoupling a certain output voltage vector using a coupling error model based on ε-SVR is less than 0.001 s. This means that the decoupling frequency is higher than 1,000 Hz. The speed of testing process based on ε-SVR is fast.

Summarizing, the calculation results of the decoupling methods demonstrate that our decoupling method based on our coupling error model and ε-SVRs has high reliability and fast running speed when no gross errors exist in the calibration data. In the next section, we will artificially add gross errors to the calibration data and analyze the decoupling method's robustness to gross errors.

Now we simulate the decoupling algorithm's reaction to gross errors. The gross errors in calibration data are also called outliers. As the quantity of gross errors in calibration data is always small, in our research, only one gross error is generated in one dimension while the calibration data of other dimensions remain the same. Here we take the first round of cross validation for example. The decoupling process is the same as Section 4.2. We artificially generate a gross error into the calibration data in X direction to test the gross error's effect on the decoupling accuracy in all dimensions. The gross error is added artificially by arbitrarily moving a point away from its initial location.

As no gross errors are introduced into the calibration data of other dimensions, the fitting results of functions in Y direction and Z direction will remain the same as shown in Figures 6 and 7. The fitting results of the non-coupling function ω_x() and corrected-coupling functions χ_xy() and χ_xz() in X direction with the gross error (plotted with red circles) are shown in Figures 8 and 9. The inverse function of ω_x() is obtained as: (14) f x = ω x − 1 ( u x ) = 74.193 u x

Comparing Figure 9 with Figure 7, when the obtained calibration data contain a gross error (outlier), the ε-SVR is not sensitive to the gross error and performs with high reliability, which correspond to the analysis in Section 2.2.2. Consequently, the proposed decoupling algorithm based on ε-SVR displays a strong robustness to the gross error. The calculated interference errors of test data I and test data II under the gross error's effect are listed in Table 5. Comparing Table 5 with Table 3, when the gross error is artificially introduced, γ_x, γ_y and γ_z after decoupling by ε-SVR of both test data I and test data II seem almost unchanged. The comparison of interference errors also confirms that ε-SVR is robust to gross errors.