1. Introduction
Force/torque sensors are widely applied in manipulators to obtain feedback on force and torque, which can help researchers achieve close-loop control. When working in extreme conditions, for instance, in space [
1] or in deep-sea [
2,
3] applications, F/T sensors will suffer severe characteristic drift due to the wide range of ambient temperature. The characteristic drifts will cause bias between measured values and real values, consequently decreasing the precision of measurements. Thus, the effects of temperature should be compensated for to alleviate and even eliminate the characteristic drifts of F/T sensors.
The impacts of temperature on strain gauge sensors are omnifarious. The thermal effects of strain gauges and the different dilatation coefficients between strain gauges and elastic bodies are two major factors which cause characteristic drifts. Hardware and software methods are two ways to compensate for the impacts of temperature. Compensating methods focusing on hardware aim to eliminate the temperature’s impact on the regulating circuit [
4,
5,
6]. However, compensating on the circuit is usually costly and lacks flexibility. More importantly, it cannot meet the requirement of precision. Thus, hardware compensating methods are used as a kind of auxiliary means in practice [
7]. Software compensating methods aim to build inverse models between ambient temperature and thermal output. Sensors compensate for the characteristic drift according to the thermal output obtained by inverse models. Software compensation methods are powerful, flexible, and easily applied to various sensors. In view of these advantages, scholars proposed lots of software compensating methods, such as least square method (LSM), support vector machine (SVM) [
8,
9], and all kinds of artificial neutral networks (ANN) [
10,
11,
12]. LSM is convenient and intuitive, so researchers commonly apply it to solve linear and low rank non-linear fitting problems. However, the compensating performances of LSM are not as good as expected because rank non-linear error is usually involved in characteristic drifts. In practice, LSM is still applied to temperature compensation due to its simplicity when the calculating capacity is limited by the process unit size. The applications of ANN have been more and more extensive in recent years due to their prominent nonlinear fitting ability, and scholars proposed massive compensation methods based on ANN [
13,
14,
15]. However, compensating methods based on ANN suffer from two issues: low convergence rate and complex network structures. SVM derives from statistical learning theory and structural risk minimization (SRM). Convex optimization algorithms, such as quadratic programs, are adopted to solve SVM. Suykens J.A.K. and Vandewalle J. reformulated standard SVM and proposed the least-square support vector machine (LSSVM) [
16]. LSSVM inherits remarkable abilities for solving nonlinear and high rank regression problems from SVM and provides some improvements, including adopting a binary norm in the objective function and converting the optimization problem to solve the linear Karush–Kuhn–Tucker (KKT) condition. Zhang et al. [
17] proposed a LSSVM error compensation model to accurately estimate the state of health (SOH) of lithium-ion batteries. Tan et al. [
18] predicted the thermal error of machine tool spindles using the segment fusion least-square support vector machine (SF-LSSVM) and improved the prediction precision by up to 51%.
Though LSSVM is powerful in nonlinear fitting, its performance is significantly influenced by the choice of its parameters. Obtaining parameters of LSSVM by blind search is a costly task; therefore, various optimization algorithms are adopted by scholars, such as the Cuckoo search algorithm (CSA), particle swarm optimization (PSO), and the sparrow search algorithm (SSA) [
19,
20,
21]. The whale optimization algorithm (WOA) is a meta-heuristic optimization algorithm which is inspired by the bubble-net hunting behavior of humpback whales [
22]. The WOA obtains competitive performances in global exploration, has high flexibility compared with other optimization algorithms, and owns a simple structure and few parameters, which make it easy to be adopted in many fields. Although the WOA can efficiently locate the latent positions of the solution in the exploration phase, it suffers from a low convergence rate in the exploitation phase and has a tendency for being trapped in local optima [
23,
24]. Several measures are taken to offset these shortcomings of WOA, and embedding a high-efficiency local algorithm into WOA is gaining widespread traction. Tong [
25] designed a hybrid algorithm framework for WOA, and verified the framework by embedding differential evolution (DE) and the backtracking search optimization algorithm (BSA) into WOA. Nadim et al. [
26] hybridized WOA and DE to solve multi-objective virtual machine scheduling in cloud computing. Based on the aforesaid idea, this paper enhances the exploitative ability of the WOA by adopting the stimulated annealing algorithm (SA).
Ensemble method is a series of machine learning methods; boosting and bagging are representatives. They follow the "sampling–training–combining" workflow [
27]. Ensemble methods are in the spotlight of the machine learning field for their remarkable improvements in accuracy compared to single learners. Bagging adopts bootstrap sampling to obtain training data and combines the output of each learner by the most common strategy, such as voting for classification and averaging for regression [
28]. Bagging can reduce the variance of base learners, especially when those base learners are unstable. In view of improving the performance of any classifier, Joossens et al. proposed trimmed bagging, which only averages outputs from the best base learners, rather than all of them, when applying the bagging method. LSSVM is a stable learning method, and the turbulence of training set can have a subtle impact on it. The standard bagging algorithm works not well, and LSSVM acts as the base learner; therefore, trimmed bagging is adopted in the proposed method.
In this study, an ensemble temperature compensation approach is proposed which adopts LSSVM as the base learner and trimmed bagging as the ensemble framework. To achieve the optimal performance of LSSVM, the hybrid whale optimization algorithm (hWOA) was used to configure parameters in LSSVM, which introduced the simulated annealing algorithm (SA) to WOA. In addition, inverse quoted error (inv QE) was taken to weight aggregate outputs of base learners for its extensive usage in practice. Finally, a temperature compensation model was established by ensemble hWOA-LSSVM based on trimmed bagging.
4. Experiment
All experiments were conducted on a six-axis force/torque sensor designed by the Institute of Intelligent Machines (IIM), Chinese Academy of Sciences (CAS). The F/T sensor consists of strain gauges and a novel double E-shape elastic body. The rated ranges of the F/T sensor are N, N, and N·m.
4.1. Calibration and Decoupling
A calibration experiment was conducted to obtain the transfer expression from outer stimuli to voltage signal responded bthe y sensor. Assuming
indicates the output vector, which consists of the voltage of each dimension, and
indicates the measured load vector, the transfer expression can be presented by the following equation:
where
W is the weight matrix, which is also called the calibration matrix, and
B is the bias vector.
The basic procedures for multi-axis F/T sensor calibration apply a series of known loads which increase from minimum to maximum rated range with a certain step. The voltage responded by the sensor is recorded at each sample point. The above procedures were repeated for three times in this calibration experiment, and all record data were used for calibration and decoupling. Configurations of calibration experiment are shown in
Table 1, and the environment temperature and humidity are 25 °C and 60%, respectively.
After all calibration procedures completed, the least-squares (LS) algorithm was adopted to calculate the calibration matrix
W and bias vector
B, which are as follows:
Finally, the transfer expression could be obtained by substituting
W and
B into Equation (
29).
4.2. High–Low Temperature Experiment
A high–low temperature experiment was conducted to analyze the measurement error of sensors caused by temperature drift and obtain the data for the training model. Wang shows that exerting loads on F/T sensors makes no difference to the temperature drift phenomenon [
35]; therefore, the sensors were not loaded with any force/toque during the temperature experiment.
The six-axis F/T sensor was placed in a high–low temperature chamber and run at 5V DC voltage in the experiment. The temperature in the chamber was varied from −30 to 70 °C, and kept for 2 h at thirteen temperature sampling points (marked as
): −30, −20, −10, 0, 10, 20, 25, 30, 35, 40, 50, 60, and 70 °C. The gathering module sampled about 450 outputs (marked as
) of the F/T sensor during each
and transmitted them to the PC for processing and storage. The temperature experiment configuration is demonstrated in
Figure 2.
The sensor measurement error
is convenient for comparing, which can be defined as follows:
where
is the measured value of the F/T sensor under T °C;
denotes the measured value under the temperature of calibration, which means 25 °C here;
denotes the full scale of the corresponding dimension.
Measurement errors caused by temperature in all six dimensions before compensation are illustrated in
Figure 3.
As is shown in
Figure 3, all dimensions of the F/T sensor suffer characteristic drift, which consists of both linear and nonlinear components. The relation between
and
presents more nonlinear features when ambient temperature is higher than 50 °C. In addition,
,
,
, and
have positive correlations with
and
; and
and
have negative correlations with them.
Overall, , , and have less and lower linearity than , , and . had the lowest , which is no more than , and had the largest , which reached at −30 °C. The measurement error of the F/T sensor caused by temperature variation was manifest; therefore, temperature compensation is vital for six-axis F/T sensors to meet the requirements of space manipulator control.
4.3. F/T Sensor Temperature Compensation
4.3.1. Model Training
The original dataset gathered in the temperature experiment was divided into 80% and 20% for training and testing, respectively. To evaluate the proposed model better, several algorithms, including standard support vector regression (Std-SVR), LSSVM optimized by particle swarm optimization (PSO-LSSVM), LSSVM optimized by the standard whale optimization algorithm (WOA-LSSVM), and the RBF neural network optimized by the standard whale optimization algorithm (WOA-RBFNN) were compared.
In the RBFNN, the number of neurons in the hidden layer was 13, and the centers of RBF function were set to , and 70, which correspond to the 13 sampling points in the aforesaid high–low temperature experiment.
The parameters for all algorithms were as identical as possible; all the parameters are demonstrated in
Table 2. Additionally, the mean square error (MSE) was used as the fitness function to evaluate the performances of algorithms.
4.3.2. Compensation
We compensated for the characteristic drift by connecting the trained temperature compensation model to outputs of the F/T sensor. To be specific, as is shown in
Figure 4, the multi-axis F/T sensor changed force and torque into voltage outputs, and then the compensation model calculated thermal outputs (which could be either positive or negative) according to the voltage outputs and current temperature. Compensation outputs can be obtained by subtracting thermal outputs from raw outputs of the sensor. Finally, the measured values were obtained by substituting the compensation outputs into the transfer Expression (
29).
4.4. Compensation Results and Analysis
The compensation results of various dimensions on the training set and testing set are shown in
Table 3 and
Table 4, respectively. As demonstrated in
Table 3, in the training set, PSO-LSSVM performed best in four dimensions (
,
,
,
), and EaW-LSSVM had the worst fitness in the other two dimensions (
,
). The ensemble model EhW-LSSVM performed slightly better than the single model WOA-LSSVM in the training set. The performance of WOA-RBFNN was average overall, and Std-SVM was inferior to other algorithms.
In the testing set, as
Table 4 shows, the EhW-LSSVM gained the lowest fitness in four dimensions (
,
,
,
), and the PSO-LSSVM performed better in
and
. The performance of the EhW-LSSVM was better than that of the single learner WOA-LSSVM in most dimensions. In addition, the fitness of all dimensions except
obtained by the WOA-RBFNN deteriorated greatly in the testing set, which indicates that the RBFNN suffers severe over-fitting under the same parameter configuration as the other algorithms.
We can draw the following inferences by analyzing the temperature compensation results: first, the EhW-LSSVM shows competitive predictive ability because of its reliable performance on the testing set and training set. Second, using the bagging ensemble method, the ensembled model did not show worse performance in training than the single learner, but obtained better predicting ability.
The best fitness of
in each iteration obtained on the training set was taken as an example to evaluate the search convergence characteristics. Additionally, the convergence curves of EhW-LSSVM (average on all base learners), PSO-LSSVM, and WOA-LSSVM are demonstrated in
Figure 5. As depicted in
Figure 5a,c, the hybrid whale optimization algorithm jumps out of local optima but drops in fitness sometimes. It is inferred that, benefiting from the hybridization of the SA, the hWOA has a remarkable ability to jump out of local optima and still retains good global searching ability. In addition, the convergence characteristic of PSO in
Figure 5b shows that the PSO converges too early and has a poor ability to get out of local optima.
After compensating with the EhW-LSSVM, the measurement errors caused by temperature variation are demonstrated in
Figure 6. Overall, the measurement errors of all dimensions decreased dramatically after compensation by the EhW-LSSVM.
and
had the best compensating performances, whose maximum measurement errors were less than
and
, respectively. The performances of
and
were somewhat inferior to those of other dimensions, and the maximum absolute measurement error of
was still less than
. In addition, the measurement errors of all dimensions showed no noticeable changes with temperature variation. Above all, the six-axis F/T sensor suffered from the temperature drift negligibly and met the request of cosmic operation after compensation by the EhW-LSSVM.
5. Conclusions
Our novel temperature compensating method consisting of LSSVM optimized by hybrid WOA and improved trimmed bagging was presented in this work for eliminating the characteristic drift of six-axis force/torque sensors in cosmic space. In addition, simulated annealing (SA) is applied to WOA to cover the shortage in exploiting. Furthermore, the optimal trim portion of trimmed bagging is determined by an adaptive trimming strategy, which automatically adjusts the trim portion according to the performances of base learners. Cross-validation and inverse quoted error are utilized to evaluate the model more accurately.
A high–low temperature experiment was conducted to investigate the impacts of temperature variation on six-axis F/T sensors and provide data for model training. The compensating results indicate that EhW-LSSVM possesses excellent predicting ability and dramatically decreased the measurement errors of six-axis F/T sensors to a level of . The hybrid WOA showed better ability than standard WOA during the process with search optimal parameters. In addition, the adaptive trimmed bagging lifted the effect of a single model in the testing set while losing no accuracy in the training set. According to temperature compensating results and comparisons with other algorithms, the EhW-LSSVM algorithm is a feasible and competitive temperature compensating method for six-axis F/T sensors.
The compensating performance of the EhW-LSSVM is satisfactory, but the complexity of its structure is also high. In future research, we will aim to reduce the model’s complexity and try to integrate the presented EhW-LSSVM into compact six-axis F/T sensors.