1. Introduction
Artificial muscle, as one of the most promising smart materials, has attracted great attention for its potential applications in intelligent robots, artificial organs and biomedical devices in recent decades [
1,
2,
3,
4,
5]. Due to the impressive characteristics of light weight, high flexibility, super agility and long durability [
6,
7,
8,
9,
10], plentiful research results have been reported regarding the properties of artificial muscle. Li et al. [
11] proposed a novel cantilever beam artificial muscle using single-walled carbon nanotubes, which showed superfast response and ultrahigh mechanical output power density. Lu et al. [
12] put forward an artificial muscle based on multi-walled carbon nanotubes, ionic liquids and biopolymer chitosan, performing an excellent bio-compatibility. Jager et al. [
13] studied the electrically-controllable characteristics of conjugated polymers artificial muscles and applied them to the drug injection therapy. Then, Escudero et al. [
14] designed a magnetically sensitive actuator with a ball screw and rare-earth magnet coreless, which was compact enough to be used in practical artificial prosthesis, and the limitations of high-price and lower-power in conventional electrochemical polymer actuations were overcome. Recently, Kim and Kwon [
15] developed a hybrid muscle powered by skeletal muscle cells based on multi-walled carbon nanotubes. It provides potential for substantial innovation in the next generation of cell-based biohybrid microsystems.
Although the above research results look appealing, with the bionics research moving forward, researchers start to realize the rhythmic and involuntary contraction characterized by oscillations of the muscle which implies the instability of the muscle’s output response. Such a drawback definitely limits the applications of artificial muscle in clinical medicine and industrial production. Obviously, capturing the tendency of tremor behavior can provide adequate time for the suppression of the tremor, which is essential to the improvement of the artificial muscle’s output performance. Normally, the tremor behavior of artificial muscle can be characterized by the tremor occurrence times and maximum peak-to-peak values which can be predicted by applying grey systems theory.
The grey systems theory, established by Deng [
16] in 1982, focuses on uncertainty systems with small samples, which has been widely used in education, industry, agriculture, medicine and other fields [
17,
18,
19,
20,
21,
22,
23]. Grey prediction, an important part of grey systems theory, can provide scientific and reasonable forecasts on the future states of grey systems. For the requirement of solving practical problems, many improvements have been proposed by researchers to obtain accurate prediction results. According to their features and functions, grey predictions can be classified into grey sequence prediction, grey topological prediction, grey catastrophe prediction and seasonal system prediction, etc. [
24,
25,
26,
27]. Among them, grey catastrophe prediction (GCP) is essentially the prediction for the time distribution of abnormal values and it can forecast the forthcoming catastrophe moments to help relevant persons to prepare for the worst conditions in advance [
28,
29,
30,
31].
In this paper, based on the generation mechanism of tremor behavior, a novel DAGCP method is proposed to forecast the tremor occurrence time of artificial muscles. Due to the importance of occurrence times and maximum peak-to-peak values in unit time for muscle tremor behavior, the time subsequence models under different starting points and dimensions are constructed. Then, the appropriate dimension is selected based on grey correlation degree and model precision, and updated with the newly generated values to forecast the forthcoming tremor time. The prediction results are compared with actual time series values for evaluating relative error and accuracy. Furthermore, in order to prove the effectiveness and reliability of the proposed method, repeated experiments are carried out for GCP and DAGCP based on 10 groups of chitosan-based artificial muscles with different process parameters. Finally, some significant conclusions are discussed.
3. Discussion
3.1. Examination of Prediction Accuracy
To analyze the reliability of DAGCP, the accuracy of models has to be tested. The data tested comes from the China’s hydropower production from 2000 to 2015, and prediction results are given by Wang et al. [
25] by seasonal autoregressive integrated moving average (SARIMA) method and GM (1, 1) are provided for a direct comparison (see
Figure 7). It can be seen that the present method is in close agreement with the actual values and SARIMA. Thus, compared with conventional methods, DAGCP shows a fine prediction precision and reliability. The reason for excellent prediction performance is that various dimensions of time subsequences constructed based on different starting points weakens the effects of the initial value, and the selected subsequence with continuously updated values captures time-varying performance of the system. Next, this method will be used for the forecast of tremor occurrence time.
3.2. The Investigation of Tremor Times
During the response process of artificial muscle, tremor occurrence times in each time unit are extracted, which characterizes the occurrence frequency of the tremor, and can be used as an important index for the evaluation of tremor behavior.
By referencing
Table 1, the lower limiting outlier of occurrence times covered by tremor behavior was assigned as 1. Thus, the corresponding tremor data sequence within the first 65 s was:
When
i = 1, the distances between the first and subsequent other points were computed to obtain a new data subsequence:
And then
i = 2, the next subsequence was given:
Analogously, based on the dimension of the original sequence, five new data subsequences were constructed under dimensions from 4 to 8 as the candidate models. The estimated values and original values of each subsequence with respect to ordinal
k were shown in
Figure 8a–e, respectively. Meanwhile, model response, model accuracy, grey correction degree and fitting degree were calculated to evaluate the availability of the candidates (
Table 2).
Analogously, based on the dimension of the original sequence, five new data subsequences were constructed under dimensions from 4 to 8 as the candidate models. The estimated values and original values of each subsequence with respect to ordinal
k were shown in
Figure 8a–e, respectively. Meanwhile, model response, model accuracy, grey correction degree and fitting degree were calculated to evaluate the availability of the candidates (
Table 2).
From
Table 2 and
Figure 8, the results showed that the 4-D model had the highest prediction accuracy; however, it was not available for prediction because its grey correlation degree was less than 0.6 and the number of data samples was too small. Additionally, the lower correlation degree and accuracy of 5-D and 6-D models limited their applications in the prediction. Although both of them had high accuracy at the initial stage, compared with the 7-D subsequence, the model accuracy and grey correlation degree of 8-D were better as a whole, which showed a better fitting degree between the subsequence and original sequence. Therefore, the 8-D subsequence model about tremor occurrence times in unit time was used for prediction and the starting time point
q(1) was set as 2.
The estimated values and relative errors of conventional GCP and DAGCP were shown in
Table 3. Specifically, the conventional GCP was constructed by the original time sequence. It is obvious that the actual number of tremors only occurred once during 75~80 s and was affected by process parameters, experimental conditions and human factors, as well as existing contingency. Hence, it was not necessary to predict ahead in this circumstance.
It was obvious that the occurrence times in 65~70 s, 70~75 s and 80~85 s were 2, 3 and 2, respectively. As shown in
Table 3, there was a little difference between GCP and DAGCP in the relative error during 65~70 s. However, the relative error of DAGCP was smaller than GCP after 70 s. Therefore, the prediction accuracy of DAGCP was superior to that of GCP as a whole.
3.3. The Investigation of Peak-to-Peak Values
The maximum peak-to-peak values (the difference between adjacent crests and troughs) in unit time are extracted, which characterizes the severe degree of tremor, and also can be used for the evaluation of tremor behavior. Here, the lower limiting outlier was assigned as 0.2 mm. In addition, the corresponding data sequence within 65 s was:
When i = 1, a new data subsequence was: .
Thus, three new data subsequences were constructed under the dimensions from 4 to 6. The estimated values and original values were shown in
Figure 9a–c, respectively. In addition, the model response, model accuracy, grey correction degree and fitting degree were displayed in
Table 4.
From
Table 4 and
Figure 9, although the grey correlation degree had achieved the basic requirement, the prediction precision of 4-D and 5-D models limited their applications in the prediction. Among them, 6-D model had the highest model accuracy and grey correlation degree, which showed a more accurate fitting degree. Thus, the 6-D model about tremor peak-to-peak value in unit time was used for the prediction and
q(1) was also set as 2.
The estimated values and relative errors were shown in
Table 5. The actual peak-to-peak value was less than 0.2 mm during 75~80 s, which was allowed to happen. Therefore, it was not necessary to predict ahead in this circumstance.
In the above analysis, the peak-to-peak values in 65~70 s, 70~75 s and 80~85 s were 0.6 mm, 0.5 mm and 0.2 mm, respectively. As shown in
Table 5, the relative error of DAGCP was obviously less than that of GCP.
To validate the effectiveness and accuracy of the proposed method, the catastrophe prediction results of 10 groups of chitosan-based artificial muscles with different process parameters based on GCP and DAGCP were shown in
Table 6. Here, the data in the first 70 s was selected as original sequence and the last 10 s was used to assess the prediction accuracy. The lower thresholds of occurrence times and peak-to-peak values were assigned as 2 and 0.2 mm, respectively. If either of the conditions was satisfied, the tremor is considered to happen.
From the comparative analysis, the proposed DAGCP can better identify the occurrence tendency of tremor behavior and individual variation of original data, and achieve smaller relative errors. Therefore, this novel method is viable and reliable for predicting the tremor behavior of artificial muscle.
4. Conclusions
In this paper, we illustrate the tremor mechanism of artificial muscle and forecast the tremor occurrence time using GCP based methods. In order to further improve the prediction accuracy, we propose a new method called DAGCP by integrating sequence optimization mechanism and dynamic additional strategy into the conventional GCP. Its excellent prediction performance indicates that various dimension of time subsequences constructed based on different starting points weakens the effects of the initial value, and the selected subsequence with continuously updated values captures time-varying performance of the system.
As shown in the case study of artificial muscle, the novel DAGCP can earlier and more accurately forecast the forthcoming tremor time than conventional GCP. Therefore, it can be selected to provide the preparation time for parameter adjustment of artificial muscles’ online behaviors and realize the suppression of tremor behavior. In summary, this method has the following advantages: the degradation of prediction accuracy caused by the immobilization of original parameters is prevented; dynamic input, real-time update and gradual forecast of the model can be realized and widely used in the industry, agriculture and medicine, etc.