Modeling of the dynamic hysteresis in DEAP actuator using an empirical mode decomposition based long-short term memory network

Abstract

As a smart material-based actuator, the dielectric electro-active polymer (DEAP) actuator is widely considered to be a potential driving mechanism for many applications, especially in intelligent bio-inspired robotics. However, the DEAP actuator demonstrates rate-dependent and asymmetrical hysteresis phenomenon which leads to great tracking inaccuracy and even oscillatory response, severely limiting its further development. Feedforward Neural Network (FNN) model has already become a widely used method to describe this kind of strong hysteresis nonlinearity in recent years. However, the FNN has no ability to remember the historical state of long period of time which is also a very important factor to restrict hysteresis phenomenon. In this paper, a novel hybrid model, Long-Short Term Memory (LSTM) network combined with Empirical Mode Decomposition (EMD), is proposed to model the dynamic hysteresis nonlinearity in DEAP actuator. At first, the original control signal sequence is preprocessed into a series of sub-sequence by the EMD method and is reshaped by one-sided dead-zone operator. Then the input space of LSTM is conducted using the original control signal, the sub-sequence, and reshaped signal. Finally, the input space and the displacement signal are applied to train the long-short term memory network. In order to verify the performance of the proposed model, the traditional artificial back propagation neural network (BPNN) model, rate-dependent Prandtl-Ishlinskii (RPI) model, and nonlinear electromechanical (NEM) model are compared from prediction accuracy. The results demonstrate that: (1) the proposed model has a higher prediction accuracy than the traditional artificial BPNN, RPI, and NEM model; and (2) the prediction accuracy of LSTM network is significantly improved by using EMD. Therefore, the long-short term memory network combined with empirical mode decomposition is a competitive method compared to the existing state-of-the-art approach.

Keywords

Dielectric electro-active polymer actuator hysteresis neural network long-short term memory network empirical mode decomposition

1. Introduction

Smart materials-based actuators are increasingly being explored for bionic robotics applications requiring lightweight, rapid response speed, silent operations, and low power consumption. Dielectric electro-active polymer (DEAP) is one kind of flexible ionic compounds consisting of three layers of polymer film laminated together, which will produce deformation under voltage excitation (Hill et al., 2017; Iskandarani and Karimi, 2012; Mun et al., 2018; Rizzello et al., 2014; Yu et al., 2015; Yun et al., 2016). The DEAP actuator has been considered particularly potential for applications in bio-inspired soft robotics (Banisadr and Chen, 2017; Case et al., 2015), such as flexible joint mechanism (Ruderman et al., 2009). However, similar to other smart materials-based actuators, the DEAP actuator also exhibits strong hysteresis nonlinearity, which has rate-dependent and asymmetrical characteristics (Hodgins et al., 2013; Truong and Ahn, 2013). Moreover, the current displacement of actuator not only depends on the current control signal, but also depends on the historical control signal time-sequence (this phenomenon is called historical-time-dependent in this paper). It has been a challenging task of major interest to effectively model this kind of hysteresis at nowadays.

With the development of research, many hysteresis models have been proposed, mainly including physics-based models and phenomenological models. On the one hand, the physics-based models are established on first principles of physics effects (Baghel and Kulkarni, 2012; Chua and Bass, 1972; Gu et al., 2016; Hoffstadt and Maas, 2015; Jiles and Atherton, 1984; Rizzello et al., 2015; Sarban et al., 2011). Jiles and Atherton (1984) proposed a first order differential equation-based static physical model which has the characteristics of less computation, stable algorithm, and clear physical meaning of parameters. Sarban et al. (2011) developed a nonlinear electromechanical model coupled an electric circuit model with a viscoelastic mechanical model, concerning the characteristics of electric circuit and material at the same time. On the other hand, the phenomenological hysteresis models such as Preisach model (Ruderman et al., 2009), PI model (Goldfarb and Celanovic, 1997), and their modifications are more efficient models. Sutor et al. (2010) presented a new continuous weight function that parameters can be easily approximated and added an analytical function to the Preisach operator to account for the forced magnetization in ferromagnetic or ferroelectric materials. PI model and its modifications are most popular because they can theoretically describe various kinds of hysteresis. It is more attractive for real-time application. As the classical PI model (Goldfarb and Celanovic, 1997) is a symmetric and rate-independent model, improvements have been made on classical PI model. The rate-independent PI model can be coupled with a superposition of weighted dead-band operators (Al Janaideh and Aljanaideh, 2018; Kuhnen, 2003), polynomial operator only with odd powers (Qin et al., 2017), or other unique operators (Al Janaideh et al., 2009; Chen et al., 2013) to characterize the asymmetrical hysteresis nonlinearity. Al Janaideh and Aljanaideh (2018) and Qin et al. (2017) updated the thresholds and weights of the backlash operators according to the input rates to account for the rate-dependence of hysteresis, respectively. However, in the aforementioned models, the number of backlash operators and thresholds were basically manually chosen, which required the handler to master much experience about hysteresis characteristics. Although high accuracy can be achieved when the number of backlash operators was large, the structural complexity and identification difficulty of model increased.

Neural network is a very powerful tool for describing complex nonlinear relationship due to its strong self-learning ability, computing ability, fault tolerance, and fast convergence (Xu et al., 2013; Yu and Xu, 2014). It has been gradually applied to hysteretic nonlinear modeling of a variety of smart material-based actuators (Asua et al., 2008; Dang and Tan, 2005; Dong et al., 2008; Jiahua et al., 2016; Li and Tian, 2018; Zhang et al., 2009; Zhao and Tan, 2006) in recent years. Zhang et al. (2009) presented a neural modeling approach for hysteresis in piezoelectric actuator, it avoids the calculations of gradient of the hysteresis against its input. Asua et al. (2008) adopted a neural network to learn the hysteresis effect of SMAs, which was then used to hysteresis compensation. Zhao and Tan (2006) proposed a hysteretic operator using a turning point and applied it as one input node of neural network to model the Preisach-type hysteresis in piezoelectric actuator. However, due to the simple fully connected structure with each layer, it cannot deal with the historical-time-dependence of hysteresis well. Recurrent neural network (RNN) is a kind of time memory neural network in which the nodes of each layer are not only fully connected with the last layer but also recurrently connected with the nodes itself. RNN is more suitable for time series prediction (Barbounis et al., 2006). However, it faces vanishing gradient and exploding gradient problems. In order to solve these problems, Long-Short Term Memory (LSTM) network was proposed (Hochreiter and Schmidhuber, 1997). Relative to RNN, the LSTM network adds three gates (input gates, output gates, and forget gates) to the hiding layer for resolving the problem that RNN cannot process long temporal sequences of data. Liu et al. (2018) presented that long-short term memory network can capture the complex hysteretic dynamics in piezoelectric actuators. Consequently, it is a more appropriate choice to model the dynamic hysteresis in DEAP actuator.

In order to obtain the time-frequency characteristics of control signal of DEAP actuator, Empirical Mode Decomposition (EMD) method is introduced in this paper (Huang et al., 1998). EMD method is an empirical signal processing algorithm used for extracting features of the nonlinear data sequence, it decomposes the random signal into sub-sequences of different characteristics or frequencies. Mi et al. (2019) used empirical mode decomposition to decompose wind speed sequences into several intrinsic mode functions and then used machine learning models to predict it. Zhou et al. (2019) proposed an Empirical Mode Decomposition and Factorization Machine based Neural Network to predict the stock market trend.

Based on the above description, this paper proposes a hybrid EMD-LSTM hysteresis model of DEAP actuator. The EMD is used to preprocess the control signal sequence and decompose it into multiple subsequences. Then, LSTM model is used to predict the displacement signal. The advantages of this method are as follows: (1) The EMD pre-processing control signal can effectively extract the frequency characteristic in the data, which is an important part of input space for the next prediction; (2) the one-sided dead-zone operator can obviously represent the asymmetrical characteristic; and (3) Compared with the artificial BPNN, RPI, and NEM model, the EMD-LSTM network has a higher accuracy for predicting the displacement of DEAP actuator.

The structure of this paper is as follows. In Section 2, the DEAP actuation system is introduced in detail. In Section 3, the theory of LSTM model, EMD method, and the one-sided dead-zone operator is given. In Section 4, the predictive performance of models is compared and discussed. The work of this study is summarized and the conclusions are given in Section 5.

2. DEAP actuation system

2.1 DEAP actuator

The rectangle DEAP that we used for experiments is from Danfoss Polypower A/s, which has a total thickness of 80 µm. In this study, the DEAP is folded and clamped in a special box to fabricate a linear displacement actuator for experimental testing. Figure 1 shows the structure of the DEAP actuator. In Figure 1(a), the area I represents the film has silver layers on both sides, the area II represents only one side has silver layer, which is used for electrode extraction, and the area III is insulated on both sides.

2.2 Experimental platform

Figure 2 shows the main devices (Figure 2(a)) and block diagram of the experimental platform (Figure 2(b)) of DEAP actuation system. Therein, (1) the computer is employed to generate the control signal and then process the measured data; (2) the DEAP actuator is applied to realize the actuation on the terminal load; (3) the high-voltage power supply (HVPS, S15-3P from Matsusada) with a fixed gain of 1000 is used to amplify the control signal for the DEAP actuator; (4) the linear variable differential transformer displacement sensor (LVDT, SDVG20 from Soway), which has a 12.5 µm resolution, is utilized to measure the real-time displacement of the terminal; and (5) the multifunction data acquisition card (Advantech PCI-1710) is applied to transmit the control signal and measured displacement signal.

2.3. Hysteresis phenomenon

The dynamic hysteresis characteristics of DEAP actuator can be described step by step according to Figure 3. In the first situation (Figure 3(a)), the control signal has a fixed frequency such as typical sinusoidal, step and triangle signal, the input-output relationship curve will just be one hysteresis loop (major loop). In the second situation (Figure 3(b)), the signal has variable amplitudes, the hysteresis loop will then contain many minor loops which are similar to the major loop. The third situation (Figure 3(c)) is that the amplitude and frequency of control signal is both variable, the hysteresis loop will be more complicated with different tilt and saturation degrees. The first two descriptions are collectively referred to as rate-independent hysteresis, and the third is called rate-dependent hysteresis.

The hysteresis of DEAP-based actuator is usually caused by the following two aspects. On the one hand, DEAP is viscoelastic material, that is, under certain sudden stress, DEAP will produce instantaneous elastic response, as well as the viscous creep. On the other hand, the nonlinearity and time delay in the drive circuit, including the voltage generator, power amplifier, and wires. Based on these analyses, many physics-based models have been established by combining a viscoelastic mechanical model with electric circuit model to describe the dynamic hysteresis (Hoffstadt and Maas, 2015; Rizzello et al., 2015; Sarban et al., 2011).

In order to demonstrate the static and dynamic characteristics of hysteresis in DEAP actuator, the response of DEAP actuator system has been measured using a set of sinusoidal control signals with different frequencies as shown in Figure 4. The thick solid purple line represents the static characteristic, which means that the terminal of DEAP actuator will finally reach a stable position when the actuator is given a fixed control signal. For example, the purple triangle on the purple line means that the actuator finally produces a displacement of 1.9 mm when it is drove by a fixed 1.2 V control voltage. In addition, the red solid line, blue dot-and-dash line, and black dashed line represent the dynamic characteristics of DEAP actuator subjects to sinusoidal excitations ( $U (t) = 1 + \sin (2 π ft - 0.5 π)$ ) with frequencies of 0.1, 0.3, and 0.5 Hz, respectively.

It is clearly shown that the voltage-displacement curve forms an asymmetrical banana-shaped hysteresis loop, and the shapes of hysteresis loops are different. The area of the hysteresis loop becomes larger and the turning point at minimum value rises with the increase in frequency of control signal. The hysteresis forms the main impediment to attaining the high-precision performance of the DEAP actuation system, the maximum error caused by the hysteresis can be as much as 15% (even 35% with the increase of frequencies) of the travel range of the DEAP actuator in general, which leads, in the best case, to reduce the motion accuracy and, in the worst case, to destabilize the control system.

3. Methodology

In this section, related theories of the proposed EMD-LSTM model are introduced firstly, including the principle of the EMD algorithm, the introduction of LSTM, and the one-sided dead-zone operator. Then the proposed EMD-LSTM model is given.

3.1. Empirical mode decomposition

In this paper, EMD is introduced to extract the time-frequency component from the control signal of DEAP actuator to represent the rate-dependent characteristic. Empirical Mode Decomposition is proposed by Huang et al. (1998) to decompose a signal into specific modes. Its particularity is that it does not use any prescribed function basis but is self-adapting accordingly to the analyzed signal. The principle of EMD is simply introduced here and more details can be found in Daubechies et al. (2011) and Huang et al. (1998).

The EMD aims to decompose a signal $u (t)$ as a sum of N + 1 Intrinsic Mode Functions (IMF) $u_{k} (t)$

u (t) = \sum_{k = 0}^{N} u_{k} (t)

(1)

An IMF is an amplitude modulated-frequency modulated function which can be written in the form

u_{k} (t) = U_{k} (t) \cos (φ_{k} (t))

(2)

where, $U_{k} (t)$ and $φ_{k} (t)$ is positive. Generally, the $U_{k}$ and $φ_{k}$ vary much slower than $u_{k}$ . The IMF $u_{k} (t)$ behaves a harmonic component.

The EMD method is a pure algorithmic method to extract IMFs. The procedures of EMD is as follows:

(1) Compute the upper envelops $\bar{u} (t)$ and lower envelops $\underline{u} (t)$ via a cubic spline interpolation from the maximum and minimum of the signal $u (t)$ ;

(2) Calculate the mean envelops $m (t)$ :

m (t) = \frac{1}{2} (\bar{u} (t) + \underline{u} (t))

(3)

(3) Compute the first candidate $r_{1} (t)$ is defined by

r_{1} (t) = u (t) - m (t)

(4)

(4) The final IMF is obtained by iterating the same process to $r_{1}$ and the subsequent $r_{k}$ :

u_{1} (t) = r_{n} (t)

(5)

(5) The next IMF is obtained by the same algorithm applied on $u (t) - u_{1} (t)$ , and the remaining IMFs is computed by repeating the above steps on the successive residues.

3.2. One-sided dead-zone model

The one-sided dead-zone operator is widely used in revised PI models to add an asymmetrical characteristic into the traditional PI model. In this paper, it is used to reshape the original control signal into corresponding asymmetrical signal. The definition of one-sided dead-zone operator is as follows:

S (x (t), r_{s}) = {\begin{matrix} max {x (t) - r_{s}, 0}; r_{s} > 0 \\ x (t); r_{s} > 0 \\ min {x (t) - r_{s}, 0}; r_{s} > 0 \end{matrix},

(6)

and the one-sided dead-zone model is obtained by a weighted superposition of one-sided dead-zone operators connected in parallel:

u_{s} (t) = w_{s} \cdot S (x (t), r_{s}),

(7)

where $S (x (t), r_{s})$ denotes one-sided dead-zone operator, the parameters $w_{s} = {[w_{s}^{1}, w_{s}^{2}, \dots, w_{s}^{n_{s}}]}^{T}$ and $r_{s} = {[r_{s}^{1}, r_{s}^{2}, \dots, r_{s}^{n_{s}}]}^{T}$ are the weights and thresholds of dead-zone operators, respectively. The parameter $n_{s}$ is the number of one-sided dead-zone operators. When $n_{s}$ is larger, the one-sided dead-zone model will be more accurate but with more computations. On the contrary, when $n_{s}$ is smaller, the model will be inaccurate but with less computations. Therefore, the number of dead-zone operators is a compromise between model accuracy and computation. It is usually determined by a lot of experiments given different $n_{s}$ . Then the parameters $r_{s}$ can be defined and $w_{s}$ can be identified by least square method.

3.3. Long-short term memory model

Long-Short Term Memory model is developed from recurrent neural network (RNN). The LSTM model is a widely accepted solution to resolve the exploding and vanishing gradients problem of RNN by introducing memory cell and cell state to control the transmission of information between the hidden layers. The structure of LSTM unit is shown in Figure 5. In a LSTM unit, there are three gates, including the input gate, forget gate, and output gate. The input gate determines how much the new information can be added into the current cell state. The forget gate determines how much the historical information should be forgotten. The output gate determines the information transforms between the cell state and the memory cell. Matrix multiplication and nonlinear summation are performed in memory units in order that the historical information can be kept remember in continuous iteration.

In time $t$ , the information firstly goes through the input gate. The value of input gate $i_{t}$ and candidate of hidden state ${\tilde{c}}_{t}$ are as follows:

i_{t} = σ (W_{i} x_{t} + U_{i} h_{t - 1} + b_{i}),

(8)

{\tilde{c}}_{t} = \tanh (W_{c} x_{t} + U_{c} h_{t - 1} + b_{c}) .

(9)

Then, the information goes through forget gate. The value of forget gate $f_{t}$ is:

f_{t} = σ (W_{f} x_{t} + U_{f} h_{t - 1} + b_{f}) .

(10)

The cell state of hidden layer can be updated by the following equation:

C_{t} = i_{t} ° {\tilde{c}}_{t} + f_{t} ° C_{t - 1},

(11)

where, “°” denotes the elementwise product. Finally, the information goes through the output gate to gain the final output of memory cell.

o_{t} = σ (W_{o} x_{t} + U_{o} h_{t - 1} + b_{o}),

(12)

h_{t} = o_{t} ° \tanh (C_{t}) .

(13)

In equations (8)–(13), the $W_{i}$ , $W_{c}$ , $W_{f}$ , and $W_{o}$ are weight matrices of hidden layers; the $U_{i}$ , $U_{c}$ , $U_{f}$ , and $U_{o}$ are weight matrices of input layers. $b_{i}$ , $b_{c}$ , $b_{f}$ , and $b_{o}$ are the corresponding bias vector. $σ (x)$ and $\tanh (x)$ are the activation function of sigmoid and hyperbolic tangent, respectively. The definition of σ(x) and tanh(x) are given as follows:

σ (x) = \frac{1}{1 + e^{- x}}

(14)

\tanh (x) = \frac{e^{x} - e^{- x}}{e^{x} + e^{- x}}

(15)

3.4. The framework of EMD-LSTM model

According to the above descriptions, the EMD method is used to decomposed the control signal into several sub-sequences, the one-sided dead-zone operator is used to reshape the control signal to asymmetrical signal. Then, the input space of LSTM for prediction of displacement of DEAP actuator, represented by $X_{t}$ , is conducted by three part: the first part is the original control signal $u (t)$ , the second part is the sub-sequence $u_{k} (t)$ , and the third part is the asymmetrical signal $u_{s} (t)$ . The training of EMD-LSTM is through back propagation through time (BPTT) mechanism. Then, the EMD-LSTM model is finally obtained. The whole framework of this model is shown in Figure 6. The EMD-LSTM network is composed of an input layer, one hidden layer, and an output layer.

4. Experimental result and discussion

In this section, the experimental data which includes the training data and the test data are firstly shown. Then, the decomposition results of EMD and identification of one-sided dead-zone model are given. After that, we introduce the input spaces of LSTM and their normalization. Finally, in order to verify the prediction performance of the EMD-LSTM model, two different input spaces are chosen and three different type of models, including BPNN, RPI (Al Janaideh and Aljanaideh, 2018), and NEM (Sarban et al., 2011) model are selected for comparison.

4.1. The experimental data

In order to increase the model’s generalization ability, the experimental data should contain enough input-output information of the DEAP actuator which covers a wide range of input rates. Therefore, four random control signals are designed manually: firstly, sinusoidal signal with full amplitudes of a period is given; secondly, a series of numerical value covers a range of [0, 2] are generated randomly and connected smoothly by interpolation. They aren’t periodic signals apparently. Since the DEAP material is sensitive to environmental disturbances, the measured displacement data would be doped with noise, resulting in a reduction of modeling accuracy. Moving average filter is processed on the displacement data, which makes the output waveform smoother, effectively reducing the effects of noise and disturbances. The experimental data used in this paper are shown in Figure 7. Figure 7(a) is the training data and Figure 7(b)–(d) are the test data, respectively.

4.2. The decomposition results

In order to represent the rate-dependent characteristic of hysteresis, the control signal of training and test data are decomposed by EMD to obtain the frequency sub-sequence $u_{k} (t)$ . The decomposition results of training and test data are shown in Figure 8.

4.3. The identification of one-sided dead-zone model

To represent the asymmetrical characteristic of hysteresis, the one-sided dead-zone model is used to reshape the control signal to obtain the reshaped signal $u_{r} (t)$ . The concrete procedure is that:

(1) Sinusoidal signal ( $U (t) = 1 + \sin (2 π f - 0.5 π)$ ) is applied to drive the DEAP actuator, and the displacement signal is obtained;

(2) Corresponding to the range of input data, the displacement data is normalized into range [0, 2];

(3) Through a lot of experiments with different $n_{s}$ , it turns out that when $n_{s} = 10$ , the one-sided dead-zone model has better accuracy and less computation. Thus, the thresholds of operator can be defined as

r_{s} = \frac{i}{10 + 1} \cdot max (u), i = 1, 2, \dots, 10

(4) Least square method is applied to identify the weights $w_{s}$ of one-sided dead-zone model.

The results of $r_{s}$ and $w_{s}$ are as follows:

r_{s} = {[0.1818, 0.3636, 0.5454, 0.7272, 0.909, 1.0908, 1.2726, 1.4544, 1.6363, 1.8181]}^{T}

w_{s} = [0.6309, - 0.4387, 0.4752, 0.2457, 0.0834, 0.3031, 0.1233, 0.2489, 0.0763, - 0.4456]^{T}

4.4. The construction of input space

Due to the one-to-many map of hysteresis nonlinearity, one-input one-output neural network model cannot describe the hysteresis. Therefore, the input space of neural network should be expanded (Zhao and Tan, 2006). In this paper, in order to verify the performance of EMD method, two different input spaces are chosen.

The first input space I is denoted $X_{I} (t)$ , and the second input space II is denoted $X_{II} (t)$ . Their constructions are:

X_{I} (t) = {u (t), u_{s} (t)},

X_{II} (t) = {u (t), u_{k} (t), u_{s} (t)} .

The first input space only includes the original control signal and reshaped signal, which means the EMD results are not used to train the neural network. The second input space includes the original control signal, the sub-sequence of EMD, and reshaped signal.

Before input into the neural network, each variable of input space should be normalized into [0, 1] to reduce the influence of variable quantity scale on network training. The formula of data normalization is:

x_{ij} = \frac{x_{ij} - min_{1 \leq i \leq N} x_{ij}}{max_{1 \leq i \leq N} x_{ij} - min_{1 \leq i \leq N} x_{ij}}

(16)

where, $i = 1, 2, \dots, M$ , $j = 1, 2, \dots, N$ . $M$ is the dimension of input space and $N$ is the number of samples.

4.5. The experimental results

In this section, the proposed EMD-LSTM model is validated by comparing with different type models, including neural network model, phenomenological model, and physics-based model. Moreover, concerning the different characteristics, model training modes of BPNN, PRI, and NEM model, the contrast experiments are divided here into two groups: (1) comparison between EMD-LSTM, LSTM, and BPNN model; and (2) comparison between EMD-LSTM, RPI, and NEM model. The models of first group are all neural network which have the same training and test mode. In the training process, the input space and displacement signal are used to train the neural network models. In the test process, the predicted value is obtained by inputting the input space of test data into the trained neural network models. On the contrary, the models of the second group are different from each other. The input of EMD-LSTM model is the input space II, while the RPI and NEM is only the control signal. Comparison results of two groups are shown in Figures 9 –20.

(1) Comparison between EMD-LSTM, LSTM and BPNN model

Figure 9 are training results of BPNN and LSTM model using the first input space $X_{I} (t)$ . In Figure 9(a), the black solid line is the measured hysteresis, the red dashed line is the prediction result of the LSTM model, and the blue dot-and-dash line is the prediction result of BPNN model. Figure 9(b) is the absolute training error of two models. It is apparently that the LSTM model have ability to predict the training displacement signal within an error between [−0.05, 0.05], while the BPNN model predicting the displacement within an error between [−0.15, 0.15]. The LSTM model has a higher prediction accuracy.

Figures 10 –12 are three test results of BPNN and LSTM model using the first input space $X_{I} (t)$ . In Figures 10(a), 11(a), and 12(a), the black solid line is the measured hysteresis, the red dashed line is the prediction result of LSTM model, and the blue dot-and-dash line is the prediction result of BPNN model. Figures 10(b), 11(b), and 12(b) are the absolute test error of two models. It can be seen from these figures that the LSTM model have generalization ability to predict the test displacement signal with smaller errors comparing with BPNN model. However, the model prediction error will increase when the frequency of the control signal increases.

Then the second input space $X_{II} (t)$ is used to train BPNN and EMD-LSTM model and the results are shown in Figure 13. In Figure 13(a), the black solid line is the measured hysteresis, the red dashed line is the prediction result of the EMD-LSTM model, and the blue dot-and-dash line is the prediction result of BPNN model. Figure 13(b) is the absolute training error of two models. It can be seen from these figures that the prediction errors of EMD-LSTM model reduce to a range of [−0.02, 0.02], while the prediction errors of BPNN model reduce to a range of [−0.1, 0.12]. The EMD method effectively improves the prediction effect of the models.

Figures 14 –16 are test results of BPNN and EMD-LSTM using the second input space $X_{II} (t)$ . In Figures 14(a), 15(a), and 16(a), the black solid line is the measured hysteresis, the red dashed line is the prediction result of the EMD-LSTM model, and the blue dot-and-dash line is the prediction result of BPNN model. Figures 14(b), 15(b), and 16(b) are the absolute test error of two models. From these figures, we can also see that the prediction accuracy of EMD-LSTM and BPNN model have been significantly improved after introduction of the EMD method.

(2) Comparison between EMD-LSTM, RPI and NEM model

In order to demonstrate the performance of the proposed EMD-LSTM model, we also compared it with existed RPI and NEM model. The input of these three models are respectively input space II, control signal and control signal, while the output of these three models are all displacement. Therefore, the training process of models are different. For EMD-LSTM model, the training process has been described. For RPI model, only control signal of training data was need to identify the parameters of model. For NEM model, a set of experimental creep data was measured to estimate the parameters of Voigt-Kelvin model, while the equipment information and the control signal of training data were used to determine the other parameters in NEM model. In test process, the trained models were validated by three sets of test data.

Figure 17 are training results of EMD-LSTM, RPI, and NEM model. In Figure 17(a), the black solid line is the measured hysteresis, the red dashed line is the prediction result of the EMD-LSTM model, the blue dot-and-dash line is the prediction result of NEM model, and the purple dot line is the prediction result of RPI model. Figure 17(b) is the absolute training error of three models. It can be seen from the figures that the RPI model and NEM model have ability to predict the test displacement signal within an error between [−0.15, 0.15], which are larger than corresponding error of EMD-LSTM model.

Figures 18 –20 are test results of EMD-LSTM, RPI, and NEM model. In Figures 18(a), 19(a), and 20(a), the black solid line is the measured hysteresis, the red dashed line is the prediction result of the EMD-LSTM model, the blue dot-and-dash line is the prediction result of NEM model, and the purple dot line is the prediction result of RPI model. Figures 18(b), 19(b), and 20(b) are the absolute test error of three models. From these figures, we can also see that the EMD-LSTM model has smaller prediction errors than RPI and NEM model.

Moreover, to evaluate the prediction performance of the proposed model, three indicators, including MAE (Mean Absolute Error), MRE (Mean Relative Error), and RMSE (Root Mean Square Error), are adopted as evaluation metrics. The definitions of these three indicators are as follows.

MAE = \frac{1}{N} \sum_{k = 1}^{N} | \hat{y} (k) - y (k) |

MRE = \frac{1}{N} \sum_{k = 1}^{N} \frac{| \hat{y} (k) - y (k) |}{y (k)}

RMSE = \frac{1}{N} \sqrt{\sum_{k = 1}^{N} {(\hat{y} (k) - y (k))}^{2}}

The results of these three indicators are shown in Table 1. The smaller values mean higher performance. Therefore, we can conclude from figures 9 to 16 and tables 1 that: (1) the prediction accuracy of the model can be significantly improved by EMD method. For example, comparing Figures 9 and 13, the MAE, MRE, and RMSE of BPNN model has been reduced from 0.0245, 0.0192 and 0.0342 to 0.022, 0.0157 and 0.0294, the MAE, MRE, and RMSE of EMD-LSTM model has been reduced from 0.022, 0.0182 and 0.0311 to 0.0186, 0.0137 and 0.0256. (2) The EMD-LSTM model has higher prediction accuracy than the traditional BPNN model. For example, from Figure 14, the MAE, MRE, and RMSE of BPNN model are 0.0299, 0.02, and 0.0441, while the corresponding indexes of EMD-LSTM model are 0.0248, 0.0178, and 0.0346.

Table 1.

Performance indicators of BPNN, LSTM, and EMD-LSTM.

		BPNN			LSTM			EMD-LSTM
		MAE	MRE	RMSE	MAE	MRE	RMSE	MAE	MRE	RMSE
Input space1	Training data	0.0245	0.0192	0.0342	0.022	0.0182	0.0311	—	—	—
	Test data 1	0.045	0.0357	0.0625	0.0282	0.022	0.0393	—	—	—
	Test data 2	0.0401	0.0387	0.0518	0.0176	0.0149	0.0238	—	—	—
	Test data 3	0.0548	0.0514	0.0709	0.0336	0.0314	0.0446	—	—	—
Input space2	Training data	0.022	0.0157	0.0294	—	—	—	0.0186	0.0137	0.0256
	Test data 1	0.0299	0.02	0.0441	—	—	—	0.0248	0.0178	0.0346
	Test data 2	0.0191	0.0182	0.0238	—	—	—	0.0128	0.0115	0.0167
	Test data 3	0.0378	0.0363	0.0492	—	—	—	0.0274	0.0244	0.0359

In the same way, the results of performance indicators of RPI, NEM, and EMD-LSTM are shown in Table 2. From Figures 17 to 20 and Table 2, we can conclude that the EMD-LSTM model has a higher prediction performance than RPI and NEM model.

Table 2.

Performance indicators of RPI, NEM, and EMD-LSTM.

	RPI			NEM			EMD-LSTM
	MAE	MRE	RMSE	MAE	MRE	RMSE	MAE	MRE	RMSE
Training data	0.034	0.0479	0.0447	0.03	0.0259	0.0405	0.0186	0.0137	0.0256
Test data 1	0.0377	0.0491	0.0492	0.025	0.0165	0.0344	0.0248	0.0178	0.0346
Test data 2	0.0469	0.0544	0.0531	0.0371	0.0349	0.0479	0.0128	0.0115	0.0167
Test data 3	0.0643	0.0476	0.0852	0.0416	0.0384	0.0548	0.0274	0.0244	0.0359

Besides the performance indicators, the training and test time of models in the paper have been recorded and shown in Table 3. It is obviously that the EMD-LSTM model has longer training time than RPI, NEM, and BPNN model. It takes 15.3854 s to train the parameters. However, the test time of EMD-LSTM model is 0.0187 s which is close to the other models. Therefore, although the training time of EMD-LSTM model is relatively long, the training process is completed offline, which will not affect the online prediction of the trained model. In a word, the EMD-LSTM model can be applied in practice.

Table 3.

Computational speed of models.

	BPNN	RPI	NEM	EMD-LSTM
Training time(s)	9.8593	1.6701	2.6581	15.3854
Test time(s)	0.0105	0.0022	0.0069	0.0187

5. Conclusion

In this paper, a hybrid EMD-LSTM model for rate-dependent and asymmetrical hysteresis of DEAP actuator has been proposed. In this model, LSTM model is a better model to deal with the time series prediction compared to the feedforward neural network, and EMD method can extract the characteristic of hysteresis. In order to test the predictive performance of the proposed model, it is compared with the traditional BPNN, RPI, and NEM model. The experimental results demonstrate that the proposed EMD-LSTM model can describe the dynamic hysteresis characteristics of DEAP actuator with higher accuracy. However, although the EMD-LSTM works well for DEAP actuators with rate-dependent hysteresis, it always exists model error due to the existence of creep effects in DEAP materials that should not be ignored in some specific situations. The compensation and control of DEAP actuator with more complicate non-linearity will be considered in the next work.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by National Natural Science Foundation of China (Grant no. 61375100 and no. 61472037).

ORCID iD

Zhaoguo Jiang

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