Modeling of pneumatic artificial muscle using a hybrid artificial neural network approach

Mechatronics xxx (2015) xxx–xxx

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Mechatronics journal homepage: www.elsevier.com/locate/mechatronics

Modeling of pneumatic artificial muscle using a hybrid artificial neural network approach Chunsheng Song a,⇑, Shengquan Xie b, Zude Zhou a, Yefa Hu a a b

School of Mechanical and Electronic Engineering, Wuhan University of Technology, Wuhan 430070, China Department of Mechanical Engineering, The University of Auckland, Auckland 1142, New Zealand

a r t i c l e

i n f o

Article history: Received 15 September 2014 Revised 13 March 2015 Accepted 27 April 2015 Available online xxxx Keywords: Pneumatic artificial muscle Dynamic modeling Hybrid approach Artificial neural network Genetic algorithm

a b s t r a c t Pneumatic Artificial Muscle (PAM) actuator has been widely used in medical and rehabilitation robots, owing to its high power-to-weight ratio and inherent safety characteristics. However, the PAM exhibits highly non-linear and time variant behavior, due to compressibility of air, use of elastic-viscous material as core tube and pantographic motion of the PAM outer sheath. It is difficult to obtain a precise model using analytical modeling methods. This paper proposes a new Artificial Neural Network (ANN) based modeling approach for modeling PAM actuator. To obtain higher precision ANN model, three different approaches, namely, Back Propagation (BP) algorithm, Genetic Algorithm (GA) approach and hybrid approach combing BP algorithm with Modified Genetic Algorithm (MGA) are developed to optimize ANN parameters. Results show that the ANN model using the GA approach outperforms the BP algorithm, and the hybrid approach shows the best performance among the three approaches. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Pneumatic Artificial Muscle (PAM) is a biomimetic device that mimics the behavior of skeletal muscles. It exhibits force–length characteristics similar to that of a human muscle. PAM has simple construction and consists of a rubber tube connected to pneumatic valves at one end. The rubber tube is housed in a sheath made-up of non-elastic and high-strength fibers. The fibers are arranged in a rhomboidal fashion, which allows a defined contraction motion in a longitudinal direction when the inner tube is inflated which results in shortening of the PAM. Consequently, force is exerted by the PAM on the environment, attached at the other end, in the axial direction. Compared to conventional actuators such as electric and hydraulic actuators, PAM draws certain advantages such as high power-to-weight and high power-to-volume ratios, low maintenance, low price, cleanliness, compliance, pliability, inherent safety, and applicability in rough environments. Air compressibility and elasticity of inner tube also plays cushioning role against unpredictable impacts. Owing to these advantages, PAM is considered an attractive and safe actuator to use in devices operating in human proximity compared to electric or hydraulic actuators. ⇑ Corresponding author. Tel.: +86 13437161368.

Recently, PAM has been regarded as a suitable alternative to hydraulic and electric actuators in medical and rehabilitation robot applications. A few examples of the successful use of PAMs in mechatronic devices for rehabilitation purposes can be found in the literature. Applications in the form of an exoskeleton exist for upper limbs [1,2] lower limbs [3], hand [4], elbow [5], and the ankle joint [6]. Unfortunately, PAM exhibits highly nonlinear pressure–length characteristics and time-variant properties due to compressibility of air, elastic-viscous properties of the inner tube and geometrically complex behaviors of the PAM shell. Rubber like behavior of the inner tube also lead to hysteresis and hence the PAM shows different characteristics during inflating and deflating. Thus, it is not easy to control them and obtain the required performance features. In view of this, previous studies have focused on methods for modeling of pneumatic muscles and controller design to improve control performance in recent years, including [7,8]. Considering that the precise modeling of PAMs can be the first step in improving the control performance of system, this paper presents the dynamic modeling of PAM. In order to identify behavior of a PAM, many models to estimate behavior of PAMs have been proposed in the past. The pioneering work in the field of PAM modeling can be classified into two aspects, analytical modeling and artificial intelligence-based modeling identification.

E-mail address: [email protected] (C. Song). http://dx.doi.org/10.1016/j.mechatronics.2015.04.021 0957-4158/Ó 2015 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Song C et al. Modeling of pneumatic artificial muscle using a hybrid artificial neural network approach. Mechatronics (2015), http://dx.doi.org/10.1016/j.mechatronics.2015.04.021

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Analytical modeling method is a common method and an early adopter. Initially, to reduce the model complexity, elastic energy contained in the inner tube was ignored and the relations between axial force, length and pressure were formulated based on the principle of virtual work [9]. Later, nonlinear characteristics of PAM were addressed which included, irregular geometric shape of the rubber tube [10,11], elastic energy of the inner tube [12], and hysteresis behaviors of PAMs. A friction model was also developed for the thread-on-thread friction in the braided shell [9,10,13]. Until recently, most researches have been focused on the static characteristics of PAMs assuming no pressure variance inside the tube during very slow motions. Chou and Hannafordpresented a simple lumped-parameter model of pneumatic circuits to estimate dynamic response of pneumatic circuits [14]. Kang and Kothera proposed a dynamic modeling of PAM [15]. The quasi-static characteristics of the PAM are modeled followed by the dynamic characteristics through spectral analysis [16]. A new approach to model the hysteresis of a basic antagonistic manipulator joint constructed by a pair of Festo fluidic muscles is present [17]. While lot of work has been done to analytically model the pneumatic muscles, the accurate prediction of their dynamic behavior could not be achieved. These analytical models still have limitations in predicting on behavior of the PAM [18]. This is due to lake of knowledge of PAM behavior in the light of its conical ends, friction between the inner tube and outer sheath, valves and fluid flow characteristics and large hysteresis. It is evident from the discussion in the preceding section that the conventional tools cannot fully comprehend the non-linear and time dependent muscle characteristics. Therefore, the artificial intelligence-based modeling identification methods are introduced and quantifiable work has been done in this direction. Ahn and Anh applied a Modified Genetic Algorithm (MGA) for optimizing parameters of a linear auto-regressive with exogenous (ARX) model of the PAM manipulator, which can be modified online with an adaptive self-tuning control algorithm. Through experimental investigation, the proposed MGA-based identification algorithm achieves excellent performance in comparison with conventional SGA and LMS methods [19]. However, the work has been done for constant loading it cannot be used for force control applications of PMA. A neural network ARX (NNARX) model has been applied to non-linear modeling and identification of the PAM manipulator using a new INCBP algorithm [19–21]. The parametric values of the ARX model have been optimized using modified GA (MGA). The MSE of this model has been reported as 0.02616 rad. Incremental back propagation algorithm used to train the NN to further reduce MSE of the model as 0.0035 radians. Prashant proposed a PAM modeling method using modified fuzzy inference mechanism. To tune the parameters of fuzzy model three approaches namely, Gradient Descent (GD) method, Genetic Algorithm (GA) and Modified Genetic Algorithm (MGA) are used. MGA based fuzzy model was found to be more accurate [22]. A novel implementation of a SOFC is proposed for the control of a single PAM. In order to assess the advantage of the intelligent adaptive control system, a comparison of the performance of three types of nonparametric control algorithms (PD, FFC and SOFC) is also presented [23]. Summarizing the above discussion, most of the research on PAM modeling has been done in no-load load or constant conditions, neglecting loads, especially the change in loads. However, in this paper, the PAM will be used for ankle rehabilitation robot application, the loads cannot be taken as constant, moreover the load may change rapidly sometimes. And also, to obtain more accuracy control performance in rehabilitation robot field, the prediction accuracy from the previous models also needs to be improved. Artificial neural networks can effectively model systems, which possess non-linearity and uncertainties [24]. In order to address

above problems of PAM modeling, this paper proposes a multilayer artificial neural network to solve the PAMs’ dynamic modeling problems. To get greater modeling accuracy, the parameters of the ANN model are optimized by three different approaches, namely, Back Propagation (BP) algorithm, Genetic Algorithm (GA) approach and hybrid approach combing with BP algorithm and Modified Genetic Algorithm (MGA). The results obtained from the three approaches are analyzed and compared in terms of mean square error (MSE) and maximum deviation of prediction pressure errors.

2. The basic characteristics of pneumatic artificial muscle PAMs converts pneumatic energy into mechanical form by transferring the pressure applied on the inner surface of its tube into the shortening action. The relationship between pressure (P), length (L) and force (F) can be written as shown below based on the principle of virtual work [14]: 2

F ¼ P  dV=dL ¼ P  D2o p=4 sin h  ½3 cos2 hk  1

ð1Þ

where k ¼ L=L0 , and L0, D0 are the initial length and the diameter of the tube respectively, h is the initial pitch angle of the braid. However, since the tube shape is not perfectly cylindrical when pressurized and large hysteresis is present in PAM, above models cannot be used in their present form, instead improved model has been built to compensate these variations. However, the model parameters are difficult to obtain because of the influence of uncertain factors, such as time-variety, nonlinearity and environment. In order to construct a neural network based model of PAM, training data is required to be obtained. The experimental set up used for this purpose shown in Fig. 1. Tests are conducted on a single PAM which is placed in a rigid hanger as shown in Fig. 1(a). Linear position sensor is positioned parallel to the PAM to record instantaneous length of PAM. A FUTEK load cell is connected to the PAM and used to measure the force dynamically. Pneumatic muscle is inflated by connecting it to the pressure supply from a compressor. The supply pressure was fixed at 2 bar and two Isonic pressure regulating valves are used to control pressure inside the PAM. These valves are capable to provide a switching frequency of 10 ms and are used to fill, leave inflated and empty the PAM actuator. As shown in Fig. 1(b), a dSPACE (DS1104) data processing system is used to provide interface to a PC allowing MATLAB and Simulink programs to be used. The dSPACE has a number of Input/Output (I/O) capabilities including serial, analogue, and digital, which are used to read data from various sensors and generator control signals. The PAM is controlled by compiling a Simulink model and downloading it to the DS1104 through the I/O interface. The dSPACE is connected to PC through RS-232 serial port. Under different loads and different pressures, the experimental data are received through serial port from various sensors. The actual behaviors of the PAM obtained from the experiment setup is shown in Figs. 2 and 3. Results from the experiments (Fig. 2) show that the characteristic between length and pressure of PMA is non-linear. The variable external loading on the PAM also affects the characteristic considerably. Moreover, from Fig. 3, the plot is some of different while inflating or deflating the PAM and a lager hysteresis exists. ANN is usually used to model complex relationships between inputs and outputs or to find patterns in data. Therefore, in this paper, a multilayered feed forward ANN is being proposed to model PAM.

Please cite this article in press as: Song C et al. Modeling of pneumatic artificial muscle using a hybrid artificial neural network approach. Mechatronics (2015), http://dx.doi.org/10.1016/j.mechatronics.2015.04.021

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Fig. 1. Experiment setup for a PAM using in an ankle rehabilitation robot. (a) Experiment setup for single PAM. (b) Overall structure of the system.

3. ANN architecture of PAM model identification Through the analysis of Section 2, the purpose of modeling is to find out a precise relationship among the parameters of length, force and pressure. However, from Figs. 2 and 3, we find that the change in load and the change in length may well influence the relationship. The ANN model developed for PAM modeling in the present work has four input variables, which are instantaneous values of length, change in length, load and change in load. The model has a single hidden layer containing five neurons; the number of neurons in the layer is roughly determined by experience formula and then it is determined to five by the method of trial and error. A hyperbolic tangent sigmoid transfer function is selected to best emulate the non-linear behavior of the PAM in hidden layer. Linear transfer function is used for the output layer as a usual practice. The detailed architecture of the ANN is shown in Fig. 4 wherein x1, x2, x3, x4 are inputs, namely, length, change in length, load and Fig. 2. Extension length of PAM when inflating.

ð1Þ

change in load. Further, wij is the input-hidden layer weight from ð2Þ

input neuron j to hidden neuron I and wki is the hidden-output weight from hidden neuron i to output neuron k. Bias weights ð1Þ

are shown by bi

Fig. 3. Extension length characteristics of PAM when inflation and deflation (100 N).

ð2Þ

and bk for hidden neuron i and output neuron

Fig. 4. Architecture of ANN model for PAM dynamic modeling.

Please cite this article in press as: Song C et al. Modeling of pneumatic artificial muscle using a hybrid artificial neural network approach. Mechatronics (2015), http://dx.doi.org/10.1016/j.mechatronics.2015.04.021

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Table 1 Whole structured parameter variables of the ANN model. w11

ð1Þ

w12

ð1Þ

w13

ð1Þ

w14

ð1Þ

w21

ð1Þ

w22

w23

w24

w31

w32

w33

ð1Þ

w34

ð1Þ

w41

ð1Þ

w42

ð1Þ

w43

ð1Þ w51

ð1Þ w52

ð1Þ w53

ð1Þ w53

ð1Þ w54

ð1Þ b1

ð1Þ

ð1Þ b2

ð1Þ

ð1Þ b3

ð1Þ

ð1Þ b4

ð1Þ b5

ð2Þ w1

ð2Þ w2

ð2Þ w3

ð2Þ w4

ð2Þ b1

k respectively. Mathematical relationship between the layers is expressed in term of these weights. According to the ANN model shown in Fig. 4, there are 31 parameter variables required to be optimized. All the variables are structured and listed in Table 1. For comparison, in the research, three training approaches of the ANN model, namely, BP, GA and hybrid approach will be developed as discussed in the following section. 4. Hybrid approach Optimum parameters of ANN using efficient optimization algorithms, is the key to achieve higher accurate modeling of PAM. As mentioned in the previous section, 31 variables of ANN are required to be optimized. There are numerous algorithms available for training ANN models such as Reinforced, Hebbian, Gradient Descent (GD) and Evolutionary Algorithm (EA). GD method is a widely used method for training ANN model [25]. Back Propagation (BP) algorithm, a type of GD method, is the most widely used approach for training ANN model. Due to simplicity, BP is a common method of training artificial neural network, however, it has some limitations, such as slow and local minimum. Because of inherent limitations of BP algorithm, it and its improved BP algorithms, such as additional momentum method, adaptive learning rate [26] are hard to resolve the problems of local optimal and sensitivity to initial values completely. Genetic Algorithm (GA) belongs to the larger class of evolutionary algorithm, which generates solutions to optimization problems using techniques inspired by natural evolution, such as mutation, selection and crossover. In theory, GA can gain global optimum solution in a certain condition, which is widely used in many areas for its favorable global searching. However, it also has some shortcomings such as premature and slow convergence. Therefore in this paper, the modified genetic algorithm are proposed to resolve the problems. We have proposed two modifications in conventional GA, first, an elitism and worst eliminated selection method has been used to ensure that the best solution does not become extinct in the process of evolution. Second, self-adjusting mutation and crossover rate method are used. However, the weak local search capabilities cannot also be solved. Therefore a hybrid approach is proposed. First, a Modified Genetic Algorithm (MGA) is proposed to improve the convergence speed and avoid premature solution. Second, in view of the high capability of BP algorithm in local search and GA in global search, a hybrid approach combing with BP algorithm and MGA is developed and proposed. MGA is used to search a global solution initially and BP algorithm is used to fine tune the solution afterwards. The fitness function (F) of MGA is defined as:

F ¼ 1=1 þ T

ð2Þ

where T is the quadratic sum of the difference (T) between reference output and actual output in Eq. (3). The quadratic sum of the difference (T) between reference output and actual output is described firstly, and it is written as:

T¼

T T X X 2 2 eðkÞ ¼ ðyd ðkÞ  ya ðkÞÞ k¼1

ð3Þ

k¼1

where k (k = 1, 2, . . . , T) is the number of output values; yd ðkÞ is the reference output values, obtained from the experiment; ya(k) is the

ð1Þ

ð1Þ

ð1Þ

ð1Þ

w44

ANN model actual output values as shown by Eq. (4), it can be written as:

ya ¼ f 2

5 X i¼1

ð2Þ wi

 f1

4  X

ð1Þ wij

 xj þ

ð1Þ bi

!! 

! þb

ð2Þ

ð4Þ

j¼1

Here, i = 1, 2, . . . , 5 is the number of the hidden neurons; j = 1, 2, 3, 4 is the number of the input neurons; f 1 is the activation function of hidden neuron; f 2 is the activation function of output neuron. The objective of training the ANN model is to minimize the difference between reference output and actual output. Thirty-one variables listed in Table 1 are grouped in a chromosome-like structure, which in turn is interpreted as the ANN model. The chromosome-like individual is coded using real number string to describe 31 variables of the ANN model. 100 individual solutions are randomly generated to form an initial population. Each variable in the individual solution is assigned for short type and the fitness accuracy of the order is 10e4, and the maximum fitness value is 0.9999, which is the fittest solution. There are two main steps of the hybrid approach. First, search the optimal weights and bias weights values using the MGA. Then, switch to the BP algorithm to fine tune the weights, when some transition conditions are satisfied. The essence of the hybrid approach is that the initial weights and bias weights values of BP algorithm are provided by MGA’s solution. The BP algorithm can find solution in global optimal path. The approach can make good use of both merits of the MGA, namely, global searching capability and BP algorithm namely, local searching capability. The rough solution is searched initially using MGA and fine tuned using the BP algorithm next. This process is repeated until a termination condition has reached. The termination conditions are, (1) Fixed number (NMax) of iterations has reached; (2) The fine tuned solution has reached the optimum solution (MSEmin). Various steps used in the whole hybrid approach are explained as below in detail. Step 1. Select the termination conditions. In the present, the approach will switch to BP algorithm if the fitness value reaches 0.9999 or the number of epochs is GMax. Step 2. Initialize a population of 100 individual solutions randomly. Step 3. Append the maximum fitness (Max (gen  1)) solution obtained from the preceding iteration into the present generation. Calculate the fitness value of each solution and sort the solutions ascending by fitness value. Then find the maximum fitness (Max (gen)) from present generation. Compare it with the termination conditions. Continue if the fitness is less than 0.9999 and the number of epochs is less than GMax, then gen = gen + 1; go to Step7 otherwise. Step 4. Save a copy of the fittest solution (Max (gen)) separately. Eliminate the worst 20 percent individual solutions and fill in by the equal numbers of random data. Step 5. Apply crossover and mutation operations to form a new generation. The crossover rate and mutation rate are changed dynamically. The top 20% epochs, the crossover rate is 0.9 and mutation rate is 0.04. The middle 40 percent epochs, the crossover rate is 0.85 and mutation rate is 0.08. The final 40% epochs, the crossover rate is 0.8 and mutation rate is 0.12. Step 6. Return to Step 3. Step 7. Switch into the BP algorithm with adaptive learning rate.

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5

Fig. 5. The process of hybrid approach.

Step 8. Calculate the MSE value of each iteration (MSE (N)). Compare it with the termination criterion. Continue if the MSE (N) is more than MSEmin and the number of iterations is less than Maximum iterations (NMax); terminate otherwise. Step 9. Forward propagation and back propagation. Step 10. Update the weights and bias weights of ANN to reduce the difference between actual output and reference output. Step 11. Return to Step 7. In order to get better understanding of the steps of the hybrid approach, the process of the hybrid approach is displayed as in Fig. 5. 5. Experimental result and discussion Experiments are carried out to validate the proposed ANN model and evaluate the performance of the BP, GA and hybrid approaches. The experimental setup as shown in Fig. 1(b) is located at he mechanic laboratory, university of Auckland, New Zealand. Because the PAM is used in an ankle robot ultimately to realize range of motion and strength training treatments for ankle injuries. Therefore, the experimental setup of the PAM was moved through a sinusoid motion trajectory. The frequency of the motion is 0.2 Hz. The amplitude of the motion is 0.1 m. The total of data are 1200,

which obtained from sensors are divided equally into three parts; one part is used for training the ANN model and the remaining parts are used for validation and testing. The ANN for PAM modeling has been discussed in Section 3. To provide accuracy model of PAM, three approaches, namely, the BP, the GA and the hybrid approach are used to train the ANN model using the training experiment data respectively. Firstly, the adaptive learning rate BP algorithm is used to train the ANN model. After 30,000 iterations, results from the training data show that the MSE to predict pressure inside of the PAM is found to be 0.0020 bars and the maximum deviation is 11.30% as shown in Fig. 6. However, from the discussion provided in Section, we know the error in pressure has an import influence on the PAM characteristic and then may affect the control performance of PAM. To investigate whether the BP algorithm has converged at a local optimal solution and there exists a better solution, the GA approach is used to train the ANN model with the same training data as BP algorithm. The initial population of GA is 100. The crossover rate of GA is 0.85 and mutation rate of GA is 0.08. After around 100 epochs, the MSE and maximum deviation are 0.0011% and 5.96% respectively as shown in Fig. 6. Finally, the hybrid approach is used to train the ANN model, after only 40 epochs of MGA and 3000 iterations of BP discussed in Section 4, the MSE and maximum deviation are only 4.9725e05 bar and 2.5%.

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Fig. 6. Prediction errors in pressure (Training Data), using three training approaches.

The ANN model 31 parameter variable values using both BP, GA and hybrid approach are listed in Tables 2–4. According to these values, we can obtain the three trained ANN models of the PAM.

The test data has been used to check up on the performance of the three trained ANN models. Results from the test data of the three trained ANN models are shown in Fig. 7. The curves of the

Table 2 Variables values of trained ANN model using BP algorithm. Variables Values Variables Values Variables Values Variables Values

ð1Þ

w12 8.9996

ð1Þ

w33 2.2143

ð1Þ

w54 18.0255

ð2Þ

w4 0.2121

w11 1.8865 w32 30.2450 w53 1.3906 w3 0.5498

ð1Þ

w13 0.6904

ð1Þ

w14 94.4721

ð1Þ

w34 33.3589

ð1Þ

b1 2.1277

ð1Þ

w41 9.0617

ð2Þ

w5 0.0683

ð1Þ

ð1Þ

w21 0.8834

ð1Þ

w42 15.7800

ð1Þ

b2 0.0984

ð2Þ

ð1Þ

w22 35.2783

ð1Þ

w43 3.8057

ð1Þ

ð1Þ

w23 1.1611

ð1Þ

w44 7.8467

ð1Þ

b3 1.9565

ð1Þ

w24 58.2131

ð1Þ

w51 9.1710

ð1Þ

b4 4.0210

b5 0.8115

ð1Þ

w31 6.7941

ð1Þ

w52 26.0014

ð2Þ

w2 0.6225

w1 0.3112

ð1Þ

ð1Þ

ð2Þ

ð2Þ

b1 1.2638

Table 3 Variables values of trained ANN model using GA approach. Variables Values Variables Values Variables Values

ð1Þ

w12 1.2597

ð1Þ

w41 1.3205

w11 0.9991 w34 1.5959 ð1Þ

b3 0.9390

ð1Þ

w13 0.6889

ð1Þ

w42 0.7246

ð1Þ

b4 1.2309

ð1Þ

w14 1.7253

ð1Þ

w43 0.8699

ð1Þ

b5 1.7410

ð1Þ

w21 0.1098

ð1Þ

w44 1.0578

ð2Þ

w2 1.1059

w1 0.6087

ð1Þ

w22 0.5190

ð1Þ

w23 0.7927

ð1Þ

w51 0.9073

ð2Þ

w3 0.0900

ð1Þ

w24 0.6427

ð1Þ

w52 1.4820

ð2Þ

ð1Þ

w31 0.5095

ð1Þ

w53 0.8407

w4 0.8914

ð2Þ

w5 1.0290

ð1Þ

w24 0.8425

ð1Þ

w53 1.1139

ð2Þ

w5 0.0118

ð1Þ

w32 1.7005

ð1Þ

w54 0.3324

ð2Þ

b1 0.0004

ð1Þ

ð1Þ

b1 0.9429

b2 0.3191

ð1Þ

w33 0.8234

ð1Þ

ð1Þ

w33 1.4898 ð1Þ

ð2Þ

Table 4 Variables values of trained ANN model using hybrid approach. Variables Values Variables Values Variables Values

ð1Þ

w12 0.2627

ð1Þ

w41 1.0881

w11 0.0586 w34 1.3249 ð1Þ

b3 0.0096

ð1Þ

w13 1.6091

ð1Þ

w42 0.2467

ð1Þ

b4 0.0338

ð1Þ

w14 0.0518

ð1Þ

w43 0.9576

ð1Þ

b5 1.2911

ð1Þ

w21 1.1907

ð1Þ

w44 0.1673

ð2Þ

w2 0.0152

w1 1.1343

ð1Þ

w22 0.3147

ð1Þ

w23 1.3745

ð1Þ

w51 1.0797

ð1Þ

w52 0.9960

ð2Þ

w3 0.0713

ð2Þ

w4 1.0319

ð1Þ

w31 0.0001

ð1Þ

w32 0.0000

ð1Þ

w54 0.8296

ð1Þ

b1 1.2710

ð2Þ

b1 1.2419

ð1Þ

ð1Þ

ð1Þ

b2 1.3957

ð2Þ

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7

Fig. 7. Predict pressure (Test Data), using three training approaches.

trained ANN model output results using BP, GA and hybrid approach are compared with the curve of reference data (experimental data) in Fig. 7. It can be seen that the GA approach has achieved better performance than the BP algorithm, and hybrid approach shows the best performance.

The prediction errors in pressure are calculated between the three trained ANN model outputs and the reference data. The deviations of three ANN models with reference data are shown in Fig. 8. It can be seen that the MSE is 0.0026 bars and the maximum deviation is 13.64%, using BP algorithm which are 0.0010 bars and 6.36% using GA approach.

Fig. 8. Prediction errors in pressure (Test Data), using three training approaches.

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The last subfigure in Fig. 8 shows the errors using hybrid approach. The MSE is 9.3398e5, which is one tenth of the GA approach. The maximum deviation is 3.28%, which is only half of the GA approach. Clearly, hybrid approach based ANN model shows improved performance compared to other approaches. The proposed approach shows improvement over a similar work by Chang and Lilly that 23,340 iterations were required to achieve a MSE of 0.0011 [27]. Three strategies were introduced in the MGA based NARX fuzzy model, and these strategies can ensure a global optimal solution, but they do not enhance the local search capabilities of GA. Their proposed best approach can provide a MD of 10% [28]. It is clear that the modeling performance for PAM using ANN model trained by hybrid approach is much better than these three strategies. 6. Conclusion This paper proposed an artificial neural network approach to model the non-linear and time variant behavior of PAM. In order to train the parameters of the ANN model, BP algorithm, GA and hybrid approach were developed. Research showed that the three trained ANN models were able to represent the relationship between force, length, and pressure of the PAM to certain degree of accuracy. The results obtained from the BP algorithm, the GA approach and the hybrid approach were analyzed in terms of their MSE, maximum deviation and convergence rate. The GA approach was found to be more accurate than the BP algorithm, and showed faster convergence rate. The hybrid approach showed the best preference among the three approaches.

[5]

[6] [7]

[8]

[9]

[10] [11]

[12]

[13] [14] [15]

[16] [17]

[18]

Acknowledgment

[19]

The authors would like to acknowledge the financial support by the National Natural Science Foundation of China (No. 51205296).

[20]

[21]

Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.mechatronics. 2015.04.021.

[22]

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Please cite this article in press as: Song C et al. Modeling of pneumatic artificial muscle using a hybrid artificial neural network approach. Mechatronics (2015), http://dx.doi.org/10.1016/j.mechatronics.2015.04.021