modeling of pneumatic artificial muscle using a hybrid artificial neural

7/26/2019 Modeling of Pneumatic Artificial Muscle Using a Hybrid Artificial Neural

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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 School of Mechanical and Electronic Engineering, Wuhan University of Technology, Wuhan 430070, Chinab 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 29 June 2015

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.

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 forcelength

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 advantagessuch 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 com-

pressibility 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 oper-

ating in human proximity compared to electric or hydraulic

actuators.

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 pressurelength

characteristics and time-variant properties due to compressibility

of air, elastic-viscous properties of the inner tube and geometri-

cally 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 fea-tures. 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 behav-

ior of a PAM, many models to estimate behavior of PAMs have been

proposed in the past. The pioneering work in the field of PAM mod-

eling can be classified into two aspects, analytical modeling and

artificial intelligence-based modeling identification.

http://dx.doi.org/10.1016/j.mechatronics.2015.04.021

0957-4158/ 2015 Elsevier Ltd. All rights reserved.

Corresponding author. Tel.: +86 13437161368.

E-mail address: [email protected](C. Song).

Mechatronics 31 (2015) 124131

Contents lists available at ScienceDirect

Mechatronics

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / m e c h a t r o n i c s
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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 prin-

ciple 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 hys-

teresis 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 charac-

teristics 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 char-

acteristics of the PAM are modeled followed by the dynamic char-

acteristics through spectral analysis [16]. A newapproach 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 pneu-

matic muscles, the accurate prediction of their dynamic behavior

could not be achieved. These analytical models still have limita-

tions 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, fric-

tion between the inner tube and outer sheath, valves and fluid flow

characteristics and large hysteresis. It is evident from the discus-

sion in the preceding section that the conventional tools cannot

fully comprehend the non-linear and time dependent muscle char-

acteristics. 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 manipu-

lator, which can be modified online with an adaptive self-tuning

control algorithm. Through experimental investigation, the pro-

posed MGA-based identification algorithm achieves excellent per-

formance in comparison with conventional SGA and LMS methods[19]. However, the work has been done for constant loading it can-

not 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 algo-

rithm[1921]. 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 algo-

rithm 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 forthe control of a single PAM. In order to assess the advantage of

the intelligent adaptive control system, a comparison of the perfor-

mance 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 condi-

tions, 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, whichpossess 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]:

F PdV=dL PD2op=4sin2h 3cos2 hk 1 1

wherek L=L0, andL0,D0are 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 uncer-

tain 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 inFig. 1.

Tests are conducted on a single PAM which is placed in a rigid

hanger as shown inFig. 1(a). Linear position sensor is positioned

parallel to the PAM to record instantaneous length of PAM. AFUTEK 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 twoIsonicpressure 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 inFig. 1(b), a dSPACE (DS1104) data processing sys-

tem 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 dig-

ital, which are used to read data from various sensors and genera-

tor control signals. The PAM is controlled by compiling a Simulinkmodel 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 tomodel PAM.

C. Song et al./ Mechatronics 31 (2015) 124131 125


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3. ANN architecture of PAM model identification

Through the analysis of Section2, the purpose of modeling is to

find out a precise relationship among the parameters of length,

force and pressure. However, fromFigs. 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 num-

ber of neurons in the layer is roughly determined by experience

formula and then it is determined to five by the method of trialand error. A hyperbolic tangent sigmoid transfer function is

selected to best emulate the non-linear behavior of the PAM in hid-

den layer. Linear transfer function is used for the output layer as a

usual practice.

The detailed architecture of the ANN is shown inFig. 4wherein

x1,x2,x3,x4 are inputs, namely, length, change in length, load and

change in load. Further, w1

ij is the input-hidden layer weight from

input neuron j to hidden neuron I andw2

ki is the hidden-output

weight from hidden neuron i to output neuron k. Bias weights

are shown by b1

i andb2

k for hidden neuroni and output neuron

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.

Fig. 2. Extension length of PAM when inflating.

Fig. 3. Extension length characteristics of PAM when inflation and deflation(100 N). Fig. 4. Architecture of ANN model for PAM dynamic modeling.

126 C. Song et al. / Mechatronics 31 (2015) 124131


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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 algo-

rithms, 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, GradientDescent (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, how-

ever, 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 evolution-

ary 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 short-

comings such as premature and slow convergence.

Therefore in this paper, the modified genetic algorithm are pro-

posed to resolve the problems. We have proposed two modifica-

tions 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 capabil-

ity of BP algorithm in local search and GA in global search, a hybrid

approach combing with BP algorithm and MGA is developed andproposed. 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 1=1 T 2

where Tis thequadratic sumof thedifference (T) between reference

output and actual output in Eq.(3).

The quadratic sum of the difference (T) between reference out-

put and actual output is described firstly, and it is written as:

TXTk1

ek2

XTk1

ydk yak2

3

wherek (k = 1, 2,. . .

,T) is the number of output values;ydkis thereference output values, obtained from the experiment;ya(k) is the

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

ten as:

ya f2X5i1

w2i

f1X4j1

w1ij

xjb1i

! !b

2

! 4

Here, i= 1, 2, . . ., 5 is the number of the hiddenneurons;j= 1, 2, 3, 4

is the number of the input neurons; f1 is the activation function of

hidden neuron; f2 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 indi-

vidual solutions are randomly generated to form an initial popula-

tion. Each variable in the individual solution is assigned for shorttype and the fitness accuracy of the order is 10e4, and the max-

imum 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 MGAs 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 conditionsare, (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 isGMax.

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 genera-

tion. 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 thanGMax, 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 cross-

over 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.

Table 1

Whole structured parameter variables of the ANN model.

w111 w

112 w

113 w

114 w

121 w

122 w

123 w

124 w

131 w

132 w

133 w

134 w

141 w

142 w

143 w

144

w151 w

152 w

153 w

153 w

154 b

11 b

12 b

13 b

14 b

15 w

21 w

22 w

23 w

24 b

21

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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 Section3. 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 trainingdata 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 perfor-

mance of PAM. To investigate whether the BP algorithm has con-

verged 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 devi-

ation 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%.

Fig. 5. The process of hybrid approach.



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The ANN model 31 parameter variable values using both BP, GA

and hybrid approach are listed inTables 24. 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 inFig. 7. The curves of the

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

Table 2

Variables values of trained ANN model using BP algorithm.

Variables w111 w

112 w

113 w

114 w

121 w

122 w

123 w

124 w

131

Values 1.8865 8.9996 0.6904 94.4721 0.8834 35.2783 1.1611 58.2131 6.7941

Variables w132 w

133 w

134 w

141 w

142 w

143 w

144 w

151 w

152

Values 30.2450 2.2143 33.3589 9.0617 15.7800 3.8057 7.8467 9.1710 26.0014

Variables w153 w

154 b

11 b

12 b

13 b

14 b

15 w

21 w

22

Values 1.3906 18.0255 2.1277 0.0984 1.9565 4.0210 0.8115 0.3112 0.6225

Variables w23 w

24 w

25 b

21

Values 0.5498 0.2121 0.0683 1.2638

Table 3

Variables values of trained ANN model using GA approach.

Variablesw

1

11 w

1

12 w

1

13 w

1

14 w

1

21 w

1

22 w

1

23 w

1

24 w

1

31 w

1

32 w

1

33Values 0.9991 1.2597 0.6889 1.7253 0.1098 0.5190 0.7927 0.6427 0.5095 1.7005 1.4898

Variables w134 w

141 w

142 w

143 w

144 w

151 w

152 w

153 w

154 b

11 b

12

Values 1.5959 1.3205 0.7246 0.8699 1.0578 0.9073 1.4820 0.8407 0.3324 0.9429 0.3191

Variables b13 b

14 b

15 w

21 w

22 w

23 w

24 w

25 b

21

Values 0.9390 1.2309 1.7410 0.6087 1.1059 0.0900 0.8914 1.0290 0.0004

Table 4

Variables values of trained ANN model using hybrid approach.

Variables w111 w

112 w

113 w

114 w

121 w

122 w

123 w

124 w

131 w

132 w

133

Values 0.0586 0.2627 1.6091 0.0518 1.1907 0.3147 1.3745 0.8425 0.0001 0.0000 0.8234

Variables w134 w

141 w

142 w

143 w

144 w

151 w

152 w

153 w

154 b

11 b

12

Values 1.3249 1.0881 0.2467 0.9576 0.1673 1.0797 0.9960 1.1139 0.8296 1.2710 1.3957

Variables b13 b

14 b

15 w

21 w

22 w

23 w

24 w

25 b

21

Values 0.0096 0.0338 1.2911 1.1343 0.0152 0.0713 1.0319 0.0118 1.2419

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trained ANN model output results using BP, GA and hybrid

approach are compared with the curve of reference data (experi-

mental 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 devi-

ations 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 devi-

ation is 13.64%, using BP algorithm which are 0.0010 bars and

6.36% using GA approach.

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

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 capabil-

ities 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 relationshipbetween 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.

Acknowledgment

The authors would like to acknowledge the financial support by

the National Natural Science Foundation of China (No. 51205296).

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.

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