Parameter estimation of pendubot model using modified differential evolution algorithm

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International Journal of Modelling and Simulation

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Parameter estimation of Pendubot model using modified differential evolution algorithm

Ngoc Son Nguyen & Duy Khanh Nguyen

Pendubot model using modified differential evolution algorithm, International Journal of Modelling and Simulation, DOI: 10.1080/02286203.2018.1525938

To link to this article: https://doi.org/10.1080/02286203.2018.1525938

Published online: 01 Oct 2018.

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Parameter estimation of Pendubot model using modi fied differential evolution algorithm

Ngoc Son Nguyen and Duy Khanh Nguyen

Faculty of Electronics Technology, Industrial University of Ho Chi Minh City, Viet Nam

ABSTRACT

Parameter estimation plays a critical role in accurately describing system behavior through

mathematical models such as a parametric dynamic model In this paper, a modi fied differential

evolution (MDE) algorithm is proposed to identify the parametric dynamic model of a Pendubot

system with friction In the MDE algorithm, the improvement is to focus on the mutation phase

with a new mutation scheme in which multi-mutation operators are used, including rand/1 and

best/1 for selecting target vectors in population The performance of the MDE algorithm is tested

on a set of fourth benchmark functions, and it is compared with the other algorithms such as a

traditional di fferential evolution (DE), a hybrid DE (HDE) algorithm and a particle swarm

optimiza-tion (PSO) The MDE algorithm is then used to identify the Pendubot ’ parameters accurately.

Experimental results demonstrate the high performance of the proposed method regarding

robustness and accuracy.

ARTICLE HISTORY Received 11 February 2018 Accepted 17 September 2018 KEYWORDS

Di fferential evolution; improved di fferential evolution; pendubot system; parameter estimation

1 Introduction

The Pendubot system has fewer actuators than the

degrees of freedom to be controlled [1], The

Pendubot system is underactuated since the angular

acceleration of the second link cannot be controlled

directly The study of Pendubot will facilitate further

research for more complicated underactuated systems

such as space robots, walking robots and underwater

robots As we know, the control performance is

affected by the strong nonlinearity and unmodeled

dynamic of the system However, almost all of the

Pendubot parameters are unknown To solve this

pro-blem, the paper [2] introduced the proposed intelligent

control scheme to control Pendubot using their

adap-tive capability However, this proposed scheme was not

able to eliminate the effect of friction So, it is necessary

to identify the dynamic model of the Pendubot with

friction

In recent years, evolutionary algorithms (EAs) are

increasingly being proposed for parameter estimation

Such methods include particle swarm optimization

(PSO) [3] and improved version [4–7], an orthogonal

learning cuckoo search algorithm [8–10], a genetic

algo-rithm [11–13], an artificial raindrop algorithm inspired

by the phenomenon of natural rainfall [14], a whale

optimization [15], and bee colony optimization [16–

18] Like as EAs, a differential evolution (or DE)

algo-rithm has been used for parameter estimation Thefirst

published article on DE appeared as a technical report of R.Storn and K.V Price in 1997 [19], Its advantages are

as follows: the simplicity and straightforwardness of implementation, better performance, fewer parameters involved, and low space complexity, had made DE as one of the most powerful tools in thefield of optimiza-tion Chin et al [20] used the DE algorithm to identify the parameters of the two-diode model of PV module In paper [21], the horizontal multilayer soil model para-meters, such as a number of layers, in addition to the resistivity and thickness of each layer, were optimized by the DE algorithm Sarmah et al [22] used the DE algo-rithm for simultaneously estimating six operating para-meters of a hybrid SOFC–GT–ST plant Orkcu et al [23] used the DE algorithm to enhance the parameter estima-tion accuracy of a three-parameter Weibull distribuestima-tion Gao et al [24] proposed a novel inversion mechanism of the extreme functional model via the DE algorithms to exactly identify time delays fractional order chaos sys-tems Garcia et al [25] used DE algorithm for the esti-mation of regression coefficients for the two multivariable regression models The use of these accu-rate models for the estimation of the maximum power would allow estimating the electric production of a concentrating photovoltaic power plant Erdbrink et al [26] introduced the proposed DE algorithm to identify the coefficients of second-order differential equations of self-excited vibrations Marcic et al [27] used the DE

CONTACT Ngoc Son Nguyen nguyenngocson@iuh.edu.vn Faculty of Electronics Technology, Industrial University of Ho Chi Minh City, Viet Nam https://doi.org/10.1080/02286203.2018.1525938

algorithm to identify the electric, magnetic, and

mechanical subsystem parameters of a line-start interior

permanent magnet synchronous motor Upadhyay et al

[28] proposed the improved version of DE technique

called DE with Wavelet Mutation (DEWM) for the

infinite impulse response (IIR) system identification

problem Son et al [29] proposed the hybrid DE

(HDE) to optimally generate the best weights of the

neural networks for modeling and identifying the

hys-teresis inverse model of the shape memory alloys

actua-tor Ayala et al [30] proposed the improved DE

algorithm for the parameter identification of one diode

model equivalent circuit of solar cell modules for real

data acquired in different temperature conditions Y

Wang et al [31] focused on the geometrical error

mod-eling and parameter identification of a 10

degree-of-freedom (DOF) redundant serial– parallel hybrid

inter-sector welding/cutting robot (IWR) using a DE

algorithm

Motivated by the above perspectives, in the paper

[32], the author proposed a newly modified DE (MDE)

and its application for training the neural networks

The improvement of MDE algorithm focuses on the

mutation phase in which multi-mutation operators are

used, including rand/1 and best/1 The modification

that aims to equalize between global exploration and

local exploitation capacitiesfinds global potential

opti-mum solutions In this paper, the MDE algorithm is

continuously proposed to identify the parametric

dynamic model of a Pendubot system with friction

To verify the performance of MDE algorithm, first it

is tested on a set of fourth benchmark functions, and it

is compared with the other algorithms such as the

traditional DE a HDE algorithm and a PSO The

MDE algorithm is then applied to identify the

Pendubot parameter Experimental results prove the

high performance of the proposed method regarding

robustness and accuracy

The rest of the paper is organized as follows.Section

2introduces a MDE algorithm.Section 3 presents the

performance of MDE algorithm tested on the fourth

benchmark functions The performance and efficiency

of the proposed method are evaluated by comparing

with the conventional DE algorithm, a HDE algorithm

and a PSO Section 4 presents the experimental

Pendubot system and the resulting parameter of the

Pendubot system obtained using MDE algorithm

Finally, the conclusion is given in Section 5

2 Modified differential evolution algorithm

R Storn and K.V Price first investigated the DE

algo-rithm in 1997 [19], Up to now, it is becoming popular

and powerful stochastic population-based optimization algorithms In this section, the proposed MDE algo-rithm used in [32] is introduced Where the improve-ment is focused on the mutation phase with a new mutation scheme which is called adaptive mutation scheme with multi-mutation operators

2.1 The adaptive mutation scheme with multi-mutation operators

It is known that the DE performance is significantly influenced by components such as vector generation strategies (i.e mutation and crossover operations), control parameters (i.e mutant factor F, crossover control parameter CR) [19] In these components, the mutation operator is known as an important factor which strongly impacts on the searching abil-ity of the algorithm Therefore, there are many dif-ferent mutation operators have been proposed for many different purposes such as ‘rand/1’, ‘rand/2’,

‘best/1’, ‘best/2’, etc However, in the DE technique, for a particular problem only one operator is used to search the solution Thus, it cannot fully inherit good characteristics of all operators Consequently, the convergence rate or quality of the solution of the algorithm can be not good

From the investigation of the effect of the muta-tion operators on the efficiency and robustness of the

DE algorithm, Qin et al [33] pointed out that the mutation operators usually possess the opposite properties For instance, the mutation operator

‘rand/1’ often brings strong exploration capability of the search domain, but has slow convergence speed While the mutation operator ‘best/1’ usually pos-sesses the fast convergence speed, but is easily trapped into a local optimum Consequently, using only one mutation operator as in the original DE may lead to some restrictions like slow convergence and be stuck into a local optimum

Based on the above analyses, in this work, the muta-tion phase of the DE is modified by means of combin-ing two mutation strategies rand/1 and best/1 together

to create trial vectors instead of only using one muta-tion operator or rand/1 or best/1 as the standard DE The modification aims to equalize between global exploration and local exploitation capacities The novel mutation scheme is described as follows:

if (rand [0,1] > threshold)

vi ¼ xr 1 þ Fðxr 2  xr 3 Þ else

vi ¼ xbest þ Fðxr 1  xr 2 Þ end

From the above mechanism, it can be recognized that

for each target vector, only one of the two mutation

operators is applied for creating the current trial vector,

depending on a uniformly distributed random value

within the range [0,1] For each target vector, if the

random value is bigger than a threshold, the rand/1 is

performed Otherwise, the best/1 is employed With

this scheme, at any particular generation, the

explora-tion and exploitaexplora-tion abilities of the algorithm can be

guaranteed Therefore, the proposed strategy can

sig-nificantly enhance the quality of optimal solution and

the convergence of the algorithm It should be noted

that the setting of the threshold is important, which

can directly influence on the search capabilities of the

algorithm For example, if the threshold is quite large,

the algorithm can produce convergence slowly due to

the trend of employing the rand/1, while if the

thresh-old is quite small, the algorithm can be stuck in a local

solution due to the trend of using the best/1 Using a

trial-and-error procedure, we realize that the threshold

of 0.3 is an adequate value that can well balance

between the searchability and the convergence of the

algorithm in this study The scale factors F is randomly

generated in the interval [0.4, 1.0] instead of being

fixed as in the original DE This aims to create the

variety of searching directions for the both cases of

the rand[0,1] (rand[0,1] > threshold and rand[0,1]

≤ threshold)

2.2 Pseudocode of MDE algorithm

Using the above new mutation mechanism, the detail of

the proposed MDE algorithm is summarized asTable 1

Where GEN is the maximum number of iterations; and

randint(1,D) is a function which returns a uniformly

distributed random integer number between 1 and D

3 Test on benchmark functions

The performance and effectiveness of MDE algorithm

are tested on the fourth Benchmark functions as

Table 2, and then it is compared with another

algo-rithm such as PSO, a conventional DE algoalgo-rithm, and a

HDE algorithm

All simulation results are performed by Matlab

ver-sion 2013b on Intel Core i3 computer with a clock rate of

2.53GHz and 2.00GB of RAM Each algorithm runs 10

times Table 3 gives parameters used in optimization,

where the parameters of DE and PSO algorithm based

on SwarmOps in [34], an HDE algorithm based on [29]

For the Benchmark function problems in Table 2,

the best and the average fitness values for all runs are

reported in Table 4 Figure 1 shows the convergence

rate of MDE, HDE, DE, and PSO in the optimization

of the Benchmark functions over 10 runs

Based on the above results, we see that the MDE algorithm yields superior results compared with DE, HDE, and PSO algorithm For example, in the case of optimization for the Ackley function, the mean error is 6.58e-6 and the standard deviation is 2.93e-6, while for the HDE, DE, and PSO algorithms, the mean error is 1.31e-4, 5.49e-4, 0.0273 and standard deviations are 1.32e-4, 5.00e-4, 0.0379, respectively The smaller stan-dard deviation (StdDev) shows that the proposed algo-rithm is more robust than the other methods Moreover, MDE algorithm can get better results in a shorter time in comparison with the HDE and PSO algorithms

4 Applying for the pendubot parameter estimation

4.1 Dynamic of the pendubot system

The Pendubot system represents planar two degree-of-freedom (2-DOF) robotic arms in the vertical plane with an actuator at the shoulder and no actuator at the elbow The Pendubot system structure is presented

inFigure 2 Where, m1and m2are the masses of links 1 and 2, respectively l1 and l2are the lengths of links 1 and 2, respectively d and d are the distances to the

Table 1.Pseudocode of MDE algorithm

1 Begin

2 Generate the initial population

3 Evaluate the fitness J ¼ 1

N

P N n¼1 ε 2 ðnÞ of each in the population

4 For G = 1 to GEN do

5 For i = 1 to NP do

6 jrand = randint(1,D)

7 F = rand[0:4; 1:0], CR = rand[0:7; 1:0]

8 For j = 1 to D do

9 If rand[0, 1 ] < CR or j = = jrand then

10 If rand[0, 1 ] > threshold then

11 Select randomly r 1 Þr 2 Þr 3 Þi

12 u i;j;Gþ1 ¼ x r1;j;G þ Fðx r2;j;G  x r3;j;G Þ

14 Select randomly r 1 Þr 2 Þbest Þi; "i 2

1 ; :::; NP

15 u i;j;Gþ1 ¼ x best;j;G þ Fðx r1;j;G  x r2;j;G Þ

18 u i;j;Gþ1 ¼ x i;j;G

20 End for

21 If f ~ Ui;Gþ1

 f ~X i;G

  then

22 ~X i;Gþ1 ¼ ~ Ui;Gþ1

23 Else

24 ~X i;Gþ1 ¼ ~ X i;G

25 End if

26 End for

27 End for

28 K ết thúc

centers of mass of links 1 and 2, respectively I1and I2

are the moments of inertia of links 1 and 2,

respec-tively b1 and b2 are the friction of links 1 and 2,

respectively.θ1is the angle that link 1 makes with the

horizontal, and θ2 is the angle that link 2 makes with

link 1.τ is the torque supplied to the link 1

We determine the nonlinear dynamic equations of

the Pendubot system using the Lagrange method Based

on [35], the dynamic equations of Pendubot system

were expressed as follows

Mð θÞ€θ þ Vðθ; _θÞ_θ þ GðθÞ ¼ τ  b 1 _θ 2

b 2 _θ 2

(1) where,

Mð θÞ ¼ M11 M12

M21 M22

¼ P1 þ P2þ 2P3cosð θ 2 Þ P2þ P3cosð θ 2 Þ

P2þ P3cosð θ 2 Þ P2

Vðθ; _θÞ ¼ V11 V12

V21 V22

¼ P3 _θ 2 sinð θ 2 Þ P3ð_θ 1 þ _θ 2 Þ sinðθ 2 Þ

P3_θ 1 sinð θ 2 Þ 0

Gð θÞ ¼ G11

G21

¼ P4 cosð θ 1 Þ þ P5cosð θ 1 þ θ 2 Þ

P5cosð θ 1 þ θ 2 Þ

P1¼ m1d2þ m2l2þ I1; P 2 ¼ m2d2þ I2; P 3 ¼ m2l1d2; P 4

¼ m1d1þ m2l1; P 5 ¼ m2d2 (2)

4.2 Pendubot parameters estimation

Based on the dynamic of Pendubot system (1), we see that some parameters are unknown These parameters have an important role in the designed advanced con-trol In this section, the proposed adaptive DE algo-rithm is used for identifying the seven parameters

w1; ::::; w7

½ T¼ P½ 1; P2; P3; P4; P5; b1; b2T in (1) using the energy theorem, which can be written as

t2

t 1uT_θdt ¼ Eðt2Þ  Eðt1Þ (3) Where u is the vector of torque applied at the joints E (ti) is the total energy at time ti, E(ti) = K(ti) + P(ti) [35], Substituting (1) into (3), we have

t2

t 1hðτ  b1_θ1Þ_θ1þ ðb2θ2:Þ θ2:idt ¼ Eðt2Þ  Eðt1Þ (4)

We denote,

ε ¼ t2

t 1

ðτ  b 1 _θ 1 Þ_θ 1 þ ðb 2 θ 2

:

Þ θ 2 :

dt  Eðt ½ 2 Þ  Eðt 1 Þ  (5) Therefore, thefitness function can be defined as

J ¼ 1 N

XN n¼1

The optimization goal is to minimize thefitness func-tion J When thefitness function converges to zero, we will achieve the best estimation values of the Pendubot

Table 2.The Benchmark functions

Sphere ½ 100; 100  n

f1 ð Þ ¼ x P n

1 x 2 i Griewank ½ 600; 600  n

f2 ð Þ ¼ 1 þ x 1

4000

P n i¼1

x 2

i  Q n i¼1 cos   p x i ffii

f3 ð Þ ¼ 20 exp 0:2 x

ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n

P n i¼1 x 2 i

 exp  1 n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P n i¼1 cos 2 ð πxi Þ

þ 20 þ exp Rastrigin ½ 5:12; 5:12  n

f4 ð Þ ¼ 10n þ x Pn

i¼1 x 2

i  10 cos 2πxi ð Þ

Table 3.The parameters of PSO, DE, HDE and MDE algorithms

Generations, GEN Acceptable error

4000 1e-5 PSO Population size, s

Inertia weight, w Particle ’s best weight, c1 Swarm ’s best weight, c2

149 –0.3236 –0.1136 3.9789

DE – HDE Population size, NP

Mutant factor, F Crossover factor, CR Learning rate, η

18 0.6714 0.5026 0.01 MDE Population size, NP

Mutant factor, F Crossover factor, CR

50 [0.4,1]

[0.7,1]

Table 4.Results obtained for the Benchmark function problems in

Sphere Best

Worst

Average

StdDev

Time (s/run)

2.30e-5 0.0028 8.06e-4 8.35e-4 0.1910

3.09e-6 9.60e-6 6.59e-6 2.46e-6 0.0763

2.21e-8 7.58e-6 2.85e-6 2.44e-6 0.2091

4.38e-7 5.56e-6 3.03e-6 1.82e-6 0.0837 Griewank Best

Worst

Average

StdDev

Time (s/run)

0.0081 0.0811 0.0461 0.0260 0.1961

2.41e-6 0.0020 5.48e-4 6.44e-4 0.1453

7.76e-6 2.91e-4 7.19e-5 1.06e-4 0.5689

1.55e-7 7.48e-6 2.56e-6 2.83e-6 0.3522 Ackley Best

Worst

Average

StdDev

Time (s/run)

0.0026 0.1291 0.0273 0.0379 0.2263

3.54e-5 0.0012 5.49e-4 5.00e-4 0.1791

5.12e-7 4.68e-4 1.31e-4 1.32e-4 0.5263

2.15e-6 9.97e-6 6.58e-6 2.93e-6 0.1797 Rastrigin Best

Worst

Average

StdDev

Time (s/run)

1.41e-5 0.9950 0.1159 0.3116 0.1455

8.64e-6 0.5830 0.0609 0.1835 0.1020

1.50e-6 2.89e-4 4.99e-5 8.79e-5 0.2884

2.22e-6 8.88e-6 4.87e-6 2.28e-6 0.1182

parameters Table 5 shows the pseudocode of MDE

algorithm used in the identification process

4.3 Experiment setup of the pendubot system

A general configuration, the schematic diagram of

the Pendubot system and a photograph of the

experi-mental system are shown in Figure 3 The hardware

includes the STM32F407 board which provides

PWM signals u(t) to control the DC motor through the H-Bridge board The two angle encoder sensors

Table 5.Pseudocode of MDE algorithm in Pendubot’s parameter estimation

1 Begin

2 Generate the initial population

3 ~x i;G ¼ w 1;i;G ; ::::; w 7;i;G

¼ P 1;i;G ; P 2;i;G ; P 3;i;G ; P 4;i;G ; P 5;i;G ; b 1;i;G ; b 2;i;G

4 Evaluate the fitness J ¼ 1

N

P N n¼1 ε 2 ðnÞof each in the population

5 For G = 1 to GEN do

6 For i = 1 to NP do

7 j rand = randint(1,D)

8 F = rand[0:4; 1:0], CR = rand[0:7; 1:0]

9 For j = 1 to D do

10 If rand[0, 1 ] < CR or j = = j rand then

11 If rand[0, 1 ] > threshold then

12 Select randomly r 1 Þr 2 Þr 3 Þi

13 u i;j;Gþ1 ¼ x r1;j;G þ Fðx r2;j;G  x r3;j;G Þ

15 Select randomly r 1 Þr 2 Þbest Þi; "i 2 1; :::; NP f g

16 u i;j;Gþ1 ¼ x best;j;G þ Fðx r1;j;G  x r2;j;G Þ

17 End if

18 Else

19 u i;j;Gþ1 ¼ x i;j;G

20 End if

21 End for

22 If J ~ u i;Gþ1

 J ~x i;G

  then

23 ~X i;Gþ1 ¼ ~ Ui;Gþ1

24 Else

25 ~X i;Gþ1 ¼ ~ X i;G

26 End if

27 End for

28 End for

29 K ết thúc

10 -5

10 0

10 5

Sphere

10 -5

10 0

Ackley

Generations

10 -5

10 0

Griewank

10 -5

10 0

Rastrigin

Generations

Magenta dotted line: PSO Cyan dashed line: DE Red solid line: HDE Blue dash-dot line: MDE

Figure 1.Convergence rate of MDE, HDE, DE, and PSO in optimization over 10 runs

g

x y

τ

Figure 2.Pendubot system

are used to measure the output angles of the two

joints

4.4 Estimation results

In this section, we study the effectiveness and

perfor-mance of our proposed MDE algorithm for identifying

the Pendubot parameters All of the simulations were

performed by Matlab version 2013b on an Intel Core i3

computer with a clock rate of 2.53GHz and 2.00GB of RAM The procedure for identifying the parameter of the Pendubot is given below:

First, the experimental input–output dataset that

is used for identifying the Pendubot parameters based on the MDE algorithm is collected from the real Pendubot system Figure 4 shows the torque input applied to the Pendubot system and the responding position output collected The torque is

y

Driver MCU

STM32F407

PWM

Encoder 1

Encoder 2

2 Matlab/Embedded Coder

Laptop

24VDC

DC motor

(a)

(b)

Figure 3.(a) Schematic diagram of experimental setup (b) Photograph of the experimental Pendubot system

a pulse of the 5 s period with 5% pulse width; each

pulse is several random numbers from 0 to 1 values

and 0.01 s interval Where dataset from

(0–900)[sam-ple] is used for estimating the Pendubot parameters;

Dataset from (901–1800)[sample] is used for

validat-ing the Pendubot parameters

Assuming that the dataset has been acquired, the

second step is to select a model structure The

pro-posed MDE algorithm is used to identify the

Pendubot parameters Table 6 shows the MDE

para-meters of the algorithm used in the identification

process

The estimation and validation process are conducted

to identify the Pendubot parameters The procedure is

run 10 times.Table 7 gives the performance results of

the MDE algorithms in identifying the Pendubot dynamic parameters The results from Table 6 show that the parametric values of the Pendubot system are precisely identified.Table 8tabulates the resulted para-meter values of the Pendubot system These parapara-meters will be used to design the controller in the experimen-tal system

6 Conclusion

In this paper, the performance of MDE algorithm is tested on the Benchmark function and is compared with other algorithms such as DE, HDE, and PSO algorithm The results show that the proposed MDE algorithm can improve the performance in comparison with a conventional DE algorithm, HDE algorithm, and better than PSO algorithm And then, the MDE algorithm is applied for identifying the Pendubot dynamic parameters based on experiment input–out-put training data In the future work, the author will use these identified parameters to propose a swing up and balance controller scheme for the Pendubot system

Figure 4.Dataset for Pendubot parameters estimation

Table 6.The MDE parameters in identification

MDE A number of generations

Population size, NP

4000 30

Table 7.The performance of the MDE in identification

MSE Training Validation Method Best Worst Average Average

MDE 2.37e-3 2.39e-3 2.37e-3 3.03e-3

Table 8.The resulted parameters of Pendubot system

0.002207 0.000400 0.000101 0.022447 0.003326 0.006592 0.00009

This research is funded by Industrial University of Ho Chi

January, 2018

Disclosure statement

No potential conflict of interest was reported by the authors

Funding

This work was supported by Industrial University of Ho Chi

in January, 2018]

Notes on contributor

Son Nguyen received his M.Sc and PhD degrees in the

Faculty of Electrical and Electronics Engineering (FEEE)

from Ho Chi Minh City University of Technology in 2012

and 2017, respectively He is currently a Lecturer and

Vice-Dean of the Faculty of Electronics Technology, Industrial

University of Ho Chi Minh City, Viet Nam His current

research interests include intelligent control, robotics,

iden-tification of nonlinear systems, and the internet of things

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