evolutionary algorithm for identification of a flexible

Evolutionary Algorithm for identification of a Flexible Single-Link System

ALI ABDULHUSSAIN MADHLOOM AL-KHAFAJI 1AND INTAN Z. MAT DARUS 1 1Department of Applied Mechanicsand Design,

Faculty of Mechanical Engineering, Universiti Teknologi Malaysia 81310, Johor Bahru

MALAYSIA [email protected]; [email protected]

Abstract-This study presents an investigation into the dynamic modelling of a flexible single-link manipulator systemusing differential evolutionary technique (DE) and Particle Swarm Optimizationtechnique (PSO).The input-output data of the system were first acquired through the simulation study using finite element method(FDM) and Lagrangian approach. A bang-bang torque was applied as an input and the dynamic response of the system was investigated. Performance of the simulation study is compared with the manipulator resonant frequencies obtained experimentally from the literature. Next, an appropriate model structure are chosen and optimized using an Intelligent parametric identification technique (DE) and(PSO). One Step Ahead (OSA) prediction, correlation tests and mean squared error (MSE) have been performed for validation and verification of the obtained model in characterizing the manipulator system. Furthermore, an unseen data used to observe the prediction ability of the model.A comparative assessment of the two models in characterizing the manipulator system is presented in time and frequency domains. Results demonstrate the advantages of (DE) over (PSO) in parametric modeling. Key-Words: -differential evolutionary algorithm, Particle Swarm Optimization, Flexible single-link Manipulator, System Identification, finite element method, dynamic modelling. 1 Introduction Modern flexible link manipulator systemsare widely used in many areas such as the space vehicles, medical, defense and automation industries. The flexible link manipulator system, comparing with rigid link manipulator system offers many advantages for example low inertia, light weight, few powerful actuators, cheaper construction, fast in response, safer operation, higher payload carrying capacity and longer reach. In general, the flexibility in modern robot manipulators system is an unwanted feature since it causes a serious of vibration problem when subjected to disturbance forces. Therefore the control of the flexible link manipulator robot becomes much more complex than rigid link robots. Accordingly, there is a growing need to develop suitable control strategies for such kind of systems [1],[2], [3]. To control the vibration of a flexible link system effectively, it is often required to obtain an accurate or approximate model of the system structure, which will results in good control. Thus, finding an appropriate model of a dynamic system, such as a flexible link manipulator system, would be the key to design an effective controller to suppress the vibration in the flexible link manipulator system [1]. To avoid solving dynamics of the flexible single link manipulator, it can be modelled using System identification approaches [4], [5], [6]. System identification is the process of developing an accurate or approximate dynamic model of a physical system using a set of input-output data pairs collected at input-

output terminals of the system using experimental or simulation study. many parametric and nonparametric estimation techniques have been employed in identification of flexible link manipulator such as least mean square (LMS) and recursive least squares (RLS) [7],[8] genetic algorithm (GA) [7],[9], particle swarm optimization (PSO) [10],[11], neural networks (NNs) [12],[13], bacterial foraging algorithms (PFA) [14]. It can be noted from the literature that the use of DE for modelling manipulators has not been reported yet. 2 The Flexible Manipulator System For the mathematical modeling of a flexible single-link manipulator, a finite element method and Lagrangian approach is utilized [15],[16],[17]. The development of the algorithm can be divided into three main steps:

1. FEM analysis 2. State space representation 3. Obtaining the result

The overall approach involves treating the link of the manipulator as an assemblage of n elements of lengthl . For each of these elements the kinetic energy iT and

potential energy iV are computed in terms of a suitably selected system of n generalized variables q and their rate of change q . In this study, a flexiblesingle-link manipulator system is considered as shown in Fig.1.

Manufacturing Engineering, Automatic Control and Robotics

ISBN: 978-960-474-371-1 41

mailto:[email protected]

mailto:[email protected]

Fig. 1. Schematic diagram of the flexible manipulator

system The link has been modelled as a pinned-free flexible beam. The pinned end of the flexible beam of length Lis attached to the hub with inertia HI , where the input torque )(tτ is applied at the hub by a motor and payload mass pM is attached at the free end. E , Iand ρ represent the Young Modulus, second moment of inertia and mass density per unit length of the flexible manipulator respectively. XY axis and 00YX axis represent the stationary and moving coordinate respectively. Both axes lie in a horizontal plane and all rotation occurs about a vertical axis. For a small angular displacement )(tθ and small elastic deflection,

t)w(x, the overall displacement t)y(x, of a point along the link at a distance x from the hub can be defined as a function of both the rigid rotation angle

)(tθ of the hub and flexible displacement t)w(x, of the beam measured from the line Thus the total displacement t)y(x, is :

t)w(x,+)(x=t)y(x, tθ using the standard finite element method to solve dynamic problems, leads to the well-known equation:

(t)Q (x)N=t)(x,w nnn where (x)Nn and (x)Qn represent the shape function and nodal displacement respectively.

Define, ∑=

1-N

1ii-x=s l , where is the length of the ith

element. the kinetic energy of an element can be expressed as follows:

ininTini ][Q][M][Q

21=T , ..N…1,2,=i

and the potential energy due to the elasticity of the FE can be obtained as

∑=

=N

iiP

1nin

Tin ]i[Q][K][Q

21

The total kinetic and potential energy can be written as

nnTnp

N

iim QMQTTTT

21

1

=++= ∑=

Similarly, the potential energy due to the elasticity of the FE can be obtained as

nnTn

N

ii QKQPP

21

1==∑

= 2.1 Dynamic equation of the link The Lagrangian of the link can be derived as followed:

nnTnnn

Tn QKQQMQPTL

21

21

−=−= (1)

A further detail on dynamic modeling can be found in[15],[16],[17]. By using Lagrange equation, the dynamic equations of a flexible manipulator can be derived utilizing the equation as followed:

τbQKQDQM ′=′′+′′+′′ (2) where QKM ′′′ ,, are mass matrix , rigidity matrix and general coordinates when substituting boundary conditions, respectively. [ ]Tb 001 =′ . τ is input torque. D′ is damping matrix and we choose the linear type Reyleigh damping and )(tQ is the nodal displacement given as

]w w[=Q(t) n11 nθθθ The KDM ′′′ ,, matrices are of size 11 *mm and

[ ] τTb 001 =′ has 1*1m size and 121 += nm .For simplicity 0=′D and 0)0( =Q The state-space form of the equation of motion is

BuAvv += (3) DuCvy += (4)

where

−−=

−− DMKM

IA

mm

11

110

,

=− eM

Bm

1

1*10

[ ]110 mm IC = , [ ]1*120 mD =

10m is an 11 *mm null matrix, 1mI is an 11 *mm identity

matrix, 1*120 m is an 1*1m null vector, and the vector e

is the first column of the identity matrix. [ ]Tu 00 τ= T

nn11n11 ] uu u u [=v θθθθ

(5)

3 Differential Evolution Algorithm Evolutionary algorithms are used effectively for solving difficult optimization problems, such as nonlinear global optimization problems which subjected to multiple nonlinear constraints[18]. The differential evolution algorithm(DE), proposed by Storn and Price is a population-based stochastic optimization algorithm developed to solve nonlinear and multi-modal global optimization problems. DE is an efficient, robust, and effective algorithm. It uses mutation and crossover operations as searching mechanisms and selection operations to direct the search to the most promising regions of the search space. New candidate individuals are made by combining the target individual and several other


ISBN: 978-960-474-371-1 42

individuals by randomly choosing from the same population, where the candidate with a better fitness value replaces the target individual[19]. The above stages can be outlined as shown in Fig 2.

Fig 2. Working Principle of DE

4 Particle Swarm Optimization Particle Swarm Optimization (PSO) is a population based, stochastic optimization technique introduced by Kennedy and Eberhart [20]. The operating procedure of a PSO can be described through the stages shown as:

1. Start 2. Initial Setup

• Number of dimensions (d) • Upper and lower bounds [ub,lb] • number of particles (n) • Maximum Iteration • Define hypothesis function • Define the cost function

3. Initial Population • Create position vectors of random (n)

particles of dimension (d) without violating [ub , lb]

• Create velocity vectors of random (n) particles of dimension (d)

4. Update Population • Loop from 1: (Max Generation) or (stop

criterion) • Evaluate cost of each particle using • Update the pbest and gbest positions and

values • Calculates the new velocity and positions

for all swarm's elements

• End Loop 5. Display Output Results 6. End

PSO is initialized with a group of random particles, ‘fly’ in the search space of an optimization problem. Particles are updated with two ‘best’ values every iteration. The first one is called pbest, which is the best position a particle has visited so far and memorizes it. Another ‘best’ value is the global best or gbest, obtained so far by any particle in the population. Using their memories of the best positions of pbestand gbest, particle is then accelerated toward those two best values by updating the particle position and velocity using the following set of equations:

1))-(tx-(p randc+1))-(tx-(p randc+1)-(twv(t)v

idgd2

idid1 id=id

)(idv+1)-(t idx=(t)idx t

and are the current velocity and position vector of the i-th particle in the d-dimensional search space respectively. and are acceleration coefficients usually and rand is the random number between 0 and 1. is the inertia serves as memory of the previous direction, preventing the particle from drastically changing direction. High value of promote global exploration and exploitation, while low value of w leads to a local search. The common approach is to provide balance between global and local search by linearly decrease during the search process. Decreases the inertia over time can be expressed as:

tw

wwwtw

end

endstartstart

−−=)(

where and are the starting and end point of inertia weight defined as linearly decrease from 0.9 to 0.25 and is the maximum number of time step the swarm is allowed to search. Start w end w max T. 5 Implementation and Results To study the dynamic behavior of the flexible manipulator system, a computer program was written within MATLAB environment to simulate the state space matrices derived from the mathematical modeling done above. A thin aluminum alloy with the specifications shown in Table 1 is considered [3].

Table1: Physical parameters of the flexible manipulator

Components value

Length 960 mm Width 19.008 mm Thickness 3.2004 mm Mass density per unit volume 2710 The second moment of inertia 5.1924 The young modulus 71* The hub inertia 5.86*

Evaluate the objective function of trail

Initialize - generate a target vector

Mutation to generate a mutant vector

Selection operation

Is terminating criteria satisfied?

Define the search space

Evaluate the objective function of target

0Crossover to generate a trail vector

Stop: output the final result

Yes

No


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For simplicity purposes, the effect of mass payload is neglected. Throughout this simulation, a bang-bang torque input with an amplitude ±0.3 Nm was applied at the hub of the manipulator as shown in Fig 3. The response of the flexible manipulator at the hub angle and end-point residual is monitored for duration of 3.0 seconds with sampling time 0.37 ms and is observed and recorded as shown in Fig 4(a),(b) and Fig 5(a),(b) in both time and frequency domain respectively. The first three resonant mode captured for end-point residual and hub angle response is at 11.59 Hz, 31,88 Hz and 58.93 Hz is compared with the experimental first three modes of vibration of the manipulator which obtained at at 11.67 Hz, 36.96 Hz and 64.22 Hz respectively (3). These, as noted, closely match with the corresponding experimental value with small percentage error that 0.6% for first mode, 13.7% for second mode and 8.2% for third mode.

Fig. 3.bang-bang input torque

(a) Time domain

(b) Frequency domain

Fig. 4. Hub angle response

(a) Time domain

(b) Frequency domain

Fig. 5. End-point residual response 5.1 Modeling of Hub Angle Using DE A MATLAB program has been created based on the DEalgorithm, and was used to estimate the parameters of ARX model based on the torque input and hub angle output data obtained from the simulation study of the flexible manipulator system, as described in the earlier sections of this study. Since there was no a priori knowledge regarding the suitable order of the model, the structure realization was performed by a trial-and-error method. Randomly selected parameters were optimized for different, arbitrarily chosen order to fit into the system by applying the working mechanism of DE based on One Step-Ahead prediction. The best result was achieved with model order = 4, that mean, , = 2 for 4000 data length, which was used to find the parameters. From the work carried out, for the model order = 4, it was found that satisfactory results were achieved with the following set of parameters:

Population Initial Range: [-3; 3] Number of Generations: 2800 Population Size: 10

The data set, comprising 7986 data points, was divided into two sets of 4000 and 3986 data points respectively. The first set was used to estimate the model parameters whilst the second set was used to validate the model. Both output and estimated outputs of the hub angle in time and frequency domains are plotted in Figs6 and 7respectively. The error between actual and predicted DE output are plotted in Fig 8. The division between the trained data and the unseen


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data is indicated as a vertical line located at point 4000 as shown in Fig 6and Fig 8. The best mean square error of DE algorithm is . Using the proposed identification procedure, the parameters of the model were estimated as follows:

Fig. 6. Output and estimated outputs of the hub angle

in time domain using DE


in frequency domain using DE

Fig. 8. Error between actual and predicted output for

hub angle using DE

The output and estimated outputs of the hub angle in time and frequency domains and error show that the model was able to mimic the measured output well. The pole-zero diagram and correlation tests are depicted in Fig 9 and Fig 10 respectively. It is noticed that the model was stable, the poles of the transfer

function were inside the unit circle and the zero was outside the unit circle indicating non-minimum phase behavior. The correlation functions were carried out for 1000 samples to determine the effectiveness of the DE-based model. The results were found to be within 95% confidence level thus confirmed the accuracy of the results.

Fig. 9. Pole-zero diagram of hub angle using DE

Fig. 10. Correlation tests for hub angle using DE

5.2 Modeling End-point residual Using DE A MATLAB program has been created based on the DEalgorithm, and was used to estimate the parameters of ARX model based on the torque input and end-point residual output data obtained from the simulation study of the flexible manipulator system, as described in the earlier sections of this study. Since there was no a priori knowledge about the suitable model order of the flexible manipulator system, the structure realization was performed using a heuristic method. Randomly selected parameters were optimized for different, arbitrarily chosen order to fit into the system by applying the working mechanism of DE based on One Step-Ahead prediction. The best result was achieved with model order = 6, that mean, , = 3 for 4000 data length, which was used to find the parameters. For the best model order, the DE was designed with 10 individuals in each iteration with maximum number of iterations was set to


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6000. The data set, comprising 7986 data points, was divided into two sets of 4000 and 3986 data points respectively. The first set was used to compute the model parameters whilst the second set was used to validate the model. In addition to that, the best mean square error of DE algorithm is . Using the proposed identification procedure, the parameters of the model were estimated as follows:

Both output and estimated output of end-point residual in time and frequency domains are plotted in Figs. 11 and12 respectively and the error between actual and predicted DE output are plotted in Fig 13. The division between the trained data and the unseen data is indicated as a vertical line located at point 60000 as shown in Fig 11 and 13. The pole-zero diagrams and correlation tests are depicted in Fig. 14 and Fig. 15 respectively, indicating that all models were stable with non-minimum phase behavior. The correlation test functions results confirm that the models are acceptable.

Fig. 11. Output and estimated outputs of end-point

residual in time domain using DE

Fig. 12.Output and estimated outputs of end-point

residual in frequency domain using DE


end-point residual using DE

Fig. 14. Pole-zero diagram of end-point residual using

DE

Fig. 15. Correlation tests of end-point residual using DE

5.3 Modeling Hub Angle Using PSO Investigations were then carried out with the PSO algorithm to modelling the hub angle using the same input-output data and same model order achieved for hub angle using DE. The PSO was designed with 50 individuals in each iteration with maximum number of iterations was set to 30000. The best mean square error of PSO algorithm is . Using the proposed identification procedure, the parameters of the model were estimated as follows:


ISBN: 978-960-474-371-1 46

Both output and estimated output of end-point residual in time and frequency domains are plotted in Figs.16 and 17 respectively and the error between actual and predicted DE output are plotted in Fig 18. The division between the trained data and the unseen data is indicated as a vertical line located at point 60000 as shown in Fig 16 and Fig 18. The pole-zero diagrams and correlation tests are depicted in Fig. 19 and Fig. 20 respectively, indicating that all models were stable with non-minimum phase behavior. The correlation test functions results confirm that the models are acceptable.


in time domain using PSO


in frequency domain using PSO

Fig.18. Error between actual and predicted output for

hub angle using PSO

Fig. 19. Pole-zero diagram for hub angle using PSO

Fig. 20. Correlation tests for hub angle using PSO

5.4 Modeling End-point residual Using PSO Investigations were then carried out with the PSO algorithm to modelling the end-point residual using the same input-output data and same model order achieved for end-point residual that used in DE. The PSO was designed with 150 individuals in each iteration with maximum number of iterations was set to 30000. The best mean square error of PSO algorithm is

. Using the proposed identification procedure, the parameters of the model were estimated as follows:

-0.448655649457146 -2.375043245413949 -0.036951869515860

Both output and estimated output of end-point residual in time and frequency domains are plotted in Figs. 21 and 22 respectively and the error between actual and predicted DE output are plotted in Fig 23. The division between the trained data and the unseen data is indicated as a vertical line located at point 60000 as shown in Fig 21 and 23. The pole-zero diagrams and


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correlation tests are depicted in Fig. 24 and Fig. 25 respectively, indicating that the model is unstable. The correlation test functions results showed that the models are biased.


residual in time domain using PSO


residual in frequency domain using PSO


end-point residual using PSO

Fig. 24. Pole-zero diagram for end-point residual using

PSO

Fig. 25Correlation tests for end-point residual using

PSO 6Comparative Assessments The overall comparative performance of DE and PSO modelling approaches in terms of the mean-squared error, stability and correlation tests is summarised in Table 2. From Table 2, it can be seen that the performance of DE modelling technique gives better approximation to the system response compared to PSO modelling technique. It is also noted that the stability and the correlation test results by DE is better than PSO especially for the end point residual modelling . However, a major advantage of the PSO is that the algorithm is simple. Table 2 also shows that the performance of DE in terms of mean-squared error are better than the PSO with the same model structure. Therefore, high performance computing power could provide better solutions in the real time implementation of the DE based identification.

Table 2Overall comparative assessments Modelling domain

MSE Stability Correlation tests

End point residual

DE stable unbiased

PSO unstable biased

hub angle DE stable unbiased

PSO stable biased

7 Conclusion In this work, CS and PSO have been adopted for modelling a single-link flexible manipulator system. Input-output data pairs have been collected from the simulation study and used in developing linear models of the system from input torque to hub-angle and end-point acceleration. The performances of CS-ARX and PSO-ARX models have been assessed through many validation tests. It has been demonstrated that the CS and PSO modelling technique has performed well in


ISBN: 978-960-474-371-1 48

approximating the system response and CS is better than PSO in modelling a single-link flexible manipulator system. Acknowledgment The authors wish to thank the Ministry of Higher Education (MOHE) and Universiti Teknologi Malaysia (UTM) for providing the research grant and facilities. This research is supported by UTM Research University Grant Vote No. 04H17. References: [1] S. K. Dwivedy and P. Eberhard, Dynamic analysis

of a flexible manipulator, a literature review, Mech. Mach. Theory., Vol. 41, pp. 749-777, 2006.

[2] W.J. Book, Modeling, design and control of flexible link manipulator arms: a tutorial review”, Proceedings of IEEE Conference on Decision and Control, pp. 500-506, 1990.

[3] Supriyono, Heru, “Novel Bacterial Foraging Optimisation Algorithms with Application to Modelling and Control of Flexible Manipulator Systems”, Ph.D. Thesis Dept. Automatic Control and Systems Eng, Sheffield Univ, 2012

[4] Shaheed, M.H. and Tokhi, M.O, Dynamic modelling of a single-link flexible manipulator: parametric and non-parametric approaches. Robotica, 20, pp. 93 –109, 2002

[5] Ljung, L, System identification: theory for the user, (2nd edition. Prentice hall. 1999)

[6] I. Z. Mat Darus and A. A. M. Al-Khafaji,. Nonparametric Modelling of a Rectangular Flexible Plate Structure, International Journal of Engineering Application of Artificial Intelligence, 25, pp. 94-106. 2012

[7] Md Zain, B.A., Tokhi, M.O. and Abd. Latiff, I. ,Dynamic modelling of a single link flexible manipulator using particle swarm optimisation. The Second International Conference on Control, Instrumentation and Mechatronic Engineering (CIM09), Malacca, Malaysia, June 2 -3, 2009a.

[8] Shaheed, M.H. and Tokhi, M.O.. Dynamic modelling of a single-link flexible manipulator: parametric and non-parametric approaches. Robotica, 20, pp. 93 –109,2002

[9] Shaheed, M. H., Tokhi, M. O., Chipperfield, A. J. and Azad, A. K. M, Modelling and open-loop control of a single-link flexible manipulator with genetic algorithms. Journal of Low Frequency Noise, Vibration and Active Control, 20(1), pp. 39 – 55.

[10] Alam, M. S. and Tokhi, M.O, Dynamic modelling of a single-link flexible manipulator system: a particle swarm optimisation approach. Journal of Low Frequency Noise, Vibration and Active Control, 26 (1), pp. 57 – 72. 2007

[11] Md Zain, B.A., Tokhi, M.O., and MdSalleh, S. (2009b). Dynamic modelling of a single link flexible manipulator using parametric techniques with genetic algorithms. Proceedings of Third UK Sim European Symposium on Computer Modelling and Simulation, Athens, Greece, November 25 – 27, 2009, pp. 373 – 378. 2009

[12] Talebi, H.A., Patel, R.V. and Asmer, H., Dynamic modeling of flexible-link manipulators using neural networks with application to the SSRMS. Proceedings of the 1998 IEEE/RSJ International Conference onIntelligent Robots and Systems, Victoria, B.C., Canada October 1998, pp. 673 – 678 (1998).

[13] Shaheed, M.H. and Tokhi, M.O. Dynamic

modelling of a single-link flexible manipulator: parametric and non-parametric approaches. Robotica, 20, pp. 93 –109,2002.

[14] H. Supriyono, M.O. Tokhi and B.A. Md Zain, Modelling of flexible manipulator systems using bacterial foraging algorithms, Proceedings of CLAWAR 2011: The 14th International Conference on Climbing and Walking Robots and the Support Technologies for Mobile Machines (CLAWAR2011), Paris, France, 6-8 September 2011, pp. 157-165. 2011

[15] Yanqing Gao, “flexible manipulators: modeling, analysis and optimum designs”, Ph.D. Thesis department of mining and geological engineering, the university of arizona (2010)

[16] M.O. Tokhi, Z. Mohamed & A.K.M. Azad. (). Finite Difference and Finite Element Simulation of a Flexible Manipulator Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 1997

[17]Tahmina Zebin M. S. Alam Modeling and Control of a Two-link Flexible Manipulator using Fuzzy Logic and Genetic Optimization Techniques, journal of computers, vol. 7, no. 3, march 2012

[18]. Michalewicz, Z. and Shoenauer, M., Evolutionary Algorithms for Constrained Parameter Optimization Problems. Evolutionary Computation 4(1):1-32, 1996.

[19]Ching-Hung Lee, Che-Ting Kuo and Hao-Han Chang (2012)”Performance enhancement of the differential evolution algorithm using local search and a self-adaptive scaling factor” International Journal of Innovative, Computing, Information and Control, Volume 8, Number 4, April 2012

[20] M. Settles. (2005). An Introduction to Particle Swarm Optimization. Department of Computer Science, University of Idaho.


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evolutionary algorithm for identification of a flexible

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