Parameter estimation of Pendubot model using modified differential evolution algorithm Full Terms & Conditions of access and use can be found at http //www tandfonline com/action/journalInformation?jo[.]
International Journal of Modelling and Simulation ISSN: 0228-6203 (Print) 1925-7082 (Online) Journal homepage: http://www.tandfonline.com/loi/tjms20 Parameter estimation of Pendubot model using modified differential evolution algorithm Ngoc Son Nguyen & Duy Khanh Nguyen To cite this article: Ngoc Son Nguyen & Duy Khanh Nguyen (2018): Parameter estimation of 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 Submit your article to this journal Article views: View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=tjms20 INTERNATIONAL JOURNAL OF MODELLING AND SIMULATION https://doi.org/10.1080/02286203.2018.1525938 ARTICLE Parameter estimation of Pendubot model using modified differential evolution algorithm Ngoc Son Nguyen and Duy Khanh Nguyen Faculty of Electronics Technology, Industrial University of Ho Chi Minh City, Viet Nam ABSTRACT ARTICLE HISTORY Parameter estimation plays a critical role in accurately describing system behavior through mathematical models such as a parametric dynamic model In this paper, a modified 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 differential evolution (DE), a hybrid DE (HDE) algorithm and a particle swarm optimization (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 Received 11 February 2018 Accepted 17 September 2018 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 problem, the paper [2] introduced the proposed intelligent control scheme to control Pendubot using their adaptive 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 algorithm [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) algorithm has been used for parameter estimation The first CONTACT Ngoc Son Nguyen nguyenngocson@iuh.edu.vn © 2018 Informa UK Limited, trading as Taylor & Francis Group KEYWORDS Differential evolution; improved differential evolution; pendubot system; parameter estimation 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 the field of optimization 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 parameters, 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 algorithm for simultaneously estimating six operating parameters of a hybrid SOFC–GT–ST plant Orkcu et al [23] used the DE algorithm to enhance the parameter estimation accuracy of a three-parameter Weibull distribution 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 systems Garcia et al [25] used DE algorithm for the estimation of regression coefficients for the two multivariable regression models The use of these accurate 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 Faculty of Electronics Technology, Industrial University of Ho Chi Minh City, Viet Nam N S NGUYEN AND D K NGUYEN 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 hysteresis inverse model of the shape memory alloys actuator 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 modeling and parameter identification of a 10 degree-offreedom (DOF) redundant serial – parallel hybrid intersector 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 capacities finds global potential optimum 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 introduces a MDE algorithm Section 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 presents the experimental Pendubot system and the resulting parameter of the Pendubot system obtained using MDE algorithm Finally, the conclusion is given in Section Modified differential evolution algorithm R Storn and K.V Price first investigated the DE algorithm in 1997 [19], Up to now, it is becoming popular and powerful stochastic population-based optimization algorithms In this section, the proposed MDE algorithm used in [32] is introduced Where the improvement 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 multimutation 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 ability of the algorithm Therefore, there are many different 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 mutation 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 possesses 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 mutation phase of the DE is modified by means of combining two mutation strategies rand/1 and best/1 together to create trial vectors instead of only using one mutation 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 ẳ xr1 ỵ Fxr2 xr3 ị else end vi ẳ xbest ỵ Fxr1 xr2 ị INTERNATIONAL JOURNAL OF MODELLING AND SIMULATION 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 exploration and exploitation abilities of the algorithm can be guaranteed Therefore, the proposed strategy can significantly 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 threshold 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 as Table Where GEN is the maximum number of iterations; and randint(1,D) is a function which returns a uniformly distributed random integer number between and D 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 algorithm such as PSO, a conventional DE algorithm, and a HDE algorithm All simulation results are performed by Matlab version 2013b on Intel Core i3 computer with a clock rate of 2.53GHz and 2.00GB of RAM Each algorithm runs 10 times Table 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 Figure shows the convergence Table Pseudocode of MDE algorithm Begin Generate the initial population N P ε2 ðnÞ of each in the population Evaluate the fitness J ¼ N1 n¼1 For G = to GEN For i = to NP jrand = randint(1,D) F = rand[0:4; 1:0], CR = rand[0:7; 1:0] For j = to D If rand[0,1] < CR or j = = jrand then 10 If rand[0,1] > threshold then 11 Select randomly r1 ịr2 ịr3 ịi 12 ui;j;Gỵ1 ẳ xr1;j;G ỵ Fxr2;j;G xr3;j;G ị 13 Else 14 Select randomly r1 Þr2 Þbest Þi; "i f1; :::; NPg 15 ui;j;Gỵ1 ẳ xbest;j;G ỵ Fxr1;j;G xr2;j;G ị 16 17 18 End if Else ui;j;Gỵ1 ẳ xi;j;G 19 20 21 End if Endfor X i;G then If f ~ U i;Gỵ1 f ~ ~ ~ Xi;Gỵ1 ẳ U i;Gỵ1 Else ~ Xi;G Xi;Gỵ1 ẳ ~ End if End for End for Kt thúc 22 23 24 25 26 27 28 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 standard deviation (StdDev) shows that the proposed algorithm 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 Applying for the pendubot parameter estimation 4.1 Dynamic of the pendubot system The Pendubot system represents planar two degree-offreedom (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 in Figure Where, m1 and m2 are the masses of links and 2, respectively l1 and l2 are the lengths of links and 2, respectively d1 and d2 are the distances to the N S NGUYEN AND D K NGUYEN Table The Benchmark functions Functions Sphere Range ½100; 100 Griewank ½600; 600n Equation n P f1 xị ẳ xi2 n ẵ30; 30n Ackley n Q f2 xị ẳ ỵ xi2 cos pxii iẳ1s! s! iẳ1 n n P P f3 x ị ẳ 20 exp 0:2 n1 xi2 exp n1 cos2xi ị ỵ 20 ỵ exp 4000 n P iẳ1 Rastrigin iẳ1 n P f4 xị ẳ 10n ỵ xi2 10 cos2xi ị ẵ5:12; 5:12n iẳ1 _ _ ỵ Gị ẳ b1 _ Mị ỵ V; ị b2 _ Table The parameters of PSO, DE, HDE and MDE algorithms Methods PSO DE – HDE MDE Parameters Dimension, n Generations, GEN Acceptable error Population size, s Inertia weight, w Particle’s best weight, c1 Swarm’s best weight, c2 Population size, NP Mutant factor, F Crossover factor, CR Learning rate, η Population size, NP Mutant factor, F Crossover factor, CR Value 4000 1e-5 149 –0.3236 –0.1136 3.9789 18 0.6714 0.5026 0.01 50 [0.4,1] [0.7,1] Table Results obtained for the Benchmark function problems in Table Hàm Best Worst Average StdDev Time (s/run) Griewank Best Worst Average StdDev Time (s/run) Ackley Best Worst Average StdDev Time (s/run) Rastrigin Best Worst Average StdDev Time (s/run) Sphere PSO 2.30e-5 0.0028 8.06e-4 8.35e-4 0.1910 0.0081 0.0811 0.0461 0.0260 0.1961 0.0026 0.1291 0.0273 0.0379 0.2263 1.41e-5 0.9950 0.1159 0.3116 0.1455 DE 3.09e-6 9.60e-6 6.59e-6 2.46e-6 0.0763 2.41e-6 0.0020 5.48e-4 6.44e-4 0.1453 3.54e-5 0.0012 5.49e-4 5.00e-4 0.1791 8.64e-6 0.5830 0.0609 0.1835 0.1020 HDE 2.21e-8 7.58e-6 2.85e-6 2.44e-6 0.2091 7.76e-6 2.91e-4 7.19e-5 1.06e-4 0.5689 5.12e-7 4.68e-4 1.31e-4 1.32e-4 0.5263 1.50e-6 2.89e-4 4.99e-5 8.79e-5 0.2884 MDE 4.38e-7 5.56e-6 3.03e-6 1.82e-6 0.0837 1.55e-7 7.48e-6 2.56e-6 2.83e-6 0.3522 2.15e-6 9.97e-6 6.58e-6 2.93e-6 0.1797 2.22e-6 8.88e-6 4.87e-6 2.28e-6 0.1182 (1) where, MðθÞ ¼ _ ¼ Vðθ; θÞ GðθÞ ¼ M11 M12 ¼ M21 M22 V11 V12 V21 G11 G21 ẳ P1 ỵ P2 ỵ 2P3 cos2 ị P2 ỵ P3 cos2 ị " P2 P2 þ P3 cosðθ2 Þ # _ _ _ P3 θ2 sin2 ị P3 ỵ ị sin2 ị ẳ V22 P3 θ_ sinðθ2 Þ P4 cosðθ1 ị ỵ P5 cos1 ỵ ị P5 cos1 ỵ ị P1 ẳ m1 d21 ỵ m2 l12 þ I1 ; P2 ¼ m2 d22 þ I2 ; P3 ¼ m2 l1 d2 ; P4 ¼ m1 d1 þ m2 l1 ; P5 ¼ m2 d2 (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 control In this section, the proposed adaptive DE algorithm is used for identifying the seven parameters ẵw1 ; ::::; w7 T ẳ ½P1 ; P2 ; P3 ; P4 ; P5 ; b1 ; b2 T in (1) using the energy theorem, which can be written as _ ẳ Et2 ị Eðt1 Þ t1 uT θdt t2 (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 h : : i t2 t1 ðτ b1 _ ị_ ỵ b2 ị dt ¼ Eðt2 Þ Eðt1 Þ (4) We denote, centers of mass of links and 2, respectively I1 and I2 are the moments of inertia of links and 2, respectively b1 and b2 are the friction of links and 2, respectively θ1 is the angle that link makes with the horizontal, and θ2 is the angle that link makes with link τ is the torque supplied to the link 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 t2 h : : i ε ¼ b1 _ ị_ ỵ b2 ị dt ẵEt2 ị Et1 ị t1 (5) Therefore, the fitness function can be defined as Jẳ N 1X nị N nẳ1 (6) The optimization goal is to minimize the fitness function J When the fitness function converges to zero, we will achieve the best estimation values of the Pendubot INTERNATIONAL JOURNAL OF MODELLING AND SIMULATION Griewank Sphere Conv ergenc e rate 105 10 Magenta dotted line: PSO Cyan dashed line: DE Red solid line: HDE Blue dash-dot line: MDE 100 10-5 10-5 Conv ergenc e rate 1000 2000 Ackley 3000 4000 100 100 10-5 10-5 1000 4000 3000 2000 Generations 1000 1000 2000 Rastrigin 3000 4000 2000 3000 Generations 4000 Figure Convergence rate of MDE, HDE, DE, and PSO in optimization over 10 runs Table Pseudocode of MDE algorithm in Pendubot’s parameter estimation y Begin Generate the initial population ~ xi;G ¼ w1;i;G ; ::::; w7;i;G ¼ P1;i;G ; P2;i;G ; P3;i;G ; P4;i;G ; P5;i;G ; b1;i;G ; b2;i;G N P ε ðnÞof each in the population Evaluate the fitness J ¼ N1 n¼1 For G = to GEN For i = to NP jrand = randint(1,D) F = rand[0:4; 1:0], CR = rand[0:7; 1:0] For j = to D 10 If rand[0,1] < CR or j = = jrand then 11 If rand[0,1] > threshold then 12 Select randomly r1 ịr2 ịr3 ịi 13 ui;j;Gỵ1 ẳ xr1;j;G þ Fðxr2;j;G xr3;j;G Þ 14 Else 15 Select randomly r1 Þr2 Þbest Þi; "i f1; :::; NPg 16 ui;j;Gỵ1 ẳ xbest;j;G ỵ Fxr1;j;G xr2;j;G ị g x τ Figure Pendubot system parameters Table 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 experimental system are shown in Figure The hardware includes the STM32F407 board which provides 17 18 19 20 21 22 End if Else ui;j;Gỵ1 ẳ xi;j;G End if End for If J ~ ui;Gỵ1 J ~ xi;G then ~ U i;Gỵ1 23 Xi;Gỵ1 ẳ ~ 24 Else ~ Xi;G 25 Xi;Gỵ1 ẳ ~ 26 End if 27 End for 28 End for 29 Kết thúc PWM signals u(t) to control the DC motor through the H-Bridge board The two angle encoder sensors N S NGUYEN AND D K NGUYEN y Laptop RS232 24VDC Matlab/Embedded Coder MCU STM32F407 PWM Driver τ Dir θ1 x DC motor Encoder Encoder (a) (b) Figure (a) Schematic diagram of experimental setup (b) Photograph of the experimental 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 performance 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 shows the torque input applied to the Pendubot system and the responding position output collected The torque is INTERNATIONAL JOURNAL OF MODELLING AND SIMULATION Figure Dataset for Pendubot parameters estimation a pulse of the s period with 5% pulse width; each pulse is several random numbers from to values and 0.01 s interval Where dataset from (0–900)[sample] is used for estimating the Pendubot parameters; Dataset from (901–1800)[sample] is used for validating the Pendubot parameters Assuming that the dataset has been acquired, the second step is to select a model structure The proposed MDE algorithm is used to identify the Pendubot parameters Table shows the MDE parameters 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 gives the performance results of Table The MDE parameters in identification Method MDE Parameters A number of generations Population size, NP VALUE 4000 30 Table The performance of the MDE in identification MSE Training Method MDE Best 2.37e-3 Worst 2.39e-3 Validation Average 2.37e-3 Average 3.03e-3 Table The resulted parameters of Pendubot system P1 P2 P3 P4 P5 b1 b2 0.002207 0.000400 0.000101 0.022447 0.003326 0.006592 0.00009 the MDE algorithms in identifying the Pendubot dynamic parameters The results from Table show that the parametric values of the Pendubot system are precisely identified Table tabulates the resulted parameter values of the Pendubot system These parameters will be used to design the controller in the experimental system 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–output 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 8 N S NGUYEN AND D K NGUYEN Acknowledgments This research is funded by Industrial University of Ho Chi Minh City, Viet Nam under grant number 04/HĐ-ĐHCN in 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 Minh City, Viet Nam [Under grant number 04/HĐ-ĐHCN 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 ViceDean of the Faculty of Electronics Technology, Industrial University of Ho Chi Minh City, Viet Nam His current research interests include intelligent control, robotics, identification of nonlinear systems, and the internet of things References [1] Spong MW, Block DJ The pendubot: a mechatronic system for control research and education in: Decision and Control, 1995, Proceedings of the 34th IEEE Conference on; New Orleans, LA, USA; Vol 1, IEEE; 1995, pp 555–556 [2] Ma XQ, Su CY A new fuzzy approach for swing up control of pendubot in: American Control Conference, 2002 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Parameter estimation of Pendubot model using modified differential evolution algorithm Ngoc Son... 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 parameters, such as a number of layers, in addition... rate of 2.53GHz and 2.00GB of RAM Each algorithm runs 10 times Table gives parameters used in optimization, where the parameters of DE and PSO algorithm based on SwarmOps in [34], an HDE algorithm