A Fuzzy Logic Controller tuned with Imperialist competitive algorithm for 2 DOF robot trajectory control with help Scaling factor

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1 94 RESEARCH JOURNAL OF FISHERIES AND HYDROBIOLOGY AENSI Publisher All rights reserved IN: Open Access Journal A Fuzzy Logic Controller tuned with Imperialist competitive algorithm for DOF robot trajectory control with help Scaling factor 1 Morteza Bagheri and Hasan Fatehi Marj 1 Department of electrical and Control, Kerman Science and Research Branch, Islamic Azad University, sirjan, iran Assistant professor of control engineering Department of electrical engineering Faculty of engineering Vali-e-Asr University of Rafsanjan. Address For Correspondence: Hasan Fatehi Marj, Assistant professor of control engineering Department of electrical engineering Faculty of engineering Vali-e-Asr University of Rafsanjan. h.fatehi@vru.ac.ir ABSTRACT Using from optimization technologies in a DOF planar robot controlling is a new idea. Until now, different techniques are used for controlling and optimization methods, is designating decision variable values for minimizing or maximizing of bests function and this is insupportably when the number of memberships functions increased or system of dynamic be slow. In this Paper by Imperialist Competitive Algorithm and Imperialist and Fuzzy logic, functions of optimization output membership for controlling a DOF planar robot are designated. In this way, we used of Imperialist Competitive algorithm for attaining to best parameters in Fuzzy logic Sugeno method. bests function in this aspect is like a set of different bests for controlling robot. The main best in this article is using from Imperialist Competitive algorithm as a method for automating planning and reaching to the functions regulation. Even though scientific observations in this algorithm and its regulation by scaling factor with Fuzzy control is used for a robot trajectory control. Keyword: Fuzzy logic control, degrees of freedom, Robot trajectory control, ICA Received: November 014 Accepted: 8 March 015 Published: 1 April 015 INTRODUCTION The system of robot arm is always used in industry. And it has many usages, for example: using robots in automatic production lines, leading and discharging load in transportation industry like airplane and ships, automatic painting, displacing of radioactive materials, exploring the depth of sea, space trips and military using from this aspect the system of robot arm is completely complex and nonlinear. This system doesnt have any fixed structure because of many reasons like: changing load or friction between junctions. And is always unsure. This uncertainly is always seen in its model. This uncertainty in the system of robot arm cause unstable operation and makes its controlling hard. Because of this matter different controllers are planned. Among controlling methods, Fuzzy control is efficient and practicable. Logic Fuzzy that is based on Fuzzy sets, invented by Lotfizadeh professor for investigating uncertainty and carelessly in knowledge (Zadeh, L.A., 1975). A Fuzzy system use of Fuzzy sets for describing chart (mapping) relationship between output and input Fuzzy of sets permit that output interpolate from between many rules. Evolutionary algorithms are function optimizations / Population algorithm. And can used in a wide range that has function. But decision variables that optimize function, are unknown. And contain engineering optimizations, timing planning, Evolutionary hardware and even though art. The title of this paper (article) is brought in the Section based on dynamic model. In the 3 Section main concepts and definitions of Fuzzy logic investigated in the control field. In the 4 Section, we talk a bout Imperialist Competitive algorithm. And experimental results in Section 5 and conclusion in Section 6. - Dynamic model of the planar robot: RESEARCH JOURNAL OF FISHERIES AND HYDROBIOLOGY, 10(9) May 015, Pages: Morteza Bagheri and Hasan Fatehi Marj, 015

2 95 The dynamic equations of a robot arm with n degree(s) of freedom can be stated using the Lagrangian method or Newton- Oiler method in the form of the degree two nonlinear equation (Zafer Bingül, Og uzhan Karahan, 011): M ( q) q C ( q, q )q G ( q) W ( q, q, q ) In the above equation, W ( q, q, q ) is a known function of robot dynamics and is a vector including unknown parameters of robot dynamics. q [q 1,...,q ] T n is the position vector of the robot joints and q [q 1,...,q ] T n and q [q 1,...,q ] T n are velocity and acceleration vectors, respectively. M (q) is the matrix of inertial n*n and C ( q, q ) is the matrix of n*n of the coriolis and centrifugal forces and G (q) is the vector of gravity. The M (q) matrix is an asymmetric and definite positive matrix. The two joint robot have been indicated in the Figure 1. As seen in the Figure, q 1 is the position of the first joint with respect to the horizontal plane and q is the position of the second joint with respect to the first joint. Fig. 1: Model of a two axis robot. The dynamic equation of the two joint robot which has been brought in many references and articles is expressed as follows: 1 m l ( q 1 q ) m l1l c ( q 1 q ) (m1 m )l1 q 1 (1) m l l s q m l l s q q m l gc (m m ) l gc m l l c q m l l s q m l gc m l ( q q ) The above equations are in the general form according the equation (1) which the M (q), C(q, q. ) and G (q) matrixes for it are as follows: ( m1 m) l1 ml ml1l c ml ml1l c M ( q) ml ml1l c ml ml1l1q s ml1l1q s C ( q, q ) ml1l1q s 0 ( m1 m) gl1c 1 mgl c1 Gq ( ) mgl c 1 c cos( q ), c cos( q ), s sin( q ), s sin( q ),c cos( q q ) and In the above equations m 1 and m are the mass of the arm and l 1 and l are the length of the arm and g is the gravity of the earth. Also, if m 3 is considered as the revulsive load on the second arm and focused on the end point of the second arm, then the system model will be introduced as the equation (1) with the following matrixes (Ahmadi, Mohammad Ali, 011): M M ( q) M C C ( q, q ) C M 11 1 M C 0

3 96 G1 Gq ( ) G That in it, M ( m m m ) l ( m m ) l ( m m ) l l c M ( m m ) l ( m m ) l l c M M 1 1 M ( m m ) l 3 C ( m m ) l l q s C ( m m ) l l q s C ( m m ) l l q s G ( m m m ) gl c ( m m ) gl c ( 3) c 1 G m m gl 3. Fuzzy control: Lotfizadeh introduce Fuzzy sets for overcoming to the certainty in knowledge in That possible using of logic forms, by the way that true and false shows a degree of fact. He extend Fuzzy sets for modeling uncertainty that seen in the planning 1 type Fuzzy sets to the known Fuzzy sets ( type) 10year (Zadeh, L.A., 1965; Mendel, J.M. and R.I.B. John, 00). Hater. In last decade, logic Fuzzy used in different aspects line regression, systems modeling, control, for classifying patterns. Among them control field is one of the best (more related). Fig. : The structure of a Fuzzy Logic Controller. FLC contains 3 main components: Fuzzifier, induction motor and Defuzzification. Blocking graph (FLC) is shown in the Figure. (Lughofer, E., 011) blocks define for output and input variables in the Fuzzy part. In most cases, output error means the difference between output of process and reference signal. And its changes contain systems` inputs. The center of knowledge is its heart. And has many rules that controller is able to reach to ideal bests by using them. Most of the time these rules are if. And act as a chart from input Fuzzy variables to the output Fuzzy variables. Induction motor is as a brain of a Fuzzy controller that able to copy human decision making based on Fuzzy concepts. This part is made by using data base rules, Fuzzy output and Fuzzy input. In Defuzzification, we obtain the real amount at output. There are different methods for Defuzzification, and the center of graceity method is the most usable one. By using Fuzzy logic from one Fuzzy system that has inputs and output use for controlling every arm. Systems inputs means the real difference between the angle and amount and real amount and derivation error, too. Output is based on input momentum to the joint. For facilitating Fuzzy system, we used Sugeno Fuzzy system (zero level) this kind of system is so similar to the moment Fuzzy system and just instead of considering membership functions in output, we use individual values. Membership functions that consider for input variables to the both Fuzzy controller are from Gaussian from in the range [-1.1] have been that normalized.

4 97 Fig. 3: Membership function for error(joint-1). Fig. 4: Membership function for error derivation(joint-1). Fig. 5: fuzzy control surfaces(joint-1).

5 98 Fig. 6: Membership function for error (Joint-). Fig. 7: Membership function for derivation(joint-). Fig. 8: Fuzzy control surfaces(joint-). Membership functions on the output as well as Singular values Uniformly In the interval [-1.1] Figure 6 is intended

6 99 Fig. 9: Membership functions for output variables(joint-1 and Joint-). At first, Fuzzy system inputs make Fuzzy by the membership functions that considered for them and then based on rule sets that considered for Fuzzy system, approximative reasoning done on the inputs and produce considered output Fuzzy system. Because controllers contain inputs and 5 membership functions then a rule base considered for them that has 5 rules. Rule base that we consider is a standard rule base and in many papers consider as a controller and is shown in table 1 (Hoffmann, F. and G. Pfister, 013). Table 1: Fuzzy rule base. e NB PB e NB NB NB NB PB PB PB PB A general schematic of controlling system in matlab software simulink is seen below. Fig. 10: Simulink model of the FLC for robot control. using from scale factor is an important point in this schematic, that cause e,e. variables put in [-1,1] interval and is considered for inputs to the Fuzzy system. Another important point is using from MATLAB Fcn that benefits from mfile (dyn.m) and robot equation is used in it.

7 100 Table : Robot parameters in simulation. Parameter Values 0.8m 0.4m 0.1kg 0.1kg 9.8 We regulate (set) simulation time on 5 seconds and step time on You see the simulation result below. Fig. 11: Tracing the first arm reference. As you see in the above Figures, controller cant control arm system and follows specified reference. This problem is arised because Fuzzy controller parameters as membership functions and scale factors dont regulate good. And regulating and optimizing these parameters is time consuming, insupportably and sometimes impossible and for solving this problem, we can use Evolutionary algorithms as Imperialist Competitive (Bonarini, A., 01). 4- Imperialist Competitive algorithm(ica): Figure 13 shows Imperialist Competitive algorithm (Atashpaz-Gargari, C., 007). As other Evolutionary algorithms, this algorithm begin by many random population, that every of them called a country a number the best population elements choose as imperialist. Other population considered as colony. colonial people based on their power, take, these colonies toward themselves with special process. General power of every emperor depend on both part: imperialist country (a main center) and it`s colonies. In math, this dependency is modeled by defining emperor`s power and plus colonies power. Atashpaz and his colleagues imperialist Competitive begin as the first emperor formed. Every emperor can`t act successfully and increase his power, will delete from this score (Jalalizadeh, 011). So permanence of an empire depends an it`s power to absorb emperors colonies from another group. As a result in imperialist Competitive process, the power of bigger empire increased and waked empire will deleted. Emperors should improve their colonies to increase their power (Lian, K. Chaoyong Zhang, 011). As time passed, colonies close to emperors and a kind of convergence formed between them. It`s final boundary is when that there be a unit emperor in world. And it`s colonies that are close to his country.

8 Experimental results: Optimization results for Fuzzy controller of robot arm in this Section, we pay attention to the controller optimization by Imperialist Competitive algorithms by using scale factors and give results. In this paper, we use from Imperialist Competitive algorithm for optimizing centers and extending membership functions on inputs and Fuzzy rules and scale factors. For optimizing controller we can use from different criterions. In this paper we use of error square average criterion as best function. As told each Fuzzy system has 5 membership functions on each input aspect (finally 0 free parameters for optimization) and 5 on the output (5 parameters) then 5 free parameters are assigned for each Fuzzy system. More than this we use 3 scale factors and free Fuzzy systems because robot arm is from its DOF, (kind) we use of free Fuzzy system. There are 56 optimization parameters. And as a result, countries contain 56 aspects (Imperialist Competitive algorithm) To compare algorithm, we assign the number and repeated process = 60 and the number of population = 15 Fig. 1: Tracing the Second arm reference. As you see in the Figure, Imperialist Competitive algorithm contain better final value. And another important point is that Imperialist Competitive algorithm get slowly at first and after 15 repeatation, has better process. Now, in bellow Figure bring, obtained results from reference tracing of both robot joint by controller: controller FLC and FIC- ICA. Controller has acceptable answers from reference tracing point of view. At last for comparing exact operation of controls, we used from ITAE(Integral Time Absolute Error) and IAE(Integral Absolute Error) criterions what are for tracing error and report them in the below tables. Reported values in the table shows that optimized controller by Imperialist competition algorithm (ITAE, IAE) are better than other controllers. Table 3: Comparing controlled criterion for the first joint. Joint 1 ITAE IAE Table 4: Comparing controlled criterion for the second joint. Joint ITAE IAE FLC FLC ICA-FLC ICA-FLC

9 10 Fig. 13: The flowchart of the proposed algorithm. Fig. 14: Graph of the target function algorithms.

10 103 Fig. 15: Comparing controller reference tracing (first joint). Fig. 16: Comparing controller reference (two joint). 6) Conclusion: In this research we study about fuzzy systems and in other Section discuss about systems and talk about its advantage and its disadvantage. Fuzzy systems lack of one matter that is need to an expert for planning fuzzy systems for special operation. As told, regulation fuzzy controllers as membership functions for controlling. Special system is time consuming. For solving this problem, there are many different ways. One of this ways (methods) is using of complementary algorithms for regulating and optimizing fuzzy controllers and in the 4 th part we discuss about Evolutionary algorithms and Imperialist Competitive algorithm.. Finally, equations of robot arm with DOF show in MATLAB software condition and we study about regulating and optimizing fuzzy system for robot arm system by using Imperialist Competitive algorithm. The results were acceptable and optimized fuzzy controller do tracing of reference joints well. And show that optimized control by Imperialist Competitive algorithm has better result.

11 104 REFERENCES Ahmadi, Mohammad Ali, 011. Prediction of Asphaltene Precipitation using Artificial Neural network Optimized by Imperialist Competitive Algorithm journal of petroleum Exploration and Production technologies, 1(1). Atashpaz-Gargari, C., Esmaeil and Lucas, 007. Imperialist competitive algorithm: an algorithm for optimization inspired by imperialistic Competitive, in Evolutionary Computation, IEEE Congress on, pp CEC 007. Bonarini, A., 01. Evolutionary learning of fuzzy rules: Competitive and cooperation, Fuzzy Model. Paradig. Pract., pp Hoffmann, F. and G. Pfister, 013. Evolutionary design of a fuzzy knowledge base for a mobile robot, Int. J. Approx. Reason., 17(4): Jalalizadeh, Saeed, Saeed Behzadpoor and Mohmmad Hashemi, 011. PID Design for AVR system by PSO and Imperialist Campetitive Algorithms first international Conference on Computer Science, Engineering and information Technology (CCSEIT-011). Lian, K. Chaoyong Zhang, Liang Gao and Xinyu Li, 011. Integrated process Planning and Scheduling using an imperialist Competitive Algorithm, International journal of production research. Lughofer, E., 011. Evolving Fuzzy Systems - Methodologies, Advanced Concepts and Applications. Springer. Mendel, J.M. and R.I.B. John, 00. Type- fuzzy sets made simple, Trans. Fuz Sys., 10(): Zadeh, L.A., Fuzzy sets, Inf. Control, 8(3): Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning I, Inf. Sci. (Ny)., 8(3): Zafer Bingül, Og uzhan Karahan, 011. A Fuzzy Logic Controller tuned with PSO for DOF robot trajectory control Expert Systems with Applications, 38: