Application of hedge algebras for controlling mechanisms of relative manipulation - Phan Bui Khoi

Vietnam Journal of Science and Technology 55 (5) (2017) 572-586 DOI: 10.15625/2525-2518/55/5/8841 572 APPLICATION OF HEDGE ALGEBRAS FOR CONTROLLING MECHANISMS OF RELATIVE MANIPULATION Phan Bui Khoi1, Nguyen Van Toan2 1Hanoi University of Science and Technology, No.1 DaiCoViet Street, Hai Ba Trung, Ha Noi, Viet Nam 2Korea Institute of Science and Technology, 5 Hwarang-ro 14-gil, Seongbuk-gu, Seoul, Republic of Korea Email: khoi.phanbui@hust.edu.vn1, toan70411hd91@gmail.com2 Recieved: 7 November 2016; Accepted for publication: 18 July 2017 ABSTRACT This paper presents a method for controlling a mechanism of relative manipulation (MRM robot) that based on an algebraic approach to linguistic hedges in the fuzzy logic. The proposed model of MRM robot is introduced as a two-component mechanism of which two serial robots co-operate with each other to realize technological manipulations. MRM robot has complex structure; therefore, the robot system's mathematical equations describing dynamical behaviors are voluminous. Furthermore, the external components affect MRM robot's dynamics that are difficult to determine adequately and exactly. Applying the well-known methods (based on dynamical equations) such as PD/PID, computed torque algorithm etc. for robot control is difficult, especially with MRM robot. By dint of the human-like inference mechanism, designing controller via the fuzzy logic can overcome the mentioned drawbacks. However, the linguistic variables in the fuzzy logic are not represented by any physical values; and hence, the comparison among linguistic variables is unable. Moreover, the composition of fuzzy relations and the defuzzification use approximation functions which trigger error in data process. Hedge Algebras (HA) gives favorable conditions to restrict the fuzzy logic's drawbacks because the linguistic labels in Hedge Algebras are represented by the semantic values; and, the composition of fuzzy relations and the defuzzification are processed by simple interpolation and mapping functions. The obtained results from HA controller are compared to the obtained results from two methods which are fuzzy controller and computed torque controller. Keywords: mechanism of relative manipulation (MRM robot), hedge algebras. 1. INTRODUCTION The mechanism of relative manipulation (MRM) robot includes two serial/parallel robots which has complex structure [1, 2]; thus, the robot's dymical equations are complicated [3, 4] and [5]. It causes the difficulty when applying classical algorithms for robot control. In [6], the fuzzy logic is applied to control MRM robot that overcomes the mentioned problem. Besides, Application of Hedge Algebras for controlling mechanisms of relative manipulation 573 the obtained results from fuzzy controller are compared with the obtained results from computed torque controller to evaluate the reliability. The comparison shows the benefits when applying the fuzzy logic to control MRM robot. However, the application of fuzzy logic still exists several limitations, namely the fuzzy logic's linguistic variables cannot be compared with each other, the use of fuzziness measure and composition of fuzzy relations is still complicated; the defuzzification must be processed by approximation functions; and hence, it leads to the error in implementation process. With specific features, Hedge Algebras (HA) can overcome aboved drawbacks. Firstly, the linguistic labels in HA are represented by the physical values (namely the semantic values of each linguistic label is calculated by the semantically quantifying mapping), so they can be compared with each other. Moreover, the input and output do not need the defuzzification because the semantically quantifying mapping of hedge algebras maps input/output's physical values to the value domain [0, 1] to process; and therefore, to calculate the physical values of controller's outputs, we just need to map their semantic value (belongs to [0, 1]) to their physical domain. The calculation of output's semantic value is conducted by an interpolation function. The product operator and the average operator are usually used to transform m-demension SAM table (the table of the semantic relationship between inputs and outputs) to 2-dimension SAM table, and then building the interpolation function. By doing this, the composition of fuzzy relations, the fuzziness measure and the defuzzification are avoided. As a results, the complexity is reduced and the accuracy is enhanced. Several papers applied HA in control and obtained reasonable results, as [7 - 16]. However, applying HA to robot control is relatively new. This paper proposes a method for controlling a mechanism of relative manipulation that based on an algebraic approach to linguistic hedges in fuzzy logic. To evaluate the proposed method, the obtained results from HA controller are compared to the obtained results from two methods which are presented in [6]. 2. MRM ROBOT'S KINEMATIC AND DYNAMICAL MODEL Figure 1 presents 5-DOF MRM robot model which is used for welding (or lazer machining), including 3-DOF tool-robot (T), 2-DOF jig-robot (B), and technological object (W), (as presented in [6]). The tool-robot consists of unmovable base L10, movable links are denoted by L11, L12 and L13. The jig-robot consists of unmovable base L20 and movable links L21, L22. 2.1 MRM robot's manipulation motion The motion of the links: Links L11 and L12 rotate around axes z10 and z11 respectively which are performed by corresponding angles Ө11 and Ө12; link L13 translates along axis z12 which is performed by d13. Links L21 and L22 rotate around axes z20 and z21 respectively which are performed by corresponding angles Ө21 and Ө22. The adequately generalized coordinates of MRM robot is presented by: [ ] [ ]T T1 5 11 12 13 21 22q ,..,q , ,d , ,= = θ θ θ θq (2.1) MRM robot's kinematic parameters are taken as given in [6] and presented in Table 1. Parameters d11, a11, a12 are kinematic dimensions of the tool-robot, Ө20, d20, a20, α20, are Denavit-Hartenberg parameters describing the position and the orientation of the jig-robot's base frame in the general base frame. Phan Bui Khoi, Nguyen Van Toan 574 Table 1. Parameters describe the robot structure. d11 a11 a12 Ө20 d20 a20 α20 d22 0.82(m) 0.33(m) 0.33(m) 0 0.3(m) 0.45(m) -900 0.1(m) During the implemention process of a technological manipulation, link L13 brings the tool performing the relative motion to the workpiece (set-up on movable jig-table L22). The welding is conducted by the tool's manipulation motions which work on the workpiece. These motions are represented through robot's manipulation coordinates in the moving space. The manipulation coordinates are determined by the position and the orientation of Catersian coordinates x13y13z13(on link L13) in Cartesian coordinates x22y22z22 (on movable jig-table L22). The robot's manipuation coordinates are presented by: [ ] TT 22 22 22 22 22 221 2 6 p13 p13 p13 p13 p13 p13p ,p ,..., p x , y , z , , , = = α β η p (2.2) The bottom right indexes of elements are represented by index of L13, the upper left indexes are represented by index of L22. Figure 1. MRM robot – workpiece. θ1 θ1 d13 d11 z10 z11 z12, z13 z21, z22 z20 θ22 θ21 L1 L11,a11 L12,a12 L13 L20 L21 L22 T B W x13 x22 Application of Hedge Algebras for controlling mechanisms of relative manipulation 575 Depending on the technological object and the kind of technological manipulation, the elements of the vector p are determined as the function of time or numeric data. As mentioned, to compare the proposed method with the computed torque controller and the fuzzy controller (as presented in [6]), we consider the technological manipulation that MRM robot welded tube part to machine part. Figure 2. Welding path is realized by MRM robot. The method of kinematic investigation in [6] is used for this paper; and, the welding process is shown in Figure 2. Machine part 2 is on jig-robot’s Table 1 which is welded with tube 3 following butt-weld 4, with parameters: - The path of the butt-weld 4 is intersection of tube surface with machine part’s surface 5, which is a parallel plane with y22 axis and slope an angle γ to x22 axis. - The tube axis z is perpendicular to the surface 5. - Center O of welding path’s circle have coordinates (xo, yo, zo) in the frame x22y22z22, radius of the tube is r. - The axis of welding tool (welding gun) z13 is coplanar and sloping an angle λ with tube axis. - The velocity vh of welding tool’s head along the welding path which is given based on the welding technics. Table 2 shows the parameters appearing in this application which are the same in [6] exept for vh (in [6], vh = 0.2 m/s). Table 2. Parameters describe the machining object. γ xo yo zo r λ vh 300 0.1(m) 0.1(m) 0.1(m) 0.07(m) 450 0.0275(m/s) z22 z13 x22 z r γ λ O 3 1 2 4 5 1. Jig-robot’s table 2. Machine part 3. Tube 4. Welding path 5. Machine part’s surface Phan Bui Khoi, Nguyen Van Toan 576 By virtue of technological object, the technical requirements for welding manipulation as aboved and parameters in Table 2; the relative position of welding gun need to be determined, namely the elements of p (2.2). The kinematics and plan motion for robot is investigated by applying the same method in [2] and [6]. The obtained trajectory will be the input data (desired trajectory) for the controller. These data are saved to files Position.txt, Velocity.txt, Acceleration.txt which contain robot's trajectory, including corresponding position, velocity and acceleration. 2.2. MRM robot's dynamical equations As given in [6], the robot's dynamical equations are writen in the matrix form: ( ) ( , ) ( )+ + + =M q q C q q G q Q Uɺɺ ɺ (2.3) where ( )5 T TTi i Ti Ri ci Ri i=1 5x5 (q) J m J +J Θ J =     ∑M (2.4) ( ) [ ] ( ) ( ) 5 T kj lj kl 1 2 5 j k l l k jk,l 1 m m1 m , c ,c ,..,c , c k,l; j q q , k,l; j 2 q q q =  ∂ ∂ ∂ = = = + −  ∂ ∂ ∂  ∑C q qɺ ɺ ɺ (2.5) ( ) [ ]T1 2 5 j j g ,g ,..,g , g q ∂Π = = ∂ G q (2.6) The values mi, JTi and JRi are the mass, the translational Jacobian, and the rotational Jacobian respectively of link I; Θ ci is the inertia tensor of the i th link about center of mass Ci which is expressed in the frame with origin at the ith link’s center of mass; i = 1,..,5; Π is the potential energy of the system; , ,q q qɺ ɺɺ are vectors of the joint positions (2.1), velocities, accelerations, respectively; (k, l; j) is Christoffel notations; mkl (k,l = 1,.., 5) are elements of the mass matrix M(q). The vector U is the expression of the calculated force/torque which matchs the programed motion of the robot [17, 18]. [ ]T1 2 5U , U ,.., U=U (2.7) The vector Q is the expression of the generalized force of friction, the disturbance as well as other non-conservative forces applying on the robot. [ ]T1 2 5Q ,Q ,..,Q=Q (2.8) In general, it is hard to build up exactly the robot’s dynamical equations because of the difficulty of their determiniation. To solve these drawbacks, a method based on the fuzzy logic which is proposed in [6] for controlling MRM robot. However, the fuzzy method still exists some shortcomings. This paper proposes a method based on HA to overcome above drawbacks and restrict the shortcoming of the fuzzy controller. We assume that Q can be determined exactly to have comparable base between the methods. The vector Q is the combination of friction and disturbance forces. It depends on the generalized position and the generalized velocity: Application of Hedge Algebras for controlling mechanisms of relative manipulation 577 T fr 1 5[q ,...,q ]= µQ ɺ ɺ (2.9) T dis 1 2 3 4 4 5[sin(q )+1,cos(q )+1,sin(q ),sin(q )cos(q ),sin(2q )]= δQ (2.10) then fr dis= +Q Q Q (2.11) with µ = 0.003 and δ = 0.5. Based on this condition, the computed torque method is conducted to obtain the clear control results. Q is not used for the fuzzy method and the HA method. The obtained results from three methods will be compared with each other to evaluate the HA controller. 3. HEDGE ALGEBRAS CONTROLLER The technical requirements are described in the previous section which are the position, orientation and the velocity of the welding gun in welding technics. To control the welding gun following the desired welding path in jig-table's Cartesian coordinates, MRM robot's joints are controlled following their trajectory which are received from the inverse kinematics. Throughout this paper, qi(t) and q (t)iɺ present actuated joint’s position and velocity of MRM robot,
(t), (t)diqɺ present desired position and velocity. The controller is designed so that qi(t) and (t)qiɺ follow
(t) and (t)diqɺ . By doing this, the welding gun follows the desired welding path in the Cartesian coordinate system of the jig- robot’s table. In other words, the control purpose is to adjust the driving torque at actuated joints so that the position error e(t) and the velocity error  (t) are small as desire. The position error and the velocity error are computed by: d i i i d i i i e (t) q (t) q (t) e (t) q (t) q (t)  = −  = −ɺ ɺ ɺ (3.1) In [6], the fuzzy controller is proposed for controlling MRM robot which aims to reduce the difficulty of the dynamical identification. In addition, the computed torque controller is used to compare with the fuzzy controller. This paper uses these methods to get the results which are used to compared with the obtained results from proposed controller. The control rules and the parameters of fuzzy controller and computed torque controller are similarly chosen in [6]. 3.1. Background of hedge algebras An algebra structure called hedge algebras - HA which is defined AX = (X, G, H, ≤) therein X = Dom(X) is a term-set of a linguistic variable X, G = {0, c-, W, c+, 1} is the set of the primary terms of linguistic variables (W is neutral element), H = {h-, h+} is a set of unary operations representing linguistic hedges and “≤” is a partially ordering relation in X.[12], [13]. If X and H are linearly ordered sets then AX is the linear hedge algebras. By dint of the action-effect of the hedges h ∈H on element x ∈ X , we observe H(x) is a set of all elements hx ∈ X and 1...nhx h h x= with 1,...,nh h ∈ H ; and therefore, H(x) is a linearly ordered set [14]. Phan Bui Khoi, Nguyen Van Toan 578 If X is created from G and hedges therein G is a linearly ordered set, then X is also a linearly ordered set. Moreover, with u and v belong to X, if and, (v) v (u)u v u< ∉ ∉H H , then (u) (v)<H H . In addition, if 1...nu h h x= and 1... (m n)mv h h x= < , then u is more special than v. Elements in H are comparable; and, if h k≠ and hx kx= then x is a fixed point [13], [14], [15]. Fuzziness measure: a given fuzziness measure of element τ called fm(τ), τ ∈ X that the values of fm(τ) always belong to [0, 1] [16]. (1) fm(τ) = 0, if τ is clear. (3.2) (2) If h is a hedge and τ is a fuzzy value then hτ is more featured than τ, so we have fm(hτ) <fm(τ) and fm(hτ) = µ(h)fm(τ), with ∀τ ∈ X. (3.3) (3) If c+, c– are two primary terms in X, then fm(c+) + fm(c–) = 1. (3.4) Semantically quantifying mapping: By fm is a fuzziness measure of X, semantically quantifying mapping υ: X → [0,1] is determined [12]: (1) ʋ(W) = θ = fm(c-), ʋ(c-) = θ – αfm(c-) = βfm(c-), ʋ(c+) =θ + αfm(c+) (3.5) (2) ʋ(hjx) = ʋ(x)+ Sign(hjx). ∑ fm hx   ω hxfm hx (3.6) where ω(hx) = [1 + Sign(hjx)Sign(hphjx)(β-α)], and j ∈ {−q ≤ j ≤ p&j ≠ 0} = [−q...p]. ∑ μ h = α and ∑ μ h = β, with α, β > 0 and α + β = 1. W is neutral element. In the fuzzy logic, the composition of fuzzy relations is demonstrated by m dimensional FAM table (the table of the fuzzy rules among the inputs and outputs), we will use the semantically quantifying mapping to convert FAM table to semantic table SAM. 3.2. Applying hedge algebras for controlling MRM robot Step 1: Determining inputs/outputs The inputs of hedge algebras controller including d d(t), (t), (t), (t)q q q qɺ ɺ which are desired trajectory, velocity and real trajectory, velocity of MRM robot. The output is adjusted torque u(t) at actuated joints so that (t), (t)q qɺ follow d d(t), (t)q qɺ , in other words, e and   are small as desire: d i i iu τ τ= − here in  is real driving torque set at the ith actuated joints to ei and !" come to 0;  is theoretical driving torque set at the ith actuated joints to ei and !" come to 0. Now, we can consider controller's inputs of e(t) and  (t), output of u(t) [ ]1 5,..., Te e=e [ ]1 5,..., Te e=de ɺ ɺ [ ]1 5,..., u Tu=u Application of Hedge Algebras for controlling mechanisms of relative manipulation 579 where , ,i i ie e uɺ are position error, velocity error and ajusted torque at the ith actuated joint of MRM robot, respectively. Approximately physical domain of inputs/outputs are later used in Table 6. Step 2: Choosing hedge parameters Owning to the number of linguistic terms in the fuzzy controller, we chose hedge parameters so that each input/output , ,i i ie e uɺ will be demostrated by five linguistic labels with five semantic values. This gives the same condition when designing two controllers for the purpose of comparing the proposed controller and the fuzzy controller. The set of the primary terms of linguistic variables and hedges are chosen such that if applying hedges to the set of the primary terms of linguistic variables, then we observe five linguistic values which present for each input/ output , ,i i ie e uɺ of each joint: G = {0, S, W, B, 1} with S= Small; B = Big. H = {H-, H+} with H- = L; H+ = V => q=1, p =1 herein L = Little; V= Very. Now, each input/output , ,i i ie e uɺ are presented by one hedge algebras which is created from the set of the primary terms of the linguistic variable G and the set of hedges H through a linearly ordering relation. However, we just use the partition k = 2 when creating linguistic values in each hedge algebras which aims to compare with the fuzzy controller. So, the linguistic terms in hedge algebras are presented by X = {VS, LS, W, LB, VB}. In the fuzzy controller, the linguistic values are the same for inputs/ outputs, but their physical domains are different. In the same way, with each hedge algebras present and, i i ie e uɺ , their linguistic values are the same; and therefore, they will have the same semantic values. However, those values just describe semantics in individual input/ output's physical domains; and hence, by mapping their semantic values to their real physical domain, we observe different values for different inputs/ outputs. The values 0 and 1 in G present elements which have the smallest semantic value and the biggest semantic value, respectively; and, they are fixed points. The values 0 and 1 will be used when processing the input/ output by merging operators to avoid the loss of the information. By virtue of G and H, the fuzziness of the primary terms and hedges are measured through (3.2), (3.3) and (3.4). fm(S) = # = 0,5 . µ(L) = µ(V) = 0,5 → α = β = 0,5 and fm(B) = 1- fm(S) = 0,5. In [6], the fuzzy terms are AL, AN, Z, DN and DL of which A, D, Z stand for negative, positive and zero, respectively; whereas N and L stand for small and big, respectively. Now, the fuzzy terms are transformed to HA-terms: AL => VS AN => LS Z => W DN => LB DL => VB Step 3: The semantic representation of linguistic labels The usual fuzzy rule base is converted into a linguistic rule base to convert the ordinary linguistic labels into HA-terms. By virtue of above parameters, (3.5) and (3.6), the FAM table is transfomed to SAM table. In fuzzy controller, the composition of fuzzy relations is presented via Table 3 [6], including linguistic values which describe inputs/ outputs , ,i i ie e uɺ of controller. However, in hedge algebras, a set of rules is demonstrated by semantic Table 4 (m-SAM table), including the semantic values of the linguistic labels which describe , ,i i ie e uɺ . If inputs ,i ie eɺ Phan Bui Khoi, Nguyen Van Toan 580 belong to linguistic domains, with semantic value in [0, 1], then the controller will respond the output signal ui belong to the linguistic domain, with suitable semantic value to adjust ,i ie eɺ to 0. Table 3. FAM table (GTNN: linguistic values). GTNN AL AN Z DN DL AL DL DL DL DN Z AN DL DN DN Z AN Z DL DN Z AN AL DN DN Z AN AN AL DL Z AN AL AL AL Table 4. m-SAM table (SV: semantic values). SV υ(VS) = 0,125 υ(LS) = 0,375 υ(W) = 0,5 υ(LB) = 0,625 υ(VB) = 0,875 υ(VS) = 0,125 0,875 0,875 0,875 0,625 0,5 υ(LS) = 0,375 0,875 0,625 0,625 0,5 0,375 υ(W) = 0,5 0,875 0,625 0,5 0,375 0,125 υ(LB) = 0,625 0,625 0,5 0,375 0,375 0,125 υ(VB) = 0,875 0,5 0,375 0,125 0,125 0,125 Table 5. 2-SAM table. m-dimentional SAM table presents the semantic relationship between the input and the output; and, it represents a super surface in 3-D space. Merging operator will be used to exchange m dimensional SAM table to 2-dimensional SAM table which aims to simplify the interpolation. This paper use Product operator. Table 5 represents 2-dimensional SAM table which is a curve in 2-D space R2. The horizontal axis is the integration of the semantic values of the linguistic labels which demonstrate ei and !" and the vertical axis is the semantic value of the linguistic label which describes ui. We can interpolate polynomial easily. In addition, a set of rules is chosen to be symmetry about semantics, so there is no point which has the similar horizontal degree with other ones. No case in interpolation should be left out, if input's physical value is on the left of linguistic value AL's physical domain then its semantic value is υ(0)= 0. If input's physical value is on the right of linguistic value DL's physical domain then its semantic value is υ(1)= 1 0 1 0.0156 0.875 0.0469 0.875 0.0625 0.875 0.0781 0.625 0.1094 0.5 0.1406 0.625 0.1875 0.625 0.2344 0.5 0.3281 0.375 0.25 0.5 0.3125 0.375 0.4375 0.125 0.3906 0.375 0.5469 0.125 0.7656 0.125 1 0 Application of Hedge Algebras for controlling mechanisms of relative manipulation 581 Step 4: Determining output's physical value The controller's outputs are adjusted torques at actuated joints; however, the interpolation based on 2-dimensional SAM table which gives the semantic values of the linguistic variables. We need to determine real physical values of outputs via their semantics. To receive outputs' real physical values, we map their semantic values from [0, 1] to their individual physical domains. The simulink model for HA-controller is presented in Figure 3. Figure 3. Simulink model for HA controller. herein Position.mat and Velocity.mat are desired position and velocity which are obtained from kinematic problem and trajectory planning. Physical Domain is approximate domains of inputs and outputs, Torque.mat is approximate torque. HAs Control is controller via hedge algebras; Robot is the robot model, all are programmed in M-File which are inserted into Simulink by Matlab Function. 3. SIMULATION RESULTS The proposed model of MRM robot is described in section (2); and, its kinematic and dynamical parameters are shown in Table 7. In this paper, the mass, the center of mass and the inertia tensor are received from design software. By doing this, the proposed model of MRM robot is more accurate and closer to real robot model (because robot's links are not homogeneous and its cross sections are not unchanged in reality). Based on the technical requirements in section (2.1) and robot's kinematics and dynamics, we obtain the results in Figure 4 - Figure 11 by applying above control methods. Table 6. Physical domains of inputs/output of HA controller. Joint ei ! i ui 1 [-0.002, 0.002] (rad) [-0.06, 0.06] (rad/s) [-42, 41.73] (N.m) 2 [-0.002, 0.002] (rad) [-0.09, 0.09] (rad/s) [-47,46.08] (N.m) 3 [-0.0001, 0.0001] (m) [-0.08, 0.08] (m/s) [-54.6, 55] (N) 4 [-0.0001, 0.0001] (rad) [-0.05, 0.05] (rad/s) [-68, 68] (N.m) 5 [-0.0001, 0.0001] (rad) [-0.03, 0.03] (rad/s) [-8.0, 8.1] (N.m) Phan Bui Khoi, Nguyen Van Toan 582 Table 7. Kinematic and dynamical parameters of MRM robot. Link 11 12 13 21 22 Weight(kg) 8 8 3.5 50 20 Coordinates of links' barycenter on local Cartesian coordinate system x(m) -0.165 -0.165 0 0 0.068845 y(m) 0 0 0 0 0.022064 z(m) 0 0 0.2 -0.077511 0.107575 Inertia tensor (kg.m2) Ixx 0.002818 0.002716 0.041672 1.36253 0.204889 Iyy 0.108298 0.115332 0.041672 1.146869 0.188627 Izz 0.108825 0.115847 0.000166 1.861916 0.307898 Ixy 0 0 0 0 0.005476 Ixz 0.000381 -0.00002 0 0 0.008992 Iyz 0 0 0 0 -0.002919 Figure 4. Graph of desired and real trajectory of MRM robot's 5 joints. Figure 5. Graph of position error of the joint 11 (e1). Application of Hedge Algebras for controlling mechanisms of relative manipulation 583 Figure 6. Graph of position error of the joint 12 (e2). Figure 7. Graph of position error of the joint 13 (e3). Figure 8. Graph of position error of the joint 21 (e4). Phan Bui Khoi, Nguyen Van Toan 584 Figure 9. Graph of position error of the joint 22 (e5). Figure10. Desired and real welding path in Cartesian coordinate system x22y22z22 of jig-robot’s table (Fig. 1). Figure 11. Position error between calculated welding path and real welding path Application of Hedge Algebras for controlling mechanisms of relative manipulation 585 The desired and real welding path, received by torque, fuzzy and HAs controller overlap each other because the position error is very small (Fig. 10). Figure 11 shows the position error of each path. 5. CONCLUSION The recevied results show that the proposed method is more accurate than the classical fuzzy method. Moreover, the application of HA in control will be easier to understand and simpler than the use of fuzzy logic. It brings several benefits for controlling engineering systems. This paper used 5 linguistic values for the fuzzification of inputs and outputs. To compare HA method with the fuzzy controller in the same condition, we just use the partition k=2 for inputs/output. 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