APPLICATION OF THE TAKAGI-SUGENO FUZZY CONTROLLER FOR SOLVING THE ROBOTS' INVERSE KINEMATICS PROBLEM UDC

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1 FACTA UNIVERSITATIS Series: Mechanics, Automatic Control and Robotics Vol.3, N o 5, 3, APPLICATION OF THE TAKAGI-SUGENO FUZZY CONTROLLER FOR SOLVING THE ROBOTS' INVERSE KINEMATICS PROBLEM UDC Vesna Ranković, Ilija Nikolić Faculty of Mechanical Engineering, Sestre Janji} 6, 34 Kragujevac, Serbia and Montenegro, inikolic@tt.yu Abstract. For maing from external to internal coordinates the Takagi-Sugeno controller is used, imlemented within the framework of adative network, which is similar, by its architecture, to RBF neural network, so it is also called the neuro-fuy system. During the learning rocess, the arameters of the membershi functions of the rimary fuy sets and arameters of the consequences of the Takagi-Sugeno controller were adated. The results are resented for the straight-line trajectory tracking, with the constant velocity of the grier and with the triangular velocity rofile.. INTRODUCTION Deficiencies of analytical and numerical methods of solving the inverse kinematics (IK) roblem led to searching of the new aroaches to maing from external to internal coordinates. A large number of aers are devoted to alication of the non-recurrent (or feed forward) neural networks for solving the IK. In [6] is used the two-layered neural network (in literature also known as Radial Basis Function RBF neural network). The dynamic rocedure was alied for learning the network. Neurons in the invisible layer have Gaussian activation functions, while the outut layer consists of a single neuron whose activation threshold is being adated. As it is shown, the main deficiency of the RBF networks, trained by learning data sets, which enhance the art of the working sace, lies in fact that their hidden layers contain large number of neurons. The RBF neural networks cannot realie maing from external to internal coordinates in real time for high grier velocities. Fuy logic was first alied for solving the IK of the lanar four-segment redundant robot in aer [4]. This aer was used as a basis for develoing the fuy logical controller (FLC), resented in aer [5], where the FLC is alied for maing from the external to internal coordinates for lanar two-segment robot and for robots of the R T T y and R R y R y Received Aril 3,

2 4 V. RANKOVIĆ, I. NIKOLIĆ minimal configuration. The biggest deficiency of such an FLC structure is the long comutational time, which is the consequence of the large number of stes during the trajectory generation. In addition, in design of logical fuy controller, esecially roblematic is defining of the rule base and selection of the membershi functions of the rimary fuy sets. Numerous researchers have tried to automatie the modeling rocess of the fuy controller. Because of such investigations the adative fuy controllers aeared. Adating mechanism decides which changes should be erformed in the fuy controller, so it would obtain the desired outut, namely it would minimie the error. This mechanism is similar to the training mechanism in neural networks. Different adative systems were develoed. The most frequently used adating mechanism erforms the modification in the hase of controlling rules, changes of the weighting functions of the joined fuy rules to controlling rules in the rule base, modification of the rimary fuy sets or the selection of the defuyfication methods. Based on available literature, it can be concluded that the different training methods were alied most frequently to the Takagi-Sugeno controller. In this work is used the Takagi-Sugeno controller imlemented within the framework of the adative network for solving the IK roblem. In the second section of the aer is resented the basic structure of the controller, and is briefly described the rocedure used for training. In the third section are resented results of maing from external to internal coordinates for the three-segment robot of the R R y R y minimal configuration. The fourth section contains the concluding remarks.. TAKAGI-SUGENO FUZZY CONTROLLER AND THE LEARNING ALGORITHM The Takagi-Sugeno controller with M inuts and one outut is shown in Figure. The fuy controller has M+ linguistic variables: M inut ones and one outut variable. Linguistic variables x i are A i, A i,, A ni, M. For the system with M inuts and one outut, the set of linguistic rules is defined in the form: R : if x is A and x is A and x M is A M, then f = x + x + + M + c R : if x is A and x is A and x M is A M, then f = x + x + + M + c R : if x is A and x is A and x M is A km, then f k = k x + k x + + km + c k R : if x is A n and x is A n and x M is A nm, then f = x + x + + M + c The number of linguistic rules is = n M. The outut from Takagi-Sugeno controller from Figure is: where: y = u f () u i = u i i u = µ ( x )* µ ( x )...* µ ( x ) M u i i M u = µ ( x)* µ ( x)...* µ M( xm)... (3) ()

3 Alication of the Takagi-Sugeno Fuy Controller for Solving the Robots' Inverse Kinematics Problem 4 * denotes certain T-norm. uk = µ ( x )* µ ( x)...* µ M( xm)... u = µ ( x )* µ ( x )...* µ ( x ) n n nm M x x x M A a,b,c u u A a,b,c µ µ µ M x x x M uf x, x,... x M A M,c M a M,b M u u uf xx,,... x M x x x M A a,b,c µ A µ µ M u k u k k a,b,c u f x, x,... x k k M x x A M a M,b M,c M y x M x x x M A n,c n a n,b n A n µ n µ n u u u f x, x,... x M a n,b n,c n µ nm A nm a nm, b nm, c nm Fig.. Takagi-Sugeno controller with M inuts and one outut. If the membershi functions are taken in the Gaussian form then:

4 4 V. RANKOVIĆ, I. NIKOLIĆ µ ij =,,,, b ij i = n j = M xj a ij + c ij Consequences functions of the fuy rules are of the form:. (4) M f = x + c (5) i ij j i j = Substituting () into () the controller outut is obtained as: y = ui fi i u = i or, substituting (5) into (6), the outut of thetakagi-sugeno controler is: (6) M y = u x + c. (7) i ij j i i j u = = i In [], for adating arameters (a, a,, a M, b, b,, b M, c, c,, c M ) is used the iterative rocedure, which is based on method of decreasing gradients. The arameters of the f i functions (,,, M, c,,,, M, c,, k, k,, km, c k,,,, M, c ) are being adated by the least squares method. The total number of arameters for adating is: 3nM + n M (M+). The learning method requires set of data for training P = {,,, r ). Each element of the set, k = (x k, y k ) is defined by the inut vector x k = (x k x k x Mk ) and desired resonse y k. If the FLC arameters correction is erformed after resenting all the samles, then the error sum of squares is: or: where: r k k = ε= ε (8) r ( yk yk ) (9) k = ε= y = u f k ik ik uik M ik ij jk i j = () f = x + c () u =µ ( x )* µ ( x )...* µ ( x ) k k k M Mk uk =µ ( x k)* µ ( xk)...* µ M( xmk)... ()

5 Alication of the Takagi-Sugeno Fuy Controller for Solving the Robots' Inverse Kinematics Problem 43 ukk =µ ( x k )* µ ( xk )...* µ M ( xmk )... uk =µ n( xk )* µ n( xk )...* µ nm ( xmk ) Minimiation of the function given with (8) can be realied through the following iterative rocedure for arameters adating: ε aq ( l+ ) = aq ( l) ηa aq ε bq ( l+ ) = bq ( l) ηb bq (3) ε cq ( l+ ) = cq ( l) η cq From equations (3) one obtains: r yk aq ( l+ ) = aq ( l) ηa ( yk yk ) k = aq (4) where: yk uik u ik = fik yk a q aq a q u where: where: ik r b ( l+ ) = b ( l) η ( y y ) y k q q b k k (5) k = bq y u u b b b k ik ik = fik yk q q q u ik r c ( l+ ) = c ( l) ( y y ) y k q q η c k k (6) k = cq y u u c c c k ik ik = fik yk q q q u ik Parameters of the f i functions are being determined by the least square method. From equations () and (5) one obtains that the resonse of the fuy controller to the k-th element is: yk = uk( xk + xk +... MxMk + c ) + uk( x k + xk +... MxMk + c ) +... (7) u ( x + x +... x + c ) k k k M Mk The aim of learning is that the real resonse is equal to the desired one. If the adating is done after resenting of all samles, then:.

6 44 V. RANKOVIĆ, I. NIKOLIĆ ux ux! u xm u ux ux! uxm u! ux ux! uxm u ux ux! uxm u! " "! " " " "! " "! urx r u rxr! u rxmr u r urx r urxr! urxmr ur! " M c u x u x! u xm u y ux ux! uxm u y " = " "! " " M " ur x r ur xr! ur xm ur c yr " " M c Equation (8) is resented in the form: UP x x = Y (9) From (9), by the least squares method, the arameters vector P x is being determined: T T x x x x (8) P = ( U U ) U Y () The structure of the fuy system from Figure is similar to the structure of the RBF network. In [3] is given the detailed comarison of these two systems. It was concluded that there exist the functional equivalency if: The number of invisible units of the RBF network is equal to number of linguistic rules in fuy system For the fuy system of Figure holds f i = const The Gaussian functions are selected as the membershi rimary fuy sets for the system of Figure For determination of the T-norm in fuy systems, one uses the algebraic roduct The activation threshold of the outut neuron for the RBF network is ero. Since the structure of the fuy system of Figure is similar to the structure of the RBF neural network, and taking into account that there exist functional equivalency, the adative fuy system is frequently called the neuro-fuy system.

7 Alication of the Takagi-Sugeno Fuy Controller for Solving the Robots' Inverse Kinematics Problem APPLICATION OF THE TAKAGI-SUGENO FUZZY CONTROLLER TO SOLVING THE PROBLEM OF ROBOTS' INVERSE KINEMATICS In aer [5] the FLC is alied to solving the roblem of inverse kinematics. In all controllers, the fuy reasoning of the first tye was alied, where the Mamdani minimum rule is alied as a function in the imlication hase. The biggest deficiency of such FLC structures is long comutational time. Esecially roblematic is definition of the base rule and selection of the membershi functions of the rimary fuy sets. In this aer to solving of the IK is alied the Takagi-Sugeno controller shown in Figure. Simulations were done for the robot shown in Figure (the R R y R y configuration). The robot segments' lengths are: l =.3 m, l = l 3 = m. It is taken that there are constraints in joints (θ = θ = θ 3 = [ π/, π/]). The inut and the outut variables of the FLC for solving the IK are shown in Figure 3. Simulations were erformed with the rogram ackage MATLAB, by using the fuy toolbox. θ 3 θ l L 3 l y x θ x Fig.. The RRR robot of minimal configuration FLC θ x FLC θ x FLC 3 θ 3 Fig. 3. Illustration of the inut and the outut variables of the Takagi-Sugeno FLCs for solving the IK of the R R y R y robot. In the first examle, the grier was moving along the straight-line trajectory, from the oint A(.4.7) to oint B( ), with the triangular velocity rofile. The time taken for erforming the motion is t max = s. In this examle by the data set for training of the FLCs is enhanced only one straight-line trajectory. The learning seeds η a, η b and η c were taken according to [4] in the form: k η a =η b =η c = ε ε ε + + aq bq c q

8 46 V. RANKOVIĆ, I. NIKOLIĆ where k is the ste. The ste sie is variable, and so does the convergence rate. According to [], if k is small, the convergence is slow, and several iteraqtions would be necessary, to achieve satisfactory accuracy. If k is big, the convergence would, at the beginning of the learning rocess, be fast, but the algorithm will oscillate. The ste k changes according to the following rules: If the error is decreasing in the four consecutive eochs, k increases for %. If the error in one eoch decreases, then in the next one k decreases for %. In the first examle for the FLC is k =., and for FLC and FLC 3 is k =.. The fuy artitioning of the inut variables of the FLC is realied by selection of rimary fuy sets. Selected are the Gaussian membershi functions, since the best results are achieved with them. The total number of the fuy rules is. In the learning rocess the membershi functions arameters of the rimary fuy sets () were adated as well as arameters of the Takagi-Sugeno controller (). In Figure 4 are resented the membershi functions of the rimary fuy sets of variables x and y before learning, and in Figure 5 are resented the membershi functions after learning. µ(x) A A µ(y) A A µ(x) x y Fig. 4. Membershi functions of variables x and y before learning. A A µ(y) A A x y Fig. 5. Membershi functions of variables x and y after learning the FLC. In Figure 6 is resented the variation of error, given by exression (8), as a function of the number of eochs. E 8 X Eoch Fig. 6. Error variation during the learning rocess of the FLC. Parameters of the fi functions, i =,4, are given in Table.

9 Alication of the Takagi-Sugeno Fuy Controller for Solving the Robots' Inverse Kinematics Problem 47 In FLC to each inut variable were assigned two membershi functions. The controller has total of 4 rules. The total number of arameters, which are being adated is 4, out of which are the remise arameters and are the consequences arameters. In Figure 7 are shown the membershi functions of the rimary fuy sets of variables x i and before learning, and in Figure 8 are shown the membershi functions after learning. The variation of error with the number of eochs is shown in Figure 9. µ(x ) µ() A A A A µ(x ) x Fig. 7. Membershi functions of variables x i and before learning. A A A µ() A x Fig. 8. Membershi functions of variables x i and after learning the FLC. E x Eoch Fig. 9. Error variation during the learning rocess of the FLC. Parameters of the fi functions, i =,4, for FLC are given in Table. FLC 3 has 4 rules (two membershi functions were assigned to each inut). The membershi functions of the rimary fuy sets of variables x i and, rior to the adatation rocess, are shown in Figure 7, while the figure shows them after the learning rocess. In Figure is illustrated the variation of error, given with exression (8) as a function of the number of eochs. µ(x ) A A µ() A A x Fig.. Membershi functions of variables x i and after learning the FLC 3.

10 48 V. RANKOVIĆ, I. NIKOLIĆ E 3 x Eoch Fig.. Error variation during the learning rocess of the FLC 3. Parameters of the fi functions, i =,4, for FLC 3 are given in Table. In Figure is given the variation of the interior coordinates during executing the working task, and in Figure 3 is given the variation of the total error of tracking the straight-line trajectory. Table. Parameters of consequences of the Takagi-Sugeno controllers FLC FLC FLC c c c c θ [ ] θ i rad.4. θ θ. θ T s [ ] Fig.. Variation of the internal coordinates along the trajectory From Figure 3 can be seen that the maximum total error of the straight-line trajectory tracking, in the first examle is less than mm. The simulation results show

11 Alication of the Takagi-Sugeno Fuy Controller for Solving the Robots' Inverse Kinematics Problem 49 that the alication of the neuro-fuy controllers, with the membershi functions, shown in Figures 5, 8 and, and with arameters of the consequences functions given in Table, give the satisfactory results. The FLCs from this examle have smaller number of linguistic rules than the number of the invisible units of the RBF networks, generated for solving the same roblem, shown in [6]. -5 x 5.5 em [ ] T s Fig. 3. The total tracking error [] In the second examle the grier was moving along the straight-line trajectory, form the oint A( ) to oint B(.6.3.3), with the seed equal to v =. m/s, and the training data set enhanced the art of the working area x = [;.7], y = [; ] and = [;,8]. The training set redicted that the trajectories can be realied both over the ositive or negative values of the internal coordinate θ 3. The FLC determines the internal coordinate θ, the FLC θ for ositive values of θ 3, and FLC 3 generates the coordinate θ for negative values of θ 3, and FLC 4 θ 3. Selection of the FLC, which would determine the coordinate θ, is realied based on the robot's initial osition. In FLC to each inut variable are assigned three membershi functions. The controller has total of 9 rules. The total number of arameters, which are being adated is 45, out of which 8 are the remise arameters, and 7 are the consequences arameters. Fuy artitioning of the inut variables FLC, FLC 3 and FLC 4 is realied by selection of 5 rimary fuy sets for each variable. The membershi functions are selected as Gaussian. The total number of the fuy rules of each controller is 5. In the learning rocess, the arameters of the membershi functions of the rimary fuy sets were adated (3) as well as the arameters of the consequences of the Takagi-Sugeno controller (75). In Figures 4 and 5 are shown the membershi functions of the rimary fuy sets of variables x and y, for FLC, and variables x and for FLC, FLC 3 and FLC 4, rior to learning. In Figures 6 to 9 are resented the membershi functions after learning. µ(x) A A A 3 µ(y)a A A x y Fig. 4. Membershi functions of variables x and y before learning.

12 5 V. RANKOVIĆ, I. NIKOLIĆ µ(x ) A A A 3 A 4 A 5 µ() A A A 3 A 4 A x Fig. 5. Membershi functions of variables x and i before learning. µ(x) A A A 3 µ(y)a A A x y Figure 6. Membershi functions of variables x and y after learning the FLC. In Figures to 3 are shown error variations as a function of number of eochs for all four networks. µ(x ) A A A 3 A 4 A 5 µ() A A A 3 A 4 A x Fig. 7. Membershi functions of variables x and i after learning the FLC. µ(x ) A A A 3 A 4 A 5 µ() A A A 3 A 4 A x Fig. 8. Membershi functions of variables x and i after learning the FLC 3. µ(x ) A A A 3 A 4 A 5 µ() A A A 3 A 4 A x Fig. 9. Membershi functions of variables x and i after learning the FLC 4.

13 Alication of the Takagi-Sugeno Fuy Controller for Solving the Robots' Inverse Kinematics Problem 5 E X Eoch Fig.. Error variation during the learning rocess of the FLC. E X Eoch Fig.. Error variation during the learning rocess of the FLC. E 3 3 X Eoch Fig.. Error variation during the learning rocess of the FLC 3. E 4 6 X Eoch Fig. 3. Error variation during the learning rocess of the FLC 4. Parameters of functions fi, for FLC, FLC, FLC 3 and FLC 4 are given in Table. In Figure 4 is resented variation of the internal coordinates during the task execution, and in Figure 5 is given the total error variation of the straight-line trajectory tracking.

14 5 V. RANKOVIĆ, I. NIKOLIĆ θ i.8 [ rad ].6 θ θ 3.4. θ T s [ ] Fig. 4. Variation of the internal coordinates along the trajectory Table. Consequences arameters FLC FLC FLC 3 FLC c c c c " " " " " c c c Maximum total error of the trajectory tracking in the second examle is less than. mm. -3 x. em [ ] T s [] Fig. 5. The total tracking error

15 Alication of the Takagi-Sugeno Fuy Controller for Solving the Robots' Inverse Kinematics Problem CONCLUSION Results of both simulations, resented in this aer, show that the alication of the neuro-fuy system to solving IK gives satisfactory results. Advantage of alication of the Takagi-Sugeno controllers, imlemented within the adative network, related to solving IK by alication od the FLCs roosed in [4], lies in the fact that the ste between inter-oints on the straight-line trajectory, does not affect the accuracy of tracking. The tracking error is smaller for the case of solving the IK by RBF neural networks []. However, the main deficiency of networks trained by data sets for learning, which enhance the art of the working sace, is the fact that their hidden layers contain large number of neurons. The FLCs from the second examle have smaller number of linguistic rules than the number of the invisible units of the RBF networks generated for solving the same roblem. For solving the IK during the straight-line motion along the ath from oint A( ) to oint B(.6.3.3) with the velocity v =. m/s, the number of neurons of the hidden layer of the first RBF network is, while the number of rules FLC is 9. The RBF and RBF 3 have 86 and 74 neurons, resectively, and FLC and FLC 3 have 5 rules, each. Simulated FLCs have larger number of arameters, which are being adated. Weights of connections between invisible units and oututs, in RBF networks are constant values, while the consequences functions for Takagi-Sugeno controllers are linear functions of inuts into the controller. In realiation of fast trajectories, advantage for solving IK should be given to Takagi- Sugeno controllers with resect to the RBF neural networks [6]. Due to large number of neurons, RBF neural networks cannot realie maing from the external to internal coordinates in real time for high grier velocities. If it is necessary to realie trajectory with an error less than mm, then the data set for training should enhance only trajectories, which the grier tracks during the task execution. REFERENCES. Godjevac, J., (997) "A method of the design of neuro-fuy controllers; an alication in robot learning," Doctorial dissertation, EPFL, Lausanne, Switerland.. Jang, S. R., (993) "ANFIS: Adative-Network-Based Fuy Inference System," IEEE Trans. on Systems, Man and Cybernetics, vol. 3, no. 3, Jang, S. R., C. T. Sun, (993) "Functional Equivalence between Radial basis Function Networks and Fuy Inference Systems," IEEE Trans. on Neural Networks, vol. 4, no., Nedungadi, A., (995) "Alication of Fuy Logic to Solve the Robot Inverse Kinematic Problem," in Proc. of Fourth World Conference Robotics Research, Pittsburgh, PA, USA, Nikolić, I., M. Janković, (998) "Fuy Controller for Solving the Inverse Kinematics of Industrial Robots", Proceedings of the XXVII ETRAN Conference, Vrnjačka Banja, Volume 3, Ranković, V., I., Nikolić, () "Solving of the Robots' Inverse Kinematics by Two-Layer Neural Network", Proceedings of XLV ETRAN Conference, Bukovička Banja, Volume 3,. 7-3.

16 54 V. RANKOVIĆ, I. NIKOLIĆ PRIMENA TAKAGI-SUGENO FUZZY KONTROLERA ZA RE[AVANJE PROBLEMA INVERZNE KINEMATIKE ROBOTA Vesna Rankovi}, Ilija Nikoli} Za reslikavanje i soljašnjih u unutrašnje koordinate je korišćen Takagi-Sugeno kontroler, imlementiran u adativnu mre`u, koji je o arhitekturi sličan RBF neuronskoj mre`i, a se takodje naiva neuro fuy sistemom. Tokom rocesa u~enja su adatirani arametri funkcija riadnosti rimarnih fuy skuova i arametri osledica Takagi-Sugeno kontrolera. Prikaani reultati se odnose na ra}enje ravolinijske trajektorije sa konstantnom brinom hvataljke i trougaonim rofilom brine.