The Chaotic Robot Prediction by Neuro Fuzzy Algorithm (2) = θ (3) = ω. Asin. A v. Mana Tarjoman, Shaghayegh Zarei

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1 The Chaotc Robot Predcton by Neuro Fuzzy Algorthm Mana Tarjoman, Shaghayegh Zare Abstract In ths paper an applcaton of the adaptve neurofuzzy nference system has been ntroduced to predct the behavor of a chaotc robot. The chaotc moble robot mples a moble robot wth a controller that ensures chaotc motons. Chaotc moton s characterzed by the topologcal transtvty and the senstve dependence on ntal condtons. We have used the controller such that the total dynamcs of the moble robot s represented by the Arnold equaton, whch s known to show the chaotc behavor of non-compressve perfect flud. Then we have used the adaptve neuro fuzzy nference system for predctng of ths chaotc moble robot. We propose to predct the behavor of the chaotc moble robot by usng an adaptve neuro-fuzzy nference system. Ths system s functonally smlar to fuzzy nference systems whch based on hybrd learnng rule and also are more quck and accurate than methods used neural networks or Kalman flter. Key words: ANFIS-chaotc moble robot- hybrd learnng rule I. INTRODUCTION Where ( [m], y [m]) s the poston of the robot and θ [rad] s the angle of the robot. Fg.. Moble robot In order to generate chaotc motons of the moble robot, we employ the Arnold equaton, whch s wrtten as follows: Chaos characterzes one of mysterous rch behavors of nonlnear dynamcal systems. Many research efforts have been pad to establsh the mathematcal theory behnd chaos. Ths paper ntroduces a chaotc moble robot that the chaotc behavor s acheved by desgnng a controller whch ensures chaotc moton. A moble robot wth such characterstcs may fnd ts applcatons as a patrol robot or a cleanng robot n a closed room, floor, or buldng. The senstve dependence on ntal condton also yelds a favorable nature as a patrol robot snce the scannng trajectory becomes hghly unpredctable. A. Chaotc moble robot wth the Arnold equaton As the mathematcal model of moble robots, we assume a two wheeled moble robot as shown n Fg.. Let the lnear velocty of the robot v [m/s] and the angular velocty ω [rad/s] be the nputs to the system. The state equaton of the moble robot s wrtten as follows [] : cosθ y& snθ & θ 0 0 v 0 ω M. Tarjoman s wth the Abhar Islamc Azad Unversty, Zanjan, Iran (emal: mana_tarjoman@ yahoo.com). Sh. Zare s wth the Tehran Islamc Azad Unversty (central branch), Tehran, Iran (e-mal: shaghayeghzare@yahoo.com). () & & Asn Bsn C sn + C cos + Acos + B cos Where A, B, and C are constants. B. Integraton of the Arnold equaton () In order to ntegrate the Arnold equaton nto the controller of the moble robot, we defne and use the followng state varables: & Dy& + C cos & D& + B sn () θ Where B, C and D are constants. Substtutng () nto (), we obtan a state equaton on, and as follows: & Dv + C cos & Dv + B sn (4) ω We now desgn the nputs as follows: A v D (5) ω C sn + B cos

2 Consequently, the state equaton of the moble robot becomes: Asn + C cos & & & y& B C sn sn + Acos + B cos v cos vsn Equaton (6) ncludes the Arnold equaton. The Arnold equaton behaves chaotcally or not, dependng upon the ntal states. We choose the ntal states of the Arnold equaton such that the trajectory should behave chaotcally. The whole states evolve n a 5-D space accordng to (6), whch ncludes a -D subspace of the Arnold flow. Fg. shows an eample of moton of the moble robot wth the ntroduced controller, obtaned by numercal smulaton. (6) In the followng sectons, the structure of the neuro-fuzzy nference system and predcton of the chaotc robot trajectory by usng the ANFIS algorthm wll be eplaned, respectvely. Fnally, the result of smulaton and concluson can be seen. II. ADAPTIVE NEURO-FUZZY INFERENCE SYSTEM In ths secton, we descrbe a class of adaptve network that are functonally equvalent to fuzzy nference systems. The propose archtecture s referred to as ANFIS [], whch stands for adaptve network-based fuzzy nference system. We descrbe how to decompose the parameter set to facltate the hybrd learnng rule for ANFIS archtecture representng both the Sugeno and Tsukamoto fuzzy models. A. ANFIS archtecture For smplcty, we assume that the fuzzy nference system under consderaton has two nput and y and output z. For a frst-order Sugeno fuzzy model, a common rule set wth two fuzzy f-then rules s the followng []: Rule : If s A and y s B, then f p +q y+r, Rule : If s A and y s B, then f p +q y+r. Fgure llustrate the reasonng mechansm for ths Sugeno model; the correspondng equvalent ANFIS archtecture s shown n fgure 4, where nodes of the same layer have smlar functons, as descrbed net Fg. Chaotc moble robot trajectory Here we propose to predct the behavor of a chaotc robot by usng an adaptve neuro-fuzzy nference system. Ths system s functonally smlar to fuzzy nference systems whch based on hybrd learnng rule and also are more quck and accurate than methods used neural networks or Kalman flter. Adaptve neural fuzzy nference system (ANFIS) s an dea by combnng the fuzzy nference system wth neural network. The fuzzy nference system s used wdely n fuzzy control, t can number rules by leadng nto a new deal of membershp functon to deal wth structural knowledge. ANFIS fully makes use of the ecellent characterstcs of neural network and fuzzy nference system. ANFIS can approach all nonlnear system wth less tranng data and qucker weakenng speed and hgher precson. Fg. Sugeno fuzzy model, wth two nputs and two fuzzy f-then rules has been shown.

3 For convenence, outputs of ths layer are called normalzed frng strengths. Layer 4. Every node I n ths layer s an adaptve node wth a node functon ( p + q y r ) O w f w + () 4, Where ϖ s a normalzed frng strength from layer and {p, q r } s the parameter set of ths node. Parameters n ths layer are referred to as consequent parameters. Layer 5. The sngle node n ths layer s a fed node labeled Σ, whch computes the overall output as the summaton of all ncomng sgnals. w f O 5, w f () w Fg. 4 Structure of ANFIS algorthm Layer. Every node I n ths layer s an adaptve node wth a node functon O ( ), for, (7) or O, µ A, B ( y µ ), for, 4 (8) Where (or y) s the nput to node and A (or B - ) s a lngustc label (such as small or large ) assocated wth ths node. In other words, s the membershp grade of a fuzzy set A (A, A, B or B ) and t specfes the degree to whch the gven nput (or y) satsfes the quantfer A. Here the membershp functon for A can be any approprate parameterzed membershp functon, such as the generalzed bell functon: µ A ( ) (9) b c + a Where {a, b, c } s the parameter set. As the values of these parameters change, the bell-shaped functon vares accordngly, thus ehbtng varous forms of membershp for fuzzy set A. Parameters n ths layer are referred to as premse parameters. Layer. Every node n ths layer s a fed node labeled Π, whose output s the product of all the ncomng sgnals: ( ) ( y) o, w µ A µ B,, (0) Each node output represents the frng strength of a rule. In general, any other T-norm operators that perform fuzzy AND can be used as the node functon n ths layer. Layer. Every node n ths layer s a fed node labeled N. The -th node calculates the rato of the -th rule s frng strength to the sum of all rule s frng strengths. w O, w,, () w + w B. Hybrd Learnng Algorthm From ANFIS archtecture shown n the fgure 4, we observe that the values of the premse parameters are fed, the overall output can be epressed as a lnear combnaton of the consequent parameters. In symbols, the output f n the fgure 4 can be rewrtten as w w f f + f w + w w + w w p + q y + r + w p + q y + r (4) ( ) ( ) ( w ) p + ( w y) q + ( w ) r + ( w ) p + ( w y) q + ( w ) r Whch s lnear n the consequent parameters p,q,r,p,q and r. The learnng algorthm for ANFIS s a hybrd algorthm whch s a combnaton between gradent descent and leastsquares method. More specfcally, n the forward pass of the hybrd learnng algorthm, node outputs go forward untl layer 4 and the consequent parameters are dentfed by the least-squares method. In the backward pass, the error sgnals propagate backward and the premse parameters are updated by gradent descendent. Table summarzes the actvtes n each pass. The consequent parameters are dentfed optmal under the condton that the premse parameters are fed. Accordngly, the hybrd approach converges much faster snce t reduced the search space dmensons of the orgnal pure back propagaton method. TABLE : LEARNING PARAMETERS OF THE ANFIS ALGORITHM Forward pass Backward pass Premse parameters Fed Gradent descent Consequent Least-Squares parameters estmator Fed Sgnals Node Outputs Error sgnals III. PREDICTION THE BEHAVIOR OF THE CHAOTIC ROBOT USING ANFIS ALGORITHM In ths part the applcaton of the ANFIS algorthm n

4 predctng the behavor of the chaotc robot and ts net values s consdered. The goal of the task s to use past values of the dynamc up to tme t to predct the value at some pont n the future t+p. The standard method for ths type of predcton s to create a mappng from D ponts of the dynamc spaced D apart that s, [(t-(d-) ),..,(t- ),(t)], to a predcted future value (t+p). In our smulaton, the values D4 and P6 were used. We have etracted 000 nput-output data pars of the followng format from the robot trajectory and used ANFIS algorthm for predcton of net values: [ ( t 8), ( t ), ( t 6), ( t); ( t + 6)] The frst 500 pars were used as tranng data set for ANFIS, whle the remanng 500 pars were the checkng data set for valdatng the dentfed ANFIS. For predctng of the chaotc robot behavor, we have consdered t separately n two as X and Y and used the ANFIS algorthm at each of these aes []. The number of membershp functons assgned to each nput of the ANFIS was set to two that have been selected bell shaped, so the number of rules s 6. In net secton, the results of smulaton consstng of error dagram, man trajectory and ANFIS predcton of the chaotc robot wll be revewed [4]. Fg. 6 chaotc robot trajectory predcton error n X drecton B. Result of smulaton n Y drecton: IV. SIMULATION RESULTS In ths part the results of smulaton n X drecton and then n Y drecton are shown. A. Result of smulaton n X drecton: Fg. 7 chaotc robot trajectory n Y drecton (blue : man trajectory, red : ANFIS output) Fg. 5 chaotc robot trajectory n X drecton (blue : man trajectory, red : ANFIS output) Fg. 8 chaotc robot trajectory predcton error n Y drecton

5 V. CONCLUSION In ths paper, the predcton of the behavor of a chaotc robot by usng an adaptve neuro-fuzzy nference system accomplshed. We etracted 000 nput-output data pars. The frst 500 pars were used as tranng data set for ANFIS, whle the remanng 500 pars were the checkng data set for valdatng the dentfed ANFIS. For predctng the chaotc robot behavor, we have consdered t separately n two aes X and Y and used the ANFIS algorthm at each of these aes. The result of ANFIS and predcton error of the consdered robot shows that ths algorthm s more accurate n compare wth methods used neural networks or Kalman flter for predctng [5]. REFERENCES [] Yoshhko Nakamura and Aknor Sekguch, The Chaotc Moble Robot, IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 7, NO. 6, DECEMBER 00 [] Jang S. R., Sun C. T. and Mtzutan E., Neurofuzzy and Soft Computng, Prentce Hall, 998. New York. [] Cuevas E., Zaldvar D. and Rojas R., Intellgent Trackng, Techncal Report B-- 0, Free Unverstät Berln, November, 00. [4] Fuzzy logc Toolbo, Mathworks, 999, New York. [5] Tae-Wan, Km, Mackey-Glass tme seres predcton usng RNN and ANFIS, Fuzzy & Intellgent System, ntellgent Multmeda Lab., Dept. of Computer & Communcaton Engneerng,, June, 007, POSTECH, South Korea.