Journa of Theoretica and Appied Inforation Technoogy 3 st Deceber. Vo. 46 No. 5 - JATIT & LLS. A rights reserved. ISSN: 99-8645 www.jatit.org E-ISSN: 87-395 TRACKING CONTROL FOR WHEELED OBILE ROBOTS USING NEURAL NETWORK ODEL ALGORITH CONTROL YUANLIANG ZHANG Schoo of echanica Engineering, Huaihai Institute of Technoogy, Lianyungang 5, Jiangsu, China ABSTRACT This paper proposed a neura network ode agorith contro (NNAC) ethod for the path tracking contro of differentiay steered wheeed obie robots (WRs) subject to nonhoonoic constraints. The ode agorith contro (AC) is a one-step-ahead predictive controer, in which the contro aw is obtained by iniizing the output error. The AC ethod needs the we known ode of the controed syste to design the controer. But soeties it is difficut to get the accurate ode of the controed syste or the ode of the controed syste is not good because of the inside and outside disturbances and unknown paraeters. Neura network has the abiity to estiate the noninear syste s ode and its corresponding inverse ode. In this paper the NNAC ethod which cobines two neura networks and AC ethod is proposed to do the path tracking contro of a wheeed obie robot. Soe siuations are conducted to show the perforance and feasibiity of the proposed contro strategy. In these siuations the WR is controed to track two difference reference paths such as the circuar shape path and the 8 shape path. Keywords: Neura Network, ode Agorith Contro, obie Robot. INTRODUCTION The differentiay steered wheeed obie robot which possesses the advantages of high obiity, high traction with pneuatic tires, and a sipe whee configuration is a kind of nonhoonoic syste []. Nonhoonoic systes ost coony arise in finite diensiona echanica systes where constraints are iposed on the otions that are not integrabe. In the case of the differentiay driven WRs, the constraints invoved are based on the assuption that there is no sipping of the whees. There are two fundaenta statuses in controing a obie robot: posture stabiization and path tracking. The ai of posture stabiization is to stabiize the robot to a reference point, whie that of path tracking is to have the robot foow a reference path. For WRs, path tracking is easier to achieve than posture stabiization. This coes fro the assuption that the whees ake perfect contact with the ground, resuting in nonhoonoic constraints. The tracking contro of nonhoonoic obie robots ais at getting the to track a given tie varying path (reference path). It is a fundaenta otion contro probe and has been intensivey investigated in the robotic doain [-5]. Feipe et a. [6] proposed an adaptive controer for autonoous obie robot trajectory tracking. Ye [7] proposed a tracking contro approach for nonhoonoic obie robots by integrating the neura network into the backstepping technique. Sun [8] designed a trajectory tracking controer for the obie robot based on poe assignent approach. Park et a. [9] soved the point stabiization of obie robots via state-space exact feedback inearization. This paper proposed a NNAC ethod for the path tracking contro of a WR. AC is a one-step ahead predictive controer, in which the contro aw is obtained fro the iniization of the output error at tie k + r []. The cosed-oop AC ethod sti requires a pant ode which is sufficienty accurate to obtain good contro perforance. However, in practice, because of the inside and outside disturbance and the unknown paraeters the ode of the WR is not good. Neura networks have the abiity to earn the syste characteristics through noninear apping to represent arbitrary noninear functions and their inverse functions and provide a strong degree of robustness because of their abiity to exhibit faut toerance. AC contro ethod can be designed basing on this neura network ode. And neura networks can iprove the adaptabiity of the contro schee through both off-ine and on-ine 794
Journa of Theoretica and Appied Inforation Technoogy 3 st Deceber. Vo. 46 No. 5 - JATIT & LLS. A rights reserved. ISSN: 99-8645 www.jatit.org E-ISSN: 87-395 weight adaptation. We propose in this paper a AC ethod using back propagation (BP) neura networks to do the path tracking contro of a WR. Siuations are conducted to show the perforance and feasibiity of the proposed contro ethod.. AC CONTROL ETHOD Consider the noninear systes described by a discrete-tie state-space ode of the for: x ( k + ) = Φ[ x ( k), u( k r)] y ( k) = h[ x ( k)] () where x denotes the state variabes, u denotes the input, y represents the output to be controed and the subscript is added to indicate the estiates of x and y obtained in the ode siuations and differentiate the easured x and y. Φ( x, u) is an anaytic vector function on X U, and h( x) is an anaytic scaar function on X. Here, r is the reative order of the syste (), i.e. r is the saest nuber of saping periods after which the anipuated input u affects the output y. The onine siuation of the ode described by Eq. () can be used to predict the future changes in the output y as foows: y k y k h x k h x k... ( + ) ( ) = [ ( )] [ ( )] y k y k h x k h x k ( + ) ( ) = [ ( )] [ ( )] y k r y k h x k h x k ( + ) ( ) = [ ( )] [ ( )] y k r y k h x k u k ( + ) ( ) = [ Φ[ ( ), ( )]} h[ x ( k)] Here the foowing notation is used: h ( x) = h( x) h x h [ x u ] r ( ) = Φ (, ), =,..., Syste () has the reative order r eans that h h ( Φ( x, u) ) u ( Φ( x, u) ) u =, =,,..., r ; Furtherore, the foowing reations wi hod: () (3) (4) y( k + ) = h [ x( k)], y k + r = h Φ x k u k =,..., ( ) { [ ( ), ( )]} (5) Therefore, r is the saest nuber of saping periods after which the anipuated input u affects the output y. With a finite reative h order r, u agebraic equation: ( Φ( x, u) ) ipies that the r h [ Φ ( x, u)] = y (6) is ocay sovabe in u. The corresponding ipicit function is denoted by: u( k) = Ψ ( x( k), y( k + r)) (7) and is assued to be we-defined and unique on X h( X ). When the predicted changes shown in Eq. () are added to the easured output signa y, one obtains the foowing cosed-oop predictions of the output: yˆ( k + ) = y( k) + h [ x ( k)] h[ x ( k)] yˆ( k + ) = y( k) + h [ x ( k)] h[ x ( k)]... yˆ( k + r ) = y( k) + h [ x ( k)] h[ x ( k)] yˆ( k + r) = y( k) + h [ Φ[ x ( k), u( k)]] h[ x ( k)] (8) where yˆ( k) represents a prediction of the output y( k ). Here a desirabe vaue, y d the ( k + r) th tie step is designed by:, of the output at y ( k + r) = ( α) y ( k) + α yˆ ( k + r ) (9) d sp where α is a tunabe fiter paraeter such that < α< and y sp is the set-point vaue. Equation (9) is referred to as the reference trajectory in the AC iterature. One can derive a noninear AC controer by requesting that the output prediction atch the reference path in the sense of iniizing the perforance index of Eq. (): in[ y ( k r) yˆ ( k r)] u( k ) d + + () Considering Eqs. (8) and (9), this becoes 795
Journa of Theoretica and Appied Inforation Technoogy 3 st Deceber. Vo. 46 No. 5 - JATIT & LLS. A rights reserved. ISSN: 99-8645 www.jatit.org E-ISSN: 87-395 in{( α) e( k) h { Φ[ x ( k), u( k)]} u( k ) + αh [ x ( k)] + ( α ) h[ x ( k)]} () where e(k)= y ( k) y( k). sp In the absence of input constraints, this iniization probe is triviay sovabe. Input u( k ) is the soution of the noninear agebraic equation: ( ( ( ), ( ))) ( ( ), ( )) h x k u k b x k e k Φ = () where b( x, e) αh ( x) ( α) ( h( x) e) = + +. Recaing the definition of Ψ (Eq. (7)), the soution can be represented as: ( ( )) u( k) = Ψ x ( k), b x ( k), e( k) (3) Therefore, the derived contro aw is given by Eq. (3). 3. AC USING NEURAL NETWORKS Two neura networks, NN and NN, are epoyed in deriving the NNAC ethod. In this proposed contro schee, the neura network NN is trained as the ode of the controed syste and the neura network NN is trained to produce the contro inputs. In this paper, we adopt the BP neura network and use the NNARX (AutoRegressive Externa input) ode structure as the structure of the two neura networks. For this ode structure, the inputs of the neura network are the past data of the syste output and input of the syste. Figure shows the configuration of the BP neura network. Figure depicts the NNARX ode structure. The paraeters a and c are the nubers of past data of the syste output and contro input, respectivey. x is the estiated syste s output, u is the syste s input, x is the neura network s output and r is the reative order of the noninear syste. ϕ ϕ p p j x Figure : Configuration Of The BP Neura Networks x( k ) x k ( a) u( k r) u( k r c) x j x ( k ) Figure : The NNARX ode Structure The neura network NN is used to estiate the ode of the controed syste and the neura network NN is trained as the inverse ode of the controed syste. For the controed syste the experient is first conducted to get the training data and test data for the neura networks NN and NN. In the experient the input signa u wi be designed carefuy in order to cover a of the operating range of the controed syste. After doing the experient a series of input signas u and their corresponding controed syste s output x are obtained. The haf of the data gotten in the experient is used as the training data of the neura networks NN and NN, and the rest of the data is used as the test data. Based on the NNARX ode structure the inputs of the neura network NN are the present saping tie syste input and severa past saping ties syste output and input. The error signa used to adjust the weights of the neura network NN is the difference between the controed syste s output and the neura network s output. Figure 3 shows the structure used to train the neura network NN. The inputs of the neura network NN are the present saping tie syste s output and severa past saping ties 796
Journa of Theoretica and Appied Inforation Technoogy 3 st Deceber. Vo. 46 No. 5 - JATIT & LLS. A rights reserved. ISSN: 99-8645 www.jatit.org E-ISSN: 87-395 syste output and input. The error signa used to adjust the weights of the neura network NN is the difference between the controed syste s input and the neura network s output. Figure 4 shows the structure used to train the neura network NN. generate x ( k ) ; and the neura network NN is trained to estiate the function of Ψ. Then the contro input can be generated by using the neura networks NN and NN. The diagra of the proposed contro syste is shown in Figure 5. u( k r) Controed Syste + x( k) y sp b x ( k ), e( k) u y E x q q q q NN Neura Network x ( ) k Figure 3: Structure Used To Train The Neura Network Nn q E u( k ) x( k + r) + Figure 5: Principe Of The Nnac Contro Syste 4. SIULATION The proposed NNAC contro ethod can be used to do the path tracking contro of the WR. Consider a nonhoonoic WR with two differentiay steered whees, as shown in Figure 6. This WR has two driving whees (radius r ) and one caster. Point H ( xh, yh ) defines the intersection of the axis of syetry with the driving whee axis, and is assued to be the origin of the coordinate frae { X H, Y H }. Point C( xc, yc ) is the center of ass of the obie robot. Length c is the distance between point H and point C, and is the ength of the rear whee axis. q u '( k) X H q q 4 q 5 Figure 4: Structure Used To Train The Neura Network Nn For AC ethod the contro input can be cacuated using Eq. (3). In Eq. (3) the function Ψ is obtained fro Eq. (7) and x ( k) coes fro Eq. (). Anaogizing to the AC ethod presented in Section, the NNAC ethod presented in Section 3 uses the two neura networks, NN and NN, to do the contro job. In this case it is assued that the ode and inverse ode of the controed syste are hard to obtain. This eans that Eq. () and Eq. (7) are unknown. In this case the neura network NN is trained to estiate the controed syste s ode Eq. () to Y H H C r Figure 6: Sketch Of A Nonhoonoic Wr First the neura networks NN and NN are trained to estiate the ode and inverse ode of the WR. Then appy these two neura networks to the NNAC ethod described in Section 3. After obtaining the NNAC controer, it is used to do the path tracking contro of a WR. We siuated c θ 797
Journa of Theoretica and Appied Inforation Technoogy 3 st Deceber. Vo. 46 No. 5 - JATIT & LLS. A rights reserved. ISSN: 99-8645 www.jatit.org E-ISSN: 87-395 the proposed contro aw to deonstrate its effectiveness. First a circuar reference path is used to do the siuation. The tracking perforance of the NNAC controer for the circuar reference path is shown in Figure 7. And Figure 8 shows the inear and anguar veocities of the obie robot in this case..8.6.4. reference path tracking path y ().5 reference path tracking path -. -.4 -.6.5 -.8 y () -.5 - -.8 -.6 -.4 -...4.6.8 x () Figure 9: Tracking Perforance In The Case Of 8 Shape Reference Path - -.5 -.5 - -.5.5.5 x () Figure 7: Tracking Perforance In The Case Of Circuar Reference path Next we choose the reference path as an 8 shape path. The tracking perforance of the NNAC controer for the 8 shape reference path is shown in Figure 9. And Figure shows the inear and anguar veocities of the obie robot in this case..5 4 v (/s) w (rad/s) 3 3 4 5 6 4 3-3 4 5 6 Figure 8: Linear And Anguar Veocities In The Case Of Circuar Reference Path v (/s) w (/s).5 3 4 5 6 7 8.5.5 -.5-3 4 5 6 7 8 Figure : Linear And Anguar Veocities In The Case Of 8 Shape Reference Path Fro the accurate perforance of the siuation it can be seen that the proposed NNAC contro ethod can provide a good contro resut for the path tracking of the WRs. 5. CONCLUSION In this paper we study the path tracking probe of the wheeed obie robots subject to nonhoonoic constrains. The NNAC contro ethod is proposed for tracking contro of the 798
Journa of Theoretica and Appied Inforation Technoogy 3 st Deceber. Vo. 46 No. 5 - JATIT & LLS. A rights reserved. ISSN: 99-8645 www.jatit.org E-ISSN: 87-395 wheeed obie robot. AC is a one-step-ahead predictive controer, in which the contro aw is obtained by iniizing the output error. Two neura networks, NN and NN, are introduced to cooperate with the AC contro ethod. The neura network NN was trained as the ode of the controed syste and the neura network NN was trained to generate the contro inputs. The nuerica siuation is conducted to show the proise of the proposed NNAC contro ethod in ters of tracking perforance. In the siuation a circuar reference path and an 8 shape reference path are used. ACKNOWLEDGEENTS This paper is supported by Huaihai Institute of Technoogy Bringing in Taent Research Starting Fund project (KQ5). REFERENCES: [] Yuin Zhang, Daehie Hong, Jae H. Chung, Steven A. Veinsky, Dynaic ode based robust tracking contro of a differentiay steered obie robot, Proceedings of the 998 Aerican Contro Conference, IEEE Conference Pubishing Services, June -6, 998, pp. 85-855. [] Oid ohareri, Rached Dhaouadi, Ahad B. Rad, Indirect adaptive tracking contro of a nonhoonoic obie robot via neura networks, Neurocoputing, Vo. 88, No. 5,, pp. 54-66. [3] George P. oustris, Spyros G. Tzafestas, Switching fuzzy tracking contro for obie robots under curvature constraints, Contro Engineering Practice, Vo. 9, No.,, pp. 45-53. [4] K.D. Do, Output-feedback foration tracking contro of unicyce-type obie robots with iited sensing ranges, Robotics and Autonoous Systes, Vo. 57, No., 9, pp. 34-47. [5] arkus auder, Robust tracking contro of nonhoonoic dynaic systes with appication to the bi-steerabe obie robot, Autoatica, Vo. 44, No., 8, pp. 588-59. [6] Feipe N. artins, Wanderey C. Ceeste, Ricardo Carei, ário Sarcinei-Fiho, Teodiano F. Bastos-Fiho, An adaptive dynaic controer for autonoous obie robot trajectory tracking, Contro Engineering Practice, Vo. 6, No., 8, pp. 354-363. [7] Jun Ye, Tracking contro for nonhoonoic obie robots: integrating the anaog neura network into the backstepping technique, Neurocoputing, Vo. 7, No. 6-8, 8, pp. 3373-3378. [8] Shui Sun, Designing approach on trajectorytracking contro of obie robot, Robotics and Coputer-Integrated anufacturing, Vo., No., 5, pp. 8-85. [9] Kyucheo Park, Hakyoung Chung, Jang Gyu Lee, Point stabiization of obie robots via statespace exact feedback inearization, Robotics and Coputer-Integrated anufacturing, Vo. 6, No. 5,, pp. 353-363. [] Qijun Xia, ing Rao, Jixin Qian, ode agorith contro for paper achines, Proceedings of Second IEEE Conference on Contro Appications, IEEE Conference Pubishing Services, Septeber 3-6, 993, pp. 3-8. 799