Bond Graph Modeling of a Robot Manipulator

Transcription

1 ème Congrès Franças de Mécanque Lon, 4 au 8 Août 5 Bond Graph Modeng of a Robot Manpuator F.Z.BAGHLI, L. EL BAKKALI Modeng and Smuaton of Mechanca Sstems Laborator, Abdemaek Essaad Unverst, Facut of Scences, BP., M hannech, 93, Tetuan, Morocco Abstract bagh.fatmazahra@gma.com In ths paper we propose a new approach for modeng a robot arm based on Bond Graph methodoog. The proposed method based on the transfer of energ between sstem components and on the descrpton of the vector veoct reaton of a movng pont n a rotatng sstem. Our sstem s a doube freedom robot manpuator drvng b an eectrc actuator, can be effcent modeed and soved b ths mutdscpnar approach. Résumé Dans ce trava, nous proposons une nouvee approche de modésaton d'un robot manpuateur basé sur a méthodooge Bond Graph. Cette méthode est basée sur e transfert d'énerge entre es composants du sstème en utsant des ens de pussance. Kewords Bond Graph, Robot manpuator, Modeng. Introducton A good modeng of the specfc manpuator needs an effcent method to descrbe a behavors of sstem. The Newton-Euer technque and Lagrange s technque are the most methods used for dnamc modeng; these technques cacuate a vector contanng the force or torque requred at each jont to attan a specfed trajector of jont postons, veoctes and acceeratons. The man dsadvantages of the above modeng technques are ther compet and ack of versatt []. The Bond Graph technque deveoped snce the 96's represent a powerfu approach to modeng robotc manpuators and mechansms [][].It s a graphca representaton that depcts the nteracton between eements of the sstem aong wth ther cause and effect reatonshps. The use of Bond Graph to descrbe a behavors of robotc manpuators can he deveoped based on knematc reatonshps between the tme rates of jont varabes and the generazed Cartesan veoctes (transatona and anguar veoctes) [].Ths effcent method can be used to obtan more nformaton such as the power requred to drve each jont actuator, or the power nteracton at the nterface wth the envronment, Such nformaton can aso be used to stud the performances ke stabt, precson of the manpuator sstem.

2 ème Congrès Franças de Mécanque Lon, 4 au 8 Août 5 In ths work the stud s etended to a hgh non near, mutpe nputs mutpe outputs (MIMO) sstem, ths stud s ustrated b the two arms manpuator. The am of ths work s to descrbe a behavors of our sstem b usng bond Graphs.. The Bond Graph. Bonds The basc eement of bond graphs s the energ bond (Fgure ), n ths method, power conssts of two varabes whch are known as generazed effort generazed fow denoted b e and f respectve; these two varabes are necessar and suffcent to descrbe the energetc transfers nsde the sstem. The phsca meanng of the effort and fow varabes depends upon the phsca doman the bond represents. A undrectona sem headed arrow shows ths energ nterchange (the arrow on the bond denotes the drecton of postve energ fow). Tabe gves eampes of the effort and fow varabes for mechanca and eectrca domans. e f Fgure Energ bond Doman Effort (e) Fow (f) Transaton Mechancs Force Veoct Rotatona Mechancs Torque Anguar veoct Eectrct Votage Current Tabe Effort and Fow varabes n some phsca domans.. Components There are four tpes of components abeed S, C, I, R, ths eementar components are cassfed b ther energetc behavor (energ dsspaton, energ storage, etc.) and defne how the effort and fow varabes on the bond reate to each other. The tabe show the eementar component of bond graph. Component Smbo Tpe of eement Actve eements Passve eements Eampe n transaton mechanc doman Eampe n rotatona mechanc doman Eampe n eectrc doman Se Effort source Force source Torque source Votage source Sf Fow source Veoct source Anguar veoct source Current source R Dsspaton Damper Rot. Damper Resstor I Inerta Rot. Inerta Inductor C Storage Compance Rot. Compance Capactor Tabe Basc Bond Graph eements.3. Junctons Components are connected together usng two tpes of junctons a or common effort juncton and a or common fow juncton. The juncton has the foowng propertes a bonds mpngng upon t have the same effort varabe and a fows on attached bonds sum to zero. Smar the juncton has the propertes a bonds mpngng upon t have the same fow varabe and a effort on attached bonds sum to zero.

3 ème Congrès Franças de Mécanque Lon, 4 au 8 Août 5 e f f e e3 e f3 e f f e e3 e f3 e=e=e3 ; f+f-f3= f=f=f3 ; e+e-e3= a) -Juncton b) -Juncton Fgure Iustraton of juncton.4 Connectng mechanca and eectrca domans To transfer between phsca domans the abt to mutp must be ncuded and bond graphs provde two means of accompshng ths the Transformer TF and the Grator G (TF or G are energ conservng). 3. Descrpton of the robot manpuator In ths secton, geometrc and knematc modes are used for modeng the behavor of a robot manpuator wth DOF. The parameters of the sstem are jont and operatona postons, the frst aows mong ts geometr and the second determnes the poston and the orentaton of the end effector M. In Fgure 3 a schema s gven of -nk rgd arm n whch each artcuaton s drven ndvdua b an eectrc actuator G q M g G q G G Fgure 3 Structure of manpuator robot of two degree of freedom The postons and the veoctes of the centers of mass of the two nks are descrbed b foowng equatons as shown n fgure

4 ème Congrès Franças de Mécanque Lon, 4 au 8 Août 5 G cos q G sn q G cosq cosq q G sn q sn q q vg sn q vg cosq v sn q q q vg cosq cosq q sn G ) ) 3) 4) 5) 6) 7) 8) 4. Bond graph for a rotatng arm The bond graph for the frst arm s derved from epressons of the veoctes of the center of mass 5 and 6 The transformers are used to convert the anguar veoct to a near veoct and the dnamcs can be ntroduced b addng I eement to the arm as shown n fgure 4. The base of the second arm s not fed n space but depends on the veoct of ts attachment pont to the frst arm, therefore, the deveopment of the frst and the second arm based on the epressons of the veoctes of center mass of the second nk. The compete bond graph of our sstem s shown n fgure 5 Where r sn( q ) (9) r cos( q ) () r sn( q ) () 3 r cos( q ) () 4 I m VG I J TF r g ω G q TF r Se Se -mg VG I m Fgure 4 Bond graph of the frst nk

5 ème Congrès Franças de Mécanque Lon, 4 au 8 Août 5 Lnk Lnk Sf VG VG VM I L dcmotor I Jm I m I L dcmotor I Jm I m Se U U VL VR R R E G k ω ω ωm f ω R b TF k ω a I J TF r TF r ω I m TF r TF r Se U U VL VR R R E G k ω ω ωm f ω R b TF k ω a TF r3 I J ω TF r3 TF r4 TF r4 I m Sf VG VG VM Se - m g Se - m g Fgure 5 Bond graph of the two arm manpuator The juncton equatons and eements are ustrated n Tabes I and II. Tabe. I. The juncton s characterstc Jonctons e e3 e4 e8 e e4 e6 f5 f6 f7 f f3 f4 f8 f f4 f6 e6 e7 e5 e e6 e7 e9 e5 f f f5 f6 f7 f f9 e e f9 f8 e8 e9 e e3 e f f f3 e5 e6 e7 f6 f7 f5 e3 k * e8 TF k f8 k * f3 e6 k4 * e5 TF k4 f5 k4 * f5 Joncton e3 e e5 f5 f3 f Jonctons «TF» e8 k * e7 TF k f k * f e9 k5 * e TF k5 f k * f Tabe.II. The equatons of eements e e e4 f f4 f TF k 3 TF k 6 e k * e f k * f e k * e f k * f I e I dt Eement P e m g Se e S e P e m g 7 I e7 I dt e dt 7 I e7 I7 The torque apped to move nk can be obtaned from the bond graph b the Eq. () and the eterna torque apped to move the second nk b the Eq. ().

6 ème Congrès Franças de Mécanque Lon, 4 au 8 Août 5 e e3 e4 e8 e e4 e6 e e6 e7 e9 e5 (3) (4) From the bond graph mode and the precedent juncton equatons and eements we can formuate a set of manpuator robot ferenta equatons n the foowng matr form M( q) M ( q) q C ( q, q ) C ( q, q ) q G ( q) M ( q) M ( q) q C( q, q) C ( q, q) q G ( q) (5) Where The eements of the nerta matr M (q) n the terms of the parameters of the robot manpuator are gven b M ( q) I I m m m m c c c c M ( q) M ( q) I m m c c c M ( q) I mc Cj( q, q )(, j,) The matr eements centrfuga and Coros force are C ( q, q) mc q s C ( q, q ) m s ( q q ) c C( q, q) m cq s C ( q, q ) Fna the eements of the vector of gravtatona torques G (q) are gven b G ( q) ( m m ) gc c mg cc G ( q) m gc c The equatons of eterna torque gven b bond graph approach correspond to those obtaned usng Lagrange methods, as ustrated n Eq. (8). Ths ndcates that, the mode deveoped capture the essenta aspects of rgd bod dnamcs of the robot manpuator. 5. Concuson A new approach s used n ths paper for modeng a robot arm. From the Bond graph mode we can formuate the ferenta equatons, ths ast one descrbe a behavours of the sstem. References [] Anand Vaz, Harmesh Kansa and An Snga, Some Aspects n the Bond Graph Modeng of Robotc Manpuators Anguar Veoctes from Smboc Manpuaton of Rotaton Matrces, IEEE 3. [] Darna Hroncováa, Patrk Šargaa, Aeander Gmterkoa, Smuaton of mechanca sstem wth two degrees of freedom wth Bond Graphs and MATLAB/Smunk, Proceda Engneerng 48 ( ) 3 3. [3] Wofgang Borutzk, Bond Graph Methodoog Deveopment and Anass of Mut-dscpnar Dnamc Sstem Modes, Sprnger. [4] F. Z.Bagh, L. E bakka, Modeng and Anass of the Dnamc Performance of a Robot Manpuator drvng b an Eectrca Actuator Usng Bond Graph Methodoog, Internatona Journa of Mechanca & Mechatroncs Engneerng IJMME-IJENS Vo4 No4.