BOLETIM TECNICO DA PETROBRAS VOL. 61, 2017

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1 BOEIM ECNICO DA PEROBRAS VO. 6, 27 Guest Edtors: Hao WANG, Mn DUAN Copyrght 27, PB Publshng ISBN: ISSN: EISSN: Study of Space Manpulator Moton Analyss Based on Mathematcal Modelng Method / Estudo da Análse de Movmento do Manpulador Espacal Baseado no Método de Modelagem Matemátca * & Zongwen Ba College of Mathematcs and Computer Scence, Yan an Unversty, Yan an 76, Chna School of Physcs and Electronc Informaton, Yan an Unversty, Yan an 76, Chna he Unversty of exas at Austn, Austn 733, USA ABSRAC: For the defect of the movement trajectory offset caused by the space manpulator whch s mult-dmensonal, strong couplng and hgh degree of freedom exsts the naccurate moton trajectory plannng n the process of movng, ths paper puts forward to use the prncple of fuzzy control on the bass of mathematcal modelng, and analyzes and researches the moton of space manpulator. Frstly, the four-degree-of-freedom (M-R) modular manpulator s used as the mathematcal modelng model and the knematc constrant object to explan the postve knematcs and nverse knematcs modelng algorthm; then, the D-H algorthm s used to represent and model the manpulator, and a postve knematc model wth jont angle as varable s obtaned, and ths paper uses the analytcal method of matrx nverse multplcaton to solve the complete analytc soluton of nverse knematcs of manpulator; fnally, the correctness of postve knematcs model and knematc nverse soluton are verfed by fuzzy control smulaton and experment. he verfcaton result shows that usng the modelng method can carry out precse postonng for the end actuator of the manpulator, whch provdes a theoretcal bass for the moton analyss of the space manpulator. Keywords: Modular manpulator; D-H algorthm; Postve knematcs prncple; Fuzzy control prncple. RESUMO: Para o defeto do deslocamento da movmentação do movmento causado pelo manpulador espacal que é multdmensonal, o acoplamento forte e alto grau de lberdade exste o planejamento da trajetóra de movmento mprecso no processo de movmentação, este documento propõe usar o prncípo de fuzzy controle com base na modelagem matemátca, e analsa e pesqusa o movmento do manpulador espacal. Em prmero lugar, o manpulador modular de quatro graus de lberdade (M-R) é usado como o modelo de modelagem matemátca e o objeto de restrção cnemátca para explcar o algortmo de modelagem de cnemátca e algortmo de cnemátca nversa postvo; então, o algortmo D-H é usado para representar e modelar o manpulador e é obtdo um modelo cnemátco postvo com ângulo artcular como varável, e este trabalho usa o método analítco de multplcação nversa da matrz para resolver a solução analítca completa da cnemátca nversa do manpulador; Por fm, a correção do modelo de cnemátca postva e a solução nversa cnemátca são verfcadas por smulação de controle dfuso e experênca. O resultado da verfcação mostra que o uso do método de modelagem pode realzar um posconamento precso para o atuador fnal do manpulador, que fornece uma base teórca para a análse de movmento do manpulador espacal. Palavras-chave: manpulador modular; Algortmo D-H; Prncípo cnemátco postvo; Prncípo de controle dfuso. SEPEMBER 27 3

. Introducton he dea of mathematcal modelng s wdely used n the analyss of hgh degree of freedom of moton equpment, such as the moton analyss gat plannng of bped robot, ndustral robot and space

2 . Introducton he dea of mathematcal modelng s wdely used n the analyss of hgh degree of freedom of moton equpment, such as the moton analyss gat plannng of bped robot, ndustral robot and space manpulator [].Wth the rapd development of scence and technology and computer technology, mathematcal model has been wdely used n varous felds of socety. Mathematcs plays an rreplaceable role n ths development process, and tself has also been an unprecedented development [2]. Due to the rapd development and popularzaton of computers, the ablty of mathematcs to solve practcal problems s enhanced. he permeaton of mathematcs to dfferent areas, such as socety, economy and nature, expands the contact surface of mathematcs and practce. When usng mathematcal knowledge and computer to solve the practcal problems n varous dscplnes and socal producton, the mathematcal model whch s used to descrbe the problem must be establshed through the analyzng and researchng the practcal problems, then use mathematcal theory and method or programmng calculaton to analyze the model and get the result, and then return to solve the practcal problem [3]. A large number of facts show that masterng mathematcs knowledge s only a necessary condton for applyng mathematcs to solve practcal problem. In the process of realzng mathematcs as a knd of techncal functon, the cultvaton of the skll usng mathematcs to solve practcal problem s manly the cultvaton of mathematcal modelng ablty. Robot technology has become a hot topc snce the 2st century because t has played an rreplaceable role n both lfe servce and engneerng ndustry, and n the aerospace feld [4]. Manpulator s a smplfed robot or automatc control devce, whch s used n the feld of non-manual or nconvenent manual operaton and has a crucal poston n the modern ndustry. he space manpulator can be dvded nto sngle degree of freedom, two degrees of freedom and multple degrees of freedom accordng to the degree of freedom [5], n researches, four degrees of freedom and sx degrees of freedom are used as study objects, for example, Bejng Unversty of echnology studed the moton control mechansm of sx-degree-of-freedom lunar manpulator, and carred out the automated fuzzy control of each jont angle n the course of ts moton [6] ; Space manpulators can be dvded nto lfe-servce manpulators, manpulators operatng under specal workng condtons, and manpulators for other applcatons accordng to dfferent applcatons. At present, the development of mechancal engneerng technology n Chna s extremely rapd, and the applcaton feld of space manpulator s also extended to all aspects of lfe, however, the moton analyss of space manpulator stll has the dsadvantages of low ntellgence, low moton control accuracy and the operatng envronment s greatly lmted [7], so the transformaton technology ready to come out. In ths paper, based on the dea of mathematcal modelng, the knematcs, nverse knematcs and DH algorthm are used to plan the moton trajectory and jont angle sequence of the four-degree-of-freedom manpulator and t uses the fuzzy control prncple, regards C++ as the basc programmng language, carres out smulaton experment for the four-degrees-of-freedom manpulator structure on the Open Inventor. Fnally, the moton analyss results of the manpulator are processed by MAAB, based on the mathematcal modelng, the moton of the space manpulator s analyzed, and the precson membershp functon of the manpulator moton plan after fuzzy processng s obtaned. 2. he establshment of mathematcal model Establshment of Postve Knematcs Modelng Postve knematcs modelng prncples are appled to many areas of automated ntellgent control, not lke robot walkng modelng, all modelngs are based on some enttes or some modelngs that are obvous n lfe [8]. In ths paper, four degrees of freedom of the manpulator are acted as the mathematcal modelng object and mathematcal constrants prototype. he desgn dea s as followng: Frst, to establsh four lnk models whch nclude four rotaton vector and create 4*4 column matrx vector; hen, on the premse that the reference coordnates must be determned to get the resultng rotaton matrx to solve the entre space arm jont angle changes; Fnally, each two jont coordnates are made comparatve study to fnd the relatonshp between them [9]. Fgure. Four-degree-of-freedom manpulator. 3 BOEIM ECNICO DA PEROBRAS

3 Wrst heart Fgure 2. D-H coordnate system. he robotc arm studed n ths paper s a four-degree-of-freedom manpulator wth four rotaton jonts, as shown n Fgure. Accordng to the structural characterstcs of the robot arm, each rotaton jont s adjusted to the calbraton zero poston, and the ntal poston of the manpulator s defned. Usng the D-H representaton to establsh the coordnate system and the coordnate system of each jont (D-H coordnate system s as shown n Fgure 2): wrst poston of the manpulator s the camera box poston, jonts to 3 rotaton drecton ntersect wth each other so as to control the poston of camera box whch decouples the jonts 4 to 6 that control the orentaton of the camera box. Snce the establshment of the coordnate system s not unque, and the D-H matrx s dfferent n dfferent coordnate systems, and the end pose s dfferent. However, for the same base coordnate system, the end pose under the same D-H matrx s the same. Consderng the problem of computatonal complexty, the base coordnate system used n ths paper s as the same as the jont coordnate system. Accordng to the D-H system structure Fgure 2, we can derve the poston of the manpulator s coordnate system where the wrst(vector C) s. a+ Qa 2+ QQ 2 a3+ QQQ 2 3a4 = c () here nto, λ = cosα, µ = snα, c = [ X Y Z ], b = [ abµ bλ ], [ cos sn ] a = a a b cos λsn µ sn Q = sn λ cos µ sn µ λ c c c he above equaton (2) represents the rotaton relatonshp between adjacent coordnate systems., b, a, respectvely, are the arm of the jont angle, offset, length and twst angle parameters. X c, Y c, Z c, respectvely, are the expected locaton of the wrst. Accordng to the defnton of the rotatng coordnate system: a = Qb (3) he equaton () can be turned nto: (2) 2( ) Q b + Q b+ QQ b = Q c b (4) he transformaton matrx of the adjacent jont coordnate system s: = A = Rot( z, ) rans(,, d ) rans( a,,) Rot( x, α ) (5) n n+ n+ n+ n+ n+ n+ A C S Cα S Sα a C S C Cα C Sα a S n+ n+ n+ n+ n+ n+ n+ n+ n+ n+ n+ n+ n+ n+ n+ = Sαn+ Cαn+ dn+ he D-H parameter s substtuted nto the above equaton (6) to calculate the adjacent jont transformaton matrx, and the postve knematcs equaton of the manpulator s obtaned. he total transformaton matrx of the end effector of the robot relatve to the base s: nx ox ax px ny oy ay p y 4 = AAAA = (7) nz oz az pz In ths formula, 4 s the the poston matrx of the end of the manpulator n the base coordnate system. nx ny n z s the drecton vector n the base coordnate system of x 4 axs whch belongs to the actuator axs of the end of manpulator. ox oy o z s the drecton vector n the base coordnate system of a x s whch belongs to the actuator axs of the end of manpulator. ax ay a z s the drecton vector n the base coordnate system of Za axs whch belongs to the actuator axs of the end of manpulator. px py p z s the poston of actuator n the end of the manpulator n the base coordnate system. he Inverse Knematcs Model As descrbed above, nverse knematcs solutons are solved by nverse knematcs based on the known forward knematcs of the manpulator. In ths paper, the end of the manpulator s two degrees of freedom module and the structure s combned by a swayng degree of freedom and a rotatng degree of freedom. Because the rotatng jont does not affect the poston of the end of the grpper n the base coordnate system, only three jont angles are ncluded n the nverse soluton. he nverse knematcs soluton s solved usng the analytcal method, accordng to the left multplcaton equaton of A (6), we can get: A 4 = 4 = AAA (8) hen, nc x + ns y oc x + os y ac x + as y pc x + ps y d nz oz az p z ns x nc y os x oc y as x ac y ps x pc y (9) cos () 2+ 3 sn( 2 + 3) dc 2 2 d4 sn( 2 + 3) sn( 2 3) cos 2+ 3 ds 2 2 d4 cos 2+ + ()() + 3 = SEPEMBER 27 32

4 et (, j ) be the element of lne and column j of the left matrx of the equaton, and (, j) R be the element of lne and column j of the rght matrx of the equaton. Accordng to equaton (9),we can get (3, 4) = (3, 4) and then ps x pc y =. Above all, we can get = arctan( p / p ) or = + 8 o. so can have up to two solutons. Accordng to equaton (9), we can get (2, 3) = (2, 3), (2, 4) = (2, 4) R. hen we can get az = cos( 2 + 3), d2 S2 + d3 cos( 2 + 3). hen, ds s 2 + da 4 z = pa. So, 2 = arcsn( pz da 3 z d2) or 2 = o. So 2 can have up to two solutons. Accordng to equaton (9), (2, 3) = (2, 3). hen,we can get az = cos( 2 + 3), that s, 23 = herefore, 23 = arccos a z or 23 = arccos a 8 o z + ; 3 = From ths we can see, 23 can have up to two solutons, 2 can have up to two solutons. herefore, 3 can have up to four solutons. Snce the nverse knematcs soluton of the manpulator s not unque, the choce of the nverse knematcs of the manpulator n practcal applcaton needs to be determned accordng to the quadrant of the end of the manpulator and the trajectory of the specfc actuator []. he structure of the robot tself and the restrcted workng envronment results to lmtaton of jont angels. he lmtaton of dfferent angles are o o o o o [ 9,9 ], 2 [,9 ], 3 [ 2,2 ]. 3. Establshment of the vrtual smulaton system Introducton In the VC development envronment, the vrtual realty smulaton system of four - degree - of - freedom manpulator was bult by usng Open Inventor. Postve knematcs s used to verfy the correctness of nverse knematcs. Based on the MFC framework and Open Inventor lbrary to develop the software, the angle of the nverse knematcs wth the arm control software s used n the Vsual Studo development platform. he Open Inventor language s smlar to the VRM modelng language, and the mechancal model s bult n the smulated three-dmensonal space. Each component and structure s programmed n a smple language, such as the materal, shape, sze, and texture of the model. he physcal model establshed n the smulaton envronment can be rotated accordng to the requred angle. he control of the model s done by the communcaton between the PC and the personal PC through the communcaton I / O nterface, and t s controlled n the smulaton nterface. he parameter lbrary on the rght defnes the propertes of the jont angle. From the above research process, we can see that the man purpose of ths study s: () postve knematcs R R y x R and D-H algorthm practce; (2) the applcaton of nverse knematcs to fnd the moment of the poston curve. Knematc Verfcaton he smulated smulaton nterface n Fgure 3 s the ntal poston of the knematc model of the manpulator, that s, when all jont angles are degree. One of the rgd bodes n the space has sx degrees of freedom, and t s necessary to comprehensvely defne an object n space by usng sx ndependent nformaton to descrbe the poston of the object orgn n the reference coordnate system relatve to the three reference axs. Fgure 3. Knematcs smulaton system of four-degree-of-freedom manpulator. he smulaton platform s desgned accordng to the geometrcal relatonshp of each module of the manpulator, whch s calculated by the D-H parameter table of the manpulator. x =, y = d = 256.5, z = d2 + d3 = = 47. he data of the ntal poston state of the manpulator s the same, so the calculated poston matrx of postve knematcs s correct. he nverse knematc verfcaton process s as follows: Snce the gven end poston has an unreachable poston, the desred jont poston s calculated by usng the postve knematc smulaton envronment and adaptng the gven jont angle 3, and the angle of each jont s obtaned by usng the nverse knematcs algorthm to carry out practcal experments. Fgure 5 shows the actual poston of the end of the nverse knematc manpulator. When the actual M-Arm manpulator was used, the 2 sets of end grppers were randomly selected from a large number of expermental data and the data of the poston reached. From the data we can see that the maxmum poston devaton does not exceed mm, and the results show that the knematc nverse soluton s correct and relable. Intellgent Fuzzy Processng o verfy the accuracy of analyss procedure track of the space manpulator wth the fuzzy PID ntellgence algorthm on the bass of knematc, and n ths process, ntellgent fuzzy 33 BOEIM ECNICO DA PEROBRAS

5 processng whch has strong robustness and random searchng optmzaton computng technology n overall stuaton wll be made good use of. PID possesses the followng advantages: smple constructon, good stablty, hgh relablty and so on, thus t s especally sutable for the determnstc control system n constructng a precse mathematcal model. Wth the rapd development of control theory and technology, PID stll accounts 9% n the ndustral process control area now. et () = rt () yt () PID controller s a lnear controller, t forms control devaton n accordance wth constant value r(t) and actual output value y(t): et () = rt () yt () () he moton controllng regularty s fulflled when the space manpulator s n a condton of contnuous moton: t de() t u( t) = kp et () + etdt () D + I dt () In the above formula, k p s the proportonalty coeffcent, I s the ntegral tme constant, D s the dfferentatng tme constant. he moton controllng regularty s fulflled when the space manpulator s n a condton of dscontnuous moton: k ek ( ) ek ( ) u( k) = k p e( k) + e( j) + D I j= (2) ( ) ( ) (3) uk ( ) = k ek ( ) ek ( ) + kek ( ) + k ek ( ) 2 ek ( ) + ek ( 2) p d In the above formula, k = kp I, ki = kp, kd = kp D, s samplng perod, k s the samplng number. 4. Analyss of expermental results In analyzng the accuracy of manpulator s moton track, fuzzy controller adopts the form of stream nput and three-polar output wth e and ec as the nput language varables and k p, k and k d as the output language varables. In the quanttatve feld of the nput language varable: set 7 fuzzy subsets: {} whch stands respectvely negatve bg, negatve medum, negatve small, zero, postve small, postve medum and postve bg. hese 7 subsets correspond wth the quantfcaton 7 ranks of the random error e and the rate of change of random error, whch are expressed as {-3, -2, -,,, 2, 3}. he sample trangle membershp functon or gauss membershp functon and the nput functon of the fuzzy control on the moton jont angle of the four-degree-freedom manpulator are shown as Fgure 4. Degree of membershp Degree of membershp e ec Fgure 4. he nput functon of the fuzzy control on the moton jont angle of the four-degree-freedom manpulator. Degree of membershp Degree of membershp Degree of membershp kp k kd Fgure 5 he output functon of the fuzzy control on the moton jont angle of the four-degree-freedom manpulator. SEPEMBER 27 34

6 In the quanttatve feld of the output language varable: set 7 fuzzy subsets: {} whch stands respectvely negatve bg, negatve medum, negatve small, zero, postve small, postve medum and postve bg. he 7 ranks of k p, k and k d, and the output functon of the fuzzy control on the moton jont angle of the four-degree-freedom manpulator are shown as Fgure Concluson he thess frstly proceeds a mathematcal model on the moton process of the four-degree-freedom space manpulator. hen the moton track and jont angle sequence of the four-degree-freedom space manpulator are planed on the bass of knematcs, nverse knematcs and D-H algorthm. At last, accordng to the mathematcal model and the analyss of space manpulator s moton, the accuracy membershp functon of the manpulator s moton plan after beng fuzzy processed s ganed on the bass of C++ programmng language, the smulaton experment on Open Inventor and the analyss results of MAAB. As the output results show, usng the method of mathematcal modelng to analyze space manpulator s moton can mprove the accuracy of space manpulator s moton track and provde a theoretcal bass for the moton analyss of robot and manpulator wth close couplng and multvarant ablty whch possesses a favorable practcal sgnfcance. 6. Acknowledgement Project: he scentfc research project of Yan an Unversty (No. YDK25-47). References [] Q. Y. Jang, J. X. Xe, J. Ye, Mathematcal Modelng (Fourth Edton). Bejng: Hgher Educaton Press, 23. [2] W. Song, Robotcs - Knematcs, Dynamcs and Control. Bejng: Scence Press, 24. [3] D. B. Zhao, J. Q. Y, Introducton to Omn-drectonal Moble Robots. Bejng: Scence Press, 23. [4] R. Fu, H. H. Ju, Real-tme and hgh-accurate nverse knematc control algorthm for decoupled sx-dof manpulator, Computer Measurement & Control, vol. 8, no. 7, pp , July 2. [5] H. C. Jang, S. R. u, Inverse knematcs analyss for 6 degree-of-freedom modular manpulator, Journal of Zhejang Unversty (Engneerng Scence), vol. 44, no. 7, pp , July 22. [6] Y. B., Z.. Jn, Knematc bass analyss of a new 3-DOF hybrd manpulator, Mechancal Drve, vol. 3, no. 6, pp. 6-7, June 24. [7] F. Fang, Smple Applcatons of Ordnary Dfferental Equatons to Mathematcal Modelng. Bejng: Bejng Unversty of echnology, 24. [8] S. Q. Zhu, S. G. u, X. Y. Wang, me-optmal and jerk-contnuous trajectory plannng algorthm for manpulators, Journal of Mechancal Engneerng, vol. 46, no. 3, pp , Aprl 23. [9] G. Z. an, X. Xu, Real-tme and accurate hand path trackng and jont trajectory plannng for ndustral robots, Journal of Central South Unversty (Natural Scence Edton), vol., no., pp. 2-7, January 25. [] S. Y. Yu, S. Q. Kuang Suqong, Nested fuzzy adaptve genetc algorthm, Control Engneerng of Chna, vol. 7, no., pp , January BOEIM ECNICO DA PEROBRAS

Analysis on the Workspace of Six-degrees-of-freedom Industrial Robot Based on AutoCAD

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