Modelling and Analysis of Planar Robotic Arm Dynamics Based on An Improved Transfer Matrix Method for Multi-body Systems

Transcription

1 he 4th FoMM World ongress, ape, awan, ctober 5-3, 5 D Number:.6567/FoMM.4H.W.S3.46 Modellng and Analyss of Planar Robotc Arm Dynamcs Based on An mproved ransfer Matrx Method for Mult-body Systems W. hen H.L X.Z.Zhang anjn Key Laboratory of the Desgn and ntellgent Academy of Mltary Medcal Scences ontrol of the Advanced Mechatroncal System anjn, hna anjn, hna Abstract: An mproved transfer matrx method for mult-rgd-body systems s presented based on the structural characterstcs of the manpulator and the relatons between moton transfer and force transfer. hus, a new model applcable to seral robots s obtaned by usng of dscrete tme transfer matrx method. Based on the mproved method, a unfed model for a manpulator wth n DF s derved. hs model contans both knematc and dynamc characterstcs of the system. Accordng to the moton transformaton relaton between the jont space and the operatonal space of the manpulator, the soluton and the procedure are developed for both forward and nverse knematcs of the unfed model. he smulaton results of a DF planar manpulator show the rghtness of the modellng method and exhbt ts superorty to other methods. Keywords: Mult-body system, Planar manpulator, An mproved transfer matrx method, Dynamcs modelng, Forward knematcs, nverse knematcs. ntroducton Wth the rapd progress of scence and technology, robots are frequently beng appled n areas such as ndustral automaton, mltary applcatons, semconductor manufacture and space exploraton. Seral jont robots are constructed by smulatng the human forearm. her advantages nclude flexblty of moton, a compact structure and hgh versatlty. hey are the most commonly used structures n robots, ncludng well-known products such as the Programmable Unversal Machne for Assembly (PUMA) and the Selectve omplance Artculated Robot Arm (SARA) []. A seral jont robot s a complex system wth multple degrees of freedom, multple varables, a hgh degree of non-lnearty and mult-parameter couplng. he knematcs and dynamcs modellng and analyss of a robot are the foundaton of the study. he Denavt-Hartenberg (D-H) method s a general method for the study of robot knematcs[]. Based on the D-H method, Yang W.L.[3] studed the knematcs modellng of a notched contnuum manpulator ncludng the mechancs-based forward knematcs and the curve-fttng-based nverse knematcs. n order to establsh the forward knematcs model by usng D-H procedure, the complant contnuum manpulator featurng the hyper-redundant degrees of freedom s smplfed nto fnte dscrete jonts. Al-Mashhadany Y.. [4] presents a cogntve archtecture for soluton of nverse knematcs problem (KP) of 6-DF elbow manpulator wchen@tjut.edu.cn, lhao4_989@6.com, 56787@sohu.com wth sphercal wrst by Locally Recurrent Neural Networks (LRNNs) and smulated the soluton by usng MALAB/Smulnk. Whle common dynamcs modellng methods for manpulators nclude the Lagrange method, the Newton-Euler equaton and the Kane equaton. Korayem M.H.[5] studed the path plannng dynamc model of three jonts weghtlftng robot usng Lagrange method. XE L.M and HEN L.[6] analyed the knematcs and dynamcs of the system by use of momentum conservaton. Wth the dea of an augmentaton approach, the dynamc equatons can be lnearly dependent on a group of nertal parameters. Allan J.F.[7] studed the geometrcal model of the new manpulator, an analytcal soluton for the nverse knematc equatons, the dynamc model used to compute torques/forces at the actuators, and smulatons performed wth MALAB and AA. Yaskevch N. [8] presented a new recursve algorthm based on Kane equatons matrx form for closed form dynamcs equatons calculaton for smple knematcal chan. However, n the studes of mathematcal models of mult-jont seral robotc arms, almost all methods are amed at system knematcs or dynamcs separately. And these two types of the models can hardly be unfed n the descrpton of one mathematcal model. n fact, the knematcs and dynamcs usually are nterrelated each other wthn the study of a manpulator. n ths paper, an mproved dscrete tme transfer matrx method for mult-body systems s frst presented and a mathematcal model establshed of a mult-jont seral robotc arm n the planar moton accordng to ts structural characterstcs. he equatons thus developed descrbe both the knematcs and the dynamcs. Next, based on ths unfed model, and by settng approprate boundary condtons, a soluton method s developed. Fnally, numercal smulaton s used to valdate the effectveness of the method.. he mproved Dscrete me ransfer Matrx Method for Mult-body Systems he dscrete tme transfer matrx method for mult-body systems s a knd of dynamcs method based on the classc transfer matrx method. t s applcable to general mult-body systems [9]. ts modellng prncple s to perform the lnearaton and the dscretaton of the dynamcs equatons of the components n the tme doman by usng a numercal ntegraton method. Moreover, the prncple assembles of the mult-body system transfer equatons and transfer matrces to calculate system moton by gradual ntegraton. herefore ths approach has the characterstcs of low matrx order, low computaton complexty, hgh

2 modellng flexblty, etc. A mult-jont seral manpulator s a typcal mult-body system, whch has features of tme varablty, non-lnearty and wde range of moton. n ths paper, a unfed knematcs and dynamcs model of a planar manpulator wth n rotaton jonts s developed usng the mproved dscrete tme transfer matrx method for mult-body systems. he modellng theory of ths method s explaned brefly n the followng secton. A. Modellng theory Accordng to the modellng dea of the dscrete tme transfer matrx method for mult-body systems, the state vector descrbng the planar moton of a rgd body s defned as[9] x, y,, m, q x, q,] () [ y where x and y are poston s of the connectng pont wth respect to the nertal reference system, s the angular dsplacement of the rgd body rotatng around -axs, m s the of the connectng pont nternal moment at the -axs, and q and q are s of x nternal force at the connectng pont n the nertal system. y o y o ( ) y θ Fg.. A Rgd Body wth Planar Moton As shown n Fg., a rgd body wth planar moton has one nput end and one output end; the nput end s, output end s, and the centre of mass s. n the dagram, the nerta system s oxy; n the body-fxed system x y, whose orgn s fxed at nput end, the of the output end s (b, b ), the s of the centre of mass are ( c, c ). Because the system s rgd body, the body-fxed system o x y s obtaned by rotatng the translatonal system θ degrees as follows[9] x y x y x y x y x y y () x y x y x y c c c c c c b c b s c s c s c b s b c where s sn and c cos. When the above method s used to deduce the transfer x c x x (3) (4) relaton between the state vectors of moton for the nput end and output end of the rgd body, the numercal ntegraton method should be determned frst to perform dscretaton and lnearaton on the non-lnear terms. ommon numercal ntegraton methods nclude Euler s method, the Newmark-β method, the Wlson-θ method, and the Houbolt method. n ths study, the Newmark-β method s used to perform the dscretaton on the non-lnear functons, such as sn θ and cos θ, n the model [9]. Because the effects of an external force and an external moment exerted on a rgd body are equvalent to an external force and a moment exerted at the rgd body mass centre, the rgd body s treated as f the external forces and the external moments are all exerted at the centre of mass. he moton equaton for the centre of mass s as follows mx my q q x, y, q q x, y, f f x, y, where m s the mass of the rgd body, x and y are the poston s of the centre of mass relatve to the nertal system, q x, and q y, are the nput end nternal forces, q x, and q y, are the output end nternal forces, and f x, and f y, are the external forces exerted at the centre of mass of the rgd body. he moment of momentum theorem s appled to the rgd body as follows J mx y my x m m m qx, y qy, x fx, y f y, x where J s the absolute moment of momentum of the rgd body relatve to the orgn of the body-fxed system ; J s the rotaton nerta of the rgd body relatve to ; and s the absolute angular acceleraton of the rgd body. Equaton (4) s lneared, equaton (3) s substtuted n equaton (5), and then lnearaton s performed on the temporary equaton. he temporary equaton s substtuted nto equaton (6). he lneared result s expressed n matrx format, resultng n the transfer equaton of a rgd body wth one nput end and one output end (5) (6) U (7) where and are state vectors, as defned n equaton (), and the transfer matrx U s [9] U u4 ma u 4 ma y x my mx ( t ( t u 43 ( t ) ) ( t ) ) u 45 u 46 bg bg b G bg u47 u 57 u67 (8) he above development yelds the relatons between rgd bodes wth planar moton. he matrx n equaton (8)

3 descrbes the geometrcal relaton, the moton relaton and the force relaton between rgd bodes [9]. he frst three lnes (-3) n the matrx descrbe the knematcs relatons of the rgd bodes wth planar moton, and the next three lnes (4-6) represent the force and moment stuatons durng the rgd body moton,.e., the rgd body dynamcs. Y 5 n- j n n+ B. he mproved dscrete tme transfer matrx method n the case of a mult-jont seral planar manpulator, the motor provdng the drvng force s located at jont, causng the angular dsplacements before and after jont + to dffer, as shown n Fg.. f the angular dsplacement of jont s θ, then the angular dsplacement of jont + adds θ' on the bass of the angular dsplacement θ of the prevous jont. herefore, the angle θ' s the rotaton angle of jont + relatve to jont. Hence, when calculatng the angular dsplacement of the transfer matrx, θ +=θ +θ' should be satsfed. θ + + θ' θ + Fg.. Dagram of Jont Moton Angular Dsplacement he exact dfference n ths angular dsplacement results n dfferent values of jont angular dsplacements for the nput end state vector of the (+)-th rgd body and the output end state vector of the -th rgd body. n the transfer matrx, ths relaton can be descrbed as where U U.. he mproved Dscrete me ransfer Matrx Method for Mult-body Systems f a planar robotc arm wth n rotatng jonts s consdered as the study subject, and the jonts and lnks of the manpulator are treated as rgd bodes n plane moton wth one nput end and one output end, then the entre robot s a chaned system composed of n rgd components, as shown n Fg. 3. n the dagram,, 3,, n- denote the jonts of the arm,, 4,, n represent the lnks of the arm, and n+ s end effector of the arm. (9) 3 4 Fg. 3. An n-jont Seral Planar Robotc Arm A. Jont model Under normal crcumstances, the rotatng jont of a arm s drven by a motor. n the model, the jont s smplfed as a central rgd body, whose nput end s the centre poston and the output end s the entre crcumference of the central rgd body. he central rgd body has rotaton moton only,.e., no translatonal moton. Accordng to the descrpton n secton., every jont experences a sudden change n angular dsplacement compared wth the prevous lnk connected to t; thus the calculated transfer matrx s multpled by U shown n equaton (9), n the front, and the transfer equaton s where j,3,,n. j U U () j, j j j j, j B. Lnk model he lnk of the robotc arm s a rgd body wth planar moton. n the model, t s a homogeneous lnk that s fxed to the jont and moves wth t. ts nput end, closer to the base, s the jont s output end; and ts output end, s the end farther from the base. A lnk of the robot experences rotaton moton around the jont and transport-nduced translatonal moton, resultng n a transfer equaton as follows where j,4,, n. U () j, j j j, j. verall model of the robotc arm Accordng to above analyss, a seral robotc arm wth the planar moton s composed of n jonts and n lnks. When a jont and a lnk are vewed as ndependent rgd bodes, the entre system comprses n rgd bodes. Because these rgd bodes are all connected n seres, accordng to the mproved transfer matrx method of mult-rgd-body systems, the overall transfer matrx of the robot s the product of the transfer matrces of n rgd bodes,.e., the unfed knematcs and dynamcs model of the entre robotc arm s where he n,n Uall, () U all s the overall transfer matrx of the system. can be expanded as U all X

4 U all U U UU U UU U U (3) n n n n 4 3 U where U k ( k ~ n ) s determned by equaton (8) and U k ( k ~ n) s determned by equaton (9). As descrbed above, equatons () and (3) compose the n-jont unfed knematc and dynamc model for a seral robotc arm. V. Soluton of the Model n the study of robot knematcs, f the jont varables are known, the computaton of poston and atttude of the end effector s the forward knematcs; conversely, for nverse knematcs, the end effector pose s known and the jont varables are the soluton. Whle n the study of dynamcs, f the external forces are known, the soluton for system parameters s forward dynamcs; conversely, when external forces are to be determned, t s the nverse dynamcs. n the case of the manpulator, the task of nverse dynamcs analyss s to determne the drvng moment to generate a certan movement, whch s mportant both for the dynamcs and the control of robots[]. n the case of the mult-jont seral arm shown n Fg., knematcs study mples two possble stuatons: one s to obtan the nformaton of the end effector by usng known jont moton law,.e., the forward knematc soluton; and the other s to obtan the jont varables from the known moton of the end effector, whch s an nverse knematc soluton. Usng the mult-body system dscrete tme transfer matrx modellng method n any of the above stuatons requres boundary condtons for equaton () to be set accordng to known and unknown condtons. hen, through transfer matrx teraton, unknown parameters for all the state vectors are determned. hese parameters contan all the poston and force nformaton for the jonts and lnks of the arm; thus, the soluton procedure s completely dfferent from the soluton for pure knematcs. Solutons for both stuatons above mentoned contan the nverse dynamcs problem analyss,.e., the combnaton of the knematcs problem and the nverse dynamcs problem. he two dfferent stuatons are descrbed n the followng. A. Soluton for forward knematcs usng the model f the moton law s gven for each jont of the manpulator shown n Fg., the angular dsplacement of each rgd component n the system s a known parameter due to a fxed connecton of the jont and the lnk. herefore, the system boundary condtons are,,,, m n, n x,, n,n, q, y x,, q n,n y,,,, n,n,,,, n, n (4) As shown n equaton (4), the angular dsplacements of rgd body and the end effector,.e.,, and n, n, are known parameters. he nput end dsplacements x and y of rgd body are always equal to and the nternal forces q and q, and the nternal moment m, are x, y, unknown parameters. For the end effector n+, the translatonal dsplacements xn, n and yn, n are unknown parameters, and the nternal force and nternal moment are equal to. he unknown parameters n vectors, and n, n can be calculated by substtutng equaton (4) nto equaton (). hen, wth the transfer equaton for each rgd component, the state vector of the connectng pont for each jont and lnk n the system can be obtaned. Furthermore, through lnearaton, the speed, the acceleraton, the angular velocty and the angular acceleraton of each pont can be determned. Detals of the modellng and soluton procedure are shown n Fg. 4. start Dsassemble system;ntale parametersθ,etc.; determne,,, n+,n alculate component x, y,x,y, determne each component U and =U Assemble and abtan U all, n,n+ =U all, Unknown parameter n n,n+ and, at tme t Unknown parameter n state vectors,,3 n,n- at tme t t m End m= t m < m=m+ Fg. 4. Soluton Procedure for the Unfed Model and Forward Knematcs of A Manpulator B.Soluton for nverse knematcs and model f the moton law for the end effector of the manpulator shown n Fg. s known, then the prmary focus of the soluton s to determne the moton law for the jonts. hs s because the moton and the force for every pont n the rgd body,.e., every element n the transfer matrx, are functons of the angular dsplacement parameter. Here boundary condton n equaton (4) ncludng the angular dsplacement of the nput end of jont,,, the nternal forces q x, and q y,, and the nternal moment m, are all unknown parameters. he rotaton angle n, n of the end effector s also an unknown parameter. he translatonal dsplacements xn, n and yn, n are known parameters, and the nternal force and nternal moment are both equal to. he soluton procedure s to substtute the system ntal boundary condton n equaton (4) and to calculate the ntal angular dsplacements of each jont at the moment of ntalaton. hen, accordng to the procedure shown n Fg.4, the known condton from equaton () s substtuted to determne all nformaton of the state vector for the robotc arm. n addton, the parameters of the

5 transfer matrx at ths moment and the boundary condton at the next moment are used to calculate the jont angular dsplacement and the state vectors of the unknown parameters at the next moment. he state vectors for all the jonts and lnks connectng ponts n the system at all moments can be obtaned by these teratons. Furthermore, through lnearaton, the speed, acceleraton, angular velocty and angular acceleraton for every pont are determned. he modellng and soluton procedures are shown n Fg.5. start Dsassemble system;ntale parameters; determne,,, Assemble and Uabtan all, =U all n,n+ n+,n alculate component x,y,x, y, determne each component U and =U, central rgd body (, 3) lnk (, 4) mass m (kg) =,3 radus r (m) =,3 mass centre (x,y )(m) =,3 nput (x,y ) (m) =,3 output (x,y ) (m) =,3 mass centre rotaton nerta J (kg.m ) =,3. (, ) (, ) (r, ).5 mass m j (kg) j=,4 length l j (m) j=,4 mass centre (x j,y j ) (m) j=,4 nput (m) (x j,y j )j=,4 output (x j,y j ) (m) j=,4 mass centre rotaton nerta j (kj g.m ) j=,4 (.5, ) (, ) (, ) /3 ABLE. he l parameters of a DF manpulator n ABLE, m s the mass of the rgd body. r s the radus of the central rgd body. l ndcates the length of the lnk. (x,y ), (x,y ) and (x,y ) are the mass centre, nput and output of rgd body respectvely. J s the mass centre rotaton nerta of the rgd body. Unknown parameter n and at tme n,n+, m=m+ m= Unknown parameter n state vectors,,3 n,n- at tme t t parallel to the nerta system x-axs, then the alculate rotaton angles of jonts andmoton law for jont and jont s n each state t vector ; at tme accordng to boundary condton + at tme t t m < t t m d t t sn t End d t Fg. 5. Soluton Procedure for nverse Knematcs and nverse Dynamcs (5) of A Manpulator V. Numercal Smulaton and Analyss o valdate the mproved mult-body system dscrete tme transfer matrx method, numercal smulatons are performed on a DF manpulator wth planar moton as shown n Fg.6. he structural parameters of the arm are shown n ABLE. Based on the moton stuatons of the arm, two stuatons are smulated. A. Soluton for forward knematcs model and numercal smulaton he soluton for the forward knematcs model s llustrated by the procedure shown n Fg. 4. At ths moment, the jont moton stuaton of the manpulator s known and the pose and dynamcs parameters of the end effector are unknown. f the arm ntal poston s two lnks n a straght lne, d where (=, ) s the expected value of the jont moton, d d 3, s the overall duraton of the moton, and are the moton ntalsaton and moton termnaton delay tmes, respectvely, = ms, =5s and '= ms..5 Y X Fg. 6. Dagram of A DF Manpulator wth Planar Moton m m Fg. 7. Robotc Arm Moton onfguraton Dagram

6 Dsplacement n the x-drecton x/m x x x3 x4 x_robot x_robot x3_robot x4_robot Fg. 8. Dsplacement of the Rgd omponents for the Manpulator n the x-drecton Notably, when calculatng the speed x, x, x, y, y, and y n equatons (3) and (4) at every pont, the mplct functon θ of tme t should be calculated durng the soluton because θ of every jont s a functon of tme t. Fg.7 shows the confguraton dagram of jont moton for the entre manpulator. Fg.8 and Fg.9 are dsplacements curves n the x- and y-drectons, respectvely. For manpulator rgd bodes wth -4 output ends, the data are calculated by the mproved transfer matrx method, and shown as x, x, x 3, x 4 and y, y, y 3, y 4 n the dagrams. o valdate ths method, the dagrams also show dsplacements curves for each component calculated by the D-H method, shown as x _robot, x _robot, x 3_robot, x 4_robot and y _robot, y _robot, y 3_robot, y 4_robot n the dagrams. alculaton results from the two methods match perfectly. Dsplacement n the y-drecton y/m y y y3 y4 y_robot y_robot y3_robot y4_robot Fg. 9. Dsplacement of the Rgd Bodes of the Manpulator n the y-drecton Fg. shows the angular dsplacement for the moton of each rgd body of manpulator. Jont and lnk have dentcal rotaton angular dsplacements, and jont 3 and lnk 4 have dentcal rotaton angular dsplacements. Fg. shows the nternal moments for all the rgd bodes. n ths dagram, m and m 3 are the drvng moments exactly to mplement moton n jonts and 3. Fg. and Fg.3 are nternal forces for the rgd bodes n the x- and y-drectons, respectvely. he data show that, the rgd body stress the maxmal forces q x and q y, whch are equal to the values exerted on the nput end of rgd body. Angular dsplacements / o Fg.. Angular Dsplacements of the Rgd Bodes of the Manpulator nternal moments m / N.m m m m 3 m Fg.. nternal Moments of the Rgd Bodes of the Manpulator nternal forces n the x-drecton qx/n q x q x q x3 q x Fg.. nternal Forces of the Rgd Bodes of the Manpulator n the x-drecton nternal forces n the x-drecton q y /N q y q y q y3 q y Fg. 3. nternal fforces of the Rgd Bodes of the Manpulator n the y-drecton

7 B. Soluton of nverse knematcs and Numercal smulaton he procedure for the nverse knematc numercal soluton s shown n Fg.5. At ths moment, the pose of the end effector s known, but the jont moton and ts dynamcs parameters are unknown. n ths case, the orgnal pont of the moton of the end effector s (.3, ), movement n the x-drecton s fxed,.e., x x start xend. 3, and moton n the y-drecton s defned as y.t.t.t.t t.( y start. ).8. y y y (6) where y start ; s, 4s, and 5s. Fg.4 shows the moton confguraton dagram of the manpulator. Fg.5 shows the dsplacement, speed and acceleraton of the end effector n the y-drecton. Fg.6 and Fg.7 show dsplacements of each rgd component n the x- and y-drectons, respectvely. he results agree wth the calculatons performed usng the robotcs method, whch valdates the rghtness of the mult-body system transfer matrx method. Fg.8 shows the angular dsplacements for every rgd body moton. he data show that the jont and lnk have consstent angular dsplacement moton; the jont 3 and lnk 4 have consstent angular dsplacement moton. Fg.9 shows the nternal moments for all the rgd bodes. n the Fg.9, m and m 3 are two jonts drvng moments. Fg. and Fg. show the nternal forces of all the rgd bodes n the x- and y-drectons, respectvely. he data show that rgd body are exerted on the maxmal forces q and q. m Dsplacement, speed and acceleraton of the robotc arm end n the y-drecton m Fg. 4. Manpulator Moton onfguraton Dagram y_end v_end a_end Fg. 5. Dsplacement, Speed and Acceleraton of the Manpulator End n the y-drecton x y Dsplacement n the x-drecton x/m x x x3 x4 x_robot x_robot x3_robot x4_robot Fg. 6. Dsplacement of the Rgd Bodes of the Manpulator n the x-drecton Dsplacement n the y-drecton y/m y y y3 y4 y_robot y_robot y3_robot y4_robot Fg. 7. Dsplacement of the Rgd Bodes of the Manpulator n the y-drecton Angular dsplacements / o Fg. 8. Angular Dsplacements of the Rgd Bodes n the Manpulator nternal moments m / N.m Fg. 9. nternal Moments of the Rgd Bodes n the Manpulator m m m 3 m 4

8 nternal forces n the x-drecton q x /N 3 - q x q x q x3 q x Fg.. nternal Forces on the Rgd Bodes of the Manpulator n the x-drecton nternal forces n the x-drecton q y /N.4. q y.6 q y. q y3.8.4 q y Fg.. nternal Forces on the Rgd Bodes of the Manpulator n the y-drecton V. onclusons n hs study, a seral manpulator wth multple degrees of freedom s consdered as the subject. Based on an mproved dscrete tme transfer matrx method for mult-body systems, a new unfed model s developed, ncorporatng both knematcs and dynamcs. A soluton method s also provded for ths model. he soluton s tested on a DF arm through a numercal smulaton. he followng conclusons are reached: () Accordng to the structural characterstcs of a seral robot and the relaton between moton transfer and force transfer, the mult-body system transfer matrx method s mproved to obtan a mult-rgd-body dscrete tme transfer matrx equaton applcable to a robotc arm. By usng ths equaton, an n-jont seral planar mult-body system dynamcs model for a manpulator s derved. hs model covers both the knematcs and dynamc characterstcs of the system; () Usng ths unfed model, and applyng the two dfferent moton stuatons,.e., jont moton and end effector moton, the correspondng boundary condtons are set. A soluton method and procedure for the forward knematcs problem and the nverse knematcs problem are proposed. n addton, a soluton method and procedure for the nverse dynamcs of the model are proposed; (3) he knematcs and dynamcs numercal smulaton of a DF planar manpulator s completed. he results show that dsplacements n the x- and y-drectons and the angular dsplacements of the rgd bodes n the arm agree wth calculated results performed by a classc robotcs method. hs proves the valdty and feasblty of the model. he model soluton also yelds characterstcs of the dynamcs, whch llustrates the superorty of the proposed model to prevous methods. n summary, the mproved dscrete tme transfer matrx method for mult-body systems proposed n ths paper provdes a new method and a new approach for robot modellng. A mult-body system model of a multple degrees-of-freedom robotc arm based on ths method unfes the study of system knematcs and dynamcs, and provdes a theoretcal foundaton for the further study of robot knematcs and dynamcs characterstcs. he results from the numerc smulaton of the DF planar manpulator llustrated the valdty and feasblty of the proposed modellng method and the superorty of the model over exstng methods. Acknowledgment he authors would lke to acknowledge the fnancal support of the natonal scence foundaton of chna (347, ), hna Postdoctoral Scence Foundaton (Mltary System: M57), and anjn Key Laboratory of the Desgn and ntellgent ontrol of the Advanced Mechatroncal System. References [] Wang X.B. Research on Structure System Dynamcs and haracterstcs of Engneerng Mechancal Arm. Zhejang(PR hna): Zhejang Unversty, 4. [] Xong Y.L., Dng H. And Lu E.. Robotcs. Bejng(PR hna): Machnery ndustry Press, 993. [3] Yang W.L, Dong W. and Du Z.J. Knematcs Modelng for a Knematc-mechancs ouplng ontnuum Manpulator. n nternatonal onference on Manpulaton, Manufacturng and Measurement on the Nanoscale (3M-NAN), ape(w), pp , 4. [4] Al-Mashhadany Y.. nverse Knematcs Problem (KP) of 6-DF Manpulator by Locally Recurrent Neural Networks (LRNNs). n nternatonal onference on Management and Servce Scence (MASS), Wuhan(PR hna), pp.-5,. [5] Korayem M.H., Abbas Esfeden R. and Nekoo S.R. Path Plannng Algorthm n Wheeled Moble Manpulators Based on Moton of Arms. Journal of Mechancal Scence and echnology, 9(4): , 5. [6] Xe L.M. and hen L. Robust and adaptve composte control of space manpulator system wth bounded torque nputs. Engneerng Mechancs, 3(3): , 3 (n hnese). [7] Allan J.F., Lavoe S., Reher S. and Lambert G. Knematc and Dynamc Analyss of A Novel 6-DF Seral Manpulator for Underground Dstrbuton Power Lnes. n EEE/RSJ nternatonal onference on ntellgent Robots and Systems. San Francsco, A(USA), pp ,. [8] Yaskevch N. Recursve Algorthm for Smulaton of Sngle han Manpulator Dynamcs. n EEE/ASME nternatonal onference on Mechatronc and Embedded Systems and Applcatons,Sengalla, Ancona(taly), pp.-5, 4. [9] Ru X.., Yun L.F. and Lu Y.Q., etal. ransfer matrx method of mult-body system and ts applcaton. Scence Press (PR hna): 8. [] Hong J.Z. omputatonal dynamcs of mult-body systems. Hgher Educaton Press (PR hna): 999.