Kinematic Analysis of a Novel 3-DOF Parallel Robot with 4 Limbs

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Arnold Morris
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1 Proceengs of the 7th WSES Internatonal Conference on Robotcs Control & Manufacturng Technolog Hanghou Chna prl Knematc nalss of a Novel -DF Parallel Robot wth Lmbs WNG ZHNGFEI QIN XINF JI SHIMING WN YUEHU PN YN The ME Ke Laborator of Mechancal Manufacture an utomaton Zhejang Unverst of Technolog Hanghou Zhejang CHIN whongf@mal.h.j.cn hqanf@6.com jshmng@jut.eu.cn wanuehua@jut.eu.cn bstract: novel -DF spatal parallel robot wth four entcal PRP R knematc lmbs s propose whch belongs to reunant actuate parallel robot an ts knematcs nclung moton propert nverse problem forwar problem an workspace are stue. fter a short escrpton of the novel archtecture then ts knematc moelng s bult through the D-H parametrc notons an the coornate transformaton technque whch verfes that ths novel parallel robot has pure spatal translatonal moton. Follow these analss the nverse an forwar knematc problems are solve wth analtcal close-form ts nverse problem has onl one soluton for each actuate jonts an ts forwar problem has onl one preferre soluton to certan assembl manner. Fnall a case stu s anale b numercal metho nclung the etermnaton of workspace smulaton of the nverse an forwar knematc solutons. Ke-Wors: Reunant ctuate Parallel Robot D-H notatons Knematcs Workspace Introucton The parallel robots (PRs) offer obvous avantages over seral ones such as hgh structural stffness poston accurac an goo namc performance. However the PRs have smaller an rregular shape workspace lower etert. Most of the PRs propose n the lterature have a non-reunant structure. To overcome some of the foregong savantages reunant actuate parallel robots (RPRs) have been esgne []. The RPRs refers to the use of more actuators than are strctl necessar to control the robot but wthout ncreasng the moblt. In general actuators reunanc can be obtane ether b replacng passve jonts of an estng PR wth actve ones or b ntroucng atonal lmbs n an estng sstem. Some researchers have nvestgate ths subject n PRs. Most of ther stues have focuse on planar RPRs [-5] but a few spatal ones [6-]. The -DF reunant actuate spatal translatonal parallel robots (RSTPRs) have been foun n [8-]. Josh an Tsa [] performe a etale comparson between a -UPU an the so-calle Trcept Robot [89] regarng the knematc workspace an stffness propertes of the mechansms. Complete knematc moelng an Jacoban analss of such robots have not receve much attenton so far an are stll regare as an nterestng problem n parallel robotcs research. In ths paper a novel RSTPR s propose whch has a smmetrcal archtecture wth four entcal PRP R topologcal lmbs so an a lmb acts as reunant one when controllng. The novel RSTPR allows fnng analtcal close-form solutons for both nverse an forwar knematc problems. It s of great mportance notce that the forwar knematcs soluton s a ke element n close loop poston control of PRs. tonall t has regular shape workspace whch s an avantage compare wth the other PRs n the lteratures. Descrpton of the Novel RSTPR The CD moel of the novel RPRs s shown n Fg.. It s comprse of the base an the movng platform wth a grpper b means of four entcal lmbs whch has PRP R topolog from the base to movng platform along the lmb. The movng platform s a square. In lterature such archtecture s conventonall calle -PRP R to ncate the sequence of the jonts n the lmbs. For brevt P an R respectvel ncate prsmatc an revolute pars an P stans for a combne jont of a planar -bar parallelogram loop. Its prototpe s shown n Fg.. From Fg. we can see that each lmb of the novel RPR possesses one prsmatc par (P) as actuate jont whch s connecte to the base two revolute pars (R) an one combne jont of a planar -bar parallelogram loop (P ). n the whole the four actuate P jonts are parallel mutuall ther aes are normal to the base an the four connecte ponts of P jonts on the base are verte of a square. The shape of the movng platform s a square. For one of lmbs

Proceengs of the 7th WSES Internatonal Conference on Robotcs Control & Manufacturng Technolog Hanghou Chna prl 5-7 7 a P jont poste n between the frst R jont whch s ajacent P jonts an the last R jont

Z l X a Z Z l Z X θ X θ l θ X θ 5 5 Z 7 Z 6 Z Z 5 7 6 l b X 7 θ 7 X θ 6 X 6 Fg.. The CD moel of the RSTPR Fg.

2 Proceengs of the 7th WSES Internatonal Conference on Robotcs Control & Manufacturng Technolog Hanghou Chna prl a P jont poste n between the frst R jont whch s ajacent P jonts an the last R jont whch s ajacent the movng platform the two R jonts aes are parallel an these aes an the R pars aes of vertees of P jont are perpencular. Z l X a Z Z l Z X θ X θ l θ X θ 5 5 Z 7 Z 6 Z Z l b X 7 θ 7 X θ 6 X 6 Fg.. The CD moel of the RSTPR Fg.. The coornate sstems an the parameters of the lnks an jonts of the -th lmb X 5 coornate sstem of each lmb pont s conce to central of the base the recton of X as s towars an the Z as s normal to the base. tonall we sgn that the character enotes the number of the lmbs.e.. nalssng the -th lmb we arrange the coornate sstems as shown n Fg. ( ). Fg.. protpe of the RSTPR The knematc moelng. The jonts coornate sstems an the D-H parameters s a matter of convenence to escrble the geometrcal parameters of the lnks an jonts n a lmb startng from the base for an lmb we number the lnks sequentall from to n an jonts from to n here n7. Followng Denavt & Hartenberg (D-H) conventon an rules a Cartesan coornate sstem s attache to each lnk. Frstl we assgn a global coornate sstem on the base -XYZ as Table. The geometrcal parameters of the four lmbs No. a j β j j θ j -a θ l π/ l π/ θ l θ 5 l π/ θ 5 π-θ 6 b π/ θ 6 π-θ 7 θ 7 π-θ Note: θ (-)π/. Where all the parameters are shown n Fg.. Base on D-H parametrc notaton for the coornate sstems the lnks an jonts parameters on the four lmbs can be tabulate from Table n whch a j enotes the parallel translaton of orgns along j an j enotes the parallel translaton along j-. The parameter of β j enotes the rotatons from j- to j aroun j an θ j enotes the ratatons from j- to j aroun j-. Here enotes the number of lmbs an j... 7 enotes the number of jonts on a lmb.. The knematc moelng The general matr form of the transformaton

3 Proceengs of the 7th WSES Internatonal Conference on Robotcs Control & Manufacturng Technolog Hanghou Chna prl relatonshps between the local coornate sstems of two ajacent lnks can be erve as follows cα cα sα cα () T + sα sα cα + cα where sα cα an are sn(α) sn(θ) cos(α) an cos(θ) respectvel. B the parameters n Table an the relatonshp of the coornate transformaton () for the frst lmb.e. the D-H transformaton matrces of the all lnks are obtane. Then the poston matr of grpper on the movng platform wth respect to the global coornate sstem s obtane n terms of the frst lmb: 7 e l l l + () where j an j enote sn(θ j ) an cos(θ j ) respectvel e a-b-l. Smlarl for other three lmbs the poston matrces of the grpper on the movng platform are obtane. 7 e l l l l + () e + l+l l () 7 l + l 7 l e + l l + l + (5) Base on the close-chan of parallel robot the poston matrces of the grpper on the movng platform erve from each knematc lmb are equvalent.e (6) s the knematc moelng of the propose PR. Then b ()~(6) we scover that the partton of the orentatonal elements s an entt matr for an lmbs. bvousl when the movng platform moves there s onl pure spatal translatonal moton. In other wors the moton propert of the propose PR has -DF pure translatonal movement so s calle the reunant actuate spatal translatonal parallel robot. Knematc nalss. Inverse Knematcs Suppose that the structural parameters are gven the purpose of the nverse knematcs s to solve the actuate an passve jonts varables from a gven pose of the grpper on the movng platform. For the nverse knematcs of the propose RSTPR the poston coornates ( ) of the grpper are gven but the slng stance of actuate P jonts an all rotatonal jont varables θ j are unknown. B the prevous secton for the propose RSTRPR wth pure spatal translatonal moton we neen t concern the rotatonal pose of the movng platform. From the last column element of each poston matr of the grpper on the movng platform whch ncates the poston thus we get three equatons. Solvng three equatons smultaneousl the actuate varables () are obtane wth corresponng to the gven ( ). For the frst lmb obtan the three equatons from the () : e l l l l + (7) Where -π/ θ j π/ for j. The nterval of θ j for are the same n the followng analses. Conserng the nterval of θ an θ solvng (7) els ± ( l + l ) ( e) (8) Meanwhle we get the epressons of θ an θ from cos θ ( e ) /(l + l ) (9) () l / Smlarl for the secon lmb from the () els ± ( l + l ) ( e) () cos θ ( e ) /(l + l ) () () l / For the thr lmb els

4 Proceengs of the 7th WSES Internatonal Conference on Robotcs Control & Manufacturng Technolog Hanghou Chna prl () ± ( l + l ) ( e) cos θ ( e + ) /(l + l ) (5) (6) l / Then for the forth lmb els ± ( l + l ) ( e) (7) co ( e + ) /(l + l ) (8) (9) l / It can be observe that there are two solutons theoretcall for each actuate varable whch are mrrorng about one plane hence there are totall steen sets of possble solutons for a gven poston of the movng platform. ll possble confguratons are shown n Fg.. In ths paper to enhance the stffness of the manpulator the confguratons from (b) to (o) are elmnate. In the two left the confguraton (a) s corresponng to case of + n (8) () () an (7) an for (p) f -. Here b the moel shown n Fg. the confguraton (a) s chosen thus the propose RSTPR has onl one nverse knematc soluton for each actuate jont. (a) (b) (c) () (e) (f) (g) (h) () (j) ( e) + ( ) (l + l ) () For the thr lmb obtans ( + e) + ( ) (l + l ) () For the forth lmb obtans ( + e) + ( ) (l + l ) () Then the forwar knematc problem s solve b foun solutons of a set of equatons ()~(). Subtractng () from () an () from () respectvel els k + () k k + (5) k where k ( ) / e k ( ) / e k ( ) / e k ( ) / e. Substtutng () an (5) nto () obtan a fourth-egree polnomal n sngle unknown. (6) where the coeffcents epen on the nput varables an geometrcal parameters of the robot B + 6l ( k l ) ( B B + 6 k k l ) B B + B + 6k l BB B B ( k + k + + e l l k e) B ( kk + k k ke ) an B ( k + k + ). Equaton (6) proves at most four solutons for n the comple fel. For an of them a unque value for an ma be obtane va () an (5) n sequence. Thus there are total of sets of an. (k) (l) (m) (n) (o) (p) Fg.. ll possble confguratons of the RSTPR. Forwar Knematcs The purpose of forwar knematcs s to fn the poston of the grpper on the movng platform when the actuate jont varables are gven. For the frst lmb conserng the -π/ θ π/ an -π/ θ π/ from (7) obtan a secon-orer algebra equaton wth respect to unknown varables through elmnatng of θ anθ. ( e) + ( ) (l + l ) () Smlarl for the secon lmb obtans 5 case stu In orer to llustrate the erve nverse an forwar knematc solutons a case stu s mplemente to entf the confguratons an workspace of the robot through a numercal metho. The geometrcal parameters of a RSTPR are amm b5mm l l mm l 5mm an the mamal stroke of actuate P jonts s 5mm respectvel. 5. Inverse knematc smulaton Base on the erve analtcal close-form solutons of nverse knematcs n the prevous secton we can entf the nput values an the unknown varables θ an θ (l ) at the gven poston. Suppose that gung the grpper on the movng platform along a spatal path

5 Proceengs of the 7th WSES Internatonal Conference on Robotcs Control & Manufacturng Technolog Hanghou Chna prl The trajectores of the actuate jont The schemng path of platform The nverse postons solutons an trajectores of actuate jont - The path of the movng platform (mm) (a) Fg Ponts ponts Fg. 6. The nverse knematc smulaton whch as shown n Fg. 5 (a).e. a helcal path the startng pont of schemng path locates at ( -)mm the raus an ptch of hel are mm an mm respectvel an the en pont s at ( 6.5)mm. Then applng the erve analtcal solutons n prevous secton we can fn the corresponng actuate jonts trajectores shown n Fg. 5 (b). - (mm) (a) (b) (mm) (b) The forwar knematc smulaton Workspace mm mm mm (mm) (mm) -5 (mm) -8 () (mm) - (mm) From the forwar knematc solutons n prevous secton (6) proves at most four solutons for n the comple fel an ma be obtane va () an (5) thus we can get four sets of possble solutons for an. lthough the number of solutons s conserabl much t can be shown that onl one soluton feasble an the preferre soluton can be etermne b the assembl manner. In orer to llustrate the erve forwar knematc solutons takng the mentone geometrcal parameters above let an. Then the polnomal (6) becomes Forwar knematc smulaton (a) -D vew (7) whch has four solutons for where suppose that the values of have no meanng that are an.95 respectvel. It s clear to see that an are brush off. The s onl preferre soluton n terms of the reference confguraton of Fg.. Then n orer to verf further other confguratons take a trgonometrc sne curve as each actuate jont nput trajector whch are shown n Fg. 6 (a).e. 6 6 sn(πt) 6 6 sn(πt/) sn(πt) an 6 +6 sn(πt/) respectvel. Base on above analss an the forwar knematc solutons n prevous secton we obtan the path of the grpper on the movng platform shown n Fg. 6 (b) Fg (b) Top-vew The workspace of a RSTPR Workspace The workspace of a PR s one of the most mportant aspects to reflect ts performance an t s necessar to anale the shape an volume of the workspace for enhancng applcatons of PRs. The reachable workspace bounar an volume of a RSTPR are etermne b a numercal metho n terms of ts nverse knematc solutons n prevous secton. Equaton ()~() represent the workspace of the -th () lmb whch s a set of clners wth the ra of (l+l). The robot workspace can be 7

6 Proceengs of the 7th WSES Internatonal Conference on Robotcs Control & Manufacturng Technolog Hanghou Chna prl erve geometrcall b the ntersecton of the four lmbs workspace. ssume that an s the ntal confguraton of a RSTPR. To etermne the bounar of the workspace we aopt a smpl numercal search metho that s so-calle the laere left-rght search algorthm (LLRS). The algorthm central ea that s the entre workspace s ve nto man laers along as each laer s ve nto some sectors an the sectors raus are etermne va nverse knematcs an some constrant of jonts so the sectors bounar an volume are etermne. The workspace volume s appromatel calculate as the sum of all sectors volume. Takng the mentone geometrcal parameters above the workspace an volume of the RSTPR are shown n Fg. 7. It s obvous that there s regular shape whch s a cubcal appromatel. tonall t has the workspace avantage along as f the actuaton stroke s long enough. 6 Concluson In ths paper a novel reunant actuate PR wth four entcal PRP R topolog lmbs s propose. fter a short escrpton of the novel archtecture ts knematc moelng s bult through the D-H parametrc notons an the coornate transformaton technque whch verf that ths novel PR has pure spatal translatonal moton. Follow these analss the nverse an forwar knematc problem are solve wth analtcal close-form. Fnall a case stu s anale b numercal metho nclung the etermnaton of workspace smulaton of the nverse an forwar knematc solutons. We can conclue that: () The well known D-H parametrc notons an the coornate transformaton technque can be apple n the analss of moton propertes of a PR. () The propose RSTPR has onl one nverse knematc soluton wth analtcal close-form for each actuate jont. lso ts forwar knematc soluton has onl one feasble soluton b the assembl manner. These avantages are ke ssues for path an trajector plannng an real tme control. () The workspace of novel RSTPR has regular shape whch s a cubcal appromatel an t has the workspace avantage along as s for the future works of ths novel RSTPR we propose the namc moelng an control ssues for polshng applcaton. cknowlegment The authors apprecate the fun support from Natonal Natural Scence Founaton of Chna (No ) an Zhejang Provnce Natural Scence Founaton (No. M599). References: [] Merlet J. P. Reunant Parallel Manpulators Laborator Robotcs an utomaton Vol. 8 No. 996 pp. 7. [] Km S. peratonal Qualt nalss of Parallel Manpulators wth ctuaton Reunanc Proc. of the IEEE Int. Conf. on Robotcs an utomaton lbuquerque New Meco 997 pp [] Kock S. an Schumacher W. Parallel - Manpulator wth ctuaton Reunanc for Hgh-Spee an ctve-stffness pplcatons Proc. of the IEEE Int. Conf. on Robotcs an utomaton Leuven Belgum 998 pp. 95. [] Lu G. F. et al. nalss an Control of Reunant Parallel Manpulators Proc. of the IEEE Int. Conf. on Robotcs an utomaton Seoul Korea pp [5] Lao H. et al. Sngulart nalss of Reunant Parallel Manpulators IEEE Int. Conf. on Sstems Man an Cbernetcs pp. -. [6] Sajaan H. an Taghra H. D. Knematc analss of the hraulc shouler: a -DF reunant parallel manpulator IEEE Int. Conf. on Mechatroncs & utomaton Nagara 5 pp. -6. [7] Km J. et al. Desgne an nalss of a Reunantl ctuate Parallel Mechansm for Rap Machnng IEEE Trans. Robotcs an utomaton Vol.7 No. pp. -. [8] Sclano B. The Trcept Robot: Inverse Knematcs Manpulablt nalss an Close-loop Drect Knematcs lgorthm. Robotca Vol pp [9] Josh S.. an Tsa L. W. The knematcs of a class of -DF -legge parallel manpulators SME J. Mech. Desgn Vol. 5 No. pp [] Josh S.. an Tsa L. W. comparson stu of two -DF parallel manpulators: one wth three an the other wth four supportng legs IEEE Trans. on Robotcs & utomaton Vol. 9 No. pp. -9.

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