Advances in Military Technology Vol. 10, No. 1, June 2015

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1 AM Avance n Mltary echnology Vol., No., June 5 Mechancal an Computatonal Degn for Control of a -PUS Parallel Robot-bae Laer Cuttng Machne R. Zavala-Yoé *, R. Ramírez-Menoza an J. Ruz-García ecnológco e Monterrey, Ecuela e Ingenería y Cenca, Calle el Puente, Ejo e Hupulco, 438 Mexco Cty, Mexco. he manucrpt wa receve on December 4 an wa accepte after revon for publcaton on 9 June 5. Abtract: A -PUS parallel robot moelle, egne an controlle n term of Newton Euler equaton n orer to be mplemente numercally. Drect an nvere knematc a well a rect an nvere ynamc are analye an olve. Drect knematc olve by ntroucng a novel numerc-geometrc metho here referre to a metho of arc. Next, rect an nvere ynamc problem are olve offerng avantage tratonal metho o not. wo type of controller were mplemente n orer to get a ere performance. wo 3D robot egn are alo hown. Numercal computaton an mulaton were evelope n MALAB. he whole egn an control converge to a laer cuttng machne applcaton whch gven at the en of the ocument. Keywor: Parallel robot, Moellng, Control, Laer applcaton.. Introucton. Laer procee are wely ue n cvl an non-cvl nutre. Markng an cuttng wth laer are typcal applcaton of th. Snce parallel robot have hgh payloa an are mechancally robut [9, ], t propoe here to egn one of thee robot for th effect. Current parallel robot come from the Stewart-Gough platform whch wa egne n the evente an ha been ubject of numerou tue. However, lttle ha been nvetgate about the fferent varant of t knematc chan, n th cae the - PUS robot. Of the few who have tue th robot, mot have one t n a very general way [, 5, 7,, ]. One avantage that th moel (-PUS) ha over the tratonal -UPS that the actuator reman fxe on the bae. In ome cae, th mplfe the * Correponng author: ecnológco e Monterrey, Campu Cua e Méxco, Puente, Ejo e Hupulco. Méxco 438. Phone: , fax: , E-mal: rzavalay@tem.mx.

2 3 R. Zavala-Yoé, R. Ramírez-Menoza an J. Ruz-García contructon of the robot an, on the other han, caue part of the payloa to be upporte by the reacton force of the floor. hu, the actuator requre le loa capacty. In orer to olve the rect knematc problem, a novel numercal metho preente. h metho bae on a mplfe geometry of the robot n uch a way that each par of lmb form a trangle. he locu of the top vertex of th trangle an arc. he algorthm (metho of arc) preente, a well a ome reult of t mplementaton. Invere knematc problem nclue too. On the other han, nvere ynamc euce propong an mprovement n a egn gven n []. Next, rect ynamc olve. Fnally, the controlle robot performance llutrate by example of a robot-bae laer cuttng machne.. Invere Knematc A t well known, a parallel robot cont of a fxe bae, a et of lmb an an en effector referre to a the platform. Recall alo that parallel robot nomenclature bae n the type of jont whch conttute the mechanm. hu, -PUS mean that our manpulator ha x lmb, each of them contng of a prmatc (P) plu a unveral (U) plu phercal (S) jont. In robot knematc, there are two man ue to eal wth: rect or forwar knematc an nvere knematc. Fnng the poton an orentaton of the platform gven the poton/length of the actuator, lea to olve the rect knematc problem. he nvere knematc problem cont of obtanng the poton/length of the actuator (alo known a the confguraton of the robot) gven a ere poton an orentaton of the platform. For parallel manpulator, uch a the -PUS robot, the ffculty to obtan the nvere an rect knematc nverte wth repect to eral robot [], [9]. Gven a certan poton an orentaton p of the en effector, t relatvely mple to calculate the length of each artculaton wth a mple um of vector. Fg. an equaton an gve an example of th. In Fg., vector p repreent the ere poton of the en effector. Vector a a known parameter of the robot. he magntue of vector b alo a parameter of the robot, an t orentaton gven by the ere orentaton of the platform; the vector gong from A to D. By referencng all th vector to the ame coornate ytem A, we can fn the magntue of each vector r a r = A p + A b A a. where p = [p x p y p z ] an the magntue of r repreent the poton/length of the actuator [9, ]. Recall that,, an are the Euler angle []. Workng out the latter equaton yel the cloe form of the nvere knematc for th manpulator: j,..., ; p b a p b a p b a r p b a j j j j { x, y, z}. j j j b a However, gven the free nature of the unveral an phercal jont, f the poton/length of the actuator known, t not eay to calculate the platform poton. Many reearcher have tue the rect knematc of the Stewart-Gough platform. Snce the analytc oluton requre olvng a th egree equaton, mot of the artcle that talk about th ubject menton teratve numerc metho to olve the problem. It ha been hown that becaue of the freeom poee by the jont, gven the length of the actuator t poble to fn 4 poton of the platform that atfy equaton () [3, 4,, ]. Neverthele, mot of the tue refer to the -UPS platform, an very few [9, ] analye t varant -PUS. Compare wth [4].. ()

3 33 33 Mechancal an Computatonal Degn for Control of a -PUS Parallel Robot-bae Laer Cuttng Machne 3. Sngularty Conton Whle knematc analy gve u a relatonhp between the poton of the platform an the length of the actuator, ometme t mportant to know the relatonhp between the velocty of the actuate lmb an the angular velocty of the platform, epecally whle tuyng the ynamc of the manpulator. he Jacoban matrx gve u that relatonhp, an that why t mportant to analye t. Coner Fg.. Fg. a) Geometry of a general -PUS manpulator. b) Frt CAD egn. he output velocty vector gven by v p ω b x, where v p the pont P velocty an b the angular velocty of vector b. he cloe loop knematc equaton gven by DB AD OA PB OP. () akng tme ervatve of the latter equaton yel r b p r ω b ω v. (3) Defnng a unt vector ˆ n the ame recton a, a unt vector n the ame recton a r an takng ot prouct n the latter equaton wth that yel: b p ω b v ˆ. (4) Rewrtng th equaton for I =,, n term of matrce prouce ˆ ˆ ˆ ˆ b p ω v b b b. (5) he latter equaton mple that a parallel robot wll preent two kn of ngular confguraton whch wll are when the Jacoban matrce, hown n equaton (), become ngular. Drect knematc ngularte come from et(j x ) = an nvere knematc ngularte come from et(j q ) =.

4 34 R. Zavala-Yoé, R. Ramírez-Menoza an J. Ruz-García J x b b, ˆ ˆ b J q ˆ 3 ˆ 3. () An algorthm to compute numercally the contrant ecrbe above gven next. Algorthm. Drect an Invere Knematc Sngularte. Step. Fx xmn, eltax, xmax; ymn,eltay,ymax; zmn,eltaz,zmax. Step. For x=xmn:eltax:xmax. Step 3. For y=ymn:eltay:ymax. Step 4. For z=zmn:eltaz:zmax. Step 5. Compute vector b accorng to the actual poton b = f (x,y,z). Step. Compute nvere knematc = f (x,y,z,b) wth equaton (). Step 7. Determne vector for that poton, = f (,x,y,z). Step 8. Obtan J x = f (b,) wth equaton (). Step 9. Calculate an tore et(j x ) for that pont. Store thee pont n p Jx. Step. Obtan J q = f (,) wth equaton (). Step. Calculate an tore et(j q ) for that pont. Store thee pont n p Jq. Step. En z, En y, En x. Step 3. Plot the pont p Jx (x,y,z) obtane n tep 9 n graent colour. Step 4. Íem for p Jq (x,y,z) obtane n tep. he graph an nterpretaton are gven n ecton Forwar Knematc: Numercal Algorthm. A mentone above, when we fn the poton an orentaton of the platform gven the poton/length of the actuator, we are olvng the rect or forwar knematc problem. he frt thng that mut be one to etablh the fferent coornate ytem that wll be ue: O at the centre of the bae, P at the centre of the platform, A at the begnnng of each actuator, wth the x ax pontng n the recton of the movement. D an B are the en of the th leg, beng D the pont that move along the ral an B the pont fxe on the platform. he metho propoe n th paper aume that the platform a trangle n whch there are only three connecton pont for the leg, one n each vertex, o that each pont ha two leg connecte to t (ee Fg. ). h archtecture known a -3. he general ea of the propoe metho that ung the correponng tranformaton matrce, three arc correponng to the poble locaton of the three vertce of the trangle are generate. So, the tance between each pont of each arc an each pont of the other arc are calculate. After calculatng all tance, we mut fn par of pont whoe tance equal to the e of the trangle. See algorthm whch en up wth a numercal oluton (ecton 8) of the rect knematc for th propoe egn [3].

5 Mechancal 35 an Computatonal Degn for Control of a -PUS Parallel 35 Robot-bae Laer Cuttng Machne Fg. Mofe egn wth trangular bae an en effector. Algorthm. Drect knematc (metho of arc) Step Calculate the x tranformaton matrce o H A [],that tranform a coornate from ytem A to ytem O. Step Gven the ere length of the actuator, fn the pont correponng to the centre of the trangle forme by each par of leg. Step 3 For each of the three trangle, fn t heght an the angle rho between the ax of the two actuator. Step 4 Fn an plot all the pont of that arc. Step 5 Fn, ave an plot the tance between each pont of arc an each pont of arc. Step Repeat lat tep ung arc an 3, repeat lat tep ung arc an 3. Step 7 Fn an ave each par of pont (, ) whoe tance equal to the length of the platform. Step 8 Repeat lat tep ung arc an 3, repeat lat tep ung arc an 3. Step 9 Fn the par of pont (, 3) whoe pont ext on the lt generate urng tep 8 Step Fn the par of pont (, 3) that ext on the lt generate urng tep. Step Save an plot the three pont an the oluton (platform). 5. Invere an Drect Dynamc For the ynamc moel, there are two fferent problem to olve. he rect ynamc erve the acceleraton prouce n the en effector, ue to a gven et of actuator force. he nvere ynamc erve the force neee n the actuator for a ere acceleraton trajectory n the en effector. Lttle ha been tue on the ynamc of parallel robot, an even le on that of the -PUS, [8-,, ]. In general, for parallel robot, the nvere ynamc not very complex, but the rect ynamc ffcult to obtan ue to poble extence of unknown reacton force n the pave jont. 5. Invere Dynamc Coner Fg. 3. Let G be the platform centre of ma. he platform ha a ma of m an nerta matrx I. he acceleraton of pont G enote by a G. Let C be the pont of applcaton of external force F an moment M. he force f whch act on pont B can be ecompoe on two component. he force f that goe along the man ax of the lmb,

6 3 R. Zavala-Yoé, R. Ramírez-Menoza an J. Ruz-García along unt vector an the force perpencular to, ue to nerta, whch wll be enote f N. We then have Fg. 3 Force exerte on the en-effector. f f f. (7) In [9] a general ynamc moel for the -PUS robot evelope, bae on the ynamc of the -UPS moel an conerng that N f τ ˆ (8) where ˆ an are the unt vector pontng along the axe of the actuator an the lmb, repectvely. However, th moel oe not coner the reacton force exerte by the bae an the actuator retrcton. he conequence of th omon that every tme the lmb are n a vertcal poton, the effect of the actuator force on the platform become zero (note the ot prouct n the equaton), an nce thee are the only force conere n the moel, the platform woul fall own. o prove th, the moel ue n [9] wa mplemente n an algorthm whch apple a contant force, an a mulaton wa run (ee ecton of Numercal Smulaton an [3], []). Now, n orer to euce the nvere ynamc of the robot we procee a follow. Frt, we coner only the force exerte by the lmb on the platform accorng to (8). Let F N (3 ) be the reultant force of all f N, =,, an M N t reultant moment aroun pont C. Let vector c be the vector that goe from pont C to pont B. Let f = f be the force that each lmb exert on the platform. he equaton of equlbrum for the ytem are: f F M f F, (9) N c M N. () Let an N the crew vector for the external force an the reultant of the lmb, repectvely. F(3) FN (3) σ, σ N. () M (3) M N (3) () ()

7 37 37 Mechancal an Computatonal Degn for Control of a -PUS Parallel Robot-bae Laer Cuttng Machne Rewrtng (9) an () n matrx form we have: N N f M F c M F. () We ee that the frt term of the rght e of () rectly relate wth the Jacoban matrx J x o that f we efne a vector f = [f,f,...,f ] we can rewrte equaton () a: N x σ f J σ (3) where f the vector forme by the magntue of each one of the force that the lmb exert on the platform. Moment M G apple on the platform wth repect to pont G F e M M G (4) where e the vector that goe from G to C; that the moment arm of F. Let a G be the lnear acceleraton vector of pont G. he Newton Euler equaton for the platform can be wrtten a follow: G m m a g F, (5) Iω ω Iω M G () where g the gravty acceleraton vector an the angular velocty of the platform. Developng algebracally the latter equaton an efnng e* we obtan an expreon for, for any vector x, a follow: - e * that, * 3 3 e e e e e e x e x e, (7) H GW σ, (8), * * * m m m m m m e v g G H ω Iω e v g e I e. (9) hee matrce have a phycal meanng: G correpon to the nerta matrx an H the vector of force ue to gravty. By balancng (8) an (3) we have: N x σ f J H W G. () Wth th we have erve the relaton between the ervatve of the twt over C (whch correpon to the acceleraton) an the force exerte on the platform by the lmb. Workng out th equaton we can fn new expreon for N, whch n matrx form expree a: U W σ N, () * * *, * * * r I r I r I r I L c L U K c K. ()

8 38 R. Zavala-Yoé, R. Ramírez-Menoza an J. Ruz-García Subttutng () n () yel an olvng for f, whch correpon to the force that the lmb nee to exert on the platform n orer to generate certan acceleraton W : G W J H U f J x x. (3) Each f pont n the recton of the correponng unt vector. Neglectng frcton on the unveral an prmatc jont, Fg. 3 how that th force f forme by three component: f ˆ kˆ nˆ. (4) where: magntue of the force exerte by thelnear actuator. ˆ kˆ nˆ R N magntue of magntue of unt vector n therecton of lnear actuator. vertcal unt vector, pontng upwar. kˆ ˆ, vector on the xy plane, perpencu lar to ˆ. R the vertcal reacton force N the vertcal reacton force exerte by theactuator' retrcton. exerte by thebae. Note that ˆ, nˆ, kˆ are orthogonal an correpon to the x, y, an z recton of the coornate ytem place on A. For that reaon, the three rght-e member of equaton (4) correpon to the x, y an z component of force f expree on coornate ytem A. he rotaton matrx A R O can be erve from the contructon parameter of the robot. A A f RO f N. (5) R Equaton (3) an (5) then complete the nvere ynamc of the robot, wth whch we can olve the force neee on the actuator (an the reacton force of the bae) for a ere acceleraton trajectory. 5. Drect Dynamc he rect ynamc more complcate. Even though t mple to olve (3) an get an expreon of the acceleraton n term of the force exerte by the lmb (equaton ), a een before uch force are mae up of three component, of whch only one known. G J f G H U W. () x Jut a the nvere knematc relatvely mple for parallel robot, but rect knematc complex becaue we o not know the value of the pave jont, we can ee that the rect ynamc n th cae mae ffcult ue to the lack of nformaton of the reacton force. We know, however, that thee force exerte on pont D prouce no work nce there no placement n thoe recton. he Vrtual Work Prncple tell u that we can wrte an equaton ung the fferental placement of each of the

9 Mechancal 39 an Computatonal Degn for Control of a -PUS Parallel 39 Robot-bae Laer Cuttng Machne pont n the mechanm, ue to the force apple on each pont, an the total um of the work prouce by each force n that cae zero. In other wor, aumng that the force on the platform oppoe an cancel the force on the actuator, we then have that for fferental (tranlatonal an rotatonal) placement x = [ P ], the total work prouce [, 5]: F P τ. (7) M Θ We can ue Jacoban matrce to relate the fferental placement on the platform to fferental placement on the actuator. J x J, (8) x q x J. (9) Subttutng (8) n () an after ome algebrac evelopment we obtan n (3) the rect ynamc moel of the robot W G J τ G H. (3). Knematc Control A mentone n ecton, the nvere knematc of th -PUS manpulator gven by equaton (). So, gven a ere poton p, the reult wll be an actuator vector = [ ] whch ay how long each actuator mut expan or contract to acheve the ere poton. It eay to ee that n orer to generate a ere trajectory, a et of poton p requre. A a conequence, a knematc poton control (open loop) can be one by programmng equaton (). he correponng experment were one an are gven n ecton Dynamc Control Equaton (3) ha to be contructe on lne urng numercal mulaton n orer to mplement a cloe loop ytem wth a PD controller (Fg. 4). he block wth a lttle robot mage repreent uch equaton (3). A a conequence of the ymmetrc archtecture of the robot, we choe matrce K p = k p I an K D = k I, k p, k +, wth I the entty matrx of orer x, ee Fg. 3. Secton 8 prove reult. Algorthm 3. PD controller. Step Fx poton, velocty an acceleraton ntal conton. Step Solve nvere knematc wth algorthm an obtan vector b an. Step 3. Fx t_ntal, t, t_fnal; Whle t t_fnal o tep 5 to tep 9. Step 4. Chooe the ere a _e, obtan Jacoban matrce an error vector _err, _err. Step 5. Compute matrce, U an G, H olvng equaton (9) an (), repectvely. Step. Compute τ K perr KD err. Step 7. Determne acceleraton n term of force wth equaton (3). Step 8. Upate velocte: W ( t) W ( t ) W ( t ) t

10 4 R. Zavala-Yoé, R. Ramírez-Menoza an J. Ruz-García Step 9: Upate poton: x( t) x( t ) W ( t) t Step. Solve nvere knematc (eq.) an obtan vector b an for the new poton. Fg. 4 Complete cloe loop control ytem for the -PUS parallel robot. 8. Numercal Smulaton In th ecton, we prove the mulaton correponng to the nvere an rect knematc cae, rect an nvere ynamc problem a well a knematc an ynamc control. 8. Invere Knematc. h cae llutrate wth two example. he frt one coner a fxe ere poton gven by p = [p x p y p z ] = [ ]. he reultng actuator vector yele to be = [ ]. he econ example reulte to be more nteretng. A ere trajectory tore n a long vector p n orer to wrte own a letter M. See Fg. 5. Fg. 5 Wrtng own letter M nee thee trajectore of the x actuator. 8. Invere Knematc Sngularte. Matrx J q ha full rank wthn a regon cloe to the orgn of the workpace. So, for any gven pont n the workpace, a longer the tance from the orgn, a cloer J q to looe rank. he followng fgure llutrate the value of et(j q ) n the workpace.

11 Mechancal 4 an Computatonal Degn for Control of a -PUS Parallel 4 Robot-bae Laer Cuttng Machne Fg. Value of et(j q ). It poble to plot a urface aocate to the latter fgure where et(j q ) =. h urface repreent the workpace of the manpulator. See Fg. 7. Fg. 7 Surface where et(j q ) =. 8.3 Drect Knematc he reult for the rect knematc are preente here. Fg. 8a how the arc generate by each par of leg. he tance between each par of pont of arc an alo hown. he horzontal plane correpon to the length of the e of the platform an the croe repreent the par of pont whoe tance equal to the ame one a the e of the platform. he whte ot the fnal oluton foun by algorthm (Secton 3). Fg. 8b how the bae of the manpulator along wth the ral of the actuator. Here three reultng poton of the platform are hown for a gven actuator vector [3].

12 4 R. Zavala-Yoé, R. Ramírez-Menoza an J. Ruz-García Fg. 8 a) Dtance between pont belongng to arc an. b) Reultant poton of the platform. In [] there a very general methoology to obtan the rect knematc of any parallel robot ung an teratve algorthm. hat algorthm wa mplemente for the robot analye n th paper; however, the geometrc metho propoe n th paper npre by a oluton preente by [] gave better reult than thoe obtane by the teratve metho bae on the work preente n []. It mportant to note that the metho work becaue the confguraton -3 mplfe each par of leg, a one leg wth only one egree of freeom. h a very ueful technque when workng wth parallel manpulator. We can alo ee that n th type of manpulator t very common for the rect knematc to have many oluton, o n orer to plan a trajectory t very mportant to know the prevou poton. 8.4 Drect Knematc Sngularte. Proceeng n th cae, a t wa one for the nvere knematc ngularte, a fgure obtane whch how et(j x ). Fg. 9 Numercal value of et(j x ) n the workpace Darker regon appear a cloer to the centre of the workpace we are. h mean that J x loe rank when the platform cloe to the bae. h happen when lmb an moton axe are collnear. Obervng Fg. 9, t euce that x, y, an z houl be lmte to x,y [ 5, 5], z [4, 7] cm for all Euler angle equal to zero. However, n orer to keep

13 Mechancal 43 an Computatonal Degn for Control of a -PUS Parallel 43 Robot-bae Laer Cuttng Machne the platform wthn ome fxe value lke thoe, t mportant to etermne what poton correpon to the actuator parameter. h ue olve by the rect knematc. 8.5 Invere Dynamc A t wa explane n Secton 5., the reacton force exerte by the robot bae an actuator retrcton were not taken nto account n [9]. Hence, when the robot tanng up (completely vertcal poton), t fall own. o llutrate th fact, the moel conere n [9] wa mplemente n an algorthm whch apple a contant nput force. Smulaton reveale the reultant behavour jut ecrbe. Acceleraton an poton trajectore ecrbe that every tme the vector are n a vertcal poton, the actuator force have no effect on the platform, an thu t fall own. A t oe, vector tart long vertcalty, nce the ot prouct ( ) re, an the actuator force have an effect on the platform an t re agan. In orer to mprove the moel, the reacton force of the actuator retrcton were taken nto account. See plot n []. 8. Drect or Forwar Dynamc Snce forwar ynamc neee to olve the control problem (ee Dynamc Control Secton), the mulaton for open loop forwar ynamc were omtte but ee []. 8.7 Knematc Control Smulaton an Moel propoe It wa explane n Secton (Knematc Control) that an open loop poton control can be numercally mplemente by programmng equaton (),.e. the nvere knematc vector equaton. Wth all the backgroun evelope o far, t wa poble to create everal anmate 3D egn n Vual Natran. wo of them are hown n Fg.. After refnng the whole moellng an control egn, a trangle hape wa choen for the bae an en effector. See alo Secton 4. Fg. Early tage 3D rener moel egn. 8.8 Dynamc Control. he control loop wa egne to have a goo regulaton performance. Laer cuttng are one va ere of tep nput. A mple PD controller wa propoe to control the robot platform n the jont pace for each actuator. Snce the robot preent a ymmetrc archtecture, all the controller were choen entcal. he reult of the PD controller performance are ecrbe next for a tep nput of ampltue 7cm. he platform poton n z (platform heght) behave a ncate n Fg a an b.

14 44 R. Zavala-Yoé, R. Ramírez-Menoza an J. Ruz-García he reultng teay tate error n the actuator wa.5cm, about 7% of the fnal value. he controller gan were k p = 3 an k = 3. Fg. a) PD controller tep repone of the platform. b) Steay tate error. In orer to mprove the latter performance, a PD plu gravty compenaton (PD+G) wa alo mplemente. So, tep n algorthm 3 wa replace by the followng comman: Step : Compute τ K K J mg p err D err In th cae the teay tate error wa equal to zero after.75 econ of rang tme. Fg. a)pd+g controller tep repone b) Steay tate repone. 8.9 Laer Cuttng Machne applcaton egn A t well known an a t wa mentone n the Abtract an Introucton, parallel robot have a hgher payloa than ther eral counterpart. Mountng a laer cuttng evce uner the robot platform woul not mofy all the mathematcal moellng evelope untl now. he nerta term o not change a lot a a reult of the nherent lght weght of the laer evce, a well a the above mentone payloa. For ntance, coner to cut a pece followng an M hape a llutrate n Fg. 3. If uch a tak evelope wth knematc (ee Fg. 5) or by ynamcal control (ee Fg. ) the goal wll be accomplhe. Snce the robot archtecture ymmetrc, the actuator an the PD+G controller are entcal among them. A a conequence, the cuttng proce wll uccee a euce from the ntal egn untl the cloe loop control.

15 Mechancal 45 an Computatonal Degn for Control of a -PUS Parallel 45 Robot-bae Laer Cuttng Machne Fg.3 Parallel robot-bae laer cuttng machne numercal prototype. 9. Concluon In th paper, a novel egn of a -PUS parallel robot wa gven. Bee, a new metho to olve the forwar knematc wa preente. In aton, nvere an forwar ynamc problem were olve n orer to mplement a control ytem for the nvere ynamc moel. Sngularte were foun numercally an a PD an a PD+G controller were egne for th robot. Both controller preente a goo performance, although PD+G wa efntely better. he controller performance wa goo enough for the preent purpoe an a a reult of ther relatve mplcty t wa not neceary to eal wth numercal problem a tffne, convergence, algebrac loop, etc., a wth nonlnear/ntellgent controller. Hgh payloa an PD help to eal wth varyng nerta on the platform or below t a a laer cutter evce. Reference [] ÁNGELES, J. Fun. Rob.Mech.Sytem. New York: Sprnger, USA, 3, 54 p. [] BRUYNINCKX, H. Parallel Robot. he Robotc WEBook. [on lne], Augut 5.< _parallel_manpulator.pf>. [3] CRUZ, P. an FERREIRA R, Knematc Moelng of Stewart-Gough Platform. In ICINCO 5 Rob. Autom. Barcelona: Span, Sprnger,, p [4] GAO, X; et al. Generalze Stewart-Gough Platform an her Drect Knematc. IEEE ran. on Robotc, 5, vol., no., p [5] HOPKINS, B. an WILLIAMS R. Knematc, egn an control of the -PSU platform. he Inutral Robot. Befor,, vol. 9, no. 5, p [] JAFARI, F. an MCINROY, J. Orthogonal Gough-Stewart Platform for Mcromanpulaton. IEEE ran. on Rob. Autom., 3, vol.9, no.4,p [7] KHALIL, W. an GUEGAN, S. Invere an Drect Dynamc Moelng of Gough- Stewart Robot. IEEE ran. on Rob., 4, vol., no. 4, p [8] LING-FU, K. an SHI-HUI, Z. Reearch on a Novel Parallel Engravng Machne an t Key echnologe. Int. J. of A. Rob. Syt, 4, vol., no., 4, p [9] MERLE, J. Parallel Robot. Dorrecht: Sprnger,, 4 p. [] SAI, L. Robot Analy. New York: Wley, p. [] ARACIL, R. Robot Paralelo. Rev. Iberoam.,, vol.3, no., p. -8.

16 4 R. Zavala-Yoé, R. Ramírez-Menoza an J. Ruz-García [] YIU, Y. K. an H. CHENG. On the Dynamc of Parallel Manpulator. In Proc. of the IEEE Int. Conf. Rob. & Aut. Seoul: Korea, IEEE,, p [3] RUIZ-GARCÍA, J. an ZAVALA-YOÉ, R. an CHAPARRO-ALAMIRANO, D. an RAMIREZ-MENDOZA, R. Drect Knematc of a -PUS Parallel Robot Ung a Numerc-Geometrc Metho, In Proc. of the IEEE Int. Conf. Mech., Elec. Auto. Eng. Morelo: Méxco, IEEE, 3, p [4] CHAPARRO, D. an ZAVALA-YOE, R. an RAMIREZ-MENDOZA, R. Knematc an Workpace Analy of a Parallel Robot Ue n Securty Applcaton, In Proc. of the IEEE Int. Conf. Mech. Elec. Auto. Eng., Morelo: Méxco, IEEE, 3, p [5] HIBBELER, R. C. Eng. Mech. New Jerey: Prentce Hall, 998, 35 p. [] RUIZ-GARCÍA, J., ZAVALA-YOÉ, R. CHAPARRO-ALAMIRANO, D., RAMIREZ-MENDOZA, R. Drect an Invere Dynamc of a -PUS Parallel Robot, In Proc. IEEE Int. Conf.Mech. Elec. Auto. Eng. Morelo: Méxco, IEEE, 3, p. -.