Novel Cable-Suspended RoboCrane Support

Transcription

1 Novel Cable-Supene RoboCrane Support Robert L. William II Ohio Univerity Athen, Ohio R.L. William II, 5, Novel Cable-Supene RoboCrane Support, Inutrial Robot: An International Journal, (4): 6-. Keywor: RoboCrane, able-upene robot, able-riven, wire-atuate, elf-ontaine bae mobility. Contat information: Robert L. William II Aoiate Profeor Department of Mehanial Engineering 59 Stoker Center Ohio Univerity Athen, OH Phone: (74) Fax: (74) URL:

2 Novel Cable-Supene RoboCrane Support Robert L. William II Ohio Univerity, Athen, Ohio Inutrial Robot: An International Journal 5 KEYWORDS RoboCrane, able-upene robot, able-riven, wire-atuate, elf-ontaine bae mobility. ABSTRACT The NIST RoboCrane i a able-upene robot with the potential to reue the iavantage of onventional rane. One weakne of the RoboCrane i the nee for at leat three fixe rigi upport point for the ix overhea able onnetion. In many potential appliation, thee rigi upport point are imply not available. Thi artile preent a new RoboCrane upport onept bae on rigi member, able atuation, an able upenion. It i elf-ontaine an provie mobility for the require ix overhea able onnetion, thu extening the workpae of the exiting RoboCrane. The artile preent the RoboCrane upport onept overview, followe by kinemati an tati analyi, plu a ae tuy of a peifi eign. Contat information: Robert L. William II, Aoiate Profeor Department of Mehanial Engineering 59 Stoker Center, Ohio Univerity Athen, OH Phone: (74) Fax: (74) williar4@ohio.eu URL:

3 . INTRODUCTION Conventional rane for ontrution an argo tranfer appliation have the following iavantage: non-rigi upport; low payloa-to-weight ratio; low reitane to win; inaurate ontrol of loa; they are only ue to lift an oarely poition loa; limite remote, autonomou apabilitie; worker are in a hazarou area; an at any given loation only one egree of freeom i ontrolle by the rane (i.e., the length of the lift able between the boom an objet); human worker are require with tag line to maintain the loa remaining five egree of freeom. Thi i ineffiient, human have limite trength, an it i angerou. To improve upon thee iavantage, the RoboCrane wa evelope at NIST [-]. The RoboCrane i an inverte Stewart Platform wherein a moving platform i ontrolle in ix egree of freeom via ix ative able an winhe. Not only an RoboCrane provie lift, but the remaining five egree of freeom are atively ontrolle to be tiff an table (over a limite range of motion an orientation). Thi onept wa extene by NIST for a tiff, table unerwater work platform, wherein the platform may be ontrolle to be tationary even if urrouning ea are not [4]. The NIST RoboCrane i hown in Figure a; it require three rigi overhea able upport point (not hown in Figure a) for hanging pair of able. The NIST RoboCrane ha been implemente for large airraft e-painting for the U.S. Air Fore (ee Figure b); again, three rigi overhea able upport point are require.

4 Figure a. RoboCrane at NIST Figure b. RoboCrane for De-Painting (Figure are both ourtey of the NIST Intelligent Sytem Diviion.) Inpire by the NIST RoboCrane, many reearher have been involve with able-upene robot. A few of thee have foue on able-upene rane evie. Aria et al. [5] evelope a even egree of freeom, three-able upene rane-type robot (the remaining freeom are an XY overhea gantry, plu top an bottom turntable) for an automobile aembly line, intene for heavy prout aembly. Mikula an Yang [6] preent a three-able rane eign for a lunar ontrution appliation, off-loaing maive moule from a laning ite, moving them, an ontruting them into an operational bae. Viomi et al. [7] evelope ontrution automation tehnology wherein Stewart platform rane (i.e. RoboCrane) are entral. Shanmugaunram an Moon [8] preent a ynami moel of a parallel link rane with poitioning an orientation apabilitie, with unilateral able ontraint. Yamamoto et al. [9] propoe a rane-type parallel mehanim with three ative able for hanling heavy objet. Shiang et al. [] preent a parallel four-able poitioning rane for offhore loaing an unloaing of argo veel uner high ea. A primary iavantage of the NIST RoboCrane i that ix overhea rigi attahment point are require for the ix ative able. In many potential appliation thee rigi attahment point are iffiult or imply impoible to provie. Therefore, thi artile introue a elf-ontaine, eployable, 4

5 able-upene RoboCrane upport ytem. Not only oe it provie the require rigi able attahment point, but it alo provie mobility for thee point, thu extening the workpae of the fixe-rigi-point RoboCrane. Potential appliation inlue ontrution, earh an reue, an other eployable, elf-ontaine able-upene robot appliation. The onept wa originate by Dr. Jame Albu of NIST an it wa evelope uring the author abbatial at NIST in Gaitherburg, MD. Thi artile preent the onept eription, followe by kinemati equation an boom tati equation for ontrol of the RoboCrane upport ytem. We onlue with a uggete eign, for whih we preent kinemati, workpae, an tati reult. 5

6 . ROBOCRANE SUPPORT SYSTEM DESCRIPTION The role of the RoboCrane upport ytem i to provie rigi, elf-ontaine, eployable, moving overhea able attahment point for the RoboCrane. A hown in Figure a, the upport onept onit of rigi equilateral triangle C C C 4 hinge to rigi boom C B at point C 4. Boom C B artiulate relative to the fixe tanar umpter bae via a univeral joint at point B. Boom C B i atively ontrolle by able length L B an L B, via motor an able reel at point B an B, repetively, poitioning point C with repet to the bae. The bae oorinate frame {} i aligne a hown; it origin i plae in the enter of the bottom bak orner of the umpter. The moving, rigi RoboCrane overhea able onnetion point are C, C, an C 4. A hown in Figure b, two ative RoboCrane able meet at eah point. The RoboCrane ative able length are L i, i =,,,6. Sine a goo eal ha been publihe on the RoboCrane eign, kinemati, an ontrol, [e.g. -4], thi artile foue only on the RoboCrane upport ytem of Figure a. C C L C L C L C C 4 L C C 4 C L B C l C B L B L L L 6 L 5 L L 4 C B 4 X Y B Z B Figure a. RoboCrane Support Conept Figure b. Support with RoboCrane 6

7 A key apet of thi elf-ontaine RoboCrane upport ytem i that the onfiguration of the overhea equilateral triangle upporting the ix RoboCrane able i maintaine by two fixe-length, paive able. Both able are fixe to the umpter at point B 4, pa over pulley at point C, an are fixe to moving equilateral triangle point C an C. A able L B an L B atively move point C, thee paive able move an rigily upport the equilateral triangle paively, whih in turn upport the RoboCrane. At any intant uring motion, the two paive able an be een a two length on either ie of the pulley, L C between point C an point C an C, an two iential length l between moving point C an fixe point B 4. With thi eign, having a revolute joint at C 4 (whoe axi i aligne with X in the nominal onfiguration when the boom i in the Y Z plane), both able portion L C are the ame length for all motion, whih enure that the Z omponent of C an C are alway the ame (though ifferent from the Z omponent of C 4 in general). Thi paive motion ontrol for the equilateral triangle enable a pantograph-like motion. L C an l both hange uring motion, but their um i ontant, et by eign to keep the equilateral triangle a horizontal a poible uring all motion (ine the exiting RoboCrane uually aume horizontal overhea able upport point). Thu, the RoboCrane upport omprie two ative egree of freeom L B an L B plu one paive egree of freeom (whih attempt to maintain horizontality for the equilateral triangle). Further, the umpter may be eploye for aitional mobility. At any umpter loation, Figure how two ro with footpa to ounterat the moment exerte by the weight an external loa. We eire the equilateral triangle portion of the ytem to be a horizontal a poible for all motion, large workpae, an able tenion to be a mall a poible, (but they mut remain in tenion for all motion). Thee fator are ompeting; we preent a reommene eign in Setion 5, inluing kinemati, workpae, an tati analyi reult. 7

8 . ROBOCRANE SUPPORT SYSTEM KINEMATICS Thi etion preent kinemati equation an olution for the motion of the RoboCrane upport ytem inluing the boom C B an equilateral triangle C C C 4. The onept an kinemati equation inlue a paive pantograph-like near-horizontal mehanim for equilateral triangle C C C 4. Kinemati relate the Carteian poition an orientation (poe) of the RoboCrane upport ytem to the paive joint angle an ative able length. Figure how a kinemati iagram of the ytem, onnete to the umpter frame at point B via a univeral joint allowing boom yaw ( ) an pith ( ). Note thi referene poition with the boom along the groun efine both angle an to be zero. The boom an be oniere to be a R erial robot onnete to the umpter with joint angle an, plu angle moving the triangle with repet to the boom. Note,, an are not ontrolle iretly but via two ative able L B an L B, an two paive able L C + l. Frame {B } i the ame a {}, but loate at point B. i the length of boom C B, i the length from boom bae point B to the equilateral triangle hinge point C 4, an i the equilateral triangle ie. Moving point C 5 i the mipoint of C C. The boom pith angle houl be kept away from the groun poition beaue thi approahe a ingularity where able L B an L B are oplanar with the boom; in thi ingularity infinite fore woul be require to move the boom. Z C X C C Y C Z C 4 X C h X, Z B Y, X B Z B,, B C 5 Figure. RoboCrane Support Kinemati Diagram C 8

9 9 The Devavit-Hartenberg (DH) parameter [] for thi erial robot are given in Table I. Note a joint angle offet i require for i= ine the X an X axe are not aligne in the zero poition. Table I. RoboCrane Support Denavit-Hartenberg Parameter i α i- a i- i i The homogeneou tranformation matrie relating frame {C i }, 5 =,,4, i (whoe origin are point C i an whoe orientation i iential to that of {}), to the worl frame {} an be foun: [ ] ( ) [ ][ ] T T T,, i C i C = 5 =,,4, i [ ] [ ] { } = i C i C I T () The (ontant) relative poition vetor { } C i for ue in () are eaily foun by geometry. Subtituting the DH parameter an { } C i into () yiel: [ ] = 4 C T () where we have ue the abbreviation i i o = an i i in =, =, i ; alo ( ) o + = an ( ) in + =. The require { } C i poition vetor are: { } = C { } = h h h C { } + + = h h h C

10 { C } = { } 4 h C 5 = + h () + h Now, given value for,, an, it i eay to evaluate the abolute poition of moving point C i with repet to {} uing the above formula. However, thee erial angular value will not be known beaue it woul inreae ot an omplexity unneearily to a angle ening to the paive univeral joint at B an paive revolute joint at C 4. Intea, we have two hoie: ) For invere poe kinemati (alulate the ative able length given the eire Carteian poe), the upper boom point C i peifie at eah intant (it an be moving). It i onvenient to peify C via angle an (ine C i ontraine by the length ), uing the firt expreion of (). Then we alulate the two require able length L B an L B : L B = C B B = C B L (4) The paive revolute angle will be etermine after the forwar poe kinemati ae below ine it i the ame for both option. ) For forwar poe kinemati (alulate the Carteian poe given the ative able length), the two able length L B an L B are known from their winh angular feebak meaurement. Upper boom point C i alulate given thee two able length. From Figure a, point C i the interetion of three phere: fixe boom raiu entere at B, raiu L B entere at B, an raiu L B entere at B. The interetion of three phere i the bai for the forwar poe kinemati olution of variou NIST-evelope able evie; thi olution may be foun in []. Point C i foun uing that metho: C i foun from the interetion of three phere: ( B,L B ), ( B,L B ), an ( B, ). Note the orering of the three phere i very important to avoi algorithmi ingularitie []. The above orering i the only ueful poibility ine point B an B alway have the ame Z

11 oorinate. Given C we an alulate paive univeral joint angle an from an invere C = P P P in (): poition kinemati olution of the expreion for { } { } T x y Px = P tan Py = in z (5) Now we an etermine ; it i one in the ame manner for both of the above ae. Figure 4a how a ie view of the ytem onept. Thi ie view how a planar repreentation of the paive equilateral triangle pantograph able; l = CB4 are the real variable portion of the two paive pantograph able, an l = CC5 i a virtual variable able repreenting the planar projetion of the able portion L C (ee Figure a an viualize the plane C C C ; l biet thi triangle). z l = L C 4 ( ) ( ) l h h + = o (6) i negative in (6) ue to it efinition in Figure an 4a. Thi pantograph mehanim will be eigne to attempt to maintain the equilateral triangle a near horizontal a poible for all motion. It an be exat only at one angle. If the triangle i perfetly horizontal, =. We nee the (atual) able length L C for etermination. The two paive pantograph able (running from B 4, over pulley at C, onneting to point C an C ) are of fixe length L = l + L. nom C Therefore, LC = Lnom l. Let u fix L nom by eign, requiring the equilateral triangle to be exatly horizontal ( = ) at a nominal value of the boom pith angle ( nom ) an for the entral value of the yaw angle, ; nom houl be in the mile of the allowable range, or ome other nominal, nom = often-ue onfiguration. At the nominal onfiguration we have L = l + L, where nom nom Cnom lnom = CnomB4. From (6), we have L Cnom = lnom + ; the nominal virtual pantograph able 4

12 length i =. The nominal loation of C an C 5 are foun by ubtituting = l nom CnomC5nom an = nom into the firt an lat expreion of (). Finally, given,, an from (5) an (6) we an alulate the poition vetor for all moving point C i from (), for general onfiguration. Thi approah wa implemente in Matlab to emontrate the near-horizontal pantograph-like mehanim in a Y Z planar projetion of the D ytem (ee Figure 4b). C l 5 C 5 h C 4 l Z B 4 5 Z X Y Figure 4a. Sie View Z 5 Figure 4b. Horizontal Mehanim Demontration Y 5

13 4. ROBOCRANE SUPPORT SYSTEM BOOM STATICS A tanar D tati moel ha been erive an implemente for the RoboCrane upport ytem. The etail are not given in thi artile, but are available from the author. Thi etion highlight a potential problem of lak able uring boom ontrol. Thi will never be a problem for the paive pantograph-like able tenion ine the horizontality pantograph mehanim eign guarantee poitive tenion for both ie (on either ie of the pulley at point C, in the abene of extreme ownwar ynami motion) of the paive able, ine the loa at C an C will maintain tenion via gravity at all time. However, negative able tenion i a potential problem for the ative boom-guiing able with length L B an L B. A een in Figure a, boom-moving able L B an L B are atively ontrolle to iniretly ontrol the yaw angle an pith angle (ee Figure ). Thee able an lift the boom from horizontal to vertial ( ABS = 9 ), although the extreme houl be avoie ue to nearingularity onition, leaing to higher tenion near the groun an near vertial. The problem we now preent i with the yaw angle ; if thi angle i ommane to a value that i too large, one of the boom-moving able will require impoible puhing fore to maintain tati equilibrium. Figure 5 how the top view of thee boom-moving able. When the X Y projetion of boom B C beome ollinear with the X Y projetion of able L B, we have reahe the poitive limit on (by ymmetry, the equal, negative limit on our when B C beome ollinear with L B ). The value of the limiting angle, ± LIMIT, i epenent on the loation of B an B, plu umpter imenion B an B. LIMIT i not epenent on boom length or boom pith angle (from (5) i inepenent of vertial omponent). From the ollinear projetion ontraint an Figure 5, we an alulate ± LIMIT :

14 B ± = ± LIMIT tan (4) B ( ) f From Figure 5 we ee that the larget LIMIT will reult from moving point B an B a far forwar in the umpter Y iretion a poible. f i the fration of B along the Y iretion where point B an B are plae (the f ae i hown in Figure 5; f orrepon to B an B = plae at the forwar ege of the umpter). Figure 6 how LIMIT for f fration from to, for a tanar-ize (ee Setion 5) umpter. We ee that ± LIMIT inreae with inreaing f, a preite from the geometry of Figure 5. Figure 6 verifie that for large limit on we mut move the point B an B a far forwar a poible ( f ). If f, the theoretial limit on i = = = ± 9, allowing for a large range of motion. All motion houl be kept well away from the peifi ± LIMIT for any given eign, to afely avoi the lak able problem an the reulting atatrophi lo of ontrol. However, f alo orrepon to a ignifiant inreae in tati able tenion ue to approahing tati ingularitie an a lo of half the moment arm for lifting the boom (ompare to f ); therefore in thi artile, we hooe f, = = plaing point B an B on the umpter bak top urfae (a een in Figure 5), yieling = ±5.. A ompromie i poible, ay hooing f =. 5 for = ±68., balaning ± LIMIT ± LIMIT a larger workpae with wore kinemati horizontality an wore tati tenion. = 4

15 B B L B B Y X 9 8 B 7 + LIMIT B Limit (eg) L B C Figure 5. LIMIT Determination Figure 6. LIMIT f (fration of full B ), Varying B an B Loation 5

16 5. OVERALL ROBOCRANE SUPPORT SYSTEM DESIGN Thi etion preent a reommene goo overall RoboCrane upport ytem eign oniering the ompeting kinemati horizontality, workpae, an tati iue (more etail are available from []). For analyi an eign purpoe, we have aopte the following parameter in thi artile (in SI unit). B = 6.96 Dumpter length (') B =.48 Dumpter with (8') B =.48 Dumpter height (8') M = 7.6 Length of moment reiting ro (5') f = Fration of B along Y for mounting B an B = 8.88 Boom length C B (6') (eign variable) C 4 B length along boom to equilateral triangle onnetion (eign variable) Equilateral triangle ie C C = CC4 = C4C B 4 y (eign variable) Bae mounting loation along Y for pantograph able Deign traeoff were tuie, partiularly between goo horizontality an large workpae area. Goo horizontality alway mean that the equilateral triangle will be mall, not even extening the working range to point C, let alone beyon it a eire. Goo horizontality eign an alo exit near eign where the eire motion range are partially reue ue to imaginary kinemati olution. Goo X Y workpae area eign mean large triangle (with not the bet horizontality), mounte low on the par (with poor aoiate Z height). There i alo a traeoff oniering only workpae, between high X Y workpae area an aeptable Z height harateriti: large X Y area an have poor height harateriti, even ipping the triangle into the groun! Goo eign for Z height harateriti (proper vertial learane, minimum variation in Z over all motion) an be aoiate with poor X Y area. Stati ha the leat effet in hooing goo eign; the important able tenion magnitue were fairly teay over a wie range of varying parameter. However, tati wa in one 6

17 ene the mot important analyi beaue it pointe out that ome eign mut be avoie beaue they require negative able tenion uring part of the range of motion. We i not attempt global optimal eign of the ytem ubjet to horizontality, workpae, an tati iue; rather, in thi etion we propoe a goo, aeptable eign for a pratial mahine. Thi applie to a eire range of motion: = [ ], 45 an [ ], 9 horizontality, workpae, an tati i: 45 ( ) =, = =, an B 4 y =. A goo eign oniering Figure 7 how the kinemati horizontality reult (7a abolute an 7b relative); Figure 8 how the X Y workpae projetion (8a) an aoiate workpae Z height (8b); an Figure 9 how the tati reult for thi goo eign, for all motion = [ ], 45 an [ ], = (eg) Ieal = ABS (eg) Figure 7a. Angle Angular Differene from Horizontal (eg) = (eg) Figure 7b. Horizontality Deviation Figure 7a an 7b how that the horizontality aoiate with our hoen eign i not very goo at the extreme; motion near nom = 6 maintain horizontality muh better. The ahe line in 7

18 Figure 7a i the eire perfet abolute horizontality, an ieal relative value for Figure 7b are zero. Note that perfet horizontality i only poible at one loation, at the nominal nom = 6, with =. Thi eign ha a relatively large X Y workpae area a hown in Figure 8a. The aoiate Z height hown in Figure 8b are aeptable (at leat they are above the groun for all motion; they ip into the groun for ome eign); however, the Z variation i rather large, another ot of large X Y workpae area (reue horizontality i alo aoiate with large X Y workpae area). The tati reult in Figure 9, howing ative tenion t B an paive tenion t C, are typial of a wie range of poible eign; tati oe not make muh ifferene in eign. Ative tenion t B i alway le than t B for < 45 ; ue to ymmetry they wap role for 45 <, o Figure 9 over the highet tenion for all motion. In thi artile, we aume that iential N weight at vertially own at eah of the four point {C i }, i =,,,4 (one on the boom tip to inlue a poible lifting able there, the other three on the equilateral triangle vertie to provie a imple moel for RoboCrane able tenion). The hoen eign oe not require any negative able tenion for the entire eign range of motion..8.7 Z Height (fration of ) = 5 45 Figure 8a. Workpae Projetion (eg) Figure 8b. Aoiate Z Height 8

19 5 Cable Tenion (N).5 x t B t C = (eg) Figure 9. Stati Reult Z (m) X (m) Y (m) Figure. Example Deign Figure how a D Matlab renition of the hoen eign, in the motion onfiguration = an = nom = 6. Again, the eign of Figure i not a global optimum, but a goo pratial eign bae on traeoff an pratial onieration. The mathematial tool in thi artile an be ue to hek eign aniate for peifi automate ontrution appliation. 6. CONCLUSION Thi artile ha introue a new RoboCrane upport ytem bae on a ombination of rigi member an able-upene tehnology, for extening the potential of RoboCrane in variou appliation. The upport ytem itelf ha a total of three egree of freeom: the boom i riven by two ative able, an a rigi equilateral triangle hinge to the boom i kept nearly horizontal via pantograph-like able, repreenting a paive egree of freeom. The role of the equilateral triangle i 9

20 to provie a rigi, elf-ontaine, mobile overhea upport for the ix ative RoboCrane able attahment point. Preente were kinemati, workpae, an tati analye for the propoe ytem, inluing a peifi reommene eign. Thee analye alo form the bai for ontrol equation. Forwar an invere kinemati olution were preente, plu analyi of the horizontality of the paive pantographlike able mehanim (ine the onventional RoboCrane aume horizontal overhea able onnetion point). Workpae wa etermine, both planar projetion workpae an the aoiate vertial height. Stati analye involve alulating ative an paive able tenion, plu internal joint fore. Traeoff were foun amongt the variou analye; for example, a eign for goo equilateral triangle horizontality lea to a poor workpae, an vie vera. Stati wa not a major fator in hooing goo eign, exept for one all-important harateriti: tati analyi expoe onition when impoible, negative able tenion are require for ertain motion an onfiguration. Thee mut be avoie ompletely by eign, to avoi atatrophi lo of ontrol. Alo, tati will be very important in eigning the ytem to reit all loa, epeially at the pivot point C 4. Thi RoboCrane upport ytem onept how promie a a elf-ontaine, mobile, rigi upport ytem for a variety of eployable able-upene robot appliation where rigi upport are not available. ACKNOWLEDGEMENTS The author gratefully aknowlege upport for thi work from the NIST Intelligent Sytem Diviion, via Grant #7NANBH. The novel RoboCrane upport ytem onept originate with Dr. Jame S. Albu, Senior NIST Fellow, an eign uggetion were ontribute by Roger V. Botelman of NIST.

21 REFERENCES [] Jame S. Albu, 989, Cable Arrangement an Lifting Platform for Stabilize Loa Lifting, U.S. Patent 4,88,84, November 8, 989. [] N.G. Dagalaki, J.S. Albu, B.-L. Wang, J. Unger, an J.D. Lee, 989, Stiffne Stuy of a Parallel Link Robot Crane for Shipbuiling Appliation, Journal of Offhore Mehanial an Arhitetural Engineering, (): 8-9. [] J.S. Albu, R. Botelman, an N.G. Dagalaki, 99, The NIST ROBOCRANE, Journal of Roboti Sytem, (5): [4] R.V. Botelman, J.S. Albu, an A.M. Watt, 996, Unerwater Work Platform Support Sytem, U.S. Patent 5,57,596, April 6, 996. [5] T. Aria, H. Oumi, an H. Yamaguhi, 99, Aembly Robot Supene by Three Wire with Seven Degree of Freeom, MS9-87, th International Conferene on Aembly Automation, SME, Dearborn, MI. [6] M.M. Mikula Jr. an L.-F. Yang, 99, Coneptual Deign of a Multiple Cable Crane for Planetary Surfae Operation, NASA Tehnial Memoranum 44, NASA LaRC, Hampton, VA. [7] B.V. Viomi, W.D. Mihalerya, an L.-W. Lu, 994, Automate Contrution in the ATLSS Integrate Builing Sytem, Automation in Contrution, (): 5-4. [8] A.P. Shanmugaunram an F.C. Moon, 995, Development of a Parallel Link Crane: Moeling an Control of a Sytem with Unilateral Cable Contraint, ASME International Mehanial Engineering Congre an Expoition, San Franio CA, DSC 57-: [9] M. Yamamoto, N. Yanai, an A. Mohri, 999, Invere Dynami an Control of Crane- Type Manipulator, IEEE/RSJ International Conferene on Intelligent Robot an Sytem, : 8-. [] W.-J. Shiang, D. Cannon, an J. Gorman, 999, Dynami Analyi of the Cable Array Roboti Crane, IEEE International Conferene on Roboti an Automation, Detroit MI, 4: [] J.J. Craig, 989, Introution to Roboti: Mehani an Control, Aion Weley Publihing Co., Reaing, MA. [] R.L. William II, J.S. Albu, an R.V. Botelman, 4, D Cable-Bae Carteian Metrology Sytem, Journal of Roboti Sytem, (5): [] R.L. William II,, "NIST Sabbatial Report:. Self-Containe Automate Contrution Crane Sytem", ubmitte to Dr. Jame S. Albu, NIST Fellow, NIST Grant #7NANBH, May.