Self-Contained Automated Construction Deposition System

Transcription

1 Self-Containe Automate Contrution Depoition Sytem Robert L. William II Ohio Univerity, Athen, Ohio Jame S. Albu an Roger V. Botelman NIST, Gaitherburg, MD Automation in Contrution (an International Reearh Journal) : 9-47, 4 Keywor: automate ontrution, epoition, mobility, metrology, RoboCrane, able-upene, elf-ontaine, forwar poe kinemati Abtrat: Thi artile preent a novel autonomou ytem onept for automate ontrution of houe an other builing via epoition of onrete an imilar material. The overall ytem onit of a novel able-upene mobility ubytem (a elf-ontaine extenion of the RoboCrane), a epoition nozzle ubytem, a metrology ubytem, an a material upply ubytem. Thi artile foue mainly on the kinemati an tati analyi for ontrol of the elf-ontaine able-upene mobility ubytem. We alo preent alternate eign onept for the mobility ytem. The purpoe of the Carteian metrology ytem i to provie an outer-loop ontroller to provie the require Carteian poe motion epite unertaintie an unmoele effet uh a able treth, wear, an flexibility, plu win loa. Correponing author 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:

2 . INTRODUCTION Conventional ontrution rane that an be een at any ontrution ite have the following harateriti: non-rigi upport; low payloa-to-weight ratio (inluing ounterweight); low reitane to win; inaurate ontrol of loa; 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 uneirable harateriti, the RoboCrane wa evelope at NIST [, 6, ]. 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 alo 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 for a tiff, table unerwater work platform, wherein the platform may be ontrolle to be tationary even if urrouning ea are not [4]. 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. [] 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 [8] 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. [] evelope ontrution automation tehnology wherein Stewart platform rane (i.e. RoboCrane) are entral. Shanmugaunram an Moon [9] preent a ynami moel of a parallel link rane with poitioning an orientation apabilitie, with unilateral able ontraint. Yamamoto et al. [4] propoe a rane-type parallel mehanim with three ative able for hanling

3 heavy objet. Shiang et al. [] preent a parallel four-able poitioning rane for offhore loaing an unloaing of argo veel uner high ea tate. A novel proe for epoition of material in ontrution appliation i uner evelopment by Khohnevi [7]. William et al. [] preent ynami moeling an ontrol for able-bae robot, enuring only poitive tenion uring all motion. The RoboCrane ha great potential a an automate ontrution robot ytem; however, it major rawbak i that i require rigi overhea able upport point, whih may not exit at mot ontrution ite. Therefore, thi artile introue an eonomial, elf-ontaine, movable-bae ontrution rane for teleoperate an/or autonomou ontrution appliation. Compare to exiting ommerial an propoe ontrution rane, the ytem i novel beaue it ombine onventional rigi rane member with RoboCrane-type able upenion an atuation onept to provie a rigi, lightweight, long-reah, overhea platform. Thi new onept ha the potential to are the hortoming lite above for onventional rane ytem. Alo, the elf-ontaine eign provie the require rigi overhea able onnetion point. Thi artile firt preent the overall ytem onept, an then mainly foue on the kinemati equation for ontrol of the elf-ontaine mobility ubytem. We alo onier quai-tati analyi to avoi onfiguration requiring negative atuating able tenion. Alternate eign onept are alo preente. Lat, we iu the propoe ontroller to enure uffiient auray epite real-worl iue uh a able flexibility an win fore... OVERALL SYSTEM DESCRIPTION AND APPLICATION An Automate Contrution Sytem onept i uner evelopment at NIST, bae on elfontaine extenion of the RoboCrane. The overall ytem onept, hown ontruting a builing in Figure, inlue 4 major omponent (ee Figure ): mobility ytem (elf-ontaine able-upene rane), material epoition ytem (lip-form tool), metrology ytem, an material upply ytem. The overall onept of automate ontrution of builing uing free-form fabriation i novel an integrate thee 4 ytem into an avane ontrution ytem apable of manually or autonomouly fabriating

4 6 Lb. onrete per / u ft ~ Lb onrete wall an truture while uner manual or omputer ontrol, repetively. The metrology ytem an be a non-ontat (e.g. laer-bae) or ontat ytem (e.g. tring-pot-bae []) that meaure the relative loation of the material epoition tool to a known loation for aurate plaement of material from the epoition tool. The material upply ytem fee the epoition tool with onrete or other material. It an be a ement truk or hopper, a hown in Figure, with a pump to move material to the tool. The material upply an the metrology ubytem will employ ommerial prout where poible. Figure how an experiment at NIST in (manual) epoition of onrete, builing a portion of a wall via a prototype epoition nozzle with lip-form tool. Suh epoition nozzle are alo available ommerially (e.g. *. One goo hoie for the Carteian metrology ytem i a erie of three non-ontat laer aime at the rane epoition ytem (Figure ). Thi olution provie an aurate 6-of poe meaurement that i inepenent of the rive train enoer for able length feebak. mobility ytem (rane) an material epoition ytem builing Mobility Sytem RoboCrane Truk or Hopper of PreMixe Conrete (or other material) flexible tube pump Hopper ~4 u ft, ~ gal apaity Figure. Automate Contrution Sytem Metrology Sytem front Depoition Sytem groun Material Supply Sytem Figure. Automate Contrution Subytem * The ientifiation of any ommerial prout or trae name oe not imply enorement or reommenation by Ohio Univerity or NIST. * The ientifiation of any ommerial prout or trae name oe not imply enorement or reommenation by Ohio Univerity or NIST. 4

5 Figure. NIST Conrete Depoition Experiment Now we preent an etimate of the time require to buil a nominal houe (5. x 9. x 6. m (5 x x ft)), via manual metho in thi paragraph, followe by the automate ytem in the following paragraph. Typial of toay ontrution metho are labor-intenive blok an brik plaement with mortar joint. Auming tanar blok imenion. x. x.4 m (8 x 8 x 6 in), there will be blok per layer an layer in the houe. Auming e for laying an mortaring eah blok, hour i require to lay the blok for the entire houe. Then auming a pot-proe tuo appliation time of. e/m ( e/ft ), an aitional.7 hour i require for urfae finihing. Thu, we etimate almot hour ( hour, 4 minute) i require for bloking the houe via onventional manual metho. Now we etimate the time require for builing an equivalent truture via onrete epoition uing the propoe automate ontrution ytem, intea of manual blok-laying. Current onrete lip-form tehnology (e.g. * enable large blok-ize (ro etion of. x. m (5 x 8 in)) epoition of onrete at a rate of 4.9 m/min (6 ft/min). For the ame nominal 5. x 9. x 6. m houe, uing thi epoition ro etion, 48.8 m (6 ft) travel i require per row, with 48 row require. Therefore, the total epoition travel mut be 4.9 m (768 ft), an the total epoition time i thu 8 hour. Sine the automate ytem an be eigne to apply the eire finih a the wall are being ontrute, no aitional time i require for finihing. Thu, aoring to our etimate, the 5

6 propoe automate approah will require jut over one-thir the time of onventional metho. The automate ytem ha the aitional benefit of little to no human uperviion require after etup. Not hown in our imple etimate are other require proee uh a reinforement between layer (one reinforement proe i explaine in [7]). Alo not hown in the etimate are embee proee that oul be intalle uring the wall-buil proe, uh a water piping an heat ut, plu eletrial, phone, internet, an other utilitie. With ingle blok-ize layer, thee utilitie oul be intalle within the onrete layer; we an alo evelop an autonomou ual-wall approah for thi.. MOBILITY SYSTEM Thi etion preent the eription, kinemati, an tati for the elf-ontaine ableupene robot of Figure. Thi i the mobility ubytem of our overall automate ontrution ytem onept.. Mobility Sytem Deription Figure 4 how the NIST elf-ontaine mobility ubytem onept. Thi robot i intene to be a veratile, eonomi, aurate tool for the ontrution inutry. The ytem i upporte by a tanar ontrution ite umpter, fitte with moment reiting upport ro to reit tipping; the umpter i rotate by a mall angle φ B to move the tipping point forwar from the front ege of the umpter. In Figure 4, the fixe bae frame ha fixe able onnetion point B, B, B, an B 4. The moving mat (boom C B plu hinge equilateral triangle C C C 4 ) i onnete to the bae umpter via a univeral joint (allowing pithing an yawing) at B. The moving mat i artiulate via able of length L B an L B, whoe ative atuating winhe are mounte at point B an B. Therefore, vertial able upport point C an move to inreae the ytem workpae in a elf-ontaine manner. Point B an B are aume to lie on the right an left umpter ie, but an be mounte anywhere along thee ie, an above the top of the umpter a hown. Two paive, fixe-length able upport the equilateral triangle, attahe from fixe point B 4 over pulley at point C to moving point C an C. 6

7 The vertie of the moving platform are P, P, an P, an point P i the entroi of the moving platform. The epoition nozzle tip N i loate at the origin of moving frame {N}. Sine RoboCranetype evie have limite rotation, the entire nozzle i rotate via a turntable attahe to the moving platform, with rotary variable θ N. The worl oorinate frame {} i aligne with the floor at the bak ege of the umpter a hown; thi frame i really hien in the view of Figure 4. The length of the nine aitional ative able are L i, i =,,, 9. A hown in Figure 4, able onnet C to P, able onnet C to P, able onnet C to P, able 4 onnet C to P, able 5 onnet C to P, able 6 onnet C to P, able 7 onnet C to P, able 8 onnet C 4 to P, an able 9 onnet C 4 to P. Cable length,, an are ontrolle by ative winhe mounte at B ; therefore the line B C i atually three (artiulating ine C an move) able, paing from eah winh over point C to moving platform point P i, i =,, ; thee three are the heavy-lift able. Cable 4 an 5 are ontrolle by ative winhe mounte at C, able 6 an 7 are ontrolle by ative winhe mounte at C, an able 8 an 9 are ontrolle by ative winhe mounte at C 4. Cable 4-9 provie table, rigi ontrol of all ix egree-of-freeom, in onjuntion with the heavy-lift able. The onept of Figure 4 an be een a an irregular RoboCrane with moving, elf-ontaine, vertial upport point C, C, C, an C 4, ontrolle by able through 9. Thi RoboCrane, however, i overatuate (three more able than the minimum number of ix able for a eiling-mounte RoboCrane an 6-of operation). Point C an C allow the moving platform workpae to exten beyon vertial point C. To maintain ontrol in all motion, all able tenion mut remain poitive at all time. In orer to provie a more table tanar ontrution ite umpter bae, we tilt the bae by a mall angle φ B about the X axi on the bak bottom orner of the umpter (ee Figure 4). Thi move the tipping point of the ytem from the front bottom orner of the umpter to the en of the moment reiting ro of length M. A hown in Figure 5, the mehanim analogy for thi umpter tipping i a 7

8 lier-rank mehanim, where the rank B pivot about X, the oupler i M (hinge at the bak top orner of the umpter), an the foot pa (preaing out the weight over the groun) i the lier, onnete to M with a pin joint. From thi lier-rank analogy, given the eire umpter tilting angle φ B, we an alulate the variable φ M an y M (ame for both umpter ie, at ifferent X loation). C L C L C L L L C C 4 L B L 4 B L 8 L 9 L B L 5 P X N P L 6 P Z N θ N N C L 7 P B B X Y B 4 Z B Y N φ B Figure 4. NIST Automate Contrution Sytem Diagram φ y M M = o = in M B oφb φb M ( φm + φb ) B in φb () M B B B φ M off z φ B Y B Z B y M off y Figure 5. Slier-Crank Analogy for Dumpter Tipping 8

9 The umpter tilting in Figure 5 oul be atuate by able an winhe, one et on either ie of the umpter, where the motor an winh i mounte to the umpter an the other en of the able i mounte to the oupler M (or vie-vera).. Mat Subytem Kinemati Thi etion preent the kinemati analyi for the mat portion of the NIST Automate Contrution Sytem. We wih the equilateral triangle to be a horizontal a poible for all motion. The role of the mat i to provie elf-ontaine mobility for the RoboCrane-like portion of the automate ontrution ytem (ee Figure 4). A hown in Figure 6, the mat onit of equilateral triangle C C C 4 hinge via a revolute joint to boom C B at point C 4. C L C C L C C 4 L B l B C B B 4 L B B h B B φ B Figure 6. Mobility Sytem Mat The key apet of thi mat onept i that the onfiguration of the equilateral triangle portion 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 move point C, thee paive able move an upport the equilateral triangle portion paively, whih in turn upport ix of the RoboCrane-like able for moving the automate ontrution platform. At any intant uring motion, the two paive able an be een a two length on either ie of the pulley, 9

10 L C between point C an point C an C, repetively, an 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 an X B in the nominal onfiguration when the boom i in the Y Z plane), both able portion L C are guarantee to be (theoretially) 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 portion of the mat 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. Figure 7 how a kinemati iagram of the mat, onnete to the umpter frame at point B via a univeral joint allowing yaw (θ ) an pith (θ ). Thi referene poition efine both angle to be zero. The mat an be oniere to be a R erial robot onnete to the umpter with joint angle θ an θ, plu angle θ moving the triangle with repet to the boom. θ, θ, an θ are not ontrolle iretly but via ative able L B an L B, an paive able L C + l. The origin of frame {B } i mounte to point B ; the orientation of {B } i iential to that of {B} (whih i the ame a {}, but rotate by φ B ). A een in Figure 7, i the length of boom C B, i the length from boom bae point B to the equilateral triangle onnetion point C 4, an i the equilateral triangle ie (with height h ). Moving point C 5 i the mipoint of C C.

11 Z C X C C Y C Z θ C 4 X C h X, Z B θ Y, X B Z B,, θ B C 5 C Figure 7. Mat Kinemati Diagram The pith angle θ houl be kept well away from the groun poition beaue thi approahe a ingularity where able L B an L B are ollinear with the mat; in thi ingularity infinite fore woul be require to move the mat (ue to the ytem eign, the ingularity atually our unergroun, but high fore are require a the mat approahe the groun). In etup, the boom will mot likely be lifte by thee able from the groun; thu, thi i another reaon to mount point B an B above the umpter top by h B (thi effetively move the ingularity further from the groun). The Denavit-Hartenberg (DH) parameter [5] for thi erial robot are given in Table. Note a joint angle offet of 9 i require for i= (i.e. θ + 9 ) ine the X B an X axe are not aligne in the zero poition. Note alo that, in the onvention of [5], the only mat length parameter to appear in the DH table i, beaue the lat ative frame i {}, entere at C 4. The other pertinent length mut be inlue at the next tage of mat kinemati. In Table, θ i are the only variable, while the remaining DH parameter are ontant. The homogeneou tranformation matrie relating frame {C i }, i =,,4, 5 (whoe origin are point C i an whoe orientation i iential to that of {}), to the worl frame {B } an be foun uing ymboli omputer kinemati from the DH parameter an homogenou tranformation relationhip.

12 Table. Mat Denavit-Hartenberg Parameter i α i- a i- i θ i + 9 θ 9 θ θ [ ] ( ) [ ] ( ) [ ][ ] T T T T, i i C B B C θ θ θ = 5 =,,4, i () Note in () we have taken avantage of the oneutive parallel Ẑ an Ẑ axe; in uh ae we expet funtion of ( ) θ θ + to implify the kinemati equation via um-of-angle formula. The lat tranform [ ] T i C for ue in () are obtaine by uing ientity for the orientation an the ontant relative poition vetor { } C i. Subtituting the DH parameter an the ontant tranform into () yiel: [ ] = 4 B C T () where we have ue the abbreviation i i θ o = an i i θ in = ; alo ( ) o θ θ + = an ( ) in θ θ + =. The formula for { } B C 4 i the fourth olumn, firt three row of (). Other poition vetor are: { } = B C { } = h h h B C (4) { } + + = h h h B C { } + + = 5 h h h B C

13 B The orientation aoiate with { C } i not epenent on θ : = C R (5) B [ ] Now, given value for θ, θ, an θ, it i eay to evaluate the abolute poition of moving point C i with repet to {B } uing the above formula. Ultimately all vetor will be repreente in the {} frame uing B C T i B Ci = T T. However, thee erial angular value θ, θ, an θ 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, the upper mat point C i peifie at eah intant (it an be moving). It i onvenient to peify C via angle θ an θ (ine C i ontraine by the length ), uing the firt expreion of (.5). If we wih to peify the pith angle a an abolute (horizontallyreferene) angle, we nee to firt alulate the relative pith angle uing the umpter tilting angular offet: θ = θ ABS φb. Then we an eaily alulate the two require able length L B an L B uing the Euliean norm of the appropriate vetor ifferene a given below: L B = C B B = C B L (6) ) For forwar poe kinemati, the two able length L B an L B are known from their winh angular feebak meaurement. Upper mat point C i alulate given thee two able length. From Figure. an., point C i the interetion of three phere: fixe mat raiu entere at B, raiu L B entere at B, an raiu L B entere at B. The interetion of three phere i alo the bai for the forwar poe kinemati olution of the nine-able RoboCrane-like evie. Thi olution i preente in []. C i foun from the interetion of three phere with thee enter an raii: ( B,L B ), ( B,L B ), an ( B, ). Note thi orering of the three phere i very important to avoi

14 ingularitie []. Given B C = B T C C we next alulate thi vetor with repet to {B }: [ ] ; then we an alulate paive univeral joint angle θ an θ from an invere poition kinemati olution of B the expreion for { } = { P P P } T C in (4): x y z θ Px = P tan Py θ = in z (7) Note that the reulting θ in (7) i a relative angle, with repet to {B }; the abolute (horizontallyreferene) pith angle mut take umpter tilting angle φ B into aount: θ θ + φ ABS = B. Now we an etermine θ ; it i one in the ame manner for both of the above ae. Note that θ i efine to be a relative angle (with repet to boom C B ) an hene require no offet like θ. Figure 8 how a ie view of the mat arrangement. Thi ie view how a planar repreentation of the paive equilateral triangle pantograph able; l = CB4 are the real 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 6 an viualize the plane C C C ; l biet thi triangle): l = L C 4. Then, uing the law of oine: ( ) ( ) l h h + θ = o (8) θ i negative in (8) ue to it efinition in Figure 7 an 8. Thi pantograph mehanim i eigne to attempt to maintain the equilateral triangle a near horizontal a poible for all motion. It an be exat only at one θ angle, but we wih it to be loe at all other onfiguration. If the triangle i perfetly horizontal, the following onition i met: θ =. θ ABS Now, we till nee to alulate the (atual) able length L C for θ etermination. The two paive pantograph able (running from B 4, over pulley at C, onneting to point C an C ) are of 4

15 fixe length, L + nom = l LC. Therefore, L C L nom l =. Let u fix L nom by eign, requiring the equilateral triangle to be exatly horizontal ( θ = ) at a nominal value of the abolute pith angle θ ABS (θ ABSnom ) an for the entral value of the yaw angle, θ ; θ ABSnom houl be in the mile of the nom = allowable θ ABS range, or ome other nominal, often-ue onfiguration. At the nominal onfiguration we have L = l + L, where lnom = CnomB4 nom nom Cnom. We alo have L = l nom ; the 4 Cnom + nominal virtual pantograph able length i l nom CnomC5nom =. The nominal loation of C an C 5 are foun by ubtituting θ = an relative angle θ nom = θ ABSnom φb into the firt an lat expreion of (4): B { B C } = { } nom nom nom C C 5nom = nom + hφ B (9) nom hφb l C 5 h C 4 l θ θ ABS θ Z B 4 X B Z φ B Y Figure 8. Mat Sie View Finally, given θ, θ, an θ from (7) an (8) we an alulate the poition vetor for point C i from (4), for general onfiguration; we an then tranform thee to {}. Our urrent mobility ytem parameter are (m unit) M = 7. 6, = 4. 84, =8. 999, = 6.96, off y = offz =. 696, an φ B =. We aume a tanar 6.96 x.48 x.48 bae umpter; point B an B are at the front of the umpter, mounte h =. 48 from the umpter top. Figure 9 how a erie of mat motion in the Y Z plane for our mat eign. 5 B

16 Figure how the horizontality reult for our mat eign, over all motion. Mat pith angle θ ABS i the inepenent variable, while familie of urve are given for ifferent θ value (,5,,45 ); in thi manner, one plot over all motion. Note that in all reult (inluing tati later), the motion i ymmetri with repet to ± θ. Figure how the θ reult (negative for eay omparion to the mat pith angle θ ) for all motion. For horizontality, we eire θ =, whih i the ahe (Ieal) line in Figure. We an θ ABS ee that thi i atifie (theoretially) only at θ θ 48, for θ. Away from thi ABS = ABSnom = = onition, the θ reult eviate ignifiantly from the eire ahe line. 7 Z θ (eg) Ieal θ = Y θ ABS (eg) Figure 9. Paive Horizontal Mehanim Demontration Figure. Mat Angle θ Figure a how the X Y workpae an Figure b how it aoiate C C Z height, for our mat eign for all motion. In analye with ifferent mat eign we iovere that goo horizontality i aoiate with poor X Y workpae an vie vera, emontrating traeoff between performane meaure in mat eign. 6

17 5 Z Height (m) Figure a. Mat X Y Workpae Projetion 5 θ = θ ABS (eg) Figure b. Aoiate Z Height. Mobility Sytem Kinemati The invere poe kinemati problem i tate: Given the require nozzle tip poe [ T] N an the eire poition of upper mat point C, alulate the eleven able length L i, i =,,, 9 an L B an L B an θ N. The olution to thi problem may be ue a the bai for a poe ontrol heme. For the automate ontrution ytem, invere poe kinemati i eaier to olve than forwar poe kinemati: given the poe of the nozzle tip {N} tool, we firt peify θ N aoring to epoition tak requirement. Then we an fin moving platform able onnetion point P, P, an P ; then the invere poe olution onit imply of alulating the able length uing the Euliean norm of the appropriate vetor ifferene between the variou moving an fixe able onnetion point. The invere poe kinemati olution yiel a unique loe-form olution, an the omputation requirement are not emaning. A preente in Setion., the firt tep in the invere poe kinemati olution are to peify C via angle θ an θ, uing (4), alulate L B an L B uing (6), alulate θ from (8), an then alulate the remaining moving able-onnetion point C i uing (,4). With an alternate teleoping boom eign, it i poible to eaily peify C iretly an then alulate angle θ an θ. 7

18 Given [ T] N an θ N, we then alulate the moving platform poe [ T] onnetion point P, P, an P : [ ] [ ][ ] T T P P N = T N P an then moving able, where P N T i a funtion of θ N an the nozzle poition with repet to the moving platform. The vetor poition of point P i with repet to {} are: P { P } [ T]{ P } i = i =,, () P i Note we mut augment eah poition vetor in (.6) with a in the fourth row. The fixe relative vetor { P P i } are from platform geometry. Given the moving able onnetion point P, P, an P in {} from (), we an to fin the nine unknown able length. The invere poe kinemati olution i the Euliean norm of the appropriate vetor ifferene a hown below: L = P C L = P C L = P C L 4 = P C L 5 = P C 6 = P C L 7 = P C L 8 = P C4 L 9 = P C4 L () The forwar poe kinemati olution i require for imulation an enor-bae ontrol of the NIST Automate Contrution Mobility Sytem. The forwar poe kinemati problem i tate: Given the eleven able length L i, =,,, 9 i, an L B, L B, an θ N, alulate the nozzle tip poe [ T] N. For thi ytem, forwar poe kinemati i not a traight-forwar a invere poe kinemati. However, unlike mot parallel robot forwar poe kinemati problem, there exit a loe-form olution, an the omputation requirement are not emaning. There are multiple olution, but generally the orret olution for the automate ontrution ytem an be eaily etermine. A preente in Setion., the firt tep in the forwar poe kinemati olution are to alulate C given L B an L B uing the interetion of three phere, alulate the paive univeral joint angle θ an θ from (7), alulate θ from (8), an then alulate the remaining moving able-onnetion point C i uing (,4). The remaining forwar poe kinemati olution onit of fining the interetion point 8

19 of three given phere; thi mut be one three aitional time in the following equene, one for eah moving platform able onnetion point P i. Let u refer to a phere a a vetor enter point an alar raiu r: (,r).. P i foun from the interetion of: ( C,L 4 ), ( C 4,L 8 ), ( C,L ).. P i foun from the interetion of: ( P, p ), ( C 4,L 9 ), ( C,L ).. P i foun from the interetion of: ( P, p ), ( P, p ), ( C,L ). The etaile olution for the interetion of three phere i preente in []; that referene alo preent iuion on imaginary olution, ingularitie, an multiple olution. Now let u finih the forwar poe kinemati olution. Given Pi, we an alulate the orthonormal rotation matrix [ R] P iretly, uing the efinition that eah olumn of thi matrix expree one of the XYZ unit vetor of {P} with repet to {} [5]. Thee olumn are alulate a follow, from moving platform geometry. X ˆ P = P P where P 4 i the mipoint of P P. Given Pi Finally, ue [ T] [ T][ T] N P P N P P = to alulate the nozzle tip poe. ˆ P P4 Y P = Zˆ P Xˆ P Yˆ P P P4 = () an [ P R], we then have [ P T], an [ ] [ ][ ] P T = T T P P P There are two olution to the interetion point of three given phere []; therefore, the forwar poe kinemati problem yiel a total of 4 = 6 mathematial olution ine we mut repeat the algorithm four time for the NIST Automate Contrution Mobility Sytem. It i generally traightforwar to etermine the orret olution uing logi in the forwar poe kinemati oftware. Figure how a nominal poe for our mobility ytem eign Matlab imulation. i i i. 9

20 Z X Y Figure. Mobility Sytem Matlab Moel.4 Mat Subytem Stati Thi etion iue the moel for quai-tati tenion-bae ontrol of the able-upene ontrution ytem. Firt we onier mat-moving tenion, then the overall mat tati moel, followe by imulation reult..4. Mat-Moving Tenion. Now we onier a ruial iue for moving the mat. If the yaw angle θ i ommane to a value that i too large, one of the mat moving able will require an impoible puhing fore. Figure how the top view of thee mat-moving able plu boom. When the X Y projetion of mat B C beome ollinear with the X Y projetion of able L B, we have reahe the poitive limit on θ (by ymmetry, the equal, negative limit on θ our when B C i ollinear with L B ). We an alulate ± θ LIMIT a follow: B ± θ = ± LIMIT tan () ( ( ) + ) + ( + ) B f off y oφb B hb offz in φb

21 f i the fration along the Y B iretion where point B an B are mounte to the umpter. Figure 4 how θ LIMIT for f fration from to an ifferent umpter tipping angle value φ B =,,5,7. B L B +θ LIMIT B Y X θ Limit (eg) φ B = 5 7 B 5 C Figure. We ee that L B θ LIMIT Determination Figure 4. LIMIT f (fration of full ) B θ for Different φ B, f ± θ LIMIT inreae (whih i goo) with inreaing f in all ae; alo ± θ LIMIT ereae (whih i ba) with inreaing φ B in all ae. Any peifi eign i a ingle point on Figure 4; the plot verify that for large limit on θ we mut move the point B an B a far forwar a poible ( f = ). However, thi aue a lo of half the moment arm for lifting the mat (ompare to f = ); thi i why we alo raie B an B off the umpter in the Z B iretion an aitional h B amount, to reover the original moment arm for able L B an L B. All motion houl be kept well away from the peifi ± θ LIMIT for any given eign, to afely avoi the lak able problem an the reulting atatrophi lo of ontrol. For our eign with φ B =, the theoretial θ limit i ± Mat Stati Moel. The mat tati problem i tate: given external loa at point C i, i =,,,4, plu the ytem onfiguration, alulate the tenion in all able. Internal joint fore between rigi member are alo unknown. We aume all rane member are weightle an all able are in tenion. Now we outline our general D mat tati olution. For eah of the free boy iagram (not hown ue to lak of pae) of the equilateral triangle C C C 4, pantograph pulley, an boom B C,

22 we an write two D vetor equation of tati equilibrium: F = an M =. Therefore, we have a total of ix alar equation time three moving boie, for eighteen equation. However, we only have twelve unknown, alar able tenion t C (the ame for both paive pantograph able), t B, an t B (the ative mat-moving able tenion), plu three D vetor internal fore unknown, between the equilateral triangle an boom, between the pulley an boom, an between the boom an bae umpter. Thu, for olution, we ignore ix of the tati equation; if we follow the metho now eribe, the unknown may be foun member by member. Firt, for the equilateral triangle, ue only the M = alar equation, in {} oorinate; thi equation yiel the unknown t C, the tenion in both pantograph able C C an C C. Then we ue all three fore omponent in the F = vetor equation to fin the internal fore of the boom ating on the equilateral triangle at the pin joint loate at C 4. Next uing the pantograph pulley free boy iagram, we quikly onlue that the two pantograph able tenion between C an B 4 are iential to the two t C previouly foun (from M = for the pulley). Then we ue all three fore omponent in the F = vetor equation to fin the internal fore of the boom ating on the pulley at the C pin joint. Finally, the remaining unknown t B an t B are foun from the boom free boy iagram an M =, in {} oorinate. In thi ae, the x moment omponent yiel = an an be ignore; imultaneou olution of the linear equation reulting from the y an z moment omponent yiel t B an t B. Now, thi omplete the outline of the tati olution for eign purpoe in thi artile; for ompletene, one oul ue the F = vetor equation to fin the internal fore of the boom ating on the bae at the univeral joint loate at B, for eign of the univeral joint. The two pantograph able tenion t C are guarantee to remain in tenion, by the eign of the pantograph portion of the ytem (exept in the ae of ertain extreme ynami motion own, whih mut be avoie); boom-moving able tenion t B an t B an beome lak uner quai-tati onition, iue in the following ubetion along with tati eign plot. z z

23 .4. Mat Stati Reult. Out of the nine unknown (three alar tenion an three - omponent vetor fore) olve in the previou ubetion, we will now preent atual tenion t C an t B for our mat eign. The internal joint fore an alo be important for ytem eign, in izing the member to hanle the tre. Alo, ue to ymmetry, atual tenion t B an t B have ymmetri behavior with regar to θ. The tati reult for our mat eign are hown in Figure 5. x 4 t B Cable Tenion (N).5.5 θ = 5 45 t C θ (eg) ABS Figure 5. Mat Stati Reult We aume that iential N weight at vertially own at eah of the four point C i, i =,,,4, for all motion. Real paive able tenion t C o not hange muh for either variation in θ or θ ; the magnitue generally tay below 5 N. Ative mat able tenion t B tay relatively ontant (ereaing lightly), a θ inreae, for a given θ. Again, the partner tenion t B i le than t B for poitive θ, with a imilar hape a t B in Figure.; for negative θ, t B i iential to the t B hown in Figure.. The magnitue of t B i generally muh greater than that of t C, at leat an orer of magnitue greater. Thi i ue to a longer moment arm to the loa for t B ompare to that of t C. 4. ALTERNATE MOBILITY SYSTEM Alternate eign are poible for our mobility ytem onept (ee Figure 4), epening on peifi appliation, workpae reah requirement, loa, an other onieration. In thi etion we onier four apet in Figure 5 that an be moifie:. The primary mat boom C B an be rigi or

24 teleoping;. There an be three heavy lift able L, L, L, or a ingle heavy lift able;. The pantograph-like mehanim for attempting to maintain horizontal RoboCrane upport point C C C 4 an be ative or paive; an 4. We an ue an equilateral triangle upport C C C 4 a hown in Figure 4, or a mat with ro par an jib. The eign onept of Figure 4 how a rigi primary mat boom, three heavy lift able, paive pantograph-like mehanim, an equilateral triangle upport. By ontrat, Figure 6 below how a teleoping primary mat boom, a ingle heavy lift able L L, ative pantograph-like mehanim, an mat with ro par an jib. Thee eign feature may be mixe an mathe a eire to aomplih peifi eign goal for variou ontrution appliation. In Figure 6, the ative pantograph-like able i ontrolle by a motor an winh at B 4 ; it pae over a pulley above point C an onnet to moving point C 4. It i theoretially poible in thi ae to enure that plane C C C 4 i alway horizontal in the worl frame. The jib i a rigi link C C 4 that i hinge via revolute joint at point C. The ingle heavy lift able onnet to the entroi P of the moving platform, run over a pulley (not hown) at point C, an i atuate by a heavy lift motor at point B. The primary mat C B teleope; the en member C C C i a rigi ro-hape member. For all eign, we an ue the ame bae umpter with tilting, the ame mat-moving able, an the ame moving platform with turntable, plu nozzle or tool. The role of the Figure 6 mat i again to provie elf-ontaine mobility for the RoboCrane-like portion of the automate ontrution mobility robot. Jib tip point C 4 i atively ontrolle by a variable able length L jib = l + l, onneting C 4 to B 4 over a pulley, ontrolle to enure that virtual ioele triangle C C C 4 i horizontal for all motion. 4

25 C C L L L B L C B 7 L 6 C 4 L 9 L 8 P Z N X N Y N P N L 5 P L 4 B X B P Y θ N L B B 4 B Z φ B Figure 6. Alternate NIST Self-Containe Mobility Sytem 5. PROPOSED CONTROLLER CONCEPT To atify ommane Carteian trajetorie, we propoe the following ontroller. Given a erie of ommane Carteian poe, we ue the invere poe kinemati olution of Setion. to alulate the require able length (all mat an moving platform able) at eah ontrol tep. Eah of thee ommane able length will be ahieve at a high ontrol upate rate (ay Hz) via the motor an able reel, with joint enoer an a rotary-to-linear mapping for atual able length feebak. Aoring to NIST RoboCrane harware experiene, able length ening uing enoer in the loa path oe not yiel uffiient auray for the ontrution tak. Therefore, we alo propoe a Carteian metrology ytem (a non-ontat laer-bae 6-of ytem inepenent of the robot) to provie an outer-loop ontroller, that an run lower (ay Hz) in orer to provie a ervo to reue error in the Carteian poe, ue to real-worl iue uh a moeling unertaintie, able treth, wear, an flexibility, plu win loa. 5

26 In future work we will implement thi ontroller, along with the equation of thi artile, in a Matlab/Simulink imulation, to etermine a baeline ontroller eign for real-worl appliation. We will alo moel able flexibility an imulate real-worl iturbane uh a win loa to tet the robutne of the propoe ontroller. However, ue to NIST harware implementation an teting experiene, we believe that our propoe ontroller with inepenent Carteian metrology-bae ervo to tak auray will be uffiient, even in non-laboratory ontrution environment. 6. CONCLUSION Thi artile ha preente two alternate eign onept for a novel automate ontrution ytem bae on material epoition. We fou mainly on the elf-ontaine, able-upene mobility ytem. The NIST RoboCrane ha been evelope a a tiff, table rane evie that ontrol all ix egree-of-freeom of the loa. However, the tanar RoboCrane require rigi overhea upport point for the ix able. Thi work i an attempt to exten the RoboCrane to a mobile, elf-ontaine able-upene rane that provie it own rigi overhea able upport point. We preente the overall ytem onept, an then erive kinemati equation for ontrol of the able-upene mobility ubytem. We alo oniere tati analyi for the mobility ytem mat, an ientifie an alulate motion limit for avoiing negative able tenion uring operation. We preente alternate mobility ytem onept for the elf-ontaine RoboCrane. The Carteian metrology ytem provie a mean to ahieve Carteian trajetorie in the fae of unertaintie an unmoele effet uh a able treth, wear, an flexibility, plu win loa. Inutrial robot olve thi problem by being bulky, tiff, an heavy, with large motor an low payloa to weight ratio. Exiting ontrution rane ytem have tiff boom but alo employ winging able that o not ontrain all ix egree of freeom. The onept of thi artile provie a lightweight ytem 6

27 with able-upene atuation that an provie tiffne in all ix egree of freeom. The metrology ytem will enable aurate ontrol epite real-worl unertaintie an iturbane. ACKNOWLEDGEMENTS The firt author gratefully aknowlege upport for thi work from the NIST Intelligent Sytem Diviion, via Grant #7NANBH. REFERENCES [] J.S. Albu, R. Botelman, an N.G. Dagalaki, 99, The NIST ROBOCRANE, Journal of Roboti Sytem, (5): [] Jame S. Albu, 989, Cable Arrangement an Lifting Platform for Stabilize Loa Lifting, U.S. Patent 4,88,84, November 8, 989. [] 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. [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] J.J. Craig, 989, Introution to Roboti: Mehani an Control, Aion Weley Publihing Co., Reaing, MA. [6] 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. [7] B. Khohnevi,, Automate Contrution uing Contour Crafting Appliation on Earth an Beyon, 9 th International Sympoium on Automation an Roboti in Contrution, Gaitherburg, MD: [8] 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. [9] 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-: [] 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: [] B.V. Viomi, W.D. Mihalerya, an L.-W. Lu, 994, Automate Contrution in the ATLSS Integrate Builing Sytem, Automation in Contrution, (): 5-4. [] R.L. William II, J.S. Albu, an R.V. Botelman,, Cable-Bae Metrology Sytem for Sulpting Aitane, ASME Deign Tehnial Conferene, 9 th Deign Automation Conferene, Chiago, IL, September -6. [] R.L. William II, P. Gallina, an J. Vaia,, "Planar Tranlational Cable-Diret-Driven Robot", Journal of Roboti Sytem, (): 7-. [4] 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-. 7