Feasibility of Coordinate Measuring System Based on Wire Driven Robot

Transcription

1 Feasblty of Coordnate Measurng System Based on Wre Drven Robot J-Hu Zhou, Qng-Song Cao, Fa-Xong Sun, and Lan B College of Mechancal and Electrcal Engneerng, East Chna Jaotong Unversty, Nanchang , Chna Abstract. The coordnate measurng machne systems (CMMS) are wdely used for art nsectons n manufacturng lants, whch mostly nclude brdge CMMS and horzontal CMMS drven by motor and ortable CMMS oerated manually. Wre-drven arallel knematc manulators (PKMs) have advantages of flexblty, accuracy and large worksace. Ths aer nvestgates the feasblty of the coordnate measurng system based on wre-drven PKM. The forward and nverse oston solutons are obtaned by MATLAB smulaton through the knematc analyss of the manulator. The controllable worksace wth tenson condtons and stffness condtons are obtaned by means of the statc analyss. Analyss results show that the manulator have hgh recson, hgh seed and large worksace, these advantages just meet the measure requrements of coordnate measurng systems. Fnally, a rmary exerment s resented to show that the measurng latform of ths manulator can move smoothly and ts measurement error s relatvely low. The feasblty of takng the wre-drven PKM as coordnate measurng system has been valdated by theoretcal and exermental study. Ths research maybe lays a foundaton for the further alcaton of wre-drven PKM n the feld of coordnate measurng. Keywords: Wre-drven arallel knematc manulators (PKMs), Coordnate measurng system, Feasblty. 1 Introducton As one of the most ndsensable metrologcal nstruments for nsectons and qualty control n manufacturng lants, coordnate measurng machne systems (CMMS) are wdely used n many felds such as mechancal manufacturng, automotve ndustry, electronc ndustry, aerosace ndustry and natonal defense ndustry. The frst CMM was develoed by the Ferrant Comany of Scotland n the 1950s, although ths machne only had 2 axes. The frst 3-axs models began aearng n the 1960s and comuter control debuted n the early 1970s. After the 1970s, CMMS have made rad develoment [1]. Wth the develoment of modern ndustry, fast and accurate measurement s an mortant research subject [2]. At the same tme, novel tye mechansms need to be desgned. Wre-drven arallel knematc manulators (PKMs) ossess a number of romsng advantages over the conventonal rgd-lnk manulators, such as smle and lghtweght mechancal structure, hgh-loadng caacty, large worksace, low moment D. L, Y. Lu, and Y. Chen (Eds.): CCTA 2010, Part IV, IFIP AICT 347, , IFIP Internatonal Federaton for Informaton Processng 2011

2 Feasblty of Coordnate Measurng System Based on Wre Drven Robot 451 nerta and hgh seed moton. The early study on wre-drven PKMs s the alcaton n cargo handlng. Over the years, a number of dfferent wre-drven PKMs desgned for a wde varety of alcatons n large-scale manufacturng such as weldng, cuttng, grndng, assembly, fxturng, ant strng, and machnng. An examle of wre drven robots that are currently n use s the NIST Robot Crane[3]. The NIST Robot Crane s a large-worksace robot for antng and mantanng arcraft whch s also sutable for materal handlng n warehouses and storage facltes [4]. Recently, sx degrees of freedom wre-drven PKMs n the alcaton of a large rado telescoe s resented [5]. The otental for cable robots are gven to be used n a varety of alcatons, the studes on the wre-drven PKMs wll be romoted. The novel three degrees of freedom (DOF) coordnate measurng machne drven by four wres s roosed n ths aer. Based on the knematc analyss of the manulator, t s confrmed that ths manulator have large worksace, hgh moment recson and low moment error after the smulaton of MATLAB. A rmary exerment s resented to show that the movng latform of ths manulator can move smoothly, ts measurement error s relatvely low and measurement recson s relatvely hgh. The feasblty of wre-drven PKM used as coordnate measurng system s verfed by theoretcal analyss and exerment. Ths study wll lay an mortant role n ushng forward the alcaton of wre-drven PKM n coordnate measurng feld. 2 Overal Desgn The basc functon of CMMS s to gan the actual shae of a work ece. The actual shae of the work ece s obtaned by robng the surface of the work ece at dscrete measurng onts whch can then be analyzed va regresson algorthms for the work ece of features. Every measurng ont s exressed n terms of ts measured coordnates. Based on ths rncle of measurement, the overall lan desgn of the measurng system s resented n fgure 1. Fg. 1. Block dagram of the system

3 452 J.-H. Zhou et al. The measurng system ncludes four man comonents: the man structure, measurng system, touch trgger robe and electrcal control system. The man structure s an 1160mm 550mm 1000mm PKMs drven by four wres. The accurate control of the measurng latform s obtaned through usng ste motor to control the lengths of the wres. The ste motor controlled by ulse-to-ste s generated by a 51 mcrocontroller. As one of the most mortant art, the touch trgger robe greatly nfluences measurement accuracy and recson. The touch trgger robe named TP60 s used n ths aer whch has a srng loaded ruby ball stylus. The dfferent functons can be realzed by the electrcal control system through the way of mcrocontroller rogrammng. The control sgnal can be mort from keyboard or from 51 mcrocontroller and PC seral communcatons. After recevng the sgnal, the mcrocontroller jum to the corresondng rocedure to realze the dfferent functons, such as dslay coordnates on the LED nxe tube, the measurng latform s rogrammed to movng u and rght. As the robe touched the surface of the comonent, the stylus deflected and smultaneously the mcrocontroller sent nformaton of coordnate (X.Y.Z) to the PC comuter. Measurng element of the target such as the shae of the comonent, the shae of the comonent and oston was obtaned by fttng the nformaton of coordnate (X.Y.Z). 3 Theores Analyss 3.1 Structure Analyss The schematc dagram of comletely restraned wre-drven PKM s shown n fgure 2. Fg. 2. PKM drven of four wres The measurng latform s connected to statc latform through four drvng cables, AA, BB, CC, DD. The length of each cable s denoted as L ( = 1, 2, 3, 4). The cable s connected to the statc latform at ont A, B, C, and D. The cable s connected to the measurng latform at ont A, B, C and D. The center of ABCD and A B C D exressed n O and P. Make coordnate system O-xyz become the statc coordnate system. Let coordnate system O-xyz becomes the movng coordnate

4 Feasblty of Coordnate Measurng System Based on Wre Drven Robot 453 system. The unt vector n the drecton of the cable s e. The manulator has only 3 Cartesan degrees-of-freedom(x, y, and z). In Fgure 2, the desgn arameters for the PKM drven by 4 wres are 2a (statc latform length), 2b (statc latform wdth), 2c (measurng latform length), and 2d (measurng latform wdth). t s the cable tenson aled for the cable. r s the oston vector from the orgn of P to the cable connecton. For statc equlbrum the sum of external forces and moments exerted on the measurng latform by the cables must equal to the resultant external wrench exerted on the envronment. F=AT (1) Where T= ( t 1 L t 4 ) s the vector of scalar cable forces, F s the resultant external wrench vector exerted on the envronment by the measurng latform, A = [ e1l e4] s the statcs jacoban matrx. L= ( L 1 e 1 LL 4 e 4 ) T s the vector of cable length. V s the cable translatonal velocty aled for the cable, exressed n V. A s the transose matrx of A matrx. By T means of the method of the vector analyss, the followng matrx form can be obtaned L= A T V (2) The analytcal exresson of cable forces s obtaned va the frst equaton. In other words, to nvert (1) we adat the well-known artcular and homogeneous soluton T= A + F + ( E A + A) q (3) Where A + s the Moore Penrose seudo nverse of A, A + F s the mnmum norm soluton of (1), q s an arbtrary 4-vector, E s the 4 4 dentty matrx. 3.2 Poston Analyss Inverse Poston Analyss In statc coordnate system, A B C D s exressed n KKKK, whle n movng ' ' ' ' coordnate system, t s exressed n KKKK The coordnates (X, Y, Z ) s the oston of P relatng the statc coordnate system. Accordng to the closed vector aroach, the followng equaton can be descrbed K = K + P (4) Therefore, the length of each cable can be calculated as follows c a X c a X L1 = K1 K 5 = b d Y, L2 = K2 K6 = d b Y 0 Z 0 Z a c X a c X L3 = K3 K 7 = d b Y, L1 = K4 K8 = b d Y 0 Z 0 Z '

5 454 J.-H. Zhou et al. The nverse oston solutons can be exressed as L1 = L1 = ( c a X ) + ( b d Y ) + Z L2 = L2 = ( c a X ) + ( d b Y ) + Z L3 = L3 = ( a c X ) + ( d b Y ) + Z L4 = L4 = ( a c X ) + ( b d Y ) + Z (5) Forward Poston Analyss The relatonsh between the length of each cable and ( X ' Y' Z ) can be obtaned va nvert equaton L = L, so the forward oston solutons can be exressed as 2 2 L1 L4 X = 4 ( a c) 2 2 L2 L1 Y = 4 ( b d) Z = L ( a c X ) ( b d Y ) L L = L L Whatever the oston and ose of the moble latform s, the length of each cable must meet the Eq. (6). The oston and ose of the measurng latform corresondng to certan cable lengths thus are determned va the forward oston analyss of arallel wre-drven robot. 3.3 Worksace Analyss Controllable Worksace The controllable worksace s defned as a set of oston and orentaton for the latform, whch meet equlbrum of force and torque, and the tenson of each cable s requred to be ostve [6]. The controllable worksace of the manulators can be determned by checkng the tenson of each cable on the bass of the rncle of vector closure. Therefore, the manulators to meet the rncle of vector closure should be roved frstly. Accordng to the rncle of vector closure, n a 3-dmensonal real sace, the arbtrary vector v s closed only f vector v has at least 4 vectors (w 1, w 2, w 3 and w 4 ) satsfyng the followng two condtons. Condton 1 s that n the four vector (w 1... w 4 ), 4 any three vectors are lnearly ndeendent; Condton 2 s that α w = 0 (for any value of should meet α > 0 ). Let unt vector e exressed n w. Take the cable tenson = 1 (6) t exressed nα. Assume the resultant external wrench exerted on the envronment s zero ( the gravty of each roe s gnored), the followng matrx can be exressed: AT=0. Therefore,

6 Feasblty of Coordnate Measurng System Based on Wre Drven Robot = 1 α w = 0 can be obtaned through the equaton of 4 te = 1 = 0. Because the mechansms does not roduce sngular, n the four vector (w 1... w 4 ), any three vectors are lnearly ndeendent. The manulators to meet the rncle of vector closure can be roved through the above analyss. + + Accordng to the equaton of T= A F + ( E A A) q, f the tenson soluton n the above equaton can always be made ostve for all ostve homogeneous solutons regardless of the external dsturbance F, the boundary of the worksace can therefore be generated by lettng F=0. Therefore, the tenson of each cable can be ostve f each element of ( E A + A) q s ostve. So the controllable worksace bound can be exressed as follow. ( E A + A) > 0 (7) Worksace wth Tenson Condtons The worksace wth tenson condtons s defned as a set of oston and orentaton under the condtons of the controllable worksace n whch the tensons t n the cables must range between a re-tenson t mn and a maxmum tenson t max and the gravty or elastcty of each roe s gnored. Therefore, the worksace wth tenson condtons can be determned by the tensons n each cable and the external forces and torques exerted on the envronment. Assume the external forces and torques exerted on the envronment s zero, n other words F=0. Wth k=t max /t mn s defned as crtcal tenson factor, the worksace wth tenson condtons can be calculated as follows max= 1L 4 B K mn= 1L 4 B (8) + B = ( E A A) > 0 Where B = ( E A + A) > 0 s the 4 4 matrx Worksace wth Stffness Condtons The worksace wth tenson condtons for n DOF PKMs s defned as a set of oston and orentaton under the condtons of the controllable worksace n whch the ose greater than n ostve egenvalues of stffness matrx s not less than stffness coeffcents K [6]. Therefore, the stffness matrx should be calculated frstly. The stffness matrx of drven by m (m n+1) wres s obtaned by Verhoeven [7]. 0 T K = K AΛ A (9) 0 0 Where K = k L s the er unt cable length stffness, k s the cable stffness aled to the cable, L 0 s the orgnal cable length aled to the cable, matrx A must meet the condton of F = AT, wth A T s the transosed matrx of matrx A, 1 0 Λ= dag( L (1 + K t )) s a dagonal matrx. The oston and ose of the moble

7 456 J.-H. Zhou et al. latform can corresondng to certan cable lengths and tenson, L and t are the certan cable lengths and tenson. 1 0 Generally, K t s small enough whch can be gnored [8]. The manulators rgdty erformance s determned by geometrcal arrangement of wres and the oston and orentaton of measurng latform. Accordng to formula (9) we can conclude that K s a symmetrc ostve semdefnte matrx whch have n ostve egenvalues λ K ( = 12L n ) defntely. Therefore, the worksace wth tenson condtons must satsfy the followng constrants max ( E A A) > Analyss of Knematc Error = 1L 4λK K Analyss of Error Moton The knematc error for measurng latform movng from one lace to another can be obtaned va lus the error n each coordnate drecton resectvely. Ths artcle scatters a secfc sace nto dfferent onts, and uses the knematcs recson analyss n calculatng the varaton of the oston and ose of measurng latform corresondng to allowable ulse-to-ste n dfferent ont. Steer motor can transform the dgtal nut ulse revolvng or the straght lne ncrease actvty electromagnetsm functonal element. When steng drve to receve a ulse, t drves steer motor rotate n the drecton set by a fxed ont of vew, then the cable length can be changed n a defnte value Δ L. The ermeter of the roller for ste motor s 60 mm. So ΔL can be calculated as Δ L = 60 = mm (11) 360 SN SN Among them, SN s the subdvson number of steer motor. So the mnmum change of each cable length s ΔL. The measurng latform wth movng left and rght s X, movng forward and backward rd s Y and movng u and down s Z. When measurng latform movng u and down, 51 mcrocontrollers gve synchronous control of all cable s seed, the cable length can change ΔL every tme. The knematc error wll not roduced only n the condton of x=0 and y=0. When measurng latform movng left and rght, 51 mcrocontrollers gve synchronous control of the frst cable and the second cable s seed and synchronous control of the thrd cable and the fourth cable s seed. Note Δ L1 = m Δ L s the change of the frst and the second cable length, note Δ L2 = n Δ L s the change of the thrd and the fourth cable length. The knematc error wll roduced n the condton of y 0. In ths aer, m and n are lmted to the same ranges, m, n [0, 20], and let them be the ostve nteger. The value of m and n can be obtaned n the condton of the knematc error n the Z ordnate drecton s the (10)

8 Feasblty of Coordnate Measurng System Based on Wre Drven Robot 457 mnmum count. When measurng latform movng forward and backward, 51 mcrocontrollers gve synchronous control of the frst cable and the fourth cable s seed and synchronous control of the second cable and the thrd cable s seed. Note Δ L1 = m Δ L s the change of the frst and the fourth cable length, note Δ L2 = n Δ L s the change of the second and the thrd cable length. The method of gan the value of m and n s smlar wth the analyss of movng left and rght. Wth dfferent oston and ose of the measurng latform, the knematc error and accuracy wll change. So the relatonsh between the knematc error, accuracy and some arameters should be analyzed by MATLAB smulatrn. Usng comuter smulaton, we can conclude that 1) Lttler the sze of measurng latform, lower the error for all movement and the accuracy for movng u and down, hgher the accuracy for movng rght, left, forward and backward. Therefore, the sze of measurng latform can be made as small as ossble but t has a lower lmt. 2) The oston and ose n the drecton of z s the crtcal element for the accuracy of movng u and down, and lttler the count of z, hgher the accuracy. 3) If the error of movng rght, left, forward and backward remaned to lower, the orgnal oston x, y should be ostve and reman low around zero, the oston z and movng dstance h should be reman low The Best Worksace A best worksace B whch coverng x, y (-100,100) and z (-924, -724) s obtaned by comuter smulaton. In ths worksace, the knematc accuracy n the drecton of coordnate x range from mm to mm; the knematc accuracy n the drecton of coordnate y range from mm to mm; the knematc accuracy n the drecton of coordnate z range from mm to mm. Therefore, the knematc accuracy of measurng latform s relatvely hgh. If the movng dstance s 100 mm, the knematc error n the drecton of coordnate x range from 0mm to mm; the knematc error n the drecton of coordnate y range from 0 mm to mm; the knematc error n the drecton of coordnate z range from mm to mm. Therefore, the knematc error of measurng latform s relatvely low. The above analyzed results show that the manulators have hgh knematcs recson and low knematcs error n the worksace B, these advantages just meet the measure requrements of coordnate measurng systems. 4 Exerment Ths art attemts to verfy the ratonalty and feasblty of adotng wre-drven PKM as coordnate measurng system through exerments. As some lmtatons, the desgned robe s relaced by some arts n the followng exerment. Therefore, checkng the robe contacts wth the work ece s determned by vsual nsecton. Besdes, the knematcs error of the measurng latform s dscussed, t ncludes the errors of the robe, the dstorton of each roe, external condtons and so on. The materal object hotograhy of CMM s shown n fgure 3.

9 458 J.-H. Zhou et al. Fg. 3. The materal object hotograhy of the desgned CMMS In the exerment, the sze of the two rectangular objects s measured n the best workng sace. The exermental data are n the followng table, the error can be obtaned va the comarson of measure and actual sze. Intal Coordnate Values Termnal Coordnate Values Table 1. The record of the exermental data (mm) Measure Sze Actual Sze Error Intal Coordnate Values Termnal Coordnate Values Measure Sze Actual Sze Error Exermental data show that the measurement error of the desgned CMMS s u to a few mllmeters, whch s somewhat of bg. The man factors for the measurement error are human factors, such as the vsual error n checkng the robe contacts wth the work ece and reacton tme of human. If the desgned robe s used n the exerment, the measurement error should be greatly reduced and t would meet the measure requrements of coordnate measurng systems. Through the observaton of

10 Feasblty of Coordnate Measurng System Based on Wre Drven Robot 459 the exerment we can conclude that the measurement latform can moved smoothly even n the condton of fast movng. The feasblty of adotng wre-drven PKM as coordnate measurng system s verfed once agan. 5 Concluson The forward and nverse oston solutons are frstly obtaned by means of the statc analyss and the controllable worksace, worksace wth tenson and stffness condtons are resented n ths aer. Then the knematcs recson and error analyss of the wre-drven PKM s gven by MATLAB smulaton based on the knematc analyss of the manulator. The best worksace n whch the manulators have hgh knematcs recson and low knematcs error s obtaned. These advantages meet the measure requrements of coordnate measurng systems exactly. Therefore, the theores analyss verfes the feasblty of adotng wre-drven PKM as coordnate measurng system. Furthermore, the overall scheme of CMMS based on wre-drven PKM s ut forward n detals. Fnally, a rmary exerment s resented to show that the measurng latform of ths manulator can move smoothly and ts measurement error s relatvely low. The feasblty of adotng wre-drven PKM as coordnate measurng system s verfed both by theoretcal and exermental study. References 1. Lu, Z., N, X.: Present state and develong trend of three-coordnate measurng machnes. Machnery 42(480), (2004) 2. Zhang, G.: The develoment tendency of coordnate measurng machnes. Chna Mechancal Engneerng 11, (2000) 3. Edward, A., Roger, B., Ncholas, D.: Summary of Modelng and Smulaton for NIST RoboCrane Alcatons. In: Deneb Internaton Smulaton Conference and Technology Showcase, Detrot, MI (1997) 4. Goodwn, K.: RoboCrane Constructon of Brdges. In: Transformaton Research Record, Transformaton Research Board, (1997) 5. Zheng, Y.Q., Lu, X.W.: Research survey and develoment tendency of wre-drven arallel manulators. Chna Mechancal Engneerng 14(9), (2003) 6. Verhoeven, R., Hller, M., Tadoroko, S.: Works-ace, stffness, sngulartes and classfcaton of tendon-drven Stewart latforms. In: Proceedngs of the 6th Internatonal Symosum on Advances n Robot Knematcs, (1998) 7. Mng, A., Hguch, T.: Study on Multle Degree of Freedom Postonng Mechansms Usng Wres (Part I): Concet, Desgn and Control. Internatonal of the Jaan Socety for Precson Engneerng 29(6), (1994) 8. Lu, X.W., Zheng, Y.Q.: Knematc analyss of a 6-dofwre-drven arallel knematc manulator. Chnese Journal of Mechancal Engneerng 38(sulement), (2002)