Design of a 3-6 Hexapod Platform Sensor Using Forward Kinematics

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1 Procdings of th World Congrss on Enginring and Computr Scinc 5 Vol I WCECS 5, Octobr -3, 5, San Francisco, USA Dsign of a 3-6 Hapod Platform Snsor Using Forward Kinmatics Hongliang Shi, Yu Sh and Xuchao Duan Abstract This papr prsnts a dsign of a 3-6 hapod platform for motion masurmnt. This hapod platform is capabl of dtrmining th displacmnts in 6 dgrs of frdom DOF b masuring th lngth changs of th struts. asd on th gomtric laout, a inmatic modling of th platform is dscribd in this papr. Th forward inmatics is drivd for th mchanism to stablish th rlationship of th output and input: th displacmnts of th top platform, and th lngth changs of th struts. asd on th drivd modl, an algorithm of 6DOF motion masurmnt is proposd for th platform snsor basd on two angl sstms. A cas stud is prsntd for th discussion of th forward inmatics basd masurmnt algorithm, and th application of th platform snsor. Ind Trms masurmnt algorithm, snsor, Stwart platform, hapod, mtrolog, inmatics. I. INTRODUCTION A paralll mchanism is formd b conncting a functional bod to a rfrnc bod through two or mor lmnts []. Paralll mchanism is widl usd in motion positioning dvic with high prcision. A lot of prior wor b othr rsarchrs has bn don rgarding dsign, inmatic modling of paralll mchanisms [], [3], [4], [5]. Culpppr and Andrson [6] dsignd and calibratd a monolithic spatial compliant nano-manipulator. Chn and Culpppr [7] dsignd and calibratd a si-ais micro-scal nanopositionr. Dagalais t al. [8] drivd th inmatic modl of a paralll robot lin cran. Chao t al. [9] prsntd a novl mthod for inmatic calibration of a planar paralll flur positionr. Varadarajan and Culpppr [] conductd th calibration of a dual-purpos positionr-fitur, which has 6 dgr-of-frdom DOF. Shi t al. [] did inmatic calibration of a hapod nanopositionr. Chn and Hsu [] drivd th inmatic modl of a tripod machin tool. Although a lot of wor has bn don in dsign, manufacturing, and calibration of hapod platform and paralll mchanisms [3], [4], thr is littl wor on th dsign of paralll mchanisms for motion masurmnt. Olarra t al. [5] dsignd a hapod mchanism, drivd th inmatic modls and did th calibrations. Nubiola and onv [6] built a 6DOF paralll masurmnt sstm. Nanua t al. [7] drivd a inmatic modl of a 3-6 Stwart platform. Th modling and masurmnt algorithm of th dsigns ar critical for th accurac in masuring 6DOF. In this papr, w dsign a 3-6 hapod platform for 6DOF displacmnt masurmnt. Th platform snsor is capabl Manuscript rcivd Jul, 5; rvisd Aug 9, 5. Hongliang Shi is with th Mchanical Enginring and Arospac Dpartmnt, Th Ohio Stat Univrsit, Columbus, OH 43, USA. Corrsponding author. -mail: shi.347@osu.du. Yu Sh is with th Mchanical Enginring and Arospac Dpartmnt, Th Ohio Stat Univrsit, Columbus, OH 43, USA. Xuchao Duan is with K Laborator of Elctronic Equipmnt Structur Dsign, Xidian Univrsit, China. Fig.. Top pla orm Mo on s c Strut A Strut Sphrical joint as stag Dsign of a 3-6 Hapod Platform of masuring th displacmnt of th top platform b mans of th data of th struts lngths. A inmatic modling is dscribd basd on th topolog and gomtric laout of th dsign. A forward inmatic solution is analzd to build an algorithm of two situations for masuring th displacmnt of th top platform. Discussion and Comparison ar proposd for guiding th us of th masurmnt algorithm. Th rst of th papr is organizd as follows. Sction II illustrats th dsign and th topolog of th 3-6 hapod platform. In Sction III, w prsnt th inmatic modl of th platform. In sction IV, th forward inmatics masurmnt algorithm is proposd for this motion snsor dsign, basd on two angl sstms. Sction V is th cas stud and discussion of th forward inmatics basd masurmnt algorithm. Sction VI is th conclusion of th papr. II. DESIGN OF A 3-6 STEWART PLATFORM In this sction, w prsnt th dsign of a platform snsor basd on th 3-6 Stwart Platform. Th dsign principl and th topolog of th platform ar dscribd. A. Dsign of a 3-6 Stwart Platform In ordr to obtain th displacmnt in 6DOF, w dcid to us a hapod dsign. Th paralll Stwart Platform dsign has th advantag of high prcision with limitd worspac and is widl usd in man applications. Howvr, it is not fficint to obtain th solution of forward inmatics. In ordr to obtain th forward solution, w choos a 3-6 hapod platform instad 6-6 hapod platform. Th struts ar indpndnt in 6-6 hapod platform whil ach two struts ar connctd on th sam top points in a 3-6 hapod platform. As shown in Fig., th platform is composd of thr main parts: bas stag, struts, and top platform. Th bas stag is fid to ground. Th moving top platform is th load-carring part, which is usd as th nd ffctor for masuring. Th strut is composd of two sparat sgmnts dfind as strut Slot ISSN: Print; ISSN: Onlin WCECS 5

2 Procdings of th World Congrss on Enginring and Computr Scinc 5 Vol I WCECS 5, Octobr -3, 5, San Francisco, USA 3 3 A 4 4 r a β i o α i Fig.. 6 Schmatic Drawing of th Hapod Platform. 5 r b A 3 5 A and strut, shown in Fig.. Strut A is a tub with a long slot. On nd of th strut A is installd with a sphrical joint, which is connctd to th top platform. Strut is placd into strut A. On nd of th strut is also installd with a sphrical joint which is connctd to th bas stag. A motion stic of strut is trudd from th slot of th strut A. Th motion stic is usd to attach translational displacmnt snsor.. Topolog Analsis As shown in Fig., w build th global coordinat sstm on th top cntr of th top platform at th hom position. Th bas stag is full constraind in 6DOF. Th struts ar connctd to bas stag with si sphrical joints. ottom 6 sphrical joints ar indpndnt btwn ach othr and ach has 3DOF of rotation. Th top platform is connctd to 3 sphrical joints. Th si struts ar connctd in paralll to th top platform at th thr joints. Each two struts ar connctd to on joint indpndntl from ach othr. This mans that ach strut also has 3DOF at th top sphrical joint. This stting can b considrd as si indpndnt sphrical joints whil ach two joints gomtricall ovrlap at th sam plac. III. KINEMATIC MODEL In this sction, w illustrat th inmatic modling of th hapod mchanism. Th gomtric laout is dscribd and thn th inmatic modl is drivd. A. Gomtric Laout For convninc, w dfin th following paramtrs for dscribing th gomtr of th inmatic modl. Th lngth of ach strut is dnotd as L i, which is dfind as th distanc btwn th cntrs of th two sphrical joints locatd rspctivl at th nd of th strut A i and i. At th original position of th hapod mchanism, w dnot th positions of th thr cntr points at th top platform and th si at th bas stags b A i, and b i, rspctivl. L i is th original lngth of th strut. Fig. 3 shows th gomtrical rlationship of th nin points. Th top thr points ar smmtricall placd at th cornrs of a triangl. Th bottom si points ar also smmtricall placd at th cornrs of a hagon. Th variation of th location of th Fig. 3. Gomtric Laout of th Joints of th Top Platform and th ottom Stags. bottom si points could b polar smmtrical, such as th ampl shown in Fig.. W dnot th distanc btwn th intrscting points of th struts at th bottom stag as. For th top platform, th distanc btwn th intrscting points of th struts is. Ths points on th moving platform and th stags can b dscribd in th global coordinat fram as 6 r a A i = [R z π/3], i =,..., 6 t r b i = [R z π/3], i =,..., 6 H t whr [R ] is th 3-b-3 rotation matri about th z ais. H dnots th hight btwn th top surfac of th bottom stag and th bottom surfac of th top platform. t is th thicnss of th top platform. r a and r b ar th radii of th strut attachmnt points bottom plats in hom position[8], [9]. H is th hight of th hapod mchanism from th bottom stag to th top plan of th platform and it is drivd as th unnown b solving quation A T A L =. t is th thicnss of th top platform.. Kinmatic Modl As shown in Fig. 4, lin d is dfind as th shortst distanc btwn point and vctor. Thus, d is prpndicular to and th intraction of thm is C [7]. As shown in Fig. 5, th vctor d is C. Go through C, w draw a vctor, which lis in th plan of bas stag, and is prpndicular to th z ais of th global fram.in gomtric math, i is dfind as i = z u i, i =,, 3 whr vctor u i is th normalizd vctor of, 4 3, 6 5. Th ar dnotd as ISSN: Print; ISSN: Onlin WCECS 5

3 Procdings of th World Congrss on Enginring and Computr Scinc 5 Vol I WCECS 5, Octobr -3, 5, San Francisco, USA z r r d c L z r r d r ψ L c φ Fig. 4. Kinmatic Modling of th 3-6 Hapod Mchanism. u = u = 4 3 u 3 = 6 5 whr vctor and ar dnotd as 3 = 4 = 6 5 = 4 3 = = T 5 Th lngth d i is dfind b th quations d = L Sinϕ d = L 4 Sinϕ d 3 = L 6 Sinϕ 3 6 whr ϕ i is drivd b th and L i. L i dnots th changd lngth of th strut whn th top platform movs to a nw position. ϕ = ArcCos +L L ϕ = ArcCos ϕ 3 = ArcCos L +L4 L 3 L 4 +L 6 L 5 L 6 As shown in Fig. 5, th position of point A i is drivd b a srial chain of vctor C i, and th projction of d i on i and z. C i is th position vctor of point C i in th global coordinat. 7 A i = C i + d i zsinψ i + d i i Cosψ i 8 whr ψ i is th angl btwn d i and i. Sinc th top platform is a rigid bod, th distanc btwn two points is a constant. Aftr w obtain th positions of A i, w can build thr constraint quations, E : A = E : A 3 A = 9 E 3 : A 3 = whr A = A A T A A. Fig. 5. Drivation of th Kinmatic Modl. IV. MEASUREMENT ALGORITHM In this sction, w prsnt th masurmnt algorithm basd on th forward inmatics. A. Forward Kinmatics Rgarding th modling of th paralll mchanisms, it is asir to obtain th invrs inmatic solution than th forward inmatic solution. In th invrs inmatics, th position of th top platform is th input and th output is th lngths of th struts. In th forward inmatics, w tr to obtain th position of th top platform in 6DOF givn th lngths of th struts. In th forward inmatics, th output is th displacmnt of th top platform which is givn b a 4 b 4 transformation matri [T ], furthr dfind b a rotation matri [R] and a vctor v. Th transformation matri [T ] is unnown whil th lngths ar nown valus in th forward inmatics. Howvr, th ar govrnd b th constraint Eq. 9. Aftr solving Eq. 9 for ψ i, w can obtain th positions of th sphrical joints of th top platform at th nw position b substitut ψ i into Eq. 8. Th vctor v is drivd b th cntr of th thr points. Th rotation matri [R] is 3 i A v = i 3 [R] = [ A A 3 ] [v v v] [ A A A 3]. Rotation Matri As w all now, th rotation matri can b obtaind b th product of thr rotation matrics [R θ ], [R θ ], [R z θ z ]. asd on th squnc of th rotation about th as, w could obtain 6 combinations. [R] = [R ] [R ] [R z ], [R ] [R z ] [R ], [R ] [R ] [R z ], [R ] [R z ] [R ], [R z ] [R ] [R ], [R z ] [R ] [R ]. W can driv th rotational angls θ, θ, θ z, in th forward inmatics givn th drivd [R] and th squnc of th rotations. Furthrmor, th rotation of a coordinat can also b dfind b th Tait-ran angls with aw, pitch, and roll. ISSN: Print; ISSN: Onlin WCECS 5

4 Procdings of th World Congrss on Enginring and Computr Scinc 5 Vol I WCECS 5, Octobr -3, 5, San Francisco, USA Hr, w dnot thm as θy, θp, θr, rspctivl. In ordr to build th rlationship btwn th rotation matri and angls, w dfin th rotation matri in a gnral wa with 9 lmnts. a a a3 [R] = a a a3 3 a3 a3 a33 Th angls θy, θp, θr ar govrnd b 3 quations. a = Cos θy a = Cos θp a33 = Cos θr 4 Du to th proprt of th rotation matri, th lmnts ar also govrnd b 6 constraint quations. a + a + a3 = a + a + a3 = a3 + a3 + a33 = a3 = a a3 a a3 a3 = a3 a a3 a a33 = a a a a 5 Fig. 6. Givn th rotation matri [R], w can also obtain th angls θy, θp, θr basd on Eq. 4 in th forward inmatics. Howvr, du to th rang of th Cosin function, multipl solutions will b obtaind. Thus, w nd Eq. 5 to furthr valuat th solutions. Multipl solutions ar introducd also bcaus of th potntial smmtrical laout of th mchanism. It is undtrmind if th valus of th strut lngths ar th onl givn information. Som mor information or assumptions ar ndd to dtrmin th positions and angls. For ampl, if th platform is usd to continuousl masur small displacmnts, w can appl th minimum nrg mthod to find th uniqu solution. Masurmnt Algorithm of Forward Kinmatics. δ θz C. Motion Masurmnt of Forward Kinmatics Th masurmnt algorithm of th forward inmatics is illustratd flowchart of Fig. 6. Hr, δi ar th nown in th and v ar unnowns. valus. θ, θ, θz or θy, θp, θr W tr to calculat th rror of th rotational displacmnts and th rror of th translational displacmnts µ. Rcord th masurmnt data δi of th changd lngths of th struts. Calculat th total strut lngth Li. Li = Li + δi 6 3 Substitut Li as Li, and build th forward inmatic modl b Eq Solv Eq. 9 for ψi. i b substitut th solutions obtaind from 5 Calculat A stp 4 to Eq. 8. i b Eq. and Eq.. 6 Driv [R ] and v from A 7 Dtrmin th rotational angls of th masurmnt objct: θ, θ, θz or θy, θp, θr. 8 Calculat θ, θ, θz b solving Eq.. Or calculat θy, θp, θr b Eq Obtain th idal targt rotational angls of th objct: θ, θ, θz or θy, θp, θr. Obtain th targt translational displacmnt of th objct v. ISSN: Print; ISSN: Onlin Fig. 7. Pur Rotation along z Ais. Th rrors of th rotational displacmnt ar calculatd b ϵ = θ θ 7 whr ϵ includs ϵ, ϵ, ϵz or ϵy, ϵp, ϵr. Th rrors of th translational displacmnt ar calculatd b µ µ µ = = v v 8 µz V. CASE STUDY AND DISCUSSION An ampl of th pur rotation along th z ais is shown in Fig. 7. Assum that th platform is targtd to purl rotat with an angl θz, our goal is to masur th motion accurac of th targt objct. In th forward inmatic masurmnt algorithm, th rror is calculatd b Eq. 7. WCECS 5

5 Procdings of th World Congrss on Enginring and Computr Scinc 5 Vol I WCECS 5, Octobr -3, 5, San Francisco, USA ϵ z = θ z θ z 9 Th discussion of th masurmnt algorithm is shown as following: Forward inmatic algorithm dirctl shows th rrors in 6DOF. Th cas stud shows that it can valuat th masurmnt according to th singl DOF. In th forward inmatics, w firstl nd th masurmnt data and thn do a sris of calculation basd on th data. For ampl, th targt objct is programmd to mov to a list of idal targt positions. Th hapod snsor follows th objct movmnts and rcords a list of ral data, δi. asd on th ral data, w driv th ral positions. Th targt ral primntal positions and th idal targt positions can b usd to calculat th rrors. 3 In th invrs inmatics, w can calculat th thortical solution δ i bfor th masurmnt. For ampl, a list of th targt positions is cratd. asd on th list, w calculat a list of δ i bfor th masurmnt. During th masurmnt, w can calculat th rror immdiatl aftr obtaining th masurmnt data. This fficint algorithm can also b programmd to a ral-tim algorithm. 4 oth algorithms rquir calculation of rotation matri and translational vctor, howvr th calculation is don in th diffrnt stps. 5 According to th rquirmnt of th masurmnt, w choos th appropriat algorithm. [8] N. G. Dagalais, J. S. Albus,.-L. Wang, J. Ungr, and J. D. L, Stiffnss stud of a paralll lin robot cran for shipbuilding applications, Journal of Offshor Mchanics and Arctic Enginring, vol., no. 3, pp , 989. [9] D. Chao, G. Zong, R. Liu, and J. Yu, A novl inmatic calibration mthod for a 3DOF flur-basd paralll mchanism, in Procding of 6 IEEE/RSJ Intrnational Confrnc on Intllignt Robots and Sstms. IEEE, Oct. 6, pp [] K. M. Varadarajan and M. L. Culpppr, A dual-purpos positionrfitur for prcision si-ais positioning and prcision fituring: Part i. modling and dsign, Prcision Enginring, vol. 3, no. 3, pp , 7. [] H. Shi, H.-J. Su, N. Dagalais, and J. A. Kramar, Kinmatic modling and calibration of a flur basd hapod nanopositionr, Prcision Enginring, vol. 37, no., pp. 7 8, 3. [] J.-S. Chn and W.-Y. Hsu, Dsign and analsis of a tripod machin tool with an intgratd cartsian guiding and mtrolog mchanism, Prcision Enginring, vol. 8, no., pp , 4. [3] H. Shi and H.-J. Su, Worspac of a flur hapod nanopositionr, in Procdings of ASME IDETC/CIE,, chicago, Illinois, August -5. [4] H.-J. Su, H. Shi, and J. Yu, Analtical complianc analsis and snthsis of flur mchanisms, in Procdings of ASME IDETC/CIE,, washington, DC, August 8-3. [5] A. Olarra, J. Alln, and D. Aint, Eprimntal valuation of a spcial purpos miniatur machin tool with paralll inmatics architctur: Fr lg hapod, Prcision Enginring, vol. 38, no. 3, pp , 4. [6] A. Nubiola and I. A. onv, Absolut robot calibration with a singl tlscoping ballbar, Prcision Enginring, vol. 38, no. 3, pp , 4. [7] P. Nanua, K. Waldron, and V. Murth, Dirct inmatic solution of a stwart platform, Robotics and Automation, IEEE Transactions on, vol. 6, no. 4, pp , Aug 99. [8] H. Shi, H.-J. Su, and N. Dagalais, A stiffnss modl for control and analsis of a MEMS hapod nanopositionr, Mchanism and Machin Thor, vol. 8, pp , 4. [9] H. Shi, Modling and analsis of compliant mchanisms for dsigning nanopositionrs, Ph.D. dissrtation, Th Ohio Stat Univrsit, Columbus, OH,USA, 3. VI. CONCLUSION In this papr, w prsnt a dsign of a 3-6 hapod platform snsor for masuring th displacmnts in 6DOF. An analtical modl is dscribd basd on th gomtric laout and inmatic modling of th mchanism. asd on th forward inmatics, w driv a masurmnt algorithm according to two angls sstms for th hapod platform snsor. A cas stud and discussion ar proposd for th application of th snsor, and th appropriat choic of th masurmnt algorithms according to th diffrnt situations. REFERENCES [] H.-J. Su, H. Shi, and J. Yu, A smbolic formulation for analtical complianc analsis and snthsis of flur mchanisms, ASME Journal of Mchanical Dsign, vol. 34, no. 5, p. 59,. [] H. Shi and H.-J. Su, An analtical modl for calculating th worspac of a flur hapod nanopositionr, ASME Journal of Mchanisms and Robotics, vol. 5, no. 4, p. 49, 3. [3] Q. Xu, Nw flur paralll-inmatic micropositioning sstm with larg worspac, Robotics, IEEE Transactions on, vol. 8, no., pp , April. [4] Y. Li and Q. Xu, A totall dcoupld pizo-drivn z flur paralll micropositioning stag for micro/nanomanipulation, Automation Scinc and Enginring, IEEE Transactions on, vol. 8, no., pp , April. [5] H. Shi, X. Duan, and H.-J. Su, Optimization of th worspac of a MEMS hapod nanopositionr using an adaptiv gntic algorithm, in Robotics and Automation ICRA, 4 IEEE Intrnational Confrnc on, Ma 4, pp , hong ong, Ma 3-Jun 7. [6] M. L. Culpppr and G. Andrson, Dsign of a low-cost nanomanipulator which utilizs a monolithic, spatial compliant mchanism, Prcision Enginring, vol. 8, no. 4, pp , 4. [7] S.-C. Chn and M. L. Culpppr, Dsign of a si-ais micro-scal nanopositionrhfl, Prcision Enginring, vol. 3, no. 3, pp , 6. ISSN: Print; ISSN: Onlin WCECS 5