Kinematics of pantograph masts

Transcription

1 Abstract Spacecraft Mechansms Group, ISRO Satellte Centre, Arport Road, Bangalore , Flght Dynamcs Dvson, ISRO Satellte Centre, Arport Road, Bangalore Department of Mechancal Engneerng, Indan Insttute of Scence, Bangalore Emal:astava@mecheng.sc.ernet.n Knematcs of pantograph masts B..Nagaraj, R. andyan and Ashtava Ghosal Ths paper deals wth the knematcs of pantographs masts, whch have wdespread use as deployable structures n space. They are overconstraned mechansms wth degree-of-freedom (d.o.f), evaluated by the Grubler-Kutzbach formula, as less than one. In ths paper, we present a numercal algorthm to evaluate the d.o.f of pantograph masts by obtanng the null-space of a constrant Jacoban matrx. In the process we obtan redundant jonts and lnks n the masts. We also present a method based on symbolc computaton to obtan the closed-form knematc equatons of trangular and box shaped pantograph masts and obtan the varous confguratons such masts can attan durng deployment..0 Introducton Deployable masts used n space are prefabrcated structures that can be transformed from a closed compact confguraton to a predetermned expanded form n whch they are stable and can carry loads. Deployable / foldable mast have one or more nternal mechansms [-] and ther d.o.f as evaluated by the Grubler-Kutzbach crteron often turns out to be less than []. In ths paper we study the knematcs of deployable masts made up of pantograph mechansms or scssor lke element (SLE). An SLE n two dmensonal form has straght rods of equal length connected by pvots n the mddle. The assembly has one d.o.f and the basc model can be folded and deployed freely. Three dmensonal masts are created wth SLE n such a way that they form a structural unt whch n plan vew s a normal polygon wth each sde beng an SLE. The polygon can be equlateral trangle, square or normal n-sded polygon. By combnng several of these normal polygon shaped unts, structures of varous geometrc confguratons can be created []. Actve cables control the deployment and prestress the pantograph and passve cables are pre-tensoned n the fully deployed confguraton. These cables have the functon of ncreasng the stffness when fully deployed. The whole system deploys synchronously. The knematcs of pantograph masts can be studed by use of relatve coordnates[5], reference pont coordnates (as n the software package ADAMS) or Cartesan coordnates[6]. In ths paper Cartesan coordnates, also called natural / basc coordnates, have been used. Ths method uses the constant dstance condton for two or more basc ponts of the same lnk. Usng untary vectors the method can be extended to spatal mechansms. The man advantage of usng Cartesan coordnates s that the constrant equatons are quadratc as opposed to transcendental equatons, and the number of varables tends to be (on average) n between relatve coordnates and reference pont coordnates. In an earler study, the foldablty equatons were formulated for SLEs based on geometrc approach[7].

2 The equatons of moton for the SLE masts were obtaned and solved numercally usng Cartesan coordnates [8]. To the best of our knowledge, no attempt has been made by prevous researchers n obtanng the closed-form soluton for these masts. Closed-form equatons, are expected to consderably reduce computaton tme and allow us to obtan dfferent confguratons a mast can attan whch helps n better desgn of the system. In ths paper the closed form knematc equatons are derved for the trangular and box mast usng symbolc software MATHEMATICA[9]. Typcal deployable masts have large number of lnks and jonts. The d.o.f of these masts, as evaluated by the Grubler-Kutzback crtera, gves numbers less than one and hence, the d.o.f formula do not gve a true number. Other methods such as screw theory and graph representaton have been proposed by varous researches to evaluate the correct d.o.f [0-]. The concept of usng frst and hgher order dervatves of constrant equatons has been used for under constraned structural systems [] to evaluate moblty and state of self stress. In ths paper, we use the natural coordnates and the dervatves of the constrant equatons to obtan the correct d.o.f of deployable masts. We also present an algorthm to dentfy redundant jonts / lnks n a mast whch leads to ncorrect d.o.f from the Grubler-Kutzback crtera..0 Knematc descrpton of the mechansm The smplest planar SLE s shown n Fgure. The revolute jont n the mddle connect the two lnks of equal length. The assembly has one degree of freedom nternal mechansm. The SLE remans stress free durng the foldng and extendng process. The trangular mast and box mast are presented n Fgure and Fgure respectvely. Revolute k a c d b l j Revolute Slder Sphercal Fxed Sphercal Z Y 6 5 SLE Revolute Sphercal jont Fxed sphercal jont Y Fgure.. Basc module of SLE Fgure. Trangular SLE mast Fgure. Box SLE mast.0 Knematc Modellng 5 Z Revolute SLE Modelng of three dmensonal mechansms wth natural coordnates [6] can be carred out such that the lnks must contan suffcent number of ponts and unt vectors so that ther moton s completely defned. A pont shall be located on those jonts n whch there s a common pont to the two lnks. A unt vector must be postoned on jonts that have rotaton or translatonal axs. All ponts of nterest whose poston are to be consdered as a prmary unknown varable can lke wse be defned as basc ponts. In the natural coordnate system the constrant equatons orgnate n the form of rgd

3 constrants of lnks and knematc jont constrants. Rgd constrants of lnk: Ths mposes a constant dstance condton between two natural coordnates and j of the lnk. Ths s gven by r r L 0 () j j j where, rj T {( ),( Y Y ),( Z Z )} and K,Y K and Z K, K = or j, are the coordnates at j j j basc ponts or j.. constrants: These constrants descrbe the relatve moton n accordance wth knematc jonts that lnk them. The knematc constrants correspondng to sphercal jont s automatcally satsfed when adjacent lnks share a basc pont. Revolute jont s formed when two adjacent lnks share a basc pont and an unt vector. The constrant equatons can also be formulated for the slder par[6]... Scssor lke element : Scssor lke element (SLE) or pantograph s shown n the Fgure. Lnk j and k are coplanar and can rotate around the pvot. It s assumed that the two lnks are not equal and pvot s not n the mddle. The poston vectors must fulfll the followng geometry condtons where, m and b a d c j k l 0 () a b a b c d c d m, j, k, s the poston vector consstng of coordnates of basc ponts.. Boundary constrants: The boundary constrants to exclude the global moton, need to be defned. If the basc pont s fxed, ts coordnates are zero. If pont Q moves along a plane perpendcular to Z axs, ts Z coordnate s zero. These equatons are wrtten as p = 0=Y p =Z p =Z q (). Constrant equatons : The rgd constrant equatons, jont constrants and boundary f,,..., t 0 for j to n () constrants can be wrtten as j n c n whch n c represents the total number of constrant equatons ncludng rgd body condtons, jont constrants due to SLE, slder, revolute par and boundary constrants and n s the number of Cartesan coordnates of the system. Dervatve of the constrant equatons, wth respect to tme gve the Jacoban matrx, whch can be symbolcally wrtten as B 0 (5) Snce, equaton (5) s homogeneous, one can obtan a non-null f the dmenson of the null-space of [ ] ( n c n) B s at least one. The exstence of the null-space mples that the mechansm possess a d.o.f along the correspondng [6].

4 The deployable masts wll have large number of knematc pars and lnks. It s useful to estmate the mnmum number of knematc pars wthout losng the desred moton of the system. The dmenson of null space bass s useful n estmatng the d.o.f and dentfyng the redundant knematc pars..5 Numercal Algorthm: The man steps n the numercal algorthm are as follows: ) Add the dervatve of the constrant equatons one at a tme n the followng order arsng out of length constrants arsng out of Slder/SLE constrants ) At each step we evaluate dmenson of null space of [B]. If dmenson of null space of [B] doesn't decrease when a constrant s added t s redundant. ) Boundary constrants are added last and the dmenson of null space of [B] s evaluated. If dmenson of null space does not decrease after addng a boundary constrant, then correspondng constrant s redundant. v) The fnal dmenson of the null space of [B] s the degree of freedom of the system. In choosng the basc ponts the fnte dmensons of jonts are not taken nto account. The sphercal jonts are taken at the ntersecton pont of two adjacent lnks. In ths formulaton the ntal folded confguraton s not consdered as t can have many sngular confguratons and hence does not gve the true d.o.f of the system. The basc ponts of ntermedate confguraton s taken for the evaluaton of Jacoban matrx..0 Closed form soluton for trangular mast The numercal algorthm presented above does not gve the closed-form expressons for drect and nverse knematcs of masts. To obtan them we have to use the orgnal constrant equatons (not n ts dervatve form) and attempt to obtan the mnmal set of constrant equatons and elmnate unwanted varables. Elmnaton of varables from a set of nonlnear equatons s known to be an extremely hard problem and the dffculty ncreases wth the number and complexty of each equaton n the set. We have used a symbolc computaton software, MATHEMATICA, to obtan closed-form solutons for some masts. The natural coordnates are useful n ths respect snce the equatons are atmost quadratc n the varables used. In ths secton, we present the approach to obtan the closedform soluton for a trangular mast shown n Fgure. For smplcty, we assume that () the lnks of SLEs are equal n length and the pvot s at the mdpont of the lnks, () the jonts to 6 are sphercal jonts, () the jonts, and are constraned to move n a plane, (v) jont s fxed, and (v) the lnks are rgd and the cables used for pre stressng does not affect the knematcs. Usng Equaton (), the length constrant equaton for lnk -5 s gven by ( ) ( Y Y ) ( Z Z ) L 0 (6)

5 Smlarly, the equatons for the other lnks can be wrtten. The SLE equatons can be obtaned by usng Equaton () as 0 (7) (8) 0 (9) It can be observed that only two of the three lnear SLE equatons are ndependent. These can be checked by reducng these lnear equatons to row reduced echelon form. By usng the assumptons () and (v), and substtutng the above equatons and observng that = 0 and Y = 0 and solvng, we get only three ndependent equatons wth fve varables. Y Z L 0 (0) ( ) ( Y Y ) Z L 0 () Y Z L 0 () Assumng,Y as known nputs, the soluton for,y, and Z can be obtaned as follows. Z L ( Y ) Y Y Y () Hence, usng Equatons () and (7) through (9), the coordnates of all the jonts of the trangular mast can be obtaned n closed form. It can be observed that for the gven, Y coordnates of jont, four confguratons are possble-two confguratons each for the postve and negatve Z coordnate respectvely. Each confguraton s the mrror mage of the trangle formed about the lne jonng the jonts and. Hence, assumng the mast moves only n the postve Z drecton, the number of knematc soluton the mast can have s. From the Equaton(), we have L ( Y ) 0. If L ( Y ) 0. The coordnates of jont le n a crcle of radus L. Ths corresponds to the fully deployed confguraton wth Z = 0. Dependng on the magntude, the jont moves along the crcle of radus L. If L ( Y ) 0, the soluton for Z s magnary. The coordnates of jont le out sde the work space of the mast. The mast cannot reach these coordnates. If L ( Y ) 0. The mast confguraton ncludes the fully folded (Z = L) and very close to the fully deployed confguraton, ( Z 0 ). In the above analyss the ndependent varables are taken as the coordnates of jont. The above method can be used by takng the coordnates of jont as ndependent varables. Alternatvely

6 Equatons () contanng and Y, can be smultaneously solved to evaluate the coordnates and Y n terms of and Y and Equaton () can be used to evaluate the coordnate Z. Box Mast : For the box mast, there are eght length constrant equatons and nne ndependent SLE equatons. We get followng closed form solutons wth, Y and as nput. The solutons wth postve Z axs are gven n Table-. Ths mast has eght solutons - four confguratons each for postve and negatve Z coordnate. Each confguraton has two folded type and two deployed type of confguratons. Table : Closed form solutons for the box mast Y Soluton () Soluton () Soluton () Soluton (v) Y Y Y Y Y Y Y Y Y Y Y Y Y Y Z L ( Y ) Z L ( Y ) Z L ( Y ) Z L ( Y ) 5.0 Results and Dscussons In ths secton the methodology for evaluatng the d.o.f descrbed n prevous secton s used for trangular and box mast. Fnally we present the deployment smulaton of a sngle trangular mast usng the closed from solutons. 5. Degree of freedom and redundancy evaluaton: The trangular mast shown n Fgure has sx rgd constrants, SLEs and fxed boundary condtons at jont-. The addtonal boundary condtons at ponts and are requred to ensure the moton n Y plane. The results of null space are presented n Table. It s observed that the null space reduces by three by addng the frst SLE. The null space reduces by only for the nd SLE and t does not change for the rd SLE. The dmenson of null space for the trangular mast s. Hence, t has one rgd body rotaton about pont and one mechansm. Table : B matrx detals for Trangular mast Contents Sze of [B] Null space Remarks Length constrants (6,8) + Boundary condtons (,8) 7 + SLE I (,8) +SLE II (7,8) (one component s redundant) + SLE III (0,8) SLE III s redundant The results of null space for box mast shown n Fgure are presented n Table. It s observed from the table that the null space reduces by three by addng each SLEs. The null space reduces by only for the rd SLE. The null space does not change for the th SLE. The dmenson of null space for the

7 box masts s. Hence, the mast has one rgd body rotaton about pont and two mechansms. It s observed from the last row that the rank does not ncrease by addng addtonal boundary condtons because three boundary condtons are suffcent to represent the moton n a plane. Table : B matrx detals for Box mast Contents Sze of [B] Null space Remarks Length constrants (8,) 6 + SLE I (,) +SLE II (,) 0 + SLE III (7,) 8 (one component s redundant) + SLE IV (0,) 8 SLE IV s redundant + Boundary condtons (,) 5 ( = Y = Z = 0) + Boundary condtons (Z = Z = Z =0) (6,) (Z s redundant) The analyss was also carred out for the hexagonal mast and smlar behavour was observed. Hence, for the n-sded SLE mast, null space reduces by three for addton of each SLE and t reduces by only for addton of (n )th SLE. The null space does not change for the addton of nth SLE. 5. Knematc smulaton for the trangular mast : The trangular mast wth fully stowed confguraton s taken as the ntal confguraton. The coordnate s vared from 0 to 0 unts n steps of unts, Y s taken as 0.0 and L = 0.0. The equatons () are solved to get the coordnates of jont as the mast deploys. The smulaton s shown n Fgure. It s observed from the fgure that as the jont moves horzontally, the two solutons of jont moves along the +60 o and 60 o lne about axs. The decrease n heght of the mast durng deployment s also shown. The smulaton were also carred out for the two trangular masts attached at the sdes. Due to page lmtatons these are not shown n ths paper. Fgure : Trajectory of jont coordnates for trangular mast

8 6.0 Conclusons In ths paper the Cartesan coordnate approach has been used to obtan the knematc equatons for the three dmensonal deployable SLE masts. The d.o.f was evaluated usng the Jacoban matrx. An algorthm was presented to dentfy the redundant knematc pars. It was observed that some of the SLEs were redundant. Hence, these masts can acheve the requred sngle d.o.f wth out these knematc pars. Ths formulaton s easy to apply for the large number of masts. The knematcs of trangular and box masts were studed n closed form and the multple solutons were evaluated. Ths method can be extended to masts of dfferent shapes and for the stacked masts. References I. A.K.S. Kwan and S. allegrno., A Cable rgdzed d pantograph. roce of th European symposum on mechansms and trbology, France, Sept, 989 (ESA-S-99, March 990). Z. You, S. allegrno, Cable stffened pantographc deployable structures, art: Trangular mast, AIAA Journal, Vol., No., pp. 8-80, Aprl A.K. Mallk, A. Ghosh and G. Dettrch, Knematc analyss and synthess of mechansms, CRC ress, 99. C. J. Gantes, J. J. Conner, R. D. Logcher and Y. Rosenfeld, Structural analyss sand desgn of Deployable structures, Computers and Structures, Vol., No. /, pp , R. S. Hartenberg and J. Denavt, Knematc synthess of lnkages. Mc Graw Hll, J. Garca De Jalon and E. Bayo, Knematc and dynamc smulaton of multbody systems: The real tme challenge, Sprnger Verlag, T. Langbecker, Knematc analyss of deployable scssor structures, Internatonal Journal of Space Structures, Vol., No., pp. -5, W. J. Chen, G. Y. Fu, J. H. Gohg, Y. L. H and S. L. Dong. Dynamc deployment smulaton for pantographc deployable masts, Mechancs of Structures and Machnes, Vol. 0, pp 9-77, Mathematca, Second Edton, S. Wolfram, Addton Wesley ublshng Co J. S. Zhao, K. Zhao and Z. J. Feng, A theory of degrees of freedom for mechansms, Mechansms and Machne theory, Vol. 9, No. 6 pp. 6-6, 00.. L.-W. Tsa, Enumeraton of knematc structures accordng to functon, CRC ress, 00.. E. N. Kuznetsov, Under constraned structural systems, Internatonal Journal of Solds and Structures, Vol., No., pp. 5-6, 988.