DETC ANALYSIS OF THREE DEGREE OF FREEDOM 6 6 TENSEGRITY PLATFORM

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1 Proceedngs of 006 DETC: ASME 006 Desgn Engneerng Techncal Conferences and Computers and Informaton n Engneerng Conference September 0-, 006, Phladelpha, Pennsylvana, USA DETC AAYSIS OF THREE DEGREE OF FREEDOM 6 6 TESEGRITY PATFORM Anton Baker Center for Intellgent Machnes and Robotcs Unversty of Florda Ganesvlle, Florda 6 Carl D. Crane III Center for Intellgent Machnes and Robotcs Unversty of Florda Ganesvlle, Florda 6 ABSTRACT The mechansm studed n ths paper s a three degree of freedom 6 6 tensegrty structure. A tensegrty structure s one that balances nternal (pre-stressed) forces of tenson and compresson. These structures have the unque property of stablzng themselves f subjected to certan types of dsturbances. The structure analyzed n ths paper conssts of two rgd bodes (platforms) connected by a total of s members. Three of the members are noncomplant constantlength struts and the other three members consst of sprngs. For typcal parallel mechansms, f the bottom platform s connected to the ground and the top platform s connected to the base by s complant leg connectors, the top platform wll have s degrees of freedom relatve to the bottom platform. However, because three of the s members connectng the two platforms are noncomplant constant-length struts, the top platform has only three degrees of freedom. The prmary contrbuton of ths paper s the analyss of the three degree of freedom tensegrty platform. Specfcally, gven the locaton of the connector ponts on the base and top platforms, the lengths of the three noncomplant constantlength struts, and the desred locaton of a pont embedded n the top platform measured wth respect to a coordnate system attached to the base, all possble orentatons of the top platform are determned. ITRODUCTIO The word tensegrty s a combnaton of the words tenson and ntegrty. Tensegrty descrbes a structural relatonshp prncple n whch structural shape s guaranteed by the closed, contnuous, tensonal behavors of the system and not by the dscontnuous compressonal member behavors. Tensegrty provdes the ablty of a structure n theory to yeld ncreasngly wthout ultmately breakng [Fuller,979]. Ths paper wll present an analyss of the geometrc propertes of platforms whch ncorporate tensegrty prncples. A platform s descrbed as any devce that has multple legs connectng a movng (top) platform to a bottom (base) platform [Abbas, Rdgeway, Adst, Crane, Duffy, 000]. It was only recently that the forward poston soluton of a platform was formulated by [Grffs and Duffy, 989]. In ther analyss, all postons and orentatons of the top platform are determned based on gven lengths of the s leg connectors. Ths platform (Fgure ) was the smplest of the geometres to solve. The formulaton of ths soluton yelded and eghth degree polynomal n the square of one defnng parameter. ater, ths soluton technque was appled to a 6 platform (Fgure ). A forward dsplacement analyss of a general 6 6 constant strut length platform (see Fgure ) showed that the soluton was n the form of a 0 th degree polynomal [Raghavan, 99]. The tensegrty structure analyzed n ths paper conssts of a specal 6 6 platform. Ths geometry was desgned to nclude the benefts of a platform and a general 6 6 platform [Grffs and Duffy, 99]. Ths specal platform makes the analyss comparable to a platform whle elmnatng mechancal nterference assocated wth the platform. The 6 6 platform analyzed n ths paper conssts of two rgd bodes connected together by three constant length noncomplant struts and three complant tes that each conssts of a sprng n seres Fgure : Platform Copyrght 006 by ASME

Fgure : 6 Platform Fgure : Marshall and Crane s Tensegrty Platform Fgure : General 6 6 Platform wth a non-complant te where the length of the non-complant te can be controlled.

For ths analyss, the bottom platform s connected to the ground and the top platform has three degrees of freedom relatve to the bottom platform.

Comparng ths devce wth a typcal parallel platform such as that n Fgure, t can be seen that three of the leg connectors have been replaced by three tenson members that consst of three complant sprngs

2 Fgure : 6 Platform Fgure : Marshall and Crane s Tensegrty Platform Fgure : General 6 6 Platform wth a non-complant te where the length of the non-complant te can be controlled. The legs are connected to the platforms wth ball and socket jonts. For ths analyss, the bottom platform s connected to the ground and the top platform has three degrees of freedom relatve to the bottom platform. MOTIVATIO Pror work (Marshall and Crane, 00) showed that t s possble to desgn a parallel platform that ncorporates tensegrty prncples. Fgure shows one verson of such a mechansm. Comparng ths devce wth a typcal parallel platform such as that n Fgure, t can be seen that three of the leg connectors have been replaced by three tenson members that consst of three complant sprngs connected n seres wth three non-complant tes. The lengths of the non-complant tes may be controlled by havng an actuator wnd the te about a drum whch s prmtvely llustrated on the base platform n Fgure. Ths acts, n effect, to allow the effectve free length of the complant tensle leg connector to be controlled. Marshall [Marshall, 00] showed that t s possble to poston and orent the top platform to a desred pose at a desred total potental energy level. In that paper t was shown how the lengths of the three leg connectors and the lengths of the three varable length non-complant tes could be determned to attan the desred pose and potental energy state. In ths paper, the three leg connectors are now replaced by three fed length struts. The new devce, whch s smlar to that n Fgure ecept that now the three compressve legs are now of fed length, s now a three degree-of-freedom system where the only varable nputs are the lengths of the three noncomplant tes that are each n seres wth a sprng of known free length and sprng constant. Here t wll be shown how the lengths of these three non-complant tes can be determned n order to poston a pont on the top platform at a desred locaton whle the mechansm s at a desred total potental energy state. The motvaton for ths work s to determne f there s a means whereby the potental energy n the system can be redstrbuted n order to reposton the top platform to some desred locaton n an energy effcent manner. The resultng system has the potental to be a very energy effcent devce. In ths paper, the lengths of the three struts are fed, so that for ths smple case, the struts wll do no work (ether postve or negatve) as the top platform moves to a new desred pont. Ths paper provdes the ntal analyss that s requred for the platform n order to consder the energy dstrbuton problem n the future. PROBEM STATEMET Ths secton presents the soluton of how to fnd all possble orentatons of the top platform wth respect the bottom platform when a pont embedded n the top platform s postoned at a desred locaton. The model used for ths analyss s shown n Fgure 5. Ponts B, B, B, and T, T, T correspond to the centers of the sphercal jonts at the bottom and top ends of the three constant length noncomplant struts whch are numbered, 5, and 8 n the fgure. Pont P s a pont that s embedded n the top platform. Fgure 5 shows two coordnate systems that are defned for ths problem. The frst s attached to the base platform wth ts orgn at pont B and X as through B. Pont B les n the T 8 z y 0 B B P 5 y B z T T Fgure 5: Knematc Model of dof Tensegrty Platform 5 Copyrght 006 by ASME

XY plane. The second coordnate system s attached to the top platform. The orgn s at pont T, ts X as passes through pont T and pont T s n the XY plane.

the three constant length noncomplant struts,, 5, and 8, the coordnates of pont P n the nd (top) coordnate system, whch mples that the lengths, 6, and 9 shown n Fgure 6 are known, the coordnates of

statement, the lengths of all ffteen lne segments shown n Fgure 6 are known.

The problem can be thought of as that of determnng all the possble relatve orentatons of the two tetrahedrons such that the three dstance constrants assocated wth the constant length struts are

3 XY plane. The second coordnate system s attached to the top platform. The orgn s at pont T, ts X as passes through pont T and pont T s n the XY plane. The precse problem statement s now presented as follows: Gven all dmensons of the top and bottom platforms,.e. the lengths 0,,,,, and 5, where the notaton refers to the length of bar, the lengths of the three constant length noncomplant struts,, 5, and 8, the coordnates of pont P n the nd (top) coordnate system, whch mples that the lengths, 6, and 9 shown n Fgure 6 are known, the coordnates of pont P n the st (base) coordnate system, whch mples that the lengths,, and 7 shown n Fgure 6 are known Fnd All possble orentatons of the top platform It s mportant to note that based on the problem statement, the lengths of all ffteen lne segments shown n Fgure 6 are known. Also t s helpful to vsualze the problem as that of havng two tetrahedrons, one defned by ponts B, B, B, and P and the other by ponts T, T, T, and P, that share the common pont P. The problem can be thought of as that of determnng all the possble relatve orentatons of the two tetrahedrons such that the three dstance constrants assocated wth the constant length struts are satsfed. IDETIFICATIO OF THREE SPHERICA FOUR-BAR MECHAISMS Ths secton begns the soluton for fndng all possble orentatons of the top platform wth respect to the bottom platform when the pont P s postoned as specfed. The analyss begns by defnng the three major planes. These planes are specal because the poston and orentaton of these planes are known. These three planes conssts of lnes (- 7-), (-7-), and (--0). Because these planes all have vertces at known ponts, the poston and orentaton can be readly determned. Several new angles wll now be defned,.e. θ, θ, θ, φ, y z T P 7 B 6 T 0 B B T Fgure 6: Known Dstances abeled 5 Fgure 7: Defnton of angles θ and θ 9 7 φ, φ, and γ, γ, γ. θ s defned as the angle between planes formed by the lnes - and -7 (Fgure 7). θ s defned as the angle between planes formed by the lnes 9-7 and -7 (Fgure 7). θ s defned as the angle between planes formed by the lnes -6 and -7 (Fgure 8). The defntons of angles φ, φ, and φ are smlar and these are shown n Fgures 8 and 9. The angle γ s defned as the angle between the planes defned by lnes -7 and -7 and s shown n Fgure 0. Ths angle s known snce the endponts of lne segments,, and 7 are known n terms of the base coordnate system. From Fgure 0 t can also be seen that φ + γ + θ = 0. () The angles γ and γ are defned n a manner smlar to γ and are also known quanttes. Further t can be shown that φ + γ + θ = 0 () and φ 6 θ Fgure 8: Defnton of angles θ and φ φ φ Fgure 9: Defnton of angles φ and φ 6 Copyrght 006 by ASME

4 9 φ + γ + θ = 0. () The analyss can proceed by recognzng that there est three sphercal four bar mechansms n the model of the devce. The output angle of one of the four-bar mechansms relates to the nput angle of the net sphercal four-bar mechansm. Fgure 7 depcts the frst sphercal four-bar mechansm. It can be seen n Fgure 6 that the known length of member mantans a constant angular relatonshp between members and 9. Smlarly Fgures 8 and 9 depcts the other two sphercal four bar mechansms snce the angular relatonshps between 6 and 9 and and 6 are constant and known. AAYSIS OF SPHERICA FOUR-BAR MECHAISM Fgure shows a sphercal four-bar mechansm. The lnks are defned by the angles α, α, α, α and the relatve angles between the lnks are defned by the angles θ through θ. A sphercal cosne law for the sphercal quadrlateral may be wrtten as (Crane and Duffy, 998) Z = c () where Z = s (X s + Y c ) + c Z (5) and where X = s s, (6) Y = -(s c + c s c ), (7) Z = c c s s c. (8) 7 γ φ θ Fgure 0: Defnton of γ S θ α S α θ α α θ S S Fgure : Sphercal Four-Bar Mechansm θ The terms s and c n equatons () through (7) are the sne and cosne of θ and the terms s j and c j are the sne and cosne of α j. Equaton () can be epanded by substtutng (5) through (7) and then rearranged to gve (A c + A ) c + A s s + A c + A 5 = 0 (9) where A = -s c s, (0) A = -s s c, () A = s s, () A = -c c s, () A 5 = c c c c. () DETERMIATIO OF PATFORM ORIETATIO Equaton () may be appled to the three sphercal mechansms to generate three equatons, one that relates θ and θ, one that relates φ and θ, and one that relates φ and φ. Equatons () through () may then be used to replace φ, φ, and φ n terms of θ, θ, and θ. The end result wll be three equatons n the three unknowns θ, θ, and θ. Table shows the varable substtuton scheme that s used to generate the three equatons n the varables θ and φ, =.. from the generc sphercal quadrlateral equaton, (6). Table : Varable Substtuton for Sphercal Mechansms Generc Quad. θ θ α - α - α - α - st Quad. θ θ α -7 α 7-9 α 9- α - nd Quad. φ θ α 7- α -6 α 6-9 α 9-7 rd Quad. φ φ α - α - α -6 α 6- Applyng the substtutons n Table to equaton (9) yelds the followng three equatons: (G c + G ) c + G s s + G c + G 5 = 0 (5) (H cosφ + H ) c + H s snφ + H cosφ + H 5 = 0 (6) (I cosφ + I ) cosφ + I snφ snφ + I cosφ + I 5 = 0 (7) where the coeffcents G through I 5 are quanttes that are epressed n terms of the constant mechansm parameters. Here the notaton s and c are used to represent the sne and cosne of θ. Equatons () through () may now be used to substtute for the angles φ n terms of θ n (5) through (7). Performng ths step and replacng the snes and cosnes of θ by the trgonometrc denttes snθ =, cosθ = + + (8) where = tan(θ )/ and regroupng gves (A 9 + A 8 + A 7 ) + (A 6 + A 5 + A ) + (A + A + A ) = 0, (9) (B 9 + B 8 + B 7 ) + (B 6 + B 5 + B ) + (B + B + B ) = 0, (0) Copyrght 006 by ASME

5 (D 9 + D 8 + D 7 ) + (D 6 + D 5 + D ) + (D + D + D ) = 0 () where the coeffcents A 9 through D are known quanttes. The problem at hand s to determne all sets of,, and that wll smultaneously satsfy the set of equatons (9), (0), and (). Values for θ may then be obtaned from each value of as θ = tan - ( ). The soluton procedure wll begn by usng Bezout s method to elmnate from equatons (9) and (0), resultng n a new equaton n and. Sylvester s method wll then be used wth ths new equaton, together wth (), to elmnate to obtan a sngle polynomal n. To start the soluton process, let = A 9 + A 8 + A 7, M = A 6 + A 5 + A, () = A + A + A and = B 9 + B 8 + B 7, M = B 6 + B 5 + B, () = B + B + B. Equatons (9) and (0) may now be wrtten as + M + = 0, () + M + = 0. (5) The condton that () and (5) have a common root for s M M = 0. (6) M M Epandng (6) and collectng terms gves V + V + V + V + V 0 = 0 (7) where V = V + V + V + V + V 0, V = V + V + V + V + V 0, V = V + V + V + V + V 0, (8) V = V + V + V + V + V 0, V 0 = V 0 + V 0 + V 0 + V 0 + V 00 and the coeffcents V j are known quantttes. The terms W, W, and W 0 are now defned as W = D 9 + D 8 + D 7, W = D 6 + D 5 + D, (9) W 0 = D + D + D. Substtutng these epressons nto () yelds W + W + W 0 = 0. (0) Sylvester s elmnaton method s then used to elmnate from the par of equatons (7) and (0). Multplyng (7) by and (0) by,, and results n s equatons that can be wrtten n matr format as W W W W W W W W 0 0 () 0 0. = W W V V V V V 0 0 V V V V V0 0 0 Ths set of s homogeneous equatons wll have a soluton only f the equatons are lnearly dependent and thus the determnant of the coeffcent matr must equal zero. Thus W W 0 0 W W 0 () 0 W W 0 0 = 0. W W V V V V V0 V V V V V0 0 Snce the terms V and W are polynomals n, epandng ths determnant yelds a 6 th degree polynomal n. Each value of obtaned by the polynomal can be substtuted back nto () to obtan correspondng values for. Once the correspondng values of have been determned for each value of, correspondng values for may be obtaned from () and (5) as M M () =, or = M M. () astly, values for the soluton sets {θ, θ, θ } are obtaned from the correspondng values of as θ = tan - ( ). The coordnates of the ponts T, T, and T may be obtaned n terms of the base coordnate system for each soluton set {θ, θ, θ }. The procedure s not presented here, but s straghtforward. Knowledge of the coordnates of these three ponts allows for the determnaton of the transformaton T that descrbes the relatve poston and orentaton of the coordnate system attached to the top platform to that attached to the base. UMERICA EXAMPE A numercal eample s presented to verfy the results. The followng nformaton was used as nput for the numercal case: Gven: P = (6.86, 0, -.7) ; pont P n st coord. system P = (.7590,, -.95) ; pont P n nd coord. system P B = (0, 0, 0) ; pont B n st coord. system P B = (0, 0, 0) ; pont B n st coord. system P B = (6.75, -5.69, 0) ; pont B n st coord. system 5 Copyrght 006 by ASME

P T = (0, 0, 0) ; pont T n nd coord. system P T = (.9500, 0, 0) ; pont T n nd coord. system P T = (0.658, -5.5675, 0) ; pont T n nd coord. system =.05 ; length of strut 5 = 9.

Table shows the calculated values for the ffteen segments shown n Fgure 6. Table : Segment engths Segment ength.9.05.86. 5 9.970 6. 7 0. 8 9.55 9.5 0 0.000 6.500 8.789 5.57.95 5 7.

Table : Results of Analyss Case Case Case Case.077 -.07.559.60 -.76 0.67 -.057 -.5 -.60.00 0.59 0.

6 P T = (0, 0, 0) ; pont T n nd coord. system P T = (.9500, 0, 0) ; pont T n nd coord. system P T = (0.658, , 0) ; pont T n nd coord. system =.05 ; length of strut 5 = ; length of strut 8 = 9.55 ; length of strut The unts of length are mmateral to ths problem as long as the same unt s used throughout the analyss. Table shows the calculated values for the ffteen segments shown n Fgure 6. Table : Segment engths Segment ength The 6 th degree polynomal n was obtaned from (). For ths case, four of the roots of ths polynomal were real. The four real cases, and the correspondng values for and are shown n Table. Table : Results of Analyss Case Case Case Case From ths nformaton, the angles θ, θ, and θ and then the coordnates of ponts T, T, and T measured n the base coordnate system were determned. Fgures through 5 show the devce n the four attanable confguratons. Fgure : Case Fgure 5: Case COCUSIO Ths study successfully analyzed a three degree of freedom tensegrty platform that ncorporates the specal 6 6 platform geometry. Ths analyss shows that f a coordnate system attached to the bottom platform s specfed, and a predetermned locaton of a pont attached to the top platform s also specfed, then the orentaton of the top platform wth respect to the coordnate system of the bottom platform can be determned. Ths analyss focused on a closed form algebrac soluton of the platform. The analyss proceeded by defnng three dependent sphercal quadrlaterals. Although teratve methods could have also gven the orentaton of the top platform, these methods would leave out crtcal nformaton. By usng teratve procedures, one would never know how many orentatons are possble of the top platform for a gven poston P. For eample, a numercal case shown n ths paper dentfed four possble orentatons of the top platform for the gven pont P. Usng an teratve procedure, the soluton of the orentaton of the top platform could possbly converge at any one of the four orentatons. The soluton of the teratve method s almost entrely based on the ntal values gven to the method. Gven ths analyss s much more rgorous than teratve methods, the nformaton obtaned about the system makes up for the addtonal effort. It was learned that there are steen possble total orentatons of the top platform. ACKOWEDGMETS The authors would lke to gratefully acknowledge the support of the Department of Energy Unversty Research Program n Robotcs (URPR). REFERECES Abbas, W., Rdgeway, S., Adst, P., Crane, C., Duffy, J., Investgaton of a Specal 6-6 Parallel Platform for Contour Mllng, ASME Journal of Manufacturng Scence and Engneerng, Vol., Feb 000, pp. -9. Crane, C., Duffy, J., Knematc Analyss of Robot Manpulators,Cambrge Unversty Press, 998 Crane, C., Screw Theory and ts Applcatons to Spatal Manpulators, n preparaton. Fgure : Case Fgure : Case 6 Copyrght 006 by ASME

7 Grffs, M. and Duffy, J., A Forward Dsplacement Analyss of a Class of Stewart Platforms, Trans. ASME, Journal of Mechansms, Transmssons, and Automaton n Desgn, Vol. 6, o.6, June 989, pp Fuller, R., SYERGETICS-Eploratons n the Geometry of Thnkng. Volumes I & II. ew York, Macmllan Publshng Co, 975, 979. Grffs, M., and Duffy, J., Method and Apparatus for Controllng Geometrcally Smple Parallel Mechansms wth Dstnctve Connectons, US Patent o. 5,79,55, January, 99 Marshall, M. and Crane, C., Desgn and Analyss of a Hybrd Parallel Platform That Incorporates Tensegrty, Proceedngs of the ASME 8 th Bennal Mechansms and Robotcs Conference, Salt ake Cty, Sep 00. Raghavan, M., The Stewart Platform of General Geometry has 0 Confguratons, Trans. of the ASME, Journal of Mechancal Desgn, Vol. 5, pp.77-8, June 99. Stewart, D., A Platform wth S Degree of Freedom, ondon, Proc. Inst. Mech Engrs., Vol. 80, pp.7-86, 965. Wlken, T. Tensegrty, Trustmark, Copyrght 006 by ASME

Pose, Posture, Formation and Contortion in Kinematic Systems

Pose, Posture, Formation and Contortion in Kinematic Systems Pose, Posture, Formaton and Contorton n Knematc Systems J. Rooney and T. K. Tanev Department of Desgn and Innovaton, Faculty of Technology, The Open Unversty, Unted Kngdom Abstract. The concepts of pose,