PARALLEL MECHANISMS WITH VARIABLE COMPLIANCE

Transcription

1 PRLLEL MECHNISMS WIH VRIBLE COMPLINCE By HYUN KWON JUNG DISSERION PRESENED O HE GRDUE SCHOOL OF HE UNIVERSIY OF FLORID IN PRIL FULFILLMEN OF HE REQUIREMENS FOR HE DEGREE OF DOCOR OF PHILOSOPHY UNIVERSIY OF FLORID 2006

2 Copyrght 2006 by Hyun Kwon Jung

3 hs dssertaton s dedcated to my wfe, Eyun Jung Lee and son, Sung Jae.

4 CKNOWLEDGMENS I would lke express my thanks to Dr. Carl D. Crane III, my academc advsor and commttee char, for hs contnual support and gudance throughout ths work. I would also lke to thank the other members of my supervsory commttee, Dr. John C. Zegert, Dr. John K. Schueller, Dr.. ntono rroyo, and Dr. Rodney G. Roberts, for ther tme, expertse, and wllngness to serve on my commttee. I would lke to thank all of the personnel of the Center for Intellgent Machnes and Robotcs for ther support and expertse. I also would lke to thank other frends of mne for provdng plenty of advce and dversons. Last but not least, I would lke to thank to my parent, parents-n-law, my wfe, and son for ther unwaverng support, love, and sacrfce. hs research was performed wth fundng from the Department of Energy through the Unversty Research Program n Robotcs (URPR), grant number DE-FG04-86NE v

5 BLE OF CONENS page CKNOWLEDGMENS... v LIS OF BLES... v LIS OF FIGURES... x BSRC...x CHPER 1 INRODUCION Motvaton Lterature Revew Problem Statement SIFFNESS MPPING OF PLNR COMPLIN MECHNISMS Sprng n a Lne Space Dervatve of Planar Sprng Wrench Jonng a Movng Body and Ground Dervatve of Sprng Wrench Jonng wo Movng Bodes Stffness Mappng of Planar Complant Parallel Mechansms n Seres Stffness Mappng of Planar Complant Parallel Mechansms n a Hybrd rrangement SIFFNESS MPPING OF SPIL COMPLIN MECHNISMS Dervatve of Spatal Sprng Wrench Jonng a Movng Body and Ground Dervatve of Sprng Wrench Jonng wo Movng Bodes Stffness Mappng of Spatal Complant Parallel Mechansms n Seres SIFFNESS MODULION OF PLNR COMPLIN MECHNISMS Parallel Mechansms wth Varable Complance Constrant on Stffness Matrx Stffness Modulaton by Varyng Sprng Parameters...60 v

6 4.1.3 Stffness Modulaton by Varyng Sprng Parameters and Dsplacement of the Mechansm Varable Complant Mechansms wth wo Parallel Mechansms n Seres Constrants on Stffness Matrx Stffness Modulaton by usng a Dervatve of Stffness Matrx and Wrench Numercal Example CONCLUSIONS...78 PPENDIX B MLB CODES FOR NUMERICL EXMPLES IN CHPER WO ND HREE...81 MPLE CODE FOR DERIVIVE OF SIFFNESS MRIX IN CHPER FOUR...98 LIS OF REFERENCES BIOGRPHICL SKECH v

7 LIS OF BLES able page 2-1 Sprng propertes of the complant couplngs n Fgure Postons of pvot ponts n terms of the nertal frame n Fgure Sprng propertes of the complant couplngs n Fgure Postons of the fxed pvot ponts of the complant couplngs n Fgure Postons and orentatons of the coordnates systems n Fgure Sprng propertes of the mechansm n Fgure Postons of pvots n ground n Fgure Postons of pvots n bottom sde of body n Fgure Postons of pvots n top sde of body n Fgure Postons of pvots n body B n Fgure Postons of pvot ponts n body E for numercal example n Postons of pvot ponts n body for numercal example n Sprng parameters wth mnmum norm for numercal example Gven optmal sprng parameters for numercal example Sprng parameters closest to gven sprng parameters for numercal example Postons of pvot ponts for numercal example Intal sprng parameters for numercal example Calculated sprng parameters for numercal example Postons of pvot ponts n body for numercal example v

8 4-10 Sprng parameters of the complant couplngs for numercal example Postons of pvot ponts for numercal example Sprng parameters wth no constrant for numercal example Sprng parameters wth body fxed for numercal example Sprng parameters wth body and body B fxed for numercal example Matlab functon lst v

9 LIS OF FIGURES Fgure page 1-1 Planar robot wth varable geometry base platform daptve vbraton absorber Parallel topology 6DOF wth adjustable complance Sprng n a lne space Sprng arrangements n a lne space. (a) parallel and (b) seres Planar complant couplng connectng body and the ground Small change of poston of P1 due to a small twst of body Planar complant couplng jonng two movng bodes Mechansm havng two complant mechansms n seres Mechansm consstng of four rgd bodes connected to each other by complant couplngs n a hybrd arrangement Spatal complant couplng jonng body and the ground Unt vector expressed n a polar coordnates system Small change of poston of P1 due to a small twst of body Spatal complant couplng jonng two movng bodes Mechansm havng two complant parallel mechansms n seres Complant parallel mechansm wth N number of couplngs Poses of the complant parallel mechansm for numercal example Poses of the complant mechansm wth body B fxed Poses of the complant mechansm wth no constrant x

10 bstract of Dssertaton Presented to the Graduate School of the Unversty of Florda n Partal Fulfllment of the Requrements for the Degree of Doctor of Phlosophy PRLLEL MECHNISMS WIH VRIBLE COMPLINCE By Hyun Kwon Jung May 2006 Char: Carl D. Crane III Major Department: Mechancal and erospace Engneerng Complant mechansms can be consdered as planar/spatal sprngs havng multple degrees of freedom rather than one freedom as lne sprngs have. he complance of the mechansm can be well descrbed by the stffness matrx of the mechansm whch relates a small twst appled to the mechansm to the correspondng wrench exerted on the mechansm. dervatve of the sprng wrench connectng two movng rgd bodes s derved. By usng the dervatve of the sprng wrench, the stffness matrces of complant mechansms whch consst of rgd bodes connected to each other by lne sprngs are obtaned. It s shown that the resultant complance of two complant parallel mechansms that are serally arranged s not the summaton of the complances of the consttuent mechansms unless the external wrench appled to the mechansm s zero. dervatve of the stffness matrx of planar complant mechansms wth respect to the twsts of the consttuent rgd bodes and the sprng parameters such as the stffness coeffcent and free length s obtaned. It s shown that the complance and the resultant x

11 wrench of a complant mechansm may be controlled at the same tme by usng adjustable lne sprngs. x

12 CHPER 1 INRODUCION 1.1 Motvaton Robots have been employed successfully n applcatons that do not requre nteracton between the robot and the envronment but requre only poston control schemes. For nstance, arc weldng and pantng belong to ths category of applcaton. here are many other operatons nvolvng contact of the robot and ts envronment. small amount of postonal error of the robot system, whch s almost nevtable, may cause serous damage to the robot or the object wth whch t s n contact. Complant mechansms, whch may be nserted between the end effecter and the last lnk of the robotc manpulator, can be a soluton to ths problem. Complant mechansms can be consdered as spatal sprngs havng multple degrees of freedom rather than one freedom as lne sprngs have. small force/torque appled to the complant mechansm generates a small dsplacement of the complant mechansm. hs relaton s well descrbed by the complance matrx of the mechansm. RCC (Remote Center of Complance) devces, developed by Whtney (1982), are one of the most successful complant mechansms. hey have a unque complant property at a specfc operaton pont and are manly used to compensate postonal errors durng tasks such as nsertng a peg nto a chamfered hole. Complant mechansms can also be employed for force control applcatons by usng the theory of Knestatc Control whch was proposed by Grffs (1991). Knestatc Control vares the poston of the last lnk of 1

13 2 the manpulator to control the poston and contact force of the dstal end of the robotc manpulator at the same tme wth the complance of the mechansm n mnd. Mechansms wth varable complance, whch s the topc of ths dssertaton, are beleved to have several advantages over mechansms havng fxed complance. Snce RCC devces typcally have a specfc operaton pont, f the length of the peg to be nserted s changed, a dfferent RCC devce should be employed to do nserton tasks unless the RCC devce has varable complance. s for force control tasks, each task may have an optmal complance. Wth varable complant mechansms, several dfferent tasks nvolvng dfferent force ranges can be accomplshed wthout havng to physcally change the complant mechansm. Varable complant mechansms also can mprove the performance of humanod robot parts such as ankles and wrsts, and anmals are beleved to have physcally varable leg complance and utlze t when runnng and hoppng (see Hurst et al. 2004). Many complant mechansms ncludng RCC devces have been desgned typcally based on parallel knematc mechansms. Parallel knematc mechansms contan postve features compared to seral mechansms such as hgher stffness, compactness, and smaller postonal errors at the cost of a smaller workspace and ncreased complexty of analyss. In ths dssertaton mechansms havng two complant parallel mechansms n a seral arrangement as well as complant parallel mechansms are nvestgated. hese mechansms may have a trade-off of characterstcs relatve to tradtonal parallel and seral mechansms. 1.2 Lterature Revew he concepts of twsts and wrenches were ntroduced by Ball (1900) n hs groundbreakng work reatse on the heory of Screws. hese concepts are employed

14 3 throughout ths dssertaton to descrbe a small (or nstantaneous) dsplacement of a rgd body and a force/torque appled to a body (Crane et al. 2006). he complance of a mechansm can be well descrbed by the stffness matrx whch s a 6 6 matrx for a spatal mechansm and a 3 3 matrx for a planar mechansm. Usng screw theory, Dmentberg (1965) studed propertes of an elastcally suspended body. Loncarc (1985) used Le groups rather than screw theory to study symmetrc spatal stffness matrces of complant mechansms assumng that the sprngs are n an equlbrum poston and derved a constrant that makes the number of ndependent elements of symmetrc 6 6 stffness matrces 20 rather than 21. Loncarc (1987) also defned a normal form of the stffness matrx n whch rotatonal and translatonal parts of the stffness matrx are maxmally decoupled. Grffs (1991) presented a global stffness model for complant parallel mechansms where he used the term global to state that the sprngs are not restrcted to an unloaded equlbrum poston. Grffs (1991) also showed that the stffness matrx s not symmetrc when the sprngs are deflected from the equlbrum postons due to an external wrench. Cblak and Lpkn (1994) decomposed a stffness matrx nto a symmetrc and a skew symmetrc part and showed the skew symmetrc part s negatve one-half the externally appled load expressed as a spatal cross product operator. Complant parallel mechansms have been nvestgated by a number of researchers to realze desred complances because of ts hgh stffness, compactness, and small postonal errors. Huang and Schmmels (1998) obtaned the bounds of the stffness matrx of complant parallel mechansms whch consst of smple elastc devces and proposed an algorthm for syntheszng a realzable stffness matrx wth at most seven

15 4 smple elastc devces. Roberts (1999) and Cblak and Lpkn (1999) ndependently developed algorthms for mplementng a realzable stffness matrx wth r number of sprngs where r s the rank of the stffness matrx. s for seral robot manpulators, Salsbury (1980) derved the stffness mappng between the jont space and the Cartesan space. Chen and Kao (2000) showed that the formulaton of Salsbury (1980) s only vald n the unloaded equlbrum pose and derved the conservatve congruence transformaton for stffness mappng accountng for the effect of an external force. Fgure 1-1. Planar robot wth varable geometry base platform (from Smaan and Shoham 2002). Fgure 1-2. daptve vbraton absorber (from Ryan et al. 1994).

16 5 Planar/spatal complant mechansms are n general constructed wth rgd bodes whch are connected to each other by smple sprngs. he stffness matrx of the mechansm depends on the geometry of the mechansm and the propertes of the consttuent sprngs such as stffness coeffcent and free length. o realze varable complant mechansms, varable geometry or adjustable sprngs have been nvestgated. Smaan and Shoham (2002) studed the stffness synthess problem usng a varable geometry planar mechansm. hey changed the geometry of the base usng sldng jonts on the crcular base (see Fgure 1-1). Ryan et al. (1994) desgned a varable sprng by changng the effectve number of cols of the sprng for adaptve-passve vbraton control (see Fgure 1-2). Fgure 1-3. Parallel topology 6DOF wth adjustable complance (from McLachlan and Hall 1999). Cantlever beam-based varable complant devces have been studed by a few researchers. Under an external force, a cantlever beam deflects and ts deflecton

17 6 depends on the length of the beam and the Young s modulus of the materal. Henre (1997) nvestgated a cantlever beam whch s flled wth magneto-rheologcal materal and changed the Young s modulus by changng the magnetc feld. McLachlan and Hall (1999) devsed a programmable passve devce by changng the length of the cantlever beam as shown n Fgure 1-3. Hurst et al. (2004) presented an actuator wth physcally varable stffness by usng two motors and analyzed t for applcaton to legged locomoton. 1.3 Problem Statement Planar/spatal complant mechansms consstng of rgd bodes whch are connected to each other by adjustable complant couplngs are nvestgated. For spatal mechansms, each adjustable complant couplng s assumed to have a sphercal jont at each end and a prsmatc jont wth an adjustable lne sprng n the mddle. For planar cases, sphercal jonts are replaced wth revolute jonts. Mechansms havng two complant parallel mechansms that are serally arranged are manly nvestgated. he complant mechansms are not restrcted to be n unloaded equlbrum confguraton and ths makes the analyss of the mechansm more complcated. Frstly a stffness mappng of a lne sprng connectng two movng bodes s derved for planar and spatal cases. he lne sprng s assumed to have a fxed stffness coeffcent and free length at ths stage. hs stffness mappng leads to the dervaton of the stffness matrx of complant mechansms consstng of rgd bodes connected to each other by lne sprngs. dervatve of the stffness matrx of a complant mechansm wth respect to the twsts of the consttuent rgd bodes and the sprng propertes such as sprng constant and free length s obtaned. Snce the complant mechansm s assumed ntally n statc

18 7 equlbrum under an external wrench, changng the sprng constants and the free lengths of the consttuent sprngs may result n the change of the resultant wrench and t may change the poston of the complant mechansm. Stffness modulaton methods, whch utlze adjustable lne sprngs and vary the poston of the robot where the complant mechansm s attached, are nvestgated to realze a desred complance and to regulate the poston of the complant mechansm.

19 CHPER 2 SIFFNESS MPPING OF PLNR COMPLIN MECHNISMS When a rgd body supported by a complant couplng moves, the deflecton and/or the drectonal change of the couplng may lead to a change of the force. In ths chapter, a planar stffness mappng model whch maps a small twst of the body nto the correspondng wrench varaton s studed. o descrbe a small (or nstantaneous) dsplacement of a rgd body and a force/torque appled to a body, the concepts of twst and wrench from screw theory are used throughout ths dssertaton (see Ball 1900 and Crane et al. 2006). Further, the notatons of Kane and Levnson are also employed (see Kane and Levnson 1985) to descrbe spatal motons of rgd bodes. Specfcally, as part of the notaton, the poston of a pont P embedded n body B measured wth respect to a reference system embedded n body wll be wrtten as r. he dervatve of the dsplacement of ths pont P (embedded n body B n terms of a reference coordnate system embedded n body ) s denoted as δ r. he dervatve of an angle of body B wth respect to a body s denoted by δ θ B and ts magntude s denoted by δθ B. he twst of a body B wth respect to a body wll be denoted by B P B P B δ D. 2.1 Sprng n a Lne Space he analyss of rgd bodes whch are constraned to move n a lne space and connected to each other by lne sprngs s presented because t s smple and ntutve and a smlar approach can be appled for planar and spatal complant mechansms. Fgure 8

20 9 2-1 llustrates a lne sprng connectng body to ground. Body s allowed to move only on a lne along the axs of the sprng. he sprng has a sprng constant k and a free length x o. he poston of body can be expressed by a scalar x and the force from the sprng by a scalar f. k x Fgure 2-1. Sprng n a lne space. he sprng force can be wrtten as f = kx ( x o ). (2.1) he relaton between a small change of the poston of body and the correspondng small force varaton can be obtaned by takng a dervatve of Eq. (2.1) as δ f = kδx. (2.2) When sprngs are arranged n parallel as shown n Fgure 2-2 (a), the resultant sprng constant k R may be derved as Eq. (2.3). δ f = k δx= kδx+ k δx R 1 2 k = k + k. (2.3) R 1 2

21 10 B k 2 k 1 k 2 x x B k 1 x (a) (b) Fgure 2-2. Sprng arrangements n a lne space. (a) parallel and (b) seres. For a seral arrangement as shown n Fgure 2-2 (b), the resultant sprng constant k R whch maps a small change of poston of body B nto a small force varaton upon body B may be wrtten as Eq. (2.4). ( ) δ f = k δx = kδx = k δx δx R B 1 2 B δx k2 = k + k 1 2 δx B kk = = kr or kr k1 k2 k1+ k2. (2.4) It s obvous from Eqs. (2.3) and (2.4) that the resultant sprng constant of sprngs n parallel s the summaton of each sprng constant and that the resultant complance of sprngs n seres s the summaton of each sprng complance. hs statement s vald for sprngs n a lne space.

22 Dervatve of Planar Sprng Wrench Jonng a Movng Body and Ground In ths secton, a dervatve of the planar sprng wrench jonng a movng body and ground, whch was presented by Pgosk (1993) and led to the stffness mappng of a planar parallel mechansm, s restated. Fgure 2-3 llustrates a rgd body connected to ground by a complant couplng. he complant couplng has a revolute jont at each end and a prsmatc jont wth a sprng n the mddle part. Body can translate and rotate n a planar space. P1 S θ P0 Fgure 2-3. Planar complant couplng connectng body and the ground. he force whch the sprng exerts on body can be wrtten as f = kl ( l o ) $ (2.5) where k, l, and l o are the sprng constant, current sprng length, and sprng free length of the complant couplng, respectvely. lso $ represents the untzed Plücker coordnates of the lne along the complant couplng whch may be wrtten as S S $ = E E = E r P0 S r P1 S (2.6)

23 12 E E E where S s the unt vector along the complant couplng and r P0 and r P1 are the poston of the pvot pont PO n the ground body and that of P1 n body, respectvely, measured wth respect to a reference coordnate system attached to ground. o obtan the stffness mappng, a small twst E δ D s appled to body and the correspondng change of the sprng force wll be obtaned. he twst E δ D may be wrtten n axs coordnates as δ D E E δ r 0 = E δφ (2.7) where E δ r s the dfferental of the poston of pont O n body whch s concdent o wth the orgn of the nertal frame E measured wth respect to the nertal frame. In addton E δφ s the dfferental of the angle of body wth respect to the nertal frame. akng a dervatve of Eq. (2.5) wth the consderaton that $ s a functon of θ n planar cases yelds δ f = kδl$ + k( l l ) δ $ lo $ = kδ l$ + k(1 ) lδθ l θ o (2.8) where S $ θ = S θ θ E E r P0 (2.9) and where S s a unt vector perpendcular to S. θ

24 13 E δ r P1 l δθ δ l P1 l δθ θ E P0 Fgure 2-4. Small change of poston of P1 due to a small twst of body. Usng screw theory, the varaton of poston P1 can be wrtten as E E E E δ P1 = δ o + δ P1 r r φ r. (2.10) It may be decomposed nto two perpendcular vectors, one along S and one along S. θ hese vectors correspond to the change of the sprng length δ l and the change of the drecton of the sprng lδθ as shown n Fgure 2-4. he change of the poston of pont P1 may thus also be wrtten as S S δr = ( δr S) S+ δr θ θ. (2.11) S = δl S + lδθ θ E E E P1 P1 P1 From Eqs. (2.10), (2.11), (2.6), and (2.7) expressons for δl and lδθ may be obtaned as δl = δr S = δr S+ δφ r S E E E E P1 o P1 = δr S+ δφ r S = E E E o P1 δ E $ D (2.12)

25 14 E S E S E E S lδθ = δ rp 1 = δ r o + δφ r P1 θ θ θ E S E E S = δr o + δφ r P1 θ θ $ E = δ D θ (2.13) and where S $ θ =. (2.14) θ E S r P1 θ ll terms of Eq. (2.14) are known. From Eqs. (2.8), (2.12), and (2.13), a dervatve of the sprng force may be wrtten as lo $ δ f = kδ l$ + k(1 ) lδθ l θ = k$$ δd lo $ $ + k(1 ) l θ θ δd = E δ D [ K ] F E E (2.15) where o [ K ] = k + k(1 ) F l $ $ $$. (2.16) l θ θ [ K F ] s the stffness matrx of a planar complant couplng and maps a small twst of body nto the correspondng varaton of the wrench. he frst term of Eq. (2.16) s always symmetrc and the second term s not. When the sprng devates from ts equlbrum poston due to an external wrench, the second term of Eq. (2.16) doesn t vansh and t makes the stffness matrx asymmetrc.

26 Dervatve of Sprng Wrench Jonng wo Movng Bodes Body B P2 S θ Body P1 E Fgure 2-5. Planar complant couplng jonng two movng bodes. In ths secton a dervatve of the sprng wrench jonng two movng bodes s derved, whch supersedes the result of the prevous secton and s essental to obtan a stffness mappng of sprngs n complcated arrangement. Fgure 2-5 llustrates two rgd bodes connected to each other by a complant couplng wth a sprng constant k, a free length l o, and a current length l. Body can move n a planar space and the complant couplng exerts a force f to body B whch s n equlbrum. he sprng force may be wrtten by where f = kl ( l o ) $ (2.17) S S $ = E = E B r P1 S r P2 S (2.18)

27 16 E E B and where S s a unt vector along the complant couplng and r P1 and r P2 are the poston vector of the pont P1 n body and that of pont P2 n body B, respectvely, measured wth respect to the reference system embedded n ground (body E). small twst of body B wth respect to an nertal frame E E δ D B s appled and t s desred to fnd the correspondng change of the sprng force. he twst E δ D B may be wrtten as δd = δd + δd (2.19) E B E B where δ D E B E B δ r o = E B δφ (2.20) δ D E E δ r o = E δφ (2.21) δ D B B δ r o = B δφ (2.22) and where the notaton from Kane and Levnson (1985) s employed as stated n the begnnng of ths chapter. For example, E δ r s the dfferental of the coordnates of B o pont O, whch s n body B and concdent wth the orgn of the nertal frame, measured wth respect to the nertal frame and E δφ s the dfferental of angle of body wth respect to the nertal frame. he dervatve of the sprng force, Eq. (2.17), can be wrtten as E δf = kδl $ + k( l l o ) δ$. (2.23) E

28 17 From the twst equaton, the varaton of the poston of pont P2 n body B wth respect to body can be expressed as B B B B δ P2 = δ o + δ P2 r r φ r (2.24) where B r P2 s the poston of P2, whch s embedded n body B, measured wth respect to a coordnate system embedded n body whch at ths nstant s concdent and algned wth the reference system attached to ground. It can also be decomposed nto two perpendcular vectors along S and S whch s a known unt vector perpendcular to S. θ hese two vectors correspond to the change of the sprng length δ l and the drectonal change of the sprng lδθ n terms of body n a way that s analogous to that shown n Fgure 2-4. hus the varaton of poston of pont P2 n body B n terms of body can be wrtten as δr = ( δr S) S+ δr B B B P2 P2 P2 S = δl S + lδθ θ S θ S θ (2.25) where S $ θ =. (2.26) θ S r P1 θ From Eqs. (2.24) and (2.25), δ l and lδθ can be obtaned as δl = δr S = δr S+ δφ r S B B B B P2 o P2 = δr S+ δφ r S = B B B o P2 B $ δ D (2.27)

29 18 B S B S B B S lδθ = δ rp2 = δ r o + δφ r P2 θ θ θ B S B B S = δr o + δφ r P2 θ θ $ B = δ D θ (2.28) where S $ θ =. (2.29) θ B S r P2 θ It s mportant to note that screw $ $ has the same drecton as but has a dfferent θ θ moment term. Only E δ $ s unknown n Eq. (2.23). It s a dervatve of the unt screw along the sprng n terms of the nertal frame and may be wrtten as E δ S δ $ = E E E δr P1 S+ r P1 δs. (2.30) E Usng an ntermedate frame attached to body, a dervatve of the drecton cosne vector may be wrtten as δs = δs+ δφ S. (2.31) E E hen, E δ $ may be decomposed nto three screws as

30 19 E δ S δ $ = E E E δr P1 S+ r P1 δs E E δs + δφ S =. (2.32) E E E δr P1 S + r P1 ( δs + δφ S) δ = + + ( δ ) E δs φ S 0 E E E E δ r P1 δ S r P1 φ S r P1 S Snce S s a functon of θ alone from the vantage of body and lδθ s already descrbed n Eq. (2.28), the frst screw n Eq. (2.32) can be wrtten as S δθ δ S θ $ 1 1 $ $ lδθ E = = = r P1 δ S E S θ l l θ θ r P1 δθ θ δd B. (2.33) s for the second screw n Eq. (2.32), E δ φ S has the same drecton wth S and a θ magntude of E δφ and thus may be wrtten as E δ S θ φ S = Eδφ. (2.34) hen the second screw n Eq. (2.32) can be expressed as S E δφ S Eδφ θ = E E r P1 ( δφ S) E S r P1 Eδφ θ. (2.35) = = θ θ $ $ Eδφ [ 0 0 1] δ D E s to the thrd screw n Eq. (2.32), δ r P1 can be decomposed nto two perpendcular E vectors along S and S respectvely and may be wrtten as θ

31 20 δr = δr + δφ r E E E E P1 o P1 E E = ( δrp 1 S) S+ δrp 1 S θ S θ (2.36) where δr S= δr S+ δφ r S E E E P1 o P1 = δr S+ δφ r S = E E o P1 E $ δ D (2.37) S S S θ θ θ E S E S = δr o + δφ r P1. θ θ (2.38) $ E = δ D θ E E E δrp 1 = δr o + δφ r P1 E By combnng Eqs. (2.36), (2.37), and (2.38) δ r P1 can be wrtten as S S δr = ( δr S) S+ δr θ θ. (2.39) E $ E S = ( $ δd ) S+ δd θ θ E E E P1 P1 P1 he thrd screw n Eq. (2.32) can now be wrtten as 0 0 E E E δ = $ S P1 ( δ ) δ r S $ D S+ D S θ θ 0 0 = $ E S = E δ $ D S δ θ θ D θ 0 = 0 $ $ δd = [ 0 0 1] δd θ θ 1 E E (2.40)

32 21 S S = k. θ snce 1( ) mong all unknowns n Eq. (2.23), δl was obtaned n Eq. (2.27) and all the terms of E δ $ were obtaned through Eqs. (2.33), (2.35), and (2.40). Hence the dervatve of the sprng force can be rewrtten as E δf = kδl$ + k( l l ) δ$ E 0 1 $ $ $ k δ k( l lo ) δ [ 0 0 1] δ 0 $ = $$ D + D + D δ l θ θ θ D θ 1 0 lo $ $ $ k k(1 ) δ k( l lo ) [ ] 0 $ = $$ + D + δ l θ θ θ θ D 1 [ K ] δ [ K ] o B B E E B E B E = F D + M δd (2.41) where o [ K ] = k + k(1 ) F l $ $ $$ (2.42) l θ θ [ K ] k( l l ) [ 0 0 1] [ 0 0 1] M θ $ $ = o θ. (2.43) It s mportant to note that [ K M ] s a functon of the external wrench. o prove t, E Eq. (2.18) s explctly expressed n a planar coordnate system and r P1 = px p y to yeld c θ $ = sθ (2.44) cθ px sθ p y

33 22 s θ $ = cθ θ sθ px cθ p y (2.45) where = cos( θ ) and sn ( θ ) c θ s θ =. By substtutng Eq. (2.45) for θ $ n Eq. (2.43), [ ] K can be expressed as M 0 0 sθ 0 0 f y KM = kl ( lo) 0 0 c θ = 0 0 fx sθ cθ 0 fy fx 0 [ ] (2.46) where = fx fy m z f s the ntal sprng wrench. s shown n Eq. (2.41), the dervatve of the sprng wrench jonng two rgd bodes depends not only on a relatve twst between two bodes but also on the twst of the ntermedate body, n ths case body, n terms of the nertal frame. [ K F ] whch maps a small twst of body B n terms of body nto the correspondng change of wrench upon body B s dentcal to the stffness matrx of the sprng assumng the body s statonary. [ K M ] s newly ntroduced from ths research and results from the moton of the base frame, n ths case body, and s a functon of the ntal external wrench. 2.4 Stffness Mappng of Planar Complant Parallel Mechansms n Seres he dervatve of the sprng wrench derved n the prevous secton s appled to obtan the stffness mappng of complant parallel mechansms n seres as shown n Fgure Body s connected to ground by three complant couplngs and body B s connected to body n the same way. Each complant couplng has a revolute jont at 1 Fgure 2-6 shows a coordnate system attached to each of three bodes for llustraton purposes. In ths analyss, the three coordnate systems are assumed to be concdent and algned at each nstant.

34 23 each end and a prsmatc jont wth a sprng n the mddle part. It s assumed that an external wrench w ext s appled to body B and that both body B and body are n statc equlbrum. he postons and orentatons of bodes and B and the sprng constants and free lengths of all consttuent sprngs are gven. he stffness matrx [ K ] whch maps a small twst of body B wth respect to the ground E δ D B nto a small wrench varaton δ w ext s desred to obtan. k 1 k 2 k 3 k k5 k6 4 Fgure 2-6. Mechansm havng two complant mechansms n seres. he statc equlbrum equaton of bodes B and can be wrtten by wext = f + f + f = f + f + f (2.47) where f are the forces from the complant couplngs.

35 24 he stffness matrx s derved by takng a dervatve of the statc equlbrum equaton, Eq. (2.47), to yeld δ w ext = [ K ] δd E B = δ f + δ f + δ f = δ f + δ f + δ f (2.48) he dervatves of sprng forces can be wrtten by Eqs. (2.49) and (2.50) snce sprngs 4, 5, and 6 connect body and ground and sprngs 1, 2, and 3 jon two movng bodes. [ KF] [ KF] [ KF] E [ K ] δ D δf + δf + δf = δd + δd + δd E E E = F RL, [ KF] [ KF] [ KF] + [ KM] + [ KM] + [ KM] B E [ K ] δd [ K ] δd δf + δf + δf = δd + δd + δd B B B δd δd δd E E E = + F RU, M RU, (2.49) (2.50) where [ K ] [ K ] F RL, 6 = = 4 [ K ] [ K ] F RU, 3 = = 1 [ K ] [ K ] M RU, 3 = 1 F F =. From Eqs. (2.49), (2.50), and (2.19) twst E δ D can be wrtten as M [ K ] = [ K ] + [ K ] δd δd δd E B E F RL, F RU, M RU, [ K ] ( ) [ K ] = δd δd + δd E B E E F RU, M RU, (2.51) 1 ([ KF] [ KF] [ KM] ) [ KF] δd = + δd. (2.52) E E B R, L RU, RU, RU,

36 25 Substtutng Eq. (2.52) for E δ D n Eq. (2.49) and comparng t wth Eq. (2.48) yelds the stffness matrx as [ K] δd = [ K ] δd E B E F RL, 1 ( ) [ ] [ ] [ ] [ ] [ ] = K K + K K K F RL, F R, L F RU, M RU, F RU, δ D E B (2.53) 1 ( ) [ ] [ ] [ ] [ ] [ ] [ ] K = K K + K K K. (2.54) F RL, F RL, F RU, M RU, F RU, It was generally accepted that the resultant complance, whch s the nverse of the stffness, of serally connected mechansms s the summaton of the complances of all consttuent mechansms (see Grffs 1991). However, the stffness matrx derved from ths research shows a dfferent result. akng an nverse of the stffness matrx Eq. (2.54) yelds [ K] [ K ] [ K ] [ K ] [ K ] [ K ] F R, L F RU, F RU, M RU, F RL, = + (2.55) he thrd term n Eq. (2.55) s newly ntroduced n ths research and t does not vansh unless the external wrench s zero. numercal example s presented to support the derved stffness mappng model. he geometry nformaton, sprng propertes of the mechansm shown n Fgure 2-6, and the external wrench w ext are gven n ables 2-1 and 2-2. [ ] w ext = N N Ncm able 2-1. Sprng propertes of the complant couplngs n Fgure 2-6. Sprng No Stffness constant k Free length l o (Unt: N/cm for k, cm for l o )

37 26 able 2-2. Postons of pvot ponts n terms of the nertal frame n Fgure 2-6. Pvot ponts E1 E2 E3 B1 B2 B3 X Y (Unt: cm) able 2-2. Contnued wo stffness matrces are obtaned. [ K 1] s from Eq. (2.54) and [ K 2] from the same equaton gnorng [ M ], K. RU N / cm N / cm N [ K1] = N / cm N / cm N N N Ncm N / cm N / cm N [ K2] = N / cm N / cm N N N Ncm he result s evaluated n the followng way: E B 1. small wrench δ w s appled n addton to w ext to body B and twsts δ D 1 and E B δ D 2 are obtaned by multplyng the nverse matrces of the stffness matrces, [ K ] 1 and [ K ] 2, respectvely, by δ w as of Eq. (2.48). Correspondng postons E B E B for body B are then determned, based on the calculated twsts δ D 1 and δ D E δ D s calculated by multplyng the nverse matrx of [ F ] RL, (2.49). he poston of body s then determned from ths twst. K by δ w as of Eq. 3. he wrench between body B and body s calculated for the two cases based on knowledge of the postons of bodes and B and the sprng parameters. he change n wrench for the two cases s determned as the dfference between the new equlbrum wrench and the orgnal. he changes n the wrenches are named δ w B,1 and B,2 K. δ w whch correspond to the matrces [ K ] 1 and [ ] 2 4. he gven change n wrench δ w s compared to δ w B,1 and δ w B,2.

38 27 he gven wrench δ w and the numercal results are presented as below. δ w = [ ] δ D E δ D E δ D E δ w δ w δ w [ ] B 3 1 = [ ] B 3 2 = E [ ] = = [ ] [ ] 5 B,1 = [ ] 5 B,2 = where δ w E s the wrench between body and ground. he unt for the wrenches s [ ] N N Ncm and that of the twsts s [ ] cm cm rad. he dfference between δ w E and δ w s small and s due to the fact that the twst was not nfntesmal. he dfference between δ w B,1 and δ w s also small and s most lkely attrbuted to the same fact. However, the dfference between δ w B,2 and δ w s not neglgble. hs ndcates that the stffness matrx formula derved n ths research produces the proper result and that the term [ M ], K cannot be neglected n Eq. (2.54). RU 2.5 Stffness Mappng of Planar Complant Parallel Mechansms n a Hybrd rrangement Fgure 2-7 depcts a complant mechansm havng complant couplngs n a seral/parallel arrangement. Each complant couplng has a revolute jont at each end and a prsmatc jont wth a sprng n the mddle part. n external wrench w ext s appled to

39 28 body and body s separately connected to body B, body C, and body D by three complant couplngs. Body B, body C, and body D are connected to ground by two complant couplngs. It s assumed that all bodes are n statc equlbrum. It s desred to fnd the stffness matrx whch maps a small twst of body n terms of ground E δ D to the correspondng wrench varaton δ w ext. he stffness constants and free lengths of all consttuent sprngs and the geometry of the mechansm are assumed to be known. he stffness matrx of the mechansm can be derved by takng a dervatve of the statc equlbrum equatons. he statc equlbrum equatons may be wrtten as where wext = f 7 + f 8 + f 9 (2.56) f 7 = f 1+ f 2 (2.57) f 8 = f 3 + f 4 (2.58) f 9 = f 5 + f 6 (2.59) w ext s the external wrench and f s the force of the -th sprng. Dervatves of Eqs. (2.56)-(2.59) can be wrtten as δ w = δ f + δ f + δ f ext = [ K ] E R δ D (2.60) δ f = δ f + δ f (2.61) δ f = δ f + δ f (2.62) ] R δ f = δ f + δ f (2.63) where [ K s the stffness matrx and E δ D s a small twst of body n terms of the nertal frame attached to the ground.

40 29 Usng Eqs. (2.15) and (2.41), Eq. (2.61) can be rewrtten as B E B [ KF] [ KM] [ K ] [ K ] [ K ] [ K ] δ f = δd + δd ( ) = δd + δd = + δd. (2.64) E B E B E B F 1 F 2 F 1 F 2 where B δ D s a small twst of body n terms of body B and E δ D B s that of body B n terms of the nertal frame. [ K ] and [ ] Eqs. (2.42) and (2.43) respectvely. he twst of body can be decomposed as F K are the matrces for -th sprng defned by M δd = δd + δd. (2.65) E E B B From Eqs. (2.64) and (2.65), E δ D B can be expressed n terms of E δ D as Eq. (2.66). [ KF] ( ) + [ KM] = ([ KF] + [ KF] ) δd δd δd δd E E B E B E B ([ KF] [ KF] [ KF] [ KM] ) [ KF] δd = + + δd (2.66) E B E By substtutng Eq. (2.66) for E δ D B n Eq. (2.64), δ f 7 can be expressed n terms of E δ D as δ ([ ] [ ] ) [ ] [ ] [ ] [ ] 1 ( ) [ ] E f 7 = KF + K 1 F K 2 F + K 1 F + K 2 F K 7 M K 7 F δd. (2.67) 7 nalogously, δ f 8 and δ f 9 can be wrtten respectvely as δ δ ([ ] [ ] ) [ ] [ ] [ ] [ ] 1 ( ) [ ] E f 8 = KF + K 3 F K 4 F + K 3 F + K 4 F K 8 M K 8 F δd (2.68) 8 ([ ] [ ] ) [ ] [ ] [ ] [ ] 1 ( ) [ ] E f 9 = KF + K 5 F K 6 F + K 5 F + K 6 F K 9 M K 9 F δd. (2.69) 9 Fnally from Eq. (2.60) and Eqs. (2.67)-(2.69), the stffness matrx can be wrtten as

41 30 1 ( F F )([ F] [ F] [ F] [ M] ) [ F] 1 ( KF KF )([ KF] [ KF] [ KF] [ KM] ) [ KF] 1 ( KF KF )([ KF] [ KF] [ KF] [ KM] ) [ KF] [ ] R [ ] [ ] [ ] [ ] [ ] [ ] K = K + K K + K + K K K numercal example of the complant mechansm depcted n Fgure 2-7 s. (2.70) presented. he four bodes are dentcal equlateral trangles whose edge length s 2 cm. Four coordnate systems, B, C, D, and are attached to body B, C, D, and, respectvely and ther postons of orgn and orentatons n terms of the nertal frame are gven n able 2-5. he sprng propertes and the postons of the fxed pvot ponts are gven n able 2-3 and able 2-4, respectvely. he external wrench s gven as 0.1 N w ext = 0.1 N. 0.2 Ncm able 2-3. Sprng propertes of the complant couplngs n Fgure 2-7. Sprng No Stffness constant k Free length l o ( Unt: N / cm for k and cm for l o ) able 2-4. Postons of the fxed pvot ponts of the complant couplngs n Fgure X Y ( Unt: cm ) able 2-5. Postons and orentatons of the coordnates systems n Fgure 2-7. Bo Co Do o X Y Φ ( Unt: cm for x, y and radans for Φ)

42 31 wo stffness matrces are obtaned. [ K 1] s from Eq. (2.70) and [ K 2] from the same equaton gnorng all [ K M ] s whch are newly ntroduced n ths research N / cm N / cm N [ K1] = N / cm N / cm N N N Ncm N / cm N / cm N [ K2] = N / cm N / cm N N N Ncm o evaluate the result, a small wrench δ w s appled to body and the statc equlbrum pose of the mechansm s obtaned by a numercally teratve method. From the equlbrum pose of the mechansm, the twst of body wth respect to ground E δ D s obtaned as 0.5 N 4 δ w = N 0.3 Ncm cm δ D = cm rad E hen the twst E δ D s multpled by both of the stffness matrces to see f they result n the gven small wrench δ w N E 4 δw1 = [ K1] δd = N Ncm

43 N E 4 δw2 = [ K2] δd = N Ncm he numercal example ndcates that [ K 1] produces the gven wrench δ w wth hgh accuracy and that [ K 2] nvolves sgnfcant errors. D w ext B Fgure 2-7. Mechansm consstng of four rgd bodes connected to each other by complant couplngs n a hybrd arrangement.

44 CHPER 3 SIFFNESS MPPING OF SPIL COMPLIN MECHNISMS akng a smlar approach adopted for planar complant mechansms, a stffness mappng of spatal complant mechansms s presented. 3.1 Dervatve of Spatal Sprng Wrench Jonng a Movng Body and Ground Fgure 3-1 depcts a rgd body and a complant couplng connectng the body and the ground. he complant couplng has a sphercal jont at each end and a prsmatc jont wth a sprng n the mddle. Body can translate and rotate n a spatal space. he wrench whch the sprng exerts on body can be wrtten as w = kl ( l o ) $ (3.1) where k, l, and l o are respectvely the sprng constant, current sprng length, and sprng free length of the complant couplng. Further, $ represents the untzed Plücker coordnates of the lne along the complant couplng whch may be wrtten by S S $ = E E = E r P0 S r P1 S (3.2) E E E where S s the unt vector along the complant couplng and r P0 and r P1 are the poston of the pvot pont PO n the ground body and that of P1 n body, respectvely, measured wth respect to a reference coordnate system attached to ground. 33

45 34 Body P1 S E P0 Fgure 3-1. Spatal complant couplng jonng body and the ground. e 3 S e 2 e 1 Fgure 3-2. Unt vector expressed n a polar coordnates system. polar coordnates system can be used to express the unt vector S (see Fgure 3-2) as sn β cosα S = sn β snα. (3.3) cos β E E It s obvous from Eqs. (3.2) and (3.3) that $ s a functon of α and β snce r P0 s fxed on ground. Hence a dervatve of the sprng wrench can be wrtten as δ w = kδ l$ + k( l l ) δ $ o lo $ $ = kδ l$ + k(1 ) lδα + lδ β l α β (3.4)

46 35 where S $ α = α S α E E r P0 (3.5) S $ β =. (3.6) β E E S r P0 β By takng a dervatve of Eq. (3.3), S and can be explctly wrtten by S α β sn β snα S = sn β cosα α 0 (3.7) cos β cosα S = cos β snα β. (3.8) sn β Snce S s not a unt vector, a unt vector S α α s ntroduced as snα S = cosα α 0 (3.9) S = sn β S. (3.10) α α Hence Eq. (2.8) can be rewrtten as l (1 o $ $ δ w = kδ l$ + k ) lsn β δα+ lδβ (3.11) l α β where

47 36 S $ α =. (3.12) α E E S r P0 α It s mportant to note that $ $ and are the untzed Plücker coordnates of the lnes α β perpendcular to S and go through the pvot pont P0. δα l sn β δα δl E 1 δ r P l sn β l β l δ β δ β Fgure 3-3. Small change of poston of P1 due to a small twst of body. In Eq. (3.11) δ l, l sn β δα, and l δβ can be consdered as the change of the sprng length and the changes of the drecton of the sprng (see Fgure 3-3). hese E values correspond to the projectons of the varaton of poston P1, δ r 1, onto the P orthonormal vectors S, S, and, respectvely. hus E δ r P1 can be rewrtten as S α β

48 37 E E E S S E S S δrp 1 = ( δrp 1 S) S+ δrp 1 + δrp 1 α α β β. (3.13) S S = δl S + lsn β δα + lδβ α β From the twst equaton, the varaton of poston P1 can be wrtten as E E E E δ P1 = δ o + δ P1 r r φ r (3.14) E where δ r 0 s the dfferental of the poston of pont O n body whch s concdent wth the orgn of the nertal frame E measured wth respect to the nertal frame. E δφ s the dfferental of the angle of body wth respect to the nertal frame. From Eqs. (2.11) and (2.10), δ l, l sn β δα, and l δβ can be expressed as δl = δr S = δr S+ δφ r S E E E E P1 o P1 = δr S+ δφ r S = E E E o P1 δ E $ D (3.15) E S E S E E S l sn βδα= δrp 1 = δr o + δφ r P1 α α α E S E E S = δr o + δφ r P1 α α $ E = δ D α E S E S E E S lδβ = δ rp 1 = δ r o + δφ r P1 β β β E S E E S = δr o + δφ r P1 β β $ = β δ D E (3.16) (3.17) where

49 38 δ D E E δ r 0 = E δφ (3.18) S $ α = α E S r P1 α (3.19) S $ β =. (3.20) β E S r P1 β It s mportant to note that $ $ and are the untzed Plücker coordnates of lnes α β perpendcular to S whch pass through the pvot pont P1 n body and E δ D s a small twst of body wth respect to ground. Substtutng Eqs. (2.12), (3.16), and (2.13) for δ l, l sn β δα, and l δβ n Eq. (3.11) yelds l o $ $ δ w = kδ l$ + k(1 ) l sn β δα + lδ β l α β l o $ $ $ $ = k$$ δd + k(1 ) + δd l α α β β = [ K ] F E E δ D E (3.21) where o [ K ] k k(1 ) F l $ $ $ $ = $$ + +. (3.22) l α α β β [ K F ] s the stffness matrx of a spatal complant couplng and maps a small twst of body nto the correspondng varaton of the wrench. he frst term of Eq. (2.16) s

50 39 always symmetrc and the second term s not. When the sprng devates from ts equlbrum poston due to an external wrench, the second term of Eq. (2.16) doesn t vansh and t makes the stffness matrx asymmetrc. hs result agrees wth the works of Grffs (1991). 3.2 Dervatve of Sprng Wrench Jonng wo Movng Bodes P2 Body B S Body P1 E Fgure 3-4. Spatal complant couplng jonng two movng bodes Fgure 3-4 llustrates two rgd bodes connected to each other by a complant couplng wth a sprng constant k, a free length l o, and a current length l. Both of body and body B can move n a spatal space and the complant couplng exerts a wrench w to body B whch s n equlbrum. he sprng wrench may be wrtten as where w = kl ( l o ) $ (3.23) S S $ = E = E B r P1 S r P2 S (3.24)

51 40 E E B and where S s a unt vector along the complant couplng and r P1 and r P2 are the poston vectors of the pont P1 n body and that of pont P2 n body B, respectvely, measured wth respect to the reference system embedded n ground (body E). It s desred to express a dervatve of the sprng wrench n terms of the twst of body B E δ D and that of body E δ D. he twst E δ D B may be expressed as B δd = δd + δd (3.25) E B E B where δ D E B E B δ r o = E B δφ (3.26) δ D E E δ r o = E δφ (3.27) δ D B B δ r o = B δφ (3.28) and where E δ r s the dfferental of pont O, whch s n body B and concdent wth B o the orgn of the nertal frame, measured wth respect to the nertal frame and E δφ B s the dfferental of angle of body B wth respect to the nertal frame. E δ r, δ r B, o o E δφ, and δφ B are defned n the same way. he dervatve of the sprng wrench n Eq. (2.17) can be wrtten as E δw = kδl $ + k( l l o ) δ$ (3.29) E and t s requred to express δ l and E δ $ n Eq. (2.23) n terms of the twsts of the bodes. From the twst equaton, the varaton of poston of pont P2 n body B wth respect to body can be expressed as

52 41 B B B B δ P2 = δ o + δ P2 r r φ r (3.30) where B r P2 s the poston of P2, whch s embedded n body B, measured wth respect to a coordnate system embedded n body whch at ths nstant s concdent and algned wth the reference system attached to ground. It can also be decomposed by projectng t onto the orthonormal vectors S, S S, and whch are defned n a smlar way as α β Eqs. (3.3), (3.9), and (3.8). hese three vectors correspond to the change of the sprng length δ l and the drectonal changes of the sprng such as l sn β δα and l δβ n terms of body n a way that s analogous to that shown n Fgure 3-3. hus the varaton of poston of pont P2 n body B n terms of body can be wrtten as S S S S ( ) α α β β. (3.31) S S = δl S + lsn β δα + lδβ α β B B B B δrp2 = δrp2 S S+ δr P2 + δrp2 From Eqs. (2.24) and (2.25), δ l n Eq. (2.23) can be obtaned as δl = δr S = δr S+ δφ r S B B B B P2 o P2 = δr S+ δφ r S = B B B o P2 δ B $ D. (3.32) In the same way, l sn β δα and l δβ can be expressed as B S B S B B S l sn βδα= δrp2 = δr o + δφ r P2 α α α B S B B S = δr o + δφ r P2 α α $ B = δ D α (3.33)

53 42 B S B S B B S l δβ = δrp2 = δr o + δφ r P2 β β β B S B B = δr o + δφ r P2 β $ = β δ D B S β (3.34) where S $ α α = B S r P2 α (3.35) S $ β =. (3.36) β B S r P2 β Now n Eq. (2.23), only E δ $ s yet to be obtaned. It s a dervatve of the unt screw along the sprng n terms of the nertal frame and may be wrtten as E δ S δ $ = E E E δr P1 S+ r P1 δs. (3.37) E Usng an ntermedate frame attached to body, a dervatve of the unt vector S can be wrtten by δs = δs+ δφ S. (3.38) E E hus E δ $ may be decomposed nto three screws as

54 43 E δ S δ $ = E E E δr P1 S+ r P1 δs E δs + δφ S = E E E δr P1 S + r P1 ( δs + δφ S). (3.39) E δs δφ S 0 = + + E E E E P1 δ P1 ( δ ) δ r S r φ S r P1 S E Snce S s a functon of α and β from the vantage of body and l sn β δα and l δβ were already descrbed n Eqs. (2.28) and (3.34), the frst screw n Eq. (2.32) can be wrtten as S S δα + δβ S S δβ α β δα δ S α β E = = + P1 δ r S E S S E E P1 δα δβ S S r + r P1 δα r P1 δβ α β α β $ 1 $ 1 = lsn βδα+ lδβ.(3.40) α l β l 1 $ $ $ $ B = + δ D l α α β β s to the second screw n Eq. (2.32), E δ φ S can be decomposed onto three orthonormal vectors along S, S S, and, respectvely, as α β S S S S δφ S = {( δφ S) S} S+ ( δφ S) + ( δφ S ) α α β β E E E E. (3.41)

55 44 From the fact that S, S S, and are unt vectors and perpendcular to each other α β (see Fgure 3-3), each dot product of Eq. (3.41) can be expressed as ( E δ ) = 0 φ S S (3.42) S δ δ S S φ S = φ S = δφ α α β ( ) E E E ( ) 0 0 δ r δ β φ β E o E = S = δ E S D S S S δφ S = δφ S = δφ β β α E E E 0 0 δ r S δ φ S α α E o E = = δ E D (3.43) (3.44) where = [ ] Hence, E δ φ S can be rewrtten as 0 0 S S δφ S= δ + δ α S D β S D β α E E E (3.45) and the second screw n Eq. (2.32) can be expressed as

56 S E S E δ δ α S D + β S D E δφ S β α = E E P1 ( δ ) r φ S 0 0 E S E S E r P1 δ + δ α S D β S D β α S S 0 α β 0 = δ + S D E S E S S r P1 β r P1 α α β 0 0 $ $ = + α S β S β α. (3.46) E E δ D E s to the thrd screw n Eq. (2.32), δ r P1 can be decomposed onto three orthonormal E δd vectors along S, S S, and, respectvely, as α β δr = δr + δφ r E E E E P1 o P1 E E E ( δ P1 ) S δ S S S = r S S+ rp 1 + δrp1 α α β β. (3.47) he frst dot product n Eq. (3.47) can be expressed as δr S = δr S+ δφ r S E E E P1 o P1 = δr S+ δφ r S. (3.48) = E E o P1 E $ δ D In the same way, the second and thrd dot products n Eq. (3.47) can be wrtten as

57 46 S S S α α α E E E δrp 1 = δr o + δφ r P1 E S E S δr o δφ r P1 = + α $ E = δ D α α (3.49) S S S β β β E E E δrp 1 = δr o + δφ r P1 E S E = δr o + δφ r P1 β $ = β δ D E S. (3.50) β E Fnally, δ r P1 S of the thrd screw n Eq. (2.32) can be expressed as $ S $ S r S = ( $ D ) S+ D + D S α α β β E E E E δ P1 δ δ δ $ E E δ S $ δ S = D S+ D S α α β β $ E E = δ S $ δ S D D α β β α (3.51) snce S, S S, and are unt vectors and perpendcular to each other (see Fgure α β 3-3). E Substtutng Eq. (3.51) for δ r P1 S of the thrd screw n Eq. (2.32) yelds

58 E E E δ = $ P1 δ S $ δ S r S D D α β β α 0 0 $ E E δ $. (3.52) = δ S D α S D β β α 0 0 S α S β α = $ $ β δ D E By replacng δ l and E δ $ n Eq. (2.23) wth Eqs. (2.27), (2.33), (2.35), and (2.40) and sortng t nto the twsts, the dervatve of the sprng wrench can be rewrtten as E δ w = kδl$ + k( l l ) δ$ E [ K ] δd [ K ] = + F o δd B E M (3.53) where o [ K ] k k(1 ) F [ K ] k( l l ) M l $ $ $ $ = $$ + + (3.54) l α α β β $ $ $ $ = o + S α α S β S S β β β α α. (3.55) It s mportant to note that [ K M ] s dentcal to the negatve of the sprng wrench expressed as a spatal cross product operator (see Featherstone 1985 and Cblak and Lpkn 1994). o prove t, all terms n Eq. (2.43) are explctly expressed n a polar E coordnate system and r P1 = px py p z to yeld

59 48 [ ] K M [ 0] [ K12] [ K12] [ K22] = (3.56) where [ ] and where [ ] 0 cβ sβ sα K12 = k l l o cβ 0 sβ cα (3.57) sβ sα sβ cα 0 [ ] ( ) 0 pxsβ sα pysβ cα pzsβ cα + p c x β K22 = kl ( lo) pxsβ sα+ pysβ cα 0 pycβ pzsβ sα (3.58) pzsβ cα pxcβ py cβ+ pzsβ sα 0 c α 0 s 3 3 zero matrx, = cos( α), and = sn( α), etc. In the same way the sprng wrench can be explctly wrtten as s α S w = kl ( lo) $ = kl ( lo) E r P1 S sc β α fx ss f β α y. (3.59) c β f z f = kl ( lo ) = = pc y β pss z β α m x m psc z β α pc x β m y pss x β α psc y β α m z By comparng Eqs. (3.57) and (3.58) wth Eq. (3.59) t s obvous that 0 fz f y K12 = k l lo fz 0 fx = f (3.60) fy fx 0 [ ] ( ) 0 mz m y K22 = mz 0 mx = m (3.61) my mx 0 [ ] where f and m are skew-symmetrc matrces representng vector multplcaton.