19 Singularity Robust Inverse Dynamics of Parallel Manipulators S. Kemal Ider Middle East Technical University Ankara,Turkey 1. Introduction Parallel manipulators have received wide attention in recent years. Their parallel structures offerbetterloadcarryingcapacityandmoreprecisepositioningcapabilityoftheend-effector compared to open chain manipulators. In addition, since the actuators can be placed closer to the base or on the base itself the structure can be built lightweight leading to faster systems (Gunawardana & Ghorbel, 1997; Merlet, 1999; Gao et al., 2002 ).Itisknownthatatkinematicsingularpositionsofserialmanipulatorsandparallel manipulators,arbitrarilyassignedend-effectormotioncannotingeneralbereachedbythe manipulatorandconsequentlyatthoseconfigurationsthemanipulatorlosesoneormore degrees of freedom. In addition, the closed loop structure of parallel manipulators gives rise toanothertypeofdegeneracy,whichcanbecalleddrivesingularity,wheretheactuators cannot influence the end-effector accelerations instantaneously in certain directions and the actuators lose the control of one or more degrees of freedom. The necessary actuator forces become unboundedly large unless consistency of the dynamic equations are guaranteed by the specified trajectory.Thepreviousstudiesrelatedtothedrivesingularitiesmostlyaimatfindingonlythe locations of the singular positions for the purpose of avoiding them in the motion planning stage (Sefrioui & Gosselin, 1995; Daniali et al, 1995; Alici, 2000; Ji, 2003; DiGregorio, 2001; St-Onge & Gosselin, 2000). However unlike the kinematic singularities that occur at workspace boundaries,drivesingularitiesoccurinsidetheworkspaceandavoidingthemlimitsthe motion in the workspace. Therefore, methods by which the manipulator can move through the drive singular positions in a stable fashion are necessary. Thischapterdealswithdevelopingamethodologyfortheinversedynamicsofparallel manipulatorsinthepresenceofdrivesingularities.Tothisend,theconditionsthatshould besatisfiedfortheconsistencyofthedynamicequationsatthesingularpositionsare derived.Forthetrajectoryoftheend-effectortoberealizablebytheactuatorsitshouldbe designedtosatisfytheconsistencyconditions.Furthermore,forfindingtheappropriate actuatorforceswhendrive singularitiestakeplace,the dynamicequationsaremodifiedby using higher order derivative information. The linearly dependent equations are replaced by themodifiedequationsintheneighborhoodsofthesingularities.Sincethelocationsofthe drivesingularitiesandthecorrespondingmodifiedequationsareknown(asderivedin Section 3), in a practical scenario the actuator forces are found using the modified equations Source: Parallel Manipulators, New Developments, Book edited by: Jee-Hwan Ryu, ISBN 978-3-902613-20-2, pp. 498, April 2008,I-Tech Education and Publishing, Vienna, AustriaOpen Access Database www.intehweb.comwww.intechopen.comParallel Manipulators, New Developments 374 inthevicinityofthesingularpositionsandusingtheregularinversedynamicequations elsewhere.Deploymentmotionsof2and3dofplanarmanipulatorsareanalyzedto illustrate the proposed approach (Ider, 2004; Ider, 2005). 2. Inverse dynamics and singular positions Consideranndegreeoffreedomparallelrobot.Letthesystembeconvertedintoanopen-treestructurebydisconnectingasufficientnumberofunactuatedjoints.Letthedegreeof freedomoftheopen-treesystembem,i.e.thenumberoftheindependentloopclosure constraints in the parallel manipulator be m-n. Let[ ]1,...,Tm = denote the joint variables of the open-tree system and[ ]1,...,Tnq q = qthe joint variables of the actuated joints. The m-n loop closure equations, obtained by reconnecting the disconnected joints, can be written as 1( ,..., ) 0i m = =1,..., - i m n(1) and can be expressed at velocity level as Gij j =0=1,..., - i m n=1,..., j m(2) where G iijj=.Arepeatedsubscriptindexinatermimpliessummationoveritsrange. Theprescribedend-effectorCartesianvariables,ix (t ) =1,..., i n representthetasksofthe non-redundant manipulator. The relations between the joint variables due to the tasks are 1( ,..., )i m if x ==1,..., i n(3) Equation (3) can be written at velocity level as Pij j ix = =1,..., i n=1,..., j m(4) where P iijjf=. Equations (2) and (4) can be written in combined form, = h(5) where T TT G P = whichisanm m matrixand T T = h x 0 .Thederivativeof equation (5) gives the acceleration level relations, = + h (6) The dynamic equations of the parallel manipulator can be written as TT G = M Z T R(7) www.intechopen.com Singularity Robust Inverse Dynamics of Parallel Manipulators 375 whereMisthem m generalizedmassmatrixandRisthevectorofthegeneralized Coriolis, centrifugal and gravity forces of the open-tree system,is the( ) 1 m- n vector of the joint forces at the loop closure joints, T is the1 nvector of the actuator forces, and each row of Z is the direction of one actuator force in the generalized space. If the variable of the jointwhichisactuatedbytheithactuatoris k ,thenfortheithrowofZ,1ikZ= and 0ijZ =for=1,..., j m ( j k ). Combining the terms involving the unknown forcesand T, one can write equation (7) as T= A M R(8) where them m matrix TAand the1 mvectorare TT T G = A Z(9) and T T T = T(10) Theinversedynamicsolutionofthesysteminvolvesfirstfinding, and fromthe kinematic equations and then finding(and hence T) from equation (8). Fortheprescribedx(t),canbefoundfromequation(6), fromequation(5)and can befoundeitherfromthepositionequations(1,3)orbynumericalintegration.However duringtheinversekinematicsolution,singularitiesoccurwhen0 = .Atthese configurations,theassignedx cannotingeneralbereachedbythemanipulatorsince,in equation(3),avectorhlyingoutsidethespacespannedbythecolumnsof cannotbe produced and consequently the manipulator loses one or more degrees of freedom. Singularitiesmayalsooccurwhilesolvingfortheactuatorforcesinthedynamicequation (8),when0 = A .Foreachdifferentsetofactuators,Zhencethesingularpositionsare different. Because this type of singularity is associated with the locations of the actuators, it iscalleddrivesingularity(oractuationsingularity).Atadrivesingularitytheassigned cannot in general be realized by the actuators since, in equation (8), a right hand side vector lyingoutsidethespacespannedbythecolumnsof TA cannotbeproduced,i.e.the actuatorscannotinfluencetheend-effectoraccelerationsinstantaneouslyincertain directions and the actuators lose the control of one or more degrees of freedom. (The system cannotresistforcesormomentsincertaindirectionsevenifallactuatorsarelocked.)The actuator forces become unboundedly large unless consistency of the dynamic equations are guaranteed by the specified trajectory. Let Gu bethe( ) ( ) m- n m- n matrixwhichiscomposedofthecolumnsof G that correspond to the variables of the unactuated joints. Since1ikZ=and0ijZ =forj k , the drive singularity condition0 = Acan be equivalently written as Gu0 = . In the literature the singular positions of parallel manipulators are mostly determined using the kinematic expression between qandxwhich is obtained by eliminating the variables www.intechopen.comParallel Manipulators, New Developments 376 oftheunactuatedjoints(Sefrioui&Gosselin,1995;Danialietal,1995;Alici,2000;Ji,2003; DiGregorio, 2001; St-Onge & Gosselin, 2000), + = J Kxq 0(11) References(Sefrioui&Gosselin,1995;Danialietal,1995;Ji,2003)namethecondition 0 = J asTypeIsingularityandthecondition0 = K TypeIIsingularity.Andin reference(DiGregorio,2001)theyarecalledinverseproblemsingularityanddirect problem singularity, respectively. Since it shows the lost Cartesian degrees of freedom, the condition0 = shown above corresponds to0 = J . For the drive singularity, equation (2) can be written as Gu u Ga= q(12) where u isthevectorof thejointvariablesof theunactuatedjointsand Ga isthematrix composed of the columns of Gassociated with the actuated joints. Since after finding ufromeqn(12)onecanfindhandhencex fromeqn(5)directly,thedrivesingularity condition0 = A (i.e. Gu0 = )givenaboveisequivalentto0 = K .Itshouldbenoted thattheidentificationofthesingularconfigurationsasshownhereiseasiersince elimination of the variables of the passive joints is not necessary. 3. Consistency conditions and modified equationsAtthemotionplanningstageoneusuallytriestoavoidsingularpositions.Thisisnot difficult as far as inverse kinematic singularities are concerned because they usually occur at the workspace boundaries (DiGregorio, 2001). In this paper it is assumed thatalways has fullrank,i.e.thedesiredmotionischosensuchthatthesystemnevercomestoaninverse kinematicsingularposition. Ontheotherhand,drive singularitiesusuallyoccurinside the workspace and avoiding them restricts the functional workspace. It is therefore important to devisetechniquesforpassingthroughthesingularpositionswhilethestabilityofthe controlforcesismaintained.Tothisend,equation(8)mustbemadeconsistentatthe singularposition.Inotherwords,sincetherowsof TA becomelinearlydependent,the samerelationmustalsobepresentbetweentherowsoftherighthandsidevector ( M R), so that it lies in the vector space spanned by the columns of TA . 3.1 Consistency conditions and modified equations when rank(A) becomes m-1 Atadrivesingularity,usuallyrankofAbecomesm-1.Letatthesingularpositionthesth row of TAbecome a linear combination of the other rows of TA . T Tsj p pjA A =1,..., p m =( ) p s , 1,..., j m =(13) where p arethelinearcombinationcoefficients(whichmaydependalsoon i ).Notice thatonlythoserowsof TA whichareassociatedwiththeunactuatedjointscanbecome www.intechopen.com Singularity Robust Inverse Dynamics of Parallel Manipulators 377 linearlydependent,hence p correspondingtotheactuatedjointsarezero.Thenforthe rows of equation (6) one must have sj j p pj j sj j s p pj j pA A M R M R = T T( )(14) Substitution of equation (13) into equation (14) yields sj j s p pj j pM R M R = ( )(15) Equation(15)representstheconsistencyconditionthat j shouldsatisfyatthesingular position.Since j areobtainedfromtheinversekinematicequations(6),thetrajectoryx must be planned in such a way to satisfy equation (15) at the drive singularity. Otherwise an inconsistenttrajectorycannotberealizedandtheactuatorforcesgrowwithoutboundsas the drive singularity is approached. Time derivative of equation (14) is sj p pj j sj p pj p pj j sj p pj jA A A A A M M + =

T T T T T( ) ( ) ( ) sj p pj p pj j s p p p pM M M R R R + + + ( )(16) Now,becauseequation(13)holdsatthesingularposition,thereexistsaneighborhoodin which the first term in equation (16) is negligible compared to the other terms. Therefore in that neighborhood this term can be dropped to yield sj p pj p pj j sj p pj jA A A M M = T T T( ) ( )sj p pj p pj jM M M + ( )s p p p pR R R + + (17) Equation (17) is the modified equation that can be used to replace the s th row of equation (8) or any other equation in the linearly dependent set. 3.2 Consistency conditions and modified equations when rank(A) becomes r , find (hence T) from equation (8). 7.Sett t t = + . If the final time is reached, stop. Otherwise continue from step 3. 4. Case studies 4.1 Two degree of freedom 2-RRR planar parallel manipulator TheplanarparallelmanipulatorshowninFigure1has2degreesoffreedom( 2 n = ). ConsideringdisconnectionoftherevolutejointatP,thejointvariablevectoroftheopen-chain system is[ ]T1 2 3 4 = . The joints at A and C are actuated, i.e.[ ]T1 2 = q . TheendpointPisdesiredtomakeadeploymentmotions(t)alongastraightlinewhose anglewithx-axisis330o = ,startingfrominitialposition0.431 moPx = ,1.385 moPy = . The time of the motion is1 s T =and its length is2.3 m L = in the positive s sense. Figure 1. Two degree of freedom 2-RRR planar parallel manipulator. The moving links are uniform bars. The fixed dimensions are labelled as or AC = , 1r AB = , 2r CD = , 3r BP = and 4r DP = .Thenumericaldataare1.75mor = , 1 2 3 41.4m r r r r = = = = , 1 26 kg m m = =and 3 44kg m m = = . The loop closure constraint equations at velocity level are G= 0 where s 2143xyoP431PA DC singular position 2www.intechopen.comParallel Manipulators, New Developments 380 1 1 3 13 2 2 4 24 3 13 4 241 1 3 13 2 2 4 24 3 13 4 24Gr s r s r s r s r s r sr c r c r c r c r c r c + = + (28) Heresini is = ,cosi ic = ,sin( )ij i js = + ,cos( )ij i jc = + .TheprescribedCartesian motion of the end point P, x can be written as ooP PP Px s t x ty s t y t+ = = + ( ) sin ( )( ) cos ( )x(29) Then the task equations at velocity level are P= x , where 1 1 3 13 3 131 1 3 13 3 130 00 0Pr s r s r sr c r c r c = + (30) The mass matrix M and the vector of the Coriolis, centrifugal and gravitational forces R are 11 1322 2413 3324 440 00 00 00 0M MM MM MM M = M(31) where

2 22 1 311 1 3 1 1 3 3( )3 3r rM m m r r r c = + + + ,23 1 3 313 3( )3 2r r r cM m = + , 2333 33rM m = 2 22 2 422 2 4 2 2 4 4( )3 3r rM m m r r r c = + + + ,24 2 4 424 4( )3 2r r r cM m = + , 2444 43rM m =(32) and 1 1 13 1 3 3 3 1 3 1 1 1 3 1 1 3 132 2 212 1 13 1 3 3 1 3 3 13 2 2 21 1 134 2 4 4 4 2 4 2 2 2 4 2 2 4 242 2 22 1 144 2 4 4 2 4 4 242 2( ) ( )( ) ( )m r r s m g r c mg r c r cRm r r s mg r c RRm r r s mg r c mg r c r cRm r r s mg r c + + + + = = + + + + R

(33) Sincethevariablesoftheactuatedjointsare 1 and 2 ,thematrixZcomposedofthe actuator direction vectors is 1 0 0 00 1 0 0 = Z(34) Then the coefficient matrix of the constraint and actuator forces, TAis www.intechopen.com Singularity Robust Inverse Dynamics of Parallel Manipulators 381 1 1 3 13 1 1 3 132 2 4 24 2 2 4 24 T3 13 3 134 24 4 241 00 10 00 0r s r s r c r cr s r s r c r cr s r cr s r c + + = A(35) Thedrivesingularitiesarefoundfrom0 = A as 1 3 2 4sin( ) 0 + = ,i.e.asthe positions when points A, B and D become collinear. Hence, drive singularities occur inside theworkspaceandavoidingthemlimitsthemotionintheworkspace.Definingapathfor the operational point P which does not involve a singular position would restrict the motion to a portion of the workspace where point D remains on one side of the line joining A and D. In fact, in order to reach the rest of the workspace (corresponding to the other closure of the closed chain system) the manipulator has to pass through a singular position. When the end point comes to0.80mds L = = , 1 3 +becomes equal to 2 4 + + , hence a drivesingularityoccurs.Atthispositionthethirdrowof TA becomes 3 4/ r r timesthe fourthrow.Then,forconsistencyofequation(8),thethirdrowoftherighthandsideof equation (8) should also be 3 4/ r rtimes the fourth row. The resulting consistency condition that the generalized accelerations must satisfy is obtained from equation (15) as 3 3 331 1 24 2 33 3 44 4 3 44 4 4r r rM M M M R Rr r r + = (36) Hence the time trajectory s(t) of the deployment motion should be selected such that at the drive singularity the generalized accelerations satisfy equation (36).An arbitrary trajectory that does not satisfy the consistency condition is not realizable. This is illustrated by considering an arbitrary third order polynomial for( ) s thaving zero initial andfinalvelocities,i.e. 2 32 33 2( )L t L tT Ts t = .Thesingularitypositionisreachedwhen 0.48s t = . The actuator torques are shown in Figure 2. The torques grow without bounds as the singularity is approached and become infinitely large at the singular position. (In Figure 2 the torques are out of range around the singular position.)For the time function s(t), a polynomial is chosen which satisfies the consistency condition at the drive singularity in addition to having zero initial and final velocities. The time dTwhen thesingularpositionisreachedandthevelocityoftheendpointPat dT ,( )P dv T canbe arbitrarily chosen. The loop closure relations, the specified angle of the acceleration of P and theconsistencyconditionconstitutefourindependentequationsforauniquesolutionof , 1,..., 4ii =

at the singular position. Hence, using iand i

at dT , the acceleration of P at dT ,( )P da T isuniquelydetermined.Consequentlyasixthorderpolynomialisselected where(0) 0, s = (0) 0, s =( ) , s T L = ( ) 0, s T =( ) ,d ds T L = ( ) ( )d P ds T v T =and( ) ( )d P ds T a T =. dTand( )P dv Tare chosen by trial and error to prevent any overshoot in s ors . The values usedare0.55 sdT = and( ) 3.0 m/sP dv T = ,yielding 2( ) 18.2 m/sP da T = .s(t)soobtainedis given by equation (37) and shown in Figure 3. www.intechopen.comParallel Manipulators, New Developments 382 2 3 4 5 6( ) 30.496 154.909 311.148 265.753 81.318 s t t t t t t = + +(37) Figure 2. Motor torques for the trajectory not satisfying the consistency condition: 1. T1, 2. T2 Furthermore,evenwhentheconsistencyconditionissatisfied, TA isill-conditionedinthe vicinityofthesingularposition,hence cannotbefoundcorrectlyfromequation(8). Deletion of a linearly dependent equation in that neighborhood would cause task violations due to the removal of a task. For this reason the modified equation (17) is used to replace the dependentequationintheneighborhoodofthesingularposition.Themodifiedequation, which relates the actuator forces to the system jerks, takes the following form. r r r rA A A A M M M Mr r r r + = + T T T T 3 3 3 331 41 1 32 42 2 31 1 24 2 33 3 44 44 4 4 4( ) ( ) 3 3 331 1 24 2 33 3 44 4 3 44 4 4r r rM M M M R Rr r r + + + (38) The coefficients of the constraint forces in eqn (38) are T T 331 41 3 1 3 13 3 2 4 244( ) ( )rA A r c r cr = + + (39a) T T 332 42 3 1 3 13 3 2 4 244( ) ( )rA A r s r sr = + + (39b) which in general do not vanish at the singular position if the system is in motion. Oncethetrajectoryischosenasabovesuchthatitrendersthedynamicequationstobe consistentatthesingularposition,thecorresponding i , i

and i

areobtainedfrom inverse kinematics, and when there is no actuation singularity, the actuator torques 1Tand www.intechopen.com Singularity Robust Inverse Dynamics of Parallel Manipulators 383 2T(along with the constraint forces 1 and 2 ) are obtained from equation (8). However in theneighborhoodofthesingularposition,equation(22)isusedinwhichthethirdrowof equation (8) is replaced by the modified equation (38). The neighborhood of the singularity where equation (22) is utilized is taken as 1 3 2 4180 1o o + < = . The motor torques necessarytorealizethetaskareshowninFigure4.Atthesingularpositionthemotor torquesarefoundas 1138.07Nm T = and 230.66Nm T = .Totestthevalidityofthe modifiedequations,whenthesimulationsarerepeatedwith0.5o = and1.5o = ,no significant changes occur and the task violations remain less than 410m. Figure 3. Time function satisfying the consistency condition. 4.2 Three degree of freedom 2-RPR planar parallel manipulator The2-RPRmanipulatorshowninFigure5has3degreesoffreedom(n=3).Choosingthe revolutejointatDfordisconnection(amongthepassivejoints)thejointvariablevectorof theopenchainsystemis[ ]T1 1 2 2 3 = ,where 1AB = and 2CD = .The link dimensions of the manipulator are labelled asa AC = ,b BD = ,c BP =andPBD = . The position and orientation of the moving platform is[ ]T3 P Px y = xwhere Px , Pyare the coordinates of the operational point of interest P in the moving platform. The velocity level loop closure constraint equations are G= 0, where 1 1 1 2 2 2 31 1 1 2 2 2 3sin cos sin cos sincos sin cos sin cosGbb = (40) The prescribed position and orientation of the moving platform,( ) t xrepresent the tasks of the manipulator. The task equations at velocity level are P= xwhere www.intechopen.comParallel Manipulators, New Developments 384 1 1 1 31 1 1 3sin cos 0 0 sin( )cos sin 0 0 cos( )0 0 0 0 1Pcc + = + (41) Figure 4. Motor torques for the trajectory satisfying the consistency condition: 1. T1, 2. T2. Figure 5. 2-RPRplanar parallel manipulator. Letthejointswhosevariablesare 1 1 2, and betheactuatedjoints.Theactuatorforce vectorcanbewrittenas[ ]T1 1 2T F F = T where 1T isthemotortorquecorrespondingto s 54G 2G5G1G13x yoP 12A C www.intechopen.com Singularity Robust Inverse Dynamics of Parallel Manipulators 385 1 ,and 1F and 2F arethetranslationalactuatorforcescorrespondingto 1 and 2 , respectively.Consideradeploymentmotionwheretheplatformmoveswithaconstant orientation given as o3320 =and with point P having a trajectory s(t) along a straight line whoseanglewithx-axisis o200 = ,startingfrominitialpositionm 0.800oPx = , m 0.916oPy = (Figure5).Thetimeofthedeploymentmotioniss 1 T = anditslengthis m 1.5 L = . Hence the prescribed Cartesian motion of the platform can be written as o3( ) ( ) sin( ) ( ) cos( )320ooP PP Px t x s ty t y s tt + = = + x(42) The link dimensions and mass properties are arbitrarily chosen as follows. The link lengths arem 1.0 AC a = = ,m 0.4 BD b = = ,m 0.2 BP c = = ,0 PBD = = .Themassesandthe centroidal moments of inertia arekg12 m = ,kg21.5 m = ,kg32 m = ,kg41.5 m = ,kg51.0 m = , kgm210.05 I= ,kgm220.03 I= ,kgm230.05 I= ,kgm240.03 I=andkgm250.02 I= .The mass centerlocationsaregivenbym1 10.15 AG g = = ,m2 20.15 BG g = = ,m3 30.15 CG g = = , m4 40.15 DG g = = ,m5 50.2 BG g = =and 50 G BD = = . ThegeneralizedmassmatrixMandthegeneralizedinertiaforcesinvolvingthesecond order velocity terms R are 11 1522 25334451 52 550 0 00 0 00 0 0 00 0 0 00 0M MM MMMM M M = M,12345RRRRR = R(43) where ijMand iRare given in the Appendix. For the set of actuators considered, the actuator direction matrix Z is 1 0 0 0 00 1 0 0 00 0 0 1 0 = Z(44) Hence, TAbecomes 1 1 1 11 1T2 2 2 22 23 3sin cos 1 0 0cos sin 0 1 0sin cos 0 0 0cos sin 0 0 1sin cos 0 0 0 b b = A(45) www.intechopen.comParallel Manipulators, New Developments 386 Since 2 2 3sin( ) b = A , drive singularities occur when 20 =or 2 3sin( ) 0 = . Noting that 2 doesnotbecomezeroinpractice,thesingularpositionsarethosepositionswhere points B, D and C become collinear. Hence, drive singularities occur inside the workspace and avoiding them limits the motion intheworkspace.Avoidingsingularpositionswhere 2 3n = ( 0, 1, 2,...) n = would restrict the motion to a portion of the workspace where point D is always on the same side of the line BC. This means that in order to reach the rest of the workspace (corresponding to the other closure of the closed chain system) the manipulator has to pass through a singular position.WhenpointPcomestom 0.662ds L = = ,adrivesingularityoccurssince 2 becomesequal to 3 + .Atthispositionthethirdandfifthrowsof TA becomelinearlydependentas 23 50T Tj jA Ab = ,1,..., 5 j = . The consistency condition is obtained as below 2 233 2 51 1 52 1 55 3 3 5( ) M M M M R Rb b + + = (46) Thedesiredtrajectoryshouldbechoseninsuchawaythatatthesingularpositionthe generalized accelerations should satisfy the consistency condition. Ifanarbitrarytrajectorythatdoesnotsatisfytheconsistencyconditionisspecified,then such a trajectory is not realizable. The actuator forces grow without bounds as the singular positionisapproachedandbecomeinfinitelylargeatthesingularposition.Thisis illustrated by using an arbitrary third order polynomial for( ) s thaving zero initial and final velocities,i.e. 2 32 33 2( )L t L tT Ts t = .Thesingularityoccurswhens 0.46 t = .Theactuatorforces are shown in Figures 6 and 7. (In the figures the forces are out of range around the singular position.) Figure 6. Motor torque for the trajectory not satisfying the consistency condition. www.intechopen.com Singularity Robust Inverse Dynamics of Parallel Manipulators 387 Figure 7. Actuator forces for the trajectory not satisfying the consistency cond.: 1. F1, 2. F2. Forthetimefunctions(t)apolynomialischosenthatrendersthedynamicequationstobe consistent at the singular position in addition to having zero initial and final velocities. The time dTwhen singularity occurs and the velocity of the end point when dt T = ,( )P dv Tcan be arbitrarily chosen. The acceleration level loop closure relations, the specified angle of the accelerationofP(o200 = ),thespecifiedangularaccelerationoftheplatform 3( 0) =

and theconsistencyconditionconstitutefiveindependentequationsforauniquesolutionof , 1,..., 5ii = at the singular position. Hence, usingandat dT , the acceleration of P at dT ,( )P da T isuniquelydetermined.Consequentlyasixthorderpolynomialisselected where(0) 0, s = (0) 0, s =( ) , s T L = ( ) 0, s T =( ) ,d ds T L = ( ) ( )d P ds T v T =and( ) ( )d P ds T a T =. Thevaluesusedfor dT and( )P dv T ares 0.62 andm s 1.7 / respectively,yielding m s2( ) 10.6 /P da T = . s(t) so obtained is shown in Figure 8 and given by equation (47). 2 3 4 5 6( ) 20.733 87.818 146.596 103.669 25.658 s t t t t t t = + +(47) Badchoicesfor dT and( )P dv T wouldcauselocalpeaksins(t)implyingbackandforth motion of point P during deployment along its straight line path. However,evenwhentheequationsareconsistent,intheneighborhoodofthesingular positions TA isill-conditioned,hence cannotbefoundcorrectlyfromequation(8).This problemiseliminatedbyutilizingthemodifiedequationvalidintheneighborhoodofthe singular position. The modified equation (17) takes the following form j jB Q = 1, 2 j =(48) where 2 21 31 51 51T T TB A A Ab b =

,2 22 32 52 52T T TB A A Ab b =

(49a) www.intechopen.comParallel Manipulators, New Developments 388 2 233 2 51 1 52 1 55 3 33 2 51 1 52 1( ) ( Q M M M M M M Mb b = + + + + 255 3 51 1 52 1 55 3) ( ) M M M Mb + + +

2 23 5 5R R Rb b + +

(49b) Figure 8. A time function that satisfies the consistency condition.Oncethetrajectoryisspecified,thecorresponding , andareobtainedfrominverse kinematics,andwhenthereisnoactuationsingularity,theactuatorforces 1T , 1F and 2F(andtheconstraintforces 1 and 2 )areobtainedfromequation(8).Howeverinthe neighborhoodofthesingularity,Aisill-conditioned.Sotheunknownforcesareobtained fromequation(22)whichisobtainedbyreplacingthethirdrowofequation(8)bythe modifiedequation(48).Theneighborhoodofthesingularpositionwhereequation(22)is utilized is taken aso o2 3180 0.5 + < = . The motor torques and the translational actuator forcesnecessarytorealizethetaskareshowninFigures9and10,respectively.Atthe singularpositiontheactuatorforcesare Nm130.31 T = , N126.3 F = and N21.61 F = .Thejoint displacements under the effects of the actuator forces are given in Figures 11 and 12. To test thevalidityofthemodifiedequationsinalargerneighborhood,whenthesimulationsare repeated with o1 = , no significant changes are observed, the task violations remaining less than 510m. 5. Conclusions A general method for the inverse dynamic solution of parallel manipulators in the presence ofdrivesingularitiesisdeveloped.Itisshownthatatthedrivesingularities,theactuator forcescannotinfluencetheend-effectoraccelerationsinstantaneouslyincertaindirections. Hence the end-effector trajectory should be chosen to satisfy the consistency of the dynamic www.intechopen.com Singularity Robust Inverse Dynamics of Parallel Manipulators 389 equations when the coefficient matrix of the drive and constraint forces, A becomes singular. Thesatisfactionoftheconsistencyconditionsmakesthetrajectorytoberealizablebythe actuators of the manipulator, hence avoids the divergence of the actuator forces. Figure 9. Motor torque for the trajectory satisfying the consistency condition Figure 10. Actuator forces for the trajectory satisfying the consistency condition: 1. F1, 2. F2 Toavoidtheproblemsrelatedtotheill-conditionoftheforcecoefficientmatrix,Ainthe neighborhoodofthedrivesingularities,amodificationofthedynamicequationsismade usinghigherorderderivativeinformation.Deletionofthelinearlydependentequationin that neighborhood would cause task violations due to the removal of a task. For this reason themodifiedequationisusedtoreplacethedependentequationyieldingafullrankforce coefficient matrix. www.intechopen.comParallel Manipulators, New Developments 390 Figure 11. Rotational joint displacements: 1. 1, 2. 2. Figure 12. Translational joint displacements: 1. 1, 2. 2. 6. References Alc,G.(2000).Determinationofsingularitycontoursforfive-barplanarparallel manipulators, Robotica, Vol. 18, No. 5, (September 2000) 569-575. Daniali,H.R.M.;Zsombor-Murray,P.J.&Angeles,J.(1995).Singularityanalysisofplanar parallelmanipulators,MechanismandMachineTheory,Vol.30,No.5,(July1995) 665-678. www.intechopen.com Singularity Robust Inverse Dynamics of Parallel Manipulators 391 DiGregorio,R.(2001).Analyticformulationofthe6-3fully-parallelmanipulators singularity determination, Robotica, Vol. 19, No. 6, (September 2001) 663-667. Gao,F.;Li,W.;Zhao,X.;Jin,Z.&Zhao,H.(2002).Newkinematicstructuresfor2-,3-,4-, and5-DOFparallelmanipulatordesigns,MechanismandMachineTheory,Vol.37, No. 11, (November 2002) 1395-1411. Gunawardana,R.&Ghorbel,F.(1997).PDcontrolofclosed-chainmechanicalsystems:an experimental study, Proceedings of the Fifth IFAC Symposium on Robot Control, Vol. 1, 79-84, Nantes, France, September 1997, Cambridge University Press, New York. Ider, S.K. (2004). Singularity robust inverse dynamics of planar 2-RPR parallel manipulators, ProceedingsoftheInstitutionofMechanicalEngineers,PartC:JournalofMechanical Engineering Science, Vol. 218, No. 7, (July 2004) 721-730. Ider,S.K.(2005).Inversedynamicsofparallelmanipulatorsinthepresenceofdrive singularities, Mechanism and Machine Theory, Vol. 40, No. 1, (January 2005) 33-44. Ji,Z.(2003)Studyofplanarthree-degree-of-freedom2-RRRparallelmanipulators, Mechanism and Machine Theory, Vol. 38, No. 5, (May 2003) 409-416. Kong,X.&Gosselin,C.M.(2001).Forwarddisplacementanalysisofthird-classanalytic3-RPRplanarparallelmanipulators,MechanismandMachineTheory,Vol.36,No.9, (September 2001) 1009-1018. Merlet,J.-P.(1999).ParallelRobotics:OpenProblems,ProceedingsofNinthInternational SymposiumofRoboticsResearch,27-32,Snowbird,Utah,October1999,Springer-Verlag, London. Sefrioui,J.&Gosselin,C.M.(1995).Onthequadraticnatureofthesingularitycurvesof planarthree-degree-of-freedomparallelmanipulators,MechanismandMachine Theory, Vol. 30, No. 4, (May 1995) 533-551. Appendix TheelementsofMandRofthe2-RPRparallelmanipulatorshowninequation(41)are given below, where im ,1,..., 5 i =are the masses of the links, iI ,1,..., 5 i =are the centroidal momentsofinertiaofthelinksandthelocationsofthemasscenters iG ,1,..., 5 i = are indicated by 1 1g AG = , 2 2g BG = , 3 3g CG = , 4 4g DG = , 5 5g BG =and 5G BD = . 2 2 211 1 1 1 2 1 2 2 5 1( ) M m g I m g I m = + + + +(A1) 15 5 1 5 1 3cos( ) M m g = (A2) 22 2 3M m m = +(A3) 25 5 5 1 3sin( ) M mg = (A4) 2 233 3 3 3 4 2 4 4( ) M mg I m g I = + + +(A5) 44 4M m =(A6) 51 5 1 5 1 3cos( ) M m g = (A7) www.intechopen.comParallel Manipulators, New Developments 392 52 5 5 1 3sin( ) M mg = (A8) 255 5 5 5M mg I = +(A9) 21 2 1 2 1 1 5 1 5 3 1 32 ( ) sin( ) R m g m g = +

m g m g m g + + +1 1 2 1 2 5 1 1[ ( ) ] cos(A10) 2 2 22 5 5 3 1 3 2 1 2 1 5 1 1 2 5 1cos( ) ( ) ( ) sin R mg m g m m m g = + +

(A11) 3 4 2 4 2 2 3 3 4 2 4 22 ( ) [ ( )] cos R m g mg m g g = + + (A12) 24 4 2 4 2 4 2( ) sin R m g mg = +

(A13) 25 5 5 1 1 1 3 1 1 1 3 3[2 cos( ) sin( ) cos( )] R mg g = + +

(A14) www.intechopen.comParallel Manipulators, New DevelopmentsEdited by Jee-Hwan RyuISBN 978-3-902613-20-2Hard cover, 498 pagesPublisher I-Tech Education and PublishingPublished online 01, April, 2008Published in print edition April, 2008InTech EuropeUniversity Campus STeP Ri Slavka Krautzeka 83/A 51000 Rijeka, Croatia Phone: +385 (51) 770 447 Fax: +385 (51) 686 166www.intechopen.comInTech ChinaUnit 405, Office Block, Hotel Equatorial Shanghai No.65, Yan An Road (West), Shanghai, 200040, China Phone: +86-21-62489820 Fax: +86-21-62489821Parallel manipulators are characterized as having closed-loop kinematic chains. Compared to serialmanipulators, which have open-ended structure, parallel manipulators have many advantages in terms ofaccuracy, rigidity and ability to manipulate heavy loads. Therefore, they have been getting many attentions inastronomy to flight simulators and especially in machine-tool industries.The aim of this book is to provide anoverview of the state-of-art, to present new ideas, original results and practical experiences in parallelmanipulators. This book mainly introduces advanced kinematic and dynamic analysis methods and cuttingedge control technologies for parallel manipulators. Even though this book only contains several samples ofresearch activities on parallel manipulators, I believe this book can give an idea to the reader about what hasbeen done in the field recently, and what kind of open problems are in this area.How to referenceIn order to correctly reference this scholarly work, feel free to copy and paste the following:S. Kemal Ider (2008). Singularity Robust Inverse Dynamics of Parallel Manipulators, Parallel Manipulators,New Developments, Jee-Hwan Ryu (Ed.), ISBN: 978-3-902613-20-2, InTech, Available from:http://www.intechopen.com/books/parallel_manipulators_new_developments/singularity_robust_inverse_dynamics_of_parallel_manipulators