Yi Lue-mail:luyi@Bo HuSchool of Mechanical Engineering,Yanshan University,Qinhuangdao,Hebei066004,China Unified Solving Jacobian/Hessian Matrices of Some Parallel Manipulators With n SPS Active Legs and a Passive Constrained LegSome parallel manipulators with n spherical joint-prismatic joint-spherical joint(SPS)-type active legs and a passive constrained leg possess a larger capability of load bearing and are simple in structure of the active leg.In this paper,a unified and simple approach is proposed for solving Jacobian/Hessian matrices and inverse/forward velocity and ac-celeration of this type of parallel manipulators.First,a general parallel manipulator with n SPS-type active legs and one passive constrained leg in various possible serial structure is synthesized,and some formulae for solving the poses of constrained force/ torque and active/constrained force matrix are derived.Second,the formulae for solving extension of active legs,the auxiliary velocity/acceleration equation are derived.Third, the formulae for solving inverse/forward velocity and acceleration and a Jacobian matrix without thefirst-order partial differentiation and a Hessian matrix without the second-order partial differentiation are derived.Finally,the procedure is applied to three par-allel manipulators with four andfive SPS-type active legs and one passive constrained leg in different serial structures and to illustrate.͓DOI:10.1115/1.2771572͔Keywords:parallel manipulator,constrained leg,kinematics,Jacobian matrix,Hessian matrix1IntroductionSome parallel manipulators with3–6degree of freedoms ͑DOFS͒have been used in many practical applications͓1,2͔. Among them,the parallel manipulators with nϽ6SPS-type active legs and one passive constrained leg in various serial structures have attracted more attention.This kind of parallel manipulator has a larger capability of load bearing and a simple structure of active leg͓3–6͔.In the aspect of kinematics,Tesar proposed a kinematic influence coefficient of mechanisms͓7,8͔.Huang et al. studied thefirst-second-order kinematic influence coefficient ma-trices which are later proved to be Jacobian/Hessian matrices͓2͔. Duffy and Rico et al.analyzed kinematics of the6-6R parallel manipulator and solved acceleration of open/closed spatial chains by means of screw theory͓9,10͔.Joshi and Tsai studied the Jaco-bian matrix for mechanisms with less than6DOF by using screw theory͓11͔.Kim and Ryu derived a homogeneous Jacobian matrix formulation by three end-effector points͓12͔.Fang and Huang solved velocity/acceleration of a3-RPS͑revolute joint-prismatic joint-spherical joint͒manipulator by using thefirst-/second-order kinematic influence coefficient matrices͓13͔.Canfield et al.ana-lyzed velocity of parallel manipulators by truss transformations ͓14͔.Lu et al.solved the velocity/acceleration and Jacobian ma-trix of some spatial parallel manipulators by using an analytic approach and a computer-aided design͑CAD͒variation geometry approach͓15–17͔.Gallardo-Alvarado et al.analyzed kinematics and singularity of a4-DOF parallel manipulator using screw theory͓18͔.Using the principle of virtual work,Tsai solved in-verse dynamics of use Stewart–Gough manipulator͓19͔.Others studied Jacobian matrix and singularity͓20,21͔.Since each of the items in Jacobian/Hessian matrices is the first-/second-order partial differentiation,this imposes difficulties on kinematics analysis of parallel manipulators.In addition,the forward pose equations of parallel manipulators generally are the implicit functions and have multisolutions.Therefore,to solve forward velocity/acceleration by means of thefirst-/second-order matrix approach is quite complex.This paper focuses on a unified and simple approach for deriv-ing Jacobian/Hessian matrices of the inverse/forward velocity andacceleration for parallel manipulator with nϽ6SPS-type active legs and one passive constrained leg in various serial structures. Two types of parallel manipulators with four,five SPS-type active legs and one passive constrained leg are presented to illustrate how to solve their Jacobian/Hessian matrices and velocity/ acceleration.2The Common Kinematics2.1A Parallel Manipulator With n SPS-Type Active Legs and One Constrained Leg in Various Serial Structure.Gener-ally,a parallel manipulator with n SPS active legs and one passiveconstrained leg r0includes a base B,a platform m,and n linearSPS active legs r i͑i=1,2,...,nϽ6͒with the linear actuators and one passive constrained leg r0in various serial structures͑see Fig.1͒.Let͕m͖be a coordinate o-xyzfixed on m at o;͕B͖be a coordinate O-XYZfixed on B at O;e i be the distance from b i to o; and E i be the distance from B i to O.Each of r i connects m at joint b i with B at joint B i,provides oneconstraint to m,and bears an active force F ai exerted on and alongr i.The constrained leg r0connects m at joint o with B at joint O,provides6-n constraints to m,and bears6-n constrained forces ortorques exerted on r0.Contributed by the Mechanisms and Robotics Committee of ASME for publica-tion in the J OURNAL OF M ECHANICAL D ESIGN.Manuscript received August5,2006;final manuscript received November15,2006.Review conducted by Qizheng Liao.Dofs of the parallel manipulators can be calculated by adopting a revised Kutzbach–Grubler equation as follows ͓1,2͔M =6͑q 0−q −1͒+͚i =1qm i −M͑1͒where q 0is the number of links;q is the number of joints;m i is the DOF of the i th joint;m 1=1for the prismatic joint P or revo-lute joint R ;m 2=2for the universal joint U or cylinder joint C ;m 2=3for the spherical joint S ;and M 0is the local redundant DOF,which have no influence on the motion of parallel mechanism.Corresponding to the number n of r i ,many possible types of constrained legs r 0in serial structure can be synthesized ͑see Table 1͒.By comparing them with each other,a prismatic joint-universal joint ͑PU ͒-type passive leg for n =3;a prismatic joint-spherical joint ͑PS ͒-type for n =4;and a universal joint-prismatic joint-universal joint ͑UPU ͒-type for n =5are considered to be bet-ter in structure because the interference among r i and r 0can be avoided easily.2.2The Geometric Constraints for Determining the Pose of Constrained Wrench.When ignoring the friction of all joints in a parallel manipulator,the workloads can be simplified as a wrench ͑F ,T ͒,which include the inertia wrench of platform and legs due to their masses,the damping wrench due to velocity,the gravity,and the external working wrench,where F is a central force and T is a central torque.The wrench ͑FT ͒is applied onto m at o ,and is variables versus time.͑FT ͒or its components ͑F X ,F Y ,F Z ,T X ,T Y ,T Z ͒are balanced by n active forces F ai ͑i =1,2...,n ͒and a constrained wrench ͑F p T p ͒exerted on r 0͑Fig.1͒.F p is composed of n 1constrained forces F pj ͑j =1,...,n 1͒;and T p is composed of n 2constrained torques T pk ͑k =1,...,n 2͒.Their numbers satisfy n +n 1+n 2=6,n =M .In a parallel manipulator with n SPS-type active legs and one constrained leg in various serial structures,the passive constrained leg may be composed of some multi-DOF joints,such as sphericaljoint S ,universal joint U ,cylindrical joint C ,prismatic joint P ,and revolute joint R .In order to determine the pose of ͑F p T p ͒,an equivalent constrained leg r e must be constructed by replacing S with three intersecting revolute joints,U with 2crossed revolute joints,and C with a prismatic joint and a revolute joint,respectively.During the movement of the parallel manipulator,since the con-strained wrench ͑F p T p ͒do not do any work,three geometric con-straints for determining the poses of the constrained wrench are obtained as follows:1.Let v re be a velocity along prismatic joint P in equivalent active leg r e ,there must be F pj v re =0,i.e.,F pj ЌP .Thus,each of the constrained forces must be perpendicular to all the prismatic joints in r e .2.Let R e be an unit vector of revolute joint R in r e ,and let ÃF pj be a torque of F pj about R ,where there must be R e ·͑ÃF pj ͒=0.Thus,each of the constrained forces must intersect or be parallel with all the revolute joints in r e .3.Let re be a rotation speed about R in r e ,where there must be T pk ·re =0,i.e.,T pk ЌR .Thus,each of the constrained torques must be perpendicular to all the revolute joints in r e .2.3The Active Force and Constrained Wrench.In order to solve active forces and constrained wrench,a set of wrench bal-ancing equations of the parallel manipulator to m at o are obtained as follows͚i =1nFai ␦i+͚j =1n 1Fpj f j=−F ,n +n 1+n 2=6͑2͚͒i =1nFai e iϫ␦i +͚j =1n 1d j ϫFpj f j+͚k =1n 2Tpk k=−Twhere f j is the unit vector of F pj ;d j is the arm vector of F pj to o ;and k is the unit vector of T pk .Equation ͑2͒is transformed asG ͓F a 1,...,F an ,F p 1,...,F pn 1,T p 1,...,T pn 2͔T =−͓F T ͔T͑3͒where G is defined as a 6ϫ6force matrix.Its formula is ex-pressed as follows G =ͫ␦1¯␦nf 1¯f n 10¯e 1ϫ␦1¯e n ϫ␦n d 1ϫf 1¯d n 1ϫf n 11¯n 2ͬ6ϫ6͑4͒Fig.1A parallel manipulator with n SPS active legs:a con-strained legTable 1Various possible serial structure of constrained leg r o for parallel manipulators with n <6SPS-type active legs n Various possible serial structure of constrained legUP UR CP CR PC PU 3PRP PPR PRR PPP RC RU RRP RPR RPP RRR SP SR UPR URR UPP URP UU PRPR PPPR PRRR PS PUP PPU PUR PRU PRRP PPRR PRPP 4PPPP RRRP RS RPU RUP RRU RUR RRPR RPRR RPRP RPPP RRRR RPC RCP RRC RCR PCP PPC PCR PRC CPR CRR CPP CRP UUP UUR UCP UCR US UPU UPC URC URPR URRR UPRR URU CCP CCR CUP CUR CS CPC CRC CPU CRU CPRR CRPR CRPP 5CRRP CRRR CRRR SC SU SPP SPR SRR SRP PPS PPCR PPRC PPRU PPPU PPPC PRPU PCC PCRR PRPC RPPC RRPU RPS RRPC RRRRR RCRRRRCRRURRRRURRUPPRRUPThus,F ai ,F pj ,T pk can be solved as follows͓F a 1,...,F an ,F p 1,...,F pn 1,T p 1,...,T pn 2͔6ϫ1T =−G −1͓F T ͔T͑5͒2.4Inverse Displacement Kinematics.The position vectors B iB of joints B i on B in ͕B ͖,the position vectors b i m of joints b i onm in ͕m ͖,and position vectors b i Bof b i in ͕B ͖can be written as ͓1,2͔B i B =΄X BiY Bi Z Bi΅,b i m =΄x biy bi z bi΅,R m B=΄x l y l z lx m y m z m x n y n z n΅,o B =΄X oY oZ o΅b i B =R m B b i m+o B͑6a ͒where R B m is a rotational transformation matrix from ͕m ͖to ͕B ͖;͑X o ,Y o ,Z o ͒are three position components of m at o in ͕B ͖;o B is a position vector of point o in ͕B ͖;and ͑x l ,x m ,x n ,y l ,y m ,y n ,z l ,z m ,z n ͒are nine orientation parameters of m ,which have the following constrained equations ͓1,2͔x l 2+x m 2+x n 2=1,y l 2+y m 2+y n 2=1z l 2+z m 2+z n 2=1,x l y l +x m y m +x n y n =0͑6b ͒x l z l +x m z m +x n z n =0,z l y l +z m y m +z n y n =0The length of active legs r i ͑i =1,2,...,n ͒and the vectors r i of r ican be solved as follows ͓1,2͔r i =͉b i B −B i B ͉,r i =͓X b i −X Bi Y b i −Y Bi Z b i −Z Bi ͔T͑7a ͒Based on the structure constraints of various parallel manipula-tors,6-M structure constrained equations can be derived.FromEqs.͑6a ͒and ͑7a ͒and 6-M structure constrained equations,Eq.͑7a ͒can be simplified as a set of common equations for solving r i as followsr i =r i ͑1,2,...,n ͒i =1,2,...,n͑7b ͒where i are n independent parameters among six pose parameters ͑X o ,Y o ,Z o ,␣,,͒,n Ͻ6.The unit vector ␦i of r i and the vector e i of the line e i can be solved as below␦i =΄␦ix␦iy ␦iz ΅=r i r i =1r i΄X b i −X Bi Y b i −Y Bi Z b i −Z Bi΅,e i =΄e ixe iy e iz΅=b i B −o B͑8͒Suppose there are two vectors ,and a skew-symmetric matrix ˆ.They must satisfy following equations ͓1,2͔=΄xy z΅,=΄x y z΅,ˆ=΄−z y z−x−yx΅ϫ=ˆ,ˆT =−ˆ͑9͒2.5Inverse Velocity/Acceleration Along the i th Active Leg.Let V =͓v ͔T be a generalizing velocity of m at o ;v i be a velocity of m at joint b i ;then there must be v i =v +Ãe i ͑i =1,2...,n ͒.A velocity v ri along r i is derived asri =i ·␦i =͑+ϫe i ͒·␦i =␦i ·+͑e i ϫ␦i ͒·=J ri V͑10͒J ri =͓␦i T ͑e i ϫ␦i ͒T ͔1ϫ6By differentiation of Eq.͑10͒with respect to time,an acceleration a ri along the i th active leg is derived as follows a ri =J ri A +J ˙riV A =͓a ͔T ,a =͓a x a a z ͔T ,=͓x z ͔T ͑11͒where J ˙ri is a 1ϫ6differentiation matrix of J ri;and A is a general acceleration of m .A differentiation ␦˙i of unit vector ␦i with re-spect to time is derived as follows␦˙i =͑i −␦i ri ͒/r i =͑+ϫe i −␦i ri ͒/r i͑12͒From Eq.͑12͒,the second item in Eq.͑11͒is expanded as J ˙ri V =͓␦˙i T ͑e ˙i ϫ␦i +e i ϫ␦˙i ͒T ͔V =␦˙i ·+͑e˙i ϫ␦i ͒·+͑e i ϫ␦˙i ͒·=␦˙i ·i +͑e ˙i ϫ␦i ͒·=␦˙i·+͑e ˙i ϫ␦i ͒+͑ϫe i ͒·␦˙i =͓͑i −␦i ri ͒·i +r i ͑ϫe iϫ␦i ͒·͔/r i =͓i 2−ri 2+r i ϫ͑ϫe i ͒·␦i ͔/r i͑13a ͒Let E 3ϫ3be a unit matrix.By means of Eqs.͑9͒and ͑10͒,the firstitem in Eq.͑13͒is derived as followsi 2=͑+ϫe i ͒·͑+ϫe i ͒=͑−e i ϫ͒·͑−e i ϫ͒=͑−eˆi ͒·͑−e ˆi ͒=͓͑E 3ϫ3−e ˆi ͔V ͒·͓͑E 3ϫ3−e ˆi ͔V ͒=͓͑E 3ϫ3−eˆi ͔V ͒T ͓͑E 3ϫ3−e ˆi ͔V ͒=V T ͫE 3ϫ3−e ˆi eˆi −eˆi 2ͬV ͑13b ͒From Eq.͑9͒,the second item in Eq.͑13a ͒is derived as followsri 2=ͩV Tͫ␦ie i ϫ␦i͓ͬͪ͑␦i T ͑e i ϫ␦i ͒T ͔V ͒=VTͫ␦i ␦i T␦i ͑e i ϫ␦i ͒T ͑e i ϫ␦i ͒␦i T ͑e i ϫ␦͒͑e i ϫ␦i ͒TͬV ͑13c ͒From Eq.͑9͒,the third item in Eq.͑13a ͒is derived as follows ϫ͑ϫe i ͒·␦i =͑ϫe i ͒·͑␦i ϫ͒=−e ˆi ·␦ˆi =T e ˆi ␦ˆi=V Tͫ03ϫ303ϫ303ϫ3͑e ˆi ␦ˆi ͒3ϫ3ͬV ͑13d ͒From Eqs.͑13a ͒–͑13d ͒,Eq.͑11͒is transformed as followsa ri =J ri A +V Th ri V ,h ri =1r i΄i h 11i h 12¯i h 16ih 21i h 22¯ih 26]]]]i h 61ih 62¯ih 66΅6ϫ6͑14͒where h ri is a 6ϫ6sub-Hessian matrix.Since each of h ri only includes simple algebraic equations without any second-order dif-ferentiation,and a Hessian matrix is simplified obviously.All items in h ri ͑i =1,2,...,n ͒are derived ͑see the Appendix ͒.2.6Inverse Velocity/Acceleration.When a parallel manipu-lator has n SPS-type active legs r i ͑i =1,2,...,n Ͻ6͒and one constrained leg,from Eqs.͑10͒and ͑14͒,its general inverse velocity/acceleration v r /a r along r i can be solved as followsr =J r V ,a r =J r A +V T H r VJ r =΄␦1T ͑e 1ϫ␦1͒T]␦n T ͑e n ϫ␦n ͒T ΅n ϫ6H r =΄h 1]h n΅,r =΄r 1]rn΅,a r =΄a r 1]a rn΅͑15͒where J r is a n ϫ6Jacobian matrix;and H r is a Hessian matrix with n layers of 6ϫ6submatrices.Since n Ͻ6and J r is not asquare matrix,neither an inverse matrix J r −1nor forward velocity/acceleration can be derived.2.7The Auxiliary Velocity/Acceleration Equations.In or-der to solve the forward velocity/acceleration of parallel manipu-lators with n Ͻ6SPS-type active legs,and one constrained leg,a common 6ϫ6Jacobian matrix and a common Hessian matrix must be created.Since the constrained wrench ͑F p T p ͒do not do any work dur-ing the movement of parallel manipulator,there must beF pj f j ·+͑d j ϫF pj f j ͒·=0,͑j =1,¯n 1͒͑16͒T pk k ·=0,͑k =1,¯n 2,n +n 1+n 2=6͒When removing F pj and T k from Eq.͑16͒,an auxiliary velocity matrix equation is derived as followsJ a V =0,J a =΄f 1T ͑d 1ϫf 1͒T]]f n 1T ͑d n 1ϫf n 1͒T01ϫ31T ]]01ϫ3n 2T ΅͑6−n ͒ϫ6͑17͒where J a is a ͑6-n ͒ϫ6auxiliary Jacobian matrix.By differentia-tion Eq.͑17͒with respect to time,an auxiliary acceleration matrix equation is derived as follows,0=J a A +J ˙a V J ˙a=΄f ˙1T ͑d ˙1ϫf 1+d 1ϫf ˙1͒T ]]f ˙n 1T ͑d ˙n 1ϫf n 1+d n 1ϫf ˙n 1͒T 01ϫ3˙1T ]]01ϫ3˙n 2T ΅͑6−n ͒ϫ6=V T H a͑18͒where H a is an auxiliary Hessian matrix.2.8The Common Velocity and Acceleration Equations.By combing Eq.͑15͒with Eqs.͑17͒and ͑18͒,respectively,the com-mon inverse velocity and acceleration ͑v c a c ͒of this type of par-allel manipulators are derived as followsc =J V ,a c =J A +V T H V c =ͫr 0ͬ6ϫ1,a c =ͫa r 0ͬ6ϫ1,J =ͫJ r J aͬ6ϫ6,H =ͫH r H aͬ͑19͒where J is a common 6ϫ6Jacobian matrix;and H is a common Hessian matrix with six layers of 6ϫ6submatrices.From Eqs.͑4͒and ͑19͒,we obtain G =J T or J =G T .Thus,J can be used to solve the active forces and constrained wrench of this type of parallel manipulators.From Eq.͑19͒,the common forward velocity/acceleration of this type of parallel manipulator is derived as followsV =J −1c ,A =J −1͑a c −V TH V ͒͑20͒In Eqs.͑14͒,͑15͒,and ͑19͒,each item in J ,J −1,H r and its sub-matrix h ri does not include any first-/second-order partial differ-entiation.Therefore,the formulae for solving inverse/forward ve-locity and acceleration of parallel manipulators are obviously simplified.The inverse/forward velocity and acceleration can be solved in the following cases:When given the forward velocity V and the acceleration A of m ,from Eq.͑15͒,the inverse velocity v r and acceleration a r can be solved.When given the common inverse velocity v c and the common acceleration a c ,from Eq.͑20͒,the forward velocity V and accel-eration A can be solved.When solving A ,a differentiation matrix of auxiliary Jacobian matrix J a must be derived.Based on thepose of constrained wrench,each differentiation item in J ˙acan be derived.3A 4-SPS/PS Parallel ManipulatorA 4-SPS/PS parallel manipulator is composed of a platform m ,a baseB ,four SPS-type active legs r i ͑i =1,2,3,4͒with the linear actuators,and one PS-type passive constrained leg r o ͑see Fig.2͒.In order to avoid singularity,let m be a rectangle link with a long side l 1,a short side l 2,four vertices b i ,and a central point o .Let B be a square with sides L i =L ,four vertices B i ,and a central point O .Each of r i connects m with B by a spherical joint S on m at b i ,a prismatic joint P along r i ,and a spherical joint S on B at B i .r o connects m with B by a prismatic joint P fixed on m at o ,a spherical joint S attached to B at O ,and there is a structure con-straint of r o Ќm .In the 4-SPS/PS parallel manipulator,the number of links is q 0=11for one platform,four cylinders,four piston rods,and one base;the number of joints is q =14for 5P ,and 9S ;and the redundancy DOF is M 0=4for four SPS-type active legs rotating about their own axes.Thus,the DOF of the 4-SPS/PS parallel manipulator is calculated as followsM =6͑q 0−q −1͒+͚i =1qm i −M=6ϫ͑11−14−1͒+͑5ϫ1+9ϫ3͒−4=43.1Inverse Kinematics.B i B of B and b i mof m ͑i =1,2,3,4͒are derived as followsB 1B =12΄L −L 0΅,B 2B =12΄L L 0΅,B 3B =12΄−L L 0΅,B 4B =12΄−L −L 0΅Fig.2A 4-SPS/PS parallel manipulator and its force situationb 1m =12΄l 1−l 20΅,b 2m =12΄l 1l 20΅,b 3m =12΄−l 1l 20΅,b 4m =12΄−l 1−l 20΅͑21͒From Eqs.͑6a ͒and ͑21͒,b i B of m are derived as followsb 1B =12΄x l l 1−y l l 2+2X ox m l 1−y m l 2+2Y o x n l 1−y n l 2+2Z o ΅,b 2B =12΄x l l 1+y l l 2+2X o x m l 1+y m l 2+2Y o x n l 1+y n l 2+2Z o΅b 3B =12΄−x l l 1+y l l 2+2X o −x m l 1+y m l 2+2Y o −x n l 1+y n l 2+2Z o΅,b 4B =12΄−x l l 1−y l l 2+2X o−x m l 1−y m l 2+2Y o −x n l 1−y n l 2+2Z o΅͑22͒When r o Ќm ,there is o B=r o z .Thus by means of Eq.͑6b ͒yieldX o /z l =Y o /z m =Z o /z n =r o ,x l 2=1−y l 2−͑X o /r o ͒2y m 2=1−x m 2−͑Y o /r o ͒2͑23͒Let R B m be defined by three Euler rotations of ͑Z ,Y 1,Z 2͒,namely,a rotation of ␣about Z axis,followed by a rotation of about Y 1axis,and a rotation of about Z 2axis,where Y 1is formed by Y rotating about Z by ␣,and Z 2is formed by Z 1rotating about Y 1by.Thus,R B m and o Bare derived as followsR m B =΄c ␣·c ·c −s ␣·s −c ␣·c ·s −s ␣·c c ␣·s s ␣·c ·c +c ␣·s −s ␣·c ·s +c ␣·c s ␣·s −s ·c s ·s c ΅,o B =r o ΄c ␣·s s ␣·s c ΅͑24͒Comparing each item in R B m and oBin Eqs.͑6͒and ͑24͒,͑x l ,y l ,z l ,x m ,y m ,z m ,x n ,y n ,z n ͒and ͑X o ,Y o ,Z o ͒can be expressed by ͑␣,,,r o ͒.From Eqs.͑6b ͒,͑7a ͒,and ͑21͒–͑24͒,r i are derived as followsr 12=͑2L 2+l 12+l 22͒/4+r o 2−͓c ͑c ␣−s ␣͒͑l 1c +l 2s ͒−͑s ␣+c ␣͒ϫ͑l 1s −l 2c ͒+2r o s ͑c ␣−s ␣͔͒L /2r 22=͑2L 2+l 12+l 22͒/4+r o 2−͓c ͑c ␣+s ␣͒͑l 1c −l 2s ͒−͑s ␣−c ␣͒ϫ͑l 1s +l 2c ͒+2r o s ͑c ␣+s ␣͔͒L /2r 32=͑2L 2+l 12+l 22͒/4+r o2−͓c ͑c ␣−s ␣͒͑l 1c +l 2s ͒−͑s ␣+c ␣͒ϫ͑l 1s −l 2c ͒−2r o s ͑c ␣−s ␣͔͒L /2r 42=͑2L 2+l 12+l 22͒/4+r o2−͓c ͑c ␣+s ␣͒͑l 1c −l 2s ͒−͑s ␣−c ␣͒ϫ͑l 1s +l 2c ͒−2r o s ͑c ␣+s ␣͔͒L /2͑25͒From Eqs.͑8͒and ͑21͒–͑25͒,␦i and e i ͑i =1,2,3,4͒are derived asfollows␦1=12r 1΄x l l 1−y l l 2+2X o −L x m l 1−y m l 2+2Y o +L x n l 1−y n l 2+2Z o΅␦2=12r 2΄x l l 1+y l l 2+2X o −Lx m l 1+y m l 2+2Y o −L x n l 1+y n l 2+2Z o΅␦3=12r 3΄−x l l 1+y l l 2+2X o +L −x m l 1+y m l 2+2Y o −L −x n l 1+y n l 2+2Z o ΅␦4=12r 4΄−x l l 1−y l l 2+2X o +L −x m l 1−y m l 2+2Y o +L −x n l 1−y n l 2+2Z o ΅e 1=12΄x l l 1−y l l 2x m l 1−y m l 2x n l 1−y n l 2΅,e 2=12΄x l l 1+y l l 2x m l 1+y m l 2x n l 1+y n l 2΅e 3=12΄−x l l 1+y l l 2−x m l 1+y m l 2−x n l 1+y n l 2΅,e 4=12΄−x l l 1−y l l 2−x m l 1−y m l 2−x n l 1−y n l 2΅͑26͒From Eqs.͑21͒–͑26͒,␦i and e i which are expressed by four poseparameters ͑␣,,,r o ͒can be derived.In the PS-type passive constrained leg r o ,a spherical joint S can be replaced by three intersecting revolute joints ͑R 1,R 2,R 3͒.Thus,an equivalent constrained leg includes three intersecting revolute joints ͑R 1,R 2,R 3͒and one prismatic joint P .Their unit vectors are determined as followsR 1=΄001΅,R 2=΄−s ␣c ␣0΅,R 3=΄c ␣·s s ␣·s c ΅,P =o B r o=΄c ␣·s s ␣·s c ΅͑27͒3.2The Common Jacobian/Hessian Matrices.Based on thegeometric constraints of the constrained wrench in Sec.2.2,in the equivalent passive constrained leg,there must be ͑F p 1Ќo B ,F p 2Ќo B ,and F p 1,and F p 2intersecting or parallel with R 1,R 2,R 3͒.Thus,both F p 1and F p 2are exerted on r o at O .By means of Eq.͑9͒,their unit vectors f 1and f 2and differentiations are deter-mined as followsf 1=x =͓x l x m x n ͔T ,f 2=y =͓y l y m y n ͔T ͑28a ͒f ˙1=x ˙=ϫx =x ˆ,f ˙2=y ˙=ϫy =y ˆThe vectors ͑d 1,d 2͒of the distances from o to F p 1and F p 2andtheir differentiations are determined as follows d 1=d 2=−o B =−r o ͓s −s ␣·c c ␣·c ͔T ,d ˙1=d ˙2=−͑28b ͒From Eqs.͑17͒,͑18͒,͑28a ͒,and ͑28b ͒,the auxiliary Jacobian/Hessian matrices are derived as followsJ a =ͫf 1T ͑d 1ϫf 1͒Tf 2T ͑d 2ϫf 2͒T ͬ2ϫ6=ͫx −o B ϫx y −o B ϫyͬ2ϫ6J ˙a =ͫf ˙1T ͑d ˙1ϫf 1+d 1ϫf ˙1͒T f ˙2T ͑d ˙2ϫf 2+d 2ϫf ˙2͒T ͬ=ͫxˆ−x ˆ−o ˆB x ˆyˆ−y ˆ−o ˆB y ˆͬ=V Tͫh a 1h a 2ͬ͑29͒where h ai ͑i =1,2͒are two auxiliary sub-Hessian matrices.Bymeans of Eq.͑9͒,h ai are derived as followsh a 1=ͫ0−xˆxˆ−o ˆB x ˆͬ6ϫ6,h a 2=ͫ0−yˆyˆ−o ˆB y ˆͬ6ϫ6From Eqs.͑19͒and ͑29͒,the common inverse velocity v c andcommon inverse acceleration a c of the 4-SPS/PS parallel manipu-lator ͒are derived as followsc =J V ,a c =J A +V T H VJ =G T =ͫJ r J aͬ=΄␦1T ͑e 1ϫ␦1͒T ␦2T ͑e 2ϫ␦2͒T ␦3T ͑e 3ϫ␦3͒T ␦4T͑e 4ϫ␦4͒Tf 1T ͑d 1ϫf 1͒T f 2T ͑d 2ϫe 2͒T΅6ϫ6,H =ͫH r H aͬ=΄h r 1h r 2h r 3h r 4h a 1h a 2΅͑30͒where J is a common Jacobian matrix and H is a common Hes-sian matrix.4A 5-SPS/UPU Parallel ManipulatorA 5-SPS/UPU parallel manipulator is composed of a platformm ,a base B ,five SPS-type active legs r i ͑i =1,2,...,5͒with the linear actuators,and one UPU-type passive constrained leg r o ͑see Fig.3͒.In order to avoid singularity,let m be a pentangle with five different sides l i ,five vertices b i ,and a center point o ;let the distance from b i to o be e i =e ;let b 2and b 4lie in the x axis and b 3in the y axis;let the angle between e 1and y be 2,and the same for the angle between e 5and y .Let B be an equilateral pentangle with five sides L i =L ,five vertices B i ,and a center point O ;let the distance from A i to O be E i =E ;let B 3lie in Y ;let the angle between E 1and Y be 2and the same for the angle between E 5and Y ;and let the angle between E 2and X be and the same for the angle between E 4and X .Each of r i connects m with B by a sphere joint S on m at b i ,a prismatic joint P ,and a sphere joint S on B at B i .The UPU-type passive leg r o connects m with B by a universal joint U m on m at o ,a P along r o ,and a universal joint U B on B at O .U B can be replaced by two cross revolute joints R B 1and R B 2.U m can be replaced by two cross revolute joints R m 1and R m 2.In addition,there are some structure constraints of ͑R B 1ЌR B 2,R m 1ЌR m 2,with R B 1being coincident with X ,R m 1being coincident with z ,R B 2ʈR m 2,R m 2Ќr o ,and R B 2Ќr o ͒.From these structure constraints,it can be verified that z intersects with the X axis at point c ͑i.e.,X ,r o ,and z lie in a plane ⌬Ooc ͒.In the 5-SPS/UPU parallel manipulator,the number of links is q 0=14for one platform,six cylinders,six piston rods,and one base;the number of joints is q =18for 6P ,2U ,and 10S ;and the redundancy DOFs is M 0=5for five SPS-type active legs rotating about their own axes.Thus,the DOF of the 5-SPS/UPU parallel manipulator is M =6͑q 0−q −1͒+͚i =1qm i −M=6ϫ͑14−18−1͒+͑6ϫ1+2ϫ2+10ϫ3͒−5=54.1Inverse Kinematics.B i B of B and b i mof m ͑i =1,...,5͒are derived as followsB 1B =E ΄s 2−c 20΅,B 2B =E ΄c s 0΅,B 3B =΄0E΅B 4B =E ΄−c s ΅,B 5B =E ΄−s 2−c 2΅͑31͒b 1m =e ΄s 2−c 20΅,b 2m =΄e00΅,b 3m =΄0e 0΅,b 4m =΄−e 00΅b 5m =e ΄−s 2−c 20΅,=18degFrom Eqs.͑6a ͒and ͑31͒,b i B of m ͑i =1,2,...,5͒are derived as followsb 1B =΄x l es 2−y l ec 2+X ox m es 2−y m ec 2+Y o x n es 2−y n ec 2+Z o΅,b 2B =΄x l e +X ox m e +Y ox n e +Z o΅b 3B =΄ey l +X oey m +Y oey n +Z o΅͑32͒b 4B =΄−x l e +X o−x m e +Y o −x n e +Z o΅,b 5B =΄−x l es 2−y l ec 2+X o−x m es 2−y m ec 2+Y o−x n es 2−y n ec 2+Z o΅Since vectors ͑o B ,z ,X ͒lie in a plane ⌬Ooc ,there must be a con-strained equation as followsΈX o Y o Z oz l z m z n 100Έ=0,i.e.,Y o =z m Z o z n͑33͒Let R mBof the 5-SPS/UPU parallel manipulator be the same as that of the 4-SPS/PS parallel manipulator.From Eqs.͑24͒and ͑33͒,o B is derived as followso B =͓X o Z o s ␣·tan Z o ͔T͑34͒From Eqs.͑6b ͒,͑7a ͒,and ͑31͒–͑34͒,r i arederivedFig.3A 5-SPS/UPU parallel manipulator and its forcesituation。
