Symbols nomenclatures $\vee$ : Join operator (in Grassmann-Cayley Algebra) $\wedge$ : Meet operator (in Grassmann-Cayley Algebra) $\cap$ : Intersection of vectors $\oplus$ : Spanning by vectors ${F}_{i}$ : Constraint wrench force of the ith limb for the Parallel Robot $\sigma$ : Permutation $\rho$ : Wrench intensity ${\Gamma }_{p,q}$ : Plücker coordinate vector of finite line passing through two distinct finite points p and q ${\Gamma }_{\infty }$ : Plücker coordinate vector of lines at infinity, passing through two points at infinity ${J}_{6×6}$ : Jacobian of Square Matrix of six columns six rows k: Number of joint ${l}_{i}$ : ith Limb or ith kinematic chain m: Number of link $\eta$ : Order of task space ${P}^{3}$ : Projective Space of 3 dimensional ${P}_{s}$ : Symbolic level of Plücker coordinates ${p}_{i}$ : Prismatic joint axis of ith limb ${\pi }_{\infty }$ : Plane at infinity in the Projective Space ${r}_{i}$ : Revolute joint axis of the ith limb ${s}_{i}$ : Spherical joint axis of the ith limb ${S}_{0}$ : Position Vector of any point on the screw axe S: Unit vector along the screw axis ${\stackrel{^}{}}_{0}$ : Zero pitch screw ${\stackrel{^}{}}_{\infty }$ : Infinite pitch screw V: Vector Space ${W}_{n}$ : nth Actuated wrench force ${\stackrel{^}{}}_{r0}$ : Wrench of zero pitch ${\stackrel{^}{}}_{r\infty }$ : Wrench of infinite pitch $\left[\underset{n\text{-points}}{\underbrace{a,\cdots ,b}}\right]$ : Superbracket which consist of n points ${S}_{0}×s$ : Moment of the screw related to the origin of the reference frame