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 i |

$\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 |

${J}_{6\times 6}$ : | Jacobian of Square Matrix of six columns six rows |

k: | Number of joint |

${l}_{i}$ : | i |

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 i |

${\pi}_{\infty}$ : | Plane at infinity in the Projective Space |

${r}_{i}$ : | Revolute joint axis of the i |

${s}_{i}$ : | Spherical joint axis of the i |

${S}_{0}$ : | Position Vector of any point on the screw axe |

S: | Unit vector along the screw axis |

${\widehat{\$}}_{0}$ : | Zero pitch screw |

${\widehat{\$}}_{\infty}$ : | Infinite pitch screw |

V: | Vector Space |

${W}_{n}$ : | n |

${\widehat{\$}}_{r0}$ : | Wrench of zero pitch |

${\widehat{\$}}_{r\infty}$ : | Wrench of infinite pitch |

$[\underset{n\text{-points}}{\underbrace{a,\cdots ,b}}]$ : | Superbracket which consist of n points |

${S}_{0}\times s$ : | Moment of the screw related to the origin of the reference frame |