Derivation of General Acceleration and Hessian Matrix of Kinematic Limbs in Parallel Manipulator by Extended Skew-Symmetric Matrixes

A general acceleration model and a Hessian matrix of the kinematic limbs in the parallel manipulators are established using new skew-symmetric matrixes. First, several extended formulas of the skew-symmetric matrixes are derived and proved. Second, the differentiations of the sub-Jacobian matrixes of the general kinematic limbs are transformed into the multiplication of the general velocity transposition of the parallel manipulator by the sub-Hessian matrixes of the kinematic limbs based on extended formulas of the skew-symmetric matrixes. Third, the formulas for solving the Hessian matrixes and the general accelerations of several typical linear kinematic limbs of the parallel manipulator are derived. Finally, the Hessian matrixes and the accelerations of the kinematic limbs of a 3-DOF RPS + UPU + SPR type parallel manipulator are derived and verified using its simulation mechanism.