New kinematics Hessian matrixes of manipulators based on Skew-symmetric matrixes theory

Abstract

• Propose novel approach for deriving Hessian matrix of PM and its generative mechanisms. • Derive Hessian matrixes of a typical PM with 3 different limbs. • Derive Hessian matrix of hybrid hand and redundant kinematic hybrid manipulator. • Derive angular accelerations and Hessian matrix of 2 typical kinematic limbs. A new approach is proposed for deriving the Hessian matrixes of parallel manipulators and their generative mechanisms using the skew-symmetric matrixes of translational/angular velocities and their combination formulae. First, several necessary skew-symmetric matrixes of the translational/ angular velocities and their combination formulae are derived. Second, the differentiations of the sub-Jacobian matrixes are transformed into a multiplication of a general velocity transposition by sub-Hessian matrixes based on the derived formulae. Third, the sub-Hessian matrixes of a typical parallel manipulator and its generative mechanisms (the hybrid hand and the redundant kinematic hybrid manipulator) as well as their kinematic limbs are derived by differentiating the sub-Jacobian matrixes based on the derived the skew-symmetric matrixes of translational/angular velocities and their combination formulae. Finally, the formulae are derived for solving the Hessian matrixes and the accelerations of the typical parallel manipulator and the generative mechanisms.