Kinematics and Dynamics Hessian Matrices of Manipulators Based on Screw Theory

The complexity of the kinematics and dynamics of a manipulator makes it necessary to simplify the modeling process. However, the traditional representations cannot achieve this because of the absence of coordinate invariance. Therefore, the coordinate invariant method is an important research issue. First, the rigid-body acceleration, the time derivative of the twist, is proved to be a screw, and its physical meaning is explained. Based on the twist and the rigid-body acceleration, the acceleration of the end-effector is expressed as a linear-bilinear form, and the kinematics Hessian matrix of the manipulator(represented by Lie bracket) is deduced. Further, Newton-Euler's equation is rewritten as a linear-bilinear form, from which the dynamics Hessian matrix of a rigid body is obtained. The formulae and the dynamics Hessian matrix are proved to be coordinate invariant. Referring to the principle of virtual work, the dynamics Hessian matrix of the parallel manipulator is gotten and the detailed dynamic model is derived. An index of dynamical coupling based on dynamics Hessian matrix is presented. In the end, a foldable parallel manipulator is taken as an example to validate the deduced kinematics and dynamics formulae. The screw theory based method can simplify the kinematics and dynamics of a manipulator, also the corresponding dynamics Hessian matrix can be used to evaluate the dynamical coupling of a manipulator.