Dynamic modeling of parallel manipulators based on Lagrange–D’Alembert formulation and Jacobian/Hessian matrices

Abstract

This paper describes the development of a dynamic model for parallel manipulators based on the Lagrange–D’Alembert equation, the Hessian matrix of the kinetic energy of the manipulator, and the Jacobian/Hessian matrices of the constraint equations. The model generates the forward and inverse dynamics formulations of parallel manipulators. First, the kinematic relations between the input actuated variables and the output variables are defined in terms of the Jacobian and Hessian matrices of the constraint equations. Then, a new form of Lagrange–D’Alembert equation is introduced and used to derive the dynamic model of parallel manipulators. In the present model, the inertia, centrifugal, and Coriolis generalized forces/torques are obtained directly from Hessian matrix of the kinetic energy of the manipulator. The developed model is simple, straightforward, and appropriate for parallel processing techniques as the elements of the Jacobian and Hessian matrices can be calculated simultaneously. The developed model is validated using the conservation of work and energy principle. A trajectory tracking control of a simple 3RRR planar parallel manipulator is used to illustrate the developed model.