Closed-form dynamics of a 3-DOF spatial parallel manipulator by combining the Lagrangian formulation with the virtual work principle

Dynamic modeling of spatial multi-degree-of-freedom (multi-DOF) parallel motion system is a challenging research because of its multi-closed-loop configuration, in contrast to serial manipulator. This study readdresses the issue of deriving explicit closed-form dynamic equations in the actuation space of non-redundant parallel mechanisms through the Lagrange modeling approach. Due to the complex relationship between passive joint variables and active joint variables, the application of the Lagrangian formulation for dynamic modeling of parallel mechanisms is considered nearly impossible. In this paper, the utilization of the virtual work principle makes the application of the Lagrangian formulation for parallel mechanisms possible and efficient. A closed-loop parallel mechanism is divided into several serial open-loop subchains. Explicit dynamic equations of each subchain can be derived by using the Lagrangian formalism straightforwardly with respect to their own local generalized coordinates. The principle of virtual work is used to combine the differential dynamic equations of the subsystems into an explicit dynamic model of the full parallel manipulator with respect to robot's active generalized coordinates. The Jacobian and Hessian matrices play a crucial role in the transformation of the dynamic equations with respect to different generalized coordinates due to the application of the virtual work principle. The introduction of the cubic Hessian matrices makes the dynamic equations more compact. The procedure of modeling for a 3-DOF spatial parallel manipulator is taken as an application example of the proposed approach. The efficiency of the proposed approach and the correctness of the dynamic model are demonstrated by simulations and experiments.