Dynamic Model of a 6-dof Parallel Manipulator Using the Generalized Momentum Approach

The dynamic model of a mechanical system relates the time evolution of its configuration (position, velocity and acceleration) with the forces and torques acting upon it. The inverse dynamic model is important for system control while the direct model is used for system simulation. Serial structure manipulator dynamic modelling is a well established subject. So, recent developments have been oriented towards the improvement of numerical efficiency enabling their use in real-time control algorithms (Lilly, 1993; Naudet, 2003; Mata, 2002; Lee, 2005; Featherstone, 2000). Parallel structure manipulators present a more complex problem, and, usually, the model algorithms cannot be generalized. When used in a real-time control framework the resulting models must be simplified as they usually demand a very high computational effort. The dynamic model of a parallel manipulator when operated in free space can be mathematically represented, in the Cartesian space, by a system of nonlinear differential equations that may be written in matrix form as ( ) ( ) ( ) f x G x x x V x x I = + ⋅ + ⋅ $ $ $ $ , (1) ( ) x I being the inertia matrix, ( ) x x V $ , the Coriolis and centripetal terms matrix, ( ) x G a vector of gravitational generalized forces, x the generalized position of the mobile platform or endeffector and f the controlled generalized force applied on the end-effector: ( ) τ x J f ⋅ = T (2)