Derivation and Calculation of the Dynamics of Elastic Parallel Manipulators

Many algorithms for the modelling and calculation of the dynamics of rigid parallel manipulators already exist, and are based on two approaches: The Newton-Euler method and the Lagrangian principle. For the Newton-Euler method the dynamics equations are generated by the complete analysis of all forces and torques of each rigid body in the robot’s structure (Featherstone & Orin, 2000, Spong & Vidyasagar, 1989). Therefore, the derivation of the equations of motion for complex systems becomes very complicated and laborious. However, due to the fact that all forces are explicitly regarded and analysed, this method supplies a very advanced understanding of the system’s dynamics. The use of the Lagrangian principle is a much more elegant and efficient procedure. A scalar function called the Lagrangian is generated, and describes the entire kinetic, potential and dissipative energy of the system in generalized coordinates. For parallel manipulators, additional equations which describe the closed kinematic loop constraints, still have to be provided. The equation of motion for the parallel structure consists thus of the system of Lagrange and algebraic equations (DAE). The Lagrangian method is very widely used in the area of parallel manipulators (Beyer, 1928, Kock, 2001). In particular, two procedures from this family are established here: Namely the Lagrangian equations of the first type (Kang & Mills, 2002, Miller & Clavel, 1992, Murray et al., 1994, Tsai, 1999) and the Lagrange-D’Alembert formulation (Nakamura, 1991, Nakamura & Ghodoussi, 1989, Park et al., 1999, Stachera, 2006a, Stachera, 2006b, Yiu et al., 2001). The use of generalized coordinates is employed in these procedures. Those being the coordinates of the active joints as well as an additional set of redundant coordinates of the passive non-actuated joints or end-effector coordinates. Active joints are the actuated joints of the machine. In the case of elastic manipulators a set of elastic degrees of freedom (DOF) will be introduced. In these generalized coordinates the energy function will be formulated. Additionally, the closed kinematic loop constraints of the parallel structure must also be considered. In the Lagrangian equations of the first type this is achieved by Lagrange multipliers. Contrary to these equations, for the Lagrange-D’Alembert formulation, the Jacobian matrices of the kinematic constraints parameterised by the nonredundant coordinates are used. The policy with the Jacobian matrix has the great advantage that the well known methods and techniques for the modelling of the manipulator’s chain dynamics, which were already applied to serial elastic robots can be