Efficient dynamic equations of 3-RPS parallel mechanism through Lagrange method

By means of an efficient method, the generalized reduced order dynamic equation for 3-RPS parallel mechanism through Lagrange method is derived. Kinematic constraints accompanying the Lagrange method for the constrained set of generalized coordinates, introduces the Lagrange multiplier into dynamical formulation. To omit the Lagrange multipliers the natural orthogonal complement matrix of kinematic constraints´ matrix should be found. To reach the natural orthogonal complement matrix, the inverse of a square matrix having order equal to the rank of a kinematic constraints´ matrix should be found. For a system having many kinematic constraints like 3-RPS, the rank of the aforementioned matrix will be high. In this research it is shown that for a 3-RPS parallel manipulator, a rearranging matrix derived from the kinematic constraints on constrained coordinates in a special way will simplify the inverse calculation. Instead of inversion of a high order matrix, only inversion of some very low order matrices should be evaluated. Therefore the natural orthogonal complement matrix can be reached without the need for inversion of a high order matrix and Lagrange multipliers can be omitted again very easily.