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

In this paper, by an efficient method the generalized reduced order dynamic equation for 3-RPS parallel mechanism through Lagrange method will be derived. Kinematic constraints accompanying Lagrange method for the constrained set of generalized coordinates, will introduce Lagrange multiplier into dynamical formulation. To omit the Lagrange multipliers the natural orthogonal complement matrix of kinematic constraints' matrix should be found. To reach to the natural orthogonal complement matrix, inverse of a square matrix having order equal to the rank of kinematic constraints' matrix should be found. For a system having many kinematic constraints like 3-R-PS the rank of the aforementioned matrix will be high In this research it will be shown that for 3-R-PS parallel manipulator 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 could be reached without the need of inversion of a high order matrix and Lagrange multipliers could be omitted again very easily.