In this paper, the problem of trajectory optimization for robotic manipulators is solved by a newly developed methodology called Discrete Mechanics and Optimal Control (DMOC). This new methodology is based on the direct discretizaion of the Lagrange-d'Alembert principle. The constraints for the objective function to be optimized are the forced discrete Euler-Lagrange equations. In this paper, DMOC is applied to a Delta robot to optimize the desired cost function for a predefined geometrical path. The challenges to use DMOC for a Parallel Kinematic Machine (PKM) are also discussed because it is the first time, to the best of author's knowledge, that DMOC is used for a PKM. The optimal results obtained from DMOC are compared with the optimal solutions obtained by other state-of-the-art techniques which show the effectiveness of this new methodology.