This paper presents a novel trajectory planning method for a flexible Cartesian robot manipulator in a point-to-point motion. In order to obtain an exact mathematical model, the parameters of the equation of motion are determined from an identification experiment. An artificial neural network is employed to generate the desired base position, and then, a particle swarm optimization technique is used as the learning algorithm, in which the sum of the displacements of the manipulator is chosen as the objective function. We show that the residual vibrations of the manipulator can be suppressed by minimizing the displacement of the manipulator. The effectiveness and validity of the proposed method are demonstrated by comparing the simulation and experimental results.