Path planning issues are often solved via constrained optimization methods but with constraints which must be satisfied over a whole interval of time or space. The use of fast numerical toolboxes implementing state-of-the-art constrained needs to discretize the continous constraints over a time grid. Thus, the obtained solution, in this way, will satisfy the constraints only for time values corresponding to the time grid. Obviously, some constraints could be violated with catastrophic consequences when dealing with, for instance, the balance of humanoid robots. In this paper we introduce a guaranteed discretization method which uses interval analysis to ensure that the constraints are satisfied over the whole time interval. We analyze numerically this method by performing a trajectory generation under constraints dedicated to the motion of the HOAP-3 humanoid robot.