This paper starts from the results reported in the article "FIR Filters for Online Trajectory Planning with Time-and Frequency-Domain Specifications", where the use of a cascade of FIR (Finite Impulse Response) filters for planning minimum-time multi-segment polynomial trajectories, i.e. trajectories composed of several polynomial segments, under constraints of velocity, acceleration, etc. is proposed. In particular, in that paper the relationship between the limits acting on the trajectory derivatives (i.e. velocity, acceleration, jerk, etc.), and the parameters of the filters is deduced, along with a set of constraints among these parameters that guarantees the time-optimality of the trajectory in the rest-to-rest case, that is with null boundary conditions on the trajectory derivatives. However, the choice of the parameters, when these conditions are not satisfied, was still an open problem, at least