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 for high order trajectories. In this paper, we show that in case the conditions are not met by the filters parameters, the optimality of the trajectory under the given kinematic bounds can be assured in any case. An algorithm for the selection of the optimal parameters for a generic nth order trajectory planner subject to n kinematic limits is provided. Additionally, the optimal combination of kinematic and frequency constraints is considered. In fact, the compliance with these two types of constraints may lead to a planner composed by a redundant number of filters and, therefore, a procedure for the selection of the minimum number of FIR filters is devised. The effectiveness of the time-optimal trajectory planner is proved by means of numerical simulations and experimental tests.
