Dynamic Filters for Online Planning Optimal Trajectories

Filters for time-optimal trajectory generation can be obtained in different ways with quite different performances and complexity levels. However, one can easily observe that the configurations of such filters are based on two main schemes: systems composed by a chain of integrators with a feedback control and systems formed by a sequence of Finite Impulse Response (FIR) filters disposed in a cascade configuration. Both trajectory generators are characterized by the order $n$ that defines the degree of continuity of the resulting trajectory and both can be designed according to a modular approach that allows to obtain the $n$-th order filter from that of $n-1$ order. In this paper, after having presented the structure and the analytical expression of the two types of trajectory filters, their common features (possibility of generating online time-optimal trajectories under constraints of velocity, acceleration, jerk, etc.) but especially the main differences are analyzed. In particular, the possible applications of the two systems are considered. Trajectory filters with a feedback loop are able to track generic input signals (and not only step functions) guaranteing the compliance of the output with the given constraints but are characterized by a complexity that limits their use to the third order. Conversely, the simple structure and the low computational cost make open-loop filters desirable for point-to point motions, even with an high degree of smoothness ($n=4$ or $5$). Moreover, the low-pass response of this type of filters allows to characterize (and to design) the output trajectory from a frequency point of view, but, on the other hand, it may cause large delays and distortions between input and the output signals.