Open-loop and closed-loop filters for optimal trajectory planning under time and frequency constraints

Dynamic filters for real-time trajectory generation can be designed in different ways with quite different levels of performances and complexity. However, one can observe that the configurations of such filters are based on two main schemes: systems composed of a chain of integrators with a feedback control and systems formed of a sequence of linear filters disposed in a cascade configuration. According to these schemes, it is possible to obtain minimum-time trajectories under constraints of velocity, acceleration, jerk and even higher derivatives. In both cases, the degree of continuity of the resulting trajectory depends on the order n of the filter, which can be designed according to a modular approach. After a short overview of the structure of the trajectory generators, pro and cons of the two approaches are analyzed. In particular, the attention is focussed on linear filters, since their structure allows a straightforward characterization of the trajectory from a frequency point of view. As a consequence, the generator can be designed by taking into account frequency constraints, besides more standard time constraints (i.e. limits on velocity, acceleration, etc.). The proposed method combines the advantages of minimum-time trajectories with those of input shaping techniques. Moreover, it is possible to prove that, under additional hypotheses, the same chain of linear filters proposed for minimum-time trajectories generation can be used for obtaining uniform B-spline curves, that are widespread in the robotic field when the interpolation of a set of given viapoints is required. In this case, the additional constraints do not allow to impose limits on the velocity or acceleration, but only to properly shape the trajectory in the frequency domain. It is therefore possible to select the trajectory/filter parameters with the purpose of suppressing residual vibrations, that may be present because elastic phenomena affecting the robotic system.