Damped Harmonic Smoother for Trajectory Planning and Vibration Suppression

In this brief, a novel filter for online trajectory generation is presented. The filter can be categorized as an input smoother since it acts on the input signal by increasing its continuity level. When fed with simple signals, as, e.g., a step input, it behaves like a trajectory generator that produces harmonic motions. Moreover, it can be combined with other filters, and in particular, with smoothers having a rectangular impulse response, in order to generate (online) more complex trajectories compliant with several kinematic constraints. On the other hand, being a filter, it possesses the capability of shaping the frequency spectrum of the output signal. This possibility can be profitably exploited to suppress residual vibration by imposing that the zeros of the filter cancel the oscillatory dynamics of the plant. For this purpose, the standard harmonic filter has been generalized in order to consider not only the natural frequency but also the damping coefficient of the plant. In this manner, the so-called “damped harmonic filter” and the related “damped harmonic trajectory” have been defined. By means of theoretical considerations, supported by experimental tests, the novel approach has been compared with the existing methods, and the advantages of its use have been proved.

I. Introduction

The growing interest for planning trajectories online has led to the development of a number of filters able to produce motion profiles with the desired degree of smoothness simply starting from basic reference signals to set the desired final position, such as step functions. For this purpose, many strategies have been proposed, including filtering and smoothing techniques by means of various kind of filters, ranging from finite impulse response filters [1]–[3] to inverse dynamics of the plant, or feedback control of a chain of integrators with bounds on velocity, acceleration, jerk, and so on, as, e.g., in [4]–[7]. In these latter works, time-optimal trajectory planners are proposed based on a closed-loop chain of integrators (whose output represents the desired trajectory) properly designed to track in the fastest possible way the reference input while remaining compliant with the given constraints. In [8], it is shown that time-optimal multisegment polynomial trajectories with constraints on the first derivatives are equivalent to the outputs of a chain of moving average filters, also known as rectangular smoothers (see Section II for a brief overview). On the other hand, in many contributions, mainly focused on a trajectory design via analytic expression optimization, the adoption of trigonometric functions is proposed with the purpose of planning motions with smoother acceleration or jerk profiles that reduce residual vibrations when applied to resonant systems (see [9]). In particular, in [10], polynomial multisegment trajectories (with constant jerk) and multisegment trajectories with square sine jerk have been experimentally compared in this respect, showing that the sine-based trajectories outperform the standard constant-jerk trajectories at the price of a noticeable increase of the motion duration.