Trajectory generation via FIR filters: A procedure for time-optimization under kinematic and frequency constraints

Abstract

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

Improving the Tracking Performance under Nonlinear Time-Varying Constraints in Motion Control Applications: From Theoretical Servo Model to Experimental Validation

“…As for other aspects, smoothness and the time-optimal effect are also valuable for automated robots and machines [49]. From the viewpoint of filtering performance, some studies of applications in FIR (finite impulse response) [50] or exponential filter [51] have been conducted. Another research study focused on jerk-continuous profile in order to generate a smooth trajectory for robotic manipulators [52].…”

On Fast Jerk-Continuous Motion Functions with Higher-Order Kinematic Restrictions for Online Trajectory Generation

“…An algorithm for computing such a function with piecewise constant jerk ("seven segment profile") was presented by the author in [4], where restrictions on velocity, acceleration, and jerk and boundary conditions regarding velocity and acceleration can be prescribed. If there are stronger requirements on vibration reduction (e.g., for restricting position errors), higher-order continuity of the motion function has proven to be effective in experimental settings [5][6][7][8][9][10][11][12]. In this contribution, we present an algorithm for computing a jerk-continuous motion function (as opposed to the piecewise-constant, discontinuous jerk function designed in [4]) in one dimension, such that symmetric restrictions regarding velocity, acceleration, jerk, and snap (the derivative of the jerk, abbreviated as sn in the sequel) are fulfilled and boundary 2 of 39 conditions for position and velocity can be arbitrarily prescribed.…”

“…This leads to a second kind of approach, which construct the motion function directly based on the given restrictions and boundary conditions. For the subsequent literature overview, we restrict ourselves to those approaches which design a jerk-continuous motion function where restrictions at least regarding velocity, acceleration and jerk can be prescribed [11,12,[26][27][28][29][30][31][32][33][34].…”

“…Many of the designed functions in Table 1 work with a fifteen-segment approach, which is the natural extension of the seven-segment profile for motion functions with continuous acceleration. Some authors [11,27,34] also allow more generally for 2 n − 1 segments when restrictions up to the n-th derivative are to be taken into account. Other approaches achieve smoother jerk functions using trigonometric and zero pieces (having less discontinuities in the snap), but this restricts the possibilities to stay at the maximum value for some period of time.…”

“…In the functions proposed by Lee and Choi [29] and Nguyen et al [27] (part b), the snap is also continuous, which, on the other hand, has the disadvantage of a slower motion. In the approach by Ezair et al [34], Nguyen et al [27] (part a) and Biagiotti and Melchiorri [11,33], the continuity order can be freely chosen (we denote this by adding a "+" in Table 1), but orders higher than 3 lead to an increase in computational effort, which can make the suitability for online trajectory generation questionable. Fang et al [30] propose the sigmoid function for going from one constant segment to another one, where all derivatives are continuous.…”