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 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.
Highlights
Motion control systems used in industry are required to be more and more responsive when unforseen events occur or manufacturing demands change
The analytical expression of the trajectory, that should take into account all the possible cases, is combined with a decision tree that determines the specific structure that the motion profile must have on the basis of the given inputs
This feature of FIR filters has been mixed with the compliance to kinematic constraints, which is typical of trajectory generators
Summary
Motion control systems used in industry are required to be more and more responsive when unforseen events occur or manufacturing demands change. The analytical expression of the trajectory, that should take into account all the possible cases, is combined with a decision tree that determines the specific structure that the motion profile must have on the basis of the given inputs (constraints and boundary conditions) This approach has been successfully used to design online optimal trajectory planners with n = 2 [Kroger et al (2006)] and n = 3 [Kroger and Wahl (2010)], but the high complexity of this method does not allow to plan higher order trajectories. The results presented so far do not guarantee the time-optimality of the trajectory for any set of kinematic constraints even in the rest-to-rest case This problem is highlighted in Biagiotti and Melchiorri (2012), where it is shown that the expressions relating the kinematic bounds with the filters parameters (which is used by all the above cited works in order to take into account the kinematic constraints) is valid only under certain conditions.
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