Abstract
In this paper, the use of FIR (Finite Impulse Response) filters for planning minimum-time trajectories for robots or automatic machines under constraints of velocity, acceleration, etc. is presented and discussed. In particular, the relationship between multi-segment polynomial trajectories, i.e. trajectories composed of several polynomial segments, each one possibly characterized by constraints on one or more specific derivatives (i.e. velocity, acceleration, jerk, etc.), and FIR filters disposed in a cascade configuration is demonstrated and exploited in order to design a digital filter for online trajectory planning. The connection between analytic functions and dynamic filters allows a generalization of these trajectories, usually obtained by second- or third-order polynomial functions (e.g. trapezoidal velocity and double S velocity trajectories), to a generic order with only a modest increase of the complexity. As a matter of fact, the computation of trajectories with higher degree of continuity simply requires additional FIR filters in the chain. Moreover, the modular structure of the planner provides a direct frequency characterization of the motion law. In this way, it is possible to define the trajectories by considering constraints expressed in the frequency-domain besides the classical time-domain specifications, such as bounds on velocity, acceleration, and so on. Two examples illustrate the main features of the proposed trajectory planner, in particular with respect to the problems of multi-point trajectories generation and residual vibrations suppression.
Highlights
The growing need of planning trajectories online has led to the development of a number of filters able to produce motion profiles with the desired degree of smoothness starting from rough reference signals, such as step functions, which set the desired final position
The filters can be applied to reference signals different from simple step functions
In motion control of CNC machines, FIR (Finite Impulse Response) filters are generally adopted because their efficiency and the possibility to be implemented by hardware
Summary
The growing need of planning trajectories online has led to the development of a number of filters able to produce motion profiles with the desired degree of smoothness starting from rough reference signals, such as step functions, which set the desired final position Examples of these trajectory planners have been presented e.g. in Zanasi et al (2000); Zanasi and Morselli (2003) or, more recently, in Zheng et al (2009), where minimumtime trajectory planners with bounds on velocity, acceleration, and jerk have been proposed. The key point is the equivalence between time-optimal multi-segment polynomial trajectories with constraints on the first n derivatives and the output of a chain of n moving average filters In this case the filters are not used for making a given trajectory smoother but for online generating a trajectory starting from initial and final positions, to feedback controlled planners.
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