Improved Trajectory Planning using Arbitrary Power Polynomials

Abstract

Increasing demands for higher operational speeds, the need for flexible machinery and recent developments in microchip technology have made programmable machine systems an attractive alternative to conventional systems. However, some difficulties still remain for proper control of programmable systems, especially at higher speeds. These can be categorized into two groups : trajectory planning and trajectory tracking. Conventional trajectory planning methods are ineffective for general application, especially when velocity and acceleration conditions are included. There are many mathematical functions but polynomials are shown to be the most versatile for trajectory planning: however, these can give curves with unexpected oscillations, commonly called meandering. Tracking of a motion in this situation could engender severe practical problems. In this study, a new interpolation method using polynomials with arbitrarypowers is proposed to overcome this disadvantage.

Input shaping for vibration-free positioning of flexible systems

“…(2) between the design points and have a tendency to pro-Again, there will be constraints on the maximum velocduce oscillations [23]. For example, Piazzi and Visioli ity, acceleration and displacement that can be imposed showed that it was necessary to switch to another funcby the drive system.…”

“…t=T S , as suggested in reference [22], in order to obtain a smooth convergence to the end-position. However, The most popular function for trajectory selection is a polynomial function of time [22,23]. The order and the switching is done at t=T S where no errors can be tolerated as the whole purpose of the exercise is to reach coeYcients of the polynomial function can be calculated in order to satisfy the design points de ned in terms of the end-position at T S seconds and stay there without any vibration.…”

Experimental evaluation of a model reference adaptive control for a hydraulic robot: a case study

“…All the trajectories used in this study have been designed by the motion design software called "MOTDES". 36 The robot has three axes, but axes 1 and 2 were used for the implementation of example trajectories in the vertical plane (see Figure 1). The results for revolute and prismatic joints are shown in the first and second column of Figure 4 respectively.…”

Motion design and implementation for a hybrid drive system

“…These advantages make the power form of polynomials particularly appropriate for motion design. In spite of these advantages, this function may su er from unexpected oscillations between the boundary conditions at higher degrees and may cause important problems [6 ]. This side e ect may be eliminated by dividing the whole motion into smaller segments and then a number of lower-order polynomials applied to these segments.…”