On the Zeros of the Transfer Function of Flexible Link Manipulators and Their Non-Minimum Phase Behaviour

Abstract

In this article, a study of the zeros of the transfer function, between the base torque and the end-effector displacement, for flexible link manipulators is performed. The analysis is carried out on a single flexible link manipulator with the initial part of the link being rigid. This type of manipulator is referred to as a slewing single rigid—flexible link manipulator (SRFLM). A new method for finding the zeros of the transfer function of an SRFLM without using the corresponding transfer function is introduced. The changes of the locations of the zeros of an SRFLM owing to the changes in all the physical parameters (PPs) are investigated. It is shown that there are PPs where the increase in their values moves the zeros further from the imaginary axis; while by increasing the values of some other PPs the zeros become closer to the imaginary axis. Finally, there are PPs where the locations of the zeros are independent of their values. These findings will be beneficial in the design as well as control of flexible link manipulators and are among the main contributions of this work.

Truncation errors of assumed shape modeling for flexible-link manipulators

“…For the derivation of these zeros, the following theorem introduced and proven by Vakil et al 11 was employed.…”

“…From equation (6) Éðz, sÞ ¼ c 1 cosð zÞ þ c 2 sinð zÞ þ c 3 coshð zÞ þ c 4 sinhð zÞ ð 8Þ and the unknown c 1 , c 2 , c 3 , and c 4 are found by imposing the boundary conditions presented in equations (2) to (5). 11 Having Éðz, sÞ and also knowing that the shear force at the end-effector is UðL, sÞ ¼ EI:É zzz ðL, sÞ, the transfer function of the pinned-pinned counterpart of the SFLM is EIÉ zzz ðL, sÞ…”

“…where the constants A, B, C, and D, as well as the parameter are obtained by imposing the boundary conditions presented in equations (23) to (26) and using the relation  ¼ À " =J o ; details can be found in the studies of Vakil et al 11 and Bellezza et al 12 Since an SFLM is an infinite dimensional system, there are infinite mode shapes. Therefore, there are infinite…”

“…11 These transfer functions have zeros and poles that are, respectively, the roots of the numerator and denominator of the transfer function. 10 The transfer function derived based on the infinite dimensional model well represents the behavior of an experimental setup, and hence its poles and zeros are more accurate than those of the AMM.…”

“…However, AMM can considerably change the locations of zeros. 8 As an example, the zeros of an SFLM which is modeled by an infinite mode of vibration must all be real, 11 but when AMM is adopted, these zeros can be imaginary as well. 10 The transfer function for an SFLM has right-hand side (RHS) zeros in the domain S of the Laplace transform and thus has the non-minimum characteristic.…”