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Research Papers

# Robust Control of Three-Dimensional Compliant Mechanisms

[+] Author and Article Information
Erfan Shojaei Barjuei

DPIA,
University of Udine,
Via delle Scienze 206,
Udine 33100, Italy
e-mail: erfan.shojaei@uniud.it

Paolo Boscariol

DPIA,
University of Udine,
Via delle Scienze 206,
Udine 33100, Italy
e-mail: paolo.boscariol@uniud.it

Renato Vidoni

Faculty of Science and Technology,
Free University of Bozen-Bolzano,
Bolzano, BZ 39100, Italy
e-mail: renato.vidoni@unibz.it

Alessandro Gasparetto

DPIA,
University of Udine,
Via delle Scienze 206,
Udine 33100, Italy
e-mail: gasparetto@uniud.it

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received March 13, 2015; final manuscript received June 21, 2016; published online July 21, 2016. Assoc. Editor: Fu-Cheng Wang.

J. Dyn. Sys., Meas., Control 138(10), 101009 (Jul 21, 2016) (14 pages) Paper No: DS-15-1110; doi: 10.1115/1.4034019 History: Received March 13, 2015; Revised June 21, 2016

## Abstract

Position control and vibration damping of flexible-link mechanisms are still challenging open issues in robotics. Finding solutions for these problems can lead to improvement in the operation and accuracy of the manipulators. In this paper, the synthesis of robust controllers based on $H∞$ loop shaping and $μ$-synthesis for both position control and vibration damping in a spatial flexible L-shape mechanism with gravity is presented. The design of the controllers is based on the evaluation of an uncertainty model which takes into account a $±20%$ uncertainty in the elasticity and mass density of the links. The response of each controller is tested also in the presence of external disturbances with the aid of highly accurate numerical simulations; furthermore, a comparison between the robust performances of synthesized controllers is presented in order to show the effectiveness of synthesized control systems.

## Figures

Fig. 1

Kinematic definition of the ERLS

Fig. 2

The mechanism built for the experimental validation of the model

Fig. 3

FEM discretization: elastic displacements

Fig. 4

Comparison between linear and nonlinear impulsive response: angular position q

Fig. 5

Comparison between linear and nonlinear impulsive response: elastic displacement u25

Fig. 6

Comparison between linear and nonlinear impulsive response: percentage error on angular position q

Fig. 7

Comparison between linear and nonlinear impulsive response: elastic displacement u25 and absolute linearization error

Fig. 8

Frequency response of the linear model from input torque τ to output angular position q with ±20% uncertainty on mass density m

Fig. 9

Frequency response of the linear model from input torque τ to output u25 (elastic displacement) with ±20% uncertainty on mass density m

Fig. 10

Frequency response of the linear model from input torque τ to output angular position q with ±20% uncertainty on Young's modulus E

Fig. 11

Frequency response of the linear model from input torque τ to output u25 with ±20% uncertainty Young's modulus E

Fig. 12

Frequency response of the linear model from input torque τ to angular position q with ±20% uncertainty on elastic modulus E and mass density m

Fig. 13

Frequency response of the linear model from input torque τ to output u25 with ±20% uncertainty on Young's modulus E and mass density m

Fig. 14

Hankel singular values of the linear model

Fig. 15

Frequency response of the full and reduced-order model—system with torque as input and angular position q as output

Fig. 16

Frequency response of the full and reduced-order model—system with torque as input and nodal displacement u25 as output

Fig. 17

System description for general H∞ synthesis

Fig. 18

System description for H∞ loop shaping

Fig. 19

The basic framework used for μ-synthesis with emphasis on analysis and synthesis

Fig. 20

H∞ loop-shaping controller: step response of the angular position q with ±20% uncertainty on the elastic constant E and mass density m

Fig. 21

H∞ loop-shaping controller: step response of the elastic displacement u25 with ±20% uncertainty on the elastic constant E and mass density m

Fig. 22

External disturbance on angular position measurement

Fig. 23

H∞ loop-shaping controller: step response of the angular position q with parametric uncertainty and measurement disturbance

Fig. 24

H∞ loop-shaping controller: step response of the elastic displacement u25 with parametric uncertainty and measurement disturbance

Fig. 25

μ-synthesis controller: block diagram

Fig. 26

μ-synthesis controller: step response of the angular position q with ±20% uncertainty on elastic constant and mass density

Fig. 27

μ-synthesis controller: step response, elastic displacement u25 with ±20% uncertainty on the elastic constant and mass density

Fig. 28

μ-synthesis controller: step response of the angular position q with measurement disturbance and ±20% uncertainty on the elastic constant E and mass density m

Fig. 29

μ-synthesis controller: step response of the nodal displacement u25 with measurement disturbance and ±20% uncertainty on the elastic constant E and mass density m

Fig. 30

Robust performance of the synthesized controllers evaluated through μ-analysis

Fig. 31

Frequency response of the H∞ and μ-synthesis controllers

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