Elsevier

Advances in Space Research

Volume 56, Issue 10, 15 November 2015, Pages 2312-2322
Advances in Space Research

Back-stepping control of two-link flexible manipulator based on an extended state observer

https://doi.org/10.1016/j.asr.2015.07.036Get rights and content

Highlights

  • We study trajectory tracking control for a two-link flexible manipulator system.

  • An extended state observer is designed to estimate the uncertain variables.

  • A back-stepping controller is designed for the nonlinear manipulator system.

  • We prove convergence of extended state observer and the closed-loop system.

Abstract

In this paper, we consider trajectory tracking control of a two-link flexible manipulator model in space. Two variables of joint angle and elastic deformation are partly decoupled by a nonlinear decoupling feedback control method. An extended state observer is introduced to estimate nonlinear terms of the two-link flexible manipulator system. Based on a back-stepping method, a nonlinear controller is designed for the flexible manipulator system. Finally, some simulation results are given to demonstrate the effectiveness of the developed techniques in this paper.

Introduction

In recent years, many manual works are replaced by robots with the development of modern technology in medical, industrial production, military, aerospace industry and so on (Hua, 2011, Hua and Yang, 2012). As an important part of robotics, the anthropomorphic manipulators are used to carry out and move loads in specialized jobs (Hua, 2013). Due to deficient energy consumption and operation speed, steel manipulators are replaced by flexible manipulators gradually (Wai and Lee, 2004). To complete the tasks more accurately, modeling and control for a flexible manipulator have been received much attention recently (Zhang et al., 2005, Zhang and Liu, 2012). Real-time adaptive control has been exploited to deal with variable payloads for a two-link flexible manipulator (Pradhan and Subudhi, 2012). Moreover, some problems have been solved for controlling a two-link flexible manipulator with changeable payload at free-end (Zhang and Liu, 2013). Furthermore, an adaptive model predictive approach has been presented on tip position control for a flexible manipulator in Pradhan and Subudhi (2014). However, many difficulties in flexible manipulators are also overcome hardly, such as high nonlinearity, various uncertainties and elastic deformation (Pereira et al., 2010). Hence, there are a lot of space to be improved on this issue, which motivates us to make an effort in this paper.

Extended state observer (ESO) is an important part of active disturbance rejection control technology (Han, 2009). The ESO is not dependent on specific mathematical models of disturbances (Xia et al., 2012). The effects of disturbances also need not to be measured directly (Xia et al., 2014). These advantages of ESO make it to be suitable on estimating the nonlinearities of flexible manipulator systems. Thereby, all the properties of high nonlinearity, various uncertainties and elastic deformation are regarded as inner disturbances in flexible manipulators. The ESO estimates the inner and outer disturbances of uncertain systems via a special feedback mechanism. An ESO has been used for micro-electro-mechanical systems to deal with immeasurable internal dynamics and external disturbances (Zheng et al., 2009). The ESO has also been adopted to estimate uncertainty and external disturbance of spacecraft systems in Xia et al. (2011). The target acceleration for attitude control of a missile system has been investigated by an ESO in Xia et al., 2011, Zhu et al., 2013. On trajectory tracking control of a flexible-joint robotic system, an ESO is designed to estimate state vector and uncertainties in Talole et al. (2010). However, to the best of our knowledge, very few results are available on control of two-link flexible manipulator via an ESO. This problem is important and challenging in both theory and practice, which motivated us carry on this research work.

In this paper, a back-stepping controller is designed to achieve accurate trajectory tracking for a two-link flexible manipulator based on an ESO. Both nonlinearities and state variables are estimated for the flexible manipulator systems by taking the advantages of the ESO. The convergence of ESO is guaranteed using an approach of self-stable region. Some simulation results are presented to illustrate the effectiveness of the control scheme.

The remainder of this paper is organized as follows. Section 2 the relevant knowledge of modeling for a two-link flexible manipulator is presented. Section 3 an ESO is designed for the nonlinear system. The convergence of the ESO is demonstrated in Section 4. Section 5 an controller is designed by the back-stepping control. Simulation results are given in Section 6 and conclusion is given in Section 7.

The main objectives are as listed:

  • (i)

    This paper studies the trajectory tracking control for a two-link flexible manipulator system which contains nonlinear terms and uncertainties.

  • (ii)

    An extended state observer is designed to estimate the uncertain variables and a back-stepping approach is proposed to design controller for the nonlinear system.

  • (iii)

    Both the convergence of extended state observer and the stability of closed-loop system are proved by the method of self-stable region and back-stepping, respectively.

Notation: In the following, if not explicitly stated, matrices are assumed to have compatible dimensions. RN+M is the column vector with (N+M)-dimension. Note that sign(e) denotes a sign function on e, i.e.,sign(e)=-1,e<0.0,e=0.1,e>0.Moreover, sat(h) is a saturation function on h. It is satisfied with the following piecewise function:sat(h)=-1,h-1.h,h<1.1,h1.

Section snippets

Problem formulation

The structure diagram of a two-link flexible manipulator is given in Fig. 1.

The coordinate system of the flexible manipulator is chosen based on an imaginative rigid manipulator, please refer to Fig. 2.

In Fig. 2, ω1 and ω2 express the elastic deformations of L1 and L2, respectively. Because of the flexibility of links, deformation will appear in the process of movement. In order to analysis simplify, we just consider the elastic deformation. The axial deformation and shear deformation are

Design of extended state observer

In this paper, the ESO is used to estimate both internal dynamics and external disturbances in real time.

In system (5), assuming the nonlinear and uncertainties term -A-1(Bx1+C) is continuously differentiable and bounded. Letting -A-1(Bx1+C) be an extended state x3, system (5) is rewritten as followsy=x1ẋ1=x2ẋ2=x3+A-1τẋ3=f(t)where f(t) is the derivative of state x3. For system (6), a third-order nonlinear ESO is designed ase1=z1-yż1=z2-β1e1ż2=z3-β2fal(e1,α1,δ)+A-1τż3=-β3fal(e1,α2,δ)where z

Convergence of extended state observer

In order to analysis the convergence of the error system (8), the approach of self-stable region is introduced.

Definition 1

Assume R is a region in state space, and the origin is its vertex. If the region is satisfied with the condition that all state trajectories, which remain in it, will eventually converge to the origin after a certain time, then the region R is called a self-stable region of the system.

For the error system (8), assume the following two regions asR2={(e1,e2,e3):|h2(e1,e2)|g1(e1)}R3={(e1,

Back-stepping controller design

The control objective is to make the output of the joint angle θ to track the desired reference input.

Define the following error variablesγ1=x1d-x1γ2=σ1-x2where x1d is the given input, σ1 is a virtual control variable. Furthermore, the error system can be definedγ̇1=ẋ1d-x2γ̇2=σ̇1-x3-A-1τThe controller τ is given asτ=A(-z3+x¨1d+c1γ̇1+γ1+c2γ2).

Theorem 4

Consider the closed-loop system (10) with the error feedback controller (11). By choosing appropriate variables c1 and c2 large enough in controller (11),

Simulation results

In the following, we provide simulation results to demonstrate the effectiveness of the proposed methods in this paper. All the constant parameters of the two-link flexible manipulator model in Fig. 2 are given in Table 1.

According to the proposed results in Theorem 3, the parameters of the ESO are designed as followsβ1=500120,β2=1000001000,β3=100010.

Let sampling step length be h=0.01. In the two nonlinear functions fal(e,α,δ), choose α1=0.5,α2=0.25, and δ=h. According to the proposed results

Conclusion

In this paper, based on the analysis of the nonlinear system of two-link flexible manipulator, we make full use of the advantage of ESO to estimate the uncertain variables. A self-stable region approach has been adopted to prove the convergence of the ESO. Furthermore, the method of back-stepping control has been used to design the controller for the nonlinear system. Some simulation results demonstrate the feasibility of the method which is provided in this paper.

Acknowledgements

The authors would like to thank the anonymous reviewers for their detailed comments which helped to improve the quality of the paper. The work was supported by the National Natural Science Foundation of China under Grant 61203023 and 61573301, the Natural Science Foundation of Hebei Education Department under Grant Q2012060 and the Hebei Provincial Natural Science Fund under Grand F2013203092 and E2014203122.

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