Back-stepping control of two-link flexible manipulator based on an extended state observer
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. is the column vector with -dimension. Note that denotes a sign function on e, i.e.,Moreover, is a saturation function on h. It is satisfied with the following piecewise function:
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, and express the elastic deformations of and , 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 is continuously differentiable and bounded. Letting be an extended state , system (5) is rewritten as followswhere is the derivative of state . For system (6), a third-order nonlinear ESO is designed aswhere
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 as
Back-stepping controller design
The control objective is to make the output of the joint angle to track the desired reference input.
Define the following error variableswhere is the given input, is a virtual control variable. Furthermore, the error system can be definedThe controller is given as Theorem 4 Consider the closed-loop system (10) with the error feedback controller (11). By choosing appropriate variables and 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 follows
Let sampling step length be . In the two nonlinear functions , choose , and . 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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