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Back-stepping control of two-link flexible manipulator based on an extended state observer
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Back-stepping control of two-link flexible manipulator based on an extended state observer Hongjiu Yang a,⇑, Yang Yu a, Yuan Yuan b, Xiaozhao Fan a b
a Institute of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China School of Automation Science and Electrical Engineering, Beihang University, 100191, China
Received 21 March 2015; received in revised form 3 July 2015; accepted 27 July 2015
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. Ó 2015 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: Flexible manipulator; Trajectory tracking control; Extended state observer; Back-stepping control
1. 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 ⇑ Corresponding author. Tel.: +86 13488687266.
E-mail addresses: [email protected] (H. Yang), [email protected] (Y. Yu), [email protected] (Y. Yuan), [email protected] (X. Fan).
(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
http://dx.doi.org/10.1016/j.asr.2015.07.036 0273-1177/Ó 2015 COSPAR. Published by Elsevier Ltd. All rights reserved.
Please cite this article in press as: Yang, H., et al. Back-stepping control of two-link flexible manipulator based on an extended state observer. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.07.036
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H. Yang et al. / Advances in Space Research xxx (2015) xxx–xxx
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) and 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:
Moreover, satðhÞ is a saturation function on h. It is satisfied with the following piecewise function: 8 > < 1; h 6 1: satðhÞ ¼ h; jhj < 1: > : 1; h P 1:
2. 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, x1 and x2 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 omitted in the model. Then the link is modeled as an Euler–Bernoulli beam which is satisfied with the proper boundary conditions at the actuated joint. Based on vibration theory, the elastic deformation xi ðxi ; tÞ is described as: xi ðxi ; tÞ ¼
li X
/ij ðxi Þqij ðtÞ;
i ¼ 1; ; N
j¼1
where qij ðtÞ is a modal coefficient associated with the assumed spatial mode shape /ij ðxi Þ of link i. Accordingly, a set of Lagrangian generalized coordinates is given by ðh; qÞ 2 RN þM , where
(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 þM is the column vector with ðN þ MÞ-dimension. Note that signðeÞ denotes a sign function on e, i.e., 8 > < 1; e < 0: signðeÞ ¼ 0; e ¼ 0: > : 1; e > 0:
Fig. 1. Structure of the two-link flexible manipulator in space.
Please cite this article in press as: Yang, H., et al. Back-stepping control of two-link flexible manipulator based on an extended state observer. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.07.036
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3
Substituting (4) into (2), it is obtained that A€h þ Bh_ þ C ¼ s
ð5Þ
1 where A ¼ M 11 M 12 M 1 22 M 21 , B ¼ D11 M 12 M 22 D21 and 1 1 C ¼ ðD12 M 12 M 22 D22 Þq_ M 12 M 22 G2 þ G1 .
3. 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 A1 ðBx1 þ CÞ is continuously differentiable and bounded. Letting A1 ðBx1 þ CÞ be an extended state x3 , system (5) is rewritten as follows 8 y ¼ x1 > > > < x_ ¼ x 1 2 ð6Þ 1 > _ x ¼ x 3 þA s > > 2 : x_ 3 ¼ f ðtÞ
Fig. 2. Coordinate of the two-link flexible manipulator in space.
T
h ¼ ½ h1 ; . . . ; hN h iT q ¼ q11 ; . . . ; q1;l1 ; . . . ; qN 1;l1 ; . . . ; qN ;lN PN and hi denotes the ith link variable, M ¼ i¼1 li is the total number of flexible variables which are used to describe the robot deformation. Based on the Lagrange assumed mode method under the condition on small deflection (Li et al., 2005), the dynamic equation of the two-link flexible manipulator is written as " # " # € s h h_ M ð1Þ þD þG¼ 0 € q q_ T
where h ¼ ½h1 ; h2 is the vector of deflection angle, T T q ¼ ½q11 ; q12 ; q21 ; q22 is the modal coefficient, s ¼ ½s1 ; s2 is the joint torque vector, M is the generalized mass matrix which is positive definite, symmetric and time-varying, D is the matrix containing centrifugal and coriolis terms, G is the gravity matrix, and M 11 ðh; qÞ M 12 ðh; qÞ M 11 M 12 M¼ ¼ M 21 ðh; qÞ M 22 ðh; qÞ M 21 M 22 " # _ qÞ D12 ðh; h; _ qÞ D11 D12 D11 ðh; h; D¼ ¼ _ qÞ D22 ðh; h; _ qÞ D21 D22 D21 ðh; h; G1 ðh; qÞ G1 G¼ ¼ G2 ðh; qÞ G2
where f ðtÞ is the derivative of state x3 . For system (6), a third-order nonlinear ESO is designed as 8 e 1 ¼ z1 y > > > < z_ 1 ¼ z2 b1 e1 ð7Þ > z_ 2 ¼ z3 b2 falðe1 ; a1 ; dÞ þ A1 s > > : z_ 3 ¼ b3 falðe1 ; a2 ; dÞ where z1 ; z2 and z3 are states of observer (7), b1 > 0; b2 > 0 and b3 > 0 are adjustable parameters, falðe1 ; a1 ; dÞ and falðe1 ; a2 ; dÞ are nonlinear functions. The general organization on function falðe; a; dÞ is given as ( e jej 6 d 1a ; falðe; a; dÞ ¼ d a jej signðeÞ; jej > d where 0 < a < 1; d is a small number which is used to express the length of the linear part. Considering systems (6) and (7), we get the following error system as 8 e 1 ¼ z1 y > > > > < e_ 1 ¼ e2 b1 e1 ð8Þ > e_ 2 ¼ e3 b2 falðe1 ; a1 ; dÞ > > > : e_ 3 ¼ f ðtÞ b3 falðe1 ; a2 ; dÞ It is known that the point of balance is zero when f ðtÞ ¼ 0. That is, states z1 ; z2 and z3 in (7) are approximate to state x1 ; x2 and x3 in (6) after a period of time later.
Substituting M; D and G into (1), we have h þ M 12 € M 11 € q þ D11 h_ þ D12 q_ þ G1 ¼ s
ð2Þ
h þ M 22 € M 21 € q þ D21 h_ þ D22 q_ þ G2 ¼ 0
ð3Þ
Through (3) transformation, the following formula on €q is given as € _ € _ þ G2 q ¼ M 1 22 ½M 21 h þ D21 h þ D22 q
ð4Þ
4. 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
Please cite this article in press as: Yang, H., et al. Back-stepping control of two-link flexible manipulator based on an extended state observer. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.07.036
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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 R2 ¼ fðe1 ; e2 ; e3 Þ : jh2 ðe1 ; e2 Þj 6 g1 ðe1 Þg R3 ¼ fðe1 ; e2 ; e3 Þ : jh3 ðe1 ; e2 ; e3 Þj 6 g2 ðe1 ; e2 Þg where h2 ðe1 ; e2 Þ ¼ e2 b1 e1 þ k 1 g1 ðe1 Þsignðe1 Þ h3 ðe1 ; e2 ; e3 Þ ¼ e3 b2 falðe1 Þ b1 ðe2 b1 e1 Þ þk 2 g2 ðe1 ; e2 Þsatðh2 =g1 Þ in which g1 ðe1 Þ and g2 ðe1 ; e2 Þ are two arbitrary continuous positive definite functions with g1 ð0Þ ¼ 0 and g2 ð0; 0Þ ¼ 0. Note that both k 1 and k 2 are two constants which are satisfied with k 1 > 1 and k 2 > 1. For notational simplicity, let h2 ¼ h2 ðe1 ; e2 Þ, h3 ¼ h3 ðe1 ;e2 ;e3 Þ; g1 ¼ g1 ðe1 Þ, g2 ¼ g2 ðe1 ;e2 Þ; g1 s ¼ g1 ðe1 Þsignðe1 Þ and g2 s ¼ g2 ðe1 ;e2 Þsatðh2 =g1 Þ. Remark 1. In this paper, there exist two points which can’t be differentiable for function falðe; a; dÞ. Since the above factor doesn’t have an effect on the following theoretical _ 1 ¼ dfalðe1 ;a1 ;dÞ. Moreover, analysis, we define that fal de1 _ 1 > 0 means that falðe1 ; a1 ; dÞ is also a monotonous fal increasing function. Theorem 1. Considering the error system (8), the trajectories of e1 and e2 converge to the origin after a certain time, if ðe1 ; e2 Þ 2 R2 . That is, the observed states z1 and z2 converge to the actual states x1 and x2 , respectively. Proof 1. Letting V h2 g1 ¼ ðh22 g21 Þ=2, for the reason of ðe1 ; e2 Þ 2 R2 , we get h2 g 1 < 0 and g1 k 1 g1 s < e2 b1 e1 < g1 k 1 g1 s: Defining a continuous positive definite function as V 0 ¼ e21 =2, the derivative of V 0 is given as
2 ðk 1 þ 1Þ dg1 g2 > jh2 j k 2 1 de1 holds. That is, the observed states z1 ; z2 and z3 converge to the actual states x1 ; x2 and x3 , respectively. Proof 2. Defining V h3 g2 ¼ ðh23 g22 Þ=2, for the reason of ðe1 ; e2 ; e3 Þ 2 R3 , we get h3 < g 2 : The derivative of V h2 g1 is given as following dg s V_ h2 g1 ¼ h2 h_ 2 g1 g_ 1 ¼ h2 h3 k 2 g2 s þ k 1 1 ðh2 k 1 g1 sÞ de1 dg1 ðh2 k 1 g1 sÞ: g1 de1 As V h3 g2 < 0; V h2 g1 P 0, i.e., h3 k 2 g2 6 ðk 2 1Þg2 , there exists dg s V_ h2 g1 6 jh2 jðk 2 1Þg2 þ k 1 h2 1 ðh2 k 1 g1 sÞ de1 dg1 þ g1 ðh2 k 1 g1 sÞ de1 2 dg 1 2 6 jh2 jðk 2 1Þg2 þ h2 ðk 1 þ 1Þ jh2 j < 0: de1 So the trajectories of e1 and e2 converge into the region R2 . By Theorem 1, the trajectories converge to the origin ultimately. For the error system (8), if e1 ! 0 and e2 ! 0, then we have e3 ! 0 easily when f ðtÞ ¼ 0, i.e., z1 ! x1 ; z2 ! x2 and z3 ! x3 after a certain time. h Theorem 3. The trajectory of the error system (8) in the region is outside R3 . Letting g2 ¼
k 3 jh2 j;
jh2 j P g1
k 3 jg1 j; jh2 j < g1
2 dg1 where k 3 is a constant and k 3 > ðkk12þ1Þ if the following 1 de1 ,
V_ 0 ¼ e1 e_ 1
inequation
¼ e1 ðe2 b1 e1 Þ < e1 ðg1 k 1 g1 sÞ
_ 1 b fal dg s b1 > k 2 k 3 þ 2 b1 k 2 k 1 k 2 1 þ k 22 k 3 de1 k3 1 jfal2 j dg1 s @g2 2 _ þ b3 b2 k 1 fal1 þ k 1 k 2 k 3 þ k3 g1 de1 @e2 1 @g2 @g2 k 1 @g2 @g2 þ þ ðb1 k 2 k 3 Þ þ þ b1 k 3 @e1 @e2 k 3 @e1 @e2
< je1 jðk 1 1Þg1 < 0: Hence, the trajectories e1 and e2 converge to the origin after a certain time, i.e., z1 ! x1 and z2 ! x2 , respectively. h Theorem 2. Considering the error system (8), the trajectories of e1 ; e2 and e3 which stay in R3 and outside R2 converge to the origin after a certain time, if the following inequation
ð9Þ
holds, then the trajectories outside region R3 of system (8) converge into R3 .
Please cite this article in press as: Yang, H., et al. Back-stepping control of two-link flexible manipulator based on an extended state observer. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.07.036
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Proof 3. The derivative of V h3 g2 is given as follows
@h3 @h3 @h3 e_3 þ e_2 þ e_1 V_ h3 g2 ¼ h3 h_3 g2 g_2 ¼ h3 @e3 @e2 @e1 @g2 @g e_2 þ 2 e_1 g2 @e2 @e1 _ 1 e_1 b1 ðe_2 b1 e_1 Þ þ k 2 @g2 s e_1 þ @g2 s e_2 ¼ h3 e_3 b2 fal @e1 @e2 @g2 @g e_1 þ 2 e_2 g2 @e1 @e2 _ 1 ðh2 k 1 g1 sÞ b1 ðh3 k 2 g2 sÞ ¼ h3 b3 fal2 b2 fal @h2 @h2 ðh2 k 1 g1 sÞ þ ðh3 þ b1 ðh2 k 1 g1 sÞ k 2 g2 sÞ þk 2 k 3 e1 @e2 @g2 @g ðh2 k 1 g1 sÞ þ 2 ðh3 þ b1 ðh2 k 1 g1 sÞ k 2 g2 sÞ g2 @e1 @e2 _ 1 þ b1 k 2 k 3 þ k 1 k 2 k 3 g_ 1 s k 2 k 2 h2 ¼ h3 ðb1 þ k 2 k 3 Þh3 þ b2 fal 2 3 jfal2 j 2 _ _ þ b2 fal1 k 1 k 1 k 2 k 3 g1 s g1 s þ b3 g1 @g2 @g2 @g2 @g2 h2 g2 h3 þ þ b1 k2 k3 @e2 @e1 @e2 @e2 @g2 @g k 1 þ b1 2 g1 s : @e1 @e2
From V h3 g2 P 0, we have jh3 j P g2 P maxfk 3 jh2 j; k 3 g1 g. The following inequality is obtained as _ 1 þ k 1 k 2 k 3 g_ 1 s V_ h3 g2 6 jh3 j ðb1 þ k 2 k 3 Þjh3 j þ jb1 k 2 k 3 b2 fal jh3 j jfal2 j _ 1 k 1 k 2 k 2 k 3 g_ 1 s jh3 j k 22 k 23 j þ b3 þ b2 fal 1 k k3 g1 3 @g @g @g @g jh3 j þ jh3 j 2 jh3 j þ 2 þ b1 2 k 2 k 3 2 @e2 @e1 @e2 @e2 k 3 @g @g jh3 j þk 1 2 þ b1 2 : @e1 @e2 k 3 Then, we have V_ h3 g2 < 0, when b1 , b2 and b3 meet the condition as Theorem 3. Therefore, all the trajectories of the error system (8) converge into the region R3 . h Through the analysis of Theorem 1, Theorem 2 and Theorem 3, there exists appropriate parameters b1 , b2 and b3 such that V_ h3 g2 < 0 holds when ðe1 ; e2 ; e3 Þ – ð0; 0; 0Þ. Hence, R3 is the self-stable region of the error system (8), i.e., ðe1 ; e2 ; e3 Þ converge to the origin after a certain time. Therefore, if the proper parameters b1 ; b2 and b3 is satisfied with the inEq. (9), then the observed state z1 ; z2 and z3 converge to x1 , x2 and x3 , respectively. 5. Back-stepping controller design
5
c1 ¼ x1d x1 c 2 ¼ r1 x 2
where x1d is the given input, r1 is a virtual control variable. Furthermore, the error system can be defined c_ 1 ¼ x_ 1d x2 ð10Þ c_ 2 ¼ r_ 1 x3 A1 s The controller s is given as s ¼ Aðz3 þ €x1d þ c1 c_ 1 þ c1 þ c2 c2 Þ:
ð11Þ
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), the closed-loop system (10) is stable. That is, the output of the joint angle h converge to the desired reference input hd . Proof 4. We design a back-stepping controller for system (10) as follows: Step 1: Select the first Lyapunov function as follows 1 V 1 ¼ c21 : 2 The derivative of the first Lyapunov function is given as V_ 1 ¼ c1 c_ 1 ¼ c1 ð_x1d x2 Þ ¼ c1 ð_x1d r1 þ c2 Þ ¼ c1 x_ 1d c1 r1 þ c1 c2
ð12Þ
Design the virtual control as r1 ¼ x_ 1d þ c1 c1
ð13Þ
where c1 > 0 is a constant design parameter. Substituting (13) into (12), we have V_ 1 ¼ c1 c21 þ c1 c2 : If c2 ¼ 0, then the first subsystem is stable. Step 2: Select the second Lyapunov function as follows 1 V 2 ¼ V 1 þ c22 : 2 The derivative of the second Lyapunov function is given as V_ 2 ¼ c1 c2 c1 c21 þ c2 c_ 2 ¼ c1 c21 þ c1 c2 þ c2 ðr_1 x_ 2 Þ ¼ c1 c21 þ c1 c2 þ c2 ðr_1 x3 A1 sÞ: Then design the controller as s ¼ Aðz3 þ r_1 þ c1 þ c2 c2 Þ
The control objective is to make the output of the joint angle h to track the desired reference input. Define the following error variables
then we have V_ 2 ¼ c1 c21 c2 c22 þ c2 e3 :
Please cite this article in press as: Yang, H., et al. Back-stepping control of two-link flexible manipulator based on an extended state observer. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.07.036
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For the reason of that the ESO (7) estimates the states of the system effectively, we assume e3 0. Moreover, the error c2 is bounded in practical. Therefore, we get V_ 2 < 0 if let c1 and c2 be large enough. That is, by choosing appropriate variables c1 and c2 , system (10) is stable. h
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 5 0 1000 0 10 0 b1 ¼ ; b2 ¼ ; b3 ¼ : 0 120 0 1000 0 10
6. Simulation results
Let sampling step length be h ¼ 0:01. In the two nonlinear functions falðe; a; dÞ, choose a1 ¼ 0:5; a2 ¼ 0:25, and d ¼ h. According to the proposed results in Theorem 4, the parameters of the back-stepping controller are designed as follows 170 0 170 0 c1 ¼ ; c2 ¼ 0 100 0 100
In the following, we provide simulation results to demonstrate the effectiveness of the proposed methods in Table 1 The parameters in the two-link flexible manipulator model. Parameters
First link
Second link
Length of link Rotational inertia Density Stiffness
L1 ¼ 1:1 m J 1 ¼ 1:0 Kg m2 q1 ¼ 2:0 Kg=m EI 1 ¼ 2 N m2
L2 ¼ 2:0 m J 2 ¼ 0:1 Kg m2 q2 ¼ 1:0 Kg=m EI 2 ¼ 2 N m2
The given angle trajectories of the two-link manipulator are oscillation curve and cosine curve, respectively. The tracking trajectories and estimations of the proposed control system are shown in Figs. 3 and 4. Figs. 5 and 6 show the tracking errors of the two links. The error curves
2 θ
1d
θ
1
1
θ1 of ESO
Angles
0
−1
−2
−3
−4
0
50
100
150
Time Fig. 3. Trajectory tracking and estimation of h1 .
25 θ2d θ
20
2
θ2 of ESO
Angles
15
10
5
0
−5
−10
0
50
Time
100
150
Fig. 4. Trajectory tracking and estimation of h2 .
Please cite this article in press as: Yang, H., et al. Back-stepping control of two-link flexible manipulator based on an extended state observer. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.07.036
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7
0.25 θ1d−θ1
0.2 0.15 0.1
Angle
0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25
0
50
100
150
Time Fig. 5. Tracking error of h1 .
10 θ
−θ
2d
2
8
Angle
6
4
2
0
−2
0
50
100
150
Time Fig. 6. Tracking error of h2 .
0.15
0.1
0
z
31
0.05
−0.05
−0.1
−0.15
−0.2 0
50
100
150
Time Fig. 7. Estimation of the first link.
Please cite this article in press as: Yang, H., et al. Back-stepping control of two-link flexible manipulator based on an extended state observer. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.07.036
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H. Yang et al. / Advances in Space Research xxx (2015) xxx–xxx 0.4 0.3 0.2
z
32
0.1 0 −0.1 −0.2 −0.3 −0.4
0
50
100
150
Time Fig. 8. Estimation of the second link.
6000
4000
τ1
2000
0
−2000
−4000
−6000
0
50
100
150
Time Fig. 9. Control input s1 of the first link.
1
x 10
5
0.5
τ
2
0 1000 −0.5 0 −1
−1.5
−1000
0
20
40
60
80
50
100
100
120
140
150
Time Fig. 10. Control input s2 of the second link.
demonstrate the effective of our control strategy. The nonlinear terms x3 of the system (5) are estimated by ESO as shown in Fig. 7 and Fig. 8. In addition, the signals of con-
trol inputs are shown in Figs. 9 and 10. The simulation results show the convergence of ESO and that the output angles track the trajectories of the input values accurately.
Please cite this article in press as: Yang, H., et al. Back-stepping control of two-link flexible manipulator based on an extended state observer. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.07.036
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9
8 θ
1d
θ1
6 4
Angles
2 0 −2 −4 −6 −8 −10
0
50
100
150
Time Fig. 11. Trajectory tracking of h1 .
15 θ
2d
θ
2
10
Angles
5
0
−5
−10
−15
0
50
100
150
Time Fig. 12. Trajectory tracking of h2 .
1.5
x 10
4
1
τ1
0.5
0
−0.5
−1
−1.5
0
50
100
150
Time Fig. 13. Control input s1 of the first link.
In order to illustrate the effectiveness of the control approach which we designed, PD is introduced. The results of the tracking for PD have been shown as Figs. 11 and 12. The corresponding signals of control inputs are shown in Figs. 13 and 14. There appears phase difference in Figs. 11
and 12, i.e., PD can not tracking the reference quickly and accurately. In order to further verify the rapidity and accuracy of our strategy, a same reference of step signal is given to links. The simulation results of tracking as shown
Please cite this article in press as: Yang, H., et al. Back-stepping control of two-link flexible manipulator based on an extended state observer. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.07.036
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H. Yang et al. / Advances in Space Research xxx (2015) xxx–xxx 12000 10000 8000 6000
τ2
4000 2000 0 −2000 −4000 −6000
0
50
100
150
Time
Fig. 14. Control input s2 of the second link.
12 θd Back−stepping PD
10
Angles
8
6
4
2
0
0
50
100
150
Time Fig. 15. Trajectory tracking of h. 10 Back−stepping PD 8
Angle
6
4
2
0
−2
0
50
100
150
Time Fig. 16. Trajectory tracking error of h.
Fig. 15. The errors are also shown in Fig. 16. In addition, the control inputs are shown in Fig. 17. Remark 2. When reference is a oscillation curve, PD controller can not tracking it precisely as shown in Figs. 11 and 12. When reference is a step curve, there are 2 3 s for reaching to steady state for the control strategy in this
paper without overshoot, but there are about 20 25 s for reaching to steady state with about 1 overshoot for PD control shown as in Fig. 15. The comparison of the two approaches indicates that the tracking effect are better based on the ESO and back-stepping controller than PD controller.
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1.5
x 10
11
5
Back−stepping PD 1
τ
0.5
0
−0.5
−1
0
50
100
150
Time Fig. 17. Control input s.
7. 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. References Han, J., 2009. From PID to active disturbance rejectioncontrol. IEEE Trans. Ind. Electron. 56, 900–906. Hua, C., 2011. Teleoperation over the internet with/without velocity signal. IEEE Trans. Instrum. Meas. 60, 4–13. Hua, C., 2013. A new coordinated slave torque feedback control algorithm for network-based teleoperation systems. IEEE/ASME Trans. Mechatron. 18, 764–774. Hua, C., Yang, Y., 2012. Bilateral teleoperation design with/without gravity measurement. IEEE Trans. Instrum. Meas. 61, 3136–3146. Li, Y., Liu, G., Hong, T., Liu, K., 2005. Robust control of a two-link flexible manipulator with quasi-static deflection compensation using neural networks. Intell. Rob. Syst. 44, 263–276.
Pereira, E., Aphale, S.S., Feliu, V., Moheimani, S.O.R., 2010. Integral resonant control for vibration damping and precise tip-positioning of a single-link flexible manipulator. IEEE/ASNE Trans. Mechatron. 16, 232–240. Pradhan, S.K., Subudhi, B., 2012. Real-time adaptive control of a flexible manipulator using reinforcement learning. IEEE Trans. Autom. Sci. Eng. 9, 237–249. Pradhan, S.K., Subudhi, B., 2014. Nonlinear adaptive model predictive controller for a flexible manipulator. IEEE Trans. Control Syst. Technol. 22, 1754–1768. Talole, S.E., Kolhe, J.P., Phadke, S.E., 2010. Extend-state-observer-based control of flexible-joint system with experimental validation. IEEE Trans. Ind. Electron. 57, 1411–1419. Wai, R., Lee, M., 2004. Intelligent optimal control of single-link flexible robot arm. IEEE Trans. Ind. Electron. 51, 201–220. Xia, Y., Zhu, Z., Fu, M., Wang, S., 2011. Attitude tracking of rigid spacecraft with bounded disturbances. IEEE Trans. Ind. Electron. 58, 647–659. Xia, Y., Zhu, Z., Fu, M., 2011. Back-stepping sliding mode control for missile systems based on an extended state observer. IET Control Theory Appl. 5, 93–102. Xia, Y., Liu, B., Fu, M., 2012. Active disturbance rejection control for power plant with a single loop. Asian J. Control 14, 239–250. Xia, Y., Chen, R., Pu, F., Dai, L., 2014. Active disturbance rejection control for drag tracking in mars entry guidance. Adv. Space Res. 53, 853–861. Zhang, L., Liu, J., 2012. Observe-based partial differential equation boundary control for a flexible two-link manipulator in task space. IET Control Theory Appl. 6, 2120–2133. Zhang, L., Liu, J., 2013. Adaptive boundary control for flexible two-link manipulator based on partial differential equation dynamic model. IET Control Theory Appl. 7, 43–51. Zhang, X., Xu, W., Nair, S.S., Chellaboina, V., 2005. PDE modeling and control of a flexible two-link manipulator. IEEE Trans. Control Syst. Technol. 13, 301–312. Zheng, Q., Dong, L., Lee, D.H., Gao, Z., 2009. Active disturbance rejection control for MEMS gyroscopes. IEEE Trans. Control Syst. Technol. 17, 1432–1438. Zhu, Z., Xu, D., Liu, J., Xia, Y., 2013. Missile guidance law based on extended state observer. IEEE Trans. Ind. Electron. 60, 5882–5891.
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ScienceDirect Advances in Space Research xxx (2015) xxx–xxx www.elsevier.com/locate/asr
Back-stepping control of two-link flexible manipulator based on an extended state observer Hongjiu Yang a,⇑, Yang Yu a, Yuan Yuan b, Xiaozhao Fan a b
a Institute of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China School of Automation Science and Electrical Engineering, Beihang University, 100191, China
Received 21 March 2015; received in revised form 3 July 2015; accepted 27 July 2015
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. Ó 2015 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: Flexible manipulator; Trajectory tracking control; Extended state observer; Back-stepping control
1. 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 ⇑ Corresponding author. Tel.: +86 13488687266.
E-mail addresses: [email protected] (H. Yang), [email protected] (Y. Yu), [email protected] (Y. Yuan), [email protected] (X. Fan).
(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
http://dx.doi.org/10.1016/j.asr.2015.07.036 0273-1177/Ó 2015 COSPAR. Published by Elsevier Ltd. All rights reserved.
Please cite this article in press as: Yang, H., et al. Back-stepping control of two-link flexible manipulator based on an extended state observer. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.07.036
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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) and 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:
Moreover, satðhÞ is a saturation function on h. It is satisfied with the following piecewise function: 8 > < 1; h 6 1: satðhÞ ¼ h; jhj < 1: > : 1; h P 1:
2. 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, x1 and x2 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 omitted in the model. Then the link is modeled as an Euler–Bernoulli beam which is satisfied with the proper boundary conditions at the actuated joint. Based on vibration theory, the elastic deformation xi ðxi ; tÞ is described as: xi ðxi ; tÞ ¼
li X
/ij ðxi Þqij ðtÞ;
i ¼ 1; ; N
j¼1
where qij ðtÞ is a modal coefficient associated with the assumed spatial mode shape /ij ðxi Þ of link i. Accordingly, a set of Lagrangian generalized coordinates is given by ðh; qÞ 2 RN þM , where
(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 þM is the column vector with ðN þ MÞ-dimension. Note that signðeÞ denotes a sign function on e, i.e., 8 > < 1; e < 0: signðeÞ ¼ 0; e ¼ 0: > : 1; e > 0:
Fig. 1. Structure of the two-link flexible manipulator in space.
Please cite this article in press as: Yang, H., et al. Back-stepping control of two-link flexible manipulator based on an extended state observer. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.07.036
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3
Substituting (4) into (2), it is obtained that A€h þ Bh_ þ C ¼ s
ð5Þ
1 where A ¼ M 11 M 12 M 1 22 M 21 , B ¼ D11 M 12 M 22 D21 and 1 1 C ¼ ðD12 M 12 M 22 D22 Þq_ M 12 M 22 G2 þ G1 .
3. 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 A1 ðBx1 þ CÞ is continuously differentiable and bounded. Letting A1 ðBx1 þ CÞ be an extended state x3 , system (5) is rewritten as follows 8 y ¼ x1 > > > < x_ ¼ x 1 2 ð6Þ 1 > _ x ¼ x 3 þA s > > 2 : x_ 3 ¼ f ðtÞ
Fig. 2. Coordinate of the two-link flexible manipulator in space.
T
h ¼ ½ h1 ; . . . ; hN h iT q ¼ q11 ; . . . ; q1;l1 ; . . . ; qN 1;l1 ; . . . ; qN ;lN PN and hi denotes the ith link variable, M ¼ i¼1 li is the total number of flexible variables which are used to describe the robot deformation. Based on the Lagrange assumed mode method under the condition on small deflection (Li et al., 2005), the dynamic equation of the two-link flexible manipulator is written as " # " # € s h h_ M ð1Þ þD þG¼ 0 € q q_ T
where h ¼ ½h1 ; h2 is the vector of deflection angle, T T q ¼ ½q11 ; q12 ; q21 ; q22 is the modal coefficient, s ¼ ½s1 ; s2 is the joint torque vector, M is the generalized mass matrix which is positive definite, symmetric and time-varying, D is the matrix containing centrifugal and coriolis terms, G is the gravity matrix, and M 11 ðh; qÞ M 12 ðh; qÞ M 11 M 12 M¼ ¼ M 21 ðh; qÞ M 22 ðh; qÞ M 21 M 22 " # _ qÞ D12 ðh; h; _ qÞ D11 D12 D11 ðh; h; D¼ ¼ _ qÞ D22 ðh; h; _ qÞ D21 D22 D21 ðh; h; G1 ðh; qÞ G1 G¼ ¼ G2 ðh; qÞ G2
where f ðtÞ is the derivative of state x3 . For system (6), a third-order nonlinear ESO is designed as 8 e 1 ¼ z1 y > > > < z_ 1 ¼ z2 b1 e1 ð7Þ > z_ 2 ¼ z3 b2 falðe1 ; a1 ; dÞ þ A1 s > > : z_ 3 ¼ b3 falðe1 ; a2 ; dÞ where z1 ; z2 and z3 are states of observer (7), b1 > 0; b2 > 0 and b3 > 0 are adjustable parameters, falðe1 ; a1 ; dÞ and falðe1 ; a2 ; dÞ are nonlinear functions. The general organization on function falðe; a; dÞ is given as ( e jej 6 d 1a ; falðe; a; dÞ ¼ d a jej signðeÞ; jej > d where 0 < a < 1; d is a small number which is used to express the length of the linear part. Considering systems (6) and (7), we get the following error system as 8 e 1 ¼ z1 y > > > > < e_ 1 ¼ e2 b1 e1 ð8Þ > e_ 2 ¼ e3 b2 falðe1 ; a1 ; dÞ > > > : e_ 3 ¼ f ðtÞ b3 falðe1 ; a2 ; dÞ It is known that the point of balance is zero when f ðtÞ ¼ 0. That is, states z1 ; z2 and z3 in (7) are approximate to state x1 ; x2 and x3 in (6) after a period of time later.
Substituting M; D and G into (1), we have h þ M 12 € M 11 € q þ D11 h_ þ D12 q_ þ G1 ¼ s
ð2Þ
h þ M 22 € M 21 € q þ D21 h_ þ D22 q_ þ G2 ¼ 0
ð3Þ
Through (3) transformation, the following formula on €q is given as € _ € _ þ G2 q ¼ M 1 22 ½M 21 h þ D21 h þ D22 q
ð4Þ
4. 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
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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 R2 ¼ fðe1 ; e2 ; e3 Þ : jh2 ðe1 ; e2 Þj 6 g1 ðe1 Þg R3 ¼ fðe1 ; e2 ; e3 Þ : jh3 ðe1 ; e2 ; e3 Þj 6 g2 ðe1 ; e2 Þg where h2 ðe1 ; e2 Þ ¼ e2 b1 e1 þ k 1 g1 ðe1 Þsignðe1 Þ h3 ðe1 ; e2 ; e3 Þ ¼ e3 b2 falðe1 Þ b1 ðe2 b1 e1 Þ þk 2 g2 ðe1 ; e2 Þsatðh2 =g1 Þ in which g1 ðe1 Þ and g2 ðe1 ; e2 Þ are two arbitrary continuous positive definite functions with g1 ð0Þ ¼ 0 and g2 ð0; 0Þ ¼ 0. Note that both k 1 and k 2 are two constants which are satisfied with k 1 > 1 and k 2 > 1. For notational simplicity, let h2 ¼ h2 ðe1 ; e2 Þ, h3 ¼ h3 ðe1 ;e2 ;e3 Þ; g1 ¼ g1 ðe1 Þ, g2 ¼ g2 ðe1 ;e2 Þ; g1 s ¼ g1 ðe1 Þsignðe1 Þ and g2 s ¼ g2 ðe1 ;e2 Þsatðh2 =g1 Þ. Remark 1. In this paper, there exist two points which can’t be differentiable for function falðe; a; dÞ. Since the above factor doesn’t have an effect on the following theoretical _ 1 ¼ dfalðe1 ;a1 ;dÞ. Moreover, analysis, we define that fal de1 _ 1 > 0 means that falðe1 ; a1 ; dÞ is also a monotonous fal increasing function. Theorem 1. Considering the error system (8), the trajectories of e1 and e2 converge to the origin after a certain time, if ðe1 ; e2 Þ 2 R2 . That is, the observed states z1 and z2 converge to the actual states x1 and x2 , respectively. Proof 1. Letting V h2 g1 ¼ ðh22 g21 Þ=2, for the reason of ðe1 ; e2 Þ 2 R2 , we get h2 g 1 < 0 and g1 k 1 g1 s < e2 b1 e1 < g1 k 1 g1 s: Defining a continuous positive definite function as V 0 ¼ e21 =2, the derivative of V 0 is given as
2 ðk 1 þ 1Þ dg1 g2 > jh2 j k 2 1 de1 holds. That is, the observed states z1 ; z2 and z3 converge to the actual states x1 ; x2 and x3 , respectively. Proof 2. Defining V h3 g2 ¼ ðh23 g22 Þ=2, for the reason of ðe1 ; e2 ; e3 Þ 2 R3 , we get h3 < g 2 : The derivative of V h2 g1 is given as following dg s V_ h2 g1 ¼ h2 h_ 2 g1 g_ 1 ¼ h2 h3 k 2 g2 s þ k 1 1 ðh2 k 1 g1 sÞ de1 dg1 ðh2 k 1 g1 sÞ: g1 de1 As V h3 g2 < 0; V h2 g1 P 0, i.e., h3 k 2 g2 6 ðk 2 1Þg2 , there exists dg s V_ h2 g1 6 jh2 jðk 2 1Þg2 þ k 1 h2 1 ðh2 k 1 g1 sÞ de1 dg1 þ g1 ðh2 k 1 g1 sÞ de1 2 dg 1 2 6 jh2 jðk 2 1Þg2 þ h2 ðk 1 þ 1Þ jh2 j < 0: de1 So the trajectories of e1 and e2 converge into the region R2 . By Theorem 1, the trajectories converge to the origin ultimately. For the error system (8), if e1 ! 0 and e2 ! 0, then we have e3 ! 0 easily when f ðtÞ ¼ 0, i.e., z1 ! x1 ; z2 ! x2 and z3 ! x3 after a certain time. h Theorem 3. The trajectory of the error system (8) in the region is outside R3 . Letting g2 ¼
k 3 jh2 j;
jh2 j P g1
k 3 jg1 j; jh2 j < g1
2 dg1 where k 3 is a constant and k 3 > ðkk12þ1Þ if the following 1 de1 ,
V_ 0 ¼ e1 e_ 1
inequation
¼ e1 ðe2 b1 e1 Þ < e1 ðg1 k 1 g1 sÞ
_ 1 b fal dg s b1 > k 2 k 3 þ 2 b1 k 2 k 1 k 2 1 þ k 22 k 3 de1 k3 1 jfal2 j dg1 s @g2 2 _ þ b3 b2 k 1 fal1 þ k 1 k 2 k 3 þ k3 g1 de1 @e2 1 @g2 @g2 k 1 @g2 @g2 þ þ ðb1 k 2 k 3 Þ þ þ b1 k 3 @e1 @e2 k 3 @e1 @e2
< je1 jðk 1 1Þg1 < 0: Hence, the trajectories e1 and e2 converge to the origin after a certain time, i.e., z1 ! x1 and z2 ! x2 , respectively. h Theorem 2. Considering the error system (8), the trajectories of e1 ; e2 and e3 which stay in R3 and outside R2 converge to the origin after a certain time, if the following inequation
ð9Þ
holds, then the trajectories outside region R3 of system (8) converge into R3 .
Please cite this article in press as: Yang, H., et al. Back-stepping control of two-link flexible manipulator based on an extended state observer. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.07.036
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Proof 3. The derivative of V h3 g2 is given as follows
@h3 @h3 @h3 e_3 þ e_2 þ e_1 V_ h3 g2 ¼ h3 h_3 g2 g_2 ¼ h3 @e3 @e2 @e1 @g2 @g e_2 þ 2 e_1 g2 @e2 @e1 _ 1 e_1 b1 ðe_2 b1 e_1 Þ þ k 2 @g2 s e_1 þ @g2 s e_2 ¼ h3 e_3 b2 fal @e1 @e2 @g2 @g e_1 þ 2 e_2 g2 @e1 @e2 _ 1 ðh2 k 1 g1 sÞ b1 ðh3 k 2 g2 sÞ ¼ h3 b3 fal2 b2 fal @h2 @h2 ðh2 k 1 g1 sÞ þ ðh3 þ b1 ðh2 k 1 g1 sÞ k 2 g2 sÞ þk 2 k 3 e1 @e2 @g2 @g ðh2 k 1 g1 sÞ þ 2 ðh3 þ b1 ðh2 k 1 g1 sÞ k 2 g2 sÞ g2 @e1 @e2 _ 1 þ b1 k 2 k 3 þ k 1 k 2 k 3 g_ 1 s k 2 k 2 h2 ¼ h3 ðb1 þ k 2 k 3 Þh3 þ b2 fal 2 3 jfal2 j 2 _ _ þ b2 fal1 k 1 k 1 k 2 k 3 g1 s g1 s þ b3 g1 @g2 @g2 @g2 @g2 h2 g2 h3 þ þ b1 k2 k3 @e2 @e1 @e2 @e2 @g2 @g k 1 þ b1 2 g1 s : @e1 @e2
From V h3 g2 P 0, we have jh3 j P g2 P maxfk 3 jh2 j; k 3 g1 g. The following inequality is obtained as _ 1 þ k 1 k 2 k 3 g_ 1 s V_ h3 g2 6 jh3 j ðb1 þ k 2 k 3 Þjh3 j þ jb1 k 2 k 3 b2 fal jh3 j jfal2 j _ 1 k 1 k 2 k 2 k 3 g_ 1 s jh3 j k 22 k 23 j þ b3 þ b2 fal 1 k k3 g1 3 @g @g @g @g jh3 j þ jh3 j 2 jh3 j þ 2 þ b1 2 k 2 k 3 2 @e2 @e1 @e2 @e2 k 3 @g @g jh3 j þk 1 2 þ b1 2 : @e1 @e2 k 3 Then, we have V_ h3 g2 < 0, when b1 , b2 and b3 meet the condition as Theorem 3. Therefore, all the trajectories of the error system (8) converge into the region R3 . h Through the analysis of Theorem 1, Theorem 2 and Theorem 3, there exists appropriate parameters b1 , b2 and b3 such that V_ h3 g2 < 0 holds when ðe1 ; e2 ; e3 Þ – ð0; 0; 0Þ. Hence, R3 is the self-stable region of the error system (8), i.e., ðe1 ; e2 ; e3 Þ converge to the origin after a certain time. Therefore, if the proper parameters b1 ; b2 and b3 is satisfied with the inEq. (9), then the observed state z1 ; z2 and z3 converge to x1 , x2 and x3 , respectively. 5. Back-stepping controller design
5
c1 ¼ x1d x1 c 2 ¼ r1 x 2
where x1d is the given input, r1 is a virtual control variable. Furthermore, the error system can be defined c_ 1 ¼ x_ 1d x2 ð10Þ c_ 2 ¼ r_ 1 x3 A1 s The controller s is given as s ¼ Aðz3 þ €x1d þ c1 c_ 1 þ c1 þ c2 c2 Þ:
ð11Þ
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), the closed-loop system (10) is stable. That is, the output of the joint angle h converge to the desired reference input hd . Proof 4. We design a back-stepping controller for system (10) as follows: Step 1: Select the first Lyapunov function as follows 1 V 1 ¼ c21 : 2 The derivative of the first Lyapunov function is given as V_ 1 ¼ c1 c_ 1 ¼ c1 ð_x1d x2 Þ ¼ c1 ð_x1d r1 þ c2 Þ ¼ c1 x_ 1d c1 r1 þ c1 c2
ð12Þ
Design the virtual control as r1 ¼ x_ 1d þ c1 c1
ð13Þ
where c1 > 0 is a constant design parameter. Substituting (13) into (12), we have V_ 1 ¼ c1 c21 þ c1 c2 : If c2 ¼ 0, then the first subsystem is stable. Step 2: Select the second Lyapunov function as follows 1 V 2 ¼ V 1 þ c22 : 2 The derivative of the second Lyapunov function is given as V_ 2 ¼ c1 c2 c1 c21 þ c2 c_ 2 ¼ c1 c21 þ c1 c2 þ c2 ðr_1 x_ 2 Þ ¼ c1 c21 þ c1 c2 þ c2 ðr_1 x3 A1 sÞ: Then design the controller as s ¼ Aðz3 þ r_1 þ c1 þ c2 c2 Þ
The control objective is to make the output of the joint angle h to track the desired reference input. Define the following error variables
then we have V_ 2 ¼ c1 c21 c2 c22 þ c2 e3 :
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For the reason of that the ESO (7) estimates the states of the system effectively, we assume e3 0. Moreover, the error c2 is bounded in practical. Therefore, we get V_ 2 < 0 if let c1 and c2 be large enough. That is, by choosing appropriate variables c1 and c2 , system (10) is stable. h
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 5 0 1000 0 10 0 b1 ¼ ; b2 ¼ ; b3 ¼ : 0 120 0 1000 0 10
6. Simulation results
Let sampling step length be h ¼ 0:01. In the two nonlinear functions falðe; a; dÞ, choose a1 ¼ 0:5; a2 ¼ 0:25, and d ¼ h. According to the proposed results in Theorem 4, the parameters of the back-stepping controller are designed as follows 170 0 170 0 c1 ¼ ; c2 ¼ 0 100 0 100
In the following, we provide simulation results to demonstrate the effectiveness of the proposed methods in Table 1 The parameters in the two-link flexible manipulator model. Parameters
First link
Second link
Length of link Rotational inertia Density Stiffness
L1 ¼ 1:1 m J 1 ¼ 1:0 Kg m2 q1 ¼ 2:0 Kg=m EI 1 ¼ 2 N m2
L2 ¼ 2:0 m J 2 ¼ 0:1 Kg m2 q2 ¼ 1:0 Kg=m EI 2 ¼ 2 N m2
The given angle trajectories of the two-link manipulator are oscillation curve and cosine curve, respectively. The tracking trajectories and estimations of the proposed control system are shown in Figs. 3 and 4. Figs. 5 and 6 show the tracking errors of the two links. The error curves
2 θ
1d
θ
1
1
θ1 of ESO
Angles
0
−1
−2
−3
−4
0
50
100
150
Time Fig. 3. Trajectory tracking and estimation of h1 .
25 θ2d θ
20
2
θ2 of ESO
Angles
15
10
5
0
−5
−10
0
50
Time
100
150
Fig. 4. Trajectory tracking and estimation of h2 .
Please cite this article in press as: Yang, H., et al. Back-stepping control of two-link flexible manipulator based on an extended state observer. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.07.036
H. Yang et al. / Advances in Space Research xxx (2015) xxx–xxx
7
0.25 θ1d−θ1
0.2 0.15 0.1
Angle
0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25
0
50
100
150
Time Fig. 5. Tracking error of h1 .
10 θ
−θ
2d
2
8
Angle
6
4
2
0
−2
0
50
100
150
Time Fig. 6. Tracking error of h2 .
0.15
0.1
0
z
31
0.05
−0.05
−0.1
−0.15
−0.2 0
50
100
150
Time Fig. 7. Estimation of the first link.
Please cite this article in press as: Yang, H., et al. Back-stepping control of two-link flexible manipulator based on an extended state observer. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.07.036
8
H. Yang et al. / Advances in Space Research xxx (2015) xxx–xxx 0.4 0.3 0.2
z
32
0.1 0 −0.1 −0.2 −0.3 −0.4
0
50
100
150
Time Fig. 8. Estimation of the second link.
6000
4000
τ1
2000
0
−2000
−4000
−6000
0
50
100
150
Time Fig. 9. Control input s1 of the first link.
1
x 10
5
0.5
τ
2
0 1000 −0.5 0 −1
−1.5
−1000
0
20
40
60
80
50
100
100
120
140
150
Time Fig. 10. Control input s2 of the second link.
demonstrate the effective of our control strategy. The nonlinear terms x3 of the system (5) are estimated by ESO as shown in Fig. 7 and Fig. 8. In addition, the signals of con-
trol inputs are shown in Figs. 9 and 10. The simulation results show the convergence of ESO and that the output angles track the trajectories of the input values accurately.
Please cite this article in press as: Yang, H., et al. Back-stepping control of two-link flexible manipulator based on an extended state observer. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.07.036
H. Yang et al. / Advances in Space Research xxx (2015) xxx–xxx
9
8 θ
1d
θ1
6 4
Angles
2 0 −2 −4 −6 −8 −10
0
50
100
150
Time Fig. 11. Trajectory tracking of h1 .
15 θ
2d
θ
2
10
Angles
5
0
−5
−10
−15
0
50
100
150
Time Fig. 12. Trajectory tracking of h2 .
1.5
x 10
4
1
τ1
0.5
0
−0.5
−1
−1.5
0
50
100
150
Time Fig. 13. Control input s1 of the first link.
In order to illustrate the effectiveness of the control approach which we designed, PD is introduced. The results of the tracking for PD have been shown as Figs. 11 and 12. The corresponding signals of control inputs are shown in Figs. 13 and 14. There appears phase difference in Figs. 11
and 12, i.e., PD can not tracking the reference quickly and accurately. In order to further verify the rapidity and accuracy of our strategy, a same reference of step signal is given to links. The simulation results of tracking as shown
Please cite this article in press as: Yang, H., et al. Back-stepping control of two-link flexible manipulator based on an extended state observer. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.07.036
10
H. Yang et al. / Advances in Space Research xxx (2015) xxx–xxx 12000 10000 8000 6000
τ2
4000 2000 0 −2000 −4000 −6000
0
50
100
150
Time
Fig. 14. Control input s2 of the second link.
12 θd Back−stepping PD
10
Angles
8
6
4
2
0
0
50
100
150
Time Fig. 15. Trajectory tracking of h. 10 Back−stepping PD 8
Angle
6
4
2
0
−2
0
50
100
150
Time Fig. 16. Trajectory tracking error of h.
Fig. 15. The errors are also shown in Fig. 16. In addition, the control inputs are shown in Fig. 17. Remark 2. When reference is a oscillation curve, PD controller can not tracking it precisely as shown in Figs. 11 and 12. When reference is a step curve, there are 2 3 s for reaching to steady state for the control strategy in this
paper without overshoot, but there are about 20 25 s for reaching to steady state with about 1 overshoot for PD control shown as in Fig. 15. The comparison of the two approaches indicates that the tracking effect are better based on the ESO and back-stepping controller than PD controller.
Please cite this article in press as: Yang, H., et al. Back-stepping control of two-link flexible manipulator based on an extended state observer. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.07.036
H. Yang et al. / Advances in Space Research xxx (2015) xxx–xxx
1.5
x 10
11
5
Back−stepping PD 1
τ
0.5
0
−0.5
−1
0
50
100
150
Time Fig. 17. Control input s.
7. 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. References Han, J., 2009. From PID to active disturbance rejectioncontrol. IEEE Trans. Ind. Electron. 56, 900–906. Hua, C., 2011. Teleoperation over the internet with/without velocity signal. IEEE Trans. Instrum. Meas. 60, 4–13. Hua, C., 2013. A new coordinated slave torque feedback control algorithm for network-based teleoperation systems. IEEE/ASME Trans. Mechatron. 18, 764–774. Hua, C., Yang, Y., 2012. Bilateral teleoperation design with/without gravity measurement. IEEE Trans. Instrum. Meas. 61, 3136–3146. Li, Y., Liu, G., Hong, T., Liu, K., 2005. Robust control of a two-link flexible manipulator with quasi-static deflection compensation using neural networks. Intell. Rob. Syst. 44, 263–276.
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Please cite this article in press as: Yang, H., et al. Back-stepping control of two-link flexible manipulator based on an extended state observer. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.07.036