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Finite Time Control Design for Bilateral Teleoperation System With Position Synchronization Error Constrained Yana Yang, Changchun Hua, and Xinping Guan
Abstract—Due to the cognitive limitations of the human operator and lack of complete information about the remote environment, the work performance of such teleoperation systems cannot be guaranteed in most cases. However, some practical tasks conducted by the teleoperation system require high performances, such as tele-surgery needs satisfactory high speed and more precision control results to guarantee patient’ health status. To obtain some satisfactory performances, the error constrained control is employed by applying the barrier Lyapunov function (BLF). With the constrained synchronization errors, some high performances, such as, high convergence speed, small overshoot, and an arbitrarily predefined small residual constrained synchronization error can be achieved simultaneously. Nevertheless, like many classical control schemes only the asymptotic/exponential convergence, i.e., the synchronization errors converge to zero as time goes infinity can be achieved with the error constrained control. It is clear that finite time convergence is more desirable. To obtain a finite-time synchronization performance, the terminal sliding mode (TSM)-based finite time control method is developed for teleoperation system with position error constrained in this paper. First, a new nonsingular fast terminal sliding mode (NFTSM) surface with new transformed synchronization errors is proposed. Second, adaptive neural network system is applied for dealing with the system uncertainties and the external disturbances. Third, the BLF is applied to prove the stability and the nonviolation of the synchronization errors constraints. Finally, some comparisons are conducted in simulation and experiment results are also presented to show the effectiveness of the proposed method. Index Terms—Position error constrained, teleoperation, terminal sliding mode (TSM), time delay.
I. I NTRODUCTION TELEOPERATION system is commonly referred to as the interconnection of five elements: a human operator that exerts force on a local manipulator connected through a
A
Manuscript received November 26, 2014; revised February 11, 2015; accepted February 18, 2015. This work was supported in part by the Hundred Excellent Innovation Talents Support Program of Hebei Province, in part by the Applied Basis Research Project under Grant 13961806D, in part by the Top Talents Project of Hebei Province, and in part by the National Natural Science Foundation of China under Grant 61290322, Grant 61273222, Grant 61322303, Grant 61473248, and Grant 61403335. This paper was recommended by Associate Editor P. Shi. Y. Yang and C. Hua are with the Institute of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China (e-mail: [email protected]). X. Guan is with the Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2015.2410785
communication channel to a remote manipulator that interacts with an environment. The application of such a system spans multiple fields, the most illustrative being space, underwater, medicine, and, in general, tasks with hazardous environments [1]. Communications often involve large distances or impose limited data transfer between the local and remote sites. Such situations can result in substantial delays. It is well known that the time-delay affects the overall stability of the system [2]. Along this direction, many control schemes were proposed for time delay compensation. The passive method was developed to overcome the instability of teleoperation system caused by constant time delay in [3]. The wave variable idea was proposed in [4] to deal with the time delay issue. A new scattering method was developed in [5] to improve the haptic feedback fidelity. In [6], a simple proportional plus derivative controller was proposed, and a delay-dependent stability condition was deduced. In addition, Hua and Liu [7] proposed delay-dependent linear matrix inequality stability criteria for the closed-loop teleoperation system under asymmetrical time-varying delays. In [8], a transparency performance improvement method was proposed for teleoperation system and master–slave controllers were designed. A new control scheme was proposed for teleoperation system with time-varying delay and input constraint [9]. Even though the above control methods can guarantee the stability of the closed-loop system, the transient-state performance is not considered. For some applications demanding satisfactory transient-state performances, the classical control methods are not up to these tasks. It has been early recognized that transient-state control performance is very important and deserves further research [10]. In most cases, the transient-state performance can be guaranteed by setting some constraints on the system errors [11]–[16]. By setting constraints on the system errors, the transient-state performances: high convergence speed, small overshoot, and an arbitrarily predefined small errors can be provided [11]. The barrier Lyapunov function (BLF) is a novel concept used for dealing with the control problems with constraints [12], [13]. In [12], the output constraint problem was addressed by using a BLF for the single-input single-output (SISO) nonlinear systems in strict feedback form. The time-varying output constraint satisfaction was guaranteed by employing the asymmetric time-varying BLF in [13] for strict feedback nonlinear systems. The output constrained problem had been considered for some practical
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systems [14]–[16]. However, in above literatures, only the exponential convergence can be obtained which means that the errors converge to zero as time goes infinity. In most cases, the asymptotic/exponential convergence cannot satisfy the requirement of high convergence speed from some practical applications. To achieve more precise and faster convergence performances, the finite-time control design will be considered in this paper by using terminal sliding mode control (TSMC) method. Different from the classic linear sliding mode (LSM) which only can provide asymptotical convergence [17], TSMC is an effective finite time control approach [18]. Similar to the LSM technique, strong robustness with respect to uncertain dynamics can be obtained. Moreover, the tracking error converges to zero in finite time [19]. However, the premier papers about TSMC encountered singularity problem [20]. To deal with the singularity problem, a globally continuous nonsingular TSMC method was presented in [21]. Different from [21], an indirect approach was applied to avoid the singularity problem for spacecraft [22]. Afterwards, fuzzy logic system and a continuous nonsingular terminal sliding mode (TSM) were combined to control a teleoperation system in [23]. In [24], the finite time coordination problem was investigated for multiagent systems with TSM. It should be noticed that the transient-state performance cannot be guaranteed just by applying finite time control. Recently, neural networks (NNs) have provided strong alternative to develop universal model free controllers [25]–[28]. The most useful property of NN in control is their ability to approximate arbitrary linear or nonlinear mapping through learning. The robust synchronization problem for 2-D discrete-time coupled dynamical networks was investigated in [29]. State estimation problem for complex networks with uncertain inner coupling and incomplete measurements was studied in [30]. In [31], the NN was applied controlling nonlinear interconnected systems with output-feedback. The adaptive NN (ANN) was used for controlling single-master-multiple-slaves teleoperation in [32]. In this paper, a new TSM control approach with new error transformed variables is designed to provide some superior performances: higher synchronization speed, more accurate convergence, finite time synchronization, safety operating, and so on. The advantages of this paper can be summarized as follows. First, a fast nonsingular TSM-based finite time control scheme is designed for nonlinear teleoperation system with time delay. Compared with the existing control works: P + d control, PD + d control, directed force feedback control, and adaptive control for teleoperation, finite time control can provide some superior control performances: higher convergence speed, more precision convergence, finite time convergence, and stronger robustness. In most cases, the finite time control can meet the requirements of speed and precision for many real applications of teleoperation. Second, the synchronization error constrained is considered in this paper. The synchronization error constrained is significant for real applications. On one hand, it provides a good steady-state performance: the errors will be bounded
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by an arbitrarily predefined small constraint. On the other hand, it provides a good transient-state performance: the errors change in a bounded area and the overshoot can be limited. Finally, the system model parameters and external disturbances are not necessary to be known. The NNs are applied to estimate the system uncertainties. Moreover, the parameter adaptive method is used to eliminate the effect from the NN estimation error and the bounded external disturbances. This paper is organized as follows. Section II presents some preliminary knowledge for the model of the teleoperation and the radial basis function (RBF) NN. In Section III, new NNs based finite time control schemes are given. Moreover, the performance analysis process is also presented in this section. Additionally, simulation and experiment results are shown in Section IV to verify the effectiveness of the proposed control method. Finally, Section V concludes with a summary of the obtained results. II. P RELIMINARY A. Dynamics Models of Master and Slave The Euler–Lagrange equations for n-link master and slave manipulators are given as ⎧ ⎪ ⎪ Mm (qm )¨qm + Cm (qm , q˙ m )˙qm + Gm (qm ) ⎨ + Bm (˙qm ) = τm + Fh (1) M qs + Cs (qs , q˙ s )˙qs + Gs (qs ) ⎪ s (qs )¨ ⎪ ⎩ + Bs (˙qs ) = τs − Fe where i = m/s stands for the master and slave manipulators, respectively; qi (t) ∈ Rn is the vector of joint displacement; q˙ i (t) ∈ Rn is the vector of joint velocity; q¨ i (t) ∈ Rn is the vector of joint acceleration; Mi (qi ) ∈ Rn×n is the positive-definite inertia matrix; Ci (qi , q˙ i ) ∈ Rn×n is the matrix of centripetal and Coriolis torques; Gi (qi ) ∈ Rn is the gravitational torque; Bi (˙qi ) ∈ Rn is the unknown bounded external disturbances, i.e., Bi (˙qi ) ≤ B¯ i , B¯ i is an unknown positive constant; Fh , Fe ∈ Rn are the human-operator force and the environment force, respectively; and τi ∈ Rn is the applied control torque. The robot dynamics given in (1) has the following useful properties [7]–[9]. Property 1: The inertia matrix Mi (qi ) is a symmetric positive-definite function, and there exist positive constants mi1 and mi2 such that mi1 I ≤ Mi (qi ) ≤ mi2 I. I is an identity matrix with the corresponding dimension. Property 2: For all qi , x, y ∈ Rn×1 , there exists a positive scalar ai such that Ci (qi , x)y ≤ ai xy. Property 3: There exists positive scalar μGi such that Gi (qi ) ≤ μGi . Throughout this paper, unless otherwise stated, by a vector norm, the vector two-norm is meant and a matrix norm, the induced matrix two-norm is meant. In reality, similar to many engineering applications, it is impossible or very difficult to obtain an exact dynamics model of the master or slave manipulators, due to the presence of large flexibility, Coulomb friction, wear, and so on.
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So, we have Mi (qi ) = Moi (qi ) + Mi (qi ); Ci (qi , q˙ i ) = Coi (qi , q˙ i ) + Ci (qi , q˙ i ); and Gi (qi ) = Goi (qi )+Gi (qi ); here Moi (qi ), Coi (qi , q˙ i ), and Goi (qi ) are the nominal parts, whereas Mi (qi ), Ci (qi , q˙ i ), and Gi (qi ) represent the uncertain parts in the system model. Thus, the dynamics models can be rewritten as ⎧ Mom (qm )¨qm + Com (qm , q˙ m )˙qm + Gom (qm ) ⎪ ⎪ ⎨ = τm + Pm (qm , q˙ m , q¨ m ) + Fh (2) M qs + Cos (qs , q˙ s )˙qs + Gos (qs ) ⎪ os (qs )¨ ⎪ ⎩ = τs + Ps (qs , q˙ s , q¨ s ) − Fe
Define the position synchronization errors between the master and slave as follows:
where Pm (qm , q˙ m , q¨ m ) = −Mm (qm )¨qm − Cm (qm , q˙ m )˙qm − Gm (qm ) − Bm (˙qm ) ∈ Rn and Ps (qs , q˙ s , q¨ s ) = −Ms (qs )¨qs − Cs (qs , q˙ s )˙qs − Gs (qs ) − Bs (˙qs ) ∈ Rn .
e˙ m = q˙ m − q˙ s (t − Ts ) e˙ s = q˙ s − q˙ m (t − Tm ).
B. RBF NNs The motivation of using NNs is to take advantage of NNs global approximation capabilities to handle nonlinearies. Based on the universal approximation theorem, a wide range of nonlinear functions can be estimated by a NN with sufficient neurons. RBF NN is an artificial NN that uses RBFs as activation functions. RBF networks are used widely for function approximation and system control for its simpleness and effectiveness. The RBF NN approximates a continuous function f (X) : Rq → Rp can be expressed as follows: f (X) = W T ϕ(X) + (X)
where Ci and bi are the center and the width of the ith neuron. With the universal approximation property of NNs, for any continuous function f (X), there exists a NN such that f (X) = W ϕ(X) + (X), (X) ≤ N
(5)
where W ∗ is the ideal weight in the approximation; (X) is the approximation error; and N is an upper bound of the approximation error (X). C. New TSM In this section, a new nonsingular fast TSM surface will be defined with new error transformed variables. To streamline the presentation, below, the definition needed in the subsequent analysis is presented as follows. Definition 1: T (6) sig(ζ )a = |ζ1 |a1 sign(ζ1 ), . . . , |ζn |an sign(ζn ) where ζ = [ζ1 , ζ2 , . . . , ζn ]T ∈ Rn ; a1 , a2 , . . . , an > 0 and sign(·) being the standard signum function.
(7)
where Tm represents the signal transmission time delay from the master side to slave side and Ts stands for the transmission time delay in the inverse direction. Then, the velocity errors are given as follows:
It should be noticed that in this paper the i = m/s and j = 1, 2, . . . , n. To achieve some superior performances, some constraints are set for the joint synchronization errors as − kamj (t) < emj (t) < kbmj (t) −kasj (t) < esj (t) < kbsj (t)
(8)
where kamj (t), kbmj (t), kasj (t), and kbsj (t) : R+ → R are positive time-varying boundaries, additionally, −kamj (t) < kbmj (t) and −kasj (t) < kbsj (t). Then, the error transformed variables are defined as
(3)
where X ∈ x ⊂ Rq is the input vector; W ∈ Rn×p is the weight matrix; n > 1 is the number of the neurons; and ϕ(X) = [ϕ1 (X), ϕ2 (X), . . . , ϕn (X)]T , with ϕi (X) being the RBF functions, where the Gaussian RBF function is X − Ci 2 ϕi (X) = exp − , i = 1, 2, . . . , n (4) 2b2i
∗T
em = qm − qs (t − Ts ) es = qs − qm (t − Tm )
emj (t) emj (t) , ξbmj (t) = kamj (t) kbmj (t) esj (t) esj (t) , ξbsj (t) = . ξasj (t) = kasj (t) kbsj (t)
ξamj (t) =
(9)
In the following, the time argument is dropped from above defined variables for presentation compactness. With the new error transformed variables, we can further have that
ξmj = pm emj ξbmj + 1 − pm emj ξamj
ξsj = ps esj ξbsj + 1 − ps esj ξasj
(10)
1 if x > 0 . 0 if x ≤ 0 Differentiating (10) with respect to the time t, yields
where p(x) =
1 − pmj pmj k˙ bmj k˙ amj emj + emj e˙ mj − e˙ mj − ξ˙mj = kbmj kbmj kamj kamj
1 − psj psj k˙ bsj k˙ asj ξ˙sj = esj + esj e˙ sj − e˙ sj − kbsj kbsj kasj kasj (11) where pmj = pm (emj ) and psj = ps (esj ). To streamline the presentation, the following definitions are given as ξbi = [ξbi1 , ξbi2 , . . . , ξbin ]T , ξai = [ξai1 , ξai2 , . . . , ξain ]T , ξi = [ξi1 , ξi2 , . . . , ξin ]T , pi (ei ) = diag(pi1 (ei1 ), pi2 (ei2 ), . . . , pin (ein )), and si = [si1 , si2 , . . . , sin ]T . With above
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defined error transformed variables, the new nonsingular fast TSM surfaces are designed as sm = ξ˙m + αm1 sig(ξm )γm1 + αm2 βm (ξm ) = ξ˙m + λm (ξm ) ss = ξ˙s + αs1 sig(ξs )γs1 + αs2 βs (ξs ) = ξ˙s + λs (ξs )
(12)
where αm1 , αs1 , αm2 , and αs2 are positive constants, γm1 > 1, γs1 > 1; with βm (ξm ) = [βm (ξm1 ), . . . , βm (ξmn )]T and βs (ξs ) = [βs (ξs1 ), . . . , βs (ξsn )]T , the βm (ξmj ) and βs (ξsj ) are defined as follows: ⎧ γm2 sig ξmj if s¯mj = 0 ⎪ ⎪
⎨ or s¯mj = 0, ξmj > μm (13) βm ξmj = 2 ⎪ k ξ + km2 sign(ξmj )ξmj ⎪ ⎩ m1 mj if s¯mj = 0, |ξm | ≤ μm and
⎧ sig(ξsj )γs2 if s¯sj = 0 ⎪ ⎪
⎨ or s¯sj = 0, ξsj > μs βs (ξsj ) = 2 ⎪ ⎪ ks1 ξsj + ks2 sign(ξ
sj )ξsj ⎩ if s¯sj = 0, ξsj ≤ μs
(14)
where 0 < γm2 < 1 and 0 < γs2 < 1; s¯mj = ξ˙mj + αm1 sig(ξmj )γm1 + αm2 sig(ξmj )γm2 , s¯sj = ξ˙sj + αs1 sig(ξsj )γs1 + γm −1 γm −2 αs2 sig(ξsj )γs2 ; km1 = (2 − γm2 )μm 2 , km2 = (γm2 −1)μm 2 , γs −1 γs −2 ks1 = (2 − γs2 )μs 2 , and ks2 = (γs2 − 1)μs 2 ; and μm and μs are small positive constants. The derivative of the sliding mode surface is
s˙ij = ξ¨ij + λ ij ξij 2 2k˙ bij k¨ bij eij 2k˙ bij e˙ ij pij − + 2 eij = e¨ ij − kbij kbij kbij kbij 2 2k˙ aij
1 − pij k¨ aij eij 2k˙ aij e˙ ij + − + 2 eij + λ ij ξij e¨ ij − kaij kaij kaij kaij (15) where λ ij (ξij ) denotes the derivative of λij (ξij ). Remark 1: The choice for synchronization error constraints has great significance for the control performances. The values of the constraints can determine the minimum convergence speed, maximum overshoot, and the maximum convergence errors. In this paper, to guarantee satisfactory synchronization performances, the kamj (t), kbmj (t), kasj (t), and kbsj (t) are defined as decaying functions of time like exponential functions. Remark 2: It should be noticed that, in teleoperation system, the slave manipulator completes task in remote side while the master set in local side. Due to the cognitive limitations of the human operator and lack of complete information about the remote environment, the control performance cannot be guaranteed, and worse still it will lead to collision between the slave manipulator and object. By guaranteeing the synchronization performance, the absolute safety can be obtained, which is significant for the teleoperation system.
Remark 3: Compared with the LSM, the TSM offers some superior properties such as higher precision, more robust, higher speed, and finite-time convergence. However, the classic TSM faces the singularity problem. To deal with the singularity problem, the switching method is used in this paper. The sliding mode of the system is switched between the TSM and LSM, i.e., when a singularity appears, the sliding mode is switched from TSM to LSM. Then, the sliding mode is switched back from LSM to TSM as soon as the system trajectory passes the singularity area. The choice of the ki1 and ki2 make the switching smooth. III. M AIN R ESULTS In this section, new finite time controllers will be designed for the master and slave manipulators with the sliding mode surfaces defined in the previous section. The new finite time controllers not only can provide the finite time synchronization performance, but also can guarantee the synchronization errors of the system moving in the constraints. A. Controller Design In this paper, a new TSM surface is proposed with the new error transformed variables. Then with the new nonsingular fast TSM designed in above section, ANN-based finite time controllers are designed for the master and slave as follows: τm = Mom (qm )¨qs (t − Ts ) + Com (qm , q˙ m )˙qm + Gom (qm ) + τ¯m1 + τ¯m2 τs = Mos (qs )¨qm (t − Tm ) + Cos (qs , q˙ s )˙qs + Gos (qs ) + τ¯s1 + τ¯s2 .
(16)
The specific definitions of τ¯m1 and τ¯s1 are given as em sm ξ˙mT τ¯m1 = Mom (qm ) − em − λ m (ξm ) 1 − ξmT ξm ξm 2 2k˙ bm 2k˙ bm e˙ m k¨ bm em − pm − − + 2 em kbm kbm kbm 2 2k˙ am e˙ m k¨ am em 2k˙ am − + 2 em − (1 − pm ) − kam kam kam em sig(sm )ρ em − Km2
− Km1 sm (ρ−1)/2 ξm ξm 1 − ξ T ξm m
(17)
and
τ¯s1 = Mos (qs ) −
ss ξ˙sT es es − λ s (ξs ) T 1 − ξs ξs ξs 2 ¨ ˙ 2k˙ bs kbs es 2kbs e˙ s − ps − − + 2 es kbs kbs kbs 2 2k˙ as e˙ s k¨ as es 2k˙ as − (1 − ps ) − − + 2 es kas kas kas es sig(ss )ρ es − Ks1 ss − Ks2
(18) (ρ−1)/2 ξs ξs 1 − ξ T ξs s
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Fig. 1.
Teleoperation system with new controller.
where Km1 = diag(Km11 , Km12 , . . . , Km1n ), Ks1 = diag(Ks11 , Ks12 , . . . , Ks1n ), Km2 = diag(Km21 , Km22 , . . . , Km2n ), and Ks2 = diag(Ks21 , Ks22 , . . . , Ks2n ) are positive definite diagonal matrices. Considering the system uncertainties and the external disturbances, the τ¯m2 and τ¯s2 are designed as follows:
sTm 1 − ξmT ξm T pm 1 − pm −1 Mom × + (qm ) kbm kam T s s ˆ sT ϕs (Xs ) − φˆ s τ¯s2 = −W 1 − ξsT ξs T ps 1 − ps −1 Mos × + (qs ) kbs kas (19)
τ¯m2
5
ˆ mT ϕm (Xm ) − φˆ m = −W
−1 ˆ −1 ˆ where φˆ m = 1/2δm1 ψm and φˆ s = 1/2δs1 ψs ; δm1 and δs1 are 2 2 ; positive constants; ψm = mN and ψs = sN mN and sN are the upper bounds of the NN approximation errors, i.e., m (Xm ) ≤ mN and s (Xs ) ≤ sN ; Xm = [qTm , q˙ Tm , q¨ Tm ]T and Xs = [qTs , q˙ Ts , q¨ Ts ]T . The teleoperation system with the new control algorithm is shown in Fig. 1. The NT denotes nonlinear transformation. Remark 4: In this paper, the finite time control design is investigated for the nonlinear teleoperation system with synchronization error constrained. Different from the error constrained control proposed for high-order SISO system, the two-order multiple-input multiple-output (MIMO) nonlinear systems are considered as the master and the slave, respectively. With the MIMO control systems, the controller design becomes more complex. Moreover, the expressions for derivation and analysis are more complicated because there are so many derivations for vectors of variables such as the term −pm 2k˙ bm e˙ m /kbm , which is a vector likes [−pm (em1 )2k˙ bm1 e˙ m1 /kbm1 , . . . , −pm (emn )2k˙ bmn e˙ mn /kbmn ]T . To avoid repetition explanations and save space, this expression will not be explained any more. Remark 5: In this paper, the TSM-based finite time control method and the BLF-based position synchronization error constrained control method are combined to provide some super control performances. The control terms (17) and (18)
are used to counteract the effect from the known nonlinear terms. Moreover, the (19) is designed to deal with the system uncertainties and the unknown external disturbances. Different from the classical TSM finite time control, the term −Ki1 si ei /ξi − Ki2 sig(si )ρ /(1 − ξiT ξi )( ρ−1)/2 ei /ξi is designed to provide the finite time synchronization performance. Remark 6: Different from [23], the new adaptive term φˆ m sm is used to replace the previous term ψˆ m sign(sm ). Compared with the previous term, the new adaptive term is continuous. Therefore, smooth control torque can be obtained. B. Performance Analysis In this section, the rigorous prove process will be presented. Theorem 1: For the teleoperation system (2) with the human operator and remote environment insert forces are zero, i.e., Fh = 0 and Fe = 0 in the presence of system uncertainties and external disturbances, if the nonsingular fast terminal sliding mode (NFTSM) manifold is chosen as (12), the NN tuning law and adaptive tuning law are designed as follows: · sTi pi 1 − pi −1 ˆ ˆi Moi (qi ) − δi2 W + W i = i ϕi (Xi ) kai 1 − ξiT ξi kbi 2 · sTi pi i 1 − pi −1 M ψˆ i = + (q ) − δi3 ψˆ i i oi 2δi1 1 − ξiT ξi kbi kai (20) where i , i , δi1 , δi2 , and δi3 are positive constants. With the continuous NFTSM controllers designed as (16), then: 1) all signals of the closed-loop system are bounded; 2) the time-varying position synchronization error constraints are never violated; 3) the NFTSM manifold sij converges to region i in finite ¯ i in finite time time, the ξij finally converge into region i = min(i1 , i2 ) ¯ i = max μi ,
γi 1
i , 2αi1
γi2
i 2αi2
= Hi /2K¯ i1 , i2 = where i1 ρ T Hi (1 − ξi ξi )(ρ−1)/2 /2K¯ i2 ; K¯ i1 = λmin (Ki1 ) and K¯ i2 = λmin (Ki2 ); λmin (A) denotes the minimum
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eigenvalue of A; and Hi is a positive constant, which will be defined later. Proof: Let us consider the Lyapunov function candidates 1 sT si i 2 1 − ξiT ξi i={m,s} 1 1 −1 ˜ 2 ˜ iT −1 ˜ V2 = tr W W i + i ψ i i 2 2
V1 =
(21) (22)
ˆ i and ψ˜ i = ψi − ψˆ i . ˜ i = Wi − W where W The derivation of the above function V1 is given as follows:
sTi s˙i ξiT ξ˙i sTi si +
2 . 1 − ξiT ξi 1 − ξiT ξi i={m,s}
(23)
With the differentiation of the sliding mode surface, we have
sTi ξ¨i + λ i (ξi ) ξiT ξ˙i sTi si ˙ V1 = +
2 1 − ξiT ξi 1 − ξiT ξi i={m,s} 2 2k˙ bi sTi pi 2k˙ bi e˙ i k¨ bi ei = − + 2 ei e¨ i − kbi kbi 1 − ξiT ξi kbi kbi i={m,s} 2 sTi 2k˙ ai 1 − pi 2k˙ ai e˙ i k¨ ai ei + − + 2 ei e¨ i − kai kai kai 1 − ξiT ξi kai +
sTi ξ T ξ˙i sTi si λ i (ξi ) + i 2 . T 1 − ξi ξi 1 − ξiT ξi
(24)
Applying the system model (2) with Fh = 0, Fe = 0, yields −1 (q )(τ + P (q , q ¨ m) − e¨ m = q¨ m − q¨ s (t − Ts ) = Mom m m m m ˙ m, q Com (qm , q˙ m )˙qm − Gom (qm )) − q¨ s (t − Ts ) and e¨ s = q¨ s − q¨ m −1 (q )(τ + P (q , q (t − Tm ) = Mos ¨ s ) − Cos (qs , q˙ s )˙qs − s s s s ˙ s, q Gos (qs )) − q¨ m (t − Tm ). Employing the new controller (16) and the NN approximation Pi (qi , q˙ i , q¨ i ) = Wi∗T ϕi (Xi ) + i (Xi ), yields sTi 2k˙ bi e˙ i pi −1 − M V˙ 1 = τ ¯ (q ) i i1 oi kbi 1 − ξiT ξi kbi i={m,s} 2 2k˙ bi k¨ bi ei − + 2 ei kbi kbi sTi 1 − pi 2k˙ ai e˙ i −1 + × M − τ ¯ (q ) i i1 oi kai kai 1 − ξiT ξi 2 2k˙ ai k¨ ai ei − + 2 ei kai kai
sTi ξiT ξ˙i sTi si
λ (ξ ) + i
2 i 1 − ξiT ξi 1 − ξiT ξi
sTi pi 1 − pi −1 Moi + + (qi ) Wi∗T ϕi (Xi ) T k k 1 − ξi ξi bi ai sTi pi 1 − pi −1 + + i (Xi )) + Moi (qi )τ¯i2 . kai 1 − ξiT ξi kbi (25) +
sTi Ki1 si sT Ki2 sig(si )ρ − i (ρ+1)/2 T 1 − ξi ξi 1 − ξiT ξi i={m,s} sTi pi 1 − pi −1 Moi + + (qi )Wi∗T ϕi (Xi ) kai 1 − ξiT ξi kbi sTi pi 1 − pi −1 ˆ iT ϕi (Xi ) Moi + − (qi )W kai 1 − ξiT ξi kbi sTi pi 1 − pi −1 Moi + + (qi )i (Xi ) kai 1 − ξiT ξi kbi 2 sT p 1 − p i i ˆ −1 i − M + (q ) φi i oi 1 − ξiT ξi kbi kai sT Ki1 si sT Ki2 sig(si )ρ = − i T − i (ρ+1)/2 1 − ξi ξi 1 − ξiT ξi i={m,s} sTi pi 1 − pi −1 ˜ iT ϕi (Xi ) Moi + + (qi )W kai 1 − ξiT ξi kbi sTi pi 1 − pi −1 Moi + + (qi )i (Xi ) kai 1 − ξiT ξi kbi 2 sT p 1 − p i i ˆ −1 i − + (26) M (q ) φi . i oi 1 − ξiT ξi kbi kai
V˙ 1 =
i={m,s}
V˙ 1 =
Further, with the designed τ¯m1 (17), τ¯s1 (18), and τ¯m2 , τ¯s2 (19), we have −
Since i (Xi ) ≤ iN and for any constants x and y have xy ≤ δx2 /2 + y2 /2δ, where δ is a positive constant. Then, we obtain that sT Ki1 si sT Ki2 sig(si )ρ − i T − i V˙ 1 ≤ (ρ+1)/2 1 − ξi ξi 1 − ξiT ξi i={m,s} sTi pi 1 − pi −1 ˜ iT ϕi (Xi ) Moi + + (qi )W kai 1 − ξiT ξi kbi 2 sT pi 1 − pi −1 i Moi (qi ) φˆ i − + T 1 − ξi ξi kbi kai 2 T pi 1 1 2 1 − pi si −1 M + δi1 + iN + (q ) i . oi 2 2δi1 1 − ξiT ξi kbi kai (27) 2 , With the definitions for φˆ i = 1/2δi1 ψˆ i and ψˆ i = iN we have sT Ki1 si sT Ki2 sig(si )ρ 1 − i T − i V˙ 1 ≤ (ρ+1)/2 + δi1 T 2 1 − ξi ξi 1 − ξi ξi i={m,s} T si pi 1 − pi −1 ˜ iT ϕi (Xi ) Moi + + (qi )W T kai 1 − ξi ξi kbi 2 T pi 1 1 − pi ˜ si −1 M + + (28) (q ) ψi . i oi 2δi1 1 − ξiT ξi kbi kai
The derivation of V2 is given as follows: · · −1 T ˜ ˆ ˜ i i W ˆ i − −1 −tr W V˙ 2 = i ψi ψ i . i={m,s}
(29)
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With the NN tuning law and the adaptive tuning law (20), we have sTi ˜ iT ϕi (Xi ) V˙ 2 = −tr W 1 − ξiT ξi i={m,s} pi 1 − pi −1 Moi (qi ) × + kbi kai 2 T pi 1 1 − pi si ˜ −1 − + (q ) M i ψ oi i 2δi1 1 − ξiT ξi kbi kai δ δ ˜ iT i2 W ˆ i + ψ˜ i i3 ψˆ i . + tr W (30) i i Thus, we can obtain that V˙ = V˙ 1 + V˙ 2 sT Ki1 si sT Ki2 sig(si )ρ = − i T − i (ρ+1)/2 1 − ξi ξi 1 − ξiT ξi i={m,s} δi3 δi1 T δi2 ˆ ˜ + tr Wi Wi + ψ˜ i ψˆ i + i i 2
(31)
ˆ i = Wi − W ˜ i and ψˆ i = ψi − ψ˜ i , we have further, with W T ˜ ˜ iT δi2 /i Wi − W ˜ iT δi2 /i W ˜ i ) and ˜ tr(Wi δi2 /i (Wi − Wi )) = tr(W T T T ˜ i δi2 /i Wi ) ≤ tr(W ˜ i δi2 δi4 /2i W ˜ i + Wi δi2 /2δi4 i Wi ). tr(W Similarly, ψ˜ i δi3 /i ψˆ i ≤ δi3 δi5 /2i ψ˜ i2 + δi3 /2δi5 i ψˆ i2 also can be obtained. Therefore, we have sT Ki1 si sT Ki2 sig(si )ρ V˙ ≤ − i T − i (ρ+1)/2 1 − ξi ξi 1 − ξiT ξi i={m,s} δ δ δ δi1 ˜ i + WiT i2 i4 Wi ˜ iT i2 W + +W 2 2i δi4 2i δi2 δi3 δi3 δi5 2 δi3 2 T 2 ˜i ˜i+ −W W ψ˜ + ψˆ − ψ˜ i 2i δi5 i 2i i i i sT Ki1 si sT Ki2 sig(si )ρ = − i T − i (ρ+1)/2 1 − ξi ξi 1 − ξ T ξi i={m,s} i
T
δi1 ˜ i ηi1 W ˜ i + tr WiT ηi2 Wi − tr W 2 − ηi3 ψ˜ i2 + ηi4 ψi2
+
(32)
where ηi1 = δi2 (2δi4 − 1)/2ii δi4 , ηi2 = δi2 δi4 /2i , ηi3 = δi3 (2δi5 − 1)/2i δi5 , and ηi4 = δi3 δi5 /2i ; δi4 and δi5 are two positive constants and chosen satisfying δi4 > 1/2 and δi5 > 1/2. Finally, we can obtain that V˙ ≤ −V +
(33)
where = min(2K¯ m1 , 2K¯ s1 , R), R = min(2ηm1 m , 2ηm3 m , 2ηs1 s , 2ηs3 s ), and = δm1 /2 + δs1 /2 + tr(WmT ηm2 Wm ) + tr(WsT ηs2 Ws ) + ηm4 ψm2 + ηs4 ψs2 . Then, based on the boundedness theorem in Khalil [22], all signals in the closed-loop system are bounded. Furthermore, we have |ξmj | < 1 and |ξsj | < 1, which implies that the timevarying error constraint is never violated. Next, the finite time synchronization performance will be proved. −1 (qi )Pi (qi , q˙ i , q¨ i ) + Define i = (pi /kbi + 1 − pi /kai )Moi −1 (pi /kbi + 1 − pi /kai )Moi (qi )τ¯i2 . The boundedness of system
7
signals shows that i is bounded and i ≤ Hi with Hi a positive constant. Next, the finite time synchronization performance is analyzed. Consider the Lyapunov candidate function V1 V1 =
1 sT si i . 2 1 − ξiT ξi
(34)
i={m,s}
The derivative of V1 is sT Ki1 si sT Ki2 sig(si )ρ sTi V˙ 1 = − i T − i + i (ρ+1)/2 1 − ξi ξi 1 − ξiT ξi 1 − ξiT ξi i={m,s} si 2 Hi ¯ ≤ − Ki1 − 2si 1 − ξiT ξi i={m,s}
(ρ−1)/2 Hi 1 − ξiT ξi si ρ+1 − K¯ i2 −
(ρ+1)/2 . 2si ρ 1 − ξiT ξi (35) It follows that if si > i , it obtains that V˙ 1 + (1+ρ)/2 σ1 V1 + σ2 V1 ≤ 0, σ1 = min(2K˜ m1 , 2K˜ s1 ), and σ2 = min(2K˜ m2 2(ρ+1)/2 , 2K˜ s2 2(ρ+1)/2 ), where K˜ i1 = K¯ i1 −Hi /2si and K˜ i2 = K¯ i2 − Hi (1 − ξiT ξi )(ρ−1)/2 /2si ρ . In addition, i is a small region containing the origin with large enough Ki1 and Ki2 . Next, the convergence region of ξij will be derived. Because of si ≤ i for the case ξij ≥ μi , i = m, s, j = 1, 2, . . . , n, then we get
γ
γ ξ˙ij + αi1 sig ξij i1 + αi2 sig ξij i2 = ij , ij ≤ i . (36) Then, we have
γ ij
γi sig ξij i1 ξ˙ij + αi1 − 2sig ξij 1
γ ij
γi sig ξij i2 = 0. + αi2 − 2 2sig ξij
(37)
Therefore, as long as αi1 − ij /2sig(ξij )γi1 > 0 and αi2 − ij /2sig(ξij )γi2 > 0, the ξij will converge to the region |ξij | ≤ ¯ i in finite time. This completes the proof. IV. S IMULATION AND E XPERIMENT A. Simulation on Teleoperation of Two-DOF Planar Manipulators In order to show the effectiveness of the proposed control scheme, some simulations are conducted in this section, in which two identical two-degree of freedom (DOF) serial links manipulators are chosen as the local and remote manipulators. Thus, the same controller parameters are set for the master and slave. The specific parameters of the teleoperation dynamics model and the controller parameters are shown in Table I. In Table I, the nominal mass and length are expressed as m1 , m2 , l1 , and l2 . The actual values are expressed as m ˜ 1, m ˜ 2 , ˜l1 , and ˜l2 . The initial joint position of the master and slave are set as qm (0) = [0.2π 0.12π ]T and qs (0) = [0.1π 0.12π ]T and the velocity of master and the slave manipulators are set as q˙ m = [0 0]T and q˙ s = [0 0]T .
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TABLE I S YSTEM PARAMETERS AND C ONTROLLER PARAMETERS
Fig. 2.
Fig. 3.
Fig. 4.
Joint position errors at the slave side with controller (38).
Fig. 5.
Joint positions of the slave with controller (38).
Fig. 6.
Joint position errors at the master side with controller (16).
Human-force input F.
Joint position errors at the master side with controller (38).
The time delays are set as Tm = 600 ms and Ts = 600 ms. To obtain some superior work performances, the synchronization error constraints are set as kaij = (1.5 − 0.05) exp(−1.6t) + 0.05 and kbij = (1.5 − 0.05) exp(−1.6t) + 0.05, j = 1, 2, i = m, s. To simplify the presentation, the same constraints are set on the different joint synchronization errors. To illustrate the control performance of the proposed controller, comparisons will be conducted. We carry out the comparison with the general nonsingularity terminal sliding mode controller, which is designed as follows:
τm = Mom (qm ) q¨ s (t − Ts ) − λ m (em ) − Km1 sm − Km2 sig(sm )ρ + Com (qm , q˙ m )˙qm 1 ˆ mT ϕm (Xm ) − φˆ m sm + Gom (qm ) − W 2σm1
τs = Mos (qs ) q¨ m (t − Tm ) − λ s (es ) − Ks ss − Ks sig(ss )ρ + Cos (qs , q˙ s )˙qs 1 ˆ sT ϕs (Xs ) − φˆ s ss . + Gos (qs ) − W (38) 2σs1 The human insert force is shown in Figs. 2–4 which illustrate the position tracking errors at the master side and slave side with controllers (38), respectively. The joint positions of the slave are shown in Fig. 5. With the controller (16), the position synchronization errors with the constraints at the master
side and the slave side are shown in Figs. 6 and 7. Then in Fig. 8, the joint positions of the slave are given. Comparing Figs. 3 and 4 with Figs. 6 and 7, it is obvious that with the controller (16) proposed in this paper, the synchronization errors are always bounded by the decaying boundaries. Moreover, by comparing Figs. 5 and 8, it can be seen that the positions of the slave links is always restricted in a boundary. This can help the slave to avoid having collision with the remote object. B. Experiment on Teleoperated Pair of Three-DOF PHANToM Manipulator The teleoperation system for the experiments consists of two PHANToM Premium 1.5 A robots (SensAble Technologies, Inc.), which have three-DOF (see Fig. 9). The two robotic manipulators are connected by two computers that are connected via the Internet network. The network environment can be set in the network-simulator block. The controller is established with the MATLAB software.
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Fig. 7.
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Joint position errors at the slave side with controller (16). Fig. 11. Position errors with the constraint with Tm = 200 ms and Ts = 200 ms.
Fig. 8.
Joint positions of the slave with controller (16). Fig. 12. Control torque at the slave side with Tm = 200 ms and Ts = 200 ms.
Fig. 9.
Teleoperation system.
Fig. 13. Joint positions of the master and the slave with Tm = 600 ms and Ts = 600 ms.
Fig. 10. Joint positions of the master and the slave with Tm = 200 ms and Ts = 200 ms.
The constraints are set as kaij = (2.5 − 0.05) exp(−0.2t) + 0.05 and kbij = (2.5 − 0.05) exp(−0.2t) + 0.05, i = m, s, j = 1, 2, 3. In the experiment, we set the master follows a desired trajectory as qmd = 0.1 + 0.2 sin(0.2t), and the slave tracks the trajectory of the master. First, we set the time delay as Tm = 200 ms and Ts = 200 ms. The joint positions are shown in Fig. 10. The joint position error with the constraints
Fig. 14. Position errors with the constraint with Tm = 600 ms and Ts = 600 ms.
are shown in Fig. 11. The control torques for the three joints are shown in Fig. 12. Then, the time delays are set as Tm = 600 ms and Ts = 600 ms. The joint positions are shown in Fig. 13.
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Fig. 15. Control torque at the slave side with Tm = 600 ms and Ts = 600 ms.
The joint position errors with the constraints are shown in Fig. 14. The control torques for the three joints are shown in Fig. 15. As we can see from Figs. 10 and 13, the slave tracks the trajectory of the master in a high speed with initial tracking errors. Then, the slave will always track the trajectory of the master. In Figs. 11 and 14, the ebs1–ebs3 stand for the ξs1 –ξs3 . As we can see in these two figures, the values of ξs1 –ξs3 are always smaller than 1, which means that the synchronization errors satisfy −kamj (t) < emj (t) < kbmj (t) and −kasj (t) < esj (t) < kbsj (t). Therefore, the position synchronization error constraints are not violated. V. C ONCLUSION This paper has studied the issues associated with the finite time synchronization control of bilateral teleoperation system with position synchronization errors constrained. The proposed control algorithm not only imbibes advantages of both methods: 1) finite time control; 2) BLF-based control method, but also provides additional advantage. Both transient-state performance and steady-state performance can be guaranteed in this paper. In simulation, by comparing the new controller with the NFTSM controller, we show that the proposed BLF-based NFTSM controller can achieve super transient-state performances. Finally, the experiments are conducted with different time delays to show the effectiveness of the proposed methods. R EFERENCES [1] P. F. Hokayem and M. W. Spong, “Bilateral teleoperation: An historical survey,” Automatica, vol. 42, no. 12, pp. 2035–2057, 2006. [2] F. B. Li, P. Shi, L. G. Wu, and X. Zhang, “Fuzzy-model-based-stability and nonfragile control for discrete-time descriptor systems with multiple delays,” IEEE Trans. Fuzzy Syst., vol. 22, no. 4, pp. 1019–1025, Aug. 2014. [3] R. J. Anderson and M. W. Spong, “Bilateral control of teleoperators with time delay,” IEEE Trans. Autom. Control, vol. 34, no. 5, pp. 494–501, May 1989. [4] G. Niemeyer and J.-J. E. Slotine, “Stable adaptive teleoperation,” IEEE J. Ocean. Eng., vol. 16, no. 1, pp. 152–162, Jan. 1991. [5] Y. Ye and P. X. Liu, “Improving haptic feedback fidelity in wave variable-based teleoperation oriented to telemedical applications,” IEEE Trans. Instrum. Meas., vol. 58, no. 8, pp. 2847–2855, Aug. 2009. [6] D. Lee and M. W. Spong, “Passive bilateral teleoperation with constant time delay,” IEEE Trans. Robot., vol. 22, no. 2, pp. 269–281, Apr. 2006. [7] C. C. Hua and X. P. Liu, “Delay-dependent stability criteria of teleoperation systems with asymmetric time-varying delays,” IEEE Trans. Robot., vol. 26, no. 5, pp. 925–932, Oct. 2010. [8] C. C. Hua and Y. N. Yang, “Bilateral teleoperation design with/without gravity measurement,” IEEE Trans. Instrum. Meas., vol. 61, no. 12, pp. 3136–3146, Dec. 2012.
[9] F. Hashemzadeh, I. Hassanzadeh, and M. Tavakoli, “Teleoperation in the presence of varying time delays and sandwich linearity in actuators,” Automatica, vol. 49, no. 9, pp. 2813–2821, 2013. [10] R. Kelly, “A tuning procedure for stable PID control of robot manipulators,” Robotica, vol. 13, pp. 141–148, Feb. 1995. [11] P. B. Charalampos and A. R. George, “Adaptive control with guaranteed transient and steady state tracking error bounds for strict feedback systems,” Automatica, vol. 45, pp. 532–538, Feb. 2009. [12] K. B. Ngo, R. Mahony, and Z. P. Jiang, “Integrator backstepping using barrier functions for systems with multiple state constraints,” in Proc. 44th IEEE Conf. Decis. Control, Seville, Spain, 2005, pp. 8306–8312. [13] K. P. Tee, S. S. Ge, and E. H. Tay, “Control of nonlinear systems with time-varying output constraint,” Automatica, vol. 47, pp. 2511–2516, Nov. 2011. [14] Z. Zhao, W. He, and S. S. Ge, “Adaptive neural network control of a fully actuated marine surface vessel with multiple output constraint,” IEEE Trans. Control Syst. Technol., vol. 22, no. 4, pp. 1536–1543, Jul. 2014. [15] X. Jin and J. X. Xu, “Iterative learning control for output-constrained systems with both parametric and nonparametric uncertainties,” Automatica, vol. 49, pp. 2508–2516, Aug. 2013. [16] Y. M. Li, S. C. Tong, and T. S. Li, “Adaptive fuzzy output-feedback control for output constrained nonlinear systems in the presence of input saturation,” Fuzzy Sets Syst., vol. 248, pp. 138–155, Aug. 2014. [17] F. B. Li, L. G. Wu, P. Shi, and C. C. Lim, “State estimation and sliding mode controller for semi-Markovian jump systems with mismatched uncertainties,” Automatica, vol. 51, pp. 385–393, Jan. 2015. [18] S. T. Venkataraman and S. Gulati, “Control of nonlinear systems using terminal sliding modes,” ASME J. Dyn. Syst. Meas. Control, vol. 115, pp. 554–560, Sep. 1993. [19] F. Y. Chen, R. Hou, B. Jiang, and G. Tao, “Study on fast terminal sliding mode control for a helicopter via quantum information technique and nonlinear fault observer,” Int. J. Innov. Comput. Inf. Control, vol. 9, no. 8, pp. 3437–3447, 2013. [20] Y. Feng, X. H. Yu, and F. L. Han, “On nonsingular terminal slidingmode control for nonlinear systems,” Automatica, vol. 49, no. 6, pp. 1715–1722, 2014. [21] S. H. Yu, X. H. Yu, B. J. Shirinzadeh, and Z. H. Man, “Continuous finite-time control for robotic manipulators with terminal sliding mode,” Automatica, vol. 41, no. 11, pp. 1957–1964, 2005. [22] K. F. Lu and Y. Q. Xia, “Adaptive attitude tracking control for rigid spacecraft with finite-time convergence,” Automatica, vol. 49, no. 12, pp. 3591–3599, 2013. [23] Y. N. Yang, C. C. Hua, and X. P. Guan, “Adaptive fuzzy finite-time coordination control for networked nonlinear bilateral teleoperation system,” IEEE Trans. Fuzzy Syst., vol. 22, no. 3, pp. 631–641, Jun. 2014. [24] G. Masood and G. N. Sergry, “Finite-time coordination in multiagent systems using sliding mode control approach,” Automatica, vol. 50, pp. 1209–1216, Apr. 2014. [25] S. C. Tong, T. Wang, and Y. M. Li, “Adaptive neural network output feedback control for stochastic nonlinear systems with unknown deadzone and unmodeled dynamics,” IEEE Trans. Cybern., vol. 44, no. 6, pp. 910–921, Jun. 2014. [26] B. Chen, K. F. Liu, and X. P. Liu, “Approximation-based adaptive neural control design for a class of nonlinear systems,” IEEE Trans. Cybern., vol. 44, no. 5, pp. 610–619, May 2014. [27] Q. K. Shen, B. Jiang, and P. Shi, “Novel neural networks-based fault tolerant control scheme with fault alarm,” IEEE Trans. Cybern., vol. 44, no. 11, pp. 2190–2201, Nov. 2014. [28] P. Shi, Y. Q. Zhang, and R. K. Agarwal, “Stochastic finite-time state estimation for discrete time-delay neural networks with Markovian jumps,” Neurocomputing, vol. 151, pp. 168–174, Mar. 2015. [29] J. Liang, Z. Wang, X. Liu, and P. Louvieris, “Robust synchronization for two-dimensional discrete-time coupled dynamical networks,” IEEE Trans. Neural Netw. Learn. Syst., vol. 23, no. 6, pp. 942–953, Jun. 2012. [30] B. Shen, Z. Wang, D. Ding, and H. Shu, “H∞ state estimation for complex networks with uncertain inner coupling and incomplete measurements,” IEEE Trans. Neural Netw. Learn. Syst., vol. 24, no. 12, pp. 2027–2037, Dec. 2013. [31] C. C. Hua and X. P. Guan, “Output feedback control for time delay nonlinear interconnected systems using neural networks,” IEEE Trans. Neural Netw., vol. 19, no. 4, pp. 673–688, Apr. 2008. [32] Z. J. Li and C. Y. Su, “Neural-adaptive control of single-mastermultiple-slaves teleoperation for coordinated multiple mobile manipulators with time-varying communication delays and input uncertainties,” IEEE Trans. Neural Netw. Learn. Syst., vol. 24, no. 9, pp. 1400–1413, Sep. 2013.
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Yana Yang received the B.Sc. degree in electrical engineering from Yanshan University, Qinhuangdao, China, in 2011, where she is currently pursuing the Ph.D. degree. Her current research interests include nonlinear control systems and teleoperation systems design.
Changchun Hua received the Ph.D. degree in electrical engineering from Yanshan University, Qinhuangdao, China, in 2005. He was a Research Fellow with the National University of Singapore, Singapore, from 2006 to 2007. From 2007 to 2009, he was with Carleton University, Ottawa, ON, Canada, funded by the Province of Ontario Ministry of Research and Innovation Program. From 2009 to 2010, he was with the University of Duisburg-Essen, Essen, Germany, funded by the Alexander von Humboldt Foundation. He is currently a Full Professor with Yanshan University. He has been involved in over ten projects supported by the National Natural Science Foundation of China, the National Education Committee Foundation of China, and other important foundations. His current research interests include nonlinear control systems, control systems design over network, teleoperation systems, and intelligent control. He has authored/co-authored over 80 papers in mathematical and technical journals, and conferences.
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Xinping Guan received the B.S. degree in mathematics from Harbin Normal University, Harbin, China, in 1986, and the M.S. degree in applied mathematics and the Ph.D. degree in electrical engineering, both from the Harbin Institute of Technology, Harbin, in 1991 and 1999, respectively. He is currently a Professor with the Department of Automation, Shanghai Jiao Tong University, Shanghai, China. He is a Cheung Kong Scholars Programme Special Appointment Professor. His current research interests include functional differential and difference equations, robust control and intelligent control for time-delay systems, chaos control and synchronization, and congestion control of networks. He has (co)authored over 200 papers in mathematical and technical journals, and conferences.
1
Finite Time Control Design for Bilateral Teleoperation System With Position Synchronization Error Constrained Yana Yang, Changchun Hua, and Xinping Guan
Abstract—Due to the cognitive limitations of the human operator and lack of complete information about the remote environment, the work performance of such teleoperation systems cannot be guaranteed in most cases. However, some practical tasks conducted by the teleoperation system require high performances, such as tele-surgery needs satisfactory high speed and more precision control results to guarantee patient’ health status. To obtain some satisfactory performances, the error constrained control is employed by applying the barrier Lyapunov function (BLF). With the constrained synchronization errors, some high performances, such as, high convergence speed, small overshoot, and an arbitrarily predefined small residual constrained synchronization error can be achieved simultaneously. Nevertheless, like many classical control schemes only the asymptotic/exponential convergence, i.e., the synchronization errors converge to zero as time goes infinity can be achieved with the error constrained control. It is clear that finite time convergence is more desirable. To obtain a finite-time synchronization performance, the terminal sliding mode (TSM)-based finite time control method is developed for teleoperation system with position error constrained in this paper. First, a new nonsingular fast terminal sliding mode (NFTSM) surface with new transformed synchronization errors is proposed. Second, adaptive neural network system is applied for dealing with the system uncertainties and the external disturbances. Third, the BLF is applied to prove the stability and the nonviolation of the synchronization errors constraints. Finally, some comparisons are conducted in simulation and experiment results are also presented to show the effectiveness of the proposed method. Index Terms—Position error constrained, teleoperation, terminal sliding mode (TSM), time delay.
I. I NTRODUCTION TELEOPERATION system is commonly referred to as the interconnection of five elements: a human operator that exerts force on a local manipulator connected through a
A
Manuscript received November 26, 2014; revised February 11, 2015; accepted February 18, 2015. This work was supported in part by the Hundred Excellent Innovation Talents Support Program of Hebei Province, in part by the Applied Basis Research Project under Grant 13961806D, in part by the Top Talents Project of Hebei Province, and in part by the National Natural Science Foundation of China under Grant 61290322, Grant 61273222, Grant 61322303, Grant 61473248, and Grant 61403335. This paper was recommended by Associate Editor P. Shi. Y. Yang and C. Hua are with the Institute of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China (e-mail: [email protected]). X. Guan is with the Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2015.2410785
communication channel to a remote manipulator that interacts with an environment. The application of such a system spans multiple fields, the most illustrative being space, underwater, medicine, and, in general, tasks with hazardous environments [1]. Communications often involve large distances or impose limited data transfer between the local and remote sites. Such situations can result in substantial delays. It is well known that the time-delay affects the overall stability of the system [2]. Along this direction, many control schemes were proposed for time delay compensation. The passive method was developed to overcome the instability of teleoperation system caused by constant time delay in [3]. The wave variable idea was proposed in [4] to deal with the time delay issue. A new scattering method was developed in [5] to improve the haptic feedback fidelity. In [6], a simple proportional plus derivative controller was proposed, and a delay-dependent stability condition was deduced. In addition, Hua and Liu [7] proposed delay-dependent linear matrix inequality stability criteria for the closed-loop teleoperation system under asymmetrical time-varying delays. In [8], a transparency performance improvement method was proposed for teleoperation system and master–slave controllers were designed. A new control scheme was proposed for teleoperation system with time-varying delay and input constraint [9]. Even though the above control methods can guarantee the stability of the closed-loop system, the transient-state performance is not considered. For some applications demanding satisfactory transient-state performances, the classical control methods are not up to these tasks. It has been early recognized that transient-state control performance is very important and deserves further research [10]. In most cases, the transient-state performance can be guaranteed by setting some constraints on the system errors [11]–[16]. By setting constraints on the system errors, the transient-state performances: high convergence speed, small overshoot, and an arbitrarily predefined small errors can be provided [11]. The barrier Lyapunov function (BLF) is a novel concept used for dealing with the control problems with constraints [12], [13]. In [12], the output constraint problem was addressed by using a BLF for the single-input single-output (SISO) nonlinear systems in strict feedback form. The time-varying output constraint satisfaction was guaranteed by employing the asymmetric time-varying BLF in [13] for strict feedback nonlinear systems. The output constrained problem had been considered for some practical
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systems [14]–[16]. However, in above literatures, only the exponential convergence can be obtained which means that the errors converge to zero as time goes infinity. In most cases, the asymptotic/exponential convergence cannot satisfy the requirement of high convergence speed from some practical applications. To achieve more precise and faster convergence performances, the finite-time control design will be considered in this paper by using terminal sliding mode control (TSMC) method. Different from the classic linear sliding mode (LSM) which only can provide asymptotical convergence [17], TSMC is an effective finite time control approach [18]. Similar to the LSM technique, strong robustness with respect to uncertain dynamics can be obtained. Moreover, the tracking error converges to zero in finite time [19]. However, the premier papers about TSMC encountered singularity problem [20]. To deal with the singularity problem, a globally continuous nonsingular TSMC method was presented in [21]. Different from [21], an indirect approach was applied to avoid the singularity problem for spacecraft [22]. Afterwards, fuzzy logic system and a continuous nonsingular terminal sliding mode (TSM) were combined to control a teleoperation system in [23]. In [24], the finite time coordination problem was investigated for multiagent systems with TSM. It should be noticed that the transient-state performance cannot be guaranteed just by applying finite time control. Recently, neural networks (NNs) have provided strong alternative to develop universal model free controllers [25]–[28]. The most useful property of NN in control is their ability to approximate arbitrary linear or nonlinear mapping through learning. The robust synchronization problem for 2-D discrete-time coupled dynamical networks was investigated in [29]. State estimation problem for complex networks with uncertain inner coupling and incomplete measurements was studied in [30]. In [31], the NN was applied controlling nonlinear interconnected systems with output-feedback. The adaptive NN (ANN) was used for controlling single-master-multiple-slaves teleoperation in [32]. In this paper, a new TSM control approach with new error transformed variables is designed to provide some superior performances: higher synchronization speed, more accurate convergence, finite time synchronization, safety operating, and so on. The advantages of this paper can be summarized as follows. First, a fast nonsingular TSM-based finite time control scheme is designed for nonlinear teleoperation system with time delay. Compared with the existing control works: P + d control, PD + d control, directed force feedback control, and adaptive control for teleoperation, finite time control can provide some superior control performances: higher convergence speed, more precision convergence, finite time convergence, and stronger robustness. In most cases, the finite time control can meet the requirements of speed and precision for many real applications of teleoperation. Second, the synchronization error constrained is considered in this paper. The synchronization error constrained is significant for real applications. On one hand, it provides a good steady-state performance: the errors will be bounded
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by an arbitrarily predefined small constraint. On the other hand, it provides a good transient-state performance: the errors change in a bounded area and the overshoot can be limited. Finally, the system model parameters and external disturbances are not necessary to be known. The NNs are applied to estimate the system uncertainties. Moreover, the parameter adaptive method is used to eliminate the effect from the NN estimation error and the bounded external disturbances. This paper is organized as follows. Section II presents some preliminary knowledge for the model of the teleoperation and the radial basis function (RBF) NN. In Section III, new NNs based finite time control schemes are given. Moreover, the performance analysis process is also presented in this section. Additionally, simulation and experiment results are shown in Section IV to verify the effectiveness of the proposed control method. Finally, Section V concludes with a summary of the obtained results. II. P RELIMINARY A. Dynamics Models of Master and Slave The Euler–Lagrange equations for n-link master and slave manipulators are given as ⎧ ⎪ ⎪ Mm (qm )¨qm + Cm (qm , q˙ m )˙qm + Gm (qm ) ⎨ + Bm (˙qm ) = τm + Fh (1) M qs + Cs (qs , q˙ s )˙qs + Gs (qs ) ⎪ s (qs )¨ ⎪ ⎩ + Bs (˙qs ) = τs − Fe where i = m/s stands for the master and slave manipulators, respectively; qi (t) ∈ Rn is the vector of joint displacement; q˙ i (t) ∈ Rn is the vector of joint velocity; q¨ i (t) ∈ Rn is the vector of joint acceleration; Mi (qi ) ∈ Rn×n is the positive-definite inertia matrix; Ci (qi , q˙ i ) ∈ Rn×n is the matrix of centripetal and Coriolis torques; Gi (qi ) ∈ Rn is the gravitational torque; Bi (˙qi ) ∈ Rn is the unknown bounded external disturbances, i.e., Bi (˙qi ) ≤ B¯ i , B¯ i is an unknown positive constant; Fh , Fe ∈ Rn are the human-operator force and the environment force, respectively; and τi ∈ Rn is the applied control torque. The robot dynamics given in (1) has the following useful properties [7]–[9]. Property 1: The inertia matrix Mi (qi ) is a symmetric positive-definite function, and there exist positive constants mi1 and mi2 such that mi1 I ≤ Mi (qi ) ≤ mi2 I. I is an identity matrix with the corresponding dimension. Property 2: For all qi , x, y ∈ Rn×1 , there exists a positive scalar ai such that Ci (qi , x)y ≤ ai xy. Property 3: There exists positive scalar μGi such that Gi (qi ) ≤ μGi . Throughout this paper, unless otherwise stated, by a vector norm, the vector two-norm is meant and a matrix norm, the induced matrix two-norm is meant. In reality, similar to many engineering applications, it is impossible or very difficult to obtain an exact dynamics model of the master or slave manipulators, due to the presence of large flexibility, Coulomb friction, wear, and so on.
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3
So, we have Mi (qi ) = Moi (qi ) + Mi (qi ); Ci (qi , q˙ i ) = Coi (qi , q˙ i ) + Ci (qi , q˙ i ); and Gi (qi ) = Goi (qi )+Gi (qi ); here Moi (qi ), Coi (qi , q˙ i ), and Goi (qi ) are the nominal parts, whereas Mi (qi ), Ci (qi , q˙ i ), and Gi (qi ) represent the uncertain parts in the system model. Thus, the dynamics models can be rewritten as ⎧ Mom (qm )¨qm + Com (qm , q˙ m )˙qm + Gom (qm ) ⎪ ⎪ ⎨ = τm + Pm (qm , q˙ m , q¨ m ) + Fh (2) M qs + Cos (qs , q˙ s )˙qs + Gos (qs ) ⎪ os (qs )¨ ⎪ ⎩ = τs + Ps (qs , q˙ s , q¨ s ) − Fe
Define the position synchronization errors between the master and slave as follows:
where Pm (qm , q˙ m , q¨ m ) = −Mm (qm )¨qm − Cm (qm , q˙ m )˙qm − Gm (qm ) − Bm (˙qm ) ∈ Rn and Ps (qs , q˙ s , q¨ s ) = −Ms (qs )¨qs − Cs (qs , q˙ s )˙qs − Gs (qs ) − Bs (˙qs ) ∈ Rn .
e˙ m = q˙ m − q˙ s (t − Ts ) e˙ s = q˙ s − q˙ m (t − Tm ).
B. RBF NNs The motivation of using NNs is to take advantage of NNs global approximation capabilities to handle nonlinearies. Based on the universal approximation theorem, a wide range of nonlinear functions can be estimated by a NN with sufficient neurons. RBF NN is an artificial NN that uses RBFs as activation functions. RBF networks are used widely for function approximation and system control for its simpleness and effectiveness. The RBF NN approximates a continuous function f (X) : Rq → Rp can be expressed as follows: f (X) = W T ϕ(X) + (X)
where Ci and bi are the center and the width of the ith neuron. With the universal approximation property of NNs, for any continuous function f (X), there exists a NN such that f (X) = W ϕ(X) + (X), (X) ≤ N
(5)
where W ∗ is the ideal weight in the approximation; (X) is the approximation error; and N is an upper bound of the approximation error (X). C. New TSM In this section, a new nonsingular fast TSM surface will be defined with new error transformed variables. To streamline the presentation, below, the definition needed in the subsequent analysis is presented as follows. Definition 1: T (6) sig(ζ )a = |ζ1 |a1 sign(ζ1 ), . . . , |ζn |an sign(ζn ) where ζ = [ζ1 , ζ2 , . . . , ζn ]T ∈ Rn ; a1 , a2 , . . . , an > 0 and sign(·) being the standard signum function.
(7)
where Tm represents the signal transmission time delay from the master side to slave side and Ts stands for the transmission time delay in the inverse direction. Then, the velocity errors are given as follows:
It should be noticed that in this paper the i = m/s and j = 1, 2, . . . , n. To achieve some superior performances, some constraints are set for the joint synchronization errors as − kamj (t) < emj (t) < kbmj (t) −kasj (t) < esj (t) < kbsj (t)
(8)
where kamj (t), kbmj (t), kasj (t), and kbsj (t) : R+ → R are positive time-varying boundaries, additionally, −kamj (t) < kbmj (t) and −kasj (t) < kbsj (t). Then, the error transformed variables are defined as
(3)
where X ∈ x ⊂ Rq is the input vector; W ∈ Rn×p is the weight matrix; n > 1 is the number of the neurons; and ϕ(X) = [ϕ1 (X), ϕ2 (X), . . . , ϕn (X)]T , with ϕi (X) being the RBF functions, where the Gaussian RBF function is X − Ci 2 ϕi (X) = exp − , i = 1, 2, . . . , n (4) 2b2i
∗T
em = qm − qs (t − Ts ) es = qs − qm (t − Tm )
emj (t) emj (t) , ξbmj (t) = kamj (t) kbmj (t) esj (t) esj (t) , ξbsj (t) = . ξasj (t) = kasj (t) kbsj (t)
ξamj (t) =
(9)
In the following, the time argument is dropped from above defined variables for presentation compactness. With the new error transformed variables, we can further have that
ξmj = pm emj ξbmj + 1 − pm emj ξamj
ξsj = ps esj ξbsj + 1 − ps esj ξasj
(10)
1 if x > 0 . 0 if x ≤ 0 Differentiating (10) with respect to the time t, yields
where p(x) =
1 − pmj pmj k˙ bmj k˙ amj emj + emj e˙ mj − e˙ mj − ξ˙mj = kbmj kbmj kamj kamj
1 − psj psj k˙ bsj k˙ asj ξ˙sj = esj + esj e˙ sj − e˙ sj − kbsj kbsj kasj kasj (11) where pmj = pm (emj ) and psj = ps (esj ). To streamline the presentation, the following definitions are given as ξbi = [ξbi1 , ξbi2 , . . . , ξbin ]T , ξai = [ξai1 , ξai2 , . . . , ξain ]T , ξi = [ξi1 , ξi2 , . . . , ξin ]T , pi (ei ) = diag(pi1 (ei1 ), pi2 (ei2 ), . . . , pin (ein )), and si = [si1 , si2 , . . . , sin ]T . With above
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defined error transformed variables, the new nonsingular fast TSM surfaces are designed as sm = ξ˙m + αm1 sig(ξm )γm1 + αm2 βm (ξm ) = ξ˙m + λm (ξm ) ss = ξ˙s + αs1 sig(ξs )γs1 + αs2 βs (ξs ) = ξ˙s + λs (ξs )
(12)
where αm1 , αs1 , αm2 , and αs2 are positive constants, γm1 > 1, γs1 > 1; with βm (ξm ) = [βm (ξm1 ), . . . , βm (ξmn )]T and βs (ξs ) = [βs (ξs1 ), . . . , βs (ξsn )]T , the βm (ξmj ) and βs (ξsj ) are defined as follows: ⎧ γm2 sig ξmj if s¯mj = 0 ⎪ ⎪
⎨ or s¯mj = 0, ξmj > μm (13) βm ξmj = 2 ⎪ k ξ + km2 sign(ξmj )ξmj ⎪ ⎩ m1 mj if s¯mj = 0, |ξm | ≤ μm and
⎧ sig(ξsj )γs2 if s¯sj = 0 ⎪ ⎪
⎨ or s¯sj = 0, ξsj > μs βs (ξsj ) = 2 ⎪ ⎪ ks1 ξsj + ks2 sign(ξ
sj )ξsj ⎩ if s¯sj = 0, ξsj ≤ μs
(14)
where 0 < γm2 < 1 and 0 < γs2 < 1; s¯mj = ξ˙mj + αm1 sig(ξmj )γm1 + αm2 sig(ξmj )γm2 , s¯sj = ξ˙sj + αs1 sig(ξsj )γs1 + γm −1 γm −2 αs2 sig(ξsj )γs2 ; km1 = (2 − γm2 )μm 2 , km2 = (γm2 −1)μm 2 , γs −1 γs −2 ks1 = (2 − γs2 )μs 2 , and ks2 = (γs2 − 1)μs 2 ; and μm and μs are small positive constants. The derivative of the sliding mode surface is
s˙ij = ξ¨ij + λ ij ξij 2 2k˙ bij k¨ bij eij 2k˙ bij e˙ ij pij − + 2 eij = e¨ ij − kbij kbij kbij kbij 2 2k˙ aij
1 − pij k¨ aij eij 2k˙ aij e˙ ij + − + 2 eij + λ ij ξij e¨ ij − kaij kaij kaij kaij (15) where λ ij (ξij ) denotes the derivative of λij (ξij ). Remark 1: The choice for synchronization error constraints has great significance for the control performances. The values of the constraints can determine the minimum convergence speed, maximum overshoot, and the maximum convergence errors. In this paper, to guarantee satisfactory synchronization performances, the kamj (t), kbmj (t), kasj (t), and kbsj (t) are defined as decaying functions of time like exponential functions. Remark 2: It should be noticed that, in teleoperation system, the slave manipulator completes task in remote side while the master set in local side. Due to the cognitive limitations of the human operator and lack of complete information about the remote environment, the control performance cannot be guaranteed, and worse still it will lead to collision between the slave manipulator and object. By guaranteeing the synchronization performance, the absolute safety can be obtained, which is significant for the teleoperation system.
Remark 3: Compared with the LSM, the TSM offers some superior properties such as higher precision, more robust, higher speed, and finite-time convergence. However, the classic TSM faces the singularity problem. To deal with the singularity problem, the switching method is used in this paper. The sliding mode of the system is switched between the TSM and LSM, i.e., when a singularity appears, the sliding mode is switched from TSM to LSM. Then, the sliding mode is switched back from LSM to TSM as soon as the system trajectory passes the singularity area. The choice of the ki1 and ki2 make the switching smooth. III. M AIN R ESULTS In this section, new finite time controllers will be designed for the master and slave manipulators with the sliding mode surfaces defined in the previous section. The new finite time controllers not only can provide the finite time synchronization performance, but also can guarantee the synchronization errors of the system moving in the constraints. A. Controller Design In this paper, a new TSM surface is proposed with the new error transformed variables. Then with the new nonsingular fast TSM designed in above section, ANN-based finite time controllers are designed for the master and slave as follows: τm = Mom (qm )¨qs (t − Ts ) + Com (qm , q˙ m )˙qm + Gom (qm ) + τ¯m1 + τ¯m2 τs = Mos (qs )¨qm (t − Tm ) + Cos (qs , q˙ s )˙qs + Gos (qs ) + τ¯s1 + τ¯s2 .
(16)
The specific definitions of τ¯m1 and τ¯s1 are given as em sm ξ˙mT τ¯m1 = Mom (qm ) − em − λ m (ξm ) 1 − ξmT ξm ξm 2 2k˙ bm 2k˙ bm e˙ m k¨ bm em − pm − − + 2 em kbm kbm kbm 2 2k˙ am e˙ m k¨ am em 2k˙ am − + 2 em − (1 − pm ) − kam kam kam em sig(sm )ρ em − Km2
− Km1 sm (ρ−1)/2 ξm ξm 1 − ξ T ξm m
(17)
and
τ¯s1 = Mos (qs ) −
ss ξ˙sT es es − λ s (ξs ) T 1 − ξs ξs ξs 2 ¨ ˙ 2k˙ bs kbs es 2kbs e˙ s − ps − − + 2 es kbs kbs kbs 2 2k˙ as e˙ s k¨ as es 2k˙ as − (1 − ps ) − − + 2 es kas kas kas es sig(ss )ρ es − Ks1 ss − Ks2
(18) (ρ−1)/2 ξs ξs 1 − ξ T ξs s
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Fig. 1.
Teleoperation system with new controller.
where Km1 = diag(Km11 , Km12 , . . . , Km1n ), Ks1 = diag(Ks11 , Ks12 , . . . , Ks1n ), Km2 = diag(Km21 , Km22 , . . . , Km2n ), and Ks2 = diag(Ks21 , Ks22 , . . . , Ks2n ) are positive definite diagonal matrices. Considering the system uncertainties and the external disturbances, the τ¯m2 and τ¯s2 are designed as follows:
sTm 1 − ξmT ξm T pm 1 − pm −1 Mom × + (qm ) kbm kam T s s ˆ sT ϕs (Xs ) − φˆ s τ¯s2 = −W 1 − ξsT ξs T ps 1 − ps −1 Mos × + (qs ) kbs kas (19)
τ¯m2
5
ˆ mT ϕm (Xm ) − φˆ m = −W
−1 ˆ −1 ˆ where φˆ m = 1/2δm1 ψm and φˆ s = 1/2δs1 ψs ; δm1 and δs1 are 2 2 ; positive constants; ψm = mN and ψs = sN mN and sN are the upper bounds of the NN approximation errors, i.e., m (Xm ) ≤ mN and s (Xs ) ≤ sN ; Xm = [qTm , q˙ Tm , q¨ Tm ]T and Xs = [qTs , q˙ Ts , q¨ Ts ]T . The teleoperation system with the new control algorithm is shown in Fig. 1. The NT denotes nonlinear transformation. Remark 4: In this paper, the finite time control design is investigated for the nonlinear teleoperation system with synchronization error constrained. Different from the error constrained control proposed for high-order SISO system, the two-order multiple-input multiple-output (MIMO) nonlinear systems are considered as the master and the slave, respectively. With the MIMO control systems, the controller design becomes more complex. Moreover, the expressions for derivation and analysis are more complicated because there are so many derivations for vectors of variables such as the term −pm 2k˙ bm e˙ m /kbm , which is a vector likes [−pm (em1 )2k˙ bm1 e˙ m1 /kbm1 , . . . , −pm (emn )2k˙ bmn e˙ mn /kbmn ]T . To avoid repetition explanations and save space, this expression will not be explained any more. Remark 5: In this paper, the TSM-based finite time control method and the BLF-based position synchronization error constrained control method are combined to provide some super control performances. The control terms (17) and (18)
are used to counteract the effect from the known nonlinear terms. Moreover, the (19) is designed to deal with the system uncertainties and the unknown external disturbances. Different from the classical TSM finite time control, the term −Ki1 si ei /ξi − Ki2 sig(si )ρ /(1 − ξiT ξi )( ρ−1)/2 ei /ξi is designed to provide the finite time synchronization performance. Remark 6: Different from [23], the new adaptive term φˆ m sm is used to replace the previous term ψˆ m sign(sm ). Compared with the previous term, the new adaptive term is continuous. Therefore, smooth control torque can be obtained. B. Performance Analysis In this section, the rigorous prove process will be presented. Theorem 1: For the teleoperation system (2) with the human operator and remote environment insert forces are zero, i.e., Fh = 0 and Fe = 0 in the presence of system uncertainties and external disturbances, if the nonsingular fast terminal sliding mode (NFTSM) manifold is chosen as (12), the NN tuning law and adaptive tuning law are designed as follows: · sTi pi 1 − pi −1 ˆ ˆi Moi (qi ) − δi2 W + W i = i ϕi (Xi ) kai 1 − ξiT ξi kbi 2 · sTi pi i 1 − pi −1 M ψˆ i = + (q ) − δi3 ψˆ i i oi 2δi1 1 − ξiT ξi kbi kai (20) where i , i , δi1 , δi2 , and δi3 are positive constants. With the continuous NFTSM controllers designed as (16), then: 1) all signals of the closed-loop system are bounded; 2) the time-varying position synchronization error constraints are never violated; 3) the NFTSM manifold sij converges to region i in finite ¯ i in finite time time, the ξij finally converge into region i = min(i1 , i2 ) ¯ i = max μi ,
γi 1
i , 2αi1
γi2
i 2αi2
= Hi /2K¯ i1 , i2 = where i1 ρ T Hi (1 − ξi ξi )(ρ−1)/2 /2K¯ i2 ; K¯ i1 = λmin (Ki1 ) and K¯ i2 = λmin (Ki2 ); λmin (A) denotes the minimum
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eigenvalue of A; and Hi is a positive constant, which will be defined later. Proof: Let us consider the Lyapunov function candidates 1 sT si i 2 1 − ξiT ξi i={m,s} 1 1 −1 ˜ 2 ˜ iT −1 ˜ V2 = tr W W i + i ψ i i 2 2
V1 =
(21) (22)
ˆ i and ψ˜ i = ψi − ψˆ i . ˜ i = Wi − W where W The derivation of the above function V1 is given as follows:
sTi s˙i ξiT ξ˙i sTi si +
2 . 1 − ξiT ξi 1 − ξiT ξi i={m,s}
(23)
With the differentiation of the sliding mode surface, we have
sTi ξ¨i + λ i (ξi ) ξiT ξ˙i sTi si ˙ V1 = +
2 1 − ξiT ξi 1 − ξiT ξi i={m,s} 2 2k˙ bi sTi pi 2k˙ bi e˙ i k¨ bi ei = − + 2 ei e¨ i − kbi kbi 1 − ξiT ξi kbi kbi i={m,s} 2 sTi 2k˙ ai 1 − pi 2k˙ ai e˙ i k¨ ai ei + − + 2 ei e¨ i − kai kai kai 1 − ξiT ξi kai +
sTi ξ T ξ˙i sTi si λ i (ξi ) + i 2 . T 1 − ξi ξi 1 − ξiT ξi
(24)
Applying the system model (2) with Fh = 0, Fe = 0, yields −1 (q )(τ + P (q , q ¨ m) − e¨ m = q¨ m − q¨ s (t − Ts ) = Mom m m m m ˙ m, q Com (qm , q˙ m )˙qm − Gom (qm )) − q¨ s (t − Ts ) and e¨ s = q¨ s − q¨ m −1 (q )(τ + P (q , q (t − Tm ) = Mos ¨ s ) − Cos (qs , q˙ s )˙qs − s s s s ˙ s, q Gos (qs )) − q¨ m (t − Tm ). Employing the new controller (16) and the NN approximation Pi (qi , q˙ i , q¨ i ) = Wi∗T ϕi (Xi ) + i (Xi ), yields sTi 2k˙ bi e˙ i pi −1 − M V˙ 1 = τ ¯ (q ) i i1 oi kbi 1 − ξiT ξi kbi i={m,s} 2 2k˙ bi k¨ bi ei − + 2 ei kbi kbi sTi 1 − pi 2k˙ ai e˙ i −1 + × M − τ ¯ (q ) i i1 oi kai kai 1 − ξiT ξi 2 2k˙ ai k¨ ai ei − + 2 ei kai kai
sTi ξiT ξ˙i sTi si
λ (ξ ) + i
2 i 1 − ξiT ξi 1 − ξiT ξi
sTi pi 1 − pi −1 Moi + + (qi ) Wi∗T ϕi (Xi ) T k k 1 − ξi ξi bi ai sTi pi 1 − pi −1 + + i (Xi )) + Moi (qi )τ¯i2 . kai 1 − ξiT ξi kbi (25) +
sTi Ki1 si sT Ki2 sig(si )ρ − i (ρ+1)/2 T 1 − ξi ξi 1 − ξiT ξi i={m,s} sTi pi 1 − pi −1 Moi + + (qi )Wi∗T ϕi (Xi ) kai 1 − ξiT ξi kbi sTi pi 1 − pi −1 ˆ iT ϕi (Xi ) Moi + − (qi )W kai 1 − ξiT ξi kbi sTi pi 1 − pi −1 Moi + + (qi )i (Xi ) kai 1 − ξiT ξi kbi 2 sT p 1 − p i i ˆ −1 i − M + (q ) φi i oi 1 − ξiT ξi kbi kai sT Ki1 si sT Ki2 sig(si )ρ = − i T − i (ρ+1)/2 1 − ξi ξi 1 − ξiT ξi i={m,s} sTi pi 1 − pi −1 ˜ iT ϕi (Xi ) Moi + + (qi )W kai 1 − ξiT ξi kbi sTi pi 1 − pi −1 Moi + + (qi )i (Xi ) kai 1 − ξiT ξi kbi 2 sT p 1 − p i i ˆ −1 i − + (26) M (q ) φi . i oi 1 − ξiT ξi kbi kai
V˙ 1 =
i={m,s}
V˙ 1 =
Further, with the designed τ¯m1 (17), τ¯s1 (18), and τ¯m2 , τ¯s2 (19), we have −
Since i (Xi ) ≤ iN and for any constants x and y have xy ≤ δx2 /2 + y2 /2δ, where δ is a positive constant. Then, we obtain that sT Ki1 si sT Ki2 sig(si )ρ − i T − i V˙ 1 ≤ (ρ+1)/2 1 − ξi ξi 1 − ξiT ξi i={m,s} sTi pi 1 − pi −1 ˜ iT ϕi (Xi ) Moi + + (qi )W kai 1 − ξiT ξi kbi 2 sT pi 1 − pi −1 i Moi (qi ) φˆ i − + T 1 − ξi ξi kbi kai 2 T pi 1 1 2 1 − pi si −1 M + δi1 + iN + (q ) i . oi 2 2δi1 1 − ξiT ξi kbi kai (27) 2 , With the definitions for φˆ i = 1/2δi1 ψˆ i and ψˆ i = iN we have sT Ki1 si sT Ki2 sig(si )ρ 1 − i T − i V˙ 1 ≤ (ρ+1)/2 + δi1 T 2 1 − ξi ξi 1 − ξi ξi i={m,s} T si pi 1 − pi −1 ˜ iT ϕi (Xi ) Moi + + (qi )W T kai 1 − ξi ξi kbi 2 T pi 1 1 − pi ˜ si −1 M + + (28) (q ) ψi . i oi 2δi1 1 − ξiT ξi kbi kai
The derivation of V2 is given as follows: · · −1 T ˜ ˆ ˜ i i W ˆ i − −1 −tr W V˙ 2 = i ψi ψ i . i={m,s}
(29)
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With the NN tuning law and the adaptive tuning law (20), we have sTi ˜ iT ϕi (Xi ) V˙ 2 = −tr W 1 − ξiT ξi i={m,s} pi 1 − pi −1 Moi (qi ) × + kbi kai 2 T pi 1 1 − pi si ˜ −1 − + (q ) M i ψ oi i 2δi1 1 − ξiT ξi kbi kai δ δ ˜ iT i2 W ˆ i + ψ˜ i i3 ψˆ i . + tr W (30) i i Thus, we can obtain that V˙ = V˙ 1 + V˙ 2 sT Ki1 si sT Ki2 sig(si )ρ = − i T − i (ρ+1)/2 1 − ξi ξi 1 − ξiT ξi i={m,s} δi3 δi1 T δi2 ˆ ˜ + tr Wi Wi + ψ˜ i ψˆ i + i i 2
(31)
ˆ i = Wi − W ˜ i and ψˆ i = ψi − ψ˜ i , we have further, with W T ˜ ˜ iT δi2 /i Wi − W ˜ iT δi2 /i W ˜ i ) and ˜ tr(Wi δi2 /i (Wi − Wi )) = tr(W T T T ˜ i δi2 /i Wi ) ≤ tr(W ˜ i δi2 δi4 /2i W ˜ i + Wi δi2 /2δi4 i Wi ). tr(W Similarly, ψ˜ i δi3 /i ψˆ i ≤ δi3 δi5 /2i ψ˜ i2 + δi3 /2δi5 i ψˆ i2 also can be obtained. Therefore, we have sT Ki1 si sT Ki2 sig(si )ρ V˙ ≤ − i T − i (ρ+1)/2 1 − ξi ξi 1 − ξiT ξi i={m,s} δ δ δ δi1 ˜ i + WiT i2 i4 Wi ˜ iT i2 W + +W 2 2i δi4 2i δi2 δi3 δi3 δi5 2 δi3 2 T 2 ˜i ˜i+ −W W ψ˜ + ψˆ − ψ˜ i 2i δi5 i 2i i i i sT Ki1 si sT Ki2 sig(si )ρ = − i T − i (ρ+1)/2 1 − ξi ξi 1 − ξ T ξi i={m,s} i
T
δi1 ˜ i ηi1 W ˜ i + tr WiT ηi2 Wi − tr W 2 − ηi3 ψ˜ i2 + ηi4 ψi2
+
(32)
where ηi1 = δi2 (2δi4 − 1)/2ii δi4 , ηi2 = δi2 δi4 /2i , ηi3 = δi3 (2δi5 − 1)/2i δi5 , and ηi4 = δi3 δi5 /2i ; δi4 and δi5 are two positive constants and chosen satisfying δi4 > 1/2 and δi5 > 1/2. Finally, we can obtain that V˙ ≤ −V +
(33)
where = min(2K¯ m1 , 2K¯ s1 , R), R = min(2ηm1 m , 2ηm3 m , 2ηs1 s , 2ηs3 s ), and = δm1 /2 + δs1 /2 + tr(WmT ηm2 Wm ) + tr(WsT ηs2 Ws ) + ηm4 ψm2 + ηs4 ψs2 . Then, based on the boundedness theorem in Khalil [22], all signals in the closed-loop system are bounded. Furthermore, we have |ξmj | < 1 and |ξsj | < 1, which implies that the timevarying error constraint is never violated. Next, the finite time synchronization performance will be proved. −1 (qi )Pi (qi , q˙ i , q¨ i ) + Define i = (pi /kbi + 1 − pi /kai )Moi −1 (pi /kbi + 1 − pi /kai )Moi (qi )τ¯i2 . The boundedness of system
7
signals shows that i is bounded and i ≤ Hi with Hi a positive constant. Next, the finite time synchronization performance is analyzed. Consider the Lyapunov candidate function V1 V1 =
1 sT si i . 2 1 − ξiT ξi
(34)
i={m,s}
The derivative of V1 is sT Ki1 si sT Ki2 sig(si )ρ sTi V˙ 1 = − i T − i + i (ρ+1)/2 1 − ξi ξi 1 − ξiT ξi 1 − ξiT ξi i={m,s} si 2 Hi ¯ ≤ − Ki1 − 2si 1 − ξiT ξi i={m,s}
(ρ−1)/2 Hi 1 − ξiT ξi si ρ+1 − K¯ i2 −
(ρ+1)/2 . 2si ρ 1 − ξiT ξi (35) It follows that if si > i , it obtains that V˙ 1 + (1+ρ)/2 σ1 V1 + σ2 V1 ≤ 0, σ1 = min(2K˜ m1 , 2K˜ s1 ), and σ2 = min(2K˜ m2 2(ρ+1)/2 , 2K˜ s2 2(ρ+1)/2 ), where K˜ i1 = K¯ i1 −Hi /2si and K˜ i2 = K¯ i2 − Hi (1 − ξiT ξi )(ρ−1)/2 /2si ρ . In addition, i is a small region containing the origin with large enough Ki1 and Ki2 . Next, the convergence region of ξij will be derived. Because of si ≤ i for the case ξij ≥ μi , i = m, s, j = 1, 2, . . . , n, then we get
γ
γ ξ˙ij + αi1 sig ξij i1 + αi2 sig ξij i2 = ij , ij ≤ i . (36) Then, we have
γ ij
γi sig ξij i1 ξ˙ij + αi1 − 2sig ξij 1
γ ij
γi sig ξij i2 = 0. + αi2 − 2 2sig ξij
(37)
Therefore, as long as αi1 − ij /2sig(ξij )γi1 > 0 and αi2 − ij /2sig(ξij )γi2 > 0, the ξij will converge to the region |ξij | ≤ ¯ i in finite time. This completes the proof. IV. S IMULATION AND E XPERIMENT A. Simulation on Teleoperation of Two-DOF Planar Manipulators In order to show the effectiveness of the proposed control scheme, some simulations are conducted in this section, in which two identical two-degree of freedom (DOF) serial links manipulators are chosen as the local and remote manipulators. Thus, the same controller parameters are set for the master and slave. The specific parameters of the teleoperation dynamics model and the controller parameters are shown in Table I. In Table I, the nominal mass and length are expressed as m1 , m2 , l1 , and l2 . The actual values are expressed as m ˜ 1, m ˜ 2 , ˜l1 , and ˜l2 . The initial joint position of the master and slave are set as qm (0) = [0.2π 0.12π ]T and qs (0) = [0.1π 0.12π ]T and the velocity of master and the slave manipulators are set as q˙ m = [0 0]T and q˙ s = [0 0]T .
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TABLE I S YSTEM PARAMETERS AND C ONTROLLER PARAMETERS
Fig. 2.
Fig. 3.
Fig. 4.
Joint position errors at the slave side with controller (38).
Fig. 5.
Joint positions of the slave with controller (38).
Fig. 6.
Joint position errors at the master side with controller (16).
Human-force input F.
Joint position errors at the master side with controller (38).
The time delays are set as Tm = 600 ms and Ts = 600 ms. To obtain some superior work performances, the synchronization error constraints are set as kaij = (1.5 − 0.05) exp(−1.6t) + 0.05 and kbij = (1.5 − 0.05) exp(−1.6t) + 0.05, j = 1, 2, i = m, s. To simplify the presentation, the same constraints are set on the different joint synchronization errors. To illustrate the control performance of the proposed controller, comparisons will be conducted. We carry out the comparison with the general nonsingularity terminal sliding mode controller, which is designed as follows:
τm = Mom (qm ) q¨ s (t − Ts ) − λ m (em ) − Km1 sm − Km2 sig(sm )ρ + Com (qm , q˙ m )˙qm 1 ˆ mT ϕm (Xm ) − φˆ m sm + Gom (qm ) − W 2σm1
τs = Mos (qs ) q¨ m (t − Tm ) − λ s (es ) − Ks ss − Ks sig(ss )ρ + Cos (qs , q˙ s )˙qs 1 ˆ sT ϕs (Xs ) − φˆ s ss . + Gos (qs ) − W (38) 2σs1 The human insert force is shown in Figs. 2–4 which illustrate the position tracking errors at the master side and slave side with controllers (38), respectively. The joint positions of the slave are shown in Fig. 5. With the controller (16), the position synchronization errors with the constraints at the master
side and the slave side are shown in Figs. 6 and 7. Then in Fig. 8, the joint positions of the slave are given. Comparing Figs. 3 and 4 with Figs. 6 and 7, it is obvious that with the controller (16) proposed in this paper, the synchronization errors are always bounded by the decaying boundaries. Moreover, by comparing Figs. 5 and 8, it can be seen that the positions of the slave links is always restricted in a boundary. This can help the slave to avoid having collision with the remote object. B. Experiment on Teleoperated Pair of Three-DOF PHANToM Manipulator The teleoperation system for the experiments consists of two PHANToM Premium 1.5 A robots (SensAble Technologies, Inc.), which have three-DOF (see Fig. 9). The two robotic manipulators are connected by two computers that are connected via the Internet network. The network environment can be set in the network-simulator block. The controller is established with the MATLAB software.
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Fig. 7.
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Joint position errors at the slave side with controller (16). Fig. 11. Position errors with the constraint with Tm = 200 ms and Ts = 200 ms.
Fig. 8.
Joint positions of the slave with controller (16). Fig. 12. Control torque at the slave side with Tm = 200 ms and Ts = 200 ms.
Fig. 9.
Teleoperation system.
Fig. 13. Joint positions of the master and the slave with Tm = 600 ms and Ts = 600 ms.
Fig. 10. Joint positions of the master and the slave with Tm = 200 ms and Ts = 200 ms.
The constraints are set as kaij = (2.5 − 0.05) exp(−0.2t) + 0.05 and kbij = (2.5 − 0.05) exp(−0.2t) + 0.05, i = m, s, j = 1, 2, 3. In the experiment, we set the master follows a desired trajectory as qmd = 0.1 + 0.2 sin(0.2t), and the slave tracks the trajectory of the master. First, we set the time delay as Tm = 200 ms and Ts = 200 ms. The joint positions are shown in Fig. 10. The joint position error with the constraints
Fig. 14. Position errors with the constraint with Tm = 600 ms and Ts = 600 ms.
are shown in Fig. 11. The control torques for the three joints are shown in Fig. 12. Then, the time delays are set as Tm = 600 ms and Ts = 600 ms. The joint positions are shown in Fig. 13.
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Fig. 15. Control torque at the slave side with Tm = 600 ms and Ts = 600 ms.
The joint position errors with the constraints are shown in Fig. 14. The control torques for the three joints are shown in Fig. 15. As we can see from Figs. 10 and 13, the slave tracks the trajectory of the master in a high speed with initial tracking errors. Then, the slave will always track the trajectory of the master. In Figs. 11 and 14, the ebs1–ebs3 stand for the ξs1 –ξs3 . As we can see in these two figures, the values of ξs1 –ξs3 are always smaller than 1, which means that the synchronization errors satisfy −kamj (t) < emj (t) < kbmj (t) and −kasj (t) < esj (t) < kbsj (t). Therefore, the position synchronization error constraints are not violated. V. C ONCLUSION This paper has studied the issues associated with the finite time synchronization control of bilateral teleoperation system with position synchronization errors constrained. The proposed control algorithm not only imbibes advantages of both methods: 1) finite time control; 2) BLF-based control method, but also provides additional advantage. Both transient-state performance and steady-state performance can be guaranteed in this paper. In simulation, by comparing the new controller with the NFTSM controller, we show that the proposed BLF-based NFTSM controller can achieve super transient-state performances. Finally, the experiments are conducted with different time delays to show the effectiveness of the proposed methods. R EFERENCES [1] P. F. Hokayem and M. W. Spong, “Bilateral teleoperation: An historical survey,” Automatica, vol. 42, no. 12, pp. 2035–2057, 2006. [2] F. B. Li, P. Shi, L. G. Wu, and X. Zhang, “Fuzzy-model-based-stability and nonfragile control for discrete-time descriptor systems with multiple delays,” IEEE Trans. Fuzzy Syst., vol. 22, no. 4, pp. 1019–1025, Aug. 2014. [3] R. J. Anderson and M. W. Spong, “Bilateral control of teleoperators with time delay,” IEEE Trans. Autom. Control, vol. 34, no. 5, pp. 494–501, May 1989. [4] G. Niemeyer and J.-J. E. Slotine, “Stable adaptive teleoperation,” IEEE J. Ocean. Eng., vol. 16, no. 1, pp. 152–162, Jan. 1991. [5] Y. Ye and P. X. Liu, “Improving haptic feedback fidelity in wave variable-based teleoperation oriented to telemedical applications,” IEEE Trans. Instrum. Meas., vol. 58, no. 8, pp. 2847–2855, Aug. 2009. [6] D. Lee and M. W. Spong, “Passive bilateral teleoperation with constant time delay,” IEEE Trans. Robot., vol. 22, no. 2, pp. 269–281, Apr. 2006. [7] C. C. Hua and X. P. Liu, “Delay-dependent stability criteria of teleoperation systems with asymmetric time-varying delays,” IEEE Trans. Robot., vol. 26, no. 5, pp. 925–932, Oct. 2010. [8] C. C. Hua and Y. N. Yang, “Bilateral teleoperation design with/without gravity measurement,” IEEE Trans. Instrum. Meas., vol. 61, no. 12, pp. 3136–3146, Dec. 2012.
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Yana Yang received the B.Sc. degree in electrical engineering from Yanshan University, Qinhuangdao, China, in 2011, where she is currently pursuing the Ph.D. degree. Her current research interests include nonlinear control systems and teleoperation systems design.
Changchun Hua received the Ph.D. degree in electrical engineering from Yanshan University, Qinhuangdao, China, in 2005. He was a Research Fellow with the National University of Singapore, Singapore, from 2006 to 2007. From 2007 to 2009, he was with Carleton University, Ottawa, ON, Canada, funded by the Province of Ontario Ministry of Research and Innovation Program. From 2009 to 2010, he was with the University of Duisburg-Essen, Essen, Germany, funded by the Alexander von Humboldt Foundation. He is currently a Full Professor with Yanshan University. He has been involved in over ten projects supported by the National Natural Science Foundation of China, the National Education Committee Foundation of China, and other important foundations. His current research interests include nonlinear control systems, control systems design over network, teleoperation systems, and intelligent control. He has authored/co-authored over 80 papers in mathematical and technical journals, and conferences.
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Xinping Guan received the B.S. degree in mathematics from Harbin Normal University, Harbin, China, in 1986, and the M.S. degree in applied mathematics and the Ph.D. degree in electrical engineering, both from the Harbin Institute of Technology, Harbin, in 1991 and 1999, respectively. He is currently a Professor with the Department of Automation, Shanghai Jiao Tong University, Shanghai, China. He is a Cheung Kong Scholars Programme Special Appointment Professor. His current research interests include functional differential and difference equations, robust control and intelligent control for time-delay systems, chaos control and synchronization, and congestion control of networks. He has (co)authored over 200 papers in mathematical and technical journals, and conferences.
