1 Introduction
Bilateral teleoperation systems are one of the most wellknown robotic systems which extend human operators’ intelligence and manipulation skills to the remote environments. A typical singlemastersingleslave teleoperation system composes of five parts: a human operator, a master manipulator, communication channel, a slave manipulator and a task environment. The master is directly operated by a human operator to manipulate the slave in the task environment, and the signals (positions, velocities or interaction forces) from the slave are sent back to the master to improve the manipulation performance. Recent years have witnessed considerable advances in the control studies of teleoperation systems, owing to their broad engineering applications in telesurgery, space exploration, nuclear operation, undersea exploration, and so forth.
In practice of teleoperation, there usually exist modeling uncertainties caused by inaccurate parameters of links, unknown load, etc., in the master and slave robots. Besides, the slave robot often interacts with unknown environments, which also leads to uncertainties in the robot manipulators as well. Adaptive control is an effective technique for handling structured uncertainties and has obtained widespread applications in manipulators, where some recent results can be found in Islam2011adaptive ; li2014decentralised ; dehghan2015adaptive ; Li2017adaptive
. Classical adaptive control usually uses tracking error feedback to update the adaptive estimates. The use of the tracking error is motivated by the need to cancel crossterms in the closedloop tracking error system within a Lyapunovbased analysis. However, these methods can only guarantee the state tracking of the system, while the parameter convergence to their true values is kind of problematic. It is well known that the convergence of the parameters to their true values can improve system performance with accurate online identification, exponential tracking and robust adaption without parameter drift. However, these features are not guaranteed unless a condition of persistent excitation (PE) is satisfied
Pan2016TAC . It is well known that the PE condition is very stringent and often infeasible in practical control systems pan2016ccdc .In another line, due to the nature of the longdistance data transmission, communication time delays should not be neglected in the control of teleoperation systems liu2017stability ; liu2016generalized ; Liu2017comparison . The masterslave synchronization yan2017 and stability analysis of teleoperation systems with various kinds of time delays, such as constant delays Spong1989_classical , timevarying delaysHua2010_LMI ; li2015guaranteed ; yang2015_syn ; SARRAS2014_JFI ; zhai2015_IJRNC ; zhai2016_inputsaturation , or stochastic delayskang2013_stodelay ; li2013_stodelay ; li2016_unknownconstraints , and so on, have been hot topics in the study of teleoperation systems in recent years. To handle parameter uncertainties and communication delays in a unified framework, classical adaptive control with sliding mode is introduced into the control of teleoperation systems. For example, Chopra et. al. chopra2008brief proposed an adaptive controller for teleoperation systems with constant time delays and without using scattering transformation. Nuño et. al Nuno2010improvedsyn pointed out the limitation of Chopra’s results in chopra2008brief and proposed a more general adaptive controller for nonlinear teleoperation systems with constant delays, and further with timevarying delays SARRAS2014_JFI . However, these classical adaptive control schemes only achieve asymptotic convergence of masterslave tracking errors, while parameter convergence is seldom considered. With the motivation of using more information to update the parameter estimates to obtain an improved tracking performance, composite adaptive control has been used in teleoperation systems kim2013apt ; chen2016adaptive . However, in the works kim2013apt ; chen2016adaptive , the communication delays were assumed to be constant, which is unrealistic in applications. Moreover, the PE condition or a relaxed sufficient excitation (SE) condition was still required for parameter convergence in these works.
Motivated by the abovementioned facts, in this paper, a new composite adaptive controller is designed for teleoperation systems with timevarying delays. This paper has some unique features and key contributions over the existing works in the following ways. First, contrary to the existing works which guarantee the boundedness of the parameter estimation errors chopra2008brief ; Nuno2010improvedsyn ; SARRAS2014_JFI , this paper achieves convergence of parameters to their true values when the communication delays are timevarying, which then gives rise to an improvement of system performance. Second, a new prediction error is designed to guarantee the parameter convergence, and neither PE or SE condition is required, thus the proposed control scheme is more practical in real applications. Third, the derivatives of the timevarying communication delays are not needed in the controller formulation, which thus makes the controller easier to implement since the information of the delays’ derivatives is not easy to obtain in real applications.
The arrangement of this paper is as follows. In Section 2 the system modeling and some preliminaries are given. In Section 3, the adaptive control with parameter convergence is proposed. Section 4 summarizes the stability results of the closedloop system, while new tracking measures are proposed in Section 5
to evaluate the delayed tracking performance. A simple teleoperation system composed of two robots with two degrees of freedom is given in Section
6 as an example to show the effectiveness of the proposed method. Finally, the summary and conclusion of this paper is given in Section 7.Notations: Throughout this paper, the superscript stands for matrix transposition,
is used to denote the identity matrix with appropriate dimensions.
represents a block matrix which is readily referred by symmetry. denotes thedimensional Euclidean space with the vector norm
, is the set of all real matrices. anddenote the maximum and the minimum eigenvalue of matrix
, respectively. For any function , the norm is defined as , and the square of the norm as . The and spaces are defined as the sets and , respectively.2 Problem Formulation and Preliminaries
Consider teleoperation systems described as follows:
(1)  
(2) 
where are the joint positions, velocities and accelerations of the master and slave devices with or representing the master or the slave robot manipulator respectively. Similarly, represents the mass matrix, embodies the Coriolis and centrifugal effects, is the control force, and finally are the external forces applied to the manipulator endeffectors. Each robot in (1) and (2) satisfies the structural properties of robotic systems, i.e., the following properties Nuno2011_tutorial , Spong2006_book :

The inertia matrix is a symmetric positivedefinite function and is lower and upper bounded. i.e., , where are positive scalars.

The matrix
is skew symmetric.

For all , there exists a positive scalar such that .

The equations of motion of link robot can be linearly parameterized as
(3) where is a matrix of known functions called regeressor, and is a vector of unknown parameters.
In this paper, we assume that the master and the slave are coupled with a communication network with timevarying time delays. Hence, the following standard assumption is used.
Assumption 1.
There exist positive constants and such that the variable communication timedelays satisfies
(4)  
(5) 
3 Adaptive Control Design
Suppose the positions of the master and the slave are available for measurement and are transmitted through the delayed network communication. Let denote the position errors by
(6)  
(7) 
and the velocity errors by
(8)  
(9) 
and then
(10)  
(11) 
we define the following auxiliary variables:
(12)  
(13) 
where are positive definite matrices. By using Property P4, letting
(14)  
for , the following control laws for the master and the slaves are proposed:
(15)  
(16) 
where is the estimate of , .
Substituting the control law (1516) into the teleoperation dynamics (12), we obtain the following dynamics for :
(17) 
where .
Remark 2.
Compared with the existing work sarras2014adaptive which considers timevarying communication delays for teleoperation systems, does not depend on the derivative of position error in the linearparametrization term (14) in this paper, which means that the time derivative of the communication delays are not required to formulate the matrix . This makes the controller more suitable for real applications since the values of the time delays and the derivatives of the time delays are not obtainable in real applications.
A straightforward choice of the adaptive law was first proposed by J. J. Slotine slotine1991applied and has been widely used in adaptive control of teleoperation systems Nuno2011_tutorial ; Nuno2010improvedsyn ; sarras2014adaptive . However, it is pointed that this adaptive law cannot guarantee accurate estimations of parameters. In order to achieve convergence of parameters to their true values, the estimation error should be introduced into the control design. However, the value of is not obtainable since the true value of is not available, and thus a prediction error or its filtered counterpart is used to improve the tracking performance. However, the use of or still needs the PE condition to make the system exponential stable. In the following, we introduce an auxiliary variable such that ,where is a designed lower bounded positivedefinite matrix, to adaptive control of the teleoperation system. Thus, the following adaptive laws are proposed for the master and the slave,
(18)  
(19)  
(20)  
(21) 
where
(22)  
(23) 
and and are two positive constants specifying the lower bound of the norm of and the maximum forgetting rate slotine1991applied , is a positive constant. and are two constant positive definite matrices. From Eqs. (20) and (21), one can show that , , and .
The coefficient is given by
(24) 
where is a constant.
Remark 3.
By (2223), it is easy to see that the prediction error is related to the regressor , which requires the information of joint acceleration. To avoid this, the adaptive law (1821) with filtered torques and filtered regressor could be used. The filtered prediction errors of estimated parameters are defined as
(25)  
(26) 
where is the filtered forces , i.e.,
and can be calculated without acceleration terms by convolving both sides of (3) by a filter kim2013design .
4 Stability Analysis
Denote , , , and define the new state which take values in ,
The following theorem summarizes the stability result of the consider teleoperation system when it is in free motion.
Theorem 4.
Consider the bilateral teleoperation system (12) controlled by (1516) together with the updating law (1821) under the communication channel satisfying Assumption 1, if there exist positivedefinite matrices such that the following linear matrix inequality (LMI) holds, respectively:
(27) 
with
then the following claims hold if the teleoperation system is in free motion, that is, :

all the signals are bounded and the position errors, velocities and the estimation errors asymptotically converge to zero, that is, .

the estimation errors converge into a specified domain within a given time.
Proof.
Defining the following function:
(28) 
It is obvious that is positive definite and radially unbounded with regard to and . By using the Property P2, the derivative of along the trajectory of system (17) is
where , when . Now we give the following Lyapunov functional
(29) 
with
(30)  
(31)  
(32)  
(33) 
where , . Obviously, .
When the external forces , by (12), the derivative of along with the trajectory of system (17) is given by
It is noted that the position error can be expressed as
(34) 
where , , hence the time derivative of along with the trajectory of system (17) is given by
Calculating the time derivative of , one has that
by Jensen’s inequality.
The derivative of is given by
By (36), we have that and . Hence, all the signals are bounded. Furthermore, by (29) and (35), one has that , , , . Thus by (1011), one has that . Now invoking Barbalat’s Lemma, we conclude that . Similarly, we have that by Barbalat’s Lemma. Thus, by (34), we arrive at .
Furthermore, the boundedness of implies that . Thus by the dynamic model (12) and the Properties P1, P3, we otbain that . Hence involing Barbalat’s Lemma again, we arrive at that .
Now we show that the parameter estimation error approaches to zero as . Note that the parameter adaption law (18) implies that
Similarly, the conclusion that is guaranteed by using Barbalat’s Lemma. By now, Claim 1 is established.
To illustrate the transient performance of the teleoperators, we start from the convergence of estimation errors . Obviously, Let . The time derivative of is given by
Thus if , we have . This implies that is always negative when with . So the parameter error will converge to a sphere , where , ,, within a given time. The proof of Claim 2 is concluded.
∎
Remark 5.
Compared to the existing works chopra2008brief ; Nuno2010improvedsyn ; SARRAS2014_JFI , the proposed control scheme guarantees the convergence of parameters to their true values, while neither the PE or SE condition is required. This is accomplished by the boundedness of the matrix in the newdefined prediction error .
When the external forces are not zero, we have the following result.
Proposition 6.
Consider the bilateral teleoperation system (12) controlled by (1516) together with the updating law (1821) under the communication channel satisfying Assumption 1, if there exist positivedefinite matrices such that the following LMI holds, respectively:
(36) 
with
then when the external forces satisfy that , all the signals are bounded and the position errors, velocities and the estimation errors asymptotically converge to zero, that is, . Moreover, the estimation errors converge into a specified domain within a given time.
Proof.
Choose the Lyapunovfunctional candidate as follows:
(37) 
where are defined in (30), (31), (32), and (33), respectively.Thus we have
Integrating both sides of (4) from to , we have
which implies that . Thus following the same line of reasoning in the proof of Theorem 4, we can obtain the conclusion. ∎
5 New tracking performance measures
In this section, we propose new tracking performance measures for bilateral teleoperation systems. A requirement for bilateral teleoperation is that the slave should follow the master’s motion. Specifically, when the slave is in free motion, there is no environmental force between the slave and the environment, and hence the slave should follow the master’ motion very tightly. However, in terms of the tracking performance, it should be noted that the tracking performance is related to the slave’s desired position, i.e., the master’s position, and it’s actual position, i.e., the slave’s position. Hence it is naturally to define a ratio between the position error and the slave’s desired position as the position tracking performance measure , i.e.,
(38) 
where representing the th joint. For simplicity, we assume that when . Similarly, when there is contact between the environment and the slave robot, the force tracking performance is related a ratio between the force error and the contact force, that is,
(39) 
where , and when .
In the presence of communication, the measures should be modified as
(40)  
(41) 
where .
In summary , we define performance measure indexes for each joint
(42)  
(43) 
Obviously, the smaller the indexes are, the better the tracking performance is for each joint.
6 Simulations
In this section, the simulation results are shown to verify the effective of the main results.
6.1 Simulation Setup
Consider a 2DOF teleoperation system with the following dynamics
(44)  
(45) 
where