I Introduction
Bilateral teleoperation systems which allow human operators to extend their intelligence and manipulation skills to remote environments are widely used in applications such as telesurgery, space exploration, nuclear operation, underwater exploration [1]. A typical bilateral teleoperation system with a configuration of singlemastersingleslave (SMSS) involves two robots which exchange position, velocity and/or haptic information through a networked communication channel. However, for some complex tasks, teleoperation of one slave robot may fail in completing such tasks where multiple manipulators in cooperation are required. Hence, teleoperation of multiple slave robots has emerged to cope with a new set of applications incompatible with SMSS configurations [2, 3, 4]. Teleoperation of multiple slaves can complete multiple tasks in a shorter time, covering largescale areas, and with the ability to adapt to single point failures more easily, and hence effectively encompass a broader range of surveillance tasks, military operations, and rescue missions, and so on [5]. Teleoperation of multiple slaves can be manipulated by one human operator through one master, or by multiple human operators through multiple masters. In this paper, we restrict our attention to the former one, that is, teleoperation systems with singlemastermultipleslaves (SMMS) configurations.
It should be noted that due to the distance of teleoperation, communication time delays are inevitable. The classical approaches to deal with delayed bilateral teleoperation systems are passivitybased approaches [6], which are mostly based on scattering theory [7] and wave variable formalism [8]. Some other passivitybased controllers relying on damping injection [9], and adaptive control [10, 11] were also developed recently. For the passivitybased control methods, the assumption that both of the human operator and the environment be passive was imposed and thus it is restrictive. To remove the passivity assumption about external forces, inputtostate stability/inputtooutput stability (ISS/IOS) theory is introduced into the control design and stability analysis of teleoperation systems. By applying ISS/IOS theory, Polushin et. al. [12, 13] firstly designed PDbased controllers for teleoperation systems, and proved the stability of the closedloop system with communication delays by constructing two inputtostate stable subsystems. However, in [12, 13] the positions of the master and the slave will exactly converge to the origin, which, however, should not be expected in applications of teleoperation systems. In other words, position synchronization between the master and the slave robots are expected, which do not necessarily imply that the positions should converge to the origin. To overcome this limitation, Zhai et. al. [14] investigated a new IOS framework based on stateindependent IOS for nonlinear teleoperation systems with asymmetric timevarying delays, where a switched filterbased control method was developed. Some other advanced control strategies like predictive control [15], optimal control [16, 17], intelligent control based on fuzzy logic [18, 19]
[20, 21, 22], prescribedperformancebased control [23, 24], etc., have also been developed to deal with other various aspects of teleoperation systems with delays, such as finitetime stability [25, 23], guaranteed synchronization performance [16, 23, 24, 26], input saturation [27, 28, 29], model uncertainties [30, 31], brainmachineinterfacebased teleoperation [32]. However, most of these results are for SMSS teleoperation systems. Some but very limited ones considered teleoperation systems with SMMS or multiplemastersmultipleslaves (MMMS) configurations. Sirouspour [33] firstly studied the problem of MMMS teleoperation, while communication delays were neglected. Based on twosubsystemsdecomposition method, [34] investigated adaptive neural network control for SMMS teleoperation systems with time delays and input deadzone uncertainties. [35] addressed fuzzy control of MMMS with asymmetric timevarying delays and model uncertainties.Like many works on networked systems [36, 37, 38], communication bandwidth limitation should not be neglected in system design and synthesis. In many SMMS teleoperation systems with spatially distributed slaves, the output of multiple slaves can not be transmitted to the master simultaneously because of bandwidth limitation in the communication network. Thus, it is desirable to stipulate that there can be only a limited number of communication slaves to access the network at the same time. Actually, in many practical cases, the communication is orchestrated by a scheduling rule called a protocol, by which the network sources can be properly scheduled. Specifically, in some practical teleoperation systems, only one sensing node (master or slave) is allowed to transmit its data over the communication network at a time, even though there are lots of nodes in the considered systems. However, to the best knowledge of the authors, very few works consider this limitation for teleoperation systems.
Along with another line, one commonly used assumption in teleoperation design is that the data transmission between the master(s) and the slave(s) is continuous in time, which is very restrictive in real applications. The continuoustime information exchanges are quite energyconsuming since the communication channels are always occupied in high frequencies [39, 40], and thus, this will increase the design and implementation cost as well. In fact, communications are likely to occur over a digital network in practice, such that the information is exchanged at discrete time intervals. Thus it is desirable to provide new results for teleoperation systems with discretetime data transmission.
Motivated by the aforementioned observations, this paper aims to solve the synchronization problem for a class of SMMS teleoperation systems under scheduling communication with discretetime information exchanges and timevarying communication delays. Thus, the existing continuoustime controller and stability criteria for teleoperation systems are inapplicable for solving the considered problem. At each sampling instant, only one slave is allowed to transmit its current information to the master side over communication network. The data transmission through the communication channel is discrete, and thus the transmitted signals are kept constant during the sampling period. An explicit expression of the designed controller is given. Only the samples of the position variables of the remote manipulators at discrete time instants are needed, and thus the amount of transmitted synchronization information greatly reduces and the efficiency of bandwidth usage increases. This makes the tracking of teleoperation systems more efficient and useful in reallife applications. Two kinds of scheduling protocols, i.e., RoundRobin (RR) scheduling protocol and TryOnceDiscard (TOD) scheduling protocol, are provided and employed. By constructing LyapunovKrasovskii functionals, some efficient stability criteria in terms of linear matrix inequalities (LMIs) are obtained for the closedloop SMMS teleoperation systems under scheduling communication network.
The rest of this paper is organized as follows. The problem formulation and some preliminaries are given in Section II. Controllers for SMMS teleoperation systems with scheduling network and the stability analysis for the closedloop system under RR and TOD scheduling protocols, respectively, are provided in Section III. Then, some simulation results are given in Section IV for illustration. Finally, Section V concludes the paper.
Notations: Throughout this paper, the superscript stands for matrix transposition. denotes the
dimensional Euclidean space with vector norm
, is the set of all real matrices. represents a block matrix which is readily referred by symmetry. represents the set of nonnegative integers while is the set of positive integers. 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.Ii Problem Formulation and Preliminaries
This paper is concerned with bilateral teleoperation control of multiple manipulators by a master manipulator over scheduling communication as shown in Fig. 1. The dynamics of the SMMS teleoperation system consisting of a single
degree of freedom (DOF) master manipulator and
() DOF coordinated slave manipulators can be described as follows:(1) 
(2) 
where are the joint positions, velocities and acceleration measurements of the master/th slave devices with , respectively. represent mass matrices, embody Coriolis and centrifugal effects. are the control forces, and finally are external forces applied to the manipulators. Each robot in (1) and (2) possess the structural property of robotic systems, i.e., the following properties [6], [41] with , respectively:

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

If and are bounded, the time derivative of is bounded.
It is assumed that the data is transmitted from the master to the slaves and from the slaves to the master over delayed communication with variable and symmetric delays, but only the data of one manipulator can be transmitted from the local side to the remote side at one time due to the bandwidth limitation of the communication network. Thus the backward communication channel is orchestrated by a scheduling rule called a protocol. This framework is described in Fig. 2.
Suppose the sampling at the master and at the slaves are synchronous, and the sampling instants , , are a sequence of monotonously increasing constants and satisfy , for and . At each sampling instant , only the sensors at one slave robot are allowed to transmit the sensed information over the communication network according to some scheduling protocols. Denote by the time needed to transmit the sampling data at the instant to the remote side. As depicted in Fig. 2, the master’s output is sampled at time instant , this sampled information reaches the slave side and updates the ZeroOrder Hold (ZOH) at time instant . Similarly, if the output of one of the slaves is sampled at time instant , this information updates the ZOH in the master side at time instant . The sampling intervals and communication delays are assumed to have certain bounds which are precisely stated in Assumption 1.
Assumption 1.
There exist positive constants and such that the sampling intervals and communication delays hold for all :

,

.
Note that in Assumption 1 the communication delays are not required to be small with . As in [38] and [42], we allow the communication delays to be nonsmall provided that the old sample cannot get to the remote side after the most recent one. The time span between the instant and the current sampling instant is bounded with .
Remark 2.
Note that in this paper we assume that the signals are transmitted only at each sampling instants, thus the communication delays can be depicted as piecewiseconstant functions. This assumption implies discretetime information exchange between the master and the slaves, which is quite different from the previous works. Discretetime information exchange would improve the communication efficiency and is energysaving since the communication channel is not required to be occupied in high frequencies. Furthermore, discretetime information exchange is more practical in real applications.
Remark 3.
Actually, the range of communication delays may vary in an interval with nonzero lower bound in practice. In this paper, for simplification of analysis, we assume that the lower bound of the communication delays is zero. Our results can be easily extended to the case with nonzero lower bounded delays. Some studies for interval communication delays can be found in [43, 44].
For the SMMS teleoperation system (12), we assume that the slave robots must maintain a distance and orientation from the formation’s geometric center at all time. Denote by for the th robot’s distance from the formation’s center , we assume and for all . Furthermore, we assume that
This paper aims to offer a stable bilateral control framework that guarantees masterslaves synchronization under scheduling communication. In summary, two control objectives are provided:

Position synchronization with coordinated motion between the master and the slaves should be achieved: the slave robots follow the master’s command while maintaining a relative distance with respect to the formation’s center at all the time, . In other words, the position errors sastify , .

Force tracking between the master and the slaves should be guaranteed, that is, the contribution of environmental forces should be reflected to the operator under steadystate conditions, i.e., .
Iii Controller Design and Stability Analysis
Suppose that the positions and velocities of the master and the slaves are available for measurement. In this paper, we allow only the positions of the local manipulators to be transmitted to the remote side. Since only one slave’s position is scheduled to be active to update with the current information at each sampling instant , the updating law of the most recently received position information at the master side is given as follows:
(3) 
where is the active scheduled slave at the sampling instant and will be determined by some scheduling protocols provided later.
As described earlier in Section II, we suppose that the control inputs from the remote side of the teleoperation system are generated by ZOH devices, and the controllers and the ZOH devices update their outputs as soon as they receive the new data, then for , the P+dlike controllers are proposed as follows:
(4)  
(5) 
where , is provided in (3), , , .
Remark 4.
Note that in (45) only the position sampling signals need to be transmitted to the remote side through the communication channel. Thus the amount of transmitted synchronization information greatly reduces and the efficiency of bandwidth usage increases, which makes the teleoperation systems more efficient and useful in applications.
In the following, the stability analysis of the teleoperation system (12) under the control (45) is provided. Specifically, the teleoperation system under RR and TOD protocols is studied, respectively.
Iiia RR protocol
Under RR scheduling protocol, the measurements of the slaves are transmitted one after another periodically, that is, for each , the active slave to access the communication network shall satisfy
(6) 
while repeating of the active node index within a circulation is prohibited, that is
(7) 
Thus, for , the master side is represented as
According to the timedelay approach [39], denote for an arbitrary given , we have
Therefore, when all the measurements are transmitted at least once, i.e., for , the master controller (4) under RR protocol (67) can be represented as
(8)  
Similarly, the slave controller (5) under RR protocol (67) can be rewritten by denoting :
(9)  
where
Substituting (89) into (12), we obtain the closedloop teleoperation system with the following dynamics for :
(10) 
where , . The initial condition for (10) has the form of , , . The following theorem summaries the main results for the stability of the closedloop system (10) under RR scheduling protocol (67).
Theorem 5.
Consider the closedloop teleoperation system (10) with the RR scheduling protocol (67). If there exist , , such that the following LMIs
(11) 
with are satisfied, then the following claims hold:

if , then for all .

if , then all the signals are bounded for all , and , which implies that and as .

the force tracking is guaranteed as the teleoperation system is in steadystate, i.e., .
Proof.
To develop the stability condition for the closedloop system (10) under RR protocol (67), we use the following LyapunovKrasovskii functional for :
(12) 
where
The derivative of along with the trajectory of system (10) for is with
Note that
(13) 
and
(14) 
and hence one has
(15) 
where is given in (11) and
(16) 
We first consider the case that for . The LMIs (11) yield , which implies that for , . Furthermore, by (15), we have
(17) 
where , then integrating both sides of (17) from to , we have
(18) 
Clearly, and for all and thus for all . The fact that is radially unbounded with respect to shows that for . Now by (13) and (14), the closedloop dynamics (10) and Properties P1P4, we know that . Thus by Barbalat’s Lemma, we have . Furthermore, by (10), one has that for any ,
thus for since for . Thus by Barbalat’s Lemma. Now by (10), it can be established that as , and thus by (13) and (14), we have as .
If , that is, there exist positive constants such that , then the derivative of the Lyapunov functional (12) is given by
(19) 
Integrating both sides of (19) from to , we have
(20) 
For all , one has , which yields that
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