Finite-time convergence for bilateral teleoperation systems with disturbance and time-varying delays

This paper addresses the robust ﬁnite-time sliding mode control algorithms for nonlinear bilateral teleoperators in the presence of variable time delays and disturbances. In this paper, the designs of terminal sliding control are proposed to guarantee the ﬁnite-time convergence of not only the sliding variables but also the coordination errors on position tracking. The proposed control algorithm is able to ensure ﬁnite-time convergence without requiring the relative velocities and power signal in the communication channel. Additionally, the ﬁnite-time controller is developed with additional terms of sign and exponential function to tackle the variable time delay between two sides in bilateral teleopera-tors. The Lyapunov synthesis principle and additional lemmas are exploited to ensure the ﬁnite-time stability of the closed-loop system in the presence of delays and disturbance. The situations of free-motion, passive, and non-passive human/environment forces are considered for ﬁnite-time convergence without the need of acceleration signals. Finally, numerical examples and experimental studies are carried out to illustrate the effectiveness of the proposed controllers.


INTRODUCTION
Bilateral teleoperators have been developed for decades starting from Raymond C. Goertz who initiated the use of teleoperation system on handling radioactive material. Nowadays, bilateral teleoperation systems, which can provide force/position/video feedback from the remote side to the human operator, have lead to extensive applications on space exploration, robotic telesurgery, and rehabilitation [1]. In the aspects of control and automation, the challenges in bilateral teleoperators are to develop appropriate controllers for the sake of ensuring stability, position tracking, feedback in the presence of dynamic uncertainties, external disturbances, and long-distance communication delays [2][3][4][5][6][7][8]. Recently, the development of control algorithms to deal with actuator saturation, full-state constraint, and finite-time control has become an emerging topic in the communities of control and robotics [8,9]. For bilateral teleoperation systems, [1] presents the solution of finite-time convergence based on the considering terminal sliding mode control as well as integrating more the exponential terms into controller. Moreover, the terminal sliding mode control technique is also employed to handle the finite-time convergence problem [10] and combining with Barrier Lyapunov Function (BLF) based position synchronization error control design. It should be noted that the BLF technique is the key technique for handling the full-state constraints in dynamical systems. Additionally, it is worth emphasizing that the full-state constraints problem was mentioned in robot systems [11]; however, this work eliminated the derivative of system matrix and the disadvantage was dealt with adaptive neural control law in [12]. Furthermore, the difficulties of the these control designs lie in the fact that they employed neural networks to approximate dynamic uncertainties [1,10]. Therefore, it requires a large amount of computation in implementing the previous solutions [1,10].
Several approaches in studying bilateral teleoperation systems were pointed out in recent times for the issues of disturbance, dynamic uncertainties, and time delays [4,5,8,13]. In [14], the saturation function was inserted into the controller to handle the actuator saturation without any back correction part. To guarantee the effective suppression of undesirable time-delayed estimation error, the compensation term has positive timevarying switching gains that gives a reason for overall system stability [15]. The filtered linear and nonlinear position/force mappings were investigated to obtain the appropriate controller in [16]. The control design under the consideration of Laplacian domain and Hamiltonian approach was mentioned in Teleoperation stabilization [17].
The identification and compensation technique of Shunt dynamics [18] and a two-layer consideration using passivity and transparency [19] were considered to handle the problem of time delays in bilateral teleoperation systems. Regarding the elimination of gravity measurement, the traditional Lyapunov method was mentioned in [20]. Additionally, the traditional passivity-based control has been considered in recent times by adding more passive element to guarantee the passive systems [21]. The extensions of bilateral teleoperation systems were mentioned as teleoperating cyber-physical systems were considered to find the appropriate control design including classical nonlinear control [22], neural networks, Linear Matrix Inequalities (LMIs) [5], and distributed formation control [23].
The time delay problem in bilateral teleoperation systems plays an important role among the design of various control techniques. The methods to handle the effect of time-delays have been considered in general case [24][25][26]. In order to tackle this disadvantage, the wave form variable theory along with scattering theory were developed in [8,27]. The idea of this method is to establish the corresponding electrical circuit of bilateral teleoperators obtaining the appropriate control scheme. Optimal control approach was mentioned in [8] based on the linearization technique and solve offline the Ricatti equation. Hence, the method in [8] is difficult for dynamic uncertainties due to analytically solving Ricatti equation.
The approach for time-delay effect under the consideration of Lyapunov-Krasovskii functionals with the advantage of large scope of application has been determined in [2,4,22]. It should be emphasizing that the proposed controllers in [7,28] only consider the situation of constant time delay. In the case of variable time delays, the work in [2,29] proposed the control design based on the appropriate Lyapunov candidate function with integral term and additional lemmas. In addition, the motionforce control was presented by using LMI technique to handle the case of variable time delays [30]. A sliding-mode-based controller was proposed in [31] with time-delay and disturbance observer using exponential functions to solve the variable time delay and estimate dynamic uncertainties. A different approach using the switching control technique was proposed in [32][33][34] to overcome the challenges of varying time-delay and actuator saturation. For the disadvantage of disturbance, the interval observer was implemented by adding more a new state variable with the corresponding estimations of upper, lower bounds in linear systems [35] and nonlinear systems [36].
In this paper, we present comprehensive solutions to tackle the challenges in the above contents, which are external disturbance, dynamic uncertainties, and time delays, in a finitetime manner. Two novel finite-time convergence based sliding mode control algorithms are proposed for bilateral teleopera-tion systems in this work with the main contributions as follows: 1) The terminal sliding mode control law with disturbances is proposed for the direction of finite-time convergence of sliding surface and trajectory tracking control objective. Different from the work also be mentioned robust adaptive controller and finite-time convergence in [1,27,28,37], thanks to employing the property of bilateral teleoeprations, the approximate neural network is eliminated in the proposed control scheme. 2) In the case of variable time delays, a proposed controller using the sign function term is not only mentioned to eliminate the approximate neural network but also handle the finite-time control problem. It is worth noting that the finite-time convergence is mentioned in both sliding variables and tracking errors of two proposed controllers. Additionally, the cases of non-passive and bounded human/environmental force are considered in the proposed controllers.
The rest of this paper is organized as follows. Section 2 formulates a teleoperation system with appropriate assumptions under time-varying delay and uncertainties. In Section 3, the two proposed control algorithm are described and proved. The performance of the developed control scheme is demonstrated in Section 4 via numerical examples and experiments on robotic manipulators. Finally, conclusion remarks are discussed in Section 5.

Preliminary and problem statement
The bilateral teleoperator considered in this paper is composed of a local robot, collocated with a human operator, and a remote robot, executing tasks in the remote environment. By describing the robots in the bilateral teleoperator with the dynamic models being given by Euler-Lagrange equations [7], we have that where, for i ∈ {l , r}, i ∈ ℝ n are the input vectors of the applied torques on the local and remote manipulators, respectively, q i ∈ ℝ n is the general coordinates, that is, the vectors of joint variables, M i (q i ) ∈ ℝ n×n are the inertia matrices, C i (q i ,q i ) ∈ ℝ n×n are the centripetal Coriolis matrices, and g i (q i ) ∈ ℝ n are the gravitational vectors. It is remarkable to note that the torque, h ∈ ℝ n , on the local robot is exerted by human, and the torque, e ∈ ℝ n , denotes the torque exerted from the remote environment. These are the input torques commanding the motion of the local and remote robots in a teleoperator. By considering the model of Euler-Lagrange equations, the following properties can be employed for designing the controller in this paper that Property 1. For a vector, i ∈ ℝ n , the robot dynamics in (1) can be written as parameterizable linearization such that M i (q i )̇i is a vec-tor of system parameters, and Y i (q i ,q i , i ,̇i ) ∈ ℝ n×q is a time-varying matrix to be independent with parameters vector Θ i ∈ ℝ q .

Property 2. Under an appropriate definition of the matrix C
Property 3. M i (q i ) is a symmetric positive-definite matrix to be bounded by the inequality such that i I n ≤ M i (q i ) ≤ i I n with i , i being two positive numbers.
The control objective of this paper is to design applied control inputs l and r such that the tracking problem in bilateral teleoperators is satisfied in both sides despite variable time delays and human, environmental torques h , e . The external torques could be various depending on the application in practice. By denoting s l and s r as the output of the local and remote robots, respectively, the human and environmental torques considered in this paper are Passive torque: where c h , c e are positive constants depending on the initial conditions of the bilateral teleoperators, and h , k h , k e are positive constants with respect to the type of input torques from the human and remote environment. In addition to the external forces exerted on the teleoperators, the signals transmitted between the local and the remote robots could be subjected to unreliable communication network. Therefore, the presence of time delays is not neglectable. By considering the signals transmitted from the local robot (remote robot) to the other sides subject to variable time delays T l (t ) (T r (t )), the coordination (tracking) errors can be given as where 0 ≤ T i (t ) ≤T i < ∞ are continuous and bounded time-varying delays; under the formulation, we denote the upper bound of the round-trip delays asT ∶=T l +T r . By taking the time-derivative of the coordination errors, we geṫ where f i (t ) = 1 −Ṫ i (t ), i = l , r are two positive scalar functions. The aforementioned delays are considered to satisfy the condition that −T l di ≤Ṫ i (t ) < 1, where 0 < T l di ≤ ∞ is the negative of the lower bound of the derivative of the variable time delays, T i (t ). It is noted that in a delay-free case, the coordination errors are given as e l (t ) = q r (t ) − q l (t ) = −e r (t ) anḋ e l (t ) =q r (t ) −q l (t ) = −̇e r (t ).
For the sake of developing the finite-time convergence control for bilateral teleoperators, the following instructive lemmas are required such that is an open neighborhood of the origin. Then, the system is finite-time stable. If U 0 = U , the system is globally finite-time stable, and the system can converge to neighborhood Ω in finite-time T , which can be computed as where 0 < < 1, and max{⋅} denotes the maximum value of the variable.

FINITE-TIME CONTROLLER FOR BILATERAL TELEOPERATORS
In this section, the main results of this paper for the finite-time controller of bilateral teleoperators by using terminal slidingmode control and the additional term of sign function in various cases are addressed. First, the finite-time control is developed for bilateral teleoperators under the influence of passivity human and environmental force (2). The disturbed teleoperator with delay-free case is first investigated by utilizing the terminal sliding-model control method and robust algorithm for dynamic uncertainties (Theorem 1). Subsequently, the issue of variable time delays in communication network is studied with the design of a novel controller to ensure finitetime convergence of the teleoperation systems (Theorem 2). The extensions of the bilateral teleoperators under different human/environmental forces, as described in (3) and (4), are also studied.

Finite-time control for teleoperators with disturbance
In order to guarantee finite-time convergence of both the sliding variables and the coordination errors, the terminal sliding variable is designed with the use of sign function. Additionally, the uncertainty of external disturbances are coped with by using the robust control algorithm. The convergence analysis of the sliding variables and tracking errors are realized sequentially under the advantage of terminal sliding-mode control technique.
For the bilateral teleoperators, the terminal sliding variables for i = {l , r} is considered as where > 0, 0 < < 1, and the sign function is given as The function (x) with constant and x = {x 1 , … , x n } ∈ ℝ n is given as where sign( . It can be seen that the function i = (e i ) is continuous so that the derivative of the sliding variables (11) is expressed aṡ Based on the design of sliding surface with the sign function, we first state a lemma which is required in the proof of the main results in this paper.

Lemma 6.
For any bounded variables = { 1 , … , n } ∈ ℝ n and a positive-definite matrix Σ ∈ ℝ n×n , there exists a positive gain such that Proof. By expanding the T + T ( ) , we have where i,max = max{ 1 , … , n }. Furthermore, based on Lemma 5, we have the estimation Therefore, we have that The control algorithm for the bilateral teleoperators with the use of Property 1 is stated as following where{ ⋅} is the estimation of the enclosed signal/matrix, and k t , k s are positive control gains. The uncertainties of dynamic parameters and disturbances are obtained by using the nominal and additional control inputs as [2] such that where Θ i0 and i0 are the nominal vectors of the uncertainties, and Θ i and i are the additional inputs given as where i , i , d i , and i are positive constants.
According to the dynamic model (1) and the proposed controller (16), the closed-loop control systems are obtained M ṙsr + C r s r + k t s r + k s (s r ) Next, we address the first results of the finite-time control for bilateral teleoperators of the proposed controller with passive force (2). (16), (19), (18) with the passive human/environmental force (2) and the equivalent nominal parameter vector Θ i 0 , bounded term i 0 and positive control coefficients i , d i , i , i . The coordination errors, e l and e r , and all signals of the closed-control system are uniformly ultimately bounded (UUB). Furthermore, the sliding variable s i (t ) and tracking errors, e i (t ), converge to the corresponding regions in finite time.

Theorem 1. Consider the nonlinear bilateral teleoperators given by (1) under the controller
Proof. Consider a positive-definite Lyapunov function for the closed-loop teleoperation system where the storage function of the passive external forces is given as with c e and c h as the constant depending on the initial conditions of the bilateral teleoperators. By taking the time-derivative of V along the trajectories of the closed-loop systems (20) and (21), we get thaṫ By utilizing the skew-symmetric property of the Euler-Lagrange equations, Property 2, the above equation becomeṡ With the use of the additional inputs, (18) and (19), and following the proof in [2], we achieve the following inequalitẏ where is a positive constant based on the control gains in (18) and (19).
Since the human/environmental forces satisfying (2) are bounded, the storage function V pass is also bounded. According to (25) and the bounded V pass , with the use of Lemma 5 we have thaṫ where min{⋅} is the minimum value of the enclose variables, and the constant Ω ′ is given as Therefore, we obtain that s l , s r are UUB based on the following estimation with > 0 that Next, the finite-time convergence is considered as following from (25) thaṫ where k t ,s ∶= min{k t , k s } and 0 < < 1. By defining Ω ′′ = Ω + sup{ V ′ pass } and invoking Lemma 6, the above equation can be rewritten aṡ Hence, based on Lemma 4,V is represented as followṡ where ′ =(1 + )∕2 < 1.
It can be seen that using Lemma 3, (31) implies the finite-time convergence of s l and s r . Furthermore, thanks to the description of terminal sliding surface (11), we can also consider the finitetime convergence of coordination errors. With the purpose of checking the ability of converging to attraction region in finite time of tracking errors e r (t ), e l (t ), we utilize the description of finite time convergence of s l , s r when s i , i = l , r are in the region Δ. Resulting from (11), without loss of generality, we consider only the remote side, i = r. The sliding variable s r gives that which leads to the error dynamics thaṫ e r = − (e r ) +q l (t ) − s r .
Considering the Lyapunov function for the error dynamics as Taking the time-derivative of (34) gives thaṫ V r = e T ṙer = − ‖e r ‖ +1 + e T r (̇q l − s r ).
Because the convergence of s r to the bounded attraction region Δ in finite time, we further obtain thaṫ where s = sup{‖̇q l ‖ + d 0 }. As 0 < +1 2 < 1, by applying Lemma 3, e r converge to the neighborhood of the origin Ω r in finite time T ft r such that where t 0 illustrates the initial time. Consequently, e r will converge to a neighborhood of the origin satisfying that This completes the proof for the finite-time convergence of bilateral teleoperators and the coordination errors. □ Remark 1. As previously stated, it is necessary to satisfy the requirement of relative velocityė i in controller [1,10]. This limits its practical applications in some circumstances because numerical derivatives might be to noisy to compute in the controller. It is obviously different from the existing methods in [1,10], the proposed controller (16) based on terminal sliding-mode control structure is first established to achieve finite time satisfactions in the absence of relative velocity. Additionally, due to the difference in choosing the Lyapunov function candidate (22) in compare with [2] by eliminating the term e T k e e, the convergence of sliding variables and tracking errors is sequentially implemented in Theorem 1.
An extension of Theorem 1 to the case with bounded external force can be stated in the next corollary by following the proof in Theorem 1 and [2].

Corollary 1.
Consider the nonlinear bilateral teleoperators described by (1) under the controller (16), (19), (18) with bounded human/environmental force (4) and the equivalent nominal parameter vector Θ i 0 , bounded term i 0 and positive control coefficients i , d i , i , i . Then, the coordination errors, e l and e r , and all signals of the closed-control system are ultimately bounded.

Finite-time control for teleoperators with time delays
The finite-time terminal sliding mode control is studied in the previous section for bilateral teleoperators with disturbance. Although the stability and finite-time convergence of both the sliding variables and coordination errors are addressed, the finite-time convergence is obtained by using the proposed controller based on the additional term of sign function, i . To avoid the use of sign function in the sliding variable, another design of sliding variables is presented in this section with the consideration of time delays in the communication network between the local and the remote robots.
Let us consider the sliding variable for i = {l , r} as where e l (t ) = q r (t − T r (t )) − q l (t ), e r (t ) = q l (t − T l (t )) − q r (t ), and e(t ) = q l (t ) − q r (t ). It is noted that comparing to (11), s i in (40) uses no sign function. The proposed controller is represented as the following equation that where ∈ (0, 1) is a real constant.
According to the dynamic model (1) eliminating the disturbances and the proposed controller (41), the closed-loop control systems are described as M ṙsr + C r s r + k t s r + k s (s r )  [38]. We assume in this paper that x( ) = ( ), ∈  and all signals belong to  2e , the extended  2 space. Consequently, we state the next result in Theorem 2. Theorem 2. Consider the nonlinear bilateral teleoperators given by (1) without disturbances, for example, l = r = 0, under the controller (41) andΘ i = Θ i0 + Θ i with the robust additional input (18). For the equivalent nominal parameter vector Θ i0 and passive force (2), the position tracking error e l , e r and the manipulator velocityq l ,̇e r , are ultimately uniformly bounded if the control coefficients satisfy the conditions that Furthermore, the convergence time of the closed-loop bilateral teleoperators is finite.
Proof. Consider the Lyapunov-Krasovskii functional given as follows By taking the time-derivative of the functional V (x t ) along the trajectory of the closed-loop system (42) and (43), we get thaṫ It can be seen that due to the additional term −k s (s i ) in proposed control scheme (41), (47). Based on the work in [2], the following estimation can be obtained such thaṫ where Ω = ∑ i={l ,r} ( i i + i d i )∕4 as defined in Theorem 1, and i are given as in (44) and (45).
For the sake of utilizing Lemma 6 for the estimation of the derivation of Lyapunov function, it is necessary to determine that all signals in the closed-loop system are bounded. The fact is that, according to the result in [2] and (48), we imply thatV (x t ) ≤ −k min ||x|| 2 + k , where k min > 0 and k = ( r r + l l )∕4 being positive constant.
Because of the finite value of terms ∫ i e i in the Lyapunov candidate function, as well as the assumption of passive force and all signals of the closed-loop system are bounded, according to (48) we obtain the following estimation by using Lemma (6) thaṫ where 1 , 2 , 3 are positive constants, and Ω ′′′ = sup{k t s T l s l + k t s T r s r + Δ ′ − Δ ′′ } with Therefore, based on Lemma 4, the inequality (49) can be represented as followṡ It can be seen that using Lemma 3, according to estimation (50) we imply the finite time convergence of not only s l , s r but also e r (t ), e l (t ). □ Remark 2. It is worth emphasizing that due to no consideration of property of BTs, the controller in [1,27,28,37] employed the Neural Networks to approximate the dynamic uncertainties and disturbance. In contrast to the existing methods in [1,27,28,37], the proposed control algorithms (16), (41) utilize Property 1 and (19) to guarantee the tracking errors and sliding variables are UUB without neural networks. The fact is that Property 1 is utilized to lump uncertainties into vector Θ i . For this reason, the adaptation laws (18,19) are proposed to complete the proposed controllers (16,41) without using neural network structure.
Remark 3. Unlike the work in [2], this work is extended to obtain finite-time convergence in not only the sliding variables but also tracking errors. Furthermore, the constraint forces h , e in (1) are considered in many related works, such as constraint forces can be computed to implement the motion/force control [39], the environmental torque is estimated by RBF neural network structure [37]. On the other hand, the Neural Networks (NN) was also employed in approximating the uncertainties including unknown parameters and disturbances in remote, local sides [37]. Moreover, the finite time tracking problem has not been discussed and the tracking problem was only considered for each sides [37]. The transmitted information required not only the trajectories q r (t ), q l (t ) but also the power signals in the communication channel [37]. However, it is obviously different from the existing methods in [37,39]; the proposed finite-time controllers handle the influence of constraint forces without estimating them.
It is worth emphasizing that although this work proposes the controllers in (16), (41) in the absence of operating torque h (t ) and environmental torque e (t ), the stability is still guaranteed in the cases of passive force (Theorem 1, 2) and non-passive force (Corollary 2). As a result, the transmitted information is only considered with trajectories q r (t ), q l (t ) and without power signals in the communication channel. Regarding the application of SMC method, authors [40] presented predictive-algorithmbased discrete time SMC for linear system. The work in [10] developed the finite-time convergence based on terminal SMC to fulfill position synchronization error control design with constant time delays. In order to deal with the challenge of variable time delays, the Lyapunov function candidate is proposed as in Equation (46) with more integral term to be utilized for variable time delay. The control scheme in [4] was developed under traditional Lyapunov technique using the position velocity to be combined error signals. In contrast to the existing methods [4], the proposed method discusses the finite time controllers for BTs with variable time delay and without relative velocity. Additionally, Lemma 3 is able to determine that the chattering phenomenon and tracking problem, convergence time, the bound of attraction region can be adjusted by coefficient in the sliding variable (40).

Corollary 2.
Consider the nonlinear bilateral teleoperators given by (1) without disturbances, for example, l = r = 0, under the controller (41) andΘ i = Θ i0 + Θ i with the robust additional input (18). For the equivalent nominal parameter vector Θ i0 and non-passive force (3), the teleoperation system is stable if the control coefficients satisfy the conditions (44) and (45). Additionally, the position tracking error e l , e r and the manipulator velocityq l ,̇e r are bounded.

Numerical examples
The proposed finite-time controllers in Sections 2.1 and 2.2 are demonstrated to confirm the efficiency via numerical simulations in this section. The local and remote robots are considered as two-degrees of freedom (2-DoF) planar robotic manipulators whose dynamics are referred to [41]. In the simulations, the teleoperation is in free-motion for t = 0 to 10 s where the human operator applies no force on the local robot. By the proposed controller, the local and remote robots coordinate to each other. After t = 10 s, the human operator exerts a spring-damper force [3, 6 43] to manipulate the local robot between different set-points with the spring and damping gains selected as 50 Nm/rad and 2 Nms/rad. For t = 10 to 25 s, the local robot is commanded to a configuration such that the x-direction of the end-effector on the remote robot contacts a wall in the remote environment. Thus, the remote robot is blocked under the environment force, and the local robot, operated by the human operator, is also under the influence of the external force. After t = 25 s, the human commands the local to another configuration where the remote robot moves away from contacting the wall.
For the control approach in Section 2.1, the control gains are considered to be identical for both robots such that = 2, k t = 20, k s = 40, and k d = 25. The bilateral teleoperators are implemented with the initial conditions q l (0) = [1.1, 0.9] T rad, q r (0) = [1.0, 1.2] T rad,̇q i (0) = 0 rad/s, andq i (0) = 0 rad/s 2 . The joint coordinates and tracking errors are illustrated in Figure 1. It shows that the proposed system is stable and able to reach the desired configuration commanded by the human operator, where the tracking errors and velocities are ultimately bounded. Furthermore, if the bilateral teleoperation system in free motion ( h = 0 and e = 0), then the tracking errors are ensured to converge to neighborhood of zero in finite time. Additionally, during hard contact, both h and e are not zero, the tracking errors are relatively large due to force reflection from the remote robot to the human operator. The human and environment torques are illustrated in Figure 2.
To highlight the improvement of convergence speed and precise tracking ability of the controller in Section 2.2, we make the comparison of the two control methods as follows: The simulation scenarios are similar to the above simulation except for that there exist the varying time delays given as Figure 3. In particular, these time-varying delays are assumed to be T l = 0.5 + 0.02sin(3t ) + 0.03sin(4t ) s, T r = 0.51 + 0.1sin(2t ) s. Additionally, in the interaction duration between the slave manipulator and the remote environment, there is a sudden disturbance affecting the slave manipulator as follows: External torques from the local human and remote environment to the robots in the simulation of a previous work [42] The simulation results of C1 are shown in the Figures 4-7, and those of C2 are shown in Figures 8-11. Firstly, the most noticeable difference between Figures 4 and 8 is that it is more easy in C2 than in C1 for the human to feel a sudden change of the external disturbances dist in the remote environment in duration 18(s) ≤ t ≤ 21(s). This confirms that the proposed  Control torques in the simulation of a previous work [42] control method provides more certain degree of transparency performance during teleoperation compared to the method in [42]. Secondly, a comparison of Figures 5 and 6 and Figures 9 and 10 reveals that C2 is better than C1. To specify, the transient response of the former is faster than that of the latter, and further the former has better capacity of coordinating position than, that is, has more precise tracking ability than, the latter as seen from the simulation. Next, the control inputs of both C1 and C2 are illustrated in Figures 7 and 11, respectively, which are bounded, smooth, and feasible. A comparison between Figures 8 and 10 points out that the tracking errors e are relatively large during the slaver robot hard contacting with the environment. This situation is due to the force reflection from the remote robot to the local robot against the human force pushing toward the desired configuration.

Experiments
To show the effectiveness and performance of the proposed finite-time teleoperator in practice, the experiments have been implemented by using two robotic manipulators, called PHAN- ToM Omni. PHANToM Omni is a haptic device with six revolute joints, six-degrees of freedom (6-DoF). In the implementation, the first three joints are fully actuated, while the three remainder joints are constraints. The proposed finite-time control program is written in C++ language with Open-Haptic API being utilizing to acquire joint position/velocity and apply the computed torques from the proposed method. The experimental setup is illustrated in [2], and the robots are connected to a personal computer (  The experiment of the proposed finite-time controller with time delays, stated in Section 2.2, is conducted under the control gains k t = 10, k d = 3, k s = 30, = 0.5, = 50, d i = i = 10, and i = 0.1. The experimental results of this control method are illustrated in Figures 15-17. Under the presence of time delay as in the simulation, the tracking errors are shown to be able to converge to a neighborhood of zero in finite time in the absence of external force. Additionally, if the human operator exerts a force on the local robot so that the tracking error between the remote and the local robots are relatively large, then the control inputs are bounded and smooth. The stable and tracking performance from the experimental results demonstrate the efficacy and performance of the proposed finite-time controller for bilateral teleoperators.

CONCLUSION
This paper has proposed two novel control methods for bilateral teleoperation systems to cope with internal uncertainties, external disturbances, and communication delays. These two methods ensure the convergence of tracking errors to a neighborhood of zero in a finite time and, respectively, correspond  to delay-free case and varying time delay. The cases of nonpassive and bounded human/environmental forces are considered under the proposed finite-time control algorithm. Additionally, to eliminate the acceleration measurements, novel sliding variables have been defined with the proof of stability and finite-time convergence. Numerical simulations and experiment results are performed to show the effectiveness of the proposed approaches. The future work of this research encompasses the study of transparency in the presence of time delays and system uncertainties with the guarantee of finite-time convergence.