Adaptive Finite-Time Control Scheme for Teleoperation With Time-Varying Delay and Uncertainties

The communication time delay and uncertain models of robotic manipulators are the major problem in the teleoperation system, which can reduce the performance and stability of the system. This article proposed a novel finite-time adaptive control scheme for position and force tracking performances of the teleoperation system. First, a combined auxiliary error system with position and force tracking errors is designed. Second, a velocity feedback filter is introduced, and a new auxiliary variable function with finite-time structure is designed for controller design. The radial basis function neural network (RBFNN) is applied to estimate the uncertain parts. Then the finite-time adaptive control scheme and adaptive laws are given. Third, based on the Lyapunov method, stability and finite-time performance are demonstrated. And finally, the simulation and experimental studies (with Phantom Ommi devices) are performed and demonstrate the effectiveness of the proposed control scheme on teleoperation position/force tracking.

substances [5], and so on. In the bilateral teleoperation system, the position and force signals are transmitted bidirectionally through the communication channel. The purpose of the control methods for the teleoperation system is to guarantee the stability of the system and implement satisfactory position tracking and force feedback performances [6].
However, some challenges often exist in controller design process of the teleoperation system. The time delay in communication channel is unavoidable, this would degrade the operating performance of the telerobotic system and even make the system instability. While, uncertainties of master/slave robots, operator, and environment dynamics, external disturbances, etc. can also make the system performance worse.
For decades, many control methods have been proposed to cope with these issues. The experimental methods [7], supervisory control [8], and software-based teleoperation method [9] are first used for the control of teleoperation systems [10]. Then the passivity-based approach is applied for control method design and stability investigation of the telerobotic system with time delays [11]. Since then, many passivity-based control methodologies have been proposed for position tracking control and bilateral control, such as scattering approach [12], [13] wave variables control [14], and improved wave variables control [15], [16]. Furthermore, some control strategies based on Lyapunov theory have been developed, including PD control [17], [18], sliding mode control [19], H ∞ control [20], output feedback control [21], and so on.
Adaptive control is a useful method to design the high performance control scheme for teleoperation system with poor structure of robot dynamic and operator/environment models. Li et al. [22] proposed a fuzzy adaptive control strategy for the teleoperation system, and multiple fuzzy logic systems are used to estimate multiple model uncertainties. In [23], the uncertainty parts in the telerobotic system are defined as a function, and the fuzzy logic systems are used to approximate this function. Yang et al. [24] proposed the prescribed performance control based on adaptive neural networks. In [25], the wave variable structure and radial basis function neural network (RBFNN) are combined for teleoperation control with time delays and model uncertainties. In [26], two different adaptive control schemes based on RBFNN are developed for position tracking in the teleoperation system. In [27], a projection mapping based on This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ saturation is introduced, and the adaptive sliding mode control laws based on the RBF neural network are designed. In [28], a novel adaptive control scheme containing the RBFNN and the robust mechanism is presented. In [29], [30], and [38], the dynamic uncertain parts are, respectively, be linearly parameterized, and the adaptive controller and adaptive laws are designed. In addition, the principle of estimating the telerobotics uncertain model by using the fuzzy logic system and neural network is based on the universal approximation theorem. And the Lyapunov method is applied to design the controllers and adaptive laws.
However, the proposed adaptive methods above can only guarantee the stability and convergence of the system when the time goes to infinity [31]. In practical applications, it is expected that systems such as machinery and robot system can be able to achieve finite-time control performance [32]. Many finite-time control methods have been developed and applied to the system control process, which illustrates the advantages of finite-time control in terms of robustness, converge precision, and antidisturbance capabilities [31], [33]. In recent years, the adaptive finite-time control-based fuzzy logic system and neural network are studied and employ to deal with the problems of nonsmooth nonlinear systems [34], uncertain nonlinear models, input delay [35], unknown backlash-like [36], and odd rational numbers powers system [37]. These successful applications fully illustrate the effectiveness of the adaptive finite-time control method. Therefore, the finite-time control approach [39]- [45] is also introduced into controller designing for providing the fast convergence, robustness, and high performance in teleoperation system. In [39] and [40], the novel fuzzy finite-time adaptive controllers are proposed for position synchronization performance in telerobotic system. The adaptive fuzzy method is employed to deal with the uncertainties in master and slave models. In [39], a new finite terminal sliding mode (TSM) is designed and the reaching time is given. In [40], the anti-windup compensator and switched filter system are introduced for finite-time controller design. Yang et al. [41] and Wang et al. [43], respectively, developed two finite-time controllers based on different nonsingular fast TSMs. Neural network is also introduced to estimate the model uncertainties, and the finite-time control scheme with error constrained [42], new integral TSM [44], and jittering free scheme [45] are, respectively, developed. It is noted that the time-varying delay is not considered in [41]- [43].
In real applications, the effective force tracking of the master and slave robots can make the system have better operational performance. Some research works have studied the force/torque tracking of the teleoperation system. In [46] and [47], the PD control and human torque compensation method are proposed for position tracking and force tracking. In [48], a feedback passivity controller is proposed, and the environment force measured by the force sensor is applied for force feedback. In [49], the torque error is introduced to construct the auxiliary function, and adaptive control scheme with position/torque tracking control are, respectively, developed. In [50]- [54], differential algorithms have been developed to estimate the environment force and external force, which are utilized for force compensating control. In [55], the SALS method is introduced to identify the parameters of dynamic and environment. In [56], the effect of time delay on model identification is considered, and a new control strategy for teleoperation control with large time delay is presented. It can be seen that the force/torque tracking control is mainly based on the measurement of the force/torque sensors and the force/environment estimation using the observer. However, using the observer to estimate the force/torque requires the accurate dynamic description of the robot. If there have the model uncertainties, it is obvious that this force/torque estimation method cannot be achieved.
Nevertheless, there still have some problems which are not considered in most of the above research works. First, many developed adaptive approaches mentioned above are implemented to provide position tracking performance, but force/torque tracking of master-slave robots is not considered (such as in [22]- [29] and [38]- [45]). Second, the position acceleration signals are introduced into the controller for control torque calculation (such as in work [22]- [24], [26]- [30], [40]- [42], [44], [45]). On the one hand, the differential of position error is applied for sliding mode surface and auxiliary variable function designing; on the other hand, the model uncertain part containing the mass inertia matrix is directly estimated by the adaptive method. In such cases, it can inevitably introduce the acceleration signals of master and slave robots. Finally, the finite-time control has not been used for position and force tracking control in existing research works.
In this article, we consider a teleoperation system with time-varying delay, parameterized uncertainties, and unknown spring-damper operator/environment models. The aim of this article is to achieve finite-time control of position and force tracking control in a teleoperation system with above conditions and challenges. That means we need to design a finite-time control structure which is different from traditional force tracking method. Furthermore, the proposed control scheme can also deal with the uncertain dynamics and timevarying delay of the teleoperation system, and has the position tracking capability as the existing work. Therefore, we construct a new auxiliary error system by using position and force tracking error, and design the finite-time adaptive control schemes. The RBFNN is applied to estimate the uncertain parts in the robot, human, and environment models. The main contributions of our work can be summarized as follows.
1) In this article, a new position-force hybrid tracking error subsystem is designed and a novel finite-time adaptive control scheme is y developed to achieve position and force tracking control in a teleoperation system. The asymmetric time-varying delays and unknown dynamics are considered in the system. Compared with the existing work with finite-time control [39]- [45], the position synchronization can be achieved and the force tracking of the master and slave robots can also be obtained. Compared with the position/force control method [23], [25], [27], [46]- [49], the finite-time control scheme can provide stronger robustness, faster and more precision convergence performance.
2) A new auxiliary variable function with finite-time structure is designed to avoid the use of the acceleration signals and smooth the velocity feedback signals. Different from existing work, the proposed auxiliary error function is composed of the hybrid error integral term, velocity feedback filter, and auxiliary velocity error, which can improve the position and force tracking performance and avoid use the acceleration and time delay information. The remainder of this article is organized as follows. The preliminary descriptions are given in Section II. The proposed finite-time control scheme is presented in Section III. The stability and finite-time analysis are discussed in Section IV. Subsequently, the simulation and system experiment results and analysis are, respectively, illustrated in Sections V and VI. Finally, this work is concluded in Section VII.

II. SYSTEM MODELING AND PRELIMINARY DESCRIPTION
In this section, we mainly describe the dynamic model of the teleoperation system, the problems addressed in our work, and some preliminary definitions and theorems.

A. Dynamic Description of Teleoperation System
The dynamic model of a typical master-slave teleoperation system with n degree of freedom (DOF) and revolute joints can be defined as follows: where q m , q s ∈ R n ,q m ,q s ∈ R n , andq m ,q s ∈ R n are the joint position, joint velocity, and joint acceleration vectors of the master and slave robot, respectively. M m (q m ), M s (q s ) ∈ R n×n are the mass inertia matrices. C m (q m , q m ), C s (q s , q s ) ∈ R n×n are the centripetal and coriolis matrices. G m (q m ), G s (q s ) ∈ R n are the gracitational vectors. B m , B s ∈ R n are the friction vectors. τ m , τ s ∈ R n are the control input torque vectors of master and slave manipulators. F h ∈ R n is the human operator torque acting on the master manipulator. F e ∈ R n is the environment torque applied to the slave manipulator. There are some well know properties on manipulator dynamic as shown below, which can be used for later theoretical analysis. Property 1: The mass inertia matrices M m (q m ) and M s (q s ) are the symmetric, positive definite, and bounded matrices, which satisfy ∃χ j,min , χ j,max > 0, 0 < χ j,min I ≤ M j ≤ χ j,max I, where j = m, s. Property 2:Ṁ m − 2C m andṀ s − 2C s are the skewsymmetricmatrix, and for j = m, s, ∀x ∈ R n the equations x T (Ṁ j − 2C j )x = 0 is always true.
In practical system, it is difficult to obtain exact dynamic parameters in (1). Thus, for j = m, s, the nominal models M jo , C jo , and G jo are introduced to describe the know and nominal parts in teleoperation dynamic. The M j , C j , and G j are used to represent the unknown parts, which we have In this article, we consider the interaction torque of human operator and environment as the passive spring-damper model, which can be described as [46], [57] where D h , D e ∈ R n×n denote the damping matrices of the operator and environment. S h , S e ∈ R n×n express the spring matrices of the operator and environment.

B. Problem Formulation and Control Object
We define the joint position tracking errors e m and e s as Similarly, we define force tracking errors F m and F s as Therefore, considering the teleoperation system dynamic (1) with uncertain model parameters, human operator and environment models (3), and varying communication time delays, the appropriate control laws τ m and τ s need to be designed to accomplish the following control objects.
1) Stability Performance: The closed-loop teleoperation system needs to be stable under time-varying delays, dynamic uncertainties, and bounded external operator torque disturbances. 2) Position Tracking Performance: Make the slave robot can track the joint position of the master robot. At the same time, the master robot can also track the joint position of the slave robot. It indicates that the position tracking error e m and e s can reach into a small neighbor. 3) Force Feedback Tracking Performance: Ensure the torque tracking error F m and F s can converge into a small neighbor of 0, which can make the human operator feel the actual torque from the environment. Although mentioned in 2), the master-to-slave position tracking can obtain the certain force feedback, but the intensity of the force feedback is uncertain and may be large or small. Therefore, it is needed to exert sufficient force feedback consistent with environment torque.

C. RBF Neural Network
In our work, the adaptive control method with the RBFNN is employed to deal with the dynamic uncertainties. The RBFNN can estimate and approximate an uncertain continuous function where W ∈ R c×a is the weight matrix, c is the node number of neural network. ϕ(x) ∈ R c is the Gaussian basis function vector. is the bounded approximation error [58]. Gaussian basis function vector ϕ(x) can be calculated by where d i ∈ R a is the center vector of Gaussian function, f is the width of the Gaussian function.

D. Preliminary Lemmas
There are two lemmas used for system stability analysis and proof.
Lemma 1 [59]: For any a 1 , a 2 , . . . , a n ∈ R and 0 < b < 1, there has the following inequality holds: Lemma 2 [60]: Considering the system dynamics aṡ The system is finite-time stable and can converge to a neighborhood in finite-time T s , if there is a positive differentiable scalar function V(x) and makes the following inequality establishes: where α, β > 0, 0 < σ < 1. The setting time T set can be described as

III. CONTROL DESIGN AND ANALYSIS
This section elaborates on the design of the proposed auxiliary error system, the auxiliary variable, and the finite-time controller. First, a novel auxiliary error system is constructed based on the position tracking error. Second, the new master and slave integral auxiliary variables are designed for the controllers design based on the auxiliary error system and angular velocity feedback filter. Third, the RBFNN is used to estimate and compensate for the uncertainty of dynamics. Finally, the finite-time control laws and adaptive laws for the master and slave robots are given.

A. Auxiliary Error System and Integral Auxiliary Variables Design
Define the torque error filters α m and α s aṡ where L m1 and L s1 are positive-definite symmetric matrices.
We combine the position tracking errors e m , e s and the torque error filters α m , α s to define and build the novel hybrid auxiliary error systems r m and r s as where η m1 , η m2 , η s1 , and η s2 are the weight values of position and torque tracking errors, they are all positive constants. Then the auxiliary variables and controllers can be designed based on r m and r s . The differential of r m and r s can be expressed aṡ Remark 1: The auxiliary error system consists of the position tracking error and force tracking error. The values of η m1 , η m2 , η s1 , and η s2 can be used to adjust the weight of position and force tracking errors in r m and r s . Therefore, by introducing the auxiliary error system, the problem of position and force tracking is transformed into the problem of making the error system r m and r s stable. On the other hand, we need to differential the auxiliary errors r m and r s to get the expression in (10). If the force tracking errors F m and F s are directly used to design the r m and r s instead of the output of force tracking error filter α m and α s , it is obvious that the F m and F s need to be differentiated when the differential of r m and r s are obtained. Therefore, it is unnecessary to introduce the additional differential signals of force tracking errors Ḟ m and Ḟ s based on force tracking error filters α m and α s . In the following auxiliary variable design, we also design the velocity feedback filter to avoid the use of accelerating signals.
The derivative of position tracking errorsė m andė s are obtained asė Obviously, the change ratesḋ m andḋ s of communication time delay, which is difficult to be obtained in the actual system, are introduced into the angular velocity errors. We define a new form of joint velocity errors e mv and e sv to facilitate and simplify theė m andė s as Then theė m ,ė s ,ṙ m , andṙ s can be written aṡ We propose and design the novel integral auxiliary variables s m and s s by coupling the auxiliary error system r m and r s , the joint velocity feedback filter ν m and ν s , and finite-timeintegral terms of r m and r s . Then we can design the suitable finite-time controller for the teleoperation system based on s m and s s . The s m and s s are defined as where E mv and E sv are the auxiliary angular velocity error functions including the joint velocity feedback filters, which can be defined as where ν m and ν s are the output of the joint velocity feedback filter from the slave side and the master side, respectively, which can be described aṡ Remark 2: Similar to the force tracking error filter α m , α s and the description in Remark 1, we also devised the velocity feedback filters ν m , ν s and novel auxiliary velocity error functions E mv , E sv to avoid the use of acceleration signals. Compared with some existing research work, similar auxiliary variables are typically designed based on velocity errors as s j =ė j + λ j β j (e j ), where j = m, s, and β j (e j ) is the function of e j . For this form of auxiliary variable, the acceleration signal is inevitably used for the designing of controllers. However, in practice, we do not want to introduce the acceleration signal to calculate the control law, as it may contain more noise. Therefore, the proposed and designed velocity feedback filter and auxiliary function in our work can avoid the acceleration signal being used.
The finite-time-integral functions ϑ m and ϑ s in s m and s s are, respectively, defined as where λ m1 , λ m2 , λ s1 , and λ s2 are the weight coefficients, and they are all positive diagonal matrices. 1 < σ m1 , σ s1 < 2 and 0 < σ m2 , σ s2 < 1. For any vector Then we can have the differential of s m and s s aṡ whereĖ mv andĖ sv areĖ ϑ m andθ s arė Next, the RBFNN is used to approximate and estimate the uncertain parts of the teleoperation dynamics.

B. Approximation of Dynamic Uncertainties
Based on the system dynamic description (1), (3), and (18)- (20) we can obtain M jṡj as (21) where for j = m, s and j = s, m.
We define μ j1 , μ j2 , and μ j3 to simplify the follow-up describtion as Then based on the dynamics and uncertain description of teleoperation system in (1)-(3), the M jṡj can be rewritten as where for j = m, f he,j = f h , D he,j = D h , and S he,j = S h . For j = s, f he,j = f e , D he,j = D e , and S he,j = S e . Then we define the uncertain part of teleoperation system P j = M j μ j1 + M j μ j2ḋj + C j μ j3 − G j −B j −D he,jqj −S he,j q j . It should be noted that the change rateḋ j is introduced into the uncertain part. In the system, the change rate of time delay may be unknown, so we define M jḋj as an uncertain part in P j , and use RBFNN to estimate it. Therefore, the input of RBFNN is set as Then, the uncertain function P j can be described as where θ j ∈ R l×n is the ideal approximation parameter matrix; l is the number of hidden layer nodes in RBFNN; j ∈ R n×1 is the approximation error and is bunded; Y(Z j ) = [y j1 (Z j ), y j2 (Z j ). ., . . . , y jl (Z j )] ∈ R l×1 is the Gaussian basis function vector of RBFNN, and can be calculated as c i ∈ R 1×5n is the center vector of Gaussian function of the ith hidden layer node; b j is the width of Gaussian function.

Remark 3:
The approximation error j is bounded for j = m, s, when using the RBFNN to estimate the dynamic uncertain part. With the assumption of bounded f he,j , it can be drawn that f he,j + j is bounded, and there exists a positive constant ω j such that f he,j + j 2 ≤ ω j . Remark 4: It is obvious that P m and P s contain the uncertain parts of the master-slave robot dynamic, the human operator model and the environment model. In our work, we consider the Lyapunov candidate function V 1 = 1/2s T m M m s m + 1/2s T s M s s s , and described the M mṡm and M sṡs which are the main items of V 1 differential result. In most of the existing work, the model uncertainty description in (2) is directly brought into the dynamic models (1) to obtain a new form of dynamics as When utilizing the RBFNN to estimate the uncertainty parts described above, the acceleration signals also need to be used. In our work, It is evident that there is no need for the acceleration signal when applying the RBFNN to approximate the P m and P s .

C. Controller Design
The block diagram of the proposed control system is shown in Fig. 1. The proposed controllers are composed of the nominal model controller, RBFNN-adaptive controller, and finite-time controller, which are given as follows. The torque error filters α m , α s , velocity feedback filters ν m , ν s , auxiliary error systems r m , r s , and auxiliary variable s m , s s have been defined above. When j = m, s, there has j = s, m, the proposed control law τ j can be designed as where K j1 , K j2 ∈ R n×n are the diagonal positive-definite matrices. a j1 , a j2 ∈ R are the positive constants.
The adaptive laws can be designed aṡ where j1 ∈ R n×n is the diagonal positive-definite matrix.

IV. STABILITY ANALYSIS
Theorem 1: Having the combined auxiliary error system in (9), force tracking error filter (8), velocity feedback filter in (16), and auxiliary variable (14), considering the teleoperation system (1) with human operator and environment model (2), dynamic uncertain (3), adaptive finite-time control law in (26), and adaptive laws in (27), the closed-loop system is asymptotically stable and having the finite-time performance, if the control parameters K j1 , K j2 , j1 are all positive-definite diagonal matrices, a j1 , a j2 , j2 , j3 are all positive constants, and 0 < σ < 1.
Then we will give the proof process of this theorem. Proof: We define the Lyapunov-Krasovskii functuin V as where for j = m, s, j = s, m First, the differentiation of V 1 can be described aṡ Based on the description in (23) and the Property 2 of the teleoperation system, it can be obtained thaṫ Therefore, with the Assumption 2, Remark 3, and Young's inequality (the detailed description of Young's inequality can be found in Appendix-A), it can be obtained thaṫ Then, with the control law in (26), it haṡ For V 2 , based on the adaptive laws in (27), we can get thaṫ For tr(θ T j −1 j1θ j ), −1 j2D j D j , and −1 j3ω jωj , the following inequalities are always established: With the presentation above, we havė For V 3 , we havė In addition, it is clear that Finally, combining with the derivation ofV 1 in (33),V 2 in (36), andV 3 in (38), we can get thaṫ is a positive constant. The λ min (•) and λ max (•) represent the minimun and maximum eigenvalues of the matrix •, respectively. It is necessary to set the approximate K j1 to make k j1 > 0, or in other words, there exists a small value of η j to make k j1 > 0.
Then define = min(k j1 , 1, 1/d j ) and = m + s , it is obvious that > 0 and > 0, we can obtainV aṡ which implies that [56] Thus, we have V(t) → / when t → ∞, and V(t) is bounded. Since V is radically consisted with s j ,θ j ,D j , and ω j , it can imply that s j ,θ j ,D j , andω j are all bounded.
The stability analysis mentioned above shows that the closed-loop system is asymptotically stable and the signals are bounded. Then the analysis of finite-time performance is represented as follows.
Taking the Lyapunov candidate function asV = V 1 + V 2 , thus similar to the (33) and (36), there existṡ We definek j1 = λ min (K j1 ),k j2 = λ min (K j2 ), and With the boundness ofθ j ,D j , andω j we have analyzed above, it is obvious that¯ j is also bounded. Thus, for j = m, s, there exists the positive constants ϒ m and ϒ s such that 0 ≤¯ m ≤ ϒ m and 0 ≤¯ s ≤ ϒ s , respectively. Then theV can be represented aṡ Then we can get thaṫ where for j = m, s Note that it is needed to set the larger values of K j1 and K j1 to make¯ 1 ,¯ 2 > 0.
We define the region j as j = min( j1 , j2 ), where implies that if s j > j , it hasV +¯ 1V +¯ 2V (σ +1/2) ≤ 0, which means the system is finite-time stable and s j can converge to the region j in finite-time. Based on the Lemma 2, the reaching time T can be given by This completes the proof. Remark 5: Based on Theorem 1, for j = m, s, the approximate control gains can be selected according to the following rules.
1) First, the filter parameters L j1 and L j2 in (8) and (16) are all positive-definite diagonal matrices. The smaller L j1 and L j2 would have smoother output of filters (the input signals are the angular velocity and force signal transmitted through the time-varying delay). 2) Then, the gains in auxiliary error system η j1 and η j2 , can be given. The values of η j1 and η j2 are all positivedefinite diagonal matrices and represent the weights of position error and force error in the error system. 3) Next, the parameters λ j1 , λ j2 , η j1 , and η j2 in auxiliary variable s j can be given according to the system output performance. In addition, they should meet λ j1 and λ j2 are positive-definite diagonal matrices, 1 < σ j1 < 2, and 0 < σ j2 < 1. 4) K j1 and K j2 are positive-definite diagonal matrices, a j1 and a j2 are positive constants. In addition, bigger K j1 , K j2 and smaller a j1 , a j2 can get the smaller convergence domain of s j . 5) j1 is the positive diagonal matrix and j2 , j3 > 0. 0 < σ < 1.

A. Stability and Tracking Performance Verification
The first simulation set is used to illustrate the stability and the position/force tracking performance of the closed-loop teleoperation system. The simulation results under above simulation conditions are shown in Figs. 3-8. Fig. 3 shows the position curves of master and slave robots and force curves of human operator and environment with time-varying delays and external human operator torque in Fig. 2. To better illustrate the effects of position and force tracking between master and slave side, Figs. 4 and 5 are given to present the position tracking and force tracking errors, respectively. It can be seen that the teleoperation system with proposed control scheme has a good position and force tracking performance. It is further verified that the position and torque tracking errors can converge into a very small neighbor with time-varying delays and uncertain dynamics.
Furthermore, based on the proof of Theorem 1, it can be obtained that s j ,θ j ,D j , andω j are all bounded, for j = m, s. Therefore, Fig. 6 shows the adaptive variablesθ j ,D j , andω j in controller. It is evident that the above adaptive parameterŝ θ j ,D j , andω j are all bounded, which can infer the boundedness ofθ j ,D j , andω j . The norm curves of the designed auxiliary variables s m and s s are also given in Fig. 7, and the boundness of s j can be verified. The control torque τ m and τ s are presented in Fig. 8. All of these results show the stability and position/force tracking performance of the teleoperation system wit our proposed control scheme.

B. Comparisons With Other Control Methods
In order to illustrate the effectiveness of the proposed control algorithm, the simulation comparisons with other related     control methods in [46]- [49] are also implemented. The comparison of simulation results is set to two parts. The study is carried out by comparing the convergence speed and the steady-state performance of tracking errors with the different control methods. The simulation settings are the same for all control methods.
In part 1, the comparison study between nonfinite-time control schemes in [46] (method 1), [47] (method 2), and this work is performed. In [46], a PD control method with force tracking compensation is investigated to get the asymptotic convergence of position and force tracking. In [47], a nonlinear-proportional plus nonlinear damping (nP+nD) controller with environment torque compensation is proposed. Fig. 9 gives the curves of position synchronization errors with method 1, method 2, and proposed methods. Fig. 10 shows the force tracking error curves with method 1, method 2, and proposed methods. From  which, it can be found that the proposed finite-time control algorithm has a faster convergence speed than the methods in [46] and [47]. This can further explain that the finitetime control structure has the faster convergence speed than infinite-time control methods.
In [48] and our previous work [49], the position and force tracking errors are used to design auxiliary variables for position and force tracking control, which has the similar idea to this work. In addition, the auxiliary variables and controller structure in [48] is different, and the controllers in [48] and [49] are not finite-time control structure. Therefore, in part 2, the comparison analysis between the control methods in [48] (method 3), [49] (method 4), and this work is performed to analyze the characteristics of them. The position and force tracking error curves of method 3, method 4, and proposed are shown in Figs. 11 and 12. It can be seen that the error convergence performance in [48] is worse than the proposed finite-time method. Meanwhile, it is shown that the error curve based on method 4 has a small oscillation in convergence process (such as e m1 , e s1 , F m1 , and F s1 ), which increases its convergence time.
To better illustrate the steady-state performance of these control methods, root mean square error (RMSE) index is used to quantify the position and force tracking errors during 10 s-20 s. The RMSE values of position tracking and force tracking are shown in Tables II and IV, respectively. One can find that the RMSEs of position and force errors with proposed control method are smaller than the other four control methods, which means the proposed controller has the better steady-state performance.

VI. EXPERIMENT ANALYSIS
In this section, the experiments are achieved to verify the validity of the proposed control method. The experimental platform is composed of two PHANToM Omni devices. These devices run in MATLAB/Simulink environment based on S-Function and OpenHaptics SDK functions. The experimental teleoperation system is shown in Fig. 13.
The PHANToM Omni robot is a small haptic device, it contains three actuated revolute joints and three passive revolute joints. We used the first three actuated joints of PHANToM Omni for the experiment. The end grips of these two robots are in the fixed state. The dynamic descriptions and the model parameters in our experiment have been borrowed from [52]. Two experimental schemes as free motion and force contact are designed to better verify the superiority of the proposed method. The time delays of the system are the same as the setting in the above simulation part. The RBFNN is used to estimate the unknown dynamic and uncertainties of the system. The width of Gaussian function are set as b j = 1.5, and the center values of Gaussian functions are c = [ − 5, −2.5, 0, 2.5, 5].

A. Free Motion
In this section, the slave robot is performed in fine free state and the operator's torque on the master robot is not introduced into the controller. Our target is to verify that the proposed control scheme is still effective when the slave robot is in the free motion. Operator does not need to feel large force feedback. Fig. 14 shows the performance of the system with dynamic unknown and varying time-delays. It is obvious that the slave robot can still track the joint position of the master robot when ignoring the operator and environmental torques. In addition, the proposed control method is compared with the other two nonfinite-time methods in [47] (case 1) and [49] (case 2). Figs. 15 and 16 show the position performance of the system in free motion with case 1 and case 2. It can be seen that the position tracking performance is weak. Especially for the second joint of slave robot, the position of master robot is unable to be well tracked. The RMSEs of position tracking errors in Figs. 14-16 are listed in Table III to better demonstrate tracking performance with different methods. It can be seen that the RMSE values of the proposed method are smaller than the other two methods. This indicates that the proposed finite-time control system can achieve more accurate and fast position synchronization when the slave robot is not in contact with the environment.

B. Force Contact
In this section, we defined the virtual operator and virtual environment models based on (3). The parameters of virtual operator and environment models are set as D h = daig(10, 10, 10), S h = daig(100, 100, 100), D e = daig (15,15,15), and S e = daig(100, 100, 100), where the magnitude of the parameter on torque is mN. Furthermore, the extra operator torque f h is designed as shown in Fig. 17. Our purpose is to employ the virtual operator and environment models to simulate the effects of the operator and environment on the teleoperation system. Real operator is not involved in the system. In this case, the teleoperation system based on these virtual torque models can reach a balanced state with the regulation of the controllers. Moreover, the extra operator torque can achieve the additional action of the operator on the    master robot, which can also make the changes in joint position, operator and environment torques. The proposed control scheme is also compared with the controllers in [47] (case 1) and [49] (case 2). Figs. 18-20 show the performance of position and force tracking with proposed control, case1, and case 2, respectively. Among these figures, the blue solid line represents the curves of the position and force of the master robot, and the red dotted line represents the curve of the position and force of the slave robot. Fig. 19 illustrates that control system in case 1 cannot guarantee accurate position synchronization and torque tracking performance under force contact condition. Figs. 18 and 20 show that the system with proposed and case 2 have better tracking performance. However, from the position and force tracking curves after exerting extra operator torques, the proposed finite-time control scheme has the faster convergence speed than the nonfinite-time control method in case 2. The RMSEs of position and force tracking errors in Figs. 18-20 are listed in Table IV. It can also be seen from the RMES values that the proposed control method has better tracking performance of position and force.

VII. CONCLUSION
In this article, the position and force tracking control problems for the teleoperation system with time-varying delay and dynamic uncertainties have been investigated. The position and force tracking errors are combined as the error systems for control scheme design. Compared to some existing research work, the proposed control method has the finitetime performance and the acceleration signals do not need to be used. The simulation experiment and practical experiment have been conducted to verify the validity of the proposed control scheme. Proof: First, a well-known lemma is introduced to assist the proof Lemma 4 [55]: For c 1 > 0, c 2 > 0, and 0 < b < 1, the following inequality is always established: Obviously, when c 1 = 0 or c 2 = 0, the above inequality is still holds . For any a 1 , a 2 , . . . , a k ≥ 0 and k ≤ 2, we have For k = 3, it is apparent that We assume that for k = n, the inequality in Lemma 1 holds, it has (a 1 + a 2 + a 3 + · · · + a n ) b ≤ a b 1 + a b 2 + a b 3 + · · · + a b n .
Based on the principle of mathematical, it is concluded that Lemma 1 is established for any positive integer n.

C. Proof of Lemma 2
Proof: A finite-time stable Lemma is described as follows. Lemma 5 [63]: Consider the nonlinear systemẋ = f (x), f (0) = 0, x ∈ R n , x(0) = x 0 , there is a positive continuously differentiable function V(x) defined in a neighborhood U ⊂ R n , ifV(x) ≤ −cV a (x), where c > 0 and 0 < a < 1, the zero solution of the system is finite-time stable. If U = R n and V(x) is also radially unbounded, the zero solution is globally finite-time stable.
It is obvious that the finite-time stability of Lyapunov function V(x) can be obtained, and the system is finite-time stable. DefineV(x) + αV(x) + βV σ (x) = 0, we have There has Then the setting time T set can be described as Therefore, based on the inequality, the setting time T set has Then the proof of Lemmas 1 and 2 are completed.