Control Schemes for Passive Teleoperation Systems over Wide Area Communication Networks with Time Varying Delay
Abstract
In this paper, the problem of time varying telecommunication delays in passive teleoperation systems is addressed. The design comprises delayed position, velocity and positionvelocity signals with the local position and velocity signals of the master and slave manipulators. Nonlinear adaptive control terms are employed locally to cope with uncertain parameters associated with the gravity loading vector of the master and slave manipulators. LyapunovKrasovskii function is employed for three methods to establish asymptotic tracking property of the closed loop teleoperation systems. The stability analysis is derived for both symmetrical and unsymmetrical time varying delays in the forward and backward communication channel that connects the local and remote sites. Finally, evaluation results are presented to illustrate the effectiveness of the proposed design for realtime applications.
Keywords
Teleoperation telerobotics LyapunovKrasovskii functional time varying delay passive systems1 Introduction
Over the past decades bilateral teleoperation control technology has been attracted by many researchers around the world, see for example, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, and many others]. The main goal in most reported control mechanism for teleoperation systems is to ensure stability and transparency of the whole closed loop system in the presence of time delay. These results can be broadly classified into two categories as passive and nonpassive teleoperation systems. The control system for nonpassive teleoperation can be found in [1, 2, 3, 4]. Polushin et al.[1, 2] developed teleoperation systems in the presence of nonpassive human and environment input forces under symmetrical time varying delay. Small gain theorem was employed to show the position and velocity synchronization errors between master and slave manipulators provided that the human and environment input forces are bounded. Most recently, authors in [3, 4] have introduced control methods for nonpassive bilateral teleoperation systems in the presence of unsymmetrical time varying delay. The stability analysis of these methods was based on using strict linear matrix inequality. Control mechanisms for passive teleoperation systems have been extensively studied by many researchers, see for example, [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. Historical survey and comparison with different teleoperation control algorithms can be traced from [5, 6]. Classical teleoperation schemes using passivity theorem and scattering approach have been proposed in [7, 8, 9, 10, 11]. Li and Song[12] developed virtual environment modeling and correction technique for force reflecting teleoperation with constant time delays.
Authors in [13] developed a control scheme for Internet based teleoperation systems using wave variable approach under time varying delays. Bate et al.[14] proposed control mechanism to reduce the reflection from wave based passive teleoperation systems. The stability analysis for wave variable method introduced in [12, 13, 14] was established for the single degree of freedom (DOF) linearized master and slave manipulators. Alise et al. [15] extended such wave variable mechanism to multipleDOF systems. In [16, 17], authors presented teleoperation systems in the presence of time varying communication delays. Lee and Spong[18] have presented proportionalderivative control scheme for passive teleoperation systems for position coordination of the master and slave manipulator. Authors in [19] have showed that the stability condition established in [18] can be obtained by using only delayed position signals based teleoperation system. However, the control design parameters require to satisfy specific conditions to ensure closed loop stability of the closed loop teleoperation system. Lee and Huang[20] developed passive setposition modulation framework for Internet based teleoperation systems with time varying delay. In [21], authors presented adaptive tracking system for uncertain nonlinear systems with time varying delays and unmodeled dynamics by using radial basis functional networks.
We can notice from the structure of the delays that most reported passive teleoperators are based on the assumption that the time delays are either constant or time varying nature. Moreover, the time delays in passive designs are assumed to be of symmetrical nature which may not be realistic in bilateral teleoperation applications. In practice, however, the data packet delays, i.e., the time required for the data packets to reach the remotely located sites and data packet returns to the local sites may be of unsymmetrical nature which varies with respect to the network load, number of the traversed nodes and distance[3, 4, 22].
In this paper, we address the stability problem of passive teleoperation systems under symmetrical and unsymmetrical time varying delays. The teleoperators are designed by coupling local and remote sites via delaying position and positionvelocity signals of the master and slave manipulators. The input algorithms also combine local position and velocity signals with nonlinear adaptive terms to cope with the parametric uncertainty associated with the gravity loading vector. In our first design, we introduce passive teleoperation systems under symmetrical time varying delays. In second design, we develop teleoperators depending on unsymmetrical nature of time varying delays. LyapunovKrasovskii function is used to analyze the asymptotic tracking property of the closed loop teleoperation systems. These stability properties are illustrated by conducting simulation studies on a 2DOF masterslave teleoperation systems.
The rest of the paper is organized as follows: Section 2 describes the dynamics of the teleoperation system. Sections 3 and 4 introduce teleoperation strategies under symmetrical and unsymmetrical time varying communication delays. A detailed convergence analysis is also given in Sections 3 and 4. Simulation example is presented in Section 5. Finally, Section 6 concludes the paper.
2 Model dynamics
2.1 Master and slave manipulator dynamics

Property 1. The matrices \({{\dot M}_m}({q_m})  2{C_m}({q_m},\,{{\dot q}_m})\) and \({{\dot M}_s}({q_s})  2{C_s}({q_s},\,{{\dot q}_s})\) are skew symmetric.

Property 2. The mass matrices M _{ m }(q _{ m }) and M _{ s }(q _{ s }) are the symmetric, bounded and positive definite satisfying the inequality ∥Mm (q _{ m })∥ ⩽ β _{ m } and ∥Ms (q _{ s })∥ ⩽ β _{ s } with β _{ m } > 0 and β _{ s } > 0.

Property 3. If the signals \({{\ddot q}_m}\) and \({{\ddot q}_s}\) are bounded, then \(\dot C_m({q_m},\,{{\dot q}_m})\) and \(\dot C_s({q_s},\,{{\dot q}_s})\) are bounded.
2.2 Interaction forces between humanmaster and between slaveenvironment
3 Algorithm design and stability analysis
In this work, we design two control strategies for passive teleoperation systems and analyze their stability properties in the presence of symmetrical and unsymmetrical time varying telecommunication delays. In our analysis, the data transmission delays in the forward communication channel from master to slave platforms and backward communication channel from slave to master platforms are defined as γ _{ dm }(t) and γ _{ ds }(t), respectively.
Using Schwartz inequality, the bound for the terms \(({{\dot q}_m}  {{\dot q}_s}(t  {\gamma _d}(t)))\) and \(({{\dot q}_s}  {{\dot q}_m}(t  {\gamma _d}(t)))\) can be manipulated as \({\mathcal A}_\alpha ^{{1 \over 2}}{{\ddot q}_m}_{2}\) and \({\mathcal A}_\alpha ^{{1 \over 2}}{{\ddot q}_s}_{2}\) Since q _{ m } and q _{ s } are bounded, then we have (q _{ m } − q _{ s }(t − γ _{ d }(t))) ∈ γ _{∞} and (q _{ s } − q _{ m } (t − γ _{ d }(t))) ∈ ℒ_{∞}. In view of our above analysis, Property 2, Property 3, the boundedness property of the parameter estimates from projection mechanism (7) and invariance theorem[26], we can conclude that, for the given ς, there exists K _{ Pm } = K _{ Ps }, K _{ Dm } and K _{ Ds } such that the position, velocity and tracking error between master and slave manipulators in (8), (9) with passive input (4), (5) under symmetrical time varying delays are asymptotically stable.
In view of Schwartz inequality, we can obtain the bound on \(\smallint _{(t  {\gamma _d}(t))}^t{{\ddot q}_m}(\eta)d\eta \) and \(\smallint _{(t  {\gamma _d}(t))}^t{{\ddot q}_s}(\eta)d\eta \) as \({\mathcal A}_\alpha ^{{1 \over 2}}{{\ddot q}_m}_{2}\) and \({\mathcal A}_\alpha ^{{1 \over 2}}{{\ddot q}_s}_{2}\) imply the bound (q _{ m } − q _{ s }(t − γ _{ ds }(t))) ∈ ℒ_{∞} and (q _{ s }−q _{ m }(t − γ _{ dm }(t))) ∈ ℒ_{∞}. Then, applying invariance theorem[26], Property 2 and Property 3 along with the parameter projection mechanism (7), we can conclude that the position, velocity and tracking error between master and slave manipulators converge to zero as the time goes to infinity.
4 Teleoperation system with delayed position and velocity signals
Applying Schwartz inequality, we can write \(\int_{(t  {\gamma _d}(t))}^t {{{\ddot q}_m}} (\eta){\rm{d}}\eta \) and \(\int_{(t  {\gamma _d}(t))}^t {{{\ddot q}_s}} (\eta){\rm{d}}\eta\) as \({\mathcal A}_\alpha ^{{1 \over 2}}{\ddot q_m}{_2}\) and \({\mathcal A}_\alpha ^{{1 \over 2}}{\ddot q_s}{_2}\). This implies that \(({{\dot q}_m}  {{\dot q}_s}(t  {\gamma _d}(t))) \in {{\mathcal L}_\infty }\) and \(({{\dot q}_s}  {{\dot q}_m}(t  {\gamma _d}(t))) \in {{\mathcal L}_\infty }\). From (17) and (18), we also have (q _{ m } − q _{ s }(t − γ _{ d }(t))) ∈ ℒ_{∞} and (q _{ m } − q _{ s }(t − γ _{ d }(t))) ∈ ℒ _{∞}. Then, in view of Property 2, Property 3, projection mechanism for the parameter estimates (7) and invariance principle[26], we can conclude that the position, velocity and tracking error signals in the closed loop teleoperators (27), (28) are bounded and asymptotically converge to zero.
We can simplify the time derivative of (36) as \(V(T)  V(0) \leqslant  {\xi _{sm}}{{\dot q}_m}_2^2  {\xi _{ss}}\dot qs{_2}\), where \({\xi _{sm}} = ({K_{Dm}} + 2{D_m} + {K_{Pm}}{\mathcal A}_{d\alpha }^2 + {\zeta _m}  {\Lambda _{dm1}})\), and \({\xi _{ss}} = ({K_{Dm}} + 2{D_s} + {K_{Pm}}{\mathcal A}_{s\alpha }^2 + {\zeta _s}  {\Lambda _{ds2}})\). This means that \(({{\dot q}_m},\,{{\dot q}_s}) \in {{\mathcal L}_2}\) and \(({q_m},\,{q_s},\,{{\dot q}_m},\,{{\dot q}_s},\,{{\dot q}_m}  {{\dot q}_s},\,{{\dot q}_s}  {{\dot q}_m},\,{q_m}  {q_s},\,{q_s}  {q_m}) \in {{\mathcal L}_\infty }\). Using the relationship \(({{\dot q}_m}  \,{{\dot q}_s}) + \smallint _{(t  {\gamma _{ds}}(t))}^t{{\ddot q}_s}(\eta)d\eta \) and \(({{\dot q}_s}  \,{{\dot q}_m}) + \smallint _{(t  {\gamma _{dm}}(t))}^t{{\ddot q}_s}(\eta)d\eta \), we can also show the bound on \(({{\dot q}_m}  {{\dot q}_s}(t  {\gamma _{ds}}(t))) \in {{\mathcal L}_\infty }\) and \(({{\dot q}_s}  {{\dot q}_m}(t  {\gamma _{dm}}(t))) \in {{\mathcal L}_\infty }\). Applying our previous analysis for (24), (25), it is also possible to guarantee (q _{ m } − q _{ s }(t − γ _{ ds }(t))) ∈ℒ _{∞} and (q _{ s } − q _{ m }(t− γ _{ ds }(t))) ∈ ℒ_{∞}. Then, using invariance theorem[26], Property 2, Property 3 and boundedness of parameter estimates from projection laws (7), we can state that the position, velocity, tracking error signals in (34) and (35) asymptotically convergent to zero.
5 Evaluation results

Remark 1. We notice from our various evaluation results that the performance with the delayed positionvelocity signal based teleoperator provides better performance than the performance achieved under delayed position signal based teleoperator. This is mainly because of the fact that the positionvelocity based teleoperator provides additional damping to the master and slave manipulator ensuring better coordination of the master and slave manipulator.

Remark 2. Note that the conditions for K _{ Pm } = K _{ Ps } can be removed by modifying V _{1} and V _{3} as \({V_1} = \dot q_m^T{M_m}({q_m}){{\dot q}_m} + {{{K_{Pm}}} \over {{K_{Ps}}}}\dot q_s^T{M_s}({q_s}){{\dot q}_s}\) and \({V_3} = {K_{Pm}}{({q_m}  {q_s})^{\rm{T}}}({q_m}  {q_s}) + {P_m}q_m^{\rm{T}}{q_m} + {{{K_{Pm}}} \over {{K_{Ps}}}}{P_s}q_s^{\rm{T}}{q_s}\).
6 Conclusions
In this paper, passive bilateral teleoperation systems have been studied with symmetrical and unsymmetrical time varying communication networks delay. A delayed position and positionvelocity signals are used to couple local and remote sites via using telecommunication channel. Nonlinear adaptive control terms have been employed locally to learn and compensate uncertain parameters associated with gravity loading vector of the master and slave manipulators. Detailed stability analysis for three different control methods has been given by using LyapunovKrasovskii function to explore asymptotic convergence property of the closed loop teleoperators for both symmetric and unsymmetrical time varying delays. Simulation results have been used to demonstrate the theoretical development of this paper.
Notes
Acknowledgements
Authors thank anonymous reviewers for their constructive comments on our original submission.
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