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International Journal of Automation and Computing

, Volume 11, Issue 1, pp 100–108 | Cite as

Control Schemes for Passive Teleoperation Systems over Wide Area Communication Networks with Time Varying Delay

  • Shafiqul Islam
  • Xiaoping P. Liu
  • Abdulmotaleb El Saddik
  • Lakmal Seneviratne
  • Jorge Dias
Article

Abstract

In this paper, the problem of time varying telecommunication delays in passive teleoperation systems is addressed. The design comprises delayed position, velocity and position-velocity 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. Lyapunov-Krasovskii 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 real-time applications.

Keywords

Teleoperation telerobotics Lyapunov-Krasovskii functional time varying delay passive systems 

1 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 multiple-DOF systems. In [16, 17], authors presented teleoperation systems in the presence of time varying communication delays. Lee and Spong[18] have presented proportional-derivative 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 set-position 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 position-velocity 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. Lyapunov-Krasovskii 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 2-DOF master-slave 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

Let us first consider the equation of motion for n-links master and slave manipulators with human and environment input forces
$$M_m(q_m)\ddot{q}_m + C_m(q_m,\dot{q}_m)\dot{q}_m + G_m(q_m) = \tau_m+F_h$$
(1)
$$M_s(q_s)\ddot{q}_s + C_s(q_s,\dot{q}_s)\dot{q}_s + G_s(q_s)=\tau_s-F_e$$
(2)
where m and s represent the master and slave systems respectively, \({\ddot q_m} \in {{\bf{R}}^n},\,{\dot q_m} \in {{\bf{R}}^n},\,{q_m} \in {{\bf{R}}^n}\) and \({{\ddot q}_s} \in {{\bf{R}}^n},\,{{\dot q}_s} \in {{\bf{R}}^n},\,{q_s} \in {{\bf{R}}^n}\) are the joint acceleration, velocity and position, M m (q m ) e R n × n and M s (q s ) ∈ R n × n are the symmetric and uniformly positive definite inertia matrices, \({C_m}({q_m},\,{{\dot q}_m}){{\dot q}_m} \in {{\bf{R}}^n}\) and \({C_s}({q_s},\,{{\dot q}_s}){{\dot q}_s} \in {{\bf{R}}^n}\) are the coriolis and centrifugal loading vectors, G m (q m ) e R n and G s (q s ) e R n is the gravitational loading vector, τ m R n and τ s R n are applied control input vector, and F h R n and F e R n denote the operational torque applied to the master by human operator and environmental torque applied to slave manipulators, respectively. The teleoperation systems (1), (2) can be written as
$$\matrix{{{{\ddot q}_m} = M_m^{ - 1}({q_m})\left[ {{\tau _m} + {F_h} - {C_m}({q_m},{{\dot q}_m}){{\dot q}_m} - {G_m}({q_m})} \right]} \cr {{{\ddot q}_s} = M_s^{ - 1}({q_s})\left[ {{\tau _s} - {F_e} - {C_s}({q_s},{{\dot q}_s}){{\dot q}_s} - {G_s}({q_s})} \right].} \cr }$$
(3)
Throughout our stability analysis, we will use the following dynamical property of master and slave manipulators[23].
  • 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 human-master and between slave-environment

The interaction forces between human and master and between slave and environment play a significant role to achieve desired transparency and stability of the closed loop teleoperation systems. These forces can be modeled as a second-order mass-spring-damper system[24]. It is also possible to use simpler models to estimate interaction forces, such as, pure stiffness or dampening-stiffness[25]. In this work, the interaction forces are estimated by using spring-damper systems as [4, 18, 19, 20, 22, 25]
$$F_h=-S_{m} q_{m}-D_{m}\dot q_{m}$$
(4)
$$F_e=S_{s} q_{s}+D_{s}\dot q_{s}$$
(5)
where S m R n × n and D m R n × n represent stiffness and damping constant matrices for the human operators and S s R n × n and D s R n × n defines stiffness and damping constants matrices for the remote environment.

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.

We first consider that the time delays in forward and backward communication channels are symmetrical as γ dm (t) = γ ds (t) = γ d (t). The input for the systems (3) is then designed by comprising delayed position signal with the local position, velocity and adaptive control term for gravity compensation as
$$\matrix{{{\tau _m} = - {K_{Pm}}\left({{q_m} - {q_s}(t - {\gamma _d}(t))} \right) - {K_{Dm}}{{\dot q}_m} + {{\hat G}_m}({q_m})} \hfill \cr {{{\dot \tilde \theta }_m} = - {\Gamma _m}{Y_m}({q_m}){{\dot q}_m}} \hfill \cr {{\tau _s} = {K_{Ps}}\left({{q_m}(t - {\gamma _d}(t)) - {q_s}} \right) - {K_{Ds}}{{\dot q}_s} + {{\hat G}_s}({q_s})} \hfill \cr {{{\dot \tilde \theta }_s} = - {\Gamma _s}{Y_s}({q_s}){{\dot q}_s}} \hfill \cr }$$
(6)
where K Pm > 0, K Dm > 0, K Ps > 0, K Ds > 0, \({Y_m}({q_m}){{\tilde \theta }_m} = \left({{{\hat G}_m}({q_m}) - {G_m}({q_m})} \right),\,{Y_s}({q_s}){{\tilde \theta }_s} = \left({{{\hat G}_s}({q_s}) - G({q_s})} \right),\,{{\tilde \theta }_m} = \left({{{\tilde \theta }_m} - {\theta _m}} \right),\,{{\tilde \theta }_s} = \left({{{\tilde \theta }_s} - {\theta _s}} \right),\,{\Gamma _m} \in {{\bf{R}}^{m \times m}}\), Γ m R m × m and Γ s R s × s are the positive definite constant matrices. The control scheme (6) is depicted in Fig. 1 with γ dm (t) = γ ds (t) = γ d (t). Parameter adaptation mechanism given above may cause discontinuous property in the parameter estimates even after parameter converges to the actual one. To ensure that the parameters θ m and θ s remain bounded over the given compact sets θ m ∈ Ω m and θ s ∈ Ω s , a projection algorithm can be used to limit the parameter estimates. For this, we define \({{\hat \Omega }_m}\) and \({{\hat \Omega }_s}\) such that \({{\hat \Omega }_m} \subset {\Omega _m}\) and \({{\hat \Omega }_s} \subset {\Omega _s}\) with Ω m = {x mi ⩽ Ω mi y mi and Ω s = {x si θ si y si }. We also define Ω δm and Ω δs as Ω δm = {x mi δ m ⩽ θ mi y mi − and Ω δs = {x si δ s θ si y si δ s } where δ s and δ s are chosen such that \({{\hat \Omega }_m} \subset {\Omega _{\delta m}}\) and \({{\hat \Omega }_s} \subset {\Omega _{\delta s}}\). Then, the projection of the parameter adaptation rule can be written as \({{\dot \hat \theta }_m} = Proj({{\hat \theta }_m},\,{\Phi _m})\) and \({{\dot \hat \theta }_s} = Proj({{\hat \theta }_s},\,{\Phi _s})\) with
$$\left[ {Proj\left( {\hat \theta _k ,\Phi _k } \right)} \right]_i = \left\{ {\begin{array}{*{20}c} {\Gamma _k \Phi _{ki} , if x_{ki} \leqslant \hat \theta _{ki} \leqslant y_{ki} or} \\ { if \hat \theta _{ki} > y_{ki} and \Phi _{ki} \leqslant 0 or} \\ {if \hat \theta _{ki} < x_{ki} and \Phi _{ki} \geqslant 0} \\ {\Gamma _k \bar \Phi _{ki} , if \hat \theta _{ki} > y_{ki} and \Phi _{ki} > 0} \\ {\Gamma _k \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Phi } _{ki} , if \hat \theta _{ki} < y_{ki} and \Phi _{ki} < 0} \\ \end{array} } \right.$$
(7)
where \({{\bar \Phi }_{ki}} = \left[ {1 + {{{y_{ki}} + {\theta _{ki}}} \over {{\delta _k}}}} \right]{\Phi _{ki}},\,{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over \Phi } }_{ki}} = \left[ {1 + {{{\theta _{ki}} - {x_{ki}}} \over {{\delta _k}}}} \right]{\Phi _{ki}}\) and Φ ki is the i-th element of the column matrix \( - Y_k^T({q_k}){{\dot q}_k}\) with k = m, s. Using (6) and (3), the closed loop teleoperators can be represented as
$$\matrix{{{{\ddot q}_m} = M_m^{ - 1}({q_m})[{K_{Pm}}{q_s}(t - {\gamma _d}(t)) - {K_{Pm}}{q_m} - {K_{Dm}}{{\dot q}_m} - } \cr {{C_m}({q_m},{{\dot q}_m}){{\dot q}_m} + {{\hat G}_m}({q_m}) - {G_m}({q_m}) - {S_m}{q_m} - } \cr {{D_m}{{\dot q}_m}]} \cr }$$
(8)
$$\matrix{{{{\ddot q}_s} = M_s^{ - 1}({q_s})[{K_{Ps}}{q_m}(t - {\gamma _d}(t)) - {K_{Ps}}{q_s} - {K_{Ds}}{{\dot q}_s} - } \cr {{C_s}({q_s},{{\dot q}_s}){{\dot q}_s} + {{\hat G}_s}({q_s}) - {G_s}({q_s}) - {S_s}{q_s} - {D_s}{{\dot q}_s}].} \cr }$$
(9)
Fig. 1

Control scheme for bilateral teleoperation systems with delayed position signals for symmetrical case γ dm (t) = γ ds (t)= γ d (t) and unsymmetrical case γ dm (t) ≠ γ ds (t)

To derive the stability condition for the systems (8) and (9), we consider the following Lyapunov-Krasovskii functional
$$V=V_1+V_2+V_3+V_4$$
(10)
with
$$\matrix{{{V_1} = \dot q_m^{\rm{T}}{M_m}({q_m}){{\dot q}_m} + \dot q_s^{\rm{T}}{M_s}({q_s}){{\dot q}_s}} \hfill \cr {{V_2} = {\Lambda _d}\int_{(t - {\gamma _d}(t))}^t {\dot q_m^{\rm{T}}} (\eta){{\dot q}_m}(\eta){\rm{d}}\eta + } \hfill \cr {{\Lambda _d}\int_{(t - {\gamma _d}(t))}^t {\dot q_s^{\rm{T}}} (\eta){{\dot q}_s}(\eta){\rm{d}}\eta } \hfill \cr {{V_3} = {K_{Pm}}{{\left({{q_m} - {q_s}} \right)}^{\rm{T}}}\left({{q_m} - {q_s}} \right) + {S_m}q_m^{\rm{T}}{q_m} + {S_s}q_s^{\rm{T}}{q_s}} \hfill \cr {{V_4} = \tilde \theta _m^{\rm{T}}\Gamma _m^{ - 1}{{\tilde \theta }_m} + \tilde \theta _s^{\rm{T}}\Gamma _s^{ - 1}{{\tilde \theta }_s}} \hfill \cr }$$
(11)
where \({\Lambda _d} = {1 \over {1 - \vert{{\dot \gamma }_d}(t)\vert}}\) with \(|{{\dot \gamma }_d}(t)| \leqslant \varsigma \prec 1\). We now take the derivative (10) along the trajectories of the closed loop teleoperation systems (8), (9). Then, we apply Property 1, adaptive control laws \({{\dot \tilde \theta }_m}\) and \({{\dot \tilde \theta }_s}\) along with projection mechanism (7), K Pm = K Ps and the following inequalities
$$\matrix{{2\dot q_m^{\rm{T}}\int_{(t - {\gamma _d}(t))}^t {\dot q_s^{\rm{T}}} (\eta){\rm{d}}\eta \leqslant \dot q_m^{\rm{T}}{{\dot q}_m} + \int_{(t - {\gamma _d}(t))}^t {\dot q_s^{\rm{T}}} (\eta){{\dot q}_s}(\eta){\rm{d}}\eta } \cr {2\dot q_s^{\rm{T}}\int_{(t - {\gamma _d}(t))}^t {\dot q_m^{\rm{T}}} (\eta){\rm{d}}\eta \leqslant \dot q_s^{\rm{T}}{{\dot q}_s} + \int_{(t - {\gamma _d}(t))}^t {\dot q_m^{\rm{T}}} (\eta){{\dot q}_m}(\eta){\rm{d}}\eta .} \cr }$$
(12)
\({\dot V}\) can be simplified as
$$\matrix{{\dot V \leqslant - {{\mathcal K}_m}\dot q_m^{\rm{T}}{{\dot q}_m} - {{\mathcal L}_s}\dot q_s^{\rm{T}}{{\dot q}_s} - \sigma \dot q_s^{\rm{T}}(t - {\gamma _d}(t)) \times } \cr {{{\dot q}_s}(t - {\gamma _d}(t)) - \sigma \dot q_m^{\rm{T}}(t - {\gamma _d}(t)){{\dot q}_m}(t - {\gamma _d}(t)) + } \cr {{K_{Pm}}\int_{(t - {\gamma _d}(t))}^t {\dot q_s^{\rm{T}}} (\eta){{\dot q}_s}(\eta){\rm{d}}\eta + {K_{Pm}} \times } \cr }$$
(13)
$$\int_{(t - {\gamma _d}(t))}^t {\dot q_m^{\rm{T}}} (\eta){\dot q_m}(\eta){\rm{d}}\eta $$
(14)
where \({\kappa _m} = (2{K_{Dm}} + 2{D_m} - {\Lambda _d} - {K_{Pm}}),\,{{\mathcal L}_s} = (2{K_{Ds}} + 2{D_s} - {\Lambda _d} - {K_{Pm}})\) and σ = Λ d (1 − ς). Let us now first apply Schwartz inequality on the last two terms and then use the bound on the time varying delays. Then, we can write \({\dot V}\) in the following simplified form
$$\matrix{{\dot V \leqslant - {{\mathcal K}_m}\dot q_m^{\rm{T}}{{\dot q}_m} - {{\mathcal L}_s}\dot q_s^{\rm{T}}{{\dot q}_s} - \sigma \dot q_s^{\rm{T}}(t - {\gamma _d}(t)){{\dot q}_s}(t - {\gamma _d}(t)) - } \cr {\sigma \dot q_m^{\rm{T}}(t - {\gamma _d}(t)){{\dot q}_m}(t - {\gamma _d}(t)) + {K_{Pm}}{{\mathcal A}_\alpha } \times } \cr {\int_{(t - {\gamma _d}(t))}^t {{{\dot q}_m}} {{(\eta)}^{\rm{T}}}{{\dot q}_m}(\eta){\rm{d}}\eta + {K_{Pm}}{{\mathcal A}_\alpha } \times } \cr {\int_{(t - {\gamma _d}(t))}^t {{{\dot q}_s}} {{(\eta)}^{\rm{T}}}{{\dot q}_s}(\eta){\rm{d}}\eta } \cr }$$
(15)
where \(|{\gamma _d}| \leqslant {{\mathcal A}_\alpha }\). Taking the integral from zero to time T, \({\dot V}\) leads to
$$\matrix{{V(T) - V(0) \leqslant - {{\mathcal K}_m}\int_0^T {\dot q_m^{\rm{T}}} {{\dot q}_m}{\rm{d}}t - {{\mathcal L}_s}\int_0^T {\dot q_s^{\rm{T}}} {{\dot q}_s}{\rm{d}}t - } \hfill \cr {\sigma \int_0^T {\dot q_s^{\rm{T}}} (t - {\gamma _d}(t)){{\dot q}_s}(t - {\gamma _d}(t)){\rm{d}}t - } \hfill \cr {\sigma \int_0^T {\dot q_m^{\rm{T}}} (t - {\gamma _d}(t)){{\dot q}_m}(t - {\gamma _d}(t)){\rm{d}}t + } \hfill \cr {{K_{Pm}}{{\mathcal A}_\alpha }\int_0^T {\int_{(t - {\gamma _d}(t))}^t {{{\dot q}_m}} } {{(\eta)}^{\rm{T}}}{{\dot q}_m}(\eta){\rm{d}}\eta {\rm{d}}t + } \hfill \cr {{K_{Pm}}{{\mathcal A}_\alpha }\int_0^T {\int_{(t - {\gamma _d}(t))}^t {{{\dot q}_s}} } {{(\eta)}^{\rm{T}}}{{\dot q}_s}(\eta){\rm{d}}\eta {\rm{d}}t.} \hfill \cr }$$
(16)
We again apply Schwartz inequality for the last two terms and then the bound on \({\dot V}\) can be calculated as \(V(T) - V(0) \leqslant - {\eta _m}||{{\dot q}_m}||_2^2 - {\eta _s}||{{\dot q}_s}||_2^2 - \sigma ||{{\dot q}_m}(t - {\gamma _d}(t))||_2^2 - \sigma ||{{\dot q}_s}(t - {\gamma _d}(t))||_2^2\) with \({\eta _m} = ({\kappa _m} - {K_{Pm}}{\mathcal A}_\alpha ^2)\) and \({\eta _s} = ({{\mathcal L}_s} - {K_{Pm}}{\mathcal A}_\alpha ^2)\). This implies that \(({{\dot q}_m},\,{{\dot q}_s},\,{{\dot q}_m}(t - {\gamma _d}(t)),\,{{\dot q}_s}(t - {\gamma _d}(t))) \in {{\mathcal L}_2}\) and \(({{\dot q}_m},\,{{\dot q}_s},\,{{\dot q}_m}(t - {\gamma _d}(t)),\,{{\dot q}_s}(t - {\gamma _d}(t))),\,{{\dot q}_m} - {{\dot q}_s},\,{{\dot q}_s} - {{\dot q}_m},\,{q_m},\,{q_s},\,{q_m} - {q_s},\,{q_s} - {q_m}) \in {{\mathcal L}_\infty }\). To show the bound on the two terms (q m qs(tγ d (t))) and (qsq m (tγ d (t))), let’s first write them as
$$\left({{q_m} - {q_s}} \right) + \int_{(t - {\gamma _d}(t))}^t {{{\dot q}_s}} (\eta){\rm{d}}\eta $$
(17)
$$\left({{q_s} - {q_m}} \right) + \int_{(t - {\gamma _d}(t))}^t {{{\dot q}_m}} (\eta){\rm{d}}\eta .$$
(18)

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 (tγ d (t))) ∈ γ and (q s q m (tγ 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.

We now design bilateral teleoperation systems where the forward and backward data packet delays are not symmetrical nature as γdm(t) ≠ γ ds (t). The input algorithms for the systems (3) are designed as
$$\matrix{{{\tau _m} = - {K_{Pm}}\left({{q_m} - {q_s}(t - {\gamma _{ds}}(t))} \right) - {K_{Dm}}{{\dot q}_m} + {{\hat G}_m}({q_m})} \cr {{\tau _s} = {K_{Ps}}\left({{q_m}(t - {\gamma _{dm}}(t)) - {q_s}} \right) - {K_{Ds}}{{\dot q}_s} + {{\hat G}_s}({q_s}).} \cr }$$
(19)
The control scheme (19) is shown in Fig. 1 with γ dm (t) ≠ γ ds (t). Using (3) and (19), the closed loop teleoperators under unsymmetrical time varying delays can be derived as
$$\matrix{{{{\ddot q}_m} = M_m^{ - 1}({q_m})[{K_{Pm}}{q_s}(t - {\gamma _{ds}}(t)) - {K_{Pm}}{q_m} - {K_{Dm}}{{\dot q}_m} - } \hfill \cr {{C_m}({q_m},{{\dot q}_m}){{\dot q}_m} + {{\hat G}_m}({q_m}) - {G_m}({q_m}) - {S_m}{q_m} - } \hfill \cr {{D_m}{{\dot q}_m}]} \hfill \cr {{{\ddot q}_s} = M_s^{ - 1}({q_s})[{K_{Ps}}{q_m}(t - {\gamma _{dm}}(t)) - {K_{Ps}}{q_s} - {K_{Ds}}{{\dot q}_s} - } \hfill \cr {{C_s}({q_s},{{\dot q}_s}){{\dot q}_s} + {{\hat G}_s}({q_s}) - {G_s}({q_s}) - {S_s}{q_s} - {D_s}{{\dot q}_s}].} \hfill \cr }$$
(20)
To analyze the convergence property of the teleoperators (20), let us use the following Lyapunov-Krasovskii function
$$V=V_1+V_2+V_3+V_4$$
(21)
where
$$\matrix{{{V_1} = \dot q_m^{\rm{T}}{M_m}({q_m}){{\dot q}_m} + \dot q_s^{\rm{T}}{M_s}({q_s}){{\dot q}_s}} \hfill \cr {{V_2} = {\Lambda _{dm}}\int_{(t - {\gamma _{dm}}(t))}^t {\dot q_m^{\rm{T}}} (\eta){{\dot q}_m}(\eta){\rm{d}}\eta + } \hfill \cr {{\Lambda _{ds}}\int_{(t - {\gamma _{ds}}(t))}^t {\dot q_s^{\rm{T}}} (\eta){{\dot q}_s}(\eta){\rm{d}}\eta } \hfill \cr {{V_3} = {K_{Pm}}{{\left({{q_m} - {q_s}} \right)}^{\rm{T}}}\left({{q_m} - {q_s}} \right) + {S_m}q_m^{\rm{T}}{q_m} + {S_s}q_s^{\rm{T}}{q_s}} \hfill \cr {{V_4} = \tilde \theta _m^{\rm{T}}\Gamma _m^{ - 1}{{\tilde \theta }_m} + \tilde \theta _s^{\rm{T}}\Gamma _s^{ - 1}{{\tilde \theta }_s}} \hfill \cr }$$
where \({\Lambda _{dm1}} = {{{K_{Dm}}} \over {1 - |{{\dot \gamma }_{dm}}(t)|}}\) and \({\Lambda _{ds2}} = {{{K_{Ds}}} \over {1 - |{{\dot \gamma }_{ds}}(t)|}}\) with \(|{{\dot \gamma }_{dm}}(t)| \leqslant \varsigma dm < 1\) and \(|{{\dot \gamma }_{ds}}(t)| \leqslant \varsigma dm < 1\). Using the Property 1, (20), adaptation laws with projection mechanism (7), K Pm = K Ps and the following inequalities
$$\matrix{{2\dot q_m^{\rm{T}}\int_{(t - {\gamma _{ds}}(t))}^t {\dot q_s^{\rm{T}}} (\eta){\rm{d}}\eta \leqslant \dot q_m^{\rm{T}}{{\dot q}_m} + \int_{(t - {\gamma _{ds}}(t))}^t {\dot q_s^{\rm{T}}} (\eta){{\dot q}_s}(\eta){\rm{d}}\eta } \cr {2\dot q_s^{\rm{T}}\int_{(t - {\gamma _{dm}}(t))}^t {\dot q_m^{\rm{T}}} (\eta){\rm{d}}\eta \leqslant \dot q_s^{\rm{T}}{{\dot q}_s} + \int_{(t - {\gamma _{dm}}(t))}^t {\dot q_m^{\rm{T}}} (\eta){{\dot q}_m}(\eta){\rm{d}}\eta .} \cr }$$
(22)
\({\dot V}\) can be represented as
$$\matrix{{\dot V \leqslant - {{\mathcal K}_{mu}}\dot q_m^{\rm{T}}{{\dot q}_m} - {{\mathcal L}_{su}}\dot q_s^{\rm{T}}{{\dot q}_s} - {\sigma _{ms}}\dot q_s^{\rm{T}}(t - {\gamma _{ds}}(t)) \times } \cr {{{\dot q}_s}(t - {\gamma _{ds}}(t)) - {\sigma _{mu}}\dot q_m^{\rm{T}}(t - {\gamma _{dm}}(t))){{\dot q}_m}(t - {\gamma _{dm}}(t)) + } \cr {\int_{(t - {\gamma _{ds}}(t))}^t {\dot q_s^{\rm{T}}} (\eta){{\dot q}_s}(\eta){\rm{d}}\eta + \int_{(t - {\gamma _{dm}}(t))}^t {\dot q_m^{\rm{T}}} (\eta){{\dot q}_m}(\eta){\rm{d}}\eta } \cr }$$
where \({K_{mu}} = (2{K_{Dm}} + 2{D_m} - {\Lambda _{dm}} - {K_{Pm}}),\,{{\mathcal L}_{su}} = (2{K_{Ds}} + 2{D_s} - {\Lambda _{ds}} - {K_{Pm}}),{\sigma _{ms}} = {\Lambda _{ds}}(1 - {\varsigma _{ds}})\), su = (2KDS + 2D s − Λ ds K Pm ), σ ms = Λ ds (1 − ς ds ) and σ mu = Λ dm (1 − ς dm ). We now use Schwartz inequality on the last two terms and then take integral from zero to time t = T. Then, \({\dot V}\) can be written as
$$\matrix{{V(T) - V(0) \leqslant - {{\mathcal K}_{mu}}\int_0^T {\dot q_m^{\rm{T}}} {{\dot q}_m}{\rm{d}}t - {{\mathcal L}_{su}}\int_0^T {\dot q_s^{\rm{T}}} {{\dot q}_s}{\rm{d}}t - } \hfill \cr {{\sigma _{ms}}\int_0^T {\dot q_s^{\rm{T}}} (t - {\gamma _{ds}}(t)){{\dot q}_s}(t - {\gamma _{ds}}(t)){\rm{d}}t - {\sigma _{mu}} \times } \hfill \cr {\int_0^T {\dot q_m^{\rm{T}}} (t - {\gamma _{dm}}(t)){{\dot q}_m}(t - {\gamma _{dm}}(t)){\rm{d}}t + {K_{Pm}}{{\mathcal A}_{d\alpha }} \times } \hfill \cr {\int_0^T {\int_{(t - {\gamma _{dm}}(t))}^t {{{\dot q}_m}} } {{(\eta)}^{\rm{T}}}{{\dot q}_m}(\eta){\rm{d}}\eta {\rm{d}}t + {K_{Pm}}{{\mathcal A}_{s\alpha }} \times } \hfill \cr {\int_0^T {\int_{(t - {\gamma _{ds}}(t))}^t {{{\dot q}_s}} } {{(\eta)}^{\rm{T}}}{{\dot q}_s}(\eta){\rm{d}}\eta {\rm{d}}t} \hfill \cr }$$
with \(|{\gamma _{dm}}| \leqslant {\mathcal A}{_{d\alpha }}\) and \(|{\gamma _{ds}}| \leqslant {\mathcal A}{_{s\alpha }}\). Applying Schwartz inequality on the last two terms, \({\dot V}\) can be simplified further
$$\matrix{{V(T) - V(0) \leqslant - {\eta _{mu}} \Vert {{\dot q}_m}\Vert_2^2 - {\sigma _{mu}}\Vert{{\dot q}_m}(t - {\gamma _{dm}}(t))\Vert_2^2 - } \hfill \cr {{\eta _{su}}\Vert{{\dot q}_s}\Vert_2^2 - {\sigma _{ms}}\Vert{{\dot q}_s}(t - {\gamma _{ds}}(t)\Vert_2^2} \hfill \cr }$$
(23)
where \({\eta _{mu}} = ({\kappa _{mu}} - {K_{Pm}}{\mathcal A}_{d\alpha }^2)\) and \({\eta _{su}} = ({{\mathcal L}_{su}} - {K_{Pm}}{\mathcal A}_{d\alpha }^2)\). This implies that \(({{\dot q}_m},\,{{\dot q}_s},\,{{\dot q}_m}(t - {\gamma _{dm}}(t)),\,{{\dot q}_s}(t - {\gamma _{ds}}(t))) \in {{\mathcal L}_2}\) and \(({{\dot q}_m},\,{{\dot q}_s},\,{{\dot q}_m}(t - {\gamma _{dm}}(t)),\,{{\dot q}_s}(t - {\gamma _{ds}}(t))) \in {{\mathcal L}_\infty }\). We can also show the bounds on the terms (q m qs(tY ds (t))) and (q s q m (tγ dm (t))). To show that, we first rewrite the term (q m q s (tγ ds (t))) as
$$\left({{q_m} - {q_s}} \right) + \int_{(t - {\gamma _{ds}}(t))}^t {{{\dot q}_s}} (\eta){\rm{d}}\eta $$
(24)
and the term (q s q m (tγ dm (t))) as
$$\left({{q_s} - {q_m}} \right) + \int_{(t - {\gamma _{dm}}(t))}^t {{{\dot q}_m}} (\eta){\rm{d}}\eta .$$
(25)

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 (tγ ds (t))) ∈ ℒ and (q s q m (tγ 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

Let us now design teleoperation systems by coupling local and remote sites by delaying position and velocity signals of the master and slave systems. The input algorithms for the systems (3) with symmetrical time varying delays can be designed as follows
$$\matrix{{{\tau _m} = - {K_{Dm}}\left({{{\dot q}_m} - {{\dot q}_s}(t - {\gamma _d}(t))} \right) + {{\hat G}_m}({q_m}) - } \hfill \cr {{K_{Pm}}\left({{q_m} - {q_s}(t - {\gamma _d}(t))} \right) - {\zeta _m}{{\dot q}_m}} \hfill \cr {{\tau _s} = {K_{Ds}}\left({{{\dot q}_m}(t - {\gamma _d}(t)) - {{\dot q}_s}} \right) - {K_{Ps}}{q_s} + {{\hat G}_s}({q_s}) - } \hfill \cr {{K_{Ps}}\left({{q_s} - {q_m}(t - {\gamma _d}(t))} \right) - {\zeta _s}{{\dot q}_s}} \hfill \cr }$$
(26)
with ζ m > 0 and ζ s > 0. The implementation block diagram of the control scheme (26) is presented in Fig. 2 with γ dm (t) = γ ds (t) = γ d (t). The closed loop teleoperation systems with input algorithms (26) can be formulated as
$$\matrix{{{{\ddot q}_m} = M_m^{ - 1}({q_m})[{K_{Dm}}{{\dot q}_s}(t - {\gamma _d}(t)) - {K_{Pm}}{q_m} - {K_{Dm}}{{\dot q}_m} - } \hfill \cr {{C_m}({q_m},{{\dot q}_m}){{\dot q}_m} + {{\hat G}_m}({q_m}) - {G_m}({q_m}) - {S_m}{q_m} - } \hfill \cr {{D_m}{{\dot q}_m} + {K_{Pm}}{q_s}(t - {\gamma _d}(t)) - {\zeta _m}{{\dot q}_m}]} \hfill \cr }$$
(27)
$$\matrix{{{{\ddot q}_s} = M_s^{ - 1}({q_s})[{K_{Ds}}{{\dot q}_m}(t - {\gamma _d}(t)) - {K_{Ds}}{{\dot q}_s} - {K_{Ps}}{q_s} - } \hfill \cr {{C_s}({q_s},{{\dot q}_s}){{\dot q}_s} + {{\hat G}_s}({q_s}) - {G_s}({q_s}) - {S_s}{q_s} - {D_s}{{\dot q}_s} + } \hfill \cr {{K_{Ps}}{q_m}(t - {\gamma _d}(t)) - {\zeta _s}{{\dot q}_s}].} \hfill \cr }$$
(28)
Fig. 2

Control scheme for bilateral teleoperation systems with delayed position and velocity signals for symmetrical time varying delay γ dm (t)= γ ds (t) = γ d (t) and unsymmetrical time varying delay γ dm (t) ≠ γ ds (t)

For the convergence analysis of the teleoperation systems (27), (28), let us employ the following Lyapunov-Krasovskii function
$$V=V_1+V_2+V_3+V_4$$
(#)
with
$$\matrix{{{V_1} = \dot q_m^{\rm{T}}{M_m}({q_m}){{\dot q}_m} + \dot q_s^{\rm{T}}{M_s}({q_s}){{\dot q}_s}} \hfill \cr {{V_2} = {\Lambda _{d1}}\int_{(t - {\gamma _d}(t))}^t {\dot q_m^{\rm{T}}} (\eta){{\dot q}_m}(\eta){\rm{d}}\eta + {\Lambda _{d2}}\int_{(t - {\gamma _d}(t))}^t {\dot q_s^{\rm{T}}} (\eta)} \hfill \cr {{{\dot q}_s}(\eta){\rm{d}}\eta } \hfill \cr {{V_3} = {K_{Pm}}{{\left({{q_m} - {q_s}} \right)}^{\rm{T}}}\left({{q_m} - {q_s}} \right) + {S_m}q_m^{\rm{T}}{q_m} + {S_s}q_s^{\rm{T}}{q_s}} \hfill \cr {{V_4} = \tilde \theta _m^{\rm{T}}\Gamma _m^{ - 1}{{\tilde \theta }_m} + \tilde \theta _s^{\rm{T}}\Gamma _s^{ - 1}{{\tilde \theta }_s}} \hfill \cr }$$
where \({\Lambda _{d1}} = {{{K_{{D_m}}}} \over {1 - |{{\dot \gamma }_d}(t))|}}\) and \({\Lambda _{d2}} = {{{K_{{D_s}}}} \over {1 - |{{\dot \gamma }_d}(t))|}}\). Using K Pm = K Ps , K Dm = K Ds , adaptive control with projection mechanism (7), Property 1, (12) and
$$\matrix{{2\dot q_m^{\rm{T}}{{\dot q}_s}(t - {\gamma _d}(t)) \leqslant \dot q_m^{\rm{T}}{{\dot q}_m} + \dot q_s^{\rm{T}}(t - {\gamma _d}(t))\dot q_s^{\rm{T}}(t - {\gamma _d}(t))} \hfill \cr {2\dot q_s^{\rm{T}}{{\dot q}_m}(t - {\gamma _d}(t)) \leqslant \dot q_s^{\rm{T}}{{\dot q}_s} + \dot q_m^{\rm{T}}(t - {\gamma _d}(t)){{\dot q}_m}(t - {\gamma _d}(t))} \hfill \cr }$$
(30)
The time derivative of (29) along the trajectories of (27), (28) can be simplified as \(V(T) - V(0) \le - {\pi _{{D_m}}}||{\dot q_m}||_2^2 - {\pi _{{D_s}}}||{\dot q_s}|{|_2}\) with \({\pi _{Dm}} = ({K_{Dm}} + 2{D_m} + {K_{Pm}}{\mathcal A}_\alpha ^2 + {\zeta _m} - {\Lambda _{d1}} - {K_{Pm}})\) and \({\pi _{Ds}} = ({K_{Dm}} + 2{D_s} + {K_{Pm}}{\mathcal A}_\alpha ^2 - {\Lambda _{d2}} - {K_{Pm}})\). This implies that \(\left({{{\dot q}_m},{{\dot q}_s}} \right) \in {{\mathcal L}_2}\) and \(\left({{q_m},{q_s},{{\dot q}_m},{{\dot q}_s},{{\dot q}_m} - {{\dot q}_s},{{\dot q}_s} - {{\dot q}_m},{q_m} - {q_s},{q_s} - {q_m}} \right) \in {{\mathcal L}_\infty }\). Now, we need to show the bounds on the terms \(\left({{{\dot q}_m} - {{\dot q}_s}(t - {\gamma _d}(t))} \right)\) and \(\left({{{\dot q}_s} - {{\dot q}_m}(t - {\gamma _d}(t))} \right)\) from the following relationship
$$\left({{{\dot q}_m} - {{\dot q}_s}} \right) + \int_{(t - {\gamma _d}(t))}^t {{{\ddot q}_s}} (\eta){\rm{d}}\eta $$
(31)
$$\left({{{\dot q}_s} - {{\dot q}_m}} \right) + \int_{(t - {\gamma _d}(t))}^t {{{\ddot q}_m}} (\eta){\rm{d}}\eta .$$
(32)

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 (tγ d (t))) ∈ ℒ and (q m q s (tγ d (t))) ∈ . 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.

Finally, we design teleoperation systems by delaying position and velocity signals with unsymmetrical time varying communication delays as
$$\matrix{{{\tau _m} = - {K_{Dm}}\left({{{\dot q}_m} - {{\dot q}_s}(t - {\gamma _{ds}}(t))} \right) - {K_{Pm}}{q_m} + {{\hat G}_m}({q_m}) + } \hfill \cr {{K_{Pm}}{q_s}(t - {\gamma _{ds}}(t)) - {\zeta _m}{{\dot q}_m}} \hfill \cr {{\tau _s} = {K_{Ds}}\left({{{\dot q}_m}(t - {\gamma _{dm}}(t)) - {{\dot q}_s}} \right) - {K_{Ps}}{q_s} + {{\hat G}_s}({q_s}) + } \hfill \cr {{K_{Ps}}{q_m}(t - {\gamma _{dm}}(t)) - {\zeta _s}{{\dot q}_s}.} \hfill \cr }$$
(33)
The control scheme is shown in Fig. 2 with γ dm (t) ≠ γ ds (t). Using (33) and (3), the closed loop teleoperation systems can be derived as
$$\matrix{{{{\ddot q}_m} = M_m^{ - 1}({q_m})[{K_{Dm}}{{\dot q}_s}(t - {\gamma _{ds}}(t)) - {K_{Pm}}{q_m} - {K_{Dm}}{{\dot q}_m} - } \hfill \cr {{C_m}({q_m},{{\dot q}_m}){{\dot q}_m} + {{\hat G}_m}({q_m}) - {G_m}({q_m}) - {S_m}{q_m} - } \hfill \cr {{D_m}{{\dot q}_m} + {K_{Pm}}{q_s}(t - {\gamma _{ds}}(t)) - {\zeta _m}{{\dot q}_m}]} \hfill \cr }$$
(34)
$$\matrix{{{{\ddot q}_s} = M_s^{ - 1}({q_s})[{K_{Ds}}{{\dot q}_m}(t - {\gamma _{dm}}(t)) - {K_{Ds}}{{\dot q}_s} - {K_{Ps}}{q_s} - } \hfill \cr {{C_s}({q_s},{{\dot q}_s}){{\dot q}_s} + {{\hat G}_s}({q_s}) - {G_s}({q_s}) - {S_s}{q_s} - {D_s}{{\dot q}_s} + } \hfill \cr {{K_{Ps}}{q_m}(t - {\gamma _{dm}}(t)) - {\zeta _s}{{\dot q}_s}].} \hfill \cr }$$
(35)
Now, for analyzing the stability of the closed loop systems (34), (35), we choose the following Lyapunov-Krasovskii functional
$$V=V_1+V_2+V_3+V_4$$
(36)
with where \({\Lambda _{dm1}} = {{{K_{Dm}}} \over {1 - |{{\dot \gamma }_{dm}}(t)|}}\,{\rm{and}}\,{\Lambda _{ds2}} = {{{K_{Ds}}} \over {1 - |{{\dot \gamma }_{ds}}(t)|}}\). We now take the derivative (36) along the trajectories of the closed loop systems (34), (35). Then, using K Pm = K Ps , K Dm = K Ds , (22), adaptation laws with parameter projection scheme (7), Property 1 and
$$\matrix{{2\dot q_m^{\rm{T}}{{\dot q}_s}(t - {\gamma _{ds}}(t)) \leqslant \dot q_m^{\rm{T}}{{\dot q}_m} + \dot q_s^{\rm{T}}(t - {\gamma _{ds}}(t))\dot q_s^{\rm{T}}(t - {\gamma _{ds}}(t))} \hfill \cr {2\dot q_s^{\rm{T}}{{\dot q}_m}(t - {\gamma _{dm}}(t)) \leqslant \dot q_s^{\rm{T}}{{\dot q}_s} + \dot q_m^{\rm{T}}(t - {\gamma _{dm}}(t)){{\dot q}_m}(t - {\gamma _{dm}}(t)).} \hfill \cr }$$
(37)

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 (tγ ds (t))) ∈ and (q s q m (tγ 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

We now show the design and evaluation process of the proposed control methods on bilateral teleoperation systems constructed by a 2-DOF master and slave manipulator. For simplicity, we only present evaluation results of the teleoperation algorithms given in (19). In our evaluation, it is assumed that the local and remote sites are connected by Internet telecommunication channel. The motion dynamics of a 2-DOF master and slave manipulator can be written as
$${M_m}({q_m}){\ddot q_m} + {C_m}({q_m},{\dot q_m}){\dot q_m} + {G_m}({q_m}) = {\tau _m} + {F_h}$$
(38)
$${M_s}({q_s}){\ddot q_s} + {C_s}({q_s},{\dot q_s}){\dot q_s} + {G_s}({q_s}) = {\tau _s} - {F_e}$$
(39)
where \(M_m (q_m ) = \left[ {\begin{array}{*{20}c} {m_{11} } & {m_{12} } \\ {m_{21} } & {m_{22} } \\ \end{array} } \right],C_m (q_m ,\dot q_m ) = \left[ {\begin{array}{*{20}c} {c_{11} } & {c_{12} } \\ {c_{21} } & {c_{22} } \\ \end{array} } \right],F_h = \left[ {\frac{{F_{h1} }} {{F_{h2} }}} \right],Fe = \left[ {\frac{{F_{e1} }} {{F_{e2} }}} \right]\) and \({G_m}({q_m}) = \left[ {\matrix{{{G_m}({q_{m1}})} \cr {{G_m}({q_{m2}})} \cr } } \right]\) with \({m_{11}} = (2{l_1}\,\cos \,{q_2} + {l_2}){l_2}{m_2} + l_1^2({m_1} + {m_2}),\,{m_{12}} = ({l_2}{m_2} + {l_1}{l_2}{m_2}\,\cos \,{q_2}),\,{m_{21}} = ({m_2}{l^2} + {m_2}{l^2}\cos {q_2}),\,{m_{22}} = {m_2}l_2^2,\,{c_{11}} = - {l_1}{l_2}{m_2}\sin \,{q_2}{{\dot q}_2},\,{c_{12}} = \, - {l_1}{l_2}\sin {q_2}({{\dot q}_2} + {{\dot q}_1}),\,{c_{21}} = {l_1}{l_2}{m_2}\sin {q_2},\,{c_{22}} = 0,\,G(q{m_1}) = g{m_2}{l_2}\,\sin \,({q_1} + {q_2}) + g({m_1} + {m_2}){l_1}\sin \,{q_1}\) and G(q m2 ) = gm 2 l 2 sin (q l + q 2). The uncertain parameters for gravity loading vectors are chosen as θ m1 = m2l2, θ m2 = m l l l and θm3 = l l m 2. We also consider that \({M_s}({q_s}) = {M_m}({q_m}),\,{C_m}({q_m},\,{{\dot q}_m}) = {C_s}({q_s},\,{{\dot q}_s}){{\dot q}_s}\). The model parameters for master and slave manipulators are chosen as m l = 15, m 2 =5, l l = 1, l 2 =0.5 and g = 9.82. In our first evaluation, the time varying delays in forward, γ dm (t), and backward, γ ds (t), internet telecommunication channels are assumed to be unsymmetrical nature as γ dm (t) = 10 + 0.8 sin(t) and γ ds (t) = 5 + 0.5cos(t). The time history of the time varying delays are given in Fig. 3. The environment and human input forces are modeled as \({F_{e1}} = {S_{s1}}{q_{s1}} + {D_{s1}}{{\dot q}_{s1}},\,{F_{e2}} = {S_{s2}}{q_{s2}} + {D_{s2}}{{\dot q}_{s2}},\,{F_{h1}} = {k_{o1}} - {S_{m1}}{q_{m1}} - {D_{m1}}{{\dot q}_{m1}},\,{F_{h2}} = {k_{o2}} - {S_{m2}}{q_{m2}} - {D_{m2}}{{\dot q}_{m2}}\) with k o1 > 0 and k o2 > 0.
Fig. 3

Unsymmetrical forward γ dm (t) and backward γ ds (t) time varying communication delays in second

In our evaluation, we consider three different types of interaction forces between slave and environment. In first case, the slave manipulators are operating in free space as F e1 = 0 and F e2 = 0. The parameters for human operator forces are chosen as k o1 = 10, k o2 = 10, S m1 = 15, S m2 = 15, D m1 = 10 and D m2 = 20. The control design parameters are selected as Γ m = I 3 × 3, Γ s = I 3 × 3, K Pm1 = KPS1} = 10, K Pm2 = KPS2} = 10, K Dm1 = K Dm2 = 150 and K Dsl = K Ds2 = 150. The evaluation results are presented in Fig. 4.
Fig. 4

Position tracking of the master and slave manipulators in radians with free remote environment space

Second, we consider that the remote slave manipulators are interacting with soft environment as S s1 =2, S s2 = 2, D s1 = 1 and D s2 = 1. The parameters of the human input forces are chosen as k o1 = 15, ko2 = 15, S m1 = 15, S m2 = 15, D m1 = 10 and D m2 = 20. The control design parameters are chosen for this evaluation as Γ m = I 3 × 3, Γs = I 3 × 3, K pm1 = K ps1 = 10, K pm2 = K ps2 = 10, K Dm1 = K Dm2 = 250, K Ds1 = K Ds2 = 150. The evaluation results are given in Fig. 5 with unsymmetrical time varying delays as depicted in Fig. 3.
Fig. 5

Position tracking of the master and slave manipulator in radians with soft interaction between remote slave and environment

In third case, we apply very hard interaction forces between remote slave manipulator and environment as S s1 = 50000, S s2 = 50 000, D s1 = 1 and D s2 = 1. Then, we choose the parameters for human and master interaction forces as k o1 = 15, k o2 = 15, S m1 = 15, S m2 = 15, D m1 =10 and D m2 = 20. The control design parameters are considered as Γm = I 3 × 3, Γs = I 3 × 3, K Pm1 = K Ps1 = 10, K Pm2 = K Ps2 = 10, K Dm1 = K Dm2 = 250, K Ds1 = K Ds2 = 150. The unsymmetrical time varying delays are kept similar to our previous evaluation as depicted in Fig. 3. With these parameters, the design strategy proposed in (20) is applied to the given teleoperation systems. The evaluation results with very hard contact forces between slave and environment are shown in Fig. 6.
Fig. 6

Position tracking of the master and slave manipulator in radians with very hard interaction between remote slave and environment

Let us now repeat the evaluation of third case with unsymmetrical random time varying delays. For our evaluation, the time derivative of the random time varying delays for the forward and backward communication channels are chosen as shown in Fig. 7. All other design parameters are kept similar to our previous case. The evaluation results are depicted in Fig. 8. The sampling time is chosen as 0.01 s. From these results, we can see that the position and tracking error of the master and slave manipulators are asymptotically stable in the presence of unsymmetrical forward and backward time varying delays. The asymptotic stability is also guaranteed in the presence of free space, soft and very hard interaction forces between remote slave and environment.
  • 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 position-velocity 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 \({V_3} = {K_{Pm}}{({q_m} - {q_s})^{\rm{T}}}({q_m} - {q_s}) + {P_m}q_m^{\rm{T}}{q_m} + {{{K_{Pm}}} \over {{K_{Ps}}}}{P_s}q_s^{\rm{T}}{q_s}\).

Fig. 7

The time derivative of unsymmetrical forward γ dm (t) and backward γ ds (t) random time varying communication delays in second

Fig. 8

Position tracking of the master and slave manipulator with random time varying delays with very hard interaction between remote slave and environment

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 position-velocity 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 Lyapunov-Krasovskii 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.

References

  1. [1]
    I. G. Polushin, P. X. Liu, C. H. Lung. A control scheme for stable force-reflecting teleoperation over IP networks. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 36, no. 4, pp. 930–939, 2006.CrossRefGoogle Scholar
  2. [2]
    I. G. Polushin, P. X. Liu, C. H. Lung. A force-reflection algorithm for improved transparency in bilateral teleoperation with communication delay. IEEE/ASME Transactions on Mechatronics, vol. 12, no. 3, pp. 361–374, 2007.CrossRefGoogle Scholar
  3. [3]
    C. C. Hua, P. X. Liu. Convergence analysis of teleoperation systems with unsymmetric time-varying delays. IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 56, no. 3, pp. 240–244, 2009.CrossRefGoogle Scholar
  4. [4]
    C. C. Hua, P.X. Liu. Teleoperation over the internet with/without velocity signal. IEEE Transactions on Instrumentation and Measurement, vol. 60, no. 1, pp. 4–13, 2011.CrossRefGoogle Scholar
  5. [5]
    P. Arcara, C. Melchiorri. Control schemes for teleoperation with time delay: A comparative study. Robotics and Autonomous Systems, vol. 38, no. 1, pp. 49–64, 2002.CrossRefzbMATHGoogle Scholar
  6. [6]
    P. F. Hokayem, M.W. Spong. Bilateral teleoperation: An historical survey. Automatica, vol. 42, no. 12, pp. 2035–2057, 2006.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    R. J. Anderson, M. W. Spong. Bilateral control of teleoperators with time delay. IEEE Transactions on Automatic Control, vol. 34, no. 5, pp. 494–501, 1989.CrossRefMathSciNetGoogle Scholar
  8. [8]
    R. J. Anderson, M. W. Spong. Asymptotic stability for force reflecting teleoperators with time delay. The International Journal of Robotics Research, vol. 11, no. 2, pp. 135–149, 1992.CrossRefGoogle Scholar
  9. [9]
    G. Niemeyer, J. J. E. Slotine. Telemanipulation with time delays. The International Journal of Robotics Research, vol. 23, no. 9, pp. 873–890, 2004.CrossRefGoogle Scholar
  10. [10]
    K. Kosuge, H. Murayama, K. Takeo. Bilateral feedback control of telemanipulators via computer network. In Proceedings of 1996 IEEE/RSJ International Conference on Intelligent Robots and Systems, IEEE, Osaka, pp. 1380–1385, 1996.Google Scholar
  11. [11]
    J. H. Park, H. C. Cho. Sliding mode control of bilateral teleoperation systems with force-reflection on the Internet. In Proceedings of 2000 IEEE/RSJ International Conference on Intelligent Robots and Systems, IEEE, Takamatsu, pp. 1187–1192, 2000.Google Scholar
  12. [12]
    H. J. Li, A. G. Song. Virtual-environment modeling and correction for force-reflecting teleoperation with time delay. IEEE Transactions on Industrial Electronics, vol. 54, no. 2, pp. 1227–1233, 2007.CrossRefGoogle Scholar
  13. [13]
    S. Munir, W. J. Book. Internet-based teleoperation using wave variables with prediction. IEEE/ASME Transactions on Mechatronics, vol. 7, no. 2, pp. 124–133, 2002.CrossRefGoogle Scholar
  14. [14]
    L. Bate, C. D. Cook, Z. Li. Reducing wave-based teleoperator reflections for unknown environments. IEEE Transactions on Industrial Electronics, vol. 58, no. 2, pp. 392–397, 2011.CrossRefGoogle Scholar
  15. [15]
    M. Alise, R. G. Roberts, D. W. Repperger, C. A. Moore, Jr., S. Tosunoglu. On extending the wave variable method to multiple-DOF teleoperation systems. IEEE/ASME Transactions on Mechatronics, vol. 14, no. 1, pp. 55–63, 2009.CrossRefGoogle Scholar
  16. [16]
    N. Chopra, M. W. Spong, S. Hirche, M. Buss. Bilateral teleoperation over the internet: The time varying delay problem. In Proceedings of the American Control Conference, IEEE, Denver, Colorado, USA, pp. 155–160, 2003.Google Scholar
  17. [17]
    Y. J. Pan, C. Canudas-de-Wit, O. Sename. A new predictive approach for bilateral teleoperation with applications to drive-by-wire systems. IEEE Transactions on Robotics, vol. 22, no. 6, pp. 1146–1162, 2006.CrossRefGoogle Scholar
  18. [18]
    D. J. Lee, M. W. Spong. Passive bilateral teleoperation with constant time delay. IEEE Transactions on Robotics, vol. 22, pp. 269–281, 2006.CrossRefGoogle Scholar
  19. [19]
    E. Nuno, R. Ortega, N. Barabanov, L. Basanez. A globally stable PD controller for bilateral teleoperators. IEEE Transactions on Robotics, vol. 24, no. 3, pp. 753–758, 2008.CrossRefGoogle Scholar
  20. [20]
    D. Lee, K. Huang. Passive-set-position-modulation framework for interactive robotic systems. IEEE Transactions on Robotics, vol. 26, no. 2, pp. 354–369, 2010.CrossRefGoogle Scholar
  21. [21]
    X. C. Shi, T. P. Zhang. Adaptive tracking control of uncertain MIMO nonlinear systems with time-varying delays and unmodeled dynamics. International Journal of Automation and Computing, vol. 10, no. 3, pp. 194–201, 2013.CrossRefGoogle Scholar
  22. [22]
    S. Islam, P. X. Liu, A. E. Saddik. Teleoperation over internet communication networks under time varying delay. In Proceedings of IEEE International Conference on Automation and Logistics, IEEE, Chongqing, China, pp. 12–17, 2011.Google Scholar
  23. [23]
    R. Kelly, V. Santibanez, A. Loria. Control of robot manipulators in joint space. Advanced Textbooks in Control and Signal Processing, New York: Springer-Verlag, 2005.Google Scholar
  24. [24]
    N. Hogan. Impedance control: An approach to manipulation, Parts 1–3. Journal of Dynamic Systems, Measurement and Control, vol. 107, no. 1, pp. 1–24, 1985.CrossRefzbMATHGoogle Scholar
  25. [25]
    D. E. Whitney. Historical perspective and state of the art in robot force control. In Proceedings of IEEE International Conference on Robotics and Automation, IEEE, St. Louis, MO, USA, vol. 2, pp. 262–268, 1985.Google Scholar
  26. [26]
    H. K. Khalil. Nonlinear Systems, New York: Maxwell Macmillan, 1992.zbMATHGoogle Scholar

Copyright information

© Science in China Press 2014

Authors and Affiliations

  • Shafiqul Islam
    • 1
    • 2
    • 8
  • Xiaoping P. Liu
    • 3
  • Abdulmotaleb El Saddik
    • 1
    • 4
  • Lakmal Seneviratne
    • 5
    • 6
  • Jorge Dias
    • 7
    • 8
  1. 1.School of Information Technology and EngineeringUniversity of OttawaOttawaCanada
  2. 2.School of Engineering and Applied SciencesHarvard UniversityCambridgeUSA
  3. 3.Department of Systems and Computer EngineeringCarleton UniversityOttawaCanada
  4. 4.Electrical EngineeringNew York UniversityUAE
  5. 5.Division of EngineeringKings CollegeLondonUK
  6. 6.Robotics and Mechanical EngineeringKhalifa University of Science & TechnologyUAE
  7. 7.Institute of Systems and RoboticsUniversity of CoimbraPortugal
  8. 8.Robotics and Electrical and Computer EngineeringKhalifa University of Science & TechnologyUAE

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