State Reconstruction of Remote Robotic System Using Delayed Output and Torque Estimation

Abstract

The state reconstruction problem for a remote robotic system with communication delay is investigated in this paper. A symmetric estimation scheme is proposed based on both input and output observers. First, a sliding-mode disturbance observer is designed to estimate the input torque of the robot on the remote side. Then, by using the received output and torque estimation subject to time delay, a novel predictive observer is proposed on the local side to reconstruct the real-time joint angles and velocities. Based on a Lyapunov approach, sufficient conditions are obtained to make sure that the estimation errors can converge exponentially to bounded regions through selecting proper observer gains. Simulation studies are conducted to verify the effectiveness of the proposed strategy. The estimation error of state reconstruction is decreased by 14% in terms of the integral square error when compared to the standard high-gain predictor, and the simulation results demonstrate the effectiveness of force estimation under disturbances and model uncertainties.

1. Introduction

Time delays are common issues in the application of remote robotic systems with long communication distances, such as space robots, underwater robots, and teleoperation systems [1,2,3,4,5]. Even a small delay can degrade the performance and stability of the robot control system [6]. A number of studies have investigated the control schemes for robotic systems with uncertainties and disturbances, such as data-based PID control [7] and fuzzy control [8]. Meanwhile, various techniques have focused on the controller design for robot systems with input delay, such as predictive controllers [9,10], adaptive controllers [11], and neural network controllers [12], which ensure the stability of the remote systems. On the other hand, however, the delay feedback signal on the local side will cause the operator to be unable to grasp the remote working status in time and thus be unable to issue the correct control instructions. Thus, it is necessary to reconstruct the state of remote robotic systems by using delayed feedback.

For nonlinear systems with output delay, high-gain observers/predictors are designated in [13,14,15] to estimate the system’s state based on delayed measurements. And this approach is extended to cases of time-varying delays in [16,17] where Lyapunov methods are applied to analyze the convergence of estimation errors. In [18,19,20], continuous-discrete time observers are applied as underlying systems by using sampled and delayed outputs. Then, cascade observers are proposed in [21,22,23], which are composed of a number of sub-observers in chains, such that the large time delays can be divided into small segmentations. Additionally, an adaptive observer is designed in [24] for heat diffusion-reaction partial differential equations with sensor delay. In [25], Immersion and Invariance techniques are used to design observers for general nonlinear systems when the output measurements are subject to constant time delays. However, the input of the system is assumed to be known in these methods. For remote robotic systems, the input torque is hard to measure and often suffers from disturbances. In addition, only the delayed estimation of input can be applied to the observer design due to the time delay.

The issues of force/torque estimations for robot control systems have been addressed in many previous studies [26,27,28]. For example, in [29,30], force estimations are applied to the controller designs for telerobotics, which consist of symmetric leader and follower robots, such that the operational performance is enhanced under time delay. For robotic systems without force sensors, the disturbance observer has been widely applied to estimate the external input of robotic systems due to its simple structure and effectiveness [31]. In [32], a globally exponentially stable observer for joint torque information is constructed using only position measurement. In [33], null-space impedance control of a kinematically redundant robot is investigated based on a disturbance observer. Recently, sliding-mode approaches have been used by disturbance observers to ensure asymptotic convergence of estimation errors [34,35,36]. However, state reconstruction based on delayed torque estimation remains an open problem.

In this paper, a symmetric state estimation scheme is proposed for a remote robot system with time delay. First, a sliding-mode disturbance observer is designed to estimate input torque on the remote (robot) side. Then, based on the delayed torque estimation and the output received on the local side, a novel predictive observer is proposed to reconstruct the real-time joint angles and velocities of the robotic system. The main novelties of this work are summarized as follows:

• The robustness to disturbances and uncertainties of the proposed sliding-mode observer is improved by applying the super-twisting algorithm [37]. The estimation error of input torque converges in finite time to 0.
• A novel predictive algorithm is proposed to design the state observer based on the Euler–Lagrange model and the delayed output. Through using Lyapunov approaches, sufficient conditions are obtained to make sure that the estimation errors can converge exponentially to bounded regions through the selection of proper observer gains.

This paper is organized as follows: The mathematical model and problem formulation are presented in Section 2. The main results, including the torque estimation and state reconstruction, are shown in Section 3. Numerical simulations are performed in Section 4 to confirm the effectiveness of the method. Finally, the results are discussed in Section 5.

2. Problem Formulation

Consider an n-degree-of-freedom (DOF) robotic system described by the following Euler–Lagrange equation:

$$M(q)\ddot{q}+C(q,\dot{q})\dot{q}+G(q)=\tau$$

(1)

where $q\in {ℝ}^n$ and $\dot{q}\in {ℝ}^n$ are the joint angles and velocities of the robots, respectively, $\tau$ represents the unknown input torque imposed by controller and external disturbances, $M(q)\in {ℝ}^{n\times n}$ is the symmetric and positive-definite inertia matrix, $C(q,\dot{q})\in {ℝ}^{n\times n}$ is the matrix of the centripetal and Coriolis torques, and $G(q)\in {ℝ}^{n\times n}$ represents the vector of the gravitational forces.

As shown in Figure 1, the control inputs (such as desired trajectories) are transferred to the remote robot through a communication channel. There is a communication delay d between the remote robot and the local control side. The objective is to obtain an estimation $\widehat{\tau }$ of the input torque $\tau$ on the remote side and then reconstruct the real-time joint angles and velocities on the local side by using the delayed signals, including $\widehat{\tau }(t-d)$, $q(t-d)$, and $\dot{q}(t-d)$. The following assumptions are useful in the design:

Assumption 1.

The system states $q$  and $\dot{q}$ are bounded.

Assumption 2.

The input torque $\tau$ and its derivative are bounded. Let$\eta \triangleq k^{-1}{M}^{-1}(q)\tau$; there are positive constants $\varphi$ and v such that

$$|\dot{\eta }+k{\lambda }_5\eta {|}_i<\varphi$$

(2)

$${‖{\displaystyle {\int }_{t-d}^t\eta (\sigma )d\sigma }‖}^2\le v$$

(3)

where ${\left|\cdot \right|}_i$ represents the absolute value of the i-th element of the vector for $i=1,\mathrm{\dots },n$; $‖\cdot ‖$ denotes the standard Euclidean norm; ${\lambda }_5$ and k are constant parameters which will be specified later.

Remark 1.

The state and input boundaries are common assumptions in many works of observer design [21,22]. In addition, the joint angle, velocity, and input torque are always bounded in practical robotic applications. Noticing that the matrix $M^{-1}(q)$ is bounded when $q$ is bounded [34], inequalities (2)–(3) are reasonable as long as $\tau$ is bounded.

3. Main Results

3.1. Force Estimation Based on Sliding-Mode Disturbance Observer

Let $q_1=q$, $q_2=\dot{q}$. According to dynamic model (1), there is

$${\dot{q}}_2=-M^{-1}(q_1)[C(q_1,q_2)q_2+G(q_1)-\tau ].$$

(4)

The sliding-mode disturbance observer is designed to estimate the input torque $\tau$ by using the super-twisting algorithm [37] as follows:

$${\dot{\widehat{q}}}_2=-M^{-1}(q_1)[C(q_1,q_2)q_2+G(q_1)]+{\lambda }_1\left|q_2-{\widehat{q}}_2\right|\mathrm{sgn}(q_2-{\widehat{q}}_2)+{\lambda }_2(q_2-{\widehat{q}}_2)+k\widehat{\eta }$$

(5)

$$\dot{\widehat{\eta }}={\lambda }_3\mathrm{sgn}(q_2-{\widehat{q}}_2)+{\lambda }_4(q_2-{\widehat{q}}_2)-{\lambda }_5\widehat{\eta }$$

(6)

where ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5$, and k are positive constants, sgn() represents the sign function, and $\widehat{\eta }$ is an estimation of $\eta \triangleq k^{-1}{M}^{-1}(q)\tau$. Thus, the estimated input torque can be obtained as

$$\widehat{\tau }=kM(q)\widehat{\eta }$$

(7)

The estimation errors of the observer are defined as $e_1=q_2-{\widehat{q}}_2$ and $e_2=\eta -\widehat{\eta }$. It can be calculated from (4)–(6) that

$${\dot{e}}_1=-{\lambda }_1\left|q_2-{\widehat{q}}_2\right|\mathrm{sgn}(q_2-{\widehat{q}}_2)-{\lambda }_2(q_2-{\widehat{q}}_2)+k{\dot{e}}_2$$

(8)

$${\dot{e}}_2=-k{\lambda }_3\mathrm{sgn}(q_2-{\widehat{q}}_2)-k{\lambda }_4(q_2-{\widehat{q}}_2)-k{\lambda }_5{e}_2+\dot{\eta }+k{\lambda }_5\eta$$

(9)

Theorem 1.

Consider dynamic systems of estimation error (8)–(9) under Assumptions 1 and 2. The gains${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5$, and $k$ can be selected with high enough values such that $e_1$ and $e_2$ converge in finite time to the origin $e_1=0$ and $e_2=0$.

Proof of Theorem 1.

Consider a Lyapunov function candidate [34]:

$$V={\displaystyle \sum _{i=1}^n{V}_i}$$

(10)

$$V_i={\displaystyle \frac{1}{2}}{{\xi }_i}^{T}Q{\xi }_i$$

(11)

where ${{\xi }_i}^T=\left[{\left|e_{1i}\right|}^{1/2}\mathrm{sgn}(e_{1i})\hspace{1em}{e}_{1i}\hspace{1em}{e}_{2i}\right]$, and $Q=\left[\begin{array}{ccc}4k{\lambda }_3+{{\lambda }_1}^2& {\lambda }_1{\lambda }_2& -{\lambda }_1\\ {\lambda }_1{\lambda }_2& 2k{\lambda }_4+{{\lambda }_2}^2& -{\lambda }_2\\ -{\lambda }_1& -{\lambda }_2& 2\end{array}\right]$.

Taking the time derivative of $V_i$ yields

$${\dot{V}}_i=-{\displaystyle \frac{1}{{\left|e_{1i}\right|}^{1/2}}}{{\xi }_i}^T{\mathrm{\Omega }}_1{\xi }_i-{{\xi }_i}^T{\mathrm{\Omega }}_2{\xi }_i-{\lambda }_1(\dot{\eta }+k{\lambda }_5\eta ){\left|e_{1i}\right|}^{1/2}\mathrm{sgn}(e_{1i})-{\lambda }_2(\dot{\eta }+k{\lambda }_5\eta )\left|e_1\right|$$

(12)

where ${\mathrm{\Omega }}_1={\displaystyle \frac{{\lambda }_1}{2}}\left[\begin{array}{ccc}2k{\lambda }_3+{{\lambda }_1}^2& 0& -{\lambda }_1\\ 0& 2k{\lambda }_4+5{{\lambda }_2}^2& -3{\lambda }_2\\ -{\lambda }_1& -3{\lambda }_2& 1\end{array}\right]$, ${\mathrm{\Omega }}_2={\lambda }_2\left[\begin{array}{ccc}k{\lambda }_3+2{{\lambda }_1}^2& 0& -{\displaystyle \frac{1}{2}}k{\lambda }_1{\lambda }_5\\ 0& k{\lambda }_4+{{\lambda }_2}^2& -{\lambda }_2\\ -{\displaystyle \frac{1}{2}}k{\lambda }_1{\lambda }_5& -{\lambda }_2& 1\end{array}\right]$.

By applying condition (2), there is

$$-{\lambda }_1(\dot{\eta }+k{\lambda }_5\eta ){\left|e_{1i}\right|}^{1/2}\mathrm{sgn}(e_{1i})-{\lambda }_2(\dot{\eta }+k{\lambda }_5\eta )\left|e_1\right|<{\lambda }_1\delta {\left|e_{1i}\right|}^{1/2}+{\lambda }_2\delta \left|e_1\right|$$

(13)

Thus, the upper bound of $\dot{V}$ is determined as

$$\dot{V}<-{\displaystyle \sum _{i=1}^n(\frac{1}{{\left|e_{1i}\right|}^{1/2}}{{\xi }_i}^T{\mathrm{\Omega }}_3{\xi }_i+{{\xi }_i}^T{\mathrm{\Omega }}_4{\xi }_i)}$$

(14)

where ${\mathrm{\Omega }}_3={\displaystyle \frac{{\lambda }_1}{2}}\left[\begin{array}{ccc}2k{\lambda }_3+{{\lambda }_1}^2-{\lambda }_1\varphi & 0& -{\lambda }_1\\ 0& 2k{\lambda }_4+5{{\lambda }_2}^2& -3{\lambda }_2\\ -{\lambda }_1& -3{\lambda }_2& 1\end{array}\right]$, ${\mathrm{\Omega }}_4={\lambda }_2\left[\begin{array}{ccc}k{\lambda }_3+2{{\lambda }_1}^2-{\lambda }_2\varphi & 0& -{\displaystyle \frac{1}{2}}k{\lambda }_1{\lambda }_5\\ 0& k{\lambda }_4+{{\lambda }_2}^2& -{\lambda }_2\\ -{\displaystyle \frac{1}{2}}k{\lambda }_1{\lambda }_5& -{\lambda }_2& 1\end{array}\right]$.

If the gains are selected with high enough values such that $Q$, ${\mathrm{\Omega }}_3$, and ${\mathrm{\Omega }}_4$ are positive-definite matrices, then there are positive constants $c_1$ and $c_2$ such that

$$\dot{V}<-c_1{V}^{{\displaystyle \frac{1}{2}}}-c_{2}V$$

(15)

Based on the comparison lemma, it can be concluded that $e_1$ and $e_2$ converge to 0 in finite time, which is less than $T={\displaystyle \frac{2V^{1/2}}{c_1}}$. The proof of Theorem 1 is completed. □

Remark 2.

It should be pointed out that the time derivative of $V_i$ in (12) is similar but not the same as the results in [37] due to the existence of an additional term, $-k{\lambda }_5{e}_2$, in error dynamic function (9). From the definition of ${\mathrm{\Omega }}_3$ and ${\mathrm{\Omega }}_4$, if ${\lambda }_1$ and ${\lambda }_3$ are sufficiently large relative to $\varphi$, then there are proper ${\lambda }_2,{\lambda }_4,{\lambda }_5$, and k such that ${\mathrm{\Omega }}_3$ and ${\mathrm{\Omega }}_4$ are positive-definite.

3.2. State Reconstruction Using Delayed Output and Torque Estimation

By using the delayed outputs $q_{1d}(t)$ and $q_{2d}(t)$ and the estimation ${\widehat{\eta }}_d(t)=\widehat{\eta }(t-d)$, the original state variables of system (1) are reconstructed by the following predictive observer:

$$\begin{array}{l}{\dot{\widehat{x}}}_1(t)={\widehat{x}}_2(t),\\ {\dot{\widehat{x}}}_2(t)=-M^{-1}({\widehat{x}}_1(t))[C({\widehat{x}}_1(t),{\widehat{x}}_2(t)){\widehat{x}}_2(t)+G(({\widehat{x}}_1(t))]+k{\widehat{\eta }}_d(t)+K_p{r}_d(t),\\ r_d(t)=q_{2d}(t)-{\widehat{x}}_{2d}(t)+\alpha (q_{1d}(t)-{\widehat{x}}_{1d}(t))-{\displaystyle {\int }_{t-d}^t{K}_p{r}_d(\sigma )+k{\widehat{\eta }}_d(\sigma )d\sigma }\end{array}$$

(16)

where ${\widehat{x}}_1(t)$ and ${\widehat{x}}_2(t)$ represent the estimations of $q_1(t)$ and $q_2(t)$, respectively, $r_d(t)$ is an auxiliary error variable, and $K_p$ is the designed constant gain, which can be expanded as $K_p=k_a+k_b+k_c$, ${\widehat{x}}_{1d}(t)={\widehat{q}}_1(t-d)$, and ${\widehat{x}}_{2d}(t)={\widehat{q}}_2(t-d)$.

Denote the estimation errors as follows:

$${\epsilon }_1(t)=q_1(t)-{\widehat{x}}_1(t)$$

(17)

$${\epsilon }_2(t)=q_2(t)-{\widehat{x}}_2(t)$$

(18)

$$\theta (t)={\displaystyle {\int }_{t-d}^t{K}_{p}r}(\sigma )d\sigma$$

(19)

$$r(t)={\epsilon }_2(t)+\alpha {\epsilon }_1(t)-\theta (t)-{\displaystyle {\int }_{t-d}^{t}k\eta (\sigma )d\sigma }$$

(20)

It should be noticed that $r(t-d)=r_d(t)$. By taking the time derivative of $r(t)$, we obtain

$$\begin{array}{cc}\hfill \dot{r}& ={\dot{q}}_2-{\dot{\widehat{q}}}_2+\alpha {\epsilon }_2+(K_p{r}_d+k{\eta }_d)-(K_{p}r+k\eta )\hfill \\ & ={\dot{x}}_{2i}+\alpha {\epsilon }_2-g({\widehat{x}}_{1i},{\widehat{x}}_{2i})-{\widehat{M}}_i^{-1}({\widehat{x}}_{1i}){\widehat{u}}_i+k{e}_{2d}-k\widehat{\eta }-K_{pi}{r}_i\hfill \\ & =N(q_1,q_2,{\widehat{x}}_{1i},{\widehat{x}}_{2i})-{\epsilon }_1+k(e_{2d}+e_2)-K_{p}r\hfill \end{array}$$

(21)

$e_2=\eta -\widehat{\eta }$ was introduced in (9), and the auxiliary vectors $N(q_1,q_2,{\widehat{x}}_{1i},{\widehat{x}}_{2i})$ are defined as

$$\begin{array}{cc}\hfill N(q_1,q_2,{\widehat{x}}_{1i},{\widehat{x}}_{2i})& =-M^{-1}(q_1)[C(q_1,q_2)q_2+G(q_1)]\hfill \\ & +M^{-1}({\widehat{x}}_1(t))[C({\widehat{x}}_1(t),{\widehat{x}}_2(t)){\widehat{x}}_2(t)+G({\widehat{x}}_1(t))]+\alpha {\epsilon }_2+{\epsilon }_1\hfill \end{array}$$

(22)

Similar to [38], the upper bound of $N$ can be derived by applying the Mean Value Theorem as follows:

$$\left|\right|N\left|\right|\le \rho (|\left|z\right||)\left|\right|z\left|\right|$$

(23)

where $z={\left[\begin{array}{ccc}{\epsilon }_1^T& {\theta }^T& r^T\end{array}\right]}^T$; $\rho (\cdot )$ is a positive invertible and a non-decreasing function.

Theorem 2.

Consider nonlinear system (1) and observer (16) under Assumptions 1 and 2. The estimation error z(t) convergences to bounded regions in the sense that

$$‖z(t)‖\le {\mu }_0{e}^{-{\mu }_{1}t}+{\mu }_2$$

(24)

where ${\mu }_0,{\mu }_1,{\mu }_2$ denote constants, provided that the following sufficient conditions are satisfied simultaneously:

$$d<{\displaystyle \frac{2\kappa \delta }{3K_p^2}}, k_a>{\displaystyle \frac{\delta }{2}}+\omega +\kappa d, k_b>{\displaystyle \frac{{\rho }^2(||z(0)||)}{4c_1}}$$

(25)

where $\kappa ,\delta ,\omega ,c_1$ are positive constants which will be specified later.

Proof of Theorem 2.

Consider the following Lyapunov function candidate:

$$V_p={\displaystyle \frac{1}{2}}{\epsilon }_1^T{\epsilon }_1+{\displaystyle \frac{1}{2}}{r}^{T}r+{\displaystyle \frac{1}{2K_p}}{\theta }^T\theta +Q+P,$$

(26)

where $Q=\kappa {\displaystyle {\int }_{t-d}^t{\displaystyle {\int }_s^t}}{\left|\right|r(\sigma )\left|\right|}^{2}d\sigma ds\le \kappa d{\displaystyle {\int }_{t-d}^t}{\left|\right|r(\sigma )\left|\right|}^{2}d\sigma ,$ and $P=\omega {\displaystyle {\int }_{t-d}^t}{\left|\right|r(\sigma )\left|\right|}^{2}d\sigma .$

Let $y={\left[{\epsilon }_1^T\hspace{1em}{r}^T\hspace{1em}{\theta }^T\hspace{1em}\sqrt{Q}\hspace{1em}\sqrt{P}\right]}^T$. It yields

$$s_1{\left|\right|y\left|\right|}^2\le V_p(y)\le s_2{\left|\right|y\left|\right|}^2,$$

(27)

where ${\lambda }_1=min\left\{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2K_p}}\right\}, {\lambda }_2=max\left\{1,{\displaystyle \frac{1}{2K_p}}\right\}$.

Taking the time derivative of V, one obtains

$$\begin{array}{l}{\dot{V}}_p={\epsilon }_1^T(r-\alpha {\epsilon }_1+\theta +{\displaystyle {\int }_{t-d}^{t}k\widehat{\eta }(\sigma )d\sigma })+r^T(N-{\epsilon }_1+k(e_{2d}+e_2)-K_{p}r)\\ \hspace{1em}\hspace{1em}+{\theta }^T(r_d-r)+\kappa d{\left|\right|r\left|\right|}^2-\kappa {\displaystyle {\int }_{t-d}^t}{\left|\right|r(\sigma )\left|\right|}^{2}d\sigma +\omega ({\left|\right|r\left|\right|}^2-{\left|\right|r_d\left|\right|}^2).\end{array}$$

(28)

The Cauchy–Schwarz inequality can be used to derive ${\left|\theta \right||}^2\le d{K}_p^2{\displaystyle {\int }_{t-d}^t}{\left|\right|r(\sigma )\left|\right|}^{2}d\sigma$. Then, by utilizing Young’s inequality (3) and (23), the upper bound of ${\dot{V}}_p$ is determined as

$$\begin{array}{l}{\dot{V}}_i\le -(\alpha -\delta ){\left|\right|{\epsilon }_1\left|\right|}^2+{\displaystyle \frac{k}{2\delta }}v+||r||\rho (||z||)||z||-(k_b+k_c){\left|\right|r_i\left|\right|}^2\\ \hspace{1em}\hspace{1em}-(k_a-{\displaystyle \frac{\delta }{2}}-\omega -\kappa d){\left|\right|r\left|\right|}^2+k\left|\right|r\left|\right|||e_{2d}+e_2||-(\omega -{\displaystyle \frac{\delta }{2}}){\left|\right|r_d\left|\right|}^2\\ \hspace{1em}\hspace{1em}-\left({\displaystyle \frac{\kappa }{d{K}_p^2}}-{\displaystyle \frac{3}{2\delta }}-{\gamma }_1^2-{\gamma }_2^2\right){\left|\right|\theta \left|\right|}^2-d({\gamma }_1^2+{\gamma }_2^2){\displaystyle {\int }_{t-d}^t}{\left|\right|r(\sigma )\left|\right|}^{2}d\sigma .\\ \hspace{1em} \le -(\alpha -\delta ){\left|\right|{\epsilon }_1\left|\right|}^2-(k_a-{\displaystyle \frac{\delta }{2}}-\omega -\kappa d){\left|\right|r\left|\right|}^2-\left({\displaystyle \frac{\kappa }{d{K}_p^2}}-{\displaystyle \frac{3}{2\delta }}-{\gamma }_1^2-{\gamma }_2^2\right){\left|\right|\theta \left|\right|}^2-{\displaystyle \frac{{\gamma }_1^2}{\kappa }}Q-{\displaystyle \frac{d{\gamma }_2^2}{\omega }}P\\ \hspace{1em}\hspace{1em}+{\displaystyle \frac{{\rho }^2(|\left|z\right||)}{4k_b}}{\left|\right|z\left|\right|}^2+{\displaystyle \frac{\delta }{2}}kv+{\displaystyle \frac{k\left|\right|e_{2d}+e_2|{|}^2}{4k_c}}\end{array}$$

(29)

By completing the squares based on the definition of z and y, the expression in (29) reduces to

$$\begin{array}{cc}\hfill {\dot{V}}_p& \le -\left[c_1-{\displaystyle \frac{{\rho }^2(||z||)}{4k_b}}\right]{\left|\right|z\left|\right|}^2-{\displaystyle \frac{{\gamma }_1^2}{\kappa }}Q-{\displaystyle \frac{d{\gamma }_2^2}{\omega }}P+{\displaystyle \frac{k}{2\delta }}v+{\displaystyle \frac{k\left|\right|e_{2d}+e_2|{|}^2}{4k_c}}\hfill \\ & \le -{\displaystyle \frac{c_2}{{\lambda }_2}}{V}_p+{\displaystyle \frac{k}{2\delta }}v+{\displaystyle \frac{k\left|\right|e_{2d}+e_2|{|}^2}{4k_c}}\hfill \end{array}$$

(30)

where $c_1=\min \left\{\alpha -{\displaystyle \frac{\delta }{2}}, k_a-{\displaystyle \frac{\delta }{2}}-\omega -\kappa d, {\displaystyle \frac{\kappa }{d{K}_p^2}}-{\displaystyle \frac{3}{2\delta }}-{\gamma }_1^2-{\gamma }_2^2\right\},$ $c_2=\min \left\{c_1-{\displaystyle \frac{{\rho }^2(||z||)}{4k_b}},{\displaystyle \frac{{\gamma }_1^2}{\kappa }},{\displaystyle \frac{d{\gamma }_2^2}{\omega }}\right\}$.

Because the convergence of $e_2$ was proved in Theorem 1, provided the conditions in (25) are satisfied, by using the comparison lemma, we can conclude that

$$V_p\le V_p(y(0))e^{-{\displaystyle \frac{c_2}{s_2}}t}+{\displaystyle \frac{s_2}{c_2}}({\displaystyle \frac{k}{2\delta }}v+{\displaystyle \frac{k\left|\right|e_{2d}(0)+e_2(0)|{|}^2}{4k_c}})$$

(31)

Then, in view of the definition of $V_p$ in (26), there exist proper constants ${\mu }_0,{\mu }_1,{\mu }_2$ such that the result in (24) can be obtained. The proof of Theorem 2 is completed. □

From (31), since $e_2(t)$ converges in finite time to 0, the observer errors ${\epsilon }_1(t)$, $r(t)$, and $\theta (t)$ can converge to bounded regions which are determined by $v$ and k. In view of the definition of $r(t)$ in (20), ${\epsilon }_2(t)$ is also bounded. Thus, we should apply a small value of k on the premise of ensuring the accuracy of torque estimation, in order to reduce the state reconstruction error. Additionally, from condition (25), the allowable delay by the observer is limited. One possible solution is to use several observers in cascade to deal with large time delays [39].

Remark 3.

The delayed auxiliary variable ${\widehat{\eta }}_d$ is used directly in observer (16) in order to simplify the calculations. If just ${\widehat{\tau }}_d$ can be received on the local side, we should use $M^{-1}(\widehat{q}){\widehat{\tau }}_d$ instead of $k{\widehat{\eta }}_d$. Under the circumstances, similar stability results of estimation errors can be obtained. Although the time delay is assumed to be constant in this paper, the proposed observer design approach can be straightforwardly applied to differentiable time-varying delays with $\dot{d}(t)<1$ [15,40].

Remark 4.

The proposed state reconstruction method can be applied to telerobotic systems with symmetric or asymmetric communication delays [29,30,41]. By using two symmetric estimation schemes on both the leader and follower sides, the time delay can be compensated through applying the reconstructed states of robots.

4. Simulations

To illustrate the performance of the proposed method, numerical simulations are performed on a 2-joint robot system with the following dynamics [41]:

$$M(q)=\left[\begin{array}{cc}{M}_{11}& M_{12}\\ M_{21}& M_{22}\end{array}\right],C(q,\dot{q})=\left[\begin{array}{cc}{C}_{11}& C_{12}\\ C_{21}& C_{22}\end{array}\right]$$

$$\begin{array}{l}{M}_{11}=(2l_a\cos q_b+l_b)l_b{m}_b+(m_a+m_b)l_a^2,\\ M_{12}=M_{21}=l_b^2{m}_b+l_a{l}_b{m}_b\cos q_b,M_{22}=l_b^2{m}_b,\\ C_{11}=-l_a{l}_b{m}_b{\dot{q}}_b\sin (q_b),C_{12}=-l_a{l}_b{m}_b({\dot{q}}_a+{\dot{q}}_b)\sin (q_b),\\ C_{21}=l_a{l}_b{m}_b{\dot{q}}_a\sin (q_b),C_{22}=0,\end{array}$$

$$G(q)=\left[\begin{array}{c}\left(m_a+m_b\right)g{L}_a\sin q_a+m_{b}g{l}_b\sin \left(q_a+q_b\right)\\ m_{b}g{L}_b\sin \left(q_a+q_b\right)\end{array}\right],$$

where the two joints are labeled as a and b, respectively, and $q={\left[\begin{array}{cc}{q_a}^T& {q_b}^T\end{array}\right]}^T$. The parameters are given as $m_a=1 \mathrm{kg},$ $m_b=0.5 \mathrm{kg},$ $l_a=0.5 \mathrm{m},$ and $l_b=0.5 \mathrm{m}$; the initial states of the robot are given as $q(0)=[\begin{array}{cc}0.1& 0.1\end{array}]\mathrm{rad}$ and $\dot{q}(0)=[\begin{array}{cc}0& 0\end{array}]\mathrm{rad}/\mathrm{s}$; and the initial estimations are ${\widehat{q}}_1(0)={\widehat{x}}_1(0)=[\begin{array}{cc}0.1& 0.1\end{array}]\mathrm{rad}$ and ${\widehat{q}}_2(0)={\widehat{x}}_2(0)=[\begin{array}{cc}0& 0\end{array}]\mathrm{rad}/\mathrm{s}.$ The input torque $\tau ={\left[{\begin{array}{cc}{{\tau }_a}^T& {\tau }_b\end{array}}^T\right]}^T$ is a PD controller designed as $\tau =30(q_r-q_1)+10({\dot{q}}_r-q_2)$, while the reference trajectory $q_r={\left[\begin{array}{cc}{q_{ra}}^T& {q_{rb}}^T\end{array}\right]}^T$ is set as $q_{ra}=\left\{\begin{array}{c}0.02t \mathrm{rad},0\le t\le 60 \mathrm{s}\\ 1.2 \mathrm{rad},t\ge 65 \mathrm{s}\end{array}\right.$ and $q_{rb}=0.5\sin (0.2t)\mathrm{rad}$. The time delay $d=2 \mathrm{s}$, and the gain parameters are designed as ${\lambda }_1=0.3,{\lambda }_2=0.3,{\lambda }_3=0.3,{\lambda }_4=0.5,{\lambda }_5=0.5$, $k=0.1$, $K_p=20$, and $\alpha =0.4$.

First, the torque estimation results of two joints (a and b) are shown in Figure 2. It can be seen that the estimations ${\widehat{\tau }}_a$ and ${\widehat{\tau }}_b$ coincide with the original torques ${\tau }_a$ and ${\tau }_b$ despite the jitter that happened due to the nature of the sliding-mode algorithm, which indicates the effectiveness of the sliding-mode disturbance observer. Then, the state reconstruction results of joint angles and velocities are given in Figure 3 and Figure 4, respectively. As can be seen, the good performance of the proposed predictive observer is realized, where the estimated states ${\widehat{x}}_1$ and ${\widehat{x}}_2$ are in correspondence with $q_1$ and $q_2$.

Next, the proposed method is tested in different situations with disturbances and model uncertainties, respectively. The disturbance input is set as a uniform random number varying from $-0.02$ to $0.02 \mathrm{N}\cdot \mathrm{m}$, and the nominal parameters used in observers (5) and (16) are given as ${m^{\prime }}_a=0.5 \mathrm{kg},$ ${m^{\prime }}_b=0.3 \mathrm{kg},$ ${l^{\prime }}_a=0.4 \mathrm{m},$ and ${l^{\prime }}_b=0.4 \mathrm{m}$, which are different from the real ones of the robot. The estimation errors $e_2$ under three different situations are compared in Figure 5, where the lines labeled “ideal” represent the results of the first simulation. It can be seen that the errors are more sensitive to disturbances than uncertainties because of the application of the sliding-mode approach. The strong robustness to model uncertainties of the proposed torque estimation method is verified. Moreover, the estimation errors of joint angles and velocities are shown in Figure 6 and Figure 7, which indicate that the proposed method can reconstruct the system state precisely when disturbances and model uncertainties exist.

Then, by setting the communication delay as $d=1,2,3 \mathrm{s}$, respectively, the reconstruction errors of joint angles are compared in Figure 8. It is obvious that the estimation error increases with the growth of the delay.

To verify the effectiveness of the proposed method, the state reconstruct results are compared with the cases without torque estimation and using the high-gain predictor [15]. First, when the parameters are set as $d=2 \mathrm{s}$ and $K_p=20$, the estimation results of the high-gain predictor are not convergent. The advantages of the proposed method in dealing with large time delays are shown. Thus, the parameters are adjusted to $d=1 \mathrm{s}$ and $K_p=1$ for all three methods, and the estimation errors of joint angles are compared in Figure 9. It can be seen that the errors without using force estimation are higher than in the other two methods, and the results of the high-gain predictor are a bit of a wobble at the beginning. In addition, the integral square errors (ISEs) of ${\epsilon }_1$ are presented and compared in

Table 1. Integral square error of ${\epsilon }_1$ compared with conventional high-gain predictor.

Method	ISE	Percentage
Proposed	0.2655	86%
Without torque estimation	0.6250	202%
High-gain predictor	0.3099	100%

, which are defined as ${\displaystyle {\int }_0^{100}{‖{\epsilon }_1(t)‖}^{2}dt}$. The results show that the estimation accuracy is improved by using the torque estimation, and the ISE of proposed method is reduced by 14% compared with the conventional high-gain predictor. The accuracy and rapidity of the proposed method in this paper are verified.

5. Conclusions

A predictive observer has been proposed to reconstruct the real-time state of remote robotic systems with time delay, while the input torque has been estimated by a sliding-mode disturbance observer. A comprehensive analysis has been carried out to investigate the fundamental properties of the proposed method. The effectiveness and robustness to disturbances and uncertainties have been verified by simulations. Future work will be focused on the observer and adaptive controller design [42] for symmetric or asymmetric telerobotics in more complex communication networks.