A simple algorithm to generate firing times for leaky integrate-and-fire neuronal model

Nomenclature

1.   Introduction

The implementation of teleoperation within robotic systems extends the ability of humans to perform specific operations personally in remote environments. Teleoperation is conducted by remote operators transmitting control signals to robotic systems through communication channels, and receiving the force interaction data from the robotic system via the same communication channels. Currently, teleoperation systems find extensive applications in diverse fields including remote surgery and deep-sea exploration ^[1,2,3].

Increasing interest has been generated in recent years for developing robotic systems with flexible manipulator joints. This is mainly due to two reasons ^[4,5]. First, the increasing complexity and refinement of industrial tasks has necessitated the implementation of elastic drive transmission systems in industrial robots, such as harmonic gear transmission, joint torque sensors, and belt pulleys ^[6]. These must be treated as flexible-joint systems to mitigate the negative impact of end effector vibrations on system stability ^[7]. Second, flexible components have been introduced in new robotic manipulator designs for obtaining more human-compatible mobility characteristics that facilitate human-robot interactions, such as high flexibility, strong adaptability, and strong robustness ^[8]. However, the flexibility of these joints renders the dynamic behavior of robotic systems increasingly intricate and challenging to model. This complexity primarily stems from nonlinearity, uncertainty, damping effects, and vibration modes of flexible joints, presenting a significant challenge in achieving satisfactory control performance in remote operation systems.

Since Spong proposed the flexible joint model in 1987, more and more control strategies have been proposed for manipulator systems with flexible joints ^[9]. Spong ^[10], Ghorbel et al. ^[11], and Chang and Daniel ^[12] designed an adaptive controller based on singular perturbation control. However, although this method can simplify the analysis, it may also oversimplify the system dynamics, potentially ignoring the key nonlinearity and complexity existing in the real scene ^[13]. Nam et al. ^[14] introduced finite time control based on sliding mode control and designed a control strategy that enables a flexible joint manipulator system to achieve effective tracking. However, the presence of buffeting reduces the tracking effect of the system. In addition, backstepping control is also a highly effective control strategy for addressing the tracking control problem in flexible joint manipulator systems. There have been many research achievements based on backstepping to solve control issues in flexible manipulator systems. Huang et al. ^[15] designed the controller of the single-link flexible manipulator system based on the reverse step method, but this control strategy is only applicable to a system with known state parameters. Cheng et al. ^[16] combined the singular perturbation control method and adaptive control method on the basis of backstep method to design a control strategy suitable for a flexible manipulator system. However, this control strategy has the disadvantage of too having complicated of a calculation. To tackle this issue, Sahu et al. ^[17] proposed a backstepping control strategy that combines extended state observer (ESO) with backstepping control. This approach not only effectively addresses the computational complexity in traditional backstepping but also efficiently handles the uncertainty and interference of the internal model, significantly improving the tracking performance of the system.

ESO is a part of active disturbance rejection control (ADRC) theory that enables real-time estimation of internal state and unmodeled dynamics of a system ^[18,19]. Recently, ESO has been widely applied, such as in tracked vehicle control ^[20], quadrotor control ^[21], DC-link voltage control ^[22], and permanent magnet synchronous linear motor (PMSLM) servo system control ^[23]. ESO can be categorized into linear and nonlinear types. Linear ESO can exhibit peaking phenomena when dealing with nonlinearities and strong couplings, thereby potentially compromising the observer's accuracy to some extent ^[24]. Consequently, in this study, nonlinear ESO was chosen to address the uncertainty in the internal model and external disturbances of the flexible-joint teleoperation system, enhancing its robustness and accuracy.

Furthermore, in real-world engineering applications, the system may face limitations due to factors such as temperature and space. If the system's operation exceeds predefined boundaries, it can adversely affect the system's performance and, in extreme cases, potentially lead to system failures, endangering the personal safety of the operators. The barrier Lyapunov function (BLF) is a standard method developed for implementing constrained control problems of this nature ^[25]. Wang ^[26] and Li et al. ^[27] used the barrier Lyapunov function to ensure that the input and output of a nonlinear system are controlled within a preset interval. Using the above research results, Zhang ^[28] and Yu et al. ^[29] applied the barrier Lyapunov function to a robot arm system with limited output, and ensured that the output of the system could be controlled within a preset interval.

While considerable progress has been made in the control of robotic systems with flexible joints, these developments have been rarely applied in the design of teleoperation systems for remote robotic control. Moreover, teleoperation systems not only must address the many uncertainties in the pose control of robotic manipulators, but also account for complexities in the actual working environments, and the inevitability of external interference. Accordingly, these issues remain poorly addressed by the current state of teleoperation system development.

In summary, to achieve effective tracking of joint position trajectories for a state-constrained flexible-joint teleoperation system, this paper proposes a control strategy based on the combination of nonlinear ESO and the Barrier Lyapunov Function. The main contributions in this work are summarized as follows: 1) The paper introduces a novel teleoperation controller that employs a nonlinear Extended State Observer (ESO) to estimate internal uncertainties related to flexible joints and external disturbances. This inclusion enhances the accuracy of the control system by actively predicting and compensating for disturbances in the input channel. 2) The research effectively addresses practical operational constraints on the system output states by leveraging the Barrier Lyapunov Function (BLF). This application ensures that the teleoperation controller operates within defined limitations, leading to more reliable and controlled system behavior. 3) The study focuses on a robotic system that combines a rigid master manipulator and a flexible slave manipulator. By enforcing constraints on the system output states using the BLF, synchronized control is achieved between these two distinct components. This synchronization is critical for precise and coordinated control in teleoperation scenarios.

The remainder of this paper is organized as follows: Section 2 presents the pertinent preliminary discussion. Section 3 elucidates the nonlinear ESO and stability analyses of the proposed teleoperation control system. Following this, Section 4 provides a detailed account of the experimental results. Finally, concluding remarks and future directions are outlined in Section 5.

2.   Preliminary discussion

The rigid master manipulator is assumed to have n degrees of freedom, and its nonlinear dynamic behavior is defined as follows.

Similarly, the flexible slave manipulator is assumed to have n degrees of freedom, and its nonlinear dynamic behavior is defined as follows ^[9].

Here, for all $ i = {\text{m}}, {\text{s}} $, $ {q_i} \in {R^n} $, $ {M_i}\left({{q_i}} \right) \in {R^{n \times n}} $, $ {C_i}\left({{{\dot q}_i}, {q_i}} \right) \in {R^{n \times n}} $, $ {f_i}\left({{q_i}} \right) \in {R^n} $, $ {B_i}\left({{q_i}} \right) \in {R^n} $, $ {G_i}\left({{q_i}} \right) \in {R^n} $, ${J_{\text{s}}} \in {R^{n \times n}}$, $ {\theta _{\text{s}}} \in {R^n} $, $ {\tau _i} \in {R^n} $, $ {\tau _{\text{c}}} \in {R^n} $, $ {\tau _{\text{h}}} \in {R^n} $, $ {\tau _{\text{e}}} \in {R^n} $, $ {\tau _{\text{f}}} \in {R^n} $, $ {\tau _{\text{c}}} = {K_{\text{s}}}\left({{\theta _{\text{s}}} - {q_{\text{s}}}} \right) $, ${{\text{K}}_{\text{s}}} \in {R^{n \times n}}$.

For convenience, we make the following substitutions in the system variables: $ {X_{{\text{m1}}}} = {q_{\text{m}}} $, $ {X_{{\text{m2}}}} = {\dot q_{\text{m}}} $, $ {X_{{\text{s1}}}} = {q_{\text{s}}} $, $ {X_{{\text{s2}}}} = {\dot q_{\text{s}}} $, $ {X_{{\text{s3}}}} = {\theta _{\text{s}}} $, and $ {X_{{\text{s4}}}} = {\dot \theta _{\text{s}}} $. Then, the following correspondences are applied.

The variables F_m, F_s, and F_f in (3) and (4) represent lumped uncertainties in the system, which contain internal uncertain parts and external perturbations, which are defined as follows.

By treating uncertainties such as ${\tau _{\text{c}}}$ as total perturbations, the design of the control system can be simplified, resulting in reduced computational load. This approach enhances the system's robustness to various disturbances while maintaining the effectiveness and flexibility of the control strategy.

3.   Controller design and stability analysis

3.1.   Nonlinear ESO design

The nonlinear ESO is applied for estimating the interference terms F_m and F_s. In this subsection, we design the observer for F_m only, and subsequently analyze its error estimation stability. The F_s and F_f estimator design and stability are entirely equivalent and are therefore not presented herein.

First, the nonlinear ESO is designed by introducing the following expansion terms:

where h_m denotes the derivative of external disturbances. Before designing the ESO, the following assumption about external disturbances is given in this paper.

Assumption 1: For the external disturbance in the nonlinear bilateral teleoperation system, there exists a positive unknown constant L such that $\left| {{h_{\text{m}}}} \right| \leqslant L$ holds.

We define the estimation error as

where Z_m1, Z_m2, and Z_m3 are the estimated values of X_m1, X_m2, and X_m3, respectively, and introduce new parametric variables

where $\varepsilon $ is an appropriate positive definite parameter. Then, the nonlinear ESO is designed in the following form:

Here, sgn(·) is the signum function that returns the sign of its argument, and $ {\beta _{{\text{m}}1}} $, $ {\beta _{{\text{m}}2}} $, and $ {\beta _{{\text{m}}3}} $ are appropriate positive definite parameters. Combining Eqs (6)–(9), the estimation error of the system can be obtained as follows.

Finally, both sides of the three expressions in error system (10) are multiplied from top to bottom by $\frac{1}{\varepsilon }$, 1, and $\varepsilon $, respectively, which yields the substitution of error system (10) as follows:

Theorem 1: For master manipulator system (1) and nonlinear error system (11), the estimation error${e_{{\text{m}}k}}\left({k = 1, 2, 3} \right)$ will converge to the neighborhood of the origin if there exist positive definite parameters$ {\beta _{{\text{m}}1}} $, $ {\beta _{{\text{m}}2}} $, and $ {\beta _{{\text{m}}3}} $ such that the following inequality is guaranteed.

where k₁ – k₉ are appropriate positive definite constants and the following inequality group holds:

Proof: The proof is based on the following Lyapunov function:

From Young's inequality, the following inequality group can be obtained:

Substituting Eq (15) into Eq (14), the following inequality can be obtained:

When positive definite constants k₁–k₉ satisfy inequality group (13), the Lyapunov function (14) is positive semidefinite.

From the converted error system (11), taking the derivative of (14) with respect to time yields the following:

According to error system (11), we can reconfigure (17) as follows:

This is further expanded as follows:

This can be transformed into the following inequality:

Therefore,

where $Q$ is shown in Eq (12), and $\xi \left(t \right)$ is applied as the following definitions and conditions:

Based on the Lyapunov stability principle, the solutions of the inequality $ Q \leqslant 0 $ represent the appropriate positive definite constants k₁–k₉ and parameters $ {\beta _{{\text{m}}1}} $, $ {\beta _{{\text{m}}2}} $, and $ {\beta _{{\text{m}}3}} $. Therefore, the estimation stability of error system (10) is assured because the estimation error converges to the neighborhood of the origin.

3.2.   Design of the BLF-based controller

In this section, the estimations of interference factors obtained by the nonlinear ESO are used for compensation in the BLF-based controller in conjunction with the constraints on the state variables. Here, the controllers are designed separately for the master and slave manipulators because these are heterogeneous systems.

For the master manipulator system (1) with the applied correspondences (3), the position and speed X_mj, j = 1, 2 of the manipulator system are assumed to be bounded in actual applications, and satisfy $\left| {{X_{{\text{m}}j}}} \right| < {k_{{\text{m}}j}}$, where ${k_{{\text{m}}j}}$ is a positive definite constant. The errors associated with X_m1 and X_m2 are assumed to be respectively definable as $ {\chi _{{\text{m}}1}} = {X_{{\text{m}}1}} - {X_{{\text{md}}}} $ and $ {\chi _{{\text{m}}2}} = {X_{{\text{m}}2}} - {\alpha _{{\text{m1}}}} $, where $ {X_{{\text{md}}}} = {q_{\text{s}}}\left({t - {T_{\text{s}}}} \right) $ is the tracking trajectory of the master manipulator, that is, the reference signal, T_s is the transmission delay of the signal from the end to the master, and $ {\alpha _{{\text{m}}1}} $ is the virtual controller. In this study, the virtual controller is an intermediate design variable used to simplify the control problem of complex systems by decomposing high-order systems into lower-order subsystems for step-by-step design and analysis. These assumptions are specified and validated as follows.

Assumption 2: Assume that appropriate positive definite constants ${k_{{\text{m}}j}}$ exist that ensure that $\left| {{\chi _{{\text{m}}j}}} \right| < {k_{{\text{m}}j}}$ is satisfied.

Assumption 3: Assume that appropriate positive definite constants $ {A_{{\text{m}}0}} $ and $ {A_{{\text{m}}1}} $ exist that can guarantee that the conditions $\left| {{X_{{\text{md}}}}} \right| \leqslant {A_{{\text{m}}0}} < {k_{m1}}$ and $\left| {{{\dot X}_{{\text{md}}}}} \right| \leqslant {A_{{\text{m1}}}}$ are satisfied.

Step 1: Select the following positive definite candidate BLF:

Taking the derivative of $ {V_{{\text{m}}1}} $ with respect to time yields the following:

Here, $ {k_{{\chi _{{\text{m}}1}}}} = \frac{{{\chi _{{\text{m}}1}}}}{{k_{{\text{m}}1}^2 - \chi _{{\text{m}}1}^2}} $. The virtual controller $ {\alpha _{{\text{m}}1}} $ is designed as follows:

Substituting (24) in (23) yields the following:

Step 2: Select the following positive definite candidate BLF:

According to (3) and (25), taking the derivative of $ {V_{\text{m}}} $ with respect to time yields the following.

Here, $ {k_{{\chi _{{\text{m}}2}}}} = \frac{{{\chi _{{\text{m}}2}}}}{{k_{{\text{m}}2}^2 - \chi _{{\text{m}}2}^2}} $. The derivative of $ {\alpha _{{\text{m}}1}} $ with respect to time is approximated by the tracking differentiator. The tracking differentiator is an algorithm for estimating signal derivatives based on PID control and improved by Han ^[18], which can accurately track and output the differential value of signals in the presence of noise. The details are as follows:

Setting $ {v_{{\text{m}}0}} = {\alpha _{{\text{m}}1}} $ indicates that $ {v_{{\text{m1}}}} $ and $ {v_{{\text{m}}2}} $ are the estimated values of $ {\alpha _{{\text{m}}1}} $ and its derivative $ {\dot \alpha _{{\text{m1}}}} $, respectively (i.e., $ {\hat \alpha _{{\text{m}}1}} = {v_{{\text{m}}1}} $ and $ {\widehat {\dot \alpha }_{{\text{m}}1}} = {v_{{\text{m}}2}} $). Therefore, the estimation error $ {\widetilde {\dot \alpha }_{{\text{m}}1}} = {\dot \alpha _{{\text{m}}1}} - {\widehat {\dot \alpha }_{{\text{m}}1}} $ is assumed to be bounded, and satisfies the condition $ \left| {{{\widetilde {\dot \alpha } }_{{\text{m}}1}}} \right| \leqslant {\mu _{{\text{m}}1}} $, where $ {\mu _{{\text{m}}1}} $ is an appropriate positive definite constant. Finally, the input controller $ {\tau _{\text{m}}} $ of the master manipulator is designed as:

where $ {\hat F_{\text{m}}} $ is the value of F_m estimated by the nonlinear ESO presented in the previous section. Therefore, (27) can be expressed as follows.

where ${\tilde F_{\text{m}}}$ is the estimation error of F_m estimated by the nonlinear ESO presented in the previous section. For the slave manipulator system (2) with the applied correspondences (4), the position, angular position, speed, and angular speed X_sj, j = 1, 2, 3, 4 of the system are assumed to be bounded in practical applications, and satisfy the condition $\left| {{X_{{\text{s}}j}}} \right| < {k_{{\text{s}}j}}$, where $ {k_{{\text{s}}j}} $ is a positive definite constant. The errors in X_s1, X_s2, X_s3, and X_s4 are assumed to be respectively definable as $ {\chi _{{\text{s}}1}} = {X_{{\text{s}}1}} - {X_{{\text{sd}}}} $, $ {\chi _{{\text{s}}2}} = {X_{{\text{s}}2}} - {\alpha _{{\text{s}}1}} $, $ {\chi _{{\text{s}}3}} = {X_{{\text{s}}3}} - {\alpha _{{\text{s}}2}} $, and $ {\chi _{{\text{s}}4}} = {X_{{\text{s}}4}} - {\alpha _{{\text{s}}3}} $, where the reference signal $ {X_{{\text{sd}}}} = {q_{\text{m}}}\left({t - {T_{\text{m}}}} \right) $ is the tracking trajectory of the slave manipulator that accounts for the transmission delay T_m of the signal from the master manipulator to the slave manipulator, and the control signal $ {\alpha _{{\text{s}}j}}, \; \; j = 1, 2, 3 $ of the virtual controller. These assumptions are specified and validated as follows.

Assumption 4: Assume that appropriate positive definite constants $ {k_{{\text{s}}j}} $ exist that ensure that $\left| {{X_{{\text{s}}j}}} \right| < {k_{{\text{s}}j}}$ is satisfied.

Assumption 5: Assume that appropriate positive definite constants $ {A_{{\text{s}}0}} $ and $ {A_{{\text{s1}}}} $ exist that ensure that the conditions $ \left| {{X_{{\text{sd}}}}} \right| \leqslant {A_{{\text{s}}0}} < {\bar k_{{\text{s}}1}} $ and $ \left| {{{\dot X}_{{\text{sd}}}}} \right| \leqslant {A_{{\text{s}}1}} $ are satisfied.

Step 1: Select the following positive definite candidate BLF:

The derivative of $ {V_{{\text{s}}1}} $ with respect to time can be obtained as follows:

Here, $ {k_{{\chi _{{\text{s}}1}}}} = \frac{{{\chi _{{\text{s}}1}}}}{{k_{{\text{s}}1}^2 - \chi _{{\text{s}}1}^2}} $, while the other terms $ {k_{{\chi _{{\text{s}}j}}}} = \frac{{{\chi _{{\text{s}}j}}}}{{k_{{\text{s}}j}^2 - \chi _{{\text{s}}j}^2}}, \; \; j = 2, 3, 4 $, will be used below. The virtual controller $ {\alpha _{{\text{s}}1}} $ is designed as follows.

Substituting (33) in (32) yields the following:

Step 2: Select the following positive definite candidate BLF:

According to Eqs (4) and (34), taking the derivative of $ {V_{{\text{s}}2}} $ with respect to time yields the following:

Again, the estimated values of $ {\alpha _{{\text{s1}}}} $ and $ {\dot \alpha _{{\text{s}}1}} $ are respectively obtained from $ {v_{{\text{s}}1}} $ and $ {v_{{\text{s2}}}} $ (i.e., $ {\hat \alpha _{{\text{s}}1}} = {v_{{\text{s}}1}} $ and $ {\widehat {\dot \alpha }_{{\text{s}}1}} = {v_{{\text{s}}2}} $) using the tracking differentiator (28) with $ {v_{{\text{s}}0}} = {\alpha _{{\text{s}}1}} $. Therefore, the estimation error $ {\widetilde {\dot \alpha }_{{\text{s}}1}} = {\dot \alpha _{{\text{s}}1}} - {\widehat {\dot \alpha }_{{\text{s}}1}} $ is assumed to be bounded and satisfies $ \left| {{{\widetilde {\dot \alpha } }_{{\text{s}}1}}} \right| \leqslant {\mu _{{\text{s}}1}} $, where $ {\mu _{{\text{s}}1}} $ is an appropriate positive definite constant. Then, the virtual controller $ {\alpha _{{\text{s}}2}} $ of the slave manipulator is designed as:

where $ {\hat F_{\text{s}}} $ is the value of F_s estimated by the nonlinear ESO. Therefore, substituting (37) in (36) yields.

where ${\tilde F_{\text{s}}}$ is the estimation error of F_s estimated by the nonlinear ESO presented in the previous section.

Step 3: Select the following positive definite candidate BLF:

According to Eqs (4) and (38), the derivative of V_s3 with respect to time can be obtained as follows:

The values of $ {\hat \alpha _{{\text{s}}2}} = {v_{{\text{s}}1}} $ and $ {\widehat {\dot \alpha }_{{\text{s}}2}} = {v_{{\text{s}}2}} $ are obtained using the tracking differentiator (28) with $ {v_{{\text{s}}0}} = {\alpha _{{\text{s}}2}} $. Therefore, the estimation error $ {\widetilde {\dot \alpha }_{{\text{s}}2}} = {\dot \alpha _{{\text{s}}2}} - {\widehat {\dot \alpha }_{{\text{s}}2}} $ is assumed to be bounded and satisfies $ \left| {{{\widetilde {\dot \alpha } }_{{\text{s}}2}}} \right| \leqslant {\mu _{{\text{s}}2}} $, where $ {\mu _{{\text{s}}2}} $ is an appropriate positive definite constant. Then, the virtual controller $ {\alpha _{{\text{s}}3}} $ of the slave manipulator is designed as follows:

Therefore, (40) can be expressed as follows:

Step 4: Select the following positive definite candidate BLF:

According to Eqs (4) and (42), the derivative of $ {V_{\text{s}}} $ with respect to time can be obtained as follows:

The values of $ {\hat \alpha _{{\text{s}}3}} = {v_{{\text{s}}1}} $ and $ {\widehat {\dot \alpha }_{{\text{s}}3}} = {v_{{\text{s}}2}} $ are obtained using the tracking differentiator (28) with $ {v_{{\text{s}}0}} = {\alpha _{{\text{s}}3}} $. Therefore, the estimation error $ {\widetilde {\dot \alpha }_{{\text{s}}3}} = {\dot \alpha _{{\text{s}}3}} - {\widehat {\dot \alpha }_{{\text{s}}3}} $ is assumed to be bounded and satisfies $ \left| {{{\widetilde {\dot \alpha } }_{{\text{s}}3}}} \right| \leqslant {\mu _{{\text{s}}3}} $, where $ {\mu _{{\text{s}}3}} $ is an appropriate positive definite constant. Then, the input controller $ {\tau _{\text{s}}} $ of the slave manipulator is designed as:

where $ {\hat F_{\text{f}}} $ is the value of F_f estimated by the nonlinear ESO. Therefore, (44) can be expressed as:

where ${\tilde F_{\text{f}}}$ is the estimation error of F_f estimated by the nonlinear ESO presented in the previous section. The control block diagram of the teleoperation system with controllers (29) and (45) is presented in Figure 1. Figure 1 delineates the advanced teleoperation control framework between a Master and Slave robot. Here, both robots employ ESOs to estimate uncertainties and a BLF-based controller for precision. The network delay in command transmission is explicitly represented, showcasing the integrated approach in handling flexible joint uncertainties. Both robots operate within full state constraints, highlighting the strategy's emphasis on synchronized and reliable robotic interactions.

Theorem 2: Assume that the teleoperation system satisfies Assumptions 2–5. Then, under the virtual controller $ {\alpha _{ij}} $, where j = 1, 2 when i = m, and j = 1, 2, 3, 4 when i = s, and the input controllers $ {\tau _{\text{m}}} $ and $ {\tau _{\text{s}}} $, all variables of the closed-loop system are bounded and do not exceed their constraints if two positive definite constants $ a $ and $ b $ exist that satisfy the following conditions.

Proof: Choose the following Lyapunov function:

When $ \left| {{\chi _{ij}}} \right| < {k_{ij}} $, positive definite constants must exist that satisfy the condition $ \frac{{k_{ij}^2}}{{k_{ij}^2 - \chi _{ij}^2}} > \frac{{\chi _{ij}^2}}{{k_{ij}^2 - \chi _{ij}^2}} $. Therefore, according to Eqs (30) and (46), the derivative of $ V $ with respect to time can be obtained as follows:

Multiplying both sides of inequality (49) by $ {e^{at}} $ yields the following form:

Furthermore, integrating (50) over the interval $\left[{0, t} \right]$ yields the following:

Given that $ {\hat F_{\text{m}}} $, $ {\hat F_{\text{s}}} $, $ {\hat F_{\text{f}}} $ and $ {\hat \alpha _{ij}} $ are bounded, $ \left| {{X_{i1}}} \right| < {k_{i1}} + {A_{i0}} < {\bar k_{i1}} $ can be obtained from $ {\chi _{i1}} = {X_{i1}} - {X_{i{\text{d}}}} $ and $ {X_{i{\text{d}}}} \leqslant {A_{i0}} $. The boundedness of $ {\chi _{i1}} $ and $ {\dot X_{i{\text{d}}}} $ ensures that $ {\alpha _{i1}} $ is bounded and $ {\alpha _{i1}} \leqslant {\bar \alpha _{i1}} $. Similarly, $ \left| {{X_{i2}}} \right| < {k_{i2}} + {\bar \alpha _{i1}} < {k_{i2}} $ can be obtained from $ {\chi _{i2}} = {X_{i2}} - {\alpha _{i1}} $, and $ \left| {{X_{{\text{s}}3}}} \right| < {k_{{\text{s}}3}} $, and $ \left| {{X_{{\text{s}}4}}} \right| < {k_{{\text{s}}4}} $ can be obtained in the same way. Furthermore, the input controller signals $ {\tau _{\text{m}}} $ and $ {\tau _{\text{s}}} $ are bounded according to their definitions in Eqs (29) and (45), respectively, where $ {\tau _{\text{m}}} $ is a function of $ {\chi _{{\text{m}}2}} $, $ {\hat F_{\text{m}}} $, and $ {\widehat {\dot \alpha }_{{\text{m}}1}} $, and $ {\tau _{\text{s}}} $ is a function of $ {\chi _{{\text{s}}4}} $, $ {\hat F_{\text{f}}} $, and $ {\widehat {\dot \alpha }_{{\text{s}}3}} $. Therefore, all signals of the system, namely $ {\tau _{\text{m}}} $, $ {\tau _{\text{s}}} $, and $ {X_{ij}} $, are bounded and meet their constraints.

4.   Experimental platform and analysis of experimental results

The experimental teleoperation platform employed for testing the effectiveness of the proposed control strategy is presented in Figure 2, and consisted of two Phantom Premium 1.5HF (SensAble Technologies, Inc.) three-degrees-of-freedom robotic arms. The Phantom Premium 1.5HF arm has three rotating joints, each consisting of three parts of a gear motor encoder, which can feel or follow the movement of the controlled object on three perpendicular axes in space. The controllers and all software were implemented in MATLAB, and the flexible joint from the slave manipulator was realized by a virtual module in MATLAB.

Angle sensors were installed at each joint on the master and slave end of the three-degree-of-freedom manipulator to measure the required angle. For the process of experimental verification, the delays were set to T_m = T_s = 200 ms. The appropriate parameters were obtained through experiments and set as: $\varepsilon = 1.05$, ${\beta _{{\text{m}}1}} = {\beta _{{\text{s}}1}} = \left[{ $$\begin{array}{*{20}{c}} {210} & 0 & 0 \\ 0 & {250} & 0 \\ 0 & 0 & {200} \end{array}$$ } \right]$, ${\beta _{{\text{m}}2}} = {\beta _{{\text{s}}2}} = \left[{ $$\begin{array}{*{20}{c}} {2.5} & 0 & 0 \\ 0 & {1.5} & 0 \\ 0 & 0 & {2.5} \end{array}$$ } \right]$, ${\beta _{{\text{m}}3}} = {\beta _{{\text{s}}3}} = \left[{ $$\begin{array}{*{20}{c}} {0.0015} & 0 & 0 \\ 0 & {0.0025} & 0 \\ 0 & 0 & {0.0025} \end{array}$$ } \right]$. The parameters of the master controller are selected as: ${K_{{\text{m}}1}} = \left[{ $$\begin{array}{*{20}{c}} 5 & 0 & 0 \\ 0 & {7.8} & 0 \\ 0 & 0 & {6.8} \end{array}$$ } \right]$, ${K_{{\text{m}}2}} = \left[{ $$\begin{array}{*{20}{c}} {0.71} & 0 & 0 \\ 0 & {0.3} & 0 \\ 0 & 0 & {0.3} \end{array}$$ } \right]$, ${k_{{\text{m}}1}} = \left[{ $$\begin{array}{*{20}{c}} {1.4} \\ {1.4} \\ {1.3} \end{array}$$ } \right]$, ${k_{{\text{m}}2}} = \left[{ $$\begin{array}{*{20}{c}} {1.5} \\ {2.8} \\ {1.5} \end{array}$$ } \right]$. The stiffness coefficient for the flexible joint of the slave manipulator was set as ${K_{\text{s}}} = \left[{ $$\begin{array}{*{20}{c}} {12} & 0 & 0 \\ 0 & {12} & 0 \\ 0 & 0 & {12} \end{array}$$ } \right]$, while the slave controller parameters were set as ${K_{{\text{s}}1}} = \left[{ $$\begin{array}{*{20}{c}} {1.1} & 0 & 0 \\ 0 & {0.9} & 0 \\ 0 & 0 & {0.9} \end{array}$$ } \right]$, ${K_{{\text{s}}2}} = \left[{ $$\begin{array}{*{20}{c}} {0.7} & 0 & 0 \\ 0 & {0.5} & 0 \\ 0 & 0 & {0.7} \end{array}$$ } \right]$, ${K_{{\text{s}}3}} = \left[{ $$\begin{array}{*{20}{c}} {3.1} & 0 & 0 \\ 0 & {0.9} & 0 \\ 0 & 0 & {0.9} \end{array}$$ } \right]$, ${K_{{\text{s}}4}} = \left[{ $$\begin{array}{*{20}{c}} {14.5} & 0 & 0 \\ 0 & {19.5} & 0 \\ 0 & 0 & {19.5} \end{array}$$ } \right]$, ${k_{{\text{s}}1}} = \left[{ $$\begin{array}{*{20}{c}} {0.4} \\ {0.5} \\ {0.5} \end{array}$$ } \right]$, ${k_{{\text{s}}2}} = \left[{ $$\begin{array}{*{20}{c}} {0.5} \\ {0.5} \\ {0.5} \end{array}$$ } \right]$, ${k_{{\text{s}}3}} = \left[{ $$\begin{array}{*{20}{c}} {0.4} \\ {0.4} \\ {0.4} \end{array}$$ } \right]$, ${k_{{\text{s}}4}} = \left[{ $$\begin{array}{*{20}{c}} {0.5} \\ {0.3} \\ {0.3} \end{array}$$ } \right]$.

In the experimental evaluation, Figures 3–5 delve into the position estimation errors of both the master and slave manipulators over three degrees of freedom. Specifically, Figure 3 showcases the master manipulator's position errors, which consistently remain below ± 0.02 rad. Figures 4 and 5, representing the slave manipulator's errors at its motors and chain rod respectively, confirm errors below ± 0.05 rad. These figures underline the nonlinear ESO's superior capability in accurately estimating the positions of both manipulators.

Figure 6, illustrating the control inputs, highlights a synchronized control approach between the master and slave manipulators. Their control signals exhibit sine wave-like consistencies, reinforcing the effectiveness of the implemented synchronization.

The master and slave manipulator positions over time are depicted in Figure 7. Notably, their positions mirror each other closely, with a mere average deviation of ± 0.02 rad as substantiated in Figure 8. Upon halting the manipulator motion after 50 seconds, the tracking error rapidly zeroes out, underscoring the controller's adeptness in position synchronization.

Figures 9 and 10 further emphasize the system's stability. Despite initially larger tracking errors, they swiftly reduce to a maximum of ± 0.05 rad and approach zero when the motion is halted after 50 seconds, indicating minimal influence from external forces.

To summarize, these experimental findings bolster the merits of the proposed control strategy. It not only achieves precise position estimations and effective synchronization, but also remains resilient against potential disturbances. This culmination holds significant promise for enhancing teleoperation systems in intricate operational landscapes.

5.   Conclusion

This research marks a significant stride in the realm of teleoperation systems specifically designed for robotic manipulators with flexible joints. Our primary focus revolves around investigating the feasibility of a pose control strategy for teleoperated systems with flexible joints. This strategy is anchored on the integration of a nonlinear Extended State Observer (ESO) with a Barrier Lyapunov Function (BLF).

The efficacy of this novel approach is convincingly demonstrated by our experimental findings. Specifically, in a master-slave operating system, the slave exhibits exceptional synchronization capabilities with high precision. The position estimation errors in the master-slave operation consistently remain within ± 0.05 radians, particularly at the chain rod where it is even less than ± 0.01 radians. Even in complex scenarios involving flexible joints, this level of precision showcases the robustness and potential of our integration strategy.

It is essential to highlight that the crux of this study was to explore the feasibility of the aforementioned control strategy. Therefore, instead of comparing it with existing methodologies, we have dedicated our efforts to comprehensively understand and elucidate its intrinsic merits.

As we pave the path forward, we envision delving deeper into enhancing the nuances of our approach. While we have not directly compared our method with industry benchmarks in this study, future endeavors might involve such comparative analyses. Nonetheless, our primary goal remains: to further elucidate, validate, and promote the unique potential and adaptability of our teleoperation strategy across diverse operational scenarios.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgements

This paper is supported by National Natural Science Foundation of China under grant No. 52305066.

Conflict of interest

The authors declare there is no conflict of interest.