Neural Networks in the Delayed Teleoperation of a Skid-Steering Robot

Abstract

Bilateral teleoperation of skid-steering mobile robots with time-varying delays presents significant challenges in ensuring accurate leader–follower coupling. This article presents a novel controller for a bilateral teleoperation system composed of a robot manipulator and a skid-steering mobile robot. The proposed controller leverages neural networks to compensate for ground–robot interactions, uncertain dynamics, and communication delays. The control strategy integrates a shared scheme between damping injection and two neural networks, enhancing the robustness and adaptability of the delayed system. A rigorous theoretical analysis of the closed-loop teleoperation system is provided, establishing conditions of control parameters to ensure stability and convergence of the coordination errors. The proposed method is validated through numerical testing, demonstrating strong agreement between theoretical outcomes and simulation results.

1. Introduction

Teleoperation systems are designed to extend the capabilities of human operators to remote locations, allowing for the safe execution of physical tasks while avoiding human exposure or transport to unsafe and/or distant environments. The human operator interacts with the workspace using a haptic device, known as the leader, which issues motion commands to the follower robot while simultaneously receiving multimodal feedback, including real-time force feedback. The operator is an integral part of the closed-loop bilateral control scheme, where stability and transparency are critical concerns [1]. These properties are typically in conflict—improving transparency often degrades stability, and vice versa. Both are strongly influenced by time delays, control architecture, and parameter tuning [2]. Teleoperation systems have a wide range of applications across various fields, including rescue and surveillance, mining, exploration, and industrial services, among others [3]. Many teleoperation-based applications can be effectively addressed using unmanned ground vehicles (UGVs) such as skid-steering mobile robots (SSMRs). This system provides low-cost maintenance, good handling, and mechanical simplicity by eliminating the need for a steering mechanism. Instead, heading control relies on differential velocities, at the expense of lateral wheel skidding [4].

One of the main challenges in robot teleoperation arises from the detrimental effects of time-varying delays [5,6]. A widely adopted approach for designing control schemes in these delayed systems, applied to manipulators, unmanned vehicles, mobile manipulators, and even humanoid robots, is based on dissipating sufficient kinetic energy, depending on the time delay, to ensure bounded coordination errors at the cost of making the system slower [7,8,9,10,11]. Furthermore, SSMRs have a kinematic mismatch with the leader, and the dynamic and underactuated nature of SSMRs, coupled with terrain interactions, introduces additional modeling uncertainties that complicate the direct application of classic teleoperation techniques. Enhancing performance requires addressing the interactions between the robot and the terrain, particularly in slippery conditions, while also accounting for the inherent uncertainties in the dynamic behavior of the robot, as discussed in [12].

One effective approach to mitigate the effects of time-varying delays, ground–robot interactions, and uncertain dynamics is the use of adaptive robust neural compensation methods, which are widely recognized for their ability to compensate for nonlinearities and external disturbances. These methods typically combine Adaptive Control [13,14,15], Robust Control [16], and neural networks [17]. Unlike conventional control methods, this approach enables online adaptation of controller parameters, and numerous studies have confirmed the advantages of neural networks in real-time approximation and dynamic adaptation, improving control accuracy and flexibility in various teleoperation contexts and enabling effective compensation of uncertainties that typically hinder system responsiveness and stability [18,19,20]. However, the optimal control structure for compensating the time-varying delays in bilateral teleoperation of skid-steering mobile robots remains an open question. Furthermore, the current state of the art does not yet provide a generalized solution for how neural networks should be integrated with traditional teleoperation schemes to control the coordination errors, taking advantage of the strengths of both strategies in the face of time delays. Table 1 presents a general comparison between the method proposed in this paper and other recent approaches within the same teleoperation framework.

In [21,22], neural networks and genetic algorithms are used to predict human commands and slippage forces, whereas, in [23], predictors are not used, but dynamics and ground–robot interactions are compensated at the remote site. Comparison items include stability test coverage, convergence characteristics, considered delays, leader and follower models, and the role of neural networks. Notably, the standardization of the specific role and function of neural networks in teleoperated control schemes remains an open challenge as it is still unclear what the network should effectively accomplish in these systems.

In this work, a neural network control approach is proposed for a delayed bilateral teleoperation system, aiming to perform time-varying delay compensation in a shared manner between damping-controlled injection and two neural networks. Additionally, the latter is employed to compensate for the unknown dynamics of both the leader and the follower, as well as uncertainties and ground–robot interactions. The main contributions of this paper are listed below:

• A neural network control scheme is designed to address unknown system dynamics and ground–robot interactions. The compensation of negative effects caused by time-varying delays is distributed between two neural networks and controlled injection of damping.
• A rigorous analysis of the bilateral closed-loop teleoperation system is provided, focusing on the control parameter conditions necessary to ensure the convergence of coordination errors despite the presence of time delays and ground–robot dynamic interactions.
• The proposed theory is validated through numerical simulations. In these tests, a two-degrees-of-freedom manipulator robot is used to represent the leader, while the follower replicates the dynamics of a skid-steering mobile robot, including ground–robot interactions. The results demonstrate the effectiveness of the control scheme in managing time delays under various operating conditions.

The content of this document is organized as follows: Section 2 presents the mathematical background. Section 3 addresses the design of the leader–follower controller and the stability analysis of the delayed system. Section 4 presents the results of numerical simulations considering different terrains and delays. Finally, Section 5 concludes the document.

Table 1. Comparison of the proposed method against previously reported NN-based strategies for delayed bilateral teleoperation of mobile robots.

Reference	Stability	Delay	Model	Neural Networks
Slawiñski et al. (2024) [23]	Lyapunov–Krasovskii Criterion, coordination error $e_s$ bounded, convergence to zero in the absence of external forces	Variable	Dynamic UGV including ground–robot interaction	One NN LSTM system to compensate unknown dynamics and ground–robot interaction
Ahmad et al. (2024) [21]	Llewellyn’s Criterion of bounded input/bounded output	Constant	Modified kinematic model for the rover subjected to longitudinal slippage	Two genetic algorithms applied as model-free predictors for estimation of slippage and commands
Abubakar et al. (2024) [22]	N/A	Constant	Modified kinematic model for the UGV on soft terrains	Two NN LSTM systems applied as predictors for estimation of slippage and commands
Proposed	Lyapunov–Krasovskii Criterion, coordination errors $e_m,e_s$ bounded, convergence to zero in the absence of external forces	Variable	Dynamic UGV including ground–robot interaction	Two NN FLNN systems to compensate unknown dynamics and ground–robot interaction as well as partially mitigate the effects of the time-varying delay

Table 1. Comparison of the proposed method against previously reported NN-based strategies for delayed bilateral teleoperation of mobile robots.

Reference	Stability	Delay	Model	Neural Networks
Slawiñski et al. (2024) [23]	Lyapunov–Krasovskii Criterion, coordination error $e_s$ bounded, convergence to zero in the absence of external forces	Variable	Dynamic UGV including ground–robot interaction	One NN LSTM system to compensate unknown dynamics and ground–robot interaction
Ahmad et al. (2024) [21]	Llewellyn’s Criterion of bounded input/bounded output	Constant	Modified kinematic model for the rover subjected to longitudinal slippage	Two genetic algorithms applied as model-free predictors for estimation of slippage and commands
Abubakar et al. (2024) [22]	N/A	Constant	Modified kinematic model for the UGV on soft terrains	Two NN LSTM systems applied as predictors for estimation of slippage and commands
Proposed	Lyapunov–Krasovskii Criterion, coordination errors $e_m,e_s$ bounded, convergence to zero in the absence of external forces	Variable	Dynamic UGV including ground–robot interaction	Two NN FLNN systems to compensate unknown dynamics and ground–robot interaction as well as partially mitigate the effects of the time-varying delay

2. Mathematical Background

In this section, the subscript $i=m,s$ is used to denote the leader and follower sides of the system, while $j=s,m$ represents the opposite side.

2.1. Dynamic Model of the Teleoperation System

The simplified dynamic model of a bilateral teleoperation system of a skid-steering mobile robot is given by a leader and a mobile follower, which are represented in [23]. The leader dynamics are represented by the following equation:

$$\begin{array}{cc}\hfill M_m\left({\mathit{q}}_m\right){\ddot{\mathit{q}}}_m+C_m({\mathit{q}}_m,{\dot{\mathit{q}}}_m){\dot{\mathit{q}}}_m+{\mathit{g}}_m\left({\mathit{q}}_m\right)& ={\mathit{\tau }}_m+{\mathit{\tau }}_h,\hfill \end{array}$$

(1)

where ${\mathit{q}}_m$, ${\dot{\mathit{q}}}_m$, and ${\ddot{\mathit{q}}}_m\in {\mathbb{R}}^n$ are vectors of the joint position, velocity, and acceleration of the leader, respectively. $M_m\left({\mathit{q}}_m\right)\in {\mathbb{R}}^{n\times n}$ is the inertia matrix, $C_m({\mathit{q}}_m,{\dot{\mathit{q}}}_m)\in {\mathbb{R}}^{n\times n}$ denotes the Coriolis and centripetal matrix, ${\mathit{g}}_m\left({\mathit{q}}_m\right)\in {\mathbb{R}}^n$ represents the gravitational vector, ${\mathit{\tau }}_m\in {\mathbb{R}}^n$ is the control input vector, and ${\mathit{\tau }}_h\in {\mathbb{R}}^n$ is the operating torque of the human operator.

Remark 1.

From now on, we suppose that $n=2$. Due to using the leader, it is possible to command the speed and angular velocity of the subactuated follower.

Furthermore, the dynamic model of the robot manipulator (1) has the following properties:

Property 1.

The inertia matrix $M_m\left({\mathit{q}}_m\right)$ is symmetric and positive-definite, and there exist positive constants such that

$${\lambda }_{min}\left\{M_m\left(\mathit{x}\right)\right\}{\parallel \mathit{x}\parallel }^2\le {\mathit{x}}^T{M}_m\left(\mathit{x}\right)\mathit{x}\le {\lambda }_{max}\left\{M_m\left(\mathit{x}\right)\right\}{\parallel \mathit{x}\parallel }^2,$$

for all $\mathit{x}\in {\mathbb{R}}^n$.

Property 2.

The matrix $C_m({\mathit{q}}_m,{\dot{\mathit{q}}}_m)$ is related to the inertia matrix $M_m\left({\mathit{q}}_m\right)$ by the expression

$${\mathit{x}}^T\left[{\displaystyle \frac{1}{2}}{\dot{M}}_m\left({\mathit{q}}_m\right)-C_m({\mathit{q}}_m,{\dot{\mathit{q}}}_m)\right]\mathit{x}=0,\forall {\mathit{q}}_m,{\dot{\mathit{q}}}_m,\mathit{x}\in {\mathbb{R}}^n,$$

and holds $\parallel C_m({\mathit{q}}_m,{\dot{\mathit{q}}}_m)\parallel <\gamma \parallel {\dot{\mathit{q}}}_m\parallel$ [24].

On the other hand, the kinematic model of the robot, shown in Figure 1, is defined by the following equation:

$$\begin{array}{c}\hfill \dot{\mathit{q}}=A\left(\mathit{q}\right){\mathit{v}}_s+B\left(\mathit{q}\right)v_y,\end{array}$$

(2)

where ${\mathit{v}}_s=\left[vw\right]$ denotes the linear and angular velocities of the mobile robot, $\mathit{q}=\left[xy\varphi \right]$ represents its pose, $v_y$ corresponds to the lateral velocity, and the matrices $A\left(\mathit{q}\right)$ and $B\left(\mathit{q}\right)$ are defined as

$$A\left(\mathit{q}\right)=\left[\begin{array}{cc}\cos \left(\varphi \right)& 0\\ \sin \left(\varphi \right)& 0\\ 0& 1\end{array}\right],B\left(\mathit{q}\right)=\left[\begin{array}{c}-\sin \left(\varphi \right)\\ \cos \left(\varphi \right)\\ 0\end{array}\right],$$

(3)

respectively, where $\varphi$ is the orientation of the mobile robot with respect to the world reference frame W, which denotes a fixed coordinate system. See Figure 1 for further details.

The dynamic model is described in detail in [12] and is given by the following equation:

$$\begin{array}{c}\hfill M_s{\dot{\mathit{v}}}_s+\mathit{N}({\mathit{v}}_s,v_y,\theta )={\mathit{u}}_s+{\mathit{f}}_e,\end{array}$$

(4)

$$\begin{array}{c}\hfill \mathit{N}=\mathit{P}{\mathit{v}}_y+\mathit{F}({\mathit{v}}_s,v_y)+g_s\left(\theta \right),\end{array}$$

(5)

where $M_s\in {\mathbb{R}}^{n\times n}$ is defined as $\left[\begin{array}{cc}m& 0\\ 0& J\end{array}\right]$, given by the mass m and inertia J of the follower system, ${\mathit{v}}_s={\left[v\omega \right]}^T$ represents the speed v and the yaw angular velocity $\mathit{\omega }$ of the skid-steering mobile robot, ${\mathit{u}}_s\in {\mathbb{R}}^{n\times 1}$ is the input signal control to the follower system, including force and torque, applied to the robot body and can be inferred from the torque applied to each motor, $\mathit{P}={\left[m\mathit{\omega }0\right]}^T$, $F\in {\mathbb{R}}^n$, and $F_y$ represent the longitudinal, angular, and lateral friction forces/torques and unmodeled dynamics, $g_s\left(\theta \right)$ is defined as the gravity magnitude on sloped terrain with inclination $\theta$ in relation to the horizontal ground plane, and ${\mathit{f}}_e\in {\mathbb{R}}^n$ is the external environmental forces representing the interaction forces between the mobile robot and the objects in its surroundings. It is important to remark that $v_y$ represents the lateral velocity due to sideslip, modeled as follows:

$$\begin{array}{c}\hfill m({\dot{v}}_y+v\omega )=F_y+f_{e_y}.\end{array}$$

(6)

where $f_{e_y}\in \mathbb{R}$ is the lateral component of the external environmental force.

2.2. Neural Network Description

In this section, a basic neural network known as a functional link neural network (FLNN) is introduced. Consider ${\mathit{f}}_i\left({\mathit{x}}_i\right)\in {\mathbb{R}}^n$ as a smooth function and let ${\mathit{x}}_i$ be restricted to a compact set $\mathcal{S}$ of ${\mathbb{R}}^n$. Then, ${\mathit{f}}_i\left({\mathit{x}}_i\right)$ can be approximated by a neural network as follows:

$${\mathit{f}}_i\left({\mathit{x}}_i\right)=W_i^T\mathit{\sigma }\left(V_i^T{\overline{\mathit{x}}}_i\right)+{\mathit{ϵ}}_i,$$

(7)

where $W_i\in {\mathbb{R}}^{L_N\times n}$ is the ideal output weights matrix, $\mathit{\sigma }\in {\mathbb{R}}^{L_N}$ is the activation function consisting of the hyperbolic tangent function, $V_i\in {\mathbb{R}}^{b\times L_N}$ is the input weights matrix, ${\overline{\mathit{x}}}_i\in {\mathbb{R}}^b$ is the input signals vector, and ${\mathit{ϵ}}_i\in {\mathbb{R}}^n$ is the approximation error. Additionally, b is the number of input signals and $L_N$ is the number of neurons in the hidden layer.

Assumption 1.

The ideal output weights of the neural network, denoted as $W_i$, are bounded such that $\parallel W_i\parallel \le W_{iM}$, where $W_{iM}$ is a known positive constant.

Assumption 2.

The approximation error of the neural network is bounded by a known positive constant ${\epsilon }_{iM}$ such that $\parallel {\mathit{ϵ}}_i\left(x_i\right)\parallel \le {\epsilon }_{iM}$ for all $x_i\in \mathcal{D}$, where $\mathcal{D}$ represents the input domain.

2.3. Mathematical Preliminaries

The time-varying delays included by the communication channel in the transmitted signals $h\left(t\right)$ from the leader to the follower and vice versa are represented by $d_k$, resulting in $h(t-d_k\left(t\right))$.

Assumption 3.

The time-varying delay $d_k$ is assumed to be continuously differentiable and uniformly bounded, satisfying $0\le d_k\le T_k$ and $|{\dot{d}}_k|\le \mu <1$, where $T_k$ and μ are known positive constants. This ensures that the delay remains bounded and there is no data loss. In addition, $k=1,2$ is used for the time delay from the follower to the leader and vice versa, respectively.

Lemma 1.

Barbalat’s lemma [25]. If the differentiable function $L\left(t\right)$ has a finite limit as $t\to \infty$, and if $\dot{L}\left(t\right)$ is uniformly continuous (or equivalently, if $\ddot{L}\left(t\right)$ is bounded), then

$$\underset{t\to \infty }{\lim }\dot{L}\left(t\right)=0.$$

3. Main Result

3.1. Control Design

In practical applications, the convergence of closed-loop trajectories for delayed bilateral teleoperation systems is a key aspect. To address this, let us define ${\mathit{e}}_m$ and ${\mathit{e}}_s\in {\mathbb{R}}^n$ as the coordination error of the teleoperation system,

$${\mathit{e}}_m={\mathit{z}}_s(t-d_1)-k_g{\mathit{q}}_m\left(t\right),$$

(8)

$${\mathit{e}}_s=k_g{\mathit{q}}_m(t-d_2)-{\mathit{z}}_s\left(t\right),$$

(9)

and ${\mathit{r}}_i\in {\mathbb{R}}^n$ as a vector, such that

$$\begin{array}{c}\hfill {\mathit{r}}_m={\Lambda }_m{\mathit{e}}_m-{\alpha }_m{\dot{\mathit{q}}}_m,\end{array}$$

(10)

$$\begin{array}{c}\hfill {\mathit{r}}_s={\Lambda }_s{\mathit{e}}_s-{\alpha }_s{\dot{\mathit{z}}}_s,\end{array}$$

(11)

where ${\Lambda }_m$ and ${\Lambda }_s$∈${\mathbb{R}}^{n\times n}$ represent a diagonal positive matrix, ${\alpha }_i$ denotes a damping coefficient, and $k_g$ is a scaling gain to map a leader position to a follower velocity reference. To apply acceleration-based damping, ${\mathit{z}}_s$ holds the following relations based on a linear observer:

$$\begin{array}{c}\hfill {\dot{\mathit{z}}}_s={\displaystyle \frac{1}{\gamma }}({\mathit{v}}_s-{\mathit{z}}_s),\end{array}$$

(12)

$$\begin{array}{c}\hfill {\ddot{\mathit{z}}}_s={\displaystyle \frac{1}{\gamma }}(\dot{{\mathit{v}}_s}-{\dot{\mathit{z}}}_s),\end{array}$$

(13)

where $\gamma$ is a positive gain parameter. Then, a neural network control is proposed as follows:

$$\begin{array}{cc}\hfill {\mathit{\tau }}_m=& K_m{\mathit{r}}_m+{\widehat{W}}_m^T\mathit{\sigma }\left(V_m^T{\overline{\mathit{x}}}_m\right)+{\delta }_m\mathbf{sign}\left({\mathit{r}}_m\right)\hfill \\ \hfill & +{\beta }_m\parallel {\dot{\mathit{q}}}_m\parallel {\mathit{r}}_m-{\Lambda }_m^{-1}{\dot{\mathit{q}}}_m,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {\mathit{u}}_s=& K_s{\mathit{r}}_s+{\widehat{W}}_s^T\mathit{\sigma }\left(V_s^T{\overline{\mathit{x}}}_s\right)+{\delta }_s\mathbf{sign}\left({\mathit{r}}_s\right)-{\Lambda }_s^{-1}{\dot{\mathit{z}}}_s,\hfill \end{array}$$

(15)

where ${\beta }_m$ is a positive gain parameter, $K_i$ and ${\delta }_i\in {\mathbb{R}}^{n\times n}$ are diagonal positive-definite matrices, ${\widehat{W}}_i^T\in {\mathbb{R}}^{n\times L_N}$ is the estimate input weights of the neural network, and $\mathbf{sign}(.)\in {\mathbb{R}}^n$ is the signum function. Figure 2 shows a block diagram of the teleoperation system, including (14) and (15), where we highlight the proposed asymmetric combination (between the leader and the follower) of damping injection, neural networks, and linear and nonlinear actions.

3.2. Closed-Loop System

In order to obtain the closed-loop dynamic bilateral teleoperation system, the time derivatives of (10) and (11) are taken:

$$\begin{array}{c}\hfill {\dot{\mathit{r}}}_m={\Lambda }_m{\dot{\mathit{e}}}_m-{\alpha }_m{\ddot{\mathit{q}}}_m,\end{array}$$

(16)

$$\begin{array}{c}\hfill {\dot{\mathit{r}}}_s={\Lambda }_s{\dot{\mathit{e}}}_s-{\alpha }_s{\ddot{\mathit{z}}}_s.\end{array}$$

(17)

Now, including the dynamics (1) and (4) into (16) and (17) considering (8) and (9), we have

$$\begin{array}{cc}\hfill {\dot{\mathit{r}}}_m=& {\Lambda }_m{\dot{\mathit{e}}}_m-{\alpha }_m{M}_m^{-1}\left({\mathit{q}}_m\right)\left({\mathit{\tau }}_m+{\mathit{\tau }}_h\right.\hfill \\ \hfill & \left.-C_m({\mathit{q}}_m,{\dot{\mathit{q}}}_m){\dot{\mathit{q}}}_m-{\mathit{g}}_m\left({\mathit{q}}_m\right))\right),\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\dot{\mathit{r}}}_s=& {\Lambda }_s{\dot{\mathit{e}}}_s-{\alpha }_s{\displaystyle \frac{1}{\gamma }}\left(\dot{\mathit{v}}-\dot{{\mathit{z}}_s}\right)\hfill \\ \hfill =& {\Lambda }_s{\dot{\mathit{e}}}_s-{\alpha }_s{\displaystyle \frac{1}{\gamma }}\left(M_s^{-1}\left(\mathit{u}-\mathit{N}+{\mathit{f}}_e\right)-\dot{{\mathit{z}}_s}\right).\hfill \end{array}$$

(19)

Using the approximation property of neural networks (7), some terms of (18) and (19) can be estimated by neural networks, respectively, as

$$\begin{array}{cc}\hfill W_m^T\mathit{\sigma }\left(V_m^T{\overline{\mathit{x}}}_m\right)+{\mathit{ϵ}}_m& ={\displaystyle \frac{1}{{\alpha }_m}}{M}_m\left({\mathit{q}}_m\right){\Lambda }_m{\dot{\mathit{e}}}_m\hfill \\ & +C_m({\mathit{q}}_m,{\dot{\mathit{q}}}_m){\dot{\mathit{q}}}_m+{\mathit{g}}_m\left({\mathit{q}}_m\right),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill W_s^T\mathit{\sigma }\left(V_s^T{\overline{\mathit{x}}}_s\right)+{\mathit{ϵ}}_s=& {\displaystyle \frac{\gamma }{{\alpha }_s}}{M}_s\left({\mathit{q}}_s\right){\Lambda }_s{\dot{\mathit{e}}}_s+\mathit{N}+{\mathit{M}}_s\dot{{\mathit{z}}_s},\hfill \end{array}$$

(21)

with the input vectors

$$\begin{array}{cc}\hfill {\overline{\mathit{x}}}_m=& {\left[1{\dot{\mathit{z}}}_s^T(t-d_1){\dot{\mathit{q}}}_m^T{\mathit{q}}_m^T{\mathit{r}}_m^T\right]}^T,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\overline{\mathit{x}}}_s=& {\left[1{\dot{\mathit{q}}}_m^T(t-d_2){\dot{\mathit{z}}}_s^T{\mathit{z}}_s^T{\mathit{r}}_s^T{\mathit{v}}_y^T\theta \right]}^T,\hfill \end{array}$$

and suppose a time-varying delay in the input signals under Assumption 3. Here, the output weight error of neural networks (20) and (21), denoted as ${\tilde{W}}_i$, is defined as the difference between the optimal output weight vector $W_i$ and the estimated output weight vector ${\widehat{W}}_i$ of the neural network. Specifically, it is expressed as follows:

$${\tilde{W}}_i=W_i-{\widehat{W}}_i.$$

(22)

Considering (20), (21), and (22) in (18) and (19), the closed-loop bilateral teleoperation system is obtained as follows:

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{{\alpha }_m}}{M}_m\left({\mathit{q}}_m\right){\dot{\mathit{r}}}_m=& {\tilde{W}}_m^T\mathit{\sigma }\left(V_m^T{\overline{\mathit{x}}}_m\right)-K_m{\mathit{r}}_m-{\delta }_m\mathbf{sign}\left({\mathit{r}}_m\right)\hfill \\ \hfill & +{\Lambda }_m^{-1}{\dot{\mathit{q}}}_m-{\beta }_m\parallel {\dot{\mathit{q}}}_m\parallel {\mathit{r}}_m+{\mathit{ϵ}}_m-{\mathit{\tau }}_h,\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\gamma }{{\alpha }_s}}{M}_s{\dot{\mathit{r}}}_s=& {\tilde{W}}_s^T\mathit{\sigma }\left(V_s^T{\overline{\mathit{x}}}_s\right)-K_s{\mathit{r}}_s-{\delta }_s\mathbf{sign}\left({\mathit{r}}_s\right)\hfill \\ \hfill & +{\Lambda }_s^{-1}{\dot{\mathit{z}}}_s+{\mathit{ϵ}}_s-{\mathit{f}}_e.\hfill \end{array}$$

(24)

Next, the system stability will be analyzed in relation to the dynamics of ${\mathit{r}}_i$.

3.3. Stability Analysis

For the following analysis of the closed-loop system (23) and (24), the torques applied by both the operator and the external environmental forces are considered bounded. The goal is to analyze the leader–follower coupling of the delayed bilateral teleoperation system.

Next, we consider the following positive-definite Lyapunov–Krasovskii functions as follows:

$$\begin{array}{cc}\hfill L_1=& {\displaystyle \frac{k_g}{2{\alpha }_m}}{\mathit{r}}_m^T{M}_m\left({\mathit{q}}_m\right){\mathit{r}}_m+{\displaystyle \frac{\gamma }{2{\alpha }_s}}{\mathit{r}}_s^T{M}_s{\mathit{r}}_s\hfill \\ & +{\displaystyle \frac{1}{2}}Tr\left\{{\tilde{W}}_m^T{\Gamma }_m^{-1}{\tilde{W}}_m\right\}+{\displaystyle \frac{1}{2}}Tr\left\{{\tilde{W}}_s^T{\Gamma }_s^{-1}{\tilde{W}}_s\right\},\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill L_2=& {\displaystyle \frac{1}{2}}{({\mathit{z}}_s-k_g{\mathit{q}}_m)}^T({\mathit{z}}_s-k_g{\mathit{q}}_m)+k_g{\int }_{-T_2}^0{\int }_{t+\theta }^t{\dot{\mathit{q}}}_m^T\left(\xi \right){\dot{\mathit{q}}}_m\left(\xi \right)d\xi d\theta \hfill \\ & +k_g{\int }_{-T_1}^0{\int }_{t+\theta }^t{\dot{\mathit{z}}}_s^T\left(\xi \right){\dot{\mathit{z}}}_s\left(\xi \right)d\xi d\theta ,\hfill \end{array}$$

(26)

where ${\Gamma }_m$, ${\Gamma }_s\in {\mathbb{R}}^{L_N\times L_N}$, and $M_s$ are diagonal positive-definite matrices and $M_m$ holds by Property 1. Differentiating (25) with respect to time, we obtain

$$\begin{array}{cc}\hfill {\dot{L}}_1=& {\displaystyle \frac{k_g}{{\alpha }_m}}{\mathit{r}}_m^T{M}_m\left({\mathit{q}}_m\right){\dot{\mathit{r}}}_m+{\displaystyle \frac{k_g}{2{\alpha }_m}}{\mathit{r}}_m^T{\dot{M}}_m\left({\mathit{q}}_m\right){\mathit{r}}_m+{\displaystyle \frac{\gamma }{{\alpha }_s}}{\mathit{r}}_s^T{M}_s{\dot{\mathit{r}}}_s\hfill \\ \hfill & +Tr\left\{{\tilde{W}}_m^T{\Gamma }_m^{-1}{\dot{\tilde{W}}}_m\right\}+Tr\left\{{\tilde{W}}_s^T{\Gamma }_s^{-1}{\dot{\tilde{W}}}_s\right\},\hfill \end{array}$$

(27)

and, by substituting the closed-loop teleoperation system (23) and (24) into (27), we have

$$\begin{array}{cc}\hfill {\dot{L}}_1=& k_g{\mathit{r}}_m^T\left({\tilde{W}}_m^T\mathit{\sigma }\left(V_m^T{\overline{\mathit{x}}}_m\right)-K_m{\mathit{r}}_m\right.-{\delta }_m\mathbf{sign}\left({\mathit{r}}_m\right)\hfill \\ \hfill & \left.+{\mathit{ϵ}}_m+{\Lambda }_m^{-1}{\dot{\mathit{q}}}_m-{\beta }_m\parallel \dot{{\mathit{q}}_m}\parallel {\mathit{r}}_m-{\mathit{\tau }}_h\right)\hfill \\ \hfill & +{\mathit{r}}_s^T\left({\tilde{W}}_s^T\mathit{\sigma }\left(V_s^T{\overline{\mathit{x}}}_s\right)-K_s{\mathit{r}}_s-{\delta }_s\mathbf{sign}\left({\mathit{r}}_s\right)\right.\hfill \\ \hfill & \left.+{\mathit{ϵ}}_s+{\Lambda }_s^{-1}{\dot{\mathit{z}}}_s-{\mathit{f}}_e\right)+{\displaystyle \frac{k_g}{2{\alpha }_m}}{\mathit{r}}_m^T{\dot{M}}_m\left({\mathit{q}}_m\right){\mathit{r}}_m\hfill \\ \hfill & -Tr\left\{{\tilde{W}}_m^T{\Gamma }_m^{-1}{\dot{\widehat{W}}}_m\right\}-Tr\left\{{\tilde{W}}_s^T{\Gamma }_s^{-1}{\dot{\widehat{W}}}_s\right\}.\hfill \end{array}$$

(28)

The following adaptation laws of the output weights, denoted by ${\widehat{W}}_m$ and ${\widehat{W}}_s$, of the neural networks are proposed:

$$\begin{array}{cc}\hfill {\dot{\widehat{W}}}_m=& k_g{\Gamma }_m\mathit{\sigma }\left(V_m^T{\overline{\mathit{x}}}_m\right){\mathit{r}}_m^T-k_g{\Gamma }_m{\eta }_m\parallel {\mathit{r}}_m\parallel {\widehat{W}}_m,\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill {\dot{\widehat{W}}}_s=& {\Gamma }_s\mathit{\sigma }\left(V_s^T{\overline{\mathit{x}}}_s\right){\mathit{r}}_s^T-{\Gamma }_s{\eta }_s\parallel {\mathit{r}}_s\parallel {\widehat{W}}_s,\hfill \end{array}$$

(30)

where ${\eta }_m$ and ${\eta }_s$ are positive constants. Next, including (29) and (30) into Equation (28) while also considering Property 2, we obtain the following:

$$\begin{array}{cc}\hfill {\dot{L}}_1\le & -k_g{\mathit{r}}_m^T{K}_m{\mathit{r}}_m-k_g{\mathit{r}}_m^T{\delta }_m\mathbf{sign}\left({\mathit{r}}_m\right)+Tr\left\{k_g{\tilde{W}}_m^T{\eta }_m\parallel {\mathit{r}}_m\parallel {\widehat{W}}_m\right\}\hfill \\ \hfill & +k_g{\mathit{r}}_m^T{\mathit{ϵ}}_m+{\mathit{r}}_m^T{\Lambda }_m^{-1}{k}_g{\dot{\mathit{q}}}_m+(-{\beta }_m+{\displaystyle \frac{\gamma }{{\alpha }_m}})k_g\parallel {\dot{\mathit{q}}}_m\parallel {\mathit{r}}_m^T{\mathit{r}}_m\hfill \\ \hfill & -{\mathit{r}}_s^T{K}_s{\mathit{r}}_s-{\mathit{r}}_s^T{\delta }_s\mathbf{sign}\left({\mathit{r}}_s\right)+Tr\left\{{\tilde{W}}_s^T{\eta }_s\parallel {\mathit{r}}_s\parallel {\widehat{W}}_s\right\}\hfill \\ \hfill & +{\mathit{r}}_s^T{\mathit{ϵ}}_s+{\mathit{r}}_s^T{\Lambda }_s^{-1}{\dot{\mathit{z}}}_s-k_g{\mathit{r}}_m^T{\mathit{\tau }}_h-{\mathit{r}}_s^T{\mathit{f}}_e.\hfill \end{array}$$

(31)

Rewritten terms, considering (10) and (11) and fulfilling ${\beta }_m>{\displaystyle \frac{\gamma }{{\alpha }_m}}$, mean that (31) holds:

$$\begin{array}{cc}\hfill {\dot{L}}_1\le & -k_g{\mathit{r}}_m^T{K}_{pm}{\mathit{r}}_m-k_g{\mathit{r}}_m^T{\delta }_m\mathbf{sign}\left({\mathit{r}}_m\right)+Tr\left\{k_g{\tilde{W}}_m^T{\eta }_m\parallel {\mathit{r}}_m\parallel {\widehat{W}}_m\right\}\hfill \\ \hfill & +k_g{\mathit{r}}_m^T{\mathit{ϵ}}_m-{\mathit{r}}_s^T{K}_s{\mathit{r}}_s-k_g{\alpha }_m{\dot{\mathit{q}}}_m^T{\Lambda }_m^{-1}{\dot{\mathit{q}}}_m-{\alpha }_s{\dot{\mathit{z}}}_s^T{\Lambda }_s^{-1}{\dot{\mathit{z}}}_s\hfill \\ \hfill & -{\mathit{r}}_s^T{\delta }_s\mathbf{sign}\left({\mathit{r}}_s\right)+Tr\left\{{\tilde{W}}_s^T{\eta }_s\parallel {\mathit{r}}_s\parallel {\widehat{W}}_s\right\}+{\mathit{r}}_s^T{\mathit{ϵ}}_s\hfill \\ \hfill & -{({\mathit{z}}_s-k_g{\mathit{q}}_m)}^T({\dot{\mathit{z}}}_s-k_g{\dot{\mathit{q}}}_m)\hfill \\ \hfill & -k_g{\dot{\mathit{q}}}_m^T{\int }_{t-d_1}^t{\dot{\mathit{z}}}_s\left(\xi \right)d\xi -k_g{\dot{\mathit{z}}}_s^T{\int }_{t-d_2}^t{\dot{\mathit{q}}}_m\left(\xi \right)d\xi -k_g{\mathit{r}}_m^T{\mathit{\tau }}_h-{\mathit{r}}_s^T{\mathit{f}}_e.\hfill \end{array}$$

(32)

Then, differentiating (26) with respect to time and considering Assumption 3, we obtain the following:

$$\begin{array}{cc}\hfill {\dot{L}}_2\le & {({\mathit{z}}_s-k_g{\mathit{q}}_m)}^T({\dot{\mathit{z}}}_s-k_g{\dot{\mathit{q}}}_m)+k_g{T}_2{\dot{\mathit{q}}}_m^T{\dot{\mathit{q}}}_m+k_g{T}_1{\dot{\mathit{z}}}_s^T{\dot{\mathit{z}}}_s\hfill \\ \hfill & -k_g{\int }_{t-d_2}^t{\dot{\mathit{q}}}_m^T\left(\xi \right){\dot{\mathit{q}}}_m\left(\xi \right)d\xi -k_g{\int }_{t-d_1}^t{\dot{\mathit{z}}}_s^T\left(\xi \right){\dot{\mathit{z}}}_s\left(\xi \right)d\xi .\hfill \end{array}$$

(33)

From (32) and (33), and by using the inequalities

$$\begin{array}{cc}\hfill -{\dot{\mathit{q}}}_m^T{\int }_{t-d_1}^t{\dot{\mathit{z}}}_s\left(\xi \right)d\xi -{\int }_{t-d_1}^t{\dot{\mathit{z}}}_s^T\left(\xi \right){\dot{\mathit{z}}}_s\left(\xi \right)d\xi \le & {\displaystyle \frac{T_1}{4}}{\dot{\mathit{q}}}_m^T{\dot{\mathit{q}}}_m,\hfill \\ \hfill -{\dot{\mathit{z}}}_s^T{\int }_{t-d_2}^t{\dot{\mathit{q}}}_m\left(\xi \right)d\xi -{\int }_{t-d_2}^t{\dot{\mathit{q}}}_m^T\left(\xi \right){\dot{\mathit{q}}}_m\left(\xi \right)d\xi \le & {\displaystyle \frac{T_2}{4}}{\dot{\mathit{z}}}_s^T{\dot{\mathit{z}}}_s,\hfill \end{array}$$

we have

$$\begin{array}{cc}\hfill \dot{L}=& {\dot{L}}_1+{\dot{L}}_2\hfill \\ \hfill \le & -k_g{\mathit{r}}_m^T{K}_m{\mathit{r}}_m-k_g{\mathit{r}}_m^T{\delta }_m\mathbf{sign}\left({\mathit{r}}_m\right)+Tr\left\{k_g{\tilde{W}}_m^T{\eta }_m\parallel {\mathit{r}}_m\parallel {\widehat{W}}_m\right\}\hfill \\ \hfill & +k_g{\mathit{r}}_m^T{\mathit{ϵ}}_m-{\mathit{r}}_s^T{K}_s{\mathit{r}}_s\hfill \\ \hfill & -{\mathit{r}}_s^T{\delta }_s\mathbf{sign}\left({\mathit{r}}_s\right)+Tr\left\{{\tilde{W}}_s^T{\eta }_s\parallel {\mathit{r}}_s\parallel {\widehat{W}}_s\right\}+{\mathit{r}}_s^T{\mathit{ϵ}}_s\hfill \\ \hfill & -k_g{\alpha }_m{\dot{\mathit{q}}}_m^T{\Lambda }_m^{-1}{\dot{\mathit{q}}}_m-{\alpha }_s{\dot{\mathit{z}}}_s^T{\Lambda }_s^{-1}{\dot{\mathit{z}}}_s\hfill \\ \hfill & +k_g{T}_2{\dot{\mathit{q}}}_m^T{\dot{\mathit{q}}}_m+k_g{T}_1{\dot{\mathit{z}}}_s^T{\dot{\mathit{z}}}_s+k_g{\displaystyle \frac{T_1}{4}}{\dot{\mathit{q}}}_m^T{\dot{\mathit{q}}}_m+k_g{\displaystyle \frac{T_2}{4}}{\dot{\mathit{z}}}_s^T{\dot{\mathit{z}}}_s-k_g{\mathit{r}}_m^T{\mathit{\tau }}_h-{\mathit{r}}_s^T{\mathit{f}}_e.\hfill \end{array}$$

(34)

Furthermore, by adding and subtracting ${\eta }_m{k}_g\parallel {\mathit{r}}_m\parallel {\displaystyle \frac{W_{mM}}{2}}$ and ${\eta }_s\parallel {\mathit{r}}_s\parallel {\displaystyle \frac{W_{mS}}{2}}$ in (34) and then grouping terms using known mathematics relations, it can be rewritten as follows:

$$\begin{array}{cc}\hfill \dot{L}\le & -k_g{\lambda }_{min}\left\{K_m\right\}{\parallel {\mathit{r}}_m\parallel }^2-{\eta }_m{k}_g\parallel {\mathit{r}}_m\parallel {\left(\parallel {\tilde{W}}_m\parallel -{\displaystyle \frac{W_{mM}}{2}}\right)}^2\hfill \\ \hfill & -{\lambda }_{min}\left\{K_s\right\}{\parallel {\mathit{r}}_s\parallel }^2-{\eta }_s\parallel {\mathit{r}}_s\parallel {\left(\parallel {\tilde{W}}_s\parallel -{\displaystyle \frac{W_{sM}}{2}}\right)}^2\hfill \\ \hfill & -{\parallel {\mathit{r}}_m\parallel }_1{k}_g\left({\lambda }_{min}\left\{{\delta }_m\right\}-{\displaystyle \frac{{\eta }_m{W}_{mM}^2}{4}}-{\epsilon }_{mM}-{\overline{\tau }}_h\right)\hfill \\ \hfill & -{\parallel {\mathit{r}}_s\parallel }_1\left({\lambda }_{min}\left\{{\delta }_s\right\}-{\displaystyle \frac{{\eta }_m{W}_{sM}^2}{4}}-{\epsilon }_{sM}-{\overline{f}}_e\right)\hfill \\ \hfill & -\left({\displaystyle \frac{{\alpha }_m}{{\Lambda }_{max}\left\{{\Lambda }_m\right\}}}-T_2-{\displaystyle \frac{T_1}{4}}\right)k_g{\parallel {\dot{\mathit{q}}}_m\parallel }^2\hfill \\ \hfill & -\left({\displaystyle \frac{{\alpha }_s}{{\Lambda }_{max}\left\{{\Lambda }_s\right\}}}-k_g{T}_1-k_g{\displaystyle \frac{T_2}{4}}\right){\parallel {\dot{\mathit{z}}}_s\parallel }^2.\hfill \end{array}$$

(35)

Here, ${\overline{f}}_e$ and ${\overline{\tau }}_h$ are the bound values of the environment and human forces, respectively. Note that, with sufficiently large values for ${\alpha }_m$ and ${\alpha }_s$, along with sufficiently large values for ${\delta }_m$ and ${\delta }_s$ such that ${\lambda }_{min}\left\{{\delta }_m\right\}>{\displaystyle \frac{{\eta }_m{W}_{mM}^2}{4}}+{\epsilon }_{mM}+{\overline{\tau }}_h$, and similarly ${\lambda }_{min}\left\{{\delta }_s\right\}>{\displaystyle \frac{{\eta }_s{W}_{sM}^2}{4}}+{\epsilon }_{sM}+{\overline{\mathit{f}}}_e$, it is satisfied that $\dot{L}\le 0$, indicating that $\dot{L}$ is semi-definite negative. In practice, the necessary values of ${\delta }_m$ and ${\delta }_s$ are quite a lot lower than the sufficient values of (35) since the neural networks used partially compensate for environmental and human forces. On the other hand, ${\eta }_m,{\eta }_s$ are positive, very small values because the main function of the second terms of (29) and (30) is to avoid some drift if the input vector to the neural network is not persistently exciting. Furthermore, if Lemma 1 is applied, it is possible to infer that $\dot{L}\left(t\right)\to 0$ as $t\to \infty$, by examining (1), (4), (16), and (17), checking that the accelerations of the leader and follower are bounded. Therefore, ${\lim }_{t\to \infty }{\mathit{r}}_m\left(t\right)=0$, ${\lim }_{t\to \infty }{\mathit{r}}_s\left(t\right)=0$, ${\lim }_{t\to \infty }{\dot{\mathit{q}}}_m\left(t\right)=0$, and ${\lim }_{t\to \infty }{\dot{\mathit{z}}}_s\left(t\right)=0$. Moreover, based on Definitions (10) and (11), the following also holds:

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }{\mathit{e}}_m\left(t\right)=0,\underset{t\to \infty }{\lim }{\mathit{e}}_s\left(t\right)=0.\end{array}$$

Based on the results obtained on boundedness and convergence, and revisiting Equation (6), it follows straightforwardly that the lateral velocity $v_y$ is bounded. However, the presence of external torques from both the operator and the environment changes the convergence behavior. In practice, if such external torques are present and are not compensated, as long as they remain bounded, the errors will also remain bounded within a certain set. This is further validated by the simulation results presented in the next section.

4. Simulation Results

4.1. System Description

Leader robot description:

For the simulation, a two-degrees-of-freedom robot manipulator, shown in Figure 3, is used as the leader robot. The elements of the dynamic model for the leader (1) systems can be expressed as follows:

$$M\left(\mathit{q}\right)=\left[\begin{array}{cc}{\theta }_1+{\theta }_2\sin \left(q_2\right)& {\theta }_3\cos \left(q_2\right)\\ {\theta }_3\cos \left(q_2\right)& {\theta }_4\end{array}\right],$$

(36)

$$C(\mathit{q},\dot{\mathit{q}})=\left[\begin{array}{cc}{\displaystyle \frac{1}{2}}{\theta }_2{\dot{q}}_2\sin \left(2q_2\right)& c_{12}\\ -{\displaystyle \frac{1}{2}}{\theta }_2{\dot{q}}_1\sin \left(2q_2\right)& 0\end{array}\right],$$

(37)

$$\mathit{g}\left(\mathit{q}\right)=\left[\begin{array}{c}0\\ -{\theta }_5\sin \left(q_2\right)\end{array}\right],$$

(38)

where $c_{12}=-{\theta }_3{\dot{q}}_2\sin \left(q_2\right)+{\displaystyle \frac{1}{2}}{\theta }_2{\dot{q}}_1\sin \left(2q_2\right)$.

Table 2. Estimated parameters of two-degrees-of-freedom robot manipulator.

Symbol	Definition	Estimation	Unit
${\theta }_1$	$m_1{L}_1^2+J_1+m_2{L}_2^2$	0.02587	Kg m2
${\theta }_2$	$m_2{l}_2^2$	0.00107	Kg m2
${\theta }_3$	$L_1{l}_2{m}_2$	0.00151	Kg m2
${\theta }_4$	$m_2{l}_2^2+J_2$	0.00111	Kg m2
${\theta }_5$	$l_2{m}_{2}g$	0.09007	N m
${\theta }_6$	$f_{v1}$	0.00090	N ms
${\theta }_7$	$f_{v2}$	0.00033	N ms
${\theta }_8$	$f_{c1}$	0.00773	N m
${\theta }_9$	$f_{c2}$	0.00543	N m

shows the estimated parameters obtained in [26] for Equations (36)–(38).

Follower robot description:

The mobile robot used in this work is a mobile skid-steering platform.

Table 3. Parameters of mobile robot.

Symbol	Definition	Value	Unit
m	Mass	31	Kg
J	Inertia	0.6621	Kg m2
r	Wheel radius	0.11	m
c	Robot width	0.31	m
g	Gravity	9.8	N/ms2

shows the estimated physical parameters of the mobile robot.

4.2. Numerical Simulation

This section presents the numerical results of the simulations developed for a bilateral teleoperation system to test the validity of the theoretical analysis derived, where the leader and the follower are represented by a robot with two degrees of freedom and a skid-steering mobile robot, respectively.

Figure 4 presents the block diagram of the proposed bilateral teleoperation system. The variables ${\mathit{\tau }}_h$, ${\mathit{\tau }}_m$, ${\mathit{u}}_s$, and ${\mathit{f}}_e$ and the delays $d_1$ and $d_2$ represent the core elements of the control and communication flow between the leader and follower. Feedback consistency is ensured by the estimator, even in the presence of time-varying delays.

We perform three tests called A, B, and C. In each test, the following considerations are taken into account: The total simulation time is 100 [s]. To evaluate the dynamic behavior of the mobile robot against different surfaces, the Coulomb friction coefficient $u_c$ and the viscous friction coefficient $u_v$ are parameterized, as shown in

Table 4. Parameters of terrain friction in simulations.

	t = (0–50) [s]	t = (50–100) [s]
$m{u}_c$	0.7	0.4
$m{u}_v$	0.6	0.3

, to compute $\mathbf{F}$ (5) and $f_{e_y}$ (6), as in [12]. These parameters model the robot–ground interaction that affects the lateral sliding velocity $v_y$. Furthermore, the initial values for simulations A and B are arbitrarily set as ${\mathit{q}}_m\left(0\right)={\left[0{\displaystyle \frac{\pi }{2}}\right]}^T$, ${\mathit{z}}_s\left(0\right)={\left[0{\displaystyle \frac{\pi }{2}}\right]}^T$, ${\dot{\mathit{q}}}_m\left(0\right)={\left[00\right]}^T$, and $\dot{{\mathit{z}}_s}\left(0\right)={\left[00\right]}^T$. In addition, the initial values of ${\widehat{W}}_m\left(0\right)=0_{L_N\times n}$ and ${\widehat{W}}_s\left(0\right)=0_{L_N\times n}$ are selected for the adaptation of the neural network and $V_m$ is a matrix given by constant values randomly selected in the range $[-1,1]$, the same as $V_s$. We consider the actions of external disturbances and the human operator as follows:

$${\mathit{\tau }}_h\left(t\right)=\left\{\begin{array}{cc}\hfill {[4 \pi /4]}^T\left[\mathrm{Nm}\right],& \mathrm{for}t>6\&t<10\hfill \\ \hfill {[-4 -\pi /8]}^T\left[\mathrm{Nm}\right],& \mathrm{for}t>30\&t<34\hfill \\ \hfill {[1 \pi /6]}^T\left[\mathrm{Nm}\right],& \mathrm{for}t>60\&t<64\hfill \\ \hfill {[0 0]}^T\left[\mathrm{Nm}\right],& \mathrm{for}\mathrm{otherwise}\hfill \end{array}\right.,$$

(39)

$${\mathit{f}}_e\left(t\right)=\left\{\begin{array}{cc}\hfill {[0.1 0.2]}^T\left[\mathrm{Nm}\right],& \mathrm{for}t>10\&t<12\hfill \\ \hfill {[0.1 0.1]}^T\left[\mathrm{Nm}\right],& \mathrm{for}t>40\&t<42\hfill \\ \hfill {[0.2 0.1]}^T\left[\mathrm{Nm}\right],& \mathrm{for}t>80\&t<82\hfill \\ \hfill {[0 0]}^T\left[\mathrm{Nm}\right],& \mathrm{for}\mathrm{otherwise}\hfill \end{array}\right.,$$

(40)

The gain selection for the teleoperation system is empirically calibrated for the nondelayed case and is shown in

Table 5. Selection of control parameters.

Leader		Follower
Gain	Value	Gain	Value
$K_{pm}$	diag{1.8,1.8}	$K_{ps}$	diag{30,30}
${\Lambda }_m$	diag{1,1}	${\Lambda }_s$	diag{1,1}
${\alpha }_m$	0.25(A)/Delay-based(B)	${\alpha }_s$	0.125(A)/Delay-based(B)
${\delta }_m$	diag{0.002,0.002}	${\delta }_s$	diag{0.001,0.001}
${\Gamma }_m$	0.1 ${\mathrm{I}}_{L_N\times L_N}$	${\Gamma }_s$	5 ${\mathrm{I}}_{L_N\times L_N}$
${\eta }_m$	5.5	${\eta }_s$	0.5
$L_N$	10

. In addition, the delays $s_0$, $s_1$, and $s_2$ described in

Table 6. Delay value for the A and B tests.

	${\mathbf{s}}_0$	${\mathbf{s}}_1$	${\mathbf{s}}_2$
$d_1$	0	0.1	0.8
$d_2$	0	0.8	0.4

are applied in the A and B tests.

4.2.1. Test A

In this test, fixed damping values $({\alpha }_m,{\alpha }_s)$ are empirically calibrated to obtain a trade-off between an adequate transient response and a moderate error for the nondelayed system. Figure 5 presents the simulation results over the first 100 [s], showing the time evolution of the leader position and follower velocity under the proposed controller for each time delay scenario considered. As is well-known in the state of the art, performance degrades as the time delay increases.

Also in Figure 5, the values of ${\mathit{r}}_m$ and ${\mathit{r}}_s$ defined in (10) and (11) are shown, as well as the control actions and velocities of the leader and follower. The neural network control helps to dynamically compensate the ground–robot interaction, but using the neural network with insufficient damping produces a poor response as the time delay increases.

4.2.2. Test B

Note that, for this scenario, the same general setting as used in the previous simulation is considered. However, instead of fixed damping, theoretical damping coefficients are applied based on the conditions achieved in (35).

Figure 6 presents the results of the numerical simulation, where it is possible to appreciate a convergent response of the coordination errors, as well as ${\mathit{r}}_m$ and ${\mathit{r}}_s$, to zero when the forces of the human operator and/or external disturbances are kept constant. Comparing Figure 5 and Figure 6 through visual inspection, we can observe that the fluctuations in the velocities, the control actions of the leader and follower, and ${\mathit{r}}_m$ and ${\mathit{r}}_s$ are qualitatively smaller than in test A, including a bigger difference as the time delay is greater, but at the cost of increasing the task completion time (lower average velocity). On the other hand, the evolution of the matrices of the neural networks is shown in Figure 7, visualizing the online adaptation capability to partially compensate the time-varying delay and ground–robot interaction.

4.2.3. Test C

Figure 8 shows the Root Mean Square (RMS) values obtained from multiple simulations of the bilateral teleoperation system considering the different communication delay conditions shown in

Table 7. Delay value for test C.

	${\mathbf{s}}_0$	${\mathbf{s}}_1$	${\mathbf{s}}_2$	${\mathbf{s}}_3$	${\mathbf{s}}_4$	${\mathbf{s}}_5$	${\mathbf{s}}_6$
$d_1$	0	0.25	0.5	0.75	0.75	1	1.15
$d_2$	0	0.2	0.03	0.5	1	1	1

. Three RMS metrics are shown: the joint velocity of the leader manipulator ${\dot{\mathit{q}}}_m$, the filtered acceleration of the mobile robot $\dot{{\mathit{z}}_s}$, and the velocity tracking error $\mathit{e}$ defined by $\mathit{e}\left(t\right)=k_g{\mathit{q}}_m\left(t\right)-{\mathit{z}}_s\left(t\right)$. The last value reflects in a direct mode as the time delay degrades the bilateral coordination of the teleoperation system. The simulations are performed using the delay levels given in Table 6 and under two damping configurations: test A (fixed damping) and test B (theoretical damping based on Lyapunov analysis). In addition, several arbitrary initial conditions are considered for both the initial position of the leader and the initial velocity of the follower.

Figure 8A shows the RMS of ${\dot{\mathit{q}}}_m$, illustrating the effect of the time delay on the system’s behavior. In the no-delay scenario ($s_0$), both control strategies perform similarly. However, as the delay increases, the configuration using Lyapunov-based theoretical damping (Setting B) leads to a significant reduction in the leader’s velocity compared to the empirically tuned strategy (Setting A). This improvement is attributed to the incorporation of a delay-dependent damping term, computed within the proposed neural network control framework. Additionally, Figure 8B shows the RMS of $\dot{\mathit{z}}$, which corresponds to the filtered acceleration of the mobile robot. A similar trend is observed, with Setting B consistently yielding lower RMS values across different latency conditions.

Additionally, Figure 8C displays the RMS of the velocity tracking error $\mathit{e}$. Setting B achieves lower coordination errors than Setting A for all evaluated delay values. Although the improvement on the error of linear velocity is small, the reduction in the tracking error of angular velocity is significative, and it increases as the communication delay becomes larger, reaching 80% in the worst case evaluated. This outcome is particularly relevant for the delayed teleoperation of skid-steering mobile robots, although it comes at the cost of a longer task completion time.

5. Conclusions

This work presented a neural network-based control strategy for the bilateral teleoperation of a skid-steering mobile robot. The approach leverages neural networks to jointly compensate for communication time delays and system dynamics, including leader, follower, and ground–robot interaction, through controlled damping injection. The proposed method improves robustness and adaptability in the presence of communication delays and model mismatches between the leader and follower systems.

The analytical results provided give design guidelines for selecting the damping coefficient as a function of communication delay within the neural control framework. The simulation results demonstrated that this delay-sensitive tuning leads to lower coordination errors compared to fixed-gain approaches, in accordance with the expected outcomes. The difference in the tracking error using the parameters laid out by the Lyapunov analysis with respect to the empirical fixed adjustment increases as the time delay increases. In the tracking error, the RMS value of the angular velocity is 80% lower if the proposed setting (Setting B) is applied in the worst tested scenario, $s_6$.

It is important to note that, in this practical implementation, real-time buffering is not required since the input vectors are composed of signals that are already available at the local and remote sites at the corresponding time steps. Therefore, no additional memory management or time-alignment mechanisms are needed during execution. However, data loss or large delays can occur. For brief communication dropouts, a local estimator such as a Kalman filter can be employed to temporarily reconstruct the missing information. However, if the communication loss exceeds a predefined safety threshold, the robot is programmed to stop its motion and alert the human operator to avoid executing further commands under uncertain conditions. The current simulation-based results are a valuable preliminary step that provides an insight into the control strategy’s behavior and supports its potential for future experimental evaluation, which should include testing in unstructured environments with both static and dynamic obstacles to further assess the system’s real-world applicability.