Precision Tracking of Industrial Manipulators via Adaptive Nonsingular Fixed-Time Sliding Mode Control

Abstract

This paper presents a novel adaptive fixed-time sliding mode control (AFxTSMC) framework for industrial manipulators. The proposed adaptive reaching law (ARL) enables rapid and stable gain reduction by leveraging the current parameter values to maintain positivity and prevent sign reversals, thereby reducing chattering. Additionally, the ARL guarantees fixed-time convergence. A singularity-free fixed-time sliding function (SF-FxTSF) ensures fast, robust, and singularity-free convergence. To enhance robustness, a modified third-order sliding mode observer (TOSMO) is integrated into the control framework. This observer estimates both internal uncertainties and external disturbances with improved estimation speed, enabling effective compensation while maintaining convergence performance. A Lyapunov-based analysis rigorously confirms the stability of the proposed method. Simulations of the SAMSUNG FARA AT2 manipulator indicate superior tracking accuracy, faster convergence, and smoother control performance compared to the three state-of-the-art methods. These results underscore the proposed method’s advantages as a robust, scalable, and high-performance control solution for industrial robotic systems.

1. Introduction

Robotic manipulators have undergone remarkable development, leading to increased accuracy and reliability across a wide range of sectors. In manufacturing, aerospace, and automotive industries, they are crucial for executing precise and repetitive operations. In specialized domains such as biotechnology, nanotechnology, and medical applications, they enable highly delicate procedures, including targeted drug administration and minimally invasive surgeries. Moreover, these systems are essential in environments that pose risks to human safety, such as outer space missions and the handling of hazardous materials in nuclear facilities. Collaborative robots, or cobots, are also gaining popularity for supporting human workers in tasks like product assembly and packaging, resulting in greater operational efficiency. In agriculture and logistics, robotic manipulators contribute to automation by handling activities like crop harvesting, planting, sorting, and delivery, thereby improving output and minimizing losses. With continuous advances in control methods, hardware systems, and sensor technologies, these systems are becoming increasingly efficient, reliable, and adaptive. Consequently, ongoing research focuses on enhancing control algorithms to meet the growing demand for more capable and versatile robotic solutions in real-world applications [1,2].

Sliding mode control (SMC) is a well-established control algorithm known for its simplicity and robust performance in the presence of uncertainties and perturbations [3]. Consequently, it has been extensively applied in numerous real-world scenarios, including quadrotor vehicles, photovoltaic battery systems, chemical chaotic reactors, and robot manipulators [4]. To handle uncertainties and perturbations, SMC employs tunable sliding parameters that drive the system states toward a predefined sliding function. Although using large values of these parameters enables the system states to approach the sliding function rapidly, it often leads to significant chattering [5], which can cause issues such as oscillations in the robot manipulator’s end effector. Reducing this chattering effect remains a challenging issue in SMC. To mitigate the chattering phenomenon in SMC control inputs, numerous enhanced control algorithms based on SMC have been proposed, including adaptive SMC (ASMC), fuzzy SMC [6], neuro-fuzzy SMC [7], and others.

ASMC employs adaptively tunable sliding gains within the reaching law to suppress chattering and enhance convergence speed [8]. With appropriately designed adaptation laws, ASMC can significantly enhance system performance while minimizing chattering. To ensure convergence and stability of the control loop, the adaptive gains must be sufficiently large to cover the full uncertainty bound while remaining strictly positive. Typically, the adaptive law consists of two components: one for increasing the adaptive parameters and another for decreasing them [9]. The increasing component is usually based on a proportional term of the sliding variable, while the decreasing component often involves an inverse term of the sliding variable. However, when the sliding variable approaches zero, this design can result in an excessive reduction of the adaptive parameters, potentially causing them to become negative—violating the positivity requirement in SMC. Moreover, a reduction in adaptive parameters may lead to increased tracking errors and reduced robustness, thereby degrading control performance. Conversely, if the sliding parameters are not decreased appropriately, chattering may persist. Increasing the control gains can enhance the system’s response speed and accelerate convergence. However, excessively high gains may surpass the physical limitations of actuators and lead to chattering—undesirable high-frequency oscillations that generate mechanical stress, signal disturbances, and potential damage to hardware components. To overcome this issue, [9] proposed an adaptive law that combines a proportional term of the sliding function with an additional exponential component to accelerate gain reduction. Although this method helps prevent the adaptive parameters from becoming excessively negative, it still does not guarantee their positivity, which remains a critical concern in ASMC design. An alternative strategy such as [10] aimed at addressing the challenges of chattering and slow adaptation involves the use of ASMCs, which are designed to achieve fast response and accurate control within a finite time. Despite their advantages, these methods still face unresolved limitations. For instance, chattering can persist due to the use of switching gains that do not decrease over time. Additionally, the trade-off between control performance and chattering suppression remains suboptimal, as it often relies on conservative estimates of system uncertainties and external disturbances. Therefore, adaptive laws in SMC design should be carefully constructed to reduce chattering and improve convergence while simultaneously preserving tracking accuracy and ensuring the positivity of adaptive parameters.

Although ASMC provides clear benefits such as reduced chattering, a simple structure, and wide applicability, many existing implementations continue to utilize linear sliding functions (LSFs) because of their simple structure and practical simplicity. However, LSFs tend to lose control effectiveness as the tracking error becomes small. This reduction in corrective influence near the target trajectory slows down the system’s response and ensures only gradual, asymptotic convergence instead of achieving finite time convergence. To resolve these issues, exponential components are added to LSFs, producing nonlinear sliding functions (NSFs) that accelerate convergence as errors diminish and ensure finite-time stability, particularly within terminal sliding mode control (TSMC) approaches [11]. Accordingly, adaptive-based TSMC has also been introduced to enhance the performance of ASMC [12]. A well-known drawback of traditional TSMC is the occurrence of singularities near zero error, which can cause the control input to diverge [13]. To overcome this obstacle, nonsingular TSMC (NTSMC) enhances the NFS to eliminate singular behavior while retaining finite-time convergence [14]. Nevertheless, NTSMC may exhibit slow convergence when the system states begin far from the desired values. To enhance response speed and improve robustness, fast TSMC (FTSMC) techniques have been proposed and have demonstrated effective performance in nonlinear system and robotic applications [15,16]. Subsequent advancements, including continuous, fast, and fractional-order versions of NTSMC, have further improved robustness, adaptability, and chattering attenuation [17,18,19]. Despite these advancements, a fundamental limitation of control methods based on TSMC persists: their convergence time inherently depends on initial conditions, which can result in slow responses when initial errors are large [20,21]. To eliminate the dependence of the settling time on the initial states, fixed-time control (FxTC) algorithms were introduced [22]. Thanks to this approach, the state variables of the system converge to the designed manifold within a predefined bound on the settling time, irrespective of the initial conditions and entirely determined by the controller design parameters, thereby offering predictable and tunable performance.

By combining the advantages of fixed-time control (FxTC) theory with the inherent robustness of SMC, fixed-time SMC (FxTSMC) was proposed, making it a widely adopted method for managing nonlinear systems [23,24,25]. These frameworks have proven particularly effective in robotic manipulator control, where reliability and fast, predictable convergence are critical. Building upon fixed-time stability theory, several FxTSMC schemes have been proposed for robot manipulators operating under such challenging conditions [26,27]. These advancements in FxTSMC have significantly contributed to the development of time-constrained tracking strategies. However, due to the inherent nature of SMC, chattering remains a persistent challenge, even within FxTSMC frameworks. To enhance convergence speed and improve applicability in handling uncertainties, nonlinearities, and external disturbances, adaptive fixed-time sliding mode control (AFxTSMC) schemes have been proposed [28]. Moreover, integrating adaptive mechanisms with FxTSMC offers a promising approach to reducing chattering while maintaining fast and predictable convergence. Despite substantial advancements in FxTC, its practical implementation in robotic manipulator tracking control still faces obstacles. One major challenge is the difficulty in handling uncertain dynamics and external perturbations, particularly in the presence of singularities, algebraic loop issues, or when an exact system model is unavailable. While FxTSMC provides strong robustness, its performance can degrade under rapidly varying uncertainties or external inputs. Moreover, its effectiveness is often limited when the system dynamics are not well known or are difficult to model accurately. To overcome these challenges, several estimation and observer-based strategies have been introduced [29,30]. These include methods based on time-delay estimation (TDE) techniques, such as adaptive time-delay control (ATDC) [31], and other approaches [26,32], as well as observer frameworks, including disturbance observers (DOs) [33], third-order sliding mode observers (TOSMOs) [34,35,36], fixed-time disturbance observers (FxTDOs) [37], fuzzy logic observers [38], and neural network observers [39]. Within these frameworks, the observer is responsible for approximating uncertain system dynamics and external perturbations, whereas SMC addresses the residual discrepancies [40]. Consequently, the approximation error from the observer can only be mitigated by selecting an appropriate sliding gain, which directly influences control precision and robustness while also helping to produce smoother control inputs. Fuzzy logic and neural network observers are particularly effective in handling complex and uncertain environments; however, they typically involve higher computational complexity and often lack guaranteed finite-time convergence [41]. FxTDOs provide convergence in fixed time but usually require acceleration measurements, which are difficult to obtain in practice [37,42]. Among the aforementioned observers, TOSMO is considered superior due to its ability to achieve finite-time convergence using only position measurements [43]. Nevertheless, its convergence speed still requires improvement to prevent potential delays in the control system.

In light of the challenges posed by system uncertainties and external disturbances, as well as the stringent requirements for tracking accuracy, there is a clear need for a fixed-time tracking control framework that ensures effectiveness, simplicity, and robustness. To meet this need, the present study proposes a novel AFxTSMC method specifically designed for robotic manipulators. The primary contributions of this work are summarized as follows:

• An adaptive law is developed to ensure rapid and stable gain reduction by utilizing the current parameter values, maintaining positivity, and preventing sign reversals. This approach effectively reduces chattering and guarantees fixed-time convergence, which is not ensured in the study [9].
• An SF-FxTSF is designed to ensure fast fixed-time convergence, robust, and singularity-free convergence.
• A modified TOSMO is integrated into the control framework to estimate both internal uncertainties and external disturbances with improved estimation speed, thereby enabling effective compensation.
• A Lyapunov-based analysis is conducted to rigorously verify the stability of the proposed AFxTSMC.
• The proposed method is evaluated via simulations conducted on the SAMSUNG FARA AT2 robot. The results highlight its enhanced capability in terms of precise trajectory tracking, rapid response, and smooth control effort, outperforming the ATDC, FTSMC, and NTSMC techniques.

The structure of the paper is as follows. In Section 2, the system dynamics, notational conventions, and necessary background are presented. Section 3 outlines the development of the proposed AFxTSMC. Section 4 discusses the simulation studies and provides a comparative performance evaluation. Concluding remarks and suggestions for future work are provided in Section 5.

2. Preliminaries and Formulation

Let $\mathit{x}\in {\mathbb{R}}^n$ be a vector, where $x_i$ denotes its i-th element.

The following functions are defined as

$$\begin{array}{cc}\hfill \mathrm{sign}\left(\mathit{x}\right)& ={[\mathrm{sign}\left(x_1\right),\dots ,\mathrm{sign}\left(x_n\right)]}^T,\hfill \\ \hfill {\lceil x_i\rfloor }^a& ={\left|x_i\right|}^a\mathrm{sign}\left(x_i\right),\hfill \\ \hfill {\lceil \mathit{x}\rfloor }^a& ={\left[\right|x_1|}^a\mathrm{sign}\left(x_1\right),\dots ,|x_n{{|}^a\mathrm{sign}\left(x_n\right)]}^T,\hfill \\ \hfill {\lceil \mathit{x}\rfloor }^{\mathit{a}}& ={\left[\right|x_1|}^{a_1}\mathrm{sign}\left(x_1\right),\dots ,|x_n{{|}^{a_n}\mathrm{sign}\left(x_n\right)]}^T,\hfill \end{array}$$

where $\mathit{a}={[a_1,\dots ,a_n]}^T$.

The notation $\parallel \mathit{x}\parallel$ denotes the Euclidean norm.

2.1. Preliminaries

Consider the following system:

$$\dot{y}=f\left(y\left(t\right)\right),y\left(0\right)=y_0,f\left(0\right)=0,y\in \Omega \subset {\mathbb{R}}^n,$$

(1)

where $f:\Omega \times {\mathbb{R}}^{+}\to {\mathbb{R}}^n$ is continuous in an open neighborhood $\Omega$ around the origin.

Definition 1

([44]). The origin of system (1) is finite-time-stable if it is Lyapunov-stable and there exists a function $T\left(y_0\right)>0$ such that ${lim}_{t\to t_0+T\left(y_0\right)}y(t,y_0)=0$, for all $y_0\in \Omega$ and $t_0\ge 0$.

Definition 2

([44]). System (1) exhibits fixed-time stability if it is finite-time-stable and there exists a uniform upper bound $T_{max}>0$ such that the settling time satisfies

$$T\left(y_0\right)\le T_{max},\forall y_0\in \Omega .$$

Lemma 1

([44]). Consider the following scalar system:

$$\dot{y}=-{\psi }_0{\lceil y\rfloor }^{{\mu }_0}-{\kappa }_0{\lceil y\rfloor }^{{\rho }_0},$$

(2)

with constants ${\psi }_0,{\kappa }_0>0$, ${\mu }_0>1$, and $0<{\rho }_0<1$. Then, the system is fixed-time-stable, and the settling time is bounded by

$$T\left(y_0\right)<T_{max}\triangleq {\displaystyle \frac{1}{{\psi }_0({\mu }_0-1)}}+{\displaystyle \frac{1}{{\kappa }_0(1-{\rho }_0)}}.$$

Lemma 2

([45]). Let $V\left(\xi \right):{\mathbb{R}}^n\to {\mathbb{R}}_{+}$ be a positive-definite continuous function for system (1), and suppose

$$\dot{V}\left(\xi \right)\le -{\chi }_1{V}^r\left(\xi \right)-{\chi }_2{V}^q\left(\xi \right)+\phi ,$$

where ${\chi }_1,{\chi }_2>0$, $r>1$, $0<q<1$, and $0<\phi <\infty$. Then, system (1) is said to be practically fixed-time-stable.

Its solution satisfies

$$\underset{t\to T}{lim}\parallel \xi \parallel \le min\left\{{\left({\displaystyle \frac{\phi }{{\chi }_1(1-\kappa )}}\right)}^{{\displaystyle \frac{1}{r}}},{\left({\displaystyle \frac{\phi }{{\chi }_2(1-\kappa )}}\right)}^{{\displaystyle \frac{1}{q}}}\right\},$$

for any $0<\kappa <1$, and the settling time is bounded by

$$T\left({\xi }_0\right)\le {\displaystyle \frac{1}{{\chi }_1\kappa (r-1)}}+{\displaystyle \frac{1}{{\chi }_2\kappa (1-q)}}.$$

Lemma 3

([46]). Consider a third-order sliding mode system of the form

$$\left\{\begin{array}{cc}\hfill {\dot{y}}_1& =-{\pi }_1{\left|y_1\right|}^{2/3}+y_2,\hfill \\ \hfill {\dot{y}}_2& =-{\pi }_2{\left|y_1\right|}^{1/3}+y_3,\hfill \\ \hfill {\dot{y}}_3& =-{\pi }_3\mathrm{sign}\left(y_1\right)+\delta \left(t\right),\hfill \end{array}\right.$$

where ${\pi }_1,{\pi }_2,{\pi }_3>0$ are positive gains, and $\delta \left(t\right)$ is a bounded perturbation satisfying $\left|\delta \left(t\right)\right|\le \overline{\delta }$. Then, there exists a finite time $T>0$, depending on the initial conditions and observer gains, such that $y_1\left(t\right)=y_2\left(t\right)=y_3\left(t\right)=0$ for all $t\ge T$.

2.2. Overview of Robot Manipulator Dynamic Modeling

The n-DOF robot manipulator dynamics, including uncertainties and external disturbances, are expressed as

$$\mathit{W}\left(\mathit{p}\right)\ddot{\mathit{p}}+\mathit{Q}(\mathit{p},\dot{\mathit{p}})\dot{\mathit{p}}+\mathit{g}\left(\mathit{p}\right)+{\mathit{f}}_r\left(\dot{\mathit{p}}\right)=\mathit{\tau }-{\mathit{\tau }}_d\left(t\right),$$

(3)

where $\mathit{p}\in {\mathbb{R}}^n$ is the joint position vector, $\mathit{W}\left(\mathit{p}\right)\in {\mathbb{R}}^{n\times n}$ is the inertia matrix, $\mathit{Q}(\mathit{p},\dot{\mathit{p}})\in {\mathbb{R}}^{n\times n}$ is the Coriolis and centrifugal matrix, $\mathit{g}\left(\mathit{p}\right)\in {\mathbb{R}}^n$ is the gravity vector, ${\mathit{f}}_r\left(\dot{\mathit{p}}\right)\in {\mathbb{R}}^n$ is the friction vector, $\mathit{\tau }\in {\mathbb{R}}^n$ is the torque signal, and ${\mathit{\tau }}_d\left(t\right)\in {\mathbb{R}}^n$ denotes the unknown disturbance.

In practice, the dynamics of manipulators involve uncertain terms and nonlinearities, such as wear, Coulomb friction, and varying payloads. To capture these uncertainties, the system components are represented as $\mathit{W}\left(\mathit{p}\right)=\widehat{\mathit{W}}\left(\mathit{p}\right)+\delta \mathit{W}\left(\mathit{p}\right)$, $\mathit{Q}(\mathit{p},\dot{\mathit{p}})=\widehat{\mathit{Q}}(\mathit{p},\dot{\mathit{p}})+\delta \mathit{Q}(\mathit{p},\dot{\mathit{p}})$, and $\mathit{g}\left(\mathit{p}\right)=\widehat{\mathit{g}}\left(\mathit{p}\right)+\delta \mathit{g}\left(\mathit{p}\right)$, where the hat terms are the nominal models, and the $\delta$ terms represent the modeling uncertainties.

Define the state vector as $\mathit{v}={\left[\begin{array}{cc}{\mathit{v}}_1^T& {\mathit{v}}_2^T\end{array}\right]}^T={\left[\begin{array}{cc}{\mathit{p}}^T& {\dot{\mathit{p}}}^T\end{array}\right]}^T$ and the control input as $\mathit{u}=\mathit{\tau }$. Then, the robot dynamics (3) can be rewritten in the following state-space form:

$$\left\{\begin{array}{cc}\hfill {\dot{\mathit{v}}}_1& ={\mathit{v}}_2,\hfill \\ \hfill {\dot{\mathit{v}}}_2& =\mathit{Z}\left(\mathit{v}\right)\mathit{u}+\mathit{B}\left(\mathit{v}\right)+\mathit{D}(\mathit{v},\delta ,{\mathit{\tau }}_d),\hfill \end{array}\right.$$

(4)

where $\mathit{Z}\left(\mathit{v}\right)={\widehat{\mathit{W}}}^{-1}\left(\mathit{p}\right)$, $\mathit{B}\left(\mathit{v}\right)=-{\widehat{\mathit{W}}}^{-1}\left(\mathit{p}\right)\left(\widehat{\mathit{Q}}(\mathit{p},\dot{\mathit{p}})\dot{\mathit{p}}+\widehat{\mathit{g}}\left(\mathit{p}\right)\right)$, and $\mathit{D}(·)=-{\widehat{\mathit{W}}}^{-1}\left(\mathit{p}\right)({\mathit{f}}_r$$\left(\dot{\mathit{p}}\right)+\delta \mathit{W}\left(\mathit{p}\right)\ddot{\mathit{p}}+\delta \mathit{Q}(\mathit{p},\dot{\mathit{p}})\dot{\mathit{p}}+\delta \mathit{g}\left(\mathit{p}\right)+{\mathit{\tau }}_d)$.

Now, define the tracking errors as

$$\begin{array}{cc}\hfill {\mathit{e}}_1& =\mathit{p}-{\mathit{p}}_d,\hfill \\ \hfill {\mathit{e}}_2& =\dot{\mathit{p}}-{\dot{\mathit{p}}}_d,\hfill \\ \hfill {\dot{\mathit{e}}}_2& =\ddot{\mathit{p}}-{\ddot{\mathit{p}}}_d,\hfill \end{array}$$

(5)

where ${\mathit{p}}_d$, ${\dot{\mathit{p}}}_d$, and ${\ddot{\mathit{p}}}_d$ denote the desired trajectories. Then, the system (4) in terms of the error variables becomes

$$\left\{\begin{array}{cc}\hfill {\dot{\mathit{e}}}_1& ={\mathit{e}}_2,\hfill \\ \hfill {\dot{\mathit{e}}}_2& =\mathit{Z}\left(\mathit{v}\right)\mathit{u}+\mathit{H}\left(\mathit{v}\right)+\mathit{D}(·),\hfill \end{array}\right.$$

(6)

where $\mathit{H}\left(\mathit{v}\right)=\mathit{B}\left(\mathit{v}\right)-{\ddot{\mathit{p}}}_d$ is a smooth nonlinear function, and $\mathit{e}={\left[\begin{array}{cc}{\mathit{e}}_1^T& {\mathit{e}}_2^T\end{array}\right]}^T\in {\mathbb{R}}^{2n}$ is the tracking error vector.

3. Control Method Design

3.1. Design of an Observer for Disturbance Estimation in Robot Manipulators

This section uses a modified TOSMO with the improved estimation speed to estimate both dynamical uncertainties and external disturbances, as described below:

$$\left\{\begin{array}{c}{\dot{\widehat{\mathit{v}}}}_1={\pi }_1{\left[{\overline{\mathit{v}}}_1\right]}^{2/3}+\varrho {\overline{\mathit{v}}}_1+{\widehat{\mathit{v}}}_2\hfill \\ {\dot{\widehat{\mathit{v}}}}_2=\mathit{Z}\left(\mathit{v}\right)\mathit{u}+\mathit{B}\left(\mathit{v}\right)+{\pi }_2{\left[{\overline{\mathit{v}}}_1\right]}^{1/3}+\varrho ({\dot{\widehat{\mathit{v}}}}_1-{\widehat{\mathit{v}}}_2)+\widehat{\mathit{D}}\hfill \\ \dot{\widehat{\mathit{D}}}={\pi }_3\mathrm{sign}\left({\overline{\mathit{v}}}_1\right).\hfill \end{array}\right.$$

(7)

Here, ${\widehat{\mathit{v}}}_1$, ${\widehat{\mathit{v}}}_2$, and $\widehat{\mathit{D}}$ denote the estimated values of ${\mathit{v}}_1$, ${\mathit{v}}_2$, and $\mathit{D}(·)$, respectively. The parameters ${\pi }_i$ $(i=1,2,3)$ are observer gains selected according to [46], and $\varrho$ is a positive design constant. By properly choosing $\varrho$, the convergence speed of the observer (7) can be significantly enhanced.

The estimation error dynamics can be obtained by subtracting Equation (7) from Equation (4), resulting in

$$\left\{\begin{array}{c}{\dot{\overline{\mathit{v}}}}_1=-{\pi }_1{\left[{\overline{\mathit{v}}}_1\right]}^{2/3}-\varrho {\overline{\mathit{v}}}_1+{\overline{\mathit{v}}}_2\hfill \\ {\dot{\overline{\mathit{v}}}}_2=-{\pi }_2{\left[{\overline{\mathit{v}}}_1\right]}^{1/3}-\varrho {\pi }_1{\left[{\overline{\mathit{v}}}_1\right]}^{2/3}-{\varrho }^2{\overline{\mathit{v}}}_1+\mathit{D}(·)-\widehat{\mathit{D}}\hfill \\ \dot{\widehat{\mathit{D}}}={\pi }_3\mathrm{sign}\left({\overline{\mathit{v}}}_1\right),\hfill \end{array}\right.$$

(8)

where ${\overline{\mathit{v}}}_1={\mathit{v}}_1-{\widehat{\mathit{v}}}_1$, and ${\overline{\mathit{v}}}_2={\mathit{v}}_2-{\widehat{\mathit{v}}}_2$.

Define $\mathit{X}={\overline{\mathit{v}}}_2-\varrho {\overline{\mathit{v}}}_1$, and using Equation (8), we obtain

$$\begin{array}{cc}\hfill \dot{\mathit{X}}& ={\dot{\overline{\mathit{v}}}}_2-\varrho {\dot{\overline{\mathit{v}}}}_1\hfill \\ \hfill & =-{\pi }_2{\left[{\overline{\mathit{v}}}_1\right]}^{1/3}-\varrho {\pi }_1{\left[{\overline{\mathit{v}}}_1\right]}^{2/3}-{\varrho }^2{\overline{\mathit{v}}}_1+\mathit{D}(·)-\widehat{\mathit{D}}-\varrho \left(-{\pi }_1{\left[{\overline{\mathit{v}}}_1\right]}^{2/3}-\varrho {\overline{\mathit{v}}}_1+{\overline{\mathit{v}}}_2\right)\hfill \\ \hfill & =-{\pi }_2{\left[{\overline{\mathit{v}}}_1\right]}^{1/3}+\mathit{D}(·)-\widehat{\mathit{D}}-\varrho {\overline{\mathit{v}}}_2.\hfill \end{array}$$

(9)

By utilizing Equation (9), the estimation error dynamics in Equation (8) can be reformulated into the following structure resembling a TOSMO:

$$\left\{\begin{array}{c}{\dot{\overline{\mathit{v}}}}_1=-{\pi }_1{\left[{\overline{\mathit{v}}}_1\right]}^{2/3}+\mathit{X}\hfill \\ \dot{\mathit{X}}=-{\pi }_2{\left[{\overline{\mathit{v}}}_1\right]}^{1/3}+\mathit{P}\hfill \\ \dot{\mathit{P}}=-{\pi }_3\mathrm{sign}\left({\overline{\mathit{v}}}_1\right)+\dot{\mathbf{\Delta }}.\hfill \end{array}\right.$$

(10)

Here, $\mathbf{\Delta }=\mathit{D}(·)-\varrho {\overline{\mathit{v}}}_2$ represents the combined disturbance term, and $\mathit{P}=\mathbf{\Delta }-\widehat{\mathit{D}}$ denotes the disturbance estimation error. It is assumed that the derivative of $\mathbf{\Delta }$ is bounded by a known positive constant, i.e., $\left|\dot{\mathbf{\Delta }}\right|\le \overline{\mathbf{\Delta }}$.

It is observed that Equation (10) shares the same structure as the system presented in Lemma 3. Therefore, by selecting an appropriate Lyapunov candidate and applying the standard finite-time stability analysis as detailed in [46], it follows that the system in Equation (10) guarantees the finite-time convergence of the variables ${\overline{\mathit{v}}}_1$, $\mathit{X}$, and $\mathit{P}$ to the origin. Consequently, the proposed observer in Equation (7) ensures that the estimation errors converge to zero within a finite time interval.

3.2. Novel Singularity-Free Fixed-Time Sliding Function

To ensure fixed-time convergence while avoiding singularities in the control input, the following novel SF-FxTSF is proposed:

$$\mathit{s}={\mathit{e}}_1+{\displaystyle \frac{1}{{\kappa }_1^{1/{\rho }_1}}}{\lceil {\psi }_1{\lceil {\mathit{e}}_1\rfloor }^{{\mu }_1}+{\dot{\mathit{e}}}_1\rfloor }^{1/{\rho }_1},$$

(11)

where the parameters satisfy ${\psi }_1>0$, ${\kappa }_1>0$, ${\mu }_1>1$, and ${\displaystyle \frac{1}{2}}<{\rho }_1<1$.

When the SF-FxTSF in (11) reaches zero, i.e., $\mathit{s}=\mathbf{0}$, the resulting error dynamics are given by

$${\dot{\mathit{e}}}_1=-{\psi }_1{\lceil {\mathit{e}}_1\rfloor }^{{\mu }_1}-{\kappa }_1{\lceil {\mathit{e}}_1\rfloor }^{{\rho }_1}.$$

(12)

Based on Lemma 1, the function defined in Equation (11) exhibits fixed-time convergence properties analogous to those in the SMC method of [47]. Consequently, the trajectories of the system described by Equation (12) reach the origin within a fixed duration that does not depend on initial states. The maximum settling time can be expressed as follows:

$$T\left({\mathit{e}}_{10}\right)<T_{max}\triangleq {\displaystyle \frac{1}{{\psi }_1({\mu }_1-1)}}+{\displaystyle \frac{1}{{\kappa }_1(1-{\rho }_1)}},$$

where ${\mathit{e}}_{10}={\mathit{e}}_1\left(0\right)$ denotes the initial tracking error.

3.3. Design of the Proposed Control Method

Let us define an auxiliary variable ${\rm Y}$ as follows: ${\rm Y}={\psi }_1{\lceil {\mathit{e}}_1\rfloor }^{{\mu }_1}+{\dot{\mathit{e}}}_1$, which implies that ${|{\rm Y}|}^{\frac{1}{{\rho }_1}-1}={\left|{\psi }_1{\lceil {\mathit{e}}_1\rfloor }^{{\mu }_1}+{\dot{\mathit{e}}}_1\right|}^{\frac{1}{{\rho }_1}-1}$.

Taking the time derivative of Equation (11) yields the dynamics of $\mathit{s}$ as

$$\dot{\mathit{s}}={\dot{\mathit{e}}}_1+\mathit{S}\left({\psi }_1{\mu }_1{\left|{\mathit{e}}_1\right|}^{{\mu }_1-1}{\dot{\mathit{e}}}_1+{\ddot{\mathit{e}}}_1\right),$$

(13)

where $\mathit{S}=\mathrm{diag}\left\{S_i\right\}$, with each diagonal entry defined as

$$S_i={\displaystyle \frac{1}{{\kappa }_1^{1/{\rho }_1}}}·{\displaystyle \frac{1}{{\rho }_1}}{\left|{{\rm Y}}_i\right|}^{\frac{1}{{\rho }_1}-1},i=1,\dots ,n.$$

This derivative forms the basis for designing the control input that guarantees fixed-time convergence of the closed-loop system.

Substituting the system dynamics from Equation (6) into Equation (13), we obtain

$$\begin{array}{cc}\hfill \dot{\mathit{s}}=& {\dot{\mathit{e}}}_1+\mathit{S}\left({\psi }_1{\mu }_1{\left|{\mathit{e}}_1\right|}^{{\mu }_1-1}{\dot{\mathit{e}}}_1+\mathit{Z}\left(\mathit{v}\right)\mathit{u}+\mathit{H}\left(\mathit{v}\right)+\mathit{D}(·)\right).\hfill \end{array}$$

(14)

Using the derivative expression in Equation (14), the control input is designed as

$$\mathit{u}={\mathit{u}}_{eq}+{\mathit{u}}_r,$$

(15)

where the equivalent control law ${\mathit{u}}_{eq}$ is defined by

$$\begin{array}{cc}\hfill {\mathit{u}}_{eq}=& -\mathit{Z}{\left(\mathit{v}\right)}^{-1}\left({\psi }_1{\mu }_1{\left|{\mathit{e}}_1\right|}^{{\mu }_1-1}{\dot{\mathit{e}}}_1+\mathit{H}\left(\mathit{v}\right)+\widehat{\mathit{D}}\right)-\mathit{Z}{\left(\mathit{v}\right)}^{-1}{\mathit{S}}^{-1}{\dot{\mathit{e}}}_1,\hfill \end{array}$$

and the term ${\mathit{u}}_r$ is expressed as

$${\mathit{u}}_r=-\mathit{Z}{\left(\mathit{v}\right)}^{-1}\left({\kappa }_2\mathbf{\Gamma }{\lceil \mathit{s}\rfloor }^{{\rho }_2}+\widehat{\mathit{\theta }}\mathrm{sign}\left(\mathit{s}\right)\right).$$

Here, ${\kappa }_2,{\psi }_2>0$ are design parameters, and $0<{\rho }_2<1$ ensures fixed-time convergence. The gain matrix $\mathbf{\Gamma }\in {\mathbb{R}}^{n\times n}$ is defined as $\mathbf{\Gamma }=\mathrm{diag}\left\{1+{\psi }_2^2{\left|s_i\right|}^{2(1-{\rho }_2)}\right\}$, and the diagonal matrix $\widehat{\mathit{\theta }}=\mathrm{diag}\left\{{\theta }_i\right\}$ contains adaptive or predefined estimates to counteract bounded uncertainties.

The structure of the designed AFxTSMC is shown in the block diagram in Figure 1.

Remark 1.

The equivalent control term ${\mathit{u}}_{eq}$, constructed from Equation (15), includes a nonlinear expression of the form ${\left|{\rm Y}\right|}^{1-\frac{1}{{\rho }_1}}{\dot{\mathit{e}}}_1$. Importantly, this term does not introduce singularities into the control input. In particular, consider the case when ${\mathit{e}}_1=\mathbf{0}$ but ${\dot{\mathit{e}}}_1\ne \mathbf{0}$. In this situation, we observe that ${\left|{\dot{\mathit{e}}}_1\right|}^{1-\frac{1}{{\rho }_1}}{\dot{\mathit{e}}}_1\ge {\dot{\mathit{e}}}_1^{2-\frac{1}{{\rho }_1}}$, and since the exponent $2-{\displaystyle \frac{1}{{\rho }_1}}$ remains strictly positive for all ${\rho }_1\in (0,1)$, the term remains bounded and continuous. Therefore, the proposed control law avoids singularities and ensures well-defined behavior across its entire domain.

Theorem 1.

For the robotic system described by Equation (3), using the disturbance estimate $\widehat{\mathit{D}}$ from the observer in Equation (7), the control strategy defined in Equation (15), which is based on the SF-FxTSF in Equation (11) and the adaptive control term ${\mathit{u}}_r$, ensures convergence of the tracking error and guarantees fixed-time stability of the closed-loop system.

Substituting the control law (15) into the system dynamics (14) yields

$$\dot{\mathit{s}}=-\mathit{S}\left({\kappa }_2\mathbf{\Gamma }{⌈\mathit{s}⌋}^{{\rho }_2}+\widehat{\theta }\mathrm{sign}\left(\mathit{s}\right)-\tilde{\mathit{D}}\right),$$

(16)

where $\tilde{\mathit{D}}=\mathit{D}-\widehat{\mathit{D}}$ denotes the disturbance estimation error.

3.4. Closed-Loop System Stability Analysis

Consider the Lyapunov function for the i-th element as

$$V_1={\displaystyle \frac{1}{2}}{s}_i^2+{\displaystyle \frac{1}{2}}{\lambda }_i{\tilde{\theta }}_i^2,$$

(17)

where $\overline{\theta }$ represents the known upper bound of the disturbance estimation error, satisfying $\left|{\tilde{D}}_i\right|\le \overline{\theta }$.

Differentiating Equation (17) with respect to time gives

$$\begin{array}{cc}\hfill {\dot{V}}_1=& s_i\left[-S_i\left({\kappa }_2{\Gamma }_i{\left|s_i\right|}^{{\rho }_2}sign\left(s_i\right)+{\theta }_{i}sign\left(s_i\right)-{\tilde{D}}_i\right)\right]-{\lambda }_i{S}_i{\tilde{\theta }}_i{\dot{\theta }}_i\hfill \\ \hfill =& S_i\left(s_i{\tilde{D}}_i-{\theta }_i\left|s_i\right|-{\kappa }_2{\Gamma }_i{\left|s_i\right|}^{{\rho }_2+1}\right)-{\lambda }_i{\tilde{\theta }}_i{\dot{\theta }}_i.\hfill \end{array}$$

(18)

The adaptive gain is updated according to

$${\dot{\theta }}_i=\left\{\begin{array}{cc}{\displaystyle \frac{1}{{\lambda }_i}}{S}_i\left(\left|s_i\right|+{\kappa }_3{\left|{\tilde{\theta }}_i\right|}^{{\rho }_3}\mathrm{sign}\left({\tilde{\theta }}_i\right)\right),\hfill & \mathrm{if}\left|s_i\right|\ge {\rho }_{\mathrm{res}},\hfill \\ -{\displaystyle \frac{{\theta }_i}{{\lambda }_i{\rho }_{\mathrm{res}}}}\left(\left|s_i\right|+{\beta }_i{\mathrm{e}}^{-|s_i|}\right),\hfill & \mathrm{if}\left|s_i\right|<{\rho }_{\mathrm{res}},\hfill \end{array}\right.$$

(19)

where ${\rho }_{\mathrm{res}}$ is a positive constant, typically chosen such that $0<{\rho }_{\mathrm{res}}<1$, to satisfy the desired control performance requirements.

Case 1: When $\left|s_i\right|\ge {\rho }_{\mathrm{res}}$, we have

$$\begin{array}{cc}\hfill {\dot{V}}_1\le & S_i\left(\overline{\theta }\left|s_i\right|-{\theta }_i|s_i|-{\kappa }_2{\left|s_i\right|}^{{\rho }_2+1}\right)-S_i\left(\left|{\tilde{\theta }}_i\right|\left|s_i\right|+{\kappa }_3{\left|{\tilde{\theta }}_i\right|}^{{\rho }_3+1}\right)\hfill \\ \hfill =& -S_i{\kappa }_2|s_i{|}^{{\rho }_2+1}-S_i{\kappa }_3{\left|{\tilde{\theta }}_i\right|}^{{\rho }_3+1}.\hfill \end{array}$$

(20)

Applying the inequalities for ${\rho }_3>1$ and $0<{\rho }_2<1$, we obtain

$${\left|{\tilde{\theta }}_i\right|}^{{\rho }_3+1}={\left({\tilde{\theta }}_i^2\right)}^{\frac{{\rho }_3+1}{2}}={\left(\frac{2V_{12}}{{\lambda }_i}\right)}^{\frac{{\rho }_3+1}{2}},$$

(21a)

$${\left|s_i\right|}^{{\rho }_2+1}={\left(s_i^2\right)}^{\frac{{\rho }_2+1}{2}}={\left(2V_{11}\right)}^{\frac{{\rho }_2+1}{2}}.$$

(21b)

Hence, Equation (20) satisfies

$${\dot{V}}_1\le S_i\left(-c_s{V}_1^{\frac{{\rho }_2+1}{2}}-c_{\theta }{V}_1^{\frac{{\rho }_3+1}{2}}+\sigma \right),$$

(22)

where $\sigma =c_s{V}_{12}^{\frac{{\rho }_2+1}{2}}+c_{\theta }{V}_{11}^{\frac{{\rho }_3+1}{2}}$, and the constants are $c_s={\kappa }_2{2}^{\frac{{\rho }_2+1}{2}},c_{\theta }={\kappa }_3{\left(\frac{2}{{\lambda }_i}\right)}^{\frac{{\rho }_3+1}{2}}.$

From Equation (22), we observe that if ${{\rm Y}}_i\ne 0$, then $S_i>0$. Consequently, the state space of $(e_{1i},{\dot{e}}_{1i})$ is divided into two distinct regions as follows:

$${\Omega }_{1i}=\left\{(e_{1i},{\dot{e}}_{1i})\mid S_i\ge 1\right\},{\Omega }_{2i}=\left\{(e_{1i},{\dot{e}}_{1i})\mid S_i<1\right\}.$$

When $S_i\ge 1$, it implies that $s_i$ will ultimately enter and remain within the bounded region $\left|s_i\right|<{\rho }_{\mathrm{res}}$ in a fixed time. The residual set is given by $\left|s_i\right|\le \sqrt{2}min\{{\left(\frac{\sigma }{c_s(1-\kappa )}\right)}^{\frac{2}{{\rho }_2+1}},$${\left(\frac{\sigma }{c_{\theta }(1-\kappa )}\right)}^{\frac{2}{{\rho }_3+1}}\}$, where $0<\kappa <1$, and the corresponding settling time is estimated as

$$T\le {\displaystyle \frac{1}{c_s\kappa \left(1-\frac{{\rho }_2+1}{2}\right)}}+{\displaystyle \frac{1}{c_{\theta }\kappa \left(\frac{{\rho }_3+1}{2}-1\right)}}.$$

(23)

From the Lyapunov function $V_1={\displaystyle \frac{1}{2}}{s}_i^2+{\displaystyle \frac{1}{2}}{\lambda }_i{\tilde{\theta }}_i^2$ and the inequality in Equation (22), we conclude that $V_1$ is bounded for all time. Hence, ${\tilde{\theta }}_i$ is also bounded. Since the true bound $\overline{\theta }$ is finite, the adaptive estimate ${\widehat{\theta }}_i={\tilde{\theta }}_i+\overline{\theta }$ is guaranteed to remain bounded as well.

Upon entering the second region ${\Omega }_{2i}$, where ${{\rm Y}}_i\ne 0$ and $0<S_i<1$, the SF-FxTSF $s_i=0$ for $i=1,\dots ,n$ remains an attracting set, as demonstrated by Equation (22). It is essential to establish that ${{\rm Y}}_i=0$ only occurs at the origin, which can be verified using the methodology from [48].

In conclusion, the SF-FxTSF $s_i=0$, for all $i=1,2,\dots ,n$, is reachable from any initial condition in the phase plane within a finite time not exceeding $T_{r\epsilon }<T_r+\epsilon \left(\tau \right)$ [48], where $\epsilon \left(\tau \right)$ is a small margin of time related to the thickness of the boundary layer. The parameter $\tau$ satisfies the inequality $|{\rm Y}|\ge \tau ={\left({\kappa }_1^{\frac{1}{{\rho }_1}}{\rho }_1\right)}^{\frac{{\rho }_1}{1-{\rho }_1}}.$

Case 2. ${\parallel \mathit{s}\parallel }_{\infty }<{\rho }_{\mathrm{res}}$. From Equation (18), the time derivative of $V_1$ satisfies

$$\begin{array}{cc}\hfill {\dot{V}}_1\le & S_i\left(\overline{\theta }\left|s_i\right|-{\theta }_i\left|s_i\right|-{\kappa }_2{\left|s_i\right|}^{{\rho }_2+1}\right)+S_i{\displaystyle \frac{|{\tilde{\theta }}_i|{\theta }_i}{{\rho }_{\mathrm{res}}}}\left|s_i\right|\hfill \\ \hfill =& -S_i{\kappa }_2{\left|s_i\right|}^{{\rho }_2+1}+S_i\left(\overline{\theta }-{\theta }_i+{\displaystyle \frac{|{\tilde{\theta }}_i|{\theta }_i}{{\rho }_{\mathrm{res}}}}\right)\left|s_i\right|.\hfill \end{array}$$

(24)

Within the region $\left|s_i\right|<{\rho }_{\mathrm{res}}$, Case 2 does not guarantee that ${\dot{V}}_1$ remains non-positive. Although $s_i\left(t\right)$ reaches this region within a fixed time, it may fluctuate around the boundary, potentially leaving and re-entering, due to the lack of assured negative definiteness of ${\dot{V}}_1$ inside the region. However, once $s_i\left(t\right)$ leaves the interval $\left|s_i\right|<{\rho }_{\mathrm{res}}$, Equation (22) ensures that ${\dot{V}}_1$ becomes negative again, which steers the system state back toward the SF-FxTSF.

When $\left|s_i\right|<{\rho }_{\mathrm{res}}$, the Lyapunov function $V_1$ is upper-bounded by

$${\displaystyle \frac{1}{2}}{s}_i^2\le V_1\le {\displaystyle \frac{1}{2}}{s}_i^2+{\displaystyle \frac{1}{2}}{\lambda }_i{\tilde{\theta }}_i^2.$$

(25)

Since $\left|s_i\right|<{\rho }_{\mathrm{res}}$, the upper bound of $V_1$ becomes

$$V_1\le {\displaystyle \frac{1}{2}}{\rho }_{\mathrm{res}}^2+{\displaystyle \frac{1}{2}}{\lambda }_i{\tilde{\theta }}_i^2.$$

(26)

Let ${\overline{\kappa }}_i^{*}=max\left\{{\lambda }_i{\tilde{\theta }}_i^2\right\}$; then,

$$V_1\le {\displaystyle \frac{1}{2}}{\rho }_{\mathrm{res}}^2+{\displaystyle \frac{1}{2}}{\overline{\kappa }}_i^{*}.$$

(27)

Hence, the ultimate bound of the sliding variable for the i-th element within the region $\left|s_i\right|<{\rho }_{\mathrm{res}}$ is given by

$$\left|s_i\right|\le \sqrt{{\rho }_{\mathrm{res}}^2+{\overline{\kappa }}_i^{*}}.$$

(28)

This result confirms that $\mathit{s}\left(t\right)$ exhibits ultimately uniform boundedness (UUB) for $t\le t_{{\rho }_{\mathrm{res}}}$. Moreover, the magnitude of its oscillation in the neighborhood of the SF-FxTSF is limited by the bound presented in Equation (28).

Remark 2.

In the AFxTSMC framework, the adaptive gains ${\theta }_i$—for $i=1,2,\dots ,n$—are regulated based on the magnitude of the SF-FxTSF. Specifically, the gains increase when $\left|s_i\right|\ge {\rho }_{\mathrm{res}}$, and they decrease when $\left|s_i\right|<{\rho }_{\mathrm{res}}$. This mechanism is carefully designed to ensure the positivity of ${\theta }_i$ throughout the control process, which is essential for maintaining stability. The adaptive law continuously tunes the update rate according to the system’s state, ensuring that the positivity of the parameter is maintained.

When $\left|s_i\right|\ge {\rho }_{\mathrm{res}}$, the adaptive gains are updated to increase, facilitating rapid convergence of the tracking error. Once the error enters the residual set, i.e., $\left|s_i\right|<{\rho }_{\mathrm{res}}$, the update rule becomes dominated by the SF-FxTSF magnitude and a corrective feedback term that includes the difference between the current gain and its nominal reference value. In this region, the exponential term ${\beta }_i{\mathrm{e}}^{-|s_i|}$ becomes relatively small, and the adaptation dynamics accelerate the gain reduction, enabling the system to enter the boundary layer $\left|s_i\right|<{\rho }_{\mathrm{res}}$ within a fixed time.

On the other hand, when $\left|s_i\right|<{\rho }_{\mathrm{res}}$ and the SF-FxTSF is close to zero, the control objective shifts toward reducing the adaptive gain to suppress chattering. However, relying solely on the term $|s_i|$ in this region may not suffice to reduce the gains quickly, especially when they are initially large. To address this issue, an exponential decay term is incorporated into the adaptation law. As $|s_i|$ becomes small, the term ${\beta }_i{\mathrm{e}}^{-|s_i|}$ becomes more influential, promoting a faster reduction in the gain. This adjustment improves steady-state performance while avoiding instability.

Moreover, since the exponential term ${\beta }_i{\mathrm{e}}^{-|s_i|}$ is inherently bounded, the adaptation rate is naturally limited. This constraint prevents overly aggressive gain variations, thereby enhancing the controller’s robustness and reducing undesirable oscillations near the SF-FxTSF, in contrast to approaches that do not incorporate an exponential decay term, such as in [9].

4. Simulation

4.1. System Configuration and Simulation Environment

This section presents the simulation results and analysis for the control schemes applied to the 3-DOF SAMSUNG FARA-AT2, Samsung Electronics, Suwon, Republic of Korea industrial manipulator. The model, shown in Figure 2, was developed using SolidWorks2018. The model’s dimensional parameters were configured to match the actual specifications of the FARA-AT2. For precise modeling of both kinematic and dynamic characteristics, the mechanical model was integrated into MATLAB/Simulink2024a via the Simscape Multibody Link toolbox. The equations of motion for the 3-DOF manipulator were derived using established dynamic modeling approaches, as described in [49,50], while specific mechanical attributes were adopted from our earlier research [26]. All simulations were executed in MATLAB/Simulink2024a using the Dormand–Prince (ODE5) solver, with a fixed integration step size of 0.001 s. This setup enables accurate and stable time-domain evaluations. The integrated modeling and simulation environment closely emulates the robot’s physical behavior, providing a dependable platform for evaluating the performance of the proposed control strategies.

The parameters necessary for modeling and dynamic calculations of the FARA-AT2 robot include the physical dimensions, mass, center of mass, and inertia for each of the three links. The link lengths are defined as $l_1=0.15\mathrm{m}$, $l_2=0.255\mathrm{m}$, and $l_3=0.41\mathrm{m}$. Correspondingly, the masses are $m_1=56.5\mathrm{kg}$, $m_2=35.6\mathrm{kg}$, and $m_3=58.9\mathrm{kg}$. The center of mass (CoM) positions, expressed in millimeters relative to each link’s coordinate frame, are given as follows: for Link 1, the CoM is located at $(-98.3,-2.9,-85.4)\mathrm{mm}$; for Link 2 at $(-5.5,0.001,-156.9)\mathrm{mm}$; and for Link 3 at $(54.6,-0.01,80.5)\mathrm{mm}$. The inertia tensors (in $\mathrm{kg}·{\mathrm{m}}^2$) are defined by their diagonal components as $I_{1xx}=0.39$, $I_{1yy}=0.59$, $I_{1zz}=0.56$ for Link 1; $I_{2xx}=0.76$, $I_{2yy}=0.44$, $I_{2zz}=0.39$ for Link 2; and $I_{3xx}=0.22$, $I_{3yy}=1.2$, $I_{3zz}=1.2$ for Link 3. These parameters are essential for accurately modeling the dynamics and mechanical behavior of the robotic system.

The robotic manipulator is required to follow a predefined joint-space reference trajectory specified as $p_{1d}=0.5cos(t/2)-0.5$, $p_{2d}=0.3cos\left(t\right)-0.3$, and $p_{3d}=0.2cos\left(t\right)-0.2$ [rad].

To evaluate the robustness and estimation capabilities of the proposed controller under uncertain conditions, both internal and external disturbances are taken into account using the friction modeling approach presented in [51]. The joint friction effects are modeled as $f_{r1}\left({\dot{p}}_1\right)=0.01\mathrm{sign}\left({\dot{p}}_1\right)+2{\dot{p}}_1$, $f_{r2}\left({\dot{p}}_2\right)=0.01\mathrm{sign}\left({\dot{p}}_2\right)+2{\dot{p}}_2$, and $f_{r3}\left({\dot{p}}_3\right)=0.01\mathrm{sign}\left({\dot{p}}_3\right)+2{\dot{p}}_3$ [N·m].

Moreover, time-varying external disturbances are introduced for each joint as ${\tau }_{d1}=4.5sin\left(t\right)+1.5sin\left(2t\right)$, ${\tau }_{d2}=6.5sin(t-\pi )-1.7cos(2t-{\displaystyle \frac{\pi }{2}})$, and ${\tau }_{d3}=5.5sin\left({\displaystyle \frac{3t}{2}}\right)+3.2cos(2t-{\displaystyle \frac{\pi }{2}})$ [N·m] to simulate dynamic environmental uncertainties.

The uncertain dynamic terms are modeled by reducing their actual values by $15\%$ such that $\widehat{\mathit{W}}\left(\mathit{p}\right)=0.85\mathit{W}\left(\mathit{p}\right)$, $\widehat{\mathit{Q}}(\mathit{p},\dot{\mathit{p}})=0.85\mathit{Q}(\mathit{p},\dot{\mathit{p}})$, and $\widehat{\mathit{g}}\left(\mathit{p}\right)=0.85\mathit{g}\left(\mathit{p}\right)$.

The initial joint positions are set to $\mathit{p}\left(0\right)={[0.2-0.10.15]}^T$.

With the above general setup, the simulation study and performance evaluation were divided into two distinct case studies:

• Example 1: This scenario evaluates the estimation capability of the proposed observer (Observer 3). Its performance is compared with that of the conventional TOSMO (Observer 1) and the fixed-time disturbance observer (FxTDO) (Observer 2), emphasizing differences in convergence speed and estimation accuracy.
• Example 2: This case assesses the tracking performance of four different control schemes. The comparison includes metrics such as tracking accuracy, peak overshoot, and steady-state error, demonstrated through MATLAB-generated plots and quantified using the RMSE metric. The analysis also considers the level of chattering in the control signals for each approach.

To quantify trajectory tracking accuracy, the following metric was employed:

• Root Mean Square Error (RMSE): This measure captures both the transient phase (0–1.5 s) and the steady-state tracking error, computed over the interval from 1.5 s to 30 s.

4.2. Performance Analysis

4.2.1. Analysis of Example 1

To assess the estimation performance of all three observers, they were implemented in conjunction with the conventional SMC, as depicted in Figure 3, to estimate the lumped uncertainties in the robot manipulator.

Observer 1 is defined as [51]

$$\left\{\begin{array}{c}{\dot{\widehat{\mathit{v}}}}_1={\pi }_1{\left[{\overline{\mathit{v}}}_1\right]}^{2/3}+{\widehat{\mathit{v}}}_2\hfill \\ {\dot{\widehat{\mathit{v}}}}_2=\mathit{Z}\left(\mathit{v}\right)\mathit{u}+\mathit{B}\left(\mathit{v}\right)+{\pi }_2{\left[{\overline{\mathit{v}}}_1\right]}^{1/3}+\widehat{\mathit{D}}\hfill \\ \dot{\widehat{\mathit{D}}}={\pi }_3\mathrm{sign}\left({\overline{\mathit{v}}}_1\right).\hfill \end{array}\right.$$

(29)

Observer 2 is defined as [52]

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\overline{\mathit{v}}}_2={\mathit{v}}_2-{\widehat{\mathit{v}}}_2\hfill \\ {\dot{\widehat{\mathit{v}}}}_2=\mathit{Z}\left(\mathit{v}\right)\mathit{u}+\mathit{B}\left(\mathit{v}\right)+\widehat{\mathit{D}}+{\xi }_1{\mathbf{\Psi }}_1\left({\overline{\mathit{v}}}_2\right)\hfill \\ \dot{\widehat{\mathit{D}}}=-{\xi }_2{\mathbf{\Psi }}_2\left({\overline{\mathit{v}}}_2\right).\hfill \end{array}\right.\end{array}$$

(30)

Here, ${\mathit{v}}_2$ is approximated by ${\widehat{\mathit{v}}}_2$. The parameters ${\xi }_1$, ${\xi }_2$, and $\gamma$ denote the observer gains. The functions are defined as ${\mathbf{\Psi }}_1\left({\overline{\mathit{v}}}_2\right)={\left[{\overline{\mathit{v}}}_2\right]}^{1/2}+\gamma {\left[{\overline{\mathit{v}}}_2\right]}^{3/2}$ and ${\mathbf{\Psi }}_2\left({\overline{\mathit{v}}}_2\right)={\displaystyle \frac{1}{2}}{\left[{\overline{\mathit{v}}}_2\right]}^0+2\gamma {\overline{\mathit{v}}}_2+{\displaystyle \frac{3}{2}}{\gamma }^2{\left[{\overline{\mathit{v}}}_2\right]}^2$.

The estimated outputs from each observer were compared simultaneously, as depicted in Figure 4. As shown in Figure 4, all three observers accurately estimated the unknown uncertain components. However, Observer 3 exhibited the fastest convergence to a stable state compared to Observers 1 and 2. Notably, Observer 3 showed no oscillation at the beginning of the convergence phase, which is critical for reducing estimation delays and thereby improving the overall responsiveness of the control system. In contrast, Observer 2 still exhibited some oscillation during this period, while Observer 1 experienced the longest duration of oscillation. It is also noteworthy that both Observers 1 and 3 required only position measurements, whereas Observer 2 required both position and velocity measurements.

Remark 3.

The convergence behaviors of Observer 1 (Equation (29)) and the modified observer (Equation (7)) differ significantly due to their structural formulations. Specifically, Equation (7) introduces an additional positive feedback term $\varrho {\overline{\mathit{v}}}_1$ and a dynamic compensation term $\varrho ({\dot{\widehat{\mathit{v}}}}_1-{\widehat{\mathit{v}}}_2)$, both of which are absent in Equation (29). These modifications are designed to accelerate the decay of the observation error ${\overline{\mathit{v}}}_1$ toward zero. As a result, the observer governed by Equation (7) exhibited a faster convergence rate compared to Equation (29), particularly during the initial transient phase when ${\overline{\mathit{v}}}_1$ was large. This enhanced convergence speed was achieved without compromising finite-time convergence properties. Consequently, the modified observer offers improved estimation performance in terms of both speed and robustness.

4.2.2. Analysis of Example 2

This subsection presents the simulation results evaluating the trajectory tracking performance of the manipulator under four distinct control strategies. The tested methods include A1: FTSMC [53], A2: NTSMC [14], A3: ATDC [31], and A4: the proposed AFxTSMC.

A1: The FTSMC is formulated as

$$\left\{\begin{array}{cc}\hfill \mathit{s}& ={\dot{\mathit{e}}}_1+{\alpha }_1{\mathit{e}}_1+{\gamma }_1{\lceil {\mathit{e}}_1\rfloor }^{{\omega }_1},\hfill \\ \hfill \mathit{u}& =-\mathit{Z}{\left(\mathit{v}\right)}^{-1}\left(\mathit{H}\left(\mathit{v}\right)+\left({\alpha }_2+{\gamma }_1{\omega }_1{\left|{\mathit{e}}_1\right|}^{{\omega }_1-1}\right){\dot{\mathit{e}}}_1+{\Lambda }_1\mathit{s}+{\kappa }_1\mathrm{sign}\left(\mathit{s}\right)\right),\hfill \end{array}\right.$$

(31)

where ${\alpha }_1$, ${\gamma }_1$, ${\Lambda }_1$, and ${\kappa }_1$ are positive gains, and $0<{\omega }_1<1$.

A2: The NTSMC is formulated as

$$\left\{\begin{array}{cc}\hfill \mathit{s}& ={\mathit{e}}_1+{\varpi }_2{(1+{\mathit{e}}_1^2)}^{\frac{j}{l}}arctan\left({\mathit{e}}_1\right){\dot{\mathit{e}}}_1^{\frac{j}{l}},\hfill \\ \hfill \mathit{u}& =\mathit{Z}{\left(\mathit{v}\right)}^{-1}({\mathit{u}}_{eq}+{\mathit{u}}_r),\hfill \\ \hfill {\mathit{u}}_{eq}& =\mathit{H}\left(\mathit{v}\right)+\frac{l}{{\varpi }_{2}j}{(1+{\mathit{e}}_1^2)}^{\frac{j}{l}-1}\left(1+\frac{2j}{l}{\mathit{e}}_{1}arctan\left({\mathit{e}}_1\right){\dot{\mathit{e}}}_1^{2-\frac{j}{l}}\right),\hfill \\ \hfill {\mathit{u}}_r& =-({\Lambda }_2\mathit{s}+{\kappa }_2\mathrm{sign}\left(\mathit{s}\right)).\hfill \end{array}\right.$$

(32)

with $j,l$ positive odd integers satisfying $1<\frac{j}{l}<2$ and ${\varpi }_2$, ${\Lambda }_2$, and ${\kappa }_2$ positive gains.

A3: The ATDC is given by

$$\begin{array}{cc}\hfill {\mathit{u}}_t& =-\overline{\mathit{W}}\left({\widehat{\mathit{N}}}_t+\alpha {\mathit{\varrho }}_{t-\iota }\right)+\overline{\mathit{W}}\left({\ddot{\mathit{p}}}_{dt}+{\mathit{K}}_d{\dot{\mathit{e}}}_{1t}+{\mathit{K}}_p{\mathit{e}}_{1t}\right)\hfill \\ \hfill & +\overline{\mathit{W}}\left({\mathbf{\Lambda }}_3{\mathit{s}}_t+\widehat{\mathit{K}}\mathrm{sign}\left({\mathit{s}}_t\right)\right),\hfill \end{array}$$

(33)

where $\overline{\mathit{W}}$ is a PD diagonal matrix, $\iota$ is the sampling time, ${\mathbf{\Lambda }}_3$ is a positive diagonal matrix, and the sliding surface is

$${\mathit{s}}_t={\dot{\mathit{e}}}_{1t}+{\mathit{K}}_d{\mathit{e}}_{1t}+{\mathit{K}}_p{\int }_0^t{\mathit{e}}_{1t}\left(\sigma \right)d\sigma ,$$

(34)

with ${\mathit{e}}_{1t}={\mathit{p}}_{dt}-{\mathit{p}}_t$. The adaptive gain $\widehat{\mathit{K}}=\mathrm{diag}({\kappa }_{1,t},\dots ,{\kappa }_{n,t})$ updates as

$${\dot{\kappa }}_{i,t}=\left\{\begin{array}{cc}{\displaystyle \frac{1}{{\lambda }_i}}\left|s_{i,t}\right|,\hfill & {\parallel {\mathit{s}}_t\parallel }_{\infty }\ge \rho ,\hfill \\ -\frac{1}{\iota }{\kappa }_{i,t}{\displaystyle \frac{|s_{i,t}|}{\rho }},\hfill & {\parallel {\mathit{s}}_t\parallel }_{\infty }\ge \rho ,\hfill \end{array}\right.$$

(35)

where ${\lambda }_i,\rho >0$.

A4: The proposed AFxTSMC (see Section 3).

Remark 4.

In the comparative simulations, the A1 and A2 controllers, based on SMC, were proposed recently and implemented without any observer components, whereas the A3 controller employed SMC combined with TDE and an adaptive mechanism. In contrast, the proposed control scheme integrates SMC with a nonlinear disturbance observer, forming a composite controller. Although the inclusion of an observer may offer a performance advantage, this comparison aims to highlight the benefits of active disturbance rejection in enhancing robustness, tracking accuracy, and chattering reduction. This evaluation approach aligns with standard practices in the literature for validating the effectiveness of composite control strategies.

Remark 5.

The control parameters for all compared methods were selected based on standard tuning principles and recommendations from related literature. Where necessary, parameters were further fine-tuned via trial-and-error simulations to balance tracking accuracy and control effort. For sliding mode-based methods (A1 and A2), the gains were conservatively chosen to exceed the estimated uncertainty bounds, ensuring robustness and convergence guarantees. For the TDE-based adaptive method (A3), parameter values were initialized based on previous works applied to the same robotic platform and then adjusted for improved performance. All methods were tested under identical initial conditions, reference signals, and disturbance profiles to ensure a fair and unbiased comparison.

The control parameters employed in the simulations for each algorithm are summarized in

Table 1. Control parameter selection.

Method	Parameter	Value
A1	${\alpha }_1,{\gamma }_1,{\kappa }_1,{\omega }_1,{\Lambda }_1$	2,2,18,0.8,7
A2	${\varpi }_2,j,l,{\kappa }_2,{\Lambda }_2$	0.6,5,3,18,7
A3	${\overline{M}}_{ii},K_{pi},K_{di}$	$4,4,4$
–	$\alpha ,\rho ,\iota ,{\Lambda }_{3i}$	$0.3,0.1,0.001,4$
A4	${\psi }_1,{\kappa }_1,{\mu }_1,{\rho }_1,{\psi }_2,{\kappa }_2,{\rho }_2$	$0.2,3,1.3,0.6,2,22,0.6$
–	${\kappa }_3,{\rho }_3,{\lambda }_i,{\beta }_i$	$0.1,1.2,0.05,0.05$
Observer	${\pi }_1,{\pi }_2,{\pi }_3,\varrho$	$5.4288,9.486,22,25$

.

Tracking Performance Evaluation and Justification

Figure 5, Figure 6 and Figure 7 and

Table 2. RMSE of tracking error for each joint during transient (0–1.5 s) and steady-state (1.5–30 s) phases.

Method	${\mathit{E}}_1\left(0\text{–}1.5\mathbf{s}\right)$	${\mathit{E}}_2\left(0\text{–}1.5\mathbf{s}\right)$	${\mathit{E}}_3\left(0\text{–}1.5\mathbf{s}\right)$	${\mathit{E}}_1\left(1.5\text{–}30\mathbf{s}\right)$	${\mathit{E}}_2\left(1.5\text{–}30\mathbf{s}\right)$	${\mathit{E}}_3\left(1.5\text{–}30\mathbf{s}\right)$
A1	$5.56\times {10}^{-2}$	$1.07\times {10}^{-1}$	$4.12\times {10}^{-2}$	$9.54\times {10}^{-5}$	$4.86\times {10}^{-4}$	$9.89\times {10}^{-4}$
A2	$6.72\times {10}^{-2}$	$1.29\times {10}^{-1}$	$4.59\times {10}^{-2}$	$1.34\times {10}^{-5}$	$9.67\times {10}^{-5}$	$7.17\times {10}^{-4}$
A3	$6.53\times {10}^{-2}$	$1.15\times {10}^{-1}$	$4.56\times {10}^{-2}$	$5.36\times {10}^{-4}$	$7.47\times {10}^{-4}$	$7.37\times {10}^{-4}$
A4	$5.55\times {10}^{-2}$	$1.08\times {10}^{-1}$	$4.34\times {10}^{-2}$	$1.43\times {10}^{-7}$	$9.50\times {10}^{-8}$	$1.09\times {10}^{-6}$

provide a comparative evaluation of the performance metrics, including convergence rate, accuracy, robustness to uncertainties and external disturbances, and other relevant criteria. Figure 5 illustrates the position tracking of the robot arm in both Cartesian (XYZ) and joint spaces under the four control strategies. While all methods successfully followed the desired trajectory despite modeled uncertainties, there are notable differences in their convergence speed and tracking accuracy results across the approaches. Figure 6 illustrates the evolution of the different sliding mode surfaces.

To further evaluate tracking performance, Figure 7 presents both the RMSE comparison and the time responses of the joint tracking errors. As shown in the enlarged view over the interval (1–30 s) and the numerical results in Table 2, Method A3 yielded the highest RMSE values across all joints, with errors of $5.36\times {10}^{-4}$, $7.47\times {10}^{-4}$, and $7.37\times {10}^{-4}$ for $E_1$, $E_2$, and $E_3$, respectively, indicating the weakest tracking performance. Method A1 also produced relatively large tracking errors, particularly in $E_3$ ($9.89\times {10}^{-4}$), although it performed slightly better than A3 in $E_1$ and $E_2$. Method A2 showed noticeable improvement by reducing the tracking errors across all joints. In contrast, the proposed Method A4 demonstrated the best tracking accuracy, achieving significantly lower RMSE values of $\mathbf{1}.\mathbf{43}\times {\mathbf{10}}^{-\mathbf{7}}$, $\mathbf{9}.\mathbf{50}\times {\mathbf{10}}^{-\mathbf{8}}$, and $\mathbf{1}.\mathbf{09}\times {\mathbf{10}}^{-\mathbf{6}}$ for $E_1$, $E_2$, and $E_3$, respectively. These values are at least two orders of magnitude smaller than those of the other methods.

Furthermore, from the enlarged view over the interval (0–1 s) in Figure 5 and Figure 7, it can be observed that Methods A1, A3, and A4 exhibited similar convergence speeds during the initial phase. This behavior can be attributed to the sliding surface designs employed in these methods, which are based on FTSMC, PID-SMC, and NFTSMC surfaces, respectively, as illustrated in Figure 6, along with the contribution of adaptive control laws in both A3 and A4. However, a more detailed comparison based on the indicators presented in Table 2 reveals that during the transient phase (0–1.5 s), A4 and A1 achieved the best overall tracking performance. For Joint 1, A4 attained an RMSE of $5.55\times {10}^{-2}\mathrm{rad}$, which is nearly identical to that of A1 ($5.56\times {10}^{-2}\mathrm{rad}$), and both methods outperformed A2 and A3. In Joint 2, A1 yielded the lowest error at $1.07\times {10}^{-1}\mathrm{rad}$, closely followed by A4 with $1.08\times {10}^{-1}\mathrm{rad}$, whereas A2 showed the highest error. For Joint 3, A1 again performed best with an RMSE of $4.12\times {10}^{-2}\mathrm{rad}$, while A4 provided comparable accuracy with $4.33\times {10}^{-2}\mathrm{rad}$. These results indicate that both A1 and A4 are effective in reducing transient tracking errors across all joints, with A4 maintaining competitive performance. In contrast, Method A2, which employed an NTSMC surface without adaptive mechanisms, demonstrated the slowest convergence speed among the four methods.

The superior performance of A4 is mainly attributed to its active robustness enhancement through the integration of a disturbance observer, the proposed sliding function, and the ARL. While Method A3 utilizes TDE to approximate and compensate for uncertainties, its estimation is based on delayed information and may degrade under fast-changing disturbances. In contrast, Method A4 employs an active disturbance observer that estimates and cancels unknown disturbances in real time, leading to more accurate and robust compensation. Meanwhile, Methods A1 and A2 rely purely on passive robustness by tuning conservative sliding gains without any uncertainty estimation mechanism. Overall, the active nature of the proposed control strategy in Method A4 significantly improves both tracking precision and robustness under uncertain and varying conditions.

Chattering Evaluation and Justification

To quantitatively assess chattering, we considered the sum of the absolute differences between consecutive control input values over the entire operation time. This metric is defined as ${\sum }_{i=1}^{N-1}\left|\Delta u_i\right|={\sum }_{i=1}^{N-1}\left|u_{i+1}-u_i\right|$, where N is the total number of discrete time steps, and $u_i$ is the control input at the i-th time step.

This sum captures the total magnitude of changes in the control signal. A high value indicates frequent and/or large variations between consecutive control inputs, which is characteristic of chattering behavior. Conversely, a lower value suggests smoother control signals with fewer abrupt changes, reflecting more stable and reliable controller performance. By analyzing this metric, we can compare different control strategies in terms of their propensity to induce chattering and evaluate their practical suitability for real-world manipulator applications.

Figure 8 illustrates the evolution of adaptive sliding gains and compares control torques among the four methods. From Figure 8a, it can be observed that during the initial convergence phase, the adaptive gains increased to compensate for the residual estimation error of the disturbance observer and to accelerated the convergence rate. Afterwards, the gains gradually decreased to suppress chattering behavior. To accelerate convergence and enhance robustness, the sliding gains in Methods A1 and A2 were conservatively set above the estimated worst-case uncertainty bounds. As a result, both A1 and A2 exhibited pronounced chattering behavior. In real-world applications, such chattering can cause several adverse effects, including increased energy consumption, reduced precision due to end-effector vibrations, and accelerated wear of mechanical components. Moreover, persistent chattering may induce structural fatigue, compromise system reliability, and shorten the operational lifespan of actuators and joints, making these methods less desirable for practical deployment. In contrast, Methods A3 and A4 generated torque signals that were much smoother than those of A1 and A2, as observed in Figure 8b and

Table 3. Sum of absolute torque differences ($\sum |\Delta \tau |$) from $t=0$ to $30\mathrm{s}$.

Method	Joint 1	Joint 2	Joint 3
A1	$5.05\times {10}^6$	$8.61\times {10}^6$	$1.67\times {10}^6$
A2	$4.90\times {10}^6$	$9.76\times {10}^6$	$1.86\times {10}^6$
A3	$1.19\times {10}^3$	$1.62\times {10}^3$	$3.16\times {10}^2$
A4	$9.96\times {10}^3$	$4.28\times {10}^3$	$6.99\times {10}^3$

. Method A3 exhibited the lowest chattering; however, it may sacrifice robustness under severe uncertainties. This method employs relatively low adaptive sliding gains, intended to compensate for the estimation errors introduced by the TDE. Similarly, Method A4 uses a moderate adaptive gain to address residual errors from the disturbance observer during the initial convergence phase. As the tracking error decreases over time, the adaptive gain is gradually reduced, resulting in smoother control performance. Such smooth control signals are essential for enhancing tracking accuracy, minimizing energy consumption, and extending the service life of electrical and mechanical components.

5. Conclusions

This paper presents a novel AFxTSMC framework for industrial robotic manipulators facing uncertainties and disturbances. The core ARL enables fast and stable sliding gain reduction by maintaining positivity and preventing sign reversals, effectively reducing chattering while ensuring fixed-time convergence. The SF-FxTSF provides fast, robust, and singularity-free convergence. Robustness is further improved by a modified TOSMO, which accelerates estimation of unknown dynamics for precise compensation without sacrificing stability. Lyapunov analysis confirms overall system stability. Simulations on the industrial robot arm demonstrate that the proposed algorithm outperforms three state-of-the-art methods in several performance metrics, including convergence rate, tracking accuracy (RMSE), and chattering reduction. Its observer converges quickly and without oscillations, reducing estimation delays and improving responsiveness. Adaptive sliding gains initially increase to counter estimation errors and then decrease to suppress chattering, balancing robustness and smoothness. The quantitative results show that the proposed methodology achieves at least two orders of magnitude better tracking accuracy than alternatives. The combination of active disturbance estimation and adaptive gain tuning maintains robustness against rapidly varying uncertainties, outperforming passive methods with conservative gain settings.

Consequently, the AFxTSMC scheme provides a reliable, adaptable, and high-performance control strategy for industrial robotic manipulators. Future research will aim at validating the method through real-world experiments and expanding its application to multi-robot coordination and practical hardware deployment.