Fixed-Time RBFNN-Based Prescribed Performance Control for Robot Manipulators: Achieving Global Convergence and Control Performance Improvement

Abstract

This paper proposes a fixed-time neural network-based prescribed performance control method (FNN-PPCM) for robot manipulators. A fixed-time sliding mode controller (SMC) is designed with its strengths and weaknesses in mind. However, to address the limitations of the controller, the paper suggests alternative approaches for achieving the desired control objective. To maintain stability during a robot’s operation, it is crucial to keep error states within a set range. To form the unconstrained systems corresponding to the robot’s constrained systems, we apply modified prescribed performance functions (PPFs) and transformed errors set. PPFs help regulate steady-state errors within a performance range that has symmetric boundaries around zero, thereby ensuring that the tracking error is zero when the transformed error is zero. Additionally, we use a singularity-free sliding surface designed using transformed errors to determine the fixed-time convergence interval and maximum allowable control errors during steady-state operation. To address lumped uncertainties, we employ a radial basis function neural network (RBFNN) that approximates their value directly. By selecting the transformed errors as the input for the RBFNN, we can minimize these errors while bounding the tracking errors. This results in a more accurate and faster estimation, which is superior to using tracking errors as the input for the RBFNN. The design procedure of our approach is based on fixed-time SMC combined with PPC. The method integrates an RBFNN for precise uncertainty estimation, unconstrained dynamics, and a fixed-time convergence sliding surface based on the transformed error. By using this design, we can achieve fixed-time prescribed performance, effectively address chattering, and only require a partial dynamics model of the robot. We conducted numerical simulations on a 3-DOF robot manipulator to confirm the effectiveness and superiority of the FNN-PPCM.

1. Introduction

Robotic manipulators are machines that are designed to manipulate objects with a high degree of precision and accuracy. Robotic manipulators typically consist of a series of joints that are linked together with actuators and sensors, allowing them to move and position objects in a precise and controlled manner. Robotic manipulators have been widely adopted across several industries, ranging from automotive manufacturing, logistics, and storage, to space and ocean exploration, among others [1,2]. This growth in the use of robotics has led to increased attention from both academia and industry. As the applications of robotic manipulators continue to expand to fields such as medical support, fire prevention, and industrial assembly, there is a growing need for improved performance. Hence, performance requirements have become more stringent and are being actively pursued in practice. However, despite the widespread use of robot manipulators, they often encounter various mechanical system issues, including state constraints, high nonlinearity, parametric variations, and other dynamical uncertainties [3]. Obtaining accurate information about the dynamical models of robot manipulators can be challenging due to their complex dynamics, and unidentified nonlinearities can significantly impact the performance and stability of the control system. When external disturbances or model uncertainties are present, the problem becomes even more challenging, and finding a useful compensation solution in the trajectory tracking control of uncertain robot manipulators remains an open problem [4].

Robot manipulators can be controlled using various strategies, including computed torque control (CTC) [5,6], PID [7], adaptive control (AC) [8], and SMC [9]. Among these, SMC is often preferred for its ability to address disturbances and achieve perturbation attenuation. Due to the use of linear sliding surfaces (LSS), classic SMCs can only provide asymptotic stability for robot manipulators, and the sliding phase of SMC can result in undesirable chattering due to unknown dynamics and the use of discontinuous control law. However, in practice, higher precision and smoother control inputs are essential. To mitigate this issue, various techniques have been developed, including approximation methods such as neural networks (NNs) [10] or fuzzy logic systems (FLSs) [11,12], observer-based controller [13], high-order SMC (HOSMC) [14], and the boundary method (BM) [15], among others. Furthermore, it is important to note that the system must be stable within a finite-time frame rather than achieving exponential stability as in various works such as [16,17,18].

NNs’ universal approximation capability has made them recent popular choices for real-world applications [19]. They can be integrated with other control system designs to create model-free approaches or remove the need for full dynamics from controller designs. Robot models are often unknown or partially known, and NN algorithms can be used for online approximation to compensate for negative effects and approximate unknown dynamics. NN-based control methods that do not use the dynamic model in the control design are called “model-free” control methods [20,21]. This approach eliminates the need to calculate the dynamic model of highly complex robotic manipulators, which can be incredibly challenging. In the typical design of model-based controllers, NN compensates for the computational errors of the dynamic model and other uncertainty components [22,23]. To combine the approximation ability of NNs with the robust features of SMC, many NN-based SMC methods have been proposed for controlling robotic manipulators [10,24,25]. Traditional SMC methods mostly rely on the system model in the control design, which may not be available or accurate in some practical cases. Therefore, integrating NNs provides a better solution for conventional SMC methods, enabling the design of SMC methods that may or may not rely on the exact system model. NN-based SMC methods that use the dynamic model of the robot manipulator in control design use NN to compensate for errors from the dynamic calculations, uncertainties, and disturbances [10,24,25,26]. The small sliding gain in the switching control law (SCL) is used to compensate for the estimation error of NN. In contrast, NN-based SMC methods that do not use the dynamic model use NN to approximate these dynamic models [21,27]. The SCL is used to handle the effects of uncertain terms, and hence, the large sliding gain is correspondingly used. In summary, each method has its own advantages and disadvantages, and the appropriate method to use depends on the requirements and actual applications available.

Finite-time control is a powerful technique for achieving high-performance control of nonlinear systems, such as robot manipulators [28] and mechanical systems [29], even in the presence of external disturbances and uncertain dynamics [30]. Two main approaches to finite-time control exist: geometric homogeneity [31,32] and Lyapunov stabilization [29,30]. The latter approach is favored for its efficiency in resolving uncertain dynamical systems and exterior perturbations. Through SMC, non-linear sliding surfaces are used to enable finite-time convergence and stability [33]. Numerous finite-time SMC techniques have been developed for the trajectory tracking of robot manipulators, including terminal SMC (TSMC) [34], integral SMC (ISMC) [35], fast terminal SMC (FTSMC) [34,36], adaptive finite control [26], and so on. However, the convergence performance of finite-time control methods may be affected by the initial conditions of the system states. To address this issue, fixed-time stability has been proposed as an extension to finite-time stability. High-performance applications may find fixed-time stabilization methods particularly attractive, as they can pre-limit the stabilization time regardless of the initial state variables of the system. In recent years, fixed-time controllers have been widely applied in various fields, such as robot manipulators [37], uncertain surface vessels [38], spacecraft [39], and other mechanical systems [40,41]. This is due to their ability to provide high-performance control even in the presence of external disturbances and uncertain dynamics.

PPC is a technique that is designed to ensure transient and steady-state performance in controlled systems, with limited tracking errors, convergence rate, and maximum overshoot [42,43,44,45,46]. To establish specific performance boundaries, a PPF is typically used. While a single PPF can define the boundary of prescribed performance and use a proportion of PPF to form the lower boundary, the two boundaries are typically unsymmetrical to each other through zero [42]. These issues can lead to challenges in selecting error transformation functions (ETFs). Therefore, it is important to carefully consider the selections to avoid any singularity problems that can negatively affect the operation of real-world applications. In these cases, a combination of PPFs can be used to effectively regulate performance within the desired range.

Motivated by the previous discussions, this study aims to explore an efficient controller that achieves fixed-time prescribed performance for robot manipulators with uncertain dynamics while accounting for external disturbances. The key contributions and innovations of this research are outlined below.

• The design of the RBFNN has effectively addressed the challenges of reducing total uncertainties, requiring only a partial dynamics model of the robot, and reducing chattering. Furthermore, by selecting the transformed errors as the input for the RBFNN, we can minimize these errors while bounding the tracking errors, resulting in a more accurate estimation compared to using tracking errors as the input for the RBFNN.
• We propose a modification to the sliding surface used in control systems by incorporating transformed errors. This modification enables us to determine the maximum range of acceptable tracking errors in the steady-state while ensuring fixed-time convergence and eliminating singularities.
• Our proposed PPFs aim to constrain position-tracking errors within a predetermined performance range. Notably, our approach ensures symmetrical boundaries for tracking errors around zero, which guarantees a zero tracking error when the transformed error is zero.
• We utilized a modified reaching law, which enables the rapid convergence of tracking error to the sliding surface within a fixed-time bound.
• The proposed control approach achieves global fixed-time convergence and prescribed performance for stabilization while providing superior precision compared to some other methods with the same form, such as SMC, TSMC, and FTSMC. Additionally, this approach comprehensively addresses the chattering problem.
• Sufficient proof was provided for the stability, non-singularity, and settling time of the suggested approaches.

This paper is structured into five main sections. The first section is the introduction, which provides an overview of the paper. The second section formulates the problem and provides some necessary preliminaries, and the third section presents the FNN-PPCM approach. Section 4 discusses the simulated performance of the FNN-PPCM on a 3-DOF robot manipulator, and Section 5 summarizes the research findings, concludes the paper, and discusses future study orientations.

2. Preliminaries and Problem Formulation

2.1. Problem Formulation

Consider a robot system [47]:

$$\mathcal{M}\left(q\right)\stackrel{..}{q}+\mathcal{C}(q,\stackrel{.}{q})\stackrel{.}{q}+\mathcal{G}\left(q\right)+{\mathcal{F}}_r\left(\stackrel{.}{q}\right)+\mathcal{D}(t,q,\stackrel{.}{q})=\tau .$$

(1)

In the system being described, the acceleration vector $\stackrel{..}{q}$, velocity vector $\stackrel{.}{q}$, and position vector $q\in R^{n\times 1}$ are represented. Additionally, the matrices $\mathcal{M}\left(q\right)\in R^{n\times n}$, $\mathcal{C}(q,\stackrel{.}{q})\in R^{n\times n}$, and the vector $\mathcal{G}\left(q\right)\in R^{n\times 1}$ correspond to the inertia, centripetal-Coriolis, and gravitational force, respectively. The interior friction vector is denoted by ${\mathcal{F}}_r\left(\stackrel{.}{q}\right)\in R^{n\times 1}$, while $\mathcal{D}(t,q,\stackrel{.}{q})$ represents the exterior disturbance vector, and the actuator torque vector is $\tau \in R^{n\times 1}$.

Equation (1) can be fully expressed in the following expression due to the existence of uncertainty terms:

$$({\mathcal{M}}_c\left(q\right)+{\mathcal{M}}_{\delta }\left(q\right))\stackrel{..}{q}+({\mathcal{C}}_c(q,\stackrel{.}{q})+{\mathcal{C}}_{\delta }(q,\stackrel{.}{\theta }))\stackrel{.}{q}+({\mathcal{G}}_c\left(q\right)+{\mathcal{G}}_{\delta }\left(q\right))+{\mathcal{F}}_r\left(\stackrel{.}{q}\right)+\mathcal{D}(t,q,\stackrel{.}{q})=\tau ,$$

(2)

where ${\mathcal{M}}_{\delta }\left(q\right),{\mathcal{C}}_{\delta }(q,\stackrel{.}{q}),{\mathcal{G}}_{\delta }\left(q\right)$, and ${\mathcal{F}}_r\left(\stackrel{.}{q}\right)$ are uncertain dynamics that always exist in the system. ${\mathcal{M}}_c\left(q\right)$, ${\mathcal{C}}_c(q,\stackrel{.}{q})$, and ${\mathcal{G}}_c\left(q\right)$ are known as nominal model.

Rewriting Equation (2) with a more concise expression:

$$\stackrel{..}{q}=\mathcal{Q}\left(q\right)\tau +\mathcal{H}(q,\stackrel{.}{q})+\varphi ,$$

(3)

where $\mathcal{Q}\left(q\right)={\mathcal{M}}_c^{-1}\left(q\right)$, $\mathcal{H}(q,\stackrel{.}{q})=-{\mathcal{M}}_c^{-1}\left(q\right)({\mathcal{C}}_c(q,\stackrel{.}{q})\stackrel{.}{q}+{\mathcal{G}}_c\left(q\right))$, and $\varphi =-{\mathcal{M}}_c^{-1}\left(q\right)$$({\mathcal{M}}_{\delta }\left(q\right)\stackrel{..}{q}+{\mathcal{C}}_{\delta }(q,\stackrel{.}{q})\stackrel{.}{q}+{\mathcal{G}}_{\delta }\left(q\right)+{\mathcal{F}}_r\left(\stackrel{.}{q}\right)+\mathcal{D}(t,q,\stackrel{.}{q}))$.

Defining $y_1=q,y_2=\stackrel{.}{q}\in R^{n\times 1}$, $y={\left[\begin{array}{cc}{y}_1& y_2\end{array}\right]}^T$, and $\mathcal{U}=\tau$, Equation (3) becomes:

$$\left\{\begin{array}{c}{\stackrel{.}{y}}_1=y_2\hfill \\ {\stackrel{.}{y}}_2=\mathcal{Q}\left(y_1\right)\mathcal{U}+\mathcal{H}\left(y\right)+\varphi \hfill \end{array}\right..$$

(4)

Let the position and velocity control errors be defined as $y_{1e}=y_1-y_{1d}$ and $y_{2e}=y_2-{\stackrel{.}{y}}_{1d}$, where $y_{1d}$ and ${\stackrel{.}{y}}_{1d}$ are denoted as the desired position and velocity, respectively. The tracking error vector is denoted as $y_e={\left[\begin{array}{cc}{y}_{1e}& y_{2e}\end{array}\right]}^T$. Thus, Equation (4) can be reconstructed as follows:

$$\left\{\begin{array}{c}{\stackrel{.}{y}}_{1e}=y_{2e}\hfill \\ {\stackrel{.}{y}}_{2e}=\mathcal{Q}\left(y_1\right)\mathcal{U}+\mathcal{H}\left(y\right)+\varphi -{\stackrel{..}{y}}_{1d}\hfill \end{array}\right..$$

(5)

Achieving precise trajectory tracking in robot systems is a complex task due to various factors, such as actuator constraints, nonlinearity in system dynamics, external disturbances, sensor noise, and uncertainty. These challenges necessitate the development of robust control strategies to ensure effective control of the system. This paper aims to propose a novel FNN-PPCM for robot systems. The proposed method achieves global fixed-time convergence and prescribed performance for stabilization and outperforms existing methodologies, such as SMC, TSMC, and FTSMC, in terms of control performance.

2.2. Preliminaries

The notation ${\lceil y\rfloor }^{{\xi }_0}={\left|y\right|}^{{\xi }_0}\mathrm{sign}\left(y\right)$ will be used throughout the paper for simplicity, where $\forall {\xi }_0\ge 0$ and $y\in R$.

The following are some introductory concepts related to finite-time stability and fixed-time stability, along with some key lemmas.

Consider the below system:

$$\stackrel{.}{y}=f(t,y\left(t\right)),y\left(0\right)=y_0,f\left(0\right)=0,y\in \mathcal{O}\subset R^n,$$

(6)

where the function $f:R^{+}\times \mathcal{O}\to R^n$ is continuous in an open neighborhood $\mathcal{O}$ of the origin, $y=0$.

Definition 1

([15]). If we assume that there exists a function of convergence time $T\left(y_0\right)>0$ such that ${lim}_{t\to t_0+T}y(t,y_0)=0$ for any initial condition $y_0$ in $\mathcal{O}$ at time $t_0$, then the origin of the system described above is (locally) stable in finite time if and only if it is Lyapunov stable.

Definition 2

([15]). The system has fixed-time global stability if it is both finite-time stable, meaning there exists a convergence time function $T\left(y_0\right)$ where ${lim}_{t\to t_0+T}y(t,y_0)=0$, and $T\left(y_0\right)$ is bounded by a positive constant $T_{max}$.

Lemma 1

([28]). The origin of the system described in Equation (6) is considered stable for a finite period of time if there are constants $0<{\rho }_0<1$ and ${\kappa }_0>0$ that satisfy $\dot{L}\left(y\right)\le -{\kappa }_0{L}^{{\rho }_0}\left(y\right)$, where $L\left(y\right)$ is a selected Lyapunov function.

Lemma 2

([48]). Consider a system:

$$\stackrel{.}{y}=-{\gamma }_0{\lceil y\rfloor }^{{\xi }_0}-{\alpha }_0{\lceil y\rfloor }^{{\omega }_0},$$

(7)

where ${\gamma }_0>0,{\alpha }_0>0,{\xi }_0>1$ and $0<{\omega }_0<1$. For the fixed-time stable system above, the convergence time $T\left(y_0\right)$ is limited by the function $T_{max}$, given by $T_{max}=\frac{1}{{\gamma }_0({\xi }_0-1)}+\frac{1}{{\alpha }_0(1-{\omega }_0)}$.

3. Control Design Synthesis

3.1. Sliding Surface Design

A sliding surface is selected, as in [49], which can avoid singularities and is defined as follows:

$$s=y_{1e}+\frac{1}{{\alpha }_1^{\frac{1}{{\omega }_1}}}{\lceil {\gamma }_1{\lceil y_{1e}\rfloor }^{{\xi }_1}+{\stackrel{.}{y}}_{1e}\rfloor }^{\frac{1}{{\omega }_1}},$$

(8)

where ${\gamma }_1>0,{\alpha }_1>0,{\xi }_1>1$, and $\frac{1}{2}<{\omega }_1<1$.

A thorough examination of how the proposed sliding surface can effectively address the challenges posed by singularity problems will be provided in a later subsection.

When the sliding surface (8) converges to its origin ($s=0$), we have:

$${\stackrel{.}{y}}_{1e}=-{\gamma }_1{\lceil y_{1e}\rfloor }^{{\xi }_1}-{\alpha }_1{\lceil y_{1e}\rfloor }^{{\omega }_1}.$$

(9)

Similar to Lemma 2, the sliding surface, in this case, has the same properties as that described in [48], which means that the system (8) is fixed-time stable. Its settling time is denoted as $T\left(y_{1e0}\right)$ and satisfies $T\left(y_{1e0}\right)<T_{max}\triangleq \frac{1}{{\gamma }_1({\xi }_1-1)}+\frac{1}{{\alpha }_1(1-{\omega }_1)}$, where $y_{1e0}=y_{1e}(t=0)$.

3.2. Controller Design

Let ${\vartheta }_1={\gamma }_1{\lceil y_{1e}\rfloor }^{{\xi }_1}+{\stackrel{.}{y}}_{1e}$, and ${\left|{\vartheta }_1\right|}^{\frac{1}{{\omega }_1}-1}=|{\gamma }_1{\lceil y_{1e}\rfloor }^{{\xi }_1}+{\stackrel{.}{y}}_{1e}{|}^{\frac{1}{{\omega }_1}-1}$. Differentiating Equation (8) with respect to time gives:

$$\stackrel{.}{s}={\stackrel{.}{y}}_{1e}+\frac{1}{{\alpha }_1^{\frac{1}{{\omega }_1}}}\frac{1}{{\omega }_1}{\left|{\vartheta }_1\right|}^{\frac{1}{{\omega }_1}-1}\left({\gamma }_1{\xi }_1{\lceil y_{1e}\rfloor }^{{\xi }_1-1}{\stackrel{.}{y}}_{1e}+{\stackrel{..}{y}}_{1e}\right).$$

(10)

Adding Equation (5) to Equation (10) obtains:

$$\stackrel{.}{s}={\stackrel{.}{y}}_{1e}+{\mathcal{A}}_1\left({\gamma }_1{\xi }_1{\lceil y_{1e}\rfloor }^{{\xi }_1-1}{\stackrel{.}{y}}_{1e}+\mathcal{Q}\left(y_1\right)\mathcal{U}+\mathcal{H}\left(y\right)+\varphi -{\stackrel{..}{y}}_{1d}\right),$$

(11)

where ${\mathcal{A}}_1=\mathrm{diag}\left\{{\mathcal{A}}_{1i}\right\},{\mathcal{A}}_{1i}=\frac{1}{{\alpha }_1^{\frac{1}{{\omega }_1}}}\frac{1}{{\omega }_1}{\left|{\vartheta }_{1i}\right|}^{\frac{1}{{\omega }_1}-1}\ge 0$, and $i=1,\dots ,n$.

Based on Equation (11), a controller is synthesized below:

$$\mathcal{U}={\mathcal{U}}_{eq}+{\mathcal{U}}_r,$$

(12)

where the two terms, ${\mathcal{U}}_{eq}$ and ${\mathcal{U}}_r$, are formulated as follows:

$${\mathcal{U}}_{eq}=-{\mathcal{Q}}^{-1}\left({\gamma }_1{\xi }_1{\lceil y_{1e}\rfloor }^{{\xi }_1-1}{\stackrel{.}{y}}_{1e}+\mathcal{H}\left(y\right)-{\stackrel{..}{y}}_{1d}\right)-{\mathcal{Q}}^{-1}{\mathcal{A}}_1^{-1}{\stackrel{.}{y}}_{1e},$$

(13)

$${\mathcal{U}}_r=-{\mathcal{Q}}^{-1}W\mathrm{sign}\left(s\right)-{\mathcal{Q}}^{-1}{\kappa }_2\mathsf{\Gamma }{\lceil s\rfloor }^{{\rho }_2},$$

(14)

where $\mathsf{\Gamma }=\mathrm{diag}\left(1+{\psi }_2^2{\left|s_i\right|}^{2(1-{\rho }_2)}\right)$, $W=\mathrm{diag}\left\{W_i\right\}$ with $W_i\ge \left|{\varphi }_i\right|$, and $i=1,\dots ,n$. ${\kappa }_2,{\psi }_2,0<{\rho }_2<1$ are constants.

Theorem 1.

If all of the uncertainties that affect the robotic system, denoted as ϕ, are known in advance, then the control inputs in Equation (12) can be used to achieve fixed-time global stability.

Remark 1.

The derivative of the sliding surface in Equation (10) contains the term ${\left|{\vartheta }_1\right|}^{1-\frac{1}{{\omega }_1}}{\stackrel{.}{y}}_{1e}$, which is interestingly non-singular. When $y_{1e}=0$ and ${\stackrel{.}{y}}_{1e}\ne 0$, we have $|{\stackrel{.}{y}}_{1e}{|}^{1-\frac{1}{{\omega }_1}}{\stackrel{.}{y}}_{1e}\ge {\stackrel{.}{y}}_{1e}^{2-\frac{1}{{\omega }_1}}$, where $2-\frac{1}{{\omega }_1}$ is a positive power term.

Remark 2.

System stabilization involves a two-stage process. First, the tracking errors are brought to the sliding surface within a bounded reaching time of $T_{r\epsilon }<T_r+\epsilon \left(\mathsf{\tau }\right)$, where $T_r<T_{max}\triangleq \frac{\pi }{2{\kappa }_2{\psi }_2(1-{\rho }_2)}$, and $\epsilon \left(\mathsf{\tau }\right)$ is defined in [50]. Then, the tracking errors are maintained along the sliding surface until they reach equilibrium, with a settling time of $T\left(y_{e0}\right)<T_{max}\triangleq \frac{1}{{\gamma }_1({\xi }_1-1)}+\frac{1}{{\alpha }_1(1-{\omega }_1)}$.

Proof of Theorem 1.

The concept of the following proof draws inspiration from a previous work [50].

We apply the control laws (12)–(14) to Equation (11) and obtain:

$$\stackrel{.}{s}={\mathcal{A}}_1\left(-W\mathrm{sign}\left(s\right)+\varphi -{\kappa }_2\mathsf{\Gamma }{\lceil s\rfloor }^{{\rho }_2}\right).$$

(15)

By selecting $L_1=\frac{1}{2}{s}_i^2$ as a candidate Lyapunov function, we can differentiate it to obtain:

$$\begin{array}{c}{\stackrel{.}{L}}_1={\mathcal{A}}_{1i}{s}_i\left(-W_i\mathrm{sign}\left(s_i\right)+{\varphi }_i-{\kappa }_2\left(1+{\psi }_2^2|s_i{|}^{2(1-{\rho }_2)}\right)|s_i{|}^{{\rho }_2}\mathrm{sign}\left(s_i\right)\right)\hfill \\ ={\mathcal{A}}_{1i}\left(-W_i|s_i|+s_i{\varphi }_i-{\kappa }_2\left(1+{\psi }_2^2|s_i{|}^{2(1-{\rho }_2)}\right)|s_i{|}^{{\rho }_2+1}\right)\hfill \\ \le {\mathcal{A}}_{1i}\left(-{\kappa }_2|s_i{|}^{{\rho }_2+1}\right)\hfill \\ \le {\mathcal{A}}_{1i}\left(-\sqrt{2}{\kappa }_2{\left(L_1\right)}^{\frac{{\rho }_2+1}{2}}\right)\hfill \end{array}.$$

(16)

From Equation (16), we can see that if ${\vartheta }_{1i}\ne 0$, then ${\mathcal{A}}_{1i}>0$. As a result, the operating state space of $(y_{1ei},{\stackrel{.}{y}}_{1ei})$ is divided into two different areas: ${\Omega }_{1i}=\left\{(y_{1ei},{\stackrel{.}{y}}_{1ei})\right|{\mathcal{A}}_{1i}\ge 1\}$ and ${\Omega }_{2i}=\left\{(y_{1ei},{\stackrel{.}{y}}_{1ei})\right|{\mathcal{A}}_{1i}<1\}$.

If the system states enter in the first area ${\Omega }_{1i}$, we can obtain:

$$\begin{array}{c}{\stackrel{.}{L}}_1\le -\sqrt{2}{\kappa }_2{\left(L_1\right)}^{\frac{{\rho }_2+1}{2}}\hfill \end{array}.$$

(17)

According to Lemma 1, since $0<{\rho }_2<1$, we have $0<\frac{{\rho }_2+1}{2}<1$, which implies that the origin $s_i$, $i=1,\dots ,n$ is stable in global finite time. With a bounded reaching time of $T_r<T_{max}\triangleq \frac{\pi }{2{\kappa }_2{\psi }_2(1-{\rho }_2)}$, the tracking errors are converged to the sliding surface. The reaching time calculation is provided in detail in the paper’s Appendix A.1, and the proof is fully confirmed.

If the system states enter the second area ${\Omega }_{2i}$ when ${\vartheta }_{1i}\ne 0$, the sliding surface $s_i=0,i=1,2,\dots ,n$ is still an attractor based on Equation (16). To complete the proof, we need to show that ${\vartheta }_{1i}=0$ is only attractive at the origin. This can be achieved using a similar technique as demonstrated in [50].

In conclusion, the sliding surface $s_i=0,i=1,2,\dots ,n$ can be converged upon from any initial condition in the phase plane within a bounded time $T_{r\epsilon }<T_r+\epsilon \left(\mathsf{\tau }\right)$[50]. $\epsilon \left(\mathsf{\tau }\right)$ is defined as a small time margin related to the boundary width, where $\mathsf{\tau }={\omega }_1^{\frac{{\omega }_1}{1-{\omega }_1}}{\alpha }_1^{\frac{1}{1-{\omega }_1}}$. □

Although the above controller’s design can achieve good performance within a fixed timeframe for the robot, accurate calculation of the robot’s dynamics equation, including unknown uncertainty components, remains a challenging task. Moreover, to maintain both steady-state and dynamic stability during the robot’s operation, it is crucial to ensure that error states are kept within a predetermined range. To address this challenge, a combination of RBFNN and PPC is applied, allowing for a direct approximation of lumped uncertainties and the achievement of a prescribed performance for the robot, respectively.

4. Proposed FNN-PPCM Design

4.1. PPC

To ensure that the tracking errors $y_e$ stay within a predetermined range, they must satisfy the following condition:

$$\begin{array}{c}\hfill -B_b\left(t\right)<y_e\mathrm{sign}\left(y_{e0}\right)<B_t\left(t\right),\end{array}$$

(18)

where $\mathbf{e}$ is Euler’s number, $y_{e0}=y_e(t=0)$, $B_t\left(t\right)=(B_0-B_{\infty }){\mathbf{e}}^{-\lambda t}+B_{\infty }$ and $B_b\left(t\right)=(B_1-B_{\infty }){\mathbf{e}}^{-\lambda t}+B_{\infty }$. These functions are smoothly decreasing, positive functions that map $R^{+}$ to $R^{+}$, and satisfy the conditions ${lim}_{t\to \infty }{B}_t\left(t\right)=B_{\infty }>0$ and ${lim}_{t\to \infty }{B}_b\left(t\right)=B_{\infty }>0$. The constants $B_0$, $B_1$, and $B_{\infty }$ satisfy $B_0>\left|y_{e0}\right|>0$, $B_0\ge B_1\ge B_{\infty }$, and the parameter $\lambda >0$ is used to tune performance bounds.

This approach introduces two separate PPFs to manage the tracking errors and maximum overshoot, namely $B_t\left(t\right)$ and $B_b\left(t\right)$, which differ from existing PPCs [42,44,45]. The upper bound $B_t\left(t\right)$ limits the possible maximum value of $y_e$ and the convergence rate, while the lower bound $B_b\left(t\right)$ limits the maximum possible overshoot and the maximum size of $y_e$. The maximum size of $y_e$ is set by $B_{\infty }$, while the maximum overshoot is set by $B_1$. Adjusting the rate at which $B_t\left(t\right)$ decreases by the parameter $\lambda$ affects the convergence rate of $y_e$. Because the bounds of the two functions $B_t\left(t\right)$ and $B_b\left(t\right)$ have been designed with the same magnitude, the performance design space will be expanded and symmetric around zero. This means that when the transformed error is zero, $y_e$ is also zero. By appropriately selecting the parameters of $B_t\left(t\right)$ and $B_b\left(t\right)$, the range of values that $y_e$ can take can be determined in advance, allowing for better control over the system’s behavior. Additionally, the proposed approach enables the easier design of ETFs without singularity issues. The prescribed performance refers to the desired behavior of the system, and it is explained in Figure 1.

The equivalent unconstrained dynamics of the constrained error dynamics are obtained using the following error transformation function (ETF):

$$\begin{array}{c}\hfill y_{1e}=\mathcal{P}\left(t\right)\mathcal{T}\left({\varrho }_1\right),\end{array}$$

(19)

where ${\varrho }_1$ is defined as a transformed error, $\mathcal{T}\left({\varrho }_1\right)$ is defined as an ETF, and

$$\begin{array}{c}\hfill \mathcal{P}\left(t\right)=\left\{\begin{array}{c}{B}_t\left(t\right)\mathrm{if}\mathrm{sign}(y_e.y_{e0})>0\hfill \\ B_b\left(t\right)\mathrm{if}\mathrm{sign}(y_e.y_{e0})<0\hfill \end{array}\right..\end{array}$$

The function $\mathcal{T}\left({\varrho }_1\right)$ is:

• a smooth and strictly increasing one;
• $-1<\mathcal{T}\left({\varrho }_1\right)<1$;
• $\mathcal{T}\left({\varrho }_1\right)=0$ if ${\varrho }_1=0$;
• $\left\{\begin{array}{c}\underset{{\varrho }_1\to -\infty }{lim}\mathcal{T}\left({\varrho }_1\right)=-1\hfill \\ \underset{{\varrho }_1\to +\infty }{lim}\mathcal{T}\left({\varrho }_1\right)=1\hfill \end{array}\right.$.

Considering all possible scenarios, the following conclusions can be drawn:

• If $y_{e0}>0$ and $y_e>0$, then $0\le \mathcal{T}\left({\varrho }_1\right)<1$ and $B_t\left(t\right)>0$. Thus, $0\le B_t\left(t\right)\mathcal{T}\left({\varrho }_1\right)<B_t\left(t\right)$.
• If $y_{e0}>0$ and $y_e<0$, then $-1<\mathcal{T}\left({\varrho }_1\right)\le 0$ and $B_b\left(t\right)>0$. Thus, $-B_b\left(t\right)<B_b\left(t\right)\mathcal{T}\left({\varrho }_1\right)\le 0$. It can be concluded that when $y_{e0}>0$, then $-B_b\left(t\right)<y_e<B_t\left(t\right)$.
• If $y_{e0}<0$ and $y_e<0$, then $-B_t\left(t\right)<B_t\left(t\right)\mathcal{T}\left({\varrho }_1\right)<0$.
• If $y_{e0}<0$ and $y_e>0$, then $0<B_b\left(t\right)\mathcal{T}\left({\varrho }_1\right)<B_b\left(t\right)$. It can be concluded that when $y_{e0}<0$, then $-B_t\left(t\right)<y_e<B_b\left(t\right)$.

By utilizing condition (18), it is possible to predict the behavior of positional error in both the transient and steady-state phases in advance.

To remove the singularity, a new ETF is introduced as follows:

$$\begin{array}{c}\hfill \mathcal{T}\left({\varrho }_1\right)=\frac{2}{\pi }arctan\left({\varrho }_1\right).\end{array}$$

(20)

${\varrho }_1$ can be derived from Equation (20), as follows:

$$\begin{array}{c}\hfill {\varrho }_1=tan\left(\frac{\pi y_{1e}}{2\mathcal{P}\left(t\right)}\right).\end{array}$$

(21)

We obtain the first and second derivatives of ${\varrho }_1$, respectively, by the following:

$$\begin{array}{c}\hfill {\dot{\varrho }}_1=\frac{\pi \left(1+{\varrho }_1^2\right)}{2\mathcal{P}\left(t\right)}\left({\stackrel{.}{y}}_{1e}-\frac{2\stackrel{.}{\mathcal{P}}\left(t\right)}{\pi }arctan\left({\varrho }_1\right)\right)\end{array}$$

(22)

and

$$\begin{array}{c}\hfill {\ddot{\varrho }}_1=\frac{\pi \left(1+{\varrho }_1^2\right)}{2\mathcal{P}\left(t\right)}\left({\stackrel{..}{y}}_{1e}-\frac{2}{\pi }\left(\stackrel{..}{\mathcal{P}}\left(t\right)arctan\left({\varrho }_1\right)+\frac{2\stackrel{.}{\mathcal{P}}\left(t\right){\stackrel{.}{\varrho }}_1}{1+{\varrho }_1^2}-\frac{2\mathcal{P}\left(t\right){\varrho }_1{\stackrel{.}{\varrho }}_1^2}{{\left(1+{\varrho }_1^2\right)}^2}\right)\right)\end{array}$$

(23)

with $\frac{\pi \left(1+{\varrho }_1^2\right)}{2\mathcal{P}\left(t\right)}>0$.

Obtaining Equations (22) and (23) are fully covered in Appendix A.2.

Referring to Equations (5), (22), and (23), the dynamics of the system subject to constraints are transformed to the corresponding unconstrained dynamics:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\stackrel{.}{\varrho }}_1={\varrho }_2\hfill \\ {\stackrel{.}{\varrho }}_2=\mathsf{\Theta }\left(\mathcal{Q}\left(y_1\right)\mathcal{U}+\mathcal{H}\left(y\right)+\varphi -{\stackrel{..}{y}}_{1d}-\overline{\mathcal{P}}\right)\hfill \end{array}\right.,\end{array}$$

(24)

where $\mathsf{\Theta }=\frac{\pi \left(1+{\varrho }_1^2\right)}{2\mathcal{P}\left(t\right)}>0$ and $\overline{\mathcal{P}}=\frac{2}{\pi }\left(\stackrel{..}{\mathcal{P}}\left(t\right)arctan\left({\varrho }_1\right)+\frac{2\stackrel{.}{\mathcal{P}}\left(t\right){\stackrel{.}{\varrho }}_1}{1+{\varrho }_1^2}-\frac{2\mathcal{P}\left(t\right){\varrho }_1{\stackrel{.}{\varrho }}_1^2}{{\left(1+{\varrho }_1^2\right)}^2}\right)$.

4.2. RBFNN

The RBFNN is composed of an input layer, a hidden layer, and an output layer. When compared to multilayer neural networks, RBFNN has a simpler architecture and achieves faster convergence. Moreover, RBFNN boasts desirable features such as online adjustment, accurate approximation, robustness to input perturbations, and the ability to classify data. These attributes make RBFNN highly suitable for deployment in real-world applications.

There is always an NN that can accurately approximate the $\varphi$ function as follows

$$\begin{array}{c}\hfill \mathcal{N}\left(\mathcal{X}\right)=\varphi \mathrm{with}\mathcal{N}\left(\mathcal{X}\right)={\phi }^T\mathsf{\Psi }\left(\mathcal{X}\right)+\mathcal{E},\end{array}$$

(25)

where the input of the RBFNN is denoted by $\mathcal{X}={\left[{\varrho }_1^T,{\stackrel{.}{\varrho }}_1^T\right]}^T$, $\mathcal{E}\in R^n$ represents the estimate error, $∥\mathcal{E}∥\le {\mathcal{E}}^{*}$ with ${\mathcal{E}}^{*}\ge 0$. The nonlinear function of the hidden nodes is represented by $\mathsf{\Psi }\left(\mathcal{X}\right)$, and the weight matrix is represented by ${\phi }^T\in R^{n\times m}$.

The estimate function of $\mathcal{N}\left(\mathcal{X}\right)$ is represented by $\widehat{\mathcal{N}}\left(\mathcal{X}\right)$, which corresponds to the output of the RBFNN, as shown below:

$$\begin{array}{c}\hfill \widehat{\mathcal{N}}\left(\mathcal{X}\right)={\widehat{\phi }}^T\mathsf{\Psi }\left(\mathcal{X}\right),\end{array}$$

(26)

where $\widehat{\phi }$ is the adaptable parameter vector.

Let $\tilde{\phi }=\phi -\widehat{\phi }$ be the estimation error of the weights. Therefore,

$$\begin{array}{c}\hfill \mathcal{N}\left(\mathcal{X}\right)-\widehat{\mathcal{N}}\left(\mathcal{X}\right)={\phi }^T\mathsf{\Psi }\left(\mathcal{X}\right)+\mathcal{E}-{\widehat{\phi }}^T\mathsf{\Psi }\left(\mathcal{X}\right)={\tilde{\phi }}^T\mathsf{\Psi }\left(\mathcal{X}\right)+\mathcal{E}.\end{array}$$

(27)

There exists a positive vector satisfying the following condition $|{\tilde{\phi }}^T\mathsf{\Psi }\left(\mathcal{X}\right)+\mathcal{E}|\le \Xi$.

We define a Gaussian function, which is selected as the nonlinear function of the hidden nodes, as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\mathsf{\Psi }\left(\mathcal{X}\right)=exp\left(\frac{-{\left(\mathcal{X}-{\eta }_l\right)}^T\left(\mathcal{X}-{\eta }_l\right)}{{\mathcal{L}}_l^2}\right),\hfill \\ l=1,2,\dots ,j.\hfill \end{array}\end{array}$$

(28)

In the network, the width and central parameter of the Gaussian function are denoted by $\mathcal{L}$ and $\eta$, respectively, while j represents the number of hidden layer nodes.

The optimal weight parameter $\phi$, obtained by minimizing the training error, is defined in the following equation:

$$\begin{array}{c}\hfill {\phi }_{\mathcal{N}}=argmin\left\{\underset{\mathcal{X}\in {\mathsf{\Theta }}_{\mathcal{X}}}{sup}\left|\mathcal{N}\left(\mathcal{X}\right)-\widehat{\mathcal{N}}\left(\mathcal{X},\widehat{\phi }\right)\right|\right\}.\end{array}$$

(29)

As a result, the NN given by Equation (26) is capable of accurately estimating any function of $\mathcal{N}\left(\mathcal{X}\right)$, as will be demonstrated in the following Lemma.

Lemma 3

([51,52]). For any real continuous function $\mathcal{N}\left(\mathcal{X}\right)$ defined on the compact set ${\mathsf{\Theta }}_{\mathcal{X}}\in R^n$ and any arbitrary constant $\mathcal{E}>0$ satisfying $\left|\mathcal{E}\right|⩽{\mathcal{E}}^{*}$ with ${\mathcal{E}}^{*}⩾0$, it is possible to construct a neural network $\widehat{\mathcal{N}}\left(\mathcal{X}\right)$ with a form similar to that of Equation (26) such that

$$\begin{array}{c}\hfill \underset{\mathcal{X}\in {\mathsf{\Theta }}_{\mathcal{X}}}{sup}\left|\mathcal{N}\left(\mathcal{X}\right)-\widehat{\mathcal{N}}\left(\mathcal{X},\widehat{\phi }\right)\right|<\mathcal{E}.\end{array}$$

(30)

Remark 3.

The input for the RBFNN is selected as $\mathcal{X}={\left[{\varrho }_1^T,{\stackrel{.}{\varrho }}_1^T\right]}^T$, which minimizes the transformed errors while bounding the tracking errors. As a result, we can obtain a more accurate estimation compared to using tracking errors as the input for the RBFNN.

4.3. Design of a Sliding Surface Based on the Transformed Error

Based on ${\varrho }_1$, the fixed-time convergence sliding surface is designed that obtains the properties of PPC as follows:

$$s={\varrho }_1+\frac{1}{{\alpha }_2^{\frac{1}{{\omega }_2}}}{\lceil {\gamma }_2{\lceil {\varrho }_1\rfloor }^{{\xi }_2}+{\stackrel{.}{\varrho }}_1\rfloor }^{\frac{1}{{\omega }_2}},$$

(31)

where ${\gamma }_2>0,{\alpha }_2>0,{\xi }_2>1$, and $\frac{1}{2}<{\omega }_2<1$.

A thorough examination of how the proposed sliding surface can effectively address the challenges posed by singularity problems will be provided in a later subsection.

When $s=0$, indicating that the sliding surface defined in Equation (31) has converged to zero, we can obtain the following:

$${\stackrel{.}{\varrho }}_1=-{\gamma }_2{\lceil {\varrho }_1\rfloor }^{{\xi }_2}-{\alpha }_2{\lceil {\varrho }_1\rfloor }^{{\omega }_2}.$$

(32)

As stated in Lemma 2, the sliding surface defined in Equation (31) shares the same properties as those proposed in a previous study [48]. Based on this, we can determine the settling time of the system under the proposed control scheme. Specifically, the settling time is given as $T\left({\varrho }_{10}\right)<T_{max}\triangleq \frac{1}{{\gamma }_2({\xi }_2-1)}+\frac{1}{{\alpha }_2(1-{\omega }_2)}$, where ${\varrho }_{10}$ is the initial value of ${\varrho }_1$.

4.4. Design of the FNN-PPCM

Set ${\vartheta }_2={\gamma }_2{\lceil {\varrho }_1\rfloor }^{{\xi }_2}+{\stackrel{.}{\varrho }}_1$, so, ${\left|{\vartheta }_2\right|}^{\frac{1}{{\omega }_2}-1}=|{\gamma }_2{\lceil {\varrho }_1\rfloor }^{{\xi }_2}+{\stackrel{.}{\varrho }}_1{|}^{\frac{1}{{\omega }_2}-1}$. Then, differentiating the sliding surface (31) according to time, we obtain:

$$\stackrel{.}{s}={\stackrel{.}{\varrho }}_1+\frac{1}{{\alpha }_2^{\frac{1}{{\omega }_2}}}\frac{1}{{\omega }_2}{\left|{\vartheta }_2\right|}^{\frac{1}{{\omega }_2}-1}\left({\gamma }_2{\xi }_2{\lceil {\varrho }_1\rfloor }^{{\xi }_2-1}{\stackrel{.}{\varrho }}_1+{\stackrel{..}{\varrho }}_1\right).$$

(33)

Inserting dynamics (24) into Equation (33) gives the following:

$$\stackrel{.}{s}={\stackrel{.}{\varrho }}_1+{\mathcal{A}}_2\left({\gamma }_2{\xi }_2{\lceil {\varrho }_1\rfloor }^{{\xi }_2-1}{\stackrel{.}{\varrho }}_1+\mathsf{\Theta }\left(\mathcal{Q}\left(y_1\right)\mathcal{U}+\mathcal{H}\left(y\right)+\varphi -{\stackrel{..}{y}}_{1d}-\overline{\mathcal{P}}\right)\right),$$

(34)

where ${\mathcal{A}}_2=\mathrm{diag}\left\{{\mathcal{A}}_{2i}\right\},{\mathcal{A}}_{2i}=\frac{1}{{\alpha }_2^{\frac{1}{{\omega }_2}}}\frac{1}{{\omega }_2}{\left|{\vartheta }_{2i}\right|}^{\frac{1}{{\omega }_2}-1}\ge 0$, and $i=1,\dots ,n$.

Based on Equation (34), a controller is synthesized below:

$$\mathcal{U}={\mathcal{U}}_{eq}+{\mathcal{U}}_r,$$

(35)

where the two terms, ${\mathcal{U}}_{eq}$ and ${\mathcal{U}}_r$, are formulated as follows:

$${\mathcal{U}}_{eq}=-{\mathcal{Q}}^{-1}\left({\mathsf{\Theta }}^{-1}{\gamma }_2{\xi }_2{\lceil {\varrho }_1\rfloor }^{{\xi }_2-1}{\stackrel{.}{\varrho }}_1+\mathcal{H}\left(y\right)+\widehat{\varphi }-{\stackrel{..}{y}}_{1d}-\overline{\mathcal{P}}\right)-{\mathcal{Q}}^{-1}{\mathsf{\Theta }}^{-1}{\mathcal{A}}_2^{-1}{\stackrel{.}{\varrho }}_1,$$

(36)

$${\mathcal{U}}_r=-{\mathcal{Q}}^{-1}\left(\Xi \mathrm{sign}\left(s\right)+\mathsf{\Gamma }{\lceil s\rfloor }^{{\rho }_2}\right),$$

(37)

where $\mathsf{\Gamma }=\mathrm{diag}\left(1+{\psi }_2^2{\left|s_i\right|}^{2(1-{\rho }_2)}\right)$, $i=1,\dots ,n$, ${\kappa }_2,{\psi }_2>0$, and $0<{\rho }_2<1$ are constants.

Theorem 2.

If all uncertain terms denoted as ϕ are approximated by an RBFNN, the proposed control law will ensure fixed-time global stability. Additionally, it guarantees both the maximum overshoot and steady-state performance of the positional error, according to prescribed criteria.

A summary of the above design is shown in Figure 2.

Remark 4.

The FNN-PPCM’s stabilization process involves two stages. During the first stage, the transformed errors are directed towards the sliding surface within a bounded reaching time of $T_{r\epsilon }<T_r+\epsilon \left(\mathsf{\tau }\right)$, where $T_r<T_{max}\triangleq \frac{\pi }{2{\kappa }_2{\psi }_2(1-{\rho }_2)}$, and $\epsilon \left(\mathsf{\tau }\right)$ is defined in [50] while ensuring that the maximum overshoot is limited. In the second stage, the transformed errors are maintained at a steady-state on the sliding surface until they reach equilibrium, with a settling time of $T\left(y_{e0}\right)<T_{max}\triangleq \frac{1}{{\gamma }_2({\xi }_2-1)}+\frac{1}{{\alpha }_2(1-{\omega }_2)}$.

Proof of Theorem 2.

The concept of the following proof draws inspiration from previous works such as [22,50].

The following equation can be obtained by using the control inputs (35)–(37) in Equation (34):

$$\begin{array}{c}\stackrel{.}{s}={\mathcal{A}}_2\mathsf{\Theta }\left(-\Xi \mathrm{sign}\left(s\right)+{\tilde{\phi }}^T\mathsf{\Psi }\left(\mathcal{X}\right)+\mathcal{E}-{\kappa }_2\mathsf{\Gamma }{\lceil s\rfloor }^{{\rho }_2}\right).\hfill \end{array}$$

(38)

After choosing the Lyapunov function $L_2=\frac{B_0}{\pi }\left(s_i^2+{\tilde{\phi }}_i^T\mathcal{K}\widetilde{{\phi }_i}\right)$, where $\widetilde{{\phi }_i}={\phi }_i-{\widehat{\phi }}_i$, it is then differentiated to obtain the following expression:

$$\begin{array}{c}{\stackrel{.}{L}}_2=\frac{2B_0}{\pi }\left(s_i{\stackrel{.}{s}}_i-{\tilde{\phi }}_i\mathcal{K}\stackrel{.}{{\widehat{\phi }}_i})\right)\hfill \\ =\frac{2B_0}{\pi }{\mathsf{\Theta }}_i{\mathcal{A}}_{2i}{s}_i\left(-{\Xi }_i\mathrm{sign}\left(s_i\right)+{\tilde{\phi }}_i^T\mathsf{\Psi }\left(\mathcal{X}\right)+{\mathcal{E}}_i-{\kappa }_2\left(1+{\psi }_2^2{\left|s_i\right|}^{2(1-{\rho }_2)}\right){\left|s_i\right|}^{{\rho }_2}\mathrm{sign}\left(s_i\right)\right)-\frac{2B_0}{\pi }{\tilde{\phi }}_i^T\mathcal{K}\stackrel{.}{{\widehat{\phi }}_i}\hfill \end{array}.$$

(39)

With the updating rule of the RBFNN being ${\stackrel{.}{\widehat{\phi }}}_i={\mathsf{\Theta }}_i{\mathcal{A}}_{2i}{\mathcal{K}}^{-1}\mathsf{\Psi }\left(\mathcal{X}\right)s_i$ with $\mathcal{K}=\mathrm{diag}\left\{{\mathcal{K}}_i\right\}$, ${\mathcal{K}}_i>0$, and $\mathcal{K}\in R^{m\times m}$, it can be inferred that:

$$\begin{array}{c}{\stackrel{.}{L}}_2=\frac{2B_0}{\pi }{\mathsf{\Theta }}_i{\mathcal{A}}_{2i}\left(-{\Xi }_i|s_i|+s_i{\tilde{\phi }}_i^T\mathsf{\Psi }\left(\mathcal{X}\right)+s_i{\mathcal{E}}_i-{\kappa }_2\left(1+{\psi }_2^2|s_i{|}^{2(1-{\rho }_2)}\right)|s_i{|}^{{\rho }_2+1}-{\tilde{\phi }}_i^T\mathsf{\Psi }\left(\mathcal{X}\right)s_i\right)\hfill \\ =\frac{2B_0}{\pi }{\mathsf{\Theta }}_i{\mathcal{A}}_{2i}\left(-{\Xi }_i|s_i|+s_i{\mathcal{E}}_i-{\kappa }_2\left(1+{\psi }_2^2|s_i{|}^{2(1-{\rho }_2)}\right)|s_i{|}^{{\rho }_2+1}\right)\hfill \\ \le \frac{2B_0}{\pi }{\mathsf{\Theta }}_{imin}{\mathcal{A}}_2\left(-{\Xi }_i|s_i|+s_i{\mathcal{E}}_i-{\kappa }_2\left(1+{\psi }_2^2|s_i{|}^{2(1-{\rho }_2)}\right)|s_i{|}^{{\rho }_2+1}\right)\hfill \\ \le {\mathcal{A}}_{2i}\left(-{\kappa }_2\left(1+{\psi }_2^2|s_i{|}^{2(1-{\rho }_2)}\right)|s_i{|}^{{\rho }_2+1}\right)\hfill \\ \le 0\hfill \end{array}.$$

(40)

Since ${\stackrel{.}{L}}_2(s\left(t\right),\tilde{\phi })\le 0$, that means $L_2(s\left(t\right),\tilde{\phi })\le L_2(s\left(0\right),\tilde{\phi })$. Hence, $s\left(t\right)$ and $\tilde{\phi }$ are bounded. Let us define a function ${\rm Y}={\mathcal{A}}_{2i}\left({\kappa }_2\left(1+{\psi }_2^2|s_i{|}^{2(1-{\rho }_2)}\right)|s_i{|}^{{\rho }_2+1}\right)\le -{\stackrel{.}{L}}_2$. By integrating the function ${\rm Y}$ over time, we can derive the following inequality:

$$\underset{0}{\overset{t}{\int }}{\rm Y}\left(t\right)dt\le L_2\left(0\right)-L_2\left(t\right)=L_2(s\left(0\right),\tilde{\phi })-L_2(s\left(t\right),\tilde{\phi }).$$

(41)

As $L_2(s\left(0\right),\tilde{\phi })$ is bounded and $L_2(s\left(t\right),\tilde{\phi })$ is a decreasing and bounded function, the boundedness of $s_i,{\tilde{\phi }}_i,i=1,\dots ,n$ is ensured. Since s depends on ${\varrho }_1$ and ${\stackrel{.}{\varrho }}_1$, the tracking error is also guaranteed to be bounded.

By defining the Lyapunov function as $L_3=\frac{B_0}{\pi }{s}_i^2$, we can establish the global fixed-time stability of the system. Hence, the time derivative of the Lyapunov function can be obtained as follows:

$$\begin{array}{c}{\stackrel{.}{L}}_3=\frac{2B_0}{\pi }{\mathsf{\Theta }}_i{\mathcal{A}}_{2i}{s}_i\left(-{\Xi }_i\mathrm{sign}\left(s_i\right)+{\tilde{\phi }}_i^T\mathsf{\Psi }\left(\mathcal{X}\right)+{\mathcal{E}}_i-{\kappa }_2\left(1+{\psi }_2^2{\left|s_i\right|}^{2(1-{\rho }_2)}\right){\left|s_i\right|}^{{\rho }_2}\mathrm{sign}\left(s_i\right)\right)\hfill \\ =\frac{2B_0}{\pi }{\mathsf{\Theta }}_i{\mathcal{A}}_{2i}\left(-{\Xi }_i|s_i|+s_i{\tilde{\phi }}_i^T\mathsf{\Psi }\left(\mathcal{X}\right)+s_i{\mathcal{E}}_i-{\kappa }_2\left(1+{\psi }_2^2|s_i{|}^{2(1-{\rho }_2)}\right)|s_i{|}^{{\rho }_2+1}\right)\hfill \\ \le \frac{2B_0}{\pi }{\mathsf{\Theta }}_{imin}{\mathcal{A}}_{2i}\left(-{\Xi }_i|s_i|+s_i{\tilde{\phi }}_i^T\mathsf{\Psi }\left(\mathcal{X}\right)+s_i{\mathcal{E}}_i-{\kappa }_2\left(1+{\psi }_2^2|s_i{|}^{2(1-{\rho }_2)}\right)|s_i{|}^{{\rho }_2+1}\right)\hfill \\ \le {\mathcal{A}}_{2i}\left(-{\Xi }_i|s_i|+s_i{\tilde{\phi }}_i^T\mathsf{\Psi }\left(\mathcal{X}\right)+s_i{\mathcal{E}}_i-{\kappa }_2\left(1+{\psi }_2^2|s_i{|}^{2(1-{\rho }_2)}\right)|s_i{|}^{{\rho }_2+1}\right)\hfill \end{array}.$$

(42)

The Gaussian function, defined above, is constrained within the range of 0 and 1, as represented by $0\le \mathsf{\Psi }\left(\mathcal{X}\right)\le 1$. It can be deduced that the condition $|{\tilde{\phi }}_i^T\mathsf{\Psi }\left(\mathcal{X}\right)+{\mathcal{E}}_i|\le {\Xi }_i$, $i=1,\dots ,n$ is fulfilled. Hence, we can achieve:

$$\begin{array}{c}{\stackrel{.}{L}}_3\le {\mathcal{A}}_2\left(-{\kappa }_2\left(1+{\psi }_2^2|s_i{|}^{2(1-{\rho }_2)}\right)|s_i{|}^{{\rho }_2+1}\right)\hfill \\ \le -{\mathcal{A}}_2\left({\kappa }_2|s_i{|}^{{\rho }_2+1}\right)\hfill \\ \le {\mathcal{A}}_2\left(-\sqrt{2}{\kappa }_2{\left(L_3\right)}^{\frac{{\rho }_2+1}{2}}\right)\hfill \end{array}.$$

(43)

It can be observed that Equation (43) is identical to the result obtained from Equation (16). Thus, we can conclude that the sliding surface $s_i=0,i=1,\dots ,n$ can be achieved from any initial condition within a bounded time $T_{r\epsilon }<T_r+\epsilon \left(\mathsf{\tau }\right)$ in the phase plane, as demonstrated in [50]. Here, $\epsilon \left(\mathsf{\tau }\right)$ is defined as a small time margin associated with the boundary width, where $\mathsf{\tau }={\omega }_2^{\frac{{\omega }_2}{1-{\omega }_2}}{\alpha }_2^{\frac{1}{1-{\omega }_2}}$. The proof has been thoroughly validated. □

5. Simulations

The effectiveness of the FNN-PPCM is demonstrated through the simulation of trajectory-tracking motion control. The simulations are conducted using MATLAB/SIMULINK and evaluate key aspects such as transient response, the convergence of sliding surface, maximum overshoot, and steady-state of the tracking errors. Other aspects generated from the proposed FNN-PPCM, such as tracking accuracy, chattering resolution, robustness to uncertainty components, and observer approximation, are also considered thoroughly through comparison with other equivalent forms, such as SMC, TSMC, and FTSMC. The 3-DOF robotic manipulator used in the simulations is based on the dynamic mathematics and kinematic design presented in previous studies [53,54], and the system parameters are selected from [43,55]. The construction of the robot is designed using both MATLAB/SIMULINK and SOLIDWORKS software, which is described in detail in [43,55]. In the simulations, Euler’s method is used to solve the differential equations with a sampling time of $t_s={10}^{-3}$.

5.1. Configuration of the Experimental System

In Appendix A.4,

Table A1. Specification of Design Parameters for a 3-DOF Robot System.

Description	Link 1	Link 2	Link 3
Link Length (m)	$l_1=0.25$	$l_2=0.7$	$l_3=0.6$
Link Weight (kg)	$m_1=33.429$	$m_2=34.129$	$m_3=15.612$
Center of Mass (mm)	$\begin{array}{c}{l}_{c1x}=0\\ l_{c1y}=0\\ l_{c1z}=-0.7461\end{array}$	$\begin{array}{c}{l}_{c2x}=0.3477\\ l_{c2y}=0\\ l_{c2z}=0\end{array}$	$\begin{array}{c}{l}_{c3x}=0.3142\\ l_{c3y}=0\\ l_{c3z}=0\end{array}$
Inertia (kg·m${}^2$)	$\begin{array}{c}\hfill I_{1xx}=0.7486\hfill \\ \hfill I_{1yy}=0.5518\hfill \\ \hfill I_{1zz}=0.5570\hfill \end{array}$	$\begin{array}{c}\hfill I_{2xx}=0.3080\hfill \\ \hfill I_{2yy}=2.4655\hfill \\ \hfill I_{2zz}=2.3938\hfill \end{array}$	$\begin{array}{c}\hfill I_{3xx}=0.0446\hfill \\ \hfill I_{3yy}=0.7092\hfill \\ \hfill I_{3zz}=0.7207\hfill \end{array}$

contains the fundamental design specifications for the robot system, such as the dimensions of the links, the position of the center of mass, and the inertia. Additionally, Figure 3 provides a visual depiction of the robot model.

The primary objective of the robot is to track a moving trajectory using its end effector, as outlined below:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathrm{X}=0.85-0.01t\hfill \\ \mathrm{Y}=0.2+0.2sin\left(0.5t\right)\hfill \\ \mathrm{Z}=0.7+0.2cos\left(0.5t\right)\hfill \end{array}\right.(m).\end{array}$$

(44)

The effectiveness and robustness of the proposed solution are evaluated through simulations in the presence of uncertain factors such as calculated dynamical errors, frictions, and external disturbances. These uncertain factors are assumed based on

Table 1. Assumed Uncertain Terms.

Assumed Uncertainties	Functions
Dynamical Errors	${\mathcal{M}}_{\delta }\left(q\right)=0.2{\mathcal{M}}_c\left(q\right)$
	${\mathcal{C}}_{\delta }\left(q,\stackrel{.}{q}\right)=0.2{\mathcal{C}}_c\left(q,\stackrel{.}{q}\right)$
	${\mathcal{G}}_{\delta }\left(p\right)=0.2{\mathcal{G}}_c\left(q\right)$
Frictions ${\mathcal{F}}_r\left(\stackrel{.}{q}\right)(\mathrm{N}·\mathrm{m})$	${\mathcal{F}}_{r1}\left(\stackrel{.}{q}\right)=0.1\mathrm{sign}\left({\stackrel{.}{q}}_1\right)+2{\stackrel{.}{q}}_1$
	${\mathcal{F}}_{r2}\left(\stackrel{.}{q}\right)=0.1\mathrm{sign}\left({\stackrel{.}{q}}_2\right)+2{\stackrel{.}{q}}_2$
	${\mathcal{F}}_{r3}\left(\stackrel{.}{q}\right)=0.1\mathrm{sign}\left({\stackrel{.}{q}}_3\right)+2{\stackrel{.}{q}}_3$
Exterior Disturbances $\mathcal{D}(\mathrm{N}·\mathrm{m})$	${\mathcal{D}}_1=4sin\left(t\right)$
	${\mathcal{D}}_2=5sin\left(t\right)$
	${\mathcal{D}}_3=6sin\left(t\right)$

.

To conduct a fair assessment of the control performance of the FNN-PPCM, it is necessary to compare it with conventional SMC and two other published methods, namely TSMC and FTSMC. The control law designs of the three aforementioned methods are briefly explained in Appendix A.3. However, due to the differences in the controllers’ structures, achieving complete fairness in the comparison is challenging. To ensure fairness as much as possible, we used the control parameters for the compared controllers from their original papers to simulate the robot. In contrast, we selected the control parameters for the FNN-PPCM through experimentation to achieve optimal performance and maximize its potential. Additionally, the robot states were initialized with identical initial conditions. The selected control parameters for the FNN-PPCM are presented in

Table 2. Control Parameters of the Proposed Algorithm.

Description	Symbol	Value
PPC	$B_0,B_1,B_{\infty },\lambda$	$\left[\begin{array}{ccc}0.34& 0.15& 0.3\end{array}\right],0.06,0.0006,8$
RBFNN	${\phi }_0,\eta ,j,\mathcal{L},{\mathcal{K}}_i$	$\mathrm{zeros}(7,3)$, $0.2\times \left[\begin{array}{ccccccc}-1.5& -1& -0.5& 0& 0.5& 1& 1.5\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ -1.5& -1& -0.5& 0& 0.5& 1& 1.5\end{array}\right]\in R^{6\times 7},7,5,0.01$
FNN-PPCM	$\begin{array}{c}{\gamma }_2,{\alpha }_2,{\xi }_2,{\omega }_2\hfill \\ {\kappa }_2,{\psi }_2,{\rho }_2,\Xi \hfill \end{array}$	$\begin{array}{c}10,2,1.4,0.8\hfill \\ 2,0.2,0.8,\mathrm{diag}(0.01,0.01,0.01)\hfill \end{array}$

.

When regulating the motion of a robot arm’s end effector to track a specified trajectory, it is crucial to assess the accuracy and robustness of its performance. This can be accomplished by analyzing the steady-state errors (SSEs) once they have converged to the origin. To calculate the SSEs, it is necessary to determine the time interval during which the system has reached steady-state. The root-mean-square method (RMSM) can then be applied to calculate the SSEs during this steady-state period. By doing so, we can assess the performance of each control method and compare their effectiveness in achieving accurate and stable trajectory tracking.

The formula of the RMSM is given by:

$$\begin{array}{c}{E}_1=\sqrt{\frac{1}{\mathrm{K}}{\displaystyle \sum _{i=1}^{\mathrm{K}}}{\left|\left(q_{d1i}-q_{1i}\right)\right|}^2};E_2=\sqrt{\frac{1}{\mathrm{K}}{\displaystyle \sum _{i=1}^{\mathrm{K}}}{\left|\left(q_{d2i}-q_{2i}\right)\right|}^2};E_3=\sqrt{\frac{1}{\mathrm{K}}{\displaystyle \sum _{i=1}^{\mathrm{K}}}{\left|\left(q_{d3i}-q_{3i}\right)\right|}^2},\hfill \end{array}$$

(45)

where the variable $\mathrm{K}$ represents the number of calculated samples. The root-mean-square errors (RMSEs) for each joint (Joint 1, Joint 2, and Joint 3) are denoted as $E_1$, $E_2$, and $E_3$, respectively. The real joint angle vector at time index i is denoted as ${[q_{1i},q_{2i},q_{3i}]}^T$, while the desired joint angle vector is represented as ${[q_{d1i},q_{d2i},q_{d3i}]}^T$.

5.2. Discussion of Performance Results

The regulation problem involves analyzing the system’s ability to maintain a desired trajectory despite the presence of external disturbances and unknown internal dynamics. This problem is common in robotics, where the robot is required to follow a trajectory while compensating for various disturbances and uncertainties, such as friction, joint flexibility, and sensor noise. The objective is to design a control system that can accurately track the desired trajectory while minimizing tracking errors and maintaining stability. A well-designed controller can also improve the robustness of the system, allowing it to operate reliably even in the presence of unforeseen disturbances or model uncertainties.

By selecting the transformed errors as the input for the RBFNN, we can minimize these errors while bounding the tracking errors. This results in a more accurate and faster estimation, which is superior to using tracking errors as the input for the RBFNN. Based on the uncertainty terms listed in Table 1, the RBFNN was able to produce a highly accurate approximation with minimal error, as shown in Figure 4. This suggests that the RBFNN is capable of generating an almost exact model of the lumped uncertain terms, which is suitable for control design. Such a model can significantly enhance the overall performance of robotic manipulators operating in the presence of disturbances and uncertainties, leading to improved tracking and stabilization performance, enhanced safety, and increased reliability of the controlled system.

Firstly, the prescribed performance of the FNN-PPCM is evaluated based on two criteria: the convergence rate and the maximum overshoot within a pre-defined bounded domain, as illustrated in Figure 5, Figure 6 and Figure 7. We will evaluate these two performance metrics in the period from the 0th second to the 2nd second. To ensure a fair comparison in terms of convergence, the same initial conditions of the system states are initialized. By utilizing the FNN-PPCM and appropriately adjusting the design parameters $B_0,B_1,B_{\infty }$, and $\lambda$, it is possible to achieve both performance indices within a specified performance domain. Nevertheless, as can be seen in the zoomed-in sections of Figure 5, Figure 6 and Figure 7, none of the other three methods meet both performance indices. The FNN-PPCM achieves better performance in terms of faster convergence and smaller maximum overshoot compared to the other controllers, indicating its superiority in regulating the robotic arm.

Secondly, the tracking error accuracy is evaluated using the RMSM given by Equation (45). The RMSM is calculated for the error variables during the sliding motion phase towards the origin between the 2nd and 20th seconds of the simulation. The resulting RMSEs are reported in

Table 3. Comparison of RMSEs from Three Controllers for Trajectory Tracking.

Control System	${\mathit{E}}_1$	${\mathit{E}}_2$	${\mathit{E}}_3$
SMC	$\begin{array}{c}\hfill 1.890\times {10}^{-4}\hfill \end{array}$	$\begin{array}{c}\hfill 3.376\times {10}^{-4}\hfill \end{array}$	$\begin{array}{c}\hfill 3.502\times {10}^{-4}\hfill \end{array}$
TSMC	$\begin{array}{c}\hfill 2.475\times {10}^{-5}\hfill \end{array}$	$\begin{array}{c}\hfill 4.672\times {10}^{-5}\hfill \end{array}$	$\begin{array}{c}\hfill 5.264\times {10}^{-5}\hfill \end{array}$
FTSMC	$\begin{array}{c}\hfill 2.383\times {10}^{-5}\hfill \end{array}$	$\begin{array}{c}\hfill 3.941\times {10}^{-5}\hfill \end{array}$	$\begin{array}{c}\hfill 5.030\times {10}^{-5}\hfill \end{array}$
FNN-PPCM	$\begin{array}{c}\hfill 5.06\times {10}^{-7}\hfill \end{array}$	$\begin{array}{c}\hfill 2.39\times {10}^{-7}\hfill \end{array}$	$\begin{array}{c}\hfill 2.737\times {10}^{-6}\hfill \end{array}$

. Figure 8 and Figure 9 depict the trajectory of the robot arm’s effective point, controlled separately using four different methods, all of which were generally successful in completing the orbital tracking of the robotic arm. Figure 10 provides a comparison of the end effector and the moving trajectory in terms of X-axis, Y-axis, and Z-axis errors. Tracking accuracy was assessed using the RMSE metric for joint errors between the actual robot trajectory and the reference trajectory.

It should be noted that the tracking errors between the actual robot trajectory and the reference trajectory at each joint are compared in Figure 5, Figure 6 and Figure 7. The proposed strategy was found to have the highest tracking accuracy and smallest SSEs, as shown in Figure 5, Figure 6 and Figure 7, Figure 10, and Table 3. Both the TSMC and FTSMC were able to effectively track the trajectory with relatively small SSEs that remained within the prescribed performance boundaries. However, the SMC occasionally resulted in SSEs that exceeded the performance boundaries. Overall, all three methods demonstrated satisfactory performance in trajectory tracking, with the FNN-PPCM exhibiting the smallest RMSE values for joint errors.

Figure 11 and Figure 12 demonstrate that the FNN-PPCM significantly reduces chattering behavior in the control torque signal in the presence of RBFNN. The FNN-PPCM exhibits a high level of robustness and achieves precise tracking accuracy. Conversely, the other three methods experience similar chattering behavior due to their use of the same sliding gain to compensate for uncertainties.

Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 demonstrate that combining SMC, PPC, and RBFNN is a highly effective solution for achieving tracking control of the robot while mitigating the negative effects of uncertainty and chattering.

6. Conclusions

This article presents a novel FNN-PPCM to address the tracking control problem of robot manipulators, even in the presence of disturbances and dynamic uncertainties. The FNN-PPCM is capable of controlling tracking errors within a prescribed performance domain, where the boundaries of these errors are symmetrical to zero. As a result, the tracking error becomes zero when the transformed error is zero. The paper introduces a new fixed-time convergence sliding surface based on the transformed error, which allows for determining the maximum allowable size of the SSEs. The method also addresses other factors such as convergence index and singularity elimination. The design procedure of the FNN-PPCM is based on fixed-time SMC combined with PPC. The FNN-PPCM integrates an RBFNN for precise uncertainty estimation, the unconstrained dynamics, and a fixed-time convergence sliding surface based on the transformed error. By combining these elements, the controller can achieve global fixed-time stability for robotic manipulators, while maintaining prescribed performance, reducing chattering, and enhancing robustness against uncertain elements. The Lyapunov theory is employed to demonstrate the closed-loop stability of the tracking method. Simulation results confirm the effectiveness and robustness of the FNN-PPCM.

Our paper focuses on the study of a robot system with matched uncertain terms, which include interior uncertain dynamics and exterior disturbances. In the future, we plan to expand the scope of our research by considering time-varying mismatched and matched uncertainties for robot systems.