Time-Synchronized Fault-Tolerant Control for Robotic Manipulators

Abstract

This article proposes a time-synchronized fault-tolerant convergence control method for n-degree-of-freedom robotic manipulators. The main challenge lies in driving the tracking errors of all joints to converge simultaneously, especially in the presence of system faults, external disturbances, and model uncertainties. We introduce a normalized sign function that guarantees the property of ratio persistence for all joints and plays a crucial role in time-synchronized convergence control. A time-synchronized convergence observer is proposed that not only adopts a time-synchronized convergence control framework but also overcomes the lumped uncertainty term, which includes the system fault components, external disturbances, and system uncertainties. A salient feature of this method is that, regardless of the initial state and various uncertainties, each component of the robot manipulator system can simultaneously converge to an equilibrium point. Simulations conducted on a two-link robotic manipulator demonstrate the notable benefits of the designed time-synchronized control method, as evidenced by the comparative results.

1. Introduction

Robotic manipulators are increasingly popular in various fields, including in industrial, medical, and agricultural applications. Consequently, the control of robotic manipulators has garnered significant interest among scholars and researchers [1,2,3,4]. Substantial contributions have been made in the field of robotic manipulator control. For example, robust control was proposed [5] using a fuzzy logic system to offset disturbances and unknown system uncertainties, the stability of the uniform ultimate boundedness (UUB) was achieved, and the tracking errors converged exponentially. The experiments verified the good performance of the method. In [6], an event-driven strategy was designed for the tracking task of a robot and reduce control energy, taking into account transient performance. To improve the learning capacity and control efficacy of robotic manipulators, intelligent control strategies employing critic learning methods and fuzzy logic are proposed in [7]. The system states proven in the above references [5,6,7] are all UUB.

To improve the system convergence time, the finite-time control of robotic manipulators was studied in [8,9,10]. Among them, a novel controller was proposed [8] involving a super-twisting method for robotic manipulators, which had robustness to nonlinear model uncertainties and measurement noise. Tracking control for a robot manipulator was proposed based on $H_{\infty }$ in [9] considering prescribed performance and uncertain inertia, which drive all states to converge in a certain time dependent on the initial values. A finite-time tracking control method was designed in [10], and the joint-drift problem was solved.

To enhance system convergence performance, fixed-time control has attracted considerable attention from researchers [11,12,13,14,15]. A controller for constrained robotic manipulators under the conditions of system outputs and system uncertainties using barrier Lyapunov function and neural networks was proposed [11]. Different from previous reinforcement learning (RL)-based finite-time control methods, ah RL-based fixed-time framework is designed in [12] to achieve optimal performance. In [13], adaptive technology was adopted to compensate for robotic manipulator uncertainties, and fixed-time stability was achieved using sliding-mode control without a singularity problem. Control strategies for robots under uncertainty, by utilizing a performance function and a predefined fixed- time method, are proposed in [14] and [15], respectively.

In addition, during robotic manipulator operations, faults such as sensor or actuator failures are common. Without appropriate proactive measures, these can cause the control system to fail. Consequently, many scholars have focused on fault-tolerant control and achieved successful results. Fuzzy logic and disturbance observers have been collaboratively used to address fault components, as demonstrated in [16,17]. Utilizing backstepping and sliding-mode technologies, UUB, finite, and fixed-time stabilities have been achieved. Robust adaptive technology to address actuator faults in robotic manipulators was proposed [5]. Additionally, fault tolerance was addressed within multiagent systems using the critic–actor approach and barrier functions, as described in [18].

However, neither unified uniformly ultimately boundedness nor finite or fixed-time control, as referred to above, can achieve simultaneous convergence performance of time-synchronized (TS) control, where the tracking errors of all joints converge to equilibrium at the exact same moment. In certain industrial sectors, aerospace engineering, and other precision operations, time-synchronized control is crucial and may determine the success or failure of a task [19]. To implement TS control in robotic manipulator systems, ref. [20] introduced a synchronous neural network strategy, ultimately achieving fixed TS control. In [21], a finite-time synchronization method was designed for primary–seconard neural networks by constructing a synchronization error system. Unlike the methods discussed in [20,21,22,23], TS control in [19] was implemented using a novel concept of ratio persistence, along with sliding-mode control. These studies show that when a system integrates “ratio persistence” with “finite-time control” or “fixed-time control”, TS stability is achieved, causing all state elements to converge simultaneously. Simulations have demonstrated the effectiveness of this concurrent convergence. The primary distinctions between the methods in [20,21,22] and those in [19] are also discussed in [24]. Subsequently, the application of TS control has been extended to MIMO systems in maritime and aerospace domains. In [25], a TS control method was developed for maritime surface vehicles that ensures all states converge to the equilibrium point simultaneously while accounting for external disturbances. To improve the transmission security of chaotic systems, [26] proposed a fixed TS control approach. The work points out that TS control can be considered as an exceptional form of finite/fixed-time control, but it requires more stringent conditions, i.e., for all state components to converge simultaneously. However, it is important to recognize certain factors that may constrain the advancement of the time-synchronized control of robotic manipulators: (1) system fault components that could render the synchronous convergence of all robotic joints unfeasible or cause abrupt control system failures; (2) as described above, the system friction matrix or external disturbances that can significantly impair TS convergence performance, with time-synchronized stability requiring stricter conditions than finite/fixed-time stability; (3) the formulation of synchronization and cross-coupling errors between primary and secondary joints in n-degree-of-freedom (DOF) manipulators, introducing complexities to the control system and the potential for additional stability issues.

Inspired by the preceding discussion and to achieve the synchronized control of manipulators while addressing the cross-coupling issue among joints, this article develops a time-synchronized control method that diverges from those presented in [20,21,22].

The following is a summary of this work’s foremost contributions:

• Compared to [21,22,23], we propose a time-synchronized control method that reduces the complexity of the controller’s structure and relaxes the criteria for establishing synchronization and cross-coupling errors. Unlike the fixed-time control approaches in [11,12,13,14,15], our method ensures that the variables across all system state degrees converge to an equilibrium simultaneously within a fixed period, regardless of the initial circumstances. The proposed method can pave the way for further investigation sof the robotic time-synchronization problem.
• Contrary to the TS control proposed in [19,25,26], our framework incorporates system fault components, rendering it more applicable to real-world engineering situations.
• A time-synchronized observer is proposed to manage lumped uncertainties, including system faults, external disturbances, and system uncertainties (EDSU). This observer not only estimates uncertainty precisely within a finite timeframe, enhancing tracking accuracy, but is also well integrated into the TS control framework. Consequently, high-precision tracking and robust performance are achieved, even in the face of actuator faults.

2. Preliminaries

Consider the following scalar system:

$$\dot{x}=f\left(x\right),$$

(1)

where $f:U\to {\mathbb{R}}^n$ is a continuous function in an open neighborhood U with origin $x=0$. $f\left(0\right)=0,x\left(0\right)=0,x\in U\subset {\mathbb{R}}^n$.

Lemma 1

([27]). The system’s origin, as characterized by (1), achieves fast finite-time stability if there exists a positive definite Lyapunov function $V\left(x\right)$ satisfying

$$\dot{V}\left(x\right)\le -\alpha V\left(x\right)-\beta V^{\gamma }\left(x\right),$$

(2)

where α, β, and γ are positive constants, and $0<\gamma <1$. The settling time $T\left(x_0\right)$, which is dependent on the initial states $x_0$, is given by

$$T\left(x_0\right)\le {\displaystyle \frac{1}{\alpha (1-\gamma )}}ln\left({\displaystyle \frac{\alpha V^{1-\gamma }\left(x_0\right)+\beta }{\beta }}\right).$$

(3)

Lemma 2

([24]). If there exists a continuous radial bounded function $V\left(x\right):{\mathbb{R}}^n⟶{\mathbb{R}}^{+}\cup \left\{0\right\}$, the following is satisfied:

$$\dot{V}\left(x\right)\le -{\gamma }_1{V}^m\left(x\right)-{\gamma }_2{V}^n\left(x\right),$$

(4)

where ${\gamma }_1$, ${\gamma }_2$, m, and n are all positive constants, $0<m<1$, and $n>1$. The system’s origin, described by (1), achieves fixed-time stability, where the settling time T is unaffected by the initial states of the system and is given by

$$T\le T_{max}={\displaystyle \frac{1}{{\gamma }_1(1-m)}}+{\displaystyle \frac{1}{{\gamma }_2(n-1)}}.$$

(5)

Definition 1

([19]). The state x of the system defined by (1) exhibits ratio persistence if

$${\displaystyle \frac{x}{\parallel x\parallel }}=\xi {\displaystyle \frac{f\left(x\right)}{\parallel f\left(x\right)\parallel }},x\ne 0,$$

(6)

where ξ represents the direction, and $\xi \in [-1,1]$.

Definition 2.

The solution of (1) achieves TS stability if, for all $x_0$, there exists a finite-time constant T, such that $x_i\left(t\right)=0(i=1,2,\cdots ,n)$ for $t\ge T$ and for all nonzero initial conditions $x_i\left(0\right)$, whereas $x_i\left(t\right)\ne 0$ for $t<T$. The time function T is called the settling time of TS stability. Furthermore, if the converge time T is dependent on the initial states of the system, then the solution of (1) shows finite TS stability; otherwise, it shows fixed TS stability.

Remark 1.

Definition 2 reveals that TS stability is achieved on the basis of finite/fixed-time stability; the essential difference between TS stability and finite/fixed-time stability is whether all elements $x_i(i=1,2,\cdots ,n)$ of the system (1) converge simultaneously.

Lemma 3

([19]). The origin of system (1) is finite TS stable if

(1) 

the equilibrium is finite-time stable;

(2) 

x shows ratio persistence.

Lemma 4.

The origin of system (1) shows fixed TS stability if

(1) 

The equilibrium shows fixed-time stability;

(2) 

x is ratio-persistent.

Definition 3.

For $x\in {\mathbb{R}}^n$, a normalized sign function ${\mathrm{sign}}_n$ is defined:

$${\mathrm{sign}}_n\left(x\right)=\left\{\begin{array}{c}{\displaystyle \frac{x}{∥x∥}},x\ne 0,\hfill \\ 0,x=0.\hfill \end{array}\right.$$

(7)

Furthermore, its exponential form is defined:

$${\mathrm{sign}}_n^{\alpha }\left(x\right)={∥x∥}^{\alpha }{\mathrm{sign}}_n\left(x\right),$$

(8)

where $\parallel ·\parallel$ represents the $L_2$ norm of ·; $\alpha >0$ is an adjustable constant.

Remark 2.

A normalized sign function is defined here differently from the traditional sign function, which plays a crucial part in the subsequent proof of ratio persistence and TS stability.

3. Main Results

3.1. Problem Description

An n-DOF robotic manipulator’s mathematical model is written as

$$\begin{array}{cc}\hfill \ddot{q}& =M^{-1}\left(q\right)(\tau -C(q,\dot{q})\dot{q}-G\left(q\right)-F\left(\dot{q}\right)-{\tau }_d)\hfill \\ \hfill & +g(t,T_f)\Phi (q,\dot{q},\tau ),\hfill \end{array}$$

(9)

where $q={[q_1,q_2,\cdots ,q_n]}^T\in {\mathbb{R}}^n$, $\dot{q}$, and $\ddot{q}$ indicate the robot’s position, velocity, and acceleration. $M\left(q\right)$ denotes the inertia matrix and has the property of $M^T\left(q\right)=M\left(q\right)$. $C(q,\dot{q})\in {\mathbb{R}}^{n\times n}$ and $G\left(q\right)\in {\mathbb{R}}^n$ represent the Coriolis-centripetal and gravitational matrix, respectively. $\tau \in {\mathbb{R}}^n$ denotes the physical input of the controller. $F\left(\dot{q}\right)$ is the unknown fraction matrix, and ${\tau }_d$ represents nonlinear term induced by EDSU. $\Phi (q,\dot{q},\tau )\in {\mathbb{R}}^n$ reveals the fault components. $g(t,T_f)$ denotes the fault time profile, which is formulated as

$$g(t,T_f)=\mathrm{diag}\left([g_1(t-T_{f1}),g_2(t-T_{f2}),\cdots ,g_n(t-T_{fn})]\right),$$

(10)

where $g_i(t-T_{fi})$, $i=1,2,\cdots ,n$ represents that the ith joint experiences a fault at time constant $T_{fi}$, and $g_i(t-T_{fi})$ is given by

$$g_i(t-T_{fi})=\left\{\begin{array}{c}0,\mathrm{if}t<T_{fi},\hfill \\ 1-e^{-a_i(t-T_{fi})},\mathrm{if}t\ge T_{fi}.\hfill \end{array}\right.$$

(11)

A positive $a_i$ represents the varying of the fault components. It reveals that when $a_i$ varies from 0 to $+\infty$, an incipient fault to an abrupt fault is introduced.

It is common for robot manipulators to malfunction in practice, such as due to the failures of sensors or actuators. Here, the gain fault and bias fault are considered. Taking these into account, the control input in (9) is transformed to

$${\tau }_c=\Omega \tau +\Delta \tau ,$$

(12)

where $\Omega ,\Delta \tau \in {\mathbb{R}}^n$ represent gain and bias faults, respectively. ${\tau }_c$ and $\tau$ denote the actual and desired control signal. Then, the fault function of $g(t,T_f)$ can be rewritten as

$$g(t,T_f)=M{\left(q\right)}^{-1}(\Omega -I)\tau +M{\left(q\right)}^{-1}\Delta \tau .$$

(13)

Letting $x_1=q$ and $x_2=\dot{q}$, system (9) can be transformed to the following format:

$$\begin{array}{cc}\hfill {\dot{x}}_1=& x_2,\hfill \\ \hfill {\dot{x}}_2=& M{\left(q\right)}^{-1}\tau +{\chi }_1+\varpi ,\hfill \end{array}$$

(14)

where ${\chi }_1=-M{\left(q\right)}^{-1}(C(q,\dot{q})x_2+G\left(q\right))$, $\varpi =M{\left(q\right)}^{-1}(-F\left(\dot{q}\right)-{\tau }_d)+g(t,T_f)\Phi (q,\dot{q},\tau )$ represent lumped unknown terms, including disturbances, friction, and actuator failures.

Remark 3.

Compared with [17], the system faults are incorporated with the disturbance and friction terms, which simplifies the state-space expressions.

The target position is denoted by $q_d\in {\mathbb{R}}^n$. Then, the tracking error $z_1$ can be calculated:

$$z_1=q-q_d.$$

(15)

The control objectives of this work were to design a fixed TS controller, such that the following statements hold:

(1) Tracking errors of the n-DOF robotic manipulator can converge to zero in a fixed time T, and T is unaffected by the system’s initial values. Meanwhile, the tracking errors of all state components converge at the same moment under the effects of actuator faults and EDSU, i.e., for $z_{1i}\left(0\right)\ne 0$, $z_{1i}\left(t\right){|}_{t\ge T}=0$ and $z_{1i}\left(t\right){|}_{t<T}\ne 0$ hold, where $z_{1i}$ is the ith element of $z_1$.

(2) All elements of the EDSU and faulty unity can be precisely estimated, and the observer is compatible with the framework for fixed TS control.

3.2. Time-Synchronized Observer Design

To compensate for lumped uncertainties including EDSU and actuator faults, system (14) is reconstructed with a positive auxiliary variable ${\gamma }_1$:

$$\begin{array}{cc}\hfill {\dot{x}}_1=& x_2,\hfill \\ \hfill {\dot{x}}_2=& -{\gamma }_{1}M{\left(q\right)}^{-1}{x}_2+M{\left(q\right)}^{-1}\tau +\chi +\varpi ,\hfill \end{array}$$

(16)

where ${\gamma }_1$ is a positive constant, $\chi =-M{\left(q\right)}^{-1}(C(q,\dot{q})x_2+G\left(q\right)-{\gamma }_1{x}_2)$.

Define $x_a$ as an auxiliary stat; then, an auxiliary system is established:

$$\begin{array}{c}\hfill {\dot{x}}_a=-{\gamma }_{1}M{\left(q\right)}^{-1}{x}_a+M{\left(q\right)}^{-1}\tau +\chi .\end{array}$$

(17)

We can determine the value of ${\dot{x}}_a$ through measurement.

Error $x_e$ and output variable y are expressed as follows:

$$x_e=x_2-x_a,$$

(18)

$$y={\gamma }_2{x}_e,$$

(19)

where ${\gamma }_2$ denotes a constant, and ${\gamma }_2>0$.

The following observer is designed to compensate for the lumped unknown term according to $x_a$ and y:

$$\begin{array}{c}\hfill \widehat{\varpi }={\displaystyle \frac{1}{{\gamma }_2}}M\left(q\right)({\gamma }_1{\gamma }_{2}M{\left(q\right)}^{-1}{\widehat{x}}_e+\dot{y}),\end{array}$$

(20)

where ${\widehat{x}}_e$ represents the observed value of $x_e$. The format of ${\widehat{x}}_e$ is

$$\begin{array}{c}\hfill {\dot{\widehat{x}}}_e=-{\gamma }_2{\gamma }_3{\widehat{x}}_e+{\displaystyle \frac{1}{{\gamma }_2}}\dot{y}+{\gamma }_{3}y-{\gamma }_4{\mathrm{sign}}_{\mathrm{n}}{\left({\tilde{x}}_e\right)}^{\beta },\end{array}$$

(21)

where ${\tilde{x}}_e={\widehat{x}}_e-x_e$, ${\gamma }_3$ and ${\gamma }_4$ are positive constants and represent the gains; $\beta$ is a constant and satisfies $0<\beta <1$. ${\mathrm{sign}}_n$ is a normalized sign function and plays a vital role in TS control.

Remark 4.

The proposed observer addresses the lumped unknown term, accounting for actuator failure, disturbances, and friction without the need for additional methods to handle actuator faults separately, thereby simplifying the controller and increasing its engineering practicality. Additionally, the assumption of upper boundedness is relaxed. Furthermore, the disturbance observer presented here exhibits advance robust performance and can adapt to various disturbances or actuator failures using the same set of parameters.

Theorem 1.

Under the effect of the time-synchronized observer (20,21), the lumped unknown term ϖ can be precisely estimated, with all state components of the observation error ${\varpi }_e=\varpi -\widehat{\varpi }$ converging to zero simultaneously.

Proof. 

Based on (16)–(18), we can obtain

$$\begin{array}{c}\hfill {\dot{x}}_e=-{\gamma }_{1}M{\left(q\right)}^{-1}{x}_e+M{\left(q\right)}^{-1}\varpi .\end{array}$$

(22)

The observe error ${\varpi }_e$ is

$$\begin{array}{cc}\hfill {\varpi }_e& =\widehat{\varpi }-\varpi \hfill \\ \hfill & ={\displaystyle \frac{1}{{\gamma }_2}}M\left(q\right)({\gamma }_1{\gamma }_{2}M{\left(q\right)}^{-1}{\widehat{x}}_e+{\gamma }_2{\dot{x}}_e)\hfill \\ \hfill & -M\left(q\right)({\dot{x}}_e+{\gamma }_{1}M{\left(q\right)}^{-1}{x}_e)\hfill \\ \hfill & ={\gamma }_1{\tilde{x}}_e.\hfill \end{array}$$

(23)

Combining (21) and (22), we have

$$\begin{array}{cc}\hfill {\dot{\tilde{x}}}_e& ={\dot{\widehat{x}}}_e-{\dot{x}}_e\hfill \\ \hfill & =-{\gamma }_2{\gamma }_3{\widehat{x}}_e+{\displaystyle \frac{1}{{\gamma }_2}}\dot{y}+{\gamma }_{3}y\hfill \\ \hfill & -{\gamma }_4{\mathrm{sign}}_{\mathrm{n}}^{\beta }\left({\tilde{x}}_e\right)-{\dot{x}}_e\hfill \\ \hfill & =-{\gamma }_2{\gamma }_3{\tilde{x}}_e-{\gamma }_4{\mathrm{sign}}_{\mathrm{n}}^{\beta }\left({\tilde{x}}_e\right).\hfill \end{array}$$

(24)

Furthermore,

$$\begin{array}{cc}\hfill {\dot{\varpi }}_e& ={\gamma }_1{\dot{\tilde{x}}}_e\hfill \\ \hfill & =-{\gamma }_2{\gamma }_3{\varpi }_e-{\gamma }_4{\gamma }_1{\mathrm{sign}}_{\mathrm{n}}^{\beta }\left({\varpi }_e\right).\hfill \end{array}$$

(25)

Choose a Lyapunov function of $x_e$:

$$\begin{array}{c}\hfill V_o={\tilde{x}}_e^T{\tilde{x}}_e.\end{array}$$

(26)

Then, the derivation of $V_o$ is

$$\begin{array}{cc}\hfill {\dot{V}}_o& =2{\tilde{x}}_e^T{\dot{\tilde{x}}}_e\hfill \\ \hfill & =2{\tilde{x}}_e^T(-{\gamma }_2{\gamma }_3{\tilde{x}}_e-{\gamma }_4{\mathrm{sign}}_n^{\beta }\left({\tilde{x}}_e\right))\hfill \\ \hfill & =-2{\gamma }_2{\gamma }_3{V}_o-2{\gamma }_4{V}_o^{{\displaystyle \frac{1+\beta }{2}}}.\hfill \end{array}$$

(27)

According to Lemma 1, state $x_e$ exhibits finite-time stability. Solving (27) yields

$$\begin{array}{c}\hfill V_o\equiv 0,{\tilde{x}}_e\equiv 0,\mathrm{if}t\ge T_o,\end{array}$$

(28)

where the settling time $T_o={\displaystyle \frac{1}{{\gamma }_2{\gamma }_3(1-\beta )}}ln{\displaystyle \frac{{\gamma }_2{\gamma }_3{V}_1{\left(t_0\right)}^{\frac{1-\beta }{2}}+{\gamma }_4}{{\gamma }_4}}$.

By calculating

$$\begin{array}{c}\hfill {\displaystyle \frac{{\dot{\tilde{x}}}_e}{\parallel {\dot{\tilde{x}}}_e\parallel }}={\displaystyle \frac{-{\gamma }_2{\gamma }_3{\tilde{x}}_e-{\gamma }_4{\mathrm{sign}}_n^{\beta }\left({\tilde{x}}_e\right)}{\parallel {\gamma }_2{\gamma }_3{\tilde{x}}_e+{\gamma }_4{\mathrm{sign}}_n^{\beta }\left({\tilde{x}}_e\right)\parallel }}={\displaystyle \frac{{\tilde{x}}_e}{\parallel {\tilde{x}}_e\parallel }}.\end{array}$$

(29)

Therefore, according to Definition 1, state variable ${\tilde{x}}_e$ shows ratio persistence. Furthermore, ${\tilde{x}}_e$ shows finite TS stability, and the elements of ${\tilde{x}}_e$ converge to zero simultaneously at $T_o$.

Based on (23), we have

$${\varpi }_e\equiv 0,\mathrm{if}t\ge T_o.$$

(30)

Thus, for $t\ge T_o$, the estimated value provided by the time-synchronized observer equals the actual value. □

Remark 5.

If the tracking error’s convergence time exceeds that of the observation error, the observation error converges first. The effects of the time-synchronization controller enable the tracking error to achieve time-synchronized performance. When the convergence time of the tracking error is less than the observation error, and if the observer shows conventional asymptotic stability and finite/fixed-time stability, due to the observer’s lack of time-synchronization stability performance, the convergence time of each state component is different, and the overall system cannot achieve TS convergence. Therefore, the existing observers are not compatible with the time-synchronization stability framework. The proposed disturbance observer can guarantee all state components convergence simultaneously; regardless of whether the observation error lags behind the convergence of tracking errors, the entire system shows TS stability, which is compatible with the framework of TS stability control.

To demonstrate the simultaneous convergence of the observer error, the proof is conducted using the contradiction method.

Based on (29), we have

$${\dot{\tilde{x}}}_e=\kappa \left(x_e\right){\tilde{x}}_e,$$

(31)

where $\kappa \left(x_e\right)={\displaystyle \frac{\parallel {\dot{\tilde{x}}}_e\parallel }{\parallel {\tilde{x}}_e\parallel }}$.

For $\forall {\tilde{x}}_{ei}\left(t\right),{\tilde{x}}_{ej}\left(t\right)\in {\tilde{x}}_e,i\ne j$, and ${\tilde{x}}_{ei}\left(t\right),{\tilde{x}}_{ej}\left(t\right)\ne 0$,

$${\displaystyle \frac{d}{dt}}{\displaystyle \frac{{\tilde{x}}_{ei}\left(t\right)}{{\tilde{x}}_{ej}\left(t\right)}}={\displaystyle \frac{{\dot{\tilde{x}}}_{ei}\left(t\right){\tilde{x}}_{ej}\left(t\right)-{\tilde{x}}_{ei}\left(t\right){\dot{\tilde{x}}}_{ej}\left(t\right)}{{\tilde{x}}_{ej}^2\left(t\right)}}.$$

(32)

Substituting ${\dot{\tilde{x}}}_{ei}\left(t\right)=\kappa \left(x_e\right){\tilde{x}}_{ei}\left(t\right)$ and ${\dot{\tilde{x}}}_{ej}=\kappa \left(x_e\right){\tilde{x}}_{ej}\left(t\right)$ into (32), we can obtain

$${\displaystyle \frac{d}{dt}}{\displaystyle \frac{{\tilde{x}}_{ei}\left(t\right)}{{\tilde{x}}_{ej}\left(t\right)}}=0.$$

(33)

Therefore, $\forall {\tilde{x}}_{ei}\left(t\right),{\tilde{x}}_{ej}\left(t\right)\in {\tilde{x}}_e$, and ${\tilde{x}}_{ei}\left(t\right),{\tilde{x}}_{ej}\left(t\right)\ne 0$,

$${\displaystyle \frac{{\tilde{x}}_{ei}\left(t\right)}{{\tilde{x}}_{ej}\left(t\right)}}=c_{ij},$$

(34)

where $c_{ij}$ is a nonzero constant. Furthermore, we have

$${\tilde{x}}_{ei}\left(t\right)=c_{ij}{\tilde{x}}_{ej}\left(t\right).$$

(35)

Based on (28), observation error ${\tilde{x}}_e$ can reach zero in a finite time. Assume that ${\tilde{x}}_{ei}\left(t\right)$ and ${\tilde{x}}_{ej}\left(t\right)$ arrive at the zero point at time $T_i$ and $T_j$, respectively, $T_i\ne T_j$. Then, according to (33), we have

$$\underset{t\to T_i^{-}}{lim}{\displaystyle \frac{d}{dt}}{\displaystyle \frac{{\tilde{x}}_{ei}\left(t\right)}{{\tilde{x}}_{ej}\left(t\right)}}=0⟹\underset{t\to T_i^{-}}{lim}{\tilde{x}}_{ei}\left(t\right)=c_{ij}{\tilde{x}}_{ej}\left(t\right).$$

(36)

As a continuous system, it can be concluded that $x_{ej}\left(T_i\right)=0$, which is contradicts the assumption that $x_{ej}$ arrives at zero at time $T_j$. So, all elements converge simultaneously.

3.3. Time-Synchronized Fault-Tolerant Control Design

For convenience, let ${\Theta }_1=z_1$ and ${\Theta }_2={\dot{z}}_1$, which, when combined with (9), (14), and (15), we can obtain

$$\begin{array}{cc}\hfill {\dot{\Theta }}_1=& {\Theta }_2,\hfill \\ \hfill {\dot{\Theta }}_2=& M{\left(q\right)}^{-1}\tau +{\chi }_1+\varpi -{\ddot{q}}_d.\hfill \end{array}$$

(37)

Construct a sliding-mode surface,

$$s={\Theta }_2+s_w.$$

(38)

where $s_w$ is the switching law and is designed as follows:

$$s_w=\left\{\begin{array}{c}{\zeta }_1{\mathrm{sign}}_n^{p_1}\left({\Theta }_1\right)+{\varrho }_1{\mathrm{sign}}_n^{g_1}\left({\Theta }_1\right),\mathrm{if}\hfill \\ s^{*}=0\mathrm{or}{s}^{*}\ne 0,\parallel {\Theta }_1\parallel >\xi \hfill \\ k_1{\Theta }_1+k_2{\mathrm{sign}}_n^4\left({\Theta }_1\right),\mathrm{if}{s}^{*}\ne 0,\parallel {\Theta }_1\parallel \le \xi ,\hfill \end{array}\right.$$

(39)

where $\xi$ represents a small constant; ${\zeta }_1$ and ${\varrho }_1$ represent positive constants. To the avoid singularity problem, $p_1$ and $g_1$ are chosen as ${\displaystyle \frac{1}{2}}<p_1<1$ and $1<g_1<4$. In order to avoid the singularity problem, $k_1$, $k_2$ satisfies

$$\begin{array}{c}\hfill {\zeta }_1(4-p_1){\parallel \xi \parallel }^{p_1-1}+{\varrho }_1(4-g_1){\parallel \xi \parallel }^{g_1-1}=3k_1,\\ \hfill {\zeta }_1(p_1-1){\parallel \xi \parallel }^{p_1-4}+{\varrho }_1(g_1-1){\parallel \xi \parallel }^{g_1-4}=3k_2.\end{array}$$

(40)

Auxiliary variable $s^{*}$ is designed as

$$s^{*}={\dot{\Theta }}_1+{\zeta }_1{\mathrm{sign}}_n^{p_1}\left({\Theta }_1\right)+{\varrho }_1{\mathrm{sign}}_n^{g_1}\left({\Theta }_1\right).$$

(41)

The fixed TS control strategy is formulated as

$$\begin{array}{cc}\hfill \tau & =-M\left(q\right)({\dot{s}}_w+{\zeta }_2\parallel s{\parallel }^{p_2-1}s+{\varrho }_2\parallel s{\parallel }^{g_2-1}s)\hfill \\ \hfill & +C(q,\dot{q})\dot{q}+G\left(q\right)-M\left(q\right)\widehat{\varpi }+M\left(q\right){\ddot{q}}_d\hfill \end{array}$$

(42)

where ${\zeta }_2$, ${\varrho }_2$, $p_2$, and $g_2$ are positive constants, $0<p_2<1$, and $g_2>1$.

Remark 6.

The proposed sliding-mode surface (38) effectively avoids the singularity issue.

Upon calculating the derivative of $s_w$, we obtain

$${\dot{s}}_w=\left\{\begin{array}{c}{\rho }_1{\Theta }_1{\Theta }_1^T{\Theta }_2+{\rho }_2{\Theta }_2,\mathrm{if}\hfill \\ s^{*}=0\mathrm{or}{s}^{*}\ne 0,\parallel {\Theta }_1\parallel >\xi ,\hfill \\ k_1{\Theta }_2+3k_2\parallel {\Theta }_1\parallel {\Theta }_1{\Theta }_1^T{\Theta }_2+k_2{\parallel {\Theta }_1\parallel }^3{\Theta }_2,\mathrm{if}\hfill \\ s^{*}\ne 0,\parallel {\Theta }_1\parallel \le \xi ,\hfill \end{array}\right.$$

(43)

where

$$\begin{array}{cc}\hfill {\rho }_1=& {\zeta }_1(p_1-1)\parallel {\Theta }_1{\parallel }^{p_1-3}+{\varrho }_1(g_1-1){\parallel {\Theta }_1\parallel }^{g_1-3}\hfill \\ \hfill {\rho }_2=& {\zeta }_1\parallel {\Theta }_1{\parallel }^{p_1-1}+{\varrho }_1{\parallel {\Theta }_1\parallel }^{g_1-1}.\hfill \end{array}$$

(44)

Theorem 2.

Consider the robotic manipulator system expressed in (9): under the effect of the time-synchronized observer (20,21) and fixed TS control (42) in sliding mode (38), all states of the tracking error can converge at the same moment $T_{total}\le T_1+T_2$, where $T_1$ and $T_2$ are given in (48) and (54), and $T_{total}$ is independent of the initial states.

Proof. 

Consider the following candidate Lyapunov function:

$$V_1={\displaystyle \frac{1}{2}}{s}^{T}s.$$

(45)

Differentiating $V_1$ and substituting (37), (38), and (42), we obtain

$$\begin{array}{cc}\hfill {\dot{V}}_1& =s^T({\dot{\Theta }}_2+{\dot{s}}_w)\hfill \\ \hfill & =-{\zeta }_2{s}^T{\mathrm{sign}}_n^{p_2}\left(s\right)-{\varrho }_2{s}^T{\mathrm{sign}}_n^{g_2}\left(s\right)+s^T(\varpi -\widehat{\varpi }).\hfill \end{array}$$

(46)

According to Theorem 1, if $T>T_o$, $\varpi =\widehat{\varpi }$ holds. Therefore, it follows that

$$\begin{array}{c}\hfill {\dot{V}}_1=-2^{{\displaystyle \frac{p_2+1}{2}}}{\zeta }_2{V}_1^{{\displaystyle \frac{p_2+1}{2}}}-2^{{\displaystyle \frac{g_2+1}{2}}}{\varrho }_2{V}_1^{{\displaystyle \frac{g_2+1}{2}}}.\end{array}$$

(47)

Hence, s shows fixed-time stability, $s=0$ for $t\ge T_1$, and settling time $T_1$ is independent of the initial states, which means that the tracking error ${\Theta }_1$ converges to the sliding-mode manifold when $t\ge T_1$. $T_1$ is given as below:

$$T_1\le {\displaystyle \frac{2^{\frac{1-p_2}{2}}}{{\zeta }_2(1-p_2)}}+{\displaystyle \frac{2^{\frac{1-g_2}{2}}}{{\varrho }_2(g_2-1)}}.$$

(48)

Furthermore,

$${\displaystyle \frac{\dot{s}}{\parallel \dot{s}\parallel }}={\displaystyle \frac{-{\zeta }_2{\mathrm{sign}}_n^{p_2}\left(s\right)-{\varrho }_2{\mathrm{sign}}_n^{g_2}\left(s\right)}{\parallel -{\zeta }_2{\mathrm{sign}}_n^{p_2}\left(s\right)-{\varrho }_2{\mathrm{sign}}_n^{g_2}\left(s\right)\parallel }}=-{\displaystyle \frac{s}{\parallel s\parallel }}.$$

(49)

According to Definition 1, s is satisfied and has ratio persistence. Based on Lemma 4, we can draw the conclusion that s shows fixed-time stability and ratio persistence; therefore, s shows fixed TS stability, and all elements of s converge simultaneously, i.e., when $t\ge T_1,s\left(i\right)\equiv 0$ holds; when $t<T_1,s\left(i\right)\ne 0$ holds, where $s\left(i\right)$ means the ith element of vector s. The proof follows a similar process to that of the time-synchronized disturbance observer, as outlined in Equations (31)–(36).

After s converges to zero, according to (41), we have

$${\dot{\Theta }}_1=-{\zeta }_1{\mathrm{sign}}_n^{p_1}\left({\Theta }_1\right)-{\varrho }_1{\mathrm{sign}}_n^{g_1}\left({\Theta }_1\right),$$

(50)

which may lead to

$${\displaystyle \frac{{\dot{\Theta }}_1}{\parallel {\dot{\Theta }}_1\parallel }}={\displaystyle \frac{{\Theta }_1}{\parallel {\Theta }_1\parallel }}.$$

(51)

Therefore, according to Definition 1, tracking error state ${\Theta }_1$ shows ratio persistence.

Next, we can construct a candidate Lyapunov function of the tracking error ${\Theta }_1$ as follows:

$$V_2={\displaystyle \frac{1}{2}}{\Theta }_1^T{\Theta }_1.$$

(52)

Differentiating $V_2$ along time and substituting (50) into it, we have

$$\begin{array}{cc}\hfill {\dot{V}}_2& ={\displaystyle \frac{1}{2}}{\Theta }_1^T{\Theta }_1\hfill \\ \hfill & ={\Theta }_1^T(-{\zeta }_1{\mathrm{sign}}_n^{p_1}\left({\Theta }_1\right)-{\varrho }_1{\mathrm{sign}}_n^{g_1}\left({\Theta }_1\right))\hfill \\ \hfill & =-2^{{\displaystyle \frac{p_1+1}{2}}}{\zeta }_1{V}_2^{{\displaystyle \frac{p_1+1}{2}}}-2^{{\displaystyle \frac{g_1+1}{2}}}{\varrho }_1{V}_2^{{\displaystyle \frac{g_1+1}{2}}}.\hfill \end{array}$$

(53)

Therefore, ${\Theta }_1$ shows fixed TS stability, and the settling time is

$$T_2\le {\displaystyle \frac{2^{\frac{1-p_1}{2}}}{{\zeta }_1(1-p_1)}}+{\displaystyle \frac{2^{\frac{1-g_1}{2}}}{{\varrho }_1(g_1-1)}},$$

(54)

which leads to the total settling time of state ${\Theta }_1$ satisfying $T_{total}\le T_o+T_1+T_2$.

Next, the simultaneous convergence performance is described.

Based on (51), one can obtain

$${\dot{\Theta }}_1=\kappa \left({\Theta }_1\right){\Theta }_1,$$

(55)

where $\kappa \left({\Theta }_1\right)={\displaystyle \frac{\parallel {\dot{\Theta }}_1\parallel }{\parallel {\Theta }_1\parallel }}$.

For $\forall {\Theta }_{1i}\left(t\right),{\Theta }_{1j}\left(t\right)\in {\Theta }_1,i\ne j$ and ${\Theta }_{1i}\left(t\right),{\Theta }_{1j}\left(t\right)\ne 0$,

$${\displaystyle \frac{d}{dt}}{\displaystyle \frac{{\Theta }_{1i}\left(t\right)}{{\Theta }_{1j}\left(t\right)}}={\displaystyle \frac{{\dot{\Theta }}_{1i}\left(t\right){\Theta }_{1j}\left(t\right)-{\Theta }_{1i}\left(t\right){\dot{\Theta }}_{1j}\left(t\right)}{{\Theta }_{1j}^2\left(t\right)}}.$$

(56)

Substituting ${\dot{\Theta }}_{1i}\left(t\right)=\kappa \left({\Theta }_1\right){\Theta }_{1i}\left(t\right)$ and ${\dot{\Theta }}_{1j}=\kappa \left({\Theta }_1\right){\Theta }_{1j}\left(t\right)$ into (56), one can obtain

$${\displaystyle \frac{d}{dt}}{\displaystyle \frac{{\Theta }_{1i}\left(t\right)}{{\Theta }_{1j}\left(t\right)}}=0.$$

(57)

Therefore, $\forall {\Theta }_{1i}\left(t\right),{\Theta }_{1j}\left(t\right)\in {\Theta }_1$ and ${\Theta }_{1i}\left(t\right),{\Theta }_{1j}\left(t\right)\ne 0$,

$${\displaystyle \frac{{\Theta }_{1i}\left(t\right)}{{\Theta }_{1j}\left(t\right)}}=C_{ij},$$

(58)

where $C_{ij}$ is a nonzero constant. Furthermore, we have

$${\Theta }_{1i}\left(t\right)=C_{ij}{\Theta }_{1j}\left(t\right).$$

(59)

Based on (53), tracking error ${\Theta }_1$ arrives at the equilibrium point of 0 in a fixed time. Assume that ${\Theta }_{1i}\left(t\right)$ and ${\Theta }_{1j}\left(t\right)$ arrive at 0 at different moments $T_i$ and $T_j$, respectively, $T_i\ne T_j$. Then, based on (57),

$$\underset{t\to T_i^{-}}{lim}{\displaystyle \frac{d}{dt}}{\displaystyle \frac{{\Theta }_{1i}\left(t\right)}{{\Theta }_{1j}\left(t\right)}}=0⟹\underset{t\to T_i^{-}}{lim}{\Theta }_{1i}\left(t\right)=C_{ij}{\Theta }_{1j}\left(t\right).$$

(60)

As a continuous system, it can be concluded that ${\Theta }_{1j}\left(T_i\right)=0$, which contradicts the assumption of ${\Theta }_{1j}$ reaching equilibrium at time $T_j$. Therefore, all state components arrive at equilibrium simultaneously. This concludes the proof. □

4. Results and Discussion

To demonstrate the proposed time-synchronized fault-tolerant control’s superiority, its performance was compared against two state-of-the-art control approaches. Controller 1 is based on neural network and barrier Lyapunov function, controller 2 is based on second-order fast sliding-mode and adaptive method. Numerical simulations were conducted on a two-DOF robotic system, depicted in Figure 1. Each link i in the system is characterized by its moment of inertia $I_i$, mass $m_i$, and length $l_i$. Parameter $l_{ci}$ denotes the distance between link i and its preceding joint $i-1$, with i being one or two. For the details of the robotic parameters, please refer to Table 1.

Figure 1. Model of a knee rehabilitation robotic manipulator. (a) Physical Structure Diagram (b) Description of each variable.

Table 1. Robotic parameters.

Parameter	Value	Unit
$m_1$	2	$\mathrm{kg}$
$m_2$	0.85	$\mathrm{kg}$
$l_1$	0.35	$\mathrm{kg}$
$l_2$	0.31	$\mathrm{kg}$
$I_1$	${\displaystyle \frac{1}{4}}{m}_1{l}_1^2$	${\mathrm{kg}·\mathrm{m}}^2$
$I_2$	${\displaystyle \frac{1}{4}}{m}_2{l}_2^2$	${\mathrm{kg}·\mathrm{m}}^2$

To demonstrate the proposed method’s effectiveness in being fault-tolerant, the following functions were used to simulate faults in the actuator:

$$\begin{array}{c}\hfill \Phi (q,\dot{q},\tau )=\left[\begin{array}{c}-0.3\tau \left(1\right),t\ge 10\\ -0.85\tau \left(2\right),t\ge 12\end{array}\right],\end{array}$$

(61)

which displays that the actuator faults occurred in joint 1 and joint 2 from the time constants of 10 s and 12 s, where joint 1 and joint 2 lose $30\%$ and $85\%$ of their effectiveness, respectively.

External disturbances were set as

$${\tau }_d=\left[\begin{array}{c}-0.8cos\left(0.01t\right)+2.5sin\left(0.1t\right)\\ 2.2cos\left(0.04t^2\right)+1.5cos\left(0.1t\right)\end{array}\right].$$

(62)

The desired trajectory was set as

$$q_d={[0.14sin\left(0.5t\right)+0.5,0.14cos\left(0.5t\right)]}^T.$$

(63)

The initial values of the joint positions $(q_1,q_2)$ were set to $(1,0)$. The control parameters were selected as ${\zeta }_1=1,{\zeta }_2=4,{\varrho }_1=0.2,{\varrho }_2=0.05,p_1=p_2=0.7,g_1=g_2=1.2,\xi =0.01$. The parameters of the observer were defined as ${\gamma }_1=0.01,{\gamma }_2=5,{\gamma }_3=15,{\gamma }_4=150,\beta =0.9$. The simulation was conducted over a period of $t=20$ seconds.

Remark 7.

The parameters were chosen based on the trial-and-error method, which does not mean that larger values indicate better control performance. On the one hand, it may influence the system’s convergence speed. On the other hand, if the value is set too large, it may cause system divergence. Firstly, the parameters chosen should satisfy the need for system stability, as shown in Equation (39), where ${\displaystyle \frac{1}{2}}<p_1<1$, $1<g_1<4$, and Equation (42), where $0<p_2<1$ and $g_2>1$. Then, in the predefined range, we chose a value according to experience. Based on the control performance and the method of dichotomy, we can determine a suitable group of values.

The tracking performance of the three controllers, subjected to system faults and EDSU, is depicted in Figure 2, Figure 3, Figure 4 and Figure 5. Figure 2 and Figure 3 specifically showcase the tracking performances of joints 1 and 2, respectively. Figure 2 shows that while the system is operating normally, e.g., $t<10$ s; without actuator faults, the tracking task can be finished by the three controllers. However, steady errors occur under the effect of controller 1, and the convergence speed is slower for controller 2. This can be attributed to friction and disturbances, which are not effectively mitigated by controllers 1 and 2. Under controller 1, the tracking precision and the converge speed of joint 2 are better than under the other two controllers, as illustrated in Figure 3 and Figure 4; the tracking errors rapidly approach zero. When actuator faults occur ($t>10$ s and $t>12$ s), the robustness of controllers 1 and 2 is weaker than that of the proposed method, which can be seen in Figure 2, Figure 3 and Figure 4. When there are faults, friction, and disturbances, controller 1, based on an NN and a barrier Lyapunov function, exhibits greater robustness compared to controller 2, which relies on sliding-mode and adaptive methods. It is clearly revealed in Figure 2 and Figure 3 that the control performance of the two joints are abrupt with controller 2 after $t>10$ s and $t>12$ s. Though the robustness of controller 1 is better than that of controller 2, we can see from Figure 4d that the tracking error of joint 1 still becomes larger than for the faults that occurred before, while the proposed method has high robustness against these influences. Furthermore, the salient features of time-synchronized convergence are illustrated in Figure 4a–d. Figure 4a,b show that with the proposed method, which is magnified at ${10}^{-3}$, the tracking errors of the two joints converge to zero simultaneously; this obvious characteristic is unique compared to existing control methods such as controller 1 and controller 2, which are shown in Figure 4c,d. Upon the occurrence of faults, the tracking errors remain unchanged with the proposed method, unlike with the other two controllers. This highlights the superior advantages of the proposed time-synchronized fault-tolerant control method, namely ,its higher precision, rapid convergence, and enhanced robustness. In addition, to make a comprehensive comparison, the integrated absolute error of $z_1$ is defined as ${\int }_0^t\left|z_1\left(t\right)\right|dt$, which is shown in a 3D bar chart in Figure 5. It shows that the proposed time-synchronized control’s tracking performance is superior to that of controllers 1 and 2. From the simulation results, we can see that under sensor or actuator faults, the time error of the proposed fault-tolerant controller can converge to ${10}^{-5}$, which can be used for precision tasks, for example, in aerospace missions.

Figure 2. The tracking performance of joint 1.

Figure 3. The tracking performance of joint 2.

Figure 4. Tracking errors of two joints: (a) time-synchronized convergence curves; (b) enlarged time-synchronized convergence curves; (c) tracking error curves for controller 1; (d) tracking error curves for controller 2.

Figure 5. Three-dimensional bar chart comparing the controllers’ tracking errors.

5. Conclusions

This article introduced a time-synchronized fault-tolerant control method for robotic manipulators, addressing the challenges posed by the time-synchronized convergence problem in the presence of system faults, friction, and external disturbances. The numerical simulations on a two-degree-of-freedom robotic system showed that the tracking errors decreased to ${10}^{-5}$, which demonstrated the efficacy of the proposed control approach, offering higher precision, faster convergence, and stronger robustness in contrast to state-of-the-art methods. The proposed control method, highlighted by the tracking errors, converged to zero simultaneously for two joints. These characteristics provide distinct advantages over other existing control methods. Moving forward, this work could be a foundation for more resilient and reliable robotic systems, paving the way for advancements in robotic applications where synchronized temporal performance is crucial.