Introduction

With the creation of new intelligent technologies, the use of robot manipulators to reduce the intensity of physical labor to complete more complex tasks has become an inevitable trend in future development. Modular robot manipulators (MRMs) are a new direction in the robotics field that have more optimal structural adaptability and flexibility than standard manipulators, hence they have gotten a lot of attention from the robotics business [1,2,3]. MRMs should not only have excellent tracking performance but also must operate safely and reliably to improve their efficiency in practical applications. During operation, the robot will interact with the environment, operating objects, and possibly even humans. If the actuator faults during the interaction, it may result in injuries and financial losses. Therefore, the researchers designed a fault-tolerant method to obtain the necessary tracking performance even when the system faults. Currently, the majority of the literature proposes a fault-tolerant control strategy that overlooks the actuator’s output capabilities [4, 5]. Because the control system may degrade some performance and may require extra control effort to adjust for the impact of the fault, actuator saturation may occur in the fault-tolerant control (FTC) of the robot.

FTC means that in the event of faults, the safe operation of the system can still be guaranteed and acceptable performance maintained [6,12]. AFTC is implemented with fault detection and fault diagnosis [13, 14]. In PFTC, the control system with a fixed structure is designed without fault detection and isolation, as well as a learning phase [15]. The different causes of fault or the different severity of the fault can lead to actuator faults with different characteristics. In practice, the actuator’s drive capability may be impaired and unable to achieve the originally expected performance due to damaged bearings, lack of lubricant, damaged transmission, etc. This type of situation can be defined as an actuator that has suffered a performance fault and the drive capacity is constrained [16,17,18,19]. Although a performance fault can cause an actuator’s drive capability to partially fail, the actuator can still drive the links of the robot to rotate, i.e., the robotic manipulator joint corresponding to the actuator is still a driveable joint. However, in actual control, there is also a kind of actuator fault that will lead to complete failure of the actuator’s driving capability, which is a complete damage fault [20, 21]. When an actuator of the robotic manipulator system is completely damaged, the actuator can no longer provide control torque to the system, and the robotic manipulator joint in which it is located becomes a free-rotating underdriven joint, and the corresponding robotic manipulator link is called an “underdriven link”. Accordingly, the manipulator link that can be driven directly by the actuator is called the “drive link”. Actuator saturation constraint is a common constraint, which means that there is a maximum value of the output torque of the actuator, and when the control signal is greater than this maximum value, the actual output torque of the actuator will remain at this maximum value [22,23,24]. In the ideal case, the maximum value of the actuator output torque is known as the rated maximum output of the actuator. However, when an actuator has a drive performance fault due to physical environmental factors, etc., there is often a large amount of friction in its drive. At this point, there will be uncertainty in the saturation constraint of the system. That is, the actual maximum torques that can be output by the actuator, both forward and reverse, are unknown, which brings a great challenge to the control.

The performance of the system mainly depends on its capacity to handle uncertainty. Many strategies have been successfully integrated into FTC systems to overcome the aforesaid challenges, according to published research results [25,26,27,28]. Among the methods mentioned, the sliding mode control (SMC) is used as a superior control method to overcome the system uncertainty, and has strong robustness [29,30,31]. Because of its simple method, fast response speed, and strong robustness, it has been widely employed. Nevertheless, traditional SMC is not ideal for every robot control due to its own constraints such as the chattering phenomenon. To avoid the limitation of traditional SMC, some researchers have proposed more advanced control methods and improved control performance in recent years [32,40,41,42]. In [40,41,42]. For nonlinear systems with full state constraints, [43] proposed a neural adaptive control technique. As far as we know, adaptive control will accomplish a satisfactory approximation effect when the estimated nonlinear function is used as the system state or the control input. Meanwhile, inadequate estimation of perturbations can lead to greater approximation errors. To address this limitation, the estimation strategies based on the disturbance observer (DO) technique have been proposed [44,45,46]. The adaptive observer is utilized in [47] to estimate uncertainties and external disturbances without the need for prior knowledge of unknown dynamics. In [48], a nonlinear disturbance observer is employed to estimate external disturbances, and the estimated value is used as reconstructed input signals for compensation when there is an uncertain disturbance. Currently, there have been many research results on ISMC, DO, and NN. Although these strategies can improve system performance when applied in practice, the performance of the system may be affected due to the limitations of the control strategy. To overcome the limitations of a single strategy, researchers have combined multiple control methods to construct DO-based ISMC controllers [49, 50], NN-DO controllers [51, 52], and other hybrid control strategies. Unfortunately, few efforts have been made in the literature to implement such a composite control method [53, 54]. The reason for this may be that the inputs of the composite control system are difficult to reconstruct, making it difficult to ensure stability and the convergence of the overall system. Furthermore, to enhance the performance of the system, it is desirable to combine the benefits of ISMC, NN, and DO. In particular, FTC systems require such control strategies to deal with unpredictable unknown nonlinear functions (uncertainties, disturbances, and faults) and need robust control effects.

Motivated by the above analysis, a novel decentralized FTC method of neural adaptive integral terminal sliding mode control (ITSMC) is proposed for the MRM system with actuator saturation. First, the dynamic of the MRM system is structured by means of joint torque feedback (JTF) techniques. Next, adaptive RBFNN and DO hybrid control strategies are integrated into the ITSMC, and the FTC is designed to decrease the effect of uncertainty and chattering. The uncertainty and interference of the system are compensated by adaptive RBFNN and DO, thereby improving the performance of the control system. Furthermore, a neuro-adaptive compensator is employed to address the saturation of actuators. To overcome the effect of residual estimation errors, further reduce the chattering phenomenon, and improve the tracking accuracy, the super-twisting algorithm (STA) was introduced. Finally, the simulation is performed to validate the effectiveness of the proposed control strategy.

The main achievements of this paper are reflected in:

  1. 1.

    A hybrid control strategy with neural adaptive sliding mode and DO is proposed to fully estimate the effects of uncertainties, disturbances, faults, and actuator saturation. Compared with the conventional SMC [29, 31] and TSMC [32, 37], the proposed FTC method designs a novel integral TSMC (ITSMC) based on adaptive RBFNN and DO, which can control the error within the allowed performance range, enhance the robustness, and reduce the chattering of the system.

  2. 2.

    Compared with the conventional FTC method [13, 14], the proposed method does not require detection and diagnosis modules and avoids time delays. In addition, a neural adaptive compensator is designed to address the shortcomings of the anti-saturation design in the current study, which not only eliminates the actuator saturation problem but also prevents the compensator ineffectiveness in the conventional design.

The remaining sections of this paper are organized as follows. “Problem formulation and preliminaries” presents the dynamics of the MRM system and preliminaries. “Decentralized fault-tolerant controller design” gives the stability analysis of the proposed neural adaptive fault tolerant controller. In “Simulation”, the results of simulations are discussed. Finally, “Conclusions” concludes the paper.

Problem formulation and preliminaries

Dynamics of the modular robot manipulator

As shown in Fig. 1, this paper considers an MRM with n degrees of freedom. Its each module is composed of rotating joint, reducer, and torque sensor. The dynamics of the system for the i-th joint can be formulated as [32, 55].

$$\begin{aligned}{} & {} {{I}_{mi}}{{\gamma }_{i}}{{\ddot{\Theta }}_{i}}+{{I}_{mi}}\sum \limits _{j=2}^{i-1}{\sum \limits _{k=1}^{j-1}{z_{mi}^\mathrm{{T}}({{z}_{k}}\times {{z}_{j}}){{{\dot{\Theta }}}_{k}}{{{\dot{\Theta }}}_{j}}}}\nonumber \\{} & {} \quad +{{I}_{mi}}\sum \limits _{j=1}^{i-1}{z_{mi}^\mathrm{{T}}{{z}_{j}}{{{\ddot{\Theta }}}_{j}}}+{{f}_{i}}({{\Theta }_{i}},{{\dot{\Theta }}_{i}})+\frac{{{\tau }_{si}}}{{{\gamma }_{i}}}={{{}}}{{\tau }_{i}}, \end{aligned}$$
(1)

where \({{I}_{mi}}\) is the rotor inertia, \({{\gamma }_{i}}\) is the reducer ratio, \({{\Theta }_{i}}\) is the joint position, \({{f}_{i}}({{\Theta }_{i}},{{\dot{\Theta }}_{i}})\) is the joint friction. \({{\tau }_{si}}\) is the coupling torque at the torque sensor position, \({{\tau }_{i}}\) is the control input. \({{z}_{mi}}\), \({{z}_{i}}\) is the unit vector along the ith joint axis of rotation.

Fig. 1
figure 1

Joint model schematic of MRM

Full size image

Uncertainty analysis

The MRM dynamics model (1) with actuator faults and disturbance, it can be described as:

$$\begin{aligned} {{\ddot{\Theta }}_{i}}= & {} {{\left( {{I}_{mi}}{{\gamma }_{i}} \right) }^{-1}}\left( \begin{aligned}&{{\tau }_{i}}-{{I}_{mi}}\sum \limits _{j=2}^{i-1}{\sum \limits _{k=1}^{j-1}{z_{mi}^\mathrm{{T}}({{z}_{k}}\times {{z}_{j}}){{{\dot{\Theta }}}_{k}}{{{\dot{\Theta }}}_{j}}}-{{\tau }_{id}}} \\&-{{I}_{mi}}\sum \limits _{j=1}^{i-1}{z_{mi}^\mathrm{{T}}{{z}_{j}}{{{\ddot{\Theta }}}_{j}}}-\frac{{{\tau }_{si}}}{{{\gamma }_{i}}}-{{f}_{i}}({{\Theta }_{i}},{{{\dot{\Theta }}}_{i}}) \\ \end{aligned} \right) \nonumber \\{} & {} +\gamma \left( t-{{T}_{f}} \right) \phi \left( {{\Theta }_{i}},{{{\dot{\Theta }}}_{i}},{{\tau }_{i}} \right) , \end{aligned}$$
(2)

where \(\phi \left( {{\Theta }_{i}},{{{\dot{\Theta }}}_{i}},{{\tau }_{i}} \right) \) stands for fault function, \({{T}_{f}}\) represents the time of the fault, \(\gamma \left( t{-}{{T}_{f}} \right) \left\{ \begin{aligned}&0,\mathop {{}}_{{}}t<{{T}_{f}} \\&1,\mathop {{}}_{^{{}}}t\ge {{T}_{f}} \\ \end{aligned} \right. \). \({{I}_{mi}}\sum \nolimits _{j=2}^{i-1}{\sum \nolimits _{k=1}^{j-1}{z_{mi}^\mathrm{{T}}({{z}_{k}}\times {{z}_{j}}){{{\dot{\Theta }}}_{k}}{{{\dot{\Theta }}}_{j}}}} \) and \({{I}_{mi}}\sum \nolimits _{j=1}^{i-1}{z_{mi}^\mathrm{{T}}{{z}_{j}}{{{\ddot{\Theta }}}_{j}}} \) represent interaction coupling terms.

For analysis and design purposes, the MRM model (2) can be rewritten as follows:

$$\begin{aligned} \begin{aligned} {{\ddot{\Theta }}_{i}}^{{}}&= B_i{{\tau }_{i}}+{{H}_{i}}\left( \Theta ,\dot{\Theta },\ddot{\Theta } \right) \\&\quad +{{F}_{i}}\left( \Theta _i,\dot{\Theta }_i,\tau _i \right) +\Upsilon _i- B_i\frac{{{\tau }_{si}}}{{{\gamma }_{i}}}, \end{aligned} \end{aligned}$$
(3)

where \({{B_i}^{{}}}{{\text {=}}^{{}}}{{\left( {{I}_{mi}}{{\gamma }_{i}} \right) }^{-1}}\) denotes the know element, \(\Upsilon _i \text {=}-{{\left( {{I}_{mi}}{{\gamma }_{i}} \right) }^{-1}}\left( {{\tau }_{id}} \right) \) denotes uncertain disturbance, \({{F}_{i}}\left( \Theta _i,\dot{\Theta }_i,\tau _i \right) = -{{\left( {{I}_{mi}}{{\gamma }_{i}} \right) }^{-1}}\left( {{f}_{i}}({{\Theta }_{i}},{{{\dot{\Theta }}}_{i}}) \right) +\gamma \left( t-{{T}_{f}} \right) \phi \left( {{\Theta }_{i}},{{{\dot{\Theta }}}_{i}},{{\tau }_{i}} \right) \) denotes the uncertainty of the system,

$$\begin{aligned}{} & {} {{H}_{i}}\left( \Theta ,\dot{\Theta },\ddot{\Theta } \right) = -{{\left( {{I}_{mi}}{{\gamma }_{i}} \right) }^{-1}} \\ {}{} & {} \quad \left( {{I}_{mi}}\sum \limits _{j=1}^{i-1}{z_{mi}^\mathrm{{T}}{{z}_{j}}{{{\ddot{\Theta }}}_{j}}}+{{I}_{mi}}\sum \limits _{j=2}^{i-1}{\sum \limits _{k=1}^{j-1}{z_{mi}^\mathrm{{T}}({{z}_{k}}\times {{z}_{j}}){{{\dot{\Theta }}}_{k}}{{{\dot{\Theta }}}_{j}}}} \right) \end{aligned}$$

denotes the interconnected dynamic coupling.

Preliminaries

Theoretically, RBFNN are capable of approximate any continuous function with arbitrary accuracy as long as they have a sufficient number of neurons. Here, the RBFNN is utilized to approximate the single-valued function \({{Q}_{j}}\left( \theta \right) \) can be approximated by

$$\begin{aligned} {{Q}_{j}}\left( \theta \right) =W_{j}^\mathrm{{T}}{{\Phi }_{j}}(\theta )+{{\varepsilon }_{j}},\;\;\;\;\;\;\;j=1,2,...,n, \end{aligned}$$
(4)

where \(\theta \) is the NN input, \(W_{j}^{{}}\) is the NN weight, and \({{\varepsilon }_{j}}\) represents the approximation error satisfying \(\left\| {{\varepsilon }_{j}}\left( \theta \right) \right\| \le {{{\varepsilon }'}_{j}}\), \({{{\varepsilon }'}_{j}}\) is a positive constant. And \({{\Phi }_{j}}(x)\) is the NN activation function. The Gaussian function is chosen as the activation function:

$$\begin{aligned} {{\Phi }_{j}}(x)=\exp \left[ \frac{{{\left\| x-{{\mu }_{j}} \right\| }^{2}}}{2\sigma _{j}^{2}} \right] ,j=1,2,...,n, \end{aligned}$$
(5)

where \({{\mu }_{j}}\) is the center of receptive field and \(\sigma _{j}^{{}}\) is the withed of \({{\Phi }_{j}}(x)\).

Lemma 1

[56] By the consideration of a nonlinear system, supposing that there exists a positive definite Lyapunov function and that its derivatives satisfy:

$$\begin{aligned} {\dot{V}}(x)\le \text { }-{{c}_{1}}V(x)+{{c}_{2}}, \end{aligned}$$
(6)

where \({{c}_{1}}\) and \({{c}_{2}}\) are positive constants, then the solution \(x\left( t \right) \) is bounded.

Decentralized fault-tolerant controller design

Neural adaptive sliding mode design

Let \({{e}_{i}}={{\Theta }_{i}}-{{\Theta }_{id}}\) be defined as the tracking error, where \({{\Theta }_{id}}\) is defined as the desired reference trajectory. The filtered tracking error signal is defined as:

$$\begin{aligned} {{\chi }_{i}}={{{\dot{e}}}_{i}}+{{\lambda }_{i}}{{e}_{i}}, \end{aligned}$$
(7)

where \(\lambda \) is a positive parameter. Substituting Eq. (6) into the derivative of (10), the following equation is obtained:

$$\begin{aligned} \begin{aligned} {{{\dot{\chi }}}_{i}}&={{{\ddot{e}}}_{i}}+{{\lambda }_{i}}{{{{\dot{e}}}}_{i}} \\&{{{}_{{}}}_{{}}}=B_i{{\tau }_{i}}+{{H}_{i}}\left( \Theta ,\dot{\Theta },\ddot{\Theta } \right) +{{F}_{i}}\left( \Theta _i,\dot{\Theta }_i,\tau _i \right) +\Upsilon _i \\&\quad -B\frac{{{\tau }_{si}}}{{{\gamma }_{i}}}-{{{\ddot{\Theta }}}_{id}}+{{\lambda }_{i}}{{{{\dot{e}}}}_{i}}. \\ \end{aligned} \end{aligned}$$
(8)

The integral terminal sliding mode hyperplane is represented as follows:

$$\begin{aligned} {{s}_{i}}={{\chi }_{i}}+\int {\left( {{\mu }_{i}}\chi _{i}^{^{{{\alpha }_{i1}}}}+{{\nu }_{i}}\dot{\chi }_{i}^{^{{{\alpha }_{i2}}}} \right) }\mathrm{{d}}\upsilon , \end{aligned}$$
(9)

where \(\mu _i, \nu _i \) are design parameters, \(1<{{\alpha }_{i2}}<2\) and \({{\alpha }_{i1}}>{{\alpha }_{i2}}\).

$$\begin{aligned} \begin{aligned}&{{{{\dot{s}}}}_{i}}={{{\dot{\chi }}}_{i}}+\mu _i \chi _{i}^{^{{{\alpha }_{i1}}}}+\nu _i \dot{\chi }_{i}^{^{{{\alpha }_{i2}}}} \\&\quad =B_i{{\tau }_{i}}+{{H}_{i}}\left( \Theta ,\dot{\Theta },\ddot{\Theta } \right) +{{F}_{i}}\left( \Theta _i,\dot{\Theta }_i,\tau _i \right) +\Upsilon _i \\&\qquad -B_i\frac{{{\tau }_{si}}}{{{\gamma }_{i}}}-{{{\ddot{\Theta }}}_{id}}+\lambda _i {{{{\dot{e}}}}_{i}}+\mu _i \chi _{i}^{^{{{\alpha }_{i1}}}}+\nu _i \dot{\chi }_{i}^{^{{{\alpha }_{i2}}}}. \\ \end{aligned} \end{aligned}$$
(10)

The uncertainty term of the system (10) is then approximated by RBFNN.

$$\begin{aligned} {{F}_{i}}\left( \Theta _i,\dot{\Theta }_i,\tau _i \right) =W_{if}^\mathrm{{T}}{{\Phi }_{if}}({{\Theta }_{i}},{{\dot{\Theta }}_{i}})+{{\varepsilon }_{if}}\;\;\;\;\;\;\;\; \left\| {{\varepsilon }_{if}} \right\| \le {{\varepsilon }_{1}},\nonumber \\ \end{aligned}$$
(11)

where \({{W}_{if}}\) stands for the ideal NN weights, \(\Phi (\cdot )\) stands for the NN activation function, \({{\varepsilon }_{if}}\), \({{\varepsilon }_{1}}\) represent the approximation error, known positive constant, respectively. \({{\hat{W}}_{if}}\) means the estimate of the NN weight \({{W}_{if}}\), \({{\hat{F}}_{i}}\left( \Theta _i,\dot{\Theta }_i,\tau _i,\hat{W}_{if} \right) \) is the estimation value of \({{F}_{i}}\left( \Theta _i,\dot{\Theta }_i,\tau _i \right) \).And \({{\hat{F}}_{i}}\left( \Theta _i,\dot{\Theta }_i,\tau _i,{{{\hat{W}}}_{if}} \right) \) is represented as:

$$\begin{aligned} {{\hat{F}}_{i}}\left( \Theta _i,\dot{\Theta }_i,\tau _i,{{{\hat{W}}}_{if}} \right) =\hat{W}_{if}^\mathrm{{T}}{{\Phi }_{if}}({{\Theta }_{i}},{{\dot{\Theta }}_{i}}). \end{aligned}$$
(12)

Define the estimation error as \({{\tilde{W}}_{if}}={{W}_{if}}-{{\hat{W}}_{if}},\)

$$\begin{aligned}{} & {} {{F}_{i}}\left( \Theta _i,\dot{\Theta }_i,\tau _i \right) -{{\hat{F}}_{i}}\left( \Theta _i,\dot{\Theta }_i,\tau _i,\hat{W}_if \right) \nonumber \\{} & {} \quad =\tilde{W}_{if}^\mathrm{{T}}{{\Phi }_{if}}({{\Theta }_{i}},{{\dot{\Theta }}_{i}})+{{\varepsilon }_{if}}. \end{aligned}$$
(13)

Assumption 1

The interconnection term \({{H}_{i}}\left( \Theta ,\dot{\Theta },\ddot{\Theta } \right) \) is bounded by

$$\begin{aligned} |{{H}_{i}}\left( \Theta ,\dot{\Theta },\ddot{\Theta } \right) |\le \sum \limits _{j=1}^{n}{{{d}_{ij}}{{E}_{j}}}, \end{aligned}$$
(14)

where \({{d}_{ij}}\ge 0,{{E}_{j}}=1+|{{s}_{j}} |+{{|{{s}_{j}} |}^{2}}\), and define \({{p}_{i}}\left( |{{s}_{i}} |\right) =n {\mathop {\max \nolimits _{ij} }}\,\left\{ {{d}_{ij}} \right\} {{E}_{i}}\).

The interconnection term are estimated as follows:

$$\begin{aligned} {{\hat{p}}_{i}}\left( |{{s}_{i}} |,{{{\hat{W}}}_{ip}} \right) ={{\hat{W}}_{ip}}\Phi \left( |{{s}_{i}} |\right) , \end{aligned}$$
(15)

where \({{\hat{W}}_{ip}}\) is an estimate of the NN weight \({{W}_{ip}}\). The estimated error is defined as \({{\tilde{W}}_{ip}}={{W}_{ip}}-{{\hat{W}}_{ip}}\).

Nonlinear disturbance observer design

To obtain the disturbance observer, combined with the RBF neural network, the following equation can be obtained:

$$\begin{aligned} {{{\dot{s}}}_{i}}= & {} B_i{{\tau }_{i}}+W_{if}^\mathrm{{T}}{{\Phi }_{if}}({{\Theta }_{i}},{{\dot{\Theta }}_{i}})+{{H}_{i}}\left( \Theta ,\dot{\Theta },\ddot{\Theta } \right) +{{\Omega }_{i}}\nonumber \\{} & {} \quad -B_i\frac{{{\tau }_{si}}}{{{\gamma }_{i}}}-{{\ddot{\Theta }}_{id}}+\lambda _i {{{\dot{e}}}_{i}}+\mu _i \chi _{i}^{^{{{\alpha }_{i1}}}}+\nu _i \dot{\chi }_{i}^{^{{{\alpha }_{i2}}}}, \end{aligned}$$
(16)

where \({{\Omega }_{i}}=\Upsilon _i+{{\varepsilon }_{if}}\) denote a new disturbance in the system. And assuming that this estimation error is bounded \(\left\| {{{\dot{\Omega }}}_{i}} \right\| \le \kappa \). \(\kappa \) is a small positive constant. To obtain the estimate of the integration uncertainty disturbance term in Eq. (16), DO is designed as follows.

$$\begin{aligned} \left\{ \begin{aligned}&{{{\hat{\Omega }}}_{i}}=\vartheta _i+\sigma _i s_i \\&\dot{\vartheta }_i=-\sigma _i \left[ \begin{aligned}&B_i{{\tau }_{i}}+{{{\hat{p}}}_{i}}\left( |{{s}_{i}} |,{{{\hat{W}}}_{ip}} \right) \mathrm{{sgn}}\left( {{s}_{i}} \right) -B_i\frac{{{\tau }_{si}}}{{{\gamma }_{i}}} \\&+\hat{W}_{if}^\mathrm{{T}}{{\Phi }_{if}}({{\Theta }_{i}},{{{\dot{\Theta }}}_{i}})-{{{\ddot{\Theta }}}_{id}}+\lambda _i {{{{\dot{e}}}}_{i}} \\&+\mu _i \chi _{i}^{^{{{\alpha }_{i1}}}}+\nu _i \dot{\chi }_{i}^{^{{{\alpha }_{i2}}}}+{{{\hat{\Omega }}}_{i}} \\ \end{aligned} \right] , \\ \end{aligned} \right. \nonumber \\ \end{aligned}$$
(17)

where \(\sigma _i \) is a positive constant.

The estimation error of DO is shown below:

$$\begin{aligned} {{\tilde{\Omega }}_{i}}={{\Omega }_{i}}-{{\hat{\Omega }}_{i}}, \end{aligned}$$
(18)

where \({{\hat{\Omega }}_{i}}\) is the value of the estimates for \({{\Omega }_{i}}\).

To make it easy to the stability analysis of DO, the weights are assumed to be ideally optimal. i.e., \({{\tilde{W}}_{ip}}=0,\tilde{W}_{iE}^{{}}=0\). The time derivative of Eq. (18) is:

$$\begin{aligned} {{\dot{\tilde{\Omega }}}_{i}}={{\dot{\Omega }}_{i}}-\sigma _i {{\tilde{\Omega }}_{i}}. \end{aligned}$$
(19)

The stability of the nonlinear disturbance observer is presented in Appendix A.

Neural network compensation control design considering actuator saturation

To solve the unknown actuator saturation, the following nonlinear function of unknown actuator saturation is defined:

$$\begin{aligned} {{\Gamma }_{i}}={{\tau }_{i}}-{{u}_{i}}, \end{aligned}$$
(20)

where \({{\Gamma }_{i}}\) is the unknown nonlinear function. \({{u}_{i}}\) is the controller input.

$$\begin{aligned} \begin{aligned} {{{\dot{s}}}_{i}}&=B_i{{u}_{i}}+B_i{{\Gamma }_{i}}+W_{if}^\mathrm{{T}}{{\Phi }_{if}}({{\Theta }_{i}},{{\dot{\Theta }}_{i}})\\&\quad +{{H}_{i}}\left( \Theta ,\dot{\Theta },\ddot{\Theta } \right) +{{\hat{\Omega }}_{i}}+\varepsilon _i\\&\quad -B\frac{{{\tau }_{si}}}{{{\gamma }_{i}}}-{{\ddot{\Theta }}_{id}}+\lambda _i {{{\dot{e}}}_{i}}+\mu _i \chi _{i}^{^{{{\alpha }_{i1}}}}+\nu _i \dot{\chi }_{i}^{^{{{\alpha }_{i2}}}}. \end{aligned} \end{aligned}$$
(21)

The neural adaptive compensation control is employed to approximate the unknown term, which can be expressed as:

$$\begin{aligned}{} & {} {{\Gamma }_{i}}=W_{i\Gamma }^\mathrm{{T}}{{\Phi }_{i\Gamma }}\left( {{\Theta }_{id}},{{{\dot{\Theta }}}_{id}} \right) +{{\varepsilon }_{i\Gamma }} \;\;\;\;\;\; \left\| {{\varepsilon }_{i\Gamma }} \right\| \le {{\varepsilon }_{2}}, \end{aligned}$$
(22)
$$\begin{aligned}{} & {} {{\hat{\Gamma }}_{i}}=\hat{W}_{i\Gamma }^\mathrm{{T}}{{\Phi }_{i\Gamma }}({{\Theta }_{id}},{{\dot{\Theta }}_{id}}), \end{aligned}$$
(23)
$$\begin{aligned}{} & {} \begin{aligned}&{{{\tilde{\Gamma }}}_{i}}=W_{i\Gamma }^\mathrm{{T}}{{\Phi }_{i\Gamma }}\left( {{\Theta }_{id}},{{{\dot{\Theta }}}_{id}} \right) -\hat{W}_{i\Gamma }^\mathrm{{T}}{{\Phi }_{i\Gamma }}({{\Theta }_{id}},{{{\dot{\Theta }}}_{id}}) \\&\;\;\;\;=\tilde{W}_{i\Gamma }^\mathrm{{T}}{{\Phi }_{i\Gamma }}({{\Theta }_{id}},{{{\dot{\Theta }}}_{id}})+{{\varepsilon }_{i\Gamma }}. \\ \end{aligned} \end{aligned}$$
(24)

The decentralized fault-tolerant controller in this paper is designed as:

$$\begin{aligned}{} & {} {{u}_{i}}={{u}_{n}}+{{u}_{s}}, \end{aligned}$$
(25)
$$\begin{aligned}{} & {} {{u}_{n}}=-{{B}^{-1}}\nonumber \\{} & {} \quad \left( \begin{aligned}&{{{\hat{p}}}_{i}}\left( |{{s}_{i}} |,{{{\hat{W}}}_{ip}} \right) \mathrm{{sgn}}\left( {{s}_{i}} \right) -B\frac{{{\tau }_{si}}}{{{\gamma }_{i}}}+\hat{W}_{if}^\mathrm{{T}}{{\Phi }_{if}}({{\Theta }_{i}},{{{\dot{\Theta }}}_{i}}) \\&-{{{\ddot{\Theta }}}_{id}}+\lambda _i {{{{\dot{e}}}}_{i}}+\mu _i \chi _{i}^{^{{{\alpha }_{1}}}}+\nu _i \dot{\chi }_{i}^{^{{{\alpha }_{2}}}}+{{{\hat{\Omega }}}_{i}}+{{{\hat{\Gamma }}}_{i}}+\hat{\varepsilon }_i \\ \end{aligned} \right) ,\nonumber \\ \end{aligned}$$
(26)
$$\begin{aligned}{} & {} \begin{aligned}&{{u}_{s}}=-{{B}^{-1}}\left( {{k}_{i1}}\sqrt{|s_i |}\mathrm{{sgn}}\left( s_i \right) +\xi _i \right) , \\&\dot{\xi }={{k}_{i2}}\mathrm{{sgn}}\left( s_i \right) , \\ \end{aligned} \end{aligned}$$
(27)

where \({{\hat{p}}_{i}}\left( |{{s}_{i}} |,{{{\hat{W}}}_{ip}} \right) \mathrm{{sgn}}\left( {{s}_{i}} \right) ,_{{}}^{{}}\hat{W}_{if}^\mathrm{{T}}{{\Phi }_{if}}({{\Theta }_{i}},{{\dot{\Theta }}_{i}})\) are provided to compensate the uncertainty term. \(k_{i1},k_{i2}\) are position constant. The \(\mathrm{{sgn}}\) is given as:

$$\begin{aligned} \mathrm{{sgn}}\left( s_i \right) \left\{ \begin{array}{ll} 1 &{} \mathrm{{if}} \quad s_i>0 \\ 0 &{} \mathrm{{if}}\quad s_i=0 \\ -1 &{} \mathrm{{if}} \quad s_i<0. \\ \end{array} \right. \end{aligned}$$
(28)

The adaptive laws are shown below:

$$\begin{aligned}&{{\dot{\hat{W}}}_{if}}=\xi {}_{if}{{s}_{i}}{{\Phi }_{if}}\left( {{\Theta }_{i}},{{{\dot{\Theta }}}_{i}} \right) , \end{aligned}$$
(29)
$$\begin{aligned}&{{\dot{\hat{W}}}_{ip}}=\xi {}_{ip}|{{s}_{i}} |{{\Phi }_{ip}}\left( |{{s}_{i}} |\right) , \end{aligned}$$
(30)
$$\begin{aligned}&{{\dot{\hat{W}}}_{i\Gamma }}=\xi {}_{i\Gamma }B{{s}_{i}}{{\Phi }_{i\Gamma }}\left( {{\Theta }_{id}},{{{\dot{\Theta }}}_{id}} \right) , \end{aligned}$$
(31)
$$\begin{aligned}&{{\dot{\hat{\varepsilon }}}_{i}}={{\Psi }_{i}}|{{s}_{i}} |. \end{aligned}$$
(32)

Theorem 1

Consider an MRM system (2) with a neural adaptive decentralized FTC (25), and the update laws (29)–(32), where the system error will converge to zero and the MRMs system is stable. The stability is given in Appendix B.

Simulation

To guarantee the effectiveness of the proposed decentralized FTC strategy, 3-DOF MRMs are used for performing simulations in this subsection.

The reference trajectories of the individual joints are as follows:

$$\begin{aligned} \begin{aligned}&{{\theta }_{1d}}=0.6\sin 1.5t+0.5\sin 0.5t, \\&{{\theta }_{2d}}=0.5\sin 0.4t+0.5\left( \cos 2t-1 \right) , \\&{{\theta }_{3d}}=0.6{\text {sin2t}}-0.4\sin t. \\ \end{aligned} \end{aligned}$$
(33)

To validate the superiority of the proposed controller, the performance of PID-SMC, NFTSMC and the proposed method was compared with each other. The rotor inertia \(I_{mi}=0.0000345\), the reduction ratio \(\gamma =101 \). The selection of controller parameters: In PID-SMC, \( k_p=230, k_i=98, k_d=10\). In NFTSMC, \(\beta =35, p=5, q=3, \lambda _i=7/5, k_i=20\). In proposed controller, \(\lambda _i=10, \mu _i=10, \nu =3, \alpha _1=3, \alpha _2=5/3\), the gain of DO \(\sigma _i=15, k_{i1}=15, k_{i2}=10\). The central value of the RBF neural network used in this paper is \([-2\;-1\;\; 0 \;\;1\;\; 2;\;\; -2\;\; -1\;\; 0 \;\;1 \;\;2]\), and the width is 3. Adaptive rate parameters \(\xi _{if}=0.003,\xi _{ip}=500, \xi _{i\Gamma }=0.003,\Psi _{i}=100\). The effectiveness of the designed perturbation observer is first analyzed. The disturbances are \(\tau _{d1}=0.3\sin (t)\), \(\tau _{d2}=0.3\sin (t)-0.2\cos ^2(t)\), \(\tau _{d3}=0.3\sin ^2(t)-0.2\cos (t)\) respectively.

Fig. 2
figure 2

Disturbance estimation curve

Full size image

The curves of the actual disturbance \(\tau _{d}\) and the output of the disturbance observer are shown in Fig. 2. From Fig. 2, it can be seen that the disturbance observer has a high estimation accuracy.

Fig. 3
figure 3

Trajectory tracking curves of three different controllers under normal operation

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Then, two sets of simulation experiments were designed to emphasize the performance of the proposed controller. First of all, the MRM operated without fault. From Figs. 3 and 4 that all three different control methods have good tracking performance. Due to the robustness of SMC to uncertainties and disturbances, the proposed method as well as NFTSMC have lower tracking errors than PID-SMC. The presented method has superior tracking capability than NFTSMC. Based on the root mean square error (RMSE) of the controller, Table 1 shows the performance of different state-of-the-art control methods.

As shown in Table 1, the tracking errors of the PID-SMC on three joints are 0.0031, 0.0023 and 0.0026, respectively. The high tracking performance of the NFTSMC is clearly shown as 0.0015, 0.0011 and 0.0012, respectively. While the RSME of the proposed method are 0.00011, 0.0.000054 and 0.000081, respectively. The control inputs of the controllers are shown in Fig. 5. Obviously, the proposed method provided a relatively smooth control effect and effectively reduces chattering. According to the above analysis, the tracking capability of the presented strategy is superior in comparison.

Fig. 4
figure 4

Tracking errors without fault

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Fig. 5
figure 5

Control inputs without fault

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Table 1 RMSE comparison of each control method
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To further test the validity of the presented FTC approach, the performance of these controllers in the event of a system fault was considered in the simulations. It is assumed that the fault function: \({{\phi }_{1}}\left( {{\theta }_{1}},{{{\dot{\theta }}}_{1}},{{\tau }_{1}} \right) =1.5\sin \left( {{\theta }_{1}} \right) \), occurred in 6 s for joint 1, and \({{\phi }_{2}}\left( {{\theta }_{2}},{{{\dot{\theta }}}_{2}},{{\tau }_{2}} \right) =0.35{{\tau }_{2}}\) occurred in 10 s for joint 2, and the joint 3 healthy operation did not fault. The tracking error of the system when the controller input fails is shown in Fig. 6. As can be seen from Fig. 6, the robustness of PID-SMC and NFTSMC to failure effects is very low. When the failure occurs at time \(t=6s\) and \(t=10s\), the stability of the system is almost destroyed.

Fig. 6
figure 6

Tracking errors with fault

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Fig. 7
figure 7

Control inputs with fault

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In the event of a fault, PID-SMC and NFTSMC are fault-tolerant controllers without saturation compensation. The proposed method employs a control law (25) consisting of a trajectory tracking controller and a saturation compensator. Figure 6 illustrates the trajectory error curve under the fault condition, and Fig. 7 demonstrates the control input under the fault condition. From Figs. 6 and 7, for joint 1, the fault (\(1.5\sin \left( {{\theta }_{1}} \right) \)) occurs at \(6\,s\), the tracking performance of PID-SMC and NFTSMC is degraded, but also remains within an acceptable range. The performance of the proposed method is superior. Joint 2 faults (\(0.35{{\tau }_{2}}\)) at 10s, the actuator has partially failed, and PID-SMC and NFTSMC are less robust to fault effects. According to Table 1, the RMSEs of the proposed method are 0.00073, 0.00055, and 0.000081, respectively. In contrast, the corresponding RMSEs of PID-SMC, and NFTSMC are (0.0286, 0.3072, 0.0028) and (0.0138, 0.2997, 0.0013). The simulation result figures given above can be seen, even though the PID-SMC-based FTC and the NFTSMC-based FTC have a degree of fault resistance. Due to the long existence of faults and the saturation of the motor, the output torque cannot meet the control requirements, leading to the control torque increasing. Under this condition, joint 2 can be damaged by excessive current to the point of causing a major accident. In comparison to the other two control methods, the presented control approach provides superior robustness and response to transient effects of faults. Therefore, the the presented control approach performs superior to PID-SMC and NFTSMC.

Conclusions

In this paper, a novel neural adaptive integral terminal sliding mode control for decentralized fault-tolerant control method is proposed for MRMs with actuator saturation. Based on JTF technology, the dynamic model of the MRMs system is established. Next, a compensated learning controller is designed by RBFNN and DO to deal with the uncertainties and external disturbances of the MRMs system. Then, STA is designed to attenuate the chattering of the MRMs system and improve the dynamic performance. In addition, an anti-saturation neural compensator is proposed for the problem of actuator saturation after a fault occurs. Finally, the stability of the MRM system is verified based on Lyapunov theory. The simulation experiments analyze and validate the effectiveness and advantages of the presented decentralized FTC approach. For future research, it is a research direction to obtain the optimal parameters to achieve the system’s stability.

In summary, compared with existing algorithms, our study considers both actuator saturation, uncertainty model, and disturbance uncertainty to solve the FTC problem of MRMs. Numerical simulation results show that the actual trajectory can achieve accurate tracking of the desired trajectory, and the actual input can be always limited to the saturation amplitude even if actuator saturation occurs.

Remark There are many reasons for generating actuator saturation, which are also briefly described in the introduction. In this paper, the controllers designed for external disturbances, actuator faults, and system uncertainties make the trajectory tracking performance and error convergence accuracy high, and all the above factors increase the control force and torque of MRMs. Therefore, it is necessary to consider control actuator saturation in our work.