Decentralized fault-tolerant control of modular robot manipulators with actuator saturation: neural adaptive integral terminal sliding mode-based control approach

A novel neural adaptive integral terminal sliding mode control for decentralized fault-tolerant control strategy, including the integral terminal sliding mode surface, the nonlinear disturbance observer, the radial basis neural network and robust controller, is presented in this paper to achieve fault-tolerant control of modular robot manipulators. First, the integral terminal sliding mode is designed for the fault-tolerant controller. Then, to boost the performance of the controlled system, the radial basis neural network and disturbance observer are introduced to approximate the nonlinear terms and disturbances. The reconstructed approximate uncertainty term is applied as compensation. Next, the super-twisting technique is employed to compensate for estimation errors to ensure stability. In addition, for the actuator saturation problem, the radial basis function neural network-based compensation control is proposed. Finally, the stability of the closed-loop robotic system is demonstrated based on Lyapunov theory. Computer simulations verified the efficiency and advantages of the presented approach.


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][7][8].FTC strategies include two 123 control strategies, active fault-tolerance control (AFTC) [9,10], and passive fault-tolerance control (PFTC) [11,12].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][33][34].A novel fast terminal sliding mode control (FTSMC) approach is designed in [35], which increases the convergence of the controlled system and overcomes the singularity problem of the system.To reduce chattering and improve the dynamic accuracy of the system, a novel approach of adaptive nonlinear fractional PID-SMC was presented in [36].Considering the external disturbance and additional uncertainty, the finite-time stability problem of the active magnetic bearing system is solved through the construction of terminal sliding mode control (TSMC) [37].TSMC is used as an effective robust control methodology that retains the advantages of classical sliding mode while allowing the system state to achieve the desired control objective within a finite time.Although TSMC has unparalleled advantages, it cannot guarantee that the control system can achieve the expected dynamic performance throughout the entire phase.To further improve the capability of SMC, some researchers have introduced the integral sliding mode control (ISMC) technique that enforces the system state to be on the sliding mode hyperplane during the initial phase, thereby eliminating the arrival stage of SMC.However, the performance of ISMC relies critically on the dynamic characteristics of the system.When system uncertainty is great, the steady-state error also increases.
According to the published literature, neural networks (NN) and fuzzy control techniques have been widely employed to approximate the unknown nonlinear function [38][39][40][41][42].In [38], the Takagi-Sugeno fuzzy neural network was utilized as an approximator to obtain approximate values.The Takagi-Sugeno fuzzy method was adopted to approximate the nonlinear model in [39].The unknown nonlinear dynamics are learned by the NN approximators 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 com-bined 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 neuroadaptive 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.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. 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."Decen-

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].
where I mi is the rotor inertia, γ i is the reducer ratio, i is the joint position, f i ( i , ˙ i ) is the joint friction.τ si is the coupling torque at the torque sensor position, τ i is the control input.z mi , z i is the unit vector along the ith joint axis of rotation.

Uncertainty analysis
The MRM dynamics model (1) with actuator faults and disturbance, it can be described as: where φ i , ˙ i , τ i stands for fault function, T f represents the time of the fault, and I mi i−1 j=1 z T mi z j ¨ j represent interaction coupling terms.For analysis and design purposes, the MRM model ( 2) can be rewritten as follows: where denotes the uncertainty of the system, 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 (θ ) can be approximated by where θ is the NN input, W j is the NN weight, and ε j represents the approximation error satisfying ε j (θ ) ≤ ε j , ε j is a positive constant.And j (x) is the NN activation function.The Gaussian function is chosen as the activation function: where μ j is the center of receptive field and σ j is the withed of 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: where c 1 and c 2 are positive constants, then the solution x (t) is bounded.

Neural adaptive sliding mode design
Let e i = i − id be defined as the tracking error, where id is defined as the desired reference trajectory.The filtered tracking error signal is defined as: where λ is a positive parameter.Substituting Eq. ( 6) into the derivative of (10), the following equation is obtained: The integral terminal sliding mode hyperplane is represented as follows: where μ i , ν i are design parameters, 1 < α i2 < 2 and α i1 > α i2 .
The uncertainty term of the system (10) is then approximated by RBFNN.
where W i f stands for the ideal NN weights, (•) stands for the NN activation function, ε i f , ε 1 represent the approximation error, known positive constant, respectively.Ŵi f means the estimate of the NN weight Define the estimation error as Assumption 1 The interconnection term H i , ˙ , ¨ is bounded by where

and define p i (|s
The interconnection term are estimated as follows: where Ŵip is an estimate of the NN weight W i p .The estimated error is defined as Wip = W i p − Ŵip .

Nonlinear disturbance observer design
To obtain the disturbance observer, combined with the RBF neural network, the following equation can be obtained: where i = ϒ i +ε i f denote a new disturbance in the system.And assuming that this estimation error is bounded ˙ i ≤ κ. κ is a small positive constant.To obtain the estimate of the integration uncertainty disturbance term in Eq. ( 16), DO is designed as follows.
where σ i is a positive constant.The estimation error of DO is shown below: where ˆ i is the value of the estimates for i .
To make it easy to the stability analysis of DO, the weights are assumed to be ideally optimal.i.e., Wip = 0, Wi E = 0.The time derivative of Eq. ( 18) is: 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: where i is the unknown nonlinear function.u i is the controller input.
The neural adaptive compensation control is employed to approximate the unknown term, which can be expressed as: The decentralized fault-tolerant controller in this paper is designed as: where pi |s i |, Ŵip sgn (s i ) , Ŵ T i f i f ( i , ˙ i ) are provided to compensate the uncertainty term.k i1 , k i2 are position con-123 stant.The sgn is given as: The adaptive laws are shown below: 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 curves of the actual disturbance τ 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.
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.
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: φ 1 θ 1 , θ1 , τ 1 = 1.5 sin (θ 1 ), occurred in 6 s for joint 1, and φ 2 θ 2 , θ2 , τ 2 = 0.35τ 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.
In the event of a fault, PID-SMC and NFTSMC are faulttolerant 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 (θ 1 )) 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 (0.35τ 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-SMCbased FTC and the NFTSMC-based FTC have a degree of fault resistance.Due to the long existence of faults and the sat- uration 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 antisaturation neural compensator is proposed for the problem of actuator saturation after a fault occurs.Finally, the sta-bility 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.

Appendix A: Stability analysis of the observer
The following candidate function is considered: From Eqs. ( 19) and (A1), and young's inequality, it can be obtained: where σ satisfies σ i > 1 2 .From Eq. (A2), it can be proved that ˜ i is bounded by Lemma 1.

Appendix B: Stability analysis of the control system
Proof Select the Lyapunov candidate function as follows: The derivative of Eq. (B3) with respect to time: Substituting the proposed decentralized fault-tolerant control law (25) into Eq.(B4), one obtains: 123 After calculating Eq. (B5), the following equation can be obtained: Equations ( 11) and ( 15) are substituted into Eq.(B6), and by considering ( 14) yields the expression: (B9) It is clear that V ≤ 0 if the selected parameter k i1 > 0 and k i2 > 0 in Eq. (B9).Based on the Lyapunov stability theory and Barabalat Lemma, the system state error e i will converge to asymptotically.

Fig. 1
Fig. 1 Joint model schematic of MRM

Fig. 3
Fig. 3 Trajectory tracking curves of three different controllers under normal operation

1 2− 1 i−k i1 |s i | 1 2−k i1 |s i | 1 2
i | W T i p i p (|s i |) + B ˜ i + εi |s i | − k i1 |s i | |s i | − k i2 |s i | − W T i p ζ −1 i p Ẇ i p − W T i ξ that |s i | ≤ |s j | ⇔ E i ≤ E j ,and the Chebyshev inequality as follows:i p |s i | i p (|s i |) − ζ −1 i p Ẇ i p |s i | − k i2 |s i | ≤ n i=1 |s i | − k i2 |s i | .
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