Event-Triggered Adaptive Neural Tracking Control of Flexible-Joint Robot Systems With Input Saturation

This paper investigates an event-triggered adaptive neural tracking control issue for flexible-joint robot (FJR) systems subject to unknown dynamic and input saturation. To enable the backstepping design framework to be implemented, the input saturation nonlinearity is replaced by a smooth function. In the control design, the dynamic surface control (DSC) and adaptive neural techniques are used to handle the “explosion of complexity” issue and unknown dynamics, respectively. Furthermore, to reduce the calculated burden caused by the adaptive neural reconstruction technique, three virtual parameters are updated by using the single-parameter-learning approach. To decrease the frequency of actuator response to the control command for reducing the mechanical wear of actuator, an event triggering mechanism is established between the control law and actuator. Finally, an event-triggered adaptive neural tracking control solution is proposed, which is endowed the advantages as: (1) it does not need any priori knowledge of FJR systems; 2) it only needs to update three unknown parameters; 3) it can reduce the transmission frequency of the control commands and the response rate of the actuator. Using the Lyapunov stability theory, the proposed event-triggered control solution ensures that all signals of the closed-loop tracking control system are bounded. Finally, the simulation results verify the effectiveness and superiority of the proposed control scheme.


I. INTRODUCTION
For the past few years, intense interests have been paid close attention to the modeling and control of flexible joint robot (FJR) systems [1]- [3], [5], [14] due to its important role in civil and military affairs. To solve the control issue of FJR systems, many effective control methods have been reported, such as uncertain observer [6], PID/PD control [7]- [9], adaptive sliding mode control [10], feedback linearisation control [11], [12], passivity control [13], singular perturbation control [14], adaptive backstepping control [15]- [17], etc. In practice, due to the modeling technique and plant's nonlinear character, the FJR systems inevitably suffer from the uncertainties. In this context, the difficulty of control design derives from the uncertain and nonlinear characteristic. For the control issue of uncertain nonlinear systems, the adaptive backstepping design framework reveals its unique The associate editor coordinating the review of this manuscript and approving it for publication was Bidyadhar Subudhi . advantage [18]. However, owing to the recursive design process, the virtual control function needs to be differentiated repeatedly such that there exists a drawback called the ''explosion of complexity'' in the design framework. To solve such a problem, Swaroop et al. [19] proposed the dynamic surface control (DSC) scheme for a class strict-feedback nonlinear systems, in which a first-order filter is introduced into the each step of the traditional backstepping design. Based on the DSC method, many significative research results have also been reported to solve the control issue of FJR systems [20]- [22].
To compensate the uncertainty of FJR systems, several effective approaches, such as extended state observer (ESO) [23]- [25], time-delay estimation [26], [27], neural network (NN) [28], [29], fuzzy logic system (FLS) [30], [31], etc. To avoid the problem of ''explosion of complexity'' in the works [29]- [31], Joo et al. [32] and Miao et al. [33] introduced a first-order filter into the backstepping design such that the differential operation of virtual control law is VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ replaced by the algebraic operation. The other implicit benefit of such an operation is that the input dimensionality of NN or FLS is reduced greatly [34]. It should be pointed out that the works in [29]- [32] surfer from the dimensional curse issue [35]. To this end, for the control issue of FJR systems, Ling et al. [36] taken the minimum learning parameter technique [34] into the DSC design framework to solve the dimensional curse issue. In practice, for any physical systems, input saturation is an inevitable problem, which ascribes the physical property of actuator. In the existing works, many effective methods have been reported to handle the input saturation issue, such as the domain of attraction [37], gain scheduling [38], model augmentation [39], auxiliary dynamic design [40], [41], smooth function substitution [42], etc. It should be pointed out that the method based on the domain of attraction requires the knowledge of plant; the gain scheduling-based method requires the mapping between system response performance and control gain, which leads to the closed-loop system incredibly complex; the model augmentation requires that the plant needs to be augmented with a first-order form, which increases the difficulty of control design. Compared with the model augmentation, the auxiliary dynamic design can simplify the control design, but it may introduce a non-smooth signal into the closed-loop system. For such an issue, a smooth function is used to replace the non-smooth saturation nonlinearity in the smooth function substitution-based method [36]. Nevertheless, the work [36] required the part model knowledge, which leads to the control solution difficulty to be implemented. Furthermore, Ding et al. [43] employed the neural approximation technique to reconstruct the unknown dynamic of FJR systems with input saturation.
Although the above methods effectively have solved the problems of unknown system dynamic, a practical problem, i.e., the actuator has a working range in the command response frequency, is not considered. In addition, owing to the sensor measurement noise, the high-frequency noise may enter into the closed-loop system, which leads to the high frequency response of the actuator and mechanical wear of actuator. Therefore, it is essential how to reduce the mechanical wear of actuator to achieve the purpose of reducing the frequency of actuator failures. For such an issue, the event-triggered control (ETC) [44], [45] method has its preponderance, in which the control commands are transmitted only if a certain event is triggered [46]- [49]. Based on this idea, the event-triggered control for FJR systems has attracted the attention of many scholars [50], [51]. In [52], an eventtriggered adaptive asymptotically tracking control solution for a single-Link Robot was proposed, in which the transmitting and computation burdens of control commands are effectively reduced by using the DSC and ETC techniques. It is should be pointed out that the event triggering mechanisms proposed in [50]- [52] only taken the measuring error of control input into account, but the control accuracy was not considered in the design. However, for the FJR systems, the control accuracy also needs to be followed in the design of event triggering mechanism.
Motivated by the previous works, an event-triggered adaptive neural tracking control scheme for FJR systems under unknown dynamics and input saturation is presented. The main contributions can be summarized as follows.
• This work proposed a DSC-based adaptive neural tracking control scheme, in which a single parameter learning and adaptive neural network techniques are used to handle the unknown dynamics and parameters. Compared with the work [29], [31], [32], our proposed control solution does not require any priori knowledge of plant.
• Different from the event triggering mechanism [50]- [52], the event triggering mechanism proposed in this work includes the internal and external trigger condition. As a result, the transmission frequency of the control commands and the execution rate of the actuator not only are drastically reduced, but also the control performance can be ensured.
• Using the single-parameter-learning idea, this work designs three adaptive laws and each one requires only one regressor, which avoids the problem of dimensional curse in [29]- [31], and overcomes the design difficulty caused by the unknown gains of FJR systems.
The rest of this paper is arranged as follows. In Section 2, the mathematical model of MSVs and the problem formulation are introduced. In Section 3, the principle of intelligent approximation using NN is presented. In Section 4, proposes the details of controller design procedures. In Section 5, the simulation results are given to show the effectiveness of the controller. In Section 6, the entire work is summarized.

II. PROBLEM FORMULATION AND PRELIMINARIES A. PROBLEM FORMULATION
According to the Euler-Lagrangian (EL) equation, the single-link FJR dynamics can be described as follows [36] ML 2q + MgL sin(q) where q,q andq denote the link position, velocity, and acceleration of FJR, respectively. q m ,q m andq m denote the rotor angular position, velocity, and acceleration, respectively. M, L and g denote the mass and length of link and the gravity acceleration, respectively. F(q) is the friction. K, J and B denote the coefficients of joint stiffness, the joint flexibility and the damping, respectively. u is the actual control input of plant.
Considering the issue of actuator's physical constraint, the control input u is subjected to input saturation, which can be described by where u c is the control command generated by control law, sgn(·) denotes the sign function, and u m is the maximum value of actuator control input. Considering the issue of actuator's physical constraint, the control input u is subjected to input saturation, which can be described by where sgn(·) denotes the sign function, and u m is the maximum value of actuator control input. Remark 1: In practice, due to the modeling technique, nonlinear characteristic, etc., it is very difficult to obtain the accurate knowledge of the model parameters of FJR systems. In addition, the friction F(q) is not modelable. For the tracking control issue, it is common that the reference trajectory y d and its derivativesẏ d andÿ d are required to known. Similar requirements are also mentioned in [2], [32], [33], [36]. Therefore, assumptions 1-2 are reasonable.
According to (4), the u is a non-smooth function with respect to u c . To facilitate the implementation of backstepping design framework, a smooth function is used to replace the non-smooth nonlinear function u, i.e., where d(u c ) is the approximate error, which satisfies According to Mean-value theorem, δ(u c ) can be written as , 0 < ι < 1, i.e., there exist a constant ω satisfying 0 < ω < . Furthermore, using (5)- (7), one can get Let From |d(u c )| ≤ D, there exists a constant ψ d satisfying |d| ≤ ψ d . In addition, according to 0 < ω < and ∈ (0, 1], there exists a constant ψ such that ψ 2 ≥ ψ can be held.

Remark 2:
From Assumption 1, we know that ψ i , i = 1, 2, is unknown, and f 2 (x 2 ) and f 4 (x 4 ) are unknown function. Nonlinear function f 2 (x 2 ) contains F(q), which is unmodelable. Thus, f 2 (x 2 ) can not be linearly parameterized. In order to handle the problem, we will utilize RBF NNs to approximate the unknown nonlinear function f 2 (x 2 ) and f 4 (x 4 ).
The control objective of this paper is to design an event-triggered adaptive NN control law u c for the single-link FJR system (1)-(2) under Assumptions 1-2 such that the actual trajectory y tracks a reference trajectory y d , while ensuring that all signals in the closed-loop single-link FJR control system are bounded.

B. PRELIMINARIES
Lemma 1 ( [40]): Any nonlinear function β(X) : R n → R defined on a compact set X ∈ R n can be estimated by the radial basis function (RBF) NN ξ * T (X), and then we have where the approximate error ε satisfies |ε| ≤ε * withε * being a positive constant. Vector ξ (X) represents the basis function. The ideal weight vector of NN is expressed as * = [ * 1 , · · · , * ] ∈ R l . > 1 is the number of node. The basis function ξ i (X) is the Gaussian error function, that is, where c i = [c i1 , · · · , c i ] T represents the center of the receptive area, and ω is the breadth of Gaussian function.

III. MAIN RESULTS
In this section, we attempt to design an event-triggered adaptive neural tracking control (ETABTC) law using DSC design framework for the single-link FJR system. The design of ETABTC is based on the following change of coordinates: where χ i is the filter version of the virtual control law α i−1 , which is produced by the following filter where χ i ∈ R is the state of filter (15), and τ is the filter design constant. The filter initial conditions satisfy χ i (0) = α i−1 (0).

A. ROBUST ADAPTIVE NEURAL TRACKING CONTROL LAW DESIGN
In this subsection, we present the design procedure of ETABTC law, and the whole design procedure contains 4 steps. At each step, the virtual/atual control laws VOLUME 10, 2022 are designed. At the step 2 and 4, the RBF NNs are applied to approximate unknown dynamics f 2 (x 2 ) and f 4 (x 4 ), respectively.
Remark 4: From (25), the unknown constant ψ 1 brings great trouble to the control design. If the conventional design method is used, an adaptive law needs to be designed to obtain the estimation valueψ 1 . Otherwise, ψ 1 needs to be assumed to be known. In this work, the single-parameter-learning method is introduced such that the difficulty in the control design is overcome dexterously. It should be pointed out that the design in this work utilizes the sign of ψ 1 adequately. Just because of this, this work does not require any any priori knowledge of FJR systems.

B. STABILITY ANALYSIS
The above design and analysis is given by the following theorem. Theorem 1: Considering the closed-loop control system consisting of the mathematical model of single-link FJR (1)-(2) subject to dynamic uncertainties and input saturation, if the assumptions 1-2 are satisfied, the adaptive control law (52) with the virtual control laws (18), (28) and (43), the adaptive laws (19), (29) and (44) and (53), and the event triggering mechanism (54) can ensure the following statements to be held.
• All signals in the closed-loop control system are bounded.
2) Zeno behavior avoidance. According to the event triggering mechanism (54), in the holding phase of u c , i.e., t ∈ [t j , t j+1 ), the actual control command u c is a constant, which implies that there is a variance between u c and u c , i.e., the control measurement error s = u c − u c . Taking the time derivative of |s| yields According to (13)- (15), (50), (52) and (53), one haṡ According to the boundedness of z 4 ,χ 4 , x 4 and u c , one can get thatż 4 is bounded from (49). From (53), it can be found that˙ 3 is bounded. In addition, from the boundedness of z 3 andχ 4 , one can get thatż 3 andχ 4 are bounded from (41), (15) and ϑ 4 = χ 4 − α 3 . Therefore,˙ u c is bounded bȳ u. For t ∈ [t  , t  +1 ), taking the fact of u c (t  ) = 0 and lim min s(t) =s being into account, one can get that the lower bound t of inter-execution intervalť = t  +1 − t  meets 43372 VOLUME 10, 2022 t ≥ t ≥s/ū c , which indicates that the Zeno behavior caused by the event triggering mechanism is avoided in this work.

IV. SIMULATION RESULTS
In this section, a single-link FJR system is employed to illuminate the effectiveness of the proposed control scheme.
Simulation results under the proposed control laws u etc and u c as shown in Figs. 1-5. Fig. 1 shows the tracking control performance under two control laws, which demonstrates that both two control laws can force the link position q to track the reference trajectory y d with satisfying performance. The result in Fig. 2 demonstrate that the tracking control accuracy of the continuous time control scheme is slightly better than the proposed ETABTC scheme, which indicates that the proposed ETABTC scheme has control energy loss due to the event triggering mechanism (54). Fig. 3 presents the control inputs under the continuous time control scheme and proposed ETABTC scheme, from which the actual control input u is bounded and reasonable. In addition, it can be observed from Fig. 3 that the actual control input u satisfies u ≤ u m ,    which indicates the effectiveness of the continuous time control scheme and proposed ETABTC scheme. Fig. 4 presents the curve ofˆ i (i = 1, 2, 3), which shows that the estimation VOLUME 10, 2022 of i is bounded. Fig. 5 draws the triggering instants and triggering times, from which it can be clearly found that the control command u etc is not transmitted indefinitely over a short period of time. Under the proposed ETABTC scheme, the actuator responses to the control commands generated by control law (52) 41 times. These results demonstrate that all signals in the closed-loop control system of FJR are bounded, as proven in Theorem 1. Therefore, the proposed ETABTC scheme is effective.
Furthermore, to clarify the superiority of the proposed ETABTC scheme, a comparison simulation is carried out between the proposed ETABTC scheme and the scheme proposed in [30]. All design parameters of the scheme proposed in [30] are taken the same as the work [30], and the initial conditions are identical to our control scheme. The simulation results are shown in Figs. 1-3, from which it can found that the scheme in [30] can also force the the link position q to track the reference trajectory y d as the same as our proposed ETABTC scheme. Although the control accuracy shown in Fig.3 under the control scheme in [30] is superior to that one of ETABTC scheme, the control scheme in [30] requires the norm parts of plant. In contrast, our proposed ETABTC scheme is more practical since no knowledge of plant is required in the control design.

V. CONCLUSION
This work develops an event-triggered adaptive neural tracking control solution for FJR systems under unknown dynamics and input saturation. A smooth function is introduced to handle the input saturation nonlinearity, and adaptive neural reconstruction techniques is used to compensate the unknown dynamics online. The DSC and single-parameter-learning techniques are used to handle the ''explosion of complexity'' and computational burden issues, respectively. In the control design, an event-triggered mechanism is established between the control law and actuator. Finally, an event-triggered adaptive neural tracking control scheme is proposed. The stability of the closed-loop system of FJR systems' tracking was proof by Lyapunov theory. Simulation results validate the effectiveness and superiority of the proposed control scheme.
In the future, the work will be extended to a more general system, such as the time-varying control gain, multiple-inputmultiple-output, non-affine system, etc. In addition, the further work is to reduce the response frequency of the actuator by one step.