Event-Triggered Adaptive Asymptotic Tracking Control of Flexible Robotic Manipulators: A Command Filtered Backstepping Method

This paper formulates an event-triggered adaptive asymptotic tracking control scheme for ﬂexible robotic manipulators via command ﬁltered backstepping method. Firstly, in the proposed design algorithm, the unknown nonlinear functions are ﬁrstly approximated by using intelligent estimation technique. Then, the “explosion of complexity” problem existing in the traditional backstepping procedure is solved by cleverly applying the command ﬁltered backstepping method. In addition, an event-triggered mechanism is adopted so that the control input is updated irregularly following the occurrence of an event. The advantages of the proposed adaptive design scheme are as follows: (i) the Barbalat’s Lemma is used to asymptotically drive the tracking error to zero; (ii) all the variables in the closed-loop system are bounded; (iii) the utilized event-triggered mechanism reduces the transmission frequency of computer and saves computer resources. Finally, the simulation results of the robotic system are given to illustrate the eﬀectiveness of our design scheme.


Introduction
Over the last few decades, the control problem of robotic manipulators has received outstanding attention, because they can be used to perform useful practical tasks requiring high precision and the performance of robustness in uncertain complex environments.So far, great progress has been made in the research of robotic manipulators and various control strategies have been proposed, see [1][2][3] and the references therein.However, it is worth pointing out that most of the above results can only guarantee bounded (non-asymptotic) stability of the robotic manipulator systems [4][5][6].This means that there is a bounded error between the output of the system and the reference signal when the time tends to infinity.Hence, how to design a control approach to solve the asymptotic output tracking problem for robotic manipulators is a significant research topic.Fortunately, [7] proposed a predictor-based controller for a high-DOF manipulator to converge joint position and velocity errors to zero.Based on the unified kinematics, a unified control method is proposed in [8] to handle the visual tracking control problem of robots with the eye-in-hand or fixed camera configuration and ensure the asymptotical convergence of image position and velocity tracking errors.As a matter of fact, the actual models of robot systems often contain completely unknown uncertainties, but this factor is not taken into account in the above-mentioned results and therefore needs to be solved urgently.
At present, a mainstream research direction for robotic manipulators is to utilize coordinate transformation to transform robotic manipulators into triangular structure models, because the backstepping method can be used for their controller design and stability analysis.As is known to all, the backstepping technique was proposed in [9], and it has become a common tool to construct adaptive controller for the lower-triangular nonlinear systems (and transformed robotic manipulator).For example, [10] proposed a control scheme for robotic manipulators to identify the originally designed virtual control signals in the framework of the backstepping control procedure.[11] presented a control method to train the unknown model dynamics of a manipulator by the traditional backstepping control and obtain the estimated model.However, in the process of backstepping iteration, repeated differentiations of virtual controls increased the computation burden, which is the so-called the problem of "explosion of complexity".As far as we know, two efficient design methods are proposed to solve the problem of "explosion of complexity" in the backstepping design, one is the dynamic surface control technology and the other is the command filtered technology [12][13][14].In addition, some important intelligent estimation techniques [15][16][17][18][19][20][21][22][23][24] have been proposed to deal with the completely unknown nonlinearities in nonlinear models.However, these effective technologies have not been fully applied to the robot systems, which inspires the exploration and research of this paper.
On the other hand, in the traditional time-triggered mechanism, the output of the controller is applied to the system at any time, which will cause the waste of network resources [25][26][27].In order to overcome the aforementioned shortcoming, an event-triggered mechanism is proposed, where the control signal is updated at the time instants according to the state trajectory of the closed-loop system.In other words, the state vector follows a condition in a time interval, which is called event condition.Only when the event condition is triggered, the output of the controller is applied to the systems.Until now, event-triggered control has attracted many researchers to explore its application potential due to its remarkable advantages.For example, a control technology called periodic event triggering is proposed in [29], where the continuous evaluation of triggering rule is avoided, and the stability of the event-triggered system is guaranteed.In [30], sliding mode technology combined with event-triggered control is applied to high-order system by the design based on reduced-order model.[31] integrated the event-triggered control into a comprehensive output-feedback mechanism such that the systems allow large uncertainties coupling to unmeasurable states.However, for all we know, few achievements are currently available for robotic manipulator systems via event-triggered control to save communication resources.
Therefore, in this paper we propose an event-triggered adaptive asymptotic tracking control scheme for flexible robotic manipulators by command filtered backstepping method.Different from the traditional time-triggered control strategy, only when the event condition is triggered, the output of the proposed controller can be applied to the system, which greatly reduces the waste of network resources.The contributions of this paper can be summarized as follow: (1) the traditional time-triggered strategy is extended to the eventtriggered strategy in this paper.Specifically, the relative threshold strategy is adopted which can adjust the threshold size according to the size of control signal to save the communication resources.(2) By applying some sophisticated mathematical reasoning and famous Barbalat's Lemma, the asymptotic tracking performance of the closed-loop system can be realized.(3) By using the command filtered backstepping technology to avoid repeated derivatives of virtual control, the "explosion of complexity" problem inherent in the traditional backstepping technology is handled.

Preliminaries and problem formulation
Consider the dynamic equation of a single-link robotic manipulator coupled with a brushed direct current motor based on a nonrigid joint (Fig. 1) as follows: where J 1 and J 2 are the inertias, q 1 is the angular positions of the link, q 2 is the motor shaft, R and L are the armature resistance and inductance respectively.i denotes the armature current, K denotes the spring constant, K t is the torque constant, u is the armature voltage, g is the acceleration of gravity, d is the position of the links center of gravity, F 1 and F 2 are the viscous friction constants, K b is the back-emf constant, M is the link mass, and N is the gear ratio.
Then, by introducing the state variables, x 1 = q 1 , x 2 = q1 , x 3 = q 2 , x 4 = q2 , x 5 = i and defining K t K = J 1 J 2 N L, the dynamic equation of the system (1) becomes the following non-strict-feedback form: where Lemma 1: Given ξ > 0 and c ∈ R, the following inequality holds: where κ is a constant and κ = 0.2785.
Lemma 2 [28]: Let zq = [z 1 , z 2 , ..., z q ] T and S(z q ) = [s 1 (z q ), ..., s l (z q )] T be the basis function vector.Then, for any positive integer m ≤ q, the following inequality holds: Neural networks: To the best of our knowledge, the radial basis function (RBF) neural networks (NNs) are considered as the most effective tool for estimating any continuous function f (Z) : R s → R over a compact set where T is the basis function vector.The basis function ϕ i (Z) is selected in the following form: where b i = [b i1 , ..., b iq ] and c are the center and the width of RBF.In this paper, for given accuracy ε > 0, if the neuron number is sufficiently large, we approximate an unknown continuous function f (Z) via the following form: where optimal weight vector θ * is chosen as 3 Neural adaptive controller design In this section, the command filtered technology will be employed to design an adaptive controller that can guarantee all signals in the closed-loop systems are bounded and asymptotically drive the tracking error to the zero.Firstly, we define the command filter as where e i and ω i are the tracking error of the command filter and output of the command filter, respectively.ω 1 = y r , y r is the desired trajectory, α i denotes the virtual controller, pi is a positive constant to be designed.Then, a 5-step adaptive command filtered backstepping control scheme will be presented.First, we introduce the compensating tracking error signal v i as where ξ i is the compensating signal of command filter.
Step 1: Choose an appropriate Lyapunov function V 1 as Then, we have Choose a compensating signal ξ1 as where c 1 > 0 and l 1 > 0 are known constants.
According to Young's inequality and Lemma 1, we have where κ is a constant and κ = 0.2785.Choose a virtual control law α 2 as By plugging ( 14) -( 16) into ( 13), we get it should be noted that w(t) is selected such that lim t→∞ ∫ t 0 w(τ )dτ ≤ w 1 ≤ +∞, where w 1 is a positive constant and w(t) is positive uniform continuous and bounded function.
Step 2: Choose an appropriate Lyapunov function V 2 as Then, we have Choose a compensating signal ξ2 as According to Lemma 1, we have From ( 7), we have According to Lemma 1 and Young's inequality, we have From Lemma 1, we get where a 2 is a known constant, |ε 2 (x 3 )| ≤ ε2 and ε2 is a positive constant.Choose a virtual control law α 3 as By plugging inequalities ( 20) -( 26) into inequality (19), we have The adaptive laws are selected as By plugging ( 28) into (27), we obtain Step 3: Choose an appropriate Lyapunov function V 3 as Then, we have Choose a compensating signal ξ3 as According to Young's inequality and Lemma 1, we have Choose a virtual control law α 4 as By plugging (33) -( 35) into (32), we get Step 4: Choose an appropriate Lyapunov function V 4 as Then, we have where a 4 is a known constant, |ε 4 (x 4 )| ≤ ε4 (x 4 ) and ε4 is a positive constant.Choose a virtual control law α 5 as By plugging inequalities (39) -( 45) into (38), it gets Adaptive laws are selected as By plugging (47) into (46), we obtain Step 5: Choose an appropriate Lyapunov function V 5 as Then, we have From ( 7), we have According to Lemma 1 and Young's inequality, we have ) 1 2a 5 θ 5 ϕ T 5 ϕ 5 + a 5 κw(t) where a 5 is a known constant, |ε 5 (x 5 )| ≤ ε5 (x 5 ) and ε5 is a positive constant.By plugging inequalities ( 52) -( 56) into (51), it can be inferred that Then, the event-triggered mechanism is designed as follows where 0 < δ < 1, m > 0, p(t) is the control law to be designed, d(t) is the measurement error.Remark 1: When t ∈ [t k , t k+1 ), the control signal satisfies the equation u(t) = p(t k ).Only when the event-triggered condition is triggered, we will get the control signal u(t) = p(t k+1 ).
Proof: According to Young's inequality, we get Combining ( 70) and (71), we obtain Integrating (72) over [0, t], we have where φ = ( ∑ i=2,4,5 Furthermore, we can get According to Barbalat lemma, we have To ensure the tracking error asymptotically converges to zero, the convergence of the ξ i should be considered.Hence, we choose the Lyapunov function V ξ as follows then, the time derivative of V ξ is In [14], the relationship between the output of the command filter and the virtual control is proposed, i.e., ||ω i+1 − α i+1 || ≤ ρ i , where ρ i is a known constant.According to (77), if the suitable l i is chosen, we can get the following inequality: where B ξ = 2 min {c i }, D ξ = √ 2 min {(l i − ρ i )} and l i > ρ i .According to the above discussion and (11) , we can get ξ 1 can converge to zero and the tracking error converges to zero asymptotically.
Remark 2: In order to eliminate the Zeno phenomenon, we need to prove that there is a t ≥ 0 such that ∀k ∈ z + , {t k+1 − t k } ≥ t.According to From ( 63), since all closed-loop signals have been proved to be bounded, we can find a constant σ > 0 such that | ṗ| ≤ σ.It is worth pointing out that d(t k ) = 0 and lim t→t k+1 d(t) = δ|u(t)| + m.According to (79), we get the lower bound of t must satisfy t ≥ δ|u(t)|+m σ .In this section, in order to show the effectiveness of the proposed controller, we consider the single-link robotic manipulator system (1) in which the parameters are selected as M = 0.5Kg, g = 9.8N /Kg, d = 0.03m, J 2 = 0.5Kgm 2 , F 2 = 1.5N ms/rad, J 1 = 0.01Kgm 2 , F 1 = 0.05N ms/rad, The simulation results are shown by Figs.2-9.Fig. 2 displays the tracking performance.Fig. 3 shows the tracking error.Figs.4-5 indicate the state variables x 2 , x 3 , x 4 , x 5 .The system adaptive laws are plotted in Figs.6-7.The actual control input u is plotted in Fig. 8.The interval of triggering events is showed in Fig. 9.It can be concluded that the proposed design scheme can achieve all the variables in closed-loop system are bounded and the tracking error converges to zero asymptotically.

Conclusions
This paper has proposed the event-triggered adaptive asymptotic tracking control design for flexible robotic manipulators via command filtered backstepping method.The adaptive controller and event-triggered mechanism have been jointly designed by a systematic approach such that the communication resources are saved.By using the intelligent estimation technique, the unknown nonlinear functions lied in the robotic system have been modeled.The "explosion of complexity" problem existing in the traditional backstepping design has been solved by adopting the command filtered backstepping method.The proposed controller not only ensures all the signals in the closed-loop system are bounded, but also asymptotically drives the tracking error to the zero.Finally, the effectiveness of the proposed design scheme is proved by the simulation results of the robotic manipulator.All the authors declare that there are no potential conflicts of interest and approval of the submission.

Figures
Figure 1 The        The trajectories of actual control input u.
The trajectories of interval of triggering events.

Fig 8 .Fig 9 .
Fig 8.The trajectories of actual control input u.
flexible single-link robotic manipulator.

Figure 2 The
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