Time-Varying Transformation-based Adaptive Tracking Control of Uncertain Robotic Systems with Event-triggered Mechanism

In this paper, an adaptive global event-triggered control scheme for uncertain robotic systems, capable of guaranteeing adjustable tracking performance, is developed. Compared with most of the existing results, the proposed control exhibits some features. Firstly, to improve the transient tracking performance without the requirement of large control effort, a time-varying scaling diagonal matrix and an error transformation are constructed so that each element of the tracking error has its own convergence rate, which can be adjusted by selecting the design parameters. Secondly, due to the consideration of decreasing the consumption of network resources, some additional diagonal matrices embedded in the new closed-loop system make the new control gain matrix not symmetric and positive definite, then the stability analysis becomes rather more challenging. To solve this issue, an alternative Lyapunov function is constructed to circumvent the above obstacles. The effectiveness of the proposed approach is verified by a 2-DOF robotic manipulator.


I. INTRODUCTION
I N order to improve the quality of products in the manufacturing industry and to safely implement the dangerous yet significant tasks in the practical systems, the rigid robot manipulators have got a broad application in many fields [1]- [5]. Therefore, the study of uncertain robots has received great attention in both research and industry communities during the past decade. There are a lot of efficient methods in the literature, such as gravity-compensation control [6], adaptive control [7], computed torque control [8], [9], sliding mode control [10] and other control methods [11]. However, for the above existing literature, the corresponding methods for uncertain robotic systems are based on fixed periodic time-driven sampling. Noting that this fixed periodic form of sampling may procedure redundant control input, then it will result in the waste of communication resources. Hence, how to effectively use the communication bandwidth has been one of the most interesting topics in the field of robotic control.
Since the event-triggered-based strategy has the ability to decrease the consumption of network resources, there are many results on event-triggered control of nonlinear/linear systems [12]- [17]. In [12], by using the hybrid system tools, an event-triggered stabilization control for a class of nonlinear systems was developed. However, the precondition of corresponding stability analysis is that the considered system must satisfy the input-to-state stability (ISS), which is difficult to identify this point. To avoid the ISS condition, in [15] an adaptive backstepping control with event-triggered rules were designed simultaneously for a class of strictfeedback nonlinear systems. Inspired by the idea of eventtriggering mechanism, some meaningful results for robotic systems have been proposed in the past decade. In [18], by utilizing the event-triggering strategy and some hybrid theory techniques, a tracking control method was proposed for mobile robots in the presence of malicious denial-ofservice attacks. In [19], a neuro-adaptive event-triggered feedback control framework for uncertain robot manipulators was developed so that the system output tracks the predefined trajectory closely. Furthermore, to solve the problem of model predictive control (MPC) optimization problem in online computation and communication resources, a novel MPC-based trajectory tracking controller for mobile robots was proposed in [20] by using the event-triggering mechanism. Nevertheless, how to improve the transient tracking performance of robotic systems is not considered in the above works.
Furthermore, some researchers focus on improving the transient tracking performance of uncertain robotic manipulators [7], [21], [22]. In [7], an improved prescribed performance control was developed for state-constrained robotic systems such that the tracking error converges to a pre-given region within a prescribed time. In [21], by relaxing the assumption on controllability of the system and the requirement for a priori knowledge on the reference trajectory, a novel prescribed performance fault-tolerant control strategy for Euler-Lagrange systems in the presence of output constraint and actuator faults was presented. In [22], by constructing an error-based transformation, an adaptive prescribed performance control approach for robotic manipulators was developed so that the tracking error is able to converge to a prespecified residual set with a pre-given decay rate. However, the implementation of the works in [7], [21], [22] must rely on the initial conditions and the corresponding results belong to semi-global results. If the reference signal or the initial state is changed, we must re-select the design parameters and re-implement the control algorithm. To relax the requirement on the initial condition, a filtered-error-based adaptive control scheme for uncertain robots was developed in [23], however, the issue of communication resources is not considered and only the transient performance for filter error (rather than the tracking error) is ensured.
Motivated by the above works, in this paper we propose an accelerated adaptive event-triggering control strategy for uncertain robotic manipulators. To improve the transient performance of the tracking error without involving large control effort, different from the normally employed filtered-errorbased methods, we propose a time-varying scaling diagonal matrix and construct a novel error transformation to facilitate the control design. Moreover, to reduce the communication resources for signal transmit, an event-triggering mechanism is utilized for control design. The contributions of this paper are summarized as: • To our best knowledge, it is the first time to design a global adaptive event-triggered control method for uncertain rigid robotic systems, which is able to guarantee the adjustable transient tracking performance without involving large initial control input; • By imposing a time-varying diagonal matrix, with the aid of error transformation, the developed control not only ensures the global result but also guarantees that the tracking error converges to an adjustable region with a pre-specified convergence rate, which is at the user's disposal; and • Due to the introduction of time-varying diagonal matrix and the event-triggering mechanism, the new virtual control gain matrix does not satisfy the positive definite property, which dramatically increases the difficulty of stability analysis. In this paper, by utilizing the information of available matrices, an alternative Lyapunov function candidate (rather than the quadratic one) is constructed, so that the stability of closed-loop systems is ensured. The remaining of this paper is given as follows. In Section II, some preliminaries and the event-triggering mechanism, as well as the control goals, are introduced, respectively. To improve the transient tracking performance and to reduce the communication resources, an adaptive control scheme for uncertain robotic systems is developed in Section III, which ensures that all signals in the closed-loop systems are globally bounded. In Section IV, a simulation study on the 2-DOF robotic manipulator is carried out, which verifies the effectiveness of the developed control scheme. Finally, we conclude this paper in Section V.
Notation: R represents the set of real numbers, R + denotes the nonnegative real numbers, R n is the set of n-dimensional real vectors, R n×n represents the set of n×n real matrices, and Z + is the positive integer set. || • || represents the Frobenius norm of matrix • and Euclidean norm of vector •, respectively.

A. DYNAMICS MODEL
The dynamic model of rigid robotic systems is described in the following form: where x 1 ∈ R m and x 2 ∈ R m denote the joint position and velocity vectors, respectively, H(x 1 ) ∈ R m×m is the inertia matrix which is symmetric and positive definite, C(x 1 , x 2 ) ∈ R m×m represents the centripetal-coriolis matrix, G g (x 1 ) ∈ R m is the gravitation vector, G τ (x 2 ) = G f + G d with G f being the static or dynamic frictional force and G d ∈ R m being the external disturbance, and u = [u 1 , u 2 , · · · , u m ] ∈ R m is the control input.
where M(·) = H −1 (x 1 ) and F(·) = M(−Cx 2 − G g − G τ ). Furthermore, the considered robot systems have the following well-known structural properties [8], [9], [24]: Property 1: The inertia matrix H and its inverse matrix M are positive definite and symmetric. In addition, there exist some positive constants λ max and λ min such that Property 2: For any x 1 and x 2 , there exist some positive constants γ 1 , γ 2 , and γ 3 such that

B. EVENT-TRIGGERING MECHANISM
To decrease the communication burden, motivated by the results in [15] and [25], we design an event-triggered mechanism in the controller-to-actuator channel, which is described as: • Control input: for k = 1, · · · , m. • Event-triggering mechanism: for ∀ j ∈ Z + , in which 0 < ϖ 1k < 1 and ϖ 2k > 0 are design parameters, t k, j+1 is the corresponding time of event triggering, v k (t) is the "virtual" control law which is a continuous signal, e k (t) is the measurement error between v k (t) and u k (t), i.e., e k (t) = v k (t) − u k (t), t k j is the update time in the k-th actuator channel, namely, if (7) is triggered, the corresponding time instant will be marked as t k, j+1 and the virtual signal v k t k, j+1 will be transmitted to the k-th actuator. In all, for ∀t ∈ t k j ,t k, j+1 , the control signal is always kept as a constant and no signal is transmitted from the controller to the actuator, i.e., u k (t) = v k t k j for ∀t ∈ t k j ,t k, j+1 . As the value of control signal must be updated to v k t k j at the triggering time instant t k j , it is seen from (7) that, for ∀t ≥ 0, one has with ρ 1k (t) and ρ 2k (t) being time-varying continuous parameters and satisfying |ρ 1k (t)| ≤ 1 and |ρ 2k (t)| ≤ 1. Moreover, there exist some positive constants ρ 1k and ρ 2k such that |ρ 1k (t)| ≤ ρ 1k and |ρ 2k (t)| ≤ ρ 2k . The similar result can also be ensured for the case u k (t) < 0. Therefore, it is reduced that which can be rewritten in another form: Then (12) becomes: where Substituting (13) into (2), one obtains which implies that η 1 is positive definite and η 2 is bounded, i.e., there exist positive constants η 1 , η 1 , and η 2 such that η 1 ≤ η 1 ≤ η 1 and η 2 ≤ η 2 .

C. OBJECTIVES AND ASSUMPTIONS
Define the position tracking error as where y d = [y d1 , y d2 , · · · , y dm ] T ∈ R m is the reference signal. The goal of this paper is to develop an event-triggered strategy for rigid robotic systems such that: 1) all signals of the closed-loop robot dynamics are globally bounded; 2) the tracking performance on tracking error can be improved by adjusting the design parameters; and 3) the Zeno behavior [26] can be avoided in the sense that there exists a finite time t * k > 0 so that the inter execution interval t k, j+1 − t k j is lower bounded by t * k , ∀ j ∈ Z + . To achieve the above control goals, the following assumptions and lemma are imposed.
Assumption 1: The desired trajectory y d and its time derivatives y (i) d , i = 1, 2, 3, are known, bounded, and piecewise continuous.
Assumption 2: The system states x 1 and x 2 are measurable.
Assumption 3: There exists an unknown positive constant Remark 2: We give a detailed explanation about Assumption 3. In this paper, G f ∈ R m denotes the static and/or dynamic frictional force and G d (t) ∈ R m is a bounded external disturbance. In the existing literatures for robotic systems, the frictional dynamic G f covers the following several types [27], [28]: with θ i (i = 1, · · · , 4) being unknown parameters. From the forms of friction in (19) and (20), it is obvious that where c i (i = 1, 2) are non-negative and bounded constants. As G d is bounded, then there exists a positive constant c 3 so that G d ≤ c 3 . Hence, G τ can be upper bounded by where γ 4 = max{c 1 +c 3 , c 2 } is an unknown parameter, which implies that Assumption 3 is reasonable.
Lemma 1: [29] Considering the dynamic system with the positive constants γ, σ , and the non-negative function h(t): then it is not difficult to prove that w(t) ≥ 0 for ∀t ≥ 0 and w(0) ≥ 0.

III. MAIN RESULT A. ERROR TRANSFORMATION
To improve the tracking performance of robotic systems, different from most existing works, by introducing a diagonal matrix β we construct the following error transformation: where β = diag{β i } ∈ R m×m is a time-varying scaling matrix and the element of β is defined as with 0 < b i f < 1 being a design parameter and the real function κ i (t) ∈ R satisfying the following properties: • The initial value of κ i (t) is 1, i.e., κ i (0) = 1; • κ i (t) > 0 is strictly monotonously decreasing with respect to (w.r.t.) time for ∀t ∈ [0, ∞) and converges to zero as t → ∞, i.e., lim i (t), k = 0, 1, 2, 3, is known, bounded, and piecewise continuous; and It is not difficult to show that many functions satisfy the above properties, i.e., κ i = exp(−t), 1 1+t 2 , 1 1+t 2 +t 3 . With such a function κ i , it is easy to prove that the time-varying function β i (t) has the following features: 1) The initial value of β i (t) is 1, i.e., β i (0) = 1; 2) β i is monotonically increasing w.r.t. time and converges to a positive constant 1 b i f as t → ∞, namely, is known, bounded, and piecewise continuous for ∀t ≥ 0. Remark 3: According to the properties of β i (t), it is readily verified that the diagonal matrix β (t) is positive definite and invertible, so is β −1 . Furthermore, its derivatives β (k) , k = 1, 2, 3, are available for control design.
Noting that E = x 1 − y d , then its time derivatives arė Then the derivatives of ζ as defined in (24) along (26) and (27) areζ andζ To facilitate the control design, different from the normally employed tracking-error-based filtered error, we introduce the following ζ -based filtered error: where λ 1 > 0 denotes a design parameter. Then the derivative of z along (28)-(29) iṡ Substituting the expression ofẋ 2 as shown in (17) into (31), one haṡ with being computable for control design.

B. CONTROL DESIGN
Due to the consideration of improving tracking performance and reducing communication burden, the diagonal matrices β and η 1 are imposed in the new system dynamics (32), it makes the normally employed quadratic function 1 2 z T z not applicable for stability analysis, then how to design an effective control algorithm and to select a proper Lyapunov function candidate to guarantee the stability is a challenging topic. This is because that, although the matrices β , M, and η 1 are all positive definite, the multiplication β Mη 1 may not be positive definite, then the traditional control methods in [30], [31], [32] for MIMO nonlinear systems are no longer available.
To circumvent the obstacles caused by the event-triggering rule and the error transformation and to decrease the difficulty of control design, instead of 1 2 z T z, we construct the following positive definite function as a part of Lyapunov function candidate, where Ω is defined as Its derivative isV Noting that κ k and its derivativeκ k are bounded, it is readily seen that ε k is bounded, then it implies that there exist some positive constants ε k and ϒ so that |ε k | ≤ ε k and ϒ ≤ ϒ. Substituting (32) into (35), one haṡ where F = z T η 1 (F + Mη 2 + β −1 ∆) + 1 2 z T ϒz is the lumped uncertainty.
According to the Properties 1-2 of robotic systems and Assumption 3, together with the definition of F, we have where a = λ max max{γ 2 , γ 3 , γ 4 } is an unknown parameter and is an available function for control design. Upon using the Young's inequality, together with Remark 1, one has Furthermore, note that Therefore, F can be upper bounded by Then (36) becomeṡ The global adaptive event-triggered controller is designed as: where c > 0 denotes a design parameter,b is the estimate of unknown parameter b and is updated by the following form: with γ and σ being positive design parameters. Furthermore, it must be emphasized that as γ z 2 Φ ≥ 0, with the aid of Lemma 1, it is guaranteed thatb(t) ≥ 0 for ∀t ∈ [0, ∞) and b(0) ≥ 0. The stability analysis of the closed-loop robotic systems under the proposed adaptive controller (46)-(47) is summarized in the following theorem.
Theorem 1: Consider the uncertain robotic system (2) with event-triggering mechanism. Suppose Assumptions 1-3 hold, if the event-triggered adaptive control scheme as designed in (46)-(47) is applied, then it is ensured that: 1) all signals in the closed-loop robot dynamics are globally bounded; 2) the tracking performance can be improved by adjusting the design parameters; and 3) the Zeno behavior is avoided.
Proof. According to the definition of adaptive controller (46), it is seen that where the fact that z T η 1 = z T η T 1 = (η 1 z) T is utilized. As M is positive definite, it leads to (η 1 z) T M(η 1 z) ≥ λ min z T η 1 η 1 z. It is seen from Remark 1 that the diagonal matrix η 1 η 1 is also positive definite, then there exists a positive constant λ η so that z T η 1 η 1 z ≥ λ η z 2 , which implies from (48) that with λ = λ min λ η . Thus, (45) becomeṡ withb = b − λb being the parameter estimate error.
To prove the stability of the considered systems, we select the following form of Lyapunov function candidate: with γ > 0. Then the time derivative of V along (50) yieldṡ

VOLUME 4, 2016
Note that b is an unknown constant, it follows thatḃ = 0, then (52) further becomeṡ Invoking the adaptive law as given in (47) into (53), one hasV By utilizing the Young's inequality, we havebb ≤ 1 with Θ = 3 4 + σ 2γλ b 2 being an unknown positive constant. As Ω is also positive definite and bounded, then there exist positive constants λ Ω and λ Ω such that Then (57) becomeṡ where Solving the equation (61) on [0,t], one has Then we guarantee the following results.
We first prove that all signals in the closed-loop systems are bounded. It is from (62) seen that V (t) is bounded, then from the definition of Lyapunov function candidate, one has z andb are bounded. Asb = b − λ minb and b and λ min are positive constants, then it is shown thatb is also bounded. In addition, note that z = λ 1 ζ +ζ , then it is not difficult to prove that ζ andζ are bounded. According to the boundedness of β andβ , together with Assumption 2, it is ensured that E andĖ are bounded, which further implies that x 1 and x 2 are bounded. As the matrices η −1 1 and ϒ are bounded, then the computational function Φ is also bounded, then from the definitions of control law and adaptive law as defined in (46)-(47), respectively, it follows that v andḃ are bounded for ∀t ≥ 0.
Next we analyze the tracking performance of each component of the tracking error. Denote z = [z 1 , · · · , z m ] T ∈ R m and ζ = [ζ 1 , · · · , ζ m ] T ∈ R m , then it follows from (30) thaṫ Solving the above equation yields Note that |z i | ≤ z ≤ z with z being a positive constant, then (64) becomes As E = [e 1 , · · · , e m ] T ∈ R m and ζ i = β i e i , then e i = β −1 i ζ i , i = 1, 2, · · · , m, which further expresses as and It is shown from (66) that the tracking error e i includes two components: the first one is b i f ζ i (t), it is seen from the above analysis that b i f ζ i (t) is always within the residual region b i f B i during the whole control process. The second one is (1 − b i f )κ i ζ i (t), the decay rate is not only governed by the term ζ i (t), but more importantly influenced by the real function κ i (t). Here we must emphasize that this function κ i is not only at user's disposal but also is independent of system initial conditions and any other design parameters. Hence, we have the freedom to choose the real function κ i properly to improve the decay rate and mode for so that the transient tracking performance of e i (t) can be significantly improved. Furthermore, it is also seen from (67) that the steady-state performance of tracking error can be improved by decreasing the design parameter b i f and the size of B i . Noting that the size of B i relies on the design parameter λ 1 and the size of z i , then it is shown that the size of z i may effect the steady-state tracking performance. Noting that (57) can be expressed asV ≤ −λ c z 2 + Θ, then it is easily proved that z will enter into and remain within the compact set , where µ is a small positive constant. As the size of z is proportional σ , and reversely proportional to c and γ, then increasing the design parameters c and γ and decreasing σ will lead to smaller z. In addition, it is seen from (65) that larger λ 1 may also lead to smaller ζ i .
Then increasing c, γ, λ 1 and decreasing σ may decrease the magnitude of z i . However, larger c might lead to larger control effort. Therefore, certain compromise between control performance and control effort should be made in practice as with many other adaptive control methods.
Finally we show that the Zeno behavior [26] is avoided.
From the definition of v(t) as shown in (46), we havev k = Asż k ,Φ, andḃ are all continuous functions, then it is indicated thatv k must be continuous. Noting thatv k is a function of all bounded closed-loop signals, then there exists a constant χ k > 0 so that |v k | ≤ χ k . According to the presented event-triggering mechanism, it follows thať e k t k j = 0 and lim t→t k, j+1ě k (t) = ϖ 1k |u k (t)| + ϖ 2k > ϖ 2k , then we obtain that there exists a lower bound of inter-execution interval t * k and it satisfies t * k ≥ ϖ 2k χ k . Therefore, the Zeno behavior is avoided. The proof is completed.
To recap the main design procedure, a control diagram that clearly illustrates the structure of the control loop is presented in Fig. III-B.
Remark 4: If the error transformation (24) is not imposed, i.e., β = I, the proposed adaptive control scheme (46)-(47) reduces to then (69) is a special case of (46)-(47) and is refereed to as the traditional event-triggered control method. Under this control scheme, although the boundedness of all signals in the closed-loop systems can be ensured, there is no any systematically guidance for improving the transient tracking performance. The normally method is to adjust the design parameters by the method of trial and error, however, this may lead to a large magnitude of control law in the initial period. In this paper, by constructing a time-varying scaling matrix and introducing a meaningful error transformation, the proposed control not only ensures the boundedness of all signals but also achieves adjustable transient performance without involving large control inputs, which is very important for the control implementation in the practical engineering systems. Remark 5: Compared with the references [33]- [36] with or without event-triggered mechanism, the main difference is that in this paper the tracking transient performance is considered, which implies that the corresponding methods are no longer applicable. In this paper, by imposing a time-varying scaling function matrix in the control design, together with an event-triggered mechanism and an alternative form of Lyapunov function, the proposed method not only circumvents the obstacle caused by the event-triggered mechanism, but also guarantees the adjustable tracking performance for uncertain robotic manipulators.
Remark 6: Compared with the existing literature about the robotic control, the main contributions and the shortcoming of this study are discussed in the following: • Due to the consideration of transient tracking performance, the original filtered-error-based control scheme is no longer applicable as the normally method on improving the transient performance is to adjust the design parameters by the method of trial and error, leading to a large magnitude of control input in the initial period and a adverse effort on the actuator. In this paper, in order to achieve the adjustable transient performance without involving large control law, by imposing a time-varying diagonal matrix, together with a error transformation, the developed global approach is able to ensure that the tracking error converges to an adjustable region with a adjustable decay rate without involving large control input; • Noting that the diagonal matrices β and η 1 are imposed in the new system dynamics (32), it makes the control gain matrix β Mη 1 not symmetric and positive definite at all, which further implies that the quadratic function 1 2 z T z is no longer applicable for stability analysis. Therefore, in this paper, different from the normally employed quadratic function, an alterative Lyapunov function candidate 1 2 z T Ωz is constructed so that not only the difficulty caused by the time-varying diagonal matrix and the event-triggering mechanism is circumvented, but also the stability of the closed-loop systems can be guaranteed. • Although the proposed control of robotic control is able to achieve the global result with adjustable tracking performance, it is seen from (65) the control precision of tracking error depends on some unknown terms, which cannot be pre-specified by the control scheme. Therefore, we will focus on studying the global prescribed performance event-triggered control of uncertain robotic systems such that the control precision can be preassigned.
The simulation results are shown in Fig. 2-Fig. 9. The evolutions of x 1k and y dk are plotted in Fig. 2-Fig. IV, respectively. The trajectories of tracking errors e 11 and e 12 are shown in Fig. IV, from which it is seen that under the proposed control, the tracking error exhibits good tracking performance. In addition, the evolutions of control inputs u 1 and u 2 are plotted in Fig. IV and Fig. IV, respectively. The trajectory of parameter estimateb is given in Fig. 7. It is seen that all signals are bounded for ∀t ≥ 0. Furthermore, the event triggered times for the event-triggering mechanism   in the controller-to-actuator channels are shown in Fig. 8 and Fig. 9, respectively, from which it is checked that the numbers of events triggered within 10s are 133 and 431.
Furthermore, to verify that the proposed control is able to adjust the transient tracking performance by choosing different real function κ i (t), three different functions for κ i (i = 1, 2) are considered, i.e., κ i (t) = 1 1+t , exp(−t), and       be adjusted, which confirms our theoretical analysis.

V. CONCLUSION
In this paper, we have developed a novel adaptive eventtriggered control strategy for uncertain robotic systems with adjustable transient tracking performance. With the aid of diagonal matrix β , the tracking error-based transformation is employed so that the tracking performance caused by the event triggering mechanism is compensated and successfully improved. Furthermore, the proposed control has also ensured the global boundedness of all closed-loop signals in robot dynamics. XIAOCHUN LIU graduated with the major in high voltage technology from Huazhong University of Science and Technology, Wuhan, China, in 1995. From 1995 to now, she is working in Hunan Railway Vocational and Technical College and focus on teaching and scientific research in the fields of electrical engineering and automatic control.
HAILONG LIU was born in 1986 in Jiangxi Province, China. He received the master's degree in control Engineering from East China Jiaotong University in 2012 and is currently an associate professor. Prior to 2013, He had been working as a control engineer at Feida Electric Equipment Co., LTD. From 2013 to now, he is working as an associate professor at Hunan Railway Vocational and Technical College. His research interests include intelligent control, industrial robotics, and artificial intelligence.
YONGCHENG ZHOU received M.S. degrees in control engineering from the Chongqing University, Chongqing in 2016, where he is currently pursuing the Ph.D. degree at the School of Automation, Chongqing University. He was a control engineer at CRRC in 2016. Since 2018, he has been a researcher and senior engineer with the Research Institute of Ruijie, Ruijie Networks Co., Ltd. His research interest includes machine learning, deep learning, cooperation control and applications, adaptive control. VOLUME 4, 2016