Adaptive Predefined Performance Neural Control for Robotic Manipulators with Unknown Dead Zone

-is paper proposes an adaptive predefined performance neural control scheme for robotic manipulators in the presence of nonlinear dead zone. A neural network (NN) is utilized to estimate the model uncertainties and unknown dynamics. An improved funnel function is designed to guarantee the transient behavior of the tracking error.-e proposed funnel function can release the assumption on the conventional funnel control. -en, an adaptive predefined performance neural controller is proposed for robotic manipulators, while the tracking errors fall within a prescribed funnel boundary.-e closed-loop system stability is proved via Lyapunov function. Finally, the numerical simulation results based on a 2-DOF robotic manipulator illustrate the control effect of the presented approach.


Introduction
Robotic manipulators have been widely utilized in industrial applications such as manufacturing industry, aerospace, and military equipment [1][2][3][4][5][6][7][8][9]. Nevertheless, the nonlinear terms include the nonlinear friction, model uncertainties, and dead zone that can reduce the control accuracy. To address this problem, the conventional PID controller was designed for robotic manipulators, but PID cannot achieve the satisfactory control performance [9]. To improve the tracking performance of robotic manipulators, a variety of control strategies were proposed for robotic systems such as adaptive control [10,11], nonlinear control [12], and backstepping control [13][14][15].
In fact, the difficulties in the control design for robotic systems mainly stem from nonlinear terms. To tackle these nonlinear terms, disturbance observer techniques were proposed to reject the unknown disturbance [1,16,17]. In [1], a new unknown dynamics estimator-(UDE-) based firstorder filter is proposed for robotic manipulators, the UDE was incorporated into control design that can effectively reject the unknown dynamics. An unknown input observer (UIO) was developed by introducing the first-order filter to estimate the unknown dynamics of servomechanisms, where the UIO had only one tuned parameter [18]. A novel nonlinear disturbance observer (NDO) was proposed for robotic manipulators in [17]. A disturbance observer (DOB) was devised for robot manipulators, where the external disturbance can be rejected by using the DOB [19]. Although the aforementioned disturbance observer-based control strategies can improve the performance of the robotic manipulator, the transient behavior is not considered in control design.
On the other hand, as neural networks (NNs) [1,[20][21][22][23][24][25] or fuzzy logic systems (FLS) [26][27][28] have been used to approximate the system uncertainties due to their approximation ability. In [25], an adaptive neural network control was proposed for robot manipulators, where the NN was utilized to approximate the unknown dead zone and system uncertainties. In [29], a contouring control method was proposed for robot manipulators and the NN was used to estimate the unknown dynamics. In [30], a NN-based terminal sliding mode control (TSMC) was designed for robot manipulators with actuator dynamics, where the NN was used to estimate the unknown actuator dynamics. A neuralfuzzy control was used to estimate the inverse dynamics; then, the approximation was incorporated into an adaptive neural-fuzzy controller to compensate the unknown dynamics of robot systems [31]. Although the aforementioned approaches can improve the control performance, the transient behavior and steady-state performance are not considered in controller design.
Recently, it is well known that the prescribed performance control (PPC) method can be used to quantitatively analyse the transient behavior [32]. e main feature is that a prescribed function with maximum overshoot and convergence rate is used to transform the original tracking error into a new error. en, the new error is used to design a controller in which the tracking error can be remained within a predefined boundary.
is control method has been used to control some systems [33][34][35][36][37][38]. In [39], an adaptive prescribed performance control was proposed for servomechanisms to improve the control performance. In [40], a modified prescribed performance function was proposed and incorporated into control design to control piezo-actuated positioning systems. A PPC was developed to control a variable stiffness actuated robot in [34]. Moreover, a funnel control (FC) as a constraint control was also proposed to guarantee the transient response [41]. e concept of FC is to construct an adjustable proportional gain τ to control the dynamics systems. e funnel control has been used to control some practice systems such as twomass systems [42], air-breathing hypersonic vehicles [43], and nonlinear dynamics systems [44]. In [45], a funnel control based on the adaptive fuzzy control was proposed to control stochastic nonlinear systems, where the fuzzy logic is utilized to approximate the unknown nonlinear dynamics. A neural network based on the adaptive control was developed for two-mass systems with backlash, where the neural network was employed to estimate the unknown backlash [46]. If a control system is with high relative degree (r ≥ 3), the funnel control may not suit this kind of systems. us, the application of funnel control is limited. Moreover, the application of funnel control in robotic manipulators cannot be found. is paper will propose a novel adaptive neural prescribed performance control method for robotic manipulators with unknown dead zone. A novel funnel variable is defined based on the tracking error. e modified funnel variable can release the assumption on the original funnel control. An echo state neural network (ESN) is adopted to estimate the unknown dynamics of robotic manipulators, and the approximation is used in control design to compensate the nonlinear dead zone. en, an adaptive control scheme for a robotic manipulator is proposed to improve the control performance. Numerical simulation demonstrates the effectiveness of the proposed control approach. e special contributions of this paper are as follows: (i) A novel funnel function is proposed based on the tracking error, and it can release the limitation on the original funnel function and is used in control design to improve the control performance (ii) A neural network is utilized to estimate the nonlinear dead zone, and the approximation is to design a controller, where the dead zone is compensated (iii) e effectiveness of the proposed control method is evaluated based on a robotic manipulator by using numerical simulations e remainder of this paper is organized as follows. Section 2 presents system description, funnel control design, and echo state neural network structure. An adaptive neural funnel controller is shown in Section 3. Numerical simulation results are given in Section 4. Finally, the paper is concluded in Section 5.

System Description.
is paper considers a n-degree-offreedom (DOF) robotic manipulator, which can be modeled as where q, _ q, and € q are the robot joint position, velocity, and acceleration, respectively; M(q) denotes the inertia matrix, C(q, _ q) represents the Coriolis/centripetal torque, including the viscous friction and nonlinear damping, G(q) is the gravity torque, τ is the control input, and d l is the unknown disturbance.
For the matrices M(q) and C(q, _ q), the following properties hold.
where M a and M b are positive constants.

Assumption 1.
e dead-zone nonlinearity (see Figure 1) can be written as where τ(t) is the control torque, d l (v) and d r (v) denote unknown smooth functions, and b l < 0 and b r > 0 denote constants. d r (τ(t)) and d l (τ(t)) can be written as Using (3) and (8), the dead zone is given as

Echo State Neural Network
Approximation. e echo state neural network is a novel NN with superior capability to approximate the unknown dynamics. e basic architecture of the ESN is shown in Figure 2. e ESN is composed of three parts: (1) K input neurons, N reservoir neurons, and L output layer. e ESN model can written as where X denotes the reservoir neuron state, C > 0 is a time constant, and a represents the leaking decay rate. W in ∈ R N×K , W ∈ R N×N , and W out ∈ R N×L denote the input weight matrix, the reservoir weight matrix, and the feedback weight matrix, respectively. e ESN can be used to approximate any continuous function f(x) over a compact domain Ω ∈ R m . e function f(x) can be expressed as where ε is the estimation error of the ESN, |ε| ≤ ε m , and W denotes the weight. erefore,

Funnel Control.
Funnel control [41] is a novel control strategy. By the error transformation, the original tracking error can be transformed into a new error dynamics. en, the new errors are used to design a control that can guarantee the control error remaining within a predefined boundary. e system S has the following properties: (i) Relative degree r � 1 or 2 (ii) Minimum phase (iii) Known high frequency gain e controller is given as where F φ (t) is the funnel function and ψ(t) denotes the scaling factor. e distance d v (t) is defined as where e(t) is the tracking error, which is defined as us, the funnel itself is defined as the set e gain τ(·) is According to [41], the boundary (see Figure 3) is where φ 0 , φ ∞ , and a are design parameters and satisfy φ 0 ≥ φ ∞ > 0 and |e(0)| < F φ (0) � φ 0 + φ ∞ . A novel funnel variable can be given as

Controller Design.
In this section, we consider the full state information, x 1 and x 2 , is measured, where (Figure 4) en, the system model can be expressed as Step 1.
e tracking error e 1 is defined as where x d is the desired trajectory. According to (15), the funnel error can be defined as  (20) is e Lyapunov function is defined as Its time derivative is e second error variable is defined as Substituting (24) into (23), one has

An intermediate control signal is chosen as
where k 1 is a design parameter.
Step 2. e time derivative of e 2 is According to (17), the second funnel error variable can be defined as e time derivative of z 2 is Figure 3: Funnel control.
Robotic manipulator system model M (q)q ·· + C (q, q · )q · + G (q) = τ + D (l) e derivative of (30) is where M(x 1 )z 2 denotes the unknown term, which can be approximated by using the NN. e actual controller can be designed as where W denotes the estimation of W, which is defined as where Γ and σ are design parameters.

Stability Analysis.
In this section, we will employ the Lyapunov function to analyse the convergence of the closedloop system.

Theorem 1.
Consider the robotic manipulators (1) with the proposed controller (32), intermediate controller (26), and adaptive law (33), then all the signals of the closed-loop system are bounded, and the tracking error can converge to the prescribed zone.

Proof. A Lyapunov function is chosen as
e time derivative of (34) is Substituting (23) and (31) into (35), one has Based on (27), (33), and adaptive law (34), one has Using Young's inequality, one has Substituting (38) into (37), we have where ρ � min 2k 1 ζ 1 , 2k 2 ζ 2 , σ and δ � (σ/2)W 2 are positive constants. From (39), we know that V is bounded by δ/ρ. erefore, all the signals of the closed-loop system are semiglobally uniformly and ultimately bounded. e parameter tuning guidelines are given as follows: (1) Select the funnel variables φ 0 , φ ∞ , and a, and they should satisfy the initial (2) Choose the control gains k 1 and k 2 , and the adaptive law parameters are Γ and σ. In general, they can be set large for the ease of fast convergence. However, practical control systems do not allow using large gains because they may produce oscillations. Hence, they can be chosen based on a trial-and-error method.

Numerical Simulation
In this section, we will employ an example to illustrate the control performance of the developed control method. A diagram of the robotic manipulator system with 2-DOF is shown in Figure 5. e robotic manipulator parameters are listed in Table 1. e system matrices M(q), C(q, _ q), and G(q) are defined as where α � 1/4m 1 + m 2 + m 3 + m 4 , β � 1/4m 3 + m 4 , and c � 1/2m 3 + m 4 , respectively. e controller parameters are given as k 1 � 5 and k 2 � 10. e adaptive parameters are Γ � 10I and σ � 0.02. e initial weight W � 0. e initial position of robotic manipulator is chosen as q 0 � [0, 0] and _ q 0 � [0, 0]. e friction term is F(q) � [15 _ q 1 + 6 sign( _ q 1 ); 15 _ q 2 + 6 sign( _ q 2 )]. e funnel function is F φi (t) � φ 0i · exp(− a i t) + φ ∞i , i � 1, 2, with φ 01 � φ 02 � 0.18, φ ∞1 � φ ∞2 � 0.01, and a 1 � a 2 � 2. e reference signals are given as q 1 d � 0.3 sin t and q 2 d � 0.3 sin t. Figures 6-8 depict the simulation results, where the output tracking performance, control actions, and ESN estimation are given. From Figure 6, we can see that the developed control approach can achieve the satisfactory control performance. In addition, we can see that the nonlinear friction can be estimated by using the echo state neural network. From these results, we find that the proposed control method improves the tracking performance of the robotic manipulator.

Conclusion
In this paper, an adaptive predefined performance control for robotic manipulators in the presence of nonlinear dead zone was proposed. A novel funnel variable was designed based on the tracking error.
e new error variable was utilized to design a controller that can guarantee the transient response. A neural network was adopted to estimate the unknown dynamics (parameter uncertainties and nonlinear dead zone), and the approximation was utilized in controller design to compensate the unknown dynamics. An adaptive controller based on funnel control was designed for the robotic manipulator. Both the transient response and  Mass of joint 2 1.5 kg m 3 Mass of link 2 2 kg m 3 Mass of actuator 1.5 kg  Data Availability e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.