RISE-Adaptive Neural Control for Robotic Manipulators With Unknown Disturbances

In this paper, a RISE-based adaptive neural network prescribed performance control is presented for the robotic manipulator with unknown disturbance. A prescribed performance function (PPF) characterizing settling time, overshoot, steady-state error, and convergence rate is presented to improve the transient performance. The unknown dynamics of the robotic manipulator are approximated by using the radial basis function neural network (RBFNN) which requires fewer adaptive parameters. The RBFNN approximation error and unknown disturbance are rejected by introducing the robust integral of the sign of the error (RISE) term. Then, an adaptive controller is designed for the robotic manipulator that can achieve precisely output tracking and guarantee the asymptotic stability of the control systems. The effectiveness of the proposed control approach is verified by simulation based on a robotic manipulator.


I. INTRODUCTION
Recently, the robotic manipulator control systems have been applied in industrial applications [1]- [7], for example, the robotic manipulators play an important role in industrial production picking due to their flexibility. However, it is difficult to achieve the desired control performance of the robotic manipulator systems because of the unknown dynamics and external disturbances existing in systems. To tackle these issues, it is well known that linear PID control [8] was used to control robotic manipulators. Nevertheless, the conventional PID algorithm can not obtain the satisfactory control accuracy [9]. To improve the control performance, several advanced control approaches were presented such as adaptive control [10]- [12], nonlinear control [13], model predictive control (MPC) [14], backstepping control [15] and sliding mode control [16].
Generally, the difficulty in control design for robotic manipulators stems from the unknown system dynamics and disturbance. To cope with these unknown dynamics, intelligent control techniques (neural network [17]- [21] and fuzzy logic control [22]- [24]) have been employed due to their learning ability. For instance, an adaptive neural control was studied for robotic manipulators with full-state feedback [25], The associate editor coordinating the review of this manuscript and approving it for publication was Valentina E. Balas . and an adaptive controller was proposed for robotic manipulator with input dead-zone and output constraint [26]. In [6], neural network based sliding mode control was developed for robotic manipulators with nonlinear friction, where the neural network was used to approximate the nonlinear friction. Moreover, the disturbance observers were utilized to estimate the disturbances [16], [27]. Although these methods can compensate the nonlinear in robotic manipulator system, the transient and steady-state performance are not considered in control design.
A prescribed performance control (PPC) method can be used to quantitatively analyse the transient and steady-state behavior of the control system [28]. The main feature of PPC is that the tracking error is transformed into a new error dynamics by using prescribed performance function (PPF), then, the new error dynamic were used to design the controller. This control approach has been applied in some fields [29]- [38]. An extended state observer based prescribed performance control was studied for servo systems with unknown disturbance [32]. In [33], an adaptive prescribed performance control combined with the high-order neural network was developed for the turntable servo system, where the PPF was used to improve control performance. In addition, PPC combined with NNs was utilized to control nonlinear systems. For example, an adaptive fuzzy control with prescribed performance was proposed to cope with the pure-feedback nonlinear systems [39]. By using NN and PPC, an adaptive neural sliding mode control with prescribed performance was presented to control nonlinear systems, where the proposed control scheme guaranteed the control performance [40]. Sliding mode observer based adaptive control was designed for space manipulators, where the PPF was used to improve the system performance [41]. An adaptive neural control was proposed to control underactuated surface vessels with unknown dynamics, PPC was used to preselect the transient and steady-state performance [41]. However, for the robotic manipulators, the system states (e.g., position and velocity) are measured by using sensors that may exist some measured errors such as unknown sensor noise. On the other hand, the approximation error of neural network or fuzzy logic control and the external disturbance can not achieve asymptotic convergence.
To achieve the asymptotic convergence, a new control approach called robust integral of the sign of the error (RISE) was developed in [42], [43]. RISE control technique can accommodate for bounded disturbance and yield asymptotic stability. Compared with the conventional SMC technique, this method can avoid the chattering phenomenon because a RISE feedback term is used in control design. This method has been applied in some fields. In [44], the RISE is used to compensate for the neural network estimation error, an adaptive RISE control is developed for MIMO nonlinear systems to achieve satisfied control performance. Adaptive RISE with NN was studied for hydraulic servo systems to achieve asymptotic tracking, where RISE was used to reject the bounded disturbance [45]. In [46], RISE based motion control is presented for servo systems, where RISE can reject the external disturbance. An adaptive control with RISE feedback term was developed to achieve asymptotic tracking, in which the RISE feedback is utilized to deal with the system uncertainties [47]. The above control methods can guarantee the steady-state error asymptotic convergence, and no literature considered the transient response in RISE methods. Thus, it is a challenging problem to obtain the transient and steady-state error asymptotic convergence of robotic manipulators.
Therefore, we will propose RISE-based adaptive neural control with PPC for robotic manipulators with uncertainties and external disturbances. The RBFNN was used to estimate the unknown dynamics (e.g., nonlinear friction and parameter uncertainties), and the estimation can be incorporated into control design to compensate them. The prescribed performance function is used to transform the tracking error into a new error dynamics, which is used to design the controller. The RBFNN approximation error and external disturbance can not achieve the asymptotic tracking. To overcome this issue, the RISE feedback term is employed to reject the approximation error and bounded disturbance that the robotic manipulator system can achieve asymptotic convergence. Besides the steady-state error can converge to zero, the developed method can also guarantee the transient behavior. The asymptotic tracking performance of robotic systems is proved by using the Lyapunov function. Finally, simulation and experimental results show the performance of the proposed adaptive controller.
The main contributions can be listed as follows: 1) A PPF is used to transform the tracking error of the robotic manipulator into a new error system, which is utilized to design the controller that can improve control performance. Moreover, the control error can be remained within a prescribed boundary. 2) Using the NN and PPF, an adaptive RISE control scheme is designed for robotic manipulator in the presence of unknown dynamics and external disturbances. The proposed method can not only guarantee the tracking errors within a predefined region but also the control errors achieve the asymptotic convergence. The rest of this paper is organized as follows. The system model, error transformation and fuzzy logic system are shown in Section II. Error transformation and RISE control design are proposed in Section III. simulation is given in Section IV. Section V shows the conclusions.

II. PROBLEM FORMULATION A. SYSTEM MODEL
A 2-degrees of freedom (DOF) robotic manipulator can be modeled as where q = [q 1 , q 2 ] T ,q andq 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, For the matrices M (q) and C(q,q), the following properties hold.
Assumption 1: The unknown external disturbance τ d and its first and second time derivatives are bounded, such that The aims are to design an adaptive robust control method that the output position q 1 can track the desired trajectory q d , and the unknown disturbance can be rejected.

B. NEURAL NETWORK APPROXIMATION
Neural network has been widely used to approximate the unknown nonlinear due to their learning ability. In this paper, a three-layer neural network is employed to estimate the unknown nonlinear. The structure of NN is shown in Fig.1.
The unknown nonlinear function f (x) is written as 97730 VOLUME 8, 2020 where W denotes the weight vector, and ε is the approximation error. (x) is the Gaussian function: where C j represents the center of the receptive field, and p j denotes the width of the Gaussian function.

C. PRESCRIBED PERFORMANCE FUNCTION
To investigate the transient behavior of servo mechanisms, a positive decreasing PPF is selected as where ϕ 0i , ϕ i (t), κ i , and κ i are the positive constants. According to [32], the tracking error e(t) = q − q d = [e 11 , e 12 ] T (q d denotes the desired trajectory) can converge to the following zone: where δ i andδ i denote the upper and lower bounds, respectively.
To cope with the constrained control problem (1), the output error e 1i (t) should be transformed into equivalent ''unconstrained'' by adopting a strictly increasing function (z 1 i) (z 1 i represents a transformed error), which must satisfy the following properties: According to the above properties, the condition (6) can be written as Thus, the inverse function of (z 1i ) can be described as To facilitate control design, the function (z 1i ) is defined as Then, the transformed error can be obtained from (9) as follows: where µ i (t) = e 1i (t)/ϕ i (t).
The time derivative of z 1i iṡ z 1i (t) = 1 2 The second derivative of z 1i (t) is where

III. CONTROL DESIGN
A RISE-based adaptive control scheme will be constructed by using RBFNN, PPF and RISE techniques. The design procedures are given as follows. The control structure is shown in Fig.2, where the RBFNN is employed to estimate the unknown nonlinear function, and RISE is utilized to compensate the external disturbance.
Substituting the estimation (19) into (18), one has Then, an adaptive controller can be designed as where µ s denotes the RISE term, which is given as (22) where k s and are the positive constants, andŴ is the estimation of W . The adaptive law iṡ Taking derivative of control law (21), one haṡ andμ s = (k s + 2)r + βsgn(z 1 ) Then, we can obtain that M (q)ż 2 = −Ṁ (q)z 2 + 1 2W (x) + (k s + 2)z 2 +βsgn(z 1 ) +Ṡ +˙ + z 1 (26) The error system can be expressed as The functionsÑ and N are given as Based on mean value theorem, one has the following inequality: where z(t) is defined as and γ ( η ) is a positive globally invertible nondecreasing function.

B. STABILITY ANALYSIS
Lemma 1: A auxiliary function L(t) is defined as where the constants β 0 and β 1 should satisfy the following conditions: Then, the following inequality can be obtained Theorem 1: Consider the robotic manipulator system (1), adaptive controller (21), and adaptive law (23), the closed-loop control system is semi-globally stable.
Proof: Let D ∈ R 5 be a domain containing, where y(t) can be defined as An auxiliary function P(t) is defined aṡ A Lyapunov function is chosen as with The function V 1 should satisfy the following inequality: where W 1 (y) = ρ 1 y 2 , W 2 (y) = ρ 2 y 2 with positive constants ρ 1 and ρ 2 . The time derivative of (37) iṡ Based on adaptive law, one has Substituting (48) into (40) yields: Using Young's inequalities, we have Then, (49) can be written aṡ where γ 3 = min{k 1 − 1/2, k 2 − 1/2 − β 1 , 1}, and c η 2 is a positive semi-definite function on a domain D.
In addition, according to definition of z 2 , we have Therefore, according to definition of (9), we can obtain that Then, we can conclude that the closed-loop control system is bounded for all t ≥ 0. This implies that the control system can asymptotically converge to zero with transient behavior.
In practical application, the parameter tuning guidelines should be given. The detailed tuning rules are summarized as follows.
• The PPC parameters are satisfied the initial conditions −δ i ϕ i (0) < e 1i (0) <δ i ϕ i (0). The parameter κ i determines the convergence of the tracking error and large κ i can obtain the good control performance but in the cost of large control actions. VOLUME 8, 2020 • The control gains k 1 and k 2 can lead to the fast convergence, nevertheless the resulting control action may be oscillated.
• The adaptive parameter can improve the parameter updating speed and the tracking performance. However, large σ may reduce the parameter adaptation speed.

IV. SIMULATION RESULTS
To validate the applicability of the proposed control algorithm. The diagram of robotic manipulator system with 2-DOF is shown in Fig.3. The system parameters are listed in Table 1.      To test the control performance of the developed control method, we have compared the developed control approach without the prescribed performance function. The control parameters are the same as our proposed control scheme. The only difference is no PPF. The desired trajectories are q 1d = q 2d = 0.3 sin(t).
The simulation results are shown in Figs.4-5, where the position tracking and tracking errors for Joint 1 and Joint 2 are given. From these figures, one can find that the control performance of the proposed adaptive neural network with PPF is better than adaptive neural network without PPF. This is mainly because the introduced PPF can improve the control performance, and the introduced RISE feedback term can reject the external disturbance.

V. CONCLUSION
The RISE -based adaptive control with prescribed performance was developed for robotic manipulators with external disturbance. A PPF was employed to improve the transient behavior and steady-state performance of the tracking error. RBFNN was utilized to approximate the unknown dynamics. The RBFNN approximation error and bounded disturbance were compensated by introducing the RISE feedback term. Using PPF and NN, an adaptive RISE control approach was designed for robotic manipulators. The proposed control method can guarantee the transient response, and achieve the asymptotic convergence. The control performance was verified based on a 2-DOF robotic manipulator via comparative simulations.