Adaptive Finite-time Dynamic Surface Neural Network Control of an Uncertain Robot with Output Constraint and Input Saturation

– In this study, a ﬁnite-time dynamic surface neural network control is developed for an uncertain n-link robot subject to input saturation and output constraints. First, a barrier Lyapunov function and a hyperbolic tangent function are applied to solve the system constraints using a dynamic surface control. Subsequently, a radial basis function neural network is utilized to handle system uncertainties. Then, a ﬁnite-time ﬁlter is employed in the design to achieve the fast convergence and a Nussbaum function is employed to optimize the design process. Finally, the simulation results show that the dynamic tracking error is proved to converging to zero, and the proposed control method is effective and never violates the constraints.


I. INTRODUCTION
In recent years, the application scenarios of manipulator systems are constantly expanding, so the research on them is increasingly deepening [1]- [3]. Due to different application scenarios, the robot system is generally subjected to constraints [4]- [8], such as dead zone [9], saturation [10], [11], hysteresis [12], [13], specified performance [14]- [16], and so forth. Saturation as a common nonlinear characteristics is broadly found in the actuator of physical systems because of the upper limit of motor torques [17]- [21]. Ignoring saturation effects can lead to system performance degradation and even instability. Output constraints widely exist in consideration of safety or performance specifications [22]. Violating output constraints will result in output performance degradation and even bring serious consequences. Therefore, the synthetic influence of output constraint and input saturation in the robot system should be considered during the process of control design.
To address the output constraint or input saturation, many researchers have proposed diverse solutions June 26, 2021 DRAFT in recent years [23]- [30]. To list some examples, in [24], [26], a saturation function was employed to describe the mathematical model of input saturation. In [27], the dead zone nonlinearity was applied to replace the saturation nonlinearity in multi-agent systems, and the hyperbolic tangent function was applied to the control law design of a flexible manipulator with input and rate constraints [30], which can effectively solve the problem that the sharp corner of saturation function was not differentiable. In [31], [32], the constrained system was transformed into the unconstrained case using the system transformation technique and the system remained to be stable. A barrier Lyapunov function (BLF) was adopted to deal with the control problems of state or output constrained systems in [2], [8], [33]- [36]. However, the above methods only resolved the issue of output constraint or input saturation and cannot be applied to the robot system simultaneously affected by output constraint and input saturation. In specific applications, constraints generally exist in multiple forms, which will pose an increased challenge to the control design and analysis.
Since the backstepping control is able to handle diverse nonlinearities during the design [37]- [42], it has become a common design aid for the control of various nonlinear systems [30], [43]- [46]. The main idea is to split the high-order system into several subsystems, and then use Lyapunov method to design the appropriate virtual control law for each subsystem to achieve the overall control object, but the biggest problem is that "term explosion" will occur in the process of frequent derivation, which makes the control law design difficult. In order to solve this issue, the dynamic surface control (DSC) was presented in [47] by designing a filter to obtain a first-order derivative approximation of the input signal based on the definition of derivative, which greatly reduced the complexity of the design process. In [48], a radial basis function neural network (RBFNN) based adaptive control approach was presented with a dynamic surface technique for stochastic nonlinear pure-feedback constrained systems. In [49], [50], an adaptive neural network based DSC was developed for nonlinear saturated systems. In [51], an adaptive DSC was presented for hypersonic vehicles with dead zone. In [52], based on fuzzy control and DSC, a dynamicscaling adaptive fuzzy tracking controller was constructed to cope with unknown nonlinearities. However, the above literatures were confined to the fixed-time DSC of nonlinear systems with input constraints, and these schemes are ineffective for the finite-time convergence [53]- [57] DSC of the robot system with output constraint and input saturation.
In this study, we tend to develop a finite-time adaptive neural network DSC for an n-link rigid robot system with output constraint and input saturation. Compared with the existing work, the main contributions are: (i) A hyperbolic tangent function and a Nussbaum function are introduced to tackle the input saturation in the robot system, and the Moore Penrose inverse term and BLF are adopted to guarantee no transgression of output constraint.
(ii) An adaptive neural network DSC with a finite-time filter is designed to approximate the unknown dynamic model, improve the system robustness, ensure a good trajectory tracking performance, and make the output of the filter track the input signal in a finite-time.
(iii) The Lyapunov method is employed to demonstrate the stability of the system, and all the trajectory tracking errors will converge to zero.

Lemma 3. [57] For the filterẋ
where x d is the output signal, x e is the input signal, α and β are positive constant, p, q are odd numbers and p > q > 0. If above conditions are true, then for any input signal x e ∈ [0, +∞], the output signal can follow the input signal in a finite-time with the convergence upper bound satisfying the following: Based on the Lagrangian function, the mathematical model of the n-linked robot under study is formulated as follows where q,q,q ∈ R n denote the position,velocity and acceleration vectors, respectively, u(v) ∈ R n is June 26, 2021 DRAFT the input of the system, and v ∈ R n is the intermediate variable. M (q), C(q,q) ∈ R n×n , and G(q) ∈ R n represnt the inertia matrix, Centripetal and Coriolis torques matrix, and gravitational force vector, respectively, with M (q) being a positive definite matrix. J(q) denotes the nonsingular Jacobian matrix, and f (t) ∈ R n denotes the vector of external disturbance.
Let x 1 = q and x 2 =q, then we obtain the following translatioṅ For the convenience of the following text, we abbreviate notations of M (q), C(q,q), G(q), J(q), and f (t) as M , C, G, J, and f , respectively.

Property 1:
The matrix M (q) is symmetric and positive definite.
In this study, the robot system subjected to the input saturation is considered and the saturation limit Hence, the hyperbolic tangent smoothing function is exploited for approximating the saturated nonlinearity designed as where v(t) is an intermediate variable, and we design auxiliary systems aṡ with c > 0, then the control design is translated into the design of ω.

A. Adaptive Neural Dynamic Surface Controller Design
Step 1: First, a position error is defined as Then, the first virtual control variable a 1 is introduced and a second error variable is defined as z 2 = x 2 − a 1 . We choose where K 1 is the gain matrix satisfying K 1 = K T 1 > 0. We choose the first Lyapunov function as The differentiation of V 1 yieldsV ).
Step 2: From (10), we obtain the followinġ The derivative of z 2 is expressed asż Now, a new error signal z 3 is given as z 2 is rewritten as Invoking the Moore Penrose inverse yields Then, we design the virtual control law a 2d as where K 2 is the gain matrix with K 2 = K T 2 > 0.
Since M , C, G, and f are uncertain, we use the RBFNN to approximate the unknown system parameters.
a 2d is then changed as whereŴ is the estimated value of the weight vector W * with the estimated error defined asW = W * −Ŵ , and S(Z) is the basis function vector.
Invoking (18) and (19), we haveŴ T S(Z) = Ca 1 + G + Mȧ 1 . Then, we can further derive W * S(Z) as with being input variables of neural networks and ε being the approximation error. And we design the updating law asẆ where Γ i is the constant gain matrix, and σ i > 0 is a small positive constant.
Consider the second Lyapunov function candidate as Differentiating (22) leads toV Step 3: Consider the Lyapunov function V 3 as In order to obtain the derivative of a 2d , the virtual control signal a 2d is designed by the finite-time first June 26, 2021 DRAFT order filter with small positive constants α 2 and β 2 aṡ In this part, a Nussbaum function is adopted to optimize the design process. The specific forms are given where γ χi > 0 is positive constant, andω is auxiliary control signal vector.

B. Stability Analysis
For the error of first-order filter, we choose the Lyapunov candidate function as Differentiating V 4 , then we havė with where η 2 is a nonnegative continuous function. According to Lemma 2, we havė In [57], the author pointed out that in DSC system, the first-order differential estimation error of filter to input signal is also very important for system stability. Then, we choose the Lyapunov function as The derivative of V 5 givesV with where ζ 2 is a nonnegative continuous function.
Choose the total Lyapunov candidate function as Differentiating V results inV First, for the Nussbuam function in the form of (26), if χ is a bounded function [58], ψ is bounded, which means that if its infinite integral exists, Based on the above analysis, we can obtainV where where λ min (•) and λ max (•) represent the minimum eigenvalue and the maximum eigenvalue of the matrix (•), respectively, and λ(•) is real. To ensure ρ > 0, α 2 must satisfy the following condition Multiplying (38) by e ρt yields d dt Integrating the above inequality gives Then we further have 1 2 Finally, we can obtain } June 26, 2021 DRAFT where D = 2(V (0) + C/ρ), and the closed-loop error signals z 1 , z 2 , andW i will remain within the compact sets Ω z1 , Ω z2 , and ΩW i , respectively. At this time, we conclude that all three signal errors will be maintained in closed sets, respectively, and all errors will converge to a neighborhood of zero under the proposed control with suitable parameter conditions.

A. Robot System
In this paper, the double joint rigid robot in [59] is used as the model, and the dynamical model description matrix of the robot system is defined as follows and G 21 = m 2 l c2 g cos (q 1 + q 2 ) . The Jacobian matrix is written as and J 11 = −l 1 sin q 1 + l 2 sin (q 1 + q 2 ) The specific parameters of the robotic system are listed in Table I, with providing the following initial states q 1 (0) = 0, q 2 (0) = 1,q 1 (0) = 1,q 2 (0) = 0.

B. Model-based control
For the model-based (MB) control, we examine the MB control designed in (18) where θ 1 = ∂a 2 /∂t, and the error comparison of the three case are shown in Figs. 8 and 9. It can seen that after applying the output constraint, the DSC and FDSC can achieve a better performance in comparison with backstepping control, and the output error under FDSC can converge faster than that under DSC.

D. Simulation Analysis
From simulation results, we can see that the presented control law (18), (19), and (27) can achieve a good trajectory tracking performance, the system constraints are never violated, compared with backstepping control (52), the complexity of the design is reduced, and the error of the system output converges faster by introducing a finite-time filter. For the control law (19), by introducing the neural network, the unknown system dynamics model can be approximated only by updating one parameter, which reduces the complexity of design and has a good performance.

V. CONCLUSION
In this paper, an adaptive finite-time dynamic surface neural network control was presented for an uncertain manipulator system with unknown dynamics and constraints. The hyperbolic tangent function and BLF were employed to eliminate the constraints, and the RBF neural networks were applied to approximate the complicated robot dynamics. By utilizing the finite-time filter in the DSC, the system achieved the fast convergence. Finally, we concluded that the derived control was able to track a desired trajectory in finite-time, and the constraints were never violated.
ACKNOWLEDGMENT Data availability statements: The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
Conflict of Interest: We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.
Funding: This study was funded by the National Natural Science Foundation of China (grant number 61803109).