Adaptive Intelligent Tracking Control of Flexible-Joint Manipulator With Full-State Constraints

Authors: st xinfeng shao
Venue: IEEE_org_ICIST_2024_Conference
Published: 2024-08-14 14:30:41

Abstract: This paper investigates the intelligent neural adaptive trajectory tracking control problem for a flexible-joint manipulator with full-state constraints. The approach uses neural networks to approximate the system's uncertain nonlinear terms and integrates backstepping recursion with a tangent-type barrier Lyapunov function to develop an intelligent adaptive control method with comprehensive state constraints. The stability of the closed-loop system is demonstrated using Lyapunov stability theory, ensuring that all state variables remain within predetermined boundaries. The proposed method's feasibility and effectiveness are ultimately validated through simulation of the flexible-joint manipulator system.

Main paper: 
 i. introduction: 
 T He manipulator is a vital component of industrial au- tomation, embodying the integration of modern and automation technologies [1] . It plays a significant role in both everyday life and various production processes. Since the mid-20th century, the rapid advancement of automation and computer technologies has established a robust foundation for the development of manipulators, leading to significant progress in related research [2] . However, traditional rigid robotic arms face several limitations, including low control precision, slow response speed, and a low load-to-weight ratio, which restrict their application in high-precision manufacturing fields such as medical technology and aerospace [3] . To overcome these limitations, flexible-joint manipulators have gained considerable attention in recent years [4] . These robotic arms offer substantial advantages in terms of adaptability, safety, and precision, thereby significantly enhancing production efficiency and workplace safety [5] . There have been many notable research outcomes related to flexible-joint manipulators, with the development of an effective control system being crucial for optimizing overall performance.The backstepping control method is highly effective for solving trajectory tracking control problems in flexible joint This work is supported in part by the National Natural Science Foundation of China (U22A2043), the Doctoral Startup Fund of Liaoning University of Technology under Grant XB2024014.robotic arms [6] . Many studies have utilized backsteppingbased approaches to address control issues in flexible robotic arms with notable results. Huang et al. [7] designed a controller for a single-link flexible robotic arm based on backstepping, but this control strategy is limited to systems with known state parameters. Liu et al. [8] combined singular perturbation control and adaptive control with backstepping to develop a control strategy for flexible dual-robot systems, but this approach is complex to compute. To address this computational complexity, Li et al. [9] combined backstepping with an instruction filter and used neural networks to approximate system uncertainties, reducing computational demands. Subsequently, Dong et al. [10] added an error compensation mechanism to eliminate potential filtering errors introduced by the filter. Zhang et al. [11] applied finite-time and stateconstrained backstepping methods to flexible joint robotic arms, achieving good results in handling trajectory tracking issues.In practical production settings, there are often stringent requirements on the operational workspace of flexible-joint robotic arms [12] . Exceeding these limits can damage the system, cause shutdowns, or jeopardize operator safety. Therefore, research into flexible-joint robotic arm systems under state constraints is of paramount importance [13] . Currently, backstepping-based control strategies are prominent and show promising control performance in the study of these systems.Motivated by the above considerations, this paper aims to investigate the adaptive NN tracking control problem for a class of flexible-joint manipulator systems with full-state constraints. Compared with the existing results, the main contributions of this paper can be summarized as:i) Modeling flexible-joint manipulator involves using neural networks to effectively handle system nonlinearities, combined with the backstepping method to provide an adaptive intelligent tracking control approach for such systems. ii) To ensure the stable and reliable operation of the flexiblejoint manipulator, a tangent-type barrier Lyapunov function is introduced to achieve safety constraint control over system states. An intelligent adaptive full-state constraints control method is proposed.The model of the flexible joint manipulator systems considered in this paper is as follows      M (q)q + C(q, q) q + G(q) + F q + K(q -q m ) = 0J qm + B qm + K(q m -q) = u y = q (1)where q, q, q ∈ R represents the manipulator displacement, velocity, and acceleration, respectively. M (q) ∈ R, G(q) ∈ R, C(q, q) ∈ R, and F ∈ R represents inertial matrix, Gravitational term, Coriolis centripetal matrix, and Joint friction coefficient matrix. q m , qm , qm ∈ R represents the rotor angle displacement, angular velocity, and angular acceleration, respectively. K ∈ R, J ∈ R, and B ∈ R represents the joint compliance, inertia term, and damping term of the joint, respectively. u and y is the system input and output, respectively.Let [x 1 , x 2 , x 3 , x 4 ] T = [q, q, q m , qm ] T , the system (1) is transformed into the following formEQUATIONwheref 2 = -M -1 (x 1 ) [C(x 1 , x 2 )x 2 + G(x 1 ) + F (x 2 ) + Kx 1 ], g 2 = M -1 (x 1 )K, f 4 = -J -1 [Bx 4 + K(x 3 -x 1 )],andg 4 = J -1 .Assumption 1 (see [14] , [15] ): There exists a positive constant g i0 such that 0 < g i0 ≤ |g i (x i )| < ∞. Without loss of generality, the assumption that 0 < g i0 ≤ g i (x i ) < ∞ holds.Assumption 2 (see [16] , [17] ): It is assumed that the reference signal y m (t) and its time derivatives ẏm , ÿm , y i) m (t) < Y i .In this section, the adaptive intelligent control design will be carried out in the backstepping recursion framework. First, define the following coordinate transformationEQUATIONBased on the above coordinate transformation, the fuzzy adaptive backstepping recursive design process is as followsStep 1: According to ( 2) and ( 3), the following equality can be obtainedEQUATIONBy using neural networks ĥ1 (Z1 | θ1 ) = θT 1 ϕ 1 (Z 1 ) to approx- imate the nonlinear function h 1 (Z 1 ) = -ẏm with Z 1 = [x 1 , y m ]T , the following can be obtainedEQUATIONwhere θ * 1 is optimal weight vector, ε 1 (Z 1 ) is NN approximation error and satisfiesEQUATIONwherek b1 = k c1 -Ȳ0 , θ1 = θ1 -θ 1 is weight parameter estimation error and θ 1 = g -1 10 θ * 1 2. From ( 4)-( 6), we haveEQUATIONBy using Young's inequality, we haveEQUATIONwhere a 1 > 0 is a design parameter.sec πz 2 1 2k 2 b1 z 1 ε 1 (Z 1 ) ≤ 1 2 sec 2 πz 2 1 2k 2 b1 z 2 1 g 10 + 1 2g10 ε * 2 1 (9)Substituting ( 8) and ( 9) into (7) , one hasEQUATIONThe virtual controller and the parameter adaptive update law are designed as followsEQUATIONwhere λ 1 , σ 1 > 0 are design parameter. Substituting (11) and ( 12) into (10) , one hasEQUATIONwhere λ * 1 = λ 1 -0.5 > 0 is a design parameter.Step 2: According to (2) and ( 3), the following equality can be obtainedEQUATION)where θ * 2 is optimal weight vector, ε 2 (Z 2 ) is NN approximation error and satisfies|ε 2 | ≤ ε * 2 with ε * 2 > 0.Construct the following tan-type barrier Lyapunov functionEQUATIONwherek b2 = k c2 -Ȳ0 , θ2 = θ2 -θ 2 is weight parameter estimation error and θ 2 = g -1 20 θ * 2 2. From ( 14)-( 16), we haveEQUATIONBy employing Young's inequality, the following inequalities are obtainedEQUATION)where a 2 > 0 is a design parameter.EQUATIONFrom inequalities ( 18) and ( 19) yieldsEQUATIONThe virtual controller and the parameter adaptive update law are designed as followsEQUATIONwhere λ 2 , σ 2 > 0 are design parameter. Substituting (21) and ( 22) into (20) , one hasV2 ≤ -sec πz 2 1 2k 2 b1 z 2 1 g 10 λ 1 * -σ 1 g 10 θ1 θ1 + a 2 1 2 + ε * 1 2 2g10 -sec πz 2 2 2k 2 b2 z 2 2 g 20 λ 2 * -σ 2 g 20 θ2 θ2 + a 2 2 2 + ε * 2 2 2g20 + sec πz 2 2 2k 2 b2 z 2 z 3 g 2 (23)where λ * 2 = λ 2 -0.5 > 0 is a design parameter.Step 3: According to (2) and (3), we haveEQUATIONm , θ1 , θ2 T. By using neural networks ĥ3 (Z 3 | θ3 ) = θT 3 ϕ 3 (Z 3 ) to approximate the nonlinear function h 3 (Z 3 ), one hasEQUATIONwhere θ * 3 is optimal weight vector, ε 3 (Z 3 ) is NN approximation error and satisfies |ε 3 | ≤ ε * 3 with ε * 3 > 0. Construct the following tan-type barrier Lyapunov functionEQUATIONwhere k b3 = k c3 -Ȳ0 , θ3 = θ3 -θ 3 is weight parameter estimation error and θ 3 = g -1 30 θ * EQUATIONBy employing Young's inequality, the following inequalities are obtainedEQUATION)where a 3 > 0 is a design parameter.EQUATION)From inequalities (28) and (29) yieldsEQUATION)The virtual controller and the parameter adaptive update law are designed as followsEQUATION)where λ 3 , σ 3 > 0 are design parameter. Substituting (31) and ( 32) into (30), one hasEQUATION)where λ * 3 = λ 3 -0.5 > 0 is a design parameter.Step 4: According to ( 2) and ( 3), we haveEQUATION. By using neural networks ĥ4 (Z 4 | θ4 ) = θT 4 ϕ 4 (Z 4 ) to approximate the nonlinear function h 4 (Z 4 ), one hasEQUATIONwhere θ * 4 is optimal weight vector, ε 4 (Z 4 ) is NN approximation error and satisfies|ε 4 | ≤ ε * 4 with ε * 4 > 0.Construct the following tan-type barrier Lyapunov functionEQUATIONwhere k b4 = k c4 -Ȳ0 , θ4 = θ4 -θ 4 is weight parameter estimation error and θ 4 = g -1 40 θ * 4 2 . From ( 24)-( 26), we haveEQUATIONBy employing Young's inequality, the following inequalities are obtainedEQUATIONwhere a 4 > 0 is a design parameter.EQUATIONFrom inequalities (38) and (39) yieldsEQUATIONThe final adaptive controller and parameter adaptive update law are designed as followsEQUATIONwhere λ 4 , σ 4 > 0 are design parameter.The above control design is summarized as the following theorem.Theorem 1: For the single-link flexible joint manipulator systems (1), Assumptions 1 and 2 hold. If the virtual controllers are given by ( 11), (21), and (31), the controller is designed by (41), and the adaptive law is designed by (42), the designed intelligent adaptive control scheme can ensure the following performance objectives: Therefore, all states of the system do not violate its pregiven constraint bounds., k 2 bj π tan πz 2 j 2k 2 bj < sec πz 2 j 2k 2 bj z 2 j holds, thus V4 ≤ -In this part, in order to further verify the rationality and effectiveness of the developed control design method, the following single-link flexible manipulator simulation example is provided.In the simulation, the desired reference trajectory at the end of the link is chosen as y m (t) = sin(0.5t) + 0.5 sin(t). The constraint boundary for the four states is chosen as k cj = 2 (j = 1, 2, 3, 4). The parameters of the single-link flexible manipulator model are given in TABLE 1 . The simulation results are shown in Figs. 1 2 3 4 5 6 7 . Fig. 1 shows the tracking curve of the system output, and Fig. 2 shows the tracking error curve. It can be seen that the tracking error is very small and converges to within a small neighborhood of zero. Figs. 3 4 5 6 show the curves of four states and constraint boundaries, and it can be seen that all states remain within the preset constraint boundaries. Fig. 7 shows the system control input curve, which verifies the effectiveness of the proposed adaptive intelligent full-state constraints control method.This paper addresses the problem of adaptive intelligent fullstate constraints control for single-link flexible manipulator systems. Utilizing neural network approximation techniques effectively handles the uncertain nonlinearities in the systems. An intelligent adaptive neural network controller has been proposed. By combining a novel barrier Lyapunov function method, a controller with a full-state constraint structure is designed, ensuring the stability of the closed-loop system. Finally, the proposed method is applied to a single-link flexible manipulator system, demonstrating its effectiveness. 
 
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Reviews: review: title: title: Accept,  but some suggestions remain
main_review: main_review: This paper introduces a tangent-type barrier Lyapunov function and  the state constraints method to achieve the intelligent neural adaptive trajectory tracking control for a flexible-joint manipulator. The derivation of this paper is adequate, but some suggestions remain:
1. Image analysis of the simulation could be more refined.
2. Grammatical descriptions could be more standardised.
title: title: Adaptive Intelligent Tracking Control of Flexible-Joint Manipulator With Full-State Constraints
main_review: main_review: This paper investigates the intelligent neural adaptive trajectory tracking control problem for a flexible-joint manipulator with full-state constraints. The topic of this paper is interesting, but the writing and the organization of this paper needs to be improved. Some comments are provided as follows:
1.	For the fluency of reading, language should be expressed consistently before and after. For example, tangent-type, flexible-joint manipulator, etc. A careful rereading is recommended.
2.	There are still many grammar errors and typos in this paper. For example, in the explanation section following (5), (6), the article is missing before a noun, a/the should be added, or the plural form should be used; in the introduction, "The" or "THE" is better than "THe".
3.	The abbreviated form should be given at the first occurrence in the article, e.g. NN.
title: title: Accept
main_review: main_review: This paper studied the adaptive intelligent tracking control for manipulator with full state constarins. The following comments should be considered in the final version.
A. The control objective of this paper should be added.
B. The definition of SGUUB should be added.
C. The authors should use the present perfect tense in the conclusion.
title: title: This paper investigates the intelligent neural adaptive trajectory tracking control problem for a flexible-joint manipulator with full-state constraints. The approach uses neural networks to approximate the system's uncertain nonlinear terms and integrates backstepping recursion with a tangent-type barrier Lyapunov function to develop an intelligent adaptive control method with comprehensive state constraints. The obtained result is valuable and can be accepted if the following problems can be clarified.
main_review: main_review: 1. This study may have different designs in comparison to some previous studies, but how are these differences significant? The author needs to highlight and clarify them in innovation point.   
2. Some remarks should be added to make the reader understand this paper well. 
3. The direction of the next work should be explained in the conclusion.
title: title: This paper is very interesting and a good work.
main_review: main_review: This paper investigates the intelligent neural adaptive trajectory tracking control problem for a flexible-joint manipulator with full-state constraints. The theory is correct and can be accepted after responding the following comments.
(1)	In the introduction, the shortages of those relevant studies are suggested to be further summarized.
(2)	In the end of Section 1, the organization of this study is suggested to be summarized.
(3)	In equation 2, check the notation “${f_2}$” and “${f_4}$”. Should they be replaced by ${f_2}({\bar x_2})$ and ${f_4}({\bar x_4})$?
(4)    The future work is missing in the Conclusion.