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Parameter-Tunable RBF Neural Network Control Facing Dual-Joint Manipulators
Parameter-Tunable RBF Neural Network Control Facing Dual-Joint Manipulators
Hindawi Journal of Robotics Volume 2022, Article ID 4231117, 10 pages https://doi.org/10.1155/2022/4231117 Research Article Parameter-Tunable RBF Neural Network Control Facing Dual-Joint Manipulators Weiying Xu HBUT Detroit Green Technology Institute, Hubei University of Technology, Wuhan 430000, China Correspondence should be addressed to Weiying Xu; firstname.lastname@example.org Received 8 July 2022; Revised 27 July 2022; Accepted 2 August 2022; Published 8 September 2022 Academic Editor: Shahid Hussain Copyright © 2022 Weiying Xu. �is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In order to improve the parameter control eƒect of the double-joint manipulator, this paper combines the RBF neural network to control the parameters of the double-joint manipulator and the command Šltering backstep impedance control method based on the RBF neural network is eƒectively applied to the multijoint manipulator. Moreover, this paper introduces the Šlter error compensation mechanism into the controller design to eliminate the inŒuence caused by the Šlter error. Finally, the eƒectiveness and superiority of the command Šltering backstep impedance control scheme of the multijoint manipulator adaptive neural network designed in this paper is veriŠed by simulation experiments. �e experimental research results verify that the parameter- tunable RBF neural network control method facing the dual-joint manipulator has a certain eƒect on the parameter control process of the dual-joint manipulator and can eƒectively improve the motion accuracy of the dual-joint manipulator. . �e hypothetical mode method usually uses Šnite-di- 1. Introduction mensional continuous modes of several orders to represent inŠnite-dimensional beams. Moreover, each order mode is �e theoretical modeling methods of robotic arms are generally based on the principle of energy conservation. For usually obtained by the product of the modal coordinates a rigid body, its energy usually includes two parts: kinetic and the modal function, and Šnally, the consecutive several energy due to motion inertia and potential energy due to order modes are added to obtain the vibration amount at any gravitational Šeld. However, for a Œexible arm, its potential point on the entire Œexible arm. Literature  has dem- energy and kinetic energy are diƒerent from those of a rigid onstrated that the use of the second-order mode can ac- manipulator. Moreover, the Œexible arm part generally in- curately reŒect the vibration oƒset of the Œexible body. �is cludes three deformations, namely, rotational deformation, research method has also been well veriŠed experimentally lateral deformation, and axial deformation . In order to and theoretically; however, this research method has certain consider the bending deformation of beams, there are deŠciencies and cannot be applied to Œexible manipulators currently four basic modeling theories of beams . Since with variable cross-sectional areas . In response to this the axial direction of the beam is generally longer than its situation, some scholars have proposed the Šnite element diameter, the Euler–Bernoulli beam theory is generally used method , and it has been shown that FE can reŒect the in many models. At the same time, since the Œexible ma- actual situation well. For the lumped mass method, it is the nipulator has inŠnite degrees of freedom, this is di”cult to simplest approximation to consider. It simply treats the achieve in modeling. �e general solution is to replace the Œexible manipulator as a spring and mass system, but this inŠnite-dimensional degrees of freedom with Šnite-di- research method is rarely used. mensional degrees of freedom and use Hamilton’s principle Traditional modeling of Œexible arms generally chooses or Lagrange dynamic equation to discretize a set of diƒer- classical deformations to derive the Lagrangian dynamic ence equations and Šnally solve them . �ere are generally equations of Œexible arms. However, in some cases, this classical deformation does not conform to the actual three current modeling methods for Œexible manipulators 2 Journal of Robotics basis for the study of the manipulator system, the dynamic situation and cannot accurately describe the dynamic be- havior of the ﬂexible arm. For example, when the ﬂexible modeling of the manipulator provides a dynamic model for the subsequent high-precision motion control of the ma- arm rotates at a high speed, the comprehensive deforma- tion theory can more accurately describe the deformation nipulator end eﬀector by making reasonable simpliﬁed form of the ﬂexible arm . So far, in describing the elastic assumptions for the actual manipulator system . (e deformation of ﬂexible arms, there are mainly three de- kinematics analysis of the manipulator system is to es- formation descriptions: classical deformation theory, tablish the relationship between the manipulator joint quadratic deformation theory, and comprehensive defor- variables and the end eﬀector pose matrix by analyzing the mation theory. Classical deformation theory is currently two parts, including forward kinematics and inverse ki- nematics . (e positive kinematics issue is to obtain the the most widely used method . Literature  uses the classical deformation theory to model the single-joint pose matrix of the end eﬀector by using the transformation matrix of the joint variables of the manipulator and in- ﬂexible manipulator and considers the inﬂuence of non- linear centrifugal force on the accuracy of the model. versely solving the value of each joint variable of the manipulator through various transformations. (e inverse Literature  uses the classical deformation theory to model a single-joint ﬂexible manipulator and designs a kinematics problem of the manipulator is the basis of the controller based on this model, and the control eﬀect is trajectory planning and trajectory tracking control of the ideal. Literature  uses the classical deformation theory joint end. However, in the process of inverse solution of the for modeling and then conducts positive feedback exper- manipulator, multiple sets of solutions are often obtained. iments and negative feedback experiments, respectively, (erefore, it is necessary to select a set of suitable solutions and ﬁnally compares and analyzes the results of theoretical according to the working conditions of the speciﬁc ma- nipulator. Trajectory planning is another important di- solutions and experiments. Analysis shows that positive feedback experiments are more robust than negative rection of manipulator research. It is based on kinematics and dynamics analysis, according to the requirements of feedback in suppressing vibration. In addition, the end of the ﬂexible manipulator contains a load to reduce the speciﬁc tasks, after a set of path points passed by the end eﬀector of the manipulator is known, through the me- natural vibration frequency of the ﬂexible manipulator. Literature  established the dynamic model of the chanical arm. (e arm inverse solution is used to obtain the Euler–Bernoulli beam using classical deformation theory corresponding joint angle of each joint. Among them, the and gave the numerical simulation results and concluded commonly used trajectory planning methods include cubic that the established dynamic model can be used as the polynomial interpolation function, quintic polynomial control equation for the design and simulation of the interpolation function, and spline interpolation function control system. Literature  established a Lagrangian . In practical engineering applications, in order to ensure the smooth operation of each joint of the manip- dynamic model of the ﬂexible arm based on the assumed modal method and Lagrange equation through classical ulator and to prevent the system from vibrating, the tra- jectory generated by the trajectory planning of the deformation theory and analyzed the physical and struc- tural parameters and external driving torque of the ﬂexible manipulator must be smooth and continuous and should manipulator. It is concluded that these parameters will have have no sudden changes. (e application of robotic arms in an impact on the dynamic characteristics of the ﬂexible various industries is more extensive. It is not just a simple manipulator. Literature  studied the theoretical mod- replacement for human work. In practical engineering, it eling problem of two ﬂexible manipulators in the plane and puts forward higher requirements for its work execution designed a controller based on the classical theoretical eﬃciency and the control accuracy of end eﬀectors. modeling problem. Literature  used the classical de- However, because the manipulator is a nonlinear, multi- formation method to study the kinematics of rigid-ﬂexible input, multioutput system, when the system model is established, there are uncertain factors such as friction coupled double rods in the plane and concluded that if the natural frequency of the ﬁrst-order system of the ﬂexible between joints, coupling, measurement error, and external disturbance. (erefore, reducing the inﬂuence of model beam is lower than that of the natural frequency of the rigid beam, the frequency diﬀerence will increase with the in- uncertainty in the robotic arm system on the control ac- crease in frequency. It increases with the increase in the curacy of the end eﬀector has always been a hot research natural frequency of the system; under the premise of topic in the ﬁeld of control at home and abroad . In certain initial conditions, the second-order natural fre- recent years, with the development of the intelligent control quency of the system will also increase with the increase in theory, some advanced control theories have also been the natural frequency of the whole system, and it shows that proposed. Although these advanced control theories can the motion and deformation process of the manipulator improve the tracking accuracy of the system to a certain will coordinate with each other when doing a large range of extent, there are still some problems to be solved. (is paper combines the RBF neural network to control motion. (ere are mutual coupling characteristics. When studying the control of the manipulator system, the parameters of the double-joint manipulator, improve the control eﬀect of the two-joint manipulator, and promote the it is ﬁrst necessary to study the dynamics modeling and kinematics analysis of the manipulator. As a theoretical work eﬃciency of the manipulator. Journal of Robotics 3 − 1 2. Command Filtering Backstep Impedance x _ � D x u − τ − C x , x x + G x . (2) 2 1 e 1 2 2 1 Control of the Adaptive Neural Network for (e command ﬁlter used in this chapter can be seen in the Dual-Joint Manipulator formula (2). In order to improve the control accuracy of the dual-joint manipulator system, the compensation signal will 2.1. Command Filtering Backstep Impedance Control of be constructed to compensate for the ﬁltering error gen- Adaptive Neural Network. In recent years, to enable robots erated by the command ﬁlter. (e compensation signal to handle increasingly complex tasks, many robots have design is as follows: installed dual-joint robotic arms with higher dexterity. When the robot provides services, the double-joint robotic ⎧ ⎪ ξ � −K ξ + ξ + x − α , ⎨ 1 1 1 2 1,c arm will physically interact with the unknown external (3) environment, so the compliance and safety of the double- ξ � 0. joint robotic arm should be fully ensured to ensure the safety of physical interaction. To adjust the physical interaction Among them, the control gain matrix is K � K > 0 and 1 1 force between the two-joint manipulator and the external x is the output signal vector of the ﬁlter. When t � 0, ξ and environment, the impedance controller of the two-joint ξ are zero vectors. manipulator has been widely used in the interactive control (e error variable is deﬁned, where x is the tracking of the robot. trajectory of the dual-joint manipulator through the im- At present, there have been many research achievements pedance relationship. (e design steps of the controller are in the backstepping impedance controller of the double- as follows: joint manipulator. However, the “computational com- Step 1: the algorithm constructs the Lyapunov function plexity” and “singularity” problems that may occur in the as V � (1/2)v v and derives V to get 1 1 1 design process of the traditional backstepping method and T T T _ _ the uncertain nonlinear terms in the dual-joint manipulator _ _ _ _ (4) V � v v � v z − ξ � v x − x − ξ . 1 1 1 1 1 1 1 2 r 1 system due to the diﬃculty of obtaining the model pa- rameters accurately deserve attention. In order to better (e virtual control law α is designed as solve these problems and make the dual-joint manipulator α � −Kz + x _ . (5) achieve better impedance control performance, this chapter 1 r introduces the adaptive neural network control technology Among them, the control gain matrix is K � K > 0. into the design of the command ﬁltering backstepping 1 1 impedance controller. Moreover, this paper constructs an By substituting formulas (3) and (5) into formula (4), adaptive neural network command ﬁltering backstep im- we get pedance controller for the dual-joint manipulator so that T T T V � v z + x + α − α − x _ − ξ � −v K v + v v . the uncertain dual-joint manipulator system can better 1 1 2 1,c r 1 1 1 1 1 2 realize the physical interaction control with the unknown (6) external environment. Compared with the existing dual- joint manipulator control scheme, this control scheme has Step 2: next, the algorithm selects the Lyapunov the following advantages: function as follows: (e dynamic equation of the double-joint manipulator system can be described as follows: (7) V � V + v D x v . 2 1 2 1 2 € _ _ D(x)x + C(x, x)x + G(x) � u − τ . (1) (e time derivative of the abovementioned formula is We deﬁne x � x and x � x _, and the model of the 1 2 obtained as follows: double-joint robotic arm system can be rewritten in the following form: T T _ _ V � V + v D x v + v D x v , 2 1 2 1 2 2 1 2 (8) u − τ − C x , x x + v e 1 2 2 1 T⎜ ⎟ T T ⎛ ⎜ ⎞ ⎟ ⎜ ⎟ _ ⎝ ⎠ � v − v K v + v D x v . 2 1 1 1 2 1 2 −G x − D x x + ξ 1 1 1,c 2 4 Journal of Robotics Substituting formulas (8) to (10) and (14) into the de- Since there are uncertain nonlinear terms in D(x ), rivative of formula (15), we get C(x , x ) and G(x ), adaptive neural network techniques T T 1 2 1 T T T T _ V � −v K v − v K v + v W S Z x _ + v W S Z x _ 1 1 1 2 2 2 2 D D D 1,c 2 C C C 1,c are used to approximate the matrix terms D(x ), C(x , x ), 1 1 2 T T T and G(x ). +v W S Z + v E − K sign v G G r r 2 1 2 G 2 (e real control law u for designing a dual-joint robotic n n T T T − σ W W − W S Z x _ v arm system is DK Dk Dk Dk DK D 1,c 2k k�1 k�1 T T n n n u � −K z + τ + W S Z x + W S Z x T T T T T 2 2 e D D 1,c C C 1,c D C − W S Z x _ v − σ W W − σ W W Ck CK C 1,c 2k CK Ck Ck GK Gk Gk (9) k�1 k�1 k�1 + W S Z − v − K sign v . G G 1 r 2 − W S Z v . Gk GK G 2k (e neural network weight adaptation law is designed as k�1 (18) (10) W � −Γ σ W + S Z x _ v , Dk Dk Dk Dk Dk D 1,c 2k (e following relation can be obtained as follows: T T (11) T W � −Γ σ W + S Z x _ v , Ck Ck Ck Ck Ck C 1,c 2k _ _ v W S Z x � W S Z x v , (19) 2 D D D 1,c Dk Dk D 1,c 2k k�1 (12) W � −Γ σ W + ϕ Z v . Gk Gk Gk Gk Gk G 2k T T _ _ v W S Z x � W S Z x v , (20) 2 C C C 1,c Ck Ck C 1,c 2k Among them, the matrix is Γ > 0,Γ > 0,Γ > 0. Dk Ck Gk k�1 σ , σ , σ are all positive numbers to improve robustness. Dk Ck GFk ∗ T T T (13) T W S Z � D + ε , D D D D v W S Z � W S Z v . (21) G G Gk G 2k 2 G Gk k�1 ∗ T W S Z � C + ε , (14) C C C C Based on Young’s inequality, the following formula can be obtained as ∗ T W S Z � G + ε . (15) G G G G 1 1 T T T T ∗T ∗ (22) −W W ≤ − W W + W W , Dk Dk Dk Dk Dk Dk Among them, ε , ε , and ε are minimal approximation 2 2 D C G error matrices. 1 1 T T T T ∗T ∗ T T T T T T (23) −W W ≤ − W W + W W , _ _ _ V � −v K v − v K v + v W S Z x + v W S Z x Ck Ck Ck Ck Ck Ck 2 1 1 1 2 2 2 2 D D D 1,c 2 C C C 1,c 2 2 T T + v W S Z + v E − K sign v . G G r r 2 2 G 2 1 1 T T T T ∗T ∗ (24) −W W ≤ − W W + W W . (16) Gk Gk Gk Gk Gk Gk 2 2 Substituting formulas (19) to (24) into formulas (18) and Note 1: within the designed control law, the control K ≥ ‖E ‖, we get rii ri parameter K should be selected as K ≥ ‖E ‖. In order r rii ri n n 1 T 1 T T T T T to ensure the stability of the method proposed in this _ V≤ − v K v − v K v − σ W W − σ W W 1 1 2 2 DK CK 1 2 Dk Dk Ck Ck 2 2 chapter when the double-joint manipulator is working, k�1 k�1 a larger value of K should be selected, but this is also r n n n 1 T 1 ∗ 1 ∗ T ∗T ∗T − σ W W + σ W W + σ W W not the best way for the system to induce chattering. GK Gk Gk DK Dk Dk CK Dk Ck 2 2 2 k�1 k�1 k�1 (erefore, the control parameter K should be changed to K � k x _ + k x + k . Among them, k ≥ ‖ε ‖, r D 1,c c 1,c G D D 1 ∗T ∗ + σ W W GK Gk Gk k ≥ ‖ε ‖, and k ≥ ‖ε ‖. C C G G k�1 ≤ a V + c . 0 0 2.2. System Stability Analysis. In this section, the Lyapunov (25) theorem will be used to determine the stability of the closed- loop system of the double-joint manipulator under the From inequality (25), it can be deduced that all control proposed control scheme. (e Lyapunov function is selected signals of the closed-loop system of the double-joint ma- as follows: nipulator are semiglobally uniformly asymptotically boun- n n ded. (e design ﬂow chart of the control method in this 1 1 T T T T −1 −1 V � V + W Γ W + W Γ W chapter can be seen in Figure 1: 2 Dk DK Dk Ck CK Ck 2 2 k�1 k�1 Note 2: the size of the control parameter determines the (17) radius of the tracking error domain. (at is, the larger T T −1 + W Γ W . Gk GK Gk the control parameter λ (K ), the smaller the radius max i k�1 of the tracking error domain. Journal of Robotics 5 x x x v 1 r 2 1 + + x z x z r 1 1,c 2 ∑ CF ∑ u - - -+ + + ∑ ∑ v 2 - - 1 2 ∑ 0 sI+K · W Dk Dk Dk sI+σ Ґ Dk Dk Ck 1 Ck Ck sI+σ Ґ Ck Ck MUX 1 Gk ^ sI+σ Ґ Gk Gk Gk Gk Dk 1,c x S Ck 1,c RBF NNs x S Gk Figure 1: . Flowchart of the controller design. ��������������� � � Furthermore, through the integral inequality (25), we n� � � � Ω � v (t) ∈ R v (t) ≤ 2(V(0) + c /a , � � v 1 1 0 0 can get c c ��������������� − a t 0 − a t 0 0 0 V(t)≤ V(0)e + 1 − e ≤ V(0) + . (26) ⎧ ⎨ � � ⎫ ⎬ 2(V(0) + c /a � � a a n 0 0 � � 0 0 Ω � v (t) ∈ R v (t) ≤ , � � v 2 2 2 ⎩ ⎭ λ D x min 1 From the abovementioned inequality, we derive ��������������� � � ⎧ ⎪ ⎫ ⎪ 1 c � �2 ⎪ ⎪ 0 ⎨ � � ⎬ � � 2(V(0) + c /a � � �v (t)� ≤ V(0) + . h 0 0 (27) 1 � � Ω � W (t) ∈ R �v (t)�≤ , 2 a DK 2 W ⎪ −1 ⎪ 0 Dk ⎪ ⎪ λ Γ ⎩ ⎭ min Dk From the abovementioned inequality, it can be known ��������������� that, under the condition that all state quantities of the ⎧ ⎪ ⎫ ⎪ ⎪ ⎪ ⎨ � � ⎬ 2(V(0) + c /a � � dual-joint manipulator system can be measured, for the h 0 0 � � Ω � W (t) ∈ R �v (t)�≤ , CK 2 W ⎪ −1 ⎪ Ck ⎪ ⎪ initial compact set, λ Γ ⎩ ⎭ min Ck (v (0), v (0), W (0), W (0), W (0)) ∈ Ω , v (t) 1 2 DK CK GK 0 1 converges into the compact set Ω . By the same ��������������� ⎧ ⎪ ⎫ ⎪ ⎪ ⎪ method, v (t) can be converged into the compact set 2 ⎨ � � ⎬ 2(V(0) + c /a � � h 0 0 � � Ω , and W (0), W (0), W (0) can be converged Ω � W (t) ∈ R �v (t)�≤ GK 2 v DK CK GK W ⎪ −1 ⎪ 2 Gk ⎪ ⎪ λ Γ ⎩ ⎭ min Gk into Ω , Ω , and Ω , respectively. W W W Dk Ck Gk (28) 6 Journal of Robotics (e model of the planar two-joint manipulator system used in this chapter is as follows: − 1 ⎧ x _ � D x u − τ − C x , x x − G x , 2 1 e 1 2 2 1 (29) y � x . m T T Here, x � [x , x ] and x � [x _ , x _ ] . Among 1 11 12 2 11 12 them, the connecting rod used in this simulation is a uni- form and regular cuboid rod; then, there is l � (l /2). ci i m According to the physical model of the two-joint ma- Obstacle nipulator, its kinetic energy equation can be calculated as 1 1 1 1 2 2 2 2 2 K(q, q _) � m l q _ + I q _ + m l q _ + m l l q _ q _ + q _ cos q _ 1 c1 1 1 1 2 1 1 2 1 c2 1 1 2 2 2 2 2 1 1 Figure 2: Plane two-joint robotic arm model. + m l q _ + q _ + I q _ + q _ . 2 c2 1 2 2 1 2 2 2 (30) ������� From inequality (27), lim ‖v (t)‖≤ 2(c /a ) is easy 1 0 0 t⟶∞ (e potential energy of the two-joint manipulator can be to obtain. (at is, for any given constant u, ������� expressed as 2(c /a )≤ μ can be obtained by adjusting the pa- 0 0 rameters a and c , and thus, lim ‖v (t)‖≤ μ can be 0 0 1 P(q) � m gl sin q + m gl sin q + l sin q + q . 1 C2 1 2 1 1 c2 1 2 t⟶∞ obtained. (31) Note 3: increasing α or decreasing c can eﬀectively ������� reduce 2(c /a ). (erefore, selecting a larger pa- According to the Euler–Lagrange equation 0 0 d/dtz(K − P)/zq _ − z(K − P)/zq � 0, the inertia matrix rameter a can make the system obtain a suﬃciently small tracking error of 0. However, because the pa- D(q) of the two-joint manipulator in the joint coordinate system, the centrifugal force and Coriolis force matrix rameters α and c both contain the parameter σ , σ , σ (k � 1, 2, . . . , n), the best control perfor- C(q, q), the gravity term matrix G(q), and the Jacobian Dk Ck Gk matrix of the two-joint manipulator in the joint coordinate mance of the system requires the designer to accu- mulate experience in simulation research. system are expressed as 3. Analysis of Simulation Results As shown in Figure 2, it is used for simulation experiments. 2 2 2 m l + m l + 2l l cos q + I + I , m l + l l cos q + I , 1 c1 2 1 1 c2 2 1 2 2 c2 1 c2 2 2 ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ D(q) � ⎣ , (32) 2 2 m l + l l cos q + I , m l + I . 2 c2 1 c2 2 2 2 c2 2 −m l l q _ sin q , −m l l q _ + q _ sin q , 2 1 c2 2 2 2 1 c2 1 2 2 22 C(q, q _) � , (33) m l l q _ sin q , 0. 2 1 c2 1 2 m l + m l g cos q + m l g cos q + q 1 c2 2 1 1 2 c2 1 2 G(q) � , (34) m l g cos q + q . 2 c2 1 2 −