A terminal sliding mode control method based on backstepping of a fuzzy reduced-order observer

Abstract

This invention discloses an end-of-line sliding mode control method based on a fuzzy reduced-order observer and backstepping. Addressing the uncertainty in the dynamic model of the robotic arm, a new system model combining the frictional force model (Dahl) and the robotic arm's dynamic model is established to progressively improve trajectory tracking accuracy and speed. A fuzzy reduced-order observer is designed to estimate the magnitude of disturbances, and torque compensation is applied to the observed disturbance values. The complex nonlinear system is then decomposed into subsystems not exceeding the system order using a backstepping method. For structural uncertainties, an improved nonsingular fast end-of-line sliding mode controller is employed, and an adaptive law is designed. The stability of the system is proven using Lyapunov. Finally, the control method is applied to the robotic arm control system, ensuring robustness under external nonlinear uncertain disturbances.

Description

Technical Field

[0001] This invention belongs to the field of high-order robotic arm trajectory tracking technology, and more specifically relates to an end-sliding mode control method based on backstepping of a fuzzy reduced-order observer. Background Technology

[0002] With the development of control theory and mechanical technology, robotic arms have been increasingly widely used in industrial fields, such as aerospace, medical, and automation. However, a robotic arm system is a complex nonlinear model, and the inability to accurately measure model parameters leads to a mismatch between modeling and simulation. Furthermore, the model is susceptible to unknown external disturbances during actual control. In current industrial applications, high-precision trajectory tracking is a major focus of discussion.

[0003] The inversion method has significant advantages in implementing robust or adaptive control for uncertain nonlinear systems, especially when disturbances are uncertain or matching conditions are not met. However, the inversion method itself lacks a good solution to the term expansion caused by the virtual control derivative and the problems arising from it. This shortcoming is particularly prominent in high-order systems. To address the term expansion problem, a dynamic sliding surface control method is adopted, using a first-order integral filter to calculate the virtual control derivative and eliminate the expansion of the differential terms. For flexible robotic arm systems with high precision requirements, non-matching disturbances, response time, and tracking error are all significant factors. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of the prior art and provide a terminal sliding mode control method based on the backstepping method of a fuzzy reduced-order observer.

[0005] This invention provides a terminal sliding mode control method based on the backstepping method of a fuzzy reduced-order observer, and applies it to a practical robotic arm. The specific design scheme is as follows:

[0006] Step 1: Construct a new dynamic model by combining the Dhal friction model of the robotic arm dynamics;

[0007] Step 2: Based on the dynamic model in Step 1, design a fuzzy reduced-order observer to predict disturbances;

[0008] Step 3: Based on the fuzzy reduced-order observer in Step 2, set an adaptive law to ensure the stability of the system;

[0009] Step 4: Set up a new type of non-singular terminal sliding surface and combine it with the disturbance estimated by the observer in the previous step. Use the inversion control method to construct the torque, and use Lyapunov to prove the stability of the system.

[0010] Furthermore, the specific steps for establishing the dynamic model of the n-DOF rotary joint rigid manipulator, which includes the Dahl friction model, in step one are as follows:

[0011]

[0012] Where: f represents frictional force, F c Where q represents the Coulomb friction force, q represents the displacement, and D0 is the stiffness coefficient. Let α be the relative velocity between the load and the contact surface, and α be a coefficient related to curvature. This model corresponds to a function of displacement and is independent of velocity, greatly reducing potential errors. This allows the friction model to be incorporated into the robotic arm's dynamics model.

[0013]

[0014] in: Let M(q) be the vector of angular displacement, angular velocity, and angular acceleration of the robotic arm joint; M(q) is the positive definite inertia matrix. The correlation matrix between centrifugal force and Coriolis force; For the friction force matrix of a traditional robotic arm, Here, G(q) is the Dhal friction force matrix; G(q) is the gravity vector; and τ is the joint control torque input vector. d Let M(q) be the uncertainty term for the robotic arm and external disturbances, where M(q) is the uncertainty term due to the external disturbances. Due to precision issues, G(q) will be replaced by M(q). G(q) decomposes M0(q), G0(q), and the uncertain part △M(q), △G(q);

[0015] Furthermore, the specific steps of step two are as follows, transforming the system into the following form:

[0016]

[0017] τ d This includes errors and uncertainties from external interference systems. This makes w1 = q, If w3 = f, then M0(q), G0(q) represents M0(w1), G0(w1); then the second-order system can be obtained as follows:

[0018]

[0019]

[0020] For w iThe estimated values, where κ1 and κ2 are the observer parameters which are positive constants, are then obtained by setting a fuzzy order reduction observer:

[0021] (s+p0) 2 =s n +κ1s n-1 +κ2

[0022] Where p0 is the observer bandwidth. Compared to traditional linearly extended observers, the gain changes faster and faster with increasing error. The pole placement technique is used in the design:

[0023]

[0024] At the same time We can conclude that:

[0025]

[0026] Observer error of fuzzy systems Design a fuzzy system using its derivative as input and bandwidth as output:

[0027] IF is A1 l1 and is A1 l2 ,

[0028] THEN y(w)is B l1l2

[0029]

[0030] Where l1 = 1, 2, 3, 4....N1; l2 = 1, 2, 3, 4...N2, the total number of rules M is N1*N2, and weight coefficient vector φ and fuzzy basis vector are introduced simultaneously.

[0031]

[0032] Wherein the definition is:

[0033]

[0034] The weighting coefficients are designed as follows:

[0035]

[0036] Where β is a positive constant, substituting the above formula yields:

[0037]

[0038] Furthermore, the specific steps of step three are as follows: First, design the sliding surface of the fast integration terminal as follows:

[0039]

[0040] All are positive definite matrices.

[0041] Substituting the previously obtained observer estimate into the error e, we get:

[0042]

[0043] This can then be written as the third-order state equation of the system:

[0044]

[0045] Furthermore, the specific steps of step four are as follows: First, the following coordinate transformation is introduced:

[0046]

[0047] Based on the inversion design method and the observation results of the fuzzy order reduction observer, the system control input can be designed as follows:

[0048]

[0049] τ=τ eq -τ aw

[0050] The present invention, by adopting the above technical solution, achieves the following beneficial effects:

[0051] (1) This invention applies the Dahl friction model to the dynamic model of the robotic arm to reflect the friction torque experienced by the robotic arm during operation. This can better simulate the operating state of the robotic arm in actual work. Based on this model, the controller can be designed to effectively improve the control accuracy.

[0052] (2) By combining the integral sliding surface with the fast terminal sliding surface, a new integral fast terminal sliding surface is proposed, which can achieve error convergence in a very short time and has a smaller steady-state error compared with other control methods.

[0053] (3) A fuzzy reduced-order observer is used to estimate lumped disturbances and joint velocities. This avoids the problem of difficulty in measuring joint velocity information, and at the same time, the feedforward compensation of disturbances by the state observer effectively avoids chattering.

[0054] (4) The torque of the control input is designed based on the inversion method, which can achieve global asymptotic stability based on Lyapunov and solve the singularity problem in fast terminal sliding mode. Attached Figure Description

[0055] Figure 1 This is a structural block diagram of the controller in this invention;

[0056] Figure 2 This is a schematic diagram of the torque model of a two-link rigid robotic arm in an embodiment of the present invention;

[0057] Figure 3 Simulation diagram of the torques at two joints of a robotic arm;

[0058] Figure 4-6 A simulation diagram showing the expected and error estimates of the position, velocity, and acceleration of two joints of a robotic arm;

[0059] Figure 7-9 To compare different algorithms, here is a simulation comparison diagram of the position and velocity of two joints of the robotic arm;

[0060] Figure 10 A schematic diagram showing the two-joint observer measuring external disturbances is presented. Detailed Implementation

[0061] The invention is further illustrated below with specific embodiments. To better illustrate the invention, the proposed control method is verified using MATLAB numerical simulation, and the results are as follows. Figures 1 to 10 As shown. The specific steps are as follows:

[0062] Step 1: Establish a dynamic model of the robotic arm that includes the Dhal friction model. The model is as follows:

[0063]

[0064] Where: f represents frictional force, Fc is Coulomb frictional force, q represents displacement, and D0 is stiffness coefficient; Let α be the relative velocity between the load and the contact surface, and α be a coefficient related to curvature. This model corresponds to a function of displacement and is independent of velocity, greatly reducing potential errors. This allows the friction model to be incorporated into the robotic arm's dynamics model.

[0065]

[0066] in: Let M(q) be the vector of angular displacement, angular velocity, and angular acceleration of the robotic arm joint; M(q) is the positive definite inertia matrix. The correlation matrix between centrifugal force and Coriolis force; For the friction force matrix of a traditional robotic arm, Here, G(q) is the Dhal friction force matrix; G(q) is the gravity vector; and τ is the joint control torque input vector. d Let M(q) be the uncertainty term of the external disturbance of the robotic arm, where M(q) is the uncertainty term. Due to precision issues, G(q) will be replaced by M(q). G(q) is decomposed into M0(q). G0(q), and the uncertain part △M(q), △G(q);

[0067] Step 2: Based on the dynamic model in Step 1, design a fuzzy reduced-order observer to predict disturbances;

[0068] The specific steps of step 2 are as follows, transforming the system into the following form:

[0069]

[0070] τ d This includes errors and uncertainties from external interference systems. This makes w1 = q, If w3 = f, then M0(q), G0(q) represents M0(w1), G0(w1); then the second-order system can be obtained as follows:

[0071]

[0072] For w i The estimated values, where κ1 and κ2 are the observer parameters which are positive constants, are then obtained by setting a fuzzy order reduction observer:

[0073] (s+p0) 2 =s n +κ1s n-1 +κ2

[0074] Where p0 is the observer bandwidth. Compared to traditional linearly extended observers, the gain changes faster and faster with increasing error. The pole placement technique is used in the design:

[0075]

[0076] At the same time We can conclude that:

[0077]

[0078] Observer error of fuzzy systems Design a fuzzy system using its derivative as input and bandwidth as output:

[0079] IF is A1 l1 and is A1 l2 ,

[0080] THEN y(w)is B l1l2

[0081]

[0082] Where l1 = 1, 2, 3, 4….N1; l2 = 1, 2, 3, 4…N2, the total number of rules M is N1*N2, and weight coefficient vector φ and fuzzy basis vector are introduced simultaneously.

[0083]

[0084] Wherein the definition is:

[0085]

[0086] The weighting coefficients are designed as follows:

[0087]

[0088] Where β is a positive constant, substituting the above formula yields:

[0089]

[0090] Step 3: Design the fast integrator sliding surface and transform the system into a third-order state equation based on this sliding surface. The fast integrator sliding surface is designed as follows:

[0091]

[0092] All are positive definite matrices.

[0093] Substituting the previously obtained observer estimate into the error e, we get:

[0094]

[0095] This can then be written as the third-order state equation of the system:

[0096]

[0097] Step 4: Set up a new type of non-singular terminal sliding surface and combine it with the disturbance estimated by the observer in the previous step. Use the inversion control method to construct the torque, and use Lyapunov to prove the stability of the system.

[0098] The specific steps of step 4 are as follows: First, the following coordinate transformation is introduced:

[0099]

[0100] Based on the inversion design method and the observation results of the fuzzy order reduction observer, the system control input can be designed as follows:

[0101]

[0102] τ=τ eq -τ aw

[0103] This invention utilizes Simulink in MATLAB 2018a to simulate and verify the non-singular fast terminal sliding mode control method based on the backstepping method of the fuzzy reduced-order observer designed in this invention for a two-joint robotic arm. The results are compared with some other control algorithms (Adaptive Inversion NFTSM Controller ABNFTMSC, Adaptive Integral Sliding Mode Controller (AISMC), Adaptive Non-Singular Fast Terminal Sliding Mode Controller (ANFTSMC), and Adaptive Fractional Non-Singular Fast Terminal Sliding Mode Controller (FO-NFTMSC)).

[0104] (1) The simulation parameters are as follows

[0105] The parameters of the robotic arm dynamics model are as follows, considering an uncertainty of 0.5 in the robotic arm model:

[0106] M(q)=M0(q)+ΔM(q)=1.5M0(q)

[0107]

[0108] G(q)=G0(q)+ΔG(q)=1.5G0(q)

[0109]

[0110]

[0111] Let the initial position and initial angular velocity of the robotic arm joints be q0 = [0,0]. T rad, The expected trajectory to be tracked is as follows:

[0112]

[0113] The interference applied to the system is as follows:

[0114]

[0115] The controller parameters are designed as follows:

[0116] parameter <![CDATA[k1]]> a b β <![CDATA[k2]]> <![CDATA[k3]]> <![CDATA[λ3]]> <![CDATA[l1]]> <![CDATA[l2]]> <![CDATA[l3]]> numerical values 4 1.2 1.3 0.0001 6 10 3 250 150 20

[0117] Figure 3The simulation diagram of the torque of the two joints of the robotic arm shows that the signals of the two robotic arms in this invention are continuous and without singularity, and there is little chattering.

[0118] Figure 4-6 The figure shows a simulation diagram of the expected and error estimation of the position, velocity, and acceleration of the two joints of the robotic arm. As can be seen from the figure, the steady-state error of the two robotic arms in this invention is very small, which reflects the advantage of high tracking accuracy of this invention.

[0119] Figure 7-9 To compare different algorithms, a simulation comparison diagram of the position and velocity of the two joints of the robotic arm is shown. As can be seen from the diagram, the control input of the two joints in this invention remains continuous, without any chattering, and the accuracy is higher.

[0120] Figure 10 The diagram shows an observer at two joints measuring external disturbances, demonstrating faster and more accurate tracking.

[0121] In summary, the control scheme designed in this invention only requires joint position information to enable the robotic arm to achieve high-precision tracking of the desired trajectory in a short time. It also exhibits strong robustness against disturbances and has global asymptotic stability.

[0122] The specific implementation examples described above are merely for the purpose of helping those skilled in the art to understand the present invention. However, the present invention is not limited to the cases described above. Any variations are obvious as long as they fall within the spirit and scope of the invention as defined and determined by the appended claims.

Claims

1. A terminal sliding mode control method based on backstepping using a fuzzy reduced-order observer, characterized in that, Includes the following steps: Step 1: Construct a new dynamic model based on the Dhal friction model of the robotic arm dynamics. The Dhal friction model is a function of displacement and is independent of velocity, including: Where: f represents frictional force, Fc is Coulomb frictional force, q represents displacement, and D0 is stiffness coefficient; Let α be the relative velocity between the load and the contact surface, and α be a coefficient related to curvature; the friction model is then incorporated into the robotic arm dynamics model: in: Let M(q) be the vector of angular displacement, angular velocity, and angular acceleration of the robotic arm joint; M(q) is the positive definite inertia matrix. G(q) is the correlation matrix between centrifugal force and Coriolis force; G(q) is the gravity vector. For the friction force matrix of a traditional robotic arm, Let τ be the Dhal friction force matrix, and τ be the joint control torque input vector. d Let M(q) be the uncertainty term of the external disturbance to the robotic arm; where, due to M(q), Due to precision issues, G(q) is replaced by M(q). G(q) is decomposed into M0(q). G0(q), and the uncertain part ΔM(q), ΔG(q); Step 2: Based on the dynamic model from Step 1, design a fuzzy reduced-order observer to predict disturbances. The observer dynamically optimizes its bandwidth using a fuzzy system: the observer error... Using its derivative as input and the observer bandwidth as output, a fuzzy system is designed, introducing the weight coefficient vector φ and the fuzzy basis vectors. pass Adjust the bandwidth and let w1 = q. If w3 = f, then M0(q) is M0(w1); the core expression of the observer is: Where p0 is the observer bandwidth; For w i The estimated value for (i = 1, 2, 3), For observer error; Step 3: Based on the fuzzy reduced-order observer in Step 2, set an adaptive law to ensure the stability of the system; Step 4: Set up a new type of non-singular terminal sliding surface and combine it with the disturbance estimated by the observer in the previous step. Use the inversion control method to construct the torque, and use Lyapunov to prove the stability of the system.

2. The terminal sliding mode control method based on the backstepping method of a fuzzy reduced-order observer according to claim 1, characterized in that, The specific steps of step two are as follows: the system is transformed into the following form, τ d Including errors and uncertainties in the external disturbance system, such that w1 = q, If w3 = f, then M0(q), G0(q) represents M0(w1), G0(w1); then the second-order system can be obtained as follows: For w i The estimated values ​​are given, where κ1 and κ2 are the observer parameters, which are positive constants. Then, a fuzzy order reduction observer is set: (s+p0) 2 =s n +κ1s n-1 +κ2 Where p0 is the observer bandwidth. Compared to traditional linearly extended observers, the gain changes faster and faster with increasing error. The pole placement technique is used in the design: At the same time We can conclude that: Observer error of fuzzy systems Design a fuzzy system using its derivative as input and bandwidth as output: IF is A1 l1 and is A1 l2 , THEN y(w)is B l1l2 Where l1 = 1, 2, 3, 4...N1; l2 = 1, 2, 3, 4...N2, the total number of rules M is N1*N2, and weight coefficient vector φ and fuzzy basis vector are introduced simultaneously. Wherein the definition is: The weighting coefficients are designed as follows: Where β is a positive constant, substituting the above formula yields:

3. The terminal sliding mode control method based on the backstepping method of a fuzzy reduced-order observer according to claim 2, characterized in that, The specific steps of step three are as follows: First, design the sliding surface of the fast integration terminal as follows: All are positive definite matrices; Substituting the previously obtained observer estimate into the error e, we get: This can then be written as the third-order state equation of the system:

4. The terminal sliding mode control method based on the backstepping method of a fuzzy reduced-order observer according to claim 3, characterized in that, The specific steps of step four are as follows: First, introduce the following coordinate transformation: Based on the inversion design method and the observation results of the fuzzy order reduction observer, the system control input can be designed as follows: τ=τ eq -t aw 。

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