An internal model control method for a robotic arm based on adaptive friction compensation

Abstract

The present invention discloses a method for controlling a manipulator internal model based on adaptive friction compensation, and relates to the field of manipulator system control; the method comprises: step 1, establishing a dynamic model of an n-degree-of-freedom manipulator system; step 2, designing an internal model, converting the position tracking and interference suppression problem of the manipulator system into a stabilization problem of an augmented system composed of the system and the internal model; step 3, designing a final controller, compensating for friction and solving the stabilization problem of the augmented system. Aiming at the situation where the manipulator system has external torque disturbances and friction in actual operation, the present invention designs an internal model controller based on adaptive friction compensation, and can suppress interference and compensate for friction of the manipulator system while allowing the system parameters to be unknown, thereby realizing high-precision position tracking control and speed tracking control of the manipulator joints.

Description

Technical Field

[0001] This invention relates to the field of robotic arm system control, specifically to a robotic arm internal mold control method based on adaptive friction compensation. Background Art

[0002] After decades of development, robotic arms can not only perform various tasks with precision and efficiency, but also replace humans in high-risk, repetitive work in harsh environments. They are currently widely used in industrial applications such as welding, assembly, and painting. However, robotic arms are also complex nonlinear systems, frequently affected by parameter uncertainties, external disturbances, and friction in practical applications. This undoubtedly reduces the system's control accuracy and may even affect its stability. Therefore, comprehensively considering the effects of parameter uncertainties, external disturbances, and friction when designing controllers to achieve high-precision control of the robotic arm is of great significance.

[0003] On the one hand, nonlinear output regulation theory has made great progress in the past few decades, and its internal model control method can conveniently solve the trajectory tracking and disturbance suppression problems of the system, and has good robustness. On the other hand, compared with other friction models, the Stribeck friction model can more accurately reflect the nonlinear characteristics of friction, and therefore it is used in this invention to describe the joint friction torque of the robotic arm. By combining the internal model control method and the adaptive friction compensation method in the robotic arm system, friction can be compensated and disturbances suppressed even when system parameters are unknown, ultimately achieving high-precision position tracking and velocity tracking performance. Summary of the Invention

[0004] To address the technical problems existing in the background technology, this invention proposes a robotic arm internal model control method based on adaptive friction compensation. Considering the external torque disturbances and friction forces present in the actual operation of the robotic arm system, an internal model controller based on adaptive friction compensation is designed. This method achieves disturbance suppression and friction compensation for the robotic arm system even when system parameters are unknown, exhibiting excellent position tracking and velocity tracking performance.

[0005] The technical solution adopted in the present invention is:

[0006] A method for controlling the inner mold of a robotic arm based on adaptive friction compensation, characterized by comprising the following steps:

[0007] Step 1: Establish the dynamic model of the n-degree-of-freedom robotic arm system;

[0008] Step 2: Design the internal model to transform the position tracking and disturbance suppression problem of the robotic arm system into the stabilization problem of the augmented system composed of the system and the internal model;

[0009] Step 3: Design the final controller to compensate for friction and solve the stabilization problem of the augmentation system.

[0010] In step 1, the dynamic model of the n-degree-of-freedom robotic arm system is established, and the process is as follows:

[0011] 1.1 The dynamic model of the n-DOF robotic arm system is established based on the Euler-Lagrange method as follows:

[0012]

[0013] in These are the joint positions and velocity vectors of the robotic arm, respectively. It is a positive definite symmetric inertial matrix. For centripetal force and Coriolis torque, For gravitational torque, For input torque, For frictional torque, External disturbances. The above-mentioned robotic arm system has the following properties:

[0014] Property 1: Matrices and The following conditions must be met:

[0015]

[0016] Property 2: The robotic arm model can undergo the following parameter linearization transformations:

[0017]

[0018] in All are known matrices, satisfying These are uncertain parameters in the robotic arm.

[0019] 1.2, Assume the expected value of the robotic arm joint position is q. d , and q d , All are continuous bounded signals, with external torque interference τ d (t)=[τ d1 (t),…,τ dn (t)] T For a finite number of step signals and sinusoidal signals, the combination takes the following form:

[0020]

[0021] where a i0 k is a constant. i Let a be the number of sinusoidal signals. ij ,ω ij ,φij These represent the amplitude, frequency, and phase of a sinusoidal signal, respectively.

[0022] 1.3, Define the tracking error e = qq d and Where α is a positive constant, we can further obtain Pick Then we have:

[0023]

[0024] 1.4, System (1) can be written in the following form:

[0025]

[0026] Combining property 2, equation (6) can be further written as:

[0027]

[0028] 1.5. The Stribeck friction model is introduced to describe the friction torque of the robotic arm joints. The rectangular formula of the friction force of the i-th joint is as follows:

[0029]

[0030] Where for i = 1, ..., n, f ci f is the Coulomb friction coefficient. si f is the static friction coefficient. vi The coefficient of viscous friction is... Let denot be the Stribeck velocity, and δ be an empirical parameter.

[0031] definition

[0032]

[0033] The frictional force of the robotic arm can then be expressed as: in Let be a known matrix, and β be an unknown parameter vector.

[0034] In step 2, an internal model is designed to transform the position tracking and disturbance suppression problem of the robotic arm system into a stabilization problem of an augmented system consisting of the system and the internal model. The process is as follows:

[0035] 2.1, for τ di Differentiating (t), i = 1, ..., n, we get:

[0036]

[0037] in All are real numbers, s i It is a positive number.

[0038] 2.2, let Then we have:

[0039]

[0040] in

[0041] 2.3, Select a pair of controllable matrices (M) i ,N i ),in For Hurwitz matrices, Then there exists a non-singular matrix T i Satisfies the following Sylvester equation:

[0042]

[0043] 2.4, define the following notation:

[0044]

[0045] Let ρ = -Tυ, Ψ = ΓT -1 According to formula (11), we can obtain:

[0046]

[0047] τ d =-Ψρ (15)

[0048] 2.5, The inner mold is designed as follows:

[0049]

[0050] 2.6 Perform the following coordinate transformation:

[0051]

[0052] Based on property 2, the following error equation is obtained:

[0053]

[0054] in

[0055] 2.7 Design the following dynamic compensator:

[0056]

[0057] 2.8, Order We can obtain:

[0058]

[0059] Then equation (7) can be written in the following form:

[0060]

[0061] in

[0062] At this point, the position tracking and interference suppression problem of the robotic arm system is transformed into the stabilization problem of the augmented system consisting of the system and the internal mold.

[0063] In step 3, the final controller is designed to compensate for friction and solve the stabilization problem of the augmentation system.

[0064] 3.1 The state feedback control law of the system is designed in the following form:

[0065]

[0066] Where K is a positive definite diagonal constant matrix. This is an estimate of θ. This is an estimate of β.

[0067] 3.2, Taking the Lyapunov equation:

[0068]

[0069] Where Δ1 is a positive definite matrix satisfying M T Δ1+Δ1M=-I, where I is the identity matrix, and κ and λ are positive constants. Let Λ1 and Λ2 be arbitrary positive definite diagonal constant matrices, then we have:

[0070]

[0071] 3.3, Since θ and β are unknown constants, Choose the adaptive law as:

[0072]

[0073] At this point, formula (24) can be simplified to the following form:

[0074]

[0075] Design Ψ to be a bounded diagonal matrix, such that there exist positive constants a1, b1 satisfying... Choose appropriate κ and K such that κ > b1, (K-a1I n If ) is a positive definite matrix, then equation (26) can be written as:

[0076]

[0077] Where a0 is a matrix (K-a1I) n The smallest eigenvalue of ) is b0 = κ - b1. Stability analysis shows that under the proposed controller, as t→∞, the closed-loop system state... Bounded. Integrating both sides of equation (27) with respect to time, we know that: According to Barbalat's lemma, we know that... Then the position tracking error e and the velocity tracking error It can eventually converge to zero.

[0078] 3.4 Based on the Lyapunov method, the final controller can be designed as follows:

[0079]

[0080] The advantages of this invention are:

[0081] This invention proposes an internal model control method for a robotic arm based on adaptive friction compensation. Addressing the external torque disturbances and friction forces present in the actual operation of the robotic arm system, an internal model controller based on adaptive friction compensation is designed. Even with unknown system parameters, this method can suppress disturbances and compensate for friction in the robotic arm system, thereby achieving high-precision position and speed tracking control of the robotic arm joints. Attached Figure Description

[0082] Figure 1 This is a schematic diagram of a two-degree-of-freedom robotic arm;

[0083] Figure 2 This is a system control block diagram;

[0084] Figure 3 For tracking curves of the joint angle position of the robotic arm;

[0085] Figure 4 Error curve for tracking the joint angle position of the robotic arm;

[0086] Figure 5 The error curve for tracking the joint angular velocity of the robotic arm; DETAILED DESCRIPTION

[0087] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments.

[0088] Example 1

[0089] This embodiment specifically uses a two-degree-of-freedom robotic arm (such as...) Figure 1 As shown in the figure, q1 and q2 represent the joint angles of the robotic arm, and g is the acceleration due to gravity.

[0090] Combine Figure 1 and Figure 2 The specific steps of the robotic arm internal mold control method based on adaptive friction compensation described in this embodiment are as follows:

[0091] In step 1, the dynamic model of the n-degree-of-freedom robotic arm system is established, and the process is as follows:

[0092] 1.1 The dynamic model of the n-DOF robotic arm system is established based on the Euler-Lagrange method as follows:

[0093]

[0094] in These are the joint positions and velocity vectors of the robotic arm, respectively. It is a positive definite symmetric inertial matrix. For centripetal force and Coriolis torque, For gravitational torque, For input torque, For frictional torque, External disturbances. The above-mentioned robotic arm system has the following properties:

[0095] Property 1: Matrices and The following conditions must be met:

[0096]

[0097] Property 2: The robotic arm model can undergo the following parameter linearization transformations:

[0098]

[0099] in All are known matrices, satisfying These are uncertain parameters in the robotic arm.

[0100] 1.2, Assume the expected value of the robotic arm joint position is q. d , and q d , All are continuous bounded signals, with external torque interference τ d (t)=[τ d1 (t),…,τ dn (t)] T For a finite number of step signals and sinusoidal signals, the combination takes the following form:

[0101]

[0102] where a i0k is a constant. i Let a be the number of sinusoidal signals. ij ,ω ij ,φ ij These represent the amplitude, frequency, and phase of a sinusoidal signal, respectively.

[0103] 1.3, Define the tracking error e = qq d and Where α is a positive constant, we can further obtain Pick Then we have:

[0104]

[0105] 1.4, System (1) can be written in the following form:

[0106]

[0107] Combining property 2, equation (6) can be further written as:

[0108]

[0109] 1.5. The Stribeck friction model is introduced to describe the friction torque of the robotic arm joints. The rectangular formula of the friction force of the i-th joint is as follows:

[0110]

[0111] Where for i = 1, ..., n, f ci f is the Coulomb friction coefficient. si f is the static friction coefficient. vi The coefficient of viscous friction is... Let denot be the Stribeck velocity, and δ be an empirical parameter.

[0112] definition

[0113]

[0114] The frictional force of the robotic arm can then be expressed as: in Let be a known matrix, and β be an unknown parameter vector.

[0115] In step 2, an internal model is designed to transform the position tracking and disturbance suppression problem of the robotic arm system into a stabilization problem of an augmented system consisting of the system and the internal model. The process is as follows:

[0116] 2.1, for τ di Differentiating (t), i = 1, ..., n, we get:

[0117]

[0118] in All are real numbers, s i It is a positive number.

[0119] 2.2, let Then we have:

[0120]

[0121] in

[0122] 2.3, Select a pair of controllable matrices (M) i ,N i ),in For Hurwitz matrices, Then there exists a non-singular matrix T i Satisfies the following Sylvester equation:

[0123]

[0124] 2.4, define the following notation:

[0125]

[0126] Let ρ = -Tυ, Ψ = ΓT -1 According to formula (11), we can obtain:

[0127]

[0128] τ d =-Ψρ (15)

[0129] 2.5, The inner mold is designed as follows:

[0130]

[0131] 2.6 Perform the following coordinate transformation:

[0132]

[0133] Based on property 2, the following error equation is obtained:

[0134]

[0135] in

[0136] 2.7 Design the following dynamic compensator:

[0137]

[0138] 2.8, Order We can obtain:

[0139]

[0140] Then equation (7) can be written in the following form:

[0141]

[0142] in

[0143] At this point, the position tracking and interference suppression problem of the robotic arm system is transformed into the stabilization problem of the augmented system consisting of the system and the internal mold.

[0144] In step 3, the final controller is designed to compensate for friction and solve the stabilization problem of the augmentation system.

[0145] 3.1 The state feedback control law of the system is designed in the following form:

[0146]

[0147] Where K is a positive definite diagonal constant matrix. This is an estimate of θ. This is an estimate of β.

[0148] 3.2, Taking the Lyapunov equation:

[0149]

[0150] Where Δ1 is a positive definite matrix satisfying M T Δ1+Δ1M=-I, where I is the identity matrix, and κ and λ are positive constants. Let Λ1 and Λ2 be arbitrary positive definite diagonal constant matrices, then we have:

[0151]

[0152] 3.3, Since θ and β are unknown constants, Choose the adaptive law as:

[0153]

[0154] At this point, formula (24) can be simplified to the following form:

[0155]

[0156] Design Ψ to be a bounded diagonal matrix, such that there exist positive constants a1, b1 satisfying... Choose appropriate κ and K such that κ > b1, (K-a1I n If ) is a positive definite matrix, then equation (26) can be written as:

[0157]

[0158] Where a0 is a matrix (K-a1I) n The smallest eigenvalue of ) is b0 = κ - b1. Stability analysis shows that under the proposed controller, as t→∞, the closed-loop system state... Bounded. Integrating both sides of equation (27) with respect to time, we know that: According to Barbalat's lemma, we know that... Then the position tracking error e and the velocity tracking error It can eventually converge to zero.

[0159] 3.4 Based on the Lyapunov method, the final controller can be designed as follows:

[0160]

[0161] The model and parameters of the robotic arm in this embodiment are as follows:

[0162]

[0163] Among them, h1, h2, h3, h4, and h5 are unknown system parameters.

[0164] The controller parameters are selected as follows:

[0165]

[0166] α=30, Λ1=200I, Λ2=120I, δ=2,q s1 =q s2 =0.5,

[0167] f c1 =0.6,f c2 =0.63,f s1 =1.38,f s2 =0.35,f v1 =7.78,f v1 =2.35.

[0168] The added external disturbance is τ d =[sin(t)-2cos(2t)] T (N·m), the desired position signal is The initial values ​​of the system's state variables are all set to 0. Based on the above controller parameters, the simulation results are as follows: Figure 3-5 shown. Figure 3 The angular position tracking curves for the two joints of the robotic arm. Figure 4 The angular position tracking error curves for the two joints of the robotic arm are shown. Figure 5The figures show the angular velocity tracking error curves for the two joints of the robotic arm. Simulation results indicate that even with uncertain parameters, external disturbances, and friction, the designed controller can still achieve high-precision position and velocity tracking performance, which is beneficial for practical engineering applications.

[0169] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A method for controlling a robot arm internal model based on adaptive friction compensation, characterized in that it comprises the following steps: Step 1: Establish the dynamic model of the n-DOF robotic arm system; Step 2: Design the internal model to transform the position tracking and disturbance suppression problem of the robot system into the stabilization problem of the augmented system consisting of the system and the internal model; Step 3: Design the final controller to compensate for friction and solve the stabilization problem of the augmented system; In step 1, the dynamic model of the n-DOF robotic arm system is established, and the process is as follows: 1.1, the dynamic model of the n-DOF manipulator system is established according to the Euler-Lagrange method as follows: in are the joint positions and velocity vectors of the robot arm, is a positive definite symmetric inertia matrix, are the centripetal force and the Coriolis torque, is the gravitational moment, is the input torque, is the friction torque, is an external disturbance; the above-mentioned robotic arm system has the following properties: Property 1: Matrix and The following conditions are met: Property 2: The following parameter linear transformations can be performed on the robotic arm model: in are all known matrices, satisfying is the uncertain parameter in the robot arm; 1.2, assuming that the expected value of the robot arm joint position is q d , and q d , are all continuous and bounded signals, and the external torque disturbance τ d (t) = [τ d1 (t),…,τ dn (t)] T It is a combination of a finite number of step signals and sinusoidal signals in the following form: where a i0 is a constant, k i is the number of sinusoidal signals, a ij ,ω ij ,φ ij are the amplitude, frequency and phase of the sinusoidal signal respectively; 1.3, define tracking error e = qq d and Where α is a positive constant, we can further get Pick Then we have: 1.4, system (1) can be written as follows: Combined with Property 2, equation (6) can be further written as: 1.5, the Stribeck friction model is introduced to describe the friction torque of the robot joint. The friction torque of the i-th joint is as follows: where for i=1,…,n, f ci is the Coulomb friction coefficient, f si is the static friction coefficient, f vi is the viscous friction coefficient, is the Stribeck velocity, δ is an empirical parameter; definition Then the friction force of the robot arm can be expressed as in is a known matrix and β is an unknown parameter vector.

2. According to claim 1, a method for controlling a manipulator based on an internal model of adaptive friction compensation is characterized in that, in step 2, an internal model is designed to transform the position tracking and interference suppression problem of the manipulator system into a stabilization problem of an augmented system composed of the system and the internal model, and the process is as follows: 2.1, for τ di (t), i=1,…,n, and then differentiate, we can get: in are all real numbers, s i is a positive constant; 2.2, let Then we have: in 2.3, select a pair of controllable matrices (M i ,N i ),in is the Hurwitz matrix, Then there exists a non-singular matrix T i The following Sylvester equation is satisfied: 2.4, define the following notation: Let ρ = -Tυ, Ψ = ΓT -1 According to formula (11), we can get: t d =-Ψρ (15) 2.5, design the inner mold as follows: 2.6, perform the following coordinate transformation: According to property 2, the following error equation is obtained: in 2.7, design the following dynamic compensator: 2.8, let We can get: Then equation (7) can be written as follows: in At this point, the position tracking and disturbance suppression problems of the robotic arm system are transformed into the stabilization problem of the augmented system consisting of the system and the internal model.

3. The method for controlling a robot arm internal model based on adaptive friction compensation according to claim 2, characterized in that: In step 3, the final controller is designed to compensate for the friction and solve the stabilization problem of the augmented system; 3.1, the state feedback control law of the designed system is as follows: where K is a positive definite diagonal constant matrix, is the estimated value of θ, is the estimated value of β; 3.2, take the Lyapunov equation: Where Δ1 is a positive definite matrix satisfying M T Δ1+Δ1M=-I, I is the unit matrix, κ, λ are positive constants, Λ1, Λ2 are any positive definite diagonal constant matrices, then: 3.3, since θ, β are unknown constants, Select the adaptive law as: At this time, formula (24) can be simplified to the following form: Design Ψ to be a bounded diagonal matrix, there exist positive constants a1, b1 that satisfy Choose appropriate κ and K so that κ>b1, (K-a1I n ) is a positive definite matrix, then equation (26) can be written as: Where a0 is the matrix (K-a1I n ), b0=κ-b1; after stability analysis, it can be seen that under the action of the proposed controller, when t→∞, the closed-loop system state s, Bounded; After integrating both sides of formula (27) over time, we can know: According to Barbalat's lemma, Then the position tracking error e and the velocity tracking error It can eventually converge to zero; 3.4, based on the Lyapunov method, the final controller can be designed as follows:

Images