Mechanical arm self-adaptive position tracking control method based on time-varying internal model

Abstract

The invention discloses a mechanical arm self-adaptive position tracking control method based on a time-varying internal model. The mechanical arm self-adaptive position tracking control method comprises the following steps that S1, a mathematical model of an n-degree-of-freedom mechanical arm is established; s2, designing a time-varying internal model, compensating external interference with unknown frequency and time varying, and converting the adaptive position tracking control problem of the mechanical arm system into a robust stabilization problem of an augmentation system composed of the mechanical arm system and the time-varying internal model; and S3, designing a state feedback controller, solving the robust stabilization problem of the augmented system, and providing a final controller. According to the mechanical arm self-adaptive position tracking control method based on the time-varying internal model, the problem of high-precision position tracking control of a mechanical arm system under the conditions that the external interference frequency is unknown and time varying is solved, and the mechanical arm self-adaptive position tracking control method has good position tracking performance and has the capacity of processing complex interference. Meanwhile, parameters of the mechanical arm are allowed to be unknown, and practical application is easy.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of mechanical arm control, in particular to a mechanical arm adaptive position tracking control method based on time-varying internal model. BACKGROUND

[0002] As a highly coupled multi-input multi-output nonlinear system, the mechanical arm has a wide range of applications in the fields of aviation, medical treatment, automobile manufacturing, etc. With the rapid development of industrial automation and intelligent manufacturing, the precise control and position tracking of the mechanical arm have become a research hotspot. In practical applications, the tracking performance of the mechanical arm is easily affected by uncertain parameters and unknown external disturbances, resulting in a decline in control accuracy. Therefore, the high-precision position tracking control of the mechanical arm is challenging.

[0003] However, at present, when the output regulation method is used to solve the high-precision position tracking control problem of the mechanical arm, it cannot achieve the suppression of unknown and time-varying frequency disturbances, which greatly limits the use range of the method. SUMMARY

[0004] The present application aims to provide a mechanical arm adaptive position tracking control method based on time-varying internal model to solve the above defects.

[0005] In order to achieve the above-mentioned purpose, the present application provides the following technical scheme:

[0006] The present application provides a mechanical arm adaptive position tracking control method based on time-varying internal model, which comprises the following steps:

[0007] S1, establishing a mathematical model of an n-degree-of-freedom mechanical arm;

[0008] S2, designing a time-varying internal model to compensate for external disturbances with unknown and time-varying frequencies, and converting the adaptive position tracking control problem of the mechanical arm system into a robust stabilization problem of an augmented system composed of the mechanical arm system and the time-varying internal model;

[0009] S3, designing a state feedback controller to solve the robust stabilization problem of the augmented system and giving a final controller.

[0010] Preferably, the S1 step specifically comprises the following steps:

[0011] S11, the mathematical model of the n-degree-of-freedom mechanical arm is specifically:

[0012]

[0013] In the formula, is the joint position of the mechanical arm, is the joint speed of the mechanical arm, for the joint acceleration of the robot arm, for the symmetric positive definite inertia matrix, for the centripetal and Coriolis force matrix, for the gravity vector, for the control torque vector, for the external disturbance vector;

[0014] It is assumed that the external disturbance vector D can be generated by a time-varying external system of the form:

[0015]

[0016] where, is the derivative of the state variable of the time-varying external system, a(t,σ,ω) is a sufficiently smooth matrix, t is time, the constant σ usually represents the size of the unknown disturbance frequency in practice, ω is the state variable of the time-varying external system, and R is a constant matrix;

[0017] S12, the mathematical model of the n-degree-of-freedom robot arm, has the following properties:

[0018] Property One: is an anti-symmetric matrix;

[0019] Property Two: For the following equation holds:

[0020]

[0021] where, is an unknown parameter vector, E(θ,x), is a known matrix.

[0022] Preferably, the S2 step, in particular the steps are as follows:

[0023] S21, define the joint reference velocity:

[0024]

[0025] where, is the reference joint velocity, is the desired joint velocity, θ d is the desired joint position, and α is a positive real number;

[0026] Taking the derivative of formula (4) gives where is the reference joint acceleration, is the desired joint acceleration;

[0027] S22, define the joint reference velocity error:

[0028]

[0029] Equation (5) can be further derived as Taking the derivative of (5) again, we have

[0030]

[0031] where s is the joint reference velocity error, is the joint reference acceleration error;

[0032] S23, combining equations (1), (3), (5) and (6), we have

[0033]

[0034] S24, for i = 1, …, n, assume there exists an integer l i , for all time t and frequency σ i , such that the ith dimension external disturbance D i satisfies:

[0035]

[0036] where, for j = 0, …, l i -1, the sufficiently smooth function b i,j (t, σ i ) and its r i order derivatives are uniformly bounded, r i = 1, …, m i , m i is a sufficiently large integer;

[0037] In addition, there exist smooth vector-valued functions and vector-valued functions satisfying

[0038] S25, for all t and σ i , we can obtain the steady-state generator in the following form:

[0039] D i = -Γ i ξ i (9),

[0040] where ξ i is the state of the steady-state generator, Γ i = [1 0 … 0 0] is a row vector, is a sufficiently smooth matrix;

[0041] Φ i (t, σi ) can be described as:

[0042] Φ i (t,σ i ) = Φ i,0 + b i (t,σ i )Γ i (10),

[0043] wherein is a constant matrix, is a smooth vector-valued function;

[0044] Since the matrix pair (Φ i,0 , Γ i ) is observable, there exists a column vector such that is a Hurwitz matrix;

[0045] S26, let G i (t,σ i ) = L i,0 + b i (t,σ i ), β i,0 (t) = 1, define wherein ψ i,j,k is the k-th component of ψ i,j , j = 0,..., l i - 1;

[0046] define wherein β i,r is a scalar function, r = 1,..., p i ;

[0047] From one obtains:

[0048]

[0049] S27, let

[0050]

[0051] wherein F i is a block-diagonal matrix, G i (t) is a sufficiently smooth matrix, P i (σ i ) is a constant matrix, is a Hurwitz matrix, is a column vector, and k = 1,..., p iis a row vector; define Φ(t,σ),F g ,G g (t),P(σ),Γ respectively as the block diagonal matrix consisting of i (t,σ i ),F i ,G i (t),P i (σ i ),Γ i ;

[0052] The time-varying internal model is designed as follows, as shown in equation (12):

[0053]

[0054] where η is the state variable of the time-varying internal model;

[0055] S28, coordinate transformation is performed where T(t,σ) is a continuously differentiable matrix, and satisfies: Further, equation (13) is obtained:

[0056]

[0057] where, Further, equation (7) can be rewritten as:

[0058]

[0059] S29, a dynamic compensator in the following form is constructed, as shown in equation (15):

[0060]

[0061] where λ is the state variable of the dynamic compensator;

[0062] coordinate transformation is performed It can be obtained that:

[0063]

[0064] Further, equation (14) can be rewritten as:

[0065]

[0066] S210, in order to handle unknown frequency σ, parameter linearization is performed on P(σ):

[0067]

[0068] where n p is a positive integer, for i=1,…,n p , P0 and are known constant matrices independent of σ, ζ i (σ) is a smooth function;

[0069] Let Then we have:

[0070] Then formula (17) can be rewritten as:

[0071]

[0072] where At this time, the adaptive position tracking problem of system (1) is transformed into the robust stabilization problem of the augmented system composed of formula (16) and formula (19).

[0073] Preferably, the S3 step, in particular, the steps are as follows:

[0074] S31, design the state feedback controller of the following form for formula (19):

[0075]

[0076] where K is a positive definite diagonal matrix, is the estimated value of ;

[0077] S32, let where Ψ is a positive definite symmetric matrix satisfying is the unit matrix, m is a positive real number, Λ is a positive definite diagonal matrix, then we have:

[0078]

[0079] S33, select the adaptive law Formula (21) can be written in the following form:

[0080]

[0081] Because P(σ) is a bounded matrix, there exist normal numbers a and b satisfying Select appropriate m and K so that m-a>0, (K-bI n ) is a positive definite matrix, then formula (22) can be written as:

[0082]

[0083] where a0=m-a is a normal number, b0 is the minimum eigenvalue of the matrix (K-bI n );

[0084] S34, the final controller is obtained in the following form:

[0085]

[0086] The present application has the advantages of:

[0087] The present application discloses a mechanical arm adaptive position tracking control method based on time-varying internal model, which solves the high-precision position tracking control problem of the mechanical arm system under the condition of unknown and time-varying external disturbance frequency by designing an adaptive controller based on time-varying internal model, and has good position tracking performance. BRIEF DESCRIPTION OF DRAWINGS

[0088] Figure 1 The mechanical arm control framework of the method of the present application;

[0089] Figure 2 The mechanical arm position tracking curve of the method of the present application;

[0090] Figure 3 The mechanical arm position tracking error curve of the method of the present application. DETAILED DESCRIPTION

[0091] The present application will be further described below in conjunction with the embodiments, and it should be noted that the embodiments are only examples and illustrations of the present application, and those skilled in the art can make various modifications or supplements or use similar ways to replace the described embodiments, as long as they do not deviate from the concept of the present application or exceed the scope defined by the present application, and they should be considered to fall within the protection scope of the present application.

[0092] Embodiment 1:

[0093] The present application discloses a mechanical arm adaptive position tracking control method based on time-varying internal model, which solves the high-precision position tracking control problem of the mechanical arm system under the condition of unknown and time-varying external disturbance frequency by designing an adaptive controller based on time-varying internal model, and has good position tracking performance.

[0094] S1, a mathematical model of an n-degree-of-freedom mechanical arm is established, and the specific steps are as follows:

[0095] S11, the mathematical model of the n-degree-of-freedom mechanical arm is specifically:

[0096]

[0097] In the formula, is the joint position of the mechanical arm, is the joint speed of the mechanical arm, is the joint acceleration of the mechanical arm, is a symmetric positive definite inertia matrix, is a centripetal force and Coriolis force matrix, is the gravity vector, is the control moment vector, is the external disturbance vector;

[0098] Suppose the external disturbance vector D can be generated by a time-varying external system of the form:

[0099]

[0100] where, is the derivative of the state variable of the time-varying external system, a(t,σ,ω) is a sufficiently smooth matrix, t is time, the constant σ usually represents the size of the unknown disturbance frequency in practice, ω is the state variable of the time-varying external system, and R is a constant matrix;

[0101] S12, the mathematical model of the n-degree-of-freedom robot arm, has the following properties:

[0102] Property One: is an anti-symmetric matrix;

[0103] Property Two: For The following equation holds:

[0104]

[0105] where, is an unknown parameter vector, E(θ,x), is a known matrix.

[0106] S2, design a time-varying internal model to compensate for the external disturbance with unknown frequency and time-varying, and convert the adaptive position tracking control problem of the robot arm system into the robust stabilization problem of the augmented system composed of the robot arm system and the time-varying internal model, the specific steps are as follows:

[0107] S21, define the joint reference velocity:

[0108]

[0109] where, is the reference joint velocity, is the desired joint velocity, θ d is the desired joint position, and α is a positive real number;

[0110] Taking the derivative of formula (4) gives where is the reference joint acceleration, is the desired joint acceleration;

[0111] S22, define the joint reference velocity error:

[0112]

[0113] Equation (5) can be further derived as Taking the derivative of (5) again, we have

[0114]

[0115] where s is the joint reference velocity error, is the joint reference acceleration error.

[0116] S23, combining equations (1), (3), (5) and (6), we have

[0117]

[0118] S24, for i = 1, …, n, assume there exists an integer l i for all time t and frequency σ i such that the ith dimension external disturbance D i satisfies:

[0119]

[0120] where for j = 0, …, l i -1, the sufficiently smooth function b i,j (t, σ i ) and its r i th derivative are uniformly bounded, r i = 1, …, m i , m i is a sufficiently large integer;

[0121] In addition, there exist smooth vector-valued functions and vector-valued functions satisfying

[0122] S25, for all t and σ i , we can obtain the steady-state generator in the following form:

[0123] D i = -Γ i ξ i (9),

[0124] where ξ i is the state of the steady-state generator, Γ i = [1 0 … 0 0] is a row vector, is a sufficiently smooth matrix.

[0125] Φi (t,σ i ) can be described as:

[0126] Φ i (t,σ i ) = Φ i,0 + b i (t,σ i )Γ i (10),

[0127] wherein is a constant matrix, is a smooth vector-valued function.

[0128] Since the matrix pair (Φ i,0 , Γ i ) is observable, there exists a column vector such that is a Hurwitz matrix.

[0129] S26, let G i (t,σ i ) = L i,0 + b i (t,σ i ), β i,0 (t) = 1, define wherein ψ i,j,k is the k-th component of ψ i,j , j = 0,..., l i - 1.

[0130] define wherein β i,r is a scalar function, r = 1,..., p i ; it follows from

[0131]

[0132] S27, let

[0133]

[0134] wherein F i is a block-diagonal matrix, G i (t) is a sufficiently smooth matrix, P i (σ i ) is a constant matrix, is a Hurwitz matrix, is a column vector, and k = 1,..., p i ​is a row vector; define Φ(t,σ),F g ,G g (t),P(σ),Γ respectively as the block diagonal matrix consisting of i (t,σ i ),F i ,G i (t),P i (σ i ),Γ i .

[0135] Design the time-varying internal model in the form as shown in equation (12):

[0136]

[0137] where η is the state variable of the time-varying internal model.

[0138] S28, perform coordinate transformation where T(t,σ) is a continuously differentiable matrix, satisfying: Further, equation (13) is obtained:

[0139]

[0140] where Further, equation (7) can be rewritten as:

[0141]

[0142] S29, construct a dynamic compensator in the form as shown in equation (15):

[0143]

[0144] where λ is the state variable of the dynamic compensator.

[0145] perform coordinate transformation , we have:

[0146]

[0147] Further, equation (14) can be rewritten as:

[0148]

[0149] S210, in order to handle unknown frequency σ, perform parameter linearization on P(σ):

[0150]

[0151] where n p is a positive integer, for i=1,…,n p , P0and are known constant matrices independent of σ, ζ i (σ) is a smooth function.

[0152] Let Then we have:

[0153] Then formula (17) can be rewritten as:

[0154]

[0155] where At this time, the adaptive position tracking problem of system (1) is transformed into the robust stabilization problem of the augmented system composed of formula (16) and formula (19).

[0156] S3, design a state feedback controller to solve the robust stabilization problem of the augmented system, and give the final controller, the specific steps are as follows:

[0157] S31, design a state feedback controller of the following form for formula (19):

[0158]

[0159] where K is a positive definite diagonal matrix, is the estimate of .

[0160] S32, let where Ψ is a positive definite symmetric matrix satisfying is the identity matrix, m is a positive real number, Λ is a positive definite diagonal matrix, then we have:

[0161]

[0162] S33, select the adaptive law Formula (21) can be written in the following form:

[0163]

[0164] Because P(σ) is a bounded matrix, there exist normal numbers a and b satisfying Select appropriate m and K so that m-a>0, (K-bI n ) is a positive definite matrix, then formula (22) can be written as:

[0165]

[0166] where a0=m-a is a normal number, b0is the minimum eigenvalue of the matrix (K-bI n ).

[0167] S34, the final controller is as follows:

[0168]

[0169] To verify the effectiveness of the above method, the three-DOF robot model is as follows:

[0170]

[0171] H 11 = a1 + 2a4cos(θ2)cos(θ2+θ3) + a2cos 2 (θ2) + a3cos 2 (θ2+θ3)

[0172] H 22 = a5 + 2a4cos(θ3), H 23 = a3 + a4cos(θ3), H 32 = H 23 , H 33 = a6

[0173]

[0174] G1 = 0, G2 = a7cos(θ2) + a8cos(θ2+θ3), G3 = a8cos(θ2+θ3),

[0175] In the formula, a1 = 0.01, a2 = 0.248, a3 = 0.032, a4 = 0.064, a5 = 0.162, a6 = 0.036, a7 = 9.016, a8 = 1.568.

[0176] The desired position is: The external disturbance is: The external system parameters are as follows:

[0177]

[0178] The controller parameters are as follows:

[0179] α = 60,

[0180]

[0181] wherein i = 1, 2, 3.

[0182] The initial position is: The initial value of the external system is: The other initial conditions of the system are 0.

[0183] Figure 1 is a mechanical arm control framework chart in the method of the application. Figure 1 As shown in the figure, a series of technical parameters are adopted, and are applied to the mechanical arm position tracking control method based on time-varying internal model disclosed in the application, and simulation results obtained are as shown in the figure. Figures 2-3 Figure 2 is a mechanical arm position tracking curve chart of the method of the application, Figure 3 is a mechanical arm position tracking error curve chart in the method of the application, as shown in the figure. Figure 2 Figure 3 It can be seen that the designed controller has good tracking performance under the condition of unknown and time-varying disturbance frequency, and verifies the practical feasibility of the application.

[0184] The application proposes a mechanical arm adaptive position tracking control method based on time-varying internal model, solves the high-precision speed tracking control problem of the mechanical arm under the condition of unknown and time-varying external disturbance frequency by designing an adaptive controller based on time-varying internal model, and has good tracking performance. Since the application can suppress the unknown and time-varying external disturbance, it has the ability to handle complex disturbance. At the same time, the application allows the mechanical arm parameters to be unknown, and is easy to be applied in practice.

[0185] The above is an exemplary description of the application, and obviously, the specific implementation of the application is not limited by the above method. As long as the method concept and technical solution of the application are adopted for such non-essential improvement, or the concept and technical solution of the application are directly applied to other occasions without improvement, they are all within the protection scope of the application.​​

Claims

1. A method for adaptive position tracking control of a robotic arm based on a time-varying internal model, characterized in that: The following steps are involved: S1. Establish a mathematical model of the n-DOF robotic arm; S2. Design a time-varying internal model to compensate for the time-varying external disturbance with unknown frequency, and transform the adaptive position tracking control problem of the manipulator system into a robust stabilization problem of the augmented system consisting of the manipulator system and the time-varying internal model. S3. Design a state feedback controller to solve the robust stabilization problem of the augmented system and give the final controller.

2. The method for adaptive position tracking control of a manipulator based on a time-varying internal model according to claim 1, characterized in that: The specific steps of the S1 step are as follows: S11. The mathematical model of the n-DOF robotic arm is specifically: Where, is the joint position of the robotic arm, is the joint velocity of the robot arm, is the joint acceleration of the robotic arm, is a symmetric positive definite inertia matrix, are the centripetal and Coriolis force matrices, is the gravity vector, is the control torque vector, is the external interference vector; Assume that the external interference vector D can be generated by a time-varying external system of the following form: Where, is the derivative of the state variable of the time-varying external system, a(t,σ,ω) is a sufficiently smooth matrix, t is time, the constant σ usually represents the magnitude of the unknown disturbance frequency in practice, ω is the state variable of the time-varying external system, and R is a constant matrix; S12. The mathematical model of the n-DOF manipulator has the following properties: Property 1: is an antisymmetric matrix; Property 2: For The following equation holds: Where, is the unknown parameter vector, E(θ,x), is a known matrix.

3. The method for adaptive position tracking control of a manipulator based on a time-varying internal model according to claim 2, characterized in that: The specific steps of the S2 step are as follows: S21. Define joint reference velocity: Where, is the reference joint velocity, is the desired joint velocity, θ d is the desired joint position, α is a positive real number; Taking the derivative of formula (4) we can get in is the reference joint acceleration, is the desired joint acceleration; S22. Define joint reference velocity error: Formula (5) can be further obtained Then, we can derive (5): Where s is the joint reference velocity error, is the joint reference acceleration error; S23, Combining equations (1), (3), (5) and (6), we can get the following formula: S24. For i=1,…,n, assume that there exists an integer l i , for all time t and frequency σ i , so that the external disturbance D in the i-th dimension i satisfy: Where, for j=0,…,l i -1, sufficiently smooth function b i,j (t,σ i ) and its i The derivatives of order are uniformly bounded, r i =1,…,m i , m i is a sufficiently large integer; Furthermore, there exists a smooth vector-valued function and vector-valued functions satisfy S25, for all t and σ i , we can get the following steady-state generator: Where, ξ i is the state of the steady-state generator, Γ i =[1 0 … 0 0] is a row vector, is a fully smooth matrix; Φ i (t,σ i ) can be described as: F i (t,s i )=Φ i,0 +b i (t,s i )C i (10), Where, is a constant matrix, is a smooth vector-valued function; Since the matrix pair (Φ i,0 ,Γ i ) is observable, there exists a column vector Make is the Hurwitz matrix; S26, let G i (t, σ i ) = L i,0 + b i (t, σ i ), β i,0 (t) = 1, define k=1,…,p i , where ψ i,j,k is ψ i,j The kth component of i -1; definition where β i,r is a scalar function, r=1,…,p i ; Depend on You can get: S27, Order Among them F i is a block diagonal matrix, G i (t) is a sufficiently smooth matrix, P i (σ i ) is a constant matrix, is the Hurwitz matrix, is a column vector, and k=1,…,p i is a row vector; define Φ(t,σ),F g ,G g (t),P(σ),Γ are Φ i (t,σ i ),F i ,G i (t),P i (σ i ),Γ i The block diagonal matrix composed of Design the time-varying internal model as follows, as shown in formula (12): Where η is the state variable of the time-varying internal model; S28. Perform coordinate transformation Where T(t,σ) is a continuously differentiable matrix that satisfies: Further formula (13) is obtained: Where, Formula (7) can be further rewritten as: S29. Construct a dynamic compensator of the following form, as shown in formula (15): Where λ is the state variable of the dynamic compensator; Perform coordinate transformation We can get: Formula (14) can be further rewritten as: S210. To process the unknown frequency σ, perform parameter linearization on P(σ): Where n p is a positive integer, for i=1,…,n p , P0 and are all known constant matrices that are independent of σ, ζ i (σ) is a smooth function; make Then we have: Then formula (17) can be rewritten as: Where, At this point, the adaptive position tracking problem of system (1) is transformed into the robust stabilization problem of the augmented system composed of Equations (16) and (19).

4. The method for adaptive position tracking control of a manipulator based on a time-varying internal model according to claim 3, characterized in that: The specific steps of the S3 step are as follows: S31. Design a state feedback controller of the following form for equation (19): Where K is a positive definite diagonal matrix, yes estimated value of; S32, order where Ψ is a positive definite symmetric matrix satisfying is the identity matrix, m is a positive real number, If Λ is a positive definite diagonal matrix, then: S33, select the adaptive law Formula (21) can be written as follows: Since P(σ) is a bounded matrix, there exist positive constants a and b that satisfy Choose appropriate m and K so that ma>0, (K-bI n ) is a positive definite matrix, then formula (22) can be written as: Where a0=ma is a positive constant, b0 is a matrix (K-bI n )’s minimum eigenvalue; S34, the final controller is obtained in the following form:

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