A non-over-parameterized layered adaptive finite-time control method for robot arms

Abstract

The application discloses a layered self-adaptive finite time control method for a mechanical arm without over-parameterization, and comprises the following steps: a dynamic model of a 2-degree-of-freedom mechanical arm system is established; a scalar linear regression equation with a real physical parameter vector as an unknown parameter is constructed; a scalar linear regression equation represented by measurable variables and real physical parameters is obtained through low-pass filtering, dynamic regression expansion, accompanying matrix transformation and screening matrix construction; a finite time parameter adaptive law is designed based on the equation under the condition of meeting an interval excitation, so that a parameter identification error converges to zero in a finite time; and a sliding mode variable is defined and a finite time control law is designed, so that a joint space tracking error converges to zero in a finite time. The application can directly estimate real physical parameters of the mechanical arm to avoid over-parameterization, and can realize fast and finite time convergence of a parameter identification error and a tracking error.

Description

Technical Field

[0001] This invention belongs to the field of robotic arm control technology, specifically a hierarchical adaptive finite-time control method for robotic arms without overparameterization. Background Technology

[0002] The dynamic model of a robotic arm system established using the Euler-Lagrange equations typically satisfies the condition of a nonlinear parametric regression form. In existing adaptive control techniques for robotic arm systems, many control methods do not directly estimate the system's true physical parameters (such as link mass, length, and moment of inertia), but rather estimate a nonlinear combination of these parameters. While this simplifies controller design, it leads to overparameterization. Overparameterization introduces several drawbacks, such as requiring stronger excitation conditions to ensure parameter estimation converges to the true value, slowing down the parameter identification process, and potentially reducing the system's convergence rate and robustness.

[0003] In recent years, the academic community has also proposed some adaptive control methods directly targeting the nonlinear parameterized model design of robotic arms. Although these methods achieve direct estimation of real physical parameters and avoid over-parameterization problems, they still have limitations in convergence performance. Specifically, these methods can usually only guarantee that the parameter identification error and spatial trajectory tracking error of the system are asymptotically convergent, that is, theoretically, the error can only converge to zero when time approaches infinity. This asymptotic convergence characteristic limits the further improvement of the dynamic performance of the robotic arm control system and cannot meet the requirements of high-precision, fast-response tasks. Summary of the Invention

[0004] To address the problems existing in the prior art, the present invention aims to provide a hierarchical adaptive finite-time control method for robotic arms without overparameterization. This method can directly estimate the real physical parameters of the robotic arm to avoid overparameterization, and can also achieve rapid and finite-time convergence of parameter identification errors and tracking errors.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0006] A hierarchical adaptive finite-time control method for a robotic arm without overparameterization includes the following steps: Step S1: Establish a dynamic model of a 2-DOF robotic arm system; Step S2: Based on the dynamic model, construct a scalar linear regression equation with real physical parameter vectors as unknown parameters; the construction method includes: designing a low-pass filter, obtaining an auxiliary regression matrix through dynamic regression expansion, and then eliminating algebraic operations and obtaining a scalar linear regression equation completely represented by measurable variables and real physical parameters through at least two adjoint matrix transformations and the constructed screening matrix; Step S3: Based on the scalar linear regression equation, under the condition of satisfying the interval excitation condition, design a finite-time parameter adaptive law so that the parameter identification error converges to zero within a finite time; Step S4: Define the tracking error and sliding mode variable, and use the parameter estimates obtained by the parameter adaptive law to design a finite-time control law so that after the parameter identification error converges, the joint space trajectory tracking error converges to zero within a finite time.

[0007] As a further improvement to the above technical solution:

[0008] The dynamic model described in step S1 is expressed as follows:

[0009]

[0010] in, It is a 2D joint angle vector. and They represent The first and second derivatives, It is a 2D control input torque; yes A generalized inertial matrix of dimension ; express The dimensional Coriolis force and centripetal force matrix; It is the potential energy function. express Jacobian matrix;

[0011]

[0012]

[0013]

[0014]

[0015]

[0016] , , , .

[0017] and These represent the mass and length of the first link of the robotic arm, respectively. and These represent the mass and length of the second link of the robotic arm, respectively. It is the gravitational constant. for The first element, for The second element, yes The first derivative, yes The first derivative, It is a nonlinear mapping The One element, .

[0018] In step S2, the dynamic model is written in the following parameterized form:

[0019]

[0020] in,

[0021]

[0022]

[0023] Is with The relevant first matrix, Is with The relevant second matrix, yes The first derivative, Represents a time variable.

[0024] The low-pass filter design in step S2 is as follows:

[0025]

[0026] in, It is a filter constant. and It is the output of the filter. yes The first derivative, yes The first derivative.

[0027] The dynamic regression extension process described in step S2 obtains the first auxiliary regression matrix by solving the following differential equation. Second auxiliary regression matrix :

[0028]

[0029] in, These are design parameters. , yes The first derivative, yes The first derivative of , where T denotes the transpose of the matrix; solving yields: .

[0030] In step S2, the method for obtaining the scalar linear regression equation with the real physical parameter vector as the unknown parameter through two adjoint matrix transformations and the constructed screening matrix is ​​as follows:

[0031] First of all, First, multiply both sides by the adjoint matrix simultaneously. Then multiply both sides by the selection matrix simultaneously. Its definition is as follows:

[0032]

[0033] get:

[0034]

[0035] in, For matrix The determinant, yes The One element, , , yes The One element, , ;

[0036] Then, according to Obtain the inverse mapping Construct the first diagonal matrix ,Will Left multiplication Generate intermediate equations:

[0037]

[0038] in,

[0039]

[0040]

[0041] Next, construct the second diagonal matrix. Make the matrix and All are measurable variables and composition:

[0042]

[0043] in, Represents a diagonal matrix;

[0044] Multiply both sides of the intermediate equation by the left side simultaneously. Then multiply by the left side at the same time The adjoint matrix The scalar linear regression equation is obtained as follows:

[0045]

[0046] in, , It is a matrix The determinant, , yes The One element, yes The One element, , , .

[0047] The finite-time parameter adaptive law designed in step S3 is as follows:

[0048]

[0049] in, Represents a symbolic function. for The estimated value, for The first derivative, These are design parameters.

[0050] In step S4: Define the spatial trajectory tracking error. and sliding mode variables :

[0051]

[0052]

[0053] in, For the desired trajectory; for The first derivative, As a parameter, and , , and It is a positive odd integer and satisfies ;

[0054] The finite-time control law is as follows:

[0055]

[0056] in, , These are design parameters. for The Frobenius norm, yes The matrix is ​​defined as follows: , , , , , , , ,in, and They represent The first and second elements, and They represent The first and second elements, , for The first derivative, for The first derivative, for The estimated value.

[0057] The interval excitation condition mentioned in step S3 is detected in real time. The system determines whether the minimum eigenvalue is greater than zero. If the condition is not met, an excitation signal is added to the system reference signal to make it meet the condition.

[0058] The actual physical parameters are the mass and length of the robotic arm link, which are directly estimated online to achieve overparameterization.

[0059] The beneficial effects of this invention are:

[0060] (1) This invention directly estimates the real physical parameters of the robotic arm system (such as the mass and length of the link) online, rather than estimating its complex nonlinear combination. This fundamentally solves the over-parameterization problem that is common in traditional adaptive control, reduces the stringent requirements on excitation conditions, and makes the parameter identification process more reliable and efficient.

[0061] (2) By designing a novel hierarchical structure, this invention combines the finite-time parameter adaptive law with the finite-time tracking control law, ensuring that not only the parameter identification error converges to zero within a finite time, but also the joint space trajectory tracking error converges to zero within a finite time. This feature significantly improves the transient response performance of the robotic arm control system, while existing technologies can usually only guarantee asymptotic convergence (convergence time is infinite).

[0062] (3) Since the error can converge quickly and in a limited time, the method of the present invention effectively shortens the transition time of the robotic arm system in response to the command, enabling the robotic arm to complete the tracking task of the predetermined trajectory more quickly and accurately, thereby improving the dynamic performance and working efficiency of the robotic arm control system as a whole. Attached Figure Description

[0063] Figure 1 This is a schematic diagram of the overall process of the present invention.

[0064] Figure 2 This is a curve showing the change in spatial trajectory tracking error according to an embodiment of the present invention.

[0065] Figure 3 Unknown physical parameters in one embodiment of the present invention and The curve showing how the estimated value changes over time.

[0066] Figure 4 Unknown physical parameters in one embodiment of the present invention and The curve showing how the estimated value changes over time. Detailed Implementation

[0067] The specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings. It should be understood that the specific embodiments described herein are for illustration and explanation only and are not intended to limit the present invention.

[0068] For ease of description, spatial relative terms such as "above," "on top of," "on the upper surface of," "above," etc., are used herein to describe the spatial positional relationship of a device or feature as shown in the figures to other devices or features. It should be understood that spatial relative terms are intended to encompass different orientations in use or operation beyond the orientation of the device as described in the figures. For example, if the device in the figures were inverted, a device described as "above" or "on top of" other devices or structures would subsequently be positioned as "below" or "under" other devices or structures. Thus, the exemplary term "above" can include both "above" and "below." The device may also be positioned in other different ways (rotated 90 degrees or in other orientations), and the spatial relative descriptions used herein will be interpreted accordingly.

[0069] A hierarchical adaptive finite-time control method for a robotic arm without overparameterization is proposed, and its design flowchart is as follows: Figure 1 As shown. The method specifically includes the following steps:

[0070] Step S1: Establish a 2-DOF robotic arm system model.

[0071] Consider a 2-DOF robotic arm system, whose dynamic model can be expressed as:

[0072] (1)

[0073] in, It is a 2D joint angle vector. and They represent The first and second derivatives, It is a 2D control input torque; yes A generalized inertial matrix of dimension ; express The dimensional Coriolis force and centripetal force matrix; It is the potential energy function. express The Jacobian matrix, in addition. , and The element is defined as follows:

[0074]

[0075]

[0076]

[0077] in, It is the gravitational constant. It is a vector of physical parameters. and They represent the first ( The mass and length of each link, for The ( ) elements, It is a nonlinear mapping The ( ) elements, The definition of is:

[0078]

[0079] in for The ( ) elements, that is , , , , , , , , .

[0080] It should be noted that in this article, a dot on a symbol represents the corresponding first derivative, and two dots on a symbol represent the corresponding second derivative; this is common knowledge.

[0081] Step S2: Construct a scalar linear regression equation with the real physical parameter vector as the unknown parameter.

[0082] To facilitate parameter estimation design, model (1) is rewritten in the following parameterized form:

[0083] (2)

[0084] in,

[0085]

[0086]

[0087] Is with The relevant first matrix, Is with The relevant second matrix, yes The first derivative, Represents a time variable.

[0088] To avoid directly using the inclusion in subsequent designs matrix Construct the following two low-pass filters:

[0089] (3)

[0090] in, It is a filter constant. and It is the output of the filter. yes The first derivative, yes The first derivative of . Through filtering, we obtain:

[0091] (4)

[0092] Based on the above equation, the following dynamic regression extension process is introduced to obtain the first auxiliary regression matrix. Second auxiliary regression matrix :

[0093] (5)

[0094] in, It is a design parameter. yes The first derivative, yes The first derivative of . By solving the above differential equation, we obtain:

[0095] (6)

[0096] Furthermore, multiply both sides of the above equation by the adjoint matrix. And then simultaneously left-multiply the matrix .in, yes The adjoint matrix, The design is as follows:

[0097] (7)

[0098] get:

[0099] (8)

[0100] in, For matrix The determinant, yes The One element, , yes The One element, .

[0101] according to The inverse mapping can be obtained. Its specific definition is as follows:

[0102] (9)

[0103] In order to eliminate The algebraic division operation in the matrix is ​​used to construct the following first diagonal matrix. :

[0104] (10)

[0105] in Represents a diagonal matrix. (The rest of the text appears to be a list or formula and doesn't translate directly.) Left multiplication This generates a new equation, called the intermediate equation:

[0106] (11)

[0107] because It is an unknown parameter vector The function is unmeasurable; construct the second diagonal matrix. Make the matrix and All are measurable variables and composition. Designed as follows:

[0108] (12)

[0109] Multiply both sides of equation (11) by the left side. and order , ,get:

[0110] (13)

[0111] Next, multiply both sides of equation (13) on the left. The adjoint matrix ,get:

[0112] (14)

[0113] in, It is a matrix The determinant, yes The One element, , yes The Each element.

[0114] Formula (14) is the constructed scalar linear regression equation.

[0115] Step S3: Design an adaptive law with finite-time parameters.

[0116] In order to achieve parameter identification, The following interval excitation assumptions need to be satisfied.

[0117] Assumption: Existence , and Make , for The identity matrix.

[0118] It should be noted that in engineering practice, a real-time detection matrix will be used. We determine whether the hypothesis is satisfied by checking if the smallest eigenvalue is greater than zero. If not, we need to artificially add some excitation-rich signals to the system reference signal to make it satisfy the hypothesis.

[0119] According to equation (14), the finite-time parameter adaptive law is designed as follows:

[0120] (15)

[0121] in, These are design parameters. Represents absolute value. Represents a symbolic function. for The estimated value, .

[0122] In this embodiment, parameter identification error This refers to the online estimation value of the robotic arm. Its true physical parameter vector The difference between them is usually denoted as , .

[0123] Definition and Related functions :

[0124] (16)

[0125] in, for The One element, , According to equation (15), we can obtain:

[0126] (17)

[0127] according to By definition, we get:

[0128] (18)

[0129] in, express 11th power.

[0130] Based on the assumptions, we can obtain ,in Since it is a constant, we have:

[0131] (19)

[0132] in for The determinant of . Then when hour,

[0133] (20)

[0134] in, It is a normal number.

[0135] From the above formula, we can see that and It will converge to zero in a finite time, with the convergence time being: .

[0136] Step S4: Design a finite-time control law.

[0137] To ensure that the joint angle tracking error is within the parameter estimation Converging to unknown parameters Subsequently, it can converge to zero in a finite amount of time. Define the spatial trajectory tracking error. and sliding mode variables ,in , , and It is a positive odd integer and satisfies , This is the expected trajectory. Let... Then there is , ,in for The elements ( ), This indicates that the first and second diagonal elements are and A diagonal matrix.

[0138] The control law is designed as follows:

[0139] (twenty one)

[0140] in, , These are design parameters. for The Frobenius norm, yes The matrix is ​​defined as follows: , , , , , , , ,in, and They represent The first and second elements, and They represent The first and second elements.

[0141] exist After that, there were At this time, we have:

[0142] (twenty two)

[0143] definition :

[0144] (twenty three)

[0145] Among them, matrix It is positive definite and has properties , If the integer is a positive positive integer, then:

[0146] (twenty four)

[0147] in, Represents the largest eigenvalue of a matrix; matrix Since it is an antisymmetric matrix, we have .

[0148] From the above formula, it can be seen that, after, and It will converge to zero in a finite time. Furthermore, from As can be seen from the definition, It will also converge to zero in a finite amount of time.

[0149] It should be noted that in the above design, when or hour, Potential singularity issues may arise. These will be avoided in subsequent implementations by: when or At that time, among them and To use smaller positive numbers set based on experience, a linear sliding mode variable is used. Replacing the sliding mode variable with the fractional power term as mentioned above, the form of the control law remains unchanged.

[0150] Step S5: Verify the performance of the control law and the adaptive law.

[0151] To verify the performance of the designed velocity tracking guidance law, it needs to be applied to a 2-DOF robotic arm system using common computer numerical calculation and simulation software such as Matlab / Simulink. If the performance of the control law and adaptive law meets the requirements, the design is complete; otherwise, the design parameters of the control law and adaptive law are adjusted based on experience or simulation results, and the performance is verified by resimulating.

[0152] In the simulation, the physical parameters related to the robotic arm system are assumed to be: , , , , ; Set state and The initial value is radian, Parameter estimation The initial value is set to The desired trajectory is set as follows: The design parameters are selected as follows: , , , , , , .

[0153] Simulation results are as follows Figures 2 to 4 As shown. Figure 2 The figure shows the curves of the spatial trajectory tracking error of the two joints of the robotic arm changing over time. , ,in, For the spatial trajectory tracking error of the first joint, This is the actual trajectory of the first joint. for The first element, For the spatial trajectory tracking error of the second joint, This is the actual trajectory of the second joint. for The first element. Figure 3 Unknown physical parameters were displayed. (Right now )and (Right now The curve showing how the estimated value of ) changes over time. Figure 4 Unknown physical parameters were displayed. (Right now )and (Right now The curve showing the estimated value of the parameter as a function of time is shown. The simulation results clearly demonstrate that, within a finite time interval, both the spatial trajectory tracking error and the parameter identification error of the system converge accurately to zero.

[0154] Finally, it is necessary to state that the above embodiments are only used to further illustrate the technical solution of the present invention in detail, and should not be construed as limiting the scope of protection of the present invention. Any non-essential improvements and adjustments made by those skilled in the art based on the above content of the present invention shall fall within the scope of protection of the present invention.

Claims

1. A hierarchical adaptive finite-time control method for a robotic arm without overparameterization, characterized in that, Includes the following steps: Step S1: Establish the dynamic model of the 2-DOF robotic arm system; Step S2: Based on the dynamic model, construct a scalar linear regression equation with the real physical parameter vector as the unknown parameter; The construction method includes: designing a low-pass filter, obtaining an auxiliary regression matrix through dynamic regression expansion, and then eliminating algebraic operations and obtaining a scalar linear regression equation that is completely represented by measurable variables and real physical parameters through at least two adjoint matrix transformations and the constructed screening matrix. Step S3: Based on the scalar linear regression equation, under the condition of satisfying the interval excitation, design a finite-time parameter adaptive law so that the parameter identification error converges to zero in a finite time. Step S4: Define the tracking error and sliding mode variable. Using the parameter estimates obtained by the parameter adaptive law, design a finite-time control law so that after the parameter identification error converges, the joint space trajectory tracking error converges to zero within a finite time.

2. The control method according to claim 1, characterized in that: The dynamic model described in step S1 is expressed as follows: ； in, It is a 2D joint angle vector. and They represent The first and second derivatives, It is a 2D control input torque; yes A generalized inertial matrix of dimension ; express The dimensional Coriolis force and centripetal force matrix; It is the potential energy function. express Jacobian matrix; ； ； ； ； ； ， ， ， ； and These represent the mass and length of the first link of the robotic arm, respectively. and These represent the mass and length of the second link of the robotic arm, respectively. It is the gravitational constant. for The first element, for The second element, yes The first derivative, yes The first derivative, It is a nonlinear mapping The One element, .

3. The control method according to claim 2, characterized in that: In step S2, the dynamic model is written in the following parameterized form: ; in, ； ； Is with The relevant first matrix, Is with The relevant second matrix, yes The first derivative, Represents a time variable.

4. The control method according to claim 3, characterized in that: The low-pass filter design in step S2 is as follows: ； in, It is a filter constant. and It is the output of the filter. yes The first derivative, yes The first derivative.

5. The control method according to claim 4, characterized in that: The dynamic regression extension process described in step S2 obtains the first auxiliary regression matrix by solving the following differential equation. Second auxiliary regression matrix : ； in, These are design parameters. , yes The first derivative, yes The first derivative of , where T denotes the transpose of the matrix; The solution yields: .

6. The control method according to claim 5, characterized in that: In step S2, the method for obtaining the scalar linear regression equation with the real physical parameter vector as the unknown parameter through two adjoint matrix transformations and the constructed screening matrix is ​​as follows: First of all, First, multiply both sides by the adjoint matrix simultaneously. Then multiply both sides by the selection matrix simultaneously. Its definition is as follows: ； get: ； in, For matrix The determinant, yes The One element, , ; yes The One element, , ; Then, according to Obtain the inverse mapping Construct the first diagonal matrix ,Will Left multiplication Generate intermediate equations: ； in, ； ； Next, construct the second diagonal matrix. Make the matrix and All are measurable variables and composition: ； in, Represents a diagonal matrix; Multiply both sides of the intermediate equation by the left side simultaneously. Then multiply by the left side at the same time The adjoint matrix The scalar linear regression equation is obtained as follows: ； in, , It is a matrix The determinant, , yes The One element, yes The One element, , , .

7. The control method according to claim 6, characterized in that: The finite-time parameter adaptive law designed in step S3 is as follows: ； in, Represents a symbolic function. for The estimated value, for The first derivative, These are design parameters.

8. The control method according to claim 7, characterized in that: In step S4: Define the spatial trajectory tracking error. and sliding mode variables : ； ； in, For the desired trajectory; for The first derivative, As a parameter, and , , and It is a positive odd integer and satisfies ; The finite-time control law is as follows: ； in, , These are design parameters. for The Frobenius norm, yes The matrix is ​​defined as follows: , , , , , , , ,in, and They represent The first and second elements, and They represent The first and second elements, , for The first derivative, for The first derivative, for The estimated value.

9. The control method according to claim 8, characterized in that: The interval excitation condition mentioned in step S3 is detected in real time. The system determines whether the minimum eigenvalue is greater than zero. If the condition is not met, an excitation signal is added to the system reference signal to make it meet the condition.

10. The control method according to any one of claims 1 to 9, characterized in that: The actual physical parameters are the mass and length of the robotic arm link, which are directly estimated online to achieve overparameterization.

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