A Fractional-Order Modeling Method for Flexible Manipulators Based on Joint Multi-Innovation Gradient

Abstract

The present invention discloses a fractional-order modeling method for a flexible manipulator based on combined multi-innovation gradient, comprising: constructing a fractional-order dynamic model of the flexible manipulator through a fractional viscous element, wherein the input data and output data of the fractional-order dynamic model are voltage data and rotation angle respectively; converting the fractional-order dynamic model into a multi-innovation fractional-order dynamic model, and constructing a multi-innovation fractional-order combined gradient according to the multi-innovation fractional-order dynamic model; constructing an error model according to the multi-innovation fractional-order dynamic model, and constructing a fractional-order recursive iteration model and a model parameter recursive iteration model according to the error model and the multi-innovation fractional-order combined gradient; performing joint interactive iterative solution on the model parameter recursive iteration model and the fractional-order recursive iteration model to obtain the identification results of the model parameters and the fractional order.

Description

Technical Field

[0001] The present invention belongs to the technical field of system modeling and parameter identification, and mainly relates to a fractional-order modeling method for a flexible manipulator based on combined multi-innovation gradient. Background Art

[0002] Fractional calculus has a history of more than 300 years. Compared with integer-order calculus, the fractional-order model established by using fractional calculus can more accurately characterize the dynamic change process of the actual physical system, such as viscoelastic material relaxation and creep, temperature diffusion, virus propagation, battery energy estimation, and optoelectronic system trajectory tracking.

[0003] With the continuous breakthrough of robot application technology, manipulators are widely used in various automated production lines such as machinery, chemical industry, electronics, and logistics. Flexible manipulators have the characteristics of high flexibility, strong adaptability to complex environments, and safe human-machine interaction, and have received extensive attention and research from scholars and institutions at home and abroad. Flexible manipulators can be divided into one-degree-of-freedom, two-degree-of-freedom, three-degree-of-freedom, and multi-degree-of-freedom flexible manipulators according to the number of degrees of freedom, and can be divided into link-flexible and joint-flexible manipulators according to the position of the flexible material. Compared with the large volume and load of rigid manipulators, flexible manipulators are smaller in volume and use lighter and cheaper manufacturing materials, enabling flexible manipulators to have higher operating speeds and larger working spaces, and flexible manipulators can have a larger working volume in the same-sized area.

[0004] The rotary flexible manipulator studied in the present invention is a single-input single-output one-degree-of-freedom flexible system. The existing system model is established on the basis of ignoring flexible vibration. However, such a model cannot accurately describe the change process of the rotation angle of the flexible manipulator and the system dynamic characteristics, resulting in a deviation in the rotation angle of the manipulator and an inability to accurately locate and obtain the target. Summary of the Invention

[0005] To solve the above technical problems, the present invention proposes a fractional-order modeling method for a flexible manipulator based on combined multi-innovation gradient to solve the problems existing in the above prior art. A fractional-order dynamic model of the flexible manipulator is constructed, and the fractional-order multi-innovation combined gradient descent algorithm is used to synchronously identify the unknown fractional order and model parameters, so as to solve the problems that the integer-order model of the existing rotary flexible manipulator cannot accurately characterize the change process of the rotation angle and the system dynamic characteristics, and the fractional-order parameter estimation method cannot accurately synchronously identify the model parameters and fractional order of the flexible manipulator. The constructed flexible manipulator model can be used as a control model for controlling the manipulator, effectively controlling the manipulator, laying a foundation for the design of advanced controllers, and providing a prerequisite guarantee for accurate positioning and accurate acquisition of the target.

[0006] To achieve the above object, the present invention provides a fractional - order modeling method for a flexible manipulator based on combined multi - innovation gradient, including:

[0007] Construct a fractional - order dynamic model of the flexible manipulator through a fractional viscous element, where the input data and output data of the fractional - order dynamic model of the flexible manipulator are the driving voltage and the rotation angle respectively;

[0008] Convert the fractional - order dynamic model into a multi - innovation fractional - order dynamic model, and construct a multi - innovation fractional - order combined gradient according to the multi - innovation fractional - order dynamic model;

[0009] Construct an error model according to the multi - innovation fractional - order dynamic model, and construct a fractional - order recurrence iteration model and a model parameter recurrence iteration model according to the error model and the multi - innovation fractional - order combined gradient;

[0010] Perform joint interactive iterative solution on the model parameter recurrence iteration model and the fractional - order recurrence iteration model to obtain the model parameter and fractional - order identification results; substitute the model parameter and fractional - order identification results into the fractional - order dynamic model, and control the flexible manipulator through the substituted fractional - order dynamic model.

[0011] Optionally, the fractional - order dynamic model is:

[0012]

[0013] where \(k\) represents the current time, \(y(k)\) is the rotation angle of the flexible manipulator, \(u(k)\) is the input voltage of the flexible manipulator, and are respectively two fractional - order viscous elements, representing the fractional - order rotation angles at times \(k - 1\) and \(k - 2\) in the past, is the fractional - order, and \(a_1,a_2,b_0\) are model parameters.

[0014] Optionally, the generation process of the multi - innovation fractional - order dynamic model includes:

[0015]

[0016] where \(Y(p,k)\) is the output data multi - innovation matrix, \(Y(p,k)=[y(k),y(k - 1),\cdots,y(k - p + 1)]^T\) T , where \(p\) is the length of the multi - innovation, and the superscript \(T\) represents matrix transpose,

[0017] \(=\) is the multi - innovation matrix jointly converted by the fractional - order viscous element and the input data,

[0018]

[0019] θ is the model parameter matrix, θ = [a1, a2, b0].

[0020] Optionally, the multi-innovation fractional joint gradient is:

[0021]

[0022] where respectively represent the joint gradient of the model parameters and the fractional order, and are respectively the estimated values of the fractional order and the model parameter matrix θ, and N is the final moment of a single joint iteration.

[0023] Optionally, the error model is:

[0024]

[0025] where e(p, k) is the error value, θ(k - 1) represents the model parameter matrix at the (k - 1)-th moment, represents the estimated value of the fractional order at the k-th moment.

[0026] Optionally, the fractional order recursive iteration model is:

[0027]

[0028] where represents the estimated value of the model parameter matrix at the k-th moment, ▽ α is the fractional gradient, α is the order of the fractional gradient, μ is the step size factor of the fractional multi-innovation gradient, Γ(·) is the gamma function, are respectively the estimated values of a1, a2, b0, ε represents a constant term, and diag represents a diagonal matrix.

[0029] Optionally, the model parameter recursive iteration model is:

[0030]

[0031] where μ1 represents the gradient step size factor, represents the partial derivative matrix, represents the fractional order at the (k - 1)-th moment.

[0032] Optionally, the process of joint interactive iteration solution includes:

[0033] Obtain the input data and output data, and perform iterative joint interaction on the model parameter recursive iteration model and the fractional order recursive iteration model according to the input data and output data until the number of interactions is reached, and obtain the identification results of the model parameters and the fractional order;

[0034] In the joint interaction, the model parameter recursive iteration model is iteratively solved to obtain the updated model parameters. The updated model parameters are substituted into the fractional order recursive iteration model for iterative solution to obtain the updated fractional order, completing one joint interaction. Based on the model parameter recursive iteration model with the updated fractional order substituted, the next joint interaction is carried out.

[0035] Optionally, after obtaining the model parameter and fractional order identification results, it further includes:

[0036] Substitute the model parameter and fractional order identification results into the fractional order dynamics model and obtain the current driving voltage. Calculate the current driving voltage through the substituted fractional order dynamics model to generate the current rotation angle of the flexible robotic arm.

[0037] Compared with the prior art, the present invention has the following advantages and technical effects:

[0038] 1. The present invention combines two fractional order gradients, uses the two gradients to interactively identify the model parameters and fractional order, and can synchronously estimate the unknown fractional order and model parameters of the flexible robotic arm.

[0039] 2. The combined fractional order gradient of the present invention combines the multi-innovation theory, uses the data at the current moment and past moments to synchronously estimate the model parameters of the flexible robotic arm, and the constructed model is more accurate.

[0040] 3. The flexible robotic arm model established by the present invention has high accuracy, can accurately describe the dynamic characteristics of the robotic arm, and this model can provide a theoretical basis for the simulation and application of advanced control technologies. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] The drawings constituting a part of this application are used to provide a further understanding of this application. The schematic embodiments of this application and their descriptions are used to explain this application and do not constitute an improper limitation to this application. In the drawings:

[0042] Figure 1 is the flowchart of the fractional order modeling method for a flexible robotic arm based on joint gradients in an embodiment of the present invention;

[0043] Figure 2 is the process of joint interaction identification of unknown parameters and fractional order of a rotating flexible robotic arm in an embodiment of the present invention;

[0044] Figure 3 is the comparison diagram between the fractional order model and the integer order model established in an embodiment of the present invention;

[0045] Figure 4 is the ITAE diagram of the actual system and the fractional order model with different joint times in an embodiment of the present invention;

[0046] Figure 5 It is the ITAE graph of the actual system with different multi-information lengths and the fractional-order model in the embodiments of the present invention. Detailed implementation manners

[0047] It should be noted that, without conflict, the embodiments in the present application and the features in the embodiments can be combined with each other. The present application will be described in detail below with reference to the drawings and in combination with the embodiments.

[0048] It should be noted that the steps shown in the flowchart of the drawings can be executed in a computer system such as a set of computer-executable instructions, and although the logical order is shown in the flowchart, in some cases, the steps shown or described can be executed in an order different from that here.

[0049] The present invention relates to a fractional-order modeling method for a flexible robotic arm based on joint multi-information gradient, including the following steps: First, a fractional-order dynamic model of the input voltage and the output rotation angle of the flexible robotic arm is constructed by using a fractional-order viscous element. Second, the real data of the input and output are collected to construct a joint gradient criterion function of the unknown fractional order and the model parameters. Then, a fractional-order model identification method for the rotating flexible robotic arm based on the fractional-order multi-information joint gradient is designed. Through the joint interactive iteration of the two gradients, the unknown parameters and the fractional order of the model can be accurately and synchronously estimated. The flexible robotic arm model established by the present invention has high accuracy and can accurately describe the dynamic characteristics of the robotic arm. This model can provide a theoretical basis for the simulation and application of advanced control technologies.

[0050] The present invention provides a fractional-order modeling method for a flexible robotic arm based on joint multi-information gradient, uses a fractional-order viscoelastic element to establish a fractional-order model of the flexible robotic arm, and combines the fractional-order multi-information joint gradient descent algorithm to estimate the unknown parameters and the fractional order of the model, so as to accurately describe the dynamic characteristics of the rotating flexible robotic arm.

[0051] In order to achieve the above object, the technical solution of the present invention is:

[0052] Step 1: Construct a fractional-order dynamic model of the input voltage data and the output rotation angle of the flexible robotic arm by using a fractional-order viscous element.

[0053] Step 2: Collect the real data of the input voltage data and the output rotation angle, and construct a joint gradient of the unknown fractional order and the model parameters.

[0054] Step 3: Design a fractional-order model identification method for the rotating flexible robotic arm based on the fractional-order multi-information joint gradient.

[0055] Step 4: Use the fractional-order multi-innovation gradient descent method to estimate the model parameters. Then, substitute the parameters into the fractional-order multi-innovation gradient descent method for the model order to identify the fractional order. Next, bring the fractional order back into the gradient descent algorithm for the model parameters to continue identifying the model parameters, and repeat the joint interaction in turn.

[0056] Step 5: Set the number of joint interactions. If the number of joint interactions is less than the set value, return to Step 4 to continue the interaction. When the number of joint interactions is equal to the set value, output the identified fractional-order model parameters and order of the rotary flexible manipulator.

[0057] Step 6: Apply the constructed fractional-order model of the flexible manipulator to design a high-precision rotation control system, which can accurately control the rotation angle of the manipulator and achieve accurate positioning and acquisition of the target.

[0058] As Figure 1 shown, the above technical solutions of the present invention are described in detail:

[0059] Step 1: Use fractional-order viscous elements to construct a fractional-order dynamic model of the input voltage and output rotation angle of the flexible manipulator.

[0060] The fractional-order dynamic model is established based on fractional-order viscous elements, and the fractional-order viscous elements extend the integer-order physical quantity to a fractional order. For a one-degree-of-freedom rotary flexible manipulator, two fractional-order elements can be used to describe the dynamic change process of the rotation angle, that is

[0061]

[0062] where \(k\) represents the current moment, \(y(k)\) is the output rotation angle of the rotary flexible manipulator, \(u(k)\) is the input voltage data of the rotary flexible manipulator, and are the two fractional-order viscous elements, representing the fractional-order rotation angles at the past \(k - 1\) moment and \(k - 2\) moment respectively, is the fractional order, and \(a_1, a_2, b_0\) are different model parameters.

[0063] Step 2: Collect the input data and output data, and construct a joint gradient criterion function for the unknown fractional order and model parameters.

[0064] Use Matlab / Simulink to build a data acquisition system for the rotary flexible manipulator. Obtain the input data and output data through the Quarc acquisition module. The input data is the driving voltage. In the present invention, the input driving voltage is 1V, the output is the rotation angle, the acquisition period is 5s, the sampling interval is 0.004s, and the data length is 1250.

[0065] Multi-innovation combines the output data at the current moment and the output data at past moments to form a multi-innovation matrix, and uses the multi-innovation matrix to estimate unknown parameters. Compared with traditional iterative methods, multi-innovation utilizes a larger amount of data, so the identification result is more accurate. According to the multi-innovation theory, the rotation angle of the rotary flexible manipulator, that is, the output data, is converted into the form of a multi-innovation matrix, and the multi-innovation matrix Y(p,k) of the output data:

[0066]

[0067] where p is the length of the multi-innovation.

[0068] Similarly, the fractional-order viscous element and the input voltage are jointly converted into the form of a multi-innovation matrix, and the multi-innovation matrix jointly converted by the fractional-order viscous element and the input data

[0069]

[0070] The unknown parameters are rewritten in the form of a matrix, and the matrix θ of the unknown model parameters:

[0071] θ = [a1, a2, b0] (4)

[0072] Then the fractional-order dynamic model of the rotary flexible manipulator can be rewritten as

[0073]

[0074] In formula (5), there are unknown fractional orders and the matrix θ of unknown model parameters. Therefore, in order to identify accurate model parameters, a multi-innovation fractional-order joint gradient is established:

[0075]

[0076] where, respectively represent the joint gradients of the model parameters and the fractional orders, and are the estimated values of the fractional order and the model parameter θ respectively, and N is the final moment of a single joint iteration.

[0077] Step 3: Design a model identification method for the rotary flexible manipulator based on fractional-order multi-innovation joint gradient descent.

[0078] Design a fractional-order multi-innovation gradient descent algorithm for the unknown parameter θ, define the dynamic change process of the rotation angle of the rotary flexible manipulator and the error of the identification model, and the error model is:

[0079]

[0080] where \(e(p,k)\) is the error value, and \(\theta(k - 1)\) represents the model parameter matrix at time \(k - 1\). represents the estimated value of the fractional order at time \(N\).

[0081] Taking the partial derivative of \(J_1(\theta)\) in Equation (6) and then performing recursive iteration in combination with Equation (7), the recursive iteration model of the fractional order can be obtained:

[0082]

[0083] where \(\nabla\) α is the fractional gradient, \(\alpha\) is the order of the fractional gradient, \(\mu\) is the step size factor of the fractional multi - innovation gradient, \(\Gamma(\cdot)\) is the gamma function. are the estimated values of \(a_1\) and \(a_2\) respectively. The \(k - 1\) and \(k - 2\) in the parentheses represent the corresponding times \(k - 1\) and \(k - 2\), \(\varepsilon=0.001\) represents the constant term, and diag represents the diagonal matrix.

[0084] Taking the partial derivative of in the formula and then performing recursive iteration in combination with Equation (7), the recursive iteration model of the model parameters can be obtained:

[0085]

[0086] where \(\mu_1\) represents the gradient step size factor. represents the partial derivative matrix. represents the fractional order at time \(k - 1\).

[0087] Step 4: When performing joint gradient identification, first initialize Set the multi - innovation length \(p = 5\) and the fractional gradient order \(\alpha=1.5\).

[0088] Secondly, use Equation (8) to identify the unknown parameters of the rotary flexible manipulator model. Through step - by - step iteration of 1250 observation data, the identification parameters at the final time after the first joint iteration can be obtained.

[0089] Then, substitute into Equation (9), and use Equation (9) to identify the unknown fractional order of the rotary flexible manipulator model. Similarly, after 1250 iterations, the updated

[0090] Finally, substitute the updated into Equation (8) for joint interaction to achieve the joint iteration of the fractional order and the model parameters. The joint interaction identification process of the unknown parameters and the fractional order of the rotary flexible manipulator is as shown in Figure 2 shown.

[0091] Step 5: Iterate formulas (8) and (9) 1250 times each, regarded as one joint interaction. Set the number of joint interactions to 3. If the number of interactions is less than 3, return to Step 4 for joint interaction. When the third interaction is completed, output the identification result. and Take the ITAE (Integral of Time multiplied by the Absolute value of the Error) as the performance index to evaluate the model accuracy, and record the results of different joint interactions in Table 1. Table 1 shows the estimated results of the model parameters and fractional orders under different numbers of joint interactions.

[0092] Table 1

[0093]

[0094] Substitute the results of the third joint identification, that is, the final identified results of the model parameters and fractional orders, into formula (1), and the fractional-order model of the rotating flexible manipulator can be obtained.

[0095] y(k) = 0.4314Δ 1.19 y(k - 1)+0.7621Δ 1.19 y(k - 2)-0.2547u(k) (10)

[0096] Step 6: Collect the input voltage u(t) data of the rotating flexible manipulator, input the input voltage data into the fractional-order model of the rotating flexible manipulator shown in formula (10), output the rotation angle of the flexible manipulator, and the dynamic change process of the rotation angle of the manipulator can be obtained. Applying this model, a high-precision rotation control system can be designed. By comparing the rotation angle with the set angle, the error of the rotation angle can be obtained. Combining the error feedback principle and the dynamic characteristics of the rotation angle, the input voltage of the manipulator can be controlled to accurately regulate the rotation angle of the manipulator and achieve the target grasping or precision machining.

[0097] The comparison results between the fractional-order model established in the embodiment of the present invention and the dynamic change process of the rotation angle of the flexible manipulator are as Figure 3 shown. It can be seen from Figure 3 that compared with the traditional integer-order model, the fractional-order model established in the present invention can more accurately characterize the dynamic change process of the rotation angle of the flexible manipulator. As the number of joint interactions increases, the ITAE gradually decreases, and the fitting effect between the model and the real data is better, as Figure 4 shown. Therefore, the effectiveness of the present invention is verified.

[0098] In addition, the influence of different multi-innovation lengths on the identification results is verified. It can be seen from Figure 5 that as the multi-innovation length increases, the ITAE gradually decreases and the identification accuracy gradually increases. Therefore, the feasibility of the method proposed in the present invention in the system identification of the rotating flexible manipulator is further verified.

[0099] The above are only the preferred specific embodiments of the present application, but the protection scope of the present application is not limited thereto. Any changes or substitutions that can be easily thought of by those skilled in the art within the technical scope disclosed in the present application should be covered by the protection scope of the present application. Therefore, the protection scope of the present application should be subject to the protection scope of the claims.

Claims

1. A fractional-order modeling method for a flexible manipulator based on combined multi-innovation gradient, characterized in that Including: Construct a fractional - order dynamic model of a flexible robotic arm through a fractional - viscous element, where the input data and output data of the fractional - order dynamic model of the flexible robotic arm are the driving voltage and the rotation angle respectively; Convert the fractional - order dynamic model into a multi - innovation fractional - order dynamic model, and construct a multi - innovation fractional - order joint gradient according to the multi - innovation fractional - order dynamic model; Construct an error model according to the multi - innovation fractional - order dynamic model, and construct a fractional - order recurrence iteration model and a model - parameter recurrence iteration model according to the error model and the multi - innovation fractional - order joint gradient; Perform joint interactive iterative solution on the model - parameter recurrence iteration model and the fractional - order recurrence iteration model to obtain the model - parameter and fractional - order identification results; Substitute the model - parameter and fractional - order identification results into the fractional - order dynamic model, and control the flexible robotic arm through the substituted fractional - order dynamic model.

2. The method according to claim 1, wherein: The fractional - order dynamic model is: where \(k\) represents the current moment, \(y(k)\) is the rotation angle of the flexible manipulator, and \(u(k)\) is the input voltage of the flexible manipulator. and are two fractional-order viscous elements respectively, representing the fractional-order rotation angles at the past \(k - 1\) and \(k - 2\) moments. is the fractional-order, and \(a_1\), \(a_2\), \(b_0\) are model parameters.

3. The method according to claim 2, wherein: The generation process of the multi - innovation fractional - order dynamic model includes: where \(Y(p,k)\) is the output data multi-innovation matrix, and \(Y(p,k)=[y(k),y(k - 1),\cdots,y(k - p+1)]\) T , where \(p\) is the length of the multi-innovation, and the superscript \(T\) represents matrix transpose. = the multi-innovation matrix jointly transformed by the fractional-order viscous element and the input data, θ is a model - parameter matrix, θ = [a1, a2, b0].

4. The method according to claim 3, wherein: The multi - innovation fractional - order joint gradient is: Among them, J1(θ), respectively represent the joint gradients of the model parameters and the fractional order, and are the estimated values of the fractional order and the model parameter matrix θ respectively, and N is the final moment of a single joint iteration.

5. The method according to claim 4, wherein: The error model is: where e(p,k) is the error value, and θ(k - 1) represents the model parameter matrix at time k - 1, which represents the estimated value of the fractional order at time k.

6. The method according to claim 5, wherein: The fractional - order recurrence iteration model is: Among them, represents the estimated value of the model parameter matrix at time k, is the fractional gradient, α is the order of the fractional gradient, μ is the step size factor of the fractional multi-innovation gradient, Γ(·) is the gamma function, are the estimated values of a1, a2, and b0 respectively, ε represents the constant term, and diag represents the diagonal matrix.

7. The method according to claim 6, wherein: The model - parameter recurrence iteration model is: where μ1 represents the gradient step factor, represents the partial derivative matrix, represents the fractional order at the (k - 1)th moment.

8. The method according to claim 1, wherein: The process of joint interactive iterative solution includes: Obtain the input data and output data, and perform iterative joint interaction on the model - parameter recurrence iteration model and the fractional - order recurrence iteration model according to the input data and output data until the number of interactions is reached to obtain the model - parameter and fractional - order identification results; In the joint interaction, perform iterative solution on the model - parameter recurrence iteration model to obtain updated model parameters, substitute the updated model parameters into the fractional - order recurrence iteration model for iterative solution to obtain updated fractional - orders, complete one joint interaction, and perform the next joint interaction based on the model - parameter recurrence iteration model with the updated fractional - orders substituted.

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