CN108656114B - Control method of flexible mechanical arm - Google Patents

Abstract

The invention discloses a control method of a flexible manipulator, which comprises the following steps: designing a slow controller by using an adaptive dynamic programming algorithm according to given position information and feedback position information in a slow time scale; Obtain the estimated value of vibration on the slow time scale; reconstruct the fast variable based on the estimated value of the vibration on the slow time scale and feedback vibration information on the fast time scale, and design the fast controller according to the fast variable using an adaptive dynamic programming algorithm ; Combine the slow controller and the fast controller to control the position and vibration of the flexible manipulator. According to the control method of the present invention, the optimal control of the position and vibration of the flexible manipulator can be realized without using a system model.

Description

Control method of flexible manipulator

technical field

The invention relates to the technical field of flexible mechanical arm control, in particular to a control method of a flexible mechanical arm.

Background technique

Flexible manipulators have been widely used in aviation, construction and other fields due to their advantages of fast movement speed, high payload-to-robot weight ratio, low manufacturing consumption, and large working space. Considering its special object structure characteristics, the motion of the flexible manipulator includes macroscopic rigid body rotation and microscopic flexible vibration, and the two are highly coupled. Moreover, the flexible manipulator has the characteristics of nonlinearity, infinite order, and parameter uncertainty. Therefore, how to improve the positioning accuracy while avoiding the vibration caused by flexibility is a challenging problem.

Based on the dynamic model of the flexible manipulator, the existing research results can be divided into two categories. One is to directly design the controller based on the rigid-flexible coupling model of the flexible manipulator. The advantage of this method is to fully consider and utilize the dynamic characteristics of the flexible manipulator. , the controller can be designed using traditional PID control, variable structure control, robust control, neural network control, fuzzy control, adaptive control and other methods. On the other hand, considering the dual time scale characteristics of flexible manipulators, the singular perturbation method is introduced into the modeling and control of complex flexible manipulator systems, and controllers are designed at different time scales. From the existing research results, it can be seen that The controller designed by this method has a simple design process and good controller performance.

Although many achievements have been made in the control of flexible manipulators, most of the control strategies are based on dynamic models. However, flexible manipulator systems have uncertainties. Therefore, using the input and system state to study the control of flexible manipulators is a hot issue. Through the retrieval of the existing literature on the control method of flexible manipulators, it is found that the model-free composite controller of flexible manipulators has been researched by using fuzzy controllers. However, the fuzzy controller needs to adjust multiple parameters at the same time, so it is difficult to achieve the optimal control performance. While the existing optimal controller using linear quadratic design has good control performance, it needs precise system parameters. Therefore, it is of great practical significance to study the model-free optimal control of flexible manipulators.

SUMMARY OF THE INVENTION

The present invention aims to solve one of the technical problems in the above technologies at least to a certain extent. Therefore, the purpose of the present invention is to propose a control method of a flexible manipulator, which can realize optimal control of the position and vibration of the flexible manipulator without using a system model.

In order to achieve the above purpose, the present invention proposes a control method for a flexible manipulator, including: designing a slow control by using an adaptive dynamic programming (ADP, Adaptive Dynamic Programming) algorithm according to a given position information and feedback position information under a slow time scale The vibration estimation is performed on the slow time scale to obtain the estimated value of the vibration quantity on the slow time scale; the fast variable is reconstructed according to the estimated value of the vibration quantity on the slow time scale and the feedback vibration information on the fast time scale, and according to For the fast variable, an adaptive dynamic programming algorithm is used to design a fast controller; the slow controller and the fast controller are combined to control the position and vibration of the flexible robotic arm.

According to the control method of the flexible manipulator according to the embodiment of the present invention, the slow controller is designed by using the adaptive dynamic programming algorithm according to the given position information and the feedback position information in the slow time scale, and vibration estimation is performed in the slow time scale to obtain Obtain the estimated value of vibration in the slow time scale, and reconstruct the fast variable according to the estimated value of the vibration in the slow time scale and feedback vibration information in the fast time scale, and use the adaptive dynamic programming algorithm to design the fast control according to the fast variable Then, the slow controller and the fast controller are combined to realize the control of the position and vibration of the flexible manipulator, so that the optimal control of the position and vibration of the flexible manipulator can be realized without using the system model.

In addition, the control method for the flexible robotic arm proposed according to the above embodiments of the present invention may also have the following additional technical features:

According to an embodiment of the present invention, the adaptive dynamic programming algorithm includes:

其中，δ_xx、I_xx、I_xu为学习过程中用来收集状态和输入信息的矩阵，Step 1: Given the initial control law u=-K ₀ x+κ, where K ₀ ∈ R _m×n is the initial gain matrix, κ is the detection noise, calculate δ _xx , I _xx , I _xu , until satisfying

Among them, δ _xx , I _xx , I _xu are the matrices used to collect state and input information in the learning process,

δ _xx =[μ(x(t ₁ ))-μ(x(t ₀ )),μ(x(t ₂ ))-μ(x(t ₁ )),...,μ(x(t _l ))-μ(x(t _l-1 )

0≤t ₀ <t ₁ <...<t _l

表示克罗内克积；in,

represents the Kronecker product;

求解P_k和K_k+1，P_k为迭代过程中求出的Riccati方程正定解，K_k为迭代过程中的反馈增益矩阵，其中，Step 2: Use the formula

Solve P _k and K _k+1 , where P _k is the positive definite solution of the Riccati equation obtained in the iterative process, and K _k is the feedback gain matrix in the iterative process, where,

γ(P _k )=[p ₁₁ p ₁₂ ... 2p _1n p ₂₂ 2p ₂₃ ... 2p _n-1 p _nn ] ^T

vec(M) represents the vectorization of matrix M, that is, vec(M _g×h )=[m ₁₁ m ₂₁ ... m _1h m _2h ... m _gh ] ^T ;

Step 3: Let k←k+1, if ||P _k -P _k-1 ||>α, α>0, go back to step 2, otherwise go to step 4;

Step 4: Set K ^* =K _k to obtain the optimal control law u=-K ^* x.

According to an embodiment of the present invention, an adaptive dynamic programming algorithm is used to design a slow controller according to the given position information and feedback position information under the slow time scale, which specifically includes:

Define the position error e _c =θ-θ _d , where θ _d is the given position information, and θ is the feedback position information;

将描述柔性机械臂的慢子系统模型重写为

其中，

上标s表示慢动态；define new variable

The slow subsystem model describing the flexible manipulator is rewritten as

in,

The superscript s indicates slow motion;

其中，Q^s＝(Q^s)^T≥0,R^s＝(R^s)^T＞0，(A^s,(Q^s)^1/2)可观；Choose an evaluation function

Among them, Q ^s =(Q ^s ) ^T ≥0, R ^s =(R ^s ) ^T >0, (A ^s ,(Q ^s ) ^1/2 ) is considerable;

为初始反馈增益矩阵，得到慢控制器

Using the adaptive dynamic programming algorithm, where x=x ^s , u=u ^s ,

is the initial feedback gain matrix to get the slow controller

According to an embodiment of the present invention, performing vibration estimation on a slow time scale to obtain an estimated vibration value on a slow time scale specifically includes:

其中，v_i代表随机误差，

为测量数据；Considering the existence of random errors, the relational expression z ^s =a+bu ^s between the estimated vibration value z ^s and u ^s in the slow time scale is written as

where v _i represents random error,

for measurement data;

求极值：Using the least squares method, the evaluation function J is defined as

Find the extreme value:

Get approximations for a and b:

Thereby, the estimated value of vibration in slow time scale is obtained

According to an embodiment of the present invention, the fast variable z ^f =k·qz ^s , where k is the minimum value in the stiffness matrix K, and q is the feedback vibration information.

According to an embodiment of the present invention, according to the fast variables, an adaptive dynamic programming algorithm is used to design a fast controller, which specifically includes:

将描述柔性机械臂的快子系统模型重写为

其中，

上标f表示快动态；define new variable

The fast subsystem model describing the flexible manipulator is rewritten as

in,

The superscript f means fast dynamic;

其中，Q^f＝(Q^f)^T≥0，R^f＝(R^f)^T＞0，(A^f,(Q^f)^1/2)可观；Choose an evaluation function

Wherein, Q ^f =(Q ^f ) ^T ≥0, R ^f =(R ^f ) ^T >0, (A ^f ,(Q ^f ) ^1/2 ) is considerable;

为初始反馈增益矩阵，得到快控制器

Using the adaptive dynamic programming algorithm, where x=x ^f , u=u ^f ,

is the initial feedback gain matrix to get the fast controller

According to an embodiment of the present invention, combining the slow controller and the fast controller to control the position and vibration of the flexible robotic arm specifically includes:

^Take _utotal =us+ ^uf as the dynamic model of the flexible manipulator

的输入，实现对所述柔性机械臂的位置和振动的控制，其中，M为正定惯性矩阵，G为非线性项。

The input of , realizes the control of the position and vibration of the flexible manipulator, where M is a positive definite inertia matrix, and G is a nonlinear term.

Description of drawings

1 is a flowchart of a control method of a flexible robotic arm according to an embodiment of the present invention;

2 is a schematic diagram of a control system of a flexible robotic arm according to an embodiment of the present invention;

跟踪K^sd收敛曲线图；FIG. 3 is a diagram according to an embodiment of the present invention.

Track the K ^sd convergence curve;

跟踪K^fd收敛曲线图；FIG. 4 is a diagram according to an embodiment of the present invention.

Track the K ^fd convergence curve;

Fig. 5 is the control method of the embodiment of the present invention and the trajectory tracking comparison diagram under the action of the fuzzy controller;

6 is a comparison diagram of a first-order modal curve under the action of a control method according to an embodiment of the present invention and a fuzzy controller;

FIG. 7 is a comparison diagram of a second-order modal curve under the action of a control method according to an embodiment of the present invention and a fuzzy controller.

Detailed ways

The following describes in detail the embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein the same or similar reference numerals refer to the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary, and are intended to explain the present invention and should not be construed as limiting the present invention.

The following describes the control method of the flexible manipulator according to the embodiment of the present invention with reference to the accompanying drawings.

First, the dynamic model of the flexible manipulator can be established by using the Lagrangian method and the hypothetical mode method:

Among them, M is the positive definite inertia matrix, G is the nonlinear term, K is the stiffness matrix, u is _always the input of the motor driving the flexible manipulator, θ is the position information, and q is the vibration information.

Based on the singular perturbation theory, the control system of the flexible manipulator can be decomposed into two subsystems, fast and slow.

Among them, the slow subsystem:

Fast subsystem:

ε＝1/k，εz＝q，β＝εK，

k为刚度矩阵K中最小值，上标s，f分别表示慢动态和快动态。in,

ε=1/k, εz=q, β=εK,

k is the minimum value in the stiffness matrix K, and the superscripts s and f represent slow dynamic and fast dynamic, respectively.

Combining slow and fast controllers together, we get:

_utotal = u ^s + u ^f (5)

Based on Tikhonov's theorem, the relationship between the state variables of the slow and fast subsystems and the state variables of the original system is as follows:

q=1/k(z ^s +z ^f )+O(ε) (7)

Most of the current flexible manipulator controller design methods are based on completely known or partially unknown dynamic models, while the present invention studies the model-free combined control problem of the flexible manipulator.

In view of the above-mentioned fast and slow subsystems decomposed based on singular perturbation theory, the present invention can design controllers at different time scales, and combine the controllers designed at different time scales to control the flexible manipulator. ,details as follows.

As shown in FIG. 1 , the control method of the flexible robotic arm according to the embodiment of the present invention includes the following steps:

S1, according to the given position information and feedback position information under the slow time scale, use the adaptive dynamic programming algorithm to design the slow controller.

Among them, the adaptive dynamic programming algorithm includes:

Step 1: Given the initial control law:

u=-K ₀ x+κ (8)

Among them, K ₀ ∈R _m×n is the initial gain matrix, κ is the detection noise, and δ _xx , I _xx , and I _xu are calculated until they satisfy:

Among them, δ _xx , I _xx , I _xu are the matrices used to collect state and input information in the learning process,

δ _xx =[μ(x(t ₁ ))-μ(x(t ₀ )),μ(x(t ₂ ))-μ(x(t ₁ )),...,μ(x(t _l ))-μ(x(t _l-1 )

0≤t ₀ <t ₁ <...<t _l

表示克罗内克积；in,

represents the Kronecker product;

Step 2: Use formula (10) to solve P _k and K _k+1 , where P _k is the positive definite solution of the Riccati equation obtained in the iterative process, and K _k is the feedback gain matrix in the iterative process.

in:

γ(P _k )=[p ₁₁ p ₁₂ ... 2p _1n p ₂₂ 2p ₂₃ ... 2p _n-1 p _nn ] ^T

vec(M) represents the vectorization of matrix M, that is, vec(M _g×h )=[m ₁₁ m ₂₁ ... m _1h m _2h ... m _gh ] ^T ;

Step 3: Let k←k+1, if ||P _k -P _k-1 ||>α, α>0, go back to step 2, otherwise go to step 4;

Step 4: Let K ^* =K _k , get the optimal control law:

u=-K ^* x (11)

As shown in Equation (2) above, the slow subsystem represents the rigid body motion of the flexible manipulator system. From the above formula (6), it can be seen that the position state of the flexible manipulator can be directly used in the design of the slow controller.

Specifically, the position error can be defined as:

e _c = θ-θ _d (12)

Among them, θ _d is the given position information, and θ is the feedback position information.

将描述柔性机械臂的慢子系统模型即上式(2)重写为：define new variable

The slow subsystem model describing the flexible manipulator, the above equation (2), is rewritten as:

in,

Choose an evaluation function:

Among them, Q ^s =(Q ^s ) ^T ≥0, R ^s =(R ^s ) ^T >0, and (A ^s , (Q ^s ) ^1/2 ) are considerable.

为初始反馈增益矩阵，得到慢控制器：Using the above adaptive dynamic programming algorithm, where x=x ^s , u=u ^s ,

For the initial feedback gain matrix, the slow controller is obtained:

令u^s＝-K^s ₀x^s+κ^s，计算

再通过

求解P^s _k和K^s _k+1。然后判断当k＞1时，是否有||P^s _k-P^s _k-1||≤α，如果否，则令k←k+1，再求解P^s _k和K^s _k+1；如果是，则

Further, for the position information θ fed back by the flexible manipulator and its first-order derivative

Let us = -K ^s ₀ ^{x s +κ s , calculate}

pass again

Solve for P ^sk and K ^s _{k +1} . Then judge whether there is ||P ^s _k -P ^s _k-1 ||≤α when k>1, if not, let k←k+1, and then solve P ^s _k and K ^s _k+1 ; if Yes, then

As shown in FIG. 2 , the slow controller designed in the embodiment of the present invention is an ADP-based slow controller, which inputs given position information θ _d and feedback position information θ , and outputs the slow parameter u ^s .

S2, perform vibration estimation on a slow time scale to obtain an estimated vibration value on a slow time scale.

According to the above formula (7), the vibration information q includes the vibration quantity z ^s on the slow time scale and the vibration quantity z ^f on the fast time scale. In order to design a fast controller, z ^s needs to be estimated first. It can be seen from the above formula (6) that the approximate structure between z ^s and u ^s is as follows:

z ^s =a+bu ^s (16)

where a and b are the parameters to be estimated. Considering the existence of random errors, the relational expression (16) between the estimated vibration value z ^s and u ^s in the slow time scale is written as:

为测量数据。where v _i represents random error,

for measurement data.

Using the least squares method, the evaluation function J is defined as:

Find the extreme value:

Get approximations for a and b:

Thus, the vibration estimate z ^s at the slow time scale is obtained:

S3, reconstruct the fast variable according to the estimated vibration value and feedback vibration information in the slow time scale in the fast time scale, and design the fast controller according to the fast variable using the adaptive dynamic programming algorithm.

The reconstructed fast variable z ^f = k·qz ^s . Among them, k is the minimum value in the stiffness matrix K, and q is the feedback vibration information shown in Figure 2.

As shown in Equation (4) above, the fast subsystem represents the flexible vibration of the flexible manipulator system.

将描述柔性机械臂的快子系统模型即上式(4)重写为：Specifically, new variables can be defined

The fast subsystem model describing the flexible manipulator, that is, the above equation (4) is rewritten as:

in,

Choose an evaluation function:

Wherein, Q ^f =(Q ^f ) ^T ≥0, R ^f =(R ^f ) ^T >0, (A ^f , (Q ^f ) ^1/2 ) is considerable.

为初始反馈增益矩阵，得到快控制器：Using the above adaptive dynamic programming algorithm, where x=x ^f , u=u ^f ,

is the initial feedback gain matrix to get the fast controller:

令z＝kq，

估计z和

再求得z^f＝z-z^s，

然后令u^f＝-K^f ₀x^f+κ^f，计算

再通过

求解P^f _k和K^f _k+1。然后判断当k＞1时，是否有||P^f _k-P^f _k-1||≤α，如果否，则令k←k+1，再求解P^f _k和K^f _k+1；如果是，则

Further, for the vibration information q fed back by the flexible manipulator and its first-order derivative

Let z=kq,

estimate z and

Then find z ^f = zz ^s ,

Then let u ^f = -K ^f ₀ x ^f +κ ^f , calculate

pass again

Solve for P ^f _k and K ^f _k+1 . Then judge whether there is ||P ^f _k -P ^f _k -1||≤α when k>1, if not, set k←k+1, and then solve P ^f _k and K ^f _k+1 ; if Yes, then

As shown in FIG. 2 , the fast controller designed in the embodiment of the present invention is an ADP-based fast controller, which inputs a fast variable zf and outputs a fast parameter uf.

S4, the slow controller and the fast controller are combined to realize the control of the position and vibration of the flexible manipulator.

的输入，并输出位置信息θ和振动信息q，实现对柔性机械臂的位置和振动的控制。As shown in Figure 2, after the slow parameters and fast parameters are obtained, u _total = u ^s + u ^f can be used as the dynamic model of the flexible manipulator

, and output the position information θ and vibration information q to realize the control of the position and vibration of the flexible manipulator.

The present invention can verify the effectiveness of the position and vibration control in the Matlab environment, and the parameters of the flexible manipulator are shown in Table 1:

Table 1

Physical parameter (unit) parameter value Boom length L(m) 1.2 Joint moment of inertia J_h(kg m²) 2 Load moment of inertia J_p(kg·m²) 0.001 Boom mass m(kg) 0.2 Load mass M(kg) 0.1 Bending stiffness EI(N m²) 60

近似相等，运用上述自适应动态规划算法，设计慢控制器。首先给定初始反馈增益矩阵

评价矩阵Q^s＝diag(1,0.1)，R^s＝I，经过有限次迭代后得到最优反馈增益矩阵

通过直接求解Raccati方程，其中A＝A^s,B＝B^s,Q＝Q^s,R＝R^s，得到最优反馈增益矩阵K^sd＝[1 2.1406]。可以看出

和K^sd近似相等，图3示出

收敛到最优值K^sd过程，可以看出，该慢控制器能够实现慢子系统的最优控制。According to the singular perturbation theory, the flexible manipulator can be decomposed into two subsystems, fast and slow, θ and

are approximately equal, using the adaptive dynamic programming algorithm described above to design a slow controller. First, the initial feedback gain matrix is given

Evaluation matrix Q ^s =diag(1,0.1), R ^s =I, the optimal feedback gain matrix is obtained after finite iterations

By directly solving the Raccati equation, where A=As , B=B ^s , Q=Q ^s , R=R ^s , the optimal feedback gain matrix K ^sd =[1 2.1406 ^] is obtained. As can be seen

and K ^sd are approximately equal, Figure 3 shows

Converging to the optimal value K ^sd process, it can be seen that the slow controller can realize the optimal control of the slow subsystem.

The least squares method is used to obtain the vibration of the flexible manipulator at the slow time scale:

评价矩阵Q^f＝diag(1,0.1,1,0.1)，R^f＝I，经过有限次迭代后得到最优反馈增益矩阵

通过直接求解Raccati方程，其中A＝A^f,B＝B^f,Q＝Q^f,R＝R^f，得到最优反馈增益矩阵K^fd＝[-0.0556-0.6739-0.0674-0.3017]。可以看出

和K^fd近似相等，图4示出

收敛到最优值K^fd过程，可以看出，该快控制器能够实现快子系统的最优控制。According to formula (7) and formula (21), the fast variable z ^f is obtained, and the above-mentioned adaptive dynamic programming algorithm is used to design a fast controller. First, the initial feedback gain matrix is given

Evaluation matrix Q ^f =diag(1,0.1,1,0.1), R ^f =I, the optimal feedback gain matrix is obtained after finite iterations

By directly solving the Raccati equation, where A=A ^f , B=B ^f , Q=Q ^f , R=R ^f , the optimal feedback gain matrix K ^fd =[-0.0556-0.6739-0.0674-0.3017] is obtained. As can be seen

and K ^fd are approximately equal, Figure 4 shows

The process of converging to the optimal value K ^fd shows that the fast controller can realize the optimal control of the fast subsystem.

In order to further verify the effectiveness of the present invention for position and vibration control, the control method of the flexible manipulator of the present invention, that is, the control effect of the combined controller based on ADP and the classical FLC (Fuzzy Logic Control, fuzzy logic control) The control effect of the combined controller is compared. As can be seen from Figure 5, under the action of the control method proposed by the present invention, the flexible robotic arm can reach the designated position faster and more accurately. Figure 6 and Figure 7 can be seen that the present invention has a The vibration of the mechanical arm has a better suppression effect.

To sum up, according to the control method of the flexible manipulator according to the embodiment of the present invention, by using the adaptive dynamic programming algorithm to design the slow controller according to the given position information and feedback position information in the slow time scale, and at the slow time scale Vibration estimation is carried out to obtain the estimated value of vibration in the slow time scale, and the fast variable is reconstructed according to the estimated value of vibration in the slow time scale and the feedback vibration information in the fast time scale, and according to the fast variable, the adaptive dynamic The planning algorithm designs the fast controller, and then combines the slow controller and the fast controller to control the position and vibration of the flexible manipulator, thereby realizing the control of the position and vibration of the flexible manipulator without using a system model. optimal control.

In the description of the present invention, it should be understood that the terms "center", "longitudinal", "lateral", "length," "width", "thickness", "upper", "lower", "front", " Back, Left, Right, Vertical, Horizontal, Top, Bottom, Inner, Outer, Clockwise, Counterclockwise, Axial , "radial", "circumferential" and other indicated orientations or positional relationships are based on the orientations or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying the indicated device or Elements must have a particular orientation, be constructed and operate in a particular orientation and are therefore not to be construed as limitations of the invention.

In addition, the terms "first" and "second" are only used for descriptive purposes, and should not be construed as indicating or implying relative importance or implying the number of indicated technical features. Thus, a feature defined as "first" or "second" may expressly or implicitly include one or more of that feature. In the description of the present invention, "plurality" means two or more, unless otherwise expressly and specifically defined.

In the present invention, unless otherwise expressly specified and limited, the terms "installed", "connected", "connected", "fixed" and other terms should be understood in a broad sense, for example, it may be a fixed connection or a detachable connection , or integrated; it can be a mechanical connection or an electrical connection; it can be a direct connection or an indirect connection through an intermediate medium, and it can be the internal connection of the two elements or the interaction relationship between the two elements. For those of ordinary skill in the art, the specific meanings of the above terms in the present invention can be understood according to specific situations.

In the present invention, unless otherwise expressly specified and limited, a first feature "on" or "under" a second feature may be in direct contact between the first and second features, or the first and second features indirectly through an intermediary touch. Also, the first feature being "above", "over" and "above" the second feature may mean that the first feature is directly above or obliquely above the second feature, or simply means that the first feature is level higher than the second feature. The first feature being "below", "below" and "below" the second feature may mean that the first feature is directly below or obliquely below the second feature, or simply means that the first feature has a lower level than the second feature.

In the description of this specification, description with reference to the terms "one embodiment," "some embodiments," "example," "specific example," or "some examples", etc., mean specific features described in connection with the embodiment or example , structure, material or feature is included in at least one embodiment or example of the present invention. In this specification, schematic representations of the above terms are not necessarily directed to the same embodiment or example. Furthermore, the particular features, structures, materials or characteristics described may be combined in any suitable manner in any one or more embodiments or examples. Furthermore, those skilled in the art may combine and combine the different embodiments or examples described in this specification, as well as the features of the different embodiments or examples, without conflicting each other.

Although the embodiments of the present invention have been shown and described above, it should be understood that the above-mentioned embodiments are exemplary and should not be construed as limiting the present invention. Embodiments are subject to variations, modifications, substitutions and variations.

Claims (2)

1. a control method of a flexible manipulator, is characterized in that, comprises: According to the given position information and feedback position information under the slow time scale, the slow controller is designed by using the adaptive dynamic programming algorithm; Perform vibration estimation on slow time scales to obtain vibration estimates on slow time scales, including: Considering the existence of random errors, the relational expression z ^s =a+bu ^s between the estimated vibration value z ^s on the slow time scale and the slow controller u ^s is written as

where v _i represents random error,

are measurement data, a and b are parameters to be estimated,

The slow controller output obtained for the ith sample; Using the least squares method, the evaluation function J is defined as

Find the extremum for the vibration at the slow time scale obtained by the ith sampling:

Get approximations for a and b:

Thereby, the estimated value of vibration in slow time scale is obtained

Reconstructing fast variables on the fast time scale according to the estimated vibration value and feedback vibration information on the slow time scale, and designing a fast controller according to the fast variables using an adaptive dynamic programming algorithm; Combining the slow controller and the fast controller realizes control of the position and vibration of the flexible robotic arm. 2 . The control method for a flexible manipulator according to claim 1 , wherein the fast variable z ^f =k qz ^s , wherein k is the minimum value in the stiffness matrix K, and q is the feedback vibration. 3 . information.

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