CN112949098B - Iterative correction method and iterative correction system for kinematic error mapping matrix - Google Patents

Abstract

The invention discloses an iterative correction method for a kinematic error mapping matrix, which includes a kinematic constraint equation establishment step, an error term introduction step, an error mapping model and matrix establishment step, a structure error acquisition step, an error mapping matrix processing step and a correction structure. Error parameter acquisition steps. Also disclosed is an iterative correction system for a kinematic error mapping matrix, including a kinematic constraint equation establishment module, an error term introduction module, an error mapping model and matrix establishment module, a structural error acquisition module, an error mapping matrix processing module, and a structural error correction module. Parameter acquisition module. The iterative correction method of the kinematic error mapping matrix and the iterative correction system thereof use the modeling error compensation matrix to solve the conflicting problem of modeling solution and modeling accuracy caused by the high-order small-quantity trade-off. At the same time, it can avoid the influence of errors and ensure the validity and accuracy of the modeling.

Description

Iterative Correction Method of Kinematic Error Mapping Matrix and Its Iterative Correction System

technical field

The invention relates to the technical field of parallel mechanisms, in particular to an iterative correction method for a kinematic error mapping matrix and an iterative correction system thereof.

Background technique

Parallel mechanism has received more and more attention because of its compact structure, high rigidity, small accumulated error, low sensitivity to input error and high precision. The end accuracy of the platform depends largely on the joint accuracy. However, due to the manufacturing dimensional error and assembly error in the actual processing and assembly process, the basic and core problem of high-precision control at this stage is how to obtain accurate information of the platform. The establishment of the error mapping model is the first step of kinematics calibration. The quality of the model establishment directly affects the identification accuracy and the final calibration accuracy.

However, when modeling errors, high-order small quantities will bring coupling terms, which makes it impossible to solve the error mapping matrix. It is necessary to discard high-order small quantities in the modeling process, which will bring modeling errors, especially in the error When the value of the item is large, the impact of directly discarding the high-order small quantity is very large, which will affect the final iteration accuracy.

SUMMARY OF THE INVENTION

In view of the above defects, the purpose of the present invention is to propose an iterative correction method and iterative correction system for the kinematic error mapping matrix, and use the modeling error compensation matrix to solve the modeling solution and modeling accuracy caused by the high-order small-quantity trade-off. The contradictory problem of , while discarding the high-order small quantities, it can avoid the influence of errors, which ensures the validity and accuracy of the modeling.

In order to achieve this purpose, the present invention adopts the following technical scheme: an iterative correction method for a kinematic error mapping matrix, comprising a step of establishing a kinematic constraint equation, a step of introducing an error term, a step of establishing an error mapping model and a matrix, a step of obtaining a structural error, Error mapping matrix processing step and correction structure error parameter acquisition step;

The steps of establishing the kinematic constraint equation are: establishing the original kinematic constraint equation of the three-axis parallel mechanism through the closed-loop vector method;

The step of introducing the error term is: performing micro-perturbation on the original kinematic constraint equation to obtain a micro-perturbation kinematic equation with an error term, wherein the error term includes a first-order epsilon and a high-order epsilon;

The error mapping model and matrix establishment steps are as follows: firstly, establish a first error mapping model that discards high-order small quantities and a second error mapping model that does not discard high-order small quantities;

Then establish a first error mapping matrix that discards high-order small quantities and a second error mapping matrix that does not discard high-order small quantities;

The step of obtaining the structural error is: performing kinematic error identification on the first error mapping model to obtain the structural error;

The error mapping matrix processing step is: firstly, substituting the structural error into the first error mapping matrix to obtain a first matrix element, and then performing a root mean square on the first matrix element to obtain a first root mean square value;

Then substitute the structural error into the second error mapping matrix to obtain a second matrix element, and then perform a root mean square on the second matrix element to obtain a second root mean square value;

Then, the ratio of the first root mean square to the second root mean square is calculated to obtain a modeling error compensation matrix;

The step of obtaining the corrected structural error parameters: performing secondary identification on the modeling error compensation matrix through an iterative algorithm to obtain the corrected structural error parameters.

For example, the three-axis parallel mechanism includes a first coaxial linear motor, a second coaxial linear motor, a third coaxial linear motor, an X-axis guide rail, a Z-axis linear guide, a Z-axis guide, a rigid rod and two wedges Rigid body parts;

The first coaxial linear motor, the second coaxial linear motor and the third coaxial linear motor are all slidably arranged on the X-axis guide rail;

The two wedge-shaped rigid body parts are respectively fixedly connected with the first coaxial linear motor and the third coaxial linear motor. The walls are oppositely arranged, and the two inclined side walls are provided with the Z-axis linear guide rail;

The Z-axis guide rail is fixedly connected to the second coaxial linear motor, the middle part of the rigid rod is slidably connected to the Z-axis guide rail, and both ends of the rigid rod are hinged with connecting parts, and the two The connecting parts are respectively slidably connected with the Z-axis linear guide rails of the two wedge-shaped rigid body parts;

The specific steps of establishing the kinematic constraint equation are: establishing the original kinematic constraint equation of the three-axis parallel mechanism by the closed-loop vector method:

zl ₁ sinα=-k ₁ (q ₀ -l ₁ cosα+l ₁ -q ₁ ),

z+l ₂ sinα=k ₂ (q ₀ +l ₂ cosα-l ₂ -q ₂ ),

x=q ₀ ,

Wherein, l ₁ and l ₂ are the distances from the middle of the rigid rod member to the two wedge-shaped rigid body members, respectively, k ₁ and k ₂ are the slopes of the inclined side walls of the two wedge-shaped rigid body members, respectively, q ₀ , q ₁ and q ₂ are the movement amounts of the second coaxial linear motor, the first coaxial linear motor and the third coaxial linear motor respectively, and x, z and α are the movement amounts of the three terminals.

It should be noted that, in the step of establishing the error mapping model and matrix, the first error mapping model is:

δx=δq ₀ ;

The second error mapping model is:

δz-(l ₁ cosα+δl ₁ cosα-k ₁ l ₁ sinα

-δk ₁ l ₁ sinα-k ₁ δl ₁ sinα-δk ₁ δl ₁ sinα)δα

=(-k ₁ cosα-δk ₁ cosα+sinα+k ₁ +δk ₁ )δl ₁

+(-l ₁ cosα-δl ₁ cosα-q ₁ +q ₀ +l ₁ +δl ₁ )δk ₁

-(k ₁ +δk ₁ )δq ₁ +(k ₁ +δk ₁ )δq ₀ ,

δz+(l ₂ cosα+δl ₂ cosα+k ₂ l ₂ sinα

+δk ₂ l ₂ sinα+k ₂ δl ₂ sinα+δk ₂ δl ₂ sinα)δα

=(k ₂ cosα+δk ₂ cosα-sinα-k ₂ -δk ₂ )δl ₂

+(l ₂ cosα+δl ₂ cosα-q ₂ +q ₀ -l ₂ -δl ₂ )δk ₂

-(k ₂ +δk ₂ )δq ₂ +(k ₂ +δk ₂ )δq ₀ ,

δx=δq ₀ ;

Among them, δ is a first-order small quantity.

Optionally, in the error mapping model and matrix establishment step, the first error mapping model is converted into a first error mapping matrix: δx=J ₁ δd;

Transforming the second error mapping model into a second error mapping matrix: δx=J ₂ δd;

in,

δd=[δd ₁ ,δd ₂ ,δd ₃ ],δd ₁ =[δq ₀ ,0,0,0] ^T ,

δd ₂ =[δq ₀ , δq ₁ , δl ₁ , δk ₁ ] ^T , δd ₃ =[δq ₀ , δq ₂ , δl ₂ , δk ₂ ] ^T .

Specifically, in the step of obtaining the corrected structural error parameter, the iterative algorithm is specifically: δd=((J ₁ J ^* ) ^T J ₁ J ^* +λE)′(J ₁ J ^* ) ^T δx,

Among them, J1 is the _first error mapping matrix, J ^* is the modeling error compensation matrix, and λE is the ridge estimation.

Preferably, an iterative correction system for kinematic error mapping matrix includes a kinematic constraint equation establishment module, an error term introduction module, an error mapping model and matrix establishment module, a structure error acquisition module, an error mapping matrix processing module and a correction structure error parameter acquisition module;

The kinematic constraint equation establishment module is used to establish the original kinematic constraint equation of the three-axis parallel mechanism through the closed-loop vector method;

The error term introduction module is configured to perform micro-perturbation on the original kinematic constraint equation to obtain a micro-perturbation kinematic equation with an error term, wherein the error term includes a first-order epsilon and a high-order epsilon;

The error mapping model and matrix building module are used to establish a first error mapping model that discards high-order small quantities and a second error mapping model that does not discard high-order small quantities;

It is also used to establish a first error mapping matrix that discards high-order small quantities and a second error mapping matrix that does not discard high-order small quantities;

The structural error acquisition module is used to perform kinematic error identification on the first error mapping model to obtain structural errors;

The error mapping matrix processing module is used for substituting the structural error into the first error mapping matrix to obtain a first matrix element, and for performing root mean square on the matrix element to obtain a first root mean square value;

is also used for substituting the structural error into the second error mapping matrix to obtain a second matrix element, and is also used for performing a root mean square on the second matrix element to obtain a second root mean square value;

is also used to calculate the ratio of the first root mean square to the second root mean square to obtain a modeling error compensation matrix;

The modified structural error parameter acquisition module is configured to perform secondary identification on the modeling error compensation matrix through an iterative algorithm to obtain modified structural error parameters.

For example, the three-axis parallel mechanism includes a first coaxial linear motor, a second coaxial linear motor, a third coaxial linear motor, an X-axis guide rail, a Z-axis linear guide, a Z-axis guide, a rigid rod and two wedges Rigid body parts;

The first coaxial linear motor, the second coaxial linear motor and the third coaxial linear motor are all slidably arranged on the X-axis guide rail;

The two wedge-shaped rigid body parts are respectively fixedly connected with the first coaxial linear motor and the third coaxial linear motor. The walls are oppositely arranged, and the two inclined side walls are provided with the Z-axis linear guide rail;

The Z-axis guide rail is fixedly connected to the second coaxial linear motor, the middle part of the rigid rod is slidably connected to the Z-axis guide rail, and both ends of the rigid rod are hinged with connecting parts, and the two The connecting parts are respectively slidably connected with the Z-axis linear guide rails of the two wedge-shaped rigid body parts;

The step of establishing the kinematic constraint equation is used to establish the original kinematic constraint equation of the three-axis parallel mechanism through the closed-loop vector method:

zl ₁ sinα=-k ₁ (q ₀ -l ₁ cosα+l ₁ -q ₁ ),

z+l ₂ sinα=k ₂ (q ₀ +l ₂ cosα-l ₂ -q ₂ ),

x=q ₀ ,

Wherein, l ₁ and l ₂ are the distances from the middle of the rigid rod member to the two wedge-shaped rigid body members, respectively, k ₁ and k ₂ are the slopes of the inclined side walls of the two wedge-shaped rigid body members, respectively, q ₀ , q ₁ and q ₂ are the movement amounts of the second coaxial linear motor, the first coaxial linear motor and the third coaxial linear motor respectively, and x, z and α are the movement amounts of the three terminals.

It should be noted that the error mapping model and the matrix establishment module are used to establish the first error mapping model:

δx=δq ₀ ;

Also used to build the second error mapping model:

δz-(l ₁ cosα+δl ₁ cosα-k ₁ l ₁ sinα

-δk ₁ l ₁ sinα-k ₁ δl ₁ sinα-δk ₁ δl ₁ sinα)δα

=(-k ₁ cosα-δk ₁ cosα+sinα+k ₁ +δk ₁ )δl ₁

+(-l ₁ cosα-δl ₁ cosα-q ₁ +q ₀ +l ₁ +δl ₁ )δk ₁

-(k ₁ +δk ₁ )δq ₁ +(k ₁ +δk ₁ )δq ₀ ,

δz+(l ₂ cosα+δl ₂ cosα+k ₂ l ₂ sinα

+δk ₂ l ₂ sinα+k ₂ δl ₂ sinα+δk ₂ δl ₂ sinα)δα

=(k ₂ cosα+δk ₂ cosα-sinα-k ₂ -δk ₂ )δl ₂

+(l ₂ cosα+δl ₂ cosα-q ₂ +q ₀ -l ₂ -δl ₂ )δk ₂

-(k ₂ +δk ₂ )δq ₂ +(k ₂ +δk ₂ )δq ₀ ,

δx=δq ₀ ;

Among them, δ is a first-order small quantity.

Optionally, the error mapping model and matrix establishment module are used to convert the first error mapping model into the first error mapping matrix: δx=J ₁ δd;

It is also used to transform the second error mapping model into a second error mapping matrix: δx=J ₂ δd;

in,

δd=[δd ₁ ,δd ₂ ,δd ₃ ],δd ₁ =[δq ₀ ,0,0,0] ^T ,

δd ₂ =[δq ₀ , δq ₁ , δl ₁ , δk ₁ ] ^T , δd ₃ =[δq ₀ , δq ₂ , δl ₂ , δk ₂ ] ^T .

Specifically, the iterative algorithm of the modified structural error parameter acquisition module is: δd=((J ₁ J ^* ) ^T J ₁ J ^* +λE)′(J ₁ J ^* ) ^T δx,

Among them, J1 is the _first error mapping matrix, J ^* is the modeling error compensation matrix, and λE is the ridge estimation.

Beneficial effects of the present invention: in the iterative correction method for the kinematic error mapping matrix, by introducing the root mean square ratio matrix in the iterative process, the modeling error introduced by discarding high-order small quantities is compensated, and the The high-order small quantities can avoid the influence of errors and ensure the validity and accuracy of the modeling.

The iterative correction method for the kinematic error mapping matrix includes the steps of establishing a kinematic constraint equation, introducing an error term, establishing an error mapping model and matrix, obtaining a structural error, processing the error mapping matrix, and obtaining a modified structural error parameter, The contradiction between modeling solution and modeling accuracy caused by high-order small-quantity trade-off is solved. The modeling error compensation matrix weighted by high-order small quantities is substituted into the established iterative formula, and its construction is corrected in the iterative process. Modulo error.

Description of drawings

1 is a flowchart of an iterative correction method in an embodiment of the present invention;

2 is a schematic structural diagram of a three-axis parallel mechanism in an embodiment of the present invention;

FIG. 3 is a schematic diagram of parameters during movement of a three-axis parallel mechanism in an embodiment of the present invention.

Among them: 1 first coaxial linear motor; 2 second coaxial linear motor; 3 third coaxial linear motor; 4X axis guide rail; 5Z axis linear guide rail; 6Z axis guide rail; 7 rigid rods; 9 Connections.

Detailed ways

Embodiments of the present invention are described in detail below, 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, only used to explain the present invention, and should not be construed as a limitation of the present invention.

In the description of the embodiments of the present invention, the terms "first" and "second" are only used for description purposes, and cannot be understood as indicating or implying relative importance or implying the number of indicated technical features. Thus, features defined as "first", "second" may expressly or implicitly include one or more of said features. In the description of the embodiments of the present invention, "plurality" means two or more, unless otherwise expressly and specifically defined.

In the description of the embodiments of the present invention, it should be noted that, unless otherwise expressly specified and limited, the terms "installed", "connected" and "connected" should be understood in a broad sense, for example, it may be a fixed connection or a It is a detachable connection, or an integral connection; it can be directly connected, or it can be indirectly connected through an intermediate medium, and it can be the internal communication of 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 embodiments of the present invention can be understood according to specific situations.

The following disclosure provides many different embodiments or examples for implementing different structures of embodiments of the invention. In order to simplify the disclosure of the embodiments of the present invention, the components and arrangements of specific examples are described below. Of course, they are only examples and are not intended to limit the invention. Furthermore, embodiments of the present invention may repeat reference numerals and/or reference letters in different instances, such repetition is for the purpose of simplicity and clarity and does not in itself indicate the relationship between the various embodiments and/or arrangements discussed . In addition, the embodiments of the present invention provide examples of various specific processes and materials, but one of ordinary skill in the art will recognize the application of other processes and/or the use of other materials.

As shown in Figure 1, an iterative correction method for a kinematic error mapping matrix includes a kinematic constraint equation establishment step, an error term introduction step, an error mapping model and matrix establishment step, a structural error acquisition step, an error mapping matrix processing step and Correction of structural error parameter acquisition steps;

The steps of establishing the kinematic constraint equation are: establishing the original kinematic constraint equation of the three-axis parallel mechanism through the closed-loop vector method;

The step of introducing the error term is: performing micro-perturbation on the original kinematic constraint equation to obtain a micro-perturbation kinematic equation with an error term, wherein the error term includes a first-order epsilon and a high-order epsilon;

The error mapping model and matrix establishment steps are as follows: firstly, establish a first error mapping model that discards high-order small quantities and a second error mapping model that does not discard high-order small quantities;

Then establish a first error mapping matrix that discards high-order small quantities and a second error mapping matrix that does not discard high-order small quantities;

The step of obtaining the structural error is: performing kinematic error identification on the first error mapping model to obtain the structural error;

The error mapping matrix processing step is: firstly, substituting the structural error into the first error mapping matrix to obtain a first matrix element, and then performing a root mean square on the first matrix element to obtain a first root mean square value;

Then substitute the structural error into the second error mapping matrix to obtain a second matrix element, and then perform a root mean square on the second matrix element to obtain a second root mean square value;

Then, the ratio of the first root mean square to the second root mean square is calculated to obtain a modeling error compensation matrix;

The step of obtaining the corrected structural error parameters: performing secondary identification on the modeling error compensation matrix through an iterative algorithm to obtain the corrected structural error parameters.

In the iterative correction method of the kinematic error mapping matrix, a root mean square ratio matrix is introduced in the iterative process to compensate for the modeling error introduced by discarding the high-order small quantities, and while the high-order small quantities are discarded It can avoid the influence of errors and ensure the validity and accuracy of the modeling.

The iterative correction method for the kinematic error mapping matrix includes the steps of establishing a kinematic constraint equation, introducing an error term, establishing an error mapping model and matrix, obtaining a structural error, processing the error mapping matrix, and obtaining a modified structural error parameter, The contradiction between modeling solution and modeling accuracy caused by high-order small-quantity trade-off is solved. The modeling error compensation matrix weighted by high-order small quantities is substituted into the established iterative formula, and its construction is corrected in the iterative process. Modulo error.

The kinematic error identification is specifically L-M iteration on the error mapping model, δx is the measured terminal error, δd is the error term to be identified, and the value of the error source in δd can be obtained through L-M iteration.

In some embodiments, as shown in FIG. 2, the three-axis parallel mechanism includes a first coaxial linear motor 1, a second coaxial linear motor 2, a third coaxial linear motor 3, an X-axis guide rail 4, and a Z-axis linear motor. Guide rail 5, Z-axis guide rail 6, rigid rod 7 and two wedge-shaped rigid body parts 8;

The first coaxial linear motor 1, the second coaxial linear motor 2 and the third coaxial linear motor 3 are all slidably arranged on the X-axis guide rail 4;

The two wedge-shaped rigid body parts 8 are respectively fixedly connected with the first coaxial linear motor 1 and the third coaxial linear motor 3, and the two wedge-shaped rigid body parts 8 are each provided with an inclined side wall, and two The inclined side walls are oppositely arranged, and the Z-axis linear guide rails 5 are provided on both of the inclined side walls;

The Z-axis guide rail 6 is fixedly connected to the second coaxial linear motor 2 , the middle of the rigid rod 7 is slidably connected to the Z-axis guide 6 , and both ends of the rigid rod 7 are hingedly connected. part 9, the two connecting parts 9 are respectively slidably connected with the Z-axis linear guide rails 5 of the two wedge-shaped rigid body parts 8;

The specific steps of establishing the kinematic constraint equation are: establishing the original kinematic constraint equation of the three-axis parallel mechanism by the closed-loop vector method:

zl ₁ sinα=-k ₁ (q ₀ -l ₁ cosα+l ₁ -q ₁ ),

z+l ₂ sinα=k ₂ (q ₀ +l ₂ cosα-l ₂ -q ₂ ),

x=q ₀ ,

Wherein, l ₁ and l ₂ are the distances from the middle of the rigid rod member 7 to the two wedge-shaped rigid body members 8 respectively, and k ₁ and k ₂ are the slopes of the inclined side walls of the two wedge-shaped rigid body members 8 respectively , q ₀ , q ₁ and q ₂ are the movement amounts of the second coaxial linear motor 2 , the first coaxial linear motor 1 and the third coaxial linear motor 3 respectively, and x, z and α are the movement amounts of the three terminals.

As shown in Figure 3, the solid line is the three-axis parallel mechanism before motion, and the dotted line is the three-axis parallel mechanism after motion. The terminal movement of the three-axis parallel mechanism in the X-axis direction is x, and the terminal movement in the Z-axis direction is z. The rigid rod 7 moves on the wedge-shaped rigid body 8, so that the The included angle between the rigid rod 7 and the direction of the X-axis is α. _l1 is the distance from the middle of the rigid rod 7 to the wedge-shaped rigid body 8 on the left, and _l2 is the distance from the middle of the rigid rod 7 to the wedge-shaped rigid body 8 on the right. k ₁ is the slope of the inclined side wall of the wedge-shaped rigid body 8 located on the left, and k ₂ is the slope of the inclined side wall of the wedge-shaped rigid body 8 located on the right. q ₀ , q ₁ and q ₂ are the movement amounts of the second coaxial linear motor 2 , the first coaxial linear motor 1 and the third coaxial linear motor 3 on the X-axis, respectively.

For example, in the error mapping model and matrix establishment step, the first error mapping model is:

δx=δq ₀ ;

The second error mapping model is:

δz-(l ₁ cosα+δl ₁ cosα-k ₁ l ₁ sinα

-δk ₁ l ₁ sinα-k ₁ δl ₁ sinα-δk ₁ δl ₁ sinα)δα

=(-k ₁ cosα-δk ₁ cosα+sinα+k ₁ +δk ₁ )δl ₁

+(-l ₁ cosα-δl ₁ cosα-q ₁ +q ₀ +l ₁ +δl ₁ )δk ₁

-(k ₁ +δk ₁ )δq ₁ +(k ₁ +δk ₁ )δq ₀ ,

δz+(l ₂ cosα+δl ₂ cosα+k ₂ l ₂ sinα

+δk ₂ l ₂ sinα+k ₂ δl ₂ sinα+δk ₂ δl ₂ sinα)δα

=(k ₂ cosα+δk ₂ cosα-sinα-k ₂ -δk ₂ )δl ₂

+(l ₂ cosα+δl ₂ cosα-q ₂ +q ₀ -l ₂ -δl ₂ )δk ₂

-(k ₂ +δk ₂ )δq ₂ +(k ₂ +δk ₂ )δq ₀ ,

δx=δq ₀ ;

Among them, δ is a first-order small quantity.

The micro-perturbation motion equation in the error term introduction step is:

z+δz-(l ₁ +δl ₁ )sin(α+δα)

=-(k ₁ +δk ₁ )[q ₀ +δq ₀ -(l ₁ +δl ₁ )cos(α+δα)+(l ₁ +δl ₁ +q ₁ +δq ₁ )],

z+δz-(l ₂ +δl ₂ )sin(α+δα)

=(k ₂ +δk ₂ )[q ₀ +δq ₀ -(l ₂ +δl ₂ )cos(α+δα)+(l ₂ +δl ₂ -q ₂ -δq ₂ )],

x+δx=q ₀ +δq ₀ ;

The first error mapping model is obtained by subtracting the original kinematic constraint equation after discarding the high-order small quantity from the micro-perturbation kinematic equation. High-order small quantities refer to the multiplication of two or more first-order small quantities δ.

It should be noted that, in the step of establishing the error mapping model and matrix, the first error mapping model is converted into a first error mapping matrix: δx=J ₁ δd;

Transforming the second error mapping model into a second error mapping matrix: δx=J ₂ δd;

Wherein, δd=[δd ₁ ,δd ₂ ,δd ₃ ],δd ₁ =[δq ₀ ,0,0,0] ^T ,

δd ₂ =[δq ₀ , δq ₁ , δl ₁ , δk ₁ ] ^T , δd ₃ =[δq ₀ , δq ₂ , δl ₂ , δk ₂ ] ^T .

The first error mapping matrix δx=J ₁ δd and the second error mapping matrix δx=J ₂ δd are both J matrices of 3*12. In the error mapping matrix processing step, the difference between discarding and not discarding is analyzed, and the single element of the first error mapping matrix and the second error mapping matrix is analyzed first, and the changes of the elements are observed.

For example, the J _2×7 elements are:

(sinα-δk ₁ -k ₁ +k ₁ cosα+δk ₁ cosα)*(l ₂ cosα+δl ₁ cosα+l ₂ k ₂ sinα+l ₂ δk ₂ sinα+δl ₂ k ₂ sinα+δl ₂ δk ₂ sinα )/(l ₁ cosα+l ₂ cosα+δl ₁ cosα+δl ₂ cosα-l ₁ k ₁ sinα+l ₂ k ₂ sinα+l ₁ δk ₁ sinα-δl ₁ k ₁ sinα+l ₂ δk ₂ sinα+δl ₂ k ₂ sinα+δl ₁ δk ₁ sinα+δl ₂ δk ₂ sinα)

Among them, δl _i δk _i is a high-order small quantity, and the item with δ is the parameter item that needs to be identified.

When this higher-order fraction is discarded, the elements of J _{2 × 7} are:

Optionally, in the step of obtaining the corrected structural error parameter, the iterative algorithm is specifically: δd=((J ₁ J ^* ) ^T J ₁ J ^* +λE)′(J ₁ J ^* ) ^T δx,

Among them, J ₁ is the first error mapping matrix, J ^* is the modeling error compensation matrix, and λE is the ridge estimation.

The modeling error compensation matrix J ^* is the introduced modeling error compensation matrix, which is a square matrix with only main diagonal elements, the number of rows is equal to the number of columns of the J matrix, and the diagonal elements are in accordance with the first mean square It is weighted by the ratio of the root and the second root mean square. The purpose is to restore the true value when it is not discarded as much as possible in each iteration. The problem.

Specifically, an iterative correction system for a kinematic error mapping matrix includes a kinematic constraint equation establishment module, an error term introduction module, an error mapping model and matrix establishment module, a structure error acquisition module, an error mapping matrix processing module, and a correction structure error. parameter acquisition module;

The kinematic constraint equation establishment module is used to establish the original kinematic constraint equation of the three-axis parallel mechanism through the closed-loop vector method;

The error term introduction module is configured to perform micro-perturbation on the original kinematic constraint equation to obtain a micro-perturbation kinematic equation with an error term, wherein the error term includes a first-order epsilon and a high-order epsilon;

The error mapping model and matrix building module are used to establish a first error mapping model that discards high-order small quantities and a second error mapping model that does not discard high-order small quantities;

It is also used to establish a first error mapping matrix that discards high-order small quantities and a second error mapping matrix that does not discard high-order small quantities;

The structural error acquisition module is used to perform kinematic error identification on the first error mapping model to obtain structural errors;

The error mapping matrix processing module is used for substituting the structural error into the first error mapping matrix to obtain a first matrix element, and for performing root mean square on the matrix element to obtain a first root mean square value;

is also used for substituting the structural error into the second error mapping matrix to obtain a second matrix element, and is also used for performing a root mean square on the second matrix element to obtain a second root mean square value;

is also used to calculate the ratio of the first root mean square to the second root mean square to obtain a modeling error compensation matrix;

The modified structural error parameter acquisition module is configured to perform secondary identification on the modeling error compensation matrix through an iterative algorithm to obtain modified structural error parameters.

The iterative correction system of the kinematic error mapping matrix is obtained through the kinematic constraint equation establishment module, the error term introduction module, the error mapping model and matrix establishment module, the structure error acquisition module, the error mapping matrix processing module, and the correction structure error parameter acquisition module. The module solves the contradiction between modeling solution and modeling accuracy caused by high-order small-quantity trade-offs, and substitutes the modeling error compensation matrix weighted by high-order small quantities into the established iterative formula, and corrects it in the iterative process. its modeling error.

Preferably, the three-axis parallel mechanism includes a first coaxial linear motor 1, a second coaxial linear motor 2, a third coaxial linear motor 3, an X-axis guide rail 4, a Z-axis linear guide rail 5, a Z-axis guide rail 6, Rigid rod member 7 and two wedge-shaped rigid body members 8;

The first coaxial linear motor 1, the second coaxial linear motor 2 and the third coaxial linear motor 3 are all slidably arranged on the X-axis guide rail 4;

The two wedge-shaped rigid body parts 8 are respectively fixedly connected with the first coaxial linear motor 1 and the third coaxial linear motor 3, and the two wedge-shaped rigid body parts 8 are each provided with an inclined side wall, and two The inclined side walls are oppositely arranged, and the Z-axis linear guide rails 5 are provided on both of the inclined side walls;

The Z-axis guide rail 6 is fixedly connected to the second coaxial linear motor 2 , the middle of the rigid rod 7 is slidably connected to the Z-axis guide 6 , and both ends of the rigid rod 7 are hingedly connected. part 9, the two connecting parts 9 are respectively slidably connected with the Z-axis linear guide rails 5 of the two wedge-shaped rigid body parts 8;

The step of establishing the kinematic constraint equation is used to establish the original kinematic constraint equation of the three-axis parallel mechanism through the closed-loop vector method:

zl ₁ sinα=-k ₁ (q ₀ -l ₁ cosα+l ₁ -q ₁ ),

z+l ₂ sinα=k ₂ (q ₀ +l ₂ cosα-l ₂ -q ₂ ),

x=q ₀ ,

Wherein, l ₁ and l ₂ are the distances from the middle of the rigid rod member 7 to the two wedge-shaped rigid body members 8 respectively, and k ₁ and k ₂ are the slopes of the inclined side walls of the two wedge-shaped rigid body members 8 respectively , q ₀ , q ₁ and q ₂ are the movement amounts of the second coaxial linear motor 2 , the first coaxial linear motor 1 and the third coaxial linear motor 3 respectively, and x, z and α are the movement amounts of the three terminals.

As shown in FIG. 3 , the terminal movement amount generated by the three-axis parallel mechanism in the X-axis direction is x, and the terminal movement amount generated in the Z-axis direction is z. The member 8 moves up, so that the included angle between the rigid rod member 7 and the direction of the X axis is α. l ₁ is the length from the middle to the left end of the rigid rod 7 , and l ₂ is the length from the middle to the right end of the rigid rod 7 . k ₁ is the slope of the inclined side wall of the wedge-shaped rigid body 8 located on the left, and k ₂ is the slope of the inclined side wall of the wedge-shaped rigid body 8 located on the right. q ₀ , q ₁ and q ₂ are the movement amounts of the second coaxial linear motor 2 , the first coaxial linear motor 1 and the third coaxial linear motor 3 on the X-axis, respectively.

In some embodiments, the error mapping model and matrix establishment module are used to establish the first error mapping model:

δx=δq ₀ ;

Also used to build the second error mapping model:

δz-(l ₁ cosα+δl ₁ cosα-k ₁ l ₁ sinα

-δk ₁ l ₁ sinα-k ₁ δl ₁ sinα-δk ₁ δl ₁ sinα)δα

=(-k ₁ cosα-δk ₁ cosα+sinα+k ₁ +δk ₁ )δl ₁

+(-l ₁ cosα-δl ₁ cosα-q ₁ +q ₀ +l ₁ +δl ₁ )δk ₁

-(k ₁ +δk ₁ )δq ₁ +(k ₁ +δk ₁ )δq ₀ ,

δz+(l ₂ cosα+δl ₂ cosα+k ₂ l ₂ sinα

+δk ₂ l ₂ sinα+k ₂ δl ₂ sinα+δk ₂ δl ₂ sinα)δα

=(k ₂ cosα+δk ₂ cosα-sinα-k ₂ -δk ₂ )δl ₂

+(l ₂ cosα+δl ₂ cosα-q ₂ +q ₀ -l ₂ -δl ₂ )δk ₂

-(k ₂ +δk ₂ )δq ₂ +(k ₂ +δk ₂ )δq ₀ ,

δx=δq ₀ ;

Among them, δ is a first-order small quantity.

For example, the error mapping model and matrix establishment module are used to convert the first error mapping model into the first error mapping matrix: δx=J ₁ δd;

It is also used to transform the second error mapping model into a second error mapping matrix: δx=J ₂ δd;

in,

δd=[δd ₁ ,δd ₂ ,δd ₃ ],δd ₁ =[δq ₀ ,0,0,0] ^T ,

δd ₂ =[δq ₀ , δq ₁ , δl ₁ , δk ₁ ] ^T , δd ₃ =[δq ₀ , δq ₂ , δl ₂ , δk ₂ ] ^T .

The first error mapping matrix is a 3*12 J matrix. In the error mapping matrix processing step, the difference between discarding and not discarding is analyzed, and the single element of the first error mapping matrix and the second error mapping matrix is analyzed first, and the changes of the elements are observed.

It should be noted that the iterative algorithm of the modified structural error parameter acquisition module is: δd=((J ₁ J ^* ) ^T J ₁ J ^* +λE)′(J ₁ J ^* ) ^T δx,

Among them, J ₁ is the first error mapping matrix, J ^* is the modeling error compensation matrix, and λE is the ridge estimation.

In the description of this specification, reference to the terms "one embodiment," "some embodiments," "exemplary embodiment," "example," "specific example," or "some examples" or the like is meant to be used in conjunction with the described embodiments. A particular feature, structure, material or characteristic described by way or example is included in at least one embodiment or example of the present invention. In this specification, schematic representations of the above terms do not necessarily refer 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.

Any description of a process or method in the flowcharts or otherwise described herein may be understood to represent a module, segment or portion of code comprising one or more executable instructions for implementing a specified logical function or step of the process , and the scope of the preferred embodiments of the invention includes alternative implementations in which the functions may be performed out of the order shown or discussed, including performing the functions substantially concurrently or in the reverse order depending upon the functions involved, which should It is understood by those skilled in the art to which the embodiments of the present invention belong.

Although the embodiments of the present invention have been shown and described above, it should be understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Variations, modifications, substitutions and alterations are made to the implementation.

Claims (2)

1. an iterative correction method of kinematic error mapping matrix, it is characterized in that: comprise kinematic constraint equation establishment step, error term introduction step, error mapping model and matrix establishment step, structure error acquisition step, error mapping matrix processing step and Correction of structural error parameter acquisition steps; The steps of establishing the kinematic constraint equation are: establishing the original kinematic constraint equation of the three-axis parallel mechanism through the closed-loop vector method; The three-axis parallel mechanism includes a first coaxial linear motor, a second coaxial linear motor, a third coaxial linear motor, an X-axis guide rail, a Z-axis linear guide, a Z-axis guide, a rigid rod and two wedge-shaped rigid bodies piece; The first coaxial linear motor, the second coaxial linear motor and the third coaxial linear motor are all slidably arranged on the X-axis guide rail; The two wedge-shaped rigid body parts are respectively fixedly connected with the first coaxial linear motor and the third coaxial linear motor. The walls are oppositely arranged, and the two inclined side walls are provided with the Z-axis linear guide rail; The Z-axis guide rail is fixedly connected to the second coaxial linear motor, the middle part of the rigid rod is slidably connected to the Z-axis guide rail, and both ends of the rigid rod are hinged with connecting parts, and the two The connecting parts are respectively slidably connected with the Z-axis linear guide rails of the two wedge-shaped rigid body parts; The specific steps of establishing the kinematic constraint equation are: establishing the original kinematic constraint equation of the three-axis parallel mechanism by the closed-loop vector method: zl ₁ sinα=-k ₁ (q ₀ -l ₁ cosα+l ₁ -q ₁ ), z+l ₂ sinα=k ₂ (q ₀ +l ₂ cosα-l ₂ -q ₂ ), x=q ₀ , Wherein, l ₁ and l ₂ are the distances from the middle of the rigid rod member to the two wedge-shaped rigid body members, respectively, k ₁ and k ₂ are the slopes of the inclined side walls of the two wedge-shaped rigid body members, respectively, q ₀ , q ₁ and q ₂ are respectively the movement amounts of the second coaxial linear motor, the first coaxial linear motor and the third coaxial linear motor, and x, z and α are the movement amounts of the three terminals; the three-axis parallel mechanism The terminal movement amount generated in the X-axis direction is x, and the terminal movement amount generated in the Z-axis direction is z. The included angle between the directions is α; The step of introducing the error term is: performing micro-perturbation on the original kinematic constraint equation to obtain a micro-perturbation kinematic equation with an error term, wherein the error term includes a first-order epsilon and a high-order epsilon; The error mapping model and matrix establishment steps are as follows: first, establish a first error mapping model that discards high-order small quantities and a second error mapping model that does not discard high-order small quantities; Then establish a first error mapping matrix that discards high-order small quantities and a second error mapping matrix that does not discard high-order small quantities; The step of obtaining the structural error is: performing kinematic error identification on the first error mapping model to obtain the structural error; The error mapping matrix processing step is: firstly, substituting the structural error into the first error mapping matrix to obtain a first matrix element, and then performing a root mean square on the first matrix element to obtain a first root mean square value; Then substitute the structural error into the second error mapping matrix to obtain a second matrix element, and then perform a root mean square on the second matrix element to obtain a second root mean square value; Then, the ratio of the first root mean square to the second root mean square is calculated to obtain a modeling error compensation matrix; The step of obtaining the modified structural error parameters: performing secondary identification on the modeling error compensation matrix through an iterative algorithm to obtain the modified structural error parameters; The iterative algorithm is specifically δd=((J ₁ J ^* ) ^T J ₁ J ^* +λE)′(J ₁ J ^* ) ^T δx, Among them, J ₁ is the first error mapping matrix, J ^* is the modeling error compensation matrix, λE is the ridge estimation; the kinematic error identification is specifically LM iteration on the error mapping model, δx is the measured terminal error, δd For the error term to be identified, the value of the error source in δd can be obtained through LM iteration; In the error mapping model and matrix establishment step, the first error mapping model is:

δx=δq ₀ ; The second error mapping model is: δz-(l ₁ cosα+δl ₁ cosα-k ₁ l ₁ sinα -δk ₁ l ₁ sinα-k ₁ δl ₁ sinα-δk ₁ δl ₁ sinα)δα =(-k ₁ cosα-δk ₁ cosα+sinα+k ₁ +δk ₁ )δl ₁ +(-l ₁ cosα-δl ₁ cosα-q ₁ +q ₀ +l ₁ +δl ₁ )δk ₁ -(k ₁ +δk ₁ )δq ₁ +(k ₁ +δk ₁ )δq ₀ , δz+(l ₂ cosα+δl ₂ cosα+k ₂ l ₂ sinα +δk ₂ l ₂ sinα+k ₂ δl ₂ sinα+δk ₂ δl ₂ sinα)δα =(k ₂ cosα+δk ₂ cosα-sinα-k ₂ -δk ₂ )δl ₂ +(l ₂ cosα+δl ₂ cosα-q ₂ +q ₀ -l ₂ -δl ₂ )δk ₂ -(k ₂ +δk ₂ )δq ₂ +(k ₂ +δk ₂ )δq ₀ , δx=δq ₀ ; Among them, δ is a first-order small quantity; In the error mapping model and matrix establishment step, the first error mapping model is converted into a first error mapping matrix: δx=J ₁ δd; Transforming the second error mapping model into a second error mapping matrix: δx=J ₂ δd; in, δd=[δd ₁ ,δd ₂ ,δd ₃ ],δd ₁ =[δq ₀ ,0,0,0] ^T , δd ₂ =[δq ₀ ,δq ₁ ,δl ₁ ,δk ₁ ] ^T ,δd ₃ =[δq ₀ ,δq ₂ ,δl ₂ ,δk ₂ ] ^T ; Wherein, the first error mapping matrix δx=J ₁ δd and the second error mapping matrix δx=J ₂ δd are both J matrices of 3*12. 2. an iterative correction system of kinematic error mapping matrix, it is characterized in that: comprise kinematic constraint equation establishment module, error term introduction module, error mapping model and matrix establishment module, structure error acquisition module, error mapping matrix processing module and Modified structural error parameter acquisition module; The kinematic constraint equation establishment module is used to establish the original kinematic constraint equation of the three-axis parallel mechanism through the closed-loop vector method; The three-axis parallel mechanism includes a first coaxial linear motor, a second coaxial linear motor, a third coaxial linear motor, an X-axis guide rail, a Z-axis linear guide, a Z-axis guide, a rigid rod and two wedge-shaped rigid bodies piece; The first coaxial linear motor, the second coaxial linear motor and the third coaxial linear motor are all slidably arranged on the X-axis guide rail; The two wedge-shaped rigid body parts are respectively fixedly connected to the first coaxial linear motor and the third coaxial linear motor. The walls are oppositely arranged, and the two inclined side walls are provided with the Z-axis linear guide rail; The Z-axis guide rail is fixedly connected to the second coaxial linear motor, the middle part of the rigid rod is slidably connected to the Z-axis guide rail, and both ends of the rigid rod are hinged with connecting parts, and the two The connecting parts are respectively slidably connected with the Z-axis linear guide rails of the two wedge-shaped rigid body parts; The step of establishing the kinematic constraint equation is used to establish the original kinematic constraint equation of the three-axis parallel mechanism through the closed-loop vector method: zl ₁ sinα=-k ₁ (q ₀ -l ₁ cosα+l ₁ -q ₁ ), z+l ₂ sinα=k ₂ (q ₀ +l ₂ cosα-l ₂ -q ₂ ), x=q ₀ , Wherein, l ₁ and l ₂ are the distances from the middle of the rigid rod member to the two wedge-shaped rigid body members, respectively, k ₁ and k ₂ are the slopes of the inclined side walls of the two wedge-shaped rigid body members, respectively, q ₀ , q ₁ and q ₂ are respectively the movement amounts of the second coaxial linear motor, the first coaxial linear motor and the third coaxial linear motor, and x, z and α are the movement amounts of the three terminals; the three-axis parallel mechanism The terminal movement amount generated in the X-axis direction is x, and the terminal movement amount generated in the Z-axis direction is z. The included angle between the directions is α; The error term introduction module is configured to perform micro-perturbation on the original kinematic constraint equation to obtain a micro-perturbation kinematic equation with an error term, wherein the error term includes a first-order epsilon and a high-order epsilon; The error mapping model and matrix building module are used to establish a first error mapping model that discards high-order small quantities and a second error mapping model that does not discard high-order small quantities; It is also used to establish a first error mapping matrix that discards high-order small quantities and a second error mapping matrix that does not discard high-order small quantities; The structural error acquisition module is used to perform kinematic error identification on the first error mapping model to obtain structural errors; The error mapping matrix processing module is used for substituting the structural error into the first error mapping matrix to obtain a first matrix element, and for performing root mean square on the matrix element to obtain a first root mean square value; is also used to substitute the structural error into the second error mapping matrix to obtain a second matrix element, and is also used to perform a root mean square on the second matrix element to obtain a second root mean square value; is also used to calculate the ratio of the first root mean square to the second root mean square to obtain a modeling error compensation matrix; The modified structural error parameter acquisition module is configured to perform secondary identification on the modeling error compensation matrix through an iterative algorithm to obtain modified structural error parameters; The iterative algorithm of the modified structural error parameter acquisition module is: δd=((J ₁ J ^* ) ^T J ₁ J ^* +λE)′(J ₁ J ^* ) ^T δx, Among them, J ₁ is the first error mapping matrix, J ^* is the modeling error compensation matrix, and λE is the ridge estimation; The kinematic error identification is specifically L-M iteration on the error mapping model, δx is the measured terminal error, δd is the error term to be identified, and the value of the error source in δd can be obtained through L-M iteration; The error mapping model and matrix establishment module are used to establish the first error mapping model:

δx=δq ₀ ; Also used to build the second error mapping model: δz-(l ₁ cosα+δl ₁ cosα-k ₁ l ₁ sinα -δk ₁ l ₁ sinα-k ₁ δl ₁ sinα-δk ₁ δl ₁ sinα)δα =(-k ₁ cosα-δk ₁ cosα+sinα+k ₁ +δk ₁ )δl ₁ +(-l ₁ cosα-δl ₁ cosα-q ₁ +q ₀ +l ₁ +δl ₁ )δk ₁ -(k ₁ +δk ₁ )δq ₁ +(k ₁ +δk ₁ )δq ₀ , δz+(l ₂ cosα+δl ₂ cosα+k ₂ l ₂ sinα +δk ₂ l ₂ sinα+k ₂ δl ₂ sinα+δk ₂ δl ₂ sinα)δα =(k ₂ cosα+δk ₂ cosα-sinα-k ₂ -δk ₂ )δl ₂ +(l ₂ cosα+δl ₂ cosα-q ₂ +q ₀ -l ₂ -δl ₂ )δk ₂ -(k ₂ +δk ₂ )δq ₂ +(k ₂ +δk ₂ )δq ₀ , δx=δq ₀ ; Among them, δ is a first-order small quantity; The error mapping model and matrix establishment module are used to convert the first error mapping model into the first error mapping matrix: δx=J ₁ δd; It is also used to transform the second error mapping model into a second error mapping matrix: δx=J ₂ δd; in, δd=[δd ₁ ,δd ₂ ,δd ₃ ],δd ₁ =[δq ₀ ,0,0,0] ^T , δd ₂ =[δq ₀ ,δq ₁ ,δl ₁ ,δk ₁ ] ^T ,δd ₃ =[δq ₀ ,δq ₂ ,δl ₂ ,δk ₂ ] ^T ; Wherein, the first error mapping matrix δx=J ₁ δd and the second error mapping matrix δx=J ₂ δd are both J matrices of 3*12.

Images