CN112949098B - Iterative correction method and iterative correction system for kinematic error mapping matrix - Google Patents
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Abstract
The invention discloses an iterative correction method of a kinematic error mapping matrix, which comprises a kinematic constraint equation establishing step, an error item introducing step, an error mapping model and matrix establishing step, a structural error obtaining step, an error mapping matrix processing step and a corrected structural error parameter obtaining step. The iterative correction system comprises a kinematic constraint equation building module, an error item introducing module, an error mapping model and matrix building module, a structural error obtaining module, an error mapping matrix processing module and a corrected structural error parameter obtaining module. The iterative correction method and the iterative correction system for the kinematic error mapping matrix solve the problem of contradiction between modeling solving and modeling precision caused by high-order small quantity rejection by using the modeling error compensation matrix, avoid the error influence caused by the high-order small quantity while omitting the high-order small quantity, and ensure the effectiveness and the accuracy of modeling.
Description
Technical Field
The invention relates to the technical field of parallel mechanisms, in particular to an iterative correction method and an iterative correction system for a kinematic error mapping matrix.
Background
The parallel mechanism receives more and more attention because of the characteristics of compact structure, high rigidity, small accumulated error, low sensitivity to input error and high precision. The end precision of the platform depends on the joint precision to a great extent, but because of the manufacturing dimension error and the assembly error in the actual processing and assembly process, the basic and core problem of the high-precision control at present is how to acquire accurate information of the platform. The establishment of the error mapping model is the first step of the kinematics calibration, and the identification precision and the final calibration precision are directly influenced by the establishment of the model.
However, when modeling is performed on errors, high-order small quantities bring coupling terms, so that an error mapping matrix cannot be solved, high-order small quantities need to be omitted in a modeling process, modeling errors are brought, particularly when the number of error terms is large, the effect of directly omitting the high-order small quantities is large, and the final iteration precision is influenced.
Disclosure of Invention
Aiming at the defects, the invention aims to provide an iterative correction method and an iterative correction system for a kinematic error mapping matrix, solves the problem of contradiction between modeling solving and modeling precision caused by high-order small quantity rejection by utilizing a modeling error compensation matrix, can avoid the error influence caused by the high-order small quantity rejection while omitting the high-order small quantity, and ensures the effectiveness and the accuracy of modeling.
In order to achieve the purpose, the invention adopts the following technical scheme: an iterative correction method of a kinematic error mapping matrix comprises a kinematic constraint equation establishing step, an error item introducing step, an error mapping model and matrix establishing step, a structural error obtaining step, an error mapping matrix processing step and a corrected structural error parameter obtaining step;
the kinematic constraint equation establishment steps are as follows: establishing an original kinematic constraint equation of the triaxial parallel mechanism by a closed-loop vector method;
the error term introduction step is as follows: performing perturbation on the original kinematic constraint equation to obtain a perturbation kinematic equation with an error term, wherein the error term comprises a first-order small quantity and a high-order small quantity;
the error mapping model and the matrix establishing steps are as follows: firstly, establishing a first error mapping model with a high-order small quantity omitted and a second error mapping model without the high-order small quantity omitted;
then establishing a first error mapping matrix with a high-order small quantity omitted and a second error mapping matrix with a high-order small quantity not omitted;
the structural error obtaining step comprises the following steps: performing kinematic error identification on the first error mapping model to obtain a structural error;
the error mapping matrix processing steps are as follows: substituting the structural error into the first error mapping matrix to obtain a first matrix element, and then executing root mean square on the first matrix element to obtain a first root mean square value;
then substituting the structural error into the second error mapping matrix to obtain a second matrix element, and then executing root mean square on the second matrix element to obtain a second root mean square value;
then calculating the ratio of the first root-mean-square to the second root-mean-square to obtain a modeling error compensation matrix;
the step of obtaining the corrected structure error parameters comprises the following steps: and performing secondary identification on the modeling error compensation matrix through an iterative algorithm to obtain a corrected structure error parameter.
For example, the three-axis parallel mechanism comprises 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 rail, a Z-axis guide rail, a rigid rod and two wedge-shaped rigid members;
the first coaxial linear motor, the second coaxial linear motor and the third coaxial linear motor are arranged on the X-axis guide rail in a sliding manner;
the two wedge-shaped rigid members are respectively and fixedly connected with the first coaxial linear motor and the third coaxial linear motor, each wedge-shaped rigid member is provided with an inclined side wall, the two inclined side walls are oppositely arranged, and each inclined side wall is provided with the Z-axis linear guide rail;
the Z-axis guide rail is fixedly connected with the second coaxial linear motor, the middle part of the rigid rod piece is connected with the Z-axis guide rail in a sliding manner, two ends of the rigid rod piece are respectively hinged with a connecting part, and the two connecting parts are respectively connected with the Z-axis linear guide rails of the two wedge-shaped rigid pieces in a sliding manner;
the kinematic constraint equation establishing step specifically comprises the following steps: establishing an original kinematic constraint equation of the three-axis parallel mechanism by a closed-loop vector method:
z-l1 sinα＝-k1(q0-l1 cosα+l1-q1)，
z+l2 sinα＝k2(q0+l2 cosα-l2-q2)，
x＝q0，
wherein l1And l2Respectively the distance, k, from the middle of the rigid rod member to the two wedge-shaped rigid members1And k2The slope of the inclined side walls of the two wedge-shaped rigid members, q0、q1And q is2The movement amounts of the second coaxial linear motor, the first coaxial linear motor and the third coaxial linear motor are respectively, and x, z and alpha are three terminal movement amounts.
It should be noted that, in the error mapping model and matrix building step, the first error mapping model is:
δx＝δq0；
the second error mapping model is:
δz-(l1 cosα+δl1 cosα-k1l1 sinα
-δk1l1 sinα-k1δl1 sinα-δk1δl1 sinα)δα
＝(-k1 cosα-δk1 cosα+sinα+k1+δk1)δl1
+(-l1 cosα-δl1 cosα-q1+q0+l1+δl1)δk1
-(k1+δk1)δq1+(k1+δk1)δq0，
δz+(l2 cosα+δl2 cosα+k2l2 sinα
+δk2l2 sinα+k2δl2 sinα+δk2δl2 sinα)δα
＝(k2 cosα+δk2 cosα-sinα-k2-δk2)δl2
+(l2 cosα+δl2 cosα-q2+q0-l2-δl2)δk2
-(k2+δk2)δq2+(k2+δk2)δq0，
δx＝δq0；
where δ is the first order fractional amount.
Optionally, in the error mapping model and matrix building step, the first error mapping model is transformed into a first error mapping matrix: δ x ═ J1δd；
Transforming the second error mapping model into a second error mapping matrix: δ x ═ J2δd；
Wherein the content of the first and second substances,
δd＝[δd1,δd2,δd3]，δd1＝[δq0,0,0,0]T，
δd2＝[δq0,δq1,δl1,δk1]T，δd3＝[δq0,δq2,δl2,δk2]T。
specifically, in the step of obtaining the corrected structure error parameter, the iterative algorithm specifically includes: δ d ═ ((J)1J*)TJ1J*+λE)′(J1J*)Tδx，
Wherein, J1Is a first error mapping matrix, J*To model the error compensation matrix, λ E is the ridge estimate.
Preferably, the iterative correction system of the kinematic error mapping matrix comprises a kinematic constraint equation building module, an error item introducing module, an error mapping model and matrix building module, a structural error obtaining module, an error mapping matrix processing module and a corrected structural error parameter obtaining module;
the kinematics constraint equation establishing module is used for establishing an original kinematics constraint equation of the three-axis parallel mechanism by a closed-loop vector method;
the error term introduction module is used for carrying out perturbation on the original kinematic constraint equation to obtain a perturbation kinematic equation with an error term, wherein the error term comprises a first-order small quantity and a high-order small quantity;
the error mapping model and matrix establishing module is used for establishing a first error mapping model with high-order small quantity omitted and a second error mapping model with no high-order small quantity omitted;
the first error mapping matrix with the high-order small quantity omitted and the second error mapping matrix with the high-order small quantity not omitted are established;
the structural error acquisition module is used for identifying kinematic errors of 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 is also used for executing root mean square on the matrix element to obtain a first root mean square value;
the second matrix element is obtained by substituting the structural error into the second error mapping matrix, and the second matrix element is subjected to root mean square to obtain a second root mean square value;
the modeling error compensation matrix is obtained by calculating the ratio of the first root mean square to the second root mean square;
and the corrected structure error parameter acquisition module is used for performing secondary identification on the modeling error compensation matrix through an iterative algorithm to obtain a corrected structure error parameter.
For example, the three-axis parallel mechanism comprises 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 rail, a Z-axis guide rail, a rigid rod and two wedge-shaped rigid members;
the first coaxial linear motor, the second coaxial linear motor and the third coaxial linear motor are arranged on the X-axis guide rail in a sliding manner;
the two wedge-shaped rigid members are respectively and fixedly connected with the first coaxial linear motor and the third coaxial linear motor, each wedge-shaped rigid member is provided with an inclined side wall, the two inclined side walls are oppositely arranged, and each inclined side wall is provided with the Z-axis linear guide rail;
the Z-axis guide rail is fixedly connected with the second coaxial linear motor, the middle part of the rigid rod piece is connected with the Z-axis guide rail in a sliding manner, two ends of the rigid rod piece are respectively hinged with a connecting part, and the two connecting parts are respectively connected with the Z-axis linear guide rails of the two wedge-shaped rigid pieces in a sliding manner;
the kinematic constraint equation establishing step is used for establishing an original kinematic constraint equation of the three-axis parallel mechanism by a closed-loop vector method:
z-l1 sinα＝-k1(q0-l1 cosα+l1-q1)，
z+l2 sinα＝k2(q0+l2 cosα-l2-q2)，
x＝q0，
wherein l1And l2Respectively the distance, k, from the middle of the rigid rod member to the two wedge-shaped rigid members1And k2The slope of the inclined side walls of the two wedge-shaped rigid members, q0、q1And q is2The movement amounts of the second coaxial linear motor, the first coaxial linear motor and the third coaxial linear motor are respectively, and x, z and alpha are three terminal movement amounts.
It is worth mentioning that the error mapping model and matrix building module is configured to build the first error mapping model:
δx＝δq0；
and is further configured to build the second error mapping model:
δz-(l1 cosα+δl1 cosα-k1l1 sinα
-δk1l1 sinα-k1δl1 sinα-δk1δl1 sinα)δα
＝(-k1 cosα-δk1 cosα+sinα+k1+δk1)δl1
+(-l1 cosα-δl1 cosα-q1+q0+l1+δl1)δk1
-(k1+δk1)δq1+(k1+δk1)δq0，
δz+(l2 cosα+δl2 cosα+k2l2 sinα
+δk2l2 sinα+k2δl2 sinα+δk2δl2 sinα)δα
＝(k2 cosα+δk2 cosα-sinα-k2-δk2)δl2
+(l2 cosα+δl2 cosα-q2+q0-l2-δl2)δk2
-(k2+δk2)δq2+(k2+δk2)δq0，
δx＝δq0；
where δ is the first order fractional amount.
Optionally, the error mapping model and matrix building module is configured to transform the first error mapping model into the first error mapping matrix:δx＝J1δd；
further for transforming the second error mapping model into a second error mapping matrix: δ x ═ J2δd；
Wherein the content of the first and second substances,
δd＝[δd1,δd2,δd3]，δd1＝[δq0,0,0,0]T，
δd2＝[δq0,δq1,δl1,δk1]T，δd3＝[δq0,δq2,δl2,δk2]T。
specifically, the iterative algorithm of the modified structure error parameter obtaining module is as follows: δ d ═ ((J)1J*)TJ1J*+λE)′(J1J*)Tδx，
Wherein, J1Is a first error mapping matrix, J*To model the error compensation matrix, λ E is the ridge estimate.
The invention has the beneficial effects that: in the iterative correction method of the kinematic error mapping matrix, the modeling error introduced by the high-order small quantity is compensated by introducing the root mean square ratio matrix in the iterative process, the high-order small quantity can be eliminated, the error influence brought by the high-order small quantity can be avoided, and the effectiveness and the accuracy of modeling are ensured.
The iterative correction method of the kinematic error mapping matrix solves the problem of contradiction between modeling solution and modeling precision caused by high-order small quantity rejection through the steps of building a kinematic constraint equation, introducing an error item, building an error mapping model and a matrix, obtaining a structural error, processing the error mapping matrix and obtaining a corrected structural error parameter, and substitutes a modeling error compensation matrix formed by high-order small quantity weighting into an established iterative formula to correct the modeling error in the iterative process.
Drawings
FIG. 1 is a flow diagram of an iterative correction method in one embodiment of the present invention;
FIG. 2 is a schematic diagram of a three-axis parallel mechanism in accordance with an embodiment of the present invention;
fig. 3 is a schematic diagram of parameters of a three-axis parallel mechanism in motion according to an embodiment of the present invention.
Wherein: 1 a first coaxial linear motor; 2 a second coaxial linear motor; 3 a third coaxial linear motor; 4X-axis guide rails; 5Z-axis linear guide rails; 6Z-axis guide rails; 7 a rigid rod member; 8 wedge-shaped rigid body members; 9 connecting part.
Detailed Description
Reference will now be made in detail to embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein like or similar reference numerals refer to the same or similar elements or elements having the same or similar function throughout. The embodiments described below with reference to the accompanying drawings are illustrative only for the purpose of explaining the present invention, and are not to be construed as limiting the present invention.
In the description of the embodiments of the present invention, the terms "first" and "second" are used for descriptive purposes only and are not to be construed as indicating or implying relative importance or implying any number of technical features indicated. Thus, features defined as "first", "second", may explicitly or implicitly include one or more of the described features. In the description of the embodiments of the present invention, "a plurality" means two or more unless specifically limited otherwise.
In the description of the embodiments of the present invention, it should be noted that, unless otherwise explicitly specified or limited, the terms "mounted," "connected," and "connected" are to be construed broadly, e.g., as being fixedly connected, detachably connected, or integrally connected; either directly or indirectly through intervening media, either internally or in any other relationship. Specific meanings of the above terms in the embodiments of the present invention can be understood by those of ordinary skill in the art according to specific situations.
The following disclosure provides many different embodiments or examples for implementing different configurations of embodiments of the invention. In order to simplify the disclosure of embodiments of the invention, the components and arrangements of specific examples are described below. Of course, they are merely examples and are not intended to limit the present invention. Furthermore, embodiments of the invention may repeat reference numerals and/or reference letters in the various examples, which have been repeated for purposes of simplicity and clarity and do not in themselves dictate a relationship between the various embodiments and/or arrangements discussed. In addition, embodiments of the present invention provide examples of various specific processes and materials, but one of ordinary skill in the art may recognize applications of other processes and/or use of other materials.
As shown in fig. 1, an iterative correction method for a kinematic error mapping matrix includes a kinematic constraint equation establishment step, an error item introduction step, an error mapping model and matrix establishment step, a structural error acquisition step, an error mapping matrix processing step, and a corrected structural error parameter acquisition step;
the kinematic constraint equation establishment steps are as follows: establishing an original kinematic constraint equation of the triaxial parallel mechanism by a closed-loop vector method;
the error term introduction step is as follows: performing perturbation on the original kinematic constraint equation to obtain a perturbation kinematic equation with an error term, wherein the error term comprises a first-order small quantity and a high-order small quantity;
the error mapping model and the matrix establishing steps are as follows: firstly, establishing a first error mapping model with a high-order small quantity omitted and a second error mapping model without the high-order small quantity omitted;
then establishing a first error mapping matrix with a high-order small quantity omitted and a second error mapping matrix with a high-order small quantity not omitted;
the structural error obtaining step comprises the following steps: performing kinematic error identification on the first error mapping model to obtain a structural error;
the error mapping matrix processing steps are as follows: substituting the structural error into the first error mapping matrix to obtain a first matrix element, and then executing root mean square on the first matrix element to obtain a first root mean square value;
then substituting the structural error into the second error mapping matrix to obtain a second matrix element, and then executing root mean square on the second matrix element to obtain a second root mean square value;
then calculating the ratio of the first root-mean-square to the second root-mean-square to obtain a modeling error compensation matrix;
the step of obtaining the corrected structure error parameters comprises the following steps: and performing secondary identification on the modeling error compensation matrix through an iterative algorithm to obtain a corrected structure error parameter.
In the iterative correction method of the kinematic error mapping matrix, the modeling error introduced by the high-order small quantity is compensated by introducing the root mean square ratio matrix in the iterative process, the high-order small quantity can be eliminated, the error influence brought by the high-order small quantity can be avoided, and the effectiveness and the accuracy of modeling are ensured.
The iterative correction method of the kinematic error mapping matrix solves the problem of contradiction between modeling solution and modeling precision caused by high-order small quantity rejection through the steps of building a kinematic constraint equation, introducing an error item, building an error mapping model and a matrix, obtaining a structural error, processing the error mapping matrix and obtaining a corrected structural error parameter, and substitutes a modeling error compensation matrix formed by high-order small quantity weighting into an established iterative formula to correct the modeling error in the iterative process.
The kinematic error identification specifically includes that L-M iteration is carried out on an error mapping model, δ x is a measured terminal error, δ d is an error item needing to be identified, and the value of an error source in δ d can be obtained through L-M iteration.
In some embodiments, as shown in fig. 2, the three-axis parallel mechanism comprises 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, a rigid rod 7 and two wedge-shaped rigid members 8;
the first coaxial linear motor 1, the second coaxial linear motor 2 and the third coaxial linear motor 3 are arranged on the X-axis guide rail 4 in a sliding manner;
the two wedge-shaped rigid members 8 are respectively and fixedly connected with the first coaxial linear motor 1 and the third coaxial linear motor 3, each wedge-shaped rigid member 8 is provided with an inclined side wall, the two inclined side walls are oppositely arranged, and each inclined side wall is provided with the Z-axis linear guide rail 5;
the Z-axis guide rail 6 is fixedly connected with the second coaxial linear motor 2, the middle part of the rigid rod 7 is connected with the Z-axis guide rail 6 in a sliding manner, two ends of the rigid rod 7 are respectively hinged with a connecting part 9, and the two connecting parts 9 are respectively connected with the Z-axis linear guide rails 5 of the two wedge-shaped rigid members 8 in a sliding manner;
the kinematic constraint equation establishing step specifically comprises the following steps: establishing an original kinematic constraint equation of the three-axis parallel mechanism by a closed-loop vector method:
z-l1 sinα＝-k1(q0-l1 cosα+l1-q1)，
z+l2 sinα＝k2(q0+l2 cosα-l2-q2)，
x＝q0，
wherein l1And l2The distance, k, from the middle of the rigid bar 7 to the two wedge-shaped rigid elements 81And k2The slopes of the inclined side walls, q, of two of said wedge-shaped rigid elements 8, respectively0、q1And q is2The 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 three terminal movement amounts.
As shown in fig. 3, the solid line represents the three-axis parallel mechanism before movement, and the dotted line represents the three-axis parallel mechanism after movement. The terminal movement amount of the three-axis parallel mechanism generated in the X-axis direction is X, the terminal movement amount generated in the Z-axis direction is Z, and the included angle between the rigid rod 7 and the X-axis direction is alpha through the movement of the rigid rod 7 on the wedge-shaped rigid member 8. l1The distance, l, from the middle of the rigid bar 7 to the left of the wedge-shaped rigid body 82The distance from the middle of the rigid rod 7 to the wedge-shaped rigid member 8 on the right. k is a radical of1The slant of the wedge-shaped rigid body member 8 on the leftSlope of the side wall, k2The slope of the sloping side wall of the wedge-shaped rigid body member 8 located on the right. q. q.s0、q1And q is2The movement amounts of the second coaxial linear motor 2, the first coaxial linear motor 1, and the third coaxial linear motor 3 in the X axis, respectively.
For example, in the error mapping model and matrix building step, the first error mapping model is:
δx＝δq0；
the second error mapping model is:
δz-(l1 cosα+δl1 cosα-k1l1 sinα
-δk1l1 sinα-k1δl1 sinα-δk1δl1 sinα)δα
＝(-k1 cosα-δk1 cosα+sinα+k1+δk1)δl1
+(-l1 cosα-δl1 cosα-q1+q0+l1+δl1)δk1
-(k1+δk1)δq1+(k1+δk1)δq0，
δz+(l2 cosα+δl2 cosα+k2l2 sinα
+δk2l2 sinα+k2δl2 sinα+δk2δl2 sinα)δα
＝(k2 cosα+δk2 cosα-sinα-k2-δk2)δl2
+(l2 cosα+δl2 cosα-q2+q0-l2-δl2)δk2
-(k2+δk2)δq2+(k2+δk2)δq0，
δx＝δq0；
where δ is the first order fractional amount.
The perturbation motion equation in the error term introduction step is as follows:
z+δz-(l1+δl1)sin(α+δα)
＝-(k1+δk1)[q0+δq0-(l1+δl1)cos(α+δα)+(l1+δl1+q1+δq1)]，
z+δz-(l2+δl2)sin(α+δα)
＝(k2+δk2)[q0+δq0-(l2+δl2)cos(α+δα)+(l2+δl2-q2-δq2)]，
x+δx＝q0+δq0；
and (4) after the high-order small quantity is removed from the perturbation kinematic equation, subtracting the original kinematic constraint equation to obtain the first error mapping model. A high order fractional amount refers to the multiplication of two and more first order fractional amounts δ.
It is worth mentioning that in the error mapping model and matrix building step, the first error mapping model is transformed into a first error mapping matrix: δ x ═ J1δd；
Transforming the second error mapping model into a second error mapping matrix: δ x ═ J2δd；
Wherein δ d is [ δ d ═ δ d1,δd2,δd3]，δd1＝[δq0,0,0,0]T，
δd2＝[δq0,δq1,δl1,δk1]T，δd3＝[δq0,δq2,δl2,δk2]T。
The first error mapping matrix δ x ═ J1δ d and the second error mapping matrix δ x ═ J2δ d are each 3 × 12J matrices. In the error mapping matrix processing step, the difference between truncation and non-truncation is analyzed, individual elements of the first error mapping matrix and the second error mapping matrix are analyzed first, and the change of the elements is observed.
For example, without rounding off higher order small quantities, J2×7The elements are as follows:
(sinα-δk1-k1+k1 cosα+δk1 cosα)*(l2 cosα+δl1 cosα+l2k2 sinα+l2δk2 sinα+δl2k2 sinα+δl2δk2 sinα)/(l1 cosα+l2 cosα+δl1 cosα+δl2 cosα-l1k1 sinα+l2k2 sinα+l1δk1 sinα-δl1k1 sinα+l2δk2 sinα+δl2k2 sinα+δl1δk1 sinα+δl2δk2 sinα)
wherein δ liδkiFor high order small quantities, the terms with δ are the parameter terms that need to be identified.
When the higher order is truncated to a small extent, J2×7The elements are as follows:
optionally, in the step of obtaining the corrected structure error parameter, the iterative algorithm specifically includes: δ d ═ ((J)1J*)TJ1J*+λE)′(J1J*)Tδx，
Wherein, J1Is a first error mapping matrix, J*To model the error compensation matrix, λ E is the ridge estimate.
Modeling error compensation matrix J*The introduced modeling error compensation matrix is a square matrix only with main diagonal elements, the row number of the square matrix is equal to the column number of a J matrix, the diagonal elements are weighted according to the ratio of the first root-mean-square to the second root-mean-square, the actual values which are not truncated are reduced as far as possible in each iteration, and lambda E is a ridge estimation which aims at solving the problem that a singular matrix is difficult to invert in the iteration process.
Specifically, the iterative correction system for the kinematic error mapping matrix comprises a kinematic constraint equation building module, an error item introducing module, an error mapping model and matrix building module, a structural error obtaining module, an error mapping matrix processing module and a corrected structural error parameter obtaining module;
the kinematics constraint equation establishing module is used for establishing an original kinematics constraint equation of the three-axis parallel mechanism by a closed-loop vector method;
the error term introduction module is used for carrying out perturbation on the original kinematic constraint equation to obtain a perturbation kinematic equation with an error term, wherein the error term comprises a first-order small quantity and a high-order small quantity;
the error mapping model and matrix establishing module is used for establishing a first error mapping model with high-order small quantity omitted and a second error mapping model with no high-order small quantity omitted;
the first error mapping matrix with the high-order small quantity omitted and the second error mapping matrix with the high-order small quantity not omitted are established;
the structural error acquisition module is used for identifying kinematic errors of 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 is also used for executing root mean square on the matrix element to obtain a first root mean square value;
the second matrix element is obtained by substituting the structural error into the second error mapping matrix, and the second matrix element is subjected to root mean square to obtain a second root mean square value;
the modeling error compensation matrix is obtained by calculating the ratio of the first root mean square to the second root mean square;
and the corrected structure error parameter acquisition module is used for performing secondary identification on the modeling error compensation matrix through an iterative algorithm to obtain a corrected structure error parameter.
The iterative correction system of the kinematic error mapping matrix solves the problem of contradiction between modeling solution and modeling precision caused by high-order small quantity rejection through the kinematic constraint equation establishing module, the error item introducing module, the error mapping model and matrix establishing module, the structural error acquiring module, the error mapping matrix processing module and the corrected structural error parameter acquiring module, and corrects the modeling error in the iterative process by substituting the modeling error compensation matrix formed by high-order small quantity weighting into the established iterative formula.
Preferably, the three-axis parallel mechanism comprises 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, a rigid rod 7 and two wedge-shaped rigid members 8;
the first coaxial linear motor 1, the second coaxial linear motor 2 and the third coaxial linear motor 3 are arranged on the X-axis guide rail 4 in a sliding manner;
the two wedge-shaped rigid members 8 are respectively and fixedly connected with the first coaxial linear motor 1 and the third coaxial linear motor 3, each wedge-shaped rigid member 8 is provided with an inclined side wall, the two inclined side walls are oppositely arranged, and each inclined side wall is provided with the Z-axis linear guide rail 5;
the Z-axis guide rail 6 is fixedly connected with the second coaxial linear motor 2, the middle part of the rigid rod 7 is connected with the Z-axis guide rail 6 in a sliding manner, two ends of the rigid rod 7 are respectively hinged with a connecting part 9, and the two connecting parts 9 are respectively connected with the Z-axis linear guide rails 5 of the two wedge-shaped rigid members 8 in a sliding manner;
the kinematic constraint equation establishing step is used for establishing an original kinematic constraint equation of the three-axis parallel mechanism by a closed-loop vector method:
z-l1 sinα＝-k1(q0-l1 cosα+l1-q1)，
z+l2 sinα＝k2(q0+l2 cosα-l2-q2)，
x＝q0，
wherein l1And l2The distance, k, from the middle of the rigid bar 7 to the two wedge-shaped rigid elements 81And k2The slopes of the inclined side walls, q, of two of said wedge-shaped rigid elements 8, respectively0、q1And q is2The 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 three terminal movement amounts.
As shown in fig. 3, the three-axis parallel mechanism has a terminal movement amount X in the X-axis direction and a terminal movement amount Z in the Z-axis direction, and the rigid rod 7 moves on the wedge-shaped rigid member 8, so that an included angle between the rigid rod 7 and the X-axis direction is α. l1The length from the middle to the left end of the rigid bar 7,/2The length from the middle to the right end of the rigid rod 7. k is a radical of1The slope, k, of the sloping side wall of the wedge-shaped rigid body member 8 on the left2The slope of the sloping side wall of the wedge-shaped rigid body member 8 located on the right. q. q.s0、q1And q is2The movement amounts of the second coaxial linear motor 2, the first coaxial linear motor 1, and the third coaxial linear motor 3 in the X axis, respectively.
In some embodiments, the error mapping model and matrix building module is configured to build the first error mapping model:
δx＝δq0；
and is further configured to build the second error mapping model:
δz-(l1 cosα+δl1 cosα-k1l1 sinα
-δk1l1 sinα-k1δl1 sinα-δk1δl1 sinα)δα
＝(-k1 cosα-δk1 cosα+sinα+k1+δk1)δl1
+(-l1 cosα-δl1 cosα-q1+q0+l1+δl1)δk1
-(k1+δk1)δq1+(k1+δk1)δq0，
δz+(l2 cosα+δl2 cosα+k2l2 sinα
+δk2l2 sinα+k2δl2 sinα+δk2δl2 sinα)δα
＝(k2 cosα+δk2 cosα-sinα-k2-δk2)δl2
+(l2 cosα+δl2 cosα-q2+q0-l2-δl2)δk2
-(k2+δk2)δq2+(k2+δk2)δq0，
δx＝δq0；
where δ is the first order fractional amount.
For example, the error mapping model and matrix building module is configured to transform the first error mapping model into the first error mapping matrix: δ x ═ J1δd；
Further for transforming the second error mapping model into a second error mapping matrix: δ x ═ J2δd；
Wherein the content of the first and second substances,
δd＝[δd1,δd2,δd3]，δd1＝[δq0,0,0,0]T，
δd2＝[δq0,δq1,δl1,δk1]T，δd3＝[δq0,δq2,δl2,δk2]T。
the first error mapping matrix is a 3 x 12J matrix. In the error mapping matrix processing step, the difference between truncation and non-truncation is analyzed, individual elements of the first error mapping matrix and the second error mapping matrix are analyzed first, and the change of the elements is observed.
It should be noted that the iterative algorithm of the modified structure error parameter obtaining module is as follows: δ d ═ ((J)1J*)TJ1J*+λE)′(J1J*)Tδx，
Wherein, J1Is a first error mapping matrix, J*To model the error compensation matrix, λ E is the ridge estimate.
In the description herein, references to the description of the terms "one embodiment," "some embodiments," "an illustrative embodiment," "an example," "a specific example" or "some examples" or the like mean that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the 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 process or method descriptions in flow charts or otherwise described herein may be understood as representing modules, segments, or portions of code which include one or more executable instructions for implementing specific logical functions or steps of the process, and alternate implementations are included within the scope of the preferred embodiment of the present invention in which functions may be executed out of order from that shown or discussed, including substantially concurrently or in reverse order, depending on the functionality involved, as would be understood by those reasonably skilled in the art of the present invention.
Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and not to be construed as limiting the present invention, and that variations, modifications, substitutions and alterations can be made in the above embodiments by those of ordinary skill in the art within the scope of the present invention.
Claims (2)
1. An iterative correction method for a kinematic error mapping matrix is characterized in that: the method comprises a kinematic constraint equation establishing step, an error item introducing step, an error mapping model and matrix establishing step, a structural error acquiring step, an error mapping matrix processing step and a modified structural error parameter acquiring step;
the kinematic constraint equation establishment steps are as follows: establishing an original kinematic constraint equation of the triaxial parallel mechanism by a closed-loop vector method;
the three-axis parallel mechanism comprises 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 rail, a Z-axis guide rail, a rigid rod piece and two wedge-shaped rigid piece pieces;
the first coaxial linear motor, the second coaxial linear motor and the third coaxial linear motor are arranged on the X-axis guide rail in a sliding manner;
the two wedge-shaped rigid members are respectively and fixedly connected with the first coaxial linear motor and the third coaxial linear motor, each wedge-shaped rigid member is provided with an inclined side wall, the two inclined side walls are oppositely arranged, and each inclined side wall is provided with the Z-axis linear guide rail;
the Z-axis guide rail is fixedly connected with the second coaxial linear motor, the middle part of the rigid rod piece is connected with the Z-axis guide rail in a sliding manner, two ends of the rigid rod piece are respectively hinged with a connecting part, and the two connecting parts are respectively connected with the Z-axis linear guide rails of the two wedge-shaped rigid pieces in a sliding manner;
the kinematic constraint equation establishing step specifically comprises the following steps: establishing an original kinematic constraint equation of the three-axis parallel mechanism by a closed-loop vector method:
z-l1sinα＝-k1(q0-l1cosα+l1-q1)，
z+l2sinα＝k2(q0+l2cosα-l2-q2)，
x＝q0，
wherein l1And l2Respectively the distance, k, from the middle of the rigid rod member to the two wedge-shaped rigid members1And k2The slope of the inclined side walls of the two wedge-shaped rigid members, q0、q1And q is2The movement amounts of the second coaxial linear motor, the first coaxial linear motor and the third coaxial linear motor are respectively, and x, z and alpha are the movement amounts of the three terminals; the three-axis parallel mechanism generates a terminal movement amount X in the X-axis direction and a terminal movement amount Z in the Z-axis direction, and the rigid rod piece moves on the wedge-shaped rigid piece to enable an included angle between the rigid rod piece and the X-axis direction to be alpha;
the error term introduction step is as follows: performing perturbation on the original kinematic constraint equation to obtain a perturbation kinematic equation with an error term, wherein the error term comprises a first-order small quantity and a high-order small quantity;
the error mapping model and the matrix establishing steps are as follows: firstly, establishing a first error mapping model with a high-order small quantity omitted and a second error mapping model without the high-order small quantity omitted;
then establishing a first error mapping matrix with a high-order small quantity omitted and a second error mapping matrix with a high-order small quantity not omitted;
the structural error obtaining step comprises the following steps: performing kinematic error identification on the first error mapping model to obtain a structural error;
the error mapping matrix processing steps are as follows: substituting the structural error into the first error mapping matrix to obtain a first matrix element, and then executing root mean square on the first matrix element to obtain a first root mean square value;
then substituting the structural error into the second error mapping matrix to obtain a second matrix element, and then executing root mean square on the second matrix element to obtain a second root mean square value;
then calculating the ratio of the first root-mean-square to the second root-mean-square to obtain a modeling error compensation matrix;
the step of obtaining the corrected structure error parameters comprises the following steps: performing secondary identification on the modeling error compensation matrix through an iterative algorithm to obtain a corrected structure error parameter;
the iterative algorithm is specifically δ d ═ ((J)1J*)TJ1J*+λE)′(J1J*)Tδx，
Wherein, J1Is a first error mapping matrix, J*For modeling error compensation matrix, λ E is ridge estimation; the kinematic error identification specifically comprises the steps of carrying out L-M iteration on an error mapping model, wherein δ x is a measured terminal error, δ d is an error item to be identified, and the value of an error source in δ d can be obtained through L-M iteration;
in the error mapping model and matrix building step, the first error mapping model is:
δx＝δq0；
the second error mapping model is:
δz-(l1cosα+δl1cosα-k1l1sinα
-δk1l1sinα-k1δl1sinα-δk1δl1sinα)δα
＝(-k1cosα-δk1cosα+sinα+k1+δk1)δl1
+(-l1cosα-δl1cosα-q1+q0+l1+δl1)δk1
-(k1+δk1)δq1+(k1+δk1)δq0，
δz+(l2cosα+δl2cosα+k2l2sinα
+δk2l2sinα+k2δl2sinα+δk2δl2sinα)δα
＝(k2cosα+δk2cosα-sinα-k2-δk2)δl2
+(l2cosα+δl2cosα-q2+q0-l2-δl2)δk2
-(k2+δk2)δq2+(k2+δk2)δq0，
δx＝δq0；
wherein δ is a first order fractional amount;
in the error mapping model and matrix building step, transforming the first error mapping model into a first error mapping matrix: δ x ═ J1δd；
Transforming the second error mapping model into a second error mapping matrix: δ x ═ J2δd；
Wherein the content of the first and second substances,
δd＝[δd1,δd2,δd3]，δd1＝[δq0,0,0,0]T，
δd2＝[δq0,δq1,δl1,δk1]T，δd3＝[δq0,δq2,δl2,δk2]T；
wherein the first error mapping matrix δ x ═ J1δ d and the second error mapping matrix δ x ═ J2δ d are each 3 × 12J matrices.
2. An iterative correction system for a kinematic error mapping matrix, characterized by: the system comprises a kinematics constraint equation establishing module, an error item introducing module, an error mapping model and matrix establishing module, a structural error acquiring module, an error mapping matrix processing module and a modified structural error parameter acquiring module;
the kinematics constraint equation establishing module is used for establishing an original kinematics constraint equation of the three-axis parallel mechanism by a closed-loop vector method;
the three-axis parallel mechanism comprises 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 rail, a Z-axis guide rail, a rigid rod piece and two wedge-shaped rigid piece pieces;
the first coaxial linear motor, the second coaxial linear motor and the third coaxial linear motor are arranged on the X-axis guide rail in a sliding manner;
the two wedge-shaped rigid members are respectively and fixedly connected with the first coaxial linear motor and the third coaxial linear motor, each wedge-shaped rigid member is provided with an inclined side wall, the two inclined side walls are oppositely arranged, and each inclined side wall is provided with the Z-axis linear guide rail;
the Z-axis guide rail is fixedly connected with the second coaxial linear motor, the middle part of the rigid rod piece is connected with the Z-axis guide rail in a sliding manner, two ends of the rigid rod piece are respectively hinged with a connecting part, and the two connecting parts are respectively connected with the Z-axis linear guide rails of the two wedge-shaped rigid pieces in a sliding manner;
the kinematic constraint equation establishing step is used for establishing an original kinematic constraint equation of the three-axis parallel mechanism by a closed-loop vector method:
z-l1sinα＝-k1(q0-l1cosα+l1-q1)，
z+l2sinα＝k2(q0+l2cosα-l2-q2)，
x＝q0，
wherein l1And l2Respectively the distance, k, from the middle of the rigid rod member to the two wedge-shaped rigid members1And k2Two wedge-shaped rigid members respectivelyThe slope of the sloped side wall of (a), q0、q1And q is2The movement amounts of the second coaxial linear motor, the first coaxial linear motor and the third coaxial linear motor are respectively, and x, z and alpha are the movement amounts of the three terminals; the three-axis parallel mechanism generates a terminal movement amount X in the X-axis direction and a terminal movement amount Z in the Z-axis direction, and the rigid rod piece moves on the wedge-shaped rigid piece to enable an included angle between the rigid rod piece and the X-axis direction to be alpha;
the error term introduction module is used for carrying out perturbation on the original kinematic constraint equation to obtain a perturbation kinematic equation with an error term, wherein the error term comprises a first-order small quantity and a high-order small quantity;
the error mapping model and matrix establishing module is used for establishing a first error mapping model with high-order small quantity omitted and a second error mapping model with no high-order small quantity omitted;
the first error mapping matrix with the high-order small quantity omitted and the second error mapping matrix with the high-order small quantity not omitted are established;
the structural error acquisition module is used for identifying kinematic errors of 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 is also used for executing root mean square on the matrix element to obtain a first root mean square value;
the second matrix element is obtained by substituting the structural error into the second error mapping matrix, and the second matrix element is subjected to root mean square to obtain a second root mean square value;
the modeling error compensation matrix is obtained by calculating the ratio of the first root mean square to the second root mean square;
the corrected structure error parameter acquisition module is used for performing secondary identification on the modeling error compensation matrix through an iterative algorithm to obtain a corrected structure error parameter;
the iterative algorithm of the modified structure error parameter acquisition module is as follows: δ d ═ ((J)1J*)TJ1J*+λE)′(J1J*)Tδx，
Wherein, J1Is a first error mapping matrix, J*For modeling error compensation matrix, λ E is ridge estimation;
the kinematic error identification specifically comprises the steps of carrying out L-M iteration on an error mapping model, wherein δ x is a measured terminal error, δ d is an error item to be identified, and the value of an error source in δ d can be obtained through L-M iteration;
the error mapping model and matrix building module is configured to build the first error mapping model:
δx＝δq0；
and is further configured to build the second error mapping model:
δz-(l1cosα+δl1cosα-k1l1sinα
-δk1l1sinα-k1δl1sinα-δk1δl1sinα)δα
＝(-k1cosα-δk1cosα+sinα+k1+δk1)δl1
+(-l1cosα-δl1cosα-q1+q0+l1+δl1)δk1
-(k1+δk1)δq1+(k1+δk1)δq0，
δz+(l2cosα+δl2cosα+k2l2sinα
+δk2l2sinα+k2δl2sinα+δk2δl2sinα)δα
＝(k2cosα+δk2cosα-sinα-k2-δk2)δl2
+(l2cosα+δl2cosα-q2+q0-l2-δl2)δk2
-(k2+δk2)δq2+(k2+δk2)δq0，
δx＝δq0；
wherein δ is a first order fractional amount;
the error mapping model and matrix building module is configured to transform the first error mapping model into the first error mapping matrix: δ x ═ J1δd；
Further for transforming the second error mapping model into a second error mapping matrix: δ x ═ J2δd；
Wherein the content of the first and second substances,
δd＝[δd1,δd2,δd3]，δd1＝[δq0,0,0,0]T，
δd2＝[δq0,δq1,δl1,δk1]T，δd3＝[δq0,δq2,δl2,δk2]T；
wherein the first error mapping matrix δ x ═ J1δ d and the second error mapping matrix δ x ═ J2δ d are each 3 × 12J matrices.
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