Method for discriminating action boundary of hole-shaft part

Abstract

The invention relates to the technical field of geometric error evaluation of hole and shaft parts in the field of machinery, and discloses a hole and shaft part action boundary discrimination method, which comprises the following steps of: deriving a motion coefficient vector for describing radius change of each measuring point based on a tiny rigid body motion model of a surface measuring point set of a measured part; and the motion coefficients are combined into a motion coefficient matrix. Furthermore, the method establishes a discriminant inequality group based on a mutual exclusion proposition. The judgment process is realized by analyzing the rank of the motion coefficient matrix and iteratively solving an equation set formed by the maximum linear independent matrix of the motion coefficient matrix, and finally, whether the fitting cylinder is the action boundary or not is determined by judging whether the inequality set has a feasible solution or not. According to the method, the problems of complicated geometric discrimination conditions, lack of universality and the like in the existing evaluation method can be solved, and a reliable basis which is easy to realize in a stylization manner is provided for determining the action boundary of the hole-shaft part.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of geometric error evaluation of hole shaft parts in the field of machinery, in particular to a hole shaft part acting boundary discrimination method. BACKGROUND

[0002] At present, hole shaft parts play an important role in mechanical manufacturing. The size error and geometric error of hole shaft parts are two core errors of mechanical products. In the tolerance system adopting maximum material requirement (MMR) or minimum material requirement (LMR), the size error allowance is allowed to compensate for the geometric error. This compensation mechanism not only meets the functional requirements such as assembly, but also reduces the processing difficulty of the parts. Therefore, the accurate discrimination of the acting boundary of the hole shaft part is the premise of realizing the functional tolerance evaluation.

[0003] In view of the above related content, the existing acting boundary evaluation mainly depends on geometric criteria. For example, the judgment of the minimum circumscribed cylinder needs to meet a set of complex geometric constraint conditions. In actual measurement, there are various methods for constructing boundary cylinders. For example, adaptive variable body method, search approximation method or rotation projection method. These methods find an ideal cylinder that contains or is contained through different ways. Other technologies use intelligent optimization algorithms to search for the optimal solution of a target function through iteration to approximately determine the boundary.

[0004] However, the above existing technologies have deficiencies in application. First, the application conditions of the geometric discrimination criteria are complex. Taking the minimum circumscribed cylinder as an example, there are as many as seven geometric discrimination criteria, and there are unlisted variants, which leads to the lack of universality of the geometric discrimination method, and it is difficult to apply in complex measurement conditions. Second, the existing construction methods such as search approximation method and rotation projection method are designed for specific geometric characteristics, and are difficult to be used as a universal discrimination method, and the applicability is limited. In addition, the evaluation method using intelligent optimization algorithm often gives an approximate optimal solution. This approximate solution does not have unique certainty, and the discrimination process lacks rigorous mathematical logic support. In the case of arbitration or high-precision verification, the result of this method is not reliable enough. At the same time, the complex geometric constraints also make it difficult to realize automation and programming.

[0005] Therefore, the present application provides a hole shaft part acting boundary discrimination method to solve the deficiencies in the prior art. SUMMARY

[0006] In view of the deficiencies of the prior art, the present application provides a hole shaft part acting boundary discrimination method, which solves the problem of complex geometric discrimination conditions and lack of universality of the existing hole shaft part acting boundary.

[0007] To achieve the above object, the present application is implemented by the following technical scheme: A hole-shaft part action boundary distinguishing method, comprising the following steps: S1, obtaining a plurality of surface measuring points of a measured shaft or hole; S2, preprocessing, establishing a coordinate system and transforming the coordinates of the surface measuring points into rectangular coordinates; S3, determining an action boundary coefficient, constructing a motion coefficient vector and combining into a motion coefficient matrix, selecting a relaxation constant, forming a distinguishing inequality group and a corresponding hyperplane equation group; S4, calculating the rank of the motion coefficient matrix and comparing with the number of the surface measuring points; S5, selecting a maximum linearly independent matrix from the motion coefficient matrix and solving the corresponding hyperplane equation group to obtain a motion trend vector solution; S6, checking whether the motion trend vector solution satisfies the distinguishing inequality corresponding to the residual motion coefficient vector; if yes, distinguishing that the fitted cylinder is not an action boundary; otherwise, proceeding to the next step; S7, checking whether all possible maximum linearly independent matrix combinations have been investigated; if yes, distinguishing that the fitted cylinder is an action boundary; otherwise, continuing to analyze the next maximum linearly independent matrix combination.

[0008] By adopting the above technical scheme, the present application converts the geometric distinguishing problem of whether a fitted cylinder is an action boundary into a mathematical logic problem of whether a linear inequality group has a feasible solution. Since the present method establishes a general kinematics model and a distinguishing inequality group, and through analyzing the rank of the boundary point set motion coefficient matrix and iterative solution checking, the complex spatial geometric constraints are converted into standardized linear algebra operations. Therefore, the present method overcomes the defect of complex conditions of the traditional geometric distinguishing method, provides a general and reliable basis for hole-shaft part action boundary distinguishing, and is easy to realize programmed analysis.

[0009] Preferably, the motion coefficient vector is constructed according to the rectangular coordinates of the surface measuring points. The motion coefficient vector contains coefficients corresponding to four rigid body motion freedom variables, including a translation along the x-axis, a translation along the y-axis, a rotation around the x-axis and a rotation around the y-axis.

[0010] Further, the motion coefficient vector is derived according to the radius variation of the surface measuring points. The radius variation is obtained by projecting the position variation vector of the surface measuring points caused by the four rigid body motion freedom variables on the unit normal vector at the surface measuring points through dot product calculation. This process establishes a linear relationship between the radius variation of each measuring point and the four rigid body motion freedom variables. The motion coefficient vector is composed of the coefficients corresponding to the four freedom variables in this linear relationship, which are calculated according to the rectangular coordinates and radial distance of the measuring points themselves.

[0011] Preferably, in the S3 step, the action boundary coefficient of the circumscribed cylinder is 1, and the action boundary coefficient of the inscribed cylinder is -1.

[0012] The discrimination inequality set is established based on a mutually exclusive negation proposition. The specific meaning of this mutually exclusive negation proposition is: if there is a small rigid body motion that makes the radius variation trend (for the circumscribed cylinder, it means moving away from the axis; for the inscribed cylinder, it means moving close to the axis) of all surface measuring points negative and less than the relaxation constant, then the fitted cylinder is not the action boundary. The discrimination inequality set formed by the method is the mathematical expression of this mutually exclusive negation proposition, and the entire discrimination process is to determine whether there is a solution to this inequality set.

[0013] Preferably, the selection of a relaxation constant less than zero in the S3 step includes: selecting a value that is less than zero and greater than the maximum value of the radius variation trend of all surface measuring points.

[0014] Preferably, the establishment of the coordinate system in the S2 step includes: establishing a specific coordinate system such that the z-axis of the specific coordinate system coincides with the axis of the fitted cylinder.

[0015] And the preprocessing in the S2 step further includes: converting the obtained surface measuring points in cylindrical coordinate form into rectangular coordinates in the specific coordinate system.

[0016] Preferably, in the S1 step, the surface measuring points are obtained using a surface coordinate measuring device in the mechanical field, and the surface coordinate measuring device in the mechanical field includes a coordinate measuring machine, an on-machine measuring device, or a three-dimensional topography measuring instrument.

[0017] Further, the specific execution mode of the S5, S6, and S7 steps is as follows: In the S5 step, the motion coefficient vector set with a number equal to the rank is selected to form the maximum linearly independent matrix, and the remaining motion coefficient vectors are denoted as the remaining motion coefficient vectors. In the S6 step, it is checked whether the motion trend vector solution makes all motion coefficient vectors corresponding to the residual motion coefficient vector satisfy the condition of being less than the relaxation constant; In the S7 step, if there is still a maximum linearly independent matrix combination that has not been examined, the next maximum linearly independent matrix combination is continuously selected and analyzed by jumping back to the S5 step.

[0018] Preferably, the hole shaft type part action boundary determination method further comprises the step of determining the fitting cylinder before the S1 step: A candidate fitting cylinder and the corresponding boundary measuring points are preliminarily determined by solving an unconstrained objective optimization problem.

[0019] Further, the objective of the unconstrained objective optimization problem is to find a group of rigid body motion fine adjustment amounts, so that the maximum distance of all surface measuring points to the axis of the fitting cylinder after adjustment is minimized.

[0020] The present application provides a hole shaft type part action boundary determination method. It has the following beneficial effects: 1. The present application converts the complex spatial geometric determination problem into the mathematical logic problem of whether the linear inequality group has a solution by establishing the mutual exclusion proposition based on the small rigid body motion and the corresponding inequality group. This method avoids the complexity and ambiguity of traditional geometric analysis, and the determination process is rigorous, the result is quantifiable and reliable.

[0021] 2. The present application constructs the motion coefficient matrix based on the general kinematic model and the surface measuring points, and the determination process does not depend on the specific geometric constraints of the part, only the rank and solution space of the matrix need to be analyzed. Therefore, this method is suitable for different specifications and types of hole shaft type parts, and has strong universality.

[0022] 3. The determination steps of the present application include constructing a matrix, calculating a rank, selecting a maximum linearly independent matrix, solving and iterating and checking, which are all standardized linear algebra operations and logical judgments. The whole process is clear in logic, and is very suitable for automatic determination through computer programming, which improves the detection efficiency. BRIEF DESCRIPTION OF DRAWINGS

[0023] Figure 1 The hole shaft type part action boundary generalization evaluation flowchart of the present application; Figure 2 The measuring point set and its degrees of freedom schematic diagram of the present application; Figure 3 The fitting cylinder and its size change trend diagram of the present application; Figure 4 The cylinder key parameter and coordinate transformation schematic diagram of the present application; Figure 5A workpiece tolerance requirement chart for the present invention. DETAILED DESCRIPTION

[0024] The technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the specification of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative work fall within the scope of protection of the present application.

[0025] Referring to the drawings in the specification of the present application, Figure 1 the present application provides a hole-shaft part action boundary discrimination method. The method converts the geometric discrimination problem of whether the fitting cylinder is an action boundary into the mathematical logic problem of whether the linear inequality system has no solution. The basic logic form of the action size is directly related to the minimum circumscribed and maximum inscribed principle of the action boundary. In combination with the corresponding relationship of the hole-shaft element, the tolerance principle, the effective boundary, the fitting cylinder and other related concepts, the basic logic form of the action boundary can be summarized as shown in the following formula: ; wherein, is an action boundary coefficient, for the circumscribed cylinder , for the inscribed cylinder ; is a motion trend vector, representing a small rigid motion of the measuring point set relative to the axis of the fitting cylinder; is a motion coefficient vector of the boundary measuring point ; is the distance change amount of the boundary measuring point to the axis of the fitting cylinder. In order to facilitate the conversion of the above formula into a programmed discrimination analysis, the present application adopts the mutually exclusive negation proposition form thereof, as shown in the following formula: ; wherein, is a relaxation constant less than 0, and satisfies . If the formula is not established, the fitting cylinder is an action boundary; otherwise, the fitting cylinder is not an action boundary. The formula can be recorded as a matrix form formula, and the inequality system is called a discrimination inequality system: ; wherein: is the number of boundary measuring points; is the motion coefficient matrix of the boundary point set, the number of rows of which is , and the number of columns of which is 4; is the coefficient vector of the radius change amount of the boundary measuring point, which is a column vector; is a column vector composed of a relaxation constant is a column vector composed of a column vector; is a motion trend vector, wherein and are translation amounts along axis and axis, is a rotation amount around axis, is a rotation amount around axis.

[0026] The criterion is that if the matrix-form formula has no solution, the fitted cylinder is the action boundary; if the matrix-form formula has a solution, the fitted cylinder is not the action boundary.

[0027] Referring to the accompanying Figure 2 and the accompanying Figure 4 , in order to establish the mathematical model required by the above-mentioned discrimination inequality group, the present application first defines a coordinate system and key basic elements. The action sizes of the hole and the shaft are irrelevant to the coordinate system in which they are located. Therefore, in order to facilitate the description of the key basic elements in the action boundary problem, i.e. the measuring point set and the fitted cylinder, the present application establishes a specific coordinate system.

[0028] As shown in the accompanying Figure 2 , in the coordinate system, the axis of the coordinate system is fixed on the axis line of the fitted cylinder. In this coordinate system, the key basic elements are defined as follows: The coordinates of the measuring point . .

[0029] The distance of the measuring point to the action boundary axis (the axis) is The calculation formula is: .

[0030] The radius of the fitted cylinder is determined by the circumscribed cylinder radius , and the inscribed cylinder radius .

[0031] The action boundary is defined as: when the circumscribed cylinder radius reaches the minimum or the inscribed cylinder radius reaches the maximum, the fitted cylinder is the action boundary.

[0032] The measuring point set has 4 degrees of freedom of rigid body motion, including: along Axis translation ,along Axis translation , around Axis rotation and around Axis rotation .

[0033] The method of this invention includes a preprocessing step. In actual measurement and evaluation, the obtained surface measuring point coordinates are in cylindrical coordinates, or the axis of the fitted cylinder is not perpendicular to... When the shafts coincide, as shown in the attached document. Figure 4 As shown, a coordinate transformation is required.

[0034] This preprocessing step converts the coordinates of the measuring point into rectangular coordinates using coordinate transformation formulas, and aligns the axis of the boundary with the coordinate axes. The axes are aligned to meet the requirements of the subsequent analysis model. This coordinate transformation is based on the principle of robot kinematic coordinate transformation, establishing a new coordinate system and determining the coordinates of the boundary measurement points in the new coordinate system.

[0035] See attached document Figure 2 After establishing the coordinate system and model described above, it is necessary to quantize the set of measurement points. Rigid body motion for each boundary measurement point The influence of position. This invention uses the principles of robot kinematics to solve the problem of adjusting the set of measurement points relative to the fitted cylinder (i.e., When determining the orientation of the axis, the measuring point directional changes Due to the motion trend vector The translation amount described in , and rotation amount , These are all minute quantities, therefore the measuring points directional changes The simplified expression can be obtained using formula (1): ; in, For measuring points The azimuth change vector; For the translational component of rigid body motion; For the rotational component of rigid body motion; and respectively along shaft and The minute translation of the axis; and They are respectively around shaft and A tiny amount of rotation of the shaft; For measuring points coordinate vector, i.e. . This orientation change is the basis for the subsequent derivation of the change of the distance of the measuring point to the axis .

[0036] Referring to the attached Figure 2 and the attached Figure 3 , the position of the boundary measuring point changes in orientation after the rigid body motion of the measuring point set, and its distance to the fitted cylindrical axis (i.e. the axis) will also change accordingly. The change in this distance (radius) is denoted by .

[0037] This radius change is the projection of the measuring point orientation change vector on the normal vector of the boundary at the measuring point . Among them, the direction of the normal vector is from the axis to the measuring point and perpendicular to the axis, and its unit vector expression is .

[0038] Therefore, can be calculated by the dot product of and . Substitute formula (1) into the dot product operation, and consider the specific form of , it can be deduced that and the four degrees of freedom variables of the rigid body motion of the measuring point set (i.e. ) have a linear relationship, as shown in formula (2): ; Where: is the change in the distance of the boundary measuring point to the fitted cylindrical axis (radius change).

[0039] is the unit normal vector of the boundary at the boundary measuring point , .

[0040] is the orientation change vector of the measuring point defined by formula (1).

[0041] is the coordinate component of the measuring point .

[0042] is the measuring point​​ to the distance of the axis, .

[0043] and are the infinitesimal translation amounts along the axis and axis, respectively.

[0044] and are the infinitesimal rotation amounts around the axis and axis, respectively.

[0045] Equation (2) establishes a linear mathematical relationship between the radius variation of each boundary measuring point and the overall rigid body motion variable , which is the key basis for subsequent construction of the discriminant inequality set.

[0046] In order to simplify the linear relationship established by equation (2) and facilitate the subsequent integration of the constraint sets of all boundary measuring points into matrix form, the present invention introduces vectorized representation. First, the 4 rigid body motion freedom variables of the measuring point in equation (2) are integrated into a motion trend vector , which is defined as: ; wherein , , , have been clarified in equation (1) and equation (2). At the same time, the coefficients corresponding to each component of the motion trend vector in equation (2) are uniformly represented as the motion coefficient vector of the measuring point , which is defined as: ; wherein , , are the coordinates of the measuring point , is the distance of the measuring point to the axis. Substituting the newly defined motion coefficient vector and motion trend vector into equation (2), the linear vector expression of can be derived, and the specific form is shown in equation (3): ; Equation (3) succinctly expresses the Radius variation of a boundary measurement point Linear relationship with the overall rigid body motion of the measurement point set. This expression is the basis for constructing the motion coefficient matrix of the final discriminant inequality group formula (6).

[0047] After establishing the kinematic model formula (3), the invention further deduces the mathematical logic of the action boundary, which is the core basis for constructing the discriminant inequality group.

[0048] The invention combines the corresponding relationship between the tolerance principles (such as MMR, LMR) of hole shaft parts, the effective boundary, the fitting cylinder (the largest inscribed cylinder, the smallest circumscribed cylinder) and other related concepts, and organizes the basic logical form of the action boundary. As shown in Tables 1, 2 and 3: Table 1: Tolerance principles and effective boundaries of hole shaft elements

[0049] Table 2: Tolerance principles and action boundaries of hole shaft elements

[0050] Table 3: Basic logical form of action boundary

[0051] The basic logical form under different conditions is as follows: For holes, when MMR (maximum entity requirement) is used, the action boundary is the largest inscribed cylinder, and the basic logical form is .

[0052] For holes, when LMR (minimum entity requirement) is used, the action boundary is the smallest circumscribed cylinder, and the basic logical form is .

[0053] For shafts, when MMR (maximum entity requirement) is used, the action boundary is the smallest circumscribed cylinder, and the basic logical form is .

[0054] For shafts, when LMR (minimum entity requirement) is used, the action boundary is the largest inscribed cylinder, and the basic logical form is .

[0055] According to the above correspondence, the basic logical form of the action size is directly related to the smallest circumscribed and largest inscribed principles of the action boundary. Therefore, the invention generalizes the above four cases into a unified basic logical form of the action boundary, as shown in formula (4): ; Where: ​This is the boundary coefficient. This coefficient is determined based on the type of boundary condition: for a circumscribed cylinder (i.e., the smallest circumscribed cylinder)... For an inscribed cylinder (i.e., the largest inscribed cylinder) .

[0056] The motion trend vector, ; For the first The motion coefficient vector of each boundary measurement point.

[0057] For the first The change in radius of each boundary measuring point.

[0058] Formula (4) means that if a fitted cylinder is the boundary of action, then for any ( Possible small rigid body motions There must exist at least one ( A boundary measurement point Its radius variation trend It is non-negative (i.e.) This means that the measuring point either moves toward the outside of the boundary or remains on the boundary.

[0059] To facilitate the conversion of the logical form defined by formula (4) into a standardized discriminant analysis, this invention uses its mutually exclusive negation form for discrimination. The mutually exclusive negation of formula (4) is shown in formula (5): ; Among them, the introduction of As a relaxation constant less than 0, and its selection must satisfy... Formula (5) means: if there exists ( A tiny rigid body motion , so that all ( Boundary measuring points radius variation trend All are negative values ​​(i.e.) This means that all boundary points move toward the inside of the fitted cylinder, so the fitted cylinder is not an effective boundary. Based on this negation, the discriminant logic is transformed into: if formula (5) is not true, then its original proposition formula (4) is true, and the fitted cylinder is an effective boundary. If formula (5) is true, then its original proposition formula (4) is not true, and the fitted cylinder is not an effective boundary. In order to achieve a standardized solution, formula (5) is written in matrix form, i.e., formula (6). This set of inequalities, formula (6), is called the discriminant inequality set: ; in: is the number of boundary measurement points.

[0060] is the motion coefficient matrix of the boundary point set, which is constructed as The matrix is a matrix, where is the product of and .

[0061] is the coefficient vector of the boundary measurement point radius variation, which is constructed as is a order column vector.

[0062] is a column vector composed of relaxation constants B, which is constructed as is a order column vector.Ψ is the motion trend vector .

[0063] Therefore, the discrimination problem of the present application is finally converted into the problem of judging whether the linear inequality group formula (6) has a feasible solution: If formula (6) has a solution, formula (5) is established, and the fitted cylinder is not the active boundary.

[0064] If formula (6) has no solution, formula (5) is not established, and the fitted cylinder is the active boundary.

[0065] Referring to the accompanying Figure 1 , the discrimination method of the present application, in specific implementation, first performs the surface measurement point acquisition step (corresponding to the first step in the accompanying Figure 1 ).

[0066] In this step, the surface measurement point set of the measured hole or shaft is acquired using a measurement device . The specific measurement device can be a coordinate measuring machine (CMM), an on-machine measurement device, or other high-precision three-dimensional topography measurement instruments.

[0067] The measurement methods include but are not limited to contact measurement (such as trigger type probe sampling) or non-contact measurement (such as optical scanning). Through measurement, a series of three-dimensional coordinate points capable of representing the actual surface topography of the measured part are acquired, and the three-dimensional coordinate point set is the basis for all subsequent calculations and discriminations.

[0068] Referring to the accompanying Figure 1 and the accompanying Figure 4 , after acquiring the surface measurement point set, the method of the present application performs the pretreatment and coordinate transformation step (corresponding to the second step in the accompanying Figure 1 ).

[0069] This step is used to handle situations where, in actual measurement and evaluation, the initial coordinate system does not satisfy the model definition (i.e., the axis of the fitted cylinder is not aligned with the model). (In cases where axes coincide). For example, the coordinates of the obtained surface measurement points are in cylindrical coordinates, or as shown in the attached figure. Figure 4 As shown, the axis of the fitted cylinder is not parallel to... Axis coincidence.

[0070] This step transforms the coordinates of the measuring points into rectangular coordinates and aligns the axis of the boundary (fitted cylinder) with the coordinate system. The axes coincide. This coordinate transformation is based on the principle of robot kinematic coordinate transformation. By establishing a new coordinate system and calculating the coordinates of the boundary measurement points in the new coordinate system, the measurement point set data can meet the analysis requirements for constructing a system of discrimination inequalities.

[0071] See attached document Figure 1 After completing the preprocessing and coordinate transformation, the method of the present invention performs the step of constructing a system of inequalities (corresponding to the appendix). Figure 1 (Step 3).

[0072] This step utilizes the boundary measurement point set that meets the model requirements obtained in the previous step (step two). And the derived kinematic model and logical relationships are used to specifically construct the set of discriminant inequalities defined by formula (6). .

[0073] The construction process includes the following: First, determine the action boundary factor based on the type of evaluation task (e.g., whether it is evaluating a hole or a shaft, and whether it is based on the maximum material requirement (MMR) or the minimum material requirement (LMR). As before, for a circumscribed cylinder, For an inscribed cylinder, .

[0074] Secondly, regarding Each of the boundary measurement points : Calculate its to Distance of axis .

[0075] According to the definition of formula (3), calculate the motion coefficient vector of the measuring point. .

[0076] Will and Multiply to obtain the corresponding value for that measurement point. vector.

[0077] Next, all of them one vector As a row vector, the stacked combination is a motion coefficient matrix of order , i.e. .

[0078] Then, a relaxation constant less than is selected. The selection of the constant must satisfy .

[0079] Finally, using the relaxation constant , a column vector of order is constructed.

[0080] Through the above construction, the matrix and the vector are obtained, combined with the unknown motion trend vector , i.e. a linear inequality system required for discrimination is formed: .

[0081] Referring to the accompanying Figure 1 , after constructing the discrimination inequality system, the method of the present application performs the rank discrimination and iteration verification steps (corresponding to the fourth step in the accompanying Figure 1 ).

[0082] This step specifically includes: calculating the rank of the motion coefficient matrix of order , denoted as . Compare with the number of boundary measurement points .

[0083] The discrimination criterion is: if , it is determined that the fitted cylinder is not the action boundary, and the process is ended; otherwise (i.e. , because is at most 4 and must be greater than or equal to ), the fifth step is performed.

[0084] Referring to the accompanying Figure 1 , after performing the fourth step discrimination as otherwise (i.e. ), this step is performed: select the largest linearly independent group (corresponding to the fifth step in the accompanying Figure 1 ).

[0085] This step is the beginning of the iteration verification. The specific implementation is: in , select linearly independent motion coefficient vectors Set into the largest linearly independent matrix At the same time, using indivual assembled The rest Motion coefficient vectors Recorded as And solve the system of linear equations. Solution .

[0086] See attached document Figure 1 After performing step five, perform this step: verify the solution (corresponding to the attached document). Figure 1 (Step 6).

[0087] The specific step is: Calculation and relaxation constant Compare.

[0088] The criterion is: if (Right now All rows satisfy If the fitted cylinder is not the boundary condition, then the process ends. Otherwise (i.e.) There is at least one line Proceed to step seven.

[0089] See attached document Figure 1 After the sixth step determines otherwise, this step is executed: loop control and final determination (see attached diagram). Figure 5 Step 7).

[0090] This step specifically involves: checking whether all [items] have been examined. One (i.e. from) Select from the boundary points The largest possible linearly independent matrix (of all combinations of ) .

[0091] The criterion is: if all have been examined If all combinations are not determined to be non-functional boundaries in step six, then the fitted cylinder is finally determined to be a functional boundary, and the process ends. Otherwise (i.e., there are still unexamined combinations), skip to step five and continue selecting and analyzing the next largest linearly independent matrix. Combine and repeat steps five and six.

[0092] In one specific embodiment, refer to the appendix. Figure 5The specific implementation step 1 (measurement object and data source) of this invention is as follows: Select an actual shaft part as the measured object, and determine its tolerance principle as either the maximum material requirement (MMR) or the minimum material requirement (LMR) according to the design requirements of the workpiece. The tolerance requirements of the measured shaft part are attached. ​ As shown.

[0093] Step 2 (initial data acquisition) of the specific embodiment of the present invention is as follows: In this embodiment, an unconstrained objective optimization problem is calculated to initially determine the candidate action boundary (fitted cylinder) and its corresponding boundary measurement points. This step is performed in the MATLAB R2022a software environment.

[0094] Specifically, it involves solving an unconstrained objective optimization problem as shown in equation (7). The goal of this equation is to find a set of rigid body motion fine-tuning quantities (translation quantities). , and rotation amount , This minimizes the maximum distance (i.e., the minimum circumscribed cylinder radius) from all measured points to the fitted cylinder axis after adjustment. Formula (7) expresses the diameter of this minimum circumscribed cylinder. .

[0095] ; To solve this optimization problem, this embodiment calls the built-in MATLAB function `fminunc` for solving unconstrained objective optimization problems. This function is used to find the minimum value of a multivariable function. The solution method is set to the built-in Gauss-Newton method, an iterative algorithm commonly used to solve nonlinear least squares problems. The initial solution of the optimization (i.e., , , , The initial value of is set to the zero vector, indicating that the iterative search starts from the unadjusted initial state of the measurement point set.

[0096] After calculation, the functional dimension (i.e., the diameter of the smallest circumscribed cylinder) of the measured shaft part was obtained as follows: mm.

[0097] After obtaining the functional dimension and the corresponding cylindrical position and orientation, the five points closest to the fitted cylindrical surface are selected from all the original surface measurement points as boundary points for subsequent discriminant analysis. The evaluation results of these five boundary points are shown in Table 4.

[0098] Table 4: Evaluation Results (mm)

[0099] Steps 3 and 4 (initial data acquisition for constructing a system of discrimination inequalities) in the specific embodiment of the present invention are as follows: Step 3: Perform model transformation, establish a suitable coordinate system, and transform the coordinates of the measured points into rectangular coordinates. If no preprocessing is required, proceed directly to Step 4. No preprocessing is required in this case.

[0100] Step 4: This step is to determine the system of inequalities. Prepare initial data. First, specify the relaxation constant. This embodiment has There are boundary measurement points, therefore specifying indivual and group them into .

[0101] Secondly, determine the action boundary coefficient. According to Table 2, this embodiment evaluates the maximum material requirement (MMR) of the shaft (geometric element), and its corresponding functional boundary is the minimum circumscribed cylinder. According to the definition of formula (4) in the invention, for a circumscribed cylinder, .

[0102] Then, construct the motion coefficient matrix. Substitute the coordinates of the five boundary measurement points obtained in step 2 (corresponding to the Gauss-Newton method) (i.e., the adjusted coordinates of measurement points numbered 2, 8, 4, 22, and 23 in Table 4) into formula (3). The motion coefficient vector corresponding to each boundary point can be obtained. .

[0103] Finally, according to formula (6) ( The definition of ) is to assign the 5 boundary points to each other. (exist In the case of The set of itself is Motion coefficient matrix of order ,Right now: ; Step 5 of the specific embodiment of the present invention (corresponding to step 4: rank determination) is as follows: Step 5: Calculate the motion coefficient matrix rank and the number of boundary measurement points Comparison. If If the fitted cylinder is not the boundary condition, then proceed to step 6; otherwise, proceed to step 6.

[0104] In this embodiment, the result obtained in step 4 1-th order matrix Perform the calculation: ; Since the calculated rank is less than the number of boundary points , the result is no, thus, go to step 6.

[0105] Steps 6, 7 and 8 of the embodiment of the invention (corresponding to the fifth, sixth and seventh steps of the method embodiment: iterative solving and checking) are as follows: Step 6: Select linearly independent motion coefficient vectors from to form the largest linearly independent matrix ; at the same time, select to form ; the remaining motion coefficient vectors are denoted as ; and solve for the solution . In this embodiment, . We select the first row of the matrix (corresponding to the point number 2) as , and the last 4 rows of the matrix (corresponding to the point numbers 8, 4, 22, 23) as . Use 4 to construct . Then, use the built-in matrix left division operator of Matlab R2022a to solve the linear equation group for the solution, and obtain: ; ; Step 7: If , the fitted cylinder is not the acting boundary; otherwise, go to step 8.

[0106] In this embodiment,

[0107] ; Compare the result with . Since -1.0959 is not less than -1 (i.e. ), the result is no, thus, go to step 8.

[0108] Step 8: If all largest linearly independent matrices have been examined, the fitted cylinder is the acting boundary; otherwise, jump to step 6 to continue selecting and analyzing the next largest linearly independent matrix.

[0109] In this embodiment,​​ Therefore, there are at most We need to examine five different combinations of maximally linearly independent matrices. .

[0110] Suppose we examine the 4th group (for example, selecting...) Lines 1, 2, 3, and 5 as The vector corresponding to the fourth row, i.e., measurement point number 22, is used as... The solution obtained by solving is .calculate Assuming the result is .

[0111] Comparison results and Since -1.0959 < -1 (i.e., ... ( ), which meets the conditions.

[0112] At this point, based on the discrimination criterion in step 7, it is concluded that the fitted cylinder constructed by the Gauss-Newton method is not the boundary of action.

[0113] The concluding summary of this invention, namely the technical problem it aims to solve, the core method it employs, and the beneficial effects it achieves, is as follows: This invention proposes a general discrimination method based on a boundary point matrix for the functional boundary determination of hole-shaft parts. This method establishes a functional boundary model for hole-shaft parts by analyzing the relative positions and distances between the hole-shaft measurement point set and the functional boundary, and performs model transformation under typical coordinate systems. The basic mathematical logic form of functional boundary determination is explored (e.g., formula (4)). ) and its corollaries (such as formula (6) ), using the boundary point matrix ( Properties of ) (such as its rank) This invention transforms logical problems into a general discrimination method. It solves the problems of complex geometric discrimination conditions and lack of universality in existing evaluation methods, providing a reliable basis for determining the functional boundaries of hole and shaft parts.

Claims

1. A method for determining the functional boundary of a hole-shaft type part, characterized in that, Includes the following steps: S1. Obtain multiple surface measurement points of the shaft or hole being measured; S2. Perform preprocessing, establish a coordinate system, and transform the coordinates of the surface measuring points into rectangular coordinates; S3. Determine the boundary coefficients, construct the motion coefficient vector and combine it into a motion coefficient matrix, select relaxation constants, and form a set of discrimination inequalities and the corresponding hyperplane equations. S4. Calculate the rank of the motion coefficient matrix and compare it with the number of surface measurement points; S5. Select a maximally linearly independent matrix from the motion coefficient matrix and solve the corresponding hyperplane equations to obtain a motion trend vector solution; S6. Verify whether the solution of the motion trend vector satisfies the discrimination inequality corresponding to the remaining motion coefficient vector; if it does, determine that the fitted cylinder is not the boundary of action; otherwise, proceed to the next step. S7. Check if all possible combinations of maximum linearly independent matrices have been examined; if so, determine that the fitted cylinder is the boundary of action; otherwise, continue analyzing the next combination of maximum linearly independent matrices.

2. The method for determining the functional boundary of a hole-shaft type part according to claim 1, characterized in that, The establishment of the coordinate system in step S2 includes: establishing a specific coordinate system such that the z-axis of the specific coordinate system coincides with the axis of the fitted cylinder; Furthermore, the preprocessing in step S2 also includes converting the obtained cylindrical coordinates of the surface measurement points into rectangular coordinates in the specific coordinate system.

3. The method for determining the functional boundary of a hole-shaft type part according to claim 1, characterized in that, The motion coefficient vector in step S3 is constructed based on the rectangular coordinates of the surface measuring points, and the coefficients contained therein correspond to the four rigid body motion degrees of freedom variables, which include: translation along the x-axis, translation along the y-axis, rotation around the x-axis, and rotation around the y-axis.

4. The method for determining the functional boundary of a hole-shaft type part according to claim 3, characterized in that, The motion coefficient vector is derived from the radius change of the surface measuring point; the radius change is obtained by calculating the orientation change vector of the surface measuring point caused by the four rigid body motion degrees of freedom variables and the projection onto the unit normal vector at the surface measuring point through dot product.

5. The method for determining the functional boundary of a hole-shaft type part according to claim 1, characterized in that, In step S3, the boundary coefficient of the action of the circumscribed cylinder is determined to be 1, and the boundary coefficient of the action of the inscribed cylinder is determined to be -1. Furthermore, the set of discrimination inequalities in step S3 is based on the following mutually exclusive negation propositions: If there exists a small rigid body motion such that the radius change trend of all surface measuring points is negative and less than the relaxation constant, then the fitted cylinder is not the working boundary.

6. The method for determining the functional boundary of a hole-shaft type part according to claim 1, characterized in that, The selection of a relaxation constant less than zero in step S3 includes selecting a value that is less than zero and greater than the maximum value of the radius variation trend of all surface measuring points.

7. The method for determining the functional boundary of a hole-shaft type part according to claim 1, characterized in that, In step S1, the surface measurement points are obtained using a surface coordinate measuring device in the mechanical field. The surface coordinate measuring device in the mechanical field includes a coordinate measuring machine, an in-machine measuring device, or a three-dimensional topography measuring instrument.

8. The method for determining the functional boundary of a hole-shaft type part according to claim 1, characterized in that, In step S5, the set of motion coefficient vectors with the same rank number is selected to form the maximum linearly independent matrix, and the remaining motion coefficient vectors are denoted as the residual motion coefficient vectors. In step S6, it is verified whether the motion trend vector solution makes the remaining motion coefficient vectors all satisfy the condition that they are all less than the relaxation constant. In step S7, by jumping back to step S5, the next maximum linearly independent matrix combination is selected and analyzed.

9. The method for determining the functional boundary of a hole-shaft type part according to claim 1, characterized in that, The method for determining the functional boundary of hole-shaft type parts also includes a step of determining the fitted cylinder before step S1: By solving an unconstrained objective optimization problem, a candidate fitting cylinder and its corresponding boundary measurement points can be initially determined.

10. The method for determining the functional boundary of a hole-shaft type part according to claim 9, characterized in that, The objective of the unconstrained objective optimization problem is to find a set of rigid body motion fine-tuning values ​​that minimizes the maximum distance from all surface measuring points to the fitted cylinder axis after adjustment.

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