Non-probabilistic reliability analysis method for positioning accuracy of rotary shaft based on ellipsoid convex model

Abstract

This invention discloses a non-probabilistic reliability analysis method for rotary axis positioning accuracy based on an ellipsoidal convex model. The invention proposes an uncertainty quantification and unified modeling method for bidirectional positioning errors of machine tool rotary axes. By representing the forward and reverse positioning error vectors using low-dimensional error patterns, it achieves a unified characterization of the main variation characteristics of the entire positioning error vector, simultaneously reflecting the overall variation law and differences in bidirectional positioning errors, facilitating subsequent uncertainty modeling and reliability analysis. Based on small-sample bidirectional positioning error test data, this invention constructs an ellipsoidal convex uncertainty domain and performs non-probabilistic envelope characterization of the error discrete range. It does not require prior assumption that the error follows a specific probability distribution, making it suitable for engineering scenarios with limited sample sizes and incomplete distribution information in machine tool rotary axis positioning accuracy tests.

Description

Technical Field

[0001] This invention relates to the field of mechanical engineering technology, specifically a method for quantifying the uncertainty and performing non-probabilistic reliability analysis of the positioning accuracy of a machine tool rotary table based on an ellipsoidal convex model. Background Technology

[0002] Multi-axis CNC machine tools are widely used in aerospace, precision mold making, and complex curved surface parts. Vertical machining centers equipped with CNC rotary tables are commonly used in practical engineering. In these machine tools, the rotary table axis serves as the main rotational degree of freedom in the workpiece coordinate system. Its rotational error is transmitted to the relative posture of the tool and workpiece through the kinematic chain, which is an important factor affecting the machine tool's volumetric accuracy and the quality of multi-axis linkage machining.

[0003] Existing methods for analyzing the positioning accuracy of machine tool rotary table shafts mostly employ repeated experimental statistics, error mean analysis, or extreme value analysis to evaluate the error level of the shaft at several angular positions. While these methods can reflect the static accuracy characteristics of the shaft to some extent, most remain at the level of describing single-point errors or single indicators, failing to characterize the overall characteristics of the entire positioning error vector as a function of angle. For forward and reverse positioning errors, their variation patterns are usually not entirely consistent; relying solely on traditional statistical comparisons is insufficient to reveal the main variation patterns and correlations within the error vector. The positioning error of a machine tool rotary table shaft exhibits significant uncertainty, stemming from the combined effects of various factors such as manufacturing and assembly deviations, transmission chain errors, friction and clearance, thermal fluctuations, and environmental disturbances during testing. In actual testing, even after repeated trials under the same operating conditions, the resulting positioning error vector still exhibits a certain degree of dispersion. This dispersion is often manifested in multiple aspects, including variations in the amplitude of the entire error vector, local peak values, and vector shape, rather than simply random fluctuations at a single angular point.

[0004] Most existing reliability analysis methods are based on probabilistic statistical models, which typically require a large sample size and pre-assume that the error follows a certain probability distribution. Machine tool rotary table positioning accuracy experiments are limited by testing costs, timelines, and on-site conditions, resulting in a limited number of samples and a significant small-sample problem. Under these conditions, the error distribution is often difficult to accurately define. Directly using reliability analysis methods based on probability distribution assumptions may lead to inconsistencies between the modeling results and actual error characteristics, thus affecting the credibility of the reliability evaluation conclusions. The convex model method provides a feasible approach for error characterization under conditions of small samples and incomplete uncertain information. This method does not rely on pre-assumptions about the error probability distribution but instead constructs a bounded uncertainty set to envelope the discrete range of the sample in the parameter space. Introducing the convex model method into the positioning accuracy analysis of machine tool rotary table helps to characterize the variation boundaries of forward and reverse positioning errors under small-sample conditions and further conduct non-probabilistic reliability analysis.

[0005] Therefore, a method for uncertainty quantification and non-probabilistic reliability analysis of the positioning accuracy of the rotary axis of a machine tool is proposed. Under the condition of not relying on the probability distribution assumption, the bidirectional positioning error data is effectively modeled, and the uncertainty domain is characterized and the reliability is determined by combining the convex model. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention provides a non-probabilistic reliability analysis method for the positioning accuracy of a rotary axis based on an ellipsoidal convex model.

[0007] The technical solution of this invention to solve the aforementioned technical problem is to provide a non-probabilistic reliability analysis method for the positioning accuracy of a rotation axis based on an ellipsoidal convex model, characterized in that the method includes the following steps: Step 1: Fix the reflector on the horizontal plane of the rotary table of the CNC machine tool, ensuring that the parallelism error between the mounting plane of the reflector and the plane of the rotary table is maintained within a reasonable range; the reflector is located within the optimal measurement range of the laser interferometer; the computer is connected to the laser interferometer; the CNC system is connected to the rotation axis of the rotary table. Step 2: The CNC system sends indexing commands to the rotary axis, causing the rotary axis to move sequentially to the position of each discrete command angle within the command angle range [0, 2π] at a fixed angular step Δα, obtaining m discrete command angles, where m = 2π / Δα; at each discrete command angle φ, positioning is completed using both forward and reverse approach methods; during the positioning process, the reflector moves synchronously with the rotary axis, and the laser interferometer collects the measurement signals of the reflector in real time, transmitting the collected measurement signals to the computer for recording, and obtaining the bidirectional positioning error test data for each discrete command angle φ on the computer; Step 3: Based on the bidirectional positioning error test data obtained in Step 2, construct the positive error matrix E respectively. + and the inverse error matrix E - They were then subjected to mean-removal processing to obtain the positively centered error matrix X. + and the reverse-centered error matrix X - Then, based on the positively centered error matrix X + and the reverse-centered error matrix X - Construct the positive covariance matrix C respectively + and the inverse covariance matrix C - Then, principal component analysis is used to extract the principal error patterns and corresponding eigenvalues ​​of each element; then, the positively centered error matrix X is... + and the reverse-centered error matrix X - Projecting the vectors onto the low-dimensional subspace formed by the corresponding principal error patterns yields the positive error pattern coefficient vector α for each set of repeated localization experiments. i + The reverse error mode coefficient vector α for each group i - Positive error mode coefficient matrix A + and the inverse error mode coefficient matrix A - This enables low-dimensional parameterized representation of bidirectional positioning error test data. Step 4: Calculate the positive error mode coefficient vector α for all groups obtained in Step 3. i + and the inverse error mode coefficient vector α of all groups i - The mean vector, covariance matrix, and envelope radius are used to construct a positive ellipsoidal convex uncertainty region Ω representing the rotation axis positioning error in the positive error mode coefficient space and the negative error mode coefficient space, respectively. + and the convex uncertainty region Ω of the inverse ellipsoid - ; Step 5: Based on the positive ellipsoidal convex uncertainty region Ω obtained in Step 4 + and the convex uncertainty region Ω of the inverse ellipsoid - With the positively oriented ellipsoidal convex uncertainty region Ω + The positive error mode coefficient vector α of a certain group in + and the convex uncertainty region Ω of the inverse ellipsoid - The inverse error mode coefficient vector α of a certain group - To optimize the vector, the mapping relationship between the error mode coefficient vector and the reconstruction error vector is combined to obtain a set of positive error mode coefficient vectors α. + and a certain set of inverse error mode coefficient vectors α - The positive positioning error Δθ corresponding to the discrete command angle φ +(φ,α + and reverse positioning error Δθ - (φ,α - Then, the positive positioning error Δθ + (φ,α + and reverse positioning error Δθ - (φ,α - We conducted point-by-point worst upper bound analysis and full-angle worst sample search to obtain the point-by-point worst upper bound envelope, full-angle worst sample vector, and corresponding worst peak value of the forward and reverse positioning errors, so as to characterize the worst case analysis results of the machine tool rotary table rotation axis positioning error under the ellipsoidal convex uncertainty domain. Step 6: Based on the positive ellipsoidal convex uncertainty region Ω obtained in Step 4 + and the convex uncertainty region Ω of the inverse ellipsoid - Combining the allowable threshold for the positioning error of the machine tool rotary table axis and the positive positioning error Δθ in step 5 + (φ,α + and reverse positioning error Δθ - (φ,α - Construct non-probabilistic reliability indices and overall reliability indices at each discrete command angle φ; then compare the envelope radius ρ of the positive ellipsoidal convex uncertainty domain obtained in step 4. + The envelope radius ρ of the convex uncertainty region of the inverse ellipsoid - By comparing the overall reliability index with the non-probabilistic reliability analysis results of the machine tool rotary table axis positioning error under uncertain set constraints, the danger angles in the corresponding directions are identified. Step 7: Based on the worst-case analysis results obtained in Step 5 and the non-probabilistic reliability analysis results obtained in Step 6, output the uncertainty quantification results of the positioning error of the machine tool rotary table axis and give the corresponding reliability judgment conclusion.

[0008] Compared with the prior art, the beneficial effects of the present invention are as follows: (1) This invention proposes an uncertainty quantification and unified modeling method for bidirectional positioning error of machine tool rotary table axis. By representing the forward and reverse positioning error vectors in a low-dimensional error pattern, it realizes a unified characterization of the main change characteristics of the entire positioning error vector. It can simultaneously reflect the overall change law and difference characteristics of bidirectional positioning error, which is convenient for subsequent uncertainty modeling and reliability analysis.

[0009] (2) Based on small sample bidirectional positioning error test data, the present invention constructs an ellipsoidal convex uncertainty domain and performs non-probabilistic envelope characterization of the error discrete range. It does not require prior assumption that the error follows a specific probability distribution and is suitable for engineering scenarios with limited sample size and incomplete distribution information in machine tool rotary axis positioning accuracy tests.

[0010] (3) By conducting point-by-point worst upper bound analysis and full-angle worst sample search, this invention can obtain both the conservative upper bound of the positioning error at each discrete command angle and the real, achievable worst complete error vector, thereby realizing a multi-level characterization of the worst case of the positioning error of the machine tool rotary table axis, which is beneficial for identifying the location of local dangerous angles and the overall most unfavorable error structure.

[0011] (4) This invention establishes a non-probabilistic reliability determination method for bidirectional positioning error of machine tool rotary axis. It can complete reliability analysis under the condition of no probability distribution assumption, and output the worst case analysis result diagram and reliability conclusion. It can be used for machine tool rotary axis accuracy evaluation and rapid analysis in engineering field. Attached Figure Description

[0012] Figure 1 This is a schematic diagram of the overall process of the present invention; Figure 2 This is a schematic diagram of the bidirectional positioning test and test scenario of the present invention; Figure 3 This is a diagram showing the sample distribution of the forward and reverse error mode coefficients and the envelope of the ellipsoidal convex domain in this invention. Figure 4 This is a diagram showing the worst-case upper bound analysis results of the bidirectional positioning error point by point according to the present invention. Figure 5 This is a search result image of the worst-case sample for bidirectional positioning error across all angles in this invention. Figure 6 This is a diagram showing the results of the non-probabilistic reliability analysis of the bidirectional positioning error of the present invention.

[0013] In the diagram, 1 is a computer, 2 is a laser interferometer, 3 is a numerical control system, 4 is a reflector, 5 is a rotary table, and 6 is a rotary axis. Detailed Implementation

[0014] Specific embodiments of the present invention are given below. These specific embodiments are only used to further illustrate the present invention in detail and do not limit the scope of protection of the present invention.

[0015] This invention provides a non-probabilistic reliability analysis method for the positioning accuracy of a rotation axis based on an ellipsoidal convex model (hereinafter referred to as the method), characterized by the following steps: Step 1: Experimental Scenario Setup: The reflector 4 is fixed to the horizontal plane of the rotary table 5 of the CNC machine tool using a high-precision positioning fixture, ensuring that the parallelism error between the mounting plane of the reflector 4 and the plane of the rotary table 5 is maintained within a reasonable range. The laser interferometer 2 is placed on an independent stable support outside the machining center. The position and pitch angle of the laser interferometer 2 are adjusted until the laser beam of the laser interferometer 2 is projected onto the surface of the reflector 4 without obstruction and forms a stable reflection. The reflector 4 is located within the optimal measurement range of the laser interferometer 2. The laser interferometer 2 is used to emit and receive laser beams and form angle measurement signals. The computer 1 is connected to the laser interferometer 2 for setting measurement parameters, triggering sampling, and receiving angle measurement data. The CNC system 3 is connected to the rotary axis 6 of the rotary table 5 for issuing discrete command angles to the rotary axis 6, controlling the movement direction of the rotary axis 6, and driving the rotary axis 6 to position itself at the set angle. Preferably, in step 1, the bottom of the independent stabilizing bracket on which the laser interferometer 2 is placed is equipped with a shock-absorbing pad, and other high-power vibration equipment in the test scene is turned off to reduce the interference of external vibration on the measurement data.

[0016] Preferably, in step 1, after the laser interferometer 2 is installed, it is confirmed that the grease lubrication system of the rotary shaft 6 is supplying oil normally, and the machine tool is started to run idle for preheating. The preheating time is not less than 30 minutes, until the temperature of the rotary shaft 6 on the control panel of the CNC system 3 is stable and there is no abnormal vibration during operation, and then the idle running is stopped.

[0017] Preferably, in step 1, before the test begins, the computer 1 is debugged and a fixed delay is set so that the measurement readings enter a steady state before sampling, ensuring that the laser interferometer 2 can accurately acquire the positioning error test data of the rotating shaft 6.

[0018] Step 2, Experimental Data Acquisition: The CNC system 3 sends indexing commands to the rotary axis 6, causing the rotary axis 6 to move sequentially to the position of each discrete command angle within the command angle range [0, 2π] at a fixed angular step Δα, obtaining m discrete command angles, where m = 2π / Δα; at each discrete command angle φ, positioning is completed using both forward approach and reverse approach methods; during the positioning process, the reflector 4 moves synchronously with the rotary axis 6, and the laser interferometer 2 collects the measurement signals of the reflector 4 in real time, and transmits the collected measurement signals to the computer 1 for recording, obtaining the bidirectional positioning error test data for each discrete command angle φ on the computer 1; Preferably, in step 2, the positioning is completed by using both forward approach and reverse approach methods. Specifically, for each discrete command angle φ, the positioning is performed by approaching clockwise and counterclockwise respectively, and n sets of repeated positioning tests are conducted under the same speed, acceleration and deceleration time and positioning criterion conditions. The computer 1 records the positioning error test data of each set to ensure the comparability of the bidirectional positioning error test data and to avoid the masking of direction-related error characteristics.

[0019] Preferably, in step 2, the positioning error of the i-th group of repeated positioning experiments under the discrete command angle φ and the direction of motion d is denoted as e. i d (φ), where i represents the group number, i=1,2,…,n; d represents the direction of motion, d=+1 represents the forward approach, d=-1 represents the reverse approach; φ represents the discrete command angle.

[0020] Step 3: Based on the bidirectional positioning error test data obtained in Step 2, construct the positive error matrix E respectively. + and the inverse error matrix E - They were then subjected to mean-removal processing to obtain the positively centered error matrix X. + and the reverse-centered error matrix X - Then, based on the positively centered error matrix X + and the reverse-centered error matrix X - Construct the positive covariance matrix C respectively + and the inverse covariance matrix C - Then, principal component analysis is used to extract the principal error patterns and corresponding eigenvalues ​​of each element; then, the positively centered error matrix X is... + and the reverse-centered error matrix X - Projecting onto the low-dimensional subspace formed by the corresponding principal error patterns yields the positive error pattern coefficient vector α for each group of repeated localization experiments (referred to as each group). i + The reverse error mode coefficient vector α for each group i - Positive error mode coefficient matrix A + and the inverse error mode coefficient matrix A - This enables low-dimensional parameterized representation of bidirectional positioning error test data. Preferably, in step 3, the positively centered error matrix X is obtained. + and the reverse-centered error matrix X - The specific steps are as follows: A31. Based on the bidirectional positioning error test data of n sets of repeated positioning tests at the discrete command angle φ, construct the positive error matrix E respectively. + and the inverse error matrix E - : (1) Positive error matrix E + and the inverse error matrix E - Each row corresponds to a set of repeated positioning experiments, and each column corresponds to bidirectional positioning error test data for a discrete command angle φ; therefore, the positive error vector e corresponding to the i-th set of repeated positioning experiments is... i+ and the inverse error vector e i - They are represented as follows: (2) Meanwhile, for the positive error matrix E + and the inverse error matrix E - The average error is calculated at each discrete command angle φ to obtain the positive average error vector. + and inverse average error vector - : (3) A32, for each group's positive error vector e i + and the inverse error vector e of each group i - After removing the mean, the positively centered error vector x for each group is obtained. i + and the reverse-centered error vector x for each group i - : (4) A33. The positively centered error vector x of all repeated localization experiments (referred to as all groups). i + Arrange the matrices to obtain the positively centered error matrix X. + ; center the error vector x of all groups in reverse order i - The reverse-centered error matrix X is obtained by arranging the matrix. - : (5) Preferably, in step 3, the positive error mode coefficient vector α of each group of repeated positioning experiments is obtained. i + The reverse error mode coefficient vector α for each group i - Positive error mode coefficient matrix A + and the inverse error mode coefficient matrix A - The specific steps are as follows: B31, Based on the positive centering error matrix X + and the reverse-centered error matrix X - Construct the corresponding covariance matrices as follows: (6) In equation (6), C + Let C represent the positive covariance matrix.- Represents the inverse covariance matrix; B32. For the positive covariance matrix C, respectively + and the inverse covariance matrix C - Principal component analysis is performed, and the principal error patterns and corresponding eigenvalues ​​are obtained through eigenvalue decomposition. Their expressions are as follows: (7) In equation (7), ψ r + and ψ r - Let λ represent the r-th positive principal error mode and the r-th negative principal error mode, respectively; r + and λ r - These represent the corresponding eigenvalues; B33. Sort the eigenvalues ​​from largest to smallest, and determine the number of main error patterns to retain based on the cumulative contribution rate. The preferred cumulative contribution rate threshold is 95% (used to determine the value of r in the r-th positive main error pattern and the r-th negative main error pattern above; for example, if there are 5 main error patterns in total, and the cumulative contribution rate of the first two main error patterns is 95%, then k...). + =2); Assume the positive principal error mode retains the first k + There are several modes, and the reverse principal error mode retains the first k... - The retained positive principal error modes form the positive principal error mode matrix Ψ. + The retained inverse principal error modes constitute the inverse principal error mode matrix Ψ. - : (8) B34. Center the error vector x of each group in a positive direction. i + Projected onto the positive principal error mode matrix Ψ + The low-dimensional subspace formed will centralize the reverse-center error vector x of each group. i - Projected onto the inverse principal error mode matrix Ψ - The low-dimensional subspace formed yields the positive error mode coefficient vector α for each group. i + and the inverse error mode coefficient vector α of each group i - : (9) Similarly, the positive error mode coefficient matrix A of n sets of repeated positioning experiments + and the inverse error mode coefficient matrix A - They are respectively: (10) Step 4: Calculate the positive error mode coefficient vector α for all groups obtained in Step 3. i + and the inverse error mode coefficient vector α of all groups i - The mean vector, covariance matrix, and envelope radius are used to construct a positive ellipsoidal convex uncertainty region Ω representing the rotation axis positioning error in the positive error mode coefficient space and the negative error mode coefficient space, respectively. + and the convex uncertainty region Ω of the inverse ellipsoid - ; Preferably, in step 4, the positive error mode coefficient space is defined as k + Dimension Space The reverse error mode coefficient space is defined as k - Dimension Space Step 3 yields the positive error mode coefficient vector α for each group. i + Satisfy α i + ∈ The inverse error mode coefficient vector α for each group i - Satisfy α i - ∈ ; Positive error mode coefficient vector α for all groups i + Constructing the positive error mode coefficient sample set S + The inverse error mode coefficient vector α for all groups i - Constructing the inverse error mode coefficient sample set S - , respectively represented as: (11) Preferably, in step 4, the positive error mode coefficient vector α of all groups is processed respectively. i + and the inverse error mode coefficient vector α of all groups i - Calculate the mean to obtain the mean vector of the positive error mode coefficients. + and the mean vector of the inverse error mode coefficients - : (12); Preferably, in step 4, the positive error mode coefficient vector α of all groups is used as the basis for the calculation. i +and the inverse error mode coefficient vector α of all groups i - Construct the corresponding covariance matrix + and - Its expression is: (13); Preferably, in step 4, to determine the envelope radius of the convex uncertainty region of the ellipsoid, the positive error mode coefficient vector α for each group is calculated. i + Relative to the mean vector of positive error mode coefficients + The Mahalanobis distance, and the inverse error mode coefficient vector α for each group. i - Relative to the mean vector of the inverse error mode coefficients - The Mahalanobis distance is expressed as: (14) In equation (14), d i + and d i - Let α represent the coefficient vector of the i-th positive error mode. i + and the i-th group of reverse error mode coefficient vector α i - Mahalanobis distance to their respective mean vectors; Then take the positive error mode coefficient vector α of all groups. i + and the inverse error mode coefficient vector α of all groups i - The maximum value of the Mahalanobis distance is taken as the envelope radius of the corresponding convex uncertain region of the ellipsoid, i.e.: (15) In equation (15), ρ + and ρ - Let represent the envelope radii of the forward ellipsoidal convex uncertainty region and the reverse ellipsoidal convex uncertainty region, respectively.

[0021] Preferably, in step 4, the positive error mode coefficient vector α obtained in step 3 is used... i + and the inverse error mode coefficient vector α i - The mean vector, covariance matrix, and envelope radius are used to construct the convex uncertainty region Ω of the positive ellipsoid. + and the convex uncertainty region Ω of the inverse ellipsoid - : (16) In equation (16), the positive error mode coefficient space is defined as k + Dimension Space The reverse error mode coefficient space is defined as k - Dimension Space .

[0022] Step 5: Based on the positive ellipsoidal convex uncertainty region Ω obtained in Step 4 + and the convex uncertainty region Ω of the inverse ellipsoid - With the positively oriented ellipsoidal convex uncertainty region Ω + The positive error mode coefficient vector α of a certain group in + and the convex uncertainty region Ω of the inverse ellipsoid - The inverse error mode coefficient vector α of a certain group - To optimize the vector, the mapping relationship between the error mode coefficient vector and the reconstruction error vector is combined to obtain a set of positive error mode coefficient vectors α. + and a certain set of inverse error mode coefficient vectors α - The positive positioning error Δθ corresponding to the discrete command angle φ + (φ,α + and reverse positioning error Δθ - (φ,α - Then, the positive positioning error Δθ + (φ,α + and reverse positioning error Δθ - (φ,α - We conducted point-by-point worst upper bound analysis and full-angle worst sample search to obtain the point-by-point worst upper bound envelope, full-angle worst sample vector, and corresponding worst peak value of the forward and reverse positioning errors, so as to characterize the worst case analysis results of the machine tool rotary table rotation axis positioning error under the ellipsoidal convex uncertainty domain. Preferably, in step 5, the positive positioning error Δθ is obtained. + (φ,α + and reverse positioning error Δθ - (φ,α - Specifically: The mapping relationship between the error mode coefficient vector and the reconstruction error vector is expressed as follows: (17) In equation (17), e + (α + ) and e - (α - ) represent the positive error mode coefficient vector α + and the inverse error mode coefficient vector α -The determined forward reconstruction error vector and backward reconstruction error vector; + It is represented as a positive average error vector. - Represented as the inverse average error vector; Ψ + Ψ represents the positive principal error mode matrix. - Represents the inverse principal error mode matrix; A set of positive error mode coefficient vectors α + ∈Ω + and a certain set of inverse error mode coefficient vectors α - ∈Ω - The forward positioning error and the reverse positioning error corresponding to the discrete command angle φ are denoted as Δθ, respectively. + (φ,α + ) and Δθ - (φ,α - ): (18) In equation (18), [Ψ + α + ] φ and [Ψ] - α - ] φ Representing vectors Ψ + α + and Ψ - α - The component corresponding to the discrete command angle φ.

[0023] Preferably, in step 5, the point-by-point worst upper bound envelope is obtained through point-by-point worst upper bound analysis. The specific steps are as follows: A51. For each discrete command angle φ, in the positively oriented ellipsoidal convex uncertainty region Ω... + and the convex uncertainty region Ω of the inverse ellipsoid - Internal relative positive positioning error Δθ + (φ,α + and reverse positioning error Δθ - (φ,α - By performing pointwise maximization, the worst-case upper bound at each discrete command angle φ is obtained: (19) In equation (19), and These represent the point-by-point worst upper bounds of the forward and reverse positioning errors at the discrete command angle φ, respectively, under the convex uncertainty domain of the ellipsoid. A52. Arrange the point-by-point worst upper bounds obtained at each discrete command angle φ in angular order to obtain the point-by-point worst upper bound envelopes of the positive positioning error. The worst-case upper bound envelope of the point-by-point positioning error Its expression is: (20).

[0024] Preferably, in step 5, the worst-case sample vector and the corresponding worst peak value are obtained through a worst-case sample search from all angles. The specific steps are as follows: B51. Full-angle worst-case search refers to calculating the worst-case vector under fixed error mode coefficients, defining the positive full-angle peak function h. + (α + ) and the inverse full-angle peak function h - (α - Its expression is: (twenty one) In the convex uncertainty region Ω of the positive ellipsoid + and the convex uncertainty region Ω of the inverse ellipsoid - The inner part performs a maximization search on the full-angle peak function to obtain the positive fixed worst-case error mode coefficient vector α. +* and the inverse fixed worst-case error mode coefficient vector α -* : (twenty two); B52. The forward-fixed worst-case error mode coefficient vector α... +* and the inverse fixed worst-case error mode coefficient vector α -* Substituting the reconstruction relation into equation (17) respectively, we obtain the worst sample vector of the positive full angle. and the worst sample vector of the reverse full angle : (twenty three) B53, respectively for the worst sample vector of the positive full angle and the worst sample vector of the reverse full angle Taking the maximum value within the entire range of discrete command angles yields the worst-case positive and worst-case negative peak values, whose expressions are as follows: (twenty four) In equation (24), and These represent the worst-case sample vectors in the positive all-angle orientation, respectively. and the worst sample vector of the reverse full angle The maximum positioning error across the entire range of discrete command angles.

[0025] Step 6: Based on the positive ellipsoidal convex uncertainty region Ω obtained in Step 4 + and the convex uncertainty region Ω of the inverse ellipsoid- Combining the allowable threshold for the positioning error of the machine tool rotary table axis and the positive positioning error Δθ in step 5 + (φ,α + and reverse positioning error Δθ - (φ,α - Construct non-probabilistic reliability indices and overall reliability indices at each discrete command angle φ; then compare the envelope radius ρ of the positive ellipsoidal convex uncertainty domain obtained in step 4. + The envelope radius ρ of the convex uncertainty region of the inverse ellipsoid - By comparing the overall reliability index with the non-probabilistic reliability analysis results of the machine tool rotary table axis positioning error under uncertain set constraints, the danger angles in the corresponding directions are identified. Preferably, in step 6, non-probabilistic reliability indices are constructed at each discrete command angle φ, and the specific steps are as follows: A61. Suppose the allowable threshold for the positioning error of the rotary axis of a machine tool is I. When the positioning error at a certain discrete command angle φ does not exceed I, it is considered that the accuracy requirement at the discrete command angle φ is met; when the positioning error at the discrete command angle φ exceeds I, it is considered that a failure has occurred at the discrete command angle φ. A62. Using the positive positioning error Δθ from step 5 respectively... + (φ,α + and reverse positioning error Δθ - (φ,α - Based on this, construct the forward and reverse failure boundaries at each discrete command angle φ: (25) A63. To characterize the safe distance from the sample center of the error mode coefficient space to the failure boundary of each discrete command angle φ, the non-probabilistic reliability indices for forward positioning error and reverse positioning error at the discrete command angle φ are defined as follows: (26) In equation (26), β + (φ) and β - (φ) represents the non-probabilistic reliability index of the forward positioning error and the reverse positioning error at the discrete command angle φ, respectively; its physical meaning is the nearest Mahalanobis distance from the sample center to the failure boundary at the discrete command angle φ in the corresponding error mode coefficient space; β + (φ) or β - The smaller (φ) is, the closer the corresponding discrete command angle φ is to the failure boundary.

[0026] Preferably, in step 6, within the entire range of discrete command angles, the minimum value of the non-probabilistic reliability index at each discrete command angle φ is taken as the overall reliability index for both forward and reverse positioning errors, i.e.: (27) In equation (27), β d + and β d - These represent the overall reliability indices for forward positioning error and reverse positioning error, respectively; the overall reliability index β d + or β d - The corresponding discrete command angle φ is determined as the danger angle for the forward positioning error and the reverse positioning error, respectively; for example, if the m-th... * The discrete command angles satisfy: or Then the discrete command angle φ m* This refers to the danger angle for positioning errors in the corresponding direction.

[0027] Preferably, in step 6, the envelope radius ρ of the positive ellipsoidal convex uncertainty region obtained in step 4 is... + The envelope radius ρ of the convex uncertainty region of the inverse ellipsoid - Respectively compared with the overall reliability index β d + and β d - The non-probabilistic reliability analysis results of the forward positioning error and the reverse positioning error are compared and determined using the following criteria: when When the positive ellipsoidal convex uncertainty region does not exceed the positive failure boundary, the positive positioning error is reliable; when When the positive ellipsoidal convex uncertainty domain intersects with or exceeds the positive failure boundary, the positive positioning error is unreliable. when When the inverted ellipsoidal convex uncertainty region does not exceed the inverted failure boundary, the inverted positioning error is reliable; when When the convex uncertainty domain of the inverted ellipsoid intersects with or exceeds the inverted failure boundary, the inverted positioning error becomes unreliable.

[0028] Step 7: Based on the worst-case analysis results obtained in Step 5 and the non-probabilistic reliability analysis results obtained in Step 6, output the uncertainty quantification results of the positioning error of the machine tool rotary table axis and give the corresponding reliability judgment conclusion.

[0029] Example 1: In step 1, the test object is the rotary axis of the machine tool rotary table configured in the SV300 vertical machining center; the laser interferometer 2 is a Renishaw XL-80. This embodiment is conducted in a constant temperature workshop, with the machine tool under stable operating conditions of no load, isothermal, and grease lubrication. Key technical parameters of the CNC machine tool rotary axis may include: continuous rotary stroke in the form of "n×360°"; resolution of 0.01"; positioning accuracy of 5"; repeatability of 3"; rated power of 5.1Kw; maximum speed of 450rpm; and maximum acceleration of 180rad / s². 2 The rated torque is 151 Nm; the locking torque is 472 Nm. This embodiment was tested in a constant-temperature workshop, with the machine tool under stable operating conditions of no load, isothermal, and grease lubrication.

[0030] In step 2, this embodiment uses an angular step size Δα=π / 6 for discrete sampling, and obtains a total of 10 sets of mutually independent bidirectional positioning error test data.

[0031] In steps 3 and 4, the principal component analysis results show that the positive error retains the first two principal error modes, and the negative error retains the first three principal error modes. Based on the obtained error mode coefficient samples, positive and negative ellipsoidal convex uncertainty regions are constructed respectively, and the corresponding results are as follows: Figure 3 As shown. Among them, Figure 3 (a) The x-axis and y-axis represent the positive first error mode coefficient and the positive second error mode coefficient, respectively. The discrete points in the figure represent the positive error mode coefficient samples of each group. Figure 3 (b) The three axes represent the inverse first error mode coefficient, the inverse second error mode coefficient, and the inverse third error mode coefficient, respectively. The discrete points in the figure represent the inverse error mode coefficient samples for each group. Figure 3 It can be seen that the bidirectional error mode coefficient samples are all effectively enveloped by the corresponding ellipsoidal convex domain, indicating that the constructed ellipsoidal convex uncertainty domain can well characterize the uncertainty distribution range of bidirectional positioning error in the error mode space.

[0032] In step 5, point-by-point worst upper bound analysis is performed in both the forward and reverse ellipsoidal convex uncertainty regions, and the corresponding results are as follows: Figure 4 As shown. Figure 4 (a) and Figure 4 (b) The horizontal axis represents the discrete command angle, and the vertical axis represents the positioning error in the corresponding direction. The dashed line represents the maximum value of the positioning error sample at each discrete command angle, and the solid line represents the worst-case upper bound envelope of the positioning error obtained based on the corresponding ellipsoidal convex uncertainty domain. Figure 4 (a) represents the worst-case upper bound analysis result of the positive positioning error point by point. Figure 4 (b) represents the worst-case upper bound analysis result of the reverse positioning error point-by-point. Figure 4It can be seen that the maximum vector of bidirectional positioning error samples is effectively covered by the corresponding point-by-point worst upper bound envelope, indicating that the point-by-point worst upper bound obtained based on the ellipsoidal convex uncertainty domain can well characterize the conservative boundary of the positioning error at each discrete command angle.

[0033] Worst-case sample searches with fixed error mode coefficients were performed in both the forward and reverse ellipsoidal convex uncertainty regions, and the corresponding results are as follows: Figure 5 As shown. Figure 5 (a) and Figure 5 (b) The horizontal axis represents the discrete command angle, and the vertical axis represents the positioning error in the corresponding direction. The dashed line represents the maximum value of the positioning error sample at each discrete command angle, and the solid line represents the worst sample vector of the full angle obtained based on the search of the convex uncertainty domain of the corresponding ellipsoid. Figure 5 (a) represents the worst-case sample analysis results of the positive positioning error across all angles. Figure 5 (b) represents the worst-case sample analysis results for the reverse positioning error across all angles. Figure 5 It is evident that the worst sample vectors of bidirectional positioning error across all angles can reflect the truly realizable worst error structure under the constraint of the convex uncertainty domain of the ellipsoid, indicating that the worst sample vectors of all angles obtained based on the fixed worst error mode coefficient vector can well characterize the overall worst-case variation characteristics of bidirectional positioning error across the entire angle range.

[0034] In steps 6 and 7, the allowable threshold for the positioning error of the machine tool rotary table axis is taken as I=3arcsec. The non-probabilistic reliability index at each discrete command angle of the bidirectional positioning error is calculated, and a reliability judgment conclusion is given based on the relevant analysis results. The corresponding results are as follows: Figure 6 As shown. Figure 6 (a) and Figure 6 (b) The horizontal axis represents the discrete command angle, the vertical axis represents the non-probabilistic reliability index in the corresponding direction, the broken line represents the non-probabilistic reliability index at each discrete command angle, and the horizontal dashed line represents the envelope radius of the corresponding ellipsoidal convex uncertainty domain. Figure 6 (a) represents the non-probabilistic reliability analysis result of the positive positioning error. Figure 6 (b) represents the non-probabilistic reliability analysis results of the reverse positioning error. Figure 6 It can be seen that the overall reliability index of the positive error reaches its minimum value at the discrete command angle φ=360°, and the overall reliability index of the negative error reaches its minimum value at the discrete command angle φ=60°, indicating that the above two points are the danger angles of the positive and negative positioning errors, respectively. At the same time, the overall reliability index of both directions is less than the envelope radius of the corresponding ellipsoidal convex uncertainty domain, indicating that the corresponding ellipsoidal convex uncertainty domain has intersected with or exceeded the failure boundary. Therefore, under the allowable threshold conditions, the set of both directions positioning errors is judged to be unreliable.

[0035] Any aspects not covered in this invention are applicable to existing technologies.

Claims

1. A non-probabilistic reliability analysis method for the positioning accuracy of a rotation axis based on an ellipsoidal convex model, characterized in that, The method includes the following steps: Step 1: The reflector (4) is fixed on the horizontal plane of the rotary table (5) of the CNC machine tool, ensuring that the parallelism error between the mounting plane of the reflector (4) and the plane of the rotary table (5) is maintained within a reasonable range; the reflector (4) is located within the optimal measurement range of the laser interferometer (2); the computer (1) is connected to the laser interferometer (2); the CNC system (3) is connected to the rotation axis (6) of the rotary table (5). Step 2: The numerical control system (3) sends an indexing command to the rotary axis (6) so that the rotary axis (6) moves sequentially to the position of each discrete command angle within the command angle range [0, 2π] according to a fixed angle step Δα, and obtains m discrete command angles, where m = 2π / Δα; at each discrete command angle φ, the positioning is completed by two methods: forward approach and reverse approach; during the positioning process, the reflector (4) moves synchronously with the rotary axis (6), the laser interferometer (2) collects the measurement signal of the reflector (4) in real time, and transmits the collected measurement signal to the computer (1) for recording, and obtains the bidirectional positioning error test data of each discrete command angle φ on the computer (1); Step 3: Based on the bidirectional positioning error test data obtained in Step 2, construct the positive error matrix E respectively. + and the inverse error matrix E - They were then subjected to mean-removal processing to obtain the positively centered error matrix X. + and the reverse-centered error matrix X - Then, based on the positively centered error matrix X + and the reverse-centered error matrix X - Construct the positive covariance matrix C respectively + and the inverse covariance matrix C - Then, principal component analysis is used to extract the principal error patterns and corresponding eigenvalues ​​of each element; then, the positively centered error matrix X is... + and the reverse-centered error matrix X - Projecting the vectors onto the low-dimensional subspace formed by the corresponding principal error patterns yields the positive error pattern coefficient vector α for each set of repeated localization experiments. i + The reverse error mode coefficient vector α for each group i - Positive error mode coefficient matrix A + and the inverse error mode coefficient matrix A - This enables low-dimensional parameterized representation of bidirectional positioning error test data. Step 4: Calculate the positive error mode coefficient vector α for all groups obtained in Step 3. i + and the inverse error mode coefficient vector α of all groups i - The mean vector, covariance matrix, and envelope radius are used to construct a positive ellipsoidal convex uncertainty region Ω representing the rotation axis positioning error in the positive error mode coefficient space and the negative error mode coefficient space, respectively. + and the convex uncertainty region Ω of the inverse ellipsoid - ; Step 5: Based on the positive ellipsoidal convex uncertainty region Ω obtained in Step 4 + and the convex uncertainty region Ω of the inverse ellipsoid - With the positively oriented ellipsoidal convex uncertainty region Ω + The positive error mode coefficient vector α of a certain group in + and the convex uncertainty region Ω of the inverse ellipsoid - The inverse error mode coefficient vector α of a certain group - To optimize the vector, the mapping relationship between the error mode coefficient vector and the reconstruction error vector is combined to obtain a set of positive error mode coefficient vectors α. + and a certain set of inverse error mode coefficient vectors α - The positive positioning error Δθ corresponding to the discrete command angle φ + (φ,α + and reverse positioning error Δθ - (φ,α - Then, the positive positioning error Δθ + (φ,α + and reverse positioning error Δθ - (φ,α - We conducted point-by-point worst upper bound analysis and full-angle worst sample search to obtain the point-by-point worst upper bound envelope, full-angle worst sample vector, and corresponding worst peak value of the forward and reverse positioning errors, so as to characterize the worst case analysis results of the machine tool rotary table rotation axis positioning error under the ellipsoidal convex uncertainty domain. Step 6: Based on the positive ellipsoidal convex uncertainty region Ω obtained in Step 4 + and the convex uncertainty region Ω of the inverse ellipsoid - Combining the allowable threshold for the positioning error of the machine tool rotary table axis and the positive positioning error Δθ in step 5 + (φ,α + and reverse positioning error Δθ - (φ,α - Construct non-probabilistic reliability indices and overall reliability indices at each discrete command angle φ; then compare the envelope radius ρ of the positive ellipsoidal convex uncertainty domain obtained in step 4. + The envelope radius ρ of the convex uncertainty region of the inverse ellipsoid - By comparing the overall reliability index with the non-probabilistic reliability analysis results of the machine tool rotary table axis positioning error under uncertain set constraints, the danger angles in the corresponding directions are identified. Step 7: Based on the worst-case analysis results obtained in Step 5 and the non-probabilistic reliability analysis results obtained in Step 6, output the uncertainty quantification results of the positioning error of the machine tool rotary table axis and give the corresponding reliability judgment conclusion.

2. The non-probabilistic reliability analysis method for the positioning accuracy of a rotation axis based on an ellipsoidal convex model according to claim 1, characterized in that, In step 1, before the test begins, the computer (1) is debugged and a fixed delay is set so that the measurement readings can enter a steady state before sampling, ensuring that the laser interferometer (2) can accurately acquire the positioning error test data of the rotating shaft (6); In step 2, the positioning is completed by two methods: forward approach and reverse approach. Specifically, for each discrete command angle φ, the positioning is performed by approaching clockwise and counterclockwise respectively, and n sets of repeated positioning tests are conducted under the same speed, acceleration and deceleration time and positioning criterion conditions. The positioning error test data of each set are recorded by computer (1).

3. The non-probabilistic reliability analysis method for the positioning accuracy of a rotation axis based on an ellipsoidal convex model according to claim 1, characterized in that, In step 3, the positively centered error matrix X is obtained. + and the reverse-centered error matrix X - The specific steps are as follows: A31. Based on the bidirectional positioning error test data of n sets of repeated positioning tests at the discrete command angle φ, construct the positive error matrix E respectively. + and the inverse error matrix E - : （1） Positive error matrix E + and the inverse error matrix E - Each row corresponds to a set of repeated positioning experiments, and each column corresponds to bidirectional positioning error test data for a discrete command angle φ; therefore, the positive error vector e corresponding to the i-th set of repeated positioning experiments is... i + and the inverse error vector e i - They are represented as follows: （2） Meanwhile, for the positive error matrix E + and the inverse error matrix E - The average error is calculated at each discrete command angle φ to obtain the positive average error vector. + and inverse average error vector - : （3） A32, for each group's positive error vector e i + and the inverse error vector e of each group i - After removing the mean, the positively centered error vector x for each group is obtained. i + and the reverse-centered error vector x for each group i - : （4） A33. The positively centered error vector x of all repeated localization experiments. i + Arrange the matrices to obtain the positively centered error matrix X. + ; the reverse-centered error vector x of all repeated localization experiments i - The reverse-centered error matrix X is obtained by arranging the matrix. - : （5）； In step 3, the positive error mode coefficient vector α for each group of repeated positioning experiments is obtained. i + The reverse error mode coefficient vector α for each group i - Positive error mode coefficient matrix A + and the inverse error mode coefficient matrix A - The specific steps are as follows: B31, Based on the positive centering error matrix X + and the reverse-centered error matrix X - Construct the corresponding covariance matrices as follows: （6） In equation (6), C + Let C represent the positive covariance matrix. - Represents the inverse covariance matrix; B32. For the positive covariance matrix C, respectively + and the inverse covariance matrix C - Principal component analysis is performed, and the principal error patterns and corresponding eigenvalues ​​are obtained through eigenvalue decomposition. Their expressions are as follows: （7） In equation (7), ψ r + and ψ r - Let λ represent the r-th positive principal error mode and the r-th negative principal error mode, respectively; r + and λ r - These represent the corresponding eigenvalues; B33. Sort the eigenvalues ​​from largest to smallest, and determine the number of main error patterns to retain based on the cumulative contribution rate; Assume the positive principal error mode retains the first k... + There are several modes, and the reverse principal error mode retains the first k... - The retained positive principal error modes form the positive principal error mode matrix Ψ. + The retained inverse principal error modes constitute the inverse principal error mode matrix Ψ. - : （8） B34. Center the error vector x of each group in a positive direction. i + Projected onto the positive principal error mode matrix Ψ + The low-dimensional subspace formed will centralize the reverse-center error vector x of each group. i - Projected onto the inverse principal error mode matrix Ψ - The low-dimensional subspace formed yields the positive error mode coefficient vector α for each group. i + and the inverse error mode coefficient vector α of each group i - : （9） Similarly, the positive error mode coefficient matrix A of n sets of repeated positioning experiments + and the inverse error mode coefficient matrix A - They are respectively: （10）。 4. The non-probabilistic reliability analysis method for the positioning accuracy of a rotation axis based on an ellipsoidal convex model according to claim 1, characterized in that, In step 4, the positive error mode coefficient vector α of all groups is processed respectively. i + and the inverse error mode coefficient vector α of all groups i - Calculate the mean to obtain the mean vector of the positive error mode coefficients. + and the mean vector of the inverse error mode coefficients - : （12）； In step 4, based on the positive error mode coefficient vector α of all groups... i + and the inverse error mode coefficient vector α of all groups i - Construct the corresponding covariance matrix + and - Its expression is: （13）； In step 4, to determine the envelope radius of the convex uncertainty region of the ellipsoid, the positive error mode coefficient vector α for each group is calculated. i + Relative to the mean vector of positive error mode coefficients + The Mahalanobis distance, and the inverse error mode coefficient vector α for each group. i - Relative to the mean vector of the inverse error mode coefficients - The Mahalanobis distance is expressed as: （14） In equation (14), d i + and d i - Let α represent the coefficient vector of the i-th positive error mode. i + and the i-th group of reverse error mode coefficient vector α i - Mahalanobis distance to their respective mean vectors; Then take the positive error mode coefficient vector α of all groups. i + and the inverse error mode coefficient vector α of all groups i - The maximum value of the Mahalanobis distance is taken as the envelope radius of the corresponding convex uncertain region of the ellipsoid, i.e.: （15） In equation (15), ρ + and ρ - Let represent the envelope radii of the forward ellipsoidal convex uncertainty region and the reverse ellipsoidal convex uncertainty region, respectively.

5. The non-probabilistic reliability analysis method for the positioning accuracy of a rotation axis based on an ellipsoidal convex model according to claim 1 or 4, characterized in that, In step 4, based on the positive error mode coefficient vector α obtained in step 3... i + and the inverse error mode coefficient vector α i - The mean vector, covariance matrix, and envelope radius are used to construct the convex uncertainty region Ω of the positive ellipsoid. + and the convex uncertainty region Ω of the inverse ellipsoid - : （16） In equation (16), the positive error mode coefficient space is defined as k + Dimension Space The reverse error mode coefficient space is defined as k - Dimension Space .

6. The non-probabilistic reliability analysis method for the positioning accuracy of a rotation axis based on an ellipsoidal convex model according to claim 1, characterized in that, In step 5, the positive positioning error Δθ is obtained. + (φ,α + and reverse positioning error Δθ - (φ,α - Specifically: The mapping relationship between the error mode coefficient vector and the reconstruction error vector is expressed as follows: （17） In equation (17), e + (α + ) and e - (α - ) represent the positive error mode coefficient vector α + and the inverse error mode coefficient vector α - The determined forward reconstruction error vector and backward reconstruction error vector; + It is represented as a positive average error vector. - Represented as the inverse average error vector; Ψ + Ψ represents the positive principal error mode matrix. - Represents the inverse principal error mode matrix; A set of positive error mode coefficient vectors α + ∈Ω + and a certain set of inverse error mode coefficient vectors α - ∈Ω - The forward positioning error and the reverse positioning error corresponding to the discrete command angle φ are denoted as Δθ, respectively. + (φ,α + ) and Δθ - (φ,α - ): （18） In equation (18), [Ψ + α + ] φ and [Ψ] - α - ] φ Representing vectors Ψ + α + and Ψ - α - The component corresponding to the discrete command angle φ.

7. The non-probabilistic reliability analysis method for the positioning accuracy of a rotation axis based on an ellipsoidal convex model according to claim 1 or 6, characterized in that, In step 5, the worst-case upper bound envelope is obtained through pointwise worst-case upper bound analysis. The specific steps are as follows: A51. For each discrete command angle φ, in the positively oriented ellipsoidal convex uncertainty region Ω... + and the convex uncertainty region Ω of the inverse ellipsoid - Internal relative positive positioning error Δθ + (φ,α + and reverse positioning error Δθ - (φ,α - By performing pointwise maximization, the worst-case upper bound at each discrete command angle φ is obtained: （19） In equation (19), and These represent the point-by-point worst upper bounds of the forward and reverse positioning errors at the discrete command angle φ, respectively, under the convex uncertainty domain of the ellipsoid. A52. Arrange the point-by-point worst upper bounds obtained at each discrete command angle φ in angular order to obtain the point-by-point worst upper bound envelopes of the positive positioning error. The worst-case upper bound envelope of the point-by-point positioning error Its expression is: （20）。 8. The non-probabilistic reliability analysis method for the positioning accuracy of a rotation axis based on an ellipsoidal convex model according to claim 1 or 6, characterized in that, In step 5, the worst-case sample vector and the corresponding worst peak value are obtained through a full-angle worst-case sample search. The specific steps are as follows: B51. Full-angle worst-case search refers to calculating the worst-case vector under fixed error mode coefficients, defining the positive full-angle peak function h. + (α + ) and the inverse full-angle peak function h - (α - Its expression is: （21） In the convex uncertainty region Ω of the positive ellipsoid + and the convex uncertainty region Ω of the inverse ellipsoid - The inner part performs a maximization search on the full-angle peak function to obtain the positive fixed worst-case error mode coefficient vector α. +* and the inverse fixed worst-case error mode coefficient vector α -* : （22）； B52. The forward-fixed worst-case error mode coefficient vector α... +* and the inverse fixed worst-case error mode coefficient vector α -* Substituting the reconstruction relation into equation (17) respectively, we obtain the worst sample vector of the positive full angle. and the worst sample vector of the reverse full angle : （23） B53, respectively for the worst sample vector of the positive full angle and the worst sample vector of the reverse full angle Taking the maximum value within the entire range of discrete command angles yields the worst-case positive and worst-case negative peak values, whose expressions are as follows: （24） In equation (24), and These represent the worst-case sample vectors in the positive all-angle orientation, respectively. and the worst sample vector of the reverse full angle The maximum positioning error across the entire range of discrete command angles.

9. The non-probabilistic reliability analysis method for the positioning accuracy of a rotation axis based on an ellipsoidal convex model according to claim 1, characterized in that, In step 6, non-probabilistic reliability indices are constructed at each discrete command angle φ. The specific steps are as follows: A61. Suppose the allowable threshold for the positioning error of the rotary axis of a machine tool is I. When the positioning error at a certain discrete command angle φ does not exceed I, it is considered that the accuracy requirement at the discrete command angle φ is met; when the positioning error at the discrete command angle φ exceeds I, it is considered that a failure has occurred at the discrete command angle φ. A62. Using the positive positioning error Δθ from step 5 respectively... + (φ,α + and reverse positioning error Δθ - (φ,α - Based on this, construct the forward and reverse failure boundaries at each discrete command angle φ: （25） A63. To characterize the safe distance from the sample center of the error mode coefficient space to the failure boundary of each discrete command angle φ, the non-probabilistic reliability indices for forward positioning error and reverse positioning error at the discrete command angle φ are defined as follows: （26） In equation (26), β + (φ) and β - (φ) represents the non-probabilistic reliability index of the forward positioning error and the reverse positioning error at the discrete command angle φ, respectively; In step 6, within the entire range of discrete command angles, the minimum value of the non-probabilistic reliability index at each discrete command angle φ is taken as the overall reliability index for both forward and reverse positioning errors, i.e.: （27） In equation (27), β d + and β d - These represent the overall reliability indices for forward positioning error and reverse positioning error, respectively; the overall reliability index β d + or β d - The corresponding discrete command angle φ is determined as the danger angle for both forward and reverse positioning errors.

10. The non-probabilistic reliability analysis method for the positioning accuracy of a rotation axis based on an ellipsoidal convex model according to claim 1, characterized in that, In step 6, the envelope radius ρ of the positively oriented ellipsoidal convex uncertainty region obtained in step 4 is... + The envelope radius ρ of the convex uncertainty region of the inverse ellipsoid - Respectively compared with the overall reliability index β d + and β d - The non-probabilistic reliability analysis results of the forward positioning error and the reverse positioning error are compared and determined using the following criteria: when When the positive ellipsoidal convex uncertainty region does not exceed the positive failure boundary, the positive positioning error is reliable; when When the positive ellipsoidal convex uncertainty domain intersects with or exceeds the positive failure boundary, the positive positioning error is unreliable. when When the inverted ellipsoidal convex uncertainty region does not exceed the inverted failure boundary, the inverted positioning error is reliable; when When the convex uncertainty domain of the inverted ellipsoid intersects with or exceeds the inverted failure boundary, the inverted positioning error becomes unreliable.

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