CN104850698A - Measuring and adjusting process-considered tolerance design method of precise machine tool - Google Patents

Abstract

The invention discloses a measuring and adjusting process-considered tolerance design method of a precise machine tool, which comprises the steps of: firstly defining a related coordination system according to a key feature plane; then determining a state space equation and a basic coefficient matrix thereof of a machine tool assembly; next, establishing a cost target function, obtaining an optimal adjustment sequence by a discrete system output regulator so as to convert the state space equation into a Gauss-Markov random process equation; then obtaining a covariance matrix, and performing iteration computation to obtain the covariance matrix after integral assembly is finished; computing a corresponding function relationship between a total precision index and a part precision index of the machine tool according to the covariance matrix after the integral assembly is finished; finally, establishing a function relation between a precision total cost and a part precision and solving to obtain an optimal part tolerance allocation result. The method can be used for accurately and reliably analyzing the tolerance of the machine tool, thus ensuring the precision performance of the machine tool.

Description

Precision machine tool tolerance design method considering measurement and adjustment process

[ technical field ] A method for producing a semiconductor device

The invention belongs to the field of mechanical design, and particularly relates to a precision machine tool tolerance design method considering a measurement and adjustment process.

[ background of the invention ]

The precision performance of the machine tool is a basic index of machine tool design and is comprehensively guaranteed through design, manufacture and assembly. However, the existing machine tool precision design in China cannot get rid of the dependence on experience, and tolerance design and assembly process planning are mainly carried out by analogy and manual inquiry methods. There is a lack of accurate and reliable tolerance analysis and assembly process planning and design tools. In order to improve the successful design, the designer can be shaped after repeated trial production of an enterprise, so that good opportunity for bringing newly developed products to the market is missed and the manufacturing cost is increased.

[ summary of the invention ]

The invention aims to provide a precision machine tool tolerance design method considering measurement and adjustment processes aiming at the problems of dependence on the experience of engineers, insufficient precision and excessive precision in the existing machine tool tolerance design process.

In order to achieve the purpose, the invention adopts the technical scheme that:

a method of designing tolerances for a precision machine tool taking into account measurement and adjustment processes, comprising the steps of:

1) simplifying the machine tool into a basic large part, determining a key characteristic surface of the basic large part of the machine tool, and establishing a local coordinate system by means of the nominal or actual orientation of the part on the basis of the key characteristic surface; meanwhile, according to the topological structure and the assembly process of the machine tool, determining a basic coefficient matrix, an adjustment coefficient matrix and an observation matrix, and establishing a state space equation;

2) determining a precision cost matrix, adjusting a cost weight matrix and a precision index weight matrix according to the market research and the processing level of a manufacturing enterprise, and establishing a cost objective function; outputting a regulator through a discrete system, combining errors of parts, and obtaining an adjustment sequence by using a Kalman filtering method;

3) rewriting a state space equation into a Gaussian Markov random process equation, obtaining a covariance matrix according to the Gaussian Markov random process equation, and then carrying out iterative solution according to an assembly process to obtain the covariance matrix after the integral assembly is finished;

4) obtaining a functional relation f of the total precision index of the machine tool and the deviation of parts by utilizing the covariance matrix after the integral assembly and combining a variance synthesis formula;

5) and establishing a functional relation W between the total precision cost and the precision index of the part, taking the functional relation as a target function, and taking the functional relation f as a constraint condition to obtain a tolerance distribution result of the part, and finishing tolerance design of the machine tool.

The large foundation part in the step 1) comprises a lathe bed, a stand column, a sliding seat, a spindle box, a saddle and a workbench, and the key characteristic surfaces comprise the installation surface, the connection surface and the tail end surface of the large foundation part.

The basic coefficient matrixes in the step 1 are A (k), B (k) and F (k), the adjusting coefficient matrix is T (k), the observation matrix is C (k), the established state space equation is shown as the formula (1),

<math> <mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <mover> <mi>X</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mi>A</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>X</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>B</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>u</mi> <mo>~</mo> </mover> <mo>*</mo> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>F</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>&omega;</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mover> <mi>y</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>C</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>X</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mover> <mi>v</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </math>

wherein A (k) is a constant matrix in the k-th assembly, F (k) is a 6 x 6 matrix for converting the error in the k-th assembly from the k-th part coordinate system to the base coordinate system,representing the total deviation of the position and the orientation of the kth part assembling characteristic surface coordinate system in the reference transmission chain relative to the nominal pose thereof,deviation of the actual pose from the nominal pose in the part coordinate system for the kth part to assemble,accumulating a 6 x 1 vector for the total deviation of the global state after the kth part is assembled with respect to the base coordinate system, B (k) is an error adjustment vectorA transformation matrix from the part coordinate system of the kth part to the basic coordinate system, C (k) is an output matrix with the r × 6 element of 0 or 1 after the kth assembly is completed,the output vector of r multiplied by 1(r is less than or equal to 6) after the k-th assembly is completed,for the noise in the k-th assembly,and k is an error adjustment vector, is more than or equal to 0 and less than or equal to N, and N is the total assembly step number or the total assembly part number.

The precision cost matrix in the step 2) is S, the cost weight matrix is adjusted to be R (k), the precision index weight matrix is Q (k), the established cost objective function J is shown as the formula (2),

<math> <mrow> <mi>J</mi> <mo>=</mo> <msup> <mover> <mi>y</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mi>N</mi> <mo>)</mo> </mrow> <mi>S</mi> <mover> <mi>y</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>N</mi> <mo>)</mo> </mrow> <mo>+</mo> <munderover> <mi>&Sigma;</mi> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <mo>[</mo> <msup> <mover> <mi>y</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>Q</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>y</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mover> <mi>u</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>R</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>u</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>]</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </math>

the adjusted sequence was calculated by the following formula:

u*(k)＝-K(k)X(k)

wherein,the superscript T represents transposition for the output vector of r multiplied by 1(r is less than or equal to 6) after the integral assembly is finished,for the kth stepMatching the output vector of r multiplied by 1(r is less than or equal to 6), k is more than or equal to 0 and less than or equal to N, N is the total step number of assembly, Q (k) is the precision index weight matrix after the assembly of the kth step is completed,describing a 6 x 1 vector matched with characteristic error adjustment in the k-th assembly process control, wherein R (k) is an adjustment cost weight matrix in the k-th assembly process, X (k) is total deviation after the k-th assembly process is completed, u (k) is an adjustment sequence, and K (k) is a Kalman gain coefficient.

The kalman gain coefficient k (k) is calculated by the following formula:

K(k)＝[R(k)+B^T(k)P(k+1)B(k)]^-1B^T(k)P(k+1)A(k)

P(k)＝Q(k)+A^T(k)P(k+1)[I+B(k)R^-1(k)B^T(k)P(k+1)]^-1A(k)

wherein R (k) is the adjustment cost weight matrix in the k-th assembling process, and B (k) is the error adjustment vectorA transformation matrix from the part coordinate system of the kth part to the base coordinate system,and (b) taking the error adjustment vector as an error adjustment vector, A (k) is a constant matrix during the k-th assembly, superscript T represents transposition, Q (k) is a precision index weight matrix after the k-th assembly is completed, I is a unit matrix, P (k) and P (k +1) are Ricatti equations of the k-th step and the k + 1-th step respectively, k is more than or equal to 0 and less than or equal to N, and N is the total assembly step number or the total assembly part number.

The Gaussian Markov random process equation in the step 3) is shown as a formula (3),

<math> <mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <msub> <mover> <mi>X</mi> <mo>~</mo> </mover> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>A</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>X</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>F</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>&omega;</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mover> <mi>X</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mo>[</mo> <mi>I</mi> <mo>-</mo> <mi>B</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>T</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>K</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>]</mo> <msub> <mover> <mi>X</mi> <mo>~</mo> </mover> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow> </math>

the covariance matrix is shown as equation (4),

D_i(k)＝A^T(k)D(k)A(k)+F(k)V(k)F^T(k)

D(k+1)＝[I-B(k)T(k)K(k)]D_i(k)[I-B(k)T(k)K(k)]^T (4)

whereinIs the total deviation after the assembly and adjustment in the k step, A (k) is a constant matrix during the assembly in the k step, the superscript T represents transposition,representing the total deviation of the position and orientation of the kth part assembly characteristic surface coordinate system in the reference transmission chain relative to the nominal pose thereof, F (k) is a 6 x 6 matrix for converting the error in the kth part assembly from the kth part coordinate system to the base coordinate system,deviation of the actual pose from the nominal pose in the part coordinate system for the kth part to assemble,accumulating a 6 x 1 vector for the total deviation of the global state after the kth part is assembled with respect to the base coordinate system, I is an identity matrix, B (k) is an error adjustment vectorA transformation matrix from the part coordinate system of the kth part to the base coordinate system,t (k) is an error adjustment vector, T (k) is an adjustment coefficient matrix, and K (k) is a Kalman gain coefficient; d_i(k) The covariance matrix of the state quantity of the kth step of the ith part, D (k) is the covariance matrix of the state quantity of the kth step, D (k +1) is the covariance matrix of the state quantity of the kth step, V (k) is the variance of part errors introduced in the kth step, k is more than or equal to 0 and less than or equal to N, and N is the total assembly step number or the total assembly part number;

and (4) iteratively calculating to obtain a covariance matrix D (N) after the integral assembly is finished.

The variance synthesis formula in the step 4) is as follows:

E_i＝D(m)+D(n)-2D(m)D(n)

deviation of total accuracy index E from partsThe functional relationship f is as follows:

<math> <mrow> <mi>E</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mover> <mi>&omega;</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </math>

wherein E is_iFor the requirement of the i-th precision index, D (m) is the value of the m-th item on the diagonal of D (N), D (n) is the value of the n-th item on the diagonal of D (N), D (N) is the covariance matrix after the whole assembly is finished, and E ═ E₁，E₂，…，E_i，…，E_H) H is the number of the total required precision, i is more than or equal to 0 and less than or equal to H.

The functional relation W in the step 5) is as follows:

<math> <mrow> <mi>W</mi> <mo>=</mo> <munderover> <mi>&Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>H</mi> </munderover> <mfrac> <msub> <mi>Q</mi> <mi>i</mi> </msub> <msub> <mi>E</mi> <mi>i</mi> </msub> </mfrac> </mrow> </math>

wherein H is the number of total required precision, i is more than or equal to 0 and less than or equal to H, E_iFor the ith accuracy index requirement, O_iThe i precision cost.

Compared with the prior art, the invention has the beneficial effects that:

the invention provides a tolerance design method of a precision machine tool considering measurement and adjustment processes, aiming at the problems that the tolerance design process of the precision machine system such as the existing machine tool depends on the experience of an engineer, the part tolerance of the precision machine tool cannot be directly obtained by distribution through a traditional extreme value method or a statistical method, and the precision of the machine tool is finally insufficient and excessive. The method comprises two parts of determination of optimal adjustment quantity of an assembly adjustment process and distribution of the whole machine precision of a machine tool, and firstly, a relevant coordinate system is defined according to a key feature plane; then determining a state space equation and a basic coefficient matrix of the machine tool assembly; secondly, establishing a cost target function, and obtaining an optimal adjustment sequence through a discrete system output regulator, so that a state space equation is converted into a Gaussian Markov random process equation; then obtaining a covariance matrix, and carrying out iterative computation to obtain the covariance matrix after the integral assembly is finished; calculating the corresponding function relation between the total precision index of the machine tool and the deviation of the parts by utilizing the covariance matrix after the integral assembly is finished; and finally, establishing a functional relation between the total precision cost and the precision indexes of the parts, and solving to obtain an optimal part tolerance distribution result. The method comprehensively considers the measuring and adjusting links in the actual assembling process, firstly obtains the optimal adjusting strategy of each step in the assembling process, and realizes the minimization of tolerance cost and adjusting cost based on the optimal adjustment, and finally realizes the comprehensive tolerance design with optimal assembling precision. The method can accurately and reliably analyze the tolerance of the machine tool, thereby ensuring the precision performance of the machine tool and having good application prospect.

[ description of the drawings ]

FIG. 1 is a flow chart of a precision machine tolerance design method provided by the present invention that takes into account measurement and adjustment processes;

fig. 2 is a schematic diagram of a key feature plane of a horizontal machining center designed in an embodiment of the present invention, wherein (a) is a specific structural diagram, and (b) is a simplified schematic diagram of (a);

FIG. 3 is a schematic diagram illustrating a coordinate system definition of a key feature plane of the horizontal machining center of FIG. 2;

FIG. 4 is a graph of deviation state quantity variation of the horizontal machining center of FIG. 2 according to an embodiment of the present invention, wherein (a) is the state quantity X₁(k) (1) with X₃(k) (3) variation during the assembly, and (b) is the state quantity X₂(k) (2) with X₄(k) (4) variations in the assembly process.

[ detailed description ] embodiments

The present invention will be described in further detail with reference to the accompanying drawings.

The invention discloses a precision machine tool tolerance design method considering measurement and adjustment processes, which aims at solving the problems of dependence on engineer experience, insufficient precision and excessive precision in the existing machine tool tolerance design process.

The flow of the precision machine tool tolerance design method considering the measurement and adjustment process provided by the invention is shown in figure 1, and the method comprises the following specific steps:

1) selecting and determining key feature facets (KCs): the machine tool is firstly simplified into basic large parts, namely a lathe bed, a column, a sliding seat, a main spindle box, a saddle, a workbench and the like, and then the mounting, connecting surface, end surface and the like of the basic large parts are selected and used as key characteristic surfaces.

Establishing a coordinate system: based on the key characteristic surface of the basic large part of the machine tool, a local coordinate system is established by means of the nominal or actual orientation of the part, the local coordinate system established by the machine tool body and the column joint surface is a basic coordinate system, and the directions of the coordinate axes of the basic coordinate system and the local coordinate system are consistent.

2) Establishing a state space equation: firstly, determining the topological structure and the assembly process of the machine tool, thereby determining the basic coefficient matrixes A (k), B (k), F (k) and the adjustment coefficient matrix T (k) and the observation matrix C (k) of the state space equation (formula (1)), and then substituting the basic data into the state space equation, namely establishing the finished state space equation.

<math> <mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <mover> <mi>X</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mi>A</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>X</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>B</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>u</mi> <mo>~</mo> </mover> <mo>*</mo> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>F</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>&omega;</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mover> <mi>y</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>C</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>X</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mover> <mi>v</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </math>

Wherein A (k) is a constant matrix in the k-th assembly, F (k) is a 6 x 6 matrix for converting the error in the k-th assembly from the k-th part coordinate system to the base coordinate system,representing the total deviation of the position and the orientation of the kth part assembling characteristic surface coordinate system in the reference transmission chain relative to the nominal pose thereof,actual pose phase during assembly for the kth partDeviation of the nominal pose in the part coordinate system; definition ofAccumulating a 6 x 1 vector for the total deviation of the global state after the kth part is assembled with respect to the base coordinate system, B (k) is an error adjustment vectorA transformation matrix from the part coordinate system of the kth part to the basic coordinate system, C (k) is an output matrix with the r × 6 element of 0 or 1 after the kth assembly is completed,the output vector of r multiplied by 1(r is less than or equal to 6) after the k-th assembly is completed,for the noise in the k-th assembly,and k is an error adjustment vector, is more than or equal to 0 and less than or equal to N, and N is the total assembly step number or the total assembly part number.

Obtaining an optimal adjustment sequence in the assembly process: firstly, determining coefficient matrixes Q (k), R (k) and S according to the market research and the processing level of a manufacturing enterprise, and then establishing a cost objective function J as shown in a formula (2):

<math> <mrow> <mi>J</mi> <mo>=</mo> <msup> <mover> <mi>y</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mi>N</mi> <mo>)</mo> </mrow> <mi>S</mi> <mover> <mi>y</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>N</mi> <mo>)</mo> </mrow> <mo>+</mo> <munderover> <mi>&Sigma;</mi> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <mo>[</mo> <msup> <mover> <mi>y</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>Q</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>y</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mover> <mi>u</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>R</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>u</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>]</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </math>

wherein,the superscript T represents transposition for the output vector of r multiplied by 1(r is less than or equal to 6) after the integral assembly is finished,is the output vector of r multiplied by 1(r is less than or equal to 6) after the assembly in the kth step, k is more than or equal to 0 and less than or equal to N, N is the total assembly step number, Q (k) is the precision index weight matrix after the assembly in the kth step is completed,describing a 6 x 1 vector matched with characteristic error adjustment in the control of the assembly process in the k step, wherein R (k) is an adjustment cost weight matrix in the assembly process in the k step, and S is a precision index weight matrix after the whole assembly is finished, namely a precision cost matrix, which is the loss of benefits caused by the reduction of precision.

Then, a discrete system output regulator is combined with part errors, a Kalman filtering method is used, and an optimal adjustment sequence is obtained through the following calculation:

u*(k)＝-K(k)X(k)

where x (k) is the total deviation after the kth assembly, u x (k) is the adjustment sequence, and k (k) is the kalman gain factor.

Find outThus, X (k) was obtained.

K(k)＝[R(k)+B^T(k)P(k+1)B(k)]^-1B^T(k)P(k+1)A(k)

P(k)＝Q(k)+A^T(k)P(k+1)[I+B(k)R^-1(k)B^T(k)P(k+1)]^-1A(k)

Wherein, I is a unit matrix, P (k) and P (k +1) are Ricitti equations of the k step and the k +1 step respectively, which can be obtained by recursive operation and have: p (n) ═ q (n).

3) Calculating a bias quantity state covariance matrix after assembly is completed: firstly, a state space equation is equivalent to a Gaussian Markov random process equation, which is expressed by the formula (3):

<math> <mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <msub> <mover> <mi>X</mi> <mo>~</mo> </mover> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>A</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>X</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>F</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>&omega;</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mover> <mi>X</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mo>[</mo> <mi>I</mi> <mo>-</mo> <mi>B</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>T</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>K</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>]</mo> <msub> <mover> <mi>X</mi> <mo>~</mo> </mover> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow> </math>

whereinIs the total deviation after the k-th assembly and adjustmentT (k) is an adjustment coefficient matrix, i.e., an adjustment feature selection matrix, whose off-diagonal elements are all 0, and it is indicated that the corresponding geometric feature can be adjusted at the k-th step by giving a diagonal element of 1, and it is indicated that the corresponding feature cannot be adjusted if 0.

Setting the mean value and the variance of the initial error state quantity to be 0, and then iteratively calculating a deviation quantity covariance matrix in the assembling process according to the assembling technological process, wherein the deviation quantity covariance matrix is expressed as a formula (4):

D_i(k)＝A^T(k)D(k)A(k)+F(k)V(k)F^T(k)

D(k+1)＝[I-B(k)T(k)K(k)]D_i(k)[I-B(k)T(k)K(k)]^T (4)

wherein D (k) is the covariance matrix of the state quantity of the k step, i.e. the variance of X (k), D (k +1) is the covariance matrix of the state quantity of the k +1 step, D_i(k) Is the covariance matrix of the state quantity of the ith part at the k step, V (k) is the variance of the part error introduced at the k step, i.e.The variance of (c). And D (N) is obtained finally through iteration, namely the covariance matrix after the integral assembly is finished.

4) Analyzing and calculating a precision index project function relation: analyzing and obtaining data in a final covariance matrix D (N) related to the precision index project according to each established key feature surface, a local coordinate system, a machine tool topological structure and the precision index project, and then utilizing a variance synthesis formula:

E_i＝D(m)+D(n)-2D(m)D(n)

wherein E is_iFor the requirement of the i-th precision index, D (m) is the value of the m-th item on the diagonal of D (N), and D (n) is the value of the n-th item on the diagonal of D (N).

By this formula, the deviation of the total accuracy index E from the parts can be obtainedThe functional relationship of (a).

<math> <mrow> <mi>E</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mover> <mi>&omega;</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mi>E</mi> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>E</mi> <mi>i</mi> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>E</mi> <mi>H</mi> </msub> <mo>)</mo> </mrow> </mrow> </math>

5) Optimizing the tolerance of the distribution parts: establishing a functional relation W between the total precision cost and the precision indexes of the parts, taking the function W as a target function, taking f as a constraint condition, and adopting modern intelligent solving methods such as a genetic algorithm and the like to obtain an optimal part tolerance distribution resultAnd finishing the tolerance design of the machine tool.

<math> <mrow> <mi>W</mi> <mo>=</mo> <munderover> <mi>&Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>H</mi> </munderover> <mfrac> <msub> <mi>Q</mi> <mi>i</mi> </msub> <msub> <mi>E</mi> <mi>i</mi> </msub> </mfrac> </mrow> </math>

Where H is the total number of required precision entries, O_iThe i precision cost.

In the design of the tolerances of a precision machine tool, a state space model is the fundamental content. The state space model not only reflects the internal state of the system, but also reveals the relationship between the internal state of the system and external input and output variables. The assembly process can be regarded as a linear discrete dynamic event system, the total deviation state of the current machine tool assembly body is regarded as a basis, error data of the current parts are used as error input and are brought into a state space equation, and then the error state in the assembly process can be obtained. And obtaining a final deviation state matrix after assembly through repeated iteration.

FIG. 1 is a flow chart of a tolerance design method of the present invention, including methods of state space modeling, Kalman filtering, etc. The tolerance assignment of the machine tool can be realized according to the tolerance design steps shown in the flow chart, and all the design flows and steps are described in detail below by taking a certain horizontal machining center as an example.

Fig. 2 is a schematic diagram of key feature surfaces of a horizontal machining center, wherein the key feature surfaces are selected mainly from an assembly joint surface, a mounting surface, a tail end surface and the like. As shown in fig. 2, in the specific selection, the moving shaft mounting surface, the bed column fitting surface, and the end surface of the assembly body (the table surface and the outer side surface of the headstock) are selected as key feature surfaces. And the matching surface of the column of the lathe bed is used as a reference surface of the datum. Table 1 shows the assembly accuracy requirements for this horizontal machining center.

TABLE 1 Assembly accuracy requirements for horizontal machining centers

In this example, there are six key feature planes, each with three angular errors. For simplicity, the machining center is simplified into a four-component assembly body consisting of a lathe bed, a stand column, a sliding seat and a workbench. The assembly features are numbered with KC0 being chosen as the reference plane. Meanwhile, the problem after simplification is a two-dimensional size problem, i.e., the angle error exists only in the plane of the paper.

The coordinate system defined by the key feature planes is shown in FIG. 3 (FIG. 3)O₀～O₄Representing the origin of the local coordinate system), the coordinate system is established based on the nominal or actual orientation of the part based on the key feature plane when defined. After assembly, the equation of state for the angular error is as follows:

<math> <mrow> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>X</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>N</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>X</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>N</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>X</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>N</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>X</mi> <mn>4</mn> </msub> <mrow> <mo>(</mo> <mi>N</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mover> <mi>&omega;</mi> <mo>~</mo> </mover> <mi>B</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>&omega;</mi> <mo>~</mo> </mover> <mi>C</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>&omega;</mi> <mo>~</mo> </mover> <mi>A</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>&omega;</mi> <mo>~</mo> </mover> <mi>D</mi> </msub> </mtd> </mtr> </mtable> </mfenced> </mrow> </math>

wherein X₁(N) Total deviation of assembled face KC1State, X₂(N) Total deviation State of assembled eigen plane KC2, X₃(N) Total deviation State of assembled eigen plane KC3, X₄(N) is the total deviation state of the characteristic face KC4 after assembly,is the angle error of the lathe bed,is the angle error of the upright post,is the angle error of the sliding seat,is the angular error of the table.

Meanwhile, the state space equation is as follows:

<math> <mrow> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>X</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>X</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>X</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>X</mi> <mn>4</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>X</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>X</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>X</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>X</mi> <mn>4</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mover> <mi>&omega;</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </math>

wherein X₁(k +1) is the total deviation of the characteristic face KC1 after the step (k +1) is assembled, X₂(k +1) is the total deviation of the characteristic face KC2 after the step (k +1) is assembled, X₃(k +1) is the total deviation of the characteristic face KC3 after the step (k +1) is assembled, X₄(k +1) is the total deviation of the characteristic face KC4 after the step (k +1) is assembled, X₁(k) For the total deviation, X, of the characteristic face KC1 after the k-th assembly₂(k) For the total deviation, X, of the characteristic face KC2 after the k-th assembly₃(k) For the total deviation, X, of the characteristic face KC3 after the k-th assembly₄(k) The total deviation of the characteristic face KC4 after the k step assembly is performed.

The assembly process varies with the mechanics of the machine tool, the capabilities of the manufacturing enterprise. Meanwhile, sometimes two or more components are assembled into a sub-assembly and then assembled with other parts. The assembly process for this problem is as follows:

(1) mounting a component A (a lathe bed) on a ground foundation;

(2) assembling the component B (upright post) and the component C (sliding seat) into a sub-assembly;

(3) mounting the sub-assembly to component a;

(4) the component D (table) is mounted on the component a.

In the first step of assembly, the mounting mating surface (KC 0: feature 0 in fig. 3, KC 3: feature 3 in fig. 3) of component B, D may be scraped or the like to accommodate the relative angular error. Since KC0 is defined as the reference plane, the adjustment matrix t (k) (0) can be written as follows:

$T\left(0\right)=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right]$

also, an adjustment cost weight matrix may be defined. Since the other components are not assembled prior to this assembly operation, the coefficients corresponding to these unassembled key features may be set to a slightly larger value, such as 1000. The adjustment cost matrix r (k) (k ═ 0) is therefore:

$R\left(0\right)=\left[\begin{array}{cccc}1000& 0& 0& 0\\ 0& 1000& 0& 0\\ 0& 0& 2& 0\\ 0& 0& 0& 1000\end{array}\right]$

in the second assembly operation (k 1), the components B and C are combined to form a sub-assembly. The feature surfaces KC2 (i.e., feature surface 2 in fig. 3) and KC1 (i.e., feature surface 1 in fig. 3) can be adjusted. Therefore, the adjustment matrix is:

$T\left(1\right)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$

assuming that the components B, C are linked by a linear motion system, the face KC1 of the motion system is more difficult to adjust than KC2, so the cost of adjustment of KC1 is greater than KC 2. The adjustment cost weight matrix of the second step is therefore:

$R\left(1\right)=\left[\begin{array}{cccc}10& 0& 0& 0\\ 0& 5& 0& 0\\ 0& 0& 1000& 0\\ 0& 0& 0& 1000\end{array}\right]$

the third step in the assembly process (k-2) is to mount the sub-assembly of component B, C to component a. Since KC1 is the bonding surface of element B, C, adjusting this feature surface would cost more, so the adjustment matrix and the adjustment cost weight matrix in the third step are:

$T\left(2\right)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right]$

$R\left(2\right)=\left[\begin{array}{cccc}20& 0& 0& 0\\ 0& 5& 0& 0\\ 0& 0& 5& 0\\ 0& 0& 0& 1000\end{array}\right]$

the fourth step (k — 3) of the assembling operation is to mount the component D on the component a. KC3 is the bonding surface of element A, D, so the adjustment matrix and the adjustment cost weight matrix are:

$T\left(3\right)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

$R\left(3\right)=\left[\begin{array}{cccc}20& 0& 0& 0\\ 0& 5& 0& 0\\ 0& 0& 20& 0\\ 0& 0& 0& 5\end{array}\right]$

during the tolerance assignment process, only the final accuracy requirement is of interest, so during the assembly process, the observation matrix C and the weight matrix S are set to a zero matrix of 4x 4.

In this assembly example, it is assumed that the accuracy requirements are the perpendicularity of KC1 and KC3, and the perpendicularity of KC2 and KC 4. The observation matrix after assembly is therefore:

$C\left(4\right)=\left[\begin{array}{cccc}1& 0& -1& 0\\ 0& 1& 0& -1\end{array}\right]$

the weight matrix is used for measuring the decrease of the profit caused by the loss of the precision, and the precision index weight matrix is set as follows:

$Q\left(4\right)=\left[\begin{array}{cc}10& 0\\ 0& 10\end{array}\right]$

therefore, the kalman gain coefficient matrix can be found as:

$K\left(0\right)=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ -0.433& -0.048& 0.433& 0.048\\ 0& 0& 0& 0\end{array}\right]$

$K\left(1\right)=\left[\begin{array}{cccc}0.167& 0.019& -0.167& -0.019\\ -0.167& 0.204& 0.167& -0.204\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$

$K\left(2\right)=\left[\begin{array}{cccc}0.100& 0.014& -0.100& -0.014\\ -0.200& 0.257& 0.200& -0.257\\ -0.400& -0.057& 0.400& 0.057\\ 0& 0& 0& 0\end{array}\right]$

$K\left(3\right)=\left[\begin{array}{cccc}0.227& 0.045& -0.227& 0.045\\ -0.182& 0.364& 0.182& 0.048\\ -0.227& -0.045& 0.227& 0.045\\ 0.182& -0.364& -0.182& 0.364\end{array}\right]$

the adjustment amount in the assembly process is determined by the Kalman gain coefficient, the adjustment matrix and the current total deviation state. Table 2 is some part error data.

TABLE 2 parts error data

With these basic data, the final deviation state can be obtained. FIG. 4 is a variation of the deviation state quantity, showing the trend of the deviation state during the assembly process under different cost weights. Wherein FIG. 4(a) is the state quantity X₁(k) (1) with X₃(k) (3) changes in the assembling process can be found, after the assembling is finished, X₁(k) And X₃(k) The values are very close and the closer the weights are. FIG. 4(b) shows the state quantity X₂(k) (2) with X₄(k) (4) variation during assembly, the results are similar to those of FIG. 4 (a).

If the error values of the components A-D are parameterized, i.e. ω_A、ω_B、ω_C、ω_DThen, in this example, the covariance of the angle error of the key feature plane can be expressed by the following equation:

<math> <mrow> <msub> <mi>D</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.037</mn> <msubsup> <mi>&omega;</mi> <mi>A</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mn>0.39</mn> <msubsup> <mi>&omega;</mi> <mi>B</mi> <mn>2</mn> </msubsup> </mrow> </math>

<math> <mrow> <msub> <mi>D</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.062</mn> <msubsup> <mi>&omega;</mi> <mi>A</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mrow> <mn>0.269</mn> <mi>&omega;</mi> </mrow> <mi>B</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mn>0.124</mn> <msubsup> <mi>&omega;</mi> <mi>C</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mn>0.167</mn> <msubsup> <mi>&omega;</mi> <mi>D</mi> <mn>2</mn> </msubsup> </mrow> </math>

<math> <mrow> <msub> <mi>D</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.078</mn> <msubsup> <mi>&omega;</mi> <mi>A</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mn>0.214</mn> <msubsup> <mi>&omega;</mi> <mi>B</mi> <mn>2</mn> </msubsup> </mrow> </math>

<math> <mrow> <msub> <mi>D</mi> <mn>4</mn> </msub> <mo>=</mo> <mn>0.067</mn> <msubsup> <mi>&omega;</mi> <mi>A</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mrow> <mn>0.250</mn> <mi>&omega;</mi> </mrow> <mi>B</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mn>0.062</mn> <msubsup> <mi>&omega;</mi> <mi>C</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mn>0.349</mn> <msubsup> <mi>&omega;</mi> <mi>D</mi> <mn>2</mn> </msubsup> </mrow> </math>

wherein D₁Is the covariance of the feature face KC1, D₂Is the covariance of the feature face KC2, D₃Is the covariance of the feature face KC3, D₄Is the covariance of the feature plane KC 4.

The verticality project covariance required by design is as follows:

<math> <mrow> <msub> <mi>D</mi> <mrow> <mn>1</mn> <mo>-</mo> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0.0075</mn> <msubsup> <mi>&omega;</mi> <mi>A</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mrow> <mn>0.0278</mn> <mi>&omega;</mi> </mrow> <mi>B</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mn>0.0069</mn> <msubsup> <mi>&omega;</mi> <mi>C</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mn>0.0083</mn> <msubsup> <mi>&omega;</mi> <mi>D</mi> <mn>2</mn> </msubsup> </mrow> </math>

<math> <mrow> <msub> <mi>D</mi> <mrow> <mn>2</mn> <mo>-</mo> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>0.0103</mn> <msubsup> <mi>&omega;</mi> <mi>C</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mn>0.0331</mn> <msubsup> <mi>&omega;</mi> <mi>D</mi> <mn>2</mn> </msubsup> </mrow> </math>

wherein D_1-3Is the pose relationship between the characteristic surfaces KC1 and KC3, D_2-4Is the pose relationship between the characteristic surfaces KC2 and KC 4.

Item D if precision is required_1-3，D_2-4Given a numerical value, the tolerances of the critical feature planes can be assigned. During distribution, the precision cost of parts can be set, and a precision cost target function is established. And then optimizing by adopting a genetic algorithm to obtain an optimized distribution result. All the precision item values are 3 multiplied by 10^-5rad, resulting part tolerance values are shown in table 3:

TABLE 3 tolerance values for parts (× 10)⁺⁵rad)

Claims (8)

1. A method of designing tolerances for a precision machine tool taking into account measurement and adjustment processes, comprising the steps of:

1) simplifying the machine tool into a basic large part, determining a key characteristic surface of the basic large part of the machine tool, and establishing a local coordinate system by means of the nominal or actual orientation of the part on the basis of the key characteristic surface; meanwhile, according to the topological structure and the assembly process of the machine tool, determining a basic coefficient matrix, an adjustment coefficient matrix and an observation matrix, and establishing a state space equation;

2) determining a precision cost matrix, adjusting a cost weight matrix and a precision index weight matrix according to the market research and the processing level of a manufacturing enterprise, and establishing a cost objective function; outputting a regulator through a discrete system, combining errors of parts, and obtaining an adjustment sequence by using a Kalman filtering method;

3) rewriting a state space equation into a Gaussian Markov random process equation, obtaining a covariance matrix according to the Gaussian Markov random process equation, and then carrying out iterative solution according to an assembly process to obtain the covariance matrix after the integral assembly is finished;

4) obtaining a functional relation f of the total precision index of the machine tool and the deviation of parts by utilizing the covariance matrix after the integral assembly and combining a variance synthesis formula;

5) and establishing a functional relation W between the total precision cost and the precision index of the part, taking the functional relation as a target function, and taking the functional relation f as a constraint condition to obtain a tolerance distribution result of the part, and finishing tolerance design of the machine tool.

2. The method for designing the tolerance of the precision machine tool in consideration of the measurement and adjustment process according to claim 1, wherein: the large foundation part in the step 1) comprises a lathe bed, a stand column, a sliding seat, a spindle box, a saddle and a workbench, and the key characteristic surfaces comprise the installation surface, the connection surface and the tail end surface of the large foundation part.

3. The method for designing the tolerance of the precision machine tool in consideration of the measurement and adjustment process according to claim 1, wherein: the basic coefficient matrixes in the step 1 are A (k), B (k) and F (k), the adjusting coefficient matrix is T (k), the observation matrix is C (k), the established state space equation is shown as the formula (1),

<math> <mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <mover> <mi>X</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mi>A</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>X</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>B</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>u</mi> <mo>~</mo> </mover> <mo>*</mo> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>F</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>&omega;</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mover> <mi>y</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>C</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>X</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mover> <mi>v</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </math>

wherein A (k) is a constant matrix in the k-th assembly, F (k) is a 6 x 6 matrix for converting the error in the k-th assembly from the k-th part coordinate system to the base coordinate system,representing the total deviation of the position and the orientation of the kth part assembling characteristic surface coordinate system in the reference transmission chain relative to the nominal pose thereof,deviation of the actual pose from the nominal pose in the part coordinate system for the kth part to assemble,accumulating a 6 x 1 vector for the total deviation of the global state after the kth part is assembled with respect to the base coordinate system, B (k) is an error adjustment vectorA transformation matrix from the part coordinate system of the kth part to the basic coordinate system, C (k) is an output matrix with the r × 6 element of 0 or 1 after the kth assembly is completed,the output vector of r multiplied by 1(r is less than or equal to 6) after the k-th assembly is completed,for the noise in the k-th assembly,and k is an error adjustment vector, is more than or equal to 0 and less than or equal to N, and N is the total assembly step number or the total assembly part number.

4. The method for designing the tolerance of the precision machine tool in consideration of the measurement and adjustment process according to claim 1, wherein: the precision cost matrix in the step 2) is S, the cost weight matrix is adjusted to be R (k), the precision index weight matrix is Q (k), the established cost objective function J is shown as the formula (2),

<math> <mrow> <mi>J</mi> <mo>=</mo> <msup> <mover> <mi>y</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mi>N</mi> <mo>)</mo> </mrow> <mi>S</mi> <mover> <mi>y</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>N</mi> <mo>)</mo> </mrow> <mo>+</mo> <munderover> <mi>&Sigma;</mi> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <mo>[</mo> <msup> <mover> <mi>y</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>Q</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>y</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mover> <mi>u</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>R</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>u</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>]</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </math>

the adjusted sequence was calculated by the following formula:

u*(k)＝-K(k)X(k)

wherein,the superscript T represents transposition for the output vector of r multiplied by 1(r is less than or equal to 6) after the integral assembly is finished,is the output vector of r multiplied by 1(r is less than or equal to 6) after the assembly in the kth step, k is more than or equal to 0 and less than or equal to N, N is the total assembly step number, Q (k) is the precision index weight matrix after the assembly in the kth step is completed,describing a 6 x 1 vector matched with characteristic error adjustment in the k-th assembly process control, wherein R (k) is an adjustment cost weight matrix in the k-th assembly process, X (k) is total deviation after the k-th assembly process is completed, u (k) is an adjustment sequence, and K (k) is a Kalman gain coefficient.

5. The precision machine tool tolerance designing method considering the measurement and adjustment process according to claim 4, wherein: the kalman gain coefficient k (k) is calculated by the following formula:

K(k)＝[R(k)+B^T(k)P(k+1)B(k)]^-1B^T(k)P(k+1)A(k)

P(k)＝Q(k)+A^T(k)P(k+1)[I+B(k)R^-1(k)B^T(k)P(k+1)]^-1A(k)

wherein R (k) is the adjustment cost weight matrix in the k-th assembling process, and B (k) is the error adjustment vectorA transformation matrix from the part coordinate system of the kth part to the base coordinate system,and (b) taking the error adjustment vector as an error adjustment vector, A (k) is a constant matrix during the k-th assembly, superscript T represents transposition, Q (k) is a precision index weight matrix after the k-th assembly is completed, I is a unit matrix, P (k) and P (k +1) are Ricatti equations of the k-th step and the k + 1-th step respectively, k is more than or equal to 0 and less than or equal to N, and N is the total assembly step number or the total assembly part number.

6. The method for designing the tolerance of the precision machine tool in consideration of the measurement and adjustment process according to claim 1, wherein: the Gaussian Markov random process equation in the step 3) is shown as a formula (3),

<math> <mrow> <mfenced open='' close=''> <mtable> <mtr> <mtd> <msub> <mover> <mi>X</mi> <mo>~</mo> </mover> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>A</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>X</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>F</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>&omega;</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mover> <mi>X</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mo>[</mo> <mi>I</mi> <mo>-</mo> <mi>B</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>T</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>K</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>]</mo> <msub> <mover> <mi>X</mi> <mo>~</mo> </mover> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow> </math>

the covariance matrix is shown as equation (4),

D_i(k)＝A^T(k)D(k)A(k)+F(k)V(k)F^T(k)

D(k+1)＝[I-B(k)T(k)K(k)]D_i(k)[I-B(k)T(k)K(k)]^T (4)

whereinIs the total deviation after the assembly and adjustment in the k step, A (k) is a constant matrix during the assembly in the k step, the superscript T represents transposition,representing the total deviation of the position and orientation of the kth part assembly characteristic surface coordinate system in the reference transmission chain relative to the nominal pose thereof, F (k) is a 6 x 6 matrix for converting the error in the kth part assembly from the kth part coordinate system to the base coordinate system,deviation of the actual pose from the nominal pose in the part coordinate system for the kth part to assemble,accumulating a 6 x 1 vector for the total deviation of the global state after the kth part is assembled with respect to the base coordinate system, I is an identity matrix, B (k) is an error adjustment vectorA transformation matrix from the part coordinate system of the kth part to the base coordinate system,t (k) is an error adjustment vector, T (k) is an adjustment coefficient matrix, and K (k) is a Kalman gain coefficient; d_i(k) The covariance matrix of the state quantity of the kth step of the ith part, D (k) is the covariance matrix of the state quantity of the kth step, D (k +1) is the covariance matrix of the state quantity of the kth step, V (k) is the variance of part errors introduced in the kth step, k is more than or equal to 0 and less than or equal to N, and N is the total assembly step number or the total assembly part number;

and (4) iteratively calculating to obtain a covariance matrix D (N) after the integral assembly is finished.

7. The method for designing the tolerance of the precision machine tool in consideration of the measurement and adjustment process according to claim 1, wherein: the variance synthesis formula in the step 4) is as follows:

E_i＝D(m)+D(n)-2D(m)D(n)

deviation of total accuracy index E from partsThe functional relationship f is as follows:

<math> <mrow> <mi>E</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mover> <mi>&omega;</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </math>

wherein E is_iFor the requirement of the i-th precision index, D (m) is the value of the m-th item on the diagonal of D (N), D (n) is the value of the n-th item on the diagonal of D (N), D (N) is the covariance matrix after the whole assembly is finished, and E ═ E₁，E₂，…，E_i，…，E_H) H is the number of the total required precision, i is more than or equal to 0 and less than or equal to H.

8. The method for designing the tolerance of the precision machine tool in consideration of the measurement and adjustment process according to claim 1, wherein: the functional relation W in the step 5) is as follows:

<math> <mrow> <mi>W</mi> <mo>=</mo> <munderover> <mi>&Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>H</mi> </munderover> <mfrac> <msub> <mi>O</mi> <mi>i</mi> </msub> <msub> <mi>E</mi> <mi>i</mi> </msub> </mfrac> </mrow> </math>

wherein H is the number of total required precision, i is more than or equal to 0 and less than or equal to H, E_iFor the ith accuracy index requirement, O_iThe i precision cost.