# US 10,838,392 B2 · Modeling and compensation method for the spindle's radial thermal drift error in a horizontal CNC lathe > Agents: this data is available as JSON from the patentgrep API and MCP server — see https://patentgrep.com/llms.txt - Number: US 10,838,392 B2 - Document ID: US10838392B2 - Status: Granted - Kind: B2 - Application number: 16/325,984 - Filed: 2017-11-06 - Published: 2020-02-27 - Granted: 2020-11-17 - Earliest claimed: 2017-11-06 - Assignees: Dalian University of Technology - Applicants: Dalian University of Technology - Inventors: Kuo Liu; Yongqing Wang; Haibo Liu; Te Li; Haining Liu; Dawei Li - Web page: https://patentgrep.com/patent/US10838392B2 ## CPC - B23Q 11/0007: {by compensating occurring thermal dilations (B23Q15/18 takes precedence)} - G05B 19/404: characterised by control arrangements for compensation, e.g. for backlash, overshoot, tool offset, tool wear, temperature, machine construction errors, load, inertia (G05B19/19, G05B19/41 take precedence) - B23Q 11/14: Methods or arrangements for maintaining a constant temperature in parts of machine tools - B23Q 15/18: Compensation of tool-deflection due to temperature or force - B23Q 17/22: for indicating or measuring existing or desired position of tool or work {(B23Q16/005 takes precedence)} - G05B 2219/35408: Calculate new position data from actual data to compensate for contour error - G05B 2219/49219: Compensation temperature, thermal displacement - B23Q: DETAILS, COMPONENTS, OR ACCESSORIES FOR MACHINE TOOLS, e.g. ARRANGEMENTS FOR COPYING OR CONTROLLING (tools of the kind used in lathes or boring machines B23B27/00); MACHINE TOOLS IN GENERAL CHARACTERISED BY THE CONSTRUCTION OF PARTICULAR DETAILS OR COMPONENTS; COMBINATIONS OR ASSOCIATIONS OF METAL-WORKING MACHINES, NOT DIRECTED TO A PARTICULAR RESULT - G05B: CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS (systems for controlling or regulating non-electric variables G05D; systems for regulating electric or magnetic variables G05F; control devices or systems insofar as characterised by mechanical features only G05G) ## Abstract The invention provides a method for modeling and compensating for the spindle's radial thermal drift error in a horizontal CNC lathe, which belongs to the field of error compensation technology of CNC machine tools. Firstly, the thermal drift error of two points in the radial direction of the spindle and the corresponding temperature of the key points are tested; then the thermal inclination angle of the spindle is obtained based on the thermal tilt deformation mechanism of the spindle, and the correlation between the thermal inclination angle and the temperature difference between the left and right sides of the spindle box is analyzed. According to the positive or negative thermal drift error of the two points that have been measured and the elongation or shortening of the spindle box on the left and right sides, the thermal deformation of the spindle is then classified and the thermal drift error model under various thermal deformation attitudes is then established. Then the influence of the size of the machine tool's structure on the prediction results of the model is analyzed. In real-time compensation, the thermal deformation attitude of the spindle is automatically judged according to the temperature of the key points, and the corresponding thermal drift error model is automatically selected to apply the compensation to the spindle. The method is used to distinguish the thermal deformation attitude of the spindle in a CNC lathe, and the thermal deformation mechanism is used to predict the radial thermal drift error of the spindle. ## Claims ### Claim 1 The invention relates to a method for modeling and compensating for the spindle's radial thermal drift error in a horizontal CNC lathe, comprising: measuring the radial thermal drift error and the temperature of the key points of the spindle of a CNC; lathe, said measuring the radial thermal drift error comprising: testing the radial thermal drift error and temperature of the spindle of a CNC lathe; using two temperature sensors respectively to measure the temperatures T1 and T2 of both the left and right sides of the spindle box, using two displacement sensors to measure the error in the X direction of the two location points of the detecting check bar clamped by the spindle; during the test, heating the spindle by rotating the spindle at a certain speed for a few hours, and then stopping the spindle for a few hours to cool down, wherein the thermal error ei of the spindle in the vertical direction produces the thermal error component ei,x in the X direction, and the thermal errors e1,x and e2,x of the spindle in the X direction are calculated according to the following formula: e2,x=sin(αxdir)×e2 (1) e1,x=sin(αxdir)×e1 (2) wherein in the above formula, αxdir is the tilt angle of the X axis of lathe, i=1 or 2, 1 indicates the right side and 2 indicates the left side; analyzing between the thermal inclination and the temperature difference of the spindle, said analyzing comprising: calculating the thermal dip angle of the spindle, after being heated, using the following formula: φ s = arctan ⁢ e 1 , x - e 2 , x sin ⁢ ⁢ ( α xdir ) × L snr ( 3 ) wherein in the above formula, φs is the thermal dip angle of the spindle and Lsnr is the distance between the two error measuring points; determining the relation diagram between the thermal dip φs of the spindle and the difference of the two temperatures ΔT is then determined, ΔT=T1−T2, analyzing the similarity of the two curves; and calculating the correlation between φs and ΔT according to the following formula: R ⁡ ( φ s , Δ ⁢ ⁢ T ) = Cov ⁡ ( φ s , Δ ⁢ ⁢ T ) Cov ⁡ ( φ s , φ s ) ⁢ Cov ⁡ ( Δ ⁢ ⁢ T , Δ ⁢ ⁢ T ) ( 4 ) wherein in the above formula, R is the correlation matrix between φs and ΔT, Cov (φs, ΔT) is the covariance matrix between φs and ΔT; setting up the error models of the spindle's radial thermal drift under different thermal deformations, and setting up comprising: according to the sign of the two error data points, e1,x and e2,x, and the extension or shortening of the spindle box on the left and right sides, dividing the thermal deformation of the spindle into three categories and ten types; then δ1 is the thermal change on the left side of the spindle box and δr is the thermal variation on the right side of the spindle box, both δ1 and δr are positive for thermal expansion and negative for thermal contraction; dcrs is the distance from the intersection point of the spindle in the initial state and the deformed spindle to the right side of the spindle box, dspl is the distance between the left and right ends of the spindle box, dss is the horizontal distance between the right end of the spindle box and the left displacement sensor, dsnr is the horizontal distance between the left displacement sensor and the right displacement sensor; when the spindle has the thermal deformation attitude that δl>δr≥0, and the check bar is close to the left displacement sensor and the right displacement sensor, the relationship between the radial thermal drift error of the spindle and the temperature is then established; the linear relationship between the thermal expansion of the left and right sides of the spindle box and the temperature is expressed by the formulas (5) and (6): δ1(t)=ζl1×(T1(t)−T1(0))+ζp2 (5) δr(t)=ζr1×(T2(t)−T2(0))+ζr2 (6) wherein in the above formulas, ζl1, ζl2, ζr1 and ζr2 are the coefficients to be identified; wherein when the spindle has the thermal deformation attitude that δl>δr≥0, and the check bar is close to the left displacement sensor and the right displacement sensor, the dcrs(t) for any time t can be calculated from formula (7): d crs ⁡ ( t ) = δ r ⁡ ( t ) δ l ⁡ ( t ) - δ r ⁡ ( t ) × d spl ( 7 ) ; and calculating the thermal drift errors, e1,x(t) and e2,x(t), of the spindle in the X direction at any time t by formulas (8) and (9): e 2 , x ⁡ ( t ) = δ l ⁡ ( t ) ⁢ d ss - δ r ⁡ ( t ) ⁢ d ss - δ r ⁡ ( t ) ⁢ d spl sin ⁢ ⁢ ( α xdir ) - 1 ⁢ d spl ( 8 ) e 1 , x ⁡ ( t ) = ( δ l ⁡ ( t ) - δ r ⁡ ( t ) ) ⁢ ( d ss + d snr ) - δ r ⁡ ( t ) ⁢ d spl sin ⁢ ⁢ ( α xdir ) - 1 ⁢ d spl ( 9 ) analyzing the influence of the influence of the size of the machine tool's structure on the predicted result of the model, said analyzing comprising: analyzing the influence of the measurement bias of dspl and dss in the thermal drift error model on the predicted results of the model; analyzing the reliability of the fluctuation value of the predicted residuals within a certain allowable deviation range using the asymptotic integration method; the expression of the function Z for the problem is described as: Z=gx(X)=δ−δa(dspl,dss) (10) wherein in the above formula, X is a random vector composed of dspl and dss, δ is the allowable deviation index, δa is the fluctuation value of the predicted residuals and is defined as: δ a ⁡ ( d spl , d ss ) = ∑ i = 1 N ⁢ ⁢  R ⁡ ( i ) - R n ⁡ ( i )  N ( 11 ) wherein in the above formulas, R is the predicted residual of dspl and dss as random variables, which is a function of dspl and dss, Rn is the predicted residual of the model when dspl and dss are true values, and N is the number of sampling points when the thermal error is measured; wherein if fx(x) is a joint probability density function of X, then the probability that the fluctuation value of the predicted residuals do not fall within a certain allowable deviation range is calculated from formula (12): pf=∫gx(x)≤0 exp [h(x)]dx (12) wherein in the above formulas, h(x)=ln fx(x); wherein if x*=(dspl*,dss*)T is a point on the surface of the limit state, at this point then h (x) is expanded into a Taylor series and taken to the quadratic term: h ⁡ ( x ) = h ⁡ ( x * ) + 1 2 ⁢ υ T ⁢ B ⁢ ⁢ υ - 1 2 ⁢ ( x - x * - B ⁢ ⁢ υ ) ⁢ B - 1 ⁡ ( x - x * - B ⁢ ⁢ υ ) ( 13 ) wherein in the above formulas: υ=∇h(x*) (14) B=[∇2h(x*)]−1 (15) wherein the limit state surface Z=gx(X)=0 is replaced by a hyperplane at point x* to realize the asymptotic integration of the probability that the fluctuation value of the predicted residual error exceeds the allowable range; wherein the First-order Second-moment Method is used to calculate the reliability index that the fluctuation value of the predicted residuals belongs in a certain allowable deviation range according to formula (16): β L = μ Z L σ Z L = [ ∇ g ⁢ ⁢ x ⁡ ( x * ) ] T ⁢ B ⁢ ⁢ υ [ ∇ g ⁢ ⁢ x ⁡ ( x * ) ] T ⁢ B ⁢ ∇ g ⁢ ⁢ x ⁡ ( x * ) ( 16 ) wherein the First-order Second-moment Method is used to calculate the failure probability index that the fluctuation value of the predicted residual belongs in a certain allowable deviation range according to formula (17): p fL = ( 2 ⁢ ) n / 2 ⁢ det ⁢ ⁢ B ⁢ f x ⁡ ( x * ) ⁢ exp ⁡ ( υ T ⁢ B ⁢ ⁢ υ 2 ) ⁢ Φ ⁡ ( - β L ) ( 17 ) wherein according to the Lagrange multiplier method for solving optimization problems, the multiplier λ is introduced; From ∂ L ⁡ ( x * , λ ) ∂ x = 0 , which is one of the stationary value conditions of the functional L (x,λ)=h(x)+λgx(x), the following result is then obtained: ∇ g ⁢ ⁢ x ⁡ ( x * ) = - 1 λ ⁢ ∇ h ⁢ ⁢ ( x * ) = - υ λ ( 18 ) wherein by substituting formula (18) into formula (16), the following result is obtained: βL=√{square root over (υTBυ)} (19) wherein by substituting formula (19) into formula (17), the following result is obtained: p fL ≈ ( 2 ⁢ π ⁢ ) ( n - 1 ) / 2 ⁢ f x ⁡ ( x * ) ⁢ det ⁢ ⁢ B υ T ⁢ B ⁢ ⁢ υ ( 20 ) , and wherein the reliability that the fluctuating value of the predicted residuals, calculated by the asymptotic integration method, falls within a certain allowable deviation range is then obtained according to (21): pr=1−pfL (21) ; and determining of the spindle's thermal deformation attitude and model, said determining comprising: judging the thermal deformation δl, δr and dσ of the two sides of the spindle box using the thermal deformation attitude of the spindle (1) which changes irregularly in the processing process; where dσ is the distance from the intersection point of the spindle in the initial state and the deformed spindle to the right side of the spindle box (2); calculating, under the various thermal deformation attitudes, the formulas of dσ from formula (22): d σ ⁡ ( t ) = d spl ×  ζ r ⁢ ⁢ 1 ⁡ ( T 2 ⁡ ( t ) - T 2 ⁡ ( 0 ) ) + ζ r ⁢ ⁢ 2 ζ l ⁢ ⁢ 1 ⁡ ( T 1 ⁡ ( t ) - T 1 ⁡ ( 0 ) ) - ζ r ⁢ ⁢ 1 ⁡ ( T 2 ⁡ ( t ) - T 2 ⁡ ( 0 ) ) + ζ l ⁢ ⁢ 2 - ζ r ⁢ ⁢ 2  ( 22 ) wherein the criteria for determining the thermal deformation attitudes of the spindle are set as follows: attitude (1): δl>δr≥0, dσ≤dss attitude (2): δr<0<δl attitude (3): δr≤δl<0 attitude (4): δl<δr<0, dss+dsnrδr≥0, dssδr≥0, dss+dsnrδr≥0 attitude (9): δl<0<δr attitude (10): δl<δr<0, dσ≤dss considering the thermal tilt of the spindle, compensating the different errors for different lengths of the workpieces-; if dwp is the distance between the processed point on the workpiece and the end face of the chuck, and ds is the distance between the left displacement sensor and the end face of the chuck; under the various thermal deformation attitudes, whether dwpds+dsnr, the amount of thermal error compensation ewp of the machined spot on the workpiece is then calculated in accordance with formula (23): e wp = e 2 , x ⁡ ( d s + d snr - d wp ) - e 1 , x ⁡ ( d s - d wp ) d snr ( 23 ) ; and inputting the thermal error's predicted value ewp is into the CNC system of the machine tool in real time to realize thermal error compensation for the spindle of the CNC lathe at any position or time.