CN107066721A - A kind of multi-axis NC Machine Tools C axle geometric error measuring systems and parameter identification method - Google Patents

A kind of multi-axis NC Machine Tools C axle geometric error measuring systems and parameter identification method Download PDF

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CN107066721A
CN107066721A CN201710226509.1A CN201710226509A CN107066721A CN 107066721 A CN107066721 A CN 107066721A CN 201710226509 A CN201710226509 A CN 201710226509A CN 107066721 A CN107066721 A CN 107066721A
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CN107066721B (en
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范晋伟
邬昌军
李中生
陶浩浩
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Beijing University of Technology
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    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/30—Circuit design
    • G06F30/36—Circuit design at the analogue level
    • G06F30/367—Design verification, e.g. using simulation, simulation program with integrated circuit emphasis [SPICE], direct methods or relaxation methods
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
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Abstract

A kind of multi-axis NC Machine Tools C axle geometric error measuring systems and parameter identification method, for the structure and feature of multi-axis NC Machine Tools C axles, according to the operation principle of ball bar, utilize theory of multi body system, establish radially, the equation of motion with geometric error and the equation of motion ideally under the tangential and three kinds of linkage patterns in axial direction, pass through the end positions variable quantity of ball bar, and state the two ends coordinate of ball bar in the same coordinate system respectively, obtain the actual range of point-to-point transmission, so as to the relation of theorize model and actual measured value, realize the identification of 8 geometric error parameters of C axles.The present invention has picked out the whole error term of rotation C axles, the coupling phenomenon existed between geometric error parameter is solved, and accurately, fast, identification precision is high, to realizing that the error identification of remaining rotary shaft of multi-axis NC Machine Tools all has great theory significance and realistic meaning.

Description

A kind of multi-axis NC Machine Tools C axle geometric error measuring systems and parameter identification method
Technical field
The present invention relates to a kind of geometric error parameter identification measuring system and method, it is adaptable to multi-axis NC Machine Tools rotary shaft C axles, by the relation of theorize mathematical modeling and actual measured value, realize the identification of geometric error parameter.
Background technology
In modern manufacturing industry, multi-axis NC Machine Tools can adjust cutting position of the cutter relative to workpiece and direction simultaneously. Processed accordingly, with respect to traditional three-axis numerical control, multi-axis NC Machine Tools have higher stock-removing efficiency and machining accuracy, in boat Processed complex part plays an important effect in the fields such as sky, space flight, the energy and national defence, is lifting China manufacture level Technological break-through mouthful.
But because multi-axis NC Machine Tools add rotary shaft, its geometric error parameter significantly increases, and mutually exist complicated Coupled relation.Therefore the identification to geometric error parameter brings bigger difficulty, while being proposed more to error identification method High requirement, it is at present, relatively fewer for the discrimination method of rotary shaft, and most of letter has been carried out to the error parameter in model Change, have ignored the coupling condition between each error parameter, cause identification precision to decline.Therefore one kind is explored accurately to miss Poor discrimination method, considers the coupled relation of error parameter, realizes the accurate recognition of multi-axis NC Machine Tools geometric error parameter comprehensively It is very necessary.
In view of the similarity of multi-axis NC Machine Tools rotary shaft, but due to C axles be in multi-axis NC Machine Tools it is a kind of relatively common Rotary shaft, therefore for C axles geometric error parameter discrimination method research have certain representativeness.
The content of the invention
For problem present in multi-axis NC Machine Tools rotary shaft discrimination method, the present invention proposes a kind of based on many body system The multi-axis NC Machine Tools C axle geometric error parameter identification methods for theory of uniting, present invention, avoiding geometric error parameter predigesting phenomenon Coupled relation between parameter, on the basis of to the error of translation shaft by compensation, it is believed that the motion of translation shaft is formed Track be ideal trajectory, using multi-shaft interlocked, using theory of multi body system, setting up under different linkage patterns has geometric error The equation of motion, so as to realize the identification of C axle geometric error parameters.
A kind of multi-axis NC Machine Tools C axle geometric error parameter identification methods based on theory of multi body system, this method includes Following steps：
(1) adjacent two-particle systems relation equation is set up
It is as shown in Figure 1 the mutual alignment situation of two movable bodies, makes { rl}={ rx ry rz 1}TRepresent P on L bodieslPoint Relative to the position array of L body coordinate systems, { P is madelh}={ xlh ylh zlh 1}TRepresent PlPosition of the point relative to I body coordinate systems Array, two-particle systems relation equation is set up according to theory of multi body system, it can thus be concluded that：
{Plh}=[SIL]p[SIL]pe[SIL]s[SIL]se{rl} (1)
In formula, [SIL]pFor relative position transformation matrix of the L body phases for I bodies, [SIL]peFor phase of the L body phases for I bodies To position error transformation matrix, [SIL]sFor relative motion transformation matrix of the L body phases for I bodies, [SIL]seIt is L body phases for I The relative motion error transformation matrix of body.
(2) C axles geometric error parameter identification method is analyzed
C axle geometric errors parameter 8 altogether, be respectively：Runout error (the δ relevant with location pointx(C), δy(C), δz (C)), top pendulum and Run-out error (εx(C), εy(C)), rolling pendulum error (εz) and the error of perpendicularity (ε unrelated with location point (C)xC, εyC)。
As shown in Fig. 2 a kind of multi-axis NC Machine Tools C axles geometric error measuring system, the measuring system includes multi-axis numerical control Lathe C axles 1, ball bar 2, cutter spindle 3, spring 4, self-balanced upper rotary 5, regulation support bar 6 and base 7；Multi-axis NC Machine Tools C Axle 1 is workbench, and the both ends of ball bar 2 connect multi-axis NC Machine Tools C axles 1 and cutter spindle 3, the end of ball bar 2 respectively One is connected to the surface of workbench, and end one is fixed on the eccentric part of workbench；The end two of ball bar 2 is directly connected to cutter master Axle 3, ball bar 2 is horizontally disposed, and cutter spindle 3 is arranged vertically；The interlude of ball bar 2 be expansion link, expansion link by with bullet Spring 4 is connected with self-balanced upper rotary 5, and self-balanced upper rotary 5 is arranged on regulation support bar 6, and regulation support bar 6 is fixed on base 7, Spring 4 and self-balanced upper rotary 5 constitute the vibration-proof structure of ball bar 2；The vertical direction position of self-balanced upper rotary 5 is by regulation support bar 6 collapsing length control.
Multi-axis NC Machine Tools C axles 1 are measured using ball bar 2, and one end of ball bar 2 is fixed on multi-axis numerical control during measurement The position that lathe C axles 1 are separately above set, one end is fixed on cutter spindle 3, the track of presetting linkage, realizes that C axles are rotated same When X-axis, Y-axis synchronous interaction, X-axis, Y-axis be level on horizontal plane to synchronously transported to the two ends of, it is ensured that ball bar 2 vertically It is dynamic, so as to reach the purpose of measure geometry error.
As shown in figure 3, PhPoint is ball bar and workbench (C axles) connection end, Ah、Bh、DhPoint is to be connected with cutter spindle End, it is assumed that axle X-axis, Y-axis error have been compensated in motion, then Ah、Bh、DhThe track that point is formed in motion process is reason Think track, workbench (C axles) has geometric error, therefore the upper point P of workbench (C axles) during due to motionhIn actual motion Ideal position P is can deviate from journeyhPoint reaches physical location Ph' point, PhAh、PhBh、PhDhBe respectively radially, it is tangential, axially measured The preferable pose of ball bar, and theoretical length is respectively dr、dt、ds, due to the presence of geometric error error, actual motion process Middle PhAh、PhBh、PhDhBecome for Ph′Ah、Ph′Bh、Ph′Dh。
Ql-xQlyQlzQlCoordinate system is L body actual motion reference frames,It is that the actual body reference of L bodies is sat Mark system, Ql-xQlyQlzQlWithBetween relation show the motion conditions of L bodies, ideal point Ph、Ah、Bh、DhIn coordinate It is Ql-xQlyQlzQlIn position be：
(3) radial direction geometric error parameter is recognized
As shown in Fig. 3 (b), it can be obtained according to formula (1), the theoretical position point P under radial modehBy there is the kinematic chain of error It is described to rest frame at pivotIn, true location point P can be obtainedh' be：
In formula, ChThe angle rotated for workbench (C axles), L is that ball bar works you ball center in pivot coordinate It is X to coordinate value, H is that ball bar works you ball center in pivot coordinate system Z-direction coordinate value.
Due to being ideal trajectory with the movement locus of cutter spindle union end, therefore according to (1) formula, missed being moved in matrix Poor parameter is set to zero, can obtain theoretical position point AhIn coordinate systemIn the location point of without motion error be：
In formula, drFor the ball bar theoretical length of radial direction.
Then existIn coordinate system, Ph' and AhDifference be：
P is tried to achieve according to formula (4)h' and AhThe distance between expression formula be：
Following equation is set up according to formula (5)：
In formula, Δ drIt is ball bar from initial position to the long variable quantity of the radial bars of h-th of position.
By (6) formula equation both sides, square abbreviation can be obtained simultaneously：
-δx(Ch)-H(εy(Ch)+εxCcosCh-εyCsinCh)=Δ dr (7)
In (7) formula, order：
Wh=εy(Ch)+εxCcosCh-εyCsinCh (8)
Take can obtaining for two groups of difference H：
-δx(Ch)-H1Wh=Δ dr1 (9)
-δx(Ch)-H2Wh=Δ dr2 (10)
Subtracting formula (9) with formula (10) can obtain：
Formula (11) substitution (9) can be obtained：
δx(Ch)=- H1Wh-Δdr1 (12)
Work as Ch=0 i.e. axle is not moved also, motion angle error εy(Ch) it is 0, it can thus be concluded that：
εxC=W0 (13)
(4) tangential geometric error parameter identification
As shown in Fig. 3 (a), it can similarly obtain,In coordinate system, Ph' and BhDifference be:
P is tried to achieve according to formula (14)h' and BhThe distance between expression formula be：
Equation is set up according to formula (15):
In formula, Δ dtIt is ball bar from initial position to the long variable quantity of the tangential bar of h-th of position.
To (16) formula both sides, square abbreviation can be obtained simultaneously：
δy(Ch)+Lεz(Ch)-H(εx(Ch)+εxCsinCh+εyCcosCh)=Δ dt (17)
In formula (17), order：
Vh=εx(Ch)+εxCsinCh+εyCcosCh (18)
Two groups of different H are taken to obtain such as following formula：
δy(Ch)+L1εz(Ch)-H1Vh=Δ dt1 (19)
δy(Ch)+L1εz(Ch)-H2Vh=Δ dt2 (20)
Subtracting formula (19) with formula (20) can obtain：
Equation below can be obtained by changing L length：
δy(Ch)+L2εz(Ch)-H1Vh=Δ dt3 (22)
Subtracting formula (19) with formula (22) can obtain：
By Vh、εz(Ch) substitute into formula (19) can obtain：
Work as Ch=0 i.e. axle is not moved also, motion angle error εx(Ch) it is 0, it can thus be concluded that：
V0=εyC (25)
Bring formula (13) and formula (25) into (8) formula, can obtain：
εy(Ch)=Wh-εxCcosCh+εyCsinCh (26)
Bring formula (13) and formula (25) into (18) formula, can obtain：
εx(Ch)=Vh-εxCsinCh-εyCcosCh (27)
(5) axial geometric error parameter identification
As shown in Fig. 3 (c), it can similarly obtain,In coordinate system, Ph' and DhDifference be:
P is tried to achieve according to formula (28)h' and DhThe distance between expression formula be:
It can be obtained according to formula (29)：
In formula, Δ dsIt is ball bar from initial position to the long variable quantity of the bar of h-th of position.
By formula (30) both sides, square abbreviation can be obtained simultaneously：
δz(Ch)-L(εy(Ch)-εyCsinCh+εxCcosCh)=Δ ds (31)
From formula (8), formula (31) is changed into：
δz(Ch)-LWh=Δ ds (32)
It is assumed that current length L is Ls, therefore obtain：
δz(Ch)=Δ ds+LsWh (32)
In formula, Δ dsIt is ball bar from initial position to the long variable quantity of the axial stem of h-th of position, LsFor L value.
So far, eight error parameters related to C axles are all picked out, the runout error (δ relevant with location pointx(C), δy (C), δz(C)) picked out respectively by formula (12), formula (24) and formula (32)；Top pendulum and Run-out error (εx(C), εy(C)) respectively Picked out by formula (27) and formula (26)；Rolling pendulum error (εz(C)) picked out by formula (23)；The perpendicularity unrelated with location point Error (εxC, εyC) picked out respectively by formula (13) and formula (25).
Compared with prior art, the present invention has advantages below：
The present invention is directed to the structure and feature of multi-axis NC Machine Tools C axles, according to the operation principle of ball bar, utilizes many body system System is theoretical, establishes the equation of motion and perfect condition with geometric error under radially, tangentially and axially three kinds of linkage patterns Under the equation of motion, by the end positions variable quantity of ball bar, and by the two ends coordinate of ball bar state respectively it is same sit In mark system, the actual range of point-to-point transmission is obtained, so that the relation of theorize model and actual measured value, realizes 8, C axles several The identification of what error parameter.The present invention has picked out the whole error term of rotation C axles, solves what is existed between geometric error parameter Coupling phenomenon, and accurately, fast, identification precision is high, to realizing that the error identification of remaining rotary shaft of multi-axis NC Machine Tools all has There are great theory significance and realistic meaning.
Brief description of the drawings
Fig. 1 is adjacent two-particle systems relation schematic diagram.
Fig. 2 is C axle ball bar instrumentation plans.
Fig. 3 is three kinds of measurement pattern movement position schematic diagrames of C axles；Wherein, (a) is tangential motion position view, and (b) is Radial motion position view, (c) is axial movement position schematic diagram.
Embodiment
The method of the invention measures realization by ball bar.The ball bar is provided with telescopic fiber rod High accuracy displacement sensor, available for the change of detection fiber rod bar length, so that the data analysis C axles for passing through collection is every several What error parameter.
The method of the invention specifically includes following steps：
Step 1, according to theory of multi body system, adjacent two-particle systems relation equation is set up；
Step 2, by the analysis to rotary axis of machine tool, the geometric error of C axles is obtained, then according to the work of ball bar Principle, obtains the radial direction of C axle geometric error parameter identifications, tangentially, axially three kinds of measurement pattern directions of motion.Ah、Bh、DhPut and be With cutter spindle connection end, during due to motion there is geometric error, therefore the upper point P of workbench (C axles) in workbench (C axles)h Ideal position P is can deviate from during actual motionhPoint reaches physical location Ph' point, PhAh、PhBh、PhDhIt is radially respectively, cuts To, the preferable pose of axially measured ball bar, and theoretical length is respectively dr、dt、ds, P during actual motionhAh、PhBh、 PhDhBecome for Ph′Ah、Ph′Bh、Ph′Dh.Reference frame and motion reference coordinate system are set up in workbench (C axles), P is obtainedh、 Ah、Bh、DhPosition coordinates.
Step 3, according to the analysis of step 2, the identification of radial direction geometric error parameter is carried out, by setting up Actual point P in coordinate systemh' obtain position equation and mathematical point AhPosition equation obtain the change of the difference between the two and radial bars length The relation of amount, picks out the runout error δ relevant with location pointx(C), the error of perpendicularity ε unrelated with location pointxC；
Step 4, according to the analysis of step 2, tangential geometric error parameter identification is carried out, by setting up Actual point P in coordinate systemh' obtain position equation and mathematical point BhPosition equation obtain the change of the difference between the two and tangential bar length The relation of amount, picks out the runout error δ relevant with location pointyAnd ε (C)z(C), the error of perpendicularity ε unrelated with location pointyC, By combining the equation that step 3 is set up, the runout error ε relevant with location point is picked outyAnd ε (C)x(C)；
Step 5, according to the analysis of step 2, axial geometric error parameter identification is carried out, by setting up Actual point P in coordinate systemh' obtain position equation and mathematical point DhPosition equation obtain the change of the difference between the two and tangential bar length The relation of amount, picks out the runout error δ relevant with location pointz(C).So far, 8 errors of C axles, which are all picked out, comes.

Claims (3)

1. a kind of multi-axis NC Machine Tools C axle geometric error parameter identification methods based on theory of multi body system, it is characterised in that：Should Method comprises the following steps：
(1) adjacent two-particle systems relation equation is set up
The mutual alignment situation of two movable bodies, makes { rl}={ rx ry rz 1}TRepresent P on L bodieslPoint is relative to L body coordinate systems Position array, make { Plh}={ xlh ylh zlh 1}TRepresent PlThe position array relative to I body coordinate systems is put, according to many body system System theory sets up two-particle systems relation equation, it can thus be concluded that：
{Plh}=[SIL]p[SIL]pe[SIL]s[SIL]se{rl} (1)
In formula, [SIL]pFor relative position transformation matrix of the L body phases for I bodies, [SIL]peFor relative position of the L body phases for I bodies Error transformation matrix is put, [SIL]sFor relative motion transformation matrix of the L body phases for I bodies, [SIL]seIt is L body phases for I bodies Relative motion error transformation matrix；
(2) C axles geometric error parameter identification method is analyzed
C axle geometric errors parameter 8 altogether, be respectively：Runout error (the δ relevant with location pointx(C), δy(C), δz(C)), run Pendulum and Run-out error (εx(C), εy(C)), rolling pendulum error (εz) and the error of perpendicularity (ε unrelated with location point (C)xC, εyC)；
PhPoint is ball bar and workbench connection end, Ah、Bh、DhPut and be and cutter spindle connection end, it is assumed that axle X-axis, Y in motion Axis error has been compensated, then Ah、Bh、DhThe track that point is formed in motion process is ideal trajectory, workbench during due to motion There is a point P in geometric error, therefore workbenchhIdeal position P is can deviate from during actual motionhPoint reaches actual bit Put Ph' point, PhAh、PhBh、PhDhIt is that radial direction, the preferable pose of tangential, axially measured ball bar, and theoretical length are respectively respectively dr、dt、ds, due to the presence of geometric error error, P during actual motionhAh、PhBh、PhDhBecome for Ph′Ah、Ph′Bh、Ph′ Dh；
Ql-xQlyQlzQlCoordinate system is L body actual motion reference frames,For the actual body reference frame of L bodies, Ql-xQlyQlzQlWithBetween relation show the motion conditions of L bodies, ideal point Ph、Ah、Bh、DhIn coordinate system Ql- xQlyQlzQlIn position be：
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(3) radial direction geometric error parameter is recognized
It can be obtained according to formula (1), the theoretical position point P under radial modehIt is described to by the kinematic chain for having error quiet at pivot Only coordinate systemIn, true location point P ' can be obtainedhFor：
<mrow> <msub> <mrow> <mo>{</mo> <msubsup> <mi>P</mi> <mi>h</mi> <mo>&amp;prime;</mo> </msubsup> <mo>}</mo> </mrow> <msub> <mi>O</mi> <mi>i</mi> </msub> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>x</mi> <mi>C</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>y</mi> <mi>C</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>x</mi> <mi>C</mi> </mrow> </msub> </mrow> </mtd> <mtd> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>y</mi> <mi>C</mi> </mrow> </msub> </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>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </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>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>&amp;epsiv;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;epsiv;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>&amp;epsiv;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>L</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>H</mi> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
In formula, ChFor workbench (C axles) rotate angle, L be ball bar work you ball center pivot coordinate system X to Coordinate value, H is that ball bar works you ball center in pivot coordinate system Z-direction coordinate value；
Due to being ideal trajectory with the movement locus of cutter spindle union end, therefore according to (1) formula, kinematic error in matrix is joined Number is set to zero, can obtain theoretical position point AhIn coordinate systemIn the location point of without motion error be：
<mrow> <msub> <mrow> <mo>{</mo> <msub> <mi>A</mi> <mi>h</mi> </msub> <mo>}</mo> </mrow> <msub> <mi>O</mi> <mi>i</mi> </msub> </msub> <mo>=</mo> <msub> <mrow> <mo>&amp;lsqb;</mo> <mi>S</mi> <mi>I</mi> <mi>L</mi> <mo>&amp;rsqb;</mo> </mrow> <mi>p</mi> </msub> <msub> <mrow> <mo>&amp;lsqb;</mo> <mi>S</mi> <mi>I</mi> <mi>L</mi> <mo>&amp;rsqb;</mo> </mrow> <mi>s</mi> </msub> <msub> <mrow> <mo>{</mo> <msub> <mi>A</mi> <mi>h</mi> </msub> <mo>}</mo> </mrow> <msub> <mi>O</mi> <mi>l</mi> </msub> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </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>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>L</mi> <mo>+</mo> <msub> <mi>d</mi> <mi>r</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>H</mi> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
In formula, drFor the ball bar theoretical length of radial direction；
Then existIn coordinate system, P 'hWith AhDifference be：
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mrow> <mo>{</mo> <msubsup> <mi>P</mi> <mi>h</mi> <mo>&amp;prime;</mo> </msubsup> <mo>}</mo> </mrow> <msub> <mi>O</mi> <mi>i</mi> </msub> </msub> <mo>-</mo> <msub> <mrow> <mo>{</mo> <msub> <mi>A</mi> <mi>h</mi> </msub> <mo>}</mo> </mrow> <msub> <mi>O</mi> <mi>i</mi> </msub> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>x</mi> <mi>C</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>y</mi> <mi>C</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>x</mi> <mi>C</mi> </mrow> </msub> </mrow> </mtd> <mtd> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>y</mi> <mi>C</mi> </mrow> </msub> </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>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </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>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>&amp;epsiv;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;epsiv;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>&amp;epsiv;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>L</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>H</mi> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </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>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>L</mi> <mo>+</mo> <msub> <mi>d</mi> <mi>r</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>H</mi> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;delta;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>L</mi> <mi> </mi> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>&amp;delta;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>+</mo> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>x</mi> <mi>C</mi> </mrow> </msub> <mo>+</mo> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>d</mi> <mi>r</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;delta;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>L</mi> <mi> </mi> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>&amp;delta;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>-</mo> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>y</mi> <mi>C</mi> </mrow> </msub> <mo>+</mo> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>d</mi> <mi>r</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>L</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>&amp;epsiv;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>x</mi> <mi>C</mi> </mrow> </msub> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>y</mi> <mi>C</mi> </mrow> </msub> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
P ' is tried to achieve according to formula (4)hWith AhThe distance between expression formula be：
<mrow> <msub> <mi>L</mi> <mrow> <msub> <mi>A</mi> <mi>h</mi> </msub> <msubsup> <mi>P</mi> <mi>h</mi> <mo>&amp;prime;</mo> </msubsup> </mrow> </msub> <mo>=</mo> <msqrt> <mrow> <msubsup> <mi>d</mi> <mi>r</mi> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <msub> <mi>d</mi> <mi>r</mi> </msub> <msub> <mi>&amp;delta;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mn>2</mn> <msub> <mi>Hd</mi> <mi>r</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <msub> <mi>Hd</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>y</mi> <mi>C</mi> </mrow> </msub> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>x</mi> <mi>C</mi> </mrow> </msub> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> </mrow> </msqrt> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
Following equation is set up according to formula (5)：
<mrow> <msqrt> <mrow> <msubsup> <mi>d</mi> <mi>r</mi> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <msub> <mi>d</mi> <mi>r</mi> </msub> <msub> <mi>&amp;delta;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mn>2</mn> <msub> <mi>Hd</mi> <mi>r</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <msub> <mi>Hd</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>y</mi> <mi>C</mi> </mrow> </msub> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>x</mi> <mi>C</mi> </mrow> </msub> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> </mrow> </msqrt> <mo>=</mo> <msub> <mi>d</mi> <mi>r</mi> </msub> <mo>+</mo> <msub> <mi>&amp;Delta;d</mi> <mi>r</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
In formula, Δ drIt is ball bar from initial position to the long variable quantity of the radial bars of h-th of position；
By (6) formula equation both sides, square abbreviation can be obtained simultaneously：
-δx(Ch)-H(εy(Ch)+εxC cos Ch-εyC sin Ch)=Δ dr (7)
In (7) formula, order：
Wh=εy(Ch)+εxC cos Ch-εyC sin Ch (8)
Take can obtaining for two groups of difference H：
-δx(Ch)-H1Wh=Δ dr1 (9)
-δx(Ch)-H2Wh=Δ dr2 (10)
Subtracting formula (9) with formula (10) can obtain：
<mrow> <msub> <mi>W</mi> <mi>h</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>&amp;Delta;d</mi> <mrow> <mi>r</mi> <mn>2</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>&amp;Delta;d</mi> <mrow> <mi>r</mi> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>H</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>H</mi> <mn>1</mn> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>
Formula (11) substitution (9) can be obtained：
δx(Ch)=- H1Wh-Δdr1 (12)
Work as Ch=0 i.e. axle is not moved also, motion angle error εy(Ch) it is 0, it can thus be concluded that：
εxC=W0 (13)
(4) tangential geometric error parameter identification
It can similarly obtain,In coordinate system, P 'hWith BhDifference be:
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mrow> <mo>{</mo> <msubsup> <mi>P</mi> <mi>h</mi> <mo>&amp;prime;</mo> </msubsup> <mo>}</mo> </mrow> <msub> <mi>O</mi> <mi>i</mi> </msub> </msub> <mo>-</mo> <msub> <mrow> <mo>{</mo> <msub> <mi>B</mi> <mi>h</mi> </msub> <mo>}</mo> </mrow> <msub> <mi>O</mi> <mi>i</mi> </msub> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>x</mi> <mi>C</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>y</mi> <mi>C</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>x</mi> <mi>C</mi> </mrow> </msub> </mrow> </mtd> <mtd> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>y</mi> <mi>C</mi> </mrow> </msub> </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>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </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>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>&amp;epsiv;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;epsiv;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>&amp;epsiv;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>L</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>H</mi> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>x</mi> <mi>C</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>y</mi> <mi>C</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>x</mi> <mi>C</mi> </mrow> </msub> </mrow> </mtd> <mtd> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>y</mi> <mi>C</mi> </mrow> </msub> </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>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </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>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>L</mi> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>d</mi> <mi>t</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>H</mi> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;delta;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>L</mi> <mi> </mi> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>&amp;delta;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>+</mo> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>x</mi> <mi>C</mi> </mrow> </msub> <mo>+</mo> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>d</mi> <mi>t</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;delta;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>L</mi> <mi> </mi> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>&amp;delta;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>+</mo> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>d</mi> <mi>t</mi> </msub> <mo>-</mo> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>y</mi> <mi>C</mi> </mrow> </msub> <mo>+</mo> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>L</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>&amp;epsiv;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>x</mi> <mi>C</mi> </mrow> </msub> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>y</mi> <mi>C</mi> </mrow> </msub> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> </mtable> </mfenced>
<mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow>
P is tried to achieve according to formula (14)h' and BhThe distance between expression formula be：
<mrow> <msub> <mi>L</mi> <mrow> <msub> <mi>B</mi> <mi>h</mi> </msub> <msubsup> <mi>P</mi> <mi>h</mi> <mo>&amp;prime;</mo> </msubsup> </mrow> </msub> <mo>=</mo> <msqrt> <mrow> <msubsup> <mi>d</mi> <mi>t</mi> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <msub> <mi>d</mi> <mi>t</mi> </msub> <msub> <mi>&amp;delta;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <msub> <mi>Ld</mi> <mi>t</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mn>2</mn> <msub> <mi>Hd</mi> <mi>t</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;epsiv;</mi> <mi>x</mi> </msub> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> <mo>+</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>x</mi> <mi>C</mi> </mrow> </msub> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>+</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>y</mi> <mi>C</mi> </mrow> </msub> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> </mrow> </msqrt> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>
Equation is set up according to formula (15):
<mrow> <msqrt> <mrow> <msubsup> <mi>d</mi> <mi>t</mi> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <msub> <mi>d</mi> <mi>t</mi> </msub> <msub> <mi>&amp;delta;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <msub> <mi>Ld</mi> <mi>t</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mn>2</mn> <msub> <mi>Hd</mi> <mi>t</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;epsiv;</mi> <mi>x</mi> </msub> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> <mo>+</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>x</mi> <mi>C</mi> </mrow> </msub> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>+</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>y</mi> <mi>C</mi> </mrow> </msub> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> </mrow> </msqrt> <mo>=</mo> <msub> <mi>d</mi> <mi>t</mi> </msub> <mo>+</mo> <msub> <mi>&amp;Delta;d</mi> <mi>t</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow>
In formula, Δ dtIt is ball bar from initial position to the long variable quantity of the tangential bar of h-th of position；
To (16) formula both sides, square abbreviation can be obtained simultaneously：
δy(Ch)+Lεz(Ch)-H(εx(Ch)+εxC sin Ch+εyC cos Ch)=Δ dt (17)
In formula (17), order：
Vh=εx(Ch)+εxC sin Ch+εyC cos Ch (18)
Two groups of different H are taken to obtain such as following formula：
δy(Ch)+L1εz(Ch)-H1Vh=Δ dt1 (19)
δy(Ch)+L1εz(Ch)-H2Vh=Δ dt2 (20)
Subtracting formula (19) with formula (20) can obtain：
<mrow> <msub> <mi>V</mi> <mi>h</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>&amp;Delta;d</mi> <mrow> <mi>t</mi> <mn>2</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>&amp;Delta;d</mi> <mrow> <mi>t</mi> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>H</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>H</mi> <mn>2</mn> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>21</mn> <mo>)</mo> </mrow> </mrow>
Equation below can be obtained by changing L length：
δy(Ch)+L2εz(Ch)-H1Vh=Δ dt3 (22)
Subtracting formula (19) with formula (22) can obtain：
<mrow> <msub> <mi>&amp;epsiv;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>&amp;Delta;d</mi> <mrow> <mi>t</mi> <mn>3</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>&amp;Delta;d</mi> <mrow> <mi>t</mi> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>L</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>23</mn> <mo>)</mo> </mrow> </mrow>
By Vh、εz(Ch) substitute into formula (19) can obtain：
<mrow> <msub> <mi>&amp;delta;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>&amp;Delta;d</mi> <mrow> <mi>t</mi> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>H</mi> <mn>1</mn> </msub> <msub> <mi>V</mi> <mi>h</mi> </msub> <mo>=</mo> <msub> <mi>&amp;Delta;d</mi> <mrow> <mi>t</mi> <mn>1</mn> </mrow> </msub> <mo>-</mo> <mfrac> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;Delta;d</mi> <mrow> <mi>t</mi> <mn>3</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>&amp;Delta;d</mi> <mrow> <mi>t</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>L</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>H</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;Delta;d</mi> <mrow> <mi>t</mi> <mn>2</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>&amp;Delta;d</mi> <mrow> <mi>t</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>H</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>H</mi> <mn>2</mn> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>24</mn> <mo>)</mo> </mrow> </mrow>
Work as Ch=0 i.e. axle is not moved also, motion angle error εx(Ch) it is 0, it can thus be concluded that：
V0=εyC (25)
Bring formula (13) and formula (25) into formula (8), can obtain：
εy(Ch)=Wh-εxC cos Ch+εyC sin Ch (26)
Bring formula (13) and formula (25) into (18) formula, can obtain：
εx(Ch)=Vh-εxC sin Ch-εyC cos Ch
(27)
(5) axial geometric error parameter identification
It can similarly obtain,In coordinate system, P 'hWith DhDifference be:
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mrow> <mo>{</mo> <msubsup> <mi>P</mi> <mi>h</mi> <mo>&amp;prime;</mo> </msubsup> <mo>}</mo> </mrow> <msub> <mi>O</mi> <mi>i</mi> </msub> </msub> <mo>-</mo> <msub> <mrow> <mo>{</mo> <msub> <mi>A</mi> <mi>h</mi> </msub> <mo>}</mo> </mrow> <msub> <mi>O</mi> <mi>i</mi> </msub> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>x</mi> <mi>C</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>y</mi> <mi>C</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>x</mi> <mi>C</mi> </mrow> </msub> </mrow> </mtd> <mtd> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>y</mi> <mi>C</mi> </mrow> </msub> </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>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </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>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>&amp;epsiv;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;epsiv;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>&amp;epsiv;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>L</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>H</mi> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> </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>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>L</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>H</mi> <mo>+</mo> <msub> <mi>d</mi> <mi>s</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;delta;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>L</mi> <mi> </mi> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>&amp;delta;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>+</mo> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>x</mi> <mi>C</mi> </mrow> </msub> <mo>+</mo> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;delta;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>L</mi> <mi> </mi> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>&amp;delta;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>-</mo> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>y</mi> <mi>C</mi> </mrow> </msub> <mo>+</mo> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <msub> <mi>&amp;epsiv;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>d</mi> <mi>s</mi> </msub> <mo>-</mo> <mi>L</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>&amp;epsiv;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>x</mi> <mi>C</mi> </mrow> </msub> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>y</mi> <mi>C</mi> </mrow> </msub> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>28</mn> <mo>)</mo> </mrow> </mrow>
P ' is tried to achieve according to formula (28)hWith DhThe distance between expression formula be:
<mrow> <msub> <mi>L</mi> <mrow> <msub> <mi>D</mi> <mi>h</mi> </msub> <msubsup> <mi>P</mi> <mi>h</mi> <mo>&amp;prime;</mo> </msubsup> </mrow> </msub> <mo>=</mo> <msqrt> <mrow> <msubsup> <mi>d</mi> <mi>s</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mn>2</mn> <msub> <mi>d</mi> <mi>s</mi> </msub> <msub> <mi>&amp;delta;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mn>2</mn> <msub> <mi>Ld</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>&amp;epsiv;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>y</mi> <mi>C</mi> </mrow> </msub> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>+</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>x</mi> <mi>C</mi> </mrow> </msub> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </msqrt> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>29</mn> <mo>)</mo> </mrow> </mrow>
It can be obtained according to formula (29)：
<mrow> <msqrt> <mrow> <msubsup> <mi>d</mi> <mi>s</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mn>2</mn> <msub> <mi>d</mi> <mi>s</mi> </msub> <msub> <mi>&amp;delta;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mn>2</mn> <msub> <mi>Ld</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;epsiv;</mi> <mi>y</mi> </msub> <mo>(</mo> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>y</mi> <mi>C</mi> </mrow> </msub> <mi>sin</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>+</mo> <msub> <mi>&amp;epsiv;</mi> <mrow> <mi>x</mi> <mi>C</mi> </mrow> </msub> <mi>cos</mi> <mi> </mi> <msub> <mi>C</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> </mrow> </msqrt> <mo>=</mo> <msub> <mi>d</mi> <mi>s</mi> </msub> <mo>+</mo> <msub> <mi>&amp;Delta;d</mi> <mi>s</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>30</mn> <mo>)</mo> </mrow> </mrow>
In formula, Δ dsIt is ball bar from initial position to the long variable quantity of the bar of h-th of position；
By formula (30) both sides, square abbreviation can be obtained simultaneously：
δz(Ch)-L(εy(Ch)-εyC sin Ch+εxC cos Ch)=Δ ds (31)
From formula (8), formula (31) is changed into：
δz(Ch)-LWh=Δ ds (32)
It is assumed that current length L is Ls, therefore obtain：
δz(Ch)=Δ ds+LsWh (32)
In formula, Δ dsIt is ball bar from initial position to the long variable quantity of the axial stem of h-th of position, LsFor L value；
So far, eight error parameters related to C axles are all picked out, the runout error (δ relevant with location pointx(C), δy(C), δz(C)) picked out respectively by formula (12), formula (24) and formula (32)；Top pendulum and Run-out error (εx(C), εy(C)) pass through respectively Formula (27) and formula (26) are picked out；Rolling pendulum error (εz(C)) picked out by formula (23)；The error of perpendicularity unrelated with location point (εxC, εyC) picked out respectively by formula (13) and formula (25).
2. a kind of multi-axis NC Machine Tools C axle geometric error parameters based on theory of multi body system according to claim 1 are distinguished Knowledge method, it is characterised in that：
This method specifically includes following steps：
Step 1, according to theory of multi body system, adjacent two-particle systems relation equation is set up；
Step 2, by the analysis to rotary axis of machine tool, the geometric error of C axles is obtained, it is then former according to the work of ball bar Reason, obtains the radial direction of C axle geometric error parameter identifications, tangentially, axially three kinds of measurement pattern directions of motion；Ah、Bh、DhPoint be with Cutter spindle connection end, during due to motion there is a point P in geometric error, therefore workbench in workbenchhIn actual motion process In can deviate from ideal position PhPoint reaches physical location P 'hPoint, PhAh、PhBh、PhDhBe respectively radially, it is tangential, axially measured Ball bar ideal pose, and theoretical length is respectively dr、dt、ds, P during actual motionhAh、PhBh、PhDhBecome for P 'hAh、 P′hBh、P′hDh；Reference frame and motion reference coordinate system are set up in workbench, P is obtainedh、Ah、Bh、DhPosition coordinates；
Step 3, according to the analysis of step 2, the identification of radial direction geometric error parameter is carried out, by setting upCoordinate Actual point P ' in systemhObtain position equation and mathematical point AhPosition equation obtain the variable quantity of the difference between the two and radial bars length Relation, picks out the runout error δ relevant with location pointx(C), the error of perpendicularity ε unrelated with location pointxC；
Step 4, according to the analysis of step 2, tangential geometric error parameter identification is carried out, by setting upCoordinate Actual point P ' in systemhObtain position equation and mathematical point BhPosition equation obtain the variable quantity of the difference between the two and tangential bar length Relation, picks out the runout error δ relevant with location pointyAnd ε (C)z(C), the error of perpendicularity ε unrelated with location pointyC, pass through The equation set up with reference to step 3, picks out the runout error ε relevant with location pointyAnd ε (C)x(C)；
Step 5, according to the analysis of step 2, axial geometric error parameter identification is carried out, by setting upCoordinate Actual point P ' in systemhObtain position equation and mathematical point DhPosition equation obtain the variable quantity of the difference between the two and tangential bar length Relation, picks out the runout error δ relevant with location pointz(C)；So far, 8 errors of C axles, which are all picked out, comes.
3. using a kind of multi-axis NC Machine Tools C axles geometric error measuring system of claim 1 methods described design, its feature exists In：
The measuring system includes multi-axis NC Machine Tools C axles (1), ball bar (2), cutter spindle (3), spring (4), self-balanced upper rotary (5), regulation support bar (6) and base (7)；Multi-axis NC Machine Tools C axles (1) are workbench, and the both ends of ball bar (2) connect respectively Multi-axis NC Machine Tools C axles (1) and cutter spindle (3) are connect, the end one of ball bar (2) is connected to the surface of workbench, end one It is fixed on the eccentric part of workbench；The end two of ball bar (2) is directly connected to cutter spindle (3), and ball bar (2) is horizontally disposed, Cutter spindle (3) is arranged vertically；The interlude of ball bar (2) be expansion link, expansion link by with spring (4) and self-balanced upper rotary (5) connect, self-balanced upper rotary (5) is arranged on regulation support bar (6), regulation support bar (6) is fixed on base (7), spring (4) and self-balanced upper rotary (5) composition ball bar (2) vibration-proof structure；The vertical direction position of self-balanced upper rotary (5) is by regulation branch The collapsing length control of strut (6).
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CN107695791A (en) * 2017-09-26 2018-02-16 西北工业大学 The general rotary shaft geometric error discrimination method unrelated with position
CN107695791B (en) * 2017-09-26 2019-02-26 西北工业大学 The general rotation axis geometric error discrimination method unrelated with position
CN109709467A (en) * 2017-10-26 2019-05-03 北京信息科技大学 A kind of automatic prober platform kinematic error compensation method
CN110109418A (en) * 2019-05-19 2019-08-09 重庆理工大学 A kind of geometric error Fast Identification Method of five face machining center of large-sized gantry
CN111451880A (en) * 2020-04-21 2020-07-28 中国工程物理研究院机械制造工艺研究所 AB double-tool pendulum five-axis magnetorheological polishing machine tool structure parameter calibration method
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CN113733155A (en) * 2021-08-12 2021-12-03 广州数控设备有限公司 Six-axis industrial robot calibration device and calibration method
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CN114905332A (en) * 2022-05-20 2022-08-16 重庆大学 Machine tool rotating shaft position-related geometric error identification method based on single-axis motion
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