JPH0966464A - Polishing method and polishing apparatus - Google Patents

Abstract

(57) Abstract: In a polishing method and apparatus, the calculation method for determining the residence time of a tool is simplified, the calculation time is shortened, labor is saved on the computer, and the machining lead time is shortened. It is possible to eliminate restrictions on conditions, prevent complication of the processing machine, expand the processing range, respond to rotary processed products, and improve the processing conditions. SOLUTION: When polishing a workpiece by a polishing driving means 101, a polishing tool is processed from a wear reduction shape h (x) and a unit removal shape f (x, u) of a polishing tool which is obtained in advance. The dwell time t (u) at each polishing point of the object is calculated by the calculating means 10
3 in the polishing method and apparatus for polishing a work piece with a polishing tool based on the residence time.
Reference numeral 03 divides the processing surface of the workpiece to be brought into contact with the polishing tool into a plurality of areas (S1, S2), and the residence time t of the polishing tool is calculated by the least square method based on Preston's rule of thumb. The configuration is such that (u) is obtained (S3, S4).

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a polishing method and a polishing apparatus, and more particularly to a polishing method and a polishing method in which a polishing tool is brought into contact with a surface to be machined of a workpiece to deplete the workpiece to a desired shape and to polish the workpiece. Regarding the device.

[0002]

2. Description of the Related Art In polishing, it is relatively easy to improve the surface roughness of a work piece, but it is difficult to improve the shape accuracy of the work piece. Machining to achieve high shape accuracy has traditionally depended on experience and skill.

By the way, as a technique for polishing the shape of a work piece with high precision, for example, Journal of Abrasive Science, VOL. 38, NO.
2,1994, MAR.p16-19, "Principle of shape accuracy by polishing and its technical problem", using a polishing tool smaller than the workpiece, keep the pressure and relative speed constant. Various methods have been proposed for controlling the wear-reduced shape by obtaining the contact time of the polishing tool with a predetermined portion of the workpiece, that is, the residence time of the polishing tool by a predetermined calculation.

As one of these methods, the difference between the surface shape data of the workpiece and the design shape data is taken as the removal shape, and the unit removal shape obtained in advance (polishing removal shape per unit time of the polishing tool) Then, there is a polishing method in which the removal shape is deconvoluted, the residence time is obtained, the polishing processing is performed, the surface shape data is measured again, and the polishing processing is repeated (see Japanese Patent Application Laid-Open No. 5-57606).

The calculation of the residence time in this polishing method is
The Fourier transform is performed three times and the inverse Fourier transform is performed once, the convolution theorem is used, and the constraint condition of the solution is sufficiently considered.

The solution will be described below. The amount of wear Δh during polishing can be expressed as Δh = η · p · v · t according to the well-known Preston's rule of thumb. Here, η is a proportional constant, p is a pressure, v is a relative speed between the polishing tool and the surface to be processed, and t is a residence time of the tool.

Assuming now that the amount of wear Δh is represented by the position x of the machined surface, Δh (x), the unit removal shape is f (x), and u is the position where the tool is retained, the amount of wear Δh (x) is , Δh (x) = f (x−u) · t (u).

When the polishing tool moves within the predetermined region A of the surface to be polished, the entire wear-reduction shape h (x) of the predetermined region A is h (x) = ∫ _A Δh (x) du _{= ∫ a f (x, u} ) becomes a · t (u) du ... formula 1. Then, in order to obtain the residence time at the tool position u, this Formula 1 may be solved for t (u).

Here, the Fourier transform of h (x):

(Equation 3) Fourier transform of f (x):

(Equation 4) Fourier transform of t (u):

(Equation 5) Then, Equation 1 is a convolution of f and t when A is ∞.

Then, according to the convolution theorem, it can be expressed as H (ωx) = F (ωx) · T (ωu) ...

Therefore, the Fourier transform of t (u) T (ω
u) is given by T (ωu) = H (ωx) / F (ωx).

By performing an inverse Fourier transform on T (ωu), the tool residence time t (u) can be obtained.

[0013]

However, in the solution method by the Fourier transform, when the calculation of the Fourier transform and the inverse Fourier transform of Formulas 1 to 5 is performed by a computer, a huge amount of calculation is required, and the calculation time is long. There is a problem that it takes a long time.

Due to the nature of the Fourier transform, the unit removal shape f (x, u) used for the calculation, the amount of wear h (x), the tool residence time t (u) and its Fourier transform T (ωx) are ±. There is a problem that the processing method is limited because it needs to be a function that converges at ∞.

Therefore, the present invention simplifies the calculation method for obtaining the tool residence time, shortens the calculation time, saves the computer labor, shortens the machining lead time, and eliminates restrictions on machining conditions. An object of the present invention is to provide a polishing method and a polishing apparatus capable of preventing the processing machine from becoming complicated, expanding the processing range, dealing with a rotary processed product, and improving the processing conditions.

[0016]

In the present invention, a polishing method for solving the above-mentioned problems is as shown in FIG. 1 (1), in which a polishing tool is brought into contact with a processing surface of an object to be processed. When polishing an object, the wear-removed shape h (x), which is the difference between the surface shape data of the workpiece and the design shape data, and the unit removal shape f (x, u) of the polishing tool, which has been obtained in advance,
In a polishing method in which a residence time t (u) at each polishing point of a workpiece of a polishing tool is calculated, and the polishing tool polishes the workpiece based on the residence time, the polishing tool is brought into contact with the polishing tool. The machined surface of the workpiece is divided into a plurality of areas (S1, S2), and h = η · v · p · t (h: wear amount,
η: proportional constant, v relative velocity, p: pressure, t residence time) Based on Preston's empirical rule, the residence time t (u) of the polishing tool is calculated by the least square method (S3).
S4).

Further, in the polishing method of the present invention, in the calculation for obtaining the residence time t (u) of the polishing tool, the wear amount Δh in polishing is represented by the position x of the machined surface, and the unit removal shape is f.
When (x, u) and the position where the tool stays are u, Δh (x) = f (x, u) · t (u) and the tool moves in the predetermined area A to be ground. the depletion shape h (x) _{when, h (x) = ∫ a} Δh (x) du = ∫ a Δh (x-u) ·
t (u) du and divide x, u in the range A into m, n (m, n
Is a positive integer, m ≧ n), and the wear amount at positions x _{1 to} x _m is

(Equation 6) And the matrix corresponding to h is Hm, and the matrix corresponding to f is F
Assuming that the matrix corresponding to m, n and t is Tn and Hm = Fm, n · Tn, the transposed matrix F ^t n, m of Fm, n is multiplied on both sides to obtain Gn = En, n · Tn (where Gn = F ^t n, m · Hm, En, n = F
^t n, m · Fm, n), and Tn = E ⁻¹ n, n · Gn, to obtain the element t (u) of Tn.

As shown in FIG. 1 (2), the polishing apparatus of the present invention includes a polishing driving means 101 for polishing a workpiece by bringing a polishing tool into contact with the surface of the workpiece, and the workpiece. Depleted shape h, which is the difference between surface shape data and design shape data
(X) and the previously determined unit removal shape f of the polishing tool
Based on (x, u), a calculation means 103 for calculating a residence time t at each polishing point of the workpiece of the polishing tool, and the above-mentioned retention time t (u) calculated by the calculation means 103. Drive control means 102 for controlling the drive of the polishing drive means 101
In the polishing apparatus provided with
The processing surface of the workpiece to which the polishing tool is brought into contact is divided into a plurality of regions, and h = η · v · p · t (h: wear amount,
η: proportional constant, v relative velocity, p: pressure, t residence time) Based on Preston's empirical rule, the residence time t of the polishing tool is calculated by the least square method.

Further, in the polishing apparatus of the present invention, the polishing calculating means represents the amount of wear Δh in polishing by the position x of the machined surface, the unit removal shape is f (x, u), and the tool remains. When the position is u, Δh (x) = f (x, u) · t (u), and the wear-removing shape h (x) when the tool moves in the predetermined area A to be polished is h (x _{) = ∫ A Δh (x)} du = ∫ A f (x, u) · t
(U) du and divide x, u in the range A into m, n (m, n
Is a positive integer, m ≧ n), and the wear loss at positions x _{1 to} x _m

(Equation 7) And the matrix corresponding to h is Hm, and the matrix corresponding to f is F
Assuming that the matrix corresponding to m, n and t is Tn and Hm = Fm, n · Tn, the transposed matrix F ^t n, m of Fm, n is multiplied on both sides to obtain Gn = En, n · Tn (where Gn = F ^t n, m · Hm, En, n = F
^t n, m · Fm, n), and Tn = E ⁻¹ n, n · Gn, to obtain the element t (u) of Tn.

According to the invention, the residence time t (u) of the tool
The calculation to obtain is a calculation for a matrix, and if the number of rows and columns of the matrix is selected appropriately, the calculation amount required for the calculation will not be so large and the calculation time of the computer will be short. Can be Further, the solution obtained by this calculation is a solution by the least square method, and is the closest to the true value.

[0021]

Embodiments of the present invention will be described below with reference to the drawings. 2 to 4 show a polishing apparatus to which the present invention is applied. In this example, as shown in FIG. 2, the polishing apparatus includes a base plate 8 provided on the upper surface of the housing 7.
In addition to the lens hold unit 9, a polishing tool drive unit 10 as a polishing drive means is provided.

The hold unit 9 has a lens holder 1 to which a lens 2 which is a workpiece is fixed. The lens holder 1 is rotatably provided around a vertical axis, and the lens holder 1 is provided. Driven pulley 1 at the lower end of the rotating shaft of
1 is provided. In addition, the lens holder 1 is provided inside the housing 7.
Is provided with a motor 12 for rotation, and a drive pulley 13 provided on the output shaft of the motor 12 and the driven pulley 11 are connected by a belt 14.

The polishing tool drive unit 10 is provided so as to be movable up and down along the guide rod 15, and is screwed to a screw rod 16 provided in parallel with the guide rod 15. The screw rod 16 is the base plate 8
Is rotatably passed through and a driven bevel gear 17 is fixed to the lower end of the screw rod 16.

A manual rotary shaft 1 orthogonal to the screw rod 16
Reference numeral 8 penetrates a side surface of the casing 7, an input bevel gear 19 meshing with the driven bevel gear 17 is fixed to an inner end of the manual rotation shaft 18, and a handle 42 is provided to an outer end of the manual rotation shaft 18. It's stuck.

The polishing tool drive unit 10 will be described below. The guide rod 15 is slidably fitted,
Moreover, the speed reducer 2 is attached to the shaft box 20 screwed to the screw rod 16.
A servomotor 22 is attached via 1. or,
A swing shaft 23 is horizontally rotatably provided on the shaft box 20, and a swing arm 24 is fixed to an end portion of the swing shaft 23 on the lens hold unit 1 side. The swing arm 24 and the output shaft of the speed reducer 21 are connected by a gear train (not shown).

The slide guide 2 is attached to the swing arm 24.
A slide plate 26 is provided so as to be able to move up and down via the switch 5. An adjusting screw 28 is rotatably attached to the upper end of the slide plate 26 via a bearing block 27, and the adjusting screw 28 is screwed to a nut block (not shown) fixed to the slide plate 26. Also, the adjusting screw 2
An adjustment band 41 is fitted on the upper end of the reference numeral 8.

A slide guide 29 is provided on the surface of the slide plate 26 facing the lens hold unit 9.
The floating plate 30 is mounted so as to be able to move up and down. A spindle unit 31 is attached to the floating plate 30, and the spindle unit 31 has a spindle shaft 32 orthogonal to the rotation axis of the lens holder 1. Further, for example, a disk-shaped polishing tool 33 is fitted to the tip of the spindle shaft 32.

A spindle drive motor 34 is attached to the upper side of the spindle unit 31 of the floating plate 30, and an output shaft (not shown) of the spindle drive motor 34 and a base end of the spindle shaft 32 are belted (not shown). Connect by.

Guides 35 are provided on both end surfaces of the slide plate 26, a balance weight 36 is slidably provided on the guide 35, and a pulley 37 is rotatably provided on the upper end of the slide plate 26. A wire 38 having one end attached to the floating plate 30 is wound around the pulley 37, and the other end of the wire 38 is attached to the balance weight 36.
To get involved. Thus, the balance weight 36 and the floating plate 30 and the units attached to the floating plate 30 are weight-balanced.

Further, an eaves plate 39 extending toward the floating plate 30 is provided at the upper end of the slide plate 26, and the eaves plate 39 is provided.
The air cylinder 40 is attached to the air cylinder 40 and the rod tip of the air cylinder 40 is fixed to the floating plate 30.

Next, the drive control means for controlling the drive of the polishing tool drive unit 10 and the calculating means for calculating the residence time of the polishing tool will be described with reference to FIG. In this embodiment, the servomotor 2 of the polishing tool drive unit 10 is used.
A motor controller 44, which is a drive controller, is connected to the motor controller 2. The motor controller 44 is connected to a main controller 44, which is, for example, a computer, which controls the entire polishing apparatus and operates as a calculator. ing.

The servo motor 22 is connected to the main controller 43.
The motor controller 44 is driven by a command from the motor controller 44. The motor controller 44 receives a rotation speed signal from a rotation detector 45 provided in the servomotor 22 and is shaken. A swing angle signal is input from a swing angle detector 46 that detects the swing angle of the moving arm 24.

In FIG. 4, reference numeral 47 is an operation plate for inputting polishing conditions and the like, and 48 is an indicator for displaying the polishing conditions or the polishing state.

The operation of the polishing apparatus according to this example will be described below. First, before starting polishing, the operation panel 47
Then, the polishing conditions are input to the main controller 43. The polishing conditions are the curved surface shape of the lens, the swing angle, and the air cylinder 40.
Of air pressure, error with respect to finished curved surface obtained by measurement before polishing, that is, target wear shape h (x), polishing tool 33
Unit removal shape f (x, u), number of processed surface divisions (m, n)
And so on.

After setting the lens 2, the adjustment handle 41 is turned to roughly match the positional relationship between the slide plate 26 and the swing arm 24 in accordance with the swing radius of the disk polishing tool 33, that is, the reference curved surface radius of the lens 2. .

The lens 12 is rotated by the motor 12 via the drive pulley 13, the belt 14 and the driven pulley 11, and while the lens 2 is rotating, the spindle drive motor 34 is driven to rotate the spindle shaft 32. The disk polishing tool 33 is brought into contact with the lens 2 via the above, and the height of the polishing tool drive unit 10, that is, the distance between the spindle shaft 32 and the swing shaft 23 is adjusted. The distance between the spindle shaft 32 and the swing shaft 23 becomes the swing radius of the polishing tool 33.

Next, compressed air is supplied to the air cylinder 40, and the disk polishing tool 33 is pressed against the lens 2 by the air cylinder 40. The contact pressure of the lens 2 of the disk polishing tool 33 is only the pressing force of the air cylinder 40 because the support mechanism supporting the disk polishing tool 33 is balanced by the balance weight 36.

Now, the relationship between the support mechanisms 30, 31, ... And the balance weight 36 will be described with reference to FIG.

The slide plate 26 swings, and the swing angle θ
Let's say At this time, considering the weight components of the weight w of the support mechanisms 30, 31, ... And the weight W of the balance weight 36 in the direction perpendicular to the lens processing surface, wc respectively.
os θ and W cos θ. Furthermore, the support mechanisms 30, 3
Assuming that there is no difference between the swing resistances of 1, ... And the swing resistance of the balance weight 36, from w = W, wcosθ = Wcosθ

Therefore, the contact pressure of the polishing tool 33 to the lens 2 is only the contact pressure of the air cylinder 40 regardless of the swinging state of the slide plate 26.

Then, the servo motor 22 is driven,
The slide plate 26 is swung through the swing shaft 23 and the swing arm 24. By the swing of the slide plate 26, the disk polishing tool 33 swings while contacting the lens 2, and the polishing point moves in the radial direction to polish the processed surface of the lens 2 into a desired curved surface. At this time, the servomotor 22 calculates the polishing tool residence time t calculated by the main controller 43.
The movement is stopped based on (u), and the polishing tool stays at a predetermined radial position for a predetermined time to polish the lens 2.

Next, the calculation of the residence time of the polishing tool in polishing will be described. This calculation is executed by the main controller 43 as described above. The theory of this operation is as follows.

The wear amount Δh at an arbitrary point in the radial direction of the lens 2 during the polishing process is represented by the following equation according to the known Preston's empirical rule as described above. Δh = η · p · v · t (where η: proportional constant, p: pressure, v: relative speed of tool and workpiece, t: tool residence time)

Assuming that the position on the workpiece is x, the position where the tool is staying is u, and the unit removal shape is f (x, u), Δh = f (x, u) · t (u ) Is.

[0045] Then, depletion distribution h when the tool is scanning the area A on the workpiece (x) is an _{h (x) = ∫ A f} (x, u) · t (u) du ... Formula 1 , X, u in the range A are sufficiently small m, n (m
≧ n), the wear amount at the positions x _{1 to} x _m is

(Equation 8) Therefore, if the matrix corresponding to h is Hm, the matrix corresponding to f is Fm, n, and the matrix corresponding to t is Tn, then Hm = Fm, n.Tn.

When both sides are multiplied by the transposed matrix F ^t n, m of Fm, n, Gn = En, nTn is obtained. However, Gn = F ^t n, m · Hm, En, n =
F ^t n, m · Fm, n.

Therefore, since Tn = E ⁻¹ n, n · Gn, the element t (u) of Tn can be obtained. Here, although there is a negative solution in the actual calculation, in that case, the solution may be constrained to 0 and recalculated.

The solution obtained by this calculation is a solution by the method of least squares, which is the closest to the true value.

FIG. 5 is a flow chart showing the procedure for calculating the residence time t (u) of the polishing tool in this example. First, f (x _1-m , u _1-n ) from the input condition, that is,
The unit removal shape at each polishing point is given. This value is obtained in advance by actual measurement or calculation (ST1).

Then, a matrix Fm, n having this f as an element
Is created (ST2), and the transposed matrix F ^t n, m of this matrix Fm, n is calculated (ST3). Next, a matrix Hm having h as an element is created from the target wear depletion shape h (x _m ) given in ST4 (ST5), and the matrix Fm, n and the matrix Hm are created.
From the left of, multiply the above F ^t n, m (ST6), F ^t n,
An inverse matrix Gn of m × Fn, m is calculated (ST7). Then, by multiplying the inverse matrix Gn from the left of F ^t n, m × Hm (ST8), T (x _1-m ), that is, the residence time t (u) of the polishing tool at each position can be obtained ( S
T9).

At this time, when a negative solution exists in T (x _1-m ), the element of Gn which gives a negative solution is fixed to 0, and ST8 to ST9 are repeated again (ST11).

Regarding the amount of calculation, the number of divisions of x and u (m,
Although n) becomes a problem, it has been experimentally proved that a sufficiently precise solution can be obtained by dividing into 10 to 100, and the calculation time was about several minutes with a commercially available personal computer.

Therefore, if the rotation speed of the lens 2, the rotation speed of the disk polishing tool 33, and the air pressure of the air cylinder 40 are kept at predetermined values, the time T (x _1-m ) can be controlled to control the lens. The curved surface can be polished into a desired state.

In the above example, the air cylinder is used as the pressing means for the polishing tool, but a spring or the like may be used. Although the cross-sectional shape (two-dimensional) of the rotationally symmetrical workpiece has been described here, a three-dimensional shape can be supported by increasing the number of matrix elements. Further, the movement of the polishing tool may be controlled so that it stays at a predetermined position for a calculated time and moves stepwise to the next place, or the staying time is converted into a moving speed to move the polishing tool. You may control so that it may shift continuously.

[0055]

EXAMPLES A first example of a polishing method and a polishing apparatus according to the present invention will be described below. 4 to 9 show setting conditions and results of this embodiment. In this example, as shown in FIG. 6, a disk-shaped rotary polishing tool 51 (diameter 50 mm)
The workpiece 52, which is glass, is polished on the side surface of the.

In this example, the measured cross-section value f (u) of the unit removal shape was as shown in FIG. Note that FIG.
Since the shape shown in (1) has a polishing time of 30 seconds, the unit removal shape used is that in which this amount of wear is converted into unit time (1 second).

Further, FIG. 8 shows the cross-sectional simulation value of the rotational wear-reduction distribution of polishing wear. In FIG. 8, shows the actual measurement value of the shape that is the input data and the target wear-removed shape, and is the predicted value of the final total wear-out distribution after polishing due to the residence time of the polishing tool calculated as the optimum, This is the estimated amount of wear at the position. The calculated residence time is shown in FIG.

The experimental results are shown in FIG. In this example, undeveloped (blank), 25 minutes after the start of polishing, 10 minutes after the start of the second processing, and 6 minutes after the completion of polishing, the deviations from the theoretical curve of the shape at the time of completion of polishing are displayed. According to this example, a satisfactory polishing result was obtained by the residence time of the polishing tool based on the calculation.

Next, another embodiment of the present invention will be described. In this example, as shown in FIG. 11 (1), the workpiece 62 is polished on the bottom surface of a disk-shaped rotary polishing tool 61 (diameter 10 mm). FIG. 11 (2) shows this rotary polishing tool 6
2 shows an outline of a unit removal shape obtained by rotating and polishing at one point on the second workpiece 62. In addition, FIG. 11 (2)
Then, the amount of wear on the workpiece is 0.5 μm. And FIG. 12 shows a residence time distribution that gives a uniform wear of 5.04 μm. FIG. 13 shows a wear reduction simulation value, and the wear distribution at each point on the workpiece 62 (in FIG. 13,
Since the printing result of the printer is shown as it is, the details of the wear distribution at each point are not clearly displayed). As a result of integration in a predetermined region, it is shown that a shape change of 5.04 μm (in FIG. 13) is achieved.
The effective range of the workpiece 62 (15 mm diameter, FIG.
The area to be polished is 21 mm to ensure that
And

[0060]

As described above, according to the present invention, the work surface of the workpiece against which the polishing tool is abutted is divided into a predetermined size, and the unit removal shape of the polishing tool and the target wear distribution are given. Since the residence time of the polishing tool is determined by the least-squares method, a sufficiently accurate solution for the residence time can be obtained in a short calculation time, and an effect that a precise polishing process can be realized is achieved.

Further, the unit removing shape may be arbitrary, and it is possible to relax restrictions on the number of revolutions of the polishing machine, speed, pressure, polishing time and the like, and it is possible to easily apply it to a rotary processed product.

[Brief description of drawings]

FIG. 1 is an explanatory view of a principle of the present invention, in which (1) shows a flow of processing of a polishing method and (2) shows a basic configuration of a polishing apparatus.

FIG. 2 is a front view showing an example of an embodiment of a polishing apparatus according to the present invention.

FIG. 3 is a side view showing a front side surface of the polishing apparatus shown in FIG.

4 is a block diagram showing a control system of the polishing apparatus shown in FIG.

5 is a flow chart showing the operation of the polishing apparatus shown in FIG.

FIG. 6 is a view showing a polishing tool used in the first embodiment of the present invention.

7 is a diagram showing measured values of a unit removal shape by the polishing tool shown in FIG.

8 is a diagram showing a cross-sectional simulation value of a wear loss distribution of polishing with the polishing tool shown in FIG.

9 is a diagram showing calculated values of residence time in the polishing tool shown in FIG.

FIG. 10 is a diagram showing a state of polishing performed with the tool shown in FIG. 7 for the residence time shown in FIG. 9;

FIG. 11 is a diagram showing a second embodiment of the present invention,
(1) is a diagram showing a polishing tool used in this example, and (2) is a diagram showing a unit removal shape of the polishing tool shown in (1).

FIG. 12 is a diagram showing calculated values of residence time in the polishing tool shown in FIG. 11 (1).

13 is a diagram showing a simulation value of a wear loss distribution of polishing with the polishing tool shown in FIG. 11 (1).

[Explanation of symbols]

 101 polishing drive means 102 drive control means 103 computing means

Claims (4)

[Claims] 1. A wear-reducing shape h (x) which is a difference between surface shape data of a work piece and design shape data when a grinding tool is brought into contact with the work surface of the work piece and the work piece is polished.
And the unit removal shape f (x,
u) and the residence time t (u) at each polishing point of the workpiece of the polishing tool, and the polishing method for polishing the workpiece with the polishing tool based on the residence time. The machining surface of the workpiece to which the tool is brought into contact is divided into a plurality of regions, and h = η · v · p · t (h: wear amount,
η: proportional constant, v relative speed, p: pressure, t residence time) Based on Preston's empirical rule, the residence time t of the polishing tool is calculated by the least square method.
A polishing method characterized by obtaining (u). 2. The calculation for obtaining the residence time of the polishing tool represents the amount of wear Δh in polishing by the position x of the processed surface,
When the unit removal shape is f (x, u) and the position where the tool is retained is u, Δh (x) = f (x, u) · t (u), and within the predetermined area A to be ground. in the depletion shape h (x) when the tool is _{moved, h (x) = ∫ a} Δh (x) du = ∫ a f (x, u) · t
(U) du, x, u in the range A are divided into m and n (m and n are positive integers, m ≧ n), and the amount of wear at the positions x _{1 to} x _m is calculated as follows. And the matrix corresponding to h is Hm, and the matrix corresponding to f is F
Let Tn be the matrix corresponding to m, n, t, and Hm = Fm, n · Tn. Multiply both sides by the transposed matrix F ^t n, m of Fm, n to obtain Gn = En, n · Tn (where , Gn = F ^t n, m · Hm, En, n =
^{F t n, m · Fm,} n), Tn = E -1 n, polishing method according to claim 1, wherein the determining from the n · Gn Tn elements t (u). 3. A polishing driving means for bringing an abrasive tool into contact with a work surface of a work to polish the work, and a wear reduction shape h which is a difference between surface shape data of the work and design shape data.
(X) and the previously determined unit removal shape f of the polishing tool
(X, u), the calculating means for calculating the residence time t (u) at each polishing point of the workpiece of the polishing tool, and the polishing drive means based on the residence time calculated by the calculating means. And a drive control means for driving and controlling the polishing tool, the calculating means divides a processing surface of the workpiece to be brought into contact with the polishing tool into a plurality of regions, and h = η · v · p · t
(H: wear loss, η: proportional constant, v relative speed, p: pressure,
t dwell time) based on Preston's empirical rule
A polishing apparatus characterized by obtaining (u). 4. The polishing calculation means represents the amount of wear Δh in polishing by the position x of the processed surface, and the unit removal shape is f (x,
u) and u is the position where the tool is staying, Δh (x) = f (x, u) · t (u), and wear when the tool moves in the predetermined area A to be ground the shape h (x), h (x ) = ∫ a Δh (x) du = ∫ a f (x, u) · t
(U) du, x and u in the range A are divided into m and n (m and n are positive integers, m ≧ n), and the wear amount at the positions x _{1 to} x _m is And the matrix corresponding to h is Hm, and the matrix corresponding to f is F
Let Tn be the matrix corresponding to m, n, t, and Hm = Fm, n · Tn. Multiply both sides by the transposed matrix F ^t n, m of Fm, n to obtain Gn = En, n · Tn (where , Gn = F ^t n, m · Hm, En, n =
^{F t n, m · Fm,} n), Tn = E -1 n, polishing apparatus according to claim 3, wherein the determining Tn elements t (u) from n · Gn.