Machine tools abstract
A method of assessing three-dimensional volumetric errors of multiaxis
machine tools in three-dimensional working space is disclosed. Each
three-dimensional volumetric error component can be systematically
measured and analyzed on the modeling of polynominal functions in
accordance with the volumetric errors and the kinematic chain in
accordence with the corresponding machine tool. The method inputs
the measured radial data performed on the three orthogonal planes,
analyzing the parametric errors such as positional, straightness,
angular, squareness, and backlash errors. The positional error components
along each of three orthogonal axes are modeled as a dimensionless
polynominal function with corresponding positional error coefficients.
The method also can assess dynamic performance of the machine tools
such errors due to the servo gain mismatch. The method employs the
kinematic ball bar to assess the volumetric errors.
Machine tools claims
What is claimed is:
1. A method of assessing three-dimensional volumetric errors in
a multiaxis machine tool, comprising:
setting up a first radial error equation according to error in
distance between commanded nominal coordinates of a spindle of the
multiaxis machine tool and actually moved coordinates of the spindle
with respect to a fixed center point to measure the volumetric errors
of the multiaxis machine tool in three-dimensional working space;
modeling first positional error components along each of three
orthogonal axes as a dimensionless polynomial function with corresponding
positional error coefficients;
modeling second positional error components due to backlash along
each of the three axes with corresponding amount of the backlash
in the three axes;
modeling third positional error components along each of the three
axes with corresponding amount of squareness errors between two
nominally orthogonal axes among the three axes;
setting up three-dimensional volumetric error equation depending
on kinematic configuration of the multiaxis machine tool;
setting up a second radial error equation by substituting the three-dimensional
volumetric error equation, the first positional error components,
the second positional error components, and the third positional
error components for the first radial error equation;
solving the second radial error equation and calculating the positional
error coefficients, the amounts of the backlash, and the amounts
of squareness by means of an approximation technique and coordinates
data.
2. A method as recited as claim 1 further including modeling straightness
error components along each of the three orthogonal axes as a dimensionless
polynomial function with second order terms and corresponding straightness
error coefficients, modeling roll error components along each of
the three orthogonal axes as a dimensionless polynomial function
with corresponding roll error coefficients, modeling pitch and yaw
error components as each derivative of the straightness error components,
and modeling error components due to servo gain mismatch along each
of the three orthogonal axes with each servo gain and velocity of
each axis.
3. A method as recited as claim 2 further including inputting automatically
measured volumetric errors into the multiaxis machine tool and compensating
the volumetric errors.
4. A method as recited as claim 1 wherein a circular calibration
technique which is dependent on measurement path in each of the
three orthogonal planes is adapted to calculate the error coefficients
and the amounts of the backlash, and the amounts of squareness.
5. A method as recited as claim 1 wherein said first positional
error components are given as
where i=1.about.N.
6. A method as recited as claim 2 wherein said straightness error
components are given as
where i=2.about.N, and wherein said roll error components are given
as
where i=1.about.N.
7. A method as recited as claim 4 wherein the measurement path
is planned before calculating the error coefficients and the amounts
of the backlash, and the amounts of squareness to assess the three-dimensional
volumetric errors.
8. A method as recited as claim 1 wherein a kinematic ball bar
is employed between the fixed center point and the spindle of the
multiaxis machine tool.
Machine tools description
FIELD OF THE INVENTION
The present invention relates to multiaxis machine tools, and more
particularly to a method of assessing three-dimensional volumetric
errors in the multiaxis machine tools to measure each component
of static and dynamic errors which have influence on precision of
the multiaxis machine tools with mutiple freedoms, for example,
computer numerical control(hereinafter referred to as "CNC")
machine tools, and three-dimensional coordiate measuring machines,
etc. and to analyze the error components and to assss the three-dimensional
volumetric errors.
Recently, development of efficient techniques for performance verification
of the multiaxis machine tools has been considered as an important
task for accuracy enhancement and quality assurance for users and
manufactures of the CNC machine tools and the coordiate measuring
machines. In order to perform precise position control and to promote
accuracy of the multiaxis machine tools, the development of efficient
techniques is directed to assessment of the three-dimensional volumetric
errors, since it is very essential to measure and analyze each error
component and to assess the three-dimensional volumetric errors.
The static errors include geometric errors, kinematic errors, and
thermal errors, etc. and the dynamic errors include errors due to
servo gain mismatch and dynamic characteristics. FIG. 1 shows the
geometric errors referred to as positional error, horizontal straightness
error, vertical straightness error, roll error, pitch error, and
yaw error.
On the other hand, a number of attempts have been made and various
techniques have been developed to assess the performance checking
of the machine tools. In order to assess the performance checking
of the machine tools, double ball bar or kinematic ball bar has
been frequently used.
Considering known prior arts, a method is described in a paper
by Bryan, J. entitled "A simple method for testing measuring
machines and machine tools" (Precision Engineering, Vol.4(2),
1982). The method employs a ball bar consisting of precision balls
and linear variable differential transducer(hereinafter referred
to as "LVDT") for checking accuracy of machine tools.
A similar method using a two-dimensional probe and a master disc
is disclosed in "Test of three-dimensional uncertainty of machine
tools and measuring machines and its relation to machine errors"
by Knapp, W. (Annals of CIRP, Vol. 32(1), pp. 459-464 1983). Another
article which is relevant for a kinematic ball bar for the parametric
error calibration of the machine tools, is entitled "On testing
coordinate measuring machines with kinematic reference standards"
by Kunzmann, H., et al. (Annals of CIRP, Vol. 32(1), pp. 465-468
1983). Further another article which is relevant for the relationship
between the ball bar measurement and the various parametric errors
for machines, is entitled "The measurement of motion errors
of NC machine tools and diagnosis of their origins by using telescoping
magnetic ball bar method" by Kakino, Y., et al. (Annals of
CIRP, Vol. 36(1), pp. 377-380 1987).
However, the aforedescribed known prior arts have the severe drawbacks
and problems that the error assessment of the multiaxis machine
tools is performed in two-dimensional work space and the only overall
error amounts are measured but each error component, which is involved
in the overall error amounts, can not be analyzed. Accordingly,
error compensation has to be repeatedly carried out by trial and
error, and moreover the error may not be accurately compensated
on account of the dependence on the overall error amounts.
SUMMARY OF THE INVENTION
The present invention is directed to overcome the drawbacks and
problems as set forth above.
It is an object of the present invention to provide a method of
assessing three-dimensional volumetric errors of a multiaxis machine
tool in 3-dimensional working space wherein each three-dimensional
error component can be systematically measured and analyzed on the
basis of the modeling of polynominal functions in accordance with
the corresponding volumetric errors and the kinematic chain of the
corresponding machine tool.
It is another object of the present invention to provide a method
of assessing three-dimensional volumetric errors of a multiaxis
machine tool in 3-dimensional working space wherein the analyzed
error components can be accurately and efficiently compensated.
It is futher object of the present invention to provide a method
of assessing three-dimensional volumetric errors of a multiaxis
machine tool in 3-dimensional working space wherein a computer aided
analysis system can be adapted for the calculation and the compensation
of the parametric error components, based on the circular measurement
using the kinematic ball bar.
According to the present invention, these objects are achieved.
There is provided a method comprising setting up a first radial
error equation in accordance to error in distance between commanded
nominal coordinates of a spindle of a multiaxis machine tool and
actually moved coordinates of the spindle with respect to a fixed
center point to measure the volumetric errors of the muiltiaxis
machine tool in three-dimensional working space; modeling first
positional error components along each of three orthogonal axes
as a dimensionless polynominal function with corresponding positional
error coefficients; modeling second positional error components
due to backlach along each of the three axes with corresponding
amount of the backlash in the three axes; modeling third positional
error components along each of the three axes with corresponding
amount of squareness errors between two nominally orthogonal axes
among the three axes; setting up three-dimensional volumetric error
equation depending on kinematic configuration of the multiaxis machine
tool; setting up a second radial error equation by substituting
the three-dimensional volumetric error equation, the first positional
error components, the second positional error components, and the
third positional error components for the first radial error eqation;
solving the second radial error eqation and calculating the positional
error coefficients, the amounts of the backlash, and the amounts
of squareness by means of an approximation method and coordinates
data.
Perferrably, the method according to the present invention further
includes modeling straightness error components along each of the
three orthogonal axes as a dimensionless polynominal function with
second order terms and corresponding straightness error coefficients,
modeling roll error components along each of the three orthogonal
axes as a dimensionless polynominal function with corresponding
roll error coefficients, modeling pitch and yaw error components
as each derivative of the straightness error components, and modeling
error components due to servo gain mismatch along each of the three
orthogonal axes with each servo gain and velocity of each axis.
The various features of novelty which characterize the invention
are pointed out with particularity in the claims annexed to and
forming a part of this disclosure.
For a better understanding of the invention, its operating advantages
and specific objects attained by its uses, reference is made to
the accompanying drawings and descriptive matter in which the preferred
embodiments of the invention are illustrated.
BRIEF DESCRIPTION OF THE DRAWINGS
In the drawings:
FIG. 1 is a schematic view explaining six geometric error components
when a machine element moves along a guideway;
FIG. 2 is a schematic front view showing an error measurement apparatus
using a kinematic ball bar which is employed to perform the method
according to the present inventon;
FIGS. 3 and 3A are a flow chart illustrating main steps of the
method according to the present inventon;
FIG. 4 is a persepective view showing schematically a column type
machining center to which the method according to the present inventon
is applied;
FIG. 5A is a view showing a set up of the kinematic ball bar to
calibrate a 360.degree. arc in XY plane;
FIG. 5B is a view showing a set up of the kinematic ball bar to
calibrate a 360.degree. arc in YZ and ZX planes;
FIG. 6A to FIG. 6C are typical plots showing actually measured
data using the kinematic ball bar along measurement path in each
of XY,YZ,and ZX planes;
FIG. 7A to FIG. 7C are typical plots showing measured data using
them kinematic ball bar along the measurement path in each of XY,YZ,and
ZX planes after compensating the error components of FIGS. 8A to
8F.
FIGS. 8A to 8F illustrate typical results of the assessment of
the machine tool volumetric errors.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
This invention will be described in further detail by way of embodiment
with reference to the accomanying drawings.
In the embodiment according to the present invention, measurement
apparatus employs a kinematic ball bar which is operatively connected
to a CNC machine tool as a multiaxis machine tool. The kinematic
ball bar may be substituted by a master disk, a master cylinder,
or a master ball and the CNC machine tool may be substituted by
a three-dimensional coordinate measuring machine.
Referring to FIG. 2 FIG. 2 shows a typical setup for an error
measurement apparatus using a kinematic ball bar 10. The kinematic
ball bar 10 which enables performance checking of the machine tool
to be assessed with high precision and high efficiency has been
frequently used. A ball bar 16 which has precision LVDTs therein
is provided with two balls 12 and 18 at both ends thereof. The ball
12 is fixed to a socket 3 which is fixedly located on a magnetic
center mount 1 and the ball 18 is attached to a magnetic tool cup
4 which is fixedly located on a spindle 2 of the machine tool. The
present embodiment uses a commercially available kinematic ball
bar with 150 mm nominal length for circular error measurement.
In FIG. 1 let O(000) be the center point of the ball 12 on the
magnetic center mount 1 and let P(X,Y,Z) which is shown in solid
line be nominal coordinates of the center point of the ball 18 attached
to the spindle 2. When the machine tool is commanded to move to
P(X,Y,Z) position, the actual position of the machine tool is assumed
P'(X',Y',Z') which is shown in alternate long and two short dashes
line. Thus, the machine geometric error can be defined as the differerce
R'-R between the two coordinates, P'(X',Y',Z') and P(X,Y,Z). That
is,
where X, Y, Z are the X,Y,Z error components at the nominal position
P(X,Y,Z) with respect to the O(000), and C is error vector. When
the error components X, Y, Z are present, the error in the distance
between the two points, R can be evaluated as,
where R is the nominal distance between the two points, O and P.
When ignoring the second order terms of error components, and remembering
the R.sup.2 =X.sup.2 +Y.sup.2 +Z.sup.2 then eq.(2) gives
Eq.(3) gives the error in the length direction of the ball bar
when the error components (X, Y, Z) are present during the machine
movement from O point to P point, and it can be used for the error
diagnosis of the machine tool. This step corresponds to step S1
in FIG. 3.
Now, parametric errors including positional errors, straightness
errors, rotational errors, squareness errors, and servo gain mismatch
errors are modeled for the machine tool. Prior to modeling parametric
errors, translational errors .delta. xi(Xj) are defined as errors
in the xi direction along the Xj axis and rotational errors Exi(Xj)
are defined as errors in xi direction along the Xj axis.
Firstly, each of positional error components is modeled as a dimensionless
polynominal function of position along each X,Y,Z axis in step S2.
The positional error components, .delta. x(X), .delta. y(Y), .delta.
z(Z) along X,Y,Z axis, which are mainly due to scale errors, lead
screw pitch errors. The positional error components are usually
defined as the difference between the actual coordinates and the
nominal coordinates along the X,Y,Z axis, and are given as
where X/R, Y/R, Z/R are the dimensioness coordinates of position
along each axis, and dxxi, dyyi, dzzi are coefficients, and i=1.about.N.
Next, each of straightness error components is modeled as a dimensionless
polynominal function of position beginning with second order terms
in step S3. The straightness error is mainly due to nonstraightness
of the guideway and due to bearing interfaces in the machine tool.
The straightness error components are defined as perpendicular deviation
along each axis, and are given as
where .delta. y(X), .delta. z(X); .delta. x(Y), .delta. z(Y); .delta.
x(Z), .delta. y(Z) are the straightness error components along X,
Y, and Z axis, respectively. Constants dyxi, dzxi, dxyi, dzyi, dxzi,
dyzi are the coefficients of the polynominal function to be determined,
and i=2.about.N.
Next, each of roll error components is modeled as a dimensionless
polynominal function of position along each X,Y,Z axis in step S4.
The roll error components Ex(X), Ey(Y), Ez(Z) are defined as angular
error in the axial direction and are given as
where exxi, eyyi, ezzi are the coefficients for the polynominal
model, and i=1.about.N.
Next, each of pitch and yaw error components is modeled as each
derivative of the corresponding straightness error profiles in step
S5. The pitch and yaw errors are the angular error components in
the perpendicular direction along each axis, and are influenced
by the guideway geometry and the bearing interfaces of the machine.
Accordingly, the pitch and yaw error components are given as ##EQU1##
where i=2.about.N.
Next, backlash errors are modeled in step S6. The backlash error,
or reversal error, is mainly caused by the backlash in the screw/gear
assembly during the motion reversal. When each amount of backlash
in the X, Y, Z axis is bx, by, bz, the errors in the X, Y, Z axis
due to the backlash can be modeled as
where sign() is the sign function returning the sign of the terms
inside the bracket, and dX/dt, dY/dt, dZ/dt are time derivatives
of position in the X, Y, Z, which are velocities. The minus sign
in eq.(8) is to give the positive backlash when the motion is changed
from the toward direction to the reverse direction.
Next, squareness errors are modeled in step S7. The squareness
error is defined as the out of squareness between the two nominally
orthogonal axes, and is mainly due to the misalignment, or misassembly
in the orthogonal axes. Letting .alpha. be the amount of nonsquareness
error of X axis from the nominal X axis (in XY plane) at the position
of distance R from the origin point O, .beta. 1 be the amount of
nonsquareness error of Z axis at the R location (in XZ plane), and
.beta. 2 be the amount of nonsquareness error of Z axis from the
nominal Z axis at R location in the XY plane, the squareness errors
are as follows,
Next, errors due to servo gain mismatch are modeled in step S8.
When the amplifier gains of the servo drives for the axis motion
are not properly matched, there exists a kind of steady state following
errors. When Ksx, Vx; Ksy, Vy; Ksz, Vz are the servo gains and velocities
of the X, Y, Z axis, the steady state following errors, X, Y, Z
between the actual positions and the command positions in the X,
Y, Z axis can be obtained from the control theory as
On the other hand, when F is the circumferential feed velocity,
Vx, Vy can be obtained as
where .theta. is angular position along the circular motion.
Applying eqs.(11)', and (11)" to eq.(10), then substituting
into eq.(3), ##EQU2## where K.sub.s =.sqroot.K.sub.SX .times.K.sub.SY
, e=(K.sub.SY -K.sub.SX)/K.sub.s, mxy (=eF/K.sub.s) is the coefficient
for the gain mismatch between X-Y axis. Similar equation can be
derived for the clockwise rotation, that is,
Eqs.(12-1) and (12-2) represent the circular error which is influenced
by the gain mismatch. Similarly, the circular errors due to the
gain mismatch between the Y-Z axis and Z-X axis are
for counter clockwise and clockwise rotation
for counter clockwise and clockwise rotation
where myz, mzx are the coefficients for the gain mismatch between
Y-Z axis and Z-X axis, respectively.
In next step S9 volumetric error equations are derived by the
kinematic chain of each machine element to perform appropriate error
diagnosis. The volumetric error equations are dependent on the kinematic
configuration of the machine tool. In this embodiment, a column
type machining center is considered as shown in FIG. 4. For the
column type machining center, the volumetric error equations are
given as
where Xp,Yp,Zp are the coordinates of the tool offset.
Next, final circular error equation is derived in step S10. The
circular error equation can be obtained by applying eq.(13) and
all error components eq.(4) to (12-4) as modeled above to eq.(3),
and the circular error of the ball bar measurement is, ##EQU3##
where Dir is a function returning plus for clockwise rotation or
minus for counter clockwise rotation depending the rotation of the
kinematic ball bar. The first terms of the polynominal functions
are considered for each parametric error component for the efficiency
of calculation, and the roll error components and the effects of
the tool offsets are not considered for the simplicity of modeling.
Next, each amount of the errors which are involved in eq.(14) is
calculated with the pre-measured data in step S11. In order to effectively
classify and analyze the error components from the ball bar measurement
data, preferably, the least squares technique is effectively applied
since output signals simultaneously involve various errors which
are not independent each other.
When Rm is the circular error to be modeled, eq.(14) can be expressed
as the linear combination of polynominal functions.
where Fi is the ith function of error modeling, and Ai is the coefficient
of the ith function, and i=1.about.m, m is the number of the considered
error factors.
Let E the sum of squares of deviation between R and Rm, then
Applying the variational principle to eq.(16) for minimizing E,
From eq.(15), .differential. (Rm)/Ai=Fi, thus eq.(17) becomes
If we set f={F1F2 . . . Fn}.sup.T, a={A1A2 . . . An}.sup.T,
then eq.(18) becomes
That is,
where C is the square matrix (f f.sup.T), and d is the column vector
(R f).
Eq.(20) is a typical linear equation for unknown matrix, and therefore
it can be solved by applying numerical methods such as Gauss elimination
method, etc.
From now on, the measurement path will be described briefly.
The measurement path must be planned in advance to synthetically
analyze the dimensional parametric errors of the machine tool in
the volumetric sense using the ball bar measurment. The method according
to the present invention employs three orthogonal two-dimensional
planes as the measurement path. Referring to FIG. 5A and FIG. 5B,
FIG. 5A shows a set up of the kinematic ball bar to calibrate a
360.degree. arc in the XY plane, and FIG. 5B shows a set of up the
kinematic ball bar to calibrate a 360.degree. arc in the YZ and
ZX plane.
After the measurement path is planned, the appropriate CNC code
is generated for the circular contouring, then it is downloaded
on the CNC machine tool.
Next, step S12 is to analyze and assess each of the measured volumetric
errors. Referring to FIG. 6A to FIG. 6C, FIG. 6A to FIG. 6C show
the raw data plot for the circular error measurement in XY, YX,
and ZX planes, respectively, where the dotted line indicates the
counter clockwise contour and the solid line does the clockwise
contour. As shown in FIG. 6A to FIG. 6C, there are considerable
errors in measured results. However, according to the prior arts,
it is difficult to individually separate each amount of the error
components from the overall errors and to analyze each error component.
Therefore, the error components can not be precisely removed as
desired.
FIGS. 8A to 8F illustrate results of the assessment of the machine
tool volumetric errors. The results are obtained by solving eq.(13)
with the least squares technique. Accordingly, the raw data as plotted
in FIG. 6A to FIG. 6C are analyzed by the developed system and the
method according to the present invention.
In next step S13 the analyzed error components are removed from
the raw data of the circular measurement. The inputting of the volumetric
errors may be performed by various techniques. The residual circular
errors are calculated and then plotted. FIG. 7A to FIG. 7C show
the residual circular errors after compensating the error components,
giving remarkably reduced error pattern. Therefore, the developed
error analysis system has been found as an efficient tool for the
error diagnosis of the machine tool based on the kinematic ball
bar measurement.
According to the present invention, a computer aided analysis system
has been developed for the parametric error components, based on
the circular error measurement using the kinematic ball bar. A new
approach has been proposed and tested such that three-dimensional
volumetric error model is effectively integrated for the ball bar
measurement, thus the three-dimensional volumetric error components
have been efficiently assessed with the ball bar measurement data.
The polynominal based error modeling has been performed for the
parametric error components of the machine tool. Then, the least
squares technique has been applied for evaluation of the coefficients
of the modeling functions. The polynominal based error modeling
combined with the least squares techinique has shown good performance
for the error analysis. Also, the developed system has been applied
to practical cases of machine tools, and has demonstrated high efficiency
for the assessment of the three-dimensional error components for
machine tools in relatively short time when compared with conventional
length and angle measuring equipments.
The invention is in no way limited to the embodiment described
hereinabove. Various modifications of disclosed embodiment as well
as other embodiments of the invention will become apparent to persons
skilled in the art upon reference to the description of the invention.
It is therefore contemplate that the appended claims will cover
any such modification or embodiments as fall within the true scope
of the invention. |