## Machine tools abstract
Control of machine tools and industrial robots is usually implemented
in a cascade structure, having not only numerous advantages but
also the fundamental disadvantage of lower dynamics in the response
to set point in comparison with a single control loop. This disadvantage
is compensated today by feed-forward control concepts which permit
ideal tracking of the controlled variable without intervention by
the controllers, apart from inaccuracies in modeling of the controlled
system or disruptions. The enormous computation complexity of these
precontrol methods in higher order systems has provided the motivation
for the development of a cartesian path control concept which is
described in the present invention. While retaining the cascade
structure, path accuracy can be improved significantly in comparison
with axial feed-forward concepts.
## Machine tools claims
What is claimed is:
1. A control method for motion control of a machine comprising:
generating, in a cartesian coordinate system, reference input variables
for the machine; performing a cartesian control of the cartesian
reference input variables in the cartesian coordinate system to
form cartesian motion values; transforming the cartesian motion
values with a kinematic transform into a machine coordinate system;
axially driving the machine as a function of motion parameters,
the motion parameters being derived from the transformed motion
values; determining actual motion values of the machine in the cartesian
coordinate system, the actual motion values belonging to at least
one of the cartesian reference input values and the cartesian motion
values; and using the actual motion values as controlled values
for the cartesian control so that the kinematic transformation into
the machine coordinate system is compensated inversely by a mechanical
system of the machine.
2. The control method according to claim 1 wherein the machine
is one of an industrial machine tool and a robot.
3. A control method for motion control of a machine, the machine
having at least one drive, comprising: generating, in a cartesian
coordinate system, cartesian position setpoints for the machine;
performing a cartesian position control of the cartesian position
setpoints in the cartesian coordinate system to form cartesian speed
values; transforming the cartesian speed values with a kinematic
transform into a machine coordinate system; specifying, as a function
of the transformed speed values, speed parameters of one of axial
speed and acceleration control for each at least one drive of the
machine; determining actual position values of the machine in the
cartesian coordinate system and belonging to the cartesian setpoints;
and using the actual position values as controlled variables for
the cartesian position control.
4. A control method for motion control of a machine having at least
one drive, comprising: generating, in a cartesian coordinate system,
cartesian position setpoints for the machine; performing a cartesian
position and speed control of the cartesian position setpoints in
the cartesian coordinate system to form cartesian acceleration values;
transforming the cartesian acceleration values with a kinematic
transform into a machine coordinate system; specifying, as a function
of the transformed acceleration values, acceleration parameters
of an axial acceleration control each at least one drive of the
machine; determining actual position values of the machine in the
cartesian coordinate system and belonging to the cartesian setpoints
and speed setpoints; determining actual speed values of the machine;
using the actual position values as controlled variables for the
cartesian position control; and using the actual speed values as
controlled variables for the cartesian speed control.
5. A control method for motion control of a machine having at least
one drive, comprising: generating, in a cartesian coordinate system,
cartesian position setpoints for the machine; performing a cartesian
position, speed and acceleration control of the cartesian position
setpoints in the cartesian coordinate system, to form one of cartesian
current setpoints and cartesian voltage setpoints; transforming
the one of cartesian current setpoints and cartesian voltage setpoint
with a kinematic transform into a machine coordinate system; specifying,
as a function of the transformed setpoints, one of current values
and voltage values for each at least one drive of the machine; determining
actual position values of the machine in the cartesian coordinate
system and belong to the cartesian setpoints and speed setpoints;
determining actual speed values of the machine; determining actual
acceleration values of the machine; using the actual position speed
values of the machine, speed setpoints and acceleration setpoints
as controlled variables for the cartesian position control; using
the actual speed values of the machine as controlled variables for
the cartesian speed control; and using the actual acceleration values
of the machine as controlled variables for the cartesian acceleration
control.
6. The control method according to claim 5 further comprising:
regulating normal error and tangential error regulated as a deviation
for cartesian position control of a path.
7. The control method according to claim 5 further comprising:
determining the actual position values by directly measuring a position
in the cartesian coordinate system at an outer end of machine kinematics,
the machine kinematics being one of on a robot gripper and on a
tool.
8. The control method according to claim 4 further comprising:
determining the actual position values by indirectly measuring a
position in the machine coordinate system; and transforming the
actual position values into the cartesian coordinate system.
9. The control method according to claim 8 further comprising:
performing an axial feedforward control to specify at least one
of speed parameters and acceleration parameters.
10. A numerical control system for motion control of a machine,
comprising: a generator generating in a cartesian coordinate system
reference input variables; a cartesian control device generating
in the cartesian coordinate system cartesian motion parameters as
a function of the reference input variables; a transformation device
for transforming, using a kinematic transformation, the reference
input signals into a machine coordinate system; an arrangement providing
motion parameters for axially driving at least one drive of the
machine with motion parameters derived from the reference input
variables; and a detector system determining actual motion values
belonging to one of the cartesian reference input variables and
the cartesian motion values in the cartesian coordinate system,
the detector system supplying controlled variables to the cartesian
control device; wherein a mechanical system of the machine is used
for inverse kinematic transformation into the machine coordinate
system.
11. A numerical control system for motion control of a machine,
comprising: a generator generating in a cartesian coordinate system
cartesian position setpoints; a cartesian position control device
generating in the cartesian coordiante system cartesian speed values
as a function of the cartesian position setpoints; a transformation
device transforming the cartesian speed values into a machine coordinate
system using a kinematic transform; an arrangement providing at
least one of an axial speed and axial acceleration parameter for
driving at least one drive of the machine, the at least one of the
axial speed and axial acceleration parameter being derived from
the transformed speed values; and a detector system detecting actual
position values of the machine and supplying the actual position
values to the cartesian position control device as controlled variables,
the actual position values being determined in the cartesian coordinate
system and belonging to the cartesian position setpoints.
12. A numerical control device for motion control of a machine,
comprising: a generator generating in a cartesian coordinate system
cartesian position setpoints; a cartesian position control device
and a cartesian speed control device, the cartesian position control
device and the cartesian speed control device generating in the
cartesian coordinate system cartesian acceleration values as a function
to the cartesian position setpoints; a transformation device transforming
the cartesian acceleration values into a machine coordinate system
using a kinematic transform; an arrangement providing acceleration
parameters for driving the machine as a function of acceleration
parameters of an axial acceleration control, the acceleration parameters
being derived from the transformed acceleration values; and a detector
system detecting actual position values of the machine and actual
speed values of the machine, and supplying the actual position values
and speed setpoints to the cartesian position control device as
controlled variables, the actual position values being determined
in the cartesian coordinate system and belonging to the cartesian
position setpoints, the detector system further supplying the actual
speed values of the machine to the cartesian speed control device
as controlled variables.
13. A numerical control system for motion control of a machine,
comprising: a generator generating in a cartesian coordinate system
cartesian position setpoints; a cartesian position control device,
a speed control device and an acceleration control device generating
in the cartesian coordinate system one of cartesian current setpoints
and cartesian voltage setpoints as a function of the cartesian position
setpoints; a transformation device transforming the one of the cartesian
current setpoints and the cartesian voltage setpoints into a machine
coordinate system using a kinematic transform; an arrangement providing
at least one of current values and voltage values for driving the
machine, the at least one of the current values and voltage values
being derived from the transformed one of the cartesian current
setpoints and cartesian voltage setpoints; and a detector system
detecting actual position values of the machine and actual speed
values of the machine, the detector system supplying the actual
position values, speed setpoints and acceleration setpoints to the
cartesian position control device as controlled variables, the actual
position values being determined in the cartesian coordinate system
and belonging to the cartesian position setpoints, the detector
system further supplying the actual speed values of the cartesian
speed control device as controlled variables, and supplying actual
acceleration values to the cartesian acceleration control device.
14. The numerical control system according to claim 11 wherein
normal error and tangential error are used as a deviation by the
cartesian position control device.
15. The numerical control according to claim 11 wherein the detector
system includes a transmitter for determining the actual cartesian
motion values, the transmitter being positioned at an outer end
of machine kinematics, the detector system further including receivers
which respond to the transmitter, the receivers being positioned
at several points outside a work space of the machine.
16. The numerical control according to claim 11 wherein the detector
system includes one or a rotary and a translatory detector for determining
the actual cartesian motion values, the detector system further
including an inverse transformation device for transforming the
actual axial motion values into the cartesian coordinate system.
17. The numerical control according to claim 10 further comprising:
an axial feedforward controller for axial specification of at least
one of the motion parameters.
## Machine tools description
FIELD OF THE INVENTION
The present invention relates to a control method and a numerical
control for controlling the motion of industrial machine tools,
robots or the like. Reference input variables for the machine are
generated in a cartesian coordinate system and subjected to a kinematic
transformation into a machine coordinate system to specify motion
parameters axially for one or more drives of the machine.
BACKGROUND INFORMATION
Control of machine tools and industrial robots is usually implemented
by purely axial control in a cascade structure which has, in addition
to numerous advantages, the basic disadvantage of lower dynamics
in response to set point changes in comparison with a single control
loop. This disadvantage is compensated today by feed-forward concepts
which permit ideal tracking of the controlled variable without intervention
by controllers, apart from disturbances or inaccuracies of the model
of the controlled system.
Another disadvantage is the enormous amount of computation power
of these feed forward controll method in higher order systems.
SUMMARY
An object of the present invention is to create an alternative
path control concept that will also significantly improve the path
accuracy, or rather, permitting a lower contour error in comparison
with axial feed-forward control concepts while also permitting a
lower contour error.
According to the present invention, this object is achieved by
providing a control method for motion control in which the cartesian
reference input variables in the cartesian coordinate system are
controlled by subjecting the resulting cartesian motion values to
a kinematic transformation into the machine coordinate system to
specify motion parameters axially for each drive. Actual motion
values determined in the cartesian coordinate system and belonging
to the cartesian reference input variables and/or cartesian motion
values are sent back as controlled variables to the cartesian control,
so that the kinematic transformation into the machine coordinate
system is compensated inversely by the mechanical system of the
machine.
The preceding object is also achieved by providing a numerical
control algrithm for motion control suitable for performing the
control method, having a reference input variable generator for
the machine in the cartesian coordinate system, a means for kinematic
transformation of the reference input variables into a machine coordinate
system and a means for axially driving of one or more drives of
the machine with motion parameters derived from the reference input
variables. A cartesian control of the cartesian reference input
variables is provided, and its resulting cartesian motion values
are transformed by the means for kinematic transformation into the
machine coordinate system in order to be specified as motion parameters
axially for each drive, with a measuring system being provided for
determining actual motion values belonging to the cartesian reference
input variables and/or cartesian motion values in the cartesian
coordinate system, the detector system supplying controlled variables
to the cartesian control, with the mechanical system of the machine
being used as the means for inverse kinematic transformation into
the machine coordinate system.
Through the present invention, the non-linear kinematic transformation,
among other things, is inversely compensated by the inverse transformation
contained in the mechanical system. Thus, the controlled system
is always linear from the standpoint of the controller if the sampling
time and thus the achievable controller dynamics are small enough.
In this way, complicated methods of acceleration feed-forward control,
in particular with elasticity in the mechanical system, can be avoided.
Since the position measured in cartesian coordinates is processed
directly without any kinematic transformation, errors due to inaccuracies
in kinematics such as thermal expansion, backlash, deformation,
etc. are conceptually excluded. A contour error of almost zero can
be achieved through a suitable characteristic of the controller
structure.
On the basis of the cartesian control concept according to the
present invention, the following different advantageous instances
of a cartesian control according to the present invention can be
created, among other things.
According to a first example embodiment of the control method and
the numerical control for motion control, a cartesian position control,
kinematic transformation and axial speed and acceleration control
are provided by performing a cartesian position control of cartesian
position setpoints by subjecting the resulting cartesian speed values
to a kinematic transformation into the machine coordinate system
and specifying them as speed parameters of an axial speed and/or
acceleration control for each drive. Actual position values determined
in the cartesian coordinate system and belonging to the cartesian
position setpoints are sent back as controlled variables to the
cartesian position control.
Another example embodiment performs a cartesian position and speed
control, subsequent kinematic transformation and axial acceleration
control by performing a cartesian position and speed control of
cartesian position setpoints by subjecting the resulting cartesian
acceleration values to a kinematic transformation into the machine
coordinate system and specifying them as acceleration parameters
of an axial acceleration control for each drive. Actual position
values determined in the cartesian coordinate system and belonging
to the cartesian position setpoints and speed setpoints are sent
back as controlled variables to the cartesian position control,
and actual speed values are sent back as controlled variables to
the cartesian speed control.
Another example embodiment permits a cartesian position, speed
and acceleration control and a subsequent kinematic transformation
of the voltage values of a converter by performing a cartesian position,
speed and acceleration control of cartesian position setpoints by
subjecting the resulting cartesian current or voltage setpoints
to a kinematic transformation into the machine coordinate system
and specifying them as the current or voltage values for each drive.
Actual position values determined in the cartesian coordinate system
and belonging to the cartesian position setpoints, speed setpoints
and acceleration setpoints are sent back as controlled variables
to the cartesian position control, actual speed values being sent
back as controlled variables to the cartesian speed control and
actual acceleration values are sent back as controlled variables
to the cartesian acceleration control.
The three example embodiments described above can be implemented
in different ways. Examples in this regard are described below.
Another example embodiment uses the normal error and the tangential
error as a deviation for cartesian position control of the path.
It is especially advantageous for the actual cartesian motion values
to be determined by direct measurement of the position in the cartesian
coordinate system, in particular at the outer end of the machine
kinematics, in particular a robot gripper or a tool.
As an alternative, the actual cartesian motion values can also
be determined by indirect axial measurement of the position in the
machine coordinate system and subsequent inverse transformation
into the cartesian coordinate system.
According to another advantageous embodiment, axial control parts
remaining in the, machine axes are equipped with known feed forward
control methods for one or more corresponding axial motion parameters
for more accurate control of a mechanical system with inherent elasticities,
for example.
Therefore, the dynamics of axial partial controls remaining in
the response to set point changes can be further utilized through
known axial feed-forward concepts as part of a cartesian path control
according to the present invention.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a vector diagram of the tracking error components for
illustration of normal error and tangential error in the cartesian
coordinate system.
FIG. 2 is a vector diagram to illustrate a an error of the actual
path from the programmed path in the natural coordinate system.
FIG. 3 shows a conventional controller structure of a position
control in the machine coordinate.
FIG. 4 shows a control structure of a (artesian path control according
to the present invention.
FIG. 5 is schematic diagram of two axes kinematics for practical
testing of the cartesian path control according to the present invention.
FIG. 6 shows an example of a contour having a C.sup.2 -steady spline.
FIG. 7 shows a resulting contour error in an example of a contour
after cartesian path control according to the present invention
and after a conventional axial control.
FIG. 8 shows the contour error in a circular path with a cartesian
path control according to the present invention and a conventional
axial control.
FIG. 9 shows a control structure for cartesian position control
according to the present invention.
FIG. 10 shows a control structure for cartesian position and speed
control according to the present invention.
FIG. 11 shows a control structure for cartesian position, speed
and acceleration control according to the present invention.
DETAILED DESCRIPTION
The concept presented below according to the present invention
is based on the fact that in machining processes, no spatial deviation
of the actual path from the programmed path can be tolerated, and
a time-delayed but spatially accurate movement along the desired
contour does not constitute an accurate path either. Therefore,
the contour error or normal error .DELTA.r.sub.n and the tangential
error .DELTA.r.sub.t are used for position control, as illustrated
in FIG. 1 instead of the deviations from the setpoint and actual
position values in the machine axes, e.g., .DELTA.x, .DELTA.y.
FIG. 1 shows a vector diagram of the tracking error components
for path control on a programmed path S. This shows a cartesian
coordinate system with the three axes x, y and z. One vector r extends
from the origin of the cartesian coordinate system to an actual
position. Another vector r.sub.F extends to the normal base point
and a third vector r.sub.w extends to the setpoint position.
This analysis permits a separation of the control dynamics from
the tangential direction and the normal direction in addition to
increasing the path accuracy.
The control structure of the cartesian path control according to
the present invention is presented and explained below on the basis
of theoretical considerations for any machine kinematics. In addition
to an estimate of complexity in comparison with the known position
control in machine axes, the efficiency of this method and of numerical
control according to the present invention is shown on the basis
of measurement results.
If possible, the variables used are combined into vectors in the
remaining figures and are shown with double lines in the structural
diagrams. Vectors in cartesian coordinates are provided with the
index `xy.` Analysis of a variable into normal and tangential components
with respect to the programmed path is indicated by the index `nt.`
In the remaining figures and discussion, the position vector is
abbreviated as r.sup.xy, the speed as v.sup.xy and the acceleration
as a.sup.xy as already shown in FIG. 1. The respective setpoints
are provided with a `w` subscript.
A description of a spatial motion in natural coordinates is explained
below.
To describe the motion of a point in space, the natural coordinate
system is also available in addition to the cartesian diagram (FIG.
1). These are the three unit vectors of the so called "Freuet
frame" which accompany point P represented by unit vector e.sub.t
in the tangential direction, e.sub.n in the direction of the main
normal and e.sub.b in the direction of the binormal. The third component
can be omitted here, because the additional considerations are limited
to planar movement sequences. However, it is readily possible to
expand the method according to the present invention to three-dimensional
sequences without any restrictions.
If such a natural coordinate system according to the diagram in
FIG. 2 places the normal base point at point P.sub.F on programmed
path S, its motion can be described as follows: ##EQU1##
The radius of curvature of programmed path S is given as .rho..sub.w,
the path traveled on the (are length) is given as s(t).
Starting from r.sub.F.sup.xy, the position vector of actual position
P on actual path I deviating from programmed path S can be given
as: ##EQU2##
After corresponding conversion, differentiation yields a speed
vector composed of normal speed and tangential speed as follows:
##EQU3##
Since the position and speed in the tangential and normal directions
are derived from one another by differentiation, a position control
in these components can be implemented in the same way as in the
cartesian system.
The procedure of kinematic transformation is explained in greater
detail below.
To obtain setpoints for axis controllers of a numerical control
system, the cartesian setpoints must be transformed into the machine
space, taking into account the given mechanical arrangement, and
the actual values of the position must be transformed in the opposite
direction.
The position vector r.sup.xy (t) of a point shown in the cartesian
coordinates can be determined from the position values in machine
coordinate G by a nonlinear transformation A, which depends on the
mechanical kinematic, as follows:
Actual values in machine coordinates, such as the position in calculation
procedure (1), are characterized below by a `G` superscript.
The time derivation of the position yields the speed in cartesian
coordinates using the chain rule as follows: ##EQU4##
Jacobi matrix ##EQU5##
is abbreviated below as J.
Conversion of the position vector from cartesian coordinates to
machine coordinates G takes place according to the equation:
and transformation of the speed vector takes place after conversion
of calculation procedure (2) through the inverse of the Jacobi matrix:
Transformation of the speed setpoint is explained in greater detail
below.
Speed setpoints can be transformed from natural coordinates into
cartesian coordinates by using a vector rotator: ##EQU6##
Angle .phi..sub.w of rotation between the cartesian coordinate
system and the natural coordinate system is defined by the speed
setpoint in the normal base point as:
In addition to the calculation of arc tan from the quotient of
.nu..sub.wy and .nu..sub.wx, their signs must also be analyzed from
the arc tan 2 function to arrive at an angle in the range [0;2..pi.[.
The computation complexity can be reduced drastically if the calculation
of the angle value is omitted and instead the sine or cosine of
angle .phi..sub.w needed for the transformation is determined directly
from the components of the speed vector according to the equations:
##EQU7##
(bzw.=and)
In this procedure, the above-mentioned analysis of signs is also
omitted.
Before presenting a possible structure of the cartesian path control
according to the present invention on the basis of the preceding
considerations, the conventional axial position control in the cascade
structure according to the present state of the art is briefly presented
below for comparison purposes.
The illustration according to FIG. 3 shows the position control
loops 9 through 12 assigned to the machine axes and provided with
a speed feed-forward 5 7. To make them symmetrical, the position
setpoints of the individual axes are delayed sent as models of the
closed speed control circuit over PT.sub.2 elements 9.
If the programmed contour is described in the form of C.sup.2 -steady
splines using a path parameter u, the arc length s can be determined
as a function of the path parameter according to the equation:
Since this is an elliptical integral, a closed solution is impossible,
so the analysis must be performed numerically, e.g., using the Euclidean
standard.
On the basis of inversion function 4 u=f.sup.-1 (s), the respective
path parameter and thus the programmed cartesian position can be
determined from the arc lenghts required by jerk-limited motion
control 1 2 3. Their time derivation 5 yields the speed setpoint
as follows: ##EQU8##
The term ##EQU9##
contained therein corresponds to the reciprocal of the integrand
of computation procedure (8), so that the programmed speed can be
calculated- from the instantaneous value of the path parameter and
speed setpoint v.sub.w of motion control 1 2 3. The cartesian
setpoint values can then be transformed by transformations 7 8
to linkage space G on the basis of computation procedures (3) and
(4).
In contrast with the conventional control in cascade structure
using the position in the machine axes as controlled variables,
the position control of the cartesian path control is based on the
normal error--i.e., the deviation from the programmed contour--and
the tangential error. The possible control structure according to
the present invention shown in FIG. 4 illustrates that velocity
and speed control 11 12 remain in the machine axes, so that, for
example, effects due to a mechanical system containing elasticities
can be handled with the known methods of feed-forward controls and
control (e.g., state control).
The normal base point is to be determined from the actual position
value (computation procedure (1)) converted into cartesian coordinates
in order to be able to compute the corresponding normal error r.sub.n
and tangential error .DELTA.s. The position of the normal base point
is determined by the fact that at this point the tangent at the
programmed contour is perpendicular to the connecting line with
respect to the actual point, so that the respective value of the
path parameter is determined from the following relationship: ##EQU10##
When using cubic splines, this means a search for the zeroes of
a 5.sup.th order polynomial, which can be achieved only numerically.
By using the newtonian method 14 the required computation power
can be minimized, especially since a good starting value is available
with the value determined in the preceding control cycle.
If the value of the path parameter belonging to the normal base
point is known, its position, the contour error, as well as the
contour length to the beginning of the path (see computation procedure
(8)) can be calculated. Difference .DELTA.S between actual path
S and programmed path Sw, which is specified by the motion control,
is sent in the tangential direction to position controller 10 which
is implemented as a P controller to guarantee approach the target
position without overshoot. In contrast with that, normal controller
17 is a PI controller because it is responsible exclusively for
correcting contour deviations. To obtain feed-forward of speed,
the speed determined by the motion control is added to the output
value of the tangential position controller.
Here again, the position setpoint must also be delayed using a
model 9 of the closed rpm control loop for balancing. The control
in the normal direction remains unaffected by these additions because
it does not have any direct influence on the response to set point
changes.
The output values of position controllers 10 17 i.e., the setpoints
for tangential and normal speed, are converted first into cartesian
system k with transformation 13 and then into the machine axes coordinates
by inverse 7 of the Jacobi matrix.
With sufficient dynamics of the speed control loop, the inverse
Jacobi matrix is compensated by the Jacobi matrix contained in the
controlled system so that from the standpoint of position controllers
10 17 the path is described approximately by the series connection
of a PT.sub.2 element 11 (speed control loop) and an integral element
12. Because the Jacobi an depends on the position, it is important
for the latter to be adapted fast enough. This requires a suitable
system clock depending on the desired traversing speed. The same
observations also apply to rotational matrix T.sup.-1 and 13 whose
"inverse" is contained in the contour and tangential error
calculation.
If the structures shown according to FIGS. 3 and 4 are compared,
it is clear that the complexity for transformations 7 8 or 7 8
13 between the different coordinate systems and the actual control
is largely identical. The complexity for the additional determination
of the normal base point, which is necessary in path control, can
be minimized by using newtonian method 14 and a favorable choice
of the initial values so that it plays only a subordinate role.
The practical investigation of cartesian path control according
to the present invention has been based on two axes kinematics with
a translatory axis A1 and a rotational axis A2 shown in the diagram
according to FIG. 5.
The main component of translatory axis A1 is a "commercially
available" feed unit V with a first motor M1 an axis A1 a
carriage SL with linear measuring system GM and end switch ES (a
first detector is assigned to motor M1), permitting a maximum travel
distance of 630 mm. A second motor M2 is mounted on carriage SL
of translatory axis A1 positioning a pivotable arm AR so that two-dimensional
motion processes are possible with the tip of the arm. A second
sensor G2 is assigned to second motor M2.
Mathematical Description of the Machine Kinematics
The position setpoints are first specified in cartesian coordinates
and consequently must be transformed to the machine coordinates.
If the variables introduced in FIG. 5 are used, the following relationship
holds: ##EQU11##
Since the investigated path control concept according to the present
invention is advantageously based on the analysis of the position
error in the normal and tangential components, the actual position
in cartesian coordinates must also be known, i.e., determined directly
(e.g., the transmitter mounted at one end of arm AR, whose signals
are detected by receivers arranged outside the work space of arm
AR). However, applications where a direct relationship of the mechanical
system (fixed point with respect to the gripper) permits an inference
as to the position of the gripper are also possible. It is also
conceivable to calculate the actual position in cartesian coordinates
from the indirect detection of position according to calculation
procedure (11):
##EQU12##
The time derivation of this equation yields the following for the
transformation of the actual speed values from the machine axes
into the cartesian coordinate system: ##EQU13##
In the opposite direction, the cartesian speed setpoints are transformed
into machine coordinates by inverse 8 of the Jacobi matrix into
joint space G as setpoints for the speed controller: ##EQU14##
The contour shown in FIG. 6 is used for comparison of cartesian
path controls according to the present invention with feed-forward
of the tangential speed and conventional control with axial speed
feed-forward. It has a C.sup.2 -steady spline. FIG. 7 shows the
time curves of the contour error in both methods (top: cartesian
control; bottom: conventional control, each in units of mm) at a
maximum path speed of v=4.5 m/min.
For interpretation of the results, it is important to know that
actual positions at the right next to the contour as seen in the
direction of movement essentially lead to positive contour errors,
so that with circular contours, for example, a reversal in the direction
of rotation of the movement leads to a change of signs with no change
in the time characteristic of the contour error.
The greatest contour inaccuracies in the form of the pronounced
peaks are to be attributed to a change in the direction of movement
of feed unit V or arm AR with both types of control. The influence
of sudden changes in load torque can be compensated by friction
pre-control.
In the course of the contour error, "excursions" which
can be attributed to the changing curvature of the programmed contour
occur at other locations. The changes in radial acceleration required
in this area despite a constant path speed also represent disturbances
for speed controller 11 12. However, their influence can be eliminated
by an acceleration feed-forward control from the existing values
for the motion guidance and the programmed path.
However, the comparative diagram in FIG. 7 shows clearly the gain
in contour accuracy of the cartesian path control according to the
present invention.
As an additional test contour, a path composed of interpolation
points arranged in a circle with a radius r.apprxeq.25 mm was selected
and was traversed at a maximum speed of v=9 m/min=150 mm/s. The
diagram according to FIG. 8 (same arrangement as in FIG. 7) again
shows a comparison of the resulting contour errors. The resulting
path deviation also confirms the superiority of the path control.
Possible instances of a cartesian path control according to the
present invention, based on the preceding variants, are presented
below on the basis of the schematic diagrams according to FIGS.
9 through 11.
FIG. 9 shows a schematic diagram of a cartesian position control
such as that explained above on the basis of FIG. 4 and the associate
variants. Cartesian position setpoints ##EQU15##
are supplied to a cartesian position controller LR which supplies
corresponding speed setpoints ##EQU16##
as motion parameters. There is a kinematic transformation from
cartesian coordinate system BKS into machine coordinate system MKS,
which yields speed setpoints ##EQU17##
for control in the machine axis. These setpoints are used to drive
an axial control R.sub.ax with a converter U, a motor M1 and a detector
system G1 assigned to it. The latter have the known design, which
is why they are not shown in detail.
Detector system G1 may be designed as a direct measurement system,
with the result that the direct cartesian position r.sup.xy of the
axis of motor M1 or of a machine element mounted on that axis is
supplied, so that this actual position value r.sup.xy can be sent
back to the input of the cartesian position controller LR.
Another possibility on which the diagram according to FIG. 9 is
based is for an indirect measurement to be performed by detector
system G1. Detector system G1 then supplies an actual position value
in machine coordinate system MKS. Therefore, there is an inverse
transformation of this actual position value into cartesian coordinate
system BKS. The resulting actual position value in cartesian coordinate
system r.sup.xy is also sent back to the input of cartesian position
controller LR.
FIG. 10 shows a schematic diagram of a possible embodiment of a
cartesian position control LR and cartesian speed control GR, having
essentially the same design as in FIG. 9 but a cartesian speed
controller GR, subordianated to the cartesian position controller,
receives the cartesian speed setpoint ##EQU18##
supplied by cartesian position controller LR and supplies the corresponding
cartesian acceleration setpoints ##EQU19##
which are then transformed from cartesian coordinate system BKS
into machine coordinate system MKS analogous to the preceding variants
in FIG. 9 in order to drive a corresponding axial control R.sub.ax.
Axial control R.sub.ax now includes only an acceleration/torque
control, because the speed control is already implemented by cartesian
speed controller GR. Axial control R.sub.ax is in turn followed
by a converter U, motor M1 and a detector system G1 in the known
design. The analysis by the detector system G1 can take place in
the same way as explained with regard to FIG. 9 but with the difference
that except for an actual cartesian position value r.sup.xy a corresponding
actual cartesian speed value .nu..sup.xy is determined and is sent
back to the input of the cartesian speed controller GR.
The diagram in FIG. 11 shows another possible embodiment with cartesian
position control LR, speed control GR and cartesian acceleration
control BR. This design corresponds essentially to that in the two
FIGS. 9 and 10 described above. Cartesian speed controller GR is
followed by another cartesian controller, namely acceleration controller
BR. This supplies current or voltage values ##EQU20##
for driving a converter U. These values are transformed from cartesian
coordinate system BKS into machine coordinate system MKS in the
same way as described above and they drive converter U as corresponding
current or voltage value ##EQU21##
This is followed in the known way by a motor M1 to which is assigned
a detector signal G1. Again in this embodiment, detector system
G1 can supply corresponding actual values through direct or indirect
measurement. In addition to actual position value r.sup.xy and actual
speed value .alpha..sup.xy, the detector system also supplies an
actual acceleration value .alpha..sup.xy which is sent back to the
input of cartesian acceleration controller BR as a controlled variable.
The efficiency of cartesian path control, the principle of which
is created according to the present invention, can be further improved
by the additions in the area of friction compensation and acceleration
feed-forward. Likewise, the system may be supplemented by a direct
cartesian measurement system to avoid measurement errors that occur
in indirect measurement because of torsion, backlash and pitch errors.
Additionally, this creates the prerequisite for achieving a high
path accuracy on the basis of the cartesian path control method
even with new types of articulated arm kinematics, where inaccuracies
in the articulated joint, sagging of the and changes in length still
cause fundamental problems today due to temperature drift. |