CN115597534A - Robot compliant cutter calibration method based on articulated measuring arm - Google Patents

Robot compliant cutter calibration method based on articulated measuring arm Download PDF

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CN115597534A
CN115597534A CN202211323218.1A CN202211323218A CN115597534A CN 115597534 A CN115597534 A CN 115597534A CN 202211323218 A CN202211323218 A CN 202211323218A CN 115597534 A CN115597534 A CN 115597534A
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coordinate system
arm
measuring
coordinate
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宋立滨
姜帅
祁辉
赖庆文
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Tsinghua University
Tianjin Institute of Advanced Equipment of Tsinghua University
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Tianjin Institute of Advanced Equipment of Tsinghua University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
    • G01B21/00—Measuring arrangements or details thereof, where the measuring technique is not covered by the other groups of this subclass, unspecified or not relevant
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
    • G01B21/00—Measuring arrangements or details thereof, where the measuring technique is not covered by the other groups of this subclass, unspecified or not relevant
    • G01B21/02—Measuring arrangements or details thereof, where the measuring technique is not covered by the other groups of this subclass, unspecified or not relevant for measuring length, width, or thickness
    • G01B21/04—Measuring arrangements or details thereof, where the measuring technique is not covered by the other groups of this subclass, unspecified or not relevant for measuring length, width, or thickness by measuring coordinates of points
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    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
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Abstract

The invention provides a calibration method of a robot compliant cutter based on an articulated measuring arm, which comprises the following steps: s1, controlling a mechanical arm to move along the direction of a coordinate axis, and recording relevant coordinates of a measuring arm and a robot; s2, controlling the mechanical arm to rotate around a fixed point, and recording relevant coordinates of the measuring arm and the robot; s3, establishing an equation set according to the coordinates of the first two steps, and solving by using a least square method to obtain a conversion matrix; and S4, calculating a calibration plane equation by using the measuring arm, controlling the mechanical arm to contact the plane, and solving according to the self pose of the robot and the plane equation to obtain the position of the tool center point. The calibration method for the flexible cutter of the robot based on the articulated measuring arm introduces a high-precision measuring instrument, has high calibration precision and is not limited by the precision of the robot; the operation difficulty is lower.

Description

Robot compliant cutter calibration method based on articulated measuring arm
Technical Field
The invention belongs to the technical field of robot calibration in the automation industry, and particularly relates to a method for calibrating a flexible cutter of a robot based on an articulated measuring arm.
Background
At present, the mechanical arm needs artifical teaching to calibrate end instrument coordinate system initial point (TCP point) before putting into operation, generally uses four-point method to calibrate, promptly: the method comprises the steps that a worker marks preset TCP points on a tool at the tail end of a mechanical arm, the mechanical arm is controlled to move four times through a mechanical arm demonstrator, the mechanical arm is guaranteed to move to the same fixed point in different postures at each time, and a mechanical arm controller collects pose data of the mechanical arm four times, so that coordinates of the TCP points of the tool at the tail end of the mechanical arm are calculated.
The calibration process needs to be completed by skilled workers, and in order to ensure high precision, the time required for calibrating a TCP point once is generally more than five minutes, which is time-consuming and labor-consuming, and the consistency of precision cannot be ensured. For a compliant cutter, the cutter is forced to displace, and the error is larger in the point aligning process. Meanwhile, the calibration precision is limited by the absolute positioning precision of the robot, so that a high-precision measuring tool needs to be introduced in the occasion of requiring a high-precision tool coordinate system.
Disclosure of Invention
In view of the above, the invention aims to provide a calibration method for a flexible cutter of a robot based on an articulated measuring arm, so as to solve the problems that the existing four-point method has higher requirements and greater difficulty for operators, and the precision completely depends on the absolute positioning precision of the robot.
In order to achieve the purpose, the technical scheme of the invention is realized as follows:
a calibration method for a robot compliant cutter based on an articulated measuring arm comprises the following steps:
s1, controlling a mechanical arm to move along the direction of a coordinate axis, and recording relevant coordinates of a measuring arm and a robot;
s2, controlling the mechanical arm to rotate around a fixed point, and recording relevant coordinates of the measuring arm and the robot;
s3, establishing an equation set according to the coordinates of the first two steps, and solving by using a least square method to obtain a conversion matrix;
and S4, calculating a calibration plane equation by using the measuring arm, controlling the mechanical arm to contact the plane, and solving according to the self pose of the robot and the plane equation to obtain the position of the tool center point.
Further, the specific method in step S1 is:
s11, selecting a fixed point on the tail end tool of the robot as a measuring point P a ，
S12, keeping the posture of the mechanical arm unchanged, controlling the mechanical arm to move along three axes of X, Y and Z, and measuring P by using the measuring arm a Measuring coordinates under a measuring arm coordinate system for five times on each axis, and recording the corresponding terminal pose of the robot; for five points of a certain axis, recording the coordinate of the measuring arm coordinate system as- M P 1 , M P 2 , M P 3 , M P 4 , M P 5 }; recording the coordinate of the tail end of the robot as a last page in the robot coordinate system R P 1 , R P 2 , R P 3 , R P 4 , R P 5 }。
Further, the specific method of step S2 is: keeping the position of the coordinate system at the tail end of the robot unchanged, only changing the posture of the robot, and measuring P by using a measuring arm a Position, recording the attitude of the end of the arm in the robot coordinate system
Figure BDA0003911347490000021
And location R P e1 , R P e2 , R P e3 , R P e4 , R P e5 And a measuring arm reading M P a1 , M P a2 , M P a3 , M P a4 , M P a5 }。
Further, the specific method in step S3 is:
first, a rotation matrix is calculated
Figure BDA0003911347490000022
For five points of any axis under the same coordinate system, a coordinate matrix is formed as follows:
Figure BDA0003911347490000023
the covariance matrix of the above coordinate matrix is
Figure BDA0003911347490000031
Mathematical expectation of coordinate X in equation (1)
Figure BDA0003911347490000032
Mathematical expectation of the coordinate Y
Figure BDA0003911347490000033
Mathematical expectation of the coordinate Z
Figure BDA0003911347490000034
For two variables M and N with desired values E (M) and E (N), the covariance Cov (M, N) is
Cov(M,N)＝E(MN)-E(M)E(N) (6)
The covariance matrix can be calculated by substituting equations (5), (6) and (7) into equation (2). And then, calculating the maximum eigenvector V of the covariance matrix by adopting a singular value decomposition method, wherein the maximum eigenvector V is the direction of the current coordinate axis. Sequentially establishing three axes X, Y and Z of the robot under a robot coordinate system to express a R X, R Y, R Z and a representation in the coordinate system of the measuring arm M X, M Y, M Z }. The two sets of coordinate vectors have the following relationship:
Figure BDA0003911347490000035
and (5) solving the equation (7) to obtain a rotation matrix:
Figure BDA0003911347490000036
subsequently calculating an offset vector from the measuring arm coordinate system to the robot coordinate system;
in step S2, a point P is measured a With the arm end coordinate P e The following relationships exist:
Figure BDA0003911347490000037
in the formula R P ai For measuring point P under ith posture of robot a The representation in the robot coordinate system is, R P ei for the representation of the robot end in the robot coordinate system in the ith pose of the robot, E t ae for the end of the arm P e To the measuring point P a The offset in the robot end coordinate system,
Figure BDA0003911347490000041
the current posture of the robot is taken as the current posture;
measurement point P a The relation between the coordinates in the robot coordinate system and the coordinates in the measuring arm coordinate system is
Figure BDA0003911347490000042
In the formula
Figure BDA0003911347490000043
In order to measure the representation of the offset vector of the arm coordinate system to the mechanical arm coordinate system under the mechanical arm coordinate system, the formula (10) is substituted into the formula (9) R P ai The following can be obtained:
Figure BDA0003911347490000044
subtracting the data of pose 2 from the data of pose 1 can obtain the following relationship:
Figure BDA0003911347490000045
by analogy, the following can be obtained
Figure BDA0003911347490000046
To find E t ae Of the best least squares solution
Figure BDA0003911347490000047
Will be provided with E t ae Brought back and calculated by the same method
Figure BDA0003911347490000048
The optimal least squares solution of.
So far, the transformation matrix from the coordinate system of the measuring arm to the coordinate system of the robot is solved
Figure BDA0003911347490000049
Figure BDA00039113474900000410
Further, the specific method in step S4 is:
selecting a fixed finish machining plane as a calibration plane, and using a measuring arm to arbitrarily take five points on the plane, wherein the coordinates of the five points are ( M x 1 , M y 1 , M z 1 )，( M x 2 , M y 2 , M z 2 )，( M x 3 , M y 3 , M z 3 )，( M x 4 , M y 4 , M z 4 )，( M x 5 , M y 5 , M z 5 ) Will convert the matrix
Figure BDA0003911347490000051
Multiplying the coordinates of the five points to obtain the representation (x) of the five points in the robot coordinate system 1 ,y 1 ,z 1 )，(x 2 ,y 2 ,z 2 )，(x 3 ,y 3 ,z 3 )，(x 4 ,y 4 ,z 4 )，(x 5 ,y 5 ,z 5 ) Calculating a plane regression equation of the five points in a robot coordinate system, wherein the specific method comprises the following steps:
the general expression of the plane equation is:
Ax+By+Cz+D＝0(C≠0)
it is transformed into the following form:
Figure BDA0003911347490000052
order to
Figure BDA0003911347490000053
Then
z＝a 0 x+a 1 y+a 2
The matrix form of the plane equation at this time can be expressed as
Figure BDA0003911347490000054
Calculating the coefficient a 0 ，a 1 ，a 2 Is obtained by least squares solution
Figure BDA0003911347490000055
From this, a planar regression equation Z = a can be obtained 0 X+a 1 Y+a 2 ；
Since the gauge head center point has a gauge head radius length from the plane, the regression equation needs to be translated in the direction of the plane normal by a gauge head radius length in the opposite direction. Setting the radius of the measuring head as r, the constant term coefficient a of the plane equation after translation 3 Is composed of
Figure BDA0003911347490000056
Equation Z = a to obtain the calibration plane 0 X+a 1 Y+a 3 Z = aX + bY + C;
controlling the robot to make the center point of the tool contact with any five points on the plane in different postures, recording the coordinates of the tail end of the robot R P t1 , R P t2 , R P t3 , R P t4 , R P t5 The sum of the gestures
Figure BDA0003911347490000057
Tool center point P of robot in different poses tcp Can be expressed as
Figure BDA0003911347490000061
In the formula R P tcpi Is the tool center point P under the i-th pose of the robot tcp The representation in the robot coordinate system is,
Figure BDA0003911347490000062
the posture of the robot at the ith time, E P tcpi tool center point P tcp A representation in the robot end coordinate system, R P ei representing the tail end of the robot in the coordinate system of the robot in the ith pose;
tool center point coordinate under different poses R P tcpi Satisfies the following equation
Figure BDA0003911347490000063
Substituting five different poses and equation (15) into an available equation set
Figure BDA0003911347490000064
To find E P tcp Of the best least squares solution
Figure BDA0003911347490000065
Compared with the prior art, the calibration method of the robot compliant cutter based on the articulated measuring arm has the following advantages:
(1) Aiming at the problems that the traditional four-point method is too high in operation difficulty, the calibration precision depends on the self precision of the robot, the compliant cutter is easy to deform under stress and the like, the method for calibrating the compliant cutter of the robot based on the articulated measuring arm calculates and obtains a conversion matrix of the measuring arm and the robot by measuring the coordinate of the measuring arm at the same point and the pose of the robot; the coordinates of subsequent points are obtained by measuring through the measuring arm, so that the calibration precision is greatly improved; in order to solve the problems that the difficulty of controlling the robot in point-to-point is high and a compliant cutter is easy to deform in the calibration process, a method for contacting the center point of a robot tool with a surface is provided, five points of the plane are measured by a measuring arm, a plane equation is calculated and converted into a robot coordinate system, the robot is controlled to contact the plane with the center point of the tool at different poses, and the equation is solved to obtain the coordinate of the center point of the tool; the alignment work of three degrees of freedom is reduced to the alignment of one degree of freedom direction, the operation difficulty is greatly reduced, and the human error is reduced.
Drawings
The accompanying drawings, which are included to provide a further understanding of the invention, illustrate embodiments of the invention and together with the description serve to explain the invention and do not constitute a limitation of the invention. In the drawings:
FIG. 1 is a flowchart illustrating an implementation of a method for calibrating a compliant tool of a robot based on an articulated measuring arm according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of a calibration system according to an embodiment of the present invention;
FIG. 3 is a schematic diagram illustrating the operation of a measuring arm probe to measure a fixed point on a tool according to an embodiment of the present invention;
FIG. 4 is an operation diagram of the tool center point contacting the calibration plane at different postures.
Detailed Description
It should be noted that the embodiments and features of the embodiments may be combined with each other without conflict.
In the description of the present invention, it is to be understood that the terms "center", "longitudinal", "lateral", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", and the like, indicate orientations or positional relationships based on those shown in the drawings, and are used only for convenience in describing the present invention and for simplicity in description, and do not indicate or imply that the referenced devices or elements must have a particular orientation, be constructed and operated in a particular orientation, and thus, are not to be construed as limiting the present invention. Furthermore, the terms "first," "second," and the like are used for descriptive purposes only and are not to be construed as indicating or implying relative importance or to implicitly indicate a number of the indicated technical features. Thus, a feature defined as "first," "second," etc. may explicitly or implicitly include one or more of that feature. In the description of the present invention, "a plurality" means two or more unless otherwise specified.
In the description of the present invention, it should be noted that, unless otherwise explicitly specified or limited, the terms "mounted," "connected," and "connected" are to be construed broadly, e.g., as meaning either a fixed connection, a removable connection, or an integral connection; can be mechanically or electrically connected; they may be connected directly or indirectly through intervening media, or they may be interconnected between two elements. The specific meanings of the above terms in the present invention can be understood by those of ordinary skill in the art through specific situations.
The present invention will be described in detail below with reference to the embodiments with reference to the attached drawings.
A robot compliant cutter calibration method based on an articulated measuring arm is disclosed, as shown in figures 1 to 4, high-precision positioning of a robot TCP is achieved by establishing a conversion matrix from a measuring arm coordinate system to a robot coordinate system, a fixed point in a four-point method is changed into a fixed plane, a TCP position coordinate is inversely solved through a plane equation and the tail end position of the robot, and calibration difficulty is reduced.
As shown in fig. 1, the method specifically comprises the following steps:
first, a fixed point is selected as a measurement point P on the end tool of the robot a As shown in fig. 3, a hexagon screw is selected, so that the measuring head can be clamped inside the screw for fixing and cannot shake.
Secondly, keeping the posture of the mechanical arm unchanged, controlling the mechanical arm to move along three axes of X, Y and Z, and measuring P by using the measuring arm a Coordinates under the measurement arm coordinate system are measured five times per axis and the corresponding robot end pose is recorded. For five points of a certain axis, recording the coordinate of the measuring arm coordinate system as- M P 1 , M P 2 , M P 3 , M P 4 , M P 5 }; recording the coordinate of the tail end of the robot as a back in the coordinate system of the robot R P 1 , R P 2 , R P 3 , R P 4 , R P 5 }。
Thirdly, keeping the position of the coordinate system at the tail end of the robot unchanged, only changing the posture of the robot, and measuring P by using the measuring arm a Position, recording the attitude of the end of the arm in the robot coordinate system
Figure BDA0003911347490000091
And location R P e1 , R P e2 , R P e3 , R P e4 , R P e5 }, and measurement arm reading M P a1 , M P a2 , M P a3 , M P a4 , M P a5 }。
And fourthly, establishing an equation set according to the coordinates of the first two steps, and calculating a conversion matrix from the coordinate system of the measuring arm to the coordinate system of the robot.
First, a rotation matrix is calculated
Figure BDA0003911347490000092
For five points of any axis under the same coordinate system, the formed coordinate matrix is as follows:
Figure BDA0003911347490000093
the covariance matrix of the coordinate matrix is
Figure BDA0003911347490000094
Mathematical expectation of coordinate X in equation (1)
Figure BDA0003911347490000095
Mathematical expectation of the coordinate Y
Figure BDA0003911347490000096
Mathematical expectation of the coordinate Z
Figure BDA0003911347490000097
For two variables M and N with desired values E (M) and E (N), the covariance Cov (M, N) is
Cov(M,N)＝E(MN)-E(M)E(N) (6)
The covariance matrix can be calculated by substituting equations (5), (6) and (7) into equation (2). And then, calculating the maximum eigenvector V of the covariance matrix by adopting a singular value decomposition method, namely the direction of the current coordinate axis. Sequentially establishing three axes X, Y and Z of the robot under a robot coordinate system to express a R X, R Y, R Z and a representation in the coordinate system of the measuring arm M X, M Y, M Z }. The two sets of coordinate vectors have the following relationship:
Figure BDA0003911347490000101
and (5) solving the formula (7) to obtain a rotation matrix:
Figure BDA0003911347490000102
offset vectors from the measuring arm coordinate system to the robot coordinate system are then calculated.
In a third step, the point P is measured a With the arm end coordinate P e The following relationships exist:
Figure BDA0003911347490000103
in the formula R P ai For measuring point P under ith posture of robot a A representation in the coordinate system of the robot, R P ei for the representation of the robot end in the robot coordinate system in the ith pose of the robot, E t ae for the end of the arm P e To the measuring point P a The offset in the robot end coordinate system,
Figure BDA0003911347490000104
is the current pose of the robot.
Measurement point P a The relation between the coordinates in the robot coordinate system and the coordinates in the measuring arm coordinate system is
Figure BDA0003911347490000105
In the formula
Figure BDA0003911347490000106
Is a representation of the offset vector of the measuring arm coordinate system to the robot arm coordinate system in the robot arm coordinate system. Substitution by bringing formula (10) into formula (9) R P ai The following can be obtained:
Figure BDA0003911347490000107
subtracting the data of pose 2 from the data of pose 1 can obtain the following relationship:
Figure BDA0003911347490000108
by analogy, the following can obtain
Figure BDA0003911347490000111
To find E t ae Of the best least squares solution
Figure BDA0003911347490000112
Will be provided with E t ae Brought back and calculated by the same method
Figure BDA0003911347490000113
The optimal least squares solution of.
So far, the transformation matrix from the coordinate system of the measuring arm to the coordinate system of the robot is solved
Figure BDA0003911347490000114
Figure BDA0003911347490000115
Fifthly, selecting a fixed finish machining plane as a calibration plane, and using a measuring arm to randomly take five points on the plane, wherein the coordinates of the five points are ( M x 1 , M y 1 , M z 1 )，( M x 2 , M y 2 , M z 2 )，( M x 3 , M y 3 , M z 3 )，( M x 4 , M y 4 , M z 4 )，( M x 5 , M y 5 , M z 5 ) Will convert the matrix M R Multiplying T with the coordinates of the five points to obtain the representation (x) of the five points in the robot coordinate system 1 ,y 1 ,z 1 )，(x 2 ,y 2 ,z 2 )，(x 3 ,y 3 ,z 3 )，(x 4 ,y 4 ,z 4 )，(x 5 ,y 5 ,z 5 ) Calculating a plane regression equation of the five points in the robot coordinate system;
the general expression of the plane equation is:
Ax+By+Cz+D＝0(C≠0)
it is transformed into the following form:
Figure BDA0003911347490000116
order to
Figure BDA0003911347490000117
Then
z＝a 0 x+a 1 y+a 2
The matrix form of the plane equation at this time can be expressed as
Figure BDA0003911347490000121
Calculating the coefficient a 0 ，a 1 ，a 2 Is obtained by least squares solution
Figure BDA0003911347490000122
From this, the planar regression equation Z = a can be derived 0 X+a 1 Y+a 2 。
Since the gauge head center point has a gauge head radius length from the plane, the regression equation needs to be translated in the direction of the plane normal by a gauge head radius length in the opposite direction. Setting the radius of the measuring head as r, the constant term coefficient a of the plane equation after translation 3 Is composed of
Figure BDA0003911347490000123
Equation Z = a to obtain the calibration plane 0 X+a 1 Y+a 3 Z = aX + bY + C may be arranged.
Sixthly, as shown in fig. 4, controlling the robot to make the center point of the tool contact with any five points on the plane in different postures, and recording the coordinates of the tail end of the robot R P t1 , R P t2 , R P t3 , R P t4 , R P t5 And attitude
Figure BDA0003911347490000124
Tool center point P of robot in different poses tcp Can be expressed as
Figure BDA0003911347490000125
In the formula R P tcpi Is the tool center point P under the i-th pose of the robot tcp A representation in the coordinate system of the robot,
Figure BDA0003911347490000126
the posture of the robot at the ith time, E P tcpi tool center point P tcp A representation in the coordinate system of the robot end, R P ei is the representation of the robot end under the robot coordinate system under the ith pose.
Tool center point coordinate under different poses R P tcpi Satisfies the following equation
Figure BDA0003911347490000127
Substituting five different poses and equation (15) into an available equation set
Figure BDA0003911347490000131
To find E P tcp Of the best least squares solution
Figure BDA0003911347490000132
Finally, it should be noted that: the above examples are intended to illustrate the process scheme of the invention, but not to limit it; although the invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art will understand that: modifications of the method solutions described in the preceding embodiments, or equivalent substitutions of some or all of the method features, are possible without departing from the scope of the method solutions of the embodiments of the present invention. The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.

Claims (5)

1. A calibration method for a robot compliant cutter based on an articulated measuring arm is characterized by comprising the following steps: the method comprises the following steps:
s1, controlling a mechanical arm to move along the direction of a coordinate axis, and recording relevant coordinates of a measuring arm and a robot;
s2, controlling the mechanical arm to rotate around a fixed point, and recording relevant coordinates of the measuring arm and the robot;
s3, establishing an equation set according to the coordinates of the first two steps, and solving by using a least square method to obtain a conversion matrix;
and S4, calculating a calibration plane equation by using the measuring arm, controlling the mechanical arm to contact the plane, and solving according to the self pose of the robot and the plane equation to obtain the position of the tool center point.
2. The method for calibrating the compliant tool of the robot based on the articulated measuring arm of claim 1, wherein the method comprises the following steps: the specific method of the step S1 is as follows:
s11, selecting a fixed point on the robot tail end tool as a measuring point P a ，
S12, keeping the posture of the mechanical arm unchanged, controlling the mechanical arm to move along three axes of X, Y and Z, and measuring P by using the measuring arm a Measuring coordinates under a measuring arm coordinate system for five times on each axis, and recording the corresponding terminal pose of the robot; for five points of a certain axis, recording the coordinate of the measuring arm coordinate system as- M P 1 , M P 2 , M P 3 , M P 4 , M P 5 }; recording the coordinate of the tail end of the robot as a back in the coordinate system of the robot R P 1 , R P 2 , R P 3 , R P 4 , R P 5 }。
3. The method for calibrating the compliant tool of the robot based on the articulated measuring arm of claim 1, wherein the method comprises the following steps: the specific method of the step S2 comprises the following steps: keeping the position of the coordinate system at the tail end of the robot unchanged, only changing the posture of the robot, and measuring P by using a measuring arm a Position, recording the attitude of the end of the arm in the robot coordinate system
Figure FDA0003911347480000011
And position
Figure FDA0003911347480000012
And measuring arm readings
Figure FDA0003911347480000013
4. The method for calibrating the compliant cutting tool of the robot based on the articulated measuring arm as claimed in claim 1, wherein: the specific method of the step S3 comprises the following steps:
first, a rotation matrix is calculated
Figure FDA0003911347480000014
For five points of any axis under the same coordinate system, the formed coordinate matrix is as follows:
Figure FDA0003911347480000021
the covariance matrix of the coordinate matrix is
Figure FDA0003911347480000022
Mathematical expectation of coordinate X in equation (1)
Figure FDA0003911347480000023
Mathematical expectation of the coordinate Y
Figure FDA0003911347480000024
Mathematical expectation of the coordinate Z
Figure FDA0003911347480000025
For two variables M and N with desired values E (M) and E (N), the covariance Cov (M, N) is
Cov(M,N)＝E(MN)-E(M)E(N) (6)
Substituting the formulas (5), (6) and (7) into the formula (2) to calculate a covariance matrix; then, calculating the maximum eigenvector V of the covariance matrix by adopting a singular value decomposition method, namely the direction of the current coordinate axis; sequentially establishing three axes X, Y and Z of the robot under a robot coordinate system to express a R X, R Y, R Z and a representation in the coordinate system of the measuring arm M X, M Y, M Z }; the two sets of coordinate vectors have the following relationship:
Figure FDA0003911347480000026
and (5) solving the formula (7) to obtain a rotation matrix:
Figure FDA0003911347480000027
subsequently calculating an offset vector from the measuring arm coordinate system to the robot coordinate system;
in step S2, a point P is measured a With the arm end coordinate P e The following relationships exist:
Figure FDA0003911347480000031
in the formula R P ai For measuring point P under ith posture of robot a The representation in the robot coordinate system is, R P ei for the representation of the robot end in the robot coordinate system in the ith pose of the robot, E t ae for the end of the arm P e To the measuring point P a The offset in the robot end coordinate system,
Figure FDA0003911347480000032
the current posture of the robot is taken;
measurement point P a The relation between the coordinates in the robot coordinate system and the coordinates in the measuring arm coordinate system is
Figure FDA0003911347480000033
In the formula
Figure FDA0003911347480000034
In order to measure the representation of the offset vector of the arm coordinate system to the mechanical arm coordinate system under the mechanical arm coordinate system, the formula (10) is substituted into the formula (9) R P ai The following can be obtained:
Figure FDA0003911347480000035
subtracting the data of pose 2 from the data of pose 1 can obtain the following relationship:
Figure FDA0003911347480000036
by analogy, the following can obtain
Figure FDA0003911347480000037
To find E t ae Best least squares solution of
Figure FDA0003911347480000038
Will be provided with E t ae Brought back and calculated by the same method
Figure FDA0003911347480000039
The optimal least squares solution of;
the transformation matrix from the coordinate system of the measuring arm to the coordinate system of the robot is determined
Figure FDA00039113474800000310
Figure FDA0003911347480000041
5. The method for calibrating the compliant cutting tool of the robot based on the articulated measuring arm as claimed in claim 1, wherein: the specific method of the step S4 is as follows:
selecting a fixed finish machining plane as a calibration plane, and using a measuring arm to arbitrarily take five points on the plane, wherein the coordinates of the five points are ( M x 1 , M y 1 , M z 1 )，( M x 2 , M y 2 , M z 2 )，( M x 3 , M y 3 , M z 3 )，( M x 4 , M y 4 , M z 4 )，( M x 5 , M y 5 , M z 5 ) Will convert the matrix
Figure FDA0003911347480000042
Multiplying the coordinates of the five points to obtain the representation (x) of the five points in the coordinate system of the robot 1 ,y 1 ,z 1 )，(x 2 ,y 2 ,z 2 )，(x 3 ,y 3 ,z 3 )，(x 4 ,y 4 ,z 4 )，(x 5 ,y 5 ,z 5 ) Calculating a plane regression equation of the five points in a robot coordinate system, wherein the specific method comprises the following steps:
the general expression of the plane equation is:
Ax+By+Cz+D＝0(C≠0)
it is transformed into the following form:
Figure FDA0003911347480000043
order to
Figure FDA0003911347480000044
Then the
z＝a 0 x+a 1 y+a 2
The matrix form of the plane equation at this time can be expressed as
Figure FDA0003911347480000045
Calculating the coefficient a 0 ，a 1 ，a 2 Is obtained by least squares solution
Figure FDA0003911347480000046
From this, a planar regression equation Z = a can be obtained 0 X+a 1 Y+a 2 ；
Because the measuring head center point has a length of the measuring head radius from the plane, the regression equation needs to be translated in the direction of the plane normal by the length of the measuring head radius in the opposite direction; setting the radius of the measuring head as r, the constant term coefficient a of the plane equation after translation 3 Is composed of
Figure FDA0003911347480000051
Equation Z = a to obtain the calibration plane 0 X+a 1 Y+a 3 Z = aX + bY + C;
controlling the robot to make the center point of the tool contact with any five points on the plane in different postures, recording the coordinates of the tail end of the robot R P t1 , R P t2 , R P t3 , R P t4 , R P t5 And attitude
Figure FDA0003911347480000052
Tool center point P of robot in different poses tcp Can be expressed as
Figure FDA0003911347480000053
In the formula R P tcpi Is the tool center point P under the ith pose of the robot tcp A representation in the coordinate system of the robot,
Figure FDA0003911347480000054
the posture of the robot at the ith time, E P tcpi tool center point P tcp A representation in the robot end coordinate system, R P ei representing the tail end of the robot in the coordinate system of the robot in the ith pose;
tool center point coordinate under different poses R P tcpi Satisfies the following equation
Figure FDA0003911347480000055
Substituting five different poses and equation (15) into an available equation set
Figure FDA0003911347480000056
To find E P tcp Of the best least squares solution
Figure FDA0003911347480000057
CN202211323218.1A 2022-10-27 2022-10-27 Robot compliant cutter calibration method based on articulated measuring arm Pending CN115597534A (en)

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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116147521A (en) * 2023-04-18 2023-05-23 菲特（天津）检测技术有限公司 Non-contact workpiece size measuring device and method

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116147521A (en) * 2023-04-18 2023-05-23 菲特（天津）检测技术有限公司 Non-contact workpiece size measuring device and method
CN116147521B (en) * 2023-04-18 2023-07-07 菲特（天津）检测技术有限公司 Non-contact workpiece size measuring device and method

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