CN115062482B - Kinematic modeling method of serial robots considering positioning errors of rotary joints - Google Patents
Kinematic modeling method of serial robots considering positioning errors of rotary joints Download PDFInfo
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Abstract
The invention discloses a kinematic modeling method of a serial robot considering a rotary joint positioning error, which comprises the following steps: according to the initial posture of the serial robots, a DH modeling theory is adopted to establish a measurement model of the serial robots; substituting the spherical center coordinate vector of the end effector of the serial robot into the measurement model; substituting the correction value of the collected angle error of the circular grating angle sensor and the length error value of the robot rod piece into the measurement model to perform error compensation; in the error-compensated measurement model, the angular positioning deviation is modeled as a function of the commanded angular position, instead of the correction value of the angular error, obtaining a first kinematic model. The method can improve modeling efficiency and calculation and simultaneously reduce the influence of the angular positioning error of the rotating shaft caused by the eccentric error of the angle sensor, the performance of the speed reducer and other factors.
Description
Technical Field
The invention relates to the technical field of robot modeling and robot kinematics calibration, in particular to a serial robot kinematics modeling method considering a rotary joint positioning error.
Background
In order to improve the absolute positioning accuracy of industrial robots, most of the research is mainly focused on DH parameter identification in robot kinematics. The DH parameter represents the position and orientation error of the axis average line of each rotation axis relative to the local coordinate system defined by the other axis. Many researchers have proposed a simple and cost-effective approach to identify DH parameters by measuring end effector position (e.g., parameter identification when the end is nominally constrained to a single point, straight line, planar or spherical surface). But the rotating shaft can generate radial, axial displacement or inclination error motion during rotation. Definition of error movement according to rotation axis in ISO 230-1: the change in position and direction of the rotating shaft relative to its axis average line is expressed as a function of the rotating shaft rotation angle. The angular positioning error of the rotary joint is considered as a main error source affecting the positioning accuracy of the tandem robots and has an amplifying effect on the influence of the positioning error of the tandem robots.
Some past studies report that even if the effect of DH parameters is compensated, the positioning error of the robot is still about 10 to 100 times that of a typical machine tool. In order to further improve the positioning accuracy of the serial robots, the non-structural parameter errors of the serial robots are compensated by adopting methods such as a neural network, and the accuracy is relatively stable only in a training data space. At present, the influence of the rotational joint angle positioning error on the positioning precision of the serial robots is not considered yet.
Therefore, how to further improve the positioning accuracy of the serial robots becomes a problem to be solved by the practitioners of the same person.
Disclosure of Invention
The invention aims to provide a kinematic calibration method of a serial robot, which takes the positioning error of a rotary joint into consideration, and the method further takes the influence of the angular positioning error of the rotary joint on the positioning precision of the serial robot into consideration on the basis of a DH model, so that the positioning precision of the serial robot is improved.
In order to achieve the above purpose, the technical scheme adopted by the invention is as follows:
the invention provides a kinematic modeling method of a serial robot considering a rotary joint positioning error, which comprises the following steps:
According to the initial posture of the serial robots, a DH modeling theory is adopted to establish a measurement model of the serial robots; the nominal value of the turning angle of each rotary joint of the serial robots in the initial gesture is zero;
Substituting the spherical center coordinate vector of the end effector of the serial robot into the measurement model; the measurement model includes: each joint variable value, rod length value, rod torsion angle value and rod offset value;
Substituting the correction value of the collected angle error of the circular grating angle sensor and the length error value of the robot rod piece into the measurement model to perform error compensation;
Modeling the angular positioning deviation as a function of a command angular position in the error-compensated measurement model, and obtaining a first kinematic model by replacing the correction value of the angular error; the angular positioning deviation is as follows: deviation caused by the command angular position of the rotation shaft and the rotation direction of the rotation shaft.
Further, in the first kinematic model, modeling a rotation axis radial error motion as a function of a command angular position, compensating an error value of the robot rod length, and obtaining a second kinematic model; the radial error motion of the rotating shaft is as follows: errors caused by roundness errors of the bearing strip, movement of the bearing balls and the mating rotational gap.
Further, a DH modeling theory is adopted to build a measurement model of the serial robot, which is as follows:
Wherein:
Wherein: θ i represents a joint variable, that is, a variation in relative position between adjacent rod members; d i represents the offset of rod i, i.e., the distance taken by the lengths l i-1 and l i of the fingers on the axis of joint i; l i denotes the rod length, i.e. the shortest distance between adjacent rotational joint axes; Alpha i represents the torsion angle of the rod, that is, any axis of the same rod moves towards the other axis to make the two lines intersect, the two lines determine a plane perpendicular to the length l i of the rod, and the plane included angle of the two lines is the matrix that the torsion angle alpha i;Ai of the rod represents the transformation from the coordinate system { i-1} of the rod i-1 to the coordinate system { i } of the rod i. Rot (z i-1,θi) represents the z-axis rotation θ i about the coordinate system of rod i-1; trans (0, d i) represents the z-axis translation d i distance along the coordinate system of rod i; Trans (l i, 0) represents the coordinate system x-axis translation l i distance along rod i; rot (x i,αi) represents an angle of rotation a i about the x-axis of the coordinate system of bar i.
Further, substituting the serial robot end effector sphere center coordinate vector into the measurement model includes:
Assuming that an end effector sphere center coordinate vector P= [ x p,yp,zp]T ] represents the position of a detection sphere center in a serial robot world coordinate system;
Substituting the homogeneous transformation matrix a i into the formula (1), the measurement model of the tandem robot is expressed as:
where x p,yp,zp represents the end effector sphere center coordinate vector.
Further, substituting the correction value of the collected angle error of the circular grating angle sensor and the error value of the length of the robot rod into a formula (3) to perform error compensation:
Where Δθ i represents the initial zero or phase angle and Δl i represents the error value of the robot lever length.
Further, the angular positioning deviation adoptsA representation; /(I)Indicating the commanded angular position of the i-th rotation axis, p i is the i-th rotation axis index commanded angular position number,/>Indicating the rotation direction of the i-th rotation axis at the commanded angular position,/>An angular velocity of the i-th rotation shaft at the command angular position;
When (when) Time,/>Assigned 1; when/>Time,/>Assigned a value of-1.
Further, θ i in the first kinematic model is replaced by Θ i (k); Θ i (k) represents the estimated angular position of the i-th rotation axis when the command angle of the i-th rotation axis is arbitrarily given asWhen the estimated angular position is calculated by linear interpolation:
Wherein, p i needs to satisfy:
delta i (k) is the interpolation weight, given by equation (7):
(6) In the formula (7): Representing any given commanded angular position of the ith rotation axis,/> Indicating the commanded angular position of the i-th rotation axis, p i is the i-th rotation axis index commanded angular position number,/>Representing the next commanded angular position of the ith rotation axis relative to the current commanded angle.
Further, the angular positioning deviation of each rotation axis is obtained by indirectly estimating the three-dimensional position of the end effector by a laser tracker;
wherein: when measuring the shaft with shorter length of the rod piece, the axis of the shaft is prolonged, and the rotation radius is increased;
Indexing the corresponding rotating shafts, and acquiring a plurality of command angle positioning stopping points on a track formed by taking the corresponding rotating shafts as circle centers and rotating radiuses;
And at each command angle positioning stopping point, measuring the three-dimensional position of the end effector by using a laser tracker to obtain the angle positioning deviation of the corresponding rotating shaft.
Further, the radial error motion of the rotating shaft is adoptedA representation; wherein/>Indicating the rotation direction of the i-th rotation axis,/>The angular velocity of the i-th rotation axis at the commanded angular position is indicated.
Further, the rotating shaft moves in radial errorFrom the linear interpolation calculation:
in the formula (9): p i is the i-th rotation axis index command angular position number, which is required to satisfy the requirement of formula (6), and delta i (k) is given by formula (7).
Compared with the prior art, the invention has the following beneficial effects:
According to the serial robot kinematic modeling method considering the rotary joint positioning error, static modeling of robot kinematics is considered, feedforward correction of a command track is used, DH parameters are used in the model, positioning errors of each rotary shaft angle are considered, and the serial robot positioning accuracy can be predicted accurately. The modeling efficiency and calculation are improved, and meanwhile, the influence of the angular positioning error of the rotating shaft, which is usually caused by the eccentric error of the angle sensor, the performance of the speed reducer and other factors, can be reduced.
Drawings
FIG. 1 is a flow chart of a serial robot kinematic calibration method taking into account rotational joint positioning errors according to an embodiment of the present invention;
Fig. 2 is a DH modeling schematic diagram based on a serial robot kinematic modeling method according to an embodiment of the present invention;
fig. 3 is a schematic diagram of measurement of a sixth axis angular deviation based on a kinematic modeling method of a tandem robot according to an embodiment of the present invention;
Fig. 4 is a schematic diagram of measurement of a first axis angle deviation based on a kinematic modeling method of a tandem robot according to an embodiment of the present invention.
Detailed Description
The invention is further described in connection with the following detailed description, in order to make the technical means, the creation characteristics, the achievement of the purpose and the effect of the invention easy to understand.
In the description of the present invention, it should be noted that the directions or positional relationships indicated by the terms "upper", "lower", "inner", "outer", "front", "rear", "both ends", "one end", "the other end", etc. are based on the directions or positional relationships shown in the drawings, are merely for convenience of describing the present invention and simplifying the description, and do not indicate or imply that the devices or elements referred to must have a specific direction, be configured and operated in the specific direction, and thus should not 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.
In the description of the present invention, it should be noted that, unless explicitly specified and limited otherwise, the terms "mounted," "provided," "connected," and the like are to be construed broadly, and may be fixedly connected, detachably connected, or integrally connected, for example; can be mechanically or electrically connected; can be directly connected or indirectly connected through an intermediate medium, and can be communication between two elements. The specific meaning of the above terms in the present invention will be understood in specific cases by those of ordinary skill in the art.
In order to achieve the aim of the invention, the invention considers static modeling of robot kinematics for feedforward correction of instruction trajectories. The model also accounts for each rotational axis angular positioning error in addition to using DH parameters. Compared to angular positioning error measurements of machine tools, angular positioning error movements of rotating tables in machine tools are typically measured by using an autocollimator or reference index table with a reference polygon, which is difficult for robots because the reference polygon or reference index table must be placed on the rotating shaft, requiring the use of a special fixture. The present invention therefore proposes the use of a laser tracker to indirectly estimate the angular positioning error of an end effector position measurement.
Referring to fig. 1, the kinematic modeling method of the tandem robot, which is provided by the invention and takes the positioning error of a rotary joint into consideration, comprises the following steps:
S10, establishing a measurement model of the serial robots by adopting a DH modeling theory according to the initial gesture of the serial robots; the nominal value of the turning angle of each rotary joint of the serial robots in the initial gesture is zero;
S20, substituting the spherical center coordinate vector of the end effector of the serial robot into the measurement model; the measurement model includes: each joint variable value, rod length value, rod torsion angle value and rod offset value;
S30, substituting the correction value of the acquired angle error of the circular grating angle sensor and the length error value of the robot rod piece into the measurement model to perform error compensation;
S40, modeling the angular positioning deviation as a function of a command angular position in the error compensated measurement model to replace a correction value of the angular error to obtain a first kinematic model; the angular positioning deviation is as follows: deviation caused by the command angular position of the rotation shaft and the rotation direction of the rotation shaft.
The angular positioning error of the rotating shaft is generally caused by the eccentric error of the angle sensor, the performance of the speed reducer, and the like. To reduce its impact, the angular positioning error is modeled in the proposed model as a function of the commanded angular position. In addition, since the positioning accuracy of the rotating shaft of the robot is often greatly affected by the eccentric error of the angle sensor, the invention designates different angular positioning deviations for different rotating directions, thereby being beneficial to improving the positioning accuracy of the serial robots.
To further increase its positioning accuracy, referring to fig. 1, the method further includes:
s50, modeling radial error motion of a rotating shaft as a function of a command angle position in a first kinematic model, and compensating the length error value of the robot rod piece to obtain a second kinematic model; the radial error motion of the rotating shaft is as follows: errors caused by roundness errors of the bearing strip, movement of the bearing balls and the mating rotational gap.
The same main cause of radial error movement for the rotating shaft is typically roundness error of the bearing band, and movement of the bearing ball and mating rotating gap results in radial error movement. When the disturbance depends on the angular position, such as the influence of gravity, which varies with the position of the rotation angle, it results in a radial error movement which, like the angular positioning error movement, is also generally a function of the angular position, so that the radial error is also modeled as a function of the commanded angular position. Compared with the traditional DH model, the kinematic model constructed by the invention further corrects the bidirectional error of the rotation angle of the DH model so as to improve the positioning accuracy of the robot without increasing the hardware cost.
The following describes the steps in detail using a6 degree of freedom serial robot as an example with reference to the accompanying drawings:
When building its measurement model, its initial position needs to be given as shown in fig. 2. In this state, the nominal value of the rotation angle of each rotary joint of the tandem robot is zero. The invention selects the vertical posture of the serial robots as the initial posture, and adopts DH modeling theory to establish the measurement model of the serial robots as follows:
Wherein:
/>
Wherein: θ i represents the joint variable, i.e., the amount of change in relative position between adjacent rods. d i represents the offset of bar i, i.e., the distance taken by finger lengths l i-1 and l i on the axis of joint i. l i denotes the rod length, i.e. the shortest distance between adjacent rotational joint axes. Alpha i represents the torsion angle of the rod, that is, any axis of the same rod moves towards the other axis to make the two lines intersect, the two lines determine a plane perpendicular to the length l i of the rod, and the plane included angle of the two lines is the matrix that the torsion angle alpha i.Ai of the rod represents the transformation from the coordinate system { i-1} of the rod i-1 to the coordinate system { i } of the rod i. Rot (z i-1,θi) represents the z-axis rotation θ i about the coordinate system of rod i-1. Trans (0, d i) represents the z-axis translation d i distance along the coordinate system of rod i. Trans (l i, 0) represents the x-axis translation l i distance along the coordinate system of rod i. Rot (x i,αi) represents an angle of rotation a i about the x-axis of the coordinate system of bar i.
Assume that the end effector sphere center coordinate vector p= [ x p,yp,zp]T ] represents the position of the probe sphere center within the tandem robot world coordinate system. Substituting the homogeneous transformation matrix a i into the formula (1), the measurement model of the tandem robot is expressed as:
where x p,yp,zp represents the end effector sphere center coordinate vector.
Since the physical zero position of the circular grating angle sensor is inconsistent with the zero position of the angle when the serial robot is assembled and the initial pose is modeled, the zero position deviation is called zero position deviation, the value needs to be corrected to zero, the correction value delta theta i can be called initial zero position or phase angle, and the length error of the ith shaft component of the robot is delta l i.
Then equation (3) may be rewritten as:
Where Δθ i represents the initial zero or phase angle, Δl i represents the robot i-th shaft length error.
The angular positioning error of the rotating shaft is generally caused by the eccentric error of the angle sensor, the performance of the speed reducer, and the like. In order to reduce its impact, in the proposed model, the angular positioning deviation is modeled as a function of the commanded angular position. In addition, since the rotational axis positioning accuracy of the robot is often significantly affected by the control positioning angle, different angular positioning deviations should be specified for different rotational directions.
For angular positioning deviationsRepresentation of/>Indicating the commanded angular position of the i-th rotation axis, p i is the i-th rotation axis index commanded angular position number,/>Indicating the rotation direction of the i-th rotation axis at the commanded angular position,/>The angular velocity of the i-th rotation axis at the commanded angular position is indicated. Wherein, when/>(Or < 0)/(Assigned +1 (or-1). When the i-th axial angular position is defined by/>At any given time, the established kinematic model is given by equation (4). Wherein θ i is replaced by Θ i (k), Θ i (k) is the estimated angular position of the i-th axis, calculated by linear interpolation:
Wherein p i must satisfy:
delta i (k) is the interpolation weight, given by equation (7):
(6) In the formula (7): Representing any given commanded angular position of the ith rotation axis,/> Indicating the commanded angular position of the ith rotation axis,/>Representing the next commanded angular position of the ith rotation axis relative to the current commanded angle.
As shown in fig. 3 and 4, a measurement is made of each axis rotational positioning deviation of the tandem robot 1. The first, second and third shafts can be directly measured, the fourth, fifth and sixth shafts are relatively short, and in order to improve the angular positioning accuracy of measuring the rotation of the fourth, fifth and sixth shafts, the fourth, fifth and sixth shafts of the serial robot are respectively connected and prolonged when the angular positioning error of the rotating shaft is measured. Taking the sixth axis as an example, when measuring the angular positioning error of the rotating shaft, the axis switching is extended as shown in fig. 3. By adding the adapter rod 5, the axis of the sixth shaft is prolonged, so that in measuring the rotational angle positioning error of the shaft, the radius of rotation is increased, thereby improving the measurement accuracy of the angular positioning error. The optimal positions of other shafts are kept, the sixth shaft of the tandem robot is indexed, the tandem robot moves bidirectionally along the track of the line 3, the solid black points 4 on the line 3 are command angle positioning stopping points, the positioning of the command angles 6 is guaranteed to be bidirectional, and the rotating range covers all the rotating space of the sixth shaft of the tandem robot. At each stopping point 4, the three-dimensional position of the end effector is measured using the laser tracker 2. The rotation angle positioning error measurement of the fourth and fifth axes of the tandem robot is similar to that of the 6 th axis, and will not be repeated.
In consideration of the influence of the self weight of the shaft on the measurement result, the invention selects the carbon fiber tube as the material of the switching rotating shaft. The density of the carbon fiber tube is mainly influenced by the constituent substances as the tubes made of other materials, the density of the carbon fiber tube is about 1.7g/cm 3, the density of the steel tube is 7.8g/cm 3, and the mass ratio of the carbon fiber tube to the steel tube with the same strength is 1:43. Therefore, the density and the quality of the carbon fiber tube are far smaller than those of the steel tube on the premise of the same strength. Besides the outstanding advantages of light weight and high strength, the carbon fiber is the most different from the metal material in that the carbon fiber belongs to a nonmetallic material, has lower electrochemical activity and very strong corrosion resistance and ageing resistance, and can also prolong the service life of the carbon fiber tube. The carbon fiber material has small thermal expansion coefficient, basically can not deform along with the change of working temperature, and ensures the stability of dimension measurement.
Fig. 4 shows the same test setup, maintaining the optimal position of the other axes, indexing the first axis of the tandem robot 1, bi-directionally moving along the line 8 trajectory, solid black dots 7 on the line 8 being command angle positioning stop points, ensuring that the command angle 9 positioning is bi-directional, and the rotational range should cover all the space where the tandem robot first axis rotates. At each stopping point 7, the three-dimensional position of the end effector is measured using the laser tracker 2. The rotational angle positioning error measurement of the second and third axes is similar to that of the first axis and will not be described again.
The same main cause of radial error movement for the rotating shaft is typically roundness error of the bearing band, and movement of the bearing ball and mating rotating gap results in radial error movement. When the disturbance depends on the angular position, such as the influence of gravity, which varies with the position of the rotation angle, it results in a radial error movement which, like the angular positioning error movement, is also generally a function of the angular position, so that the radial error is also modeled as a function of the commanded angular position.
The effects of such radial error movements can also be incorporated into the proposed model byMeaning that, similar to angular positioning error motion, radial error motion is modeled as a function of commanded angular position and direction of rotation. With this radial error motion, the kinematic model of the tandem robot is represented by equation (4), Δl i by/> Instead of this.
Can be obtained by linear interpolation:
in the formula (9): p i is the i-th rotation axis index command angular position number, which is required to satisfy the requirement of formula (6), and delta i (k) is given by formula (7).
According to the serial robot kinematic modeling method considering the rotary joint positioning error, static modeling of robot kinematics is considered, feedforward correction of a command track is used, DH parameters are used, positioning errors of each rotary shaft angle and radial error motions of the rotary shaft are considered, and the serial robot positioning accuracy can be predicted accurately. The modeling efficiency and calculation are improved, and meanwhile, the influence of the angular positioning error of the rotating shaft, which is usually caused by the eccentric error of the angle sensor, the performance of the speed reducer and other factors, can be reduced.
It will be appreciated by those skilled in the art that embodiments of the present invention may be provided as a method, system, or computer program product. Accordingly, the present invention may take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects. Furthermore, the present invention may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, magnetic disk storage, optical storage, and the like) having computer-usable program code embodied therein.
The present invention is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems) and computer program products according to embodiments of the invention. It will be understood that each flow and/or block of the flowchart illustrations and/or block diagrams, and combinations of flows and/or blocks in the flowchart illustrations and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks. These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
It will be apparent to those skilled in the art that various modifications and variations can be made to the present invention without departing from the spirit or scope of the invention. Thus, it is intended that the present invention also include such modifications and alterations insofar as they come within the scope of the appended claims or the equivalents thereof.
Claims (10)
1. A kinematic modeling method of a tandem robot considering a rotational joint positioning error, the method comprising:
According to the initial posture of the serial robots, a DH modeling theory is adopted to establish a measurement model of the serial robots; the nominal value of the turning angle of each rotary joint of the serial robots in the initial gesture is zero;
Substituting the spherical center coordinate vector of the end effector of the serial robot into the measurement model; the measurement model includes: each joint variable value, rod length value, rod torsion angle value and rod offset value;
Substituting the correction value of the collected angle error of the circular grating angle sensor and the length error value of the robot rod piece into the measurement model to perform error compensation;
Modeling the angular positioning deviation as a function of a command angular position in the error-compensated measurement model, and obtaining a first kinematic model by replacing the correction value of the angular error; the angular positioning deviation is as follows: deviation caused by the command angular position of the rotation shaft and the rotation direction of the rotation shaft.
2. The method according to claim 1, characterized in that in the first kinematic model, the rotation axis radial error motion is modeled as a function of the commanded angular position, compensating the error value of the robot lever length, obtaining a second kinematic model; the radial error motion of the rotating shaft is as follows: errors caused by roundness errors of the bearing strip, movement of the bearing balls and the mating rotational gap.
3. The method of claim 2, wherein the modeling of the series robot using DH modeling theory is:
Wherein:
Wherein: θ i represents a joint variable, that is, a variation in relative position between adjacent rod members; d i represents the offset of rod i, i.e., the distance taken by the lengths l i-1 and l i of the fingers on the axis of joint i; l i denotes the rod length, i.e. the shortest distance between adjacent rotational joint axes; Alpha i represents the torsion angle of the rod, namely, any axis of the same rod moves towards the other axis to be intersected, the two straight lines determine a plane perpendicular to the length l i of the rod, and the plane included angle of the two straight lines is the matrix that the torsion angle alpha i;Ai of the rod represents the conversion from the coordinate system { i-1} of the rod i-1 to the coordinate system { i } of the rod i; Rot (z i-1,θi) represents the z-axis rotation θ i about the coordinate system of rod i-1; trans (0, d i) represents the z-axis translation d i distance along the coordinate system of rod i; Trans (l i, 0) represents the coordinate system x-axis translation l i distance along rod i; rot (x i,αi) represents an angle of rotation a i about the x-axis of the coordinate system of bar i.
4. The method of claim 3, wherein substituting the serial robotic end effector sphere center coordinate vector into the measurement model comprises:
Assuming that an end effector sphere center coordinate vector P= [ x p,yp,zp]T ] represents the position of a detection sphere center in a serial robot world coordinate system;
Substituting the homogeneous transformation matrix a i into the formula (1), the measurement model of the tandem robot is expressed as:
where x p,yp,zp represents the end effector sphere center coordinate vector.
5. The method of claim 4, wherein the correction value of the collected angle error of the circular grating angle sensor and the error value of the length of the robot bar are substituted into formula (3) for error compensation:
where Δθ i represents the initial zero or phase angle, Δl i represents the robot ith bar length error value.
6. The method according to claim 5, wherein the angular positioning deviation is usedA representation; /(I)Indicating the command angular position of the i-th rotation axis, p i is the i-th rotation axis index command angular position number,Indicating the rotation direction of the i-th rotation axis at the commanded angular position,/>An angular velocity of the i-th rotation shaft at the command angular position;
When (when) Time,/>Assigned 1; when/>Time,/>Assigned a value of-1.
7. The method of claim 6, wherein Θ i in the first kinematic model is replaced by Θ i (k); Θ i (k) represents the estimated angular position of the i-th rotation axis when the command angle of the i-th rotation axis is arbitrarily given asWhen the estimated angular position is calculated by linear interpolation:
Wherein, p i needs to satisfy:
delta i (k) is the interpolation weight, given by equation (7):
(6) In the formula (7): Representing any given commanded angular position of the ith rotation axis,/> Indicating the commanded angular position of the i-th rotation axis, p i is the i-th rotation axis index commanded angular position number,/>Representing the next commanded angular position of the ith rotation axis relative to the current commanded angle.
8. The method of claim 7, wherein the angular positioning bias for each axis of rotation is obtained by indirectly estimating a three-dimensional position of an end effector by a laser tracker;
wherein: when measuring the shaft with shorter length of the rod piece, the axis of the shaft is prolonged, and the rotation radius is increased;
Indexing the corresponding rotating shafts, and acquiring a plurality of command angle positioning stopping points on a track formed by taking the corresponding rotating shafts as circle centers and rotating radiuses;
And at each command angle positioning stopping point, measuring the three-dimensional position of the end effector by using a laser tracker to obtain the angle positioning deviation of the corresponding rotating shaft.
9. The method of claim 7, wherein the rotating shaft radial error motion employsA representation; /(I)Indicating the rotation direction of the i-th rotation axis,/>The angular velocity of the i-th rotation axis at the commanded angular position is indicated.
10. The method of claim 9, wherein the rotating shaft radial error motionFrom the linear interpolation calculation:
Wherein: p i is the i-th rotation axis index command angular position number, which is required to satisfy the requirement of formula (6), and delta i (k) is given by formula (7).
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