CN112873199B - Robot absolute positioning precision calibration method based on kinematics and spatial interpolation - Google Patents
Robot absolute positioning precision calibration method based on kinematics and spatial interpolation Download PDFInfo
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Abstract
The invention discloses a robot absolute positioning precision calibration method based on kinematics and spatial interpolation, belonging to the field of robot control methods; firstly, respectively establishing a geometric parameter error model and a flexibility error model of the robot to obtain a quantifiable calculation error delta V; then, measuring the actual terminal pose of the robot to obtain identification experiment data, performing quantitative calculation error delta V parameter identification on the robot by adopting an EKF algorithm, and correcting the name-meaning geometric parameters of the robot by using the robot error parameters obtained by identification, namely completing the first positioning error compensation of the robot and remaining the positioning residual errors of the robot; then constructing a robot positioning residual error model; finally, performing second positioning error compensation on the robot; and finally, obtaining the expected position of the robot. The invention establishes the variable node distance interpolation algorithm considering the influence degree of the joints, can effectively improve the absolute positioning precision of the robot, and overcomes the defects of the prior method and technology in precision.
Description
Technical Field
The invention belongs to the field of robot control methods, and particularly relates to a robot absolute positioning precision calibration method based on kinematics and spatial interpolation, in particular to an absolute positioning precision error compensation method for an industrial robot for machining such as grinding and milling.
Background
Industrial robot is mostly six serial connection in space mechanism, compares in digit control machine tool, and it has the processing flexibility height, with low costs and working space advantage such as big, consequently more and more is applied to the machining field, especially has the work piece of characteristics such as many varieties small batch and structure are great. However, the absolute positioning accuracy of the industrial robot is poor (generally greater than 0.1mm), so that the position control accuracy of the robot is low, and the tail end of the robot is difficult to accurately contact with a workpiece, so that the machining accuracy is low. Because of the absolute positioning precision, the industrial robot is mostly used for extensive operation such as stacking, carrying and the like and other rough machining processes with lower precision requirements at present, and the application of the industrial robot in the field of high-precision machining is greatly limited. Therefore, how to effectively improve the absolute positioning precision of the industrial robot has important theoretical significance and engineering application value for improving the processing quality and expanding the application range of the industrial robot.
In order to effectively improve the absolute positioning accuracy of the robot, a great deal of research is carried out by a plurality of scholars and engineers at home and abroad. The calibration method of the absolute positioning accuracy of the robot is mainly divided into two types according to the control mode: full closed-loop control and semi closed-loop control. The method is mainly characterized in that a detection device is additionally arranged at the tail end of the robot, the tail end position is obtained in real time through technologies such as visual identification and the like, the tail end position is fed back to a robot control system through an integration technology, the tail end position is compared with a theoretical position, and the tail end position is corrected according to real-time deviation data. The method does not pay attention to the source of positioning errors of the robot, is simple in principle and high in precision, but is high in application cost and complex in process, and cannot be easily implemented for parts with complex shapes and in industrial fields. The latter mainly carries out analysis and quantitative research aiming at the robot positioning error source, constructs a corresponding error identification model, and obtains the deviation of the measured data and the theoretical data through external measuring equipment, thereby identifying the error between the actual model and the theoretical model of the robot and achieving the purpose of compensating the absolute positioning accuracy. Because the application is simple and the cost is low, the method is widely researched and applied. However, the robot has numerous positioning error sources, including gear gaps, mass distribution of mechanical arms, load variation, thermal effect and other non-geometric parameters with strong nonlinearity and coupling characteristics, so that it is difficult to construct an accurate mapping model. Meanwhile, most researches do not consider the structural characteristics and the motion characteristics of the articulated robot, the pose error distribution characteristics of the robot and the action rule. Therefore, by combining the structure and the spatial motion characteristics of the robot and further performing targeted multiple error compensation according to the positioning error source, the absolute positioning accuracy of the robot can be further effectively improved, and the application range of the robot in the high-precision machining field is expanded.
Disclosure of Invention
The technical problem to be solved is as follows:
in order to avoid the defects of the prior art, the absolute positioning accuracy of the robot is effectively improved, the application range of the robot in the high-precision machining field is further expanded, the invention provides a robot absolute positioning accuracy calibration method based on kinematics and spatial interpolation, the geometric structure and the spatial motion characteristic of the robot are combined, and on the basis of comprehensively considering the positioning error generation mechanism and the spatial motion anisotropic characteristic of the robot, the mode of combining geometric parameter quantitative calculation and non-geometric error interpolation is adopted.
The technical scheme of the invention is as follows: a robot absolute positioning precision calibration method based on kinematics and spatial interpolation is characterized by comprising the following specific steps:
the method comprises the following steps: based on the geometric parameters of the robot, an MD-H kinematics method is adopted to construct a geometric parameter error model of the robot, which is expressed as follows:
ΔT1＝JIΔVg
in the formula,. DELTA.T1For robot end pose error, Δ V, caused by geometric parameter errorgRepresenting the geometric parameter error of the robot, i.e. Δ Vg＝(Δdg,Δθg,Δag,Δαg,Δβg) (ii) a The geometric parameters of the robot comprise joint distance d, joint rotation angle theta, connecting rod length a, rod torsion angle alpha and y-axis torsion angle beta; j is a unit ofITo identify the Jacobian matrix, the method is specifically expressed as:
wherein M isθ,Md,Ma,Mβ,MαIs a 3 XN order partial derivative matrix of the position of the tail end of the robot considering the error of the geometric parameters, Rθ,Rβ,RαThe method is a robot tail end attitude partial derivative matrix taking geometric parameter errors into consideration in the order of 3 multiplied by N;
step two: and (3) combining the structural characteristics of the robot and stress analysis to construct a robot flexibility error model, wherein the expression is as follows:
ΔT2＝JIΔVc
ΔVc＝(0,Δθc,0,0,0)
wherein, Delta T2For robot end pose error, Δ V, caused by compliance errorcRepresenting the compliance error, Δ θ, of the robotc＝(0,Δθc2,Δθc3,0,0,0)T，ΔθciRepresenting joint angle errors due to compliance deformation;
step three: the quantifiable calculation error delta V of the robot, namely the geometric parameter error and the joint flexibility error under the action of stress, is obtained by calculation in the first step and the second step; the expression is as follows:
ΔV＝(Δd,Δθ,Δa,Δα,Δβ)＝ΔVg+ΔVc；
further obtaining the actual position and attitude error delta P-delta T of the robot end1+ΔT2＝JIΔ V, introducing a measurement coordinate system transformation matrix' E in consideration of a coordinate transformation error caused by incomplete coincidence of the measurement coordinate system and the robot base coordinate system, and solving Δ P by the following formula:
ΔP＝ΔPm-(′E-E)Pt
wherein, Δ PmIndicating the pose error obtained by measurement, PtRepresenting the theoretical terminal pose of the robot, 'E' is a measurement coordinate system transformation matrix, and E is an identity matrix;
then substituting the delta P into a geometric parameter error model and a flexibility error model of the robot, and solving to obtain a quantifiable calculation error delta V of the robot;
step four: measuring the actual terminal pose of the robot to obtain identification experiment data; secondly, identifying quantifiable calculation error delta V parameters of the robot by adopting an EKF algorithm, and correcting name-defined geometric parameters of the robot by using the robot error parameters obtained by identification, namely completing the first positioning error compensation of the robot and remaining the positioning residual errors of the robot;
step five: constructing a robot positioning residual error model based on a spatial interpolation algorithm; the expression is as follows:
wherein d isΔkRepresenting the Euclidean distance between an end point and a vertex k, wherein the total number of the vertexes of the Cartesian space grid of the robot is 8, and k is 1 and 2 … 8; p is a radical ofkRepresenting the influence weight of each vertex k of the space grid on an endpoint;
step six: based on the robot positioning residual error model constructed in the fifth step, acquiring robot joint rotation quantity delta theta for compensating the tail end positioning error by utilizing the mapping relation between the tail end displacement of the robot and the joint corner, and compensating the robot joint rotation quantity delta theta to the name meaning joint rotation quantity of the robot, namely performing the second positioning error compensation on the robot; and finally obtaining the expected position of the robot.
The further technical scheme of the invention is as follows: in said step one, Δ d is definedi,Δθi,Δai,Δαi,ΔβiI is 1 and 2 … 6, which respectively represent the joint distance error, joint rotation angle error, connecting rod length error, rod torsion angle error and y-axis torsion angle error of the ith joint;
an MD-H kinematics model is adopted to construct a transformation matrix of adjacent connecting rods of the robot, and the expression is as follows:
wherein the content of the first and second substances,
a transformation matrix representing the coordinate system i-1 to the coordinate system i, di,θi,ai,αi,βiRespectively showing the joint distance, the joint angle, the length of a connecting rod, the torsion angle of a rod piece and the torsion angle of a y axis of the ith joint;
the actual position of the robot end effector center point under the base coordinate system is expressed as:
wherein the content of the first and second substances,
a position coordinate transformation matrix from the ith joint to the (i-1) th joint under the geometric parameter error;
an error coordinate matrix from the ith joint to the (i-1) th joint;
will be provided with
Simplified to a simple linear equation:
based on the formulas (2) and (3), high-order micro quantity is further saved, and robot tail end pose error delta T caused by geometric parameter error is obtained1And finally, expressing the established robot geometric parameter error model as follows:
ΔT1＝JIΔVg (4)
the further technical scheme of the invention is as follows: in the second step, the robot is structurally characterized in that the connecting rod is a rigid body, so that the robot flexibility deformation only considers the joint deformation, and the joint deformation amount calculation method comprises the following steps:
Δθci＝Cτ (5)
wherein, Delta thetaciThe method comprises the steps that a joint corner error generated due to flexibility deformation is shown, C shows a flexibility coefficient of a joint angle, the flexibility coefficient is a constant value for a fixed joint angle, and tau represents a moment acting on the joint;
the robot joint moment comprises self gravity and processing force; the total stress on the joint i is thus obtained as:
τi＝τGi+τFi (6)
wherein, tauGiRepresenting the moment, tau, experienced by the joint i under the action of gravityFiThe moment borne by the joint i under the action of the machining force is shown;
the flexibility coefficient C is a node rigidity coefficient KqReciprocal of (a), KqThe solving formula of (2) is as follows:
F＝J-TKqJ-1Δxt (7)
wherein F is the end stress of the robot, and delta xtFor robot end displacement, KqIs a 6 x 6 diagonal stiffness matrix; by KqThe compliance coefficient C of the joint angle can be obtained by solving;
further substituting the total stress moment of the joint and the flexibility coefficient C into the formula (5) to obtain the joint rotation angle error delta theta generated by the flexibility deformationci。
The further technical scheme of the invention is as follows: in the third step, the quantifiable calculation error delta V of the robot is obtained by calculation in the first step and the second step, namely the geometric parameter error and the joint flexibility error under the stress action; the expression is as follows:
ΔV＝(Δd,Δθ,Δa,Δα,Δβ)＝ΔVg+ΔVc； (8)
the coordinate conversion error caused by incomplete coincidence of the measurement coordinate system and the robot base coordinate system needs to be considered, and a measurement coordinate system conversion matrix is introduced to describe the conversion relation of the two coordinate systems, and the conversion relation is expressed as follows:
wherein r isx,ry,rzRespectively representing the minimum rotation errors of the base coordinate system relative to the three axes of the measuring coordinate system under the actual condition of the industrial robot; dx,dy,dzRespectively representing the minimal movement error of the base coordinate system relative to the origin of the measuring coordinate system under the actual condition of the industrial robotA difference;
then establishing a robot terminal pose error delta P considering the measurement errormAnd (3) the correlation relation with the actual robot end pose error delta P:
ΔPm＝Pm-Pt＝′EP′-Pt＝′E(Pt+ΔP)-Pt＝(′E-E)Pt+ΔP+(′E-E)ΔP (10)
in the formula,. DELTA.PmRepresenting the pose error obtained by measurement, PmRepresents the actual pose, PtRepresenting the theoretical terminal pose of the robot, ' E ' is a transformation matrix of a measuring coordinate system, E is an identity matrix, and (' E-E) delta P is ignored as high-order micro quantity; thereby obtain the terminal position appearance actual error of robot:
ΔP＝ΔPm-(′E-E)Pt＝JIΔV (11)
the invention further adopts the technical scheme that: and in the fourth step, a pose coordinate measuring instrument is adopted to measure the actual terminal pose coordinate information of the robot, and the pose coordinate measuring instrument is an API laser tracker.
The invention further adopts the technical scheme that: the weight p in the step fivekIs solved as follows:
according to the error similarity principle, the reciprocal of the distance between two positions is used as a weight, the closer the distance, the larger the weight, the farther the distance, the smaller the weight, therefore, the inverse distance weighting method IDSW is adopted to solve the weight,
wherein r represents an exponential weight, generally taking the value of 1 or 2.
The further technical scheme of the invention is as follows: in the sixth step, a robot joint rotation amount Δ θ formula for compensating the end positioning error is as follows:
Δθ＝J(θ)-1Δd (13)
wherein, J (theta)-1Represents the robot Jacobian matrix J (theta)) Of inverse matrix, Δ d, i.e
Since the displacement is to be expressed, it is expressed by Δ d.
Advantageous effects
The invention has the beneficial effects that:
1. the method of the invention starts with three different main error sources of an industrial robot, and provides a robot absolute positioning precision calibration method based on kinematics and a spatial interpolation algorithm aiming at error source characteristics: aiming at the quantitative calculation errors of the robot, namely the geometric parameters and the flexibility errors, the coupling effect of the geometric parameters and the flexibility errors and the coordinate system conversion errors brought by the measurement process are considered, and an error model based on kinematics is constructed. Random noise influence is fully considered, an EKF algorithm is adopted for error parameter identification, and identification precision is improved; further aiming at the residual error of the robot, based on the spatial error similarity principle, the structure and the motion characteristics of the robot are combined in a key mode, the influence difference of the motion performance space opposite characteristics of the robot and different joints on the tail end position is comprehensively considered, a variable node distance interpolation algorithm considering the influence degree of the joints is established, the absolute positioning precision of the robot can be effectively improved, and the defects of the existing method and the existing technology in precision are overcome.
2. Based on the algorithm, the KUKA KR210 robot is used for experimental verification of a research object, and the absolute positioning accuracy of the robot is improved from 1.658mm to 0.106 mm. The method provided by the invention can effectively improve the absolute positioning accuracy of the robot and expand the application range of the robot.
Drawings
FIG. 1 is a general control flow chart for absolute positioning accuracy calibration of a construction robot according to the method of the present invention;
FIG. 2 is a schematic diagram of geometric parameters for modeling parameter errors of a construction robot implemented by the method of the present invention;
FIG. 3 is a schematic diagram of a flexibility error modeling of a construction robot implemented by the method of the invention;
FIG. 4 is a schematic diagram of the method for analyzing the distribution rule of the flexibility error of the industrial robot;
fig. 5 is a schematic diagram of the method of the invention for analyzing the spatial motion characteristics of an industrial robot.
Detailed Description
The embodiments described below with reference to the drawings are illustrative and intended to be illustrative of the invention and are not to be construed as limiting the invention.
Refer to fig. 1 to 5. The invention takes KUKA KR210 industrial robot as an example, and adopts an API T3 laser tracker as a measuring tool to measure the tail end position of the robot, so as to explain the specific steps of geometric parameter error modeling, flexibility error modeling, error parameter identification, residual error modeling and positioning error compensation of the robot absolute positioning precision calibration method based on kinematics and spatial interpolation algorithm.
S1, constructing a robot geometric parameter error model based on the MD-H kinematics method;
s2, constructing a robot flexibility error model by combining the structural characteristics and the stress analysis of the robot;
s3, obtaining a quantifiable calculation error of the robot based on the established geometric parameter error and flexibility error model of the robot, and further obtaining an actual error model of the terminal pose of the robot under consideration of the measurement error;
s4, measuring the terminal pose of the robot to obtain identification experiment data; the robot quantifiable calculation error parameter identification is carried out based on an EKF algorithm, so that the robot error parameter is obtained and the first positioning error compensation is carried out;
s5, constructing a robot positioning residual error model based on a spatial interpolation algorithm;
and S6, based on the established robot positioning residual error model, performing secondary robot positioning error compensation by using the mapping relation between the terminal micro displacement of the robot and the joint rotation angle.
The positioning precision calibration of the industrial robot respectively compensates geometric parameter errors, flexibility errors, measurement errors and residual errors.
The method comprises the following steps: and (5) geometric parameter error modeling.
And constructing a transformation matrix of adjacent connecting rods of the robot by adopting an MD-H kinematic model. The geometrical parameters of the robot mainly comprise joint distance d, joint rotation angle theta, connecting rod length a, rod torsion angle alpha and y-axis torsion angle beta. The introduction of the y-axis torsion angle beta is to prevent the singular phenomenon that when the coordinate systems of two adjacent connecting rods of the robot are in parallel positions, the small deflection of the joint axis can cause the great change of the joint distance. Therefore, the adjacent link transformation matrix based on the MD-H kinematic model can be expressed as:
wherein:
a transformation matrix representing the coordinate system i-1 to the coordinate system i, di,θi,ai,αi,βiRespectively shows the joint distance, the joint rotation angle, the connecting rod length, the rod torsion angle and the y-axis torsion angle of the ith joint.
Considering that a real robot generates certain processing errors, namely geometric parameter errors, defined as deltad in the manufacturing and assembling processi,Δθi,Δai,Δαi,ΔβiAnd respectively representing the joint distance error, the joint rotation angle error, the connecting rod length error, the rod piece torsion angle error and the y-axis torsion angle error of the ith joint. The actual position of the robot end effector center point under the base coordinate system can therefore be expressed as:
wherein
A position coordinate transformation matrix from the ith joint to the (i-1) th joint under the geometric parameter error;
as the moment of error coordinates of the i-th joint to the i-1 th jointAnd (5) arraying.
Will be provided with
Simplified to a simple linear equation:
based on the formulas (2) and (3), high-order micro-quantity is further saved, and the robot tail end pose error delta T caused by geometric parameter errors can be obtained1And finally, expressing the established geometric parameter error model of the robot as follows:
ΔT1＝JIΔVg (4)
wherein, is Δ VgRepresenting the errors of the geometrical parameters of the robot, i.e. Δ Vg＝(Δdg,Δθg,Δag,Δαg,Δβg)；JITo identify the Jacobian matrix, it can be specifically expressed as:
wherein, Mθ,Md,Ma,Mβ,MαIs a 3 XN order partial derivative matrix of the position of the tail end of the robot considering the error of the geometric parameters, Rθ,Rβ,RαThe matrix is a robot terminal attitude partial derivative matrix with 3 XN order and geometric parameter errors considered.
FIG. 2 is a schematic diagram of geometric parameters for modeling parameter errors of a construction robot implemented by the method of the present invention;
table 1 shows the parameters of the MD-H model of the KUKA KR210 industrial robot;
TABLE 1
Step two: modeling the flexibility error of the robot:
firstly, setting (1) a robot connecting rod as a rigid body, and neglecting the flexibility deformation of the rigid body; (2) under the condition of small deformation, the flexible deformation of the robot joint and the moment applied by the robot joint accord with a linear proportional relation. Therefore, the robot flexibility deformation only considers the joint deformation, and the joint deformation amount calculation method comprises the following steps:
Δθci＝Cτ (6)
wherein Δ θciThe angle error of the joint corner caused by the flexibility deformation is shown, C represents the flexibility coefficient of the joint angle, the flexibility coefficient is a constant value for a fixed joint angle, and tau represents the moment acting on the joint.
The robot joint moment mainly comes from two parts, one is self gravity, and the other is processing force. First, for the self-gravity part, the gravity of the robot is mainly from the weight of the robot link, wherein the second and third links are the heaviest. Industrial robots are typically six-axis tandem mechanisms. For ease of description, the joints from the bottom are designated as j1-j 6. The torque of j2 and j3 is therefore the largest, resulting in the largest compliance error. Since the joints 2 and 3 have a large influence on the position of the end of the robot, the gravity of other links is omitted for simplicity, and only the compliance errors of j2 and j3 are considered, so that the compliance error model can be further simplified. The compliance error under the gravitational force of j2 and j3 can be expressed as:
τG3＝G3L3 cosθ3/2 (8)
wherein tau isGThe moment borne by the joint under the action of gravity is shown, G2 and G3 are the gravity centers of the connecting rod 2 and the connecting rod 3 respectively, and L2And L3The length of the joints 2 and 3, theta2And theta3The angles of deflection of the links 2 and 3 relative to vertical and horizontal, respectively.
In terms of processing force, the following can be obtained according to the principle of virtual work:
J(q)TF＝τF (9)
where J is the Jacobian matrix, τFRepresenting the moment of the joint under the action of gravity. The expression (9) represents the mapping relation between the acting force of the tail end of the robot and the moment of each joint, so that under the condition that the force applied to the tail end of the robot is known, the force applied to each joint of the robot can be obtained through reverse calculation. Similarly, we only focus on the joint forces of the joints 2 and 3, so we can get the total stress of the joint i as:
τi＝τGi+τFi (10)
c is the reciprocal of the node stiffness coefficient K, and the solving formula of K is as follows:
F＝J-TKqJ-1Δxt (11)
wherein F is the end stress of the robot, and Δ xtFor end displacement of the robot, KqIs a 6 x 6 diagonal stiffness matrix. By KqThe compliance coefficient C of the joint angle can be obtained by solving.
Further the total stress moment tau of the jointiAnd substituting the flexibility coefficient C into the formula (6) to obtain the joint rotation angle error delta theta generated by the flexibility deformationciThus, a robot flexibility error model is obtained, and the expression is as follows:
ΔT2＝JIΔVc (12)
ΔVc＝(0,Δθc,0,0,0) (13)
wherein, Delta T2End pose error of robot, delta V, caused by compliance errorcRepresenting the compliance error, Δ θ, of the robotc＝(0,Δθc2,Δθc3,0,0,0)T，ΔθciRepresenting joint rotation angle errors due to compliance deformation;
FIG. 3 is a schematic diagram of a robot for compliance error modeling in the practice of the method of the present invention;
FIG. 4 is a schematic diagram of the method for analyzing the distribution rule of the flexibility error of the industrial robot.
Step three: modeling the actual error of the terminal pose of the robot under the consideration of the measurement error:
on the basis, the quantifiable calculation error of the robot, namely the geometric parameter error and the joint flexibility error under the action of stress can be obtained. Namely:
ΔV＝(Δd,Δθ,Δa,Δα,Δβ)＝ΔVg+ΔVc (14)
ΔVg＝(Δdg,Δθg,Δag,Δαg,Δβg) Characterizing the geometric parameter error;
ΔVc＝(0, Δθ c0,0,0) characterizes the compliance error, where Δ θc＝(0,Δθc2,Δθc3,0,0,0)T。
Based on the robot can quantitatively calculate the error, a robot terminal pose actual error model can be further obtained:
ΔT＝ΔT1+ΔT2＝JIΔV (15)
further considering coordinate conversion errors caused by incomplete coincidence of the measurement coordinate system and the robot base coordinate system, introducing a measurement coordinate system conversion matrix to describe the conversion relation of the two coordinate systems, and expressing as follows:
wherein r isx,ry,rzRespectively representing the minimum rotation errors of the base coordinate system relative to the three axes of the measuring coordinate system under the actual condition of the industrial robot; dx,dy,dzRespectively, the minimal movement error of the base coordinate system relative to the origin of the measuring coordinate system in the actual situation of the industrial robot.
Therefore, it is possible to establish the robot tip pose error Δ P in consideration of the measurement errormCorrelation with actual error Δ P:
ΔPm＝Pm-Pt＝′EP′-Pt＝′E(Pt+ΔP)-Pt＝(′E-E)Pt+ΔP+(′E-E)ΔP (17)
in the formula, delta PmRepresenting the pose error obtained by measurement, PmRepresents the actual pose, PtRepresenting the theoretical end pose of the robot. E is an identity matrix, and (` E-E) delta P is negligible as a high-order tiny quantity, so that the actual error of the terminal pose of the robot can be obtained through further deformation:
ΔP＝ΔPm-(’E-E)Pt (18)
then substituting the delta P into a geometric parameter error model and a flexibility error model of the robot, namely:
ΔP＝ΔT＝ΔT1+ΔT2 (19)
further solving to obtain robot related parameter errors;
step four: based on the error parameter identification of the EKF, the first error compensation is carried out.
In the measuring process, the positions of the robot base and the laser tracker are kept unchanged, the target ball of the laser tracker is arranged at the tail end of the robot, and the tail end poses of the robot under different joint angle motions are recorded. And the recording software adopts Spatial Analyzer software, introduces the robot body model, sets related measurement parameters and measures the coordinates of the center position of the robot end effector. In the parameter identification algorithm, random errors of the system are considered, certain random noises of the system are eliminated by adopting an EKF algorithm, and the identification precision is improved. And finally, correcting the quantifiable error parameters of the robot by using the error parameters obtained by identification, namely completing the first positioning error compensation of the robot.
Table 2 shows the geometric parameter error and the measurement coordinate system error of the KUKA KR210 robot;
TABLE 2
Step four: modeling residual errors;
according to the motion characteristics of the robot, introducing spatial motion flexibility KCI to characterize the intensity of motion change of the robot, as shown in the following formula:
wherein k (J)N) The condition numbers characterizing the Jacobian matrix are characterized, the subscripts N and tr (-) denote the normalization process of the matrix and the trace of the matrix, and m denotes the number of rows of the matrix. The larger the KCI value is, the better the flexibility of the robot is. J. the design is a squareNTo normalize the jacobian matrix, it is expressed as follows:
wherein, O3×3,I3×3J is a 3 × 3 zero matrix, an identity matrix, and a jacobian matrix, respectively. L is the characteristic length of the jacobian matrix normalization process.
Based on the method, the distribution situation of the motion spatial flexibility of the robot is obtained (shown in fig. 5). And then selecting sample points based on a variable grid interval division method of motion flexibility. It is divided into two regions of (0.1-0.3) and (0.3-1) according to the KCI value. According to the principle that the higher the flexibility is, the smaller the side length of the spatial grid is, in the region outside z0, selecting the side length of the grid of 30mm to perform spatial grid division on the working space along the x-axis direction of the robot coordinate system; in the z0 region, a 50mm grid side length was chosen to spatially grid the workspace along the x-axis direction of the robot coordinate system. And compiling an offline positioning program according to the divided theoretical coordinates of the vertexes of the cubic grid, controlling the robot to perform positioning, and measuring and recording by using a laser tracker. Then each region is divided into 30 cubic grids respectively, namely 60 spatial cubic grids are obtained in total.
And further calculating the distance between the target point and the sample point on the basis of the spatial grid. And (4) establishing a calculation method of Euclidean distance between two points by considering the influence degree of different vertexes on the terminal position.
Wherein d isΔRepresenting the Euclidean distance, ρ, between two target locationsiThe influence degree of the joint i on the position error of the tail end is represented; thetaiIs the joint angle at the target position, thetai′Is the joint angle at the sampling point.
According to the error similarity principle, the reciprocal of the distance between two positions is used as a weight, the closer the distance is, the larger the weight is, the farther the distance is, the smaller the weight is, and therefore an inverse distance weighting method (IDSW) is adopted for weight solution.
Where r represents an exponential weight, typically taking the value 1 or 2.
Finally, a robot residual error calculation method based on a spatial interpolation algorithm can be obtained:
wherein d isΔkRepresenting the Euclidean distance between an end point and a vertex k, wherein the total number of the vertexes of the Cartesian space grid of the robot is 8, and k is 1 and 2 … 8; p is a radical ofkRepresenting the influence weight of each vertex k of the space grid on an endpoint;
fig. 5 is a schematic diagram of the method of the invention for analyzing the spatial motion characteristics of an industrial robot.
Step five: robot absolute positioning accuracy error compensation
And obtaining the absolute positioning error of the tail end of the robot after the nominal geometric parameter correction through a residual error model, obtaining the robot joint rotation amount for compensating the tail end positioning error by utilizing the mapping relation (formula 26) of the micro displacement of the tail end of the robot and the joint corner, and compensating to the robot nominal joint rotation amount, namely performing the second positioning error compensation on the robot. Through the precision compensation, the robot is finally enabled to reach the expected position.
Δθ＝J(θ)-1Δd (26)
Table 3 shows the effect comparison of the method of the present invention with other compensation methods;
TABLE 3
Although embodiments of the present invention have been shown and described above, it will be understood that the above embodiments are exemplary and not to be construed as limiting the present invention, and that those skilled in the art may make variations, modifications, substitutions and alterations within the scope of the present invention without departing from the spirit and scope of the present invention.
Claims (7)
1. A robot absolute positioning precision calibration method based on kinematics and spatial interpolation is characterized by comprising the following specific steps:
the method comprises the following steps: based on the geometric parameters of the robot, an MD-H kinematics method is adopted to construct a geometric parameter error model of the robot, which is expressed as follows:
ΔT1＝JIΔVg
in the formula,. DELTA.T1For robot end pose error, Δ V, caused by geometric parameter errorgRepresenting the errors of the geometrical parameters of the robot, i.e. Δ Vg＝(Δdg,Δθg,Δag,Δαg,Δβg) (ii) a The geometric parameters of the robot comprise joint distance d, joint rotation angle theta, connecting rod length a, rod torsion angle alpha and y-axis torsion angle beta; j. the design is a squareITo identify the Jacobian matrix, the method is specifically expressed as:
wherein, Mθ,Md,Ma,Mβ,MαIs a 3 XN order partial derivative matrix of the position of the tail end of the robot considering the error of the geometric parameters, Rθ,Rβ,RαThe method is a robot tail end attitude partial derivative matrix taking geometric parameter errors into consideration in the order of 3 multiplied by N;
step two: and (3) combining the structural characteristics of the robot and stress analysis to construct a robot flexibility error model, wherein the expression is as follows:
ΔT2＝JIΔVc
ΔVc＝(0,Δθc,0,0,0)
wherein, Delta T2End pose error of robot, delta V, caused by compliance errorcRepresenting the compliance error, Δ θ, of the robotc＝(0,Δθc2,Δθc3,0,0,0)T，ΔθciRepresenting joint rotation angle errors due to compliance deformation;
step three: the error delta V which can be quantitatively calculated by the robot is obtained by the calculation of the first step and the second step, namely the geometric parameter error and the joint flexibility error under the action of stress; the expression is as follows:
ΔV＝(Δd,Δθ,Δa,Δα,Δβ)＝ΔVg+ΔVc；
further obtaining the actual position and attitude error delta P-delta T of the robot end1+ΔT2＝JIΔ V, introducing a measurement coordinate system transformation matrix' E in consideration of a coordinate transformation error caused by incomplete coincidence of the measurement coordinate system and the robot base coordinate system, and solving Δ P by the following formula:
ΔP＝ΔPm-(′E-E)Pt
wherein, Δ PmRepresenting pose errors obtained by measurement, PtRepresenting the theoretical end pose of the robot, 'E' is a transformation matrix of a measurement coordinate system, and E is a unitA matrix;
then substituting the delta P into a geometric parameter error model and a flexibility error model of the robot, and solving to obtain a quantifiable calculation error delta V of the robot;
step four: measuring the actual terminal pose of the robot to obtain identification experiment data; secondly, identifying quantifiable calculation error delta V parameters of the robot by adopting an EKF algorithm, and correcting name-defined geometric parameters of the robot by using the robot error parameters obtained by identification, namely completing the first positioning error compensation of the robot and remaining the positioning residual errors of the robot;
step five: constructing a robot positioning residual error model based on a spatial interpolation algorithm; the expression is as follows:
wherein, dΔkRepresenting the Euclidean distance between an end point and a vertex k, wherein the total number of the vertexes of the Cartesian space grid of the robot is 8, and k is 1 and 2 … 8; p is a radical ofkRepresenting the influence weight of each vertex k of the space grid on an endpoint;
step six: based on the robot positioning residual error model constructed in the fifth step, acquiring robot joint rotation quantity delta theta for compensating the tail end positioning error by utilizing the mapping relation between the tail end displacement of the robot and the joint corner, and compensating the robot joint rotation quantity delta theta to the name meaning joint rotation quantity of the robot, namely performing the second positioning error compensation on the robot; and finally obtaining the expected position of the robot.
2. The robot absolute positioning precision calibration method based on kinematics and spatial interpolation as claimed in claim 1, characterized in that: in the first step, Δ d is definedi,Δθi,Δai,Δαi,ΔβiI is 1 and 2 … 6, which respectively represent the joint distance error, joint rotation angle error, connecting rod length error, rod torsion angle error and y-axis torsion angle error of the ith joint;
an MD-H kinematics model is adopted to construct a transformation matrix of adjacent connecting rods of the robot, and the expression is as follows:
wherein the content of the first and second substances,
a transformation matrix representing the coordinate system i-1 to the coordinate system i, di,θi,ai,αi,βiRespectively showing the joint distance, the joint angle, the length of a connecting rod, the torsion angle of a rod piece and the torsion angle of a y axis of the ith joint;
the actual position of the robot end effector center point under the base coordinate system is expressed as:
wherein the content of the first and second substances,
a position coordinate transformation matrix from the ith joint to the (i-1) th joint under the geometric parameter error;
an error coordinate matrix from the ith joint to the (i-1) th joint;
will be provided with
Simplified to a simple linear equation:
based on the formulas (2) and (3), high-order micro quantity is further saved, and robot tail end pose error delta T caused by geometric parameter error is obtained1Finally establishing a geometric parameter error model table of the robotShown as follows:
ΔT1＝JIΔVg (4)。
3. the robot absolute positioning precision calibration method based on kinematics and spatial interpolation as claimed in claim 1, characterized in that: in the second step, the robot is structurally characterized in that the connecting rod is a rigid body, so that the robot flexibility deformation only considers the joint deformation, and the joint deformation amount calculation method comprises the following steps:
Δθci＝Cτ (5)
wherein, Delta thetaciThe method comprises the following steps that a joint corner error generated due to flexibility deformation is shown, C shows a flexibility coefficient of a joint angle, the flexibility coefficient is a constant value for a fixed joint angle, and tau represents a moment acting on the joint;
the robot joint moment comprises self gravity and processing force; the total stress on the joint i is thus obtained as:
τi＝τGi+τFi (6)
wherein, tauGiRepresenting the moment, tau, experienced by the joint i under the action of gravityFiThe moment borne by the joint i under the action of the machining force is represented;
the flexibility coefficient C is a node rigidity coefficient KqReciprocal of (a), KqThe solving formula of (2) is as follows:
F＝J-TKqJ-1Δxt (7)
wherein F is the end stress of the robot, and Δ xtFor end displacement of the robot, KqIs a 6 x 6 diagonal stiffness matrix; by KqThe compliance coefficient C of the joint angle can be obtained by solving;
further substituting the total stress moment of the joint and the flexibility coefficient C into the formula (5) to obtain the joint rotation angle error delta theta generated by the flexibility deformationci。
4. The robot absolute positioning precision calibration method based on kinematics and spatial interpolation as claimed in claim 1, characterized in that: in the third step, the quantifiable calculation error delta V of the robot is obtained by calculation in the first step and the second step, namely the geometric parameter error and the joint flexibility error under the stress action; the expression is as follows:
ΔV＝(Δd,Δθ,Δa,Δα,Δβ)＝ΔVg+ΔVc； (8)
the coordinate conversion error caused by incomplete coincidence of the measurement coordinate system and the robot base coordinate system needs to be considered, and a measurement coordinate system conversion matrix is introduced to describe the conversion relation of the two coordinate systems, and the conversion relation is expressed as follows:
wherein r isx,ry,rzRespectively representing the minimum rotation errors of the base coordinate system relative to the three axes of the measuring coordinate system under the actual condition of the industrial robot; dx,dy,dzRespectively representing the minimum movement error of the base coordinate system relative to the origin of the measurement coordinate system under the actual condition of the industrial robot;
then, a robot terminal pose error delta P considering the measurement error is establishedmAnd the correlation relation with the actual pose error delta P of the robot terminal:
ΔPm＝Pm-Pt＝′EP′-Pt＝′E(Pt+ΔP)-Pt＝(′E-E)Pt+ΔP+(′E-E)ΔP (10)
in the formula,. DELTA.PmRepresenting the pose error obtained by measurement, PmRepresents the actual pose, PtRepresenting the theoretical terminal pose of the robot, ' E ' is a transformation matrix of a measurement coordinate system, E is a unit matrix, and (' E-E) delta P is ignored as high-order microminiature; thereby obtain the terminal position appearance actual error of robot:
ΔP＝ΔPm-(′E-E)Pt＝JIΔV (11)。
5. the robot absolute positioning precision calibration method based on kinematics and spatial interpolation as claimed in claim 1, characterized in that: and in the fourth step, a pose coordinate measuring instrument is adopted to measure the actual terminal pose coordinate information of the robot, and the pose coordinate measuring instrument is an API laser tracker.
6. The robot absolute positioning precision calibration method based on kinematics and spatial interpolation as claimed in claim 1, characterized in that: the weight p in the fifth stepkIs solved as follows:
according to the error similarity principle, the reciprocal of the distance between two positions is used as a weight, the closer the distance is, the larger the weight is, the farther the distance is, the smaller the weight is, therefore, the inverse distance weighting method IDSW is adopted to carry out weight solution,
wherein r represents an exponential weight, and generally takes a value of 1 or 2.
7. The robot absolute positioning precision calibration method based on kinematics and spatial interpolation as claimed in claim 1, characterized in that: in the sixth step, a robot joint rotation amount Δ θ formula for compensating the end positioning error is as follows:
Δθ＝J(θ)-1Δd (13)
wherein, J (theta)-1An inverse matrix representing the robot Jacobian matrix J (theta), Δ d
Since the displacement is to be expressed, it is expressed by Δ d.
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