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Abstract: Joint zero error of articulated coordinate measuring machine is caused by assembly process error, which has a great influence on the spatial pose error of the probe tip. In this paper, based on the mathematical model of six-degree-of-freedom joint-based coordinate measuring machine, a linear calibration algorithm of joint zero deviation is established by using mathematical methods such as total differential, least-squares method and iterative algorithm, and verified by computer simulation. The correctness of the calibration algorithm.
Keywords: Coordinate measuring machine Articulated zero offset calibration
Calibration Algorithm of Joint Zero-position Deviations for 6 DOF Joint-type CMM
Ye Dong et al
Abstract :The joint zero-position deviations of joint-type CMM,which result from assembly technology errors,have an enormous influence on the pose errors of probe end. Therefore, a calibration algorithm of the joint zero-position deviations is proposed. On the basis of setting up mathematics model of 6 DOF joint-type CMM, a linear calibration algorithm by applying total differential, least square and iterative algorithm is presented. And the correctness of the calibration method is validated by the computer simulation.
Keywords :CMM joint-type zero-position deviation calibration
1. introduction
The 6-DOF articulated coordinate measuring machine is a new type of non-Cartesian coordinate measuring machine. It simulates the human joint structure and replaces the length reference with the angle reference. The six rods and one probe are connected in series through six rotary joints. One end is fixed on the base, and the other end (the probe) is free to move in space. A six degree of freedom closed spherical measuring space. Compared with the traditional Cartesian coordinate measuring machine, it has the advantages of simple mechanical structure, small size, large measuring range, flexibility and convenience, etc. It is mainly used in the field of CAD/CAM three-dimensional model surface digitization and large parts of the geometric dimensions Detection and other fields [1,2].
In the assembly process of the six-degree-of-freedom joint coordinate measuring machine, the angle deviation generated due to the non-coincidence of the zero of the angle photoelectric encoder and the theoretical zero position of the joint structure is called a joint zero position deviation. Its characteristics are: the zero deviation of each joint is different; due to the inevitable assembly process error, the joint zero deviation is larger (about ± 3 °); for each assembled joint type coordinate measuring machine, each joint zero position The bias value is fixed and is a systematic error. Due to the amplification effect of the rod length, the joint zero deviation produces a large position error at the end probe. Therefore, it is very important to calibrate the joint-type coordinate measuring machine in order to compensate for zero joint misalignment and improve the measurement accuracy.
2. mathematical model
Figure six degrees of freedom articulated
Scaling machine structure model Six degrees of freedom The articulated coordinate measuring machine can be seen as a series of open kinematic chains from the mechanical structure. The structural model is shown in the figure (the Y axis of each coordinate system is determined by the "right hand rule").
Referring to the figure, the pose of the probe local coordinate system O7-X7Y7X7 with respect to the base reference coordinate system O0-X0Y0Z0 is denoted as T07, which is a 4x4 homogeneous matrix and can be described as
T07=A01A12A23A34A45A56A67 (1)
Wherein, A i-1i (i = 1, ..., 7) is a rod member i-1 posture homogeneous transformation matrix of the lever with respect to i. In 1995, Denavit and Hartenberg proposed an analytical method for the relationship between two interconnected and relatively moving components, and gave the corresponding homogeneous transformation matrix [3].
(2)
In the formula, θi is the rotation angle of the joint, here is the variable, called the joint variable; φi is the angle between the rotation axes of the adjacent joints, where it is approximately a right angle; ai is the distance along the vertical line of the space between adjacent joint axes, where 0; di is the distance between the Z axis of the origin of the adjacent member's coordinates, which is called the bar length.
For multi-joint coordinate measuring machines, the attitude of the probe in space is not important, and the spatial position coordinates of the probe are needed. After the equations (1) and (2) are combined, the coordinate equation of the probe head position is
P=(R1R2R3R4R5R6)q7+(R1R2R3R4R5)q6+R1R2R3R4)q5
+(R1R2R3)q4+(R1R2)q3+R1q2+q1 (3)
The formula includes three coordinate component equations, which are functions of the joint variables, ie
P = F (θ1, θ2, θ3, θ4, θ5, θ6) (4)
In order to obtain the relationship between joint zero position error and probe position error, assuming that joint zero-point deviation is sufficiently small, the differential equation (4) is fully differentiated and the position error equation of the probe is approximated as [4]
(5)
Formula (5) is simply described by a matrix, ie
Δ P=J δΔδ 　　　　　 (6)
Where Δ P = (ΔPx ΔPy ΔPz)T
　　
　　 Δδ=(Δθ1 Δθ2 Δθ3 Δθ4 Δθ5 Δθ6)T
Equations (4) and (6) can be used to obtain a linear equation that describes the relationship between zero joint misalignment and probe position error.
3. The calibration algorithm to determine the zero offset value of each joint requires a series of known standard position coordinates. These standard position coordinates can be measured with a high-precision coordinate measuring machine. There are m standard position coordinates. The measuring head of the articulated coordinate measuring machine touches these standard positions respectively. The corresponding joint rotation angles are respectively obtained by the photoelectric encoder. Substituting these joint rotation angles into equation (4) respectively, the probes are calculated. The theoretical position coordinates are then compared with the standard coordinates and m probe position errors are obtained. Substituting these data into equation (6), 3×m position error equations can be obtained.
Δ Q=G Δδ 　　　　　 (7)
among them
There are six unknowns in equation (7). As long as 3×m>6, the least squares method can be used to solve the joint zero deviation.
Δδ=( G T G ) -1 G TΔ Q (8)
The calculated zero joint offset value is used as the correction of zero deviation into equation (4) to calculate the new probe position coordinates. Then the new position error and the new coefficient matrix are substituted into equation (7) and then repeated. Formula (8) calculation. After repeated iterations above, until the position error of the probe is less than the set value, an optimal solution is finally obtained, that is, the zero value of the joint closest to the actual joint.
4. Simulation check
In order to carry out the computer simulation check, the structural parameters (φi, ai, di) of the six-degrees-of-freedom coordinate measuring machine are first set, and the joint zero deviation (Δθi) is assumed. The specific values ​​are listed in Table 1.
Table 1 Structural Parameters of Six Degrees of Freedom Joint Coordinate Measuring Machine
Bar number φi(°) ai(mm) di(mm) Zero offset Δθi(°) 1 -90.1 0.01 99.85 2.4 2 90.05 0.02 151.38 -2.05 3 90 -0.01 448.6 -1.5 4 -89.9 -0.03 101.1 2.2 5 -90.05 0.01 352.2 1.2 6 89.35 0.005 99 .75 -1.8 7 0 0 150.25 0 In the calculation, three simulation calculations were repeated for the calibration algorithm. Each time, three groups of joint angle combinations are randomly selected. According to the structural parameters and joint zero position deviations in Table 1, three standard (actual) spatial coordinate vectors are calculated according to equation (4). At the same time, without considering the zero deviation, three theoretical coordinate vectors are calculated and corresponding error vectors are obtained. According to (5), (6), (7), 9 error equations can be obtained. Use equation (8) to solve the six joint zero bias values. In order to obtain more accurate data, an iterative algorithm is used to substitute the calculated joint zero offset value into the modified theoretical model of equation (4) and the above process is repeated until the spatial position error is less than a set value (here set to 0.3 mm). . The results of the three simulation calculations are listed in Table 2.Table 2 Simulation Checking Results
Number of simulation joint rotation angle combinations (θ1,..., θ6)(°) Standard coordinate vector (mm) Position error (mm) Iteration 1 iteration 2 iterations 3 iterations 4 times zero error (°) Position error (mm) Zero error (°) Position error (mm) Zero Error (°) Position error (mm) Zero error (°) Position error (mm) 1 (-90, -90, 0, 90,
0,90) 138.558
575.782
-283.282 39.968 2.406
-2.063
-1.432
2.177
1.776
-1.375 1.717 2.406
-2.063
-1.490
2.177
1.719
-1.776 1.483 2.406
-2.063
-1.490
2.177
1.203
-1.776 0.148 2.406
-2.063
-1.492
2.182
1.203
-1.781 0.146 (120,90,120,-20,
70,50) -652.092
645.842
202.687 30.043 1.201 0.955 0.206 0.206 (-90,0,0,90,
0,0) 142.57
522.883
503,83 49.592 0.732 0.801 0.244 0.232 2 (-90,10,10,90,
10,10) 61.557
434.09
594.691 49.319 2.521
-1.948
-1.432
2.464
1.146
-1.261 1.479 2.406
-2.063
-1.490
2.181
1.203
-1.781 0.212 (0,90,0,-90,
0,90) 320.85
157.437
-251.305 23.654 1.155 0.230 < < (-120,140,20,60,
10,150) -148.998
-518.998
-237.442 39.30 2.176 0.082 3 (50,-80,50,40,
-60,40) -463.903
-760.076
176.659 73.57 2.349
-2.120
-1.432
2.292
1.146
-1.432 0.603 2.406
-2.063
-1.490
2.180
1.203
-1.781 0.068 (10,80,30,150,
30,140) 186.165
160.292
290.97 26.475 0.894 0.069 < < (40,30,60,150,
90,-90) 179.32
102.388
101.179 17.161 0.898 0.122
From the simulation verification results, the following conclusions can be drawn:
(1) The smaller joint zero position deviation will cause a large probe position error, up to 73.57mm;
(2) The calibration algorithm given in this paper is correct. The calibrated joint zero offset value is similar to the set true value. The maximum error is 0.02°.
(3) The iterative algorithm is convergent and generally does not exceed 4 iterations;
(4) The results of the three simulation calculations are the same, indicating that as long as any three points are taken in the measurement space, the six joint zero deviations can be accurately and uniquely calibrated.
5. Conclusion
Based on the Denavit-Hartenberg method, this paper establishes the mathematical model of a six-degrees-of-freedom coordinate measuring machine. As can be seen from the model, the relationship between the coordinates of the probe tip position and the six joint angles is non-linear, which is quite difficult to derive from the known spatial coordinate values ​​to calculate the joint zero offset. Therefore, we used the full differential method to obtain a linear relationship between the zero error of the joint and the position error of the tip of the probe, thereby greatly simplifying the calibration process. In the case of known spatial point coordinates, the least-squares method and the finite number of iterative operations are applied to find the optimal joint zero-offset value. Finally, the correctness of the algorithm is proved by computer simulation. The calibration algorithm is also fully applicable to the calibration of other structural parameter errors. The calibration algorithm is of great significance for improving the measurement accuracy of a six-degree-of-freedom joint coordinate measuring machine.
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