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Calibration algorithm of joint zero position deviation of six degree of freedom articulated coordinate measuring machine

1. Introduction

six degree of freedom articulated coordinate measuring machine is a new non Cartesian coordinate measuring machine. It imitates the human body joint structure, replaces the length reference with the angle reference, connects six rods and a probe in series through six rotating joints, one end is fixed on the machine base, and the other end (probe) can move freely in space, forming a closed spherical measuring space with six degrees of freedom. Compared with the traditional Cartesian coordinate measuring machine, it has the advantages of simple mechanical structure, small volume, large measurement range, flexibility and convenience. It is mainly used in the fields of 3D Model Surface Digitization in CAD/CAM and on-site detection of geometric dimensions of large parts [1, 2]

the angular deviation caused by the non coincidence between the zero position of the angle photoelectric encoder and the theoretical zero position of the joint structure in the assembly process of the six degree of freedom joint coordinate measuring machine is called the joint zero position deviation. Its characteristics are: the zero position deviation of each joint is different; Due to the inevitable assembly process error, the zero position deviation of the joint is large (about 3); For each assembled joint coordinate measuring machine, the zero position deviation value of each joint is fixed, which belongs to systematic error. Due to the amplification of the rod length, the zero position deviation of the joint produces a large pose error at the end probe. Therefore, in order to compensate the joint zero position deviation and improve the measurement accuracy, it is very important to calibrate the joint coordinate measuring machine

2. Mathematical model

figure 6-DOF articulated sitting

structural model of CMM 6-DOF articulated CMM can be regarded as a series of open kinematic chains from the mechanical structure. Its structural model is shown in the figure (the Y axis of each coordinate system is determined by the right-hand rule)

as shown in the figure, the pose of the probe local coordinate system o7-x7y7x7 relative to the base reference coordinate system o0-x0y0z0 is recorded as T07, which is a 4-4 homogeneous matrix, which can be described as (1)

T07 = a01a12a23a34a45a56a67

where ai-1i (I = 1, 7) is the homogeneous pose transformation matrix of rod I relative to rod I-1. Denavit and hartenberg put forward the analysis method of the relationship between two interconnected and relatively moving components in 1995, and gave the corresponding homogeneous transformation matrix [3], that is,

(2)

for multi joint coordinate measuring machines, the attitude of the probe in space is not important, but the spatial position coordinates of the probe need to be obtained. After combining equations (1) and (2), the position coordinate equation of the probe is

+ (r1r2r3) Q4 + (r1r2) Q3 + r1q2 + Q1 (3)

the equation includes three coordinate component equations, which are functions of joint variables, that is,

in order to obtain the relationship between the zero position deviation of the joint and the position error of the probe, Assuming that the zero position deviation of the joint is small enough, the full differential of equation (4) is obtained, and the approximate position error equation of the probe is [4]

(5)

to simply describe equation (5) in matrix mode, that is,

in which p = (PX py PZ) T

= (123456) T

from equation (4) and equation (6), the linear equation describing the relationship between the zero position deviation of the joint and the position error of the probe can be obtained

3. calibration algorithm

in order to determine the zero position deviation value of each joint, a series of known standard position coordinates are required, which can be measured by a high-precision coordinate measuring machine. There are m standard position coordinates. The probe of the articulated coordinate measuring machine contacts these standard positions respectively, and the corresponding joint angles are obtained by the photoelectric encoder. These joint angles are respectively substituted into equation (4), and the theoretical position coordinates of the probe are calculated, and then compared with the standard coordinates to obtain the position errors of M probes. By substituting these data into equation (6), 3 m position error equations can be obtained, that is,

q = g (7)

, in which there are 6 unknown quantities in equation (7). As long as 3 m> 6, the joint zero position deviation can be solved by the least square method, that is,

= (gtg-1 Q (8)

the calculated joint zero position deviation value is substituted into equation (4) as the correction of zero position deviation, and the new probe position coordinates can be calculated, Then, the new position error and the new coefficient matrix are substituted into equation (7), and then the calculation of equation (8) is repeated. After the above iterative process, until the position error of the probe is less than the set value, the optimal solution is finally obtained, that is, the closest to the actual zero position deviation value of the joint

4. simulation checking

in order to carry out computer simulation checking, first set the structural parameters (I, AI, DI) of the six degree of freedom articulated coordinate measuring machine, and assume the zero position deviation (I) of the joint. The specific values are listed in Table 1

Table 1 structural parameters of six degree of freedom articulated coordinate measuring machine

in the checking calculation, the calibration algorithm is repeated for three times. Three groups of joint rotation angle combinations are randomly selected each time, and three standard (actual) spatial coordinate vectors are calculated according to formula (4) according to the structural parameters and joint zero position deviation in Table 1. At the same time, three theoretical coordinate vectors are calculated without considering the zero position deviation, and the corresponding error vectors are obtained. According to equations (5), (6) and (7), nine error equations can be obtained. Use equation (8) to solve the zero position deviation value of 6 joints. In order to obtain more accurate data, the iterative algorithm is used to substitute the calculated joint zero position deviation value into equation (4) to modify the theoretical model, and the above process is repeated until the spatial position error is less than a set value (set here as 0.3mm). The results of three simulation calculations are listed in Table 2

Table 2 Simulation checking results

from the simulation checking results, it can be concluded that stacking experiments requiring constant pressure can also be carried out. The following conclusions:

(1) a small 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 zero position deviation of the joint obtained by calibration is approximate to the set true value, and the maximum error is 0.02

(3) the iterative algorithm is convergent, generally not more than 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 zero position deviation of six joints can be accurately and uniquely calibrated

5. Conclusion

based on the denavit hartenberg method, this paper establishes the mathematical model of the six degree of freedom articulated coordinate measuring machine. It can be seen from the model that the relationship between the position coordinates of the end of the probe and the angles of the six joints is nonlinear, which is quite difficult to calculate the zero position deviation of the joint from the known spatial coordinates. Because the user can choose the appropriate specification according to the test requirements of the product, we use the total differential method to obtain the linear relationship between the zero position deviation of the joint and the position error of the end of the probe, which greatly simplifies the calibration process. When the coordinates of space points are known, the optimal zero position deviation of the joint is obtained by using the least square method and a finite number of iterative operations. 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 to improve the measurement accuracy of the six degree of freedom joint coordinate measuring machine

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