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Calibration method of robot base frame by quaternion form
WANG Wei , LIU Lidong, WANG Gang, YUN Chao
School of Mechanical Engineering and Automation, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Abstract:The accuracy pose of robot base frame with respect to the world frame needs to be calibrated under the task of multi-arm cooperation, identification of kinematics parameter and off-line program of robot. An accurately calibration method of robot base frame using quaternion form was proposed and applied. The kinematic model of robot was established by the product-of-exponential formula. 5 tool center points (TCPs), which were not coplanar, were measured with the external measuring device, and the corresponding robot joint configurations were also recorded. By considering the elements of rotational matrix corresponding to the base frame pose as the variables, a group of incompatible equations was set up, and the initial homogeneous matrix of the base frame pose was also obtained. However, due to the errors of measuring and truncation, the rotational matrix part of initial homogeneous matrix cannot satisfy the regulation of unit orthogonal matrix. Using the geometrical constraints of a quaternion, an objective equation by the form of penalty function was established, and the initial homogeneous matrix was orthogonalized. By a set of practical calibration experiments of robot, the validity of calibration method was verified, which improved the accuracy of the location of robot end.
Key words: robot     calibration     base frame     quaternion     orthogonality

Haytti模型[4]、CPC模型[5]、S模型[6]和指数积模型[7]等被广泛应用于机器人运动学参数的标定,而有关机器人基坐标系标定的研究则较少.

2 基坐标系初始标定

Rms包括9个未知数,式(4)可以分解为12个独立方程,那么式(4)为不相容方程组.对式(4)两端做转置,可得

RmsT=[x1 x2 x3],则由式(5)可得

3.2 旋转矩阵的优化

R为正交化结果时,应满足以下方程:

q*=[q0O q1O q2O q3O]取得该数值解时,其对应的旋转矩阵与初步标定的旋转矩阵距离最近,从而实现旋转矩阵的精细标定. 4 标定试验

 图 1 机器人基坐标系标定系统原理图Fig. 1 Schematic of robot base frame calibration system
4.2 试验步骤和数据

 图 2 末端坐标测量Fig. 2 Coordinate measurement of end

 p Px Py Pz 1 -460.890 1 549.517 5 -27.609 1 2 -362.291 9 390.865 8 -53.967 2 3 -296.550 1 231.185 4 -85.149 6 4 -260.442 2 78.538 5 -119.147 9 5 -463.434 9 458.011 5 -136.617 8 6 -238.384 0 167.067 7 -201.002 5 7 -218.092 9 7.843 2 -238.375 4 8 -207.299 7 158.385 2 -184.361 8 9 -392.060 1 406.649 7 -239.146 6 10 -331.767 6 249.871 2 -281.312 5 11 -308.002 4 101.785 4 -318.316 2 12 -160.459 2 94.983 5 -311.768 5 13 -294.703 1 185.589 5 -402.290 9 14 -306.393 9 349.437 7 -346.226 6 15 -263.037 1 184.986 8 -388.513 1 16 -241.509 4 29.029 0 -429.597 9 17 -408.238 0 265.099 1 -456.836 2 18 -430.845 4 420.156 7 -401.901 6 19 -231.475 4 120.404 1 -523.064 6 20 -232.738 0 -42.431 6 -558.272 0

 p q/(°) 1 -10.000,25.000,-40.00,-40.000,95.000,-160.000 2 -10.000,30.000,-30.00,-30.000,100.000,-155.000 3 -10.000,35.000,-20.00,-20.000,105.000,-150.000 4 -10.000,40.000,-10.00,-10.000,110.000,-145.000 5 -5.000,25.000,-35.000,-35.000,110.000,-140.000 6 -5.000,35.000,-15.00,-15.000,95.000,-155.000 7 -5.000,40.000,-5.000,-5.000,100.000,-150.000 8 -5.000,45.000,-20.000,-20.000,105.000,-145.000 9 0.000,25.000,-30.000,-30.000,100.000,-145.000 10 0.000,30.000,-20.000,-20.000,105.000,-140.000 11 0.000,35.000,-10.000,-10.000,110.000,-160.000 12 0.000,45.000,-15.000,-15.000,95.000,-150.000 13 5.000,30.000,-15.000,-15.000,95.000,-145.000 14 5.000,35.000,-30.000,-30.000,100.000,-140.000 15 5.000,40.000,-20.000,-20.000,105.000,-160.000 16 5.000,45.000,-10.000,-10.000,110.000,-155.000 17 10.000,25.000,-20.000,-20.000,105.000,-155.000 18 10.000,30.000,-35.000,-35.000,110.000,-150.000 19 10.000,40.000,-15.000,-15.000,95.000,-140.000 20 10.000,45.000,-5.000,-5.000,100.000,-160.000
4.3 标定结果

1) 非线性最小二乘拟合.

2) 初步标定.

3) 精细标定.

4) 标定结果对比.

 图 3 精度验证和比较Fig. 3 Verification and comparation of accuracy

5 结 论

1) 初始标定步骤中,建立待标定齐次矩阵的不相容非线性方程组,以最小二乘解作为初始标定结果.

2) 在精细标定中,采用了四元数法来描述机器人基坐标系的实际位姿,以初始标定位姿与实际位姿偏差矩阵的F范数为优化目标方程,采用拉格朗日乘子法和牛顿迭代法获得机器人基坐标系的精确标定结果.

3) 搭建了工业机器人和测量臂组成的标定试验系统,完成20个测量点的验证试验,证明经过本方法标定后,最大机器人定位误差减小为1.30 mm.

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#### 文章信息

WANG Wei, LIU Lidong, WANG Gang, YUN Chao

Calibration method of robot base frame by quaternion form

Journal of Beijing University of Aeronautics and Astronsutics, 2015, 41(3): 411-417.
http://dx.doi.org/10.13700/j.bh.1001-5965.2014.0239