US 11,654,568 B2
3D measurement model and spatial calibration method based on 1D displacement sensor
Wei Liu, Liaoning (CN); Bing Liang, Liaoning (CN); Mengde Zhou, Liaoning (CN); Kun Liu, Liaoning (CN); Yang Zhang, Liaoning (CN); Di Feng, Liaoning (CN); Xintong Jiang, Liaoning (CN); Likun Si, Liaoning (CN); and Zhenyuan Jia, Liaoning (CN)
Assigned to DALIAN UNIVERSITY OF TECHNOLOGY, Liaoning (CN)
Appl. No. 16/647,654
Filed by Dalian University of Technology, Dalian (CN)
PCT Filed May 15, 2019, PCT No. PCT/CN2019/086948
§ 371(c)(1), (2) Date Mar. 16, 2020,
PCT Pub. No. WO2020/151134, PCT Pub. Date Jul. 30, 2020.
Claims priority of application No. 201910069437.3 (CN), filed on Jan. 24, 2019.
Prior Publication US 2021/0095959 A1, Apr. 1, 2021
Int. Cl. B25J 9/16 (2006.01); B25J 19/02 (2006.01); G01B 21/04 (2006.01)
CPC B25J 9/1692 (2013.01) [B25J 9/1694 (2013.01); B25J 19/027 (2013.01); G01B 21/042 (2013.01); G01B 21/045 (2013.01)] 1 Claim
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1. A method of calibrating extrinsic parameters of a 1D displacement sensor by a 3D measurement model, wherein, firstly, a 3D measurement system based on a fixed 1D displacement sensor is established; then a spatial measurement model based on the 1D displacement sensor is established; and then based on precision pose data of a measurement plane and sensor measurement data, a spatial calibration constraint equation are established; finally, a weighted iterative algorithms is employed to calculate the extrinsic parameters of the 1D displacement sensor, the measurement origin and measurement vector, that meet precision requirements, then the calibration process is completed; finally, a precision 3D measurement model is established; the method comprising steps of:
step 1) establishing the 3D measurement system based on the fixed 1D displacement sensor:
wherein, first, the 1D displacement sensor is fixed on the frame for 3D information measurement, and its displacement measurement value is obtained through a data acquisition card; then the space calibration system is set up and the measurement plane is mounted on a hexapod, which is capable of 6-DOF motion; a workpiece coordinate system is established based on the hexapod, and the initial plane equation of the measurement plane in the workpiece coordinate system is determined as follows:
N0·(xyz1)T=0  (1)
where, N0 is the plane equation parameter; then, the above-mentioned calibration system is moved into the measurement range of the 1D displacement sensor by an extrinsic actuator, and the transformation relationship between the world coordinate system and the workpiece coordinate system is established by the extrinsic measurement system, which is used to transform subsequent calibration results to the world coordinate system;
step 2) establishing the spatial measurement model based on the 1D displacement sensor:
wherein, in the workpiece coordinate system, the measurement origin of the displacement sensor is defined as O=[Ox Oy Oz]T, the unit measurement vector is defined as t=[tx ty tz]T, and the following relation holds:
t2∥=tx2+ty2+tz2=1  (2)
the measurement origin O and the measurement vector t are the extrinsic parameters of the 1D displacement sensor to be calibrated;
from the above equations, the measurement points Pin the workpiece coordinate system are:
P=(xyz)T=O+δt  (3)
where, δ is the displacement value measured by the 1D displacement sensor; according to equation (1), at the initial position, the point P0 on the measurement plane satisfies:
N0·(P01)T=0  (4)
step 3) based on the precision pose data of the measurement plane and the measurement data of the 1D displacement sensor, establishing the spatial calibration constraint equations:
wherein, multiple pose transformations are performed, and parameters (l,m,n,α,β,γ) for each pose transformation are recorded separately, where (l,m,n) represents the displacements along the X, Y, Z axes, (α,β,γ) represents the Euler angles rotated around the X, Y, Z axes; thus, the rotation matrix R and translation matrix T are expressed as:

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after the ith pose transformation of the measurement plane, the point Pi on the measurement plane satisfies:
Ni·(Pi1)T=0  (6)
meanwhile, according to equation (5), there is:
Pi=Ri·P0+Ti  (7)
combining equations (4) and (7), we can get:

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substituting equations (3) and (8) into equation (6), the constraint equation is expressed as:

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where, O and t are the extrinsic parameters of the 1D displacement sensor to be calibrated, δi is the measured value of the 1D displacement sensor at the ith time, and the remaining variables are all known quantities; Ri and Ti are calculated from the pose transformation parameters (l,m,n,α,β,γ), and N0 has been obtained in advance;
step 4) employing a weighted iterative algorithms to calculate the extrinsic parameters of the 1D sensor that meet the precision requirements:
wherein, due to the measurement and calculation errors, equation (9) cannot be zero, and iterative optimization is required to minimize its value;
let s=(O,t)=(Ox,Oy,Oz,tx,ty,tz), equation (9) be expressed as:
min ∥f(s)∥→0  (10)
where, ε is the allowable error; due to the different contribution to the overall error of each calibration sample, it is necessary to adjust the weight of each sample to avoid generating local solutions through over-optimization; equation (10) is converted into:
fi(s)∥≤λiε  (11)
where λi (i=1, 2, . . . , M) is the weight coefficients and M is the number of samples;
the optimization objective is constructed as follows, and the set of inequalities (11) is transformed into a minimum optimization problem:

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it can be seen from the sufficient and necessary that the solutions of equations (11) and (12) are equivalent;
then, the conventional LM iterative optimization algorithm is adopted, and the optimal solution is obtained:
s*=argmins{G(s)}  (14)
through the above steps, the establishment and spatial calibration of the 3D measurement model based on the 1D displacement sensor in the workpiece coordinate system is completed, and the translation relationship between world coordinate system and workpiece coordinate system is used to transform the calibrated 3D measurement model to the world coordinate system.