一种生成新坐标系的普适三点模型及其应用

doi:10.13229/j.cnki.jdxbgxb20181228

Abstract

Abstract:

Coordinate system generation and transformation play an important role in many fields. Taking the left-hand coordinate system as an example, a clockwise transformation model is deduced based on any rotation angle for plane rectangular coordinate systems in detail. The model is also suitable for the counterclockwise transformation for the right-hand coordinate system. The study shows that the form of rotation matrix has nothing to do with the size of rotation angle, but relates to the sequences and the rotation directions of coordinate axes, which changes when one of the two sequences changes, and stays the same when both sequences change. Based on the above derivation, a universal three-point model is proposed for the generation a new spatial rectangular coordinate system. By adding the fine criterions to the rotational parameter calculation in the gradual coordinate transformation, the universal model can arbitrarily generate a new coordinate system based on non-collinear three points and realize coordinate transformation. This model breaks the distribution limitations for three points used to generate a new coordinate system in the conventional axis alignment method. Furthermore, the universal model can be applied for generating the coordinate system of control field for the calibration of a non-metric camera which has been widely used in the fields of low-altitude remote sensing and close-range photogrammetry as a photoelectric sensor. The reliability of the proposed universal model is verified.

Key words: survey engineering, generation and transformation of coordinate system, universal three-point model, axis alignment, control field for camera calibration

CLC Number:

TP391

Feng-yan WANG,Xue-xin YAN,Ming-chang WANG,Xu-qing ZHANG,Xue-feng NIU,Qing WANG. Universal three-point model of generating a new coordinate system and its application[J].Journal of Jilin University(Engineering and Technology Edition), 2020, 50(1): 324-332.

Figures/Tables 15

Fig.1

Fig.2

Table 1

Rotation matrix forms for different axis sequences and rotation directions"

	顺时针	逆时针
左手系	$c o s α s i n α - s i n α c o s α$	$c o s α - s i n α s i n α c o s α$
右手系	$c o s α - s i n α s i n α c o s α$	$c o s α s i n α - s i n α c o s α$

Table 1

Fig.3

Fig.4

Fig.5

Fig.6

Table 2

Fig.7

Fig.8

Fig.9

Table 3

Table 4

Fig.10

Table 5

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P₁		P₂		P₃		P₁	P₂	P₃
原点	X₁,Y₁,Z₁	卦限	X₂,Y₂,Z₂	卦限	X₃,Y₃,Z₃	X₁′,Y₁′,Z₁′	X₂′,Y₂′,Z₂′	X₃′,Y₃′,Z₃′
原点	0,0,0	1st	2,2,2	1st	1,3,1	0,0,0	3.464,0,0	2.8868,1.6330,0
				2nd	-4,1,1			-1.155,4.082,0
				3rd	-2,-4,2			-2.309,-4.32,0
				4th	2,-2,1			0.5774,-2.9439,0
				5th	1,3,-1			1.7321,2.8284,0
				6th	-4,1,-1			-2.3094,3.559,0
				7th	-2,-4,-2			-4.6188,-1.633,0
				8th	2,-2,-1			-0.5774,-2.9439,0
		2nd	-2,2,2	1st	1,3,1	0,0,0	3.464,0,0	1.7321，-2.8284,0
				2nd	-4,1,1			3.4641,2.4495,0
				3rd	-2,-4,2			0,4.8890,0
				4th	2,-2,1			-1.7321，-2.4495,0
				5th	1,3,-1			0.5774，-3.2660,0
				6th	-4,1,-1			2.3094,3.5590,0
				7th	-2,-4,-2			-2.3094,4.3205,0
				8th	2,-2,-1			-2.8868，-0.8165,0

点号	坐标（测量坐标系）			坐标（控制场坐标系）
点号	X/m	Y/m	Z/m	X′/m	Y′/m	Z′/m
K₁₅	0.1236	3.6481	-1.3054	0	0	0
K₁₉	1.6659	3.9391	-1.3079	1.5695	0	0
K₆₆	0.2381	4.8546	-1.3025	0.3363	1.1644	0
…	…	…	…	…	…	…
K₇₀	1.8394	5.1672	-1.3001	1.9678	1.1746	0.0005

点号	第1次测量坐标(控制场坐标系)				第2次测量坐标(控制场坐标系)			坐标差值
点号	X′/m	Y′/m		Z′/m	X′/m	Y′/m	Z′/m	ΔX′/m	ΔY′/mm	ΔZ′/mm
K₁	-0.0063	-0.0139	1.5655		-0.0065	-0.0136	1.5653	0.2	-0.3	0.2
K₂	0.4023	-0.0136	1.5676		0.4021	-0.0136	1.5673	0.2	0	0.3
K₃	0.7781	-0.0131	1.5697		0.7779	-0.0134	1.5694	0.2	0.3	0.3
K₄	1.1666	-0.0128	1.5714		1.1663	-0.0129	1.5712	0.3	0.1	0.2
K₅	1.5598	-0.0138	1.5717		1.5594	-0.014	1.5715	0.4	0.2	0.2
…	…	…	…		…	…	…	…	…	…
K₇₀	1.9679	1.175	0.0049		1.9678	1.1746	0.005	0.1	0.4	-0.1

相机参数		标定结果
主距	f_x/mm	24.4390±0.002
主距	f_y/mm	24.4416±0.002
主点偏移	c_x/mm	-0.1115±0.004
主点偏移	c_y/mm	0.1545±0.003
畸变系数	k₁	-1.7786E-04±6.9234E-07
	k₂	2.9316E-07±4.8850E-09
	p₁	-1.4317E-05±1.3790E-06
	p₂	2.0670E-06±1.4964E-07

Universal three-point model of generating a new coordinate system and its application

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