通用非线性高斯-赫尔默特模型参数估计的同伦方法

doi:10.11947/j.AGCS.2024.20240027

摘要/Abstract

摘要：

通用非线性高斯-赫尔默特模型是顾及因变量或全体变量误差的显式和隐式非线性函数平差模型的统一表达。针对在迭代初值与真值相差较大时，高斯-牛顿迭代解算法存在不收敛的问题，本文提出融合同伦方法与非线性最小二乘的通用非线性高斯-赫尔默特模型参数估计法。从引入同伦参数的非线性最小二乘平差准则出发，推导了求解通用模型参数的微分方程组和追踪同伦曲线的固定步长预测公式与牛顿校正公式，给出了隐式函数模型残差向量的近似计算公式。为避免计算立体矩阵，将克罗内克积和矩阵拉直运算引入推导过程，降低了计算微分方程组的复杂度。通过仅顾及自变量误差的距离定位、顾及卫星坐标误差和测距误差的伪距定位、顾及全体平面坐标误差的圆曲线拟合，以及顾及已知坐标误差的测边网平差4个试验，对本文方法的可行性进行了验证。试验结果表明：在设定的两组初值中，当高斯-牛顿法收敛时，本文方法也收敛；当高斯-牛顿法不收敛时，本文方法仍收敛；本文方法收敛的初值范围更大。

关键词: 非线性高斯-赫尔默特, 同伦方法, 距离定位, 伪距定位, 圆曲线拟合, 测边网

Abstract:

The generalized nonlinear Gauss-Helmert model is a unified expression of explicit and implicit nonlinear function adjustment models that consider the errors of the dependent variable or the whole variable. Aiming at the problem of non-convergence of its Gauss-Newton iterative solution algorithm when the difference between the initial value and the true value is large, the parameter estimation method of the generalized nonlinear Gauss-Helmert model that integrates the homotopy method and nonlinear least squares is proposed. Starting from the nonlinear least-squares adjustment criterion that introduces the homotopy parameter, the system of differential equations for solving the generalized model parameters and the fixed-step prediction formula for tracking the homotopy curve with the Newton's correction formula are derived, and the approximation formula for calculating the residual vector of the implicit function model is given. The complexity of computing the system of differential equations is reduced by introducing the Kronecker product and the matrix straightening operation into the derivation process in order to avoid computing the cubic matrix. The feasibility of the method is verified through three experiments, including distance positioning that only considers the error of the independent variable, pseudo-distance positioning that considers the satellite coordinate error and ranging error, trilateration network that considered the errors in the known coordinates, and circular curve fitting that considers the error of plane coordinates. The experimental results show that the new method converges to a larger range of initial values.

Key words: nonlinear Gauss-Helmert, homotopy method, distance positioning, pseudo-distance positioning, circular curve fitting, trilateration network

中图分类号:

P207

胡川, 史宗浩, 任大钦. 通用非线性高斯-赫尔默特模型参数估计的同伦方法[J]. 测绘学报, 2024, 53(11): 2178-2188.

Chuan HU, Zonghao SHI, Daqin REN. On homotopy method to parameter estimation for generalized nonlinear Gauss-Helmert model[J]. Acta Geodaetica et Cartographica Sinica, 2024, 53(11): 2178-2188.

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编号	X	Y	Z	d
1	10	20	6	22.180 0
2	8	30	0	30.958 0
3	10	-20	6	23.174 9
4	8	-30	0	30.880 0
5	-10	-20	-6	23.070 0
6	-8	-30	0	30.440 0
7	-10	20	-6	22.700 0
8	-8	30	0	30.720 0

参数	采用初值1		采用初值2
参数	高斯-牛顿法	通用同伦平差法	高斯-牛顿法	通用同伦平差法
Xu/m	不收敛	-0.214	不收敛	-0.214
Yu/m		0.124		0.124
Zu/m		0.549		0.549
		0.302		0.302
		1.078		1.078
		0.045		0.045
		4.013		4.013

参数项	采用初值1		采用初值2
参数项	高斯-牛顿法	通用同伦平差法	高斯-牛顿法	通用同伦平差法
Xu/m		-2 157 555.665		-2 157 555.665
Yu/m		4 380 372.532		4 380 372.532
Zu/m		4 081 041.770		4 081 041.770
		175 322.286		175 322.286
	不收敛	137 879 264.216	不收敛	137 879 264.215
	不收敛	293 314 174.167	不收敛	293 314 174.167
		336 342 488.338		336 342 488.338
		266 625 946.277		266 625 946.277
		240 957 927.393		240 957 927.393

点号	X	Y
1	1 651.034 5	2 546.311 1
2	1 147.604 0	2 837.157 6
3	575.052 4	2 736.103 3
4	201.365 7	2 290.801 4
5	201.407 8	1 709.210 3
6	575.008 2	1 263.745 4
7	1 147.459 4	1 162.778 5
8	1 651.067 4	1 453.676 8
9	1 849.899 3	1 999.983 2
10	1 651.360 8	2 546.350 9

参数项	高斯-牛顿法		通用同伦平差法
参数项	采用初值1	采用初值2	采用初值1	采用初值2
	1 000.009		1 000.009	1 000.009
	1 999.985		1 999.985	1 999.985
	849.993		849.993	849.993
	0.015	发散	0.015	0.015
	0.003		0.003	0.003
	0.003		0.003	0.003
	0.002		0.002	0.002