Two Approaches to Nonlinear Least Squares Adjustments

1974 ◽  
Vol 28 (5) ◽  
pp. 663-669 ◽  
Author(s):  
Allen J. Pope
Keyword(s):  
Numerical Analysis ◽  
New Brunswick ◽  
International Symposium ◽  
Least Squares ◽  
North American ◽  
Nonlinear Least Squares ◽  
Special Consideration ◽  
Geodetic Adjustment ◽  
Least Squares Adjustment ◽  
Newton Raphson

The following discussion of Newton-Raphson and Newton-Gauss applied to least squares adjustment of nonlinear conditions with parameters is abstracted from the longer paper, “Modern Trends in Adjustment Calculus”, presented at the International Symposium on Problems Related to the Redefinition of North American Geodetic Networks, Fredericton, New Brunswick. Copies are available from the author. This paper attempts a broad survey of some interesting, useful, or potentially useful, developments in today’s geodetic adjustment theory. Special consideration is given to relevant inputs from numerical analysis and statistics.

Numerical Solution Using Nonlinear Least-Squares Method for Inverse Kinematics Calculation of Redundant Manipulators

2012 ◽  
Vol 24 (2) ◽  
pp. 363-371 ◽  
Author(s):  
Shunsuke Toritani ◽  
◽  
Ruhizan Liza Ahmad Shauri ◽  
Kenzo Nonami ◽  
Daigo Fujiwara
Keyword(s):  
Least Squares ◽  
Inverse Kinematics ◽  
Joint Angle ◽  
Least Squares Method ◽  
Nonlinear Least Squares ◽  
Robotic Manipulation ◽  
Redundant Manipulators ◽  
Newton Raphson ◽  
Nonlinear Least Squares Method ◽  
User Friendly

In this paper, we present an Inverse Kinematics (IK) algorithm based on the nonlinear least-squares method for redundant manipulators. The Newton-Raphson (NR) method is a commonmethod for IK calculation of redundant manipulators. The NR method, however, causes many problems in terms of joint angle limits, singularity, and solvability. Severalmethods have therefore been proposed to solve these problems. Most, however, focus only on IK calculation performance when a desired trajectory moves outside of the workspace. A manipulator is required to move continuously, even after a desired trajectory moves outside of the workspace. It is thus also necessary to implement the IK calculation method for bringing a desired trajectory back into the workspace. In this study, we propose a user-friendly method for robotic manipulation that is capable of implementing accurate IK calculation when a desired trajectorymust be returned to the workspace.

scholarly journals The Impact of Outliers on Parameter Estimation Method for Bilinear Model

2020 ◽  
pp. 1-7
Author(s):  
Mohd Isfahani Ismail ◽  
Hazlina Ali ◽  
Sharipah Soaad Syed Yahaya
Keyword(s):  
Least Squares ◽  
Simulation Analysis ◽  
Estimation Method ◽  
Nonlinear Least Squares ◽  
Bilinear Model ◽  
Temporary Change ◽  
Level Change ◽  
Additive Outlier ◽  
Newton Raphson ◽  
The Impact

Nonlinear least squares (NLS) method along with Newton-Raphson (NR) iterative procedure is the best method to estimate parameters for bilinear model. However, the existence of outliers will affect the estimated value of the parameter and its validity can be doubtful. This statement was proven by conducting simulation analysis for the bilinear model, especially on bilinear (1,0,1,1) model without and with the existence of additive outlier (AO), innovational outlier (IO), temporary change (TC) and level change (LC) in the data. The performance of the NLS method is measured in terms of bias. Numerical results show that, in general, the NLS method performs better in estimating the parameters without the existence of AO, IO, TC or LC in the data. Keywords: bilinear model; nonlinear least squares; Newton-Raphson; additive outlier; innovational outlier; temporary change; level change

scholarly journals A Variable Projection Method for Solving Separable Nonlinear Least Squares Problems

1974 ◽  
Vol 3 (27) ◽  
Author(s):  
Linda Kaufman
Keyword(s):  
Numerical Analysis ◽  
Least Squares ◽  
Nonlinear Least Squares ◽  
Orthogonal Matrix ◽  
Linear Least Squares ◽  
Least Squares Problems ◽  
Least Squares Problem ◽  
Linear Least Squares Problem ◽  
Nonlinear Least Squares Problem ◽  
Trapezoidal Decomposition

<p>Consider the separable nonlinear least squares problem of finding ~a in R^n and ~alpha in R^k which, for given data (y_i, t_i) i=1,...,m and functions varphi_j(~alpha,t) j=1,2,...n (m&gt;n), minimize the functional</p><p>r(~a,~alpha) = ||~y - Phi(~alpha)~a||_(2)^(2)</p><p>where Phi(~alpha)_(i,j) = varphi_(j)(~alpha,t_j). This problem can be reduced to a nonlinear least squares problem involving $\mathovd{\mathop{\alpha}\limits_{\textstyle\tilde{}}}$ only and a linear least squares problem involving ~a only. the reduction is based on the results of Colub and Pereyra, <em>SIAM J. Numerical Analysis</em>, April 1973, and on the trapezoidal decomposition of Phi, in which an orthogonal matrix Q and a permutation matrix P are found such that</p><p>\begin{displaymath} Q Phi R = R &amp; S 0 &amp; 0 \end{array}\right) \begin{array}{l} \rbrace\, r \\ \mbox{} \end{array} \end{displaymath}</p><p>where R is nonsingular and upper trianular. To develop an algorithm to solve the nonlinear least squares probelm a formula is proposed for the Frechet derivation D(Phi_(2) (~alpha)) where Q i partioned into</p>

Accuracy Assessment of Nonlinear Least Squares Adjustment Parameters

2013 ◽  
Vol 353-356 ◽  
pp. 3428-3433
Author(s):  
Gui Ling Li ◽  
Fu Liang Mei
Keyword(s):  
Least Squares ◽  
Accuracy Assessment ◽  
Nonlinear Least Squares ◽  
Covariance Propagation ◽  
Different Types ◽  
Least Squares Adjustment ◽  
Existing Result

The research of variance-covariance propagation of nonlinear least squares adjusted parameters must keep pace with the development of nonlinear least squares adjustment. In this area, the existing result is inclined to the variance-covariance propagation of observation value, and the nonlinear least squares adjusted parameters precision is seldom concerned. This paper gives a variance-covariance propagation of nonlinear least squares adjusted parameters for different types of data and its calculating formula. The results of example prove that the method is effective and reliable.

scholarly journals GNSS-SNR water level estimation using global optimization based on interval analysis

2016 ◽  
Vol 6 (1) ◽  
Author(s):  
J. Reinking
Keyword(s):  
Least Squares ◽  
Interference Pattern ◽  
Interval Analysis ◽  
Correction Term ◽  
Simulated Data ◽  
Nonlinear Least Squares ◽  
Real Data ◽  
Data Set ◽  
One Dimensional ◽  
Least Squares Adjustment

AbstractThe signal-to-noise ratio (SNR) from GNSS receivers allows computing the height of a reflecting surface by analyzing the interference pattern. In classical interference pattern technique the distance between the antenna and the reflector is derived from the multipath pattern using a one-dimensional Lomb-Scargle periodogram (LSP) which permits the estimation of constant or quasi static reflector heights only. Inwaters with tidal influence some authors used one-dimensional LSP to iteratively estimate an approximate time-dependent correction term for the variable reflector height. Other authors applied nonlinear least squares adjustment that requires choosing initial parameters what might become crucial due to the multimodality of the problem. We suggest and apply an alternative approach that allows finding the global optimum of a multi-dimensional cost function of a common least squares adjustment based on interval analysis. This method reduces the computational efforts compared to LSP. The technique is demonstrated using a simulated data set derived fromreal measurements on the Weser river, Germany. Additionally, real data from a gauge in the North Sea is analyzed.

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