scholarly journals Geodesic survey and modernization of a route as the task of optimization

2014 ◽  
Vol 63 (1) ◽  
pp. 75-87
Author(s):  
Edward Nowak
Keyword(s):  
Mahalanobis Distance ◽  
Least Squares Method ◽  
Local Coordinate System ◽  
Position Vector ◽  
Similar System ◽  
B Spline ◽  
Normal Line ◽  
Polygonal Chain ◽  
Survey Point ◽  
Scalar Variable

Abstract A geodesic survey of an existing route requires one to determine the approximation curve by means of optimization using the total least squares method (TLSM). The objective function of the LSM was found to be a square of the Mahalanobis distance in the adjustment field ν. In approximation tasks, the Mahalanobis distance is the distance from a survey point to the desired curve. In the case of linear regression, this distance is codirectional with a coordinate axis; in orthogonal regression, it is codirectional with the normal line to the curve. Accepting the Mahalanobis distance from the survey point as a quasi-observation allows us to conduct adjustment using a numerically exact parametric procedure. Analysis of the potential application of splines under the NURBS (non-uniform rational B-spline) industrial standard with respect to route approximation has identified two issues: a lack of the value of the localizing parameter for a given survey point and the use of vector parameters that define the shape of the curve. The value of the localizing parameter was determined by projecting the survey point onto the curve. This projection, together with the aforementioned Mahalanobis distance, splits the position vector of the curve into two orthogonal constituents within the local coordinate system of the curve. A similar system corresponds to points that form the control polygonal chain and allows us to find their position with the help of a scalar variable that determines the shape of the curve by moving a knot toward the normal line.

scholarly journals A Robust and Adaptive Complementary Kalman Filter Based on Mahalanobis Distance for Ultra Wideband/Inertial Measurement Unit Fusion Positioning

Sensors ◽  
2018 ◽  
Vol 18 (10) ◽  
pp. 3435 ◽  
Author(s):  
Xin Li ◽  
Yan Wang ◽  
Kourosh Khoshelham
Keyword(s):  
Kalman Filter ◽  
Mahalanobis Distance ◽  
Inertial Measurement Unit ◽  
Least Squares Method ◽  
Ultra Wideband ◽  
Measurement Unit ◽  
Observation Error ◽  
System State ◽  
Inertial Measurement ◽  
The Impact

Ultra wideband (UWB) has been a popular technology for indoor positioning due to its high accuracy. However, in many indoor application scenarios UWB measurements are influenced by outliers under non-line of sight (NLOS) conditions. To detect and eliminate outlying UWB observations, we propose a UWB/Inertial Measurement Unit (UWB/IMU) fusion filter based on a Complementary Kalman Filter to track the errors of position, velocity and direction. By using the least squares method, the positioning residual of the UWB observation is calculated, the robustness factor of the observation is determined, and an observation weight is dynamically set. When the robustness factor does not exceed a pre-defined threshold, the observed value is considered trusted, and adaptive filtering is used to track the system state, while the abnormity of system state, which might be caused by IMU data exceptions or unreasonable noise settings, is detected by using Mahalanobis distance from the observation to the prior distribution. When the robustness factor exceeds the threshold, the observed value is considered abnormal, and robust filtering is used, whereby the impact of UWB data exceptions on the positioning results is reduced by exploiting Mahalanobis distance. Experimental results show that the observation error can be effectively estimated, and the proposed algorithm can achieve an improved positioning accuracy when affected by outlying system states of different quantity as well as outlying observations of different proportion.

Automatic Point Clouds Registration Based on the Method of Least Squares

2009 ◽  
Vol 419-420 ◽  
pp. 305-308
Author(s):  
Fan Wen Meng ◽  
Lu Shen Wu ◽  
Qing Jin Peng
Keyword(s):  
Coordinate System ◽  
Least Squares ◽  
Least Squares Method ◽  
Local Coordinate System ◽  
Surface Profile ◽  
Point Clouds ◽  
Fringe Projection ◽  
Method Of Least Squares ◽  
Surface Patches ◽  
Euclidean Distances

An object has to be measured to recover its 3D shape in reverse engineering applications. The object surface is sampled point by point using a fringe projection. The method of least squares is used to match overlapping surfaces to estimate transformation parameters between a local coordinate system and the template coordinate system. The Gauss–Markoff model can minimize the sum of squares of Euclidean distances between surfaces for matching arbitrarily oriented 3D surface patches. This research uses the least squares method for the registration of point clouds. A relief example shows the feasibility of the proposed method. It takes about 4 seconds for the registration of 1531209 points with the error less than 0.03mm, and the iteration number is only 20. The surface profile is complete and smooth after the registration, which can meet the requirement of surface reconstruction.

Integration of Moving Least Squares Method and Multilevel B-Spline Method for Scattered Data Approximation

2011 ◽  
Vol 291-294 ◽  
pp. 2245-2249
Author(s):  
Shi Ju Yan ◽  
Bin Ge
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Scattered Data ◽  
Moving Least Squares ◽  
Scattered Data Approximation ◽  
Approximation Accuracy ◽  
Data Approximation ◽  
B Spline ◽  
Moving Least Squares Method ◽  
Method Accuracy

For scattered data approximation with multilevel B-spline(MBS) method, accuracy could be enhanced by densifying control lattice. Nevertheless, when control lattice density reaches to some extent, approximation accuracy could not be enhanced further. A strategy based on integration of moving least squares(MLS) and multilevel B-spline(MBS) is presented. Experimental results demonstrate that the presented strategy has higher approximation accuracy.

Solitary waves of the RLW equation via least squares method

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ozlem Ersoy Hepson ◽  
Idris Dag ◽  
Bülent Saka ◽  
Buket Ay
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Least Square Method ◽  
Least Square ◽  
Spline Functions ◽  
Algebraic Equations ◽  
B Spline ◽  
B Splines ◽  
Rlw Equation ◽  
Set Up

Abstract Integration using least squares method in space and Crank–Nicolson approach in time is managed to set up an algorithm to solve the RLW equation numerically. Trial functions in the least square method consist of a combination of the quartic B-spline functions. Integration of the RLW equation gives a system of algebraic equations. The solutions consisting of a combination of the quartic B-splines are given for some initial and boundary value problems of RLW equation.

B-Spline Curve Approximation Using Feature Points

2013 ◽  
Vol 397-400 ◽  
pp. 1093-1098
Author(s):  
Xian Guo Cheng
Keyword(s):  
Error Bound ◽  
Least Squares Method ◽  
Maximum Error ◽  
Control Points ◽  
Feature Points ◽  
Shape Information ◽  
B Spline ◽  
Spline Curve ◽  
Curve Approximation ◽  
Curvature Information

This paper addresses the problem of B-spline curve approximating to a set of dense and ordered points. We choose local curvature maximum points based on the curvature information. The points and the two end points are viewed as initial feature points, constructing a B-spline curve approximating to the feature points by the least-squares method, refining the feature points according to the shape information of the curve, and updating the curve. This process is repeated until the maximum error is less than the given error bound. The approach adaptively placed fewer knots at flat regions but more at complex regions. Under the same error bound, experimental results showed that our approach can reduce more control points than Parks approach,Piegls approach and Lis approach. The numbers of control points of the curve is equal to that of the feature points after refinement.

Convergence Characteristics of Geometrically Accurate Spatial Finite Elements

2020 ◽  
Vol 16 (1) ◽  
Author(s):  
Brian Tinsley ◽  
Ahmed A. Shabana
Keyword(s):  
Finite Elements ◽  
Computer Aided Design ◽  
Eigenvalue Analysis ◽  
Reference Configuration ◽  
Position Vector ◽  
B Spline ◽  
Split Method ◽  
Stiffness Matrices ◽  
General Continuum ◽  
Aided Design

Abstract The convergence characteristics of three geometrically accurate spatial finite elements (FEs) are examined in this study using an eigenvalue analysis. The spatial beam, plate, and solid elements considered in this investigation are suited for both structural and multibody system (MBS) applications. These spatial elements are based on geometry derived from the kinematic description of the absolute nodal coordinate formulation (ANCF). In order to allow for an accurate reference-configuration geometry description, the element shape functions are formulated using constant geometry coefficients defined using the position-vector gradients in the reference configuration. The change in the position-vector gradients is used to define a velocity transformation matrix that leads to constant element inertia and stiffness matrices in the case of infinitesimal rotations. In contrast to conventional structural finite elements, the elements considered in this study can be used to describe the initial geometry with the same degree of accuracy as B-spline and nonuniform rational B-spline (NURBS) representations, widely used in the computer-aided design (CAD). An eigenvalue analysis is performed to evaluate the element convergence characteristics in the case of different geometries, including straight, tapered, and curved configurations. The frequencies obtained are compared with those obtained using a commercial FE software and analytical solutions. The stiffness matrix is obtained using both the general continuum mechanics (GCM) approach and the newly proposed strain split method (SSM) in order to investigate its effectiveness as a locking alleviation technique.

scholarly journals An active beacon-based leader vehicle tracking system

ACTA IMEKO ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 33
Author(s):  
Stanislaw Goll ◽  
Elena Zakharova
Keyword(s):  
Kalman Filter ◽  
Least Squares Method ◽  
Unscented Kalman Filter ◽  
Automatic Gain Control ◽  
Tracking System ◽  
Local Coordinate System ◽  
Estimation Method ◽  
Gain Control ◽  
Distance Measurements ◽  
Receiver Array

This article focuses on mobile robot convoying along a path travelled by a certain leader carrying the active ultrasonic beacon. The robot is equipped with the three-dimensional receiver array in order to receive both the ultrasonic wave and the RF wave marking the beginning of the measurement cycle. To increase measurement reliability, each receiver contains two independent measurement channels with automatic gain control. The distance measurements are pre-processed in order to identify the artefacts and then either remove them or replace them with the interpolated value. To estimate the position of the beacon in the robot’s local coordinate system, several methods are used, including the least squares method with subsequent exponential smoothing, the linear Kalman filter, the Rauch-Tung-Striebel smoother, the extended Kalman Filter, the unscented Kalman filter, and the particle filter. The experiments were undertaken in order to estimate the estimation method preferable for following the leader’s path.

scholarly journals New Cubic B-Spline Approximation for Solving Linear Two-Point Boundary-Value Problems

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1250
Author(s):  
Busyra Latif ◽  
Samsul Ariffin Abdul Abdul Karim ◽  
Ishak Hashim
Keyword(s):  
Boundary Value Problems ◽  
Least Squares Method ◽  
Interpolation Method ◽  
Spline Interpolation ◽  
Boundary Value ◽  
Spline Approximation ◽  
Second Order ◽  
Order Derivative ◽  
Point Boundary ◽  
B Spline

In this study, we introduce a new cubic B-spline (CBS) approximation method to solve linear two-point boundary value problems (BVPs). This method is based on cubic B-spline basis functions with a new approximation for the second-order derivative. The theoretical new approximation for a second-order derivative and the error analysis have been successfully derived. We found that the second-order new approximation was O(h3) accurate. By using this new second-order approximation, the proposed method was O(h5) accurate. Four numerical problems consisting of linear ordinary differential equations and trigonometric equations with different step sizes were performed to validate the accuracy of the proposed methods. The numerical results were compared with the least squares method, finite difference method, finite element method, finite volume method, B-spline interpolation method, extended cubic B-spline interpolation method and the exact solutions. By finding the maximum errors, the results consistently showed that the proposed method gave the best approximations among the existing methods. We also found that our proposed method involved simple implementation and straightforward computations. Hence, based on the results and the efficiency of our method, we can say that our method is reliable and a promising method for solving linear two-point BVPs.

The role of differential phase contrast in electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 176-177
Author(s):  
E.M. Waddell ◽  
J.N. Chapman ◽  
R.P. Ferrier
Keyword(s):  
Phase Contrast ◽  
Practical Method ◽  
Position Vector ◽  
Differential Phase ◽  
Pure Phase ◽  
Differential Phase Contrast ◽  
The Difference ◽  
Image Position ◽  
First Moment

Dekkers and de Lang (1977) have discussed a practical method of realising differential phase contrast in a STEM. The method involves taking the difference signal from two semi-circular detectors placed symmetrically about the optic axis and subtending the same angle (2α) at the specimen as that of the cone of illumination. Such a system, or an obvious generalisation of it, namely a quadrant detector, has the characteristic of responding to the gradient of the phase of the specimen transmittance. In this paper we shall compare the performance of this type of system with that of a first moment detector (Waddell et al.1977).For a first moment detector the response function R(k) is of the form R(k) = ck where c is a constant, k is a position vector in the detector plane and the vector nature of R(k)indicates that two signals are produced. This type of system would produce an image signal given bywhere the specimen transmittance is given by a (r) exp (iϕ (r), r is a position vector in object space, ro the position of the probe, ⊛ represents a convolution integral and it has been assumed that we have a coherent probe, with a complex disturbance of the form b(r-ro) exp (iζ (r-ro)). Thus the image signal for a pure phase object imaged in a STEM using a first moment detector is b2 ⊛ ▽ø. Note that this puts no restrictions on the magnitude of the variation of the phase function, but does assume an infinite detector.

‘Least squares’ method—theorem or law?

1980 ◽  
Vol 59 (9) ◽  
pp. 8
Author(s):  
D.E. Turnbull
Keyword(s):  
Least Squares ◽  
Least Squares Method
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