Unknown Coordinates Research Articles

BUCHBERGER ALGORITHM APPLIED TO PLANAR LATERATION AND INTERSECTION PROBLEMS

The Buchberger algorithm which is incorporated in algebraic software of MAPLE and MATHEMATICA is here used to derive expressions relating the known coordinates of planar stations P1 ∈ E2 and P2 ∈ E2 (i.e. {X1, Y1}P1 and {X2, Y2}P2 respectively) and the observables {S1, S2, T10, T12, T20, T21} to the unknown planar coordinates {X0, Y0}p0 of station P0 for planar lateration and intersection problems. In the case of lateration, the coordinates {X0, Y0}P0 of the unknown station P0 ∈ E2 are expressed as quadratic equations in terms of the measured distances {S2, S2} to the known station {P1, P2} ∈ E2 and the coordinates {X1, Y1}P1 and {X2, Y2}P2 of these known stations respectively. For intersection, the coordinates {X0, Y0}P0 of the unknown stations P0 ∈ E2 are expressed in terms of the measured directions {T10, T12, T20, T21} and the coordinates {X1, Y1}P1 } and {X2, Y2}P2 of known stations {P1, P2} ∈ E2. These expressions can easily be programmed in laptops or programmable calculators and thus making it feasible for any practitioner with these computing devices to obtain directly planar coordinates once the distances {Sl, S2} or directions {T10, T12, T20, T21} have been measured.

BUCHBERGER ALGORITHM APPLIED TO PLANAR LATERATION AND INTERSECTION PROBLEMS

The Buchberger algorithm which is incorporated in algebraic software of MAPLE and MATHEMATICA is here used to derive expressions relating the known coordinates of planar stations P1 ∈ E2 and P2 ∈ E2 (i.e. {X1, Y1}P1 and {X2, Y2}P2 respectively) and the observables {S1, S2, T10, T12, T20, T21} to the unknown planar coordinates {X0, Y0}p0 of station P0 for planar lateration and intersection problems. In the case of lateration, the coordinates {X0, Y0}P0 of the unknown station P0 ∈ E2 are expressed as quadratic equations in terms of the measured distances {S2, S2} to the known station {P1, P2} ∈ E2 and the coordinates {X1, Y1}P1 and {X2, Y2}P2 of these known stations respectively. For intersection, the coordinates {X0, Y0}P0 of the unknown stations P0 ∈ E2 are expressed in terms of the measured directions {T10, T12, T20, T21} and the coordinates {X1, Y1}P1} and {X2, Y2}P2 of known stations {P1, P2} ∈ E2. These expressions can easily be programmed in laptops or programmable calculators and thus making it feasible for any practitioner with these computing devices to obtain directly planar coordinates once the distances {Sl, S2} or directions {T10, T12, T20, T21} have been measured.

Time-Relative Positioning with a Single Civil GPS Receiver

Time-relative positioning is a recent method for processing GPS phase observations. The operational method undertaken in this paper consists of the following steps: first, recording phase observations at a station of known coordinates; second, moving the GPS receiver to an unknown station (which can be located up to a few hundred meters away, dependint on what type of transportation – e. g., walking, motorcycle – is available) while continuously observing carrier phases; and, third, recording phase observations at a second station of unknown coordinates with a single GPS receiver. To obtain the position of the unknown station relative to the first (known) station, the processing method uses combined observations taken at two different epochs and two different stations with the same receiver. For this reason, the errors that vary between two epochs must be taken into account in an appropriate way, especially errors in satellite clock corrections and ephemerides, and errors related to tropospheric and ionospheric delays. Ionospheric modeling using IONEX files (the ionospheric maps calculated by the International GPS Service) was also tested to correct L1 phase observations. This method has been used to calculate short vectors with an accuracy of a few centimeters (for a processing interval of 30 s) using a single civil GPS receiver. © 2001 John Wiley & Sons, Inc.

Estimates of the Duality Gap for Optimum Partition Problems

Four estimates of the duality gap are constructed for a continuous multiproduct optimum partition problem for a set in an n-dimensional euclidean space on a subset with unknown coordinates for the centers with constraints in the form of equalities and inequalities.

Exact solutions for the problem of source location from measured time differences of arrival

Source location using the difference in time of arrival of a signal (TDOA) at an array of receivers is used in many fields including speaker location. Using three TDOA values from four noncollinear receivers one can, in principle, solve for the unknown source coordinates in terms of the receiver locations. However, the equations are nonlinear, and in practice signals are contaminated by noise. Practical systems often use multiple receivers for accuracy and robustness, and improved S/N, and solutions must be obtained via minimization. A new family of exact solutions, for the case of four receivers located in a plane, is presented. These solutions can be evaluated using a small number of arithmetic operations. The performance of these solutions for several practical situations is examined via simulation, and possible geometries of receiver locations suggested. For the case where multiple receivers (>4) are used, a new formulation is presented, that incorporates the present solutions, imposes additional constraints on source location, and enables use of a constrained L1 optimization procedure to achieve a robust estimate of the source location. The present estimator is compared with ones from the literature, and found to be robust and accurate, and more efficient. [Work partially supported by DARPA.]

The determination of an unknown oxygen atom position in rare-earth zirconate pyrochlores by a 111 systematic-row convergent-beam electron diffraction technique

Abstract A focused probe systematic-row large angle convergent-beam electron diffraction (CBED) technique has been used to determine an unknown oxygen atom fractional coordinate in two rare-earth zirconate pyrochlores La2Zr2O7 and Sm2zr2O7 to an estimated accuracy of better than 0.005. The bond valence method suggests that the CBED-determined x values are chemically plausible. The simplicity of the CBED technique used suggests that it may be a very useful technique for determining positions of light atoms in the presence of strongly scattering heavy-metal atoms.

Flexible Multibody Simulation and Choice of Shape Functions

The approach most widely used for the modelling of flexible bodies in multibody systems has been called the floating frame of reference formulation. In this methodology the flexible body motion is subdivided into a reference motion and deformation. The displacement field due to deformation is approximated by the Ritz method as a product of known shape functions and unknown coordinates depending on time only. The shape functions may be obtained using finite-element-models of flexible bodies in multibody systems, resulting in a detailed system representation and a high number of system equations. The number of system equations of such a nodal approach can be reduced considerably using a modal representation of deformation. This modal approach, however, leads to the fundamental problem of selecting the shape functions. The floating frame of reference formulation is reviewed here using a generic flexible body model, from which the various body models used in multibody simulations may be derived by formulation of specific constraint equations. Special attention is given in this investigation to the following subjects: • The separation of flexible body motion into a reference motion and deformation requires the definition of a body reference frame, which in turn affects the choice of shape functions. Some alternatives will be outlined together with their advantages and disadvantages. • Assuming the body deformation to be small, the system equations can be linearized. This may require considering geometric stiffening terms. The problem of how to compute these terms has been solved in literature on the instability of structures under critical loads. For finite element models the geometric stiffening terms are obtained from the tangential stiffness matrix. • The generality of the flexible body model allows the definition of an object oriented data base to describe the system bodies. Such a data base includes a general interface between multibody- and finite-element-codes. • By combining eigenfunctions and static deformation modes to represent body deformation one obtains a set of so-called quasi-comparison functions. When selected properly these functions can be shown to improve the representation of stresses significantly.

Complete calibration of a stereo photogrammetric system through control points of unknown coordinates

This paper presents a new method for calibrating a video 3D stereo-photogrammetric system. The external parameters and the focal lengths of the cameras are determined from the epipolar constraint and the principal points are computed through the minimisation of a cost function carried out through an evolutionary optimisation. The method has been made more robust with a deterministic annealing procedure of the search region amplitude. Calibration is carried out by moving a rigid bar, carrying two markers on its extremities, inside the working volume. The distance between the two markers is the only measure required. Tests on real data are reported which show that the obtained accuracy is comparable to the one achieved calibrating with control points of known 3D coordinates.

Locating the crack tip of a surface-breaking crack Part I. Line crack

Four distinct algorithms to locate the crack tip of a surface-breaking crack using only the arrival time information of the first diffracted waves are described and compared. To illustrate these algorithms, a line crack in a half-plane is considered. The first two algorithms are based mainly on elementary geometric arguments, where the crack tip is formulated as the intersecting point of two ellipses (algorithm 1) and/or three circles (algorithm 2). The other two algorithms are formulated as optimization problems, where cost functions based upon the arrival time data of diffracted waves are constructed. The unknown crack tip coordinates are then determined by minimizing the cost functions through the Lagrange multiplier method (algorithm 3) or the simplex method (algorithm 4). In the numerical experiments, the exact arrival times are superimposed by Gaussian error with different levels to simulate the real extracted arrival times from experimental signals. The numerical optimization method (algorithm 4) is found to have the best performance with respect to noise, as well as for accuracy. Moreover, the recovery of the crack length is much more robust than the orientation and depth.

Active self-calibration of robotic eyes and hand-eye relationships with model identification

We first review research results of camera self-calibration achieved in photogrammetry, robotics and computer vision. Then we propose a method for self-calibration of robotic hand cameras by means of active motion. Through tracking a set of world points of unknown coordinates during robot motion, the internal parameters of the cameras (including distortions), the mounting parameters as well as the coordinates of the world points are estimated. The approach is fully autonomous, in that no initial guesses of the unknown parameters are to be provided from the outside by humans for the solution of a set of nonlinear equations. Sufficient conditions for a unique solution are derived in terms of controlled motion sequences. Methods to improve accuracy and robustness are proposed by means of best model identification and motion planning. Experimental results in both a simulated and a real environments are reported.

Closed-form solution of P4P or the three-dimensional resection problem in terms of Möbius barycentric coordinates

The perspective 4 point (P4P) problem - also called the three-dimensional resection problem - is solved by means of a new algorithm: At first the unknown Cartesian coordinates of the perspective center are computed by means of Mobius barycentric coordinates. Secondly these coordinates are represented in terms of observables, namely space angles in the five-dimensional simplex generated by the unknown point and the four known points. Substitution of Mobius barycentric coordinates leads to the unknown Cartesian coordinates (2.8)–(2.10) of Box 2.2. The unknown distances within the five-dimensional simplex are determined by solving the Grunert equations, namely by forward reduction to one algebraic equation (3.8) of order four and backward linear substitution. Tables 1.–4. contain a numerical example. Finally we give a reference to the solution of the 3 point (P3P) problem, the two-dimensional resection problem, namely to the Ansermet barycentric coordinates initiated by C.F. Gaus (1842), A. Schreiber (1908) and A.␣Ansermet (1910).

直接像及び鏡像を利用した多面体対象物の位置計算

A method in order to calculate the positions of feature points on a polyhedral object is presented. The method uses a single picture of both an object and its mirror images in a mirror or mirrors set in a field of vision. The unknown 3D coordinates of feature points on an object can be calculated by solving the simultaneous equations which are derived from the 2D coordinates of a pair of projected picture points corresponding to an object point and its mirror image point. The hidden surfaces of an object being imaged in the mirror, the positions of feature points on the hidden surface can be obtained. The results of fundamental experiments are shown.

Optimization and potential characteristics of controllable (adaptive) optical systems

1. Radiation transmission through a turbulent atmosphere using controllable optical systems is a statistical problem and requires the application of statistical optimization techniques. 2. Using the concepts and results of the adaptive Bayes approach allows one to synthesize an algorithm for controlling a transmission aperture. 3. Control algorithms were obtained for single soundings of an object with known coordinates (16) and (18), with unknown coordinates (24), and for multi-interval control (28). 4. The efficiency of the obtained algorithms and the considerations for the selection of the number of controllable subapertures was analyzed.

AREA CUTOFF BY COORDINATES

AbstractThe method of calculating areas by coordinates is commonly used in land partitioning. Depending on the case under consideration, the coordinates of one or both endpoints of the cutoff line can be determined. The first case occurs when the cutoff line passes through a given point on the boundary, while the second case occurs when the cutoff line has a given bearing. In either case, equations equal in number to the unknown coordinates are found and solved simultaneously. As a result, formulae are found for the unknown coordinates as functions of the given data. Two numerical examples demonstrate the solutions of both cases.

Fitting data to nonlinear functions with uncertainties in all measurement variables

An analytical approach is made to the nonlinear least squares problem having uncertainties in all measurements coordinates. It is shown how the unknown independent coordinates may be eliminated from the sum of the squares, thus reducing it to an ordinary minimisation problem. An algorithm is derived using analytical derivatives, but a simple procedure also allows the use of standard numerical derivatives.

Optimal design of geodetic nets, 2

Let the configuration or first-order design matrix A of a geodetic net be given and let the weight or second-order design matrix P of m observations be unknown. For some predetermined choice of the dispersion matrix Σ of the n unknown net coordinates, the unique minimum Euclidean norm solution for the weight matrix P is P = (A′)+Σ−1A+. A′ indicates the transpose and A+ the pseudoinverse of the (in general) rectangular matrix A. For a general first-order design matrix A and a given Σ, there does not exist a solution for a diagonal positive definite weight matrix P. Our solution for P belongs to the first category of an optimal design of C. R. Rao. The designs P = I and P = (A′)+Σ−1A+ yield the same best linear unbiased estimator for the unknowns, but the variance covariance matrix for these designs is different. Examples, especially the structure by G. I. Taylor and T. Karman for the homogeneous and isotropic geodetic net, illustrate theoretical results.

An r-Dimensional Quadratic Placement Algorithm

In this paper the solution to the problem of placing n connected points (or nodes) in r-dimensional Euclidean space is given. The criterion for optimality is minimizing a weighted sum of squared distances between the points subject to quadratic constraints of the form X′X = 1, for each of the r unknown coordinate vectors. It is proved that the problem reduces to the minimization of a sum or r positive semi-definite quadratic forms which, under the quadratic constraints, reduces to the problem of finding r eigenvectors of a special “disconnection” matrix. It is shown, by example, how this can serve as a basis for cluster identification.

The adjustment of intersecting chains of triangulation using a medium sized computer

A method of adjusting intersecting chains of triangulation by direct elimination, using variation of coordinates, is given which is suitable for a computer of modest storage capacity (about 4000–8000 words). The network is considered to be made up of “nodes” (where two or more chains intersect) and “links” (which connect two nodes). The normal equations for the points in each link are computed, the coefficients of the unknown coordinates of the associated nodes being treated as extra right hand sides. These equations are then solved to express the coordinates of any link station in terms of the associated nodes. The process is repeated for all links. The normal equations for the nodes are then set up and substitution made for all “link” terms.

Solutions to the Vector Tetrahedron Equation

A set of nine closed-form solutions are presented to the single, three-dimensional vector tetrahedron equation, sum of vectors equals zero. The set represents all possible combinations of unknown spherical coordinates among the vectors, assuming the coordinates are functionally independent. Optimum use is made of symmetry. The solutions are interpretable and can be evaluated reliably by digital computer in milliseconds. They have been successfully applied to position determination of many three-dimensional mechanisms.

On the Reduction of Photographic Star Positions and Proper Motions.

view Abstract Citations References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS On the Reduction of Photographic Star Positions and Proper Motions. Eichhorn, Heinrich K. Abstract It is shown that a considerable increase in systematic accuracy can be obtained in photographic position work by eliminating the systematic errors which result between the positions of the same stars as obtained from overlapping plates with different centers. Technically, a strict least-squares solution for this problem will necessitate the solution of a system where the number of unknown coordinates is equal to the total number of stars plus the number of stars times plate constants. This method lends itself particularly well to the reduction of photographic star catalogues. A rigorous solution of this system can be easily and practically obtained by iterations which will always converge. Theoretical investigations show that the accuracy to be expected from a solution of that kind can be predicted under the assumption of a smooth distribution of stars. The application of this principle (the practical application of which requires an electronic computer) to parallax work not only increases the accuracy of the parallaxes thus obtained, but also yields accurate positions and motion of the comparison stars, which are very often urgently needed for the determination of mass ratios if observation series from different instruments are to be combined in one solution. Publication: The Astronomical Journal Pub Date: 1960 DOI: 10.1086/108131 Bibcode: 1960AJ.....65R.488E full text sources ADS |