Robust Time-Difference-of-Arrival (TDOA) Localization Using Weighted Least Squares with Cone Tangent Plane Constraint
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Finding the position of a radiative source based on time-difference-of-arrival (TDOA) measurements from spatially separated receivers has been widely applied in sonar, radar, mobile communications and sensor networks. For the nonlinear model in the process of positioning, Taylor series and other novel methods are proposed. The idea of cone constraint provides a new way of solving this problem. However, these approaches do not always perform well and are away from the Cramer-Rao-Lower-Bound (CRLB) in the situations when the source is set at the array edge, the noise in measurement is loud, or the initial position is biased. This paper presents a weighted-least-squares (WLS) algorithm with the cone tangent plane constraint for hyperbolic positioning. The method adds the range between the source and the reference sensor as a dimension. So, the space-range frame is established. Different from other cone theories, this paper sets the reference sensor as the apex and finds the optimal source estimation on the cone. WLS is used for the optimal result from the measurement plane equations, a vertical constraint and a cone constraint. The cone constraint equation is linearized by a tangent plane. This method iterates through loops and updates the tangent plane, which approximates the truth-value on the cone. The proposed algorithm was simulated and verified under various conditions of different source positions and noises. Besides, some state-of-the-art algorithms were compared in these simulations. The results show that this algorithm is accurate and robust under poor external environment.Keywords:
Tangent cone
Cramér–Rao bound
Least-squares function approximation
We study the set of tangent limits at a given point to a set definable in any o-minimal structure by characterizing the set of exceptional rays in the tangent cone to the set at that point and investigating the set of tangent limits along these rays. Several criteria for determining exceptional rays will be given. The main results of the paper generalize, to the o-minimal setting and to arbitrary dimension, the main results of O'Shea--Wilson which deals with algebraic surfaces in $\mathbb R^3$.
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Third-generation direct methods programs are based on a phasing algorithm ( e.g. the tangent or the parameter shift method) and on dual space refinement techniques. The two spaces may be alternated during the phasing procedure or used in a sequential way: for example, first phase and after extend and refine. The tangent approach in SIR2011 belongs to the second category: phases are first estimated by the tangent formula, then their extension and refinement is performed in direct space via electron density modification techniques. In this article a number of new algorithms are described, with the aim of improving the SIR2011 tangent step and allowing more efficient phase extension and refinement. New criteria were chosen for defining the number of reflections to phase, for modifying the tangent weighting scheme, and for fixing the active use of the psi-0 triplets and of the quartet invariants. Each tangent trial may now be submitted to the RELAX procedure, a tool for moving to the correct position a well oriented but misplaced structural model. The resulting procedure shows surprising efficiency, testified by a wide set of applications. The experimental results have been compared with the tangent and VLD ( vive la différence ) approaches implemented in SIR2011 .
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Tangent cone
Position (finance)
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Tangent cone
Tangent vector
Cone (formal languages)
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After the detailed description of the three methods of acquiring the tangent point on the tire tip,a measurement of tire tangent plane based on stereo vision and laser projection is proposed in this paper.Firstly,the radial laser stripe is projected sequentially on the various positions on the tire tip.Secondly,through the laser stripe extraction algorithm,spatial plane fitting process and coordinate transformation algorithm,the tangent point on the tire tip is acquired.And then the tire tangent plane equation is calculated.Experiments show that the method is stable and reliable,able to have the tire tangent plane equation acquired precisely.
Tangent vector
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Projection plane
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Basis (linear algebra)
Tangent cone
Cone (formal languages)
Irrational number
Tangent vector
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The main new notions are the notions of tangent-like spaces and local monoids. The main result is the pasage from a local monoid to its tangent-like space which is a local Leibniz algebra.
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Aim To study local distribution characters of fractal sets. Methods Definition of tangent line and plane is introduced to fractal sets by generalizing angle density to bilateral angle density and coangle density. Results and Conclusion This paper gives the concept of bilaternal angle density and coangle density of fractal sets, gives definition of tangent and tangent plane of fractal sets, and discusses its existing conditions.
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Line (geometry)
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In many textbooks, the tangent plane of a surface S at a point P is defined as the set of all tangent lines of curves at P which are on S and pass through P. Then a sufficient condition for the existence of the tangent plane is followed. Suppose that S is defined by an implicit function F, if the partial derivatives of F are continuous at P, then there is a tangent plane of S at P. In this paper, we prove that if S is defined by an implicit function F, then the differentiation of F at P is sufficient to the existence of tangent plane of S at P. We also discuss some related problems in this paper.
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This paper analyses the tangent equation of space curve and the derivation process of the curved tangent plane equation in the higher-mathematics teaching-materials,and shows another method to extract the tangent equation of space curve.
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Tangent cone
Tangent stiffness matrix
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