Solution Classification for Perspective-Three-Point Problem Base on PST Method

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.475-476.1067 ◽

2013 ◽

Vol 475-476 ◽

pp. 1067-1070

Author(s):

Ming Liang Li ◽

Jian Liang Tang

Keyword(s):

Point Problem ◽

Stable Algorithm ◽

Similar Triangle ◽

Real Solutions ◽

Position And Orientation ◽

Permutation Problem ◽

New Equation ◽

Scene Object ◽

Solution Classification

The perspective-n-point (PnP) problem is originated from camera calibration. It is to determine the position and orientation of the camera with respect to a scene object from n correspondent points. And a new stable algorithm by using a geometric constraint called perspective similar triangle (PST) can give new equations to solve P3P. The PST method achieves high stability in the permutation problem and in presence of image noise. Using the complete discrimination system, we obtain the solution classification of the new equation for the P3P problem. The solution classification gives a set of formulas to determine the number of real solutions to the P3P problem. Based on the formulas, we may know whether the parameters give multiple solutions or not and are critical or not which is very important to present robust algorithm.

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A STABLE DIRECT SOLUTION OF PERSPECTIVE-THREE-POINT PROBLEM

International Journal of Pattern Recognition and Artificial Intelligence ◽

10.1142/s0218001411008774 ◽

2011 ◽

Vol 25 (05) ◽

pp. 627-642 ◽

Cited By ~ 20

Author(s):

SHIQI LI ◽

CHI XU

Keyword(s):

Classical Problem ◽

Main Idea ◽

Point Problem ◽

Numerical Instability ◽

Direct Solution ◽

Unknown Parameters ◽

Similar Triangle ◽

Danger Cylinder ◽

Geometric Singularity ◽

Permutation Problem

The perspective-three-point problem (P3P) is a classical problem in computer vision. The existing direct solutions of P3P have at least three limitations: (1) the numerical instability when using different vertex permutations, (2) the degeneration in the geometric singularity case, and (3) the dependence on particular equation solvers. A new direct solution of P3P is presented to deal with these limitations. The main idea is to reduce the number of unknown parameters by using a geometric constraint we called "perspective similar triangle" (PST). The PST method achieves high stability in the permutation problem and in the presence of image noise, and does not rely on particular equation solvers. Furthermore, reliable results can be retrieved even in "danger cylinder", a typical kind of geometric singularity of P3P, where all existing direct solutions degenerate significantly.

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Kinetostatic analysis and solution classification of a class of planar tensegrity mechanisms

Robotica ◽

10.1017/s026357471800070x ◽

2018 ◽

Vol 37 (7) ◽

pp. 1214-1224 ◽

Cited By ~ 6

Author(s):

P. Wenger ◽

D. Chablat

Keyword(s):

Static Equilibrium ◽

Equilibrium Conditions ◽

Geometric Conditions ◽

Equilibrium Configurations ◽

Groebner Base ◽

Systematic Solution ◽

Alternative Designs ◽

Interesting Alternative ◽

Solution Classification

SUMMARYTensegrity mechanisms are composed of rigid and tensile parts that are in equilibrium. They are interesting alternative designs for some applications, such as modeling musculo-skeleton systems. Tensegrity mechanisms are more difficult to analyze than classical mechanisms as the static equilibrium conditions that must be satisfied generally result in complex equations. A class of planar one-degree-of-freedom tensegrity mechanisms with three linear springs is analyzed in detail for the sake of systematic solution classifications. The kinetostatic equations are derived and solved under several loading and geometric conditions. It is shown that these mechanisms exhibit up to six equilibrium configurations, of which one or two are stable, depending on the geometric and loading conditions. Discriminant varieties and cylindrical algebraic decomposition combined with Groebner base elimination are used to classify solutions as a function of the geometric, loading, and actuator input parameters.

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Self-motions of pentapods with linear platform

Robotica ◽

10.1017/s0263574715000843 ◽

2015 ◽

Vol 35 (4) ◽

pp. 832-860 ◽

Cited By ~ 5

Author(s):

Georg Nawratil ◽

Josef Schicho

Keyword(s):

Real Solutions ◽

Kinematic Mapping ◽

Bond Theory ◽

Anchor Points ◽

Self Motion ◽

Full Classification ◽

Direct Kinematics

SUMMARYWe give a full classification of all pentapods with linear platform possessing a self-motion beside the trivial rotation about the platform. Recent research necessitates a contemporary and accurate re-examination of old results on this topic given by Darboux, Mannheim, Duporcq and Bricard, which also takes the coincidence of platform anchor points into account. For our study we use bond theory with respect to a novel kinematic mapping for pentapods with linear platform, beside the method of singular-invariant leg-rearrangements. Based on our results we design pentapods with linear platform, which have a simplified direct kinematics concerning their number of (real) solutions.

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A NEW EQUATION TO ESTIMATE GLOMERULAR FILTRATION RATE VERSUS OLD ONES FOR CLASSIFICATION OF CHRONIC KIDNEY DISEASE IN PATIENTS WITH ARTERIAL HYPERTENSION AND GLUCOSE INTOLERANCE

Journal of Hypertension ◽

10.1097/00004872-201106001-00571 ◽

2011 ◽

Vol 29 ◽

pp. e209-e210

Author(s):

V. Tolkacheva ◽

S. Villevalde ◽

E. Tyukhmenev ◽

Z. Kobalava

Keyword(s):

Chronic Kidney Disease ◽

Glomerular Filtration Rate ◽

Kidney Disease ◽

Arterial Hypertension ◽

Glomerular Filtration ◽

Filtration Rate ◽

Rate Versus ◽

Estimate Glomerular Filtration Rate ◽

New Equation

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Classification of the perspective-three-point problem, discriminant variety and real solving polynomial systems of inequalities

Proceedings of the twenty-first international symposium on Symbolic and algebraic computation - ISSAC '08 ◽

10.1145/1390768.1390782 ◽

2008 ◽

Cited By ~ 12

Author(s):

Jean-Charles Faugère ◽

Guillaume Moroz ◽

Fabrice Rouillier ◽

Mohab Safey El Din

Keyword(s):

Polynomial Systems ◽

Point Problem ◽

Discriminant Variety ◽

Systems Of Inequalities

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Forward Positional Analysis for the General 4–6 In-Parallel Platform

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1243/pime_proc_1995_209_122_02 ◽

1995 ◽

Vol 209 (1) ◽

pp. 55-67 ◽

Cited By ~ 11

Author(s):

Q Liao ◽

L D Seneviratne ◽

S W E Earles

Keyword(s):

Polynomial Equation ◽

Unknown Variable ◽

Real Solutions ◽

Positional Analysis ◽

Order Polynomial ◽

Parallel Platform ◽

New Equation ◽

Published Paper ◽

Special Case ◽

Kinematic Solution

Presented is the forward positional (kinematic) solution for the general case of the 4–6 in-parallel platform mechanism; in particular, the spherical joints of the moving and base platforms are not restricted to lie in planes, but can be freely chosen. The forward positional analysis consists of 13 equations which are reduced to a single thirty-second order polynomial equation in one unknown variable. This new equation is numerically solved and validated by substituting the 32 roots into the 13 forward positional equations. The new analysis is also used to solve an example, with a set of known results from a previously published paper, in which a special case of the 4–6 in-parallel platform is considered; the results are in exact agreement. For a number of general platform and actuator inputs a maximum of 24 real solutions have been found. One example is illustrated.

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