Complex relativity and real solutions. I: Introduction

1985 ◽  
Vol 17 (2) ◽  
pp. 111-132 ◽  
Author(s):  
C. B. G. Mcintosh ◽  
M. S. Hickman
Keyword(s):  
Real Solutions ◽  
Complex Relativity

Complex relativity and real solutions II: Classification of complex bivectors and metric classes

1985 ◽  
Vol 17 (5) ◽  
pp. 475-491 ◽  
Author(s):  
G. S. Hall ◽  
M. S. Hickman ◽  
C. B. G. McIntosh
Keyword(s):  
Real Solutions ◽  
Complex Relativity

Complex relativity and real solutions. V. The flat space background

1988 ◽  
Vol 20 (7) ◽  
pp. 647-657 ◽  
Author(s):  
C. B. G. McIntosh ◽  
M. S. Hickman ◽  
A. W. -C. Lun
Keyword(s):  
Flat Space ◽  
Real Solutions ◽  
Complex Relativity

scholarly journals On the Effect of Control-Point Spacing on the Multisolution Phenomenon in the P3P Problem

2018 ◽  
Vol 2018 ◽  
pp. 1-13
Author(s):  
M. Vynnycky ◽  
G. M. M. Reddy
Keyword(s):  
Image Analysis ◽  
Camera Calibration ◽  
Pose Estimation ◽  
Computer Animation ◽  
Solution Structure ◽  
Control Point ◽  
Quartic Equation ◽  
Real Solutions ◽  
Algebraic Analysis ◽  
And Robotics

The perspective 3-point (P3P) problem, also known as pose estimation, has its origins in camera calibration and is of importance in many fields: for example, computer animation, automation, image analysis, and robotics. One possibility is to formulate it mathematically in terms of finding the solution to a quartic equation. However, there is yet no quantitative knowledge as to how control-point spacing affects the solution structure—in particular, the multisolution phenomenon. Here, we consider this problem through an algebraic analysis of the quartic’s coefficients and its discriminant and find that there are significant variations in the likelihood of two or four solutions, depending on how the spacing is chosen. The analysis indicates that although it is never possible to remove the occurrence of the four-solution case completely, it could be possible to choose spacings that would maximize the occurrence of two real solutions. Moreover, control-point spacing is found to impact significantly on the reality conditions for the solution of the quartic equation.

Geometric Design of Symmetric 3-RRS Constrained Parallel Platforms

2004 ◽  
Author(s):  
Eric Wolbrecht ◽  
Hai-Jun Su ◽  
Alba Perez ◽  
J. Michael McCarthy
Keyword(s):  
Geometric Design ◽  
Homotopy Continuation ◽  
Total Degree ◽  
Polynomial Equations ◽  
Dimensional Synthesis ◽  
Kinematic Synthesis ◽  
Real Solutions ◽  
Parallel Platform ◽  
Three Degree Of Freedom ◽  
Serial Chains

The paper presents the kinematic synthesis of a symmetric parallel platform supported by three RRS serial chains. The dimensional synthesis of this three degree-of-freedom system is obtained using design equations for each of three RRS chains obtained by requiring that they reach a specified set of task positions. The result is 10 polynomial equations in 10 unknowns, which is solved using polynomial homotopy continuation. An example is provided in which the direction of the first revolute joint (2 parameters) and the z component of the base and platform are specified as well as the two task positions. The system of polynomials has a total degree of 4096 which means that in theory it can have as many solutions. Our example has 70 real solutions that define 70 different symmetric platforms that can reach the specified positions.

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