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Numerical algebraic geometry and algebraic kinematics. (English) Zbl 1254.13031

This article reviews the algebraic kinematics and its related topics, i.e., numerical algebraic geometry and its recent researches. This consists of five parts; Part 1 is Fundamentals of algebraic kinematics, Part 2 is Numerical algebraic geometry, Part 3 is Advanced topics, Part 4 is Frontiers and Part 5 is Conclusions. The algebraic kinematics is kinematics which is modelled as a polynomial system arising in robotics and in traditional mechanism designs. In terms of the special Euclidean group \(\mathrm{SE}(3)\), the algebraic kinematics is realized as an algebro-geometric object in \(\mathrm{SE}(3)^N\) for an appropriate positive integer \(N\), which has singularities in general. By means of methods in the numerical algebraic geometry over complex number, which is classically called transcendental algebraic geometry, the algebraic kinematics problems are solved. The techniques in the numerical algebraic geometry, which are primarily based on homotopy methods, and their related topics are properly reviewed.

MSC:

13P15 Solving polynomial systems; resultants
70G55 Algebraic geometry methods for problems in mechanics
14Q15 Computational aspects of higher-dimensional varieties
70B15 Kinematics of mechanisms and robots
70E60 Robot dynamics and control of rigid bodies
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