Closure Polynomials for Strips of Tetrahedra

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Presentation transcript:

Closure Polynomials for Strips of Tetrahedra Federico Thomas and Josep M. Porta

What’s a strip of tetrahedra?

If shared traingular face degeneres, the structure flexes …

What’s the problem to solve? Problem: Find of possible assembly modes We remove one edge We impose the distance between the endpoints We do not impose the orientations of the tetrahedra

Outline of the presentation A substitution rule to derive a scalar closure condition Deriving a closure polynomial Examples Forward kinematics of the decoupled platform Forward kinematics of a 4-4 Gough-Stewart platform

A substitution rule

Clearing radicals After iteratively applying this substitution rule, we end up with a closure condition containing nested radicals.

Example 1: The forward kinematics of the decoupled platform

Example 1: The forward kinematics of the decoupled platform

Example 1: The forward kinematics of the decoupled platform

Example 1: The forward kinematics of the decoupled platform

Example 1: The forward kinematics of the decoupled platform After clearing radicals, we obtain a polynomial of 24th degree, but we know that it should be of degree 16 to be minimal. The rod connecting P3 and P5 belongs to the shared face defined by P3, P4, and P5 which is singular when D(3,4,5)=0, that is, when After dividing the obtained polynonimial by this factor till the remainder is not null, we get

Example 2: The forward kinematics of a 4-4 platform Coplanar base and platform 24 solutions (12+12)

Numerical Example

Numerical Example After clearing radicals, a 52nd- degree polynomial in is obtained, but we know that it should be of degree 12. The rod connecting P4 and P5 belongs to two shared faces (the ones defined by P4P5P6 and P3P4P5), whose associated singular terms are After iteratively dividing the obtained polynomia lby these two terms till the remainder is not null, we get

Numerical Example

Numerical Example

Numerical Example

Conclusions The presented technique requires some symbolic manipulations. Despite this, it runs faster than all global root finders. J.M. Porta and F. Thomas, "Closed-Form Position Analysis of Variable Geometry Trusses”, submitted for publication This technique can be extended to: - multiple strips of tetrahedra, and - include orientation constraints