Dietmeier.world File Reference

A 6-6 parallel platform with 40 solutions. More...

Detailed Description

Introducion

In the early 1990's, several approaches (numerical experimentation, intersection theory, Groebner bases, resultants, and algebra) showed that there can be at most 40 different poses for the platform, given the lengths of the six legs. The question of whether it was possible to find a design with exactly 40 real poses remained open until 1998, when Dietmaier introduced a numerical method that was able to find one such case [Dietmeier 1998]. We next provide the geometric parameters of Dietmeier's platform.

This example is almost singular. There is a set of points on a one-dimensional curbe that are only $1e-8$ away from being solutions. Therefore, this is in the limit of what CuikSuite can solve.

Geometry

The geometric parameters of this parallel platform are

Coord.	A₁	A₂	A₃	A₄	A₅	A₆	B₁	B₂	B₃	B₄	B₅	B₆
	Base Points						Platform Points
x	0	1.107915	0.549094	0.735077	0.514188	0.590473	0	0.542805	0.956919	0.665885	0.478359	-0.137087
y	0	0	0.756063	-0.223935	-0.526063	0.094733	0	0	-0.528915	-0.353482	1.158742	-0.235121
z	0	0	0	0.525991	-0.368418	-0.205018	0	0	0	1.402538	0.107672	0.353913

leg

length

0.645275

1.086284

1.503439

1.281933

0.771071

References

P. Dietmeier, "The Stewart-Gough Platform of General Geometry Can Have 40 real Postures", in Advances in Robot Kinematics: Analysis and Control, J. Lenarcic and M. Husty (editors), pp. 7-16, Strobl, 29 June-4 July, 1998.

Definition in file Dietmeier.world.