Dietmeier.world File Reference

Detailed Description

[Introduction] [Geometry] [Process] [Statistics] [Results] [References]

Introduction

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

Base Points

Coord. A1 A2 A3 A4 A5 A6
x 0 1.107915 0.549094 0.735077 0.514188 0.590473
y 0 0 0.756063 -0.223935 -0.526063 0.094733
z 0 0 0 0.525991 -0.368418 -0.205018

Platform Points

Coord. B1 B2 B3 B4 B5 B6
x 0 0.542805 0.956919 0.665885 0.478359 -0.137087
y 0 0 -0.528915 -0.353482 1.158742 -0.235121
z 0 0 0 1.402538 0.107672 0.353913
leg 1 2 3 4 5 6
length 1 0.645275 1.086284 1.503439 1.281933 0.771071

Process

This example is treated following this steps (from the main CuikSuite folder):

  • Generate the equations: Execute
  • Solve the positional analysis problem: Execute
    • bin/cuik examples/ParallelPlatform/Dietmeier
  • Visualize the solutions:
    • scripts/cuikplayer examples/ParallelPlatform/Dietmeier examples/ParallelPlatform/Dietmeier

Statistics

Characteristics of the problems:

Nr. of loops 6
Nr. of links 8
Nr. of joints (including composited joints) 6
Nr. of equations (in the simplified system) 30
Nr. of variables (in the simplified system) 30

Here you have the statistics about the execution (on an Intel Core i7 at 2.9 Ghz).

Nr. of empty boxes 2657
Nr. of solution boxes 43
Solver time (s) 380

Results

This example is extremelly ill conditioned and 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.

The parameters setting for this example are carefully selected to accurately bound the 40 solutions in the form of 37 isolated solutions plus 3 solutions bounded by a cluster of two boxes (where for each one of the two boxes the error is below $1e-6$). This shows how cuik is conservative: it migth return more boxes than solutions (i.e., some of the returned boxes include no solution but points with small error). On the other hand, no solution is lost (all solutions are inside one box), unless there is a bug in the implementation.

Here you have the 40 solutions identified by the CuikSuite:

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.