Griffis-Duffy.world File Reference

Detailed Description

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

Introduction

For a general Stewart-Gough platform, once the lengths of all legs are fixed, the entire structure becomes rigid (although the same lengths may be compatible with up to 40 endplate locations). In the special case above, however, any choice for such lengths makes the endplate mobile with one degree-of-freedom [Husty and Karger 2000]. This linkage is actually a sub-case of the so-called Griffis-Duffy platform, a parallel platform invented by M. Griffis and J. Duffy [Griffis and Duffy 1993], whose special geometry allows a closed-form solution for its direct kinematic problem, with the advantage of having all of its spherical joints separated (which facilitates its physical construction).

Geometry

In [Griffis and Duffy 1993] the authors propose two types of mechanisms, referred to as the "midline to apex" and "apex to apex" embodiements. The linkage of this benchmark is a special case of the "midline to apex" embodiement. Specifically, the base and platform bodies must be equilateral triangles, and every leg connects one vertex of a triangle with an edge midpoint of the other triangle, as shown in the figure above. Provided the mechanism can actually be assembled, any leg lengths will let the one-dimensional self-motion occur.

The geometric parameters of this parallel platform are (the leg lengths computed by randomly placing the top platform):

Base Points
Coord. B1 B2 B3 B4 B5 B6
x 0 -0.5 0.5 1.5 1 0.5
y 0 0.866025 0.866025 0.866025 0 -0.866025
z 0 0 0 0 0 0
Platform Points
Coord. A1 A2 A3 A4 A5 A6
x 0 0.5 1 1.5 2 1
y 0 0.866025 1.732051 0.866025 0 0
z 0 0 0 0 0 0

leg 1 2 3 4 5 6
length 1.519640 1.922131 1.812880 1.380117 1.715536 1.714524

Process

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

  • Generate the equations:
  • Solve the positional analysis problem: Execute
    • bin/cuik examples/ParallelPlatform/Griffis-Duffy
  • Sort the solutions:
  • Visualize the solutions:
    • scripts/cuikplayer examples/ParallelPlatform/Griffis-Duffy examples/ParallelPlatform/Griffis-Duffy_1
    • scripts/cuikplayer examples/ParallelPlatform/Griffis-Duffy examples/ParallelPlatform/Griffis-Duffy_2
  • Visualize the configuration space:
    • bin/cuikplot3d examples/ParallelPlatform/Griffis-Duffy_1 3 9 11 0 gd_1.gcl
    • bin/cuikplot3d examples/ParallelPlatform/Griffis-Duffy_2 3 9 11 0 gd_2.gcl
    • geomview gd_1.gcl gd_2.gcl

If you want to trace the red point on the platform execute:

  • Generate the equations: First, ensure that the REPRESENTATION is set to LINKS (or just not set) in the parameter file (this is the default for this example)
  • Add the equations of the point to trace:
    • bin/cuikmerge _trace examples/ParallelPlatform/Griffis-Duffy examples/ParallelPlatform/Griffis-Duffy_extra
  • Solve the positional analysis problem: Execute (this will take longer than the cuik on the problem without the point to trace)
    • bin/cuik examples/ParallelPlatform/Griffis-Duffy_trace
  • Sort the solutions:
    • bin/cuiksort examples/ParallelPlatform/Griffis-Duffy_trace
  • Visualize one of the traced curves:
    • bin/cuikplot3d examples/ParallelPlatform/Griffis-Duffy_trace_1 25 26 27 0 gd_trace.gcl
  • Visualize the solutions: Eventual jumps at the end of the animation are due ot clustering and can be elimanted with smaller resolution.
    • scripts/cuikplayer examples/ParallelPlatform/Griffis-Duffy examples/ParallelPlatform/Griffis-Duffy_trace_1
    • Pause the animation and load the gd_trace.gcl file (using the geomview main menu).

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 805
Nr. of solution boxes 971
Solver time (s) 200

Results

By changing the colors of the two plotted curbes you should get something like the figure below from the first sequence of commands in the process section.

The result of second set of commands of the process section gives you a plot as (actually this is just a snapshot as the mechanism will move tracing the curbe).

References

Definition in file Griffis-Duffy.world.