Serial6RBricard.world File Reference

Detailed Description

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

Introduction

A closed kinematic chain consisting of six binary links, connected together by six revolutes is generally rigid, i.e., it can only be assembled in a finite number of different ways. However, if there are certain geometrical conditions imposed upon the relevant linkage parameters (the normal distances, the twist angles and the offsets of the revolutes on each of the links) the chain my be mobile with one degree of freedom. Such 6R loops are said to be overconstrained mechanisms, and one example is "rectangular Bricard chain" shown above [Wohlhart 1987].

The Bricard chain belongs to a larger family of overconstrained 6R loops, characterized by the fact that in every position of the linkage there is a transversal, a straight line that intersects all six revolute axes-lines. In the case of the rectangular Bricard chain, if we number the revolute axis lines from 1 to 6 consecutively, the even numbered ones and the odd numbered ones meet in all positions at points $P_{135}$ and $P_{246}$, respectively [Wohlhart 1987], and the transversal is the line defined by these two points (see the figure above).

Geometry

Denavit-Hartenberg parameters for these parameters are:

i $a_i$ $d_i$ $\alpha_i$ Interpretation
1 1 0 $\pi/2$
2 1 0 $-\pi/2$
3 1 0 $\pi/2$
4 1 0 $-\pi/2$
5 1 0 $\pi/2$
6 1 0 $-\pi/2$

Process

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

  • Generate the equations: Execute
  • Solve the positional analysis problem: Execute
    • bin/cuik examples/Serial6R/Serial6RBricard
  • Sort the solutions:
  • Animate the solutions: This
  • Plot and visualize the configuration space (you will see the two one-dimensional solution sets):
    • bin/cuikplot3d examples/Serial6R/Serial6RBricard 9 17 23 0 Serial6RBricard.gcl
    • geomview Serial6RBricard.gcl

Statistics

Characteristics of the problem:

Nr. of indep. loops 1
Nr. of links 6
Nr. of joints 6
Nr. of equations (in the simplified system) 24
Nr. of variables (in the simplified system) 24

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

Nr. of Empty boxes 583
Nr. of Solution boxes 352
Execution time (s) 40

Results

This is one of the configurations got with the instructions previously given:

and this is the projection of the configuration space

References

  • K. Wohlhart, "A New 6R Space Mechanism", 7th World Congress on the Theory of Machines and Mechanisms. Vol. 1, pp. 193-198, Sevilla, Spain, 1987.

Definition in file Serial6RBricard.world.