6 Spatial motion structures based
on Bricard linkages
6.1 Threefold- symmetric Bricard linkages and its assemblies
The Bricard linkages reviewed in Section 2.3.5 are the only 6R overconstrained linkages that are not derived from 4R, 5R or other 6R linkages.
Of a total of six types of Bricard linkages, the most suitable ones for the
purpose of constructing motion structures is the threefold- symmetric
Bricard linkage, obtained by combining the general plane- symmetric and
trihedral Bricard linkages. The geometric parameters of the linkage satisfy
the following conditions.
,
,
,
(6.1)
(i = 1, 2, ···, 6).
The linkage has threefold rotational symmetry and also three planes of
symmetry, hence its name. The configuration of this linkage is shown in
Figure 6.1. It is easy to see that threefold- symmetric Bricard linkages form
a subset of the set of plane- symmetric Bricard linkages. If, as a further
specialisation, π/2 is selected as the angle α, then relationships given by Eq.
(6.1) satisfy the geometrical conditions of trihedral Bricard linkages (Baker,
1980). Therefore, in this case, a threefold- symmetric Bricard linkage is also
a trihedral Bricard linkage.
The linkage is mobile because of the plane- symmetric property guarantees it. It is however yet to know whether the number of degrees of
freedom increases with an increase in the degree of symmetry. Because of
threefold symmetry, the six revolute variables must satisfy the following
conditions.
,
.
(6.2)