126 The Bricard linkages
revolute joints are extended and joints are connected with bars not perpendicular to the axes, see Figure 6.13, where the solid lines present the links
of the original linkage and the dashed lines present the bars in the alternative form.
For simplicity, assume that symmetry is kept in the alternative form. We
have
(6.6)
So all the bars of the alternative form have the same length, L, which is
.
(6.7)
For each given set of c and d, an alternative form for the threefoldsymmetric Bricard linkage can be obtained. The most compact folding
can be achieved if simultaneously the points A, C and E meet at a point
while the points B, D and F also meet at another point. This means that
physically the linkage becomes a bundle whose length is L. Denote θ 1
and θ 2 in this fully folded configuration as θ 1f and θ 2f , respectively. On
the other hand, when the linkage is fully expanded, points A, B, C, D, E
and F are on the same plane, i.e. the linkage is completely flattened to
form an equilateral hexagon. Take θ 1 and θ 2 in this configuration as θ 1e
and θ 2e .
The relationship among the geometric parameters of the threefoldsymmetric Bricard linkage and the design parameters of the linkage in the
alternative forms can be found geometrically. Several alternative forms
have been studied. Here only two of the most interesting ones, proposed
by Pellegrino (2002), are discussed.
Figure 6.13 The alternative form of threefold-symmetric Bricard linkage.