40 Planar double chain linkages
at a certain pivoting angle but it reappears if the angle varies. In other
words, even if the connectors could be connected to form a loop at a particular pivoting angle, the resulted assembly will be immobile because a
slight change of pivoting angle will result in reappearance of the gap.
This problem can be solved by making use of symmetry. Figure 3.5(b) is
a chain made of the same pairs but in an order which is different from that
in Figure 3.5(a). The chain has two- fold symmetry and sum of the central
angles for each quarter is π/2 so that no split occurs when four quarters are
joined together when the pivoting angle varies. Closure of the double chain
becomes possible. Moreover, the closed double chain has mobility one.
In terms of practicality, symmetry is a good way to construct a mobile
planar double chain of angulated scissor- like elements. However, other
more general solutions exist which are obtained by considering the geometry of the entire assembly using the kinematic analysis tool that was introduced in Section 2.2.
3.2 Closed double chain
3.2.1 Background
The conquest of constructing mobile closed double chains dates back to
over a century ago when Kempe (1878) first reported that under certain
geometrical circumstances an assembly of two planar 4R linkages connected together by four additional hinges could become mobile. Six classes
of such linkages known as the Kempe linkages were discovered, one of
which is shown in Figure 3.6. Subsequently Darboux and Fontené provided further proofs of mobility for the Kempe linkage. A summary of
their work can be found in Baker and Yu (1983).
(a)
(b)
Figure 3.5 (a) A chain made from angulated elements, each subtends a constant
central angle; (b) a mobile closed chain of angulated elements constructed using two-fold symmetry.
at a certain pivoting angle but it reappears if the angle varies. In other
words, even if the connectors could be connected to form a loop at a particular pivoting angle, the resulted assembly will be immobile because a
slight change of pivoting angle will result in reappearance of the gap.
This problem can be solved by making use of symmetry. Figure 3.5(b) is
a chain made of the same pairs but in an order which is different from that
in Figure 3.5(a). The chain has two- fold symmetry and sum of the central
angles for each quarter is π/2 so that no split occurs when four quarters are
joined together when the pivoting angle varies. Closure of the double chain
becomes possible. Moreover, the closed double chain has mobility one.
In terms of practicality, symmetry is a good way to construct a mobile
planar double chain of angulated scissor- like elements. However, other
more general solutions exist which are obtained by considering the geometry of the entire assembly using the kinematic analysis tool that was introduced in Section 2.2.
3.2 Closed double chain
3.2.1 Background
The conquest of constructing mobile closed double chains dates back to
over a century ago when Kempe (1878) first reported that under certain
geometrical circumstances an assembly of two planar 4R linkages connected together by four additional hinges could become mobile. Six classes
of such linkages known as the Kempe linkages were discovered, one of
which is shown in Figure 3.6. Subsequently Darboux and Fontené provided further proofs of mobility for the Kempe linkage. A summary of
their work can be found in Baker and Yu (1983).
(a)
(b)
Figure 3.5 (a) A chain made from angulated elements, each subtends a constant
central angle; (b) a mobile closed chain of angulated elements constructed using two-fold symmetry.