Structural analysis of plane linkages by Franke's condensed notation

Structural synthesis of serial platform manipulators

“…The methods of structural synthesis reported by Davies and Crossley [44], Dobrjanskyi [45], Buchsbaym [46], Freudenstein [47], Dobrjanskyi and Freudenstein [48], Manolescu [49] were based on graph theory to find the set of kinematic chains and mechanisms.…”

Kinematic Design and Description of Industrial Robotic Chains

“…But if the chains have two, three or more degrees of mobility there are two degenerate forms of chains called partial and fractionated mobilities (Manolescu, 1964). If any link of the chain is selected to be the frame of a mechanism and if one chooses any M members to be the independent driving links (called "motor" links), then the position of every remaining member is dependent on the position of all links, such a mechanism is said to have a total degree of mobility (Davies & Crossley, 1966). This is the case of structures: 2 of Table 11, and 2, 3, 4 and 6 of Table 12.…”

“…Such a mechanism possesses at least one link (with at least four joints), which may be decomposed into two or more parts, with the result that each sub-mechanism forms a closed chain. The mobility of the whole mechanism is equal to the sum of the mobilities of all sub-mechanisms (Davies & Crossley, 1966). This is the case of structures 4 of table 11; and 5 of table 12.…”

“…A double circle represents it and is noted by a capital Greek letter. Let us remember that in the molecules proposed by Davies and Crossley, based on Franke's notation (Davies & Crossley, 1966), a circle is used to represent a polygonal link (ternary, quaternary etc) and a line to represent a band joining the polygonal links either by a kinamatic joint of class 5 or by one or more binary links in a chain. Whereas in the proposed method a circle represents a contour and a line only a kinematic joint of class 5.…”

“…There exist different methods for kinematic synthesis of planar chains with simple joints (Tischler et al, 1995;Belfiore, 2000;Rao & Deshmukh, 2001): intuition and inspection (Crossley, 1964), graphs theory (Dobrjanskyi and Freudenstein, 1967;Woo, 1967). Others consist in transformation of binary chains (Mruthyunjaya, 1979;Mruthyunjaya, 1984-a;Mruthyunjaya, 1984-b;Mruthyunjaya, 1984-c), in the concept of Assur groups Manolescu, 1964;Manolescu, 1979;Manolescu, 1987), or Franke's notation (Davies & Crossley, 1966;Crossley, 1966). Recently, new methods based on genetic algorithms or neuronal networks are also used (Tejomurtula & Kak, 1999;Abo-Hamour et al, 2002;Cabrera et al, 2002;Laribi et al, 2004).…”