The question of minimax loop-size in planar homogeneous kinematic chains

A homogeneous planar kinematic chain (KC) is a KC having all its constituting smallest independent circuits (fundamental circuits) of the same size. Davies [J. Mech. 3, 87-100 (1968)] established, using graph theory, that a planar KC (homogeneous) of mobility M cannot have all fundamental circuits of size greater than M + 4. In the present paper many counterexamples to Davies's statement are presented and it is indicated that the maximum size of the circuits depends not only on the mobility of the mechanism but also on the number of links. The counterexamples given consist of, primarily, single degree of freedom (DOF) planar homogeneous KCs of loop sizes 6, 7 and 8; some zero DOF and two DOF chains are also included. The causes for the failure of Davies's theorem have been analysed in detail to show that it is valid only in the domain of planar KCs with planar graphs. Copyright (C) 1996 Elsevier Science Ltd