Joint Rotation Space and Mobility of Linkages

Abstract

The paper presents a general concept regarding the joint rotation space (JRS) of single loop planar and spherical N-bar linkages as well as its significance and applicability in multiple loop and spatial linkages. A JRS refers to the input domain of a linkage. The paper first discusses and classifies the JRS types of single loop five-bar linkages based on the types of the JRS boundary. The concept of sheets and pages of JRS is then introduced as an aid to understand the mobility of linkages. A sheet refers to the JRS of a linkage branch (or circuit). A JRS sheet may have one or more pages and each page refers to an uncertainty singularity free JRS. The concept is then generalized to any single loop planar and spherical N-bar linkages (N ≥ 3). The paper offers geometric insights to the input domain of a linkage and establishes a one-to-one correspondence between the input domain and the linkage configurations. The applications to the mobility identification of complex linkages are discussed and demonstrated with geared five-bar linkages, Stephenson six-bar linkages, as well as spatial RCRCR mechanisms. The paper presents a useful model explaining the fundamental principle regarding the formation of branches and sub-branches in Stephenson six-bar linkages as well as some other multiloop and spatial linkages. The Mobility similarity between multiloop linkages and spatial linkages is also highlighted.