Mobility Analysis of Altmann Overconstrained Linkages by Modified Grübler-Kutzbach Criterion

Developing a general mobility method/formula is a hot topic lasting for more than 150 years in kinematics. It is necessary to apply any mobility method to puzzling overconstrained mechanisms for verification of its generality. Altmann linkages are such recognized puzzling mechanisms that their mobility analysis is of important significance. A necessary condition for judging a general mobility method is that the method can be fit for Altmann linkages. Firstly, this study classes Altmann linkages into 17 types in terms of the numbers and types of kinematic pairs, and then Altmann overconstrained linkages are further classified into 4 types. Secondly, the mobility of Altmann overconstrained linkages is systematically analyzed by the Modified Grübler-Kutzbach criterion based on screw theory, where passive freedoms are defined as limb passive freedoms and mechanism passive freedoms. In addition, the full-cycle mobility is judged, which overcomes the shortcoming of instantaneous property of screw theory. It is shown that Modified Grübler-Kutzbach criterion not only obtains the correct numerical mobility, but also gives the mobility character by resolving reciprocal screws for the constraint system. This study lays the foundation of verification for the generality of Modified Grübler-Kutzbach criterion. Besides, Altmann overconstrained linkages almost comprise all kinds of modern parallel mechanisms and some classical mechanisms, which provides an important reference for mechanism mobility calculation.

Mobility of the Myard 5R Linkage Involved in “Gogu Problem”

Since the traditional Griibler-Kutzbach criterion fails in many overconstrained mechanisms, developing a general mobility formula is a hot topic lasting for more than 150 years in mechanisms. GOGU systematically investigated various mobility methods, and pointed that the methods were not fit for two kinds of paradoxical overconstrained mechanisms. The mobility on the two kinds of mechanisms is regarded as ＂Gogu problem＂. The Modified Griibler-Kutzbach criterion has solved the mobility of the second kind of mechanisms in ＂Gogu problem＂, and has developed into a systematic mobility methodology. Myard 5R linkage is one of the single-loop mechanisms involved in ＂Gogu problem＂, its joint axes are distributed in space with special geometric conditions, which increases the difficulty of mobility analysis. The study is to calculate the global mobility of the Myard 5R linkage using the mobility methodology. Firstly, the mobility methodology based on screw theory is briefly introduced. Secondly, some homogeneous transforms are performed according to the D-H parameters and the invariance of the linkage plane symmetry is revealed, which provides an idea to judge a plane-symmetric loop. The special geometric features of the axes distribution are discussed as well. Finally, the global mobility of the Myard 5R linkage is determined by the Modified Grubler-Kutzbach criterion. The results show that the methodology can be applied to more paradoxical mechanisms.