The order of local mobility of mechanisms

The mobility or degrees of freedom is a fundamental issue in mechanisms and robotics. In this work, we distinguish the global mobility and local mobilities with different orders, and derive the corresponding conditions systematically. The relations between the global mobility and the local mobilities are disclosed. We show that the rank-deficiency of Jacobian matrix is equivalent to the first-order local mobility, and the global mobility is equivalent to the infinite-order local mobility. The second-order local mobility can be considered as a point freely moving a submanifold, which shares the same curvature with all hypersurfaces defined by constraints. We further discover a novel four-bar linkage with the second-order local mobility, which validates the theoretical mobility analysis.