| In this paper, the concepts of connectivity, degrees of control, and redundancy are revisited from a pure topological viewpoint and then applied to robotics. The redundancy matrix is defined to provide designers with a useful support in the first conceptual phase of the project of a new manipulator. An algorithm for building the connectivity and the redundancy matrices for a large class of manipulators is derived and implemented in an algebraic manipulation programming language. Based on some results borrowed from graph theory, the procedure can be used to study open-loop, closed-loop, and hybrid kinematic chains. In particular, it is shown how the biconnected components of the graph corresponding to the manipulator under analysis have to be detected for a correct computation of connectivity and redundancy. Furthermore, the study of the connectivity and of the degrees of control led to the development of a full mobility test that automatically detects the type of mobility of any given robot: total, partial, or fractionated. One of the presented sample cases offers the opportunity to discover some differences between the connectivity matrix obtained by means of the new algorithm and that presented, on the same kinematic chain, in a previous one.