Mobility is a fundamental parameter of mechanisms expressing in a qualitative manner their kinematic and dynamic properties. The mobility formulae presented in literature take into consideration some of the structural entities, such as bodies, joints, constraints, closed loops, and space characteristics; however, the specific mechanical model that has traditionally been at the origin of the mobility criteria themselves is incompletely specified and, even then, only implicitly. In this paper, we propose a classification of the mobility criteria based on the known dynamic models. While all known mobility criteria have been associated with a specific dynamic model, some particular, less used dynamic models (like natural coordinates and multi-particle models) suggested new mobility criteria models. The associated mechanical model for each category of mobility criteria allows a qualitative assessment of the kinematic and dynamic sets of equations to be formulated in later stages of analysis. A simple multi-loop mechanism is taken as an example just to illustrate the mobility calculation and qualitative assessment of the kinematic and dynamic models in each case. Based on the proposed classification of the mobility formulae, an assessment is made with particular regard to their applicability to overconstrained mechanisms.

This paper presents applications of group theory tools to simplify the analysis of kinematic chains associated with mechanisms and parallel manipulators. For the purpose of this analysis, a kinematic chain is described by its properties, i.e. degrees-of-control, connectivity and redundancy matrices. In number synthesis, kinematic chains are represented by graphs, and thus the symmetry of a kinematic chain is the same as the symmetry of its graph. We present a formal definition of symmetry in kinematic chains based on the automorphism group of its associated graph. The symmetry group of the graph is associated with the graph symmetry. By using the group structure induced by the symmetry of the kinematic chain, we prove that degrees-of-control, connectivity and redundancy are invariants by the action of the automorphism group of the graph. Consequently, it is shown that it is possible to reduce the size of these matrices and thus reduce the complexity of the kinematic analysis of mechanisms and parallel manipulators in early stages of mechanisms design.

In this paper, we present a new method of enumeration of parallel manipulators with one end-effector. The method consists of enumerating all the manipulators possible with one end-effector that a single kinematic chain can originate. A very useful simplification for kinematic chain, mechanism and manipulator enumeration is their representation through graphs. The method is based on group theory where abstract structures are used to capture the internal symmetry of a structure in the form of automorphisms of a group. The concept used is orbits of the group of automorphisms of a colored vertex graph. The theory and some examples are presented to illustrate the method.

Singularities of a manipulator have been addressed repeatedly. However, the singularities and the degree(s) of freedom, as a matter of fact, are two different aspects of the mobility of a manipulator. Consequently, this paper dedicates to discussing the mobility properties through mobility space, which synchronously define the type, number and direction characteristics of the independent motions that the manipulator should execute. The mobility space of a manipulator can be obtained with reciprocal screws of the manipulator via singular value decomposition, which instantaneously depicts the singularity and mobility problems of the manipulators. Application example demonstrates that this methodology can investigate the all-sided mobility properties of parallel manipulators.