Definition: Given the importance of kinematic singularity and the vast literature on the subject it may be surprising that one rarely encounters a clear general definition of the phenomenon. To provide one is the course’s first objective: singularity is defined rigorously and in simple terms. 

Interpretation: Singularities are interpreted as silhouettes of configuration space when “seen” from the spaces of input or output coordinates of a mechanism. This provides a rich picture of singularities, and connects them to the notions of workspace boundary, workspace barrier, and reduceddexterity locus. The interpretation allows one to visualise the various physical phenomena occurring at forward and inverse singularities. 

Classification: Numerous singularity classifications exist. Since singularity is defined via instantaneous kinematics, the most fundamental taxonomy describes the types of degeneracy of the forward and inverse velocity problems. Finer distinctions exist for specific mechanism types, e.g., the important constraint singularities of parallel manipulators. When noninstantaneous properties are considered, other distinctions arise, such as between cusplike and foldlike singularities, or the existence of selfmotions. 

Computation: One of the most practicallyimportant problems of kinematic analysis is the explicit calculation of the singularity set. A general method using numerical partitioning of the ambient parameter space is outlined. Also, a powerful approach for formulating and solving symbolically the algebraic equations of the endeffector’s motionpattern and singularposes set is studied in detail. 

Avoidance: The course explores the possibility of a singularityfree workspace and the ability to escape from singularity, issues of major practical importance for the design of path planning algorithms and singularity consistent control schemes. 

Singularityset and configurationspace topology: The singularityfreeconnectivity properties of the configuration space are discussed, including the fascinating cuspidal manipulators, able to change posture while avoiding singularities. Related fundamental problems of genericity and configurationspace and singularityset topology are explored. We examine the possibility of multiple operation modes, sometimes with strikingly different platform motion patterns, connected by constraint singularities. 

Mathematical tools and formalisms: The course is a handson introduction to the various analytical and computational tools for dealing with singularities. We explore screwgeometrical techniques and Liegroupbased localanalysis methods. Algebraicgeometry formulations combined with either symbolic computation or numerical methods (branchandprune, and continuation methods) are used. Topology and differential geometry provide the basis for the definitions and formulations throughout the course. 