Kinematic Singularities of Robot Manipulators

Abstract

Kinematic singularities of robot manipulators are configurations in which there is a change in the expected or typical number of instantaneous degrees of freedom. This idea can be made precise in terms of the rank of a Jacobian matrix relating the rates of change of input (joint) and output (end-effector position) variables. The presence of singularities in a manipulator’s effective joint space or work space can profoundly affect the performance and control of the manipulator, variously resulting in intolerable torques or forces on the links, loss of stiffness or compliance, and breakdown of control algorithms. The analysis of kinematic singularities is therefore an essential step in manipulator design. While, in many cases, this is motivated by a desire to avoid singularities, it is known that for almost all manipulator architectures, the theoretical joint space must contain singularities. In some cases there are potential design advantages in their presence, for example fine control, increased load-bearing and singularityfree posture change. There are several distinct aspects to singularity analysis—in any given problem it may only be necessary to address some of them. Starting with a given manipulator architecture, manipulator kinematics describe the relation between the position and velocity (instantaneous or infinitesimal kinematics) of the joints and of the end-effector or platform. The physical construction and intended use of the manipulator are likely to impose constraints on both the input and output variables; however, it may be preferable to ignore such constraints in an initial analysis in order to deduce subsequently joint and work spaces with desirable characteristics. A common goal is to determine maximal singularity-free regions. Hence, there is a global problem to determine the whole locus of singular configurations. Depending on the architecture, one may be interested in the singular locus in the joint space or in the work space of the end-effector (or both). A more detailed problem is to classify the types of singularity within the critical locus and thereby to stratify the locus. A local problem is to determine the structure of the singular locus in the neighbourhood of a particular point. For example, it may be important to know whether the locus separates the space into distinct subsets, a strong converse to this being that a singular configuration is isolated. Typically, there will be a number of design parameters for a manipulator with given architecture—link lengths, twists and offsets. Bifurcation analysis concerns the changes in both local and global structure of the singular locus that occur as one alters design parameters in a given architecture. The design process is likely to involve optimizing some desired characteristic(s) with respect to the design parameters. 2