Singularities in Motion and Displacement Functions of Constrained Mechanical Systems

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Abstract
A general approach for the determination of singularities in motion and displacement functions of manipulators and linkages is proposed. Singularities in motion of a manipula tor occur at positions where the parameters in motions of the end-effector become dependent, or when the end-effector is at rest and some manipulator links become movable. Singular ities in motion of a linkage occur at positions where an over- constrained group of links becomes mobile while the driving link is held at rest. Both cases of singularities are determined with the condition that the rank r of the system matrix of six linear equations is less than 6. The system of linear equa tions relates the velocities in motion in joints with the veloci ties of the linkage driving link (manipulator end-effector). In the proposed approach, the elements of the system matrix can be derived directly by using the matrix representation of coordinate transformation. The requirement r < 6 provides an equation (the singularity equation) that relates some parameters of motion. The structure of the singularity equa tion determines the structure of the to-be-derived displace ment equations. The derivation of displacement equations is based on (i) modeling of the manipulator or linkage by open subchains, and (ii) determination of invariants for the sub chains. The number of configurations that can be formed by linkage links is proposed to relate to the number of multiple singularity positions. A method for determination of multiple singularity positions is proposed. The proposed approach is illustrated with examples of manipulators and linkages.