This article develops a general technique for differential analysis that can be applied to singularities of three related problems: path tracking for nonredundant robots, self-motion analysis for robots with one degree of redundancy, and displacement analysis of single-loop mechanisms. For each of these problems, the locus of displacement solutions generally forms a set of one-dimensional manifolds in the space of variable parameters. However, if singularities occur, the manifolds may degenerate into isolated points, or into curves that include bifurcations at the singular points. Higher-order equations, derived from Taylor series expansion of the matrix equation of closure, are solved to identify singularity type and, in the case of bifurcations, to determine the number of intersecting branches as well as a Taylor series expansion of each branch about the point of bifurcation. To avoid unbounded mathematics, branch expansions are derived in terms of an introduced curve parameter. The results are useful for identifying singularity type, for numerical curve tracking with continuation past bifurcations on any chosen branch, and for determining exact rate relations for each branch at a bifurcation. The noniterative solution procedure involves configuration-dependent systems of equations that are evaluated by recursive algorithm, then solved using singular value decomposition, polynomial equation solution, and linear system solution. Examples show applications to RCRCR mechanisms and the Puma manipulator.
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