DIFFERENTIAL ANALYSIS OF BIFURCATIONS AND ISOLATED SINGULARITIES FOR ROBOTS AND MECHANISMS

This article develops a general technique for differential analysis th at can be applied to singularities of three related problems: path tra cking for nonredundant robots, self-motion analysis for robots with on e degree of redundancy, and displacement analysis of single-loop mecha nisms. For each of these problems, the locus of displacement solutions generally forms a set of one-dimensional manifolds in the space of va riable parameters. However, if singularities occur, the manifolds may degenerate into isolated points, or into curves that include bifurcati ons at the singular points. Higher-order equations, derived from Taylo r series expansion of the matrix equation of closure, are solved to id entify singularity type and, in the case of bifurcations, to determine the number of intersecting branches as well as a Taylor series expans ion of each branch about the point of bifurcation. To avoid unbounded mathematics, branch expansions are derived in terms of an introduced c urve parameter. The resuls are useful for identifying singularity type , for numerical curve tracking with continuation past bifurcations on any chosen branch, and for determining exact rate relations (i.e., vel ocity, acceleration, etc.) for each branch at a bifurcation. The nonit erative solution procedure involves configuration-dependent systems of equations that are evaluated by recursive algorithm, then solved usin g singular value decomposition, polynomial equation solution, and line ar system solution. Examples show applications to RCRCR mechanisms and the Puma manipulator.