In this paper, the configuration bifurcation characteristics at the vicinities of singular points going with different input parameters are investigated. Then, with the aid of the assembly configurations at the theoretical singular points, the reasons to cause the singularities are analyzed. We find that the dimensional-utmost singularities, line vectors correlation singularities and Jacobian matrix correlation singularities can occur individually or jointly while choosing different number of input parameters. The number and the combination form of the input parameters have great influences on the complexity of the singularities and the curvature radiuses of the configuration curves. Selecting a group of adjacent input parameters, the simple configuration bifurcation and the large singularity-free input parameters zones can be obtained. And adopting multi-input parameters, the self-motion regions and the singularity avoidance errors can be reduced. These new discoveries are valuable and of significance for the trajectory design, the singularity avoidance, and the self-motion control for the parallel manipulator, as well as the parallel tools.

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