# Linear Pentapods with a Simple Singularity Variety – Part II: Computation of Singularity-Free Balls

## Abstract

The configuration space of a linear pentapod can be defined as the set of all points (*u, v, w, p*_{x} *, p*_{y} *, p*_{z}) *∈* ℝ^{6} located on the singular quadric *Γ* : *u* ^{2} + *v*^{2} + *w*^{2} = 1, where (*u, v, w*) determines determines the orientation of the linear platform and (*p*_{x} *, p*_{y} *, p*_{z}) its position. In such terminology, the set of all singular robot configurations is obtained by intersecting *Γ* with a cubic hypersurface *Σ* in ℝ^{6}, which is only quadratic in the orientation variables and position variables, respectively. We study the computational aspects of the determination of singularity-free balls under the design restrictions that *Σ* is either (1) linear in position variables or (2) linear in orientation variables. It turns out that for these pentapod designs the computation of singularity-free balls in the configuration space simplifies considerably. One can even obtain a closed form solution, which is paving the way for a real-time singularity-free path planning/optimization in the configuration space.

## Keywords

Linear pentapods Distance computation Singularity-free balls## Preview

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## Notes

### Acknowledgement

The research is supported by Grant No. P 24927-N25 of the Austrian Science Fund FWF. Moreover the first author is funded by the Doctoral College “Computational Design” of Vienna University of Technology.

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