### Motion planning algorithms for configuration spaces in the higher dimensional case

DOI: http://dx.doi.org/10.12775/TMNA.2016.030

#### Abstract

The topological approach to the motion planning problem was introduced by Farber in \cite{F} and \cite{F2}. A motion planning problem is a rule assigning a continuous path to given two configurations - initial point and desired final point of a robot. Farber introduced the notion of topological complexity which measures the discontinuity of any motion planner in a configuration space. In \cite{Ru}, Rudyak introduced higher topological complexity, the concept fully developed in \cite{B}. Higher topological complexity is related to motion planning problem which assigns a continuous path (with $n$-legs) to given $n$ configurations. More precisely, it can be understood as a motion planning algorithm when a robot travels from the initial point $A_{1}$ to $A_{2}$, then from $A_{2}$ to $A_{3}$, and this keeps going until it reaches at the desired final point $A_{n}$.

This paper is based on the work of Mas-Ku and Torres-Giese who gave an explicit motion planning algorithm for configuration spaces $F(\mathbb{R}^{2},k)$ and $F(\mathbb{R}^{n},k)$, in \cite{MT}. In the last section, we will consider the higher dimensional case in the sense of Rudyak in \cite{Ru}, and give an explicit motion planning algorithm for this case.

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#### References

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M. Farber, Instabilities of robot motion, Topology Appl. 140 (2004), 245–266.

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H. Mas-Ku and E. Torres-Giese, Motion planning algorithms for configuration spaces, Bol. Soc. Mat. Mex., DOI 10.1007/s40590-014-0046-2.

Yu. Rudyak, On higher analogs of topological complexity, Topology Appl. 157 (2010), 916–920; Erratum: Topology Appl. 157 (2010), p. 1118.

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