Critical configurations of planar robot arms

Abstract

It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, P can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or robot arms. We introduce the notion of the oriented area for an open polygonal chain, prove that critical points are exactly the cyclic configurations with antipodal endpoints and derive a formula for the Morse index of a critical configuration. MSC:52Cxx, 58E05

Coulomb Control of Polygonal Linkages

“…It should be noted that this research arose as a natural continuation of our previous joint results on Morse functions on moduli spaces of polygonal linkages [4], [5], [6]. The present paper was completed during a "Research in Pairs" session in CIRM (Luminy) in January of 2013.…”

Point Charges and Polygonal Linkages

“…We wish to add that this research arose as a natural continuation of our previous joint results on Morse functions on moduli spaces of polygonal linkages [9], [11].…”

Cross-ratios of quadrilateral linkages

“…We complement results of [7] and [11] by discussing several new aspects which emerged in course of our study of extremal problems on moduli spaces of polygonal linkages (cf. [10], [11], [12], [13], [14] ). In this context it is natural to consider polygonal linkage as a purely mathematical object defined by a collection of positive numbers and investigate its moduli spaces [4].…”