Critical configurations of planar multiple penduli

Extremal configurations of robot arms in three dimensions

“…In [5] and [8] the 2-dimensional case was treated with the signed area function on the configuration space. We recall some definitions and results.…”

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].…”

“…Two types of quadrilateral linkages are considered: (1) conventional quadrilateral linkages with the fixed lengths of the sides, and (2) quadrilateral linkages with one telescopic link [4]. Obviously, the latter concept is equivalent to the so-called planar robot 3-arm (or planar triple pendulum [14]). To unify and simplify terminology it is convenient to refer to these two cases by speaking of closed and open 4-vertex linkages.…”

“…Comprehensive results on the geometry of planar 4-bar mechanisms are presented in [7]. Some recent results may be found in [5], [11], [13].…”

Critical configurations of planar robot arms

“…The present paper is concerned with certain planar configurations of robot arms appearing as critical points of the oriented area considered as a function on the moduli space of the arm in question. This setting naturally arose in the framework of a general approach to extremal problems on configuration spaces of mechanical linkages developed in [2,4,5], which has led to a number of new results on the geometry of cyclic polygons [3,6] and suggested a variety of open problems. The approach and results of [2,5] provided a paradigm and basis for the developments presented in this paper.…”