THE CONFIGURATION SPACE OF ARACHNOID MECHANISMS NIR SHVALB, MOSHE SHOHAM, AND DAVID BLANC ABSTRACT. The configuration spaces of arachnoid mechanisms are analyzed in this paper. These mechanisms consist of k branches each of which has an arbitrary number of links and a fixed initial point, while all branches end at one common end-point. It is shown that generically, the configuration spaces of such mecha- nisms are manifolds, and the conditions for the exceptional cases are determined. The configuration space of planar arachnoid mechanisms having k branches, each with two links is analyzed for both the non-singular and the singular cases. 1. I NTRODUCTION Mechanisms and robots consist of links and joints, the actuation of which causes them to move. The type of a mechanism is described by an abstract graph which corresponds to its links and joints, and a specific embedding of this graph in the plane or in 3-space is called a configuration of the mechanism. The collection of all such embeddings forms a topological space, called the configuration space of the mechanism. For example, the configuration space of a planar mechanism with revolute joints consisting of n rods arranged serially is the n-torus. In recent years, there has been interest among mathematicians in the study of such spaces, which are of importance in motion planning – that is, moving a mechanism from one given position to another, taking into account various con- straints (see for example [MT2]). The topological properties of the configuration space provide insight into practical questions in planning such motions (see [F]) and analysis of some mechanical singularities (see [NM], and [ZFB]). The main focus had been set on the configuration spaces of a type of mecha- nism called polygonal linkage, which is simply a concatenation of links and hinged joints forming a closed chain. A substantial amount of mathematical literature on polygonal linkage’s configuration space has accumulated: Kamiyama , Tezuka and Toma studied Euler characteristics in [K], and homology groups in [KT, KTT]; Trinkle and Milgram constructed a handle-body surgery in [MT1]; and in [Ho], Holcomb studied a special parallel graph mechanism called multi-polygonal link- ages, which are three free branches identified at their initial and terminal vertices. In this paper we analyze a type of mechanism called arachnoid which, to the best of our knowledge, has never been dealt with in the literature. This kind of mecha- nism consists of multiple branches each of which has an arbitrary number of links and a fixed initial point, while all branches end at a common end-point (this type of mechanism resembles some parallel robots which are in practical use). It is shown that generically, the configuration spaces of such mechanisms are manifolds, and 1991 Mathematics Subject Classification. Primary 70G25; Secondary 57N99, 70B15. Key words and phrases. configuration space, work space, robotics, planar mechanism. 1