Polyhedra with Articulated Faces

Abstract—This paper introduces the concept of polyhe-dra with articulated faces, which can be described as poly-hedral frameworks whose faces are constrained to remainplanar. A mechanical arrangement based on a single typeof component is proposed for the construction of the poly-hedra. Then, a method is presented for the determinationof their mobility. Finally, examples are given in order toillustrate the new family of mechanical constructions. Keywords:polyhedra, articulated mechanisms, degree of freedom I. IntroductionPolyhedra have fascinated scientists and engineers forcenturies. In particular, mechanisms based on polyhe-dra have been studied by several researchers over the pastdecades [1], [2], [3]. Several types of polyhedral expand-able mechanisms were also proposed (see for instance [4],[5], [6], [7]), but the latter are outside of the scope of thispaper.Several definitions of a polyhedron can be found in theliterature [8]. Polyhedra have often been defined as solids.However,theyhavealsobeendefinedasasetofsurfaces,inwhich case they can have mobility. Using the assumptionthat the faces are rigid and are connected at the edges byrevolute joints, Cauchy proved that convex polyhedra arealways rigid [9]. However, non-convex polyhedra can bemoveable in some cases [2]. Polyhedra can also be definedasframeworks. Apolyhedralframework(PFW)isobtainedbyplacingbarsattheedges, thebarsbeingconnectedatthevertices by spherical joints. Mathematically, this is equiv-alent to constraining the distance between any two verticesconnectedbyanedgetobeconstant. Ifsomeofthefacesofa polyhedron have four sides or more, the result is an artic-ulated mechanism which generally has numerous degreesof freedom (DOFs). It can be argued that such a polyhe-dralframeworkisnotapolyhedronpersesincesomeofthefaces are no longer planar. If the faces are triangularized,i.e., the faces are constrained by additional bars in order toobtainonlytriangles, thenitcanbeshownthattheresultingframework is rigid if it is generated from a convex poly-hedron, and that almost all triangularized frameworks arerigid [1].