We propose a technique which significantly simplifies the computation of frictionless force-closure grasps of a curved planar part$P$. We use a colored projection scheme from the three-dimensional wrench space to two-dimensional screens, which allows us to reduce the problem of identifying combinations of arcs and concave vertices of$P$that admit frictionless force-closure grasps, to colored intersection searching problems in the screens. We show how to combine this technique with existing intersection searching algorithms to obtain efficient, output-sensitive algorithms to compute all force-closure grasps of$P$, where at most four hard, frictionless point contacts exert exactly four wrenches on$P$. If the boundary of$P$consists of$n$algebraic arcs of constant complexity and$m$concave vertices, we show how to compute all force-closure grasps with:four contacts along four arcs in$O(n^{8/3}\log ^{1/3}n+K)$time;

Object immobilization is important to robot hand grasping and to many manufacturing processes. A huge number of existing papers considered issues such as the analysis of grasps, the existence of immobilizing grasps for various classes of objects, and the synthesis of immobilizing grasps for twoand three-dimensional objects. However, no algorithm has been proposed to efficiently enumerate all form-closure grasps for any class of three-dimensional objects. As an initial step towards a general solution to this complex problem, we propose the first efficient algorithm for computing all form-closure grasps of a rectilinear polyhedron. Our approach is based on a decomposition of the original problem in the abstract sixdimensional wrench space into closely related subproblems in three-dimensional subspaces, and a subsequent transformation of these subproblems into two-color intersection problems on planar screens in these spaces. We then use techniques from computational geometry to efficiently solve the planar intersection problems. The resulting algorithm reports all K sets of six to seven faces of a rectilinear polyhedron that yield at least one form-closure grasp in O(n2K′log4n + K) time, where n is the number of the faces, and K′ is the size of an intermediate output. We show that K = Ω (n2K′) in the worst case.

We propose the first efficient output-sensitive algorithms for computing all form-closure grasps of a planar curved part P with at most four frictionless point contacts. The boundary of P consists of m concave vertices and n algebraic arcs with a constant degree. All our algorithms are output-sensitive, which means that their running times largely depend on the actual output size K rather than the (often much larger) maximum size of the output. More specifically, we show how to determine • all form-closure grasps with four points along four arcs in O(n8/3log1/3n + K) time, • all form-closure grasps with four points along three arcs in O(n5/2+ε+ K) time, • all form-closure grasps with one point at a concave vertex and two points along two arcs in O(n2m1/2+ε+ K) time, • all form-closure grasps with one point at a concave vertex and two points along a single arc in O(nm) or O(n3/2+ε+ K) time (depending on the size of m), • all form-closure grasps with two points at concave vertices and one point along arc in O(nm2) or O (n2+ε+ K) time (depending on the size of m), where ε is an arbitrarily small positive constant. All our algorithms rely on the geometric condition in three-dimensional wrench space, which is transformed into two-dimensional geometric intersection problems.

We study the problem of fixturing a chain of hinged objects in a given placement with frictionless point contacts. We define the notions of immobility and robust immobility - which are comparable to the second and first order immobility for a single object - to capture the intuitive requirement for the fixture of a chain of hinged objects. Robust immobility differs from immobility in that it additionally requires insensitivity to small perturbations of contacts. We show that (p+2) frictionless point contacts can immobilize any chain of p/spl ne/3 polygons without parallel edges; six contacts can immobilize any chain of three such polygons. Any chain of p arbitrary polygons can be immobilized with at most (p+4) contacts. We also show that /spl lceil/(6/5)(p+2)/spl rceil/ contacts suffice to robustly immobilize p polygons without parallel edges, and that /spl lceil/(5/4)(p+2)/spl rceil/ contacts can robustly immobilize p/spl ne/3 arbitrary polygons, and eight contacts can robustly immobilize three polygons.